Models for Tropical Climate Dynamics: Waves, Clouds, and Precipitation [1st ed.] 978-3-030-17774-4;978-3-030-17775-1

Table of contents :
Front Matter ....Pages i-xvi
Front Matter ....Pages 1-2
The Governing Equations and Dry Dynamics (Boualem Khouider)....Pages 3-22
Moisture and Moist Thermodynamics (Boualem Khouider)....Pages 23-39
Observations of Tropical Climate Dynamics and Convectively Coupled Waves (Boualem Khouider)....Pages 41-55
Introduction to Stochastic Processes, Markov Chains, and Monte Carlo Simulation (Boualem Khouider)....Pages 57-71
Front Matter ....Pages 73-73
Simple Models for Moist Gravity Waves (Boualem Khouider)....Pages 75-98
The Multicloud Model with Congestus Preconditioning (Boualem Khouider)....Pages 99-115
Convectively Coupled Equatorial Waves in the Multicloud Model (Boualem Khouider)....Pages 117-132
Convective Momentum Transport and Upscale Interactions in the MJO (Boualem Khouider)....Pages 133-162
Implementation of the Multicloud Model in an Aquaplanet Global Climate Model (Boualem Khouider)....Pages 163-181
Front Matter ....Pages 183-183
Stochastic Birth and Death Models for Clouds (Boualem Khouider)....Pages 185-228
Implementation of the SMCM in a Global Climate Model (Boualem Khouider)....Pages 229-243
SMCM in CFS: Improving the Tropical Modes of Variability (Boualem Khouider)....Pages 245-268
Back Matter ....Pages 269-303

Citation preview

Mathematics of Planet Earth Series 3

Boualem Khouider

Models for Tropical Climate Dynamics Waves, Clouds, and Precipitation

Mathematics of Planet Earth Volume 3

Series editors Ken Golden, The University of Utah, USA Mark Lewis, University of Alberta, Canada Yasumasa Nishiura, Tohoku University, Japan Joseph Tribbia, National Center for Atmospheric Research, USA Jorge Passamani Zubelli, Instituto de Matem´atica Pura e Aplicada, Brazil

Springer’s Mathematics of Planet Earth collection provides a variety of well-written books of a variety of levels and styles, highlighting the fundamental role played by mathematics in a huge range of planetary contexts on a global scale. Climate, ecology, sustainability, public health, diseases and epidemics, management of resources and risk analysis are important elements. The mathematical sciences play a key role in these and many other processes relevant to Planet Earth, both as a fundamental discipline and as a key component of cross-disciplinary research. This creates the need, both in education and research, for books that are introductory to and abreast of these developments.

More information about this series at http://www.springer.com/series/13771

Boualem Khouider

Models for Tropical Climate Dynamics Waves, Clouds, and Precipitation

123

Boualem Khouider Mathematics and Statistics University of Victoria Victoria, BC, Canada

ISSN 2524-4264 ISSN 2524-4272 (electronic) Mathematics of Planet Earth ISBN 978-3-030-17774-4 ISBN 978-3-030-17775-1 (eBook) https://doi.org/10.1007/978-3-030-17775-1 Mathematics Subject Classification (2010): 34Lxx, 35-xx, 35Pxx, 35Q30, 35Q31, 35Q40, 45Jxx, 60Jxx, 62F15, 65Cxx, 65M06, 15A16, 15A18 © Springer Nature Switzerland AG 2019 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG. The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

To my wife, Aldjia, who supported and tolerated my long hours in the office. To my children, Ines and Mohand. To the memory of my parents.

Preface

Mathematicians and physicists are no strangers to the challenging problems of the century, posed by global warming and climate change. It is even more meaningful for them to get immersed into the core of climate modeling and into improving the tools that are used for day-to-day weather forecasting and climate predictions. While enormous progress has been made in terms of improving the fidelity of these large computer models to represent the climate system, climate models are still struggling in accurately representing the mean and variability of rainfall and wind patterns associated with waves and clouds in the tropics. This volume reviews a series of mathematical models for tropical convection and convectively coupled waves, developed recently by the author and his collaborators. It blends in physical intuition, gained from observations, with the state-of-the-art mathematical tools such as Galerkin truncation, linear stability analysis, numerical methods, and stochastic processes to capture and understand the subtle physics of wave-convection interactions and cloud dynamics. It is intended for graduate students in applied mathematics and physics interested in climate modeling and tropical meteorology, in particular. It assumes prior knowledge of basic theory of partial differential equations, numerical analysis, and fluid mechanics. Exposure to meteorology and stochastic processes can be helpful but not necessary. With the warming climate threat, it is of vital importance for the human kind to keep improving the only tools we have in order to understand climate change and make faithful forecasts. Despite the world-wide coordinated efforts and subsequent improvements in our climate modeling capabilities, during the last decades, current climate models suffer from various biases that make them less reliable in terms of predicting local changes in the climate system, needed for decision-making at the regional and the continental level. The long-range predictions of natural phenomena, such as major droughts and floods, or the intensity and frequency of tropical cyclones remain very uncertain, especially in some of the highly populated tropical regions such as India and the Philippines. There is strong scientific evidence that these biases are associated with the way in which climate models treat physical processes associated with tropical precipitation and clouds. Progress in this regard is tied to a better understanding of these processes and how the associated unresolved vii

viii

Preface

convective motions interact with synoptic and planetary-scale waves. Exposing this subject to the wider audience, including mathematicians and physicists, will not only increase our chances of getting there much quicker. Moreover, organized tropical convection and convectively coupled tropical waves offer many challenging scientific and mathematical problems that deserve attention from some of the brightest minds. They involve a wide spectrum of physical phenomena ranging from the phase change of water occurring at the micro-scale to the role of the change in sign of the Coriolis force at the equator that drives very unique wave phenomena occurring at various scales and to the direct interactions of clouds with atmospheric dynamics and the climate system as a whole. Various teleconnections patterns varying on weekly to monthly time scales are known to exist between tropical wave dynamics and mid to high-latitude weather and climate variability. As such, a good understanding and better representation by climate models of these waves are very important for weather predictions all over the globe on multiple time scales ranging from days to seasons. Victoria, BC, Canada

Boualem Khouider

Acknowledgments

This volume and the contained research wouldn’t have been possible without the involvement of many collaborators, students, and post-docs. I am especially grateful to my longtime collaborator and post-doc mentor, Prof. Andrew Majda, who made me discover this fascinating subject. I have been nourished through the years by being able to interact and occasionally collaborate with some of the most famous experts in the field such as George Kiladis, Mitch Moncrieff, and Joseph Tribbia, I was standing on the shoulders of giants. Special thanks to the Banff International Research Station for giving me the opportunity to organize workshops on these topics which allowed me to broaden my horizons and interact with many people. The Pacific Institute for the Mathematical Sciences CRG on mathematical problems in climate modeling was one of the precursors for the success of this research. I acknowledge the generous support from the Natural Sciences and Engineering Research Council of Canada, the Canadian Foundation for Climate and Atmospheric Research, and the Indian Institute for Tropical Meteorology, without which, many projects in this research won’t have been even started.

ix

Contents

Part I Background and Preliminaries 1

The Governing Equations and Dry Dynamics . . . . . . . . . . . . . . . . . . . 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 The Primitive Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Equatorially Trapped Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 The Shallow Water Approximation . . . . . . . . . . . . . . . . . . . 1.4 The Vertical Normal Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Riemann Invariants and Meridional Expansion . . . . . . . . . . . . . . . . 1.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 3 6 8 8 16 18 20

2

Moisture and Moist Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Moisture Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Equation of State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2 Cloudy and Precipitating Atmosphere . . . . . . . . . . . . . . . . . 2.2 The First Law, the Second Law, and Other Thermodynamic Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Phase Change . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Conserved Moist Thermodynamic Variables . . . . . . . . . . . . . . . . . . 2.5 Processes Leading to Saturation and Formation of Clouds . . . . . . . 2.6 Dry and Moist Adiabatic Lapse Rates . . . . . . . . . . . . . . . . . . . . . . . . 2.7 Stability of Moist Air . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

23 23 24 25

3

Observations of Tropical Climate Dynamics and Convectively Coupled Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Historical Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Clouds in the Tropics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 The Madden-Julian Oscillation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Convectively Coupled Equatorial Waves . . . . . . . . . . . . . . . . . . . . . 3.5 Self-similar Multiscale Convective Systems . . . . . . . . . . . . . . . . . .

26 27 30 32 34 35 38 41 41 43 48 50 52 xi

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4

Contents

Introduction to Stochastic Processes, Markov Chains, and Monte Carlo Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Computing with Random Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 Law of Large Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.2 Monte Carlo Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.3 Inverse Transform Method . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.4 Acceptance-Rejection Method . . . . . . . . . . . . . . . . . . . . . . . 4.2 Introduction to Markov Chains and Birth-Death Processes . . . . . . 4.2.1 Discrete Time Markov Chains . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 The Poisson Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.3 Continuous Time Markov Chains . . . . . . . . . . . . . . . . . . . . . 4.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

57 57 58 58 59 60 61 61 63 64 68

Part II The Deterministic Multicloud Model 5

6

7

8

Simple Models for Moist Gravity Waves . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Wave-CISK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 A Simple Adjustment Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Mass-Flux Schemes and WISHE Waves . . . . . . . . . . . . . . . . . . . . . 5.4 Stratiform Instability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.1 Nonlinear Simulations with the Stratiform Model: The Beautiful and the Ugly . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

75 76 78 82 87

The Multicloud Model with Congestus Preconditioning . . . . . . . . . . . 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 The Model Formulation and Main Closure Assumptions . . . . . . . . 6.3 Linear Stability Analysis and the Congestus Preconditioning Instability Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Nonlinear Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

99 99 100

Convectively Coupled Equatorial Waves in the Multicloud Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Governing Equations and Method of Solution . . . . . . . . . . . . . . . . . 7.2 Uniform Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Meridional Shear Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Effect of a Vertical Shear Background . . . . . . . . . . . . . . . . . . . . . . . Convective Momentum Transport and Upscale Interactions in the MJO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 CMT in a Simulated Kelvin Wave: A Thought Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 CMT Parameterization Prototypes . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.1 Some Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.2 Stochastic Model of Majda and Stechmann . . . . . . . . . . . . .

92 97

105 111 117 118 123 127 130 133 133 135 140 140 143

Contents

xiii

8.3.3

CMT Parameterization with a Third Baroclinic Feedback and its Effect on an MJO like disturbance . . . . . . . . . . . . . . 8.4 Multiscale Waves in MJO Envelope and CMT Feedback . . . . . . . . 8.4.1 A Simple Multiscale Model with Features of CMT . . . . . . 8.4.2 Equatorial Waves in a Realistic MJO Background . . . . . . . 9

Implementation of the Multicloud Model in an Aquaplanet Global Climate Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1 Introduction: The Cumulus Parameterization Problem . . . . . . . . . . 9.2 The Multicloud Model as a Simplified Cumulus Parameterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3 Uniform Aquaplanet Simulations: MJO v.s. Convectively Coupled Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4 Warm Pool Simulations: MJO Initiation and Northward Propagation, Monsoon Climatology, and Variability . . . . . . . . . . . .

146 151 151 155 163 163 165 168 172

Part III The Stochastic Multicloud Model: SMCM 10

Stochastic Birth and Death Models for Clouds . . . . . . . . . . . . . . . . . . . 10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 A Birth-Death Model for Convective Inhibition . . . . . . . . . . . . . . . 10.2.1 The Microscopic Stochastic Model for CIN: Ising Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2.2 The Coarse Grained Mesoscopic Stochastic Model: Birth-Death Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2.3 Gillespie’s Exact Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . 10.2.4 Numerical Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3 The Stochastic Multicloud Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3.1 The Stationary Distribution, Cloud Area Fractions, and the Equilibrium Statistics of the Lattice Model . . . . . . . . . . . . . 10.3.2 Coarse Grained Birth-Death Stochastic Model and the Mean Field Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3.3 The Deterministic Mean Field Equations and Numerical Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4 Coupling the SMCM to a Cumulus Parameterization . . . . . . . . . . . 10.4.1 The SMCM in a Toy-GCM . . . . . . . . . . . . . . . . . . . . . . . . . . 10.5 Inference of SMCM Parameters from Data . . . . . . . . . . . . . . . . . . . 10.5.1 The Bayesian Inference Procedure . . . . . . . . . . . . . . . . . . . . 10.5.2 The Giga-LES Inferred Time Scales . . . . . . . . . . . . . . . . . . 10.6 SMCM with Nearest Neighbour Interactions . . . . . . . . . . . . . . . . . . 10.6.1 The Multiple Particle Hamiltonian . . . . . . . . . . . . . . . . . . . . 10.6.2 Coarse Grained Approximation . . . . . . . . . . . . . . . . . . . . . . 10.6.3 Mean Field Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.6.4 Numerical Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

185 185 189 189 192 195 196 198 201 203 206 209 210 215 217 219 221 223 225 225 226

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Contents

11

Implementation of the SMCM in a Global Climate Model . . . . . . . . 11.1 SMCM-HOMME Aquaplanet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.1 Case of a Uniform Forcing . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Role of Stratiform Heating on Organized Convection . . . . . . . . . . . 11.2.1 MJO and Kelvin Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2.2 Case of Asian Monsoon-Like Warm Pool . . . . . . . . . . . . . .

229 229 230 233 233 234

12

SMCM in CFS: Improving the Tropical Modes of Variability . . . . . 12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2 Implementation of SMCM in CFS . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2.1 Prescribed Vertical Profiles of Moistening/Drying . . . . . . . 12.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3.1 Improved Climatology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3.2 Improved Tropical Modes of Variability . . . . . . . . . . . . . . . 12.3.3 MJO and Monsoon ISO Propagation . . . . . . . . . . . . . . . . . . 12.3.4 Further Physical Aspects of the MJO Signal . . . . . . . . . . . . 12.3.5 Rainfall Event Distribution . . . . . . . . . . . . . . . . . . . . . . . . . .

245 245 247 251 252 253 254 257 263 265

Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

269

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

277

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

299

Acronyms

LCL LFC LNB CAPE CAPEl CIN GCM HOMME CFSv2 WRF CRM LES ITCZ MCMC MJO ISO ABL WIG EIG RCE CMT MCM SMCM MRG MGW MGWI BSISO MISO LPS CCEW CISK

Lifted condensation level Level of free convection Level of neutral buoyancy Convective available potential energy Low-level CAPE Convective inhibition General circulation (or Global climate) model High-order method modeling environment Climate Forcast System version 2 Weather Research and Forecasting Model Cloud resolving model Large eddy simulation Intertropical convergence zone Markov chain Monte Carlo Madden-Julian oscillation Intra-seasonal oscillation Atmospheric boundary layer Westward inertia-gravity wave Eastward inertia-gravity wave Radiative-convective equilibrium Convective momentum transport Multicloud model Stochastic multicloud model Mixed Rossby-gravity wave Moist gravity wave Moist gravity wave instability Boreal summer intra-seasonal oscillation Monsoon intra-seasonal oscillation Monsoon low-pressure system Convectively coupled equatorial wave Convective instability of the second kind xv

xvi

WISHE WTG GARP-GATE or GATE TOGA-COARE CINDY-DYNAMO

Acronyms

Wind-induced surface heat exchange (also called windevaporation feedback) Weak temperature gradient approximation Field experiment: Global Atmospheric Research Program - GARP Atlantic Tropical Experiment Field experiment: Tropical Ocean Global Atmosphere Coupled Ocean-Atmosphere Response Experiment Field experiment: Cooperative Indian Ocean Experiment on Intraseasonal Variability—Dynamics of the MJO

Part I

Background and Preliminaries

Part I introduces some basic background material which is important for the understanding of the subsequent-core chapters of this book. The expert reader can skip directly to Part II.

Chapter 1

The Governing Equations and Dry Dynamics

1.1 Introduction The equations of motion that govern atmospheric (and also oceanic) flows are based on the theory of fluid mechanics comprising the Euler and/or Navier Stokes equations which model conservation of mass, momentum, and energy [27, 11] of a Newtonian fluid such as air and water. The so-known hydrostatic primitive equations are derived from these basic laws of physics after some major simplifications or approximations taking into account the particular topology of planetary flows [61, 216, 166]. Namely, the lower atmospheric layer, known as the troposphere, where weather and important climatic systems take place, is very thin compared to the extent of Earth’s surface and as such horizontal flows (parallel to Earth’s surface) are much more important on the global scale than vertical motions. Vertical motions are much weaker on planetary and synoptic scales (∼1000 km - 5000 km) and strong vertical flows are more confined to smaller convective scales on the order of the tropospheric depth. Thus, for the flows of interest for global climate and medium- to long-range weather modelling, for instance, the vertical acceleration is neglected in the momentum equations resulting in what is known as the hydrostatic balance equation ρ 1 ∂ p , (1.1) g =− ρ0 ρ0 ∂ z where g is the gravity acceleration constant, ρ  and p are density and pressure perturbations from fixed background values ρ0 and p0 . The hydrostatic approximation can be derived by simple scaling arguments based on the above observations [48]. This equation should not be confused with the static pressure equation which states that the vertical pressure gradient, at any given height, acts as a constant reaction force balancing the weight of the air column above that point. Instead it expresses

© Springer Nature Switzerland AG 2019 B. Khouider, Models for Tropical Climate Dynamics, Mathematics of Planet Earth 3, https://doi.org/10.1007/978-3-030-17775-1 1

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4

1 The Governing Equations and Dry Dynamics

the deviations from this background due to large-scale wave motions, for example. The total air pressure and total air density satisfy,

ρ = ρ0 + ρ  , p = p0 + p , gρ0 = −

∂ p0 > 0. ∂z

Note the > inequality sign in this equation which contrasts with (1.1) where the pressure gradient can have an arbitrary sign. The quantities ρ0 and p0 are often referred to as a background stratification which is removed from (1.1) for convenience. Another important simplification, which also results from the small aspect ratio of the tropospheric layer, amounts to neglecting the variations of density in the momentum and mass conservation equations, except for the buoyancy term. This is known as the Boussinesq approximation [48, 61, etc.]. Moreover, for the diabatic flows of concern, density variations are due mostly to temperature variations except for pressure changes due to adiabatic lifting and sinking. It is thus convenient to introduce the potential temperature, θ , as the temperature a parcel of air would have if it is displaced adiabatically (without exchange of heat with its environment) to a reference pressure ps (usually the surface pressure). Using the equation of state for an ideal gas, p = Rρ T , where T is temperature and R is the gas constant, and the first law of thermodynamics, namely that the change in internal energy at constant pressure is balanced by the work done by pressure, Cp dT = ρ −1 d p, we arrive at [48, 285]  κ ps θ= T, (1.2) p where κ = RC−1 p ≈ 0.286 with C p is the heat capacity at constant pressure of dry air. If we express the ideal gas law in terms of potential temperature for both the actual environment and the background state, then we obtain 1+ or

ρ θ p p = (1 + )(1 + )(1 + )κ p0 ρ0 θ0 p0 ρ θ p = − + (1 − κ ) + h.o.t ρ0 θ0 p0

where h.o.t is the combination of all high order terms of the Taylor expansion, in terms of ρ  /ρ0 , p /p0 , θ  /θ0 ,which are assumed to be relatively small. If, in addition, we ignore the term involving p /p0 under the grounds that pressure perturbations adjust quickly due to fast sound waves, then to a first order approximation, the hydrostatic balance in (1.1) can be rewritten as g

θ 1 ∂p . = θ0 ρ0 ∂ z

(1.3)

We note that in (1.3) θ0 can be taking to be a constant independent of height, θ0 = 300 K, without aggravating the underlying approximation.

1.1 Introduction

5

θ0 The vertical gradient of the background stratification ddz defines the stability of the background state for when it is positive it relates to a situation where lighter (warmer) fluid lies on top of heavier (colder) fluid. In this case we have a stable stratification. The normalized quantity

N2 = g

1 d θ0 θ0 dz

is the square of the so-called Brunt-Va¨ısa¨ıla buoyancy frequency. When N 2 is positive, it defines the frequency at which a vertically displaced, by an infinitesimal amount, parcel of air would oscillate around its initial position. Indeed, if z is the vertical displacement of the parcel, from its original position, zi = 0, then Newton’s second law states z¨ = −g

θ0 (z) − θ0 (zi ) 1 d θ0 ⇐⇒ z¨ + N 2 z = 0, ≈ −zg θ0 θ0 dz

which is indeed the equation of motion – harmonic oscillator with the oscillation frequency N. In the free troposphere, above the neutrally stable mixed boundary layer (and below the tropopause), N is approximately constant. Unless, otherwise stated, here, we set N = 0.01 sec−1 , which is equivalent to a potential temperature increase in height of roughly 3 K km−1 , consistent with observations [61, 48]. Geophysics texts often refer to the lapse rate or dry adiabatic lapse rate (more on this in Chapter 2), denoted by Γd as being the rate at a which a parcel of (dry) air cools down as it is raised adiabatically, i.e., along the curve of constant θ . The value of Γd can thus be derived directly from (1.2) by making use of the hydrostatic equation gρ = − ddzp . The case of moist air is slightly different and will be discussed separately in the next chapter. We have  κ   ps dT g dθ −1 =κ − =⇒ Γd = C−1 0= p g ≈ 10 K km . dz p dz Cp In some textbooks the quantity S =

dT0 dz

− Cgp is called the static stability, where T0 is

the actual background or environmental temperature profile. Note that N 2 ∝ S. The given background is said to be stable if S ≥ 0 and unstable if S < 0. When S = 0 the profile is said to be neutrally stable. The latter situation is often observed in the mixed planetary boundary layer (PBL). In addition to the force of gravity, which induces important wave motions known as gravity waves, which in turn play a central role of this book, another important force is due to Earth’s rotation and is known as the Coriolis force. When the equations of motion are written in a frame of reference which rotates with Earth’s angular velocity Ω = 1 day−1 , the most significant component of the Coriolis force points in the vertical, and acts on the moving fluid as a force in the direction perpendicular to its horizontal velocity everywhere but at the equator where this effect is zero, because the rotation axis is perpendicular to Earth’s radius there. Much like the force of gravity, which induces hydrostatic balance dynamics, the Coriolis force induces

6

1 The Governing Equations and Dry Dynamics

balanced flows, where this force is balanced by the horizontal pressure gradient, towards which atmospheric and oceanic flows adjust relatively rapidly, especially in mid- and high-latitudes where this force is significant. Similarly to the hydrostatic balance, the so-called geostrophic balance relation is behind an important approximation, known as quasi-geostrophic dynamics, leading to quasi-two-dimensional flows where a single variable, known as the potential vorticity, is conserved and plays the role of relative vorticity in 2D Euler equations and fully determines the flow field [61, 166, 216]. Much like the hydrostatic balance induced oscillations, deviation from geostrophy induces synoptic to planetary scale waves, the scales at which Coriolis force is important, known by the generic name of Rossby waves, which are responsible for most of the mid-latitudes weather patterns. Because the Coriolis force vanishes at the equator, the quasi-geostrophic approximation breaks down in the tropics and leads the way to much richer and more complex dynamics. As we will see in the next sections and throughout this volume, because of this property, the equator acts as a wave-guide to a large family of waves that travel in both directions along the equator. These equatorially trapped waves interact with moist convection and water vapour, nonlinearly with each other, and with extra-tropical Rossby waves. During the early development of meteorology as a scientific discipline, tropical dynamics and tropical weather were thought to be chaotic and unpredictable but it is now widely accepted that most of tropical precipitation is associated with synoptic and planetary scale waves coupled with convection, i.e., cloud systems. The two-way coupling between these waves and convection and their contribution to global dynamics are the main topic of this volume.

1.2 The Primitive Equations In addition to the hydrostatic (1.3) and the Boussinesq approximation mentioned above, we introduce the beta-plane approximation for the equations of motion on a rotating spherical coordinate system [61]; the Earth’s surface is assumed to be flat along a certain strip surrounding the globe parallel to the equator. The effects of Earth’s curvature are ignored except for the Coriolis parameter which is assumed to take the linear form f (y) = β y, where y is the geodesic distance from the equator. The resulting equations, known as the hydrostatic equatorial beta-plane primitive equations, then read

∂v ∂v + v · ∇v + w + β yv⊥ = −∇p + Sv , ∂t ∂z ∂ p gθ = , ∂z θ0 ∂θ ∂θ ∂ θ¯ + v · ∇θ + w +w = Sθ , ∂t ∂z ∂z ∂w = 0. ∇·v+ ∂z

(1.4)

1.2 The Primitive Equations

7

Here, v = (u, v)T is the horizontal velocity field, where u is the zonal (along the equator) component, v is the meridional (North-South) component, and w is the vertical velocity. The corresponding coordinates, x, y, z, which represent longitude, latitude, and altitude, are directed from West to East, from South to North, and from bottom to top, respectively, while t > 0 is time. We fix z = 0 at the surface or bottom of the troposphere and z = H is at the top of the troposphere (H = 16 km). The operator ∇ = (∂x , ∂y ) is the horizontal gradient and X · Y is the dot product of the two-dimensional vectors X and Y . Also v⊥ = (−v, u) is the perpendicular velocity vector—a rotation by 90 degrees to the left. The scalars p and θ represent the pressure (per unit mass) and potential temperature perturbations, respectively. The equations in (1.4) have been normalized by the density, which is assumed constant except when multiplied by the gravity acceleration, according to the Boussinesq approximation. The total potential temperature, including a background that depends only on height, is given by

θ total (x, y, z,t) = θ0 + θ¯ (z) + θ (x, y, z,t) where θ0 = 300 Kelvin is a reference constant temperature and θ¯ defines the vertical ¯ profile-background stratification and satisfies N 2 = θg0 ∂∂θz > 0 where N is the BruntV¨ais¨al¨a buoyancy frequency. The first equation in (1.4) represents the conservation of horizontal momentum with β = 2Ω /R ≈ 2.2804 × 10−11 m−1 s−1 is the gradient of the Coriolis parameter at the equator. Here, Ω = 2π /24 hr−1 is the frequency of Earth’s rotation and R = 6378 km is its radius. The second equation expresses hydrostatic balance, and the third one is the energy equation written in terms of the potential temperature. The last one is the divergence free constraint which expresses conservation of mass, often called the continuity equation, under the Boussinesq approximation. The terms on the right-hand side, Sv and Sθ , are the sources and sinks of momentum and energy, respectively. In particular, Sθ represents convective heating from clouds and radiative cooling (and heating), which constitute the main sources of energy input in the tropics. The equations in (1.4) are augmented with the rigid lid boundary conditions w|z=0,H = 0.

(1.5)

While this assumption is certainly valid at the surface z = 0, it is only approximately valid at the top of the tropopause; It is merely justified by the sharp transition of the static stability there, making the stratosphere much more stable and leaving the weather flows confined to the tropospheric layer. Gravity waves of various sorts do penetrate to the stratosphere and stratospheric flows are believed to influence the climate system on longer time scales through processes such as the quasi-biennial oscillation and radiative forcing [68, 10]. However, these two-way interactions between the troposphere and the stratosphere can be ignored for our purpose. Periodic boundary conditions are naturally assumed in the x-direction with a period equalling the equatorial circumference of the globe while they are left somewhat arbitrary in the y-direction; either vanishing solutions at y −→ ±∞ or no-flow boundaries at

8

1 The Governing Equations and Dry Dynamics

y = ±L for some distance L from the equator will be considered when dealing with practical situations, case by case.

1.3 Equatorially Trapped Waves In this section we will derive some special solutions for the linearized version of the primitive equations (1.4), on the form of plane waves along the equator. They are known as the equatorially trapped waves because as they travel zonally, along the equator, they are meridionally confined to the vicinity of the equator over some distance set by a dimensional parameter known as the Rossby deformation radius. Before we make a formal mode expansion in the vertical direction in order to devise a systematic Galerkin truncation methodology expanding the equations (1.4) into a set of vertical mode equations, we will first use a method developed in [187] of transforming the primitive equations into a set of shallow water equations. Notice that the intuition behind this transformation comes from the fact that the primitive equations are meant to model planetary flows with horizontal scales of hundreds of kilometres while their vertical scales are limited by the tropospheric height of up to 16 km. Indeed, from the perspective of these planetary motions, the troposphere is a thin layer of fluid surrounding the solid Earth.

1.3.1 The Shallow Water Approximation We consider the state of rest, u = v = w = 0, with a constant background stratification, set by the buoyancy frequency N = 0.01 s−1 [61], as our equilibrium solution about which the equations of motion are linearized and simple wave solutions are sought. With the nonlinear advection terms, v · ∇v, w ∂∂ vz , v · ∇θ , w ∂∂θz , and the forcing terms, Sv , Sθ , all set to zero, we combine the three last equations to obtain an evolution equation for the pressure.    ∂ ∂ ∂w 1 ∂p = 0. (1.6) + ∂ t ∂ z N 2 (z) ∂ z ∂z  From dimensional analysis point of view, the differential operator ∂∂z N12 ∂∂z acts on the pressure perturbation p and transforms it into a dimensionless field. It is thus reasonable to look for special solution for which this operator is merely a measurement of the pressure perturbation in terms of a reference scale given by gH ∗ , which is simply the hydrostatic pressure per unit mass of a hypothetic atmospheric column of height H ∗ . In other words, H ∗ sets the scale of the pressure perturbation under consideration. Mathematically, we need to choose H ∗ so that λ = gH1 ∗ is an eigenvalue of our differential operator and the special solutions are simply the associated eigenfunctions:

1.3 Equatorially Trapped Waves

−

9

  ∂ 1 ∂p ∂p = 0, z = 0, H. = λ p, 0 ≤ z ≤ H; 2 ∂z N ∂z ∂z

(1.7)

We will postpone the choice of H ∗ and the computation of these eigenfunctions for a moment and we will first look at the implications of this simplification on their horizontal variations. For a fixed scale height H ∗ , the unforced-linearized primitive equations reduce to the shallow water-like equations,

∂u ∂p −βy v = − ∂t ∂x ∂v ∂p +βy u = − ∂t ∂y   ∂p ∂ u ∂ v + c2 + = 0, ∂t ∂x ∂y

(1.8)

where for convenience the u and v equations were written separately and the last equation is nothing but (1.6) combined with the continuity equation. For a reason √ to be clarified later, we also set c2 = gH ∗ , where c = gH ∗ is known as the gravity wave speed, according to the standard theory of shallow water equations. We note however that one of the main differences between (1.8) and the standard shallow water equations resides in the physical meaning of the constant H ∗ . While in the standard case, the equivalent of H ∗ is the mean water column height, around which surface waves oscillate and propagate, here H ∗ is typically internal to the typical tropospheric height, up to H ≈ 16 km, and only sets the vertical scale of the pressure disturbances that may not be directly seen on top of the fluid column. The associated waves are called internal waves as opposed to surface waves that are seen at the beach, for example. We now return to the eigenvalue problem (1.7) to express the value of the equivalent depth H ∗ and the phase speed c. The general solution is p(z) = cos(

mπ √ 2 mπ z ), = N λ , m = 1, 2, · · · . H H

This yields λ = π 2 m2 /(H 2 N 2 ) = (gH∗)−1 = c−2 . Thus, we have for m = 1, for example, H ∗ = 259 m and c ≈ 50 m s−1 . The integer values of n define the various internal modes also known as the baroclinic modes. More on this in the next section, where we consider a more systematic mathematical treatment and thus justification of the elliptic eigenvalue problem. 1.3.1.1 The Kelvin Wave We begin by looking at the special case when the meridional velocity is zero, v = 0. In particular, this reduces the system in (1.8) to a single equation for u alone, since p and u become constrained by the v-equation. As we will see throughout this volume, despite its simplicity the resulting solution plays a major role in tropical dynamics—

10

1 The Governing Equations and Dry Dynamics

it is perhaps the most important of all tropical waves observed in nature. The vequation yields a balance between the meridional pressure gradient and the zonal velocity, ∂p βy u = − . ∂y The remaining equations for u and p suggest a travelling wave solution along the equator. Through separation of zonal and meridional variables x and y, we let u(x, y,t) = f (x,t)φ (y), v = 0, and p(x, y,t) = g(x,t)ψ (y). Substituting this ansatz in (1.8) yields, ft φ (y) = −gx ψ (y),

β y f φ (y) = −gψy , gt ψ (y) + c2 fx φ (y) = 0. The first and third equations suggest that φ and ψ are constant multiples of each other and without loss of generality, we can assume ψ = φ . Thus, combining the two equations leads to the wave equation, ftt = c2 fxx , which has simple solutions consisting of both left going and right going waves of the form f (x,t) = h(x ± ct) and g(x,t) = ∓c h(x ± ct). The second equation then gives an equation for φ . βy φy = ± φ , c 

2 whose elementary solution takes the form, φ (y) = exp ± β2cy . The plus sign corresponds to a westward propagating wave is unbounded as |y| −→ ∞ while the minus sign yields the solution   β y2 1 u(x, y,t) = p(x, y,t) = h(x − ct) exp − . (1.9) c 2c The associated flow perturbation structure is displayed in Figure 1.1 when the 2π function h takes a wave form h(z) = sin(kz), for a fixed wavenumber k = 8000 −1 km , or an 8000 km wavelength. Here, the gravity wave speed is set to c = 50 m s−1 , corresponding to the first baroclinic mode. As expected, the main characteristics of the Kelvin wave include a perfect symmetry about the equator, eastery winds (winds pointing to the West) are aligned with a low-pressure disturbance and westerlies (winds pointing to the East) are on top of a high pressure ridge. And more importantly, the wave is confined to the vicinity of the Equator (y = 0) with both the winds and pressure anomalies being strongly attenuated as we move away from the tropical region. The wave is said to be trapped in the vicinity of the equator. The distance by which the wave gets attenuated is determined by the dimensional parameter

1.3 Equatorially Trapped Waves

11

Le = c/β ≈ 1480 km1 , when c = 50 m s−1 or an equivalent depth H ∗ ≈ 255 m. Le is known as the equatorial Rossby deformation radius. It sets a natural length scale for synoptic scale dynamics which are responsible for weather disturbances in the tropics occurring on time scales of two to ten days. 1.3.1.2 Rossby, Gravity, and Mixed Rossby-Gravity Waves We now return to the general case with v = 0, and introduce the reference scales c = −1 √ ∗ gH for velocity, Le = c/β for length, and Te = Le /c = cβ for time. When written in these units the equation in (1.8) will remain unchanged except for the constants β and c2 which will factor out leaving the same set of equations but with β and c2 being both unity. In some sense, the chosen reference scales are the scales at which the beta-effect and gravity wave dynamics have equal importance. It has to be pointed out however that different equivalent depths can result in substantially different gravity wave speeds and consequently different length and time scales.

Fig. 1.1 Pressure anomalies (contours, negative anomalies are dashed) and velocity profile (arrows) for the Kelvin wave in (1.9). Here, c = 50 m s−1 , h(z) = sin(kz), k = 2π /8000 km−1 . 1

Le =



2c/β , in some textbooks.

12

1 The Governing Equations and Dry Dynamics

As noted above, the value H ∗ ≈ 255 m (c = 50 m s−1 ) yields Le ≈ 1480 km and consequently Te = 8.2263 hours. However, for H ∗ = 64 m we get c = 25 ms−1 , Le = 1047 km, and Te = 11.6338 hours. Smaller equivalent depth leads to waves with slower propagation speeds, smaller wavelengths, and longer time scales. As can be expected this will have important implications on the interaction dynamics of waves of different equivalent depths. We seek simple wave solutions on the form u(x, y,t) = u(y) ˆ exp [i(kx − ω t)], ˆ exp [i(kx − ω t)] with i2 = −1. v(x, y,t) = v(y) ˆ exp [i(kx − ω t)], and p(x, y,t) = p(y) When this ansatz is inserted into (1.8), we get the following eigenvalue problem: −iω uˆ − β yvˆ + ik pˆ = 0 −iω vˆ + β yuˆ + pˆ = 0

(1.10)



−iω pˆ + c (vˆ + iku) ˆ = 0. 2

The primed variables denote y-derivatives. The three equations in (1.10) are combined to obtain a standalone equation for v. ˆ  2  ω β 2 y2 kβ  2 vˆ + 2 − k − 2 − vˆ = 0, c c ω

ω uˆ = iβ yvˆ + k p, ˆ 

2 2  (ω − c k ) pˆ = i c kβ yvˆ − ω c vˆ . 2

(1.11)

2 2



 2 By the change of unknowns v(y) ˆ = cH( βc y) exp − β2cy , the first equation reduces to H  − 2ξ H  + μ H = 0,

2  for H(ξ ) with μ + 1 = βc ωc2 − k2 − kωβ = Te2 ω 2 − Le2 k2 − c ωk or simply, μ + 1 =  ω 2 − k2 − ωk in the chosen non-dimensional frame of reference, and ξ = βc y is the non-dimensional distance from the equator. This is the well-known Hermite’s equation which admits an infinite number of polynomial solutions provided μ is an even non-negative integer [144]. This results in an infinite set of solutions,       β β y2 kβ c ω2 2 y exp − − k − = 2n + 1, n = 0, 1, 2, · · · vˆn = cHn , c 2c β c2 ω (1.12) The Hermite polynomials are given by Hn (ξ ) = (−1)n exp(ξ 2 )

dn exp(−ξ 2 ) dξ n

and satisfy the recursive relations Hn+1 (ξ ) = 2ξ Hn (ξ ) − Hn (ξ ), n = 0, 1, 2, · · · ,

1.3 Equatorially Trapped Waves

13

Hn (ξ ) = 2nHn−1 (ξ ), n = 1, 2, · · ·

(1.13)

The first few are given by H0 (ξ ) = 1, H1 (ξ ) = 2ξ , H2 (ξ ) = 4ξ 2 − 2, · · · The restriction to polynomial solutions for Hermite’s equation guarantees that vˆ 2 vanishes exponentially fast, as e−ξ /2 , in the same fashion as the Kelvin wave solution in (1.9). Nonetheless, the Hermite polynomials modulated by exp(−ξ 2 /2) (the functions defining the vˆn solutions in (1.12)) constitute a complete set for square integrable functions, so in principle any other physically sound solution can be expressed as a (possibly infinite) sum of such solutions. The modulated Hermite polynomials are called the parabolic cylinder functions and will be discussed in some more details below. The relationship between frequency and wavenumber in (1.12) is called the dispersion relation and it can be used to express, for instance the phase speed, cs = ω /k, and the group velocity, cg = ddkω , of the waves as functions of the wavenumber. While the phase speed determines the propagation speed of peaks and troughs of individual waves, the group velocity merely determines the speed of wave packets or rather the energy carried by the waves [61]. When the phase speed varies with k so that waves of different wavelengths travel at different phase speeds, the associated waves are said to be dispersive. For dispersive waves, the group velocity and the phase speed are distinct. The integral parameter n in (1.12) fixes the mode of meridional structure and the shape of the dispersion curves. It is often called the meridional index. For fixed values of n and k, ω is determined by solving the cubic equation in (1.12). There are three distinct solutions. For n ≥ 1, they are approximately given by2

ωr ≈ −

k k2 + 2n + 1

, ωge =



k2 + 2n + 1, ωgw = −



k2 + 2n + 1.

(1.14)

The first solution, ωr , corresponds to slowly (low-frequency) westward-moving waves known as equatorial Rossby waves by analogy to the geostrophic Rossby waves found in midlatidudes. The other two correspond, respectively, to eastward and westward moving inertio-gravity waves, they are the equivalents of Poincar´e waves [61]. The case n = 0 is somewhat special. The corresponding dispersion relation easily √ factors out to yield the three solutions: ω0 = −k and ω± = 12 k + ± 12 k2 + 4. The first solution yields a non-dispersive westward moving, Kelvin-like wave, which is discarded [216]. In fact, if this solution is inserted back into (1.11), then it yields v ≡ 0 which takes us back to the case of the Kelvin wave considered above. The two other solutions corresponding to a westward moving and an eastward moving waves, depending on the sign of k, behave like Rossby and eastward inertio-gravity waves, respectively. They are often called Yanai waves [294, 229, 253] but they also Consider the two cases when ω > 1 separately to simplify the dispersion relation to approximately −k2 − ωk = 2n + 1 and ω 2 − k2 = 2n + 1, respectively.

2

14

1 The Governing Equations and Dry Dynamics

bear the name mixed Rossby-gravity waves because they branch out accordingly depending on the sign of k. The dispersion relation for the Kelvin wave can be recovered from (1.12) by setting n = −1 to yield the single solution ω = k. To help visualize the different cases, the dispersion relation curves corresponding to the first few values of n (n = −1, 0, 1, 2, 3) are plotted in Figure 1.2. Once the meridional velocity v is fixed by the parabolic cylinder function corresponding to a given meridional mode n ≥ 0, the pressure and zonal velocity fields can be recovered through the last two equations in (1.11). The flow structure of the Yanai and the n = 1 gravity and Rossby waves are displayed in Figure 1.3. They should be compared with that of the Kelvin wave in Figure 1.1. It is worthwhile

Fig. 1.2 Dispersion relation curves of equatorially trapped waves corresponding to the first few meridional indices, n = −1, 0, 1, 2, 3, comprising the Kelvin, the Yanai (or mixed Rossby-gravity), Rossby, and inertio-gravity waves. Here, the equivalent depth H ∗ = 255 m and the frequency units are in cycles per day (CPD) and k is expressed in terms of the planetary wavenumber corresponding to a maximum wavelength P = 40000 km (k = 1), the Earth’s perimeter at the Equator.

1.3 Equatorially Trapped Waves

15

Fig. 1.3 Pressure anomalies (contours, negative anomalies are dashed) and velocity field (arrows) for eastward and westward Yanai or n = 0 eastward inertia-gravity (EIG) and mixed Rossby-gravity (MRG), n = 1 EIG, n = 1 westward inertia-gravity (WIG), and n = 1 and n = 2 Equatorial Rossby (ER) waves, as indicated. Here, c = 50 m s−1 , k = 2π /8000 km−1 .

noting that the top panels in Figure 1.3 display the typical flow structure of inertiagravity waves which is characterized by a velocity field that crosses pressure isolines at right angles while the bottom panels display the typical regime behaviour of Rossby gyre motions with velocity arrows running parallel to the pressure contours,

16

1 The Governing Equations and Dry Dynamics

away from the equator (where β y = 0). The Rossby wave flows are reminiscent of mid-latitude geostrophic motions where the pressure field acts as a stream function, while the gravity waves exhibit mainly divergence driven flows. The convergent nature of gravity wave flows is even more dramatic for westward inertia-gravity waves as we will see in the subsequent chapters. Moreover, notice that as expected the Yanai waves behave as Rossby waves when they are moving westward and as an inertia-gravity wave when they move eastward. For this reason the eastward moving branch is sometimes called the n = 0 inertia-gravity (EIG) wave. Finally notice that when n is odd the associated equatorial waves are symmetric about the equator and when n is even they are anti-symmetric.

1.4 The Vertical Normal Modes We return to the primitive equations (1.4) and assume separation of variables in the vertical direction on the basis that on the scale of interest the tropospheric column behaves as a cohesive thin shallow layer of fluid where vertically propagating waves travel infinitely fast, due to the hydrostatic approximation. ˜ y,t)ψw (z), v(x, y, z,t) = v˜ (x, y,t)φ (z), w(x, y, z,t) = w(x, p(x, y, z,t) = p(x, ˜ y,t)φ (z), θ (x, y, z,t) = θ˜ (x, y,t)ψθ (z).

(1.15)

Plugging this ansatz into the linearized and unforced version of (1.4) yields d ψw dφ 1 d θ¯ = C1 φ , = C2 ψθ , ψw = ψθ , dz dz Γ0 dz and

∂ v˜ + β y˜v⊥ = −∇ p, ˜ ∂t gθ˜ ∇ · v˜ +C1 w˜ = 0, C2 p˜ = , θ0 ∂ θ˜ + Γ0 w˜ = 0. ∂t

(1.16)

Here, C1 ,C2 are constants to be determined and Γ0 is a reference background stratification. Combining the φ and ψ equations leads to the eigenvalue problem   dφ d Γ0 d φ = 0 when z = 0, H, (1.17) − = λ φ , with dz θ¯z dz dz which is precisely equivalent to the one set forward for the pressure perturbations, based solely on physical intuition, to derive the shallow water approximation and the notion of equivalent height. The Neumann boundary conditions follow from the rigid lid conditions imposed on w.

1.4 The Vertical Normal Modes

17

For an arbitrary atmospheric background the eigenvalue problem in (1.17) can be solved numerically and the modes of vertical structure can be recovered [104, 105]3 . However, the salient features of the primitive equations (1.4) can be studied and major scientific insight can be gained by considering the case of constant stratification, θ¯z = Γ0 . In this case the vertical eigenstructure functions are simple trigonometric functions [166], πz φ (z) := φm (z) = cos(m ), H

πz m2 π 2 ψw (z) = ψθ (z) := ψm (z) = sin(m ), λm = , H H2 √ with C1 = −C2 = λm = mHπ and m = 1, 2, 3, · · · . The equations (1.16) then reduce to the shallow water equations in (1.8) with a phase speed cm = NH/(mπ ) and an N2H2 equivalent depth Hm∗ = gm 2 π 2 for m = 1, 2, · · · . As anticipated for H = 16 km and N = 0.01 s−1 , we get

c1 ≈ 50.93 m s−1 , H1∗ = 265 m, c2 = 25.46 m s−1 , H2∗ = 66 m. The vertical modes associated with m = 1, 2, · · · are known as the baroclinic or internal modes of vertical structure. In the special case of constant background stratification, the mode m = 0 is special and is associated with a flow field with zero vertical velocity and zero temperature perturbation. It is called the barotropic or external mode. The corresponding equations can be recovered directly from the primitive equations by a simple vertical average over the tropospheric depth [166]. Because the vertical velocity averages out, the barotropic flow is divergence free and can be expressed in vorticity stream function form. The equatorial beta-plane barotropic equations are given by

∂ v0 + v0 · ∇v0 + β yv⊥ 0 = −∇p0 + B + Sv ∂t ∇ · v0 = 0.

(1.18)

Here, B represents the upscale momentum transport from internal modes onto the barotropic mode. For the time being, we assume B = 0 and Sv = 0 and look for special solutions for these equations. Set v0 = (u0 , v0 ) and let ξ0 = ∂x v0 − ∂y u0 and ∂x ψ0 = v0 , ∂y ψ0 = −u0 be the associated vorticity stream function formulation. We have

∂ ξ0 ∂ ψ0 + J(ψ0 , ξ ) + β = 0, ∂t ∂x

(1.19)

where J( f , g) = ∂x f ∂y g − ∂y f ∂x g. Solutions on the form ψ0 (x, y,t) = A exp[i(kx + ly − ω t)], where k, l are respectively the zonal and meridional wavenumbers and ω 3

These numerical solutions will be utilized later in this book when the multicloud model is used as a cumulus parameterization in a climate model.

18

1 The Governing Equations and Dry Dynamics

is frequency, are possible provided ω = − k2β+lk 2 . This is the dispersion relation for planetary Rossby waves that can propagate westward and in both northward and southward towards the poles. Notice the similarity between √ this dispersion relation and those of the baroclinic Rossby waves in (1.14) with 2n + 1 playing the role of meridional wavenumber [169, 17, 116]. Importantly, while these wave solutions are often viewed as small, linear, disturbances, they happen to be also exact solutions to the full nonlinear barotropic equations in (1.18); this is because the Jacobian determinant J(ψ0 , ξ0 ) vanishes as a results of the functional dependence of ψ0 and ξ0 . In the light of this exposition, it is natural to envision a systematic expansion of the solutions of the primitive equations onto a barotropic and infinite sequence of baroclinic modes, each associated a single equivalent height Hm∗ and the √ with ∗ associated gravity wave speed cm = gHm . ∞

v = v0 +

∑ vm φm (z),

m=1

p = p0 +

∞

∞

∞

m=1

m=1

m=1

∑ pm φm (z), w = ∑ wm ψm (z), θ = ∑ θm ψm (z). (1.20)

In the linear case, each vertical wave mode m, m = 0, 1, 2, ·, can be solved independently and a solution for the primitive equations can, in principle, be obtained via the superposition principle. But in the fully nonlinear case, various modes can interact with each other through advection nonlinearity or through convection processes. This vertical mode decomposition will be the backbone of most of the studies presented in the volume as some of the most prominent features of tropical dynamics and tropical-extra tropical interactions are rooted through such multimode interactions.

1.5 Riemann Invariants and Meridional Expansion Consider the shallow water-like system√in (1.18) associated with a vertical mode m ≥ 1 and a gravity wave speed cm = gHm∗ . Here we re-derive the equatorially trapped wave solutions introduced in Section 1.3.1 using a more direct method that can be generalized to the case of a non-uniform background and applied as such in Chapters 7 and 8. Namely, we use a Galerkin projection procedure based on the parabolic cylinder functions to the full PDE system without passing through a reduction into a single high order equation for the meridional velocity. Instead, we take advantage of the availability of Riemann invariants, i.e., a change of variable solutions that transform the shallow water equations into a skew-symmetric system of PDEs for which the parabolic cylinder functions are of central importance. In particular, with this procedure the equatorial beta-plane shallow water equations reduce to a set of PDEs in (x,t) consisting of a single scalar equation for the Kelvin wave alone, a 2-by-2 system for the mixed Rossby-gravity waves, and an infinite

1.5 Riemann Invariants and Meridional Expansion

19

sequence of triads each with an eastward and a westward gravity and a Rossby wave solutions. Mathematically speaking, this demonstrates that the Galerkin projection in y and Fourier transforms in x and t can be applied to the linear beta-plane shallow water equations, in any order. Let (um , vm , pm ) be the corresponding velocity components and pressure, respectively. We introduce the Riemann invariants qm = um + c1m pm and rm = um − c1m pm . We have   ∂ qm ∂ qm ∂ β + cm + cm − y vm = 0, ∂t ∂x ∂ y cm   ∂ rm ∂ rk ∂ β − cm + cm + y vm = 0, (1.21) ∂t ∂x ∂ y cm     ∂ vm cm ∂ β β cm ∂ + + y qm − − y rm = 0. ∂t 2 ∂ y cm 2 ∂ y cm

 The differential operators L± = ∂∂y ± cβm y are the raising and lowering operators of quantum associated with the parabolic cylinder functions Dn (y) =  mechanics

 β β y2 Hn cm y exp − 2cm , n = 0, 1, 2, · · · , where the Hn ’s are the Hermite polynomials. Based on the properties in (1.13), we can show that  L− Dn (y) = −

β Dn+1 (y), n = 0, 1, 2 · · · c

 and L+ Dn (y) = 2n

β Dn−1 (y), n = 1, 2, · · · c

Dropping the subscript m for simplicity and assuming the meridional expansions, ∞

q=

∑ q˜n (x,t)Dn (y),

n=0

r=

∞

∞

n=0

n=0

∑ r˜n (x,t)Dn (y), v = ∑ v˜n (x,t)Dn (y),

we obtain [166] a sequence of decoupled PDEs, including a single linear wave equation for q˜0 ,

∂ q˜0 ∂ q˜0 + ck = 0, ∂t ∂x a two-by-two system coupling q˜1 and v˜0 ,

∂ q˜1 ∂ q˜1 + cm − ck β v˜0 = 0, ∂t ∂x ∂ v˜0 + ck β q˜1 = 0, ∂t and a triad coupling q˜n , v˜n−1 , r˜n−2 ,

20

1 The Governing Equations and Dry Dynamics

∂ q˜n ∂ q˜n + cm − cm β v˜n−1 = 0, ∂t ∂x ∂ r˜n−2 ∂ r˜n−2 − cm + 2(n − 1) cm β v˜n−1 = 0, ∂t ∂x ∂ v˜n−1 1 + n cm β q˜n − cm β r˜n−2 = 0, ∂t 2 for all n ≥ 2. Seeking plane wave solutions of the form φ˜ = φˆ exp[i(kx − ω t)] in the above equations yields the dispersion relations of the Kelvin waves, the Yanai or MRG waves, and the cubic relation in (1.12) resulting in the Rossby and the east and westward inertia-gravity waves, with meridional index n − 1. This systematic meridional expansion using the parabolic cylinder functions leading naturally to the equatorially trapped wave solutions will be exploited in the subsequent chapters to expand complex models for convectively coupled waves in order to perform adequate linear analysis.

1.6 Exercises 1. Hydrostatic Balance Consider the (unforced) rotating Boussinesq equations

∂v ∂v + v · ∇v + w + f v⊥ = −∇p, ∂t ∂z ∂w ∂v ∂p ρ + v · ∇w + w =− +g ∂t ∂z ∂z ρ0 ∂w = 0. ∇·v+ ∂z

(1.22)

Here, v = (u, v) is the horizontal velocity field, w is the vertical velocity, and p and ρ are the pressure (normalized by background density) and density perturbations with respect to the background hydrostatic pressure, p(z), ¯ and background ¯ density ρ0 + ρ¯ (z) ≈ ρ0 , respectively, such that ρ0 + ρ¯ (z) = 1g d p(z) dz while f is the Coriolis parameter which is assumed constant, for simplicity, and g is the constant gravitational acceleration. We introduce the non-dimensional variables, t  , x , y , z , v , w , p , such that t = T t  , (x, y) = L(x , y ), v = Uv , z = Hz , w = W w , p = Pp . Here, T, L,U, H,W, P are the dimensional reference scales for time, horizontal length, horizontal winds, height, vertical velocity, and pressure, respectively. Rewrite the system in (1.23) in terms of the primed (non-dimensional) variables and show that if T = L/U = W /H and P = L2 /T 2 , then a. horizontal and vertical convergence terms in the continuity equation remain in balance (i.e., of the same order of magnitude),

1.6 Exercises

21

b. the horizontal acceleration, i.e., time change and nonlinear advection of horizontal momentum is balanced by the horizontal pressure gradient, which is also balanced by the Coriolis term f v⊥ if in addition we have T = f −1 . c. If we, moreover, assume ε = HL  1, i.e., the horizontal scale is much larger than the vertical scale, then the Boussinesq equations can be written in the new non-dimensional units as

∂v ∂v + v · ∇v + w + v⊥ = −∇p, ∂t ∂z   ∂w ∂v ρ 1∂p + v · ∇v + w + g ε =− ∂t ∂z ε ∂z ρ0 ∂w = 0. ∇·v+ ∂z

(1.23)

where g = gT /U is the non-dimensionalized acceleration of gravity. d. Deduce that when ε  1, the vertical gradient of pressure is, to a first order approximation, balanced by the sole buoyancy acceleration and that the vertical acceleration (vertical momentum time derivative and advection) terms can be neglected, thus leading to the primitive equations in (1.4). 2. Consider the eigenvalue problem in (1.7). a. Show that if N 2 > 0 and constant, then this problem admits an infinite number of eigenvalues and eigensolutions given by

λn0 =

n2 π 2 nπ z ), n = 0, 1, 2, · · · , φn0 (z) = cos( H 2N2 H

b. Now let φn (z) and λn , n = 0, 1, 2, · · · denote the eigensolutions and eigenvalues of the eigenvalue problem in (1.7) in the general case of a non-constant but positive N 2 (z). Assume that the eigenvalues are non-zero. Show that the eigenfunctions φn (z), n = 0, 1, · · · are orthogonal, i.e., 1 H

 H 0

φn (z)φm (z)dz = 0 ⇐⇒ n = m.

c. Compare (qualitatively) the eigensolutions in (2a) with those obtained by Kasahara and Puri [104] for a typical tropical atmospheric stratification. Identify the barotropic and baroclinic modes. d. Possible student project. Pick a typical atmospheric stratification with a specific, non-constant, N 2 (z) profile and solve the eigenvalue problem in (1.7) numerically (using finite differences, for example, as Kasahara and Puri did) and compare to the analytic solutions in (2a). 3. Carry out the details leading to the simplification of the equation for v(y) ˆ in (1.11) into Hermite’s equation and show that the vˆn (y) solutions, i.e., the parabolic cylinder functions, are orthogonal to each other. 4. Derive the approximate dispersion relations in (1.14).

22

1 The Governing Equations and Dry Dynamics

5. Show that the equations in (1.16) are indeed the shallow water equations in (1.8) with c = cm = HN mπ , m = 1, 2, · · · . 6. Find the plane Rossby wave solutions and the corresponding dispersion relation for the barotropic mode equations in (1.18), when linearized about a state of rest and with B = Sv = 0. Show that they are also solutions to the nonlinear barotropic equations. Identify the analogies and discrepancies between baroclinic and barotropic Rossby waves. Find the meridional wavenumbers for which the barotropic Rossby wave dispersion relations match those of the first baroclinic equatorial Rossby wave meridional indices 1,2, and 3. 7. Use Matlab or an equivalent computer software to draw and discuss the vertical structure of the flow field (in terms of the horizontal and vertical velocities and temperature perturbations) of the first and second baroclinic modes. 8. Use Matlab or an equivalent computer software to plot the velocity arrows and pressure contours for a total flow composed of a first baroclinic and a second baroclinic Kelvin waves of the same wavelength, if the second baroclinic has a relative strength of 0.25 and is lagged by a phase difference φ0 . Consider the cases when φ0 = 0, φ0 = π /8, φ0 = π /4, φ0 = π /2. Consider both the horizontal and vertical structures of velocity arrows and pressure contours.

Chapter 2

Moisture and Moist Thermodynamics

2.1 Moisture Variables The most popular state variables that are used to define a thermodynamic system are the pressure, the temperature, and the composition. For our purpose, the atmospheric composition is divided into two main constituents: dry air, comprising nitrogen, oxygen, carbon dioxide, etc., and water in its various states or phases including water vapour (referred to herein as moisture), liquid water in the form of suspended cloud or rain droplets, ice crystals, and snowflakes. There are other constituents like dust and aerosols but these can be neglected in the thermodynamical context [48]. While the composition of dry air in the atmosphere can be assumed constant, for the purpose of weather and climate studies, except of course for carbon dioxide and other greenhouses gases which have important consequences on climate change, the water content is constantly changing, which in itself constitutes an important challenge in atmospheric science and so remains a major uncertainty in climate and weather forecasting models. At first, we assume that the atmosphere is unsaturated so that the liquid and ice water constituents can be ignored. There are two variables usually used to measure the water vapour content in the atmosphere, the mixing ratio, r, and the specific humidity, q, which are given by

ρv ρd ρv ρv q= = ρd + ρv ρ r=

(2.1) (2.2)

with ρd , ρv are, respectively, the densities of dry air and water vapour and ρ is the total density of the mixture. Under normal atmospheric conditions, ρv 0, (2.33) N = − Γd + T T dz dr p,T dz

36

2 Moisture and Moist Thermodynamics

where the ideal gas law, p = ρ RT , and the hydrostatic balance, d p = −ρ gdz, have been used to express the partial derivative in (2.32). If the vapour mixing ratio is neglected, then we will recover the of N 2  definition ρ g dθ g dθ dT 2 introduced previously, in Section 1.1, namely N = T −Γd + dz = θ dz ≈ θ0 dz . An efficient way to take into account the moisture effects on N 2 is through the use of the virtual potential temperature as a state variable [48, 61]. We have N 2 = θgv ddzθv . When the displaced parcel becomes saturated with water vapour, the notion of stability discussed above is no longer valid. Because the amount of water vapour that a parcel of air can hold decreases with decreasing pressure, when the lifted parcel becomes saturated, condensation will take place. The latent heat release associated with the condensation process is used to heat the parcel hence (partially) compensating for the dry adiabatic cooling. Thus, if the temperature gradient is larger than the moist lapse rate, Γs , defined in equation (2.30) the risen parcel will be restored by the buoyancy force towards its original position, otherwise it will continue to rise. Whereas, since the amount of water vapour that a sinking parcel can hold always increases, the parcel becomes unsaturated, and the conditions of stability are the same as for unsaturated air. Thus, if the temperature gradient lies between the dry and moist lapse rates, Γd < dT /dz < Γs , the atmosphere is unstable only for saturated parcels of air displaced upward. In this case we say that the atmosphere is conditionally unstable. This behaviour merits close attention. Under normal atmospheric conditions, a moist parcel will almost always become unstable if it risen sufficiently high—until saturation. However the energy necessary to achieve saturation by lifting can be very high. The atmospheric situations under which parcels can be lifted by external means or their own inertia until they reach a level at which they become positively buoyant are uncertain and can depend on various factors. This is one of the reasons—though not the most complex—why atmospheric convection is one of the most difficult problems in atmospheric science and climate modelling in particular. As pointed out previously, the level at which the risen parcel becomes saturated is called the lifted condensation level (LCL) while the level at which the parcel becomes positively buoyant is called the level of free convection (LFC). This and the above discussion about the stability of moist air is pictured in Figure 2.1. The work done by the buoyancy force on a rising parcel, when it is vertically displaced upward from its given initial position, is known as the convective available potential energy or CAPE1 for short:   Z2  α p − αa + CAPE = g dZ (2.34) αa Z1 1 CAPE is normally defined as the integral of positive buoyancy between LFC and LNB (see below). Here we make a more general introduction of this concept for the sake of completeness. Although, maybe not practical in general there is no reason why we can’t define the available potential energy of a moist parcel of air that gets entrained by an existing updraft or one that gets detrained into the environment below LFC. In fact the latter situation is exploited in Chapters 6 through 12 to introduce the notion of low-level CAPE for parcels detraining below the freezing level and lead to the formation of congestion clouds.

2.7 Stability of Moist Air

37

where Z1 and Z2 are the altitudes of the parcel at its initial and final positions, respectively. For atmospheric flows of interest, the contribution of pressure perturbation to buoyancy is negligible as the time scales of sound waves are much faster than those of atmospheric convection, we obtain

Unconditionally stable sounding

Z

Conditionally unstable sounding Moist adiabat Unconditionally unstable sounding Dry adiabat

Temperature Fig. 2.1 A cartoon of the stability of moist air. An unsaturated parcel displaced vertically from its position, which doesn’t exchange mass or heat with the environment, follows the dry adiabat. A saturated parcel on the other hand follows the moist adiabat. The dashed lines illustrate several scenarios of atmospheric soundings that are either stable for displacement of both saturated and unsaturated parcels, conditionally stable profiles that are stable for unsaturated parcels but unstable for saturated parcels (conditionally unstable), and finally those that are unstable for both unsaturated and saturated displacement.

CAPE ≈

 Z2 Z1

1 1 g 1 + r αa



d αd ds∗

 p,r∗

(s∗p − s∗a ) +



d αd dr∗

+



 p,s∗

− αa

(r∗p − ra∗ )

dZ.

An other simplification comes out by neglecting changes in density induced by water vapour, which is equivalent to ignoring the last term inside the integral. We have,  +   Z2 d αd 1 ∗ ∗ g (s − s ) dZ. CAPE ≈ αd ds∗ p p a Z1

38

2 Moisture and Moist Thermodynamics

From the Maxwell relations [48], we get     dT d αd α = = Γs ∗ ds p d p s∗ g 

dT where Γs = dZ Thus, we get

 s∗

is the moist lapse rate. CAPE ≈

 Z2 Z1

Γs (s∗p − s∗a )+ dZ.

(2.35)

Furthermore, we can express entropy in terms of equivalent potential temper ∗ − ln θ ∗ ≈ ature s = c p ln θe and the integrant can be replaced by c pΓs ln θep ea

 ∗ − θ ∗ ) , where θ is a reference temperature, typically θ = 3000 K. c pΓs θ10 (θep 0 0 ea ∗ can be approxMoreover, since θe∗ is almost conserved during adiabatic lifting, θep imated by its value at the original level, often within the mixed planetary boundary layer. This approximation/assumption is systematically used in Chapter 5 to define CAPE fluctuations when designing simplified models for clouds and moist convection. In the above derivation of CAPE, we assumed that the buoyancy of the rising parcel is positive between the levels Z1 and Z2 . However, in reality rising parcels of air often experience a region of negative buoyancy before they reach their LFC level and they stop rising when they reach the first level at which buoyancy becomes again non-positive. This latter is called the level of neutral buoyancy (LNB). In the climate modelling community CAPE refers to the above integral between LFC and LNB. Z1 = Zl f c and Z2 = Zlnb . The equivalent integral restricted to the levels below LFC where B < 0 is called CIN which is short for convective inhibition. CIN is an energy barrier the risen parcel needs to overcome before it can reach its LFC and produce deep convection, cumulus clouds, and ultimately rain. Figure 2.2 illustrates the concepts of CAPE and CIN as the areas between the saturated equivalent temperatures of the rising parcel and the environment. More discussion on CIN and CAPE will be given in Chapter 11, where a stochastic model for CIN, based on a lattice interacting particles, is presented.

2.8 Exercises 1. Derive the formula for the potential temperature in (2.14) and compare the exact value of θ and the approximation on the right of that equation for an ambient temperature of T = 290 K and mixing ratio of r = 0.01 at pressure p = 850 hPa. Assume a reference pressure of p0 = 1000 hPa. 2. Derive the saturation vapour pressure at the triple point in (2.20).

2.8 Exercises

39

3. Summarize the steps leading to the expression for the equivalent potential temperature in (2.24) and assess the given approximations. 4. Derive equation (2.29) that leads to the definition of the moist lapse rate.

Fig. 2.2 Illustration of the concepts of CAPE and CIN. The solid curve represents the equivalent potential temperature of the surrounding environment, a.k.a. the sounding. The dashed curve represents the path followed by the rising parcel of moist air. The blue line represents the dry adiabatic lapse rate, along which the unsaturated parcel cools down without any change in composition. The red line is the moist adiabat line, which the parcel follows after saturation while constantly changing composition as the excess of water vapour condenses and leaves the parcel. The dotted line is the saturation equivalent potential temperature of the environment. CAPE and CIN are proportional to, respectively, the positive and negative areas between the dashed and dotted curves, as illustrated.

Chapter 3

Observations of Tropical Climate Dynamics and Convectively Coupled Waves

3.1 Historical Notes The first evidence of equatorially trapped waves in observational records appeared in 1966 in the work of Yanai and Maruyama [294], at the same time as the theoretical work of Matsuno [187]. Yanai and Maruyama [294] found signals of wavelike motion with strong cross equatorial wind in US Navy stratospheric wind data (which were apparently used to monitor nuclear activity during the cold war) when they were looking for evidence of eddy momentum transport as a plausible energy source for the quasi-biannual oscillation (QBO) in the equatorial stratosphere [188]. These waves correspond to the mixed Rossby-gravity waves from the Matsuno theory, which are also sometimes called Yanai-Maruyama or simply Yanai waves. Two years later Wallace and Kousky [271] (see also [83]) published their work on the discovery of Kelvin waves in the tropical stratosphere, which unlike those identified earlier by Yanai and Maruyama they are characterized by dominating zonal winds in phase with pressure perturbations. They were also motivated by the search for an energy source for the QBO in the form of wave eddy momentum. Systematic use of spectral analysis for the study of equatorially trapped waves begins with the advent of data from satellites and station networks of synoptic scale extent in the Pacific Ocean during the early 1970s and late 1960s. These early works, which are conducted independently by Yanai’s group at the University of Tokyo and Wallace and his colleagues at the University of Washington in Seattle are summarized in a review paper by Wallace [270]. Spectral peaks in the 4–5 days range were identified and were associated with westward equatorially trapped MRG and Rossby waves. It has been established that those waves are baroclinic in nature as they are characterized by a wind reversal between the lower and upper troposphere above 300 hPa, and positive temperature disturbance in the upper troposphere. Madden and Julian [164] applied spectral analysis to time series of winds and pressure anomalies (∼10 years as opposed to only 4–6 months used in the studies reported in [270]) collected by an observation network at Canton Islands in the

© Springer Nature Switzerland AG 2019 B. Khouider, Models for Tropical Climate Dynamics, Mathematics of Planet Earth 3, https://doi.org/10.1007/978-3-030-17775-1 3

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3 Observations of Tropical Climate Dynamics and Convectively Coupled Waves

equatorial Central pacific. They found evidence of a much slower mode of variability with broadband spectral peaks corresponding to periods of 40–60 days. They then later established its global scale circulation cell pattern [163]. This wave disturbance is now known as the Madden-Julian oscillation or MJO for short and it has been recognized as the dominant low-frequency mode of tropical atmospheric variability [300]. While Madden and Julian [163] have rightfully suggested the link between the MJO and convection, it is the satellite images of long wave radiation (OLR) and the first international field campaign in the equatorial Atlantic (GARP-GATE) that helped identify the multiscale organization of convection and revealed the persistence of cloud clusters [201, 203, 202, 90, 185, 85, 86, 88, 92]. The work of Nakazawa [204] particularly revealed that the cloud clusters are sometimes organized into cloud superclusters, which in turn are a manifestation of synoptic scale and planetary scale wave disturbances including the Kelvin, MRG, Rossby, and westward inertio-gravity waves as well as the MJO. An example of such organization is given in Figure 3.1 which displays streaks of outgoing long wave radiation (OLR) as observed from satellite, and used as a proxy for deep convective clouds, revealing the signature of the Madden-Julian oscillation as it propagates eastward at roughly 5 m s−1 over the Indian Ocean-Western Pacific warm pool in which are embedded Kelvin waves. Kelvin waves are particularly seen to persist over the Central Pacific Ocean and other parts of the tropical belt. There are also other waves that appear to propagate westward, especially between 80°W and the Greenwich line. These are most likely African Easterly waves that are not part of the equatorially trapped waves but are believed to play a central role in Hurricane initiation [45]. A systematic link between the observed spectral signatures of these cloud superclusters and the shallow water modes of Matsuno was first established by Takayabu [252, 251] and further elucidated by Wheeler and Kiladis [283], see also [284]. As shown in Figure 3.2, Wheeler and Kiladis showed that a spectral analysis of the brightness temperature (or OLR for that matter) reveals the persistence of various spectral power peaks along the dispersion relations of equatorially trapped waves of Matsuno summarized in Figure 1.2, after removing a background red noise. We note that in Figure 3.2, branches corresponding to various equivalent heights are superimposed and that the wave spectrum has been split into symmetric and antisymmetric waves. Spectral peaks corresponding to Kelvin, mixed Rossby gravity (MGR), n = 0 inertio-gravity (which is nothing but the eastward moving MRG), n = 1 equatorial Rossby, n = 1 westward inertio-gravity (WIG) waves are evident together with an MJO peak (recognizable from its ∼ 40 days period and planetary wavelength, resulting in a slowly eastward moving signal) which appears to be isolated from the dispersion relation curves of the dry equatorially trapped waves given by the Matsuno theory in Chapter 1. So in a sense the MJO needs something extra, and cannot be explained by just the pure linear shallow water dry dynamics. We now know the answer is moisture coupling with dynamics via convection and precipitation.

3.2 Clouds in the Tropics

43

Fig. 3.1 Hovm¨oller diagram of brightness temperature averaged between 2.5°S and 7.5°N for the period of January to April 1987. This picture depicts wave-like patterns moving in both east-west directions of various sizes. Particularly highlighted are a big blurb corresponding to a slowly moving Madden-Julian oscillation “lump” of convection and streaks of convectively coupled Kelvin waves (dashed lines) sometimes embedded with the MJO. Within these synoptic scale super-cloud clusters we can see cloud clusters moving in the opposite direction. Courtesy of George Kiladis [127].

3.2 Clouds in the Tropics Figure 3.3 is a satellite image of the globe taken by the International Weather Satellite (https://www.accuweather.com/en/world/satellite) depicting instantaneous global cloud coverage among other interesting features such as ice cover at the South and North poles. One of the most striking aspects of this picture concerns the distribution of cloud cover displaying three distinct areas of cloudiness comprising the stormy regions of mid-latitudes in both hemispheres corresponding to dominant

44

3 Observations of Tropical Climate Dynamics and Convectively Coupled Waves

Fig. 3.2 OLR power spectrum for 22 years yearly observational record revealing spectral peaks of convectively coupled equatorial waves including the MJO. Top panel is for symmetric wave signals and bottom is for anti-symmetric waves. A red noise background has been removed. Courtesy of George Kiladis [283].

weather patterns in those regions and a thin region lounging the equator which constitutes the inter-tropical convergence zone (ITCZ). In between, we can distinguish cloud free regions corresponding to some of the major deserts such as the Sahara in North Africa and Middle East, Northern Australia, the Gobi desert, and Southern United States and Mexico. This cloud distribution is in fact associated with a major global mean circulation pattern known as the Hadley cell, consisting of air converging and rising over the cloudy regions of the tropics, diverging aloft and then sinking in the subtropics (around ±30°) bringing down dry air from the upper troposphere. While the sinking air motions are forcing deserts beneath them, the water vapour that rises in the tropics condenses and falls back in situ as precipitation. The corresponding mean motions relative to the mid-latitude storms and low-pressure synoptic eddies characterized by air rising at ± 45–50°and descending at the same latitudes as the tropical-Hadley cell are known as the Ferrel cells [267, 122], as depicted in Figure 3.4(A). Also as shown in Figure 3.4, the regions of rising motion (in grey) are roughly associated with the cloudy regions in Figure 3.3 while the subsidence regions (white areas) are over the deserts. A closer look at the cloud features in Figure 3.3 reveals some important differences between the mid-latitude and tropical cloud coverage. While in the midlatitudes we see mainly smooth patches, extending over very wide areas associated with eddy-like patterns, the cloud coverage in the tropics seems to be more rugged and intermittent with blobs of different sizes which are sometimes connected and sometimes isolated. There is in particular a large blob over the Indian Ocean and another over the Western Pacific which may be associated with a wave-train of two MJOs travelling together. In fact, we now know, thanks to many observational campaigns, that clouds in the tropics are organized into a hierarchy of scales ranging

3.2 Clouds in the Tropics

45

from individual clouds of roughly 1–10 km to cloud clusters of 100–500 km, to synoptic scale superclusters of 1000–5000 km, and to planetary scale envelopes of 10,000–20,000 km [183]. Each of these hierarchical entities is associated with a well-established flow pattern, starting from the individual clouds which are characterized by a convective core of rising air and the surrounding subsiding air generically referred to as updrafts and downdrafts, respectively [92, 306]. However, this dichotomy into three categories and assumed spectral gaps are rather arbitrary. Overlapping across these temporal and spatial scales often occurs in nature [232]. The cloud clusters form within mesoscale circulation systems characterized with a dominant tilted upward convective motion and a large subsidence region of stratiform anvils [89, 85, 146, 194]. The superclusters are associated with convectively coupled equatorial waves while the planetary scale cloud cluster envelopes are signatures of intra-seasonal oscillation flows such as the MJO and poleward moving monsoon ISOs. The last two are discussed in more detail in the next two sections. For decades if not centuries, the meteorological community thought that organized convection in the tropics, in the backdrop of the multiscale convective systems, is associated only with deep penetrative cumulonimbus clouds also known as hot towers that are responsible for the heavy rain in the tropics. Important corrections to this thinking came from lessons learned from the GARP-GATE1 and the TOGA-COARE field experiment [281]2 conducted during the 1970s and early 1980s, over the tropical Atlantic and the early 1990s, over the Western Pacific Ocean, respectively3 . It was almost immediately realized that stratiform clouds are also abundant in the tropics and contribute to both surface precipitation and the dynamics of mesoscale systems [87, 183]. However, it took sometime for the scientific community to understand that cumulus congestus clouds, which detrain in the mid-troposphere around the 0°C level, are also important for the dynamics of convectively coupled flow systems [156, 152, 157]. Shallow cumulus and deep cumulonimbus clouds were long been thought to be important for tropical dynamics, the former because of their radiative impact due to their abundance and their modulation of turbulent eddies induced by updrafts and downdrafts and the latter because of the associated vigorous convective flows and heavy precipitation. Johnson et al. [99] are the first to postulate the multi-modal nature of tropical cloudiness and revealed the importance of the third cumulus cloud type, namely cumulus congestus which detrain in the middle of the troposphere. They identified three stability layers within the troposphere each associated with a detrainment level of a certain class or type of clouds. Shallow cumulus clouds detrain near the boundary layer top as they are capped by the trade wind inversion layer, which constitutes an important energy barrier for convection. When the troposphere is dry, convective parcels that make it through the inversion layer (either because they have enough kick-off energy or the inversion itself has been eroded by preceding parcels) find 1

Global Atmosphere Research Program-GARP Atlantic Tropical Experiment. The Tropical Ocean Global Atmosphere Coupled Ocean Atmosphere Response Experiment. 3 Further knowledge, especially regarding the dynamics of the MJO, has been gained during the most recent Dynamo-Cindy field campaign and more will be gained with the upcoming Years of the Maritime Continent experiment. 2

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3 Observations of Tropical Climate Dynamics and Convectively Coupled Waves

Fig. 3.3 Satellite image showing various levels of cloudiness of the globe on the date of 2 Nov 2013, 18:00 UTC as indicated. We distinguish three main bands of cloudiness. Two along the mid-latitude storm tracks and a third surrounding the tropics. As opposed to mid-latitudes, clouds in the tropics appear to organized on multiple scales involving cloud clusters and superclusters, especially over the Indian Ocean and Western Pacific warm pool where synoptic to planetary scale organization can be seen. The Intertropical Convergence Zone appears a thin strip over the Eastern Atlantic. International Weather Satellite Images. Source google.com.

themselves in a hostile–dry environment and began to cool and lose buoyancy because of evaporation and detrain and dissipate before they hit the upper troposphere where they can get boosted by latent heat of freezing. The growth of deep convective towers is stopped by the tropopause and leave behind them stratiform anvils in the upper troposphere. As summarized in Figure 3.5, the three cloud types (four if we add stratiform clouds) that characterize organized convection in the tropics abundantly coexist in the tropics. The subtropics are mainly subjected to shallow cumulus clouds when it is not a complete desert (as it occurs in the Eastern Pacific Ocean off the Coast of California—one major foggy regions—there is also a lot of fog in the coastal regions of the Arabic peninsula). This concept of a trimodal nature of tropical clouds was first introduced by Johnson et al. [99], following their study of observational records from the TOGA-COARE field campaign. It corrects for a too long-standing misconception of tropical meteorology of neglecting the role of cumulus congestus cloud types. This is a slightly different view from the multicloud model idea discussed here (starting from Chapter 6), where the three cloud types refer to congestus, deep, and stratiform (instead of shallow cumulus because

3.2 Clouds in the Tropics

47

Fig. 3.4 (A) mean circulation patterns depicting the streamlines of the Hadley (red) and Ferrel cells (blue), overlaying the tropics and mid-latitudes, respectively; the other red cells are called the polar cells. The slanted cells (C) are streamlines in the moist-entropy latitude coordinate and depict moist and warm air leaving the tropics and dry and cool air sinking at the poles (the circulation is counter clock-wise along the solid lines and clock-wise around the dashed lines). (c) It depicts regions of upward motion (shaded) while the middle right panel (d) shows climatological zonalvertical wind vectors depicting the Walker cell along the tropical belt consisting of warm and moist air rising over the Indian Ocean Western-Pacific warm pool and sinking over the Eastern Pacific, coinciding with the cloud free region between 100W and 80W in (c) and Fig. 3.3. The bottom panel is a cartoon of the distribution of cloud populations within the Hadley and Walker circulations.

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3 Observations of Tropical Climate Dynamics and Convectively Coupled Waves

Fig. 3.5 A cartoon of clouds showing deep cumulonimbus towers with stratiform anvils at their tops, near the tropopause dominating the scenery within the tropical region together with cumulus congestus clouds with tops around the freezing level and shallow topped by the trade inversion layer. This cartoon is taken from [99] and summarizes the findings in that study. From [122].

of their dynamical importance. Shallow cumulus clouds are considered more or less dynamically passive background for organized tropical convective systems).

3.3 The Madden-Julian Oscillation As already anticipated the Madden-Julian oscillation is a large-scale disturbance of convection, winds, temperature, and moisture in the tropical atmosphere that propagates eastward at roughly 5 m s−1 . It originates over the central Indian Ocean, crosses the Maritime continent, and extends to the Western and, sometimes, the Central Pacific [300]. As first described by its discoverers [163], the MJO circulation pattern consists of rising motion in the convection centre and descending air during the suppressed phase. It goes through a cycle of initiation, amplification, and demise as pictured on the left panels of Figure 3.6, successively from F to E.

3.3 The Madden-Julian Oscillation

49

The two-dimensional view shown on the left of Figure 3.6 can be misleading because the MJO has a very complex three flow structure characterized by two vortex quadruples straddling the equator one in the lower troposphere and one aloft as seen on the right panel. In the lower troposphere, there are two cyclones in the lower troposphere at the west of the convection centre one on each side of the equator, funnelling in (relatively dry) air from behind through the two Rossby gyres and a strong westerly wind burst from behind, running along the equator. Two opposite anti-cyclones (typically weaker) are seen to the east of the convection centre converging in moist air to supply the convective activity. The role played by moisture convergence (the term q∇ · v) versus moisture advection (the term v · ∇q) in MJO dynamics is very subtle; while convergence supplies moisture largely compensating for the precipitation sink, moisture advection is mainly a drying process as it brings relatively dry air to the convection centre. The eastward propagation of the MJO is most likely due to the asymmetry between the strong westerlies in the back versus the easterlies in front. However, the physical parameters that are responsible for this asymmetry are not yet elucidated. This asymmetry presumably favours the formation of congestus clouds east of the convection centre which preconditions the environment and helps the convection move to the east, via the multicloud paradigm. The four vortices are reversed in the upper troposphere suggesting a first baroclinic vertical structure. However, the MJO structure is much more complex as the upper and lower tropospheric flow patterns are not exactly in phase but the MJO displays a strongly tilted structure in winds, temperature, and moisture profiles [126]. More discussion on the MJO dynamics and structure can be found in the incoming chapters, namely Chapters 9, 11, and 12. The MJO has a complex structure and lacks a counterpart from the dry equatorial wave dynamics, unlike the convectively coupled waves. The search for a simple model to explain the dynamics and propagation of the MJO is still a major problem in theoretical research in meteorology although some important progress has been made in the last decade or so [e.g. 239, 2, 272, 18]. However, some new promising ideas are emerging such as the MJO-skeleton model of Majda and Stechmann which views the MJO as a neutrally stable mode owing its existence to the planetary scale modulation of synoptic and mesoscale convective activity through the interaction of low-level moisture and precipitation [178]. A thorough overview of the MJO is found in a sequence of two review papers by Chidong Zhang [300, 302] where insight on its importance for global weather and climate is also given. We refrain from digging too much in those directions but instead we provide here the ingredients necessary for the simulation of the MJO and other tropical convective systems in coarse resolution GCMs, based on the multicloud paradigm, which is the focus of this monograph.

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3 Observations of Tropical Climate Dynamics and Convectively Coupled Waves

Fig. 3.6 Cartoons of the Madden-Julian oscillation dynamics and structure as described by Madden and Julian. From [163]. Left panel shows successive circulation patterns with air converging and rising within the convection centre as it slowly moves eastward. Sea level pressure and tropopause layer are indicated on each frame as the low sea level pressure and the tropopause height disturbances move together with the MJO wave. Right panel depicts a 3D flow chart of the MJO characterized by a quadruple vortex structure straddling the equator both at the top and bottom of the troposphere. Cyclones and anti-cyclones are indicated by the letter C and A, respectively. Arrows represent air flow direction and strength.

3.4 Convectively Coupled Equatorial Waves The equatorial convectively coupled waves are divided into a symmetric and an antisymmetric subgroups, with respect to the equator. Listed in the order of their spectral power strength, in addition to the MJO signal which is separated from the dispersion relation curves (ω = ω (k), where ω is the frequency and k the zonal wavenumber) of the dry equatorial shallow water waves of Matsuno [187], the symmetric group comprises

3.4 Convectively Coupled Equatorial Waves

51

1. a Kelvin mode peak located at around the zonal wavenumber k = 4 and a period of about 6 days, somewhat elongated in both directions, along the first quadrant between the dry Kelvin wave dispersion relations of equivalent depths 50 m and 25 m, corresponding roughly to a wave speed of about 16 m s−1 , 2. a Rossby wave peak lying on the dry M=1 Rossby dispersion curves of similar equivalent depths around zonal wavenumber 3 and a period of about 30 days, corresponding to a speed of about 5 m s−1 3. a westward gravity wave peak corresponding to the westward branches of the M=1 gravity waves located around k = −10 with a period of about 2 days, corresponding roughly to a phase speed of 20 m s−1 to 30 m s−1 . 4. The anti-symmetric part is dominated by a single signal peaking between wavenumber k = −5 and k = 10 along the Yanai dispersion relation curves. According to [283], this Yanai signal is further split naturally onto two regions located on both sides of the ω -axis. The westward part, corresponding to k < 0, bear the generic name of “mixed Rossby-gravity waves” (MRG) while those on the eastward branches are called the M = 0 eastward inertio-gravity (EIG) waves. The most significant peak in the M = 0 EIG waves lies at around wavenumber k=5 and has a period of about 3 days, corresponding to a phase speed of about 30 m s−1 . The physical structure and dynamical features of the convectively coupled waves identified above are presented in a series of papers by Kiladis and several co-authors [127, 248, 250, 284] by combining reanalysis and in situ data of brightness temperature and dynamical fields. They found that the horizontal structure and vertical wavenumber of these waves qualitatively match those of the dry shallow water waves but with a much larger equivalent depth. The convectively coupled waves thus appear to have a reduced equivalent depth compared to their dry counterparts. Moreover, the vertical structure of the Kelvin, gravity, and MRG waves exhibits a self-similar boomerang shape characterized by a front to rear vertical tilt emanating at the surface and reversed aloft above the 200 hPa level, in the moisture, temperature, and zonal velocity fields. The forward tilt above the tropopause is attributed to the vertical propagation of gravity waves resulting from a moving heat source. The example of the Kelvin wave given in Figures 3.7 and 3.8, depicts the corresponding horizontal and vertical composite structures, respectively. The heating field, confined to the tropospheric layer below 200 hPa, is characterized by a progressive deepening of shallow convection followed by deep convection and a trailing stratiform wake and is in phase with the vertical velocity maximum. Both the reduced speed and the front to rear tilt, below the 200 hPa level, for the equatorial convectively coupled waves remained for a while an open problem for theoreticians and modellers in the field. Nevertheless, it is now widely recognized that the backward tilt is primarily due to the fact that convectively coupled waves,

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3 Observations of Tropical Climate Dynamics and Convectively Coupled Waves

including the MJO, project heavily on the first two baroclinic modes of vertical structure and that there is a significant time lag between these two modes. Moreover, analysis of cloud resolving modelling results confirmed that the bulk of the kinetic energy of equatorial convectively coupled waves is carried by these first two baroclinic modes [76, 219, 263, 264]. This explains in part the success of simple convective parameterization models, utilizing a crude vertical resolution reduced to the first two baroclinic modes [184, 175, 174, 118, 119, 121, 120] in reproducing many key features of convectively coupled waves. More on this will be given in the upcoming chapters.

3.5 Self-similar Multiscale Convective Systems As it turns out the backward tilted structure, against the direction of propagation, depicted in Figure 3.8 for the Kelvin wave is common to all convectively coupled waves and the MJO [127] as well as mesoscale convective systems [182]. This is despite the fact that these systems are often embedded in each other like Russian dolls. This conundrum led Mapes et al. [182] to come up with the idea of a stretched building block hypothesis consisting of statically dominating congestus clouds in front of the waves, deep convective towers in its centre followed by stratiform anvils at the back, operating at the three main scales: mesoscale, synoptic, and planetary. In Figure 3.9 groups cartoons of various convective systems of various scales and different propagation directions and speeds are compiled. These cartoons were actually originated from different sources and reflect observations of various wave-like phenomena occurring on different scales and in different parts of world. This is a clear indication that this resemblance is due to a universally profound physical entity which govern the dynamics of convective systems via the multicloud paradigm. This apparent self-similarity is now widely recognized [127] and its analytic justification can be found in Majda [168, 167]. In each panel of Figure 3.9, we see the three cloud patterns, congestus, deep, and stratiform cloud decks, embedded within a background of shallow cumulus clouds, which form the building block for the dynamical interactions of tropical convective systems and are the basis of the multicloud model framework discussed extensively in this monograph.

3.5 Self-similar Multiscale Convective Systems

53

Fig. 3.7 Observed horizontal structure of a convectively coupled equatorial Kelvin wave as it propagates over the Western Pacific. Kelvin-wave filtered OLR (shading), pressure perturbations (contours, negative is dashed), and velocity profile (arrows). Top shows wave structure in the lower troposphere at day 0 of the composite reference point, middle shows the same profile 2 days later, and bottom panel is for the top of the troposphere at day 0. From [127].

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3 Observations of Tropical Climate Dynamics and Convectively Coupled Waves

Fig. 3.8 Observed vertical structure of Kelvin wave as it propagates over the Majuro Island over its life cycle of roughly 12 days. Filled contours of zonal wind, temperature, and specific humidity are shown as indicated. The evolution of OLR is shown at the top panel. It is characterized by a pronounced minimum at day 0 corresponding to the convection maximum. Since this is an eastward moving wave, the horizontal-time axis can be relabelled by longitude coordinates. From [127].

3.5 Self-similar Multiscale Convective Systems

55

Fig. 3.9 Cartoons of the observed cloud and flow self-similar morphology of multiscale tropical convective systems. (a) Westward moving squall line, (b) eastward moving mesoscale convective system, (c) two-day wave (moves westward), (d) Kelvin wave (moves eastward), (e) the MaddenJulian oscillation. First compiled by Mapes et al. [182].

Chapter 4

Introduction to Stochastic Processes, Markov Chains, and Monte Carlo Simulation

A major part of this book discusses the use of stochastic models for atmospheric convection, namely through the use of a stochastic model for CIN and a stochastic multicloud model which pertains to tracking the statistics of clouds of various types. These models rely on the theory of stochastic processes and Markov chains in particular. Here, we provide a brief introduction to this topic to help the reader better appreciate and comprehend the cloud models. The expert reader can skip this chapter. It is intended to readers with no or rather very little background in probability theory and stochastic processes. Nonetheless, we assume that the reader is familiar with the basic notions of probability distributions and random variables.

4.1 Computing with Random Numbers Computations with random numbers are based on the notion of Monte Carlo simulations which in essence permits the approximation on a digital computer of the basic statistics of random variables and stochastic processes whose distributions are known explicitly or implicitly hidden behind complex mathematical rules. The material presented here is meant to be only informative. The interested reader can consult the volumes by Steward [247] and Robert and Casella [210] for more in-depth discussion. The backbone of this technique is one of the strongest results in the theory of probability and statistics, namely the law of large numbers which in a nutshell asserts that the statistical properties, i.e., the moments, of random variables can be approximated by taking averages of large sample ensembles. We thus begin by recalling this law.

© Springer Nature Switzerland AG 2019 B. Khouider, Models for Tropical Climate Dynamics, Mathematics of Planet Earth 3, https://doi.org/10.1007/978-3-030-17775-1 4

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4 Introduction to Stochastic Processes, Markov Chains, and Monte Carlo Simulation

4.1.1 Law of Large Numbers Let X1 , X2 , · · · , Xn , n ≥ 2 be a sequence of independent and identically distributed (i.i.d) random variables, each having a mean μ and a standard deviation σ . The sample mean of the sequence is defined as n

1 1 X¯n = (X1 + X2 + · · · Xn ) = n n

∑ Xj.

j=1

It is easy to see that the sample mean is unbiased, i.e., the expectation of the sample mean, E[X¯n ], is precisely μ , thanks to the linearity of the expectation operator E[X¯n ] =

1 n

n

∑ E[X j ].

j=1

Moreover, because the Xi ’s are not correlated we have Var[X¯n ] =

1 n2

n

∑ Var[X j ] =

j=1

σ2 . n

As a result the sample mean converges to the population mean μ , when n goes to infinity. Consequently, X¯n can be treated as a deterministic variable, in practice, for large enough n. This can be made more precise in the probabilistic sense but we refrain from developing this here.

4.1.2 Monte Carlo Integration 

Consider the integral I = 01 g(x)dx. This integral can be thought of as the expectation E[g(U)] where U is a random variable uniformly distributed on [0, 1] (U ∼ U (0, 1)). According to the law of large numbers above the expected value I = E[g(U)] can be estimated by a sample mean. Let U1 ,U2 , · · · ,Un be a sequence of random numbers generated according to the uniform distribution U (0, 1). Then, I ≈ In =

1 n

n

∑ g(U j ).

j=1

We note that the sampling of genuinely random numbers is not possible on a digital computer. Instead, most programming languages and computing environments such as Matlab or Fortran have one or more built-in functions that can generate sequences of pseudo-random numbers. Unfortunately, not all distributions are easily sampled on a digital computer but techniques which take advantage of the availability of uniform random number generators do exist. Here we mention the two most common methods that will be used to simulate the cloud models discussed herein.

4.1 Computing with Random Numbers

59

4.1.3 Inverse Transform Method The inverse transform method, for generating pseudo-random numbers, from an arbitrary distribution, is based on the following basic statement. If X is a random variable with a probability density fX and cumulative distribution function, FX (x), then the random variable  X

U = FX (X) =

−∞

fX (x)dx

is uniformly distributed on (0, 1). Conversely, if U ∼ U (0, 1), then we have P({X ≤ x}) = P({F −1 (U) ≤ x}) = P({U ≤ F(x)}) = F(x). This is represented schematically in Figure 4.1. If we are able to invert F easily, then the inverse transform method for fX can be implemented in two easy steps. 1. Draw a uniform pseudo-random variate U ∼ U (0, 1). 2. Set X = F −1 (U). As an example, we consider the exponential distribution X ∼ exp(λ ). We have F(x) = 1 − e−λ x and for a given uniform variate, U, the inverse transform method yields 1 X = F −1 (U) = − ln(1 −U). λ

1 0.9 0.8 0.7 0.6 0.5

b

U

0.4 0.3 0.2 a

X

0.1 0 −3

−1 X

−1 X

F (b)

F (a) −2

−1

0

1

Fig. 4.1 Schematic of the inverse method: P({a ≤ U ≤ b}) =

2

P({FX−1 (a) ≤

3

X ≤ FX−1 (b)})

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4 Introduction to Stochastic Processes, Markov Chains, and Monte Carlo Simulation

4.1.4 Acceptance-Rejection Method As mentioned above, the inverse transform method is only feasible when the CDF F(x) can be easily inverted. Although one can always resort to numerical root finding techniques such as Newton’s method to invert F(x), it is not always a good idea because this can be very costly especially when we need to generate a large number of variates—to perform a Monte Carlo integration, for example. As an alternative, the acceptance-rejection method starts by finding a function t(x) such that f (x) ≤ t(x), ∀x ∈ R whose CDF is easy to invert. Once such a function t(x) is found, the acceptance-rejectionmethod consists of the three main steps listed +∞ t(y)dy. below. Let g(y) = t(y)/K where K = −∞ 1. Use the inverse transform method to generate a pseudo-random number Y corresponding to g(y). 2. Draw a uniform variate U from U (0, 1), independent of Y . 3. If U ≤ f (Y )/t(Y ), then return X = Y (accept), otherwise go to step 1 (reject). Recall that for a given (fixed) Y , the probability for a uniformly distributed random number U to satisfy U ≤ f (Y )/t(Y ) is P({U ≤ f (Y )/t(Y )}) = f (Y )/t(Y ). Therefore, the more this ratio is close to one, the better are the chances for the random number Y to be accepted and the above algorithm to be efficient. Moreover, the points Y where this ratio is close to 1 are very likely to be accepted while those with a small f (Y )/t(Y ) are very unlikely to be accepted. To gain efficiency, it is thus important to choose a function t(x) which is as close as possible to f (x). Also, it can be shown that the average number of iterations (acceptance and rejection trails) +∞ t(x)dx. to terminate the loop and return a sample X is K = −∞ If the support of f (x) is bounded, i.e., f (x) = 0 outside a bounded interval [α , β ], then a natural choice for g(x) is simply the uniform distribution on this interval. As an example we consider the beta-distribution on [0, 1]. f (x) =

xα1 −1 (1 − x)α2 −1 , B(α1 , α2 )

where B(α1 , α2 ) =

 1 0

x ∈ [0, 1]

xα1 −1 (1 − x)α2 −1 dx

For α1 = α2 = 3, we have f (x) = 30(x2 − 2x3 + x4 ), x ∈ [0, 1]. Note that its CDF is a fifth order polynomial that is not easy to invert and thus the inverse transform method would be hard to apply here. Let t(x) = max[0,1] f (x) = 30/16. Using the uniform distribution as the reference density g(x), we have the following algorithm: 1. Draw two independent uniform random variates U1 ,U2 from U (0, 1) 2. If U2 ≤ 16(U12 − 2U13 +U14 ), then accept and return X = U1 ; otherwise, reject and go back to step 1.

4.2 Introduction to Markov Chains and Birth-Death Processes

61

4.2 Introduction to Markov Chains and Birth-Death Processes A stochastic process is by definition a collection of random variables, Xt , which can be either discrete or continuous. The random variables can be either mutually correlated or uncorrelated. Markov chains lie somewhat in between the fully correlated and fully uncorrelated extremes. In this section, we will review Markov chains in general and in particular the case of the birth and death process, which will appear in the cloud models presented in the subsequent chapters where clouds of various types, or rather cloudy locations of various sorts, are viewed as dynamically evolving and interacting populations. In fact, birth and death processes are widely used in biology and computer science and many other disciplines. More details on Markov chains and the birth-death process can be found in many standard textbooks of the subject, e.g., [162, 142].

4.2.1 Discrete Time Markov Chains A discrete stochastic process Xn , n ≥ 0 is called a Markov chain if in addition it satisfies the Markov property P{Xn+1 = x j /Xn = xi , Xn−1 = xk , · · · , X0 = xl } = P{Xn+1 = x j /Xn = xi }.

(4.1)

The Markov chain is said to be stationary or homogeneous if P{Xn+1 = x j /Xn = xi } ≡ P{X1 = x j /X0 = xi }, for all n. The conditional probabilities Pi j = P{Xn+1 = x j /Xn = xi } are called the transition probabilities and the matrix P = [Pi j ]i, j≥0 is called the transition probability matrix. Two important properties of the matrix P are 1) all of its entries lie between 0 and 1 and 2) all of its rows sum to one, namely 0 ≤ Pi j ≤ 1 and ∑∞j=0 Pi j = 1. A matrix that satisfies these two properties is called a stochastic matrix. (n)

Consider the n-step transition probabilities Pi j = P{Xn+k = x j /xk = i} = P{Xn = x j /X0 = xi }. According to the definition of conditional probabilities, we have, for all i, j ≥ 0, (n+m)

Pi j =

≡ P{Xn+m = x j /X0 = xi }

∞

∞

k=0

k=0

∑ P{Xn+m = x j /Xm = xk }P{Xm = xk /X0 = xi } = ∑ Pik

Thus (n+m)

Pi j

∞

=

∑ Pik

(n) (m) Pk j , i,

(n) (m) Pk j .

j = 0, 1, 2, · · · .

k=0

These identities are known as the Chapman-Kolmogorov equations. If we let P(n) denote the n-step transition probability matrix, then it is easy to see, by

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4 Introduction to Stochastic Processes, Markov Chains, and Monte Carlo Simulation

the Chapman-Kolmogorov equations, that P(n+1) = P(n) × P and P(n) = Pn = P × P × · · · × P (n times). A Markov chain Xn is said to have a limiting distribution if the limit limn−→∞ Pinj exists for all i, j, and is independent of i. The limit π j = limn−→∞ Pinj when it exists is called the limiting distribution of Xn . As a consequence of the ChapmanKolmogorov equations, the limiting distribution satisfies ∞

π j = ∑ πi Pi j or π = π P,

(4.2)

i=0

i.e., π is a left eigenvector of the matrix P associated with the eigenvalue λ = 1. The existence of the eigenvalue λ = 1 is guaranteed by the Perron-Frobenius theorem of linear algebra. Any row vector π j , j = 0, 1, 2 · · · of real numbers which satisfies (4.2) and the two properties of a probability distribution: ∑∞ i=0 π j = 1, 0 ≤ π j ≤ 1 is called a stationary or invariant distribution of the Markov chain. The limiting distribution when it exists is an invariant distribution but the converse is not always true. Sufficient conditions for the existence of the limiting distribution and for the uniqueness of the stationary distribution are known but they are beyond the scope of this brief introduction. They involve the notion of ergodicity which intuitively amounts to saying that all the states of the chain are visited equally infinitely many times when the chain is run for an infinitely long time. The interested reader is referred to the books by S. M. Ross [162] and G. F. Lawler [142]. Time Reversible Chains and Detailed Balance Consider a stationary ergodic (i.e., that has a limiting and unique stationary distribution π j ) Markov chain with transition probabilities Pi j . Assume that the chain is run for a very long time and is in its equilibrium state, i.e., it satisfies P{Xn = x j } = π j , j = 0, 1, 2, · · · . Consider the backward process Xn , Xn−1 , Xn−2 , · · · when n goes to infinity. By some manipulations we can show that the time reversed process is also a Markov chain with the transition probabilities Qi j ≡ P{Xm = x j /Xm+1 = xi } =

πj Pji . πi

A Markov chain is said to be time reversible if Qi j = Pi j for all i, j ≥ 0. In other words, Xn is time reversible if

πi Pi j = π j Pji , i, j = 0, 1, 2, · · · .

(4.3)

The equations in (4.3) are known as the detailed balance equations. They basically state that, in the long run, the rate of transiting from state xi to x j equals the rate of transiting back, from x j to xi . As a simple example we consider the random walk process on a finite set. Imagine a person who takes random steps between the positions 0, 1, 2, · · · , M. If at time n the person finds herself in a state i, 0 ≤ i ≤ M, she will take a step randomly to the

4.2 Introduction to Markov Chains and Birth-Death Processes

63

left or to the right according to the following transition probabilities: Pi,i+1 = αi , Pi,i−1 = 1 − αi , i = 1, 2, · · · , M − 1, P0,1 = 1, PM,M−1 = 1. Here 0 < αi < 1 for i = 1, · · · , M − 1 to guarantee that the induced Markov chain is ergodic and time reversible. We set α0 = 1 and αM = 0 so that when the random walker hits 0 or M, with probability one, she will return to 1 or to M −1, respectively, at the next step. This is a bounded random walk with reflecting boundaries. While the Pi, j ’s define the transition probability matrix of the discrete time Markov chain, we can easily write down the detailed balance equations:

π0 = (1 − α1 )π1 , πi αi = πi+1 (1 − αi+1 ), i = 1, 2, · · · M − 2, πM−1 αM−1 = πM . The solution to these equations is given by 

j

αl−1 π0 = 1 + ∑ ∏ j=1 l=1 1 − αl M

−1

j

αl−1 . l=1 1 − αl

and π j = π0 ∏

For the particular case α j = 0.5, j = 1, 2, · · · , M − 1, we have π0 = πM = 1/2M and π j = 1/M, for 1 ≤ j ≤ M − 1.

4.2.2 The Poisson Process The Poisson process, Nt , is a counting process, i.e., the process of counting the number of events that occur in sequence by time t, such that P{Nt+s − Nt = k} = P{Ns − N0 = k} = P{Ns = k} = e−λ s

(λ s)k . k!

i.e., Nt is a Poisson random variable for any given t with mean λ t. The parameter λ > 0 is known as the rate of the Poisson process. We have, for all small h > 0, P{Nt+h − Nt = 1} = λ h + o(h) and P{Nt+h − Nt ≥ 2} = o(h) where o(h) is a little O of h, i.e., lim o(h)/h = 0. h−→0

Inter-Arrival Times Consider a Poisson process Nt ,t ≥ 0 with a rate λ > 0. Let T1 be the time at which the first event of Nt occurs, T2 the time spent between the first and the second events, T3 the time between the second and third events, etc. Then T1 , T2 , · · · is a sequence of independent exponentially distributed random variables with a common rate λ > 0. This follows directly from the independence of increments property and the fact that P{T1 > t} = P{Nt = 0} = e−λ t . The waiting time Sn = T1 + T2 + · · · + Tn until the nth event occurs is Gamma distributed with parameters n and λ .

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4 Introduction to Stochastic Processes, Markov Chains, and Monte Carlo Simulation

This intimate relationship between the Poisson process and the exponential random variables is due to the so-called memory less property of the exponential distribution, namely T is an exponentially distributed random variable if and only if P{T > t + s} = P{X > t}P{X > s}.

4.2.3 Continuous Time Markov Chains A stochastic process Xt where t is in (0, +∞) defined on a discrete state space x0 , x1 , x2 , · · · is called a Markov chain if it satisfies the Markov property (in continuous form): P{Xt+s = x j /Xs = xi , Xu = xu , 0 ≤ u ≤ s} = P{Xt+s = x j /Xs = xi }. If P{Xt+s = x j /Xs = xi } is independent of s, then Xt is said to be a stationary or homogeneous Markov chain. Similarly to the discrete case, the conditional probabilities Pi, j (t) = P{Xt+s = x j /Xs = xi } are called the transition probabilities and the time-dependent matrix P(t) = [Pi j (t)] is called the transition probability matrix. The continuous version of the Chapman-Kolmogorov equations, which also fellows from the elementary definition of conditional probabilities, reads ∞

Pi j (t + s) =

∑ Pik (s)Pk j (t).

k=0

It results from the conditional probability formula P{Xt+s = x j /X0 = xi } = ∑∞ k=0 P{Xt+s = x j /Xt = xk }P{Xt = xk /X0 = xi }. Notice the analogy with the discrete case. Waiting Time and Transition Rates An important property of continuous time Markov chains is one that links them directly to the Poisson process. Assume that at time t the Markov chain Xt is in state xi , i.e., Xt = xi . For a stationary stochastic process, the Markov property is equivalent to the fact that the time Ti the process stays in state xi before it makes a transition (or jump) to an another state x j = xi , i.e., Ti = inf{s > 0 such that Xt+s = xi given that Xt = xi }, is an exponential random variable. Moreover, the times Ti j , it takes the chain to make a transition from xi to x j , j = i are independent exponential random variables. By construction, we have Ti = min{Ti j , j = 0, 1, 2 · · · , j = i} and if qi j , i = j are the rates of the Ti j ’s and vi denote the rates of the waiting times Ti , i = 0, 1, 2, · · · . Then, according to the properties of exponential random variables, vi = ∑ qi j . j=i

4.2 Introduction to Markov Chains and Birth-Death Processes

65

The matrix R defined by Rii = −vi and Ri j = qi j , when i = j, is called the infinitesimal generator of the chain. As we will see below, the matrix R completely determines the Markov chain. The entries qi j are often called the transition rates of the chain. As an immediate consequence of this, we have, for sufficiently small h > 0, Pii (h) = P{Ti < h} = 1 − vi h + o(h) and Pi j (h) = P{Ti j > h} = qi j h + o(h). Kolmogorov Forward and Backward Equations By the Chapman-Kolmogorov equations, we have for h,t > 0 Pi j (t + h) − Pi j (t) =

∞

∑ Pik (h)Pk j (t) − Pi j (t) = ∑ Pik (h)Pk j (t) − [1 − Pii (h)]Pi j (t).

k=0

k=i

Dividing both sides by h and letting h −→ 0 and using the identities above yields the Kolmogorov backward equations d Pi j (t) = ∑ qik Pk j (t) − vi Pi j (t). dt k=i If instead we write Pi j (t + h) = ∑∞ k=0 Pik (t)Pk j (h), then similar manipulations yield the Kolmogorov forward equations d Pi j (t) = ∑ qk j Pik (t) − v j Pi j (t). dt k= j In matrix notation, the backward equations become P = RP while the forward equations read P = PR. The backward and forward equations provide, separately, a system of linear ordinary differential equations for the transition probabilities. It is closed with the initial conditions  1 if i = j Pi j (0) = δi j = 0 otherwise. Following the standard theory of differential equations, its solution is given by P(t) = exp(tR).

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However evaluating the matrix exponential can be problematic in practice, especially, when the matrix R is very large, i.e., when the Markov chain has a large number of states. A lot of research has been done in order to come up with practical solvers for this large ODE system. The interested reader is referred to [247]. An efficient and stable method for solving these equations in the context of the parameter inference for the stochastic multicloud model is proposed in [35] and will be discussed in Chapter 10. Limiting Distribution and Detailed Balance Similarly to the discrete case, the limiting distribution of Xt is given by Pj = lim Pi j (t), when this limit exists and is independent of i. It satisfies the steady t−→∞ state forward-equations ∞

∑ qk j Pk = v j Pj , 0 ≤ Pj ≤ 1, ∑ Pj = 1. j=0

k= j

A solution Pj , j ≥ 0 to these equations is called a stationary or an equilibrium distribution. Intuitively, these equations express the fact that, in the long run, the rate of transition away from state j, v j Pj , is balanced by the sum of the rates of all possible transitions back to state j, ∑ qk j Pk . k= j

By extension from the discrete case, the time continuous chain Xt is said to be time reversible if (4.4) Pi qi j = Pj q ji , i, j = 0, 1, 2, · · · These are the detailed balance equations for the continuous chain Xt . Intuitively, these equations express the fact that the rate of transition from state i to state j and from j to i is balanced in the long run. These equations will be the basis for constructing the stochastic cloud models in Chapters 10 and 11 in such a way that the stochastic model emulates the statistics of large Hamiltonian dynamical system. To illustrate, we consider the example of a machine that operates for an exponentially distributed random time before it breaks down at a rate μ > 0. Upon its failure the machine is sent to the repair shop where it takes an exponential time with rate λ > 0 before it is repaired and put back in service. Let Xt be the random variable that takes the value 1 if the machine is in service at time t and 0 otherwise. Then Xt is a continuous time Markov chain with the state space reduced to 0 and 1, a two state Markov chain. The infinitesimal generator is given by   −λ λ R= μ −μ and the backward equations are given by   P00 = λ [P10 − P00 ], P10 = μ [P00 − P10 ],

  P01 = λ [P11 − P01 ], P11 = μ [P01 − P11 ],

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67

with the initial conditions Pi j (0) = δi j . The solution is given by P00 (t) =

λ μ μ −(λ +μ )t u + e−(λ +μ )t ; P10 = − e ; λ +μ λ +μ λ +μ λ +μ

P01 (t) =

λ λ λ μ −(λ +μ )t − e−(λ +μ )t ; P11 = + e . λ +μ λ +μ λ +μ λ +μ

Letting t −→ ∞ leads to the limiting distribution of the chain P0 =

μ λ , P1 = . λ +μ λ +μ

Notice that detailed balanced is evidently satisfied, since λ P0 = μ P1 . The Birth and Death Process Assume that customers arrive in a shop according to a Poisson process with rate λ . The customers are served by m tellers. Upon arrival, a customer proceeds to the first available teller or waits in a queue until a teller is freed. Assume that the service time of each teller is an exponential random variable with rate μ > 0. Then, the number of customers, Xt , in the store at any given time, t ≥ 0, is a continuous time Markov chain with transition rates qn,n+1 = λ , for n ≥ 0, qn,n−1 = nμ for 1 ≤ n ≤ m, qn,n−1 = mμ for n ≥ m + 1. The rates of the waiting times between two transitions (i.e., before the next arrival or departure occurs) are given by v0 = λ , vn = λ +nμ , for 1 ≤ n ≤ m, and vn = λ +mμ if n ≥ m + 1. In queuing theory this is known as the M/M/m model. The infinitesimal generator of the chain is a tridiagonal matrix. Similar models are ubiquitous in practice. They are called birth and death processes. Xt = n is regarded as the number of population members at a given time t. The rate at which arrivals occur, qn,n+1 = λn , n ≥ 0, is called the birth rate and the rate at which departures occur, qn,n−1 = μn , n ≥ 1, is called the death rate. Notice that in the general case both arrival and departure rates are assumed to be dependent on n and that we implicitly assumed μ0 = 0. If in addition λk = 0 for a certain integer k ≥ 1, then the process becomes bounded to the states 0, 1, 2, · · · , k. If Xt = k, then new arrivals are not accepted. The steady state forward equations for the birth-death process are

λ0 P0 = μ1 P1 , (λn + μn )Pn = μn+1 Pn+1 + λn−1 Pn−1 , n = 1. Substitution of the first equation into the second, the second into the third, and so on yields λn Pn = μn+1 Pn+1 , n ≥ 0

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which are nothing but the detailed balance equations (4.4) for the birth-death process. With the constraint ∑∞j=0 Pj = 1, the solution to this equations is −1  ∞ λn−1 λn−2 · · · λ0 λn−1 λn−2 · · · λ0 Pn = P0 , P0 = 1 + ∑ . μn μn−1 μ1 μn μn−1 μ1 j=0 Therefore, a necessary condition for the birth-death process to admit a limiting distribution is ∞ λn−1 λn−2 · · · λ0 (4.5) ∑ μn μn−1 μ1 < ∞ j=0 This is guaranteed for the M/M/m process if the ratio λ /mμ < 1, i.e., the rate at which customers arrive is smaller than the rate at which they are being served. In the case of the finite state birth-death process, on the other hand, the condition (4.5) is trivially satisfied because the summation terminates at n = k since λk = 0, which makes the summation finite regardless of the ratio λ /(mμ ).

4.3 Exercises 1. Verify statistically that the average number of iterations to generate one random number using the acceptance rejection method for the beta-distribution example in the text is 30/16 ≈ 1.875. Use Monte Carlo integration based on the  acceptance-rejection method to estimate the expectation E[X] = 01 x f (x)dx. 2. In this exercise we test three different approaches to generate pseudo-normal variates, with mean zero and variance one. One is based on the central limit theorem, one is the polar method, and the third is the function randn of Matlab. a. Method Based on the Central Limit Theorem. Recall that according to the central limit theorem, if xk , k = 1, 2, · · · , n are n random numbers from a distribution with mean μ and variance σ 2 , then as n increases ∑n xk − nμ y = k=1 √ σ n approaches a normal distribution with mean zero and variance one. If the xk ’s are uniformly distributed on [0, 1], then μ = 1/2 and σ 2 = 1/12. If in addition, we choose n = 12, then we obtain a simple formula for y. 12

y=

∑ xk − 6

k=1

4.3 Exercises

69

is approximately normally distributed with mean zero and variance one. Write a short Matlab code to generate normal variates according to this formula by generating 12 uniform variates using the function rand of Matlab, sum them together, and substrate 6, e.g. >> y = sum(rand(1,12))- 6; will generate one pseudo-random number, which is approximately N (0, 1). Use this as a building block to write a Matlab code to generate sequences of normally distributed random numbers of arbitrary size. b. The Polar Method. Let x1 , x2 be two independent uniformly distributed (pseudo) random numbers on [0, 1). Then, y1 = sin(2π x1 ) −2 log(x2 ) and y2 = cos(2π x1 ) −2 log(x2 ) are normally distributed with mean zero and variance one. Write a Matlab code to generate a sequence of normally distributed random numbers of an arbitrary size, according to this algorithm. c. Generate 1000 normally distributed N (0, 1) according to each one of the algorithms above and according to the Matlab function randn and save the three sequences as three different vectors, which you may call Ncentral, Npolar, Nrandn , respectively. Then use the Matlab function hist to bin each one of three random vectors in bins of size 1. Normalize the bin numbers by the total samples (1000). Use the bar command of Matlab to plot the histogram √ 2 and plot the normal density f (x) = e−x /2 / 2π on top of each one of the histogram. Check the validity of each one of the methods above by comparing the numerical values of unit-binned histograms to the exact normal distribution. 3. Use the Chapman-Kolmogorov equation for P(n+1) to show that the limiting distribution of a Markov chain satisfies π j = ∑∞ i=0 πi Pi j for all j ≥ 0. 4. A taxi driver conducts his business in three different towns 1,2, and 3. On any given day, when he is in town 1, the probability that the next passenger he picks up is going to a place in town 1 is 0.3, the probability that the passenger is going to town 2 is 0.2 and the probability that he is going to town 3 is 0.5. When he is in town 2, the next passenger he picks up is going to a place in town 1 with probability 0.1, to town 2 with probability 0.8, and to town 3 with probability 0.1. When he is in town 3, these probabilities are 0.4 to go to towns 1 and 2 and 0.2 to go to a place in town 3. a. Argue why the underlying process of tracking the location of the taxi driver after dropping a passenger is a Markov chain. Write down the associated transition probability matrix.

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b. Given that the taxi driver is currently in town 1 and is waiting to pick up his first customer for the day, what is the probability that he picks his third customer of the day in town 2? c. In the long run, which of towns 1, 2, 3 does the taxi driver visit the most? Justify your answer. 5. Let T1 , T2 be two independent exponentially distributed random variables with rates λ > 0, μ > 0, respectively. (a)

Show that S = min(T1 , T2 ) is an exponential random variable with rate λ + μ.

(b)

Show that P{T1 < T2 } =

(c)

Show that if μ = λ then T1 + T2 is a Gamma random variable with parameters α = 2 and λ and in general if T1 , T2 , · · · , Tn are n independent and exponentially distributed random variables, with the same rate λ , then the sum Sn = T1 + T2 + · · · Tn is Gamma distributed with parameters n and λ ; fSn (x) = xn−1 λ n e−λ x /(n − 1)!.

λ λ +μ .

6. Show that the product of two stochastic matrices is a stochastic matrix. 7. A matrix P is said to be doubly stochastic if both all of its rows and all of its columns sum to 1. Show that the limiting distribution of Markov chain on a finite state space {x0 , x1 , · · · , xM } with a doubly stochastic probability transition matrix is uniform, i.e., π j = 1/M + 1 for j = 0, 1, 2, · · · , M. 8. Let X be a non-negative random variable. Show that X is exponentially distributed if and only if it satisfies the memoryless property P{X > s + t} = P{X > s}P{X > t}, for all s,t > 0. 9. Write down the forward and backward equations for a bounded birth-death process with birth rates λn and death rates μn where μ0 = 0 and λk = 0 for k ≥ 1. Give the infinitesimal generator matrix. 10. Find the transition probabilities for a birth only process, i.e., a birth-death process for which λn > 0 and μn = 0 for all n. Start with the case λn = λ , i.e., the birth rate is independent of n. 11. Let Xt be a continuous time Markov chain with state space 1,2,· · · and associated waiting rates v1 , v2 , · · · and transition rates qi j , i = j. Consider the first passage time Tk into state k, given by Tk = min{t ≥ 0, Xt = k}.

4.3 Exercises

71

Let mik be the expected first passage time from state i to state k: mik = E[Tk /X0 = i]. a. Show that vi mik = 1 + ∑ qi j m jk . j=k

b. Find m14 if Xt is a four state Markov chain with rates q1,2 = 2, q1,3 = 2, q1,4 = 1, q2,1 = 3, q2,3 = 3, q2,4 = 0, q3,1 = 0, q3,2 = 2, q3,4 = 2, q4,1 = 1, q4,2 = 0, q4,3 = 3. 12. Let Xt be a continuous time Markov chain with state space {1, 2, 3} and rates q1,2 = 1, q2,1 = 4, q2,3 = 1, q3,2 = 4, q1,3 = q3,1 = 0. Find the probability transition matrix P(t) of Xt and the limiting distribution if it exists.

Part II

The Deterministic Multicloud Model

Chapter 5

Simple Models for Moist Gravity Waves

This chapter introduces the reader to basic simple ideas for convectively coupled wave models mostly in order to put the multicloud model, which will be discussed in Chapter 6, in the context of preceding theories. It explores some of the ideas that have been proposed in the literature and their main characteristics and pitfalls using simplistic models. In our quest for the simplest possible model to explain and capture the dynamics of convectively coupled tropical waves, here we discuss a hierarchy of models for tropical convection. As pointed out already, the troposphere, which makes up the bulk of climate variability and weather processes and where tropical waves matter the most, a shallow water-like model seems to be sufficient to capture these waves. This is in fact, as seen is Chapter 1, the idea pioneered by Matsuno [187] and Gill [62], except for the fact that the early models were not two-way coupled to convection; convective heating was either ignored completely or it was assumed to be fixed/external to the fluid dynamics. Given that the barotropic mode is not directly affected by diabatic heating, the first model we consider is reduced to the first baroclinic mode of vertical structure. As the need for further complications arises, we will gradually include a thermodynamic boundary layer, and then a second baroclinic mode. In terms of convective parameterization schemes, i.e., mathematical models to close the relationship between the large-scale dynamics and convection, we begin with one of the oldest ideas known as wave-CISK. We then explore the concept of quasi-equilibrium by discussing a few models that rely on this idea. We will see that without an external destabilizer such as WISHE, convective parameterization schemes based on the quasi-equilibrium assumption tend to be stable, i.e., incapable to initiate or sustain convectively coupled waves. Two types of quasi-equilibrium schemes will be explored, adjustment schemes and mass-flux schemes. In fact the boundary layer model is needed for the massflux-type schemes because convective plumes are assumed to originate from the boundary layer and in turn they rapidly consume convective instability. The schemes

© Springer Nature Switzerland AG 2019 B. Khouider, Models for Tropical Climate Dynamics, Mathematics of Planet Earth 3, https://doi.org/10.1007/978-3-030-17775-1 5

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that are based on the sole first baroclinic mode for the free tropospheric dynamics rely on the idea that deep cumulonimbus clouds form the bulk of diabatic heating in the tropics and as such they must be the dominant dynamical players. As we will see, while the first statement is valid other cloud types that contribute significantly less in terms of total heating and precipitation can have a huge impact on the dynamics and especially on the initiation and propagation of organized convection and on convectively coupled waves in particular. We conclude this chapter by introducing the concept of stratiform instability, which requires the inclusion of a secondary heating mode and cloud type, namely stratiform cloud decks that are observed to trail deep convection. In particular, we will show how stratiform clouds can impact the large-scale dynamics in terms of both growth rates and physical structure of the moist gravity waves. This will set the ground for the introduction of the multicloud model in the next chapter, by adding a third heating mode/cloud type to count for cumulus congestus cloud decks that are observed to prevail before the onset of organized convection and lead the convection core of convectively coupled waves. This comes not only as a response to making up a model that genuinely reflects the dynamics and morphology of convectively coupled waves but it also fixes a major deficiency in the stratiform instability concept.

5.1 Wave-CISK We consider a dynamical model for the topical atmosphere with a crude vertical resolution reduced to the first vertical baroclinic mode. In other words we assume the following vertical profiles for the dynamical variables: v = v1 cos(z), w = w1 sin(z), θ = θ1 sin(z), according to the expansion in (1.20). Such models are believed to capture well the large-scale response of the tropical atmosphere to diabetic heating [187, 32, 62, 47, 207, 299, etc.]. As we will see in the subsequent chapters, while this is probably true in terms of energetics it is far from capturing the dynamics because of non-trivial wave interactions at various temporal and spatial scales with convective processes. For simplicity, we assume meridional symmetry, i.e., we consider flows that run east-west parallel to the Equator. ut − θx = 0 θx − ux = Q.

(5.1)

In (5.1), where the subscript 1 was dropped for simplicity, u is the zonal velocity component and θ is the corresponding potential temperature associated with the first baroclinic mode. Here Q is the diabatic heating due essentially to deep towering cumulus, which are believed to dominate the cumulus heating field in the tropics. However, as we will see later, although they do dominate the heating field other

5.1 Wave-CISK

77

cloud types make significant contribution in terms of destabilization, growth, and propagation of tropical waves. The main problem of convective parameterization is to come up with a closure for Q as a function of the large-scale variables, here, u and θ . The convective instability of the second kind (CISK) theory pioneered by Charney and Eliassen [23] was probably the first attempt to address this problem. They proposed such theory for hurricane growth. It is then followed by the work of Kuo and co-authors who are perhaps the first to generalize the concept of CISK to tropical convection in the more general context of convective parameterization [136, 137, 6] and also by Lindzen [158] and many others. In a nutshell, CISK assumes that, from the largescale dynamics perspective, it is the deficit between precipitation and evaporation that drives the tropical circulation. As such the difference between the two is proportional to the horizontal convergence of vertically integrated water vapour, a.k.a, the precipitable water. Thus, CISK provides a way to destabilize the large-scale atmosphere by convection which bypasses the details of small-scale convective motions. This is perhaps where the phrase “instability of the second kind” comes from, i.e., it is totally unrelated to the concept of convective stability reviewed in Chapter 4 which requires the presence of lighter air below heavier air. In other words, the atmosphere is assumed to be always in some state of quasi-equilibrium where convection responds instantaneously to large-scale convergence and converts it to diabatic heating which in turn produces vertical motion and reinforces the large-scale convergence. As we will see this leads to a catastrophic situation where all wavelength modes grow without bounds with growth rates increasing with the wavenumber. To illustrate, we set  1 H div(qv)dz, Q∝− H 0 which implies heating in regions of large-scale convergence and cooling in divergence regions. Here q is the specific humidity. In our idealized linear model (5.1), the q equation decouples and thus ignored for simplicity. Only the background moisture contributes to the linear dynamics. We assume the following closure equation which when combined with the dynamical core equations in (5.1), the resulting model can be easily analysed. (5.2) Q = −α ux , where α is an unknown proportionality factor, i.e., a parameter. Combining (5.1) and (5.2) yields the following wave equation for θ alone:

θtt = (1 − α )θxx . This is the traditional wave equation of a vibrating string. We consider simple wave solutions on the form θ (x,t) = θˆ eikx−ω t . When α ≤ 1, we have two √ stable waves travelling in the opposite directions with the reduced speed cr = ± 1 − α : θ± = θ0 (x ± cr t). Whereas when α > 1 the system becomes unstable and we have two standing modes, one grows exponentially and the other is damped: √ θ (x,t) = e± α −1kt eikx .

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The growth rate of the unstable mode increases linearly with the wavenumber, k. Shorter wavelengths grow faster than longer ones. We note also that at the bifurcation point, when the mode becomes unstable, the instability forms from the merging of the eastward and westward moving gravity waves into two standing modes, one being stable and the other unstable. This is typical to wave-CISK. Thus, despite the crudeness of this model, which is by all means far from the more involved CISK models presented in the literature, it exhibits quite well one of the most salient features of CISK. As anticipated, a major problem with the CISK theory is that it doesn’t possess a scale selective instability, i.e., an instability of distinctive largescale wavelengths from the hierarchy of organized tropical convective systems. The results obtained here are somewhat consistent with the work of [140] who used a similarly simple wave-CISK mode to interpret the results of a simplified global climate model (GCM) simulation although in their case, the unstable wave continues to propagate at a reduced speed. These quantitative differences are perhaps due to the fact that their model is based on the full primitive equations, including a spectrum of equivalent depths and vertical heating profile based on the GCM simulation. To correct for this lack of scale selection, and in order to avoid catastrophic instabilities at the grid scale for the numerical scheme, when this concept is used as a parameterization for cumulus convection, and provide a scale selective CISK in large-scale models, many modifications have been proposed and used. We can name, for example, the phase-lagged wave-CISK proposed by Davies [32] and the boundary layer frictional wave-CISK proposed and used by Wang and collaborators [275, 273, 291, 272, etc]. The phase-lagged CISK breaks the quasi-equilibrium assumption by introducing a time lag according to which cumulus heating adjusts slowly to the large-scale moisture convergence while the boundary-layer CISK uses boundary layer convergence instead of full column convergence to relax the quasiequilibrium assumption and rely on frictional dissipation to overcome the CISK instability at small scales.

5.2 A Simple Adjustment Scheme The wave-CISK theory is based on the assumption that the tropical atmosphere is always in some state of quasi-equilibrium and that convection is a somewhat homogeneous process which maintains this equilibrium. An other interpretation of this assumption amounts to saying that convection acts to restore the stability of the environment whenever it develops locally so that at large scales, only the action of the ensemble of convective elements (coherent updrafts and downdrafts) need to be considered when dealing with global climate models [8] perhaps by analogy to the statistical theory of homogeneous turbulence which may have greatly influenced the early climate modellers. It is not surprising that convection schemes, which are based on this interpretation of the quasi-equilibrium assumption, lead to dynamical models that are near

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79

equilibrium and lack the intrinsic variability which characterizes tropical convective systems [151, 155]. Perhaps the simplest way to implement this concept of rapid restoration of stability is the idea of convective adjustment first proposed by Manabe and co-authors [181] and later on refined and used by Betts and Miller [15, 14]. In a nutshell, the convective adjustment scheme consists in constantly driving the climate model towards a reference state of equilibrium, i.e., a temperature and humidity profile which is stable to rising parcels, e.g., the moist adiabat [181, 7] or an observed profile, which is typically more stable than the moist adiabat [15]. Due to their conceptual simplicity, the Manabe-Betts-type adjustment schemes were used extensively in theoretical studies based on simple analytical and numerical models [176, 56, 54, 117, 215, 244] as well as idealized climate models [55]. To illustrate, we consider the following idealized prototype convective adjustment model: ut − θx = 0

θt − ux = Q+ c − QR qt + Qux = −P + E

(5.3)

As in (5.1) we consider zonal flows above the equator with the vertical resolution reduced to the first baroclinic mode. The convective heating takes the form Qc = τ1q (q − q) ˆ − τ1 (θ − θˆ ), where qˆ and θˆ are, respectively, the reference values θ towards which the moisture and potential temperature adjust, with τq and τθ are the adjustment time scales. QR is a prescribed constant radiative cooling, P = Q+ c is the precipitation rate, and E = τ1e (q∗s − q) is the evaporation rate with τe the evaporation time scale and q∗s is the surface saturation specific humidity while Q is the background moisture stratification which is fixed to Q = 0.9 [56]. Note that unlike the CISK model presented above, the adjustment scheme equations (5.3) carry an additional-moisture equation. When the equations in (5.3) are linearized about a radiative convective equilibrium (RCE) state, i.e., a time and space homogeneous solution where the imposed radiative cooling is balanced by the convective heating and precipitation is balanced by evaporation, we obtain, ut − θx = 0 1 1 θt − ux = q − θ τq τθ   1 1 1 qt + Qux = − + q+ θ τq τe τθ We seek plane wave solutions of the form ⎛ ⎞ u ⎝ θ ⎠ = Uei(kx−ω t) q

(5.4)

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where k is the zonal wavenumber, ω ≡ ω (k) (dispersion relation) is the generalized phase, with Im(ω ) ≡ growth rate, Re(ω )/k ≡ phase speed. We obtain the following matrix eigenvalue problem: −iω U + ikAU = BU, or

ω U = (kA + iB)U with

⎤ ⎤ ⎡ 0 −1 0 0 0 0 ⎦ 1/τq A = ⎣−1 0 0⎦ , B = ⎣0 −1/τθ 0 1/τθ −(1/τq + 1/τe ) Q 0 0 ⎡

The resulting growth rate and phase speed diagrams are plotted in Figure 5.1.

Fig. 5.1 Growth rate and phase speed diagrams for the adjustment scheme for the parameters Q = 0.9, τq = τθ = 2 hours and τe = 10 days.

From Figure 5.1, we can see that for the given parameters, the adjustment scheme has the following characteristics: • It has two damped gravity waves moving in the opposite directions (with perfect symmetry between east and west, as expected from the lack of rotation and/or background wind).

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81

• The propagation speed of the gravity waves approaches ±50 m/s (the speed of first baroclinic free gravity waves c.f. Chapter 1) at small scales (large k). This is ample evidence that the adjustment scheme has little effect on the small scales, which is rather intriguing. • It has a damped standing mode, which we may call the moisture mode because it arises as a result of adding the moisture equation to the dynamical model. Such moisture modes are still playing roles, rather controversially, in the development of theories for planetary scale tropical dynamics and the MJO in particular. • The phase speed of the moist gravity waves is significantly reduced, at large scales, k ≤ 20. • There is a narrow transition region in wavenumber space characterized by three standing modes, where the gravity wave speed transitions from slow to fast (moist to dry) as the wavenumber increases. • More importantly, the whole system is stable (growth rates ≤ 0) and most of the time the waves are strongly damped except near the planetary scales where the gravity waves are almost neutral and appear to propagate at nearly 10 m s−1 . This is consistent with the fact that this scheme is known to produce planetary scale slowed down Kelvin waves in place of the MJO [55]. The diagram changes quantitatively with the change in parameters but not qualitatively. • The above features are robust. For significant parameter changes the transition gap tends to shift to planetary scale for small evaporative time scales and the slow phase speed instability band shrinks around wavenumber one for large τq and τθ values. Nonetheless, numerical tests show that the scheme remains stable for all physically plausible parameter values. The parameter Q controls the speed of the slowed down waves. Smaller Q values yield faster moving moist gravity waves [56]. As we will see in the next section, linear stability is rather a common characteristic of quasi-equilibrium schemes, which are not based on the CISK theory. In order to initiate convection and excite convectively coupled waves, quasi-equilibrium schemes are in need for an external destabilizer and wave amplifier. Various triggers or destabilizers have been proposed and used in the literature including nonlinear surface or radiative fluxes [47, 46, 205, 225, 57]. The wind induced surface heat exchange (WISHE) [46, 205] is perhaps the most popular mechanism suggested as an amplifier of convection and as an effective destabilizer of convectively coupled waves. It was first proposed for tropical cyclone intensification as an alternative to CISK theory [47, 29, 22]. The underlying assumption is that background winds supplied by the large-scale flow enhance the surface evaporation and thus trigger and maintain convection. As a result, eastward moving waves require an easterly background wind, u, ¯ which is in conflict with observation because the convective core of such waves is often dominated by westerly winds (especially the MJO over the Indian Ocean/Western Pacific warm pool).

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5.3 Mass-Flux Schemes and WISHE Waves As demonstrated in the previous section, one deficiency of quasi-equilibrium schemes is that they are linearly stable, i.e., convectively coupled waves are not expected to initiate and grow without an additional mechanism. In addition to the adjustment model discussed above, here we introduce the concept of mass-flux schemes whose convective closure requires the notion of potential energy for convection. The idea is to represent the bulk effect of an ensemble of moist parcels (plumes) originating from the surface under the influence of an energy potential due to large-scale instability [8]. For this we need to augment the first baroclinic equations with a slab boundary layer equation for the bulk equivalent potential temperature. We thus consider here what we call “a one-and-a-half layer model” for the tropical atmosphere, following the work of Majda and Shefter [176]: A full dynamical layer for the free troposphere, represented by the first baroclinic mode, on top of the thin well-mixed boundary layer near the sea surface, represented by the separate boundary layer model. The boundary layer is passive except for time variations in the equivalent potential temperature, averaged over the boundary layer (ABL) depth, θeb . We have [176] 1 ut − yv − θx = − Cv (|v|)u h 1 vt + yu − θy = − Cv (|v|)v h θt − div(v) = Qc − QR d θem 1 = D − QR dt H d θeb 1 1 = E − D. dt h h

(5.5)

In (5.5), v = (u, v) is the horizontal velocity field and θ is the potential temperature while θeb and θem are the equivalent potential temperatures in the thin boundary layer and at the middle of the troposphere, respectively. The term 1h Cv (|v|) is the coefficient of nonlinear momentum drag, Qc is the convective heating, QR = Q0R + 1 ∗ τR θ is the long wave radiative cooling, and E = Cθ (|v|)(θeb − θeb ) is the rate of ∗ evaporation from the ocean surface, where θeb is the saturation θe in the boundary layer, and D is the downdrafts mass flux. Note that downdrafts dry and cool the boundary layer and warm and moisten the upper troposphere by the same amount. As a result the system conserves the vertically integrated moist static energy, namely the quantity θez = θem + Hh θeb is conserved when the external forcing terms QR and E are set to zero. The dependence of Cθ (|v|) on the wind speed is known as the effect of the wind induced surface heat exchange [46, 205] or WISHE for short. We will see here that this term alone is capable to create unstable moist gravity waves that propagate at realistic speeds and have length scales comparable to those of convectively coupled waves as observed in nature. See Chapter 3.

5.3 Mass-Flux Schemes and WISHE Waves

83

The concept of mass-flux scheme begins with the partitioning into clear and cloudy regions, of a large horizontal area extending horizontally, in both directions, over a few hundred kilometres [8], spanning a few global climate model cells. The area is big enough to contain a sizable ensemble of rising plumes of moist air, i.e., clouds. Following [176], the vertical velocity is decomposed into convective and environmental contributions, i.e., within and outside the cloud according to the top hat approximation [305]. We have w = (1 − σc )we + σc wc where wc is the average vertical velocity within the cloud (convective, upward) and we is the vertical velocity in the environment outside the cloud and σc is the area fraction of the cloudy region (assumed small compared to the grid box). In the simple setting considered here, the convective heating is assumed to be proportional to the upward mass flux within the cloudy air, mc = σc wc (where density is set to one for simplicity). Thus, we have Qc =

α¯ σc wc , Hm

where α¯ is a dimensional constant of temperature and Hm is the middle tropospheric height [176]. The downdrafts also have environmental and convective contributions D = −[(1 − σc )w− e − σ wd ](θeb − θem ) where wd =

1 − εp wc εp

with ε p is the precipitation efficiency parameter. Thus, we obtain the closure (1 − σc )we = w − σc wc = Hm div(v) − σc wc . Finally, the momentum drag satisfies Cv (|v|) = C0 (u20 + |v|2 )1/2 where u0 is the turbulent fluctuation velocity scale and C0 is a non-dimensional constant. Similarly, we have Cθ (|v|) = Cθ0 (u20 + |v|2 )1/2 with Cθ0 = 1.2 × 10−3 . It remains to find a closure for the convective heating. In a first approach, the area fraction is fixed to σc = 0.01 [176], and think of it as a model parameter while we look at some closure options for the convective updraft velocity wc . Majda and Shefter [176] reviewed a few options that roughly summarize the state-of-the-art of cumulus parameterization ideas at that time. 1. Lagrange parcel adjustment (LPA) scheme: dwc w2 = (CAPE − c )H (wc ), dt 2H

(5.6)

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∗ ) is a crude approximawhere H is the Heaviside function. CAPE= Γs (θeb − θem tion of the fluctuations in convective available potential energy (c.f. Chapter 2) where the moist adiabatic lapse rate is set to Γs = 6 K km−1 . Here w2c /2 represents the kinetic energy of the updrafts and it is set to adjust dynamically to changes in CAPE on a time scale set by the time it takes the parcel to reach the ∗ ≈ γθ where γ = 1.7 is the ratio between top of the troposphere, H/wc , and θem the dry and moist adiabatic lapse rates. 2. The instantaneous adjustment to CAPE, ICAPE

w2c ∗ + = Γs (θeb − θeb ) 2H 3. The strict “quasi-equilibrium” scheme or simply QE scheme where we assume that the buoyancy of raised parcels is zero (B ≡ 0) at all time, i.e., convection maintains the tropical atmosphere in a neutrally stable state, where the temperature and moisture profiles are those of the moist adiabat, i.e., θeb = γθ in the present setting. Thus, the two equations for θeb and θ can be used to derive a prognostic equation for wc . 1 1 α¯ E− D=γ σc wc − γ QR + div(|v|). h h Hm The equations in (5.5) and either one of these three closure schemes for wc form three separate models for convectively coupled waves. As in the previous section, we restrict the analysis to meridionally symmetric flows where both Earth’s rotation and y-variations are ignored. We also seek plane-wave solutions for the linearized version of the resulting model. In [176], the linear analysis is performed around a radiative-convective equilibrium for a range of parameter values and more importantly for varying values of the mean boundary layer turbulent velocity u0 and the mean wind u. ¯ We refer the interested reader to the original paper [176] for the details of the procedure. The results obtained with the three schemes above, for u0 = 2 m s−1 and u¯ = −2 m s−1 , are summarized in Figure 5.2 which shows the growth rates and phase speeds as functions of the zonal wavenumber. Apart from the manifestation of super-fast waves, in the LPA model, that can become unstable in some parameter regimes [176], the three schemes capture essentially the same wave instabilities. They all show a dominant mode peaking at synoptic scales (wavenumber 20 or so corresponding to 2000 km wavelength) corresponding to a branch of moist gravity waves moving eastward, at roughly 13 m s−1 . We note that the unstable waves propagate in the direction opposite to the imposed mean wind so that the WISHE effect is dominant in front of the wave convective core, which coincides with the region of upward motion. However, unlike the LPA and ICAPE schemes which clearly display a preferred scale of instability at synoptic scale, the QE scheme appears to extend the instability towards smaller scales. While this is not as catastrophic as the wave-CISK model in Section 5.1 where the growth rate increases linearly with the wavenumber, it is still not a desired feature. Also represented in Figure 5.2 is the case of the ICAPE scheme with the downdrafts

5.3 Mass-Flux Schemes and WISHE Waves

85

turned off. In this case the growth rate becomes very large and increases drastically with wavenumber while the phase speed of the unstable wave decreases exponentially to zero. This demonstrates that downdraft is important as, intuitively, they dry and cool the boundary layer and stabilize the environment so further convection is inhibited locally and the wave is forced to move forward. 2 1.8

Growth Rate, 1/day

1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0

50

100 150 200 250 Wavenumber, Equator = 40,000 km

300

a. Growth Rates 20 18

Phase Velocity, m/s

16 14 12 10 8 6 4 2 0

0

50

150 200 250 100 Wavenumber, Equator = 40,000 km

300

b. Generalized Phase Speeds Fig. 5.2 Growth rates and phase speed diagrams of the most unstable mode for the 1.5 layer model using the LPA (long-dashed), ICAPE (solid), and QE (dash-dotted) mass-flux convection schemes. The dotted lines are for the ICAPE scheme with the downdrafts turned off. Adopted from Figure 3. of Majda and Shefter [176].

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The full model with rotation was also considered in [176] through a solution method which consists of first Galerkin projecting the linearized equations onto the few first tropical wave-modes reviewed in Chapter 1, i.e., the Kelvin, the mixedRossby gravity (MRG), and the first few inertia-gravity and Rossby waves, in terms of meridional index [187, 62, 176, 174, 120]. This method will be again employed and discussed in more detail in Chapter 7. It is found that, with an easterly mean wind, u, ¯ the system exhibits instabilities on the Kelvin and MRG wave branches. The stability results, for a wide range of easterly mean winds and ABL turbulent velocity strengths, are reported in Figure 5.3. As we can see, the instabilities of the Kelvin and MRG waves exhibit a similar behaviour. They are both stable when the mean wind is zero and the instability varies with both the strength of the mean wind and u0 . For u0 smaller than a critical value nearing u0 = 4 m s−1 , both waves are unstable for a range of mean easterlies. As u0 grows the instability range of easterly winds decreases gradually and the waves become stable when u0 exceeds its critical value. This is a clear manifestation of a balanced interplay between the ¯ and boundary layer dissipation, which increases with increasing u0 and increasing u, the WISHE effect which increases with increasing u. ¯

Fig. 5.3 Stability diagram for the ICAPE scheme with rotation showing the range of values of mean easterly wind and turbulent velocity strength for which the Kelvin and the MRG waves are unstable. Adopted from Figure 8 of Majda and Shefter [176].

5.4 Stratiform Instability

87

5.4 Stratiform Instability As demonstrated in the previous section, the main problem with the two early and unfortunately most widespread theories for the tropical convection closure problem, namely wave-CISK and WISHE theories, is that they are inadequate for convectively coupled tropical waves. The first leads to catastrophic instabilities with growth rates increasing with wavenumber without any scale selection, which is at odds with the observations that clearly suggest the existence of a hierarchy of scales for convective organization [25, 283, 182, etc.] and the second is unphysical as it operates under conditions that are not those observed in nature. The mean wind in the tropics, especially over the Indian Ocean and western Pacific where these waves are mostly observed, is actually westerly while WISHE requires easterlies to destabilize eastward propagating waves. There are other theories that invoke other instability mechanisms such as cloud radiation feedback, for instance, but it tends to excite the moisture mode which is intrinsically a standing wave and requires advection nonlinearity to “propagate” [57, 239]. A physically sound model for convectively coupled waves should be able to capture the instability as an upscale build up organization of convection through the interactions of tropical clouds with each other and with the large-scale dynamics and should not require additional mechanisms that are against the observations [86, 88, 89, 91, 185, 183, 87, 186]. In addition, the convectively coupled wave model should be able to capture the most salient features of tropical waves. As already pointed out in Chapter 3 synoptic scale convectively coupled tropical waves a.k.a cloud superclusters, for instance, are characterized by [249, 174] (see Figure 3.8): (1)An eastward phase speed of roughly 15 m s−1 . (2)A convective field on the order O(2000–3000) km. (3)Anomalously cold temperatures in lower troposphere (below 500 hPa) and warm in the upper troposphere (500–250 hPa) within and often leading the heating region, and strong updrafts. (4)The wind and pressure have a boomerang shape in the vertical characterized by an upward-westward tilt in lower troposphere (below 250 hPa), and an upwardeastward tilt above this level. (5)Anomalous increases in θeb and CAPE, and shallow convection lead the wave. (6)A trailing part dominated by stratiform precipitation. While (1) and (2) are more or less captured by the models in the previous sections, the remaining four features require a more advanced model. The features in (3) and (4) are mainly dynamical and directly related to the vertical resolution while (5) and (6) are mostly thermodynamical and involve the distribution of the cloud field which is more complex than what is conveyed by the quasi-equilibrium theory by implying an instantaneous response of deep convection to large-scale instability. A natural fashion to include deviations from the quasi-equilibrium in a cumulus parameterization that would also take into account some features of the observed structure of convectively coupled waves comes with the inclusion of diabatic heating and cooling associated with stratiform cloud decks that are observed to trail deep

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convection [185, 183, 87, 186, 249, 157]. Mapes [184] proposed the idea of stratiform instability. Unlike deep convective clouds that heat the whole tropospheric depth quasi-uniformly, stratiform clouds are believed to warm the upper troposphere through condensation and freezing of supercooled water in the wake of deep convection and cools the lower troposphere via the evaporation of stratiform rain that falls to the already dry lower troposphere. This destabilization is balanced by downdrafts induced by the sinking of cooled air which dries and cools the boundary layer. The restoration of the boundary layer θe is then accompanied by significant CAPE build up and thus the re-initiation of deep convection. As a consequence, the simplest model for convectively coupled waves cannot be reduced to the sole first baroclinic mode but should at least also include the second baroclinic mode in order to capture the dipole effect caused by upper tropospheric warming and lower tropospheric cooling due to stratiform clouds. Following the work of Mapes [184], Majda and Shefter [175] proposed and used a simple model based on the first two baroclinic modes (1.20) to perform a systematic wave analysis in the presence of stratiform clouds. While Mapes [184] used convective inhibition as a convective trigger, Majda and Shefter [175] formulated their model but using the evaporation of stratiform precipitation to the downdraft mass flux. As we will see the stratiform model of Majda and Shefter has the merit of being linearly unstable all by itself, unlike the adjustment and mass flux scheme discussed previously. We assume the following vertical expansion for the primitive equations, reduced to the first two baroclinic modes,



  v v v1 cos(z) + 2 cos(2z), = p1 p2 p

 

 w w1 ¯ w2 = sin(z) + sin(2z). θ θ1 θ2

(5.7)

The model for stratiform instability of Majda and Shefter takes the form of two systems of linear shallow water equations of the beta-plane equatorial dynamics coupled through the parameterized convective and stratiform heating rates and augmented with equations for boundary layer and mid troposphere θe to account for the effect of moisture and precipitation.

∂ v1 ¯ −1 − α ∇H θ1 + β yv⊥ 1 = −CD v1 − τD v1 , ∂t ∂ θ1 − α¯ divH v1 = S1 . ∂t ∂ v2 −1 − α¯ ∇H θ2 + β yv⊥ 2 = −CD v2 − τD v2 ∂t ∂ θ2 α¯ − divH v2 = S2 , j = 1, 2 ∂t 4

(5.8)

(5.9)

5.4 Stratiform Instability

89

Here (α¯ α¯ )1/2 ≈ 50 m s−1 is the speed of the first baroclinic gravity wave and S1 = Hd − QR,1 , S2 = Hs − QR,2 represent convective heating and radiative cooling forcing terms, where Hd and Hs are the deep convective and stratiform heating rates and QR, j = Q0R, j + τ1R θ j , j = 1, 2 are the radiative cooling terms where Q0R, j are imposed radiative cooling rates and τ1R θ j are Newtonian cooling adjustment terms, associated with the first (deep heating) and second (stratiform) baroclinic modes. The direct-deep convective heating is set to be proportional to CAPE: Hd = σc Hα¯m (CAPE+ )1/2 following the ICAPE scheme of the previous section [175] with the exception that the saturation equivalent potential temperature in the mid troposphere includes a contribution from the second mode: θem ∗ = θ1 + α2 θ2 with α2 = 0.1. The stratiform heating and boundary layer equivalent potential temperature, θeb , follow the equations: 1 ∂ Hs = (αs Hd − Hs ) ∂t τs ∂ θeb = −D + E. h ∂t

(5.10)

The stratiform heating is modelled according to the adjustment principle put forward by Mapes [184] according to which stratiform heating is continuously adjusting towards a fraction αs of deep convection with a time lag τs . Here αs = 0.25 and τs = 3 hours [184, 175]. The downdrafts include the effect of cooling induced by the evaporation of stratiform rain as well as the environmental downdrafts induced by the second baroclinic mode. Majda and Shefter [175] set  ¯ D = m+ + (σc − 1)w− e (θeb − θem ) where

1−Λ [μ ms + (1 − μ )mc ] Λ is the convective downdraft mass flux, with Λ = 0.9 is the precipitation efficiency, mc = σc wc is the deep convective part and ms = Hsαm¯ Hs is the stratiform contribution. Also, the environmental mass flux is decomposed according to m+ =

(1 − σc )we = we,1 − α2 we,2 , with we,1 = −(mc + Hm divv1 ), and we,2 = −(

Hm q2 Hm divv2 ). + α¯ 2 4

Here α2 and μ are parameters that set the contributions of the stratiform mode to convective updrafts and downdrafts. The default values are α2 = 0.1 and μ = 0.5. Moreover, the turbulent drag and the evaporation rate coefficients, CD and Cθ (WISHE) can include the effect of both the first and second baroclinic modes. 1/2 1 0  2 v + v1 + bv2 |2 CD/θ = CD/ θ u0 + |¯ h

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with CD0 = 10−3 ,Cθ = 1.2 × 10−3 and b = 0.9 is a parameter which sets the contribution from the second baroclinic mode. In [175], the two-baroclinic model with deep and stratiform heating modes is linearized and analysed around a radiative convective equilibrium state of rest; the background wind is set to zero. An importing result that stands out from this analysis is that the new model is capable of reproducing the desired scale selective instability without a background wind, i.e., without WISHE. For a physically sound parameter regime, the instability peaks at around wavelength 2000 km and is associated with moist gravity waves propagating in both east and west directions, in the case without rotation, with speeds comparable to those of observed superclusters ∼ 15 m s−1 . While the instability is robust to small changes in the convective closure parameters, one parameter in particular stands out. For small values of the convective area fractions, around σc = 0.001 and smaller, the system is stable but the growth rates increase drastically and the instability peak moves rapidly towards smaller scales when σc = 0.01, as can be seen in Figure 5.4. However, the case σc = 0.0014, which Majda and Shefter call the supercluster regime [175], displays a moderate growth rate, less than 0.2 per day, which peaks at the synoptic scale ∼ 2000 km while the case σc = 0.01 is called the cluster regime and peaks at the mesoscales ∼ 600 km. While this is a physically sensible interpretation, reflecting the extent of the area occupied by deep convection when viewed from different scales, it is not clear how one would control the parameter σc during a numerical simulation. Other strongly sensitive parameters of the stratiform model include the contribution of stratiform downdrafts parameter, μ , the precipitation efficiency parameter Λ , and the adjustment time scale τs [175]. The sensitivity to μ and τs in particular reveals of importance because of the capacity of the evaporation of stratiform rain in the wake of the deep convection to modulate the wave life cycle. While both an increase in the μ value and a decrease in τs yield an overall increase of growth rates, a small value of τs in particular tends to move the instability band to smaller scales. The model seems to be only marginally sensitive to the values of α2 and b. This is, in particular, exploited in [174] by setting both these parameters to zero and thus slightly reducing the number of parameters. A budget analysis of CAPE consisting of decomposing its time derivative into contribution from the first baroclinic, the second baroclinic, and boundary layer prognostic variables,

∂ {CAPE} = F1 −K2 Hs − Kθ θeb , ∂t with F1 contains all the first baroclinic variables while K2 and Kθ 0 are positive constants. We note that because we set α2 = b = 0, among all the second baroclinic mode variables, only Hs contributes to the CAPE budget equation. From this equation we can see that negative anomalies in stratiform heating Hs and θeb result in a positive CAPE tendency. The CAPE and various combinations of these three quantities plotted over a full wavelength for the eastward moving moist gravity wave are reported in Figure 5.5. Because the wave is moving to the east as a plane wave, the x-axis can be interpreted as time running from right to left. While

5.4 Stratiform Instability

91

Fig. 5.4 Stability diagram for stratiform model of Majda and Shefter with the standard parameter regime (see text for details) and zero background wind u¯ = 0. The growth rates of the unstablemoist-gravity branch, for σc = 0.0014 (dash) and σc = 0.01 (solid), are shown on the left panel while the phase speeds are plotted on the right panel for σc = 0.01. The branches corresponding to slowed down moist gravity waves, dry second baroclinic, and standing waves are represented by the solid, dashed, and dotted lines, respectively. Because of symmetry only the eastward branches are shown. Adopted from Majda and Shefter [175].

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the first baroclinic and the second baroclinic tendencies seem to cancel each other, it is the resulting residual that drives the total CAPE tendency while θeb acts as a damping term, which tends to restore the instability during the suppressed phase, as expected from the ICAPE closure, consistent with the quasi-equilibrium theory [8]. The positive CAPE tendency appears to undergo two stages. A longer stage to the right of x = 200 km where the positive trend is due to a contribution from −K2 Hs and a shorter period to the left of x = 200 km where F1 dominates. While the first baroclinic mode seems to contribute to the regeneration of CAPE during the wave cycle, it is evident from this picture that it is the second baroclinic and especially the negative anomalies in the stratiform downdrafts which drives the instability. This is the essence of the stratiform instability. Positive anomalies in Hs tend to cool and dry the boundary layer and reduce CAPE in the wake of deep convection while negative anomalies will automatically restore CAPE through the opposite effect. However, as we will demonstrate below in this section, while vigorous positive anomalies in Hs are expected to occur as a result of deep convection, the growth of negative anomalies will be limited because of the nature of convective heating, which is a positive only function. What is pictured in Figure 5.5 is merely an artefact of linear theory, in a realistic nonlinear simulation (and in nature) the small amplitude of the negative heating anomalies is compensated by longer periods compared to the short lived but vigorous convective events.

5.4.1 Nonlinear Simulations with the Stratiform Model: The Beautiful and the Ugly Nonlinear simulations with the stratiform model were considered in [174]. We recall the two shallow water systems in (5.9), coupled with each other through the cumulus parameterization scheme and augmented with the mid-tropospheric and boundary layer θe conservation equations. Although wind advection is neglected in all equations, the resulting system is intrinsically nonlinear and some of the nonlinear terms actually depend on the derivatives of the prognostic variables, namely the large divergence involved in the downdraft formula. For this reason, the coupled system is viewed in [174] as a fully nonlinear PDE and is discretized via a high order essentially non-oscillatory finite difference scheme, developed for HamiltonJacobi equations [209]. For simplicity, only the case of a meridionally symmetric flow above the equator, without rotation, is considered. The system is solved on the full 40,000 km equatorial ring with periodic boundary conditions and a spacial resolution Δ x = 40 km, or 1000 spacial grid points. The time integration is handled by a 3rd order Adams-Bashforth scheme using a small time step of 15 seconds to avoid numerical instabilities emanating from grid scale convection, without the use of artificial diffusion. The system is forced via a non-uniform boundary layer sat∗ , which mimics the distribution of sea urated equivalent potential temperature, θeb surface temperature along the equator, taking into account the persistence of the Indian Ocean/Western Pacific warm pool which extends over a region of roughly

5.4 Stratiform Instability

93

Fig. 5.5 Evolution of CAPE (a) fluctuation associated with the unstable moist gravity wave. Panels (b) and (c) display the total CAPE tendency and the contributions from first and second modes combined, from θeb , and from the first and second baroclinic modes separated. Since the wave is moving eastward the x-axis can be interpreted as time running from right to left. Adopted from Majda and Shefter [175].

10,000 km and centred in the middle of the computational domain as shown in the top-left panel of Figure 5.6. The initial conditions consist of the radiative convective ∗ plus a small random perturbation. equilibrium solution based on a constant θeb The equations are integrated for 440 days to reach an established statistical equilibrium—climatology characterized with wave oscillations about a radiative convective equilibrium. Some of the results are shown in Figure 5.6. The time evolution of the root mean square of the first baroclinic zonal velocity u1 is shown in the

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second panel from the top. The solution undergoes a short transient period of less than 50 days (which corresponds to the Newtonian cooling time scale τD ) before it enters its statistical equilibrium state characterized by almost periodic oscillations. The top-left panel displays the x − t contours (a.k.a a Hovm¨oller diagram) of the u1 deviations from its time mean, taken over the entire statistical equilibrium period from t = 50 days to t = 440 days. The u1 streaks are indicative of the persistence of synoptic scale waves propagating in both eastward and westward directions, at roughly 15 m s−1 , which emerge, amplify, and die as they propagate away from the centre of the domain. Note that because of the periodic boundary conditions, waves exiting the domain at the left boundary re-enter it from the right end and vice versa. The mean zonal wind (u1 and negative of u2 ) and mean stratiform heating are shown on the third and fourth panels from the top. As expected the mean heating peaks sharply in the middle at the warm pool centre consistent with observations. The mean wind components are characterized by a sharp gradient at this location indicative of low-level convergence and upper-level divergence within the heating region as illustrated in the bottom panel. It is interesting to note however that the mean total wind is nearly zero near the surface due to the perfect cancellation between the first and second baroclinic modes. The snapshot of the physical structure of the eastward propagating wave is displayed in Figure 5.7. Consistent with observations, the heating, wind, and temperature anomalies are characterized by a front to rear vertical tilt. The back of the waves is characterized by heating aloft and cooling near the bottom due to the lagging stratiform mode. The moisture is high in front of the wave and peaks down quickly as the wave passes while CAPE (not shown) peaks in front of the waves and drops down quickly after the convection peak. These are essentially the main features of convectively coupled waves listed in the beginning of this section [249, 174]. Nonetheless, the most discomforting feature comes from the fact that the waves themselves amplify and propagate away from the warm pool region while from observations we know that tropical convective systems are ubiquitous in the region over the Indian Ocean and Western Pacific (see Chapter 3 and the references therein). The main reason why the waves amplify only within the two regions on either side of the warm pool is because these are the regions where the mean wind is not zero and the WISHE effect is most significant. Consistent with the WISHE instability in Section 5.3, we see clearly eastward waves originating and amplifying within the region of low-level easterlies and westward waves over westerlies, with respect to the mean u1 component. The u2 component doesn’t contribute to WISHE since b = 0. To further confirm that the simulated waves are indeed WISHE waves and are not due to the stratiform instability as desired, we conducted another simulation but with WISHE turned off. The results are reported in Figure 5.8 where both the mean u1 component and the Hovm¨oller diagram for the deviations from this mean are displayed. While a mean circulation of comparable magnitude seems to have been reached the wave fluctuations are very weak and occur on much smaller length scales.

5.4 Stratiform Instability

Fig. 5.6 Nonlinear simulation for the stratiform model.

95

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5 Simple Models for Moist Gravity Waves

Fig. 5.7 Physical structure of waves as simulated by the stratiform model.

This discrepancy between the linear theory and the nonlinear simulations can be explained by the simple fact that for the linear solution, the negative phase of the stratiform mode produces negative stratiform heating anomalies in front of the wave which act as an instability trigger by increasing CAPE according to the CAPE budget analysis above. As already anticipated, the nonlinear solution has a much weaker negative anomaly because of the positive only nature of convective heating and as such it limits the growth of CAPE. To cope with this deficiency of the stratiform model, we present in the next chapter the multicloud model which in addition to the deep convective and stratiform clouds it carries a third cloud type, namely cumulus congestus that are observed to dominate the front of the wave. Congestus cloud decks serve to warm the lower troposphere and cool the upper troposphere just like a negative stratiform phase would do but the actual mechanism through which they

5.5 Exercises

97

Fig. 5.8 Mean (top) and Hovm¨oller diagram of the first baroclinic zonal wind as simulated with the stratiform model with WISHE turned off.

feed the convective instability is more subtle as they are associated with the complex process of moistening and preconditioning of the middle troposphere prior to deep convection. The rest of the volume will be devoted to the multicloud model in which congestus clouds play a central role.

5.5 Exercises 1. Consider the following “lagged” wave-CISK model as a modification of the CISK model in (5.1)-(5.2), following the idea of Davies [32]. ut − θx = 0

θx − ux = Q 1 Qt = − (α ux + Q) . τ

(5.11)

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Here τ is the time by which convection, Q, is assumed to lag low-level moisture convergence and α a positive parameter. a. Show that the dispersion relation for this system is given by iω (ω 2 − k2 )τ + (1 − α )k2 − ω 2 = 0. b. Use a computer software such as Matlab to solve the dispersion relation in (1a) for a small value of τ = 0.1 and discuss the stability of the propagation properties of the wave modes. Consider both the case α < 1 and the case α > 1. 2. Consider the convective adjustment system in (5.3). Let θw = θ + q/Q¯ be the socalled condensation temperature. Show that if Q¯ < 1 then this quantity decreases in the presence of precipitation. Consider the quantity 1 1 1 Q¯ E = u2 + θ 2 + θ 2. 2 2 2 1 − Q¯ w

3. 4.

5.

6.

Show that under the appropriate boundary conditions (u or θ vanishing at in∞ E dx decreases with time if finity, for example) the bulk “moist-energy” −∞ ∞ Pq dx > 0, when the evaporative forcing E and radiative cooling QR are ig−∞ nored. Reduce the convective adjustment system in (5.4) into a single equation for θ and find the associated dispersion relation. With the help of a computer software (e.g., Matlab) demonstrate that the adjustment scheme in (5.4) can grow unstable if Q¯ > 1 for sufficiently small τq . Assume that τθ = 0.1 and τe = 1. Consider the convective adjustment scheme in (5.4) with Q¯ = 0.9, τθ = τq = 0.1!and τe = 1. Plot " the wave structure of the eigensolutions (u, θ , q) = i(kx− ω t) Re (u, ˆ θˆ , q)e ˆ where ω = ω (k) is the generalized frequency and Le k = 2π l P is the dimensional wavenumber. Here, Le = c/β ≈ 1480 km is the equatorial deformation radius and P = 40, 000 km is the Earth’s circumference, and l = 1, 2, · · · . Consider the cases l = 1, 20, 47, 80, 200. Discuss the differences and similarities between the different wavenumbers. Compare the physical features of propagating and non-propagating waves. Consider the one-and-half layer model in (5.5). Write the detailed governing equations including the convective parameterization for the 2D case without rotation and with meridional symmetry (i.e., for flows above the equator). Consider each one of the LPA, ICAPE, and quasi-equilibrium closure schemes. Find the corresponding radiative convective equilibrium solutions and write down the associated linear systems. Use a computer software to reproduce the results in Figure 5.2

Chapter 6

The Multicloud Model with Congestus Preconditioning

6.1 Introduction As we saw from the previous chapter, among all the theories mentioned so far, for convectively coupled waves, the stratiform instability seems to be the most promising one. However, it suffers from a major shortcoming that the linear instability is not self-sustained without the use of WISHE, which itself is deemed unphysical as demonstrated by the nonlinear simulation presented there. There is no observational evidence of sustained easterlies in the tropical atmosphere to force eastward moving (Kelvin and MJO) waves that are ubiquitous in the tropical atmosphere, through WISHE [248, 126]. While the stratiform instability is very appealing and appears to be physically sound [184, 175], something fundamental seems to be missing. The answer to this hard question is rooted from the observations led by Lin and Johnson [156, 157] Johnson et al. [99], using data from the Tropical Ocean Global Atmosphere Coupled Ocean Atmosphere Response Experiment (TOGA COARE). These authors discovered that unlike the common belief at that time, the dynamics of organized tropical convection involves three cloud types besides shallow cumulus clouds that are omnipresent. In addition to deep and stratiform clouds, congestus clouds that do not penetrate above the freezing level (4–5 km) play a major dynamical roles. In fact, as seen in Chapter 3, propagating multiscale convective systems in the tropics were recognized to involve these three cloud types in a hierarchical manner (see [182] and references therein) with cumulus congestus dominating the front of the wave serve to heat the lower troposphere through condensation of water vapour and cool the upper troposphere through radiation and detrainment at cloud top. Deep convective cumulonimbus are found in the centre of the waves while stratiform clouds trail deep convection just as considered in the previous chapter. Deep clouds heat the entire troposphere, above the boundary layer, and stratiform clouds heat the upper troposphere and cool the lower troposphere. This hierarchy of clouds, as illustrated in the cartoon on top of Figure 6.1, forms the basic building block for the multicloud model (MCM), which is introduced in this chapter. The main dynamical mechanism, which constitutes the working hypothesis of the MCM, is © Springer Nature Switzerland AG 2019 B. Khouider, Models for Tropical Climate Dynamics, Mathematics of Planet Earth 3, https://doi.org/10.1007/978-3-030-17775-1 6

99

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6 The Multicloud Model with Congestus Preconditioning

Fig. 6.1 The three cloud types, congestus, deep, and stratiform which play a dynamical role in multiscale tropical convective systems, on top of a background of shallow cumulus clouds. The dashed lines show the three stable layers which characterize the tropical atmosphere, namely the trade wind inversion, the freezing level, and the tropopause. The red arrows show upward convecting flow and the green arrows represent downdrafts associated with environmental subsidence and or those induced by the evaporation of stratiform rain. The atmospheric boundary layer through which energy is supplied for convection due to sustained evaporation at the sea-surface is also illustrated. Adopted from [121].

as follows. When there is a potential for convection, i.e., positive CAPE (and low CIN), convecting parcels arise from the boundary layer and began to condense water in the lower troposphere and at the same time entrain dry and cold air from the environment. If the entrained air is too dry, then some of the condensed water starts to evaporate which further cools the parcel, which in turn looses buoyancy and detrains before it reaches the freezing level. However, if the entrained air is relatively moist, the parcel will continue its ascent and eventually reach the freezing level where it gets a boost through freezing (by adding even more latent heat) and the parcel happily penetrates to the top of the troposphere. This process is illustrated in Figure 6.3. As Johnson et al. put it, the tropical troposphere has indeed three stability layers, the trade wind inversion (the region of CIN) which caps shallow cumulus, the freezing level which caps cumulus congestus clouds, and the tropopause which caps deep cumulonimbus clouds. These three layers are illustrated in Figure 6.1 (see also Figure 6.5).

6.2 The Model Formulation and Main Closure Assumptions As for the stratiform model in the previous chapter, the minimal dynamical core for the multicloud model consists of the first two baroclinic modes of vertical structure forced and coupled through the heating and cooling rates associated with the three cloud types. These are essentially the main modes that are directly forced by the heating rates, Hc , Hd , Hs (defined below), associated with the three cloud types, congestus, deep, and stratiform, respectively, whose imposed profiles are illustrated in Figure 6.2.

6.2 The Model Formulation and Main Closure Assumptions

101

Fig. 6.2 Imposed vertical profiles of heating and cooling fields associated with the three cloud types. Adopted from [121].

et a_

{e

p}

Tropospheric Height

th

*

e}^

ta_{

the

theta_{ep}

Top

Moist adiabat theta_{eb}

*

{e}^

ta_ the

Moist adiabat: theta_{eb}

Tropospheric Height

Top

Surface

Surface Temperature

Temperature

Fig. 6.3 Convecting parcel rising though the troposphere. Left panels show a parcel evolving in a dry environment as it loses buoyancy and detrains below the freezing level to form congestus cloud while the right panel contrasts the case of a moist troposphere where the parcel happily raises above the freezing level and reaches its level of neutral buoyancy near the tropopause. Adopted from [120] (see also [111]).

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6 The Multicloud Model with Congestus Preconditioning

∂vj 1 + β yv⊥j − ∇θ j = −Cd (u0 )v j − v j , j = 1, 2, ∂t τW ∂ θ1 − div v1 = H¯ d + ξs H¯ c + ξs H¯ s + S1 ∂t ∂ θ2 1 − div v2 = Hc − Hs + S2 , ∂t 4

(6.1)

√

Following [121], the total precipitation is set to P = 2 π 2 (Hd + ξs Hs + ξc Hc ), and it includes contributions from deep convection, stratiform, and congestus clouds. Here ξs and ξc are parameters, with values between 0 and 1, representing the relative contributions of stratiform and congestus clouds to surface precipitation. As in the previous chapter, the total horizontal velocity and potential temperature are given in terms of the Galerkin expansions √ √ √ √ (6.2) v = 2v1 cos(z) + 2v2 cos(2z), θ = 2θ1 sin(z) + 2 2θ2 sin(2z). In (6.1), v⊥ = k× v, where k is the upward vertical unit vector, and ∇ and div are the gradient and divergence operators, respectively. The nonlinear advection terms are neglected for simplicity except for the moisture equation; they are believed to play a secondary role, though they may be important for interactions across scales as we see in the next chapters. The other source terms represent radiative cooling rates: S j = −Q0R, j − τ1R θ j , j = 1, 2. The rest of the parameters and variables are described in Table 6.1. The equations in (6.1) are supplemented by an equation for the boundary layer equivalent potential temperature, θeb , and an equation for the vertically integrated moisture content, q:

∂ θeb 1 = (E − D) ∂t hb

 ∂q 1 + div (v1 + α˜ v2 )q + Q˜ div(v1 + λ˜ v2 ) = −P + D ∂t HT

(6.3)

As before, hb ≈ 500 metres is the (fixed) height of the moist boundary layer and ˜ λ˜ , and α˜ are parameters associated HT = 16 km is the tropospheric height while Q, with a prescribed moisture background and perturbation vertical profiles. They are obtained through a systematic projection of an imposed moisture background of ˜ the form Q(z) = q0 e−z/Hq onto the first two baroclinic modes. The details of the procedure can be found in [118]. In particular, the parameter λ˜ measures the strength of moisture convergence induced by the second baroclinic mode due to congests heating, and as we will see below, it plays an important role in the dynamics of convectively coupled tropical waves and arguably, it is the backbone of the type of instability that arises from this new model. It complements the stratiform instability discussed in the previous chapter.

6.2 The Model Formulation and Main Closure Assumptions

103

Table 6.1 Parameters and symbols used in the multicloud model. While the typical values of some parameters are given here the interest reader should may consult the original papers as those values change considerably from case to case. The primes represent deviations from the RCE solution. Symbol Cd τW τR β u0 ξs , ξc

Λ

Description Turbulent boundary layer drag coefficient Rayleigh damping time scale Newtonian cooling time scale Gradient of Coriolis force at the equator Turbulence velocity scale Contribution of respectively stratiform and congestus heating to total precipitation Background moisture stratification Coefficient of second baroclinic nonlinear convergence Relative contribution of second baroclinic moisture convergence, linear term Stratiform heating adjustment coefficient Congestus heating adjustment coefficient Coefficient of θeb in Qd Coefficient of q in Qd Relative strength of buoyancy fluctuation in Qd Relative contribution of θ2 in Qd Relative strength of buoyancy fluctuation in Qc Relative contribution of θ2 in Qc Contribution of stratiform rain evaporation to downdraft Moisture switch function

Λ¯

Moisture switch function at RCE

m0

Downdraft mass-flux scale

HT Q0R,1 {1+μ [αs (1−Λ¯ )−Λ¯ αc ](θ¯eb −θ¯em )

Q¯

Background heating potential

Defined at RCE

Q˜ α˜

λ˜ αs αc a1 a2 a0 γ2 a0 γ2 μ

Value 0.001 75 days 50 days 2.28 ×10−11 m−1 s−1 2 m s−1 typically 05, 1.25, respectively 0.9 0.1 0.8 0.25 (varies) 0.1 (varies) 0.5 (varies) 1 − a1 (varies) 2 (varies) 0.1 (varies) 2 (varies) 0.1 (varies) 0.25 (varies) 1 if θeb − θem ≥ 20 K 0 if θeb − θem ≤ 10 K 0.1(θeb − θem ) − 1 if 10 K≤ θeb − θem ≤ 20 K 0.1(θ¯eb − θ¯em ) − 1 (varies)

According to the first equation in (6.3), θeb changes in response to the downdrafts, D, and the sea-surface evaporation E. When setting the closure for the forcing terms in (6.3), conservation of vertically integrated moist static energy is used as a √ design principle, namely the quantity θe z = HhbT θeb + 2 π 2 θ1  + q is conserved, in the absence of external forcing: S1 = E = 0. To ensure that the heating rates obey the physical intuition gained from observations as already discussed, the various forcing terms on the right-hand sides of the equations in (6.1) and (6.3) are formulated as follows. The stratiform and congestus heating rates, Hs and Hc , satisfy relaxation-type equations:

∂ Hs 1 = (αs Hd − Hs ) ∂t τs

(6.4)

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6 The Multicloud Model with Congestus Preconditioning

and

∂ Hc 1 = (αcΛ Qc − Hc ) ∂t τc

(6.5)

while the deep convective heating, Hd , is given by Hd = (1 − Λ )Qd .

(6.6)

The “moisture switch” function Λ controls the transition between congestus to deep convection, and it depends on the difference between boundary layer and midtropospheric equivalent potential temperatures, θeb − θem . We have ⎧ 1 if θeb − θem ≥ 20 K ⎨ 0 if θeb − θem ≤ 10 K Λ= (6.7) ⎩ Linear and continuous in between, so that when the middle troposphere is dry congestus clouds are promoted and deep convection is inhibited and when the mid troposphere is moist, deep convection is activated instead. In this fashion, deep convection will wait until the middle troposphere is moist enough to allow large-scale organization and sustain instabilities at synoptic and planetary scales of convectively coupled tropical waves, as it is demonstrated in the sections below. The diagnostic functions Qd , Qc , D, and Λ involve many nonlinear switches. We have     1     ¯ a1 θeb + a2 q − a0 (θ1 + γ2 θ2 ) , 0 Qd = max Q + τconv represents the deep convective heating potential which responds directly to atmosphere instability. It increases when the boundary layer θe is high or the midtropospheric temperature perturbation is low, which is in a sense some measure of CAPE and when moisture is high. The potential for congestus heating is set to     αc  Qc = max Q¯ + θeb − a0 (θ1 + γ2 θ2 ) , 0 τconv as such it responds to an increase in θeb or a decrease in lower tropospheric temperature anomalies. Note that Qc is not sensitive to moisture but responds only to positive CAPE. Indeed, a systematic derivation of the expression Qc given here is presented in [120] based on the concept of low-level CAPE of a diluted parcel according to the conceptual cartoon in Figure 6.3. The key idea is to express the CAPE integral (2.35) in Chapter 2 between the surface and mid-troposphere where the moist entropy (or equivalently θe ) of the rising parcel varies with height as the rising boundary layer air mixes with dry air by an amount that forces it to reach its level of neutral buoyancy below the freezing level. This systematic derivation allows, for instance, to obtain physically based values for the parameters γ2 and αc , for example, [120].

6.3 Linear Stability Analysis and the Congestus Preconditioning Instability Mechanism

105

The primed variables in the expressions of Qd and Qc represent deviations from their constant background values set by the radiative-convective equilibrium solution, which is defined below. The downdraft flux takes the form   Hs − Hc , 0 (θeb − θem ) . D = m0 max 1 + μ Q¯ In addition to a background downward mass-flux m0 , downdraft increases mainly with the increase in stratiform heating, mimicking the sinking of cold air due to the evaporation of stratiform rain, in the lower troposphere, when it falls into a dry environment. Since congestus heating induces an upward motion in the lower troposphere, it is set to counter the downdraft, when it is active. The formulation reported here follows the modified version of the MCM as presented in [121]. The congestus heating and the downdrafts are formulated differently in the seminal paper [118] where in particular the congestus heating is set to obey a delay equation with respect to downdraft. While such a formulation is sensible in terms of the life cycle of wave packets, there is no observational evidence to back it up, while the formulation of Qc presented here obeys the physical intuition and the abundant observations confirming that congestus clouds are ubiquitous in the tropics and they are not always tied to the regeneration of propagating waves. The present formulation obeys the idea that congestus heating will be activated whenever there is positive CAPE and the middle troposphere is dry. Finally, we note that most of the nonlinear advection terms are ignored but in the equations in (6.1). However, (6.3) are highly nonlinear due to the presence of the convection-source terms Hd , Hc , Hs , and D.

6.3 Linear Stability Analysis and the Congestus Preconditioning Instability Mechanism As in the previous chapter, we seek linear wave solutions around a radiative convective equilibrium (RCE)–a steady state for rest. Denoting RCE values by an overbar, the RCE solution for the equations (6.1)–(6.7) satisfies

π H¯ d + ξs H¯ c + ξs H¯ s = √ P¯ = Q1R,0 , Q2R,0 = H¯ c − H¯ s 2 2 ¯ H¯ d = (1 − Λ¯ )Q¯ H¯ s = αs H¯ d , H¯ c = αcΛ¯ Q,   Q2R,0 1 ¯ ¯ = P¯ D(θeb − θ¯em ) = m0 1 + μ ¯ HT Q 1 ¯ ¯ 1 D(θeb − θ¯em ) = (θe∗ − θ¯eb ). hb τe

(6.8)

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6 The Multicloud Model with Congestus Preconditioning

Accordingly, a radiative convective equilibrium is fully determined in terms of the three climatological variables, which control the tropospheric moist instability. Namely, we impose Q1R,0 , the background radiative cooling associated with the first baroclinic mode, the equivalent potential temperature difference θ¯eb − θ¯em , and the strength of surface evaporation θe∗ − θ¯eb . In particular the downdraft mass-flux scale m0 and the evaporative time scale τe are inferred directly from the RCE equations once the three climatological variables are prescribed, as summarized in Table 6.1. See [121] for details. The equations in (6.1)–(6.3) are linearized around this RCE background for flows above the equator, where rotation effects are ignored. Linear wave solutions, of the form U = Uˆ exp(i(kx − ω t), are then sought by solving the resulting matrix eigenvalue problem on a computer. For each fixed wavenumber k, we find the associated ˆ The effects of rotation eigenvalues ω (k) and the corresponding eigenvectors U(k). and its implications for convectively coupled equatorially trapped waves will be considered in the next two chapters. According to observations of the tropical atmosphere (see, for example, [48]), we set Q0R,1 = 1 K day−1 , θe∗ − θ¯eb = 10 K, and let θ¯eb − θ¯em vary from a moist background with a value around 11 K to an extremely ∗ − θ¯ dry environment with θ¯eb em = 19 K. The phase speeds (the real part of the eigenvalue ω (k) divided by k) and growth rates (the imaginary part of ω (k)) are reported on the two left rows of panels in Figure 6.4. Four different moist thermodynamic backgrounds are considered. For the moist background, corresponding to θ¯eb − θ¯em = 11 K, we see an instability band, 2 ≤ k ≤ 28, of two moist-gravity waves, peaking at around wavenumber k = 15 with a maximum growth rate of about 0.4 day−1 and a phase speed of roughly 17 m s−1 at their instability peak. As we move to θ¯eb − θ¯em = 14 K, the instability of the moist gravity waves reduces in both strength and extent and appears a new instability of a standing mode (zero phase speed) at the planetary wavenumber k = 1. This tendency becomes more evident at θ¯eb − θ¯em = 16 K. For reasons to be clarified later, we call this planetary scale standing wave, the congestus mode. Another regime however appears at θ¯eb − θ¯em = 19 K where both the moist gravity wave and planetary-congestus mode instabilities disappear and two new instability bands take place, a finite band spanning 10 ≤ k ≤ 25 of gravity waves moving at roughly 25 m s−1 , the speed of the second baroclinic dry gravity waves, and an infinite band corresponding to a new standing mode. While the latter bears some resemblance to the Wave-CISK instability seen in the previous chapter, we refrain from naming it as such because it has a different physical origin. Before moving further, it is important to remember that the moisture switch function Λ is set to depend explicitly on the value of θeb − θem and that when θ¯eb − θ¯em = 11 K, the heating profile is essentially taking up by deep convection and when θ¯eb − θ¯em = 19 K, deep convection is basically inhibited while congestus heating is promoted. A first insight into the physical structure of these waves is provided by the bar diagrams on the right two rows of Figure 6.4 that represent the relative strength of the dynamical and thermodynamical variables of the associated wave modes. First,

6.3 Linear Stability Analysis and the Congestus Preconditioning Instability Mechanism

107

Fig. 6.4 Left: Dispersion relations and stability diagrams for the multicloud model with congestus preconditioning. Four different atmospheric conditions are considered. From an extremely moist troposphere with θ¯eb − θ¯em = 11 K (top) to an extremely dry troposphere with θ¯eb − θ¯em = 19 K. Two intermediate states with θ¯eb − θ¯em = 14 K and θ¯eb − θ¯em = 16 K are also shown. The unstable bands are highlighted with red circles on dispersion curves. Right: Bar diagrams for one representative unstable mode corresponding to the 4 cases on the left displaying the contribution of the dynamical and thermodynamic variables to the wave dynamics. The moist gravity, the congestus standing, the moisture, and dry gravity modes are shown. Adopted from [121].

the moist gravity waves on the top panels consistently show a strong contribution from the two baroclinic velocity component, u1 , u2 (labelled as V1 ,V2 ), a strong deep convection heating, Hd , and moderate values for θeb , q, and Hs while Hc is relatively weak; only the relative contribution of Hc and Hd is significantly affected by the value of θ¯eb − θ¯em . For the planetary standing mode, at both 14 K and 16 K, while there is still strong fluid dynamics contribution from both baroclinic modes, the thermodynamics are dominated by the moisture anomaly and the congestus term dominates the heating rates. For this reason, this is called the congestus mode. The standing mode at 19 K on the other hand has very little fluid dynamics contribution and both strong moisture and deep convection contribution; the congestus heating contribution is strong but it is no longer dominating. Arguably, moisture and deep convection (i.e., precipitation) are the main drivers of this mode, for this reason it is termed as the moisture mode and it is indeed reminiscent to the mode of the same name identified in Section 5.2, but now it is unstable. Incidentally, this moisture mode manifests itself in all atmospheric models where moisture is coupled to the dynamics but it becomes unstable only when the model under consideration is taken beyond its physical limits, in an unrealistically dry environment, for example. There

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6 The Multicloud Model with Congestus Preconditioning

are, for instance, theories for the Madden-Julian oscillation [e.g. 226, 227, 239, 2] that put forward this moisture mode but they are questionable on other fronts [101]. On the other hand some promising models for the MJO involve a mode involving moisture-precipitation coupled wave but as a stable oscillation [178]. Finally, the fast gravity wave mode obtained at 19 K, together with the moisture mode, has a peculiar feature of being too much biased towards the second baroclinic mode and involves relatively little moisture and heating anomalies. It is not a coincidence that these waves move at constant speed of 25 m s−1 , the speed of the second baroclinic free/dry gravity wave. This mode seems to be simply a manifestation of that wave and as such carries very little moisture and lacks convection coupling; Its dispersion relation is hardly altered by the moist dynamics—The associated moisture and heating anomalies, as well as the feeble first baroclinic mode contribution, are likely slaved to the dry dynamics in this case, instead of being active in setting up or reducing the phase speed, for example. The physical and dynamical structure of the moist gravity waves and the two standing modes are displayed in Figure 6.5, where the u − w velocity arrows are overlaid on both the potential temperature and heating and cooling anomalies. The most striking difference between the propagating waves and the standing ones is the non-trivial front to rear tilt displayed by the moist gravity wave structure, consistent with observations and also with the stratiform instability waves seen in the previous chapter. This tilt is consistent with the leading low-level heating due to congestus heating and trailing upper-level heating due to stratiform cloud decks. Notice also the prominent low-level convergence in front of the wave due to congestus heating and the strong low-level subsidence in the wake of the heating centre due to stratiform cooling there. Also upward motion is in phase with the heating maximum. The two standing modes however show upright features. One major difference between the two is that the middle one has a second baroclinic dominated structure and is bottom heavy while the bottom one is mostly first baroclinic with a strong upright flow penetrating to the upper troposphere. The structure of the former is consistent with the dominating congestus heating while the latter corresponds mostly to deep convection. These structures are in line with the congestus and moisture modes name tags. The right two panels in Figure 6.6 exhibit the phase relationship between the dynamical and thermodynamical variables of the moist gravity wave, featuring the wave cycle. Importantly, these plots help understand the mechanics of the model instability which leads to these important waves that are the equivalents of convectively coupled Kelvin waves discussed in Chapter 3. As we can see, consistent with observations, the boundary layer θe (thick black line, top panel) peaks first in front of the wave which triggers an increase in congestus heating (red circle on bottom panel). These two are indeed in phase. Congestus heating then triggers low-level convergence which leads to an increase in moisture which appear to lag the congestus and θeb peaks by roughly one eighth wavelength, corresponding roughly to 8 hours time lag for a wave moving at 17 m s−1 . The increase in moisture coincides with a decrease in θeb − θem implying a mid-tropospheric moistening. Deep convection (black line on bottom panel) then follows this moistening episode after just a

6.3 Linear Stability Analysis and the Congestus Preconditioning Instability Mechanism

109

Fig. 6.5 Physical and dynamical structure of the typical unstable modes in the multicloud model. The (u, w) velocity arrows are overlaid on top of the potential temperature (left panels) and heating and cooling (right panels) anomalies. The moist gravity, the congestus standing, and the moisture mode are shown. Adopted from [121].

few hours and in turn depletes the moisture. Stratiform heating lags deep convection by about three hours, consistent with (6.4) and triggers downdrafts (blue triangle). We note that stratiform heating does not appear to be the main driver of downdrafts but mainly an intensifier and trigger of θeb fluctuations, the bulk values seem to be associated with θeb − θem , which is the conveyor belt of the stratiform instability as demonstrated in [179]. This is followed by a dry phase where downdrafts are the strongest, which in turn is followed by a regeneration of θeb and the wave cycle is closed. Therefore, the manifestation of the moist gravity wave instability (MGWI) is due both to the congestus preconditioning which drives moisture convergence in front of the wave and stratiform heating which amplifies downdrafts in the rear of the wave. This is consistent with the observed self-similar structure of tropical convective systems on various scales discussed in Chapter 3 [182] and we believe that the ability of this combined effect of the three cloud types to produce a prop-

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6 The Multicloud Model with Congestus Preconditioning

Fig. 6.6 Top: Wave left cycle of the moist gravity wave. The phasing of the large-scale dynamical and thermodynamical variables is displayed on the top panel while the parameterized heating/cooling and moistening/drying variables are plotted on the bottom panel. Bottom: dispersion relation for the case with second baroclinic moisture convergence set to zero displaying the instability of the moisture mode instead of moist gravity waves. Adopted from [121].

agating convectively coupled tropical disturbance operates at all these scales, from mesoscale systems to synoptic scale waves to the Madden-Julian oscillation as it is demonstrated in this volume. To demonstrate that the physical mechanisms of the MGWI are indeed as advocated above, we report on the right panels of Figure 6.6, parts of the dispersion relations associated with a few peculiar parameter regimes, first by setting the parameter λ˜ that drives low-level moisture convergence to zero and then by altering the parameter μ which sets the contribution of stratiform heating to the downdrafts. As we can see, when λ = 0 (while keeping all other parameters as in the top-left panel of Figure 6.4), the MGWI disappears completely and the instability is shifted towards the standing moisture mode-type obtained with the dry environment θ¯eb − θ¯em = 19 K. When μ = 0 instead, the whole system becomes stable and when we increase it from the standard value of μ = 0.25 to μ = 0.5, the instability of moist waves persists and becomes much stronger. The same shifting of the instability towards the standing moisture mode, as in the case with λ˜ = 0, is also seen when we set αs = 0. This confirms that the MGWI is indeed due to both the effect of stratiform clouds, mainly through the induced intensification of unsaturated downdrafts (parameter μ ) that cool and dry the boundary layer in the tail of the wave, and congestus preconditioning which supplies moisture to the front of the wave via the second baroclinic moisture convergence (parameter λ˜ ).

6.4 Nonlinear Simulations

111

6.4 Nonlinear Simulations To further demonstrate the effectiveness and robustness of the multicloud model in representing convectively coupled waves and organized convection in general, we present some results of nonlinear simulations when the model equations in (6.1) and (6.3) are integrated as an initial value problem, as done in the previous chapter for the stratiform instability model of Majda and Shefter [175]. However, unlike in the previous chapter we will not include additional mechanisms such as WISHE. As in the previous section, we consider 2d, x − z, flows above the equator where the effects of rotation are ignored so there is a perfect east-west symmetry. The equations are integrated numerically using a non-oscillatory finite difference scheme [116], on a periodic domain of 40,000 km, the circumference of the globe at the equator, over a long period of a few hundred days. We use a grid spacing of 40 km and a small time step of 2 minutes [121]. First, we consider two uniform RCE backgrounds with the intermediate dryness levels corresponding to the two values θ¯eb − θ¯em = 14 K and θ¯eb − θ¯em = 16 K, respectively. We recall from the previous section that each one of these two RCE states has two types of unstable modes, moist gravity waves moving in both directions at roundly 17 m s−1 and a planetarycongests mode instability. In the first one the MGWI dominates while in the second the congestus mode instability dominates. In both cases, the initial condition consists of the corresponding RCE solution plus a small perturbation. The solution for the moderately moist background with θ¯eb − θ¯em = 14 K is shown in Figure 6.7 in terms of the Hovm¨oller diagrams, i.e., the space-time contours, of its dynamical and thermodynamical variables in order to reveal the existence of travelling waves, for the simulation period between 100 and 200 days. The most apparent feature is the persistence of eastward moving waves that constantly circle the periodic domain visible in all panels from the red and blue strikes traversing the space-time corresponding to the movement of wave crests and troughs, respectively. The waves are moving roughly at 17 m s−1 as indicated by the dashed line following the wave crests in the Hd panel, associated indeed with the dominating MGWI from linear theory. By simply counting the number of wave crests across the domain, we can see that these are wavenumber 6 waves, which is lower than the linear instability peak wavenumber of 10. Somehow the nonlinearity shifts the instability to lower wavenumbers, i.e., towards large scales. Nonetheless, the curious thing about this simulation is the appearance of a secondary wavenumber one wave, especially visible in the θ2 , moisture, and congestus Homv¨oller plots, moving slowly at about 6 m −1 as shown by the dashed line in the Hc panel. The low-frequency wave streaks appear as envelopes that modulate the higher frequency moist gravity waves. This is undoubtedly a manifestation of the standing-congests mode instability which appears to move slowly in response to and in the direction opposite to the moist gravity waves. The fact that the wave is merely discernible in the Hc , θ2 , and Q variables confirms the congestus-preconditioning instability mechanism proposed above, with the congestus heating leading to an increase in θ2 and low-level convergence of moisture via the second baroclinic mode.

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6 The Multicloud Model with Congestus Preconditioning

Fig. 6.7 Hovm¨oller diagrams for the nonlinear simulation with the multicloud model for 2D flows above the equator. Case of a uniform background with a moderately moist RCE state θ¯eb − θ¯em = 14 K where the MGWI dominates. Adopted from [121].

6.4 Nonlinear Simulations

113

Fig. 6.8 Left: Spectral power of the nonlinear solution in Figure 6.7 showing distinct maxima corresponding to the higher frequency (synoptic) mode corresponding to the linear MGWI on the left of the Fourier domain and low-frequency mode corresponding to the wave envelopes moving in the opposite direction. The deep and congestus heating fields are shown. Right: Physical structure of the two modes obtained by filtering in Fourier space. Adopted from [121].

On the right panels of Figure 6.8, we plot the contours of the spectrum power (by performing a Fourier transform in both time and space) in the frequencywavenumber domain of the deep and congestus heating fields. Clearly, there is power on both panels associated with both the higher frequency moist-gravity waves (MGW) on the right of wavenumber zero and the low-frequency congestuspreconditioning mode with a period of about 25 days to the left. The period of the MGW varies between 4 and 5 days for the strongest congestus heating peak but the power follows a straight dispersion line towards higher frequency and higher wavenumber modes, consistent with the 17 m s−1 wave speed. It is attempting to interpret the low-frequency wave as being simply the energy modulation of the moistgravity waves moving with their group velocity. However, a quick look at the associated dispersion relation in Figure 6.4 reveals that the frequency actually increases with wavenumber (especially around wavenumber 6) implying a positive group velocity (for the eastward moving branch) and therefore the energy-modulating wave envelope would move in the direction of the wave and not in the opposite direction. This is in addition to the fact that this low-frequency mode is hardly visible in the first baroclinic velocity Hovm¨oller plot, for example, which is by far the most dominant energy wise. The right panels of Figure 6.8 show the physical and dynamical structure of the filtered low-frequency (top) and the synoptic-scale moist gravity (bottom) waves, obtained by applying a cut-off filter in the Fourier space, focusing on the spectrum peaks associated with each mode, separately. Details on the procedure can be found in [121]. The contours of the filtered potential temperature and total heating fields are overlaid on top of the u–w velocity arrows. Despite the different physical origins of these waves and the staggering difference in wavelengths, the two waves seem to exhibit qualitatively the same self-similar structure, with the backward

114

6 The Multicloud Model with Congestus Preconditioning

Fig. 6.9 Right same as Figure 6.7 but a moderately dry RCE state θ¯eb − θ¯em = 16 K where the congestus instability dominates. Here the Hovm¨oller plots are for deviations from a time mean (climatology) whose flow structure is shown on the right panels. Adopted from [121].

titled winds, temperature, and heating anomalies, as the moist gravity wave of linear theory in Figure 6.5, consistent with the observation and theory of multiscale convective systems [182, 128, 168] and in agreement with the multicloud paradigm [125, 156, 157, 99]. Note however, that the low-frequency mode in Figure 6.8 is relatively weak. We will see in Chapter 8 that mesoscale convective momentum transport can help boost the amplitude of this mode. The results of the simulation with a moderately dry background corresponding to θ¯eb − θ¯em = 16 K are shown in Figure 6.9, where the mean climatology (the time averaged solution) is represented on the three panels on the right while the Hovm¨oller plots of the deviations from this mean are shown on the left. Apparent from the left panel, especially in the Hc plot, is a standing mode response corresponding to the wavenumber one congestus mode, which dominates the linear instability in Figure 6.4. This standing mode then provides a shelter for the less unstable moist gravity waves that appear to bounce back and forth from the boundaries. Outside this sheltered region the atmosphere seems to be stable which further exemplifies the role of congestus preconditioning in sustaining convectively coupled waves. The resulting mean circulation which forms from the interaction of the congestus and the moist gravity wave modes shown on the left panel, in terms of the u − w velocity arrows overlaid on top of the potential temperature and heating and cooling anomalies curiously resembles the Walker cell-type circulation discussed in the previous chapter, although here we have a purely uniform background and this mean circulation is not driven by surface fluxes but by rather the interactions on the congestus mode and the moist gravity waves. If such behaviour has physical meaning, it would be over the arid regions of the glob if there could be enough surface moisture to sustain such circulation pattern while the free troposphere is kept dry by external processes. Nonetheless, this experiment constitutes a theoretical demonstration of the relevance of congestus cloud decks for organized tropical convection.

6.4 Nonlinear Simulations

115

Fig. 6.10 Case of a non-uniform background mimicking a warm pool like surface evaporation flux. The left panel shows the Hovm¨oller diagram of the first baroclinic velocity u1 for the period of 500 day simulation period while a blow-up on a 50-day period and only the left part of the spatial domain, including the wave pool centre. See text for details. Adopted from [119].

We end this section by showing in Figure 6.10 the simulation with a non-uniform background, where the surface flux term θe∗ − θ¯eb is set to mimic the Indian OceanWestern Pacific warm pool as in Figure 5.6. The results in Figure 6.10 were first reported in [119] and they were obtained with the first formulation of the multicloud model where, as already mentioned, the main difference with the equations reported here is in the closure for the congestus heating, where (6.5) is replaced by   ∂ Hc 1 D = αcΛ − Hc , ∂t τc HT so congestus heating lags the downdraft from the previous convective event instead of emanating directly from the atmospheric instability. Despite the lack of physical justification this formulation of the multicloud model seems to work fine at least in the context of simple 2D flows above the equator. This older version of the MCM was not pursued any further unlike the newer version which is implemented in a climate model as a cumulus parameterization. As can be seen from the Hovm¨oller plots in Figure 6.10, unlike the WISHE waves simulated in Figure 5.6, the wave activity is now concentrated within the warm pool region, in the centre of the domain, consistent with the ubiquity of propagating convective systems in the tropical atmosphere over the Indian Ocean-Western Pacific warm pool. The blow-up plot of convective heating on the right panels shows clear streaks of convective events associated with moist gravity waves moving at roughly 16 m s−1 and a hint of preconditioning envelopes moving in the opposite direction, with a speed ranging from 1.7 to 3.5 m s−1 , due to congestus heating developed in the wake of the previous events.

Chapter 7

Convectively Coupled Equatorial Waves in the Multicloud Model

As we saw in the previous chapter, the multicloud model exhibits a new kind of instability of moist gravity waves that are self-sustained, in the sense that these waves amplify naturally during nonlinear simulation without the need of an artifact such as WISHE; this wasn’t the case for the stratiform instability in Chapter 5. We also demonstrated that the multicloud instability is a combination of the stratiform instability and congestus preconditioning through low-level convergence. The latter mechanism should not be confused with congestus moistening through detrainment which may not be effective in sustaining and organizing the propagation of convectively coupled equatorial waves (CCEWs) but can indeed be important during the initiation phase [40, 268, 82, 100]. To demonstrate that the moist gravity waves seen in the previous chapter are indeed the analogs of CCEWs as observed in nature (c.f. Chapter 3), here we consider the effect of rotation and the background wind shear in the multicloud model (MCM) and show that under this combined effect, the MCM is able to capture the observed spectrum of CCEWs. The inclusion of the Coriolis effect will allow us to study, among other things, the effect of convection, namely the multicloud convection scheme, on the equatorially trapped waves introduced in Chapter 2. The effect of the background mean state is believed to be important for the destabilization and propagation of gravity waves and many previous studies (observational and theoretical) have been devoted to this important subject. The effect of the vertical and meridional shears have caught most of the attention [44, 280, 278, 301, 103, 274, 291, 17, 245, 180, 50, 80]. Thus part of this chapter will be devoted to the effect of meridional and vertical shear on the organization of tropical convection and convectively coupled equatorial waves in the context of the multicloud model, following the work of Han and Khouider [80]. In the next chapter, however, we also consider the effect of the background moisture as it impacts directly the preconditioning by low-level convergence associated with congestus heating and thus the moist gravity wave instability (MGWI) as discussed in the previous chapter, and more precisely the importance of the moisture background for the generation of mesoscale convective systems and multiscale waves in the MJO envelope [113]. © Springer Nature Switzerland AG 2019 B. Khouider, Models for Tropical Climate Dynamics, Mathematics of Planet Earth 3, https://doi.org/10.1007/978-3-030-17775-1 7

117

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7 Convectively Coupled Equatorial Waves in the Multicloud Model

7.1 Governing Equations and Method of Solution We consider the nonlinear hydrostatic Boussinesq primitive equations on an equatorial beta-plane, with rigid lid boundary conditions, introduced in Chapter 2. Following [80, and the references therein], we project these equations onto the barotropic or zeroth mode and the first and second baroclinic modes of vertical structure. √ √ V = v0 + 2 cos(z)v1 + 2 cos(2z)v2 , √ √ √ √ 2 2 sin(2z)w2 = − 2 sin(z)∇ · v1 − sin(2z)∇ · v2 , W = 2 sin(z)w1 + 2 2 √ √ P = p0 + 2 cos(z)p1 + 2 cos(2z)p2 , √ √ Θ = z + 2 sin(z)θ1 + 2 2 sin(2z)θ2 . (7.1) Here V is the (truncated) horizontal velocity, W is the vertical velocity, P is the pressure, and Θ is the total potential temperature. The barotropic and first baroclinic dynamical components solve the following three coupled two-dimensional systems of PDEs:

∂ v0 + ∇p0 + yv⊥ 0 + v0 · ∇v0 = − (v1 · ∇v1 + v2 · ∇v2 + w1 v1 + w2 v2 ) ∂t = −(div(v1 ⊗ v1 ) + div(v2 ⊗ v2 )) divv0 = 0, (7.2) √ 1 ∂ v1 2 v + v · ∇v + v · ∇v = S − − ∇θ1 + yv⊥ (v1 · ∇v2 + v2 · ∇v1 + v1 ∇ · v2 + 2v2 ∇ · v1 ) 0 1 1 0 1 1 ∂t 2 2 √ ∂ θ1 2 1 − ∇ · v1 + v0 · ∇θ1 = S1θ − (2v1 · ∇θ2 − v2 · ∇θ1 + 4θ2 ∇ · v1 − θ1 ∇ · v2 ), (7.3) ∂t 2 2

and √ ∂ v2 2 ⊥ v − ∇θ2 + yv2 + v0 · ∇v2 + v2 · ∇v0 = S2 − (v1 · ∇v1 − v1 ∇ · v1 ) ∂t 2 √ ∂ θ2 1 2 − ∇ · v2 + v0 · ∇θ2 = S2θ − (v1 · ∇θ1 − θ1 ∇ · v1 ). (7.4) ∂t 4 4 ⊥ The term yv⊥ i , i = 0, 1, 2 is the Coriolis force with (u, v) = (−v, u) and the parameter beta has been normalized to one according to the nondimensionalization units introduced previously, while Svj and Sθj represent, respectively, momentum damping and convective heating and cooling terms. Note that the hydrostatic and continuity equations give the relations:

1 θ j = −p j , w j = − ∇ · v j , j = 1, 2. j

(7.5)

7.1 Governing Equations and Method of Solution

119

Recall that in the previous chapter, both the barotropic mode and the momentum transport terms were omitted. For consistency, here we also include the advective effects in the boundary layer potential temperature (θeb ) and in the congestus (Hc ) and stratiform (Hs ) heating equations. Numerical experiments show that without these advective terms in the stratiform, congestus, and θeb equations, the multicloud model with shear exhibits “catastrophic instabilities” at small scales, i.e., non-scaleselective instabilities that do not decay with wavenumber. We recall that the stratiform/congestus mode mimics cloud decks in the upper/lower troposphere that heat the upper/lower troposphere and cool the lower/upper troposphere. We assume the following vertical structure function for the stratiform/congestus cloud types, confined to the upper/lower troposphere: √ Hs (x, y, z,t) = 2Hs (x, y,t)φ (z) √ Hc (x, y, z,t) = 2Hc (x, y,t)(−φ (π − z)), (7.6) 

where

φ (ζ ) =

sin(2ζ ) 0

π 2

ζ π 0  ζ < π2 .

We consider the conservation equations for the stratiform and congestus heating rates: ∂ Hx ∂ Hx + V · ∇Hx + w = Fx , x = s, c, ∂t ∂z where Fx , x = s, c are the production functions of stratiform and congestus, respectively, when these equations are projected onto the basis functions φ (z) and −φ (π − z), respectively, we obtain √ √ ∂ Hs 32 2 16 2 + v0 · ∇Hs = v1 · ∇Hs + w1 Hs + fs , (7.7) ∂t 15√π 15√π ∂ Hc 32 2 16 2 + v0 · ∇Hc = − v1 · ∇Hc − w1 Hc + fc . (7.8) ∂t 15π 15π Here fx stands for the projection of Fx . Similarly, by integrating over the mixed boundary layer, we obtain the governing equation for the boundary layer equivalent potential temperature √ ∂ θeb + v0 · ∇θeb = fb − 2(v1 · ∇θeb + v2 · ∇θeb ). ∂t

(7.9)

The closure formulations including the conservation equation for the vertically integrated specific humidity, q, are as in the previous chapter, they are thus not repeated here. In this chapter we are mainly interested in the linear waves and MCM instabilities in the presence of the beta and wind shear background effects. We thus linearize the equations in (7.2)–(7.9) around a radiative-convective equilibrium with a background meridional and vertical shear. For this we follow the derivation set forward

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7 Convectively Coupled Equatorial Waves in the Multicloud Model

in [80], see [180] for the case without rotation. For simplicity, we assume that the meridional shear is contained in the barotropic background wind, while the background baroclinic wind is independent of y. The vertical shear can only be associated with the baroclinic wind background by construction. As in [80], we use √ √ U(x, y, z) = u0 (y) + u1 2 cos(z) + u2 2 cos(2z), (7.10) where u¯ j , j = 0, 1, 2 represents the strength of the jth component of the background wind. The linearized equations for perturbations from the background solution are obtained by neglecting all quadratic terms leaving only those where the background wind and the perturbation act on each other. We obtain [80]: ∂ u1 ∂ u1 ∂ u0 (y) ∂ θ1 C u0 1 + u0 (y) + v1 − − yv1 = − d u1 − u1 ∂t ∂x ∂y ∂x hb τD √   2 3 ∂ u2 ∂ u1 1 ∂ v2 ∂ v1 + 3u2 + u1 + 2u2 − u1 2 2 ∂x ∂x 2 ∂y ∂y

√   ∂ u2 ∂ u2 ∂ u0 (y) ∂ θ2 2 ∂ v1 Cd u0 1 + u0 (y) + v2 − − yv2 = − u2 − u2 + u1 ∂t ∂x ∂y ∂x hb τD 2 ∂y √   ∂ v1 ∂ v1 ∂ θ1 2 ∂ v2 ∂ v1 Cd u0 1 + u0 (y) − + yu1 = − + u2 v1 − v1 − u1 ∂t ∂x ∂y hb τD 2 ∂x ∂x √   ∂ v2 ∂v ∂θ 2 ∂v C u 1 + u0 (y) 2 − 2 + yu2 = − d 0 v2 − v2 − u1 1 ∂t ∂x ∂y hb τD 2 ∂x √     ∂ θ1 ∂ θ1 ∂ u1 ∂ v1 2 ∂ θ2 ∂ θ1 + u0 (y) − + − u2 = S1 − 2u1 ∂t ∂x ∂x ∂y 2 ∂x ∂x √     ∂ θ2 ∂ θ2 1 ∂ u2 ∂ v2 2 ∂ θ1 + u0 (y) − + u1 (7.11) = S2 − ∂t ∂x 4 ∂x ∂y 4 ∂x     ∂q ∂ q ˜ ∂ u1 ∂ v1 ∂ u2 ∂ v2 ∂q 1 + u0 (y) +Q + + D − (u¯1 + α˜ u¯2 ) + λ˜ Q˜ = −P + ∂t ∂x ∂x ∂y ∂x ∂y HT ∂x   ∂ θeb ∂ θeb 1 ∗ ∂ θeb ∂ θeb D √ + u0 (y) = (θeb − θeb ) − − 2 u1 + u2 ∂t ∂x τe hb ∂x ∂x √   1 ∂ Hs ∂ Hs ∂ Hs 32 2 + u0 (y) = [αs Hd − Hs ] + u1 ∂t ∂x τs 15π ∂x √   1 ∂ Hc ∂ Hc ∂ Hc 32 2 + u0 (y) = [αcΛ Qc − Hc ] − u1 , ∂t ∂x τc 15π ∂x

where √ 2 2 P= (Hd + ξs Hs + ξc Hc ) π 1 S1 = Hd + ξs Hs + ξc Hc − Q0R,1 − θ1 τR 1 S2 = −Hs + Hc − Q0R,2 − θ2 . τR

7.1 Governing Equations and Method of Solution

121

Because the baroclinic background wind is constant in the horizontal direction, the barotropic perturbation is set to zero and not represented in (7.11) as there is no other means to drive it. Terms pertaining to the advection of the temperature background (resulting from background pressure gradient associated with the background baroclinic wind) are not included because they are small as they appear with a factor y in front of them. As demonstrated in [114], those terms induce only small quantitative changes to the results of [80] that are summarized in Section 7.4. More on this issue will be discussed in Chapter 8 where the effect of the third baroclinic mode background shear will also be considered. By further linearizing the expression for Hd , Λ Qc , and D, as in the previous chapter, we obtain a linear PDE system of the form [80]     ∂U ∂U ∂U ∂U ∂ u¯0 (y) + A1 + A2 u¯0 (y) + yR2U + R3 U = BU, + R1 ∂t ∂x ∂x ∂y ∂y where U stands for the vector solution, which contains all the diagnostic variables u j , v j , θ j , q, θeb , Hc , Hs , j = 1, 2 and A1 , A2 , R1 , R2 , R3 , B are 10 × 10 matrices representing the different forcing terms in the multicloud equations. The linearized equations are then expanded in the meridional direction in terms of the parabolic cylinder functions as done in Chapter 1. Here, however, the procedure is more involved and the equations do not break down easily into low dimensional linear systems as the triads in Chapter 1 that lead to the dry equatorially trapped waves. Here all meridional modes remain coupled to each other and from both baroclinic modes. To remedy this inconvenience we rely on a Galerkin truncation procedure, keeping only the first N meridional modes from the otherwise infinite expansion. Sensitivity experiments conducted in [120] reveal that 16 (N = 15) modes are enough to capture the main dynamical modes of the multicloud equations without suffering too much the influence of highly oscillating high meridional modes [172]. With the ansatz [120, 80] U(x, y,t) ≈

N

∑ U n (x,t)φn (y), N = 15,

(7.12)

n=0

supplemented by the following radiation conditions to filter out spurious waves introduced by the crude Galerkin truncation [172] vNi = 0, θin = − 1i uni , i = 1, 2 n = H n = H n = 0, n = N − 1, N , qn = θeb s c we obtain the following system of (136) linear PDEs in (x,t) only.

∂Un ∂Un ∂U n ∂ ∂ u0 n + A1 + A2 (u0 ) + R1 ( U) = BU n U)n + R2 (& yN U)n + R3 ( ∂t ∂x ∂x ∂ y&N ∂y n = 0, 1, ...N − 1. (7.13)

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7 Convectively Coupled Equatorial Waves in the Multicloud Model

The symbols y&N and ∂ /∂ y&N , whose precise definitions can be found in [172], are linear operators (matrices), acting on the finite dimensional subspace spanned by the 15 parabolic cylinder basis functions that approximate, respectively, the multiplication by y and the y-derivative of an arbitrary function of y expanded in terms of the parabolic cylinder functions. The operators A2 (u0 ∂∂Ux ) and R3 ( ∂∂uy0 U) are as follows. Let G = u0 (y)

∂ U N−1 ˜ = ∑ Gi φi (y), ∂x i=0

where N

N

l=1

l=1

G˜ i = G, φi (y) =

∂ ul

∑ Gl φi (yl )H l = ∑ u0 (yl ) ∂ x φi (yl )H l

and N−1

ul =

∑ U˜ k φk (yl ),

k=0

where yl and H l are the abscissas and weights of the Hermite-Gauss quadrature. Then we get G˜ i =

N

N−1

l=1

k=0

∑ u0 (yl )[ ∑

where al = H l u0 (yl ), u˜kx =

N−1 N ∂ U˜ k φk (yl )]φi (yl )H l = ∑ ∑ φi (yl )φk (yl )al u˜kx , ∂x k=0 l=1

∂ U˜ k ∂x .

So that G˜ i = ∑N−1 k=0 Aik u˜kx , where

Aik = ∑Nl=1 φi (yl )φk (yl )al = ECE T with ⎡ ⎤ ⎤ ⎡ φ0 (y1 ) · · · φ0 (yN ) u0 (y1 )H 1 · · · 0 ⎢ ⎥ ⎥ ⎢ .. .. .. .. .. .. E =⎣ ⎦. ⎦, C = ⎣ . . . . . . φN−1 (y1 ) · · · φN−1 (yN ) 0 · · · u0 (yN )H N Thus, A2 (u0

N−1 ∂U n ) = A2 G˜ n = A2 ∑ Ank u˜kx ). ∂x k=0

(7.14)

The operator R3 ( ∂∂uy0 U)n is obtained in the same fashion and is given by R3 (

N−1 ∂ u0 n U) = R3 ∑ RnkU˜ k , ∂y k=0

(7.15)

7.2 Uniform Background

123

where N

Rik =

∑ φi (yl )φk (yl )bl = EDE T , bl = H l u1 (yl ), u1 (y) =

l=1

⎤ u1 (y1 )H 1 · · · 0 ⎥ ⎢ .. .. .. D=⎣ ⎦. . . . 0 · · · u1 (yN )H N ⎡

∂ u0 (y) ∂y

The equations in (7.13) take the form:

∂ U˜ ∂ U˜ ˜ +A + RU˜ = BU, ∂t ∂x where U˜ stands for the expanded vector solution containing all the meridional coefficients of the multicloud U-variables. Linear wave solutions of the form ˆ −(kx−ω t) , U˜ = Ue where k is the zonal wavenumber and ω is the generalized frequency, are then sought, as in the previous chapter. For clarity, in what follows, we consider the following three cases separately: the case of a uniform background where the wind shear components are set to zero u¯ j = 0, j = 0, 1, 2 to look at the effect of rotation alone, the case of the barotropic wind shear alone, u¯0 = 0, u¯1 = u¯2 = 0, and the case of a vertical shear alone u¯0 = 0, u¯ j = 0, j = 1, 2. Different configurations of the barotropic wind shear will be considered. The simultaneous effect of both the barotropic or meridional shear and vertical shear on the multicloud waves remains an open research problem. The model parameters and constants used in this chapter are summarized in Table 7.1 for the sake of completeness, although most of them are the same as in the previous chapter.

7.2 Uniform Background In Figure 7.1 we report the dispersion relations, including phase speeds and growth rates, for the linearized MCM equations in (7.11) with a uniform background. The unstable modes are highlighted with circles on the frequency and phase speed diagrams. As indicated on the figure panels, the MCM with rotation exhibits instabilities on the Kelvin, n = 0 eastward inertia-gravity (EIW), and n = 1 westward inertiagravity (WIG) wave branches, at synoptic scales, with instability bands raging between roughly wavenumbers 10 and 20, corresponding to wavelengths 4000 km and 2000 km, respectively, consistent with observations. Also consistent with observa-

124

7 Convectively Coupled Equatorial Waves in the Multicloud Model

Table 7.1 Parameters of the multicloud parameterization Parameter HT hb Cd a1 a2 a0

Value 15.75 km 500 m 0.001 0.25 0.75 3

a0

1.5

α˜ λ˜ Q˜ u0 θ± Λ∗ τconv τs τc m0 τcong τR τD αs αc ∗ −θ θeb eb

0.1 0.8 0.9 2m s−1 20, 10 K 0 2h 3h 1h Determined at RCE τconv /αc = 20h 50 days 75 days 0.16 0.1 10 K

θ eb − θ em

11 K

Q0R,1 Q0R,2 Q μ

1.K/day Determined at RCE Determined at RCE 0.25

α2 γ2 γ2 ξs

0.1 0.1 2 0.5

ξc

1.25

Description Height of the troposphere Height of the boundary layer Momentum drag coefficient Relative contribution of θeb to deep convection Relative contribution of q to deep convection Coefficient of θ1 in Qd formula: inverse convective buoyancy time scale associated with deep clouds Coefficient of θ1 in Qd formula: inverse convective buoyancy time scale associated with congestus clouds Coefficient of v2 in nonlinear moisture convergence Coefficient of v2 in linear moisture convergence Background moisture stratification Strength of turbulent fluctuations Moisture switch threshold values Lower threshold of moisture switch Convective time scale Stratiform adjustment time scale Congestus adjustment time scale Downdraft mass flux reference scale Congestus heating time scale Newtonian cooling time scale Rayleigh-wind relaxation time scale Stratiform adjustment coefficient Congestus adjustment coefficient Discrepancy between boundary layer θe and its saturation value Discrepancy between boundary layer and middle troposphere θe ’s at RCE Imposed first baroclinic radiative cooling rate Second baroclinic radiative cooling rate Bulk convective heating at RCE Relative contribution of stratiform and congestus heating rates to downdrafts Relative contribution of θ2 to θem Relative contribution of θ2 to deep heating Relative contribution of θ2 to congestus heating Relative contribution of stratiform clouds to first baroclinic heating Relative contribution of congestus clouds to first baroclinic heating corresponding to fs = f c = 0.1 (KM08)

tions, the largest growth rates on the bottom panels are by far dominated by the Kelvin modes, consistent with their dominant spectral power in observations (Figure 3.2). Also, the comparison with the observed power spectrum diagram indicates that only three out of the main spectral peaks reported there have their instability analogs in Figure 7.1, the equivalents for the Madden-Julian oscillation, the westward mixed Rossby-gravity (MRG), and Rossby wave peaks are lacking. As we will see below, the instability of MRG and Rossby waves will be recovered through the

7.2 Uniform Background

125

Fig. 7.1 Dispersion relations of the multicloud model with rotation. Case of uniform background. The symmetric waves are shown on the left panels and anti-symmetric waves are on the left panels. The frequencies ℜ(ω ), phase speeds ℜ(ω )/k, and growth rates ℑ(ω ) are plotted as functions of the horizontal wavenumber, k. The unstable modes (ℑ(ω ) > 0) are highlighted with red circles on the frequency and phase speed plots.

effect of a barotropic-meridional shear. The MJO, however, is beyond the scope of the linear analysis of the MCM equations but it is captured in global circulation model (GCM) simulations presented in Chapters 10 and 14 consistent with the idea that the MJO is a planetary scale envelope of synoptic and mesoscale convective systems as conjectured by some observational studies [204, 126, 300] and was the basis for some theoretical modelling ideas [165, 178]. MJO theory is still an active

126

7 Convectively Coupled Equatorial Waves in the Multicloud Model

research area and the physical mechanisms responsible for its genesis, structure, and propagation are still in debate [141, 160]. The physical and morphological structures of the unstable waves in Figure 7.1 are reported in Figure 7.2. The strength of each basic coefficient in the Galerkin expansion is reported on the top row panels. The meridional coefficients of the physical variables, corresponding to the first and second baroclinic mode equations as well as the moisture, boundary layer θe , the congestus, stratiform, and deep heating rates are shown in various grey shadings. The darkest corresponding to the lowest order mode. The coefficients of the even basis functions φ0 , φ2 , φ4 , · · · are shown for the two symmetric waves (Kelvin and n = 1 WIG), while those of φ1 , φ3 , φ5 , · · · are reported for the anti-symmetric n = 0 EIG wave, also known as the Yanai wave. Similarly to the two-dimensional case in Chapter 6, the unstable modes are mixtures of the first and second baroclinic modes. Moreover, unlike the case of the dry equatorially trapped waves, the convectively coupled waves reported here display a more complex meridional structure in the sense that they have non-zero contributions from higher parabolic cylinder functions; the Kelvin wave, for example, involves functions of higher index than φ0 . This is, however, a mere result of the second baroclinic Kelvin wave contribution whose structure doesn’t match perfectly that of φ0 because it is associated with a shallower equivalent depth and thus a smaller equatorial Rossby deformation radius [172]. The coefficients of the meridional velocities v1 , v2 are consistently both zero for the Kelvin wave mode. More importantly, consistent with the 2d case, all the waves carry significant moisture and heating components confirming that these are indeed the convectively coupled analogs of the equatorially trapped waves reported in Chapter 1. We recall that in Chapter 6, we have instead moist gravity waves that move in both east and west directions as the analogs of the Kelvin and inertia-gravity waves in general. The horizontal structure plots reported on the second and third rows of Figure 7.2 further confirm the identity of the unstable waves, as the convectively coupled equivalents of the equatorially trapped waves. Indeed, despite the fact that they are mixtures of first and second baroclinic modes (whose meridional structures are significantly distinct—scale-wise), we can easily recognize the features of the corresponding equatorially trapped waves seen in Chapter 1, although the near surface and top of tropospheric wind vectors seem to be out-of-phase instead of being simply in opposed phases as expected from the deepest baroclinic waves. Indeed, the vertical structure plots shown on the bottom panels show that the velocity vectors and heating and cooling contours are self-similarly tilted backward (against the direction of the propagation), consistent with observations in Figures 3.7 and 3.8. This backward tilt, which is also present in the temperature and pressure anomalies, is a direct result of the organized multicloud structure of these waves, where cumulus congestus cloud decks, heating the lower and cooling the upper troposphere, lead the wave while stratiform heating with the opposite signature trail deep convection

7.3 Meridional Shear Background

127

which defines the heating centre of the wave, as already discussed in Chapter 6. The key point to note here is that this tilted vertical structure in addition of being self-similar across scales as shown in Figure 6.7, it transcends the equatorial wave spectrum as recognized in observations [127] (see Figure 3.9).

7.3 Meridional Shear Background The type of meridional shear mimicking the jet stream with maximum strong westerlies in the subtropics and weak easterlies in the tropics is shown on the top panel of Figure 7.3, with strengths varying from u0 = 5 m s−1 to u0 = 20 m s−1 . The dispersion relation of the multicloud model equations under the influence of this barotropic shear background is presented on the middle and bottom panels of Figure 7.3 when u0 = 10 m s−1 and u0 = 20 m s−1 , respectively. The first thing to notice is that at u0 = 10 m s−1 , the instability diagram still features the Kelvin and the n = 0 EIG waves as the dominant ones. However, it also seems to suggest that eastward moving waves are favoured at the expense of westward waves, since we see the emergence of unstable modes along the n = 1 and n = 2 EIG waves, while the previously unstable n = 1 WIG waves are now not unstable. This has perhaps something to do with the nature of the background wind which has dominant westerlies as it is the case of the vertical wind shear presented below. With the strong background wind (u0 = 20 m s−1 ), we see a more significant change in the instability diagram as we see a manifestation of new instabilities, namely of MRG and Rossby waves. As demonstrated in [80], the instability of MRG and Rossby waves is primarily due to the presence of the meridional shear, i.e., this is essentially a barotropic-baroclinic instability due to dry dynamics, though significantly reinforced by convection. This is consistent with the studies of African easterly waves [149, 93]. We note also that there are other instabilities that are not linked to known branches of equatorially trapped waves. They are essentially associated with the extra-prognostic equations governing the evolution of moisture, boundary layer θe , and also the stratiform and congestus heating rates. For this reason these modes are known by the generic name of a moisture mode (see Section 5.2). As already pointed out, there is a whole line of research on MJO theory based on the moisture mode as a main mechanism for the MJO [e.g., 2]. To confirm that the two new instabilities correspond indeed to observed convectively coupled equatorial waves and especially the MRG and Rossby waves, which were missing in the case with a uniform background, we report in Figure 7.4 the horizontal structure of the Kelvin, the n = 1 WIG, the MRG, and the Rossby waves. We note that the Kelvin and WIG waves are under the influence of a weak wind shear u0 = 5 m s−1 , where the WIG is still unstable. While the wave structure bear a lot of resemblances with the free equatorially trapped waves of the same name, they have also many discrepancies. There are also discrepancies between the Kelvin and WIG waves of the previous sections, corresponding to the MCM equations with a uniform background, and those evolving in a barotropic wind shear environment.

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Fig. 7.2 Physical and morphological wave structure of the unstable modes, namely the Kelvin, the n = 1 WIG , and the n = 0 EIG, respectively. The top row panels are the bar diagrams measuring the relative amplitude of the basic coefficients of each wave mode, respectively. The second and third rows show the corresponding horizontal structure where the filled contours of heating at z = 15 km and pressure near the surface with the horizontal velocity field (u, v) overlaid. The last row reports the associated vertical profiles of the temperature perturbation and the (u, w) velocity arrows, where all variables are averaged in the y-direction.

These differences include a severe deformation of the WIG wave by the wind shear and the emergence of non-trivial meridional flow in the Kelvin wave structure. The non-zero meridional flow in the Kelvin wave is due to the geostrophic unbalance induced by the meridional shear as shown in [50], while the severe deformation of the WIG wave is perhaps due to strengthening of the effective shearing as a result

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Fig. 7.3 Convectively coupled waves in the presence of a barotropic/meridional shear. The type of meridional shear background with strengths varying from u0 = 5 m s−1 to u0 = 20 m s−1 is shown on the top panel. The remaining panels report the wave instability diagrams for the cases u0 = 10 m s−1 and u0 = 20 m, separating the symmetric and anti-symmetric parts of the wave spectrum. The unstable modes are highlighted with red circles with the sizes of circles set proportional to the growth rate. The growth and phase speeds at the instability peak are indicated for each branch.

of the direction of the wave propagation which is against the two subtropical jets. In fact, a somewhat aggravated similar deformation is seen in the MRG and Rossby waves on the bottom panels, which also happen to move westward.

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Fig. 7.4 Horizontal structure of the Kelvin (top-left), the n = 1 WIG (top-right), the MRG (bottomleft), and the n = 1 Rossby (bottom-right) waves that are unstable in the MCM equations with a background barotropic shear (top: u0 = 5 m s−1, bottom: u0 = 20 m s−1)

7.4 Effect of a Vertical Shear Background Now we turn attention to the case of a vertical shear background. This is achieved by setting u¯0 = 0 in (7.10) and allowing u¯1 and u¯2 to be non-zero. We begin in Figure 7.5 by looking at the case when the first and second baroclinic components of the background wind are considered separately. With the first baroclinic shear, the main effect seems to be that positive values of u¯1 , corresponding to westerlies in the lower troposphere and easterlies aloft, result in the amplification of the instability of the westward moving waves, while negative u¯1 appears to stabilize westward waves altogether and significantly amplify eastward moving waves, namely Kelvin and n = 0 EIG waves. The effect is more dramatic under the influence of the second baroclinic shear background. In this case, we see clearly that positive u¯2 values favour eastward waves and stabilize WIG waves, while negative u¯2 values have the exact opposite effect. Comparing the first and second baroclinic cases we notice that the sign relationship is reversed. This is consistent with the fact that positive u¯1 is associated with a negative (easterly) upper level wind, while positive u¯2 value corresponds to a positive (westerly) upper level wind. We thus conjecture that in

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Fig. 7.5 Instability diagrams for the MCM with vertical shear background. Two top rows show the case of 1st baroclinic wind shear alone, while the bottom rows are with a second baroclinic wind only.

general, a positive upper level wind shear favours eastward moving wave, while the opposite case favours westward moving waves. To further investigate the above conjecture, we consider in Figure 7.6 two different combinations of the first and second baroclinic wind shear components, namely the case when the two have the same sign and same magnitude and the case when they have the same magnitude but they are of opposite sign. The first case leads to a reinforcement of the low-level shear and a weakening of the upper level shear and the latter has the exact opposite effect. Consistent with what was already observed, the sign of the upper level shear seems to control the growth and instability of con-

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Fig. 7.6 Same as Figure 7.5 but different combination of first and second baroclinic wind shears.

vectively coupled waves based on their direction of propagation. The strength of the upper level shear affects the magnitude of the growth rates as well as the wavelength of the most unstable modes. Furthermore, the case of a stronger low-level wind shear results in a new instability of moisture-like waves that grow at smaller scales. This is consistent with squall line and mesoscale system theories which state that lowlevel shear is the main driver for this type of convective disturbances. Squall lines and mesoscale systems will be discussed further in the next chapter especially with the addition of the third baroclinic mode wind shear and the variation of moisture background. The addition of a third baroclinic mode will in particular improve the resolution (or representation) of the low-level shear as observed in nature.

Chapter 8

Convective Momentum Transport and Upscale Interactions in the MJO

8.1 Introduction As already discussed, the tropical atmosphere harbours a scale hierarchy of convective wave disturbances that are often embedded in each other like Russian dolls, evolving on a wide spectrum of scales ranging from the individual cloud cell diameter to the size of planetary scale disturbances such as the Madden-Julian oscillation, see Figure 8.1a. A particularly interesting issue is to understand the way these multiscale convective systems interact with each other across temporal and spatial scales. There is enough evidence from both observations and numerical simulations that momentum transport plays a genuine role in these interactions [147, 194, 145, 287, 177, 138, 113]; the smaller embedded waves provide turbulent fluxes for the envelopes, while the larger scale envelope waves provide a background advecting wind for the smaller scale waves. In this fashion, there will be two-way interactions between the MJO and the synoptic scale CCWs, the MJO and mesoscale convective systems and squall lines, and between the CCWs and the mesoscale systems. This is illustrated schematically in Figure 8.1b. Also as already mentioned, these wave disturbances present a self-similar zonal and vertical structure with a pronounced front-to-rear vertical tilt [182, 168], which appears to be of particular significance for their ability to transport momentum upscale as reviewed in this chapter. This self-similarity which is briefly discussed in the previous chapters is well documented in various theoretical and observational works. Recall the self-similar cloud-type and flow morphology of these waves illustrated in Figure 3.9. Convective clouds not only release heat and moisture into the troposphere but they also redistribute momentum due to turbulent fluxes generated on a wide range of scales, ranging from those due to direct parcel lifting to mesoscale (100 to 500 km) and synoptic (1000 to 5000 km) scale organized circulations. This dynamical mechanism commonly known as convective momentum transport (CMT) is generally described as the process of vertical transport of horizontal momentum by orga-

© Springer Nature Switzerland AG 2019 B. Khouider, Models for Tropical Climate Dynamics, Mathematics of Planet Earth 3, https://doi.org/10.1007/978-3-030-17775-1 8

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Fig. 8.1 Two-way interactions between the MJO, CCWs, and mesoscale convective systems.

nized convection onto the environmental flow. The horizontal fluxes turn out to be negligible. The observational studies of LeMone [147] and LeMone and Moncrieff [145], for instance, show a clear relationship between the vertical transport of horizontal momentum and cumulus clouds. Using data from Tropical Ocean Global Atmosphere Coupled Ocean Atmosphere Response Experiment Intensive Observing Period [281] (TOGA-COARE IOP), Tung and Yanai [265, 266] demonstrated that kinetic energy is transferred from convection to large scales in events such as squall lines and westerly wind bursts (WWBs). Similar conclusions were reached by LeMone et al. [148] and Grabowski and Moncrieff [76]. In general, the dynamical scale of cumulonimbus is on the order of 10 km, according to observations and numerical simulations, while the accompanying convectively driven mesoscale circulation can be hundreds of kilometres or more, making it difficult to reliably estimate CMT [194, 288]. There is also an evidence that the large-scale environment modulates convective systems at all scales [146, 231, 230, 289, 280, 81, 201]. The effect of the background shear on synoptic scale convectively coupled waves has been discussed in the previous chapter in the context of the multicloud model [80]. In this chapter, we explore the two-way interactions between multiscale tropical convective systems through various simulation and modelling examples. We begin by presenting a cloud-permitting numerical simulation of a Kelvin wave, following the work

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of Khouider and Han [112], shows evidence of embedded mesoscale-scale disturbances with a non-negligible CMT effect on the parent wave. We then present a prototype parameterization models for mesoscale CMT in GCMs in an idealized test bed. We begin with the stochastic model for CMT of Majda and Stechmann [177] and follow with a deterministic version and an illustration on how mesoscale CMT can impact the simulation of the Madden-Julian oscillation-like disturbance and moist gravity waves in a toy-tropical climate model following the work of Khouider et al. [113]. We end this chapter by discussing the effect of the background shear and the background moisture, specifically associated with the MJO, on convectively coupled synoptic waves and mesoscale convective system as well as a discussion on their potential CMT feedback on the MJO itself.

8.2 CMT in a Simulated Kelvin Wave: A Thought Experiment The results presented here have appeared in Khouider and Han [112]. The interested reader is referred to that article for details. The simulation is done within the framework of the Weather Research and Forecasting (WRF) model. WRF is a very popular computer code developed by the National Center for Atmospheric Research scientists and used by the atmospheric science community for both research and actual weather forecasting purposes. It is a comprehensive atmospheric model which solves the non-hydrostatic equations of large-scale motion with a user defined resolution and several subgrid model options for the physics or unresolved dynamics. An overview can be found in the model’s website (www.wrf-model.org) and more details can be found in the research articles provided there. The specific configuration used for the present simulation is detailed in [112]. In particular, Khouider and Han use an idealized setting consisting of a square horizontal domain, centred at the equator, which is wide enough for a synoptic scale CCW to propagate parallel to the equator. It is 4500 km large with a coarse 10 km resolution without any convective parameterization. This is indeed a too coarse resolution to resolve actual clouds; typical cloud resolving modelling simulations use grids sizes on the order of 1 to 3 km but at such resolutions only small domain simulations on the order of 2000 km are possible, because of limitation in computing resources. As we know already, coarse resolution climate models rely on a cumulus parameterization to represent the subgrid effect of convection; however, at 10 km resolution we start to represent some coarse features of convection and we are essentially in the “grey” zone in the sense that we are unsure whether a convective parameterization is still useful or not. However, as demonstrated in [112], this 10 km resolution

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turns out to be enough for capturing some cloud systems and the essential dynamics of a convectively coupled Kelvin wave and that the use of one particular convective parameterization revealed to be detrimental for the targeted experiment as it yielded results that are farther away from those of the same experiment using a finer (3.3 km) resolution 2000 km × 2000 km square domain, nested in the middle of the 45002 km2 main computational domain (reported here for the sake of comparison). More details on the nesting technique can be found in the WRF documentation and in [112]. The nesting allows in addition to gain more insight into the mesoscale circulation, which is better resolved within the fine resolution part of the domain. With the use of the convective parameterization, the apparent rainfall was reduced by a factor of roughly 4 and overall wave organization was destroyed while the simulations with and without nesting and without a convective parameterization provided qualitatively and quantitatively comparable results [112]. The 450 × 450 points horizontal domain is extended with 50 levels, in the vertical up to a maximum height of 20 km, uniformly distributed in the η -coordinate (normalized pressure). The time step is 10 seconds. An ocean surface with a zonally uniform surface temperature decreases linearly from 301 K at the equator to 293 K at the northern and southern boundaries. For the solar forcing, a perpetual fall with constant insolation over the duration of the simulation is assumed. The simulations are initialized with a dry Kelvin wave perturbation and integrated for a total period of 90 days between the fictitious dates of 20 OCT 2000 and 29 JAN 2001. The boundary conditions are periodic in the zonal (x) direction and non-reflecting in the north-south (y) direction. After a transient time of roughly 30 days the solution reaches a statistical equilibrium state where convectively coupled waves form and propagate. The time evolution of the solution during the last 30 days is pictured in Figure 8.2, which shows the Hovm¨oller diagrams of the zonal wind near the surface and near the top of troposphere as well as that of the surface precipitation. The latter is shown for the simulations both with and without nesting, for the sake of comparison. All fields are averaged in the meridional direction, between 2.5° South and 2.5° North (roughly 550 km). Notice the kinks in the precipitation contours on panel (D) which mark the transition between the fine and coarse resolutions, an artifact of the grid nesting. Nonetheless, the similarity of the two simulations gives confidence that the 10-km resolution captures the main features of the physical solution. The fields displayed here are averaged in the north-south direction between 5° South and 5° North.

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The contours in Figure 8.2 depict a wave travelling eastward at roughly 17 m s−1 suggesting a convectively coupled Kelvin wave consistent with the satellite observations reported in Chapter 3 and the multicloud model results in the previous (and upcoming) chapters. The zonal winds and precipitation patterns are both quantitatively and qualitatively realistic.

Fig. 8.2 Simulation of a convectively coupled wave in WRF. (a) and (b): Hovm¨oller diagrams of zonal winds in the lower (850 hPa) and upper (250 hPa) troposphere, respectively, for the 10 km simulation without nesting. (c) and (d): Hovm¨oller diagrams of precipitation rate for the simulation with and without nesting, respectively.

A snapshot of the instantaneous horizontal (a and b) and vertical (c) structure is given in Figure 8.3, where, respectively, the (u, v) and (u, w) arrows are overlaid on top of the zonal velocity contours. We note in particular a strong region of flow convergence in the middle of the panel in Figure 8.3(a) and a region of strong divergence in panel (b), which is slightly lagging behind, by roughly 7–10 degrees. This lagging is consistent with the tilted structure of convectively coupled waves as shown in observations and by the multicloud model solution. The front-to-rear tilted vertical structure is indeed captured by the WRF simulation as illustrated in panel (d). However, while a strong zonally parallel flow is dominant at the equator, the horizontal wind patterns in (a) and (b) exhibit some important deviations from a dry Kelvin wave flow structure. Some of the meridional divergence may be as-

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sociated with the imposed gradient in the sea-surface forcing, there are substantial transient disturbances in both the meridional and zonal velocities which occur at the mesoscale of 50–500 km. This is easily confirmed by the blown up Hovm¨oller diagram of 250hPa zonal wind displayed in Figure 8.3(d). On this panel we see a clear signature of westward moving waves lasting only a few hours with wavelengths of roughly 100 km or less. These are mesoscale disturbances associated with rain bands embedded within the propagating Kelvin wave, recognizable as regions of higher precipitation in Figure 8.2(c) and (d). Such rain bands are often associated with the MJO as reported in [191], for example, who used a cloud resolving global simulation at 7 km resolution and made a careful investigation of the mesoscale CMT impact of the MJO background. In particular, they found that CMT spins up westerlies near the surface and the upper troposphere and easterlies in the midtroposphere, thus retarding the deepening of the westerly wind burst associated with the MJO active phase and therefore believed to have the tendency to slow down the eastward propagation of the MJO [191]. In a nutshell, mesoscale CMT, as diagnosed in [191], has a third baroclinic mode structure and the same will be demonstrated here for the case of mesoscale systems embedded within the simulated Kelvin wave. A running mean composite, along the characteristic speed of the propagating streaks in Figure 8.2 (c.f. Chapter 6), of the zonal wind horizontal and vertical structure and surface precipitation anomalies is plotted in Figure 8.4. As we can see the structure of a convectively coupled Kelvin wave is readily confirmed with a clear backward tilt in the vertical as in the observations in Figure 3.8. As expected the positive precipitation anomaly is in phase with low-level convergence and upperlevel divergence lags behind by roughly 5 degrees consistent with the snapshot in Figure 8.3. Though it is a delicate exercise as mentioned above, we now attempt to diagnose the CMT effect on the simulated Kelvin wave due the embedded mesoscale disturbances. Let L = 50 km be the rough measure of the wavelength of the flow disturbances associated with the mesoscale rain bands. The mesoscale averaged zonal flow is given by u(x, ¯ y, z,t) =

1 4L2

 L L −L −L

u(x + x , y + y , z,t) dx dy

and the mesoscale fluctuation wind reads u = u − u. ¯ The mesoscale vertical CMT flux of zonal momentum is defined as FCMT = −(u w )z . The Hovm¨oller plot of the CMT (meridionally averaged between ± 2.5° North/South) associated with the Kelvin wave is displayed in Figure 8.5(c).

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Fig. 8.3 Snapshot of an instantaneous flow structure and 850 hPa (a) and at 250 hPa (b).

While it fluctuates between positive and negative values, positive CMT is clearly dominating at this level. Notice also the alignment of CMT fluctuations along the Kelvin wave zonal wind anomalies in Figure 8.3(d). The dominant positive CMT is confirmed by the running mean (composite) in Figure 8.5(b). In the upper troposphere CMT is always positive on average, it thus tends to accelerate the westerly in front of the wave and decelerate the easterlies in its wake. The full vertical CMT structure given in Figure 8.5(a) suggests a mixture of third and fourth baroclinic modes somewhat consistent with the findings of [191], in the sense that there is an alternating of positive and negative CMT forcing in the vertical between the surface, middle, and upper troposphere. Surmising from Figures 8.4(d) and 8.5(a) the CMT forcing seems to enforce the Kelvin wave in front and to weaken it in the wake in the upper troposphere and the exactly opposite effect near the surface. Thus CMT seems to strengthen the vertical shear over the whole wave life cycle. In the mid-troposphere near 700 hPa, CMT is the strongest and approaches 6 m s−1 day−1 . It enforces westerlies near the wave convection centre implying a deepening of the westerly winds and thus tending to accelerate the propagation of the wave in contrast with the results of [191] for the MJO. Thus, a full understanding of CMT effect remains a largely unsolved and important research topic. Nevertheless, in

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Fig. 8.4 Horizontal (a and b) and vertical structure (d) of Kelvin wave composite zonal winds together with filtered surface precipitation (c).

the following sections we will illustrate some of its subtleties in the contest of the multicloud model.

8.3 CMT Parameterization Prototypes 8.3.1 Some Background Various attempts have been made in order to include the CMT effects from the unresolved cloud systems in GCMs [230, 303, 304, 289]. It is found that CMT improves the overall climate structure and variability in the GCMs and results in more coherent MJO signals [286, 240, 241, 79]. However, these earlier studies did not systematically include CMT effect due to organized mesoscale systems which is found to play a significant role in the TOGA-COARE data [265, 266].

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Fig. 8.5 The vertical structure of the Kelvin composite of CMT (running meaning) (a), its horizontal distribution at 250 hPa (b) and the Hovm¨oller diagram (c), averaged between 2.5° South and 2.5° North. In (c) the blue patches are for positive CMT and the contours are negative CMT. The contour interval is −6 m s−1 day−1 .

The idea of designing a CMT parameterization to account for the unresolved circulation due to mesoscale convective systems and squall lines was pioneered by Moncrieff and his collaborators [194, 288] though they used ad hoc physical assumptions to approximate the mesoscale circulation in closed form using archetypical circulation patterns. Majda and Stechmann [177] developed a Markov jump process mesoscale CMT parameterization that allows both upscale and downscale momentum transfers, based on the weak temperature gradient theory [167] and using the multicloud model as a toy GCM. In the Majda and Stechmann CMT model the subgrid redistribution of heating assumes a wave form. This is further refined in Khouider et al. [113] where both the wave form assumption for the subgrid redistribution of heating and the stochastic jump process were relaxed towards a somewhat more realistic Gaussian profile and a simple mean-field-like distribution between upscale and downscale CMT, respectively. It is found in [112] that the addition of CMT in the multicloud parameterization impacts positively both the synoptic scale waves and their planetary scale envelopes. Before plunging into the details of some of these CMT studies, here we will illustrate by a simple example introduced in [122] how the tilted structure of the

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embedded system and by ricochet stratiform/congestus heating are important for mesoscale CMT. As done above, consider the zonal momentum equation as an example and assume that the flow is divided into a mean and fluctuation parts: u = u¯ + u . Assume that the fluctuation part has a vertical structure in the form u = − f (x) cos(z) − g(x) cos(2z), where f (x) and g(x) vanish quickly enough at infinity or are periodic functions. Let w = f  (x) sin(z) + 12 g (x) sin(2z) be the vertical velocity associated with u , which can be related to the heating anomaly profile as a consequence of the weak temperature gradient approximation. To ensure positive heating we require f  (x) ≥ 0 and g (x) ≤ 0, in the case of stratiform and deep heating combination or f  (x) ≥ 0 and g (x) ≥ 0, in the case of deep and congestus heating dominated atmosphere. It is easy to see that the associated turbulent flux is given by 3 − (u w )z = − f  g [cos(z) − cos(3z)] . 4

(8.1)

Here the overbar represents a zonal average over the mesoscale domain. Therefore, the CMT resulting from the combination of deep convective and stratiform (and/or congestus) clouds forces the first and third baroclinic components of the large-scale zonal wind u¯ but not the second baroclinic mode. If in addition, we assume that g(x) = −2α f (x + x0 ), 0 < α < 1, i.e., stratiform heating is proportional to and lagging behind deep convection (or equivalently, g(x) = 2α f (x + x0 ), 0 < α < 1 for the congestus dominated case), then the CMT flux is zero if x0 = 0 or α = 0. Thus the statement that a vertical tilt, in the convective heating, is necessary to achieve a non-zero mesoscale and/or synoptic scale CMT [170] is true. For small enough x0 we have f  (x) f (x + x0 ) ≈ f  (x) f (x) + x0 [ f  (x)]2 = x0 [ f  (x)]2 , i.e., the convective momentum flux satisfies 3 −(u w )z ≈ ± α x0 [ f  (x)]2 [cos(z) − cos(3z)]. 2 For stratiform dominated disturbances (+), the last expression has the same sign as x0 near the surface, i.e., positive for westward tilted waves (x0 > 0), and negative for eastward tilted waves (x0 < 0). The full vertical structure of the CMT flux is plotted in Figure 8.6(a) for both x0 > 0 and x0 < 0. According to Figure 8.6(a), westward tilted waves accelerate large-scale westerly winds in the lower troposphere and accelerate easterlies in the upper troposphere, while eastward tilted waves do the opposite. Accordingly, observations revealed that when there is organized mesoscale squall line convection, CMT is positively correlated, thus accelerates the background shear and when convection is scattered, CMT and the background shear are negatively correlated. Since congestus heating is generally much weaker than

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deep convective heating, we focus only on CMT generated by stratiform dominated waves.

Fig. 8.6 (A) Profile of CMT flux due to eastward and westward tilted waves. (B) Time evolution of the background mean mind under the influence of stochastic CMT (from [177])

The CMT diagnostics of Tung and Yanai [266] are reported in Figure 8.7 for the cases of both scattered convection and organized mesoscale systems. The typical convective signals associated with each category are illustrated on the top (a) panels. The (b) and (c) panels show the vertical distributions of CMT in the horizontal and meridional directions (−(u w )z and −(v w )z ), respectively, and the respective background winds. We note in particular that during scattered convection events of CMT and the background winds are generally negatively correlated, while during mesoscale organized episodes zonal winds and zonal CMT are in particular positively correlated. The results are not as conclusive for the meridional winds which are themselves relatively weaker, nevertheless. However, it is worthwhile mentioning that the implication of the acceleration of the low-level westerlies and of upperlevel easterlies by organized CMT in the zonal direction is evident as predicted by the simple phenomenological model presented in the previous two paragraphs and illustrated in Figure 8.6(a). On the other hand the negative correlation of the background wind and CMT profiles during the scattered convection episodes suggests significant damping of the background shear. In fact, Tung and Yanai stressed that on average CMT is downscale, i.e., it tends to weaken the background wind shear.

8.3.2 Stochastic Model of Majda and Stechmann As pointed out by Tung and Yanai [265, 266], mesoscale CMT can accelerate or decelerate the background wind depending on whether there is mesoscale organization or not. However, organized and non-organized mesoscale convective events are hard to detected or predict from the large-scale point of view; CMT acceleration

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Fig. 8.7 CMT diagnostic from the TOGA-COARE field experiment data (source [266]) for organized (left) and scattered convection (right). Top panels show typical snaps of reflectivity (clouds) distribution in a case of scattered convection and in a case of an organized mesoscale system. The bottom panels show averaged CMT fluxes for such corresponding events in both zonal and meridional directions on top of the corresponding mean wind profiles.

and deceleration events can be regarded as being random events. To demonstrate the implications of the random occurrence of CMT acceleration and deceleration events, Majda and Stechmann [177] propose a Markov jump process to quantify this chaotic behaviour. Assuming three main states of organized convection, “dry regime”, “upright convection regime”, and “squall line regime”, they constructed a Markov process for CMT accordingly leading to acceleration or deceleration. We note that despite the chosen terminology “dry regime” refers in general to episodes of weak to no convection, while “squall line regime” refers to the dominance of mesoscale-type convection within the grid cell. Let r be an order parameter that takes the values 1, 2, 3 according to whether we are in the “dry”, “upright”, or “squall line” regimes, respectively. Transitions from state i to state j, 1 ≤ i, j ≤ 3, occur according to a continuous time Markov chain with transition rates Ri j , depending on the large-scale variables. The model has been tested in the context of the multicloud model in one dimension, i.e., neglecting the x-dependence; this is known as a single column setting in the climate modelling community. We note that besides the addition of the CMT tendencies in the velocity equations, the rest of the model equations remain as presented in Chapter 6. However, as pointed out above, CMT excites a third baroclinic velocity component, which is assumed in [177] to be a slaved variable without any feedback into the convective parameterization or the dynamics of the first two modes, on which the multicloud parameterization is built. We have [177]

8.3 CMT Parameterization Prototypes

 ⎧ ⎨ FCMT =

du j dt

145

 j = FCMT , j = 1, 2, 3 CMT

−d1U, if r = 1 if r = 2 −d2U, ⎩ κ [cos(z) − cos(3z), if r = 3.

(8.2)

Here κ is a parameter depending on the strength of the stratiform heating and its lagging distance with respect to deep convection, during an organized mesoscale episode. Using the potentials for deep and congestus heating, Qd and Qc , respectively, the mid-troposphere dryness function Λ , and the low-level shear, Δ Ulow , as the predictors for the stochastic CMT model, Majda and Stechmann set 1 R12 = H (Qd ) exp[βλ (1 − Λ ) + βQ Qd ], τ 1 R21 = exp[βλ Λ + βQ (Qd,re f − Qd )], τ 1 R23 = H (|Δ Ulow |) exp[βU |Δ Ulow | + βQ Qc ], τ 1 R21 = exp[βU (|Δ U|re f − |Δ U|low + βQ (Qc,re f − Qc )]. τ

R13 = 0,

R31 = R21 , (8.3)

Here the βx ’s, are tunable parameters and H is the Heaviside function. Note that accordingly, a transition from the dry regime to the upright convection regime is favoured when the mid-troposphere is moist and there is potential for deep convection, while a transition to the squall line regime (either from dry or upright convection regimes) is favoured when the low-level shear is significant, in either direction, according to observations and theory of squall lines [192, 194, 290]. To obtain consistent zonal and vertical wind perturbation of a tilted mesoscale wave, Majda and Stechmann assume a mesoscale heating perturbation in the form of a plane-circular wave: Sθ = cos(kx − ω t) sin(z) + α cos(k(x + x0 ) − ω t) sin(2z), where x0 is the lagging distance and α the relative strength of stratiform heating, with respect to deep convection, while k and ω are the associated wave number and frequency. Assuming a weak temperature gradient approximation under which the vertical advection perturbation of the background potential temperature stratification perfectly balance the heating perturbation, w = Sθ , and obtain the perturbation zonal velocity through divergence constraint: ux +wz = 0, which then yield the wave form expressions for the functions f and g in (8.1). To make the CMT flux dependent of the background shear and heating potential, the parameter κ (which intrinsically depends on x0 and α ) is set to ) −( QQd )2 Δ Umid if Δ Umid Δ Ulow < 0 d,re f κ= 0 if Δ Umid Δ Ulow > 0,

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8 Convective Momentum Transport and Upscale Interactions in the MJO

where Δ Ulow and Δ Umid are the maximum low-level and mid-level shear values [177]. The result of the first test of this stochastic CMT model is illustrated in Figure 8.6(b) where the large-scale wind, for the case of a single column simulation, is plotted at various days of its evolution. As we can see there is a net acceleration of low-level westerlies and upper-level easterlies, suggesting an episode of eastward moving mesoscale systems according to Figure 8.6(a). The results of a fully x-depending simulation with the stochastic CMT are reported in Figures 8.8 and 8.9. The multicloud model is run on a short periodic domain of 6000 km, a domain large enough to simulate the propagation of a synoptic scale convectively coupled gravity wave, similarly to the WRF simulation presented in the previous section. The Hovm¨oller diagrams in Figure 8.8 exhibit the evolution of a convectively coupled gravity wave, moving to the left, which grows gradually over a transient period of about 10 days before reaching an equilibrium state and perpetually circles the periodic domain. The evolution of the order parameter of convective regimes, pictured in panel b, shows that the CMT active regime and the upright convective regimes are restricted to the active phase region of the wave, consistent with the WRF simulation where the strong positive and negative CMT events are co-located with the moving streaks of precipitation maximum (see Figures 8.2 and 8.5). Rapid transitions between upright convection and squall line regimes are evident from the evolution of the third baroclinic velocity in Figure 8.8d. However, the cases with and without CMT were compared against each other, in Figure 8.9, we can see that besides the induced third baroclinic velocity, which also shows the most significant standard deviation, there is no major qualitative or quantitative changes in the solution behaviour. As we will see in the next section, this is most likely due to the lack of feedback onto the multicloud model dynamics and convective parameterization in particular from the third baroclinic mode.

8.3.3 CMT Parameterization with a Third Baroclinic Feedback and its Effect on an MJO like disturbance As noted above while the inclusion of a stochastic CMT model, due to unresolved tilted mesoscale convective systems, into the multicloud model, deemed important especially in terms of generating a significant third baroclinic flow, resulting in the amplification of the low-level and mid-level shears, its overall effect of the qualitative and quantitative features of the simulated convectively coupled gravity wave is very weak. Clearly, the modification of the background shear by CMT would feed back positively to the CMT parameterization consistently with observation but the third baroclinic flow would also induce temperature and moisture perturbations that would influence the convective parameterization which drives the simulated wave. Also, while the use of a stochastic model is largely motivated by the observed chaotic and intermittent nature of organized convection, it is always important to know whether the mean effect of the stochastic CMT is sufficient or the cross cor-

8.3 CMT Parameterization Prototypes

147

Fig. 8.8 Simulation of a convectively coupled gravity wave with the multicloud model coupled with the stochastic model for CMT of Stechmann and Majda. Hovm¨oller diagrams of deep convection (a), the stochastic order parameters (taking values from 1, 2 (grey), and 3 (black)), and first (c) and third (d) baroclinic velocity components. Notice the negative of u3 is plotted to highlight the westerly wind bursts occurring in the mid-way in the lower troposphere. Adopted from Majda and Stechmann [177].

relation terms, between the resolved and unresolved dynamics, is so significant that the mean alone will not be able to provide the correct CMT forcing. Such situation is expected if, for example, the scale separation doesn’t exist or it is not significant and it will be more so for GCMs and regional climate models with grid resolution in the so-called grey zone, between 10 and 50 km, where part of these organized systems is resolved. In this subsection, we will consider a deterministic CMT model which can be in some sense regarded as the mean-field limit of the stochastic CMT model of Majda and Stechmann [177]. This model has been introduced and tested in [113]. More importantly, here the feedback from the third baroclinic mode into the multicloud model parameterization is taken into account. So, instead of the more elaborate Markov jump, we assume a simple exponential distribution for the CMT forcing conditional on the shear strength and the convective mass-flux ∗

∗

FCMT = −e−VT /V d(v − V0 )) + (1 − e−VT /V )∂z (−v w ),

(8.4)

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8 Convective Momentum Transport and Upscale Interactions in the MJO

Fig. 8.9 Zonal structure of simulated moist gravity waves zonal wind components with and without CMT parameterization and the associated differences. The top panels (a–c) are for the time mean structures and the bottom panels (d–f) are for the associated standard deviations.

where d = (3 days)−1 is the momentum damping rate, v is the horizontal velocity, V0 is the background-mean climatology, and w is the vertical velocity. The overbar represents the running horizontal average over a large-scale area and the primes denote the subgrid mesoscale flow variables. More importantly, VT /V∗ measures the relative strength of the vertical wind shear, VT , with respect to a turbulent velocity scale V∗ . The quantity VT is a measure of the wind strength in the upper troposphere VT = |v|z=3π /4 = |v1 − v3 |, where the indices refer to the first and third baroclinic modes. As such it serves both as a measure for the vertical shear strength and to set the lag between the stratiform anvils and the organized convection core, which determines the imposed tilt. Here V ∗ sets a threshold beyond which mesoscale organization becomes significant. When VT > V∗ upscale mesoscale CMT becomes dominant. Similarly to the precedent model, the velocity perturbations v , w are obtained in closed form from WTG approximation, w = Sθ ≡ Hd (r) sin(z) + Hs (˜r)(sin(z) − 0.5 sin(2z)).

8.3 CMT Parameterization Prototypes

149

The mesoscale heating Sθ is a combination of a deep convective perturbation and 2 + y2 is the distance to the centre of deep a stratiform contribution. Here r = x convection and r˜ = (x − Lm )2 + y2 is the distance to the centre of the stratiform anvil, which is assumed to be lagging behind convection in the direction x, parallel to the wind shear direction, v1 − v3 , with a lagging distance L = VT τ , τ = 3 hours. We note that the profile sin(z) − 0.5 sin(2z) yields a top heavy heating which in essence has both deep and stratiform components. Instead of a travelling wave form, the deep and stratiform heating horizontal distributions take the shape of two Gaussian functions Hd (r) =

α Hd −r2 /(2σs2 ) Hd −r2 /(2σ 2 )  d , H (r) = e e , s 2 σs2 σd

where for convenience Hd is the multicloud deep convective heating used by the large-scale model and the parameters σd , σs , α are prescribed. Khouider et al. [113] set σd = 75 km and σs = 1.5σd eVT /V∗ . In addition to the WTG approximation, we assume that the horizontal flow, associated with the mesoscale systems, is potential. This is justified by arguing that the rotational part of the flow, when there is one, contributes to the turbulent eddies which damp the mean flow and they are accounted for by the dissipation part of the CMT forcing. The CMT flux in the direction perpendicular to the shear is identically zero. In the direction of the tilting of the system, it is given by [113]   D2 α F G F G (8.5) −(u w )z ≈ 02 ( 2 − 2 ) cos z − 3( 2 + 2 ) cos 3z , L∗ 2σ 4σv 2σ 4σv where F=

√

2ππσv e

−L2 2σv2

 e

L2 4σv2

2

L ·( 1 ) L2 1 L2 1 I1/2 ( 2 ) − e 4σv2 1+σv2 /σ 2 I1/2 ( 2 ) 4σv 4σv 1 + σv2 /σ 2 1 + σv2 /σ 2



(8.6) and   L2 ·( 1 √ −L2 L2 L2 1 L2 1 2 1+σ 2 /σ 2 ) 4 σ 2 2 v I G = − 2ππ e 2σ σ e 4σ I1/2 ( 2 ) − e ) . 1/2 ( 4σ 4σ 2 1 + σ 2 /σv2 1 + σ 2 /σv2

(8.7) Here I1/2 (.) is the Bessel function. The quantity in (8.5) is then rotated back to the frame of the computational grid to yield the CMT flux at each grid point, which is almost everywhere dissipative except at the grid points where the shear is significantly large, consistent with the WRF simulations presented in the first section, the observations, and the Markov jump model of Majda and Stechmann. As in the previous subsection, the new CMT parameterization is coupled to the two-dimensional (x − z) multicloud model large-scale equations. However, in addition to the 8 equations in (6.1)–(6.4), the multicloud model is expanded to include the third baroclinic mode, carrying both velocity and potential temperature components. In addition, the moisture equation has been modified to include the

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8 Convective Momentum Transport and Upscale Interactions in the MJO

low-level moisture convergence due to the third baroclinic mode. More importantly, the closure for the deep and congestus heating potentials and the mid-troposphere equivalent potential temperature as it appears in the downdrafts and in the moisture switch function Λ have been modified to include a contribution from θ3 . As such the third baroclinic feedback is indeed multifold. We have the following additions/modifications to the equations in (6.1)–(6.6):

∂ θ3 1 ∂ u3 1 − ( ) = − θ3 , ∂t 9 ∂x τR ∂q ∂ ˜ ∂ u2 ) + γ˜Q( ˜ ∂ u3 ), ˜ ∂ u1 ) + λ˜ Q( + (α˜ 1 u1 q + α˜ 2 u2 q + α˜ 3 u3 q) + Q( ∂t ∂x ∂x ∂x ∂x 1 + Qd = Q¯ + [a1 θe b + a2 q − a0 (θ1 + γ2 θ2 + γ3 θ3 )] , (8.8) τconv 1 Qc = Q¯ + [θe b + a2 q − a0 (θ1 + γ2 θ2 + γ3 θ3 )]+ τconv √ 2 2 θem = q + d (θ1 + α2 θ2 + α3 θ3 ). π We note that the new θ3 variable is forced only by the third baroclinic divergence and Newtonian damping, while the moisture equation has been modified to include the third baroclinic convergence terms. The u3 equation and all the remaining prognostic variable equations (namely u1 , u2 , θ1 , θ2 , θeb , Hs , Hc ) are the same as in the previous subsections, therefore not repeated here. The new parameters are as follows: α˜ 3 = −1, γ˜ = 0.9, γ˜3 = −1, γ˜3 = −1 [113]. To test, the new CMT model with third baroclinic feedback, the simulation in Figure 6.7 is repeated when the CMT model is activated. Recall that unlike the Majda and Stechmann study presented above, we consider the MCM nonlinear simulation in Figure 6.7. We recall that this simulation is run on the full 40,000 km equatorial ring and it is characterized by similar moist gravity waves circling the domain but somehow modulated by an MJO-like wave envelope moving at a slower phase speed in the opposite direction. The Hovm¨oller diagrams of u1 , u2 , q, Hc , Hd are plotted in Figure 8.10 where the original case without the CMT model is re-plotted to ease the comparison. The first striking difference between the two simulations is that while in the simulation without CMT only moisture and congestus heating appear to carry the slow moving wave signal, with the CMT model, this MJO-like envelope wave is very prominent on the right panels, with clear episodes of strengthening followed by episodes of weakening of the fast moving moist gravity waves, modulated by the wave envelope. Notice that both deep and congestus heating are inhibited during the suppressed phase of the wave envelope. Moreover, the strength of the overall solution has significantly increased especially in terms of the second baroclinic velocity, the moisture, and heating (both congestus and deep). Recall that CMT has no direct effect on the sec-

8.4 Multiscale Waves in MJO Envelope and CMT Feedback

151

ond baroclinic mode consistent with equation (8.1), however, because of the nontrivial θ3 and u3 feedback included in the heating closures and moist thermodynamic equations, congestus heating appears to be hugely increased which in return induces second baroclinic convergence of moisture, and so on. The structure of both the planetary scale envelope and the synoptic scale moist gravity filtered waves is reported in Figure 8.11. As we can see, while the two structures remain qualitatively the same as in Figure 6.7, i.e., the self-similarity being preserved, there are significant quantitative differences, especially in the planetary scale MJO-like wave. There is also an increase in wavelength for both waves. The significant strengthening of the MJO-like wave from CMT is consistent with the MJO theory of Majda and Biello [170], although they assumed that CMT is from synoptic waves, as in the simulations of Grabowski and Moncrieff [76].

8.4 Multiscale Waves in MJO Envelope and CMT Feedback Now we turn around and ask the question on how the background dynamics and moist thermodynamics climatology itself influence the organization of waves at synoptic and mesoscales. We have already seen in Chapter 7 that the background wind shear can have a great influence on the MCM instability diagram of synoptic scale waves; however, the picture remains incomplete without looking at the full picture. In particular, what happens if the background state is allowed to vary simultaneously with the embedded waves leading to two-way interactions. As seen in the previous section, CMT from tilted convective systems forces essentially the first and third baroclinic modes. Therefore it is important to revisit the study in Section 7.4 on the effect of the vertical shear on convectively coupled waves, in the context of a more realistic MJO background both in terms of its wind field and moisture profiles that are assumed to vary throughout the various phases of the MJO.

8.4.1 A Simple Multiscale Model with Features of CMT We begin by presenting a simple one-dimensional multiscale model with two wave interactions between a background wind shear and convectively coupled waves introduced by Majda and Stechmann [180] for the sake of illustration. Assume that the zonal wind shear varies on two separate time scales, the synoptic time scale, t, on the order of hours and a slower time scale, τ = ε 2t, with ε = 0.1, on the order of 30 days [180]. The time scale τ can be thought of as being intra-seasonal and t itself is synoptic scale, to mimic waves evolving within the MJO envelope [231, 127]. For simple flows above the equator, we set

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8 Convective Momentum Transport and Upscale Interactions in the MJO without CMT U1(x,t)(m/s)

U1(x,t)(m/s) S=0.6Hd

3

4

590 2

570

1

560 550

0

540 −1

530

17m/s

520

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1

550

0

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−1

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−2

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−2

17m/s

510 −3

500 0

5

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15 20 25 X(1000km)

30

without CMT U2(x,t)(m/s)

−4 0

5

1

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530 520

8

580

6

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35

10

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10

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4

580

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35

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TIME(days)

2

570

510

560

4

550 540

2

530

0

520

−1

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−2

510 −2

500 0

5

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15 20 25 X(1000km)

30

0

1 0.5

540

0

530

−0.5

520

−1

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−1.5

500 10

15 20 25 X(1000km)

30

35

TIME(days)

560

30

35

6

1.5

550

15 20 25 X(1000km)

590

580 570

10

Q(x,t)(k) S=0.6Hd

2

5

5

2.5

590

0

−4

500

35

without CMT Q(x,t)(k)

TIME(days)

3

580 TIME(days)

580

TIME(days)

590

5

580

4

570

3

560

2

550

1

540

0

530

−1

520

−2

510

−3 −4

500 0

5

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15 20 25 X(1000km)

30

35

Fig. 8.10 Hovm¨oller diagram of u1 , u2 , q, Hc , Hd for the multicloud simulation with and without the CMT model, with third baroclinic feedback. Note the difference in scale between the left and right panels. From [113].

8.4 Multiscale Waves in MJO Envelope and CMT Feedback without CMT Hc(x,t)(k/day)

Hc(x,t)(k/day) S=0.6Hd

0.1

1

590

590

0.9

580

580

0.8

570

570

0.05

560 550 540 0

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TIME(days)

TIME(days)

153

6 m/s

520 510 500 0

−0.05 5

10

15 20 25 X(1000km)

30

0.6

550

0.5

540

0.4

530

0.3

520

0.2

510

0.1

500 0

35

0.7

5m/s

560

0 5

10

15 20 25 X(1000km)

30

35

Hd(x,t)(k/day) S=0.6Hd

without CMT Hd(x,t)(k/day)

25

8 590

590

7

580

580 6

560

5

550 540

4

530

3

560 550

15

540 530

10

520

520 2

510 500 0

20

570

TIME(days)

TIME(days)

570

5

10

15 20 25 X(1000km)

30

35

1

510 500 0

5 5

10

15 20 25 X(1000km)

30

35

Fig. 8.10 (continued)

utot (x,t) = U(z, ε 2t) + u(x, z,t), where U is the background wind, varying on the intra-seasonal time scale, and u is the synoptic-wave fluctuation. A systematic asymptotic expansion in terms of powers of ε yields the following coupled equations for U(z, τ ) and u(x, z,t) [180]: dU = −(u w)z dτ du dU du +U +w +··· . dt dx dz

(8.9)

Here the dots refer to other terms representing synoptic scale dynamics. We note that this phenomenological model suggests that the synoptic waves force the background wind through the turbulent flux (u w)z , while the background wind affects the waves by horizontal advection and convergence of vertical shear. Majda and Stechmann [180] have also included the background temperature variations as they are forced by the advection nonlinearity (wθ )z .

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8 Convective Momentum Transport and Upscale Interactions in the MJO S=0.6Hd in CMT, θ, u&w planetary−scale, C.I=0.15K

no CMT, θ, u&w planetary−scale,C.I=0.015K

16

Height (KM)

Height (KM)

16

8

0

0 0

10

30

20

40

20

30

40

X (1000KM)

no CMT, θ, u&w synoptic−scale, C.I=0.1K

S=0.6Hd in CMT, θ, u&w synoptic−scale,C.I=0.15K

Height (KM)

16

8

0 10

10

0

X (1000KM)

16

Height (KM)

8

11

12

13

14

15

16

X (1000KM)

17

18

19

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8

0 10

11

12

13

14

15

16

17

18

19

20

X (1000KM)

Fig. 8.11 Zonal and vertical structure of the filtered waves for both the cases with and without CMT. Top panels the planetary scale MJO-like envelope wave and bottom panel for the synoptic scale moist gravity waves. Contours of potential temperature (dash = negative) and (u, w) velocity arrows. From [113].

Using the multicloud model for the synoptic scale dynamics, Majda and Stechmann performed a systematic projection in the vertical variable z, keeping only the first two baroclinic modes for the synoptic dynamics and first three baroclinic modes for the background wind U. Recall that two-baroclinic tilted wave CMT affects only the first and third baroclinic modes of the background wind. However, advection nonlinearities of θ can excite the fourth baroclinic mode of the background potential temperature, denoted by Θ4 . In [180], the multicloud closure equations have been modified accordingly to take into account the changes in the thermodynamic background by, for instance, including terms such as γ3Θ3 and γ4Θ4 in the expressions of Qc and Qd . The details can be found in the original article [180]. The coupled multicloud and mean background equations, projected in the vertical direction, are then integrated numerically on a 6000 km domain, for three different

8.4 Multiscale Waves in MJO Envelope and CMT Feedback

155

scenarios, one with a mean wind oscillating about a zero mean background, one an oscillation about westerly wind burst (WWB)-like regime of the MJO, i.e., an MJO active phase, and one representative of the WWB onset phase. To illustrate we report in Figure 8.12 the results of the WWB simulation. As we can see from Figure 8.12, the Hovm¨oller plots suggest complex two-way interactions between the waves and the mean wind. We have essentially phases of highly regular eastward moving waves and phases of westward moving waves connected by transition periods characterized by smaller scales disturbances moving in the direction opposite to the preceding wave. Concurrently, the background wind goes though phases where the WWB is amplified and phases of clear deceleration. We see clearly that the acceleration of the WWB and upper-level easterlies coincides with the synoptic wave moving to the right (east) and the deceleration phases coincide with the synoptic waves going to the left (westward). This is consistent with the discussion provided in the beginning of Section 8.3 and Figure 8.6. An important point to note is that the swinging back and forth between eastward and westward moving waves is controlled by the change in the background shear. It appears that, as it was indeed confirmed by linear stability analysis in [180], the pronounced WWB profile seen, for example, at time t = 1070–1080 days in Figure 8.12 results in the instability of westward moving waves, while the weaker shear at times 1040 and 1100 days seems to favour eastward waves. Since eastward moving (westward tilted) waves accelerate westerlies at low level and easterlies in the upper troposphere, and vice versa for the westward moving (eastward tilted) wave, through the CMT effect, the wave creates its own demises. In bulk part, these results are consistent with the results of Han and Khouider [80] for convectively coupled equatorial waves (on a beta-plane). According to [80], generally speaking, the upper-level easterly background shear destabilizes westward moving waves while the upper-level westerly background shear favours eastward waves. However, the work of Majda and Stechmann suggests that it is the mere weakening of the background upper-level easterly shear which favours eastward moving waves. This is somewhat reminiscent to the fact that in [80] convectively coupled Kelvin waves remain unstable in circumstances of westerly low-level shear, when the upper-level easterly shear is not overwhelmingly strong. It is also possible that the third baroclinic mode is playing some non-trivial role.

8.4.2 Equatorial Waves in a Realistic MJO Background Inspired by the crucial role played by the third baroclinic mode vertical shear and thermodynamic background, we revisit the problem of convectively coupled waves in a sheared environment discussed in Chapter 7 (see also [231]). More precisely, we will summarize the work of Khouider et al. [114] who looked at this issue in

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8 Convective Momentum Transport and Upscale Interactions in the MJO

Fig. 8.12 Two-way interactions between a background wind and an intra-seasonal oscillation (a and c) and synoptic scale convective coupled waves (b and d). Panel (a) shows the WWB acceleration phase due to eastward moving waves and panel (c) illustrates the case of decelerating WWB due to the prevalence of westward moving waves. From [180].

the specific context of an MJO background and then infer the CMT feedback on the dynamics of MJO using the results discussed in this chapter as guidelines. We consider the multicloud model on an equatorial beta-plane, linearized about a zonal wind background vertical shear with three baroclinic components, at radiative convective equilibrium (RCE) [114]:

8.4 Multiscale Waves in MJO Envelope and CMT Feedback

V(z, τ ) =

3

∑

 U j (τ ), 0 cos( jz),

157

(8.10)

j=1

where 0 ≤ z ≤ π is the height coordinate normalized so that z = 0 is at the surface and z = π is at the top of the troposphere and τ is a slow time (see below). For the purpose of the linear analysis below, it can be thought of as the MJO lag time. √   ∂ u1 ∂ θ1 2 3 ∂ u2 ∂ u1 1 ∂ v2 ∂ v1 5 ∂ u2 3 ∂ v2 U1 − − yv1 + + 3U 2 + U1 + 2U 2 + U3 + U3 ∂t ∂x 2 2 ∂x ∂x 2 ∂y ∂y 2 ∂x 2 ∂y 1 Cd u0 u1 − u1 =− hb τD √   ∂ u2 ∂ θ2 2 ∂ v1 ∂ u1 ∂ v1 1 Cd u0 u2 − u2 − − yv2 + −U 1 + 4U 3 + 3U 3 =− ∂t ∂x 2 ∂y ∂x ∂y hb τD √   Cd u0 ∂ v1 ∂ θ1 2 ∂ v2 ∂ v1 ∂ v2 1 − + yu1 + +U 2 +U 3 =− U1 v1 − v1 ∂t ∂y 2 ∂x ∂x ∂x hb τD √   1 ∂ v2 ∂ θ2 2 ∂ v1 ∂ v1 Cd u0 U1 v2 − v2 − + yu2 + +U 3 =− ∂t ∂y 2 ∂x ∂x hb τD   √   ∂ θ1 ∂ u1 ∂ v1 2 ∂ θ2 ∂ θ1 ∂ θ2 − + + 2U 1 −U 2 −2U 3 ∂t ∂x ∂y 2 ∂x ∂x ∂x √ 2 1 (−yv2U¯ 1 + 2yv1U¯ 2 + 3yv2U¯ 3 ) = Hd + ξs Hs + ξc Hc − Q0R,1 − θ1 + 8 τR   √   √ ∂ θ2 1 ∂ u2 ∂ v2 2 ∂ θ1 ∂ θ1 2 − + + −U 3 + (yv1U¯ 1 + 3yv1U¯ 3 ) U1 ∂t 4 ∂x ∂y 4 ∂x ∂x 4 1 = Hc − Hs − Q0R,2 − θ2 τR     ∂q ∂q ∂q ∂ q ˜ ∂ u1 ∂ v1 ∂ u2 ∂ v2 + α1 U 1 + α2 U 2 + α3 U 3 +Q + + λ˜ Q˜ + ∂t ∂x ∂x ∂x ∂x ∂y ∂x ∂y √ 2 2 1 =− (Hd + ξs Hs + ξc Hc ) + D π HT   ∂ θeb √ ∂ θeb ∂ θeb ∂ θeb D 1 ∗ − θeb ) − + 2 U1 +U 2 +U 3 = (θeb ∂t ∂x ∂x ∂x τe hb √  √    32 2 1 ∂ Hs 32 2 ∂ Hs ∂ Hs − + = [αs Hd − Hs ] U1 U3 ∂t 15π ∂x 21π ∂x τs √  √      ∂ Hc 32 2 ∂ Hc ∂ Hc 1 32 2 1 U1 U3 αc Λ D − Hc , (8.11) + − = ∂t 15π ∂x 21π ∂x τc HT

where the prognostic and diagnostic variables have the usual meaning [c.f. Chapter 7], the main difference is the presence of the third baroclinic background velocity component. Also, the terms yv jUk in the θ1 and θ2 equations represent the meridional advection of the background potential temperature, Θ , resulting from thermal wind balance yU = ∂Θ ∂ y . Numerical tests reveal that its effect is very limited but it is kept in the equations for the sake of completeness [114]. The linear stability results in Figure 8.13 display the effect of a strong versus a weak shear background in three different moisture environments (shown on the top panels). The first moisture profile (moisture 1) has a dry lower troposphere and moist upper troposphere; it roughly represents the conditions in the wake of deep

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8 Convective Momentum Transport and Upscale Interactions in the MJO

convection within the MJO that promotes stratiform clouds on the planetary scale. The second profile (moisture 2) mimics normal tropical conditions, which is the standard parameter regime used in the previous chapters, portraying a moist lower troposphere and relatively dry upper troposphere. The third moisture profile (moisture 3) exhibits an extreme case of an even moister low-level troposphere. We note that changes in the moisture profile amount to changes in the parameters Q˜ and λ˜ in the multicloud equations (8.11) as illustrated in Table 8.1. The panels in Figure 8.13 confirm once again that the strong upper-level easterly shear favours westward propagating synoptic waves, consistent with the results in Chapter 7 and in the previous section. However, a few important points have to be noted. First, when the shear is weak, the drier moisture profile (moisture 1) makes the equations stable, while the moister profile results in a new type of instability, namely higher frequency mesoscale-type waves that propagate in the direction of the low-level shear, according to the theory of squall lines [192]. This squall line type instability also amplifies with the increasing shear strength, as seen on the bottom panel. Note also the instability of the westward waves, in the strong shear regime, actually increases with decreasing low-level moisture background profile. This is consistent with the association of the synoptic wave instability to the upper-level shear, while the low-level shear is thought to drive squall lines [282, 214, 196, 124]. This feature of inducing mesoscale system type instability was pursued further in [124], where the multicloud model is extended to four baroclinic modes for the fluid dynamics variables and two moisture layers in the free troposphere above the boundary layer. A new parameterization of extended stratiform anvil, due to new condensation in the upper troposphere, is also introduced there. Under the appropriate background shear and background moisture profile, mimicking the eastern Pacific ITCZ, the authors discovered a multiscale instability of meso-beta and mesoalpha scale convective systems in the MCM, in addition to the prominent synoptic scale waves. Table 8.1 Moisture profiles used and implied model parameter changes Profile Figure Q˜ λ˜ Moisture 1 Moisture 2 Moisture 3 Lag−6 Lag−3 Lag−2 Lag 0 Lag+2 Lag+4 Lag+9

8.13 8.13 8.13 8.14 8.14 8.14 8.14 8.14 8.14 8.14

0.9031 0.9 1.0382 0.7677 0.9 1.0382 1.0837 0.7787 0.6811 0.6378

0.6047 0.8 0.7846 0.849 0.8 0.7846 0.7679 0.7846 0.83 0.849

For a proper investigation of the evolution of waves during and within the various phases of the MJO envelope [231], Khouider et al. [114] considered a succession of shear and moisture profiles that are directly inferred from an MJO composite. The top-left panel of Figure 8.14 shows the lag-height contours of the zonal wind

8.4 Multiscale Waves in MJO Envelope and CMT Feedback

159

MJO composite. As shown in this diagram, roughly 30 days prior to the MJO peak (day 0), low-level easterlies topped by upper-level westerlies start to peak up reaching their maximum at −5 lag time (roughly −15 days) and decaying towards the climatological background at day 0. After that, westerlies at low level, topped by upper-level easterlies, start to develop and quickly deepen to reach their maximum at +15 day lag. The top-right panel shows vertical slices of this background wind during the second half of the MJO period, between lags 1 and 10, as it projects onto the first three baroclinic modes, according to (8.10). The development, deepening, and decay of the low-level westerly wind are clear. The bottom panels of Figure 8.14 show a series of background moisture profile mimicking the evolving MJO moisture field characterized by a gradual low-level moisture leading the wave (lags −6, −3, −2) followed by an upper-level moistening, a gradual low-level drying, and then an overall dry phase marking the MJO suppressed phase (lags 4, 9, and −6). As illustrated on the bottom left panel, the transition between the active to the dry phase goes through a troposphere, which is moist below and relatively dry above to one that is moist above and dry below, due to the prevalence of stratiform anvils in the wake of deep convection.

Fig. 8.13 Typical moisture and shear profiles used (top panels) and the corresponding instability diagrams middle (weak shear) and bottom (strong shear). Left (moisture 1), middle (moisture 2), and right (moisture 3). From [114].

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8 Convective Momentum Transport and Upscale Interactions in the MJO

The zonal wind and moisture background profiles at various MJO composite lag times are then used, in consistent combinations, to force the linear multicloud equations in (8.11) and the resulting stability diagrams are reported in Figure 8.15. As expected the early stage of the suppressed phase characterized by upper-level westerlies and dry troposphere results mainly in eastward moving waves (panels (A), (B), and (C)). As we approach the active convective peak, right before the transition to low-level westerlies, when the background shear weakens and simultaneously the mid-troposphere becomes sufficiently moist, the instability of eastward synoptic waves weakens and the instability of mesoscale waves moving westward in the direction of the low-level winds develops (panel D) and then suddenly weakens and transition to eastward squall lines appears as we cross the day 0 lag, i.e., as we transition from low-level easterly shear to a low-level westerly shear (panels E and F). We note that the instability of eastward moving (Kelvin waves) persists during this transition. As the low-level westerly shear amplifies, the instability of eastward mesoscale systems amplifies (panel G). They then suddenly disappear as the troposphere dries out and the upper-level easterly shear strengthens and gives

Fig. 8.14 Lag-height MJO composite zonal wind contours (top right) and the corresponding vertical shear slices projected onto the three first baroclinic modes (top right). Evolving moisture background profile associated with the passage of MJO wave. The lag unit is roughly 3 days. From [114].

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161

rise to westward moving synoptic waves (panel H), which in turn weaken as the shear weakens (panel I). Keeping in mind that westward tilted (eastward moving waves) accelerates lowlevel westerlies and upper-level easterlies, while eastward tilted waves accelerates low-level easterlies and upper-level westerlies (Figure 8.6), the CMT implication of these multiscale waves on the MJO background can be qualitatively inferred. This is summarized graphically in Figure 8.16. As we can see while the synoptic scale waves overall re-affirm the main conclusion reached from the simple experiment in the previous subsection (Figure 8.12) that these waves create their own demise by always acting to decelerate the background that is the most favourable for them, there are a few subtleties. First we see that mesoscale systems (generically called here squall lines, although they are not necessarily as such) developing around day 0 lag propagate in the direction of the low-level background shear in which they develop, as such they would tend to have rather a positive feedback. Further, Kevin waves seem to persist during the WWB phase and thus would switch from having

(A)

(B)

(C)

(D)

(E)

(F)

(G)

(H)

(I)

Fig. 8.15 Multiscale waves in the MJO moisture and shear background represented in Figure 8.14, as simulated by the multicloud model. From top to bottom: lag −6 (A), lag −3 (B), lag −2 wind and lag −3 moisture (C), lag −2 (D), lag −1 wind and lag −2 moisture (E), lag +1 wind and lag 0 moisture (F), lag +2 wind and lag 0 moisture (G), lag +4 (H), lag +9 (I). From [114].

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8 Convective Momentum Transport and Upscale Interactions in the MJO

a negative feedback to accelerating the WWB. Overall, the deceleration of the lowlevel easterlies and upper-level westerlies by eastward moving Kelvin waves may actually help the propagation of the MJO and may play a crucial role during the initiation phase by favouring the emergence of the WWB at low-level, which is then amplified by mesoscale systems. Finally westward EIG waves would decelerate the WWB-type wind background and help in the emergence of the next MJO event by accelerating low-level easterlies again. This story is not finished yet.

Fig. 8.16 Schematic of two-interaction between the MJO and embedded multiscale waves. From [114].

Chapter 9

Implementation of the Multicloud Model in an Aquaplanet Global Climate Model

9.1 Introduction: The Cumulus Parameterization Problem Global climate and numerical weather prediction models (GCMs and NWPMs) simulate the atmospheric large-scale dynamical processes by solving the hydrostatic or non-hydrostatic primitive equations on a fixed grid with a horizontal mesh size of 25 km to 200 km. Atmospheric processes occurring at smaller scales that cannot be represented on those coarse resolutions are either neglected or represented via subgrid models known as parameterizations [246, 132]. As already demonstrated in the previous chapters convective clouds have a major impact on the atmospheric dynamics on synoptic and planetary scales that cannot be ignored. Convectively coupled waves and convective motions in general make the bulk of the atmospheric circulation in the tropical latitudes and account for the majority of precipitation which falls in this part of the globe. Among those, the Madden-Julian oscillation (MJO) [164, 163], in particular, constitutes the major source of atmospheric variability on the intra-seasonal and planetary scales and interacts with important global weather an climate patterns [300, 302]. While producing large amounts of precipitation, deep convective clouds that are up to 10 km to 15 km wide transport considerable amounts of energy upward, from the sun heated ocean (and land) surface, in the form of latent heat which is released during the condensation of liquid and ice water. Additionally, clouds interact with Earth’s radiative budget in a non-trivial fashion and their contribution to climate change remains a major research problem. The parameterization of convection and cumulus clouds in general is thus an important component of GCMs and NWPMs [246, 7]. As discussed in Chapter 2 of this volume, the widely used cumulus parameterization schemes can be divided into three main groups. Adjustment-type schemes, first introduced by Manabe and co-workers [181] and later refined by Betts and Miller [14], assume that the main dynamical role of convection is to maintain the large-scale tropospheric profiles in a state of equilibrium; a convective adjustment scheme amounts to force back the troposphere to a known background equilibrium profile. Moisture convergence schemes, first introduced by Kuo et al. [136], assume © Springer Nature Switzerland AG 2019 B. Khouider, Models for Tropical Climate Dynamics, Mathematics of Planet Earth 3, https://doi.org/10.1007/978-3-030-17775-1 9

163

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9 Implementation of the Multicloud Model in an Aquaplanet Global Climate Model

that convective heating and the resulting precipitation are proportional to moisture convergence in the lower troposphere; they are based on the idea of the existence of a convective instability of a second kind (CISK) or wave-CISK [137, 32, 237, 158], i.e., an instability other than the widely accepted convective instability purely driven by buoyancy. Mass-flux schemes, which are perhaps the most conceptually elaborate schemes, attempt to represent directly the turbulent fluxes due to convective flows occurring at the subgrid scales in terms of the associated upward and downward fluxes of air masses. There are two types of mass-flux schemes. There is the bulk mass-flux schemes of Tiedtke [259] and its refinements [208, 130] where the bulk mass due to an ensemble of cloud aggregates or types is systemically integrated following a bulk buoyant parcel from cloud base to the level of neutral buoyancy and there are spectral-type mass-flux schemes such as Arakawa and Shubert [8] and Zhang and McFarlane [305] where the bulk updraft mass flux is obtained from the averaging of an infinite number of plumes launched from the surface that detrain at a continuum spectrum of detraining levels each according to its unique entrainment rate. Despite the steady improvements and sustained sophistications, those early cumulus parameterizations have hit a brick wall in terms of their ability to represent organized tropical convection. The resulting GCM simulations, using various versions of those earlier cumulus parameterization concepts, represented poorly tropical rainfall, wind, temperature, and moisture variability at the synoptic and intra-seasonally scale. The MJO was especially notoriously hard among the climate modelling community [151, 155, 95, 160]. Significant improvements have been observed during the fifth assessment of climate models that participated in the report of the International Panel on Climate Change [95] but besides high resolution models representing explicitly convection (cloud permitting models) and superparameterization models [224], where a cloud resolving model is embedded within each GCM grid column, systematic biases reminded in coarse resolution GCMs. As seen in Chapter 5, some of the closure assumptions behind these cumulus parameterization schemes have systematic deficiencies in capturing a scale selective instability leading directly to these convectively coupled large-scale waves as a mechanism that transfers energy upscale, directly from the convective scale. This is unlike the multicloud model which is presented and analysed in the previous chapters that appears to process a congestus-moisture preconditioning instability at synoptic and planetary scales without the need of artifacts such as WISHE or CISK that have systematic problems. It is thus tempting to adopt the multicloud model as a cumulus parameterization in a host dynamical core GCM to represent the large-scale atmospheric dynamics. This is done for the first time by Khouider et al. [123] in the context of an aquaplanet GCM, i.e., an atmosphere only model overlying a globe covered with only water, i.e., no land or topography based on the high order methods modelling environment (HOMME) atmospheric general circulation model [39, 16] as the main dynamical core. Accordingly, we provide in this chapter an outline of the implementation of the multicloud model in the HOMME GCM and a summary of a few important results reported in seminal paper [123] and in two follow-up papers [3, 5] regarding the successful simulation of the MJO, convectively coupled Kelvin, Rossby, and two-day waves, and the monsoon climatology and intra-seasonal variability.

9.2 The Multicloud Model as a Simplified Cumulus Parameterization

165

HOMME is a highly scalable spectral element model of atmospheric dynamics based on a cubed sphere desensitization [256, 257, 190].

9.2 The Multicloud Model as a Simplified Cumulus Parameterization The influence of the multicloud model on the large-scale atmospheric dynamics occurs through the latent heating associated with the three cloud types, congestus, deep, and stratiform, distributed in the vertical according to judiciously chosen heating-profile basis functions. As an extension of the sine and cosine functions used in the context of the idealized crude vertical resolution model, reduced to the first two baroclinic modes discussed in the few previous chapters, we invoke the vertical normal modes of Kasahara and Puri [104] that are obtained when the method of separation of variables is applied to the linear primitive equations in (1.4) with a non-constant Brunt-V¨ais¨al¨a buoyancy frequency, N(z). This leads to the St¨urmLiouville eigenvalue problem   p d φn d 1 − = 2 φn , d p RΓ d p cn where R is the gas constant and Γ is the static stability. The eigensolutions, φn (z), n = 0, 1, 2, · · · are the vertical structure functions such that the horizontal velocity satisfies [104] (9.1) u(x, y, p,t) = ∑ un (x, y,t)φn (p), n

where un (x, y,t) = u, φn  ≡

1 pB − pT

 pB pT

u(x, y, p,t)φn (p)d p, n ≥ 0.

The vertical velocity is recovered through the incompressibility constraint

ω (x, y, p,t) = ∑ ωn (x, y,t)ψn (p) n

with

ψn (p) = −

1 pB − pT

 p pT

φn (p )d p

(9.2)

and

ωn = (pB − pT )∇ · (un ). Accordingly, the potential temperature expands in terms of the ψn ’s:

θ (x, y, p,t) = ∑ θn (x, y,t)ψn (p). n

(9.3)

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It is thus natural to use the vertical structure functions of potential temperature, associated with the first and second baroclinic modes, n = 1, 2, to set up the heating profiles for deep, congestus, and stratiform heating, respectively, consistent with both observations and cloud resolving modelling simulations [293, 78, 128, 263, 219]. The structure of the functions φn (z) and ψn (z), n = 1, 2 associated with the first and second baroclinic modes are displayed in Figure 9.1. We note that when used to from the total heating forming the temperature source or moisture sink (precipitation), the ψn functions where clipped to zero above the 200 hPa line (dashed green line in Figure 9.1) to avoid unrealistic behaviour in the upper atmosphere. We set Qc = Hd ψ˜ 1 (p) + (Hc − Hs )ψ˜ 2 (p), where Hd , Hc , Hs are, respectively, the deep convection, congestus, and stratiform heating rates and ψ˜ j , j = 1, 2 represent the truncated versions of the Kasahara and Puri vertical mode profile functions. Accordingly, the radiative cooling is set to QR = Q0R,1 ψ˜ 1 (p) + Q0R,2 ψ˜ 2 (p),

(9.4)

so that the total convective tendency of the large-scale (GCM) temperature is given by  κ p DT 1 1 = M(y) (Qc − QR ) + (T0 − T ), (9.5) Dt 1−κ pB τR where κ ≈ 2/3 is the ratio of the gas constant and the specific heat capacity at constant pressure of dry air. Here M(y) = 12 (1 − tanh(k(|y| − y0 )) is a smoothed Heaviside function of latitude, y, to restrict the cumulus heating and cooling to the tropics. The shape of the mask is defined by setting k = 10 and y0 = 30° (or 40° in the case of monsoon simulations in Section 9.4) and its structure is shown in Figure 9.1(B). The last term in (9.5) represents Newtonian cooling and serves to keep the large-scale temperature near a fixed background T0 with a relaxation time scale τR = 50 days. To keep the multicloud model as close as possible to the original idealized version, we utilize a vertically averaged moisture equation and bulk boundary layer potential temperature equation, instead of a fully resolved moisture in the vertical and a full boundary layer dynamics equations. The governing equation for the vertically averaged moisture perturbation is obtained by a systematic averaging of the moisture conservation equation when the zonal and vertical velocities are truncated to the first two baroclinic modes n = 1, 2 and the barotropic mode, represented by an over bar. We have [123] dq + ∇ · (q (u¯ + u1 + α˜ u2 )) + Q˜ 1 ∇ · u1 + Q˜ 2 ∇ · u2 = −P + E, dt

(9.6)

where Q˜ j =

 pB ˜ d Q(p) pT

dp

ψ j (p) d p, j = 1, 2.

(9.7)

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The terms P and E are, respectively, the sink and sources of water vapour in the middle of the troposphere that represent mainly surface precipitation and evaporation of rain and detrainment of cloud water. The boundary layer equivalent potential temperature solves d θeb 1 1 + u(x, y, p1 ,t) · ∇θeb = Es − D. dt h h

(9.8)

Here Es is the evaporation from the sea surface, D is the downdraft mass flux, h = 500 m is the height of the atmospheric boundary layer, which is assumed to be constant, and p1 is the pressure at the lowest grid level above the surface. Enforcing conservation of vertically integrated moist static energy, we obtain the following closure relations for precipitation and mid-tropospheric evaporation: P=

1 pB − pT

 pB pT

Qc (x, y, p,t)d p, E =

D , H

where H = 16 km is the tropospheric height. The closure equations for the congestus, deep, and stratiform heating rates, Hc , Hd , Hs , the surface latent heat flux, Es , and the downdrafts flux D, complete the description of the multicloud model parameterization. These are summarized in Table 9.1. They are essentially the same as in the original idealized multicloud model of the previous chapters. The parameter values vary from simulation to simulation and are reported in the following sections accordingly. Table 9.1 List of multicloud variables and closure equations. [Adopted from [123].] Variable Description θem Middle tropospheric θe

Λ

Moisture switch function

Hd

Deep convection

Hc

Congestus heating

Hs

Stratiform heating

D

Downdraft

Es

Surface flux

Equation θem = q + ψ1 (θ1 + α2 θ2 )

Λ = 1 if θeb − θem > 20 K Λ = 0 if θeb − θem < 10 K Linear and continuous for 10 ≤ θeb − θem ≤ 20 K

+ 1 Hd = (1 − Λ ) Q¯ + τconv (a1 θeb + a2 q + a0 (θ1 + γ2 θ2 )) dHc 1 = dt τc



  + 1   ¯ Λ αc Q + (θeb + a0 (θ1 + γ2 θ2 ) − Hc τconv 1 dHs = (αs Hd − Hs ) dt τs

D=

m0 Q¯

(1 + μ (Hs − Hc ))+ (θeb − θem ) 1 ∗ 1 Es = (θeb − θeb ) h τe

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Fig. 9.1 (A) Basis functions of vertical structure. Red areas on the ψ1,2 plots show regions of convective heating associated with the deep convection and congestus cloud types, respectively, while the blue area designates cooling aloft. The vertical profile of the stratiform forcing (not shown here) is that of −ψ2 (with a minus sign) so that we have cooling below (red area in ψ2 ) and warming aloft (blue area in the ψ2 ) see KM06, KM08a. The shear and middle jet-shear flows associated with φ1,2 are indicated by arrows. The pressure levels where ψ˜ 1,2 are truncated are shown with the horizontal dashed lines. (B) Meridional structure of the mask that limits convective heating to the tropics. (C) Vertical profile of the imposed radiative cooling for three different values of the stratiform fraction αs . From [123].

9.3 Uniform Aquaplanet Simulations: MJO v.s. Convectively Coupled Waves We first consider the case of an idealized setup where the earth surface is covered with water (an aquaplanet) whose temperature is uniformly distributed zonally— along the equator, and summarize herein some results first reported in Khouider et al. [123] (see also [122]). Based on the model design described above, the observed background profiles (soundings) for temperature and moisture from

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the Global Atmospheric Research Program (GARP) Atlantic Tropical Experiment (GATE) [75] are used to construct the vertical structure basis functions, φi, j (z), ψi, j (z), j = 1, 2, and the background moisture projections, Q˜ j , j = 1, 2. Consistent with the linear analysis results reported in [e.g., 118] and summarized in the previous chapters, the constants Q˜ j , j = 1, 2, are of central importance regarding the ability of the multicloud model to simulate large-scale convectively coupled waves, as they control the amount of moisture convergence supplied by the background climatology. After inferring their values from the GARP-GATE sounding, systematic changes in the Q˜ j , j = 1, 2 parameters were considered. Another important parameter which is varied here is the stratiform heating fraction αs which controls both the tilting of the diabatic heating and the downdraft mass flux which cools and dries the boundary layer and helps in the organization and propagation of large-scale convective disturbances via the stratiform instability mechanism. Here we present the result of two tropical climate simulations using the multicloud-HOMME aquaplanet model, one with a doubled moisture backgrounds, where the constants Q˜ j , j = 1, 2 assume values double of what is inferred from the GATE sounding and the other with the stratiform heating fraction αs = 0.5 instead of the standard value of αs = 0.25. In the first experiment with doubled moisture background constants all other parameters including αs are kept to their standard values and similarly for the second experiments in which Q˜ j , j = 1, 2 are as inferred from GATE. The effect of the parameter αs on the imposed radiative cooling, which allows an RCE solutions, is illustrated in Figure 9.1(c). The case with αs = 0.5 has in particular a more top heavy cooling profile. The results of these two experiments are summarized in Figures 9.2 to 9.4, which show, respectively, the corresponding Hovm¨oller diagrams and spectral power plots for the two solutions. Note that each simulation was run for 2000 days and only the last 500 days or 200 days are shown on the Hovm¨oller plots. As can be seen from these figures, the first simulation yields a solution consisting of a dominating Madden-Julian oscillation-like disturbance, while the second leads to a menagerie a synoptic scale convectively coupled waves consisting of Kelvin, Rossby, and westward inertio-gravity waves. The dominant MJO-like disturbance evident in zonal wind and deep heating, Hd , perturbation illustrated by the eastward moving streaks in Figure 9.2 is confirmed by the spectral diagrams shown in Figure 9.3. Here we see a dominant precipitation peak at the 40 days period and zonal wavenumber 2, yielding a propagation speed of roughly 5 m s−1 as in observations. The zonal velocity spectral diagram seems to exhibit a double peak, one near the 40 day, wavenumber 2 mark, and one with a double period and doubled wavenumber, i.e., a wavenumber one and an 80 days period. The latter is a signature of the modulation of the train of the two wavenumber 2 MJOs as can be surmised from the persisting blue streaks seen between days 1700 and 1750 on the 211 hPa wind plot in Figure 9.2. Similarly, although not shown, the second experiment in Figure 9.4 exhibits spectral peaks along the Rossby, Kelvin, and westward inertio-gravity waves [123]. This is consistent with the fact that the

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9 Implementation of the Multicloud Model in an Aquaplanet Global Climate Model

multicloud model was proven to process the key instabilities reminiscent to convectively coupled equatorial waves as discussed in Chapter 7. A better account of the ability of the multicloud parameterization to capture convectively coupled waves will be given in Chapter 11 where the stochastic version of the model is used in a fully coupled ocean-atmosphere global climate model. The horizontal structure of the MJO-filtered wave (abusively called MJO composite), obtained by using a low pass-filter in Fourier space based on the spectral power peak in Figure 9.3, is shown in Figures 9.5 and 9.6. The flow pictures show a train of two (wavenumber 2) MJO-like disturbances: one with deep heating maximum (convection centre) located between 200° and 250° longitudes and the other around 50° longitude. As can be seen the MJO-like waves have many physical features resembling the observed MJO [127, 100] discussed in Chapter 3: Zonal convergence at low level capped with divergence at upper level in phase with maximum heating, low-level warm temperature leading the wave and quadruple vortices surrounding the heating centres straddle the equator, both at low level and upper level, with anticyclones (negative vorticity in northern hemisphere and positive vorticity south of the equator) at low-level and cyclones in the upper troposphere. More importantly, as in observations, positive anomalies in boundary layer θe , a surrogate for CAPE, lead to positive mid-level moisture anomalies which in turn lead to MJO convection. During the MJO suppressed phase, minimum deep heating anomalies, congestus heating becomes prominent and especially on the northern and southern flanks of the heating centre. The congestus heating helps re-moisten the environment after the MJO convection episode and precondition it for the next MJO convection event by inducing low-level (second baroclinic) convergence as anticipated in Chapter 7. The vertical structure plots (top panels of Figure 9.7) show a clear front to rear tilt especially in temperature and zonal wind anomalies consistent with observations and with the results of Chapters 7 and 8.The bottom two panels show the climatological (time and zonal mean) zonal velocity vertical/meridional profile and thermodynamic variables. Contrary to the expected prevailing of equatorial easterlies and westerly jets in the subtropic, the simulated climatology is at odds with observations. It shows superrotating winds at the equator due to the persisting MJO waves which dominate the variability. Such equatorial superrotation has been reported in a few numerical, observational, and theoretical studies [233, 193, 300, 19]. The last references especially discuss equatorial superrotation in the MJO context. Biello et al. [19] specifically demonstrated how vertical momentum transport due to synoptic scale disturbances embedded within the MJO envelope would induce equatorial superrotation. In fact, equatorial superrotation is observed also in the mean climatology of the second experiments of Figure 9.4. Curiously, the maximum mean zonal wind on the left-bottom panel of Figure 9.7 is about 5 m s−1 , coinciding with the MJO eastward propagation speed. This is probably not just a coincidence as it may suggest that the simulated MJO-like disturbance is simply stirred by the background

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171

wind similarly to mesoscale convective systems [193, e.g.,]. However, despite the disputed existing theories [302], an MJO theory based on the background stirring wind is not in the radars of theoreticians. We close this section by mentioning that as expected the mean deep heating Hd on the right bottom panel of Figure 9.7 matches the imposed radiative cooling of 1 K day−1 with the stratiform heating levelling up to 25% of this value according to the value of αs = 0.25. The congestus heating is small as expected. The mean boundary layer θe and moisture peak are below the imposed strong moisture background.

Fig. 9.2 Hovm¨oller diagrams of low-level (left) and upper-level (middle) zonal wind and deep heating (right) averaged between 5°N and 5°S, showing MJO like disturbances propagation eastward, as simulated by the MCM-HOMME aquaplanet model in the case of a doubled moisture gradient. From [123].

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9 Implementation of the Multicloud Model in an Aquaplanet Global Climate Model

Fig. 9.3 The spectral power of upper-level zonal wind and deep heating for the MJO-like simulation in Figure 9.2. From [123].

9.4 Warm Pool Simulations: MJO Initiation and Northward Propagation, Monsoon Climatology, and Variability In this section, we modify slightly the model setting presented above, by changing the surface heat flux to incorporate a region of warm sea surface mimicking the presence of the Indian Ocean/Western Pacific warm as in Figure 6.9. Specifically, ∗ − θ¯ profile (see the boundary layer θ equation) on the form of we impose a θeb e eb a Gaussian-like in the meridional direction y and a half sine in x in the warm pool region and constant everywhere, see the bottom panel in Figure 9.8. We start with the case when the warmest point is on the equator and then move it gradual to the north to simulate the migration of the intertropical convergence zone (ITCZ) during the monsoon season.

9.4 Warm Pool Simulations: MJO Initiation and Northward Propagation. . .

173

Fig. 9.4 Same as Figure 9.2 but for the case of convectively coupled wave simulations corresponding to a doubling in the stratiform fraction parameter. The purple and white dashed lines highlight disturbance streaks corresponding to Rossby waves moving westward at 6.6 m s−1 , Kelvin waves moving eastward at 18.5 m s−1 , and inertio-gravity waves moving westward at 29 m s−1 . From [123].

The Hovm¨oller diagram of precipitation (black contours) and upper-tropospheric ∗ − θ¯ is slightly to the North of the zonal wind for the case when the maximum θeb eb equator is shown in Figure 9.8. The Hovm¨oller diagram shows streaks of precipitation moving slowly eastward, superimposed on those of westerlies, that are in the same direction and have the same speed, a signature of the westerly wind bursts that follow the MJO convection centre. The successive MJO-like disturbances are all concentrated in the warm pool region, roughly between 60°E and 180°E consistent with observations [292, 98]. To the east of the warm pool, we have elongated streaks of wind anomalies and occasional precipitation events scattered all over, with not much coherence. As confirmed by the Wheeler-Kiladis-Takayabu diagram for precipitation and near-surface and upper-level zonal wind anomalies, shown in Figure 9.9, the fast moving waves appearing to the east of the warm pool correspond in fact to Kelvin waves that are ejected by each MJO event. As highlighted on each panel, the MJO signal is dominant in all three panels of Figure 9.9, while the Kelvin wave peak is significant only in the upper-level zonal wind anomalies. It is suggested in [3] that these dry Kelvin waves serve as precursors for the succeeding MJO events by carrying large-divergent flow anomalies around the globe

174

9 Implementation of the Multicloud Model in an Aquaplanet Global Climate Model MJO composite: 827 hPa Zonal wind (m/s)

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Fig. 9.5 Zonal structure of the simulated MJO-like wave for zonal and meridional wind anomalies, temperature anomalies, and vorticity anomalies. From [123].

and trigger planetary scale convective organization when the associated divergence anomalies circle back and surmount the warm pool region. This is consistent with a few observational and modelling studies which suggested an active role of Kelvin waves in MJO initiation [127, 159, 189]. In Figure 9.10, the warm pool is moved and expanded progressively to the North mimicking the northward migration of the ITCZ during the Indian summer monsoon. The word monsoon originates from Arabic and it simply means season indicating the rainy season during which the trade winds in the Arabian sea change direction from north-easterly to south-westerly between the months of June and September, which facilitates navigation for Arab merchants doing trade with India. This change in wind direction is accompanied with large amounts of rain in that region and over the Indian continent in particular with major implication for the Indian agriculture and economy [58, 69].

9.4 Warm Pool Simulations: MJO Initiation and Northward Propagation. . .

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Fig. 9.6 Same as Figure 9.5 but the deep, stratiform, and congestus heating, boundary layer equivalent potential temperature, and mid-level moisture. From [123].

The three top panels in Figure 9.10 show the shape of the warm pool fixed at three different locations 5° North, 10° North, and 15° North which are conducted and first reported in [4]. The bottom three panels display the corresponding Hovm¨oller diagrams of precipitation (black contours) and zonal wind anomalies as simulated by the HOMME-MCM model, averaged between 0 and 15° latitudes. Similar to Figure 9.8, when the warm pool is at 5°N, we see active MJO-like events moving eastward at roughly 5 m s−1 within the warm pool region, between 60 E and 180 E. East of these regions, the waves move at faster speeds and appear to be decoupled from the highly scattered and rare rain events. When the warm pool is at 10°N (middle panel), the eastward events become highly intermittent and even more confined to the warm pool region. The dry Kelvin waves outside the warm pool region are clearer. The picture changes completely when the warm pool is at 15°N. The precipitation becomes even more confined and more chaotic and seems to be organized in bands that are much smaller in spatial and temporal extent and more frequent. Some of these waves seem to move to the west. This is a signature of synoptic scale monsoon low-pressure systems which interact with eastward and northward propagating monsoon intra-seasonal oscillations [143, 279].

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9 Implementation of the Multicloud Model in an Aquaplanet Global Climate Model MJO composite: Zonal wind anomalies averaged over [-10,10]

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The simulated climatological means are shown in Figure 9.11 where the horizontal structure of the vertical vorticity and horizontal winds are shown on the three left panels, for the three warm pool locations 5, 10, and 15°N. The establishment of a monsoon climatology is clear from the emergence of south-westerly winds (blowing in the north-west direction) north of the equator, West of 120°E. The monsoon circulation persists and strengthens as the warm pool is displaced to the North. More importantly, there is a significant positive vorticity taking place near 10°N which also strengthens with the northward displacement of the warm pool. This persistent patch of positive vorticity is known as the monsoon trough and is one of the main characteristic of the monsoon climatology. The three panels on the right show the meridional-vertical mean circulation patterns and the associated mean heating. The three simulations all display a consistent circulation pattern with air raising north of the equator where there is positive heating and sinks south of equator where there is cooling on average. There are a few differences however. The main one being the progressive widening of the main circulation cell and the weakening of the heating maximum. The common characteristics of the three simulations shown in Figure 9.11 are the main features of the Indian monsoon climatology [69].

9.4 Warm Pool Simulations: MJO Initiation and Northward Propagation. . .

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Fig. 9.8 Top: Hovm¨oller diagram of precipitation (black contours) and upper-tropospheric zonal wind (colours, m s−1 ). The numbers show regions where precipitation (or equivalently convective heating) exceeds 3 K day−1 . Bottom: warm pool structure. See text for details. From [3].

The three top panels show the lag-latitude contours of filtered precipitation anomalies obtained by compositing low-frequency precipitation filtered in Fourier space in meridional wavenumbers between ±4 and 20–100 days time periods and averaged between 70 and 140 longitudes. The data is further statistically regressed using a time-lag correlation technique to single out the most significant coherent signal within this range of time and spatial scales. A clear northward propagating features appear and are the most prominent when the warm pool is 10°N. Although not shown here, northward propagating monsoon intra-seasonal disturbances

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Fig. 9.9 Wheeler-Kiladis-Takayabu spectral diagrams for the warm pool simulation in Figure 9.8 for precipitation (left), low-level zonal wind (middle), and upper-level zonal wind (right). The purple lines are the dispersion relations of equatorially trapped wave from linear theory with the respective equivalent depth indicated on each curve. The dominant MJO peak is indicated on each panel. From [3].

are observed in the raw data of both the 10N and 15N simulations [4], while the 5N simulation is more representative of a fall-to-winter climatology in the Indian Ocean/Western Pacific warm pool which is dominated by eastward moving MJOlike events. The lag-regression analysis seems to miss the northward propagation character of precipitation events on the intra-seasonal time scales, instead the panel in Figure 9.12(c) shows a seesaw oscillation between the equatorial region between 10 S and the equator and the northern hemisphere where the maximum precipitation makes a sadden switch with a period of roughly 10–15 days. The panel (d) of Figure 9.12 shows the horizontal structure of the simulated monsoon intra-seasonal oscillation-like disturbance composited from the 10°N simulation. The structure shows positive vorticity anomalies on the northern flank of northeastward wind anomalies resembling the mean monsoon climatology displayed in Figure 9.11(c) and (d). This anomalous circulation seems to act in reinforcing the climatology. Finally, Figure 9.12(d) shows that the plot of the spectral power of precipitation for the 10N simulation has two dominant peaks: one at 50 days corresponding to northward and eastward intra-seasonal signals and one between 5 and 10 days representing various synoptic scale disturbances including both eastward and westward moving waves such as equatorial Kelvin and monsoon low-depression waves. More discussion on the Madden-Julian oscillation, convectively coupled waves, and monsoon intra-seasonal oscillations is given in Chapter 11 where simulations with the stochastic multicloud model and using a fully coupled GCM are presented.

9.4 Warm Pool Simulations: MJO Initiation and Northward Propagation. . .

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∗ − θ¯ for Fig. 9.10 Moving the warm pool to the North. The three top panels show the imposed θeb eb three different cases mimicking the warm pool at 5°N, 10°N, and 15°N. The bottom panels show the time-longitude Hovm¨oller associated precipitation (black contours) and upper-level zonal wind anomalies (colours) averaged between 0 and 15° North. From [4].

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Fig. 9.11 Left: Time mean vorticity contours (colours, ×10−6 s−1 ) and mean horizontal wind arrows (m s−1 ) for the three simulations in Figure 9.10. Right: Vertical-meridional contours of heating and cooling anomalies (K day−1 ) and (v, w) wind arrows. From [4].

9.4 Warm Pool Simulations: MJO Initiation and Northward Propagation. . .

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Fig. 9.12 (a-c) Lag-latitude plots of filtered (wave numbers ± 4 and 20–100 days) 850 hPa zonal winds (m s−1 ) and averaged between 70°and 140° East. (d) CEOF1 of 850 hPa winds (m s−1 , vectors) and relative vorticity (10−6 s−1 , shaded) anomalies for the simulation with a warm pool at 10°N. (e) Power spectra of PC1 of the CEOF1 for the simulation in (d). From [4].

Part III

The Stochastic Multicloud Model: SMCM

Chapter 10

Stochastic Birth and Death Models for Clouds

10.1 Introduction As already been stressed out, atmospheric convection is the process through which warm and moist air parcels rise from the surface, condense liquid water, and form cumulus clouds. In the tropics, moist convection constitutes the main energy source for both local and large-scale circulations. Precipitation patterns in the tropics are organized into cloud clusters and superclusters on a wide range of scales; they range from the convective cell (the cumulus cloud) of 1 to 10 km, to planetary scale waves with oscillation periods of 40 to 60 days. Due to the complex interactions between the local processes of convection and the large-scale waves, climate models fail to properly capture tropical circulation patterns and their effect on the global circulation. In a climate model, the governing equations are discretized on a coarse mesh of roughly 100 km to 200 km and the effects of processes that are not resolved on such grids are represented by a parameterization also called as subgrid model. According to the last report of the United Nations’ Intergovernmental Panel on Climate Change (IPCC), the interactions of clouds and the climate system is one of the major challenges in climate research. The phenomenon of convection is in essence due to the very simple physical mechanism of buoyancy, which was discovered thousands of years ago by Archimedes. Buoyancy is the force that pushes light fluid to rise and heavy fluid to sink. When light fluid lies over heavy fluid as in normal atmospheric and oceanic conditions the situation is stable and if a fluid parcel is displaced mechanically in the vertical it will quickly sink or rise back towards its initial position and would normally undergo an oscillatory motion around its initial position, according to the Brunt V¨ais¨al¨a buoyancy frequency as discussed in Chapters 1 and 2. When the sun heats the surface (sea or land), the air parcels near the ground become quickly warm and moist, due to the evaporation of sea or land water. Because the warm and moist air near the surface is lighter than the dry and cold air above it, a turbulent motion begins and quickly mixes the air layer near the surface; because of the strong tropospheric stratification (the large discrepancy between the air density at the surface © Springer Nature Switzerland AG 2019 B. Khouider, Models for Tropical Climate Dynamics, Mathematics of Planet Earth 3, https://doi.org/10.1007/978-3-030-17775-1 10

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and at the top of the troposphere), the day is not long enough for this mixing process to penetrate very high before sunset. The cold surface-stable conditions are quickly restored at night as the ground loses its heat to space as long-wave radiation and the cycle continues. This process is called dry convection because it does not involve the phase change of water. It mainly serves to form what is known as the planetary boundary or mixed layer where the air density and potential temperature are fairly uniform. It leaves the upper troposphere unperturbed—happy and cool. When some “lucky” parcels are able to make it beyond the mixed layer they expand and cool down because of the pressure drop and potentially become saturated with water vapour. At this point the rising parcel starts to condense liquid water and releases latent heat, which in turn warms the parcel and partially compensates for the cooling by expansion. When this diabatic heating is large enough, the parcel becomes positively buoyant and will eventually rise high enough and entrain further convection and form a cloud. The height at which a rising parcel starts to condense water is called the lifted condensation level (LCL), while the height at which a parcel becomes positively buoyant because of condensational heating is called the level of free convection (LFC). The path of a hypothetical air parcel rising from the surface is shown by the thick solid line in Figure 10.1. The thick dashed line is the environmental virtual temperature (see Chapter 2). Given that the potential energy associated with the convecting parcel is the vertical integral of the buoyancy force (which is proportional to the virtual temperature difference between the rising parcel and the environment), typically, the rising parcel needs to overcome a certain amount of negative energy before it reaches its LFC. The negative energy is known as convective inhibition (CIN) and is represented by the red area in Figure 10.1, while the positive (green) area above it is called convective available potential energy (CAPE). Observations showed that various mechanisms can contribute to provide the energy necessary for the rising parcel to overcome the CIN barrier. These include both local effects such as turbulent fluctuations in the boundary layer moisture and temperature, gust fronts, cold pools, and density currents and large-scale effects such as organized convergence and propagating waves. Due to its complexity, the effect of CIN is still very poorly understood and as such it is not very well represented in climate models. The deterministic parameterization of CIN is at best unrealistic! In the next section we discuss a simple stochastic model for CIN based on the Ising model for magnetization [21]. The same framework is then expanded in Section 10.3 to develop the stochastic multicloud model (SMCM) for organized tropical convection. The SMCM is further exposed and used in the next chapter as an extension of the (deterministic) multicloud model and implemented in general climate models. A major conundrum of atmospheric convection, especially in the tropics, resides in the mechanisms behind its self-similar organization over a wide range of scales and its interaction thereof with large-scale flows and waves. As already pointed out, organized convection involves three cloud types that interact with each other and help define a self-similar morphology for convectively coupled waves of various sizes that are often embedded in each other like Russian dolls. It is suggested in [e.g., 182] that this can be possible only if the cloud-cloud interactions occur in

10.1 Introduction

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Fig. 10.1 Left: Path followed by a hypothetical parcel of air rising from the surface through the atmospheric column (solid line). The dashed line represents the environmental virtual temperature. The green area represents the convective available potential energy (CAPE), while the red area is the negative energy (CIN) that the rising parcel needs to overcome in order to reach its level of free convection and become freely buoyant. The dotted lines show the LCL and LFC levels. Right: A cartoon of a hot-tower cumulus cloud formed by air parcels rising from the mixed boundary layer.

some stochastic manner so that when they are averaged locally or globally they would preserve their self-similar structure. This common structure consists of cumulus congestus clouds that prevail in front of the wave followed by deep convective clouds that penetrate to the upper troposphere (near the tropopause) which in turn are lagged by icy stratiform anvils that prevail in the upper troposphere. Arguably a consensual explanation for the main physical mechanisms for this behaviour is still an active research area but a more or less accepted explanation is as follows: (1) Because of the entrainment of surrounding dry air, the rising parcels lose their buoyancy typically below the freezing level and form congestus clouds. (2) As they form and dissipate in the middle of the troposphere, congestus clouds deposit and converge moisture in the horizontal and thus serve to precondition the environment for future parcels to penetrate higher and form deep convective clouds. (3) Stratiform clouds are believed to be a continuation of the deep penetrative clouds which start to dissipate from below and leave their icy anvil tops behind that continue to produce ice, thus heat. A cartoon of this three cloud-type paradigm is given on the top panel of Figure 10.2. This is the building block of the SMCM presented in Section 10.3. However, as pointed out above, the fact that multiple systems evolving on various scales possess this cloud trilogy at all scales may seem problematic when these various convective systems are embedded in each other like Russian dolls. Mapes et al. [182] invoke a hypothesis of a stretched building block in which the embedded systems in front of the planetary scale envelop would favour cumulus congestus cloud population, deep convective towers in the middle, and stratiform anvils in the back, so that their statistical average would favour an arrangement at

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the envelope scale with the same cloud trilogy. This is illustrated on the two upper panels in Figure 10.2.

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Fig. 10.2 (A) A cartoon of the three cloud types showing congestus, deep convective, and a decaying deep convective tower with a lagging large stratiform anvil, with stratiform rain falling into a dry region below it where it eventually evaporates and cools the environment (hatched area). The arrows indicate convective motion within the cloud. (B) Cloud clusters showing various levels of aggregation, dominant congestus (left), dominant deep (middle), and dominant stratiform (right), and (C) their statistical average forming the planetary scale envelope according to the stretched building block hypothesis. Inspired from [182]

10.2 A Birth-Death Model for Convective Inhibition

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10.2 A Birth-Death Model for Convective Inhibition To help understand the mathematics behind the SMCM presented in the next section, we begin here by discussing a simpler stochastic model for CIN. The stochastic models presented here rely on the theory of particle interacting systems on a lattice [150] to build cloud models from first principles, at the microscopic level and then operate a systematic coarse graining procedure to derive Markovian birth-death processes for the macroscopic cloud coverages.

10.2.1 The Microscopic Stochastic Model for CIN: Ising Model Here, we give a brief description of the microscopic lattice model, which is used as a basis for the birth-death process for CIN. The interested reader is referred to [173] for more details. It is based on the Ising model for magnetization of statistical mechanics [21]. For simplicity, we begin by ignoring the trimodality of tropical convection and assume that there is one cloud type consisting of towering cumulonimbus clouds as illustrated in Figure 10.3. As stated above, CIN is an energy barrier for spontaneous deep penetrative convection in the tropic and it is known to have important fluctuations in the horizontal on the order of 1 km to 10 km. Hence, we consider sites that are uniformly distributed on a lattice (which can be thought of as spanning one horizontal grid box of the climate model) on which we define an order parameter σI on a finite lattice Λ ⊂ {0, 1}Z .

σI (x) = 1 at a site if convection is inhibited (a CIN site) (10.1) σI (x) = 0 at a site if there is potential for deep convection (a PAC site) A cartoonist picture of this representation is shown in Figure 10.3. On the coarse grid of mesh size Δ x of a climate model, the value of CIN at a coarse mesh point, jΔ x, is given by the average

σ¯ I ( jΔ x) =

1 Δx

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Here we assume a simple 1D domain for simplicity1 . The model’s cumulus parameterization then “decides” according to this average CIN value on whether to allow deep convection to occur at that grid point or not. As explained in the introduction section, factors to overcome CIN are very complex and can be both local or external. Instead of trying to formulate the detailed physical mechanisms for CIN, the microscopic CIN sites are set to interact with each other and with the external large-scale The SMCM discussed in the next section not only extents to the case when σI takes multiple values, non-binary, but it also assumes a two-dimensional rectangular lattice.

1

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Sigma_I = 1

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Fig. 10.3 A cartoon of a deep penetrative hot-tower cloud represented at a PAC site. The order parameter takes values 0 or 1 on a given site according to whether it is a CIN site or there is potential for deep convection.

values of the deterministic flow variables according to the following probabilistic rules: A) If a CIN site is surrounded by mostly CIN sites, then it has higher probability to remain a CIN site. B) If a PAC site is surrounded by mostly CIN sites, then it has higher probability to switch to a CIN site. C) The large-scale flow, u j , supplies an external potential h(u j ) that can modify the microscopic dynamics according to whether external conditions favour CIN or PAC. Following the standard theory of the Ising model of statistical mechanics, the microscopic energy for CIN is given by the Hamiltonian Hh (σI ) = −

1 ∑ J(γ (x − y))σI (x)σI (y) − h ∑ σI (x). 2∑ x y=x x

(10.3)

Here, Hh (σI ) is the microscopic energy associated with a given configuration σI with J ≥ 0 as the symmetric interaction potential and γ defines the range of microscopic interactions J(γ r) =

N 1 U( r) L+1 L+1

with U(r) = U(−r), r ∈ R, U(r) = 0, |r| ≥ 1. As a simple example, we consider the uniform potential  U0 , if r < 1, U(r) = (10.4) 0, otherwise.

10.2 A Birth-Death Model for Convective Inhibition

191

N Here, γ = L+1 so that in the 1D setting, 2L is the number of interacting neighbouring sites and N is the total number of sites. Note that, in the absence of the external factor, h = 0, the minimum energy level is achieved when σI (x) = 1 ∀x, i.e., an all-CIN configuration, while the all-PAC configuration, σI (x) = 0, has the highest level—zero energy. If we regard the mixed boundary layer as a heat bath for CIN, so that changes in h are neglected when considering the microscopic dynamics, then according to the theory of statistical mechanics, the equilibrium distribution of the configurations σI for the Hamiltonian dynamics (that oscillate about the minimum energy level) is given by the grand canonical Gibbs measure

G(σI ) =

1 −β Hh (σI ) e , ZΛ

(10.5)

where β is a positive parameter that depends on the “temperature” of the system and ZΛ is a normalization constant (which can be very large and hard to compute!). The external potential h modifies the CIN configuration according to whether the large scales are favourable to CIN or not. The Hamiltonian in (10.3) is a monotonic function of h, this is very helpful when it comes to choosing the proper dependence on the large-scale variables. In practice, h can be represented by either the large scale subsidence of upper troposphere air, which cools and dries the boundary layer and thus increases the energy for CIN, or the large-scale fluctuations in the boundary layer temperature and moisture, which tends to destroy CIN energy. Here, we omit the discussion about the actual coupling of the stochastic CIN to an actual climate model. The interested reader is referred to the papers [173, 115, 171]. To simulate the dynamics of this high-dimensional Hamiltonian system, we make use of a Metropolis algorithm which systematically obeys the rules in A), B), and C). A configuration randomly flips at a site x  1 − σI (x) if y = x x σI (y) = (10.6) σI (y) if y = x according to a Markov jump process where the transition rates c(σI , x) are given by the Arrhenius adsorption and desorption rates  1 −β V (x) e , σI = 1 c(σ, x) = τ 1 (10.7) σI = 0 τ, for which G(σ ) in (10.5) is the invariant measure. Here V (x) ≡ H[σInew (x) = 0] − H[σIold (x) = 1] = 12 ∑z=x J(γ (x − z))σI (z) + h is the energy difference between the new configuration and the old configuration of σI , when one single CIN site is destroyed, i.e., the energy that the rising parcel needs to provide in order for it to potentially penetrate deep in the troposphere. We note that a CIN site surrounded by CIN sites has a high energy to overcome in order to become a PAC site (rule A)), while a PAC site naturally decays into a CIN site and would remain so for a long time if it is surrounded by CIN sites (rule B)). Finally, the inclusion of the external

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10 Stochastic Birth and Death Models for Clouds

potential h accounts for the large-scale influence, i.e., rule C). Here τ is a parameter that represents the time scale of CIN which is typically on the order of a few minutes to a few hours. This in effect defines a two state Markov chain at each site x of the lattice whose transition rates, q10 = e−β V (x) /τ and q01 = 1/τ , depend on the state of the neighbouring sites through the energy potential J. It amounts to constructing an ergodic Markov chain whose equilibrium distribution is the probability distribution we wish to sample. It is easy to verify in our case that, at each site, our two state Markov chain is in detailed balance with respect to the Gibbs measure: q10 G[σI (x = 1)] = q01 G[σI (x = 0)]. In practice, the climate model grid box is on the order of 100 to 200 km, which would require up to 40×40 microscopic CIN sites (in the full three-dimensional setting) that are 2 to 5 km apart. For each grid box, we thus need to sample 1600 times the Gibbs measure at each time step of the climate model. This would induce astronomical computations in addition to the already high order complexity of the large-scale model that solves for the flow components. Below, we present a systematic coarse graining strategy that permits the derivation of a single birth-death process (that counts the number of active CIN sites) on each climate model grid box.

10.2.2 The Coarse Grained Mesoscopic Stochastic Model: Birth-Death Process We now use a systematic coarse graining strategy to reduce the complexity of the microscopic CIN model. This method was first developed in [106] and used for the CIN model in [115]. The coarse grained procedure starts by the definition of a coarse grained stochastic process that tracks the total number of CIN sites in a given mesoscopic region, e.g., the climate model grid box. We introduce a coarse lattice Λc . Let m, q be two integers. The fine and coarse lattices are given by

Λ≡

1 1 Z ∩ [0, 1] and Λc ≡ Z ∩ [0, 1]. mq m

Each cell, Dk , k = 1, . . . , m, on the coarse lattice is divided into q microscopic cells: 1 Dk ≡ {1, 2, . . . , q}, ∀k = 1, . . . , m. q We introduce the coarse grained sequence of random variables (stochastic p)

ηt (k) =

∑

y∈Dk

σI,t (y).

(10.8)

10.2 A Birth-Death Model for Convective Inhibition

193

Then, the sequence η = {ηt (k)}k,t in (10.8) is a birth-death Markov process defined on the configuration space Hm,q = {0, 1, . . . , q}Λc such that ηt (k) obeys the transition probabilities Prob{ηt+Δ t (k) = n + 1|ηt (k) = n} = Ca (k, n)Δ t + o(Δ t) Prob{ηt+Δ t (k) = n − 1|ηt (k) = n} = Cd (k, n)Δ t + o(Δ t)

(10.9)

Prob{ηt+Δ t (k) = n|ηt (k) = n} = 1 − (Ca (k, n) +Cd (k, n))Δ t + o(Δ t), where the coarse grained absorption and desorption rates are given, respectively, by Ca (k, n) = Cd (k, n) =

1 [q − η (k)] τI

1 ¯ η (k)e−β V (k) , τI

(10.10)

where V¯ (k) =

¯ l)η (k) + J(0, ¯ 0) ∑ J(k,

 η (k) − 1 + h

(10.11)

l∈Λc

l=k

with J¯ as the coarse grained interaction potential. It is shown in [106] that under the assumption of long range interactions, L  1, J¯ satisfies 1 q O( ), x ∈ Dl , y ∈ Dk , k = l, L+1 L+1

¯ l) + J(x − y) = J(k, where  

¯ l) = m2 J(k,

Dl ×Dk

J(γ (r − s))drds

=

1 ∑ ∑ J(γ (x − y)) q2 x∈D k y∈Dl

=

1 1 N ∑ ∑ U( L + 1 |x − y|), k = l, q2 L + 1 x∈D k y∈Dl

(10.12)

¯ 0) = J(0,

1 1 N ∑ ∑ U( L + 1 |x − y|), q(q − 1) L + 1 x∈D l y∈D

(10.13)

and

l

y=x

194

10 Stochastic Birth and Death Models for Clouds

and the coarse grained Hamiltonian is given by

 ¯ η ) = − 1 ∑ ∑ J(k, ¯ l)η (k)η (l) − 1 J(0, ¯ 0) ∑ η (l) η (l) − 1 − h ∑ η (l). H( 2 l∈Λc k∈Λc 2 l∈Λc l∈Λc k=l

(10.14) Notice the presence of the term representing the interactions between mesoscopic (coarse grained) cells in the definition of the coarse grained Hamiltonian. The canonical Gibbs measure for the coarse grained process is given by Gm,q,β (η ) =

1 ¯ e−β H(η ) Pm,q (d η ), Zm,q,β

where Pm,q (d η ) is the prior distribution. It is easily verified that this distribution satisfies detailed balance with respect to the coarse grained adsorption desorption rates: Ca (k, η )Gm,q,β (η ) = Cd (k, η + δk )Gm,q,β (η + δk ) Cd (k, η )Gm,q,β (η ) = Ca (k, η + δk )Gm,q,β (η + δk ).

(10.15)

If the interactions between coarse grained sites are ignored, then the expres¯ sions for the coarse grained potential, V¯ , and the coarse grained Hamiltonian, H, in (10.11) and (10.14), respectively, at an isolated coarse site k with a spin η (k) simplify to ¯ 0)(η (k) − 1) + h V¯h (η (k)) ≡ H¯ h (η (k)) − H¯ h (η (k) + 1) = J(0,

 1¯ H¯ h (η (k)) = − J(0, 0)η (k) η (k) − 1 − hη (k). 2

(10.16)

According to the definition of the internal potential U in (10.4) and the definition ¯ of J(0.0) in (10.13) we have in the case where only nearest neighbour interactions (L = 1) are allowed between microscopic sites U(N|x − y|/L + 1) = 0 ⇐⇒ x = y or x = y ±

1 , N

(with 1/N being the actual mesh size on the microscopic lattice), hence ¯ 0) = J(0,

2U0 U0 = . 2(q − 1) q − 1

10.2 A Birth-Death Model for Convective Inhibition

195

Transition Probability Matrix According to the theory of Markov chains in Section 4.2 the transition probabilities, Pt (i, j), 0 ≤ i, j ≤ q, that go from a state η0 (k) = i at time t = 0 to a state ηt (k) = j at time t > 0 satisfy the forward equations Pi, j (t) = Cd ( j + 1, k)Pi, j+1 (t) +Ca ( j − 1, k)Pj, j−1 (t) 

− Ca ( j, k) +Cd ( j, k) Pi, j (t), j = 0, · · · , q.

(10.17)

The solution of this linear ODE is easily computed through the standard exponential formula; the transition matrix is given by [pt ( j, j )] = etA ,

(10.18)

where A is the tridiagonal-infinitesimal generator matrix; its upper and lower diagonals are formed by desorption and adsorption rates, Cd ( j + 1, k),Cd ( j − 1, k), respectively, while the main diagonal is the negatives of the waiting time rates, −(Cd ( j, k) +Cd ( j, k)). However, in practice, there is no need to compute this exponential matrix when using this model for the purpose of climate simulations. The birth-death Markov process can be easily simulated without having to solve directly for the transition probabilities by means of Gillespie’s exact algorithm given, which is given next.

10.2.3 Gillespie’s Exact Algorithm To simulate the birth-death process during an actual climate simulation, we use Gillespie’s exact algorithm which amounts to directly sample the exponential distribution S = min(T1 , T2 ), where T1 , T2 are, respectively, the waiting time until a birth and death occur. Another popular method used to sample Markov chains is the acceptance-rejection method in which case the transition time is fixed and then a given transition is accepted or rejected according to its probability of occurrence. If s ≤ Δ T (where again Δ T is the large-scale time step of the climate model), then we accept the transition and further classify it as a birth or a death as in the previous algorithm. This is repeated until the cumulative time reaches or exceeds Δ T . Gillespie’s Exact-Inverse-Method Algorithm 1) Given the state ηt of the process at time t, 0 ≤ t ≤ Δ T . 2) Draw a uniform random number r1 from [0, 1] and set s = − λ +1 μ ln(r1 ). 3) If s + t > Δ T , then set t = Δ T and terminate the algorithm. Otherwise (the transition is accepted) we draw a second uniform random number r2 in [0, 1].

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10 Stochastic Birth and Death Models for Clouds

4) If r2 < λ /(λ + μ ), set ηt+s = ηt + 1. Otherwise set ηt+s = ηt − 1. 5) Set t = t + s. If t < Δ T go to 1. Notice that this algorithm does not assume that at most one transition occurs at each time step, instead, it finds the exact time when the first transition occurs. This is the reason why it is called the exact algorithm.

10.2.4 Numerical Tests Here, we implement and test the Monte Carlo algorithm above for the birth and death process with the adsorption and desorption rates in (10.10) in the absence of the external field: h ≡ 0. We assume a single-uncoupled mesoscopic cell of size q. In Figures 10.4, 10.5, and 10.6 we plot the time evolution of ηt /q for both a single realization (top of the figure) and the average over 100 realizations (bottom panel) for the values β J0 = −4, 0, 2 and q = 5, q = 10, and q = 40, respectively. birth−death average over 100 realizations: q=5, τI=3 hours, h ≡ 0

birth−death single realization: q=5, τI=3 hours, h ≡ 0 1

1

β J0=2

0.9

0.9

0.8

0.8

0.7

0.7 β J0=0 η/q

η/q

0.6 0.5 0.4

0.5 0.4

0.3

0.3

0.2

β J0=–4

0.1 0

0.6

0

2

4

6

8

10

12

0.2 14

16

18

time (nondim units, T≈ 8 hours,1E+6 iter.)

20

0.1

0

2

4

6

8

10

12

14

16

18

20

time (nondim units, T≈ 8 hours,1E+6 iter.)

Fig. 10.4 Evolution in time of the random process ηt /q. top: single realization, bottom: average over 100 realizations, τI = 3 hours, q = 5

Independently on q, we observe that when β U0 is positive (attractive potential) the process tends to equilibrate around the maximum level q (CIN site), when β U0 = 0 (no local interactions) it equilibrates around the middle point q/2, and when β U0 < 0 (repulsive potential) it tends to the low CIN level 0, i.e., a PAC state. This behaviour can be explained as follows: • When β U0 > 0, the factor e−β J0 is a small positive number and so the birth (adsorption) rate dominates the (death) desorption rate except at the highest level q when it is zero. Thus, the process oscillates somewhere close to the state ηt = q.

10.2 A Birth-Death Model for Convective Inhibition

197 birth−death average over 100 realizations: q=10, τI=3 hours, h ≡ 0

birth−death single realization: q=10, τI=3 hours, h ≡ 0 1

1

β J0=2

0.9

0.8

0.7

0.7

β J0=0

β J0=0

0.6 η/q

η/q

0.6 0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1 0

β J0=2

0.9

0.8

0.1

β J0=–4 0

2

4

6

8

10

12

14

16

18

β J0=–4

0

20

0

2

time (nondim units, T≈ 8 hours,1E+6 iter.)

4

6

8

10

12

14

16

18

20

time (nondim units, T≈ 8 hours,1E+6 iter.)

Fig. 10.5 Same as in Figure 10.4 but for q = 10 birth−death average over 100 realizations: q=40, τI=3 hours, h ≡ 0

birth−death single realization: q=40, τI=3 hours, h ≡ 0 1

1 β J0=2

0.9

0.8

0.8 β J0=0

0.7

0.7 0.6 η/q

η/q

0.6 0.5

β J0=0

0.5 0.4

0.4 β J0=–4

0.3

0.3

0.2

0.2

0.1

0.1

0

β J0=2

0.9

0

2

4

6

8

10

12

14

16

18

20

time (nondim units, T≈ 8 hours,1E+6 iter.)

0

β J0=–4

0

2

4

6

8

10

12

14

16

18

20

time (nondim units, T≈ 8 hours,1E+6 iter.)

Fig. 10.6 Same as in Figure 10.4 but for q = 40

• When β U0 = 0, the two rates are comparable; the desorption rate is higher for j > q/2, whereas the adsorption rate is higher when j < q/2, and they are equal at q/2. Hence, the process oscillates around the middle q/2 where the births and deaths are balanced. • When β U0 > 0, the adsorption rate dominates except near the state j = 0, where the death rate is effectively zero. Hence, the process will oscillate in the vicinity of the low CIN level (PAC state).

198

10 Stochastic Birth and Death Models for Clouds

10.3 The Stochastic Multicloud Model This section presents a multidimensional birth-death stochastic process to capture the random interactions between the three cloud types that characterize organized tropical convection. This model has first appeared in [111] and since then it has been refined in various ways and used in multiple global climate models [37, 42, 218, 66] as well as in simple models [52, 53, 217, 36, 34, 41, 73]. By design, the stochastic multicloud model (SMCM) aims to represent the unresolved variability of organized tropical convection in a large-scale climate simulation with a typical mesh size of 100 to 200 km. We consider a horizontal grid box for the tropical troposphere, above the planetary boundary layer, of rectangular shape, divided into a lattice of n × n lattice points or sites. The parameter n is a positive integer on the order 100 or less so that the lattice sites are 1 to 5 kilometres apart, the typical scale for an individual cloud. We assume that each lattice site is either occupied by a certain cloud type (congestus, deep, or stratiform) or it is a clear sky site. A given site will switch from a given configuration to another according to some probability rules, which depend on the large-scale resolved variables. We thus construct a stochastic process at each lattice site taking the discrete values from 0 to 3 according to whether it is a clear sky site or it is occupied by a certain cloud type, as shown in Figure 10.7.

0

1

0

2

2

3

2

0

0

0

1

0

1

1

0

3

Fig. 10.7 Lattice cloud model. A given lattice site is either clear sky (0) or occupied by a congestus cloud (1), a deep convective cloud (2), or a stratiform anvil cloud (3).

Let Xti denote the state of site i of the lattice, i = 1, · · · , n × n, at time t ⎧ 0 if site i is clear sky ⎪ ⎪ ⎨ 1 if site i is occupied by a congestus cloud Xti = 2 if site i is occupied by a deep convective cloud ⎪ ⎪ ⎩ 3 if site i is occupied by a stratiform anvil.

(10.19)

10.3 The Stochastic Multicloud Model

199

Xti is a Markov chain with the transition probabilities + i , i i Plki ≡ Prob Xt+ Δ t = k/Xt = l = Rlk Δ t + o(Δ t), for l, k = 0, 1, 2, 3, and l = k and

+ i , i Plli ≡ Prob Xt+ Δ t = l/Xt = l = 1 −

3

∑

Plki ,

(10.20)

(10.21)

k=0,k=l

where Δ t > 0 is a small time increment and the Rilk ’s are prescribed transition rates. For simplicity, we ignore the direct-local interactions between sites and assume that the rates Rilk depend solely on the large-scale resolved variables according to the following intuitive rules of switching back and forth between cloud type to cloud type and from cloudy and non-cloudy sites. The case with local interactions will be discussed in Section 10.6. As for the CIN model, we assume that the lattice cloud sites evolve according to the following intuitive-probabilistic rules: 1. A clear site turns into a congestus site with high probability if CAPE is positive and the middle troposphere is dry. 2. A congestus or clear sky site turns into a deep convective site with high probability if CAPE is positive and the middle troposphere is moist. 3. A deep convective site turns into a stratiform site with high probability with a prescribed conversion rate, which may or may not depend on the state of the environment. 4. A cloudy site turns back to a clear sky with a certain probability according to a prescribed decay time scale for each cloud type. 5. It is very unlikely, during the short period of time Δ t, for a clear sky or a congestus site to turn into a stratiform site, for a deep convective or stratiform site to turn into a congestus site, nor for a stratiform site to turn into a deep convective site. Notice that the assumption that the transition rates depend only on the large-scale variables, which amounts to ignoring interactions between the lattice sites all together. This implies that the stochastic processes associated with the different sites are independent and statistically identical. Therefore, unless otherwise stated, in the remaining of the section, we drop the superscript i and consider only the generic process Xt with the transition probabilities Plk and transition rates Rlk . It follows immediately from Assumption 5 that R03 = R13 = R21 = R31 = R32 = 0.

(10.22)

200

10 Stochastic Birth and Death Models for Clouds

For fixed large-scale conditions, the stochastic process is a stationary Markov chain with the infinitesimal generator ⎡ ⎤ −R01 − R02 R01 R02 0 ⎢ −R10 − R12 R12 0 ⎥ R10 ⎥. (10.23) R=⎢ ⎣ 0 R20 − R23 R23 ⎦ R20 R30 0 0 −R30 Among all the physical quantities used to describe the state of the atmosphere, in a given large-scale numerical model, two are considered to be important for both triggering and maintaining tropical convection, i.e., for the formation and decay of the three cloud types (congestus, deep, and stratiform). These quantities are the convective available potential energy (CAPE) and the relative moisture content, i.e., moistness or rather dryness of the middle of the troposphere. In practice both CAPE and the atmospheric dryness are well defined functions of the large-scale moist thermodynamic variables. Here, we assume that both CAPE and dryness are two external parameters, denoted here by the letters C and D, respectively, normalized to take values between roughly 0 and 2. Let  1 − e−x if x > 0 Γ (x) ≡ (10.24) 0 Otherwise. Then, according to the assumptions 1, 2, 3, 4 given above, we define 1 Γ (C)Γ (D), τ01 1 R10 = Γ (D), τ10 1 R20 = (1 − Γ (C)), τ20 R01 =

1 Γ (C)(1 − Γ (D)), τ02 1 R12 = Γ (C)(1 − Γ (D)), τ12 R02 =

R23 = 1/τ23 ,

(10.25) R30 = 1/τ30 .

−1 Note, for instance, that R01 is zero when C ≤ 0 or D ≤ 0 and approaches τ01 when C and D are sufficiently large and positive, consistent with Assumption 1. Here, the τlk ’s are prescribed time scales of formation or decay of the corresponding cloud type or of conversion of cloud type l to cloud type k. There is no obvious way to choose their values. Based on physical intuition gained from observations, numerical simulations, and theory of tropical convection, the rule of thumb is that the cloud life time is on the order of hours, that the rate of cloud formation is much faster than that of their decay, and that stratiform clouds should decay much more slowly than either congestus or deep. We first consider the two extreme cases depicted in Table 10.1, to highlight some interesting features of the stochastic multicloud model parameterization. A systematic Bayesian method to infer these parameters from detailed cloud simulations and/or from radar data will be discussed and used in Section 10.5.

10.3 The Stochastic Multicloud Model

201

In (10.25), we assumed for simplicity that the stratiform generation and decay rates, R23 and R30 , are both independent of the large-scale parameters C, D. However, there is no physical reason why this should be the case and obviously, the results would be sensitive to such dependence. To illustrate this point we also consider, in addition to (10.25), an example where R23 increases slowly with CAPE, using the time scales associated with Case 2 of Table 10.1 R23 =

√ 1 Γ ( C). τ23

(10.26)

Table 10.1 Example of prescribed values of the time scale of formation or decay of each cloud type or of conversion of one cloud type to another. Time Description τ01 Formation of congestus τ10 Decay of congestus τ12 Conversion of congestus to deep τ02 Formation of deep τ23 Conversion of deep to stratiform τ20 Decay of deep τ30 Decay of stratiform

Case 1 1 hour 5 hours 1 hour 2 hours 3 hours 5 hours 5 hours

Case 2 3 hours 2 hours 2 hours 5 hours 0.5 hour 5 hours 24 hours

10.3.1 The Stationary Distribution, Cloud Area Fractions, and the Equilibrium Statistics of the Lattice Model The equilibrium distribution, Pe , of the multistate Markov chain Xt introduced above, is given by the left eigenvalue of the infinitesimal generator. We have Pe =



 1

R12 R01 R23 R12 R01 1 1 01 R , R , 1, R10R+R , + + 02 02 R +R R +R R R +R R +R 12 20 23 10 12 30 20 23 10 12 Z (10.27)

where Z is a normalization constant, so that the entries of Pe sum to one. Next, we define the area fractions σc , σd , σs occupied by clouds of type congestus, deep, or stratiform at any given time t, as the number of lattice sites for which Xt = 1, Xt = 2, Xt = 3, respectively, divided by the total number of sites N = n × n:

σc =

1 N 1 N 1 N 1{Xti =1} , σd = ∑ 1{Xti =2} , σs = ∑ 1{Xti =3} , ∑ N i=1 N i=1 N i=1

where

 1{Xti =k} =

1 if Xti = k 0 otherwise.

The clear sky area fraction is given by

σcs ≡ 1 − σc − σd − σs .

(10.28)

202

10 Stochastic Birth and Death Models for Clouds

In effect, the area fraction vector (σcs , σc , σd , σs ) is given by the probability distribution of the generic stochastic process Xt at time t. Therefore, the equilibrium distribution Pe in (10.27) yields the long-time statistical equilibrium for the filling fractions σc , σd , σs . We now use Monte Carlo to simulate the sequence of Markov chain’s Xti , i = 1, 2, · · · N, associated with each one of the lattice sites. We use Gillespie’s exact algorithm to simulate the birth-death process discussed in the previous section, where the transition times of the different sites are assumed to be independent exponential random variables. A maximum of two random numbers are thus generated at each iteration and for each lattice site, conditional on the states 0, 1, 2, 3. Recall that state 0 can change to state 1 or state 2, state 1 can go to either 0 or 2, and 2 can go to either 0 or 3, while state 3 can go only to 0. The first random number determines whether we can make a change or not and the second random number determines if we can go up or down, accordingly in the hierarchy of states. Only one random number is generated for state 3, since only one change (3 to 0) is permitted. As a test case, we let C = 0.25 and D = 0.75: a relatively moist middle troposphere with a moderate but positive CAPE value. Starting with a random initial lattice configuration, we integrate the stochastic lattice model for about 100 hours, with n = 20 and the typical time scales displayed in Table 10.1, Case 1. A snapshot (single realization at a fixed time) of the lattice state is shown in Figure 10.8(a), while the associated time series of the area fractions for each cloud type are shown in Figure 10.8(b), with the corresponding equilibrium values overlaid. Starting initially with a random lattice configuration, the cloud coverage fractions relax quickly to their corresponding equilibrium values and fluctuate around them with a significant variability of about 5% to 25% of the total area.

(A)

Time in hours =99.92

3 Stratiform

(B) 0.55 0.5

2.5

1.5 1 Congestus

Filling Fraction

2 Deep

0.45

Clear

0.4 0.35 Congestus

0.3 0.25

Stratiform

0.2 0.15

0.5 0 Clear

Deep

0.1 0.05

0

20

40

60

80

100

Time in hours

Fig. 10.8 An example of Monte Carlo simulation of stochastic multicloud model with n = 20,C = 0.25, D = 0.75, and the cloud time scales are as in Table 10.1, Case 1. (A) A snapshot picture of one typical lattice configuration and (B) time series of the total coverages associated with each cloud type with the equilibrium values overlaid (dashed lines).

10.3 The Stochastic Multicloud Model

203

10.3.2 Coarse Grained Birth-Death Stochastic Model and the Mean Field Equations Clearly, for a large number of sites of up to 100 × 100, the full Monte Carlo simulation of evolving the 100 × 100 Markov chains all at once is impractical. However, since in practice we do not need to know the microscopic configuration of the lattice but only large-scale macroscopic features such as the cloud fractions are needed. Here, we derive a multidimensional stochastic birth-death process for the three cloud species using a coarse graining methodology similar to the one used for the CIN model, though much simpler because local interactions are ignored. This exercise is straightforward in this case where local interaction was ignored. Let N = n×n be the total number of lattice sites. Let Nct be the number of congestus sites, Ndt the number of deep convective sites, and Nst the number of stratiform sites, inside the lattice, at any given time t ≥ 0. The number of clear sky sites is t = N − N t − N t − N t , by conservation of the total number of sites. Next, we comNcs c s d pute the (transition) probabilities for the numbers (random variables) Nct , Ndt , Nst to go up or down by one during the small interval of time (t,t + δ t]. We have N

i Prob{Nct+δ t ≥ k + 1/Nct = k} = ∑ Prob{Xti = 0} P01 + o(Δ t), i=1

i.e., the probability that the number of congestus sites goes up by at least one is the sum of all the probabilities that a given clear sky site will turn into a congestus site. Given that all the sites are identical, i.e., the transition probability Plki is independent of i. Moreover, as stated above, we have (see (10.28)) Nct = σct , N Nt Prob{Xti = 3} = s = σst , N Prob{Xti = 1} =

Ndt = σdt , (10.29) N Nt t Prob{Xti = 0} = cs = σcs , ∀i = 1, 2, · · · , N. N Prob{Xti = 2} =

Using the fact that the sites are independent, and the fact that the min of m exponential random variables of equal rates λ is an exponential random variable with rate mλ , we arrive at Prob{Nct+Δ t = k + 1/Nct = k} = Ncs P01 + o(Δ t) = Ncs R01 Δ t + o(Δ t). Similarly, we have N

i i Prob{Nct+Δ t = k − 1/Nct = k} = ∑ Prob{Xti = 1} (P10 + P12 ) + o(Δ t) i=1

= Nc (R10 + R12 )Δ t + o(Δ t),

(10.30)

204

10 Stochastic Birth and Death Models for Clouds N

i i Prob{Ndt+Δ t = k + 1/Ndt = k} = ∑ Prob{Xti = 0} P02 + Prob{Xti = 1}P12 + o(Δ t) i=1

= (Ncs R02 + Nc R12 )Δ t + o(Δ t), N

i i Prob{Ndt+Δ t = k − 1/Ndt = k} = ∑ Prob{Xti = 2} (P20 + P23 ) + o(Δ t) i=1

= Nd (R20 + R23 )Δ t + o(Δ t), N

i Prob{Nst+Δ t = k + 1/Nst = k} = ∑ Prob{Xti = 2} P23 + o(Δ t) i=1

= Nd R23 Δ t + o(Δ t), N

i Prob{Nst+Δ t = k − 1/Nst = k} = ∑ Prob{Xti = 3} P30 + o(Δ t) i=1

= Ns R30 Δ t + o(Δ t).

(10.31)

The stochastic processes Nxt , x = cs, c, d, s form a coupled system of birth-death Markov processes whose transition probabilities are given by (10.30) to (10.31), which can be easily evolved in time using Gillespie’s exact algorithm for which the cloud coverages are recovered according to (10.29), consistent with (10.28). In practice, we can also view this coupled birth-death system as a multistate/multivariable Markov chain undergoing one of the following seven transitions at a time: one congestus is formed from a clear sky, one deep is formed from a clear sky, one congestus is converted to deep, one deep is converted to stratiform, or one cloudy site of type 1,2, or 3 turns to a clear sky site. The associated transition probabilities are given by the original rates in (10.25) multiplied by the total number of sites that are subject to the given transition in a way which is consistent with the formulae (10.30) to (10.31). For example, the rate of transition from clear sky to congestus is Ncs R01 and the rate of conversion of congestus to deep is Nc R12 , etc. The vector (Nct , Ndt , Nst ) is in effect a multidimensional birth-death process with immigration for the three cloud populations. The birth rates are given by the spontaneous formation of congestus and deep clouds from clear sky sites, Ncs R01 and Ncs R0,2 , and the death rates are given by the natural decay rates of the three cloud types, Nc R10 , Nd R2,0 , Ns R30 . The rates of conversion from congestus to deep and from deep to stratiform, Nc R1,2 , Nd R2,3 , represent the rates of immigration from one population to another. We note that by construction of the multicloud model the birth rate of stratiform clouds and the immigration from deep to congestus, from congestus to stratiform, from stratiform to deep, and from stratiform to congestus are all set to zero. Accordingly, we can easily write down the multidimensional forward equations of this 3D birth-death process.

10.3 The Stochastic Multicloud Model

205

Let ε 1 = (1, 0, 0), ε 2 = (0, 1, 0), ε 3 = (0, 0, 1) denote the three canonical unit vectors of R3 and let Zt = (Nct , Ndt , Nst ) denote our three-dimensional birth-death process. Let i = (i, j, k) be a generic element of the state space {0, 1, · · · , N}3 and Pi,j denote the transition probability matrix of Zt . Then the backward equations are given by d Pi,j =Ncs R01 Pi+ε 1 ,j + Ncs R02 Pi+ε 2 ,j + Nc R12 Pi+ε 2 −ε 1 ,j dt + Nd R23 Pi+ε 3 −ε 2 ,j + Nc R10 Pi−ε 1 ,j + Nd R20 Pi−ε 2 ,j + Ns R30 Pi−ε 3 ,j − [Ncs (R01 + R02 ) + Nc (R10 + R12 ) + Nd (R23 + R20 ) + Ns R30 ] Pi,j i, j ∈ {0, 1, · · · N}3 .

(10.32)

We note that (10.32) is a very large system of differential equations of dimension N 3 . With a typical N = 40 × 40 sites this forms a ∼ (4 × 109 ) dimensional system. Though the solution is known closed form as the exponential of the infinitesimal matrix. Its actual computation is very difficult and even impossible using conventional methods. A sophisticated high performance computing technique is needed; although the infinitesimal generator is a very sparse matrix (only 8 entries are nonzero on each row), its exponential is a full matrix. Nonetheless, for the purpose of climate simulations, as for the case of the CIN model above, we do not actually need to compute the transition matrix Pi,j but we need only to evolve the three-dimensional process Zt using a Monte Carlo algorithm involving only the seven non-zero transition rates at each time step. We return to the backward Kolmogorov equations (10.32) in Section 10.5 and present an efficient solver for the multivariate distribution Pi,j (t), using the uniformization method to approximate the matrix exponential. The Gillespie’s exact algorithm version for simulating the 3D birth-death process, over one climate model time step [0, Δ T ], can be formulated as follows. Multidimensional Gillespie’s Exact Algorithm: 0) 1) 2) 3) 4)

Let λ = Ncs (R01 + R02 ) + Nc (R10 + R12 ) + Nd (R23 + R20 ) + Ns R30 . Let Zt = (Nc , Nd , Ns ) be the state of the system at time t, 0 ≤ t < Δ T . Generate a random number r1 uniformly from (0, 1). Set s = − λ1 ln(r1 ). If t + s > Δ T , then no transition occurs. Set t = Δ T . Stop. If t + s ≤ Δ T . Divide the interval into seven subintervals I1 , I2 , · · · , I7 of sizes Ncs R01 Ncs R02 Nc R10 Nc R12 Nd R23 Nd R20 Ns R30 , , , , , , , λ λ λ λ λ λ λ respectively.

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5) Generate a random number r2 . Select the subinterval Ik such that r2 ∈ Ik . Then make the corresponding transition as follows: – – – – – – –

If r2 ∈ I1 , then Nc = Nc + 1. If r2 ∈ I2 , then Nd = Nd + 1. If r2 ∈ I3 , then Nc = Nc − 1. If r2 ∈ I4 , then Nc = Nc − 1, Nd = Nd + 1. If r2 ∈ I5 , then Nd = Nd − 1, Ns = Ns + 1. If r2 ∈ I6 , then Nd = Nd − 1. If r2 ∈ I7 , then Ns = Ns − 1.

6) Set t = t + s. If t < Δ T go to 1. As one would expect, the dynamics of the area fractions obtained by evolving the full microscopic lattice model, described in the previous section, through the detailed description of each one of the stochastic processes, Xti , are statistically equivalent to those obtained by evolving the coarse grained birth-death processes just described. However, it is important to note that the computations are orders of magnitude cheaper in the latter case: Compare generating two random numbers and testing them against seven transition rates versus simulating each one of the n × n sites.

10.3.3 The Deterministic Mean Field Equations and Numerical Simulations As for all evolving physical quantities, we can use standard calculus to derive deterministic differential equations for the cloud coverages σc , σd , σs . According to the discussion above, we have the following three-by-three system of ODEs. In the jargon of stochastic modelling they are called mean field equations and can be obtained rigorously as the continuous limit when the number of lattice sites goes to infinity

σ˙ c = (1 − σc − σd − σs )R01 − σc (R10 + R12 ) σ˙ d = (1 − σc − σd − σs )R02 + σc R12 − σd (R20 + R23 ) σ˙ s = σd R23 − σs R30 .

(10.33)

Note that the growth and decay rates of the mean field variables in (10.33) are given, respectively, by the birth, death, and immigration rates in (10.32) that are simply normalized by the total number of sites N. This is a non-homogeneous linear system of ODEs with a unique equilibrium solution given by the stationary distribution in (10.27).

10.3 The Stochastic Multicloud Model

207

The stability properties of the mean field equations can be used to learn something about the behaviour of the stochastic system. In Figure 10.9, we plot the contours of the real and imaginary parts of the eigenvalues of the matrix (of the ode system) in (10.33) as functions of the parameters C and D, using the time scales from Table 10.1, Case 1. Recall from the standard theory of differential equations that an equilibrium point is said to be an asymptotically stable node if all the associated eigenvalues are real negative. Small perturbations of the equilibrium decay to zero and the equilibrium is recovered when t −→ ∞. If at least one eigenvalue is positive, then the equilibrium is an unstable node and small perturbations grow without bound. If the matrix admits complex eigenvalues, then the equilibrium is called an asymptotically stable spiral if the real parts of all the eigenvalues are negative. It is an unstable spiral if one complex eigenvalue has a positive real part. In the case of an asymptotically stable spiral, small perturbations of the equilibrium spiral in, towards the equilibrium position, while in an unstable spiral they spiral out, away from the equilibrium position; in the first case the solution exhibits decaying oscillations, while in the second it is characterized by oscillations that grow in amplitude. An equilibrium with pure imaginary eigenvalues is called a centre. In this case the solution is characterized by constant amplitude oscillations. As we see from Figure 10.9, the equilibrium of the mean field equations (10.33) goes from an asymptotically stable node to a stable spiral as C is increased from 0 to 2. In other words, this system bifurcates from an exponentially damped regime to an oscillatory damped regime; for large values of C, we have one real negative eigenvalue and a pair of complex conjugate eigenvalues with negative real part while for small values of C, we have three negative real eigenvalues. The imaginary part increases significantly with increasing values of C, especially for slightly moist conditions corresponding to D between 0.2 and 0.3. The damping strength is also sensitive to changes in C and D. An important non-dimensional number for the stochastic multicloud model is given by the ratio of the frequency to the damping rate, for the complex conjugate pair, plotted in Figure 10.9(E). In Figure 10.10, we display two time series of the area coverages obtained by evolving the stochastic model with D = 0.4 and the two different values of C = 0.1 and C = 1.5. According to Figure 10.9(E), the case C = 0.1 has a frequency-to-damping ratio near zero (below 0.1), while in the second case this ratio is above 0.6. As it is anticipated, the two time series are qualitatively different with the one corresponding to C = 1.5 having sharper peaks, while the one corresponding to C = 0.1 has much longer excursions. This suggests that a large frequency-to-damping ratio, in a complex conjugate pair for the mean field equations, would yield sharp and rapid oscillations, for the associated stochastic system, while a small ratio would yield smoother oscillations with much longer excursions from equilibrium.

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(A)

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Fig. 10.9 Equilibrium eigenvalues of the mean field equations. Panels (A), (B), and (C) represent the contours of the real parts of the three eigenvalues, respectively, as CAPE C (horizontal axis) and dryness D (vertical axis) are varied from 0 to 2, panel (D) shows the imaginary part of the complex conjugate pair, and panel (E) displays the ratio of the frequency over the damping rate. 0.8

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Fig. 10.10 Stochastic oscillations for both (a) when the frequency-to-damping ratio is small and (b) when it is large for the parameter values D = 0.4 and C = 0.1 and C = 1.5, respectively, and the τlk ’s are as in Table 10.1.

10.4 Coupling the SMCM to a Cumulus Parameterization

209

10.4 Coupling the SMCM to a Cumulus Parameterization As constructed, the SMCM provides a prognostic parameterization for the cloud area fractions of congestus, deep, and stratiform clouds. In principle, the SMCM can be coupled to any mass-flux based cumulus parameterization [e.g., 8, 305, 258]. As their name suggests this type of parameterizations simulates the vertical distribution of the mass flux of the convective updrafts and downdraft with the GCM grid box or column. Updrafts in particular are assumed to occupy a certain convective area σu of the grid box and the updraft mass flux Mu = σu wu , where wu is the vertical velocity of the updraft. To go around the inconvenience of not knowing both σu and wu , these early max flux cumulus schemes rely on the quasi-equilibrium assumption, where σu is assumed to be very small and that convection consumes large-scale instability (e.g., CAPE) at a time scale that is much faster than it is generated by the slowly evolving large-scale atmospheric state. The quasi-equilibrium assumption then yields a prognostic (or diagnostic [e.g., 212, 200]) closure for the updraft mass-flux terms of the large-scale GCM variables. The challenges associated with the quasi-equilibrium assumption are at least twofold. Firstly, a time scale separation between the large-scale flow and convection is not fully justified, especially concerning organized convection as the evolution of cloud populations of variable sizes responds progressively to the evolving large-scale state, low-to-mid level moisture, in particular. As such the cumulus parameterization needs to have memory in order to capture the progressive transition from shallow to deep convection [268, 135, 254] and thus fixing the area fraction for all cloud types even if they are arguably all small cannot be accurate. Secondly, the assumption that the updraft area fraction remains small breaks down when the cloud population becomes organized and important and may cover large portions of the grid column, this becomes evident for weather forecast model and for GCMs run at high resolution. The need for scale aware parameterization has been recently recognized by various investigators and resolutions at which the “scale separation” (if there is one) breaks down situated roughly between 4 km and 50 km, which is termed as grey zone [9, 77]; such resolutions are too coarse to fully resolve convection. Grids sizes as small as 100 m are needed in order to resolve all the mixing processes that occur, especially, at cloud boundaries. In essence, the SMCM can be used by any mass-flux based cumulus parameterization to overcome the quasi-equilibrium closure by re-introducing the missing subgrid variability due to the evolving cloud populations through the use of the readily available stochastic area fraction for the main cloud types which characterize organized convection. The SMCM can be an integral part of the cumulus scheme or run in parallel to inform the parameterization within each grid column as illustrated in Figure 10.11. Here, we favour the first strategy and present in detail the coupling of the stochastic birth-death model to the multicloud model parameterization presented in Part II of this monograph, in the context of a simple two

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baroclinic mode toy-GCM, in an aquaplanet GCM (HOMME), and in a full-fledged atmosphere-ocean coupled climate model, namely the National Centers for Environmental Predictions (NCEP) Climate Forecasting System (CFS). Nonetheless, some variants of the SMCM have been implemented in CGMs using the second strategy [42, 218]. More elaborate strategies on using effectively the SMCM to inform the cumulus parameterization not only to modify its closure assumption but also to improve the entrainment and detrainment rates as well as the evolving spectrum of the cloud type populations are under investigation by the author and his collaborators and will eventually appear in new research publications. We continue this section by describing the coupling of the SMCM to a toy atmospheric circulation model for the tropics and illustrate some of its few properties following the work of Frenkel et al. [52, 53]. The GCM results will be presented in the next chapters.

Stochastic Multi-cloud Model to inform cumulus parametrization: represent the missing sub-grid scale variability

Lattice Model for Convection GCM grid

Cumulus Parametrization

GCM: Large Scale Dynamics

Microscopic grid Stochastic Multi-cloud Model (for cloud area fractions) Lattice points 1-10 km apart. Occupied by a certain cloud type (congestus, deep, stratiform)or is a clear sky site.

Fig. 10.11 Implementation of the SMCM in a GCM. The left panel shows a diagram in which the SMCM is implemented in parallel with a pre-existing cumulus scheme and the right panel illustrated the division of each GCM horizontal grid cell (thick lines) into a microscopic lattice for the cloud model. Interaction across coarse-GCM cell is allowed when the SMCM with nearest neighbour interactions is utilized.

10.4.1 The SMCM in a Toy-GCM The strategy adopted in [111] and the many subsequent papers using the multicloud parameterization with prescribed heating profiles amounts to simply supersede, in the deterministic MCM in Part II, the switch function in (6.7), used to control the transition between congestus and deep convection based on the moistening of the mid-troposphere, by the area fractions of the corresponding cloud types. Accordingly, in the context of the two baroclinic mode model presented in (6.1)–(6.3), the closure equations for the heating and cooling rates associated with the three cloud types are assumed to be proportional to the corresponding area fractions [111, 52]

10.4 Coupling the SMCM to a Cumulus Parameterization

¯ c αα CAPEl+ Hm 1 (a1 θeb + a2 q − a0 (θ1 + γ2 θ2 ))]+ Hd = [σ¯ d Q¯ + τc (σd ) σ¯d 0 τc (σd ) = τ σd c 1 Hs = αs [σ¯ s Q¯ + (a1 θeb + a2 q − a0 (θ1 + γ2 θ2 ))]+ τc (σs ) σ¯s τc (σs ) = τc0 . σs

211

Hc = σc

(10.34)

Here, σ¯c , σ¯d , σ¯s are the mean (long-time average) values of the corresponding cloud area fractions and CAPEl is low-level CAPE. In essence, the cloud area fraction controls the convective time scales, τc (.), of the corresponding cloud types. While the details can be found in the cited papers, it is worthwhile mentioning a few more characteristic of the SMCM model. In [52], for instance, a few modifications have been accommodated to achieve optimal results. The low-level CAPE quantity was also used to set up the birth rate of congestus clouds, R01 . These modifications include the stratiform heating closure in (10.34) which is set proportional to CAPE [111]. With the scaled CAPE, C = CAPE/CAPE0 , low-level CAPE, Cl = CAPEl /CAPE0 , and dryness, D = θeb − θem /T0 , where CAPE0 = 200 J kg−1 and T0 = 10 K are the fixed reference values. Consistent with the approximations introduced in Chapter 5, CAPE and low-level CAPE (which is understood as the integral of positive buoyancy in the lower troposphere, below the freezing level) are given by CAPE = CAPE + R(θeb − γ (θ1 + γ2 θ2 )), CAPEl = CAPE + R(θeb − γ (θ1 + γ2 θ2 )), where CAPE and R are the dimensional constants [111]. The transition rates are as in (10.25) except for the birth rate of congestus which is set to R01 =

1 Γ (Cl )Γ (D), τ01

instead. Moreover, in [52] the time scales τ01 are scaled by a proportionality factor τgrid to allow a systematic dependence of the transition rates on the grid size. In [52], this parameter is varied between 1 hour and 16 hours and tested with the two grid resolutions of 40 km and 160 km. We set

τ01 = τ10 = τ12 = τgrid , τ02 = τ23 = τ20 = 3τgird , τ30 = 5τgrid .

(10.35)

While as in Table 10.1 the choice of the transition time scale is based solely on physical intuition, we note that according to the formulae above, the congestus time scales are assumed to be fasted and the stratiform decay time scale is the slowest, while the deep convection related time scales are intermediate. This is more or less in agreement with the time scales obtained by Peters et al. [217] who used radar data from Darwin and from Marshall Islands in the west Pacific Ocean to calibrate

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the SMCM. Nonetheless, a more systematic Bayesian strategy for inferring these parameters from data will be presented in Section 10.5. Figure 10.12 illustrates the behaviour of the SMCM when coupled to a column model version of the two-baroclinic model in (6.1)–(6.3). As we can see, the stochastic area fractions synchronize quickly with the large-scale variability and exhibit the expected transitions from congestus to deep to stratiform (third panel of Figure 10.12). As demonstrated in [52], one of the key features of the SMCM parameterization is the increase of variability and mean circulation in addition to the improved coherence and structure of both the simulated waves and mean circulation. In a series of ∗ − θ¯ mimicking the Indian Ocean/Western Pasimulations using a background θeb eb cific warm pool as in Figure 5.6 (see also Figure 6.10), Frenkel et al. [52] tested the SMCM model for various τgrid values at both high 40 km resolution and a coarse resolution of 160 km (typically used in comprehensive GCMs as in Chapter 10). While the structures of the corresponding mean circulations are consistent with the Walker cell in Figure 5.6, the wave disturbances are qualitatively similar with those in Figure 6.10 obtained with the deterministic multicloud model as they show a high degree of realism compared to the WHISHE waves discussed in Chapter 5 and display more variability when compared to those in Figure 6.10. A wave composite for the stochastic simulation is shown in Figure 10.13. We can see all the expected build in features of the multicloud model such as congestus heating prior to deep convection and progressive moisture building up prior to deep convection and so on. We also note the stochastic nature of the cloud population from the area fractions presented on the fourth panel. The sequence of congestus, deep, and stratiform cloud dynamics is clear; however, the transitions are stochastic in nature as all the cloud three types are present almost everywhere with various degrees of occurrences as in the stretched building block hypothesis [182] discussed above. Also, the convectively coupled wave structure such as the front to rear tilt in wind and temperature anomalies is evident. The mean of zonal and vertical winds and standard deviations of deep and congestus heating are reported in Table 10.2, where the SMCM results are contrasted with those obtained with the deterministic multicloud model of Chapter 5, using the same large-scale setting. Another important parameter which is varied in Table 10.2 is the number of microscopic site within each GCM grid box. As anticipated, Table 10.2 indicates that the stochasticization improves drastically both the mean and variance of the Walker circulation at both low and high resolution. Moreover, the sensitivity of the simulations to the parameters n and τgrid is evident. While increasing n decreases the variability of the heating while retaining the mean flow strength, as expected from a Monte Carlo simulation, changing τgrid is more subtle. First, we note that changing the resolution from 40 km to 160 km leads to a slight increase in the mean winds, while the standard deviation of the heating decreases significantly (especially for deep). However, the incremental increase of τgrid leads at first to an increase, leading to some critical value at τgrid = 4, and then a systematic decrease in both mean vertical velocity maximum and the standard deviation of Hd while that of Hc continuous to decrease. While this clearly

10.4 Coupling the SMCM to a Cumulus Parameterization

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Fig. 10.13 Wave composite of the typical convectively coupled wave for the stochastic model with a warm pool SST. The top four panels display the structure of the anomalies of indicated variables. The bottom two panels show the zonal-vertical velocity arrows overlaid on top of the (total) potential temperature anomalies and total heating perturbation, respectively. The composite is obtained by averaging one particular wave signal from Figures 10, 11, over the lifespan of the corresponding wave, following a reference frame moving westward (with the wave) at 17 m/s. From [52]

10.5 Inference of SMCM Parameters from Data

215

Table 10.2 Mean circulation strength and variability of heating fields for the stochastic and deterministic parameterizations under different scalings Scaling Warm pool max (U,W ) std(Hd ) std(Hc ) τgrid = 1, n=302 (10 m s−1 , 2 cm s−1 ) 2.14 K/day 2.83 K/day

Model Stochastic

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160 km τgrid = 1,n=1202

(12 m s−1 , 3 cm s−1 ) 1.34 K/day 1.89 K/day

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160 km

τgrid = 1,n=302

(12 m s−1 , 3 cm s−1 ) 1.67 K/day 2.41 K/day

Stochastic

160 km τgrid = 3, n = 302

(12 m s−1 , 4 cm s−1 ) 1.80 K/day 2.21 K/day

Stochastic

160 km τgrid = 4, n = 302

(12 m s−1 , 6 cm s−1 ) 1.96 K/day 2.07 K/day

Stochastic

160 km τgrid = 5, n = 302

(11 m s−1 , 5 cm s−1 ) 2.09 K/day 1.80 K/day

Stochastic

160 km τgrid = 6, n = 302

(11 m s−1 , 5 cm s−1 ) 1.66 K/day 1.35 K/day

Stochastic

160 km τgrid = 16, n = 302 (10 m s−1 , 3 cm s−1 ) 0.49 K/day 0.89 K/day

Deterministic 40 km

–

(4 m s−1 , 4 cm s−1 )

0.97 K/day 0.14 K/day

Deterministic 160 km

–

(5 m s−1 , 4 cm s−1 )

0.55 K/day 0.14K/day

demonstrates the sensitivity of the SMCM to the GCM resolution, the same mean climatology and variability level can be restored by the coarse resolutions when key parameterization parameters such as n and τgrid are tuned. This suggests that the SMCM is scale aware provided both the number of microscopic sites n × n and the transition time scales are carefully tuned. Indeed, as shown in [36] the sensitivity of the transition time scales inferred from data, though the Bayesian inference discussed in the next section, is indeed sensitive to the coarse graining cell size but the change in the τlk ’s is not uniform as some time scales are more sensitive than others.

10.5 Inference of SMCM Parameters from Data Regardless of how accurate the physical intuition on which a theory of any given natural phenomenon is based, there are always difficulties in measuring directly key parameters which depend heavily on how accurate the instruments used are. The difficulty can be made heavily worse when the intuition is in doubt due to the lack of physical understanding as in the case of cumulus parameterization. As already mentioned, a set of important key parameters for the SMCM that cannot be obtained by direct measurement are the transition time scales τkl , assuming that

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the Arrhenius type activation function in (10.24) and its large-scale arguments (the choice of predictors, i.e., CAPE and Dryness and their respective normalization factors) constitute a good enough guess. Indeed Dorrestijn et al. [42] used a radical approach where no actual functional dependence of the SMCM transition rates was assumed. Instead, the transition probabilities are inferred directly from data. While such heavily data driven methods can be appealing, they demand large amounts and highly accurate datasets that can be provided only through large-eddy simulations. Observations are still too coarse and too sparse for such methods to work adequately on the global scale. Peters et al. [217] found that the SMCM reproduces with great fidelity the stochastic behaviour of observed cloud and rainfall variability in two different datasets when the vertical velocity was used as a predictor instead of CAPE (see also [218]). To calibrate the tau parameters of the SMCM, Peters et al. [217] produced from radar data equilibrium distributions of the relevant cloud type area fractions in terms of the large-scale predictors (large-scale vertical velocity and dryness– saturation deficiency) and were able to visually fit, qualitatively and quantitatively, those of the SCMC by inferring the tau values, using the trial and error method. Independently on whether the large-scale predictor which is supplemented with dryness was chosen to be CAPE, low-level CAPE, or vertical velocity, the time scales they obtained are consistent with the prevailing intuition as they are all between a few minutes and a few hours (see last column of Table 10.3). One of the key findings in Peters et al. [217] is that tropical convection consists of two regimes: A regime of strong and organized convection with a large mean rain rate and a regime of weak and disorganized convection characterized by a weak mean rain rate. The first regime with strong mean rates has a very low variance, thus mainly deterministic, i.e., convection is tied to large-scale variability, while the second regime with weak mean rain rates has a large variance, thus highly stochastic. As illustrated in Figure 10.14, the SMCM captures this highly counter-intuitive behaviour very well which is at odds with linear noise models which tend to increase the variance with increasing mean that are typically used by some of the early stochastic parameterizations of convection [20, 153]. De La Chevroti`ere et al. [35] developed a Bayesian procedure to systematically infer the parameters τkl of the SMCM from data. The method was successfully used in [36] for the case of large-eddy simulation of a field campaign case study of tropical convection over the tropical Atlantic ocean, namely the Giga-LES dataset of Khairoutdinov et al. [108]. In the remainder of this section, we will summarize this methodology and highlight some of the Giga-LES inference results.

10.5 Inference of SMCM Parameters from Data

217

Fig. 10.14 Histograms of mean cloud area fractions and the corresponding standard deviations in % at two tropical observational sites (Darwin, Australia, and Kwajalein, Marshall Islands) in terms of large-scale indicators, relative humidity based dryness DRH , and vertical velocity indicator Cw (see [217] for details). Top panels are from radar observations and bottom panels are those obtained from the cloud area fraction time series as produced by the SMCM when forced with the observed large-scale indicators. Adopted from [217].

10.5.1 The Bayesian Inference Procedure The SMCM simulates the evolution of the cloud populations x = (Nc , Nd , Ns ) constrained by the large-scale atmospheric state indicators: u = (C,Cl , D). Let xt and ut be the time series of the observed cloud fractions and observed large-scale indicators, respectively. For simplicity in exposition, the seven transition time scales we wish to infer from these data are stacked in a vector

θ = (τ01 , τ10 , τ12 , τ02 , τ23 , τ20 , τ30 ).

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While the SMCM characterizes the behaviour of the future observations of X conditional on θ , a statistical inference method allows instead to deduce from observations x of X and the statistical distribution of the parameter vector θ . An elegant solution to this inverse problem is provided by the Bayesian paradigm. The Bayesian approach incorporates the initial information and residual uncertainty about the model parameters θ into a prior distribution π (θ ), which is then updated by the model likelihood function f (xt |θ ) to formulate a posterior distribution π (θ |xt ) of the parameters given the data [228]:

π (θ |xt ) = 

f (xt |θ )π (θ ) . f (xt |θ )π (θ )d θ

(10.36)

The inference problem then translates to finding the probability distribution of θ conditional on x as defined by (10.36). Where f (xt |θ ) is the probability density distribution of xt given the set of parameters θ known as the likelihood function [228]. Conditioning further on ut , the posterior is given (up to a proportionality constant) as π (θ |xt , ut ) ∝ f (xt |ut , θ )π (θ ). (10.37) The likelihood function f (xt |ut , θ ) is obtained by solving the forward equations (10.17). Because the SMCM is a Markov process the likelihood function factors out and each factor can be computed under the assumption that the largescale predictor is time independent allowing to obtain the solution in explicit form. In the Giga-LES data, cloud data is available every 15 minutes (Δ t = 15 min) and over such a short period of time, the large-scale variable u is effectively approximately constant [35]. We have [35, 36] T

f (x1:T |u1:T , θ ) = ∏ ft−1 (xt |xt−1 , ut−1 , θ ) t=1 T

= ∏ 1∗φ (dt−1 ,Nt−1 ,Nt−1 ) exp[R(ut−1 , θ )Δ t]1φ (dt ,Nt ,Nts ) . t=1

d

s

d

(10.38)

Here Δ t is the time increment between successive observations (xt−1 , ut−1 ) and (xt−1 , ut−1 ) and 1∗φ (·) is the transpose of 1φ (·) , the canonical unit vector in R|S | that takes 1 at the index corresponding to φ (·), and 0’s everywhere else, with 3 2 2 φ (d, b, c) = d6 + d2 + d3 + db − b2 + 3b 2 + c for non-negative integers 0 ≤ d ≤ N, 0 ≤ b ≤ d, and 0 ≤ c ≤ d − b, and d = Nct + Ndt + Nst . The function φ : S → N maps each such triplet in the subset S of R3 : 0 ≤ c ≤ d − b, 0 ≤ b ≤ d, 0 ≤ d ≤ N, to a counting order (address), which is needed to automate the construction of the large but sparse matrix R [35].

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219

It is easy to see that dim(R) = O(N 3 ), and thus the size and memory requirements of increase and quickly become prohibitively large with increasing cloud lattice size. A parallel version of a preconditioning technique known as the uniformization method was developed in [35] to allow for fast, numerically stable, and scalable approximations of the large matrix exponential. The sampling of the posterior distribution is done with the Monte Carlo Markov Chain technique (see [35] and the references therein).

10.5.2 The Giga-LES Inferred Time Scales The Giga-LES dataset is a 24-hr long LES of deep tropical convection over a domain of 204.8 km in both horizontal directions and about 27 km in the vertical, which uses the mean sounding and forcing observed during the GATE (GARP Atlantic Tropical Experiment) Phase III experiment over the Atlantic Inter-Tropical Convergence Zone (ITCZ) [108]. The atmospheric fields are available every 15 minutes at all 2048 × 2048 × 256 grid points of the three-dimensional space. A full description of the simulation setup including the idealized mean GATE initial thermodynamic profiles and large-scale forcing is found in [108]. The top-right panel of Figure 10.15 presents a visualization of the cloud scene over the whole 204.8×204.8 km2 domain at hour 13. The scene illustrates complex convection activity, with individual deep clouds and a mesoscale cloud system dominated by stratiform anvils, surrounded by smaller congestus and shallow clouds. The bottom-right panel shows the evolution of the domain averaged liquid and ice water, which is characterized by a spin up period of a few hours before convection gets started and starts producing convectively active and suppressed periods. Excluding a transient period of approximately 4 to 5 hours, the length of the time series available for analysis is between 76 and 80 data points. De La Chevroti`ere et al. [36] subdivided the 205 × 205 km2 Giga-LES domain into m × m subdomains (Figure 10.15 top-left), and time series of cloud populations and large-scale convection indicators are obtained for each subdomain (Figure 10.15 bottom-left). The purpose of this partitioning is twofold. It allows to increase the information content of the data by elongating the time series and permits to obtain and compare transition time scale parameters on two different coarse domain sizes, 100×100 km2 and 50×50 km2 , respectively. To obtain the cloud area fractions associated with each cloud type, the cloud condensate, shown in the bottom of Figure 10.5, is projected onto three piece-wise constant basis functions of vertical structure taking the value one in the lower troposphere and zero aloft, one in the upper troposphere and zero below, or one along the extent of troposphere, representing congestus, stratiform, and deep convective clouds, respectively. The corresponding times series as illustrated on the bottomright panel of Figure 10.15. As we can see, the cloud area fraction time series exhibits clear transitions from congestus to deep to stratiform consistent with the design principle of the SMCM.

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Fig. 10.15 Top-right: cloud scenery of the Giga-LES simulation, shown in white colour is the vertically integrated liquid water. Top-left: Partitioning of the computational domain into 4 or 16 subdomains. Bottom-right: time evolution of domain averaged cloud condensate (liquid and ice water). Bottom-left evolution of the congestus, deep, and stratiform area fractions proxied from the projection of the simulated cloud condensate onto judiciously chosen piece-wise constant basis functions for the 2 × 2 domain partition. From [36].

De La Chevroti`ere et al. [36] applied the Bayesian inference procedure above to the Giga-LES times series in Figure 10.15 for both the 2 × 2 and the 4 × 4 partitions where the sequential learning strategy was adopted; the Bayesian algorithm is run with the data in each subdomain of the partitions considered in sequence so that the posterior from the previous subdomain is the prior of the succeeding one [35]. The sequences of prior and posterior distributions of the transition time scale parameters are reported in Figure 10.16 for both the 2 × 2 and 4 × 4 domain partitions. The mean and variances of the converged parameter distributions are reported in Table 10.3 for both subdomain partitions. As we can see from Table 10.3, all the mean transition time scales appear to be smaller in the fine 4 × 4 partition but τ01 seems to increase by roughly 15% (from 27.686 hours to 31.789). Also the amount by which the majority of time scales decreases varies considerably among the τi j ’s. While the physical meaning of this behaviour is not understood, it warrants against

10.6 SMCM with Nearest Neighbour Interactions

221

the use of a simple proportionality factor τgrid varying with the GCM grid resolution. Nonetheless, it is interesting to note that the variance is consistently smaller for all parameters with the 4 × 4 partition. This is probably due to the longer time series. De La Chevroti`ere et al. [36] demonstrated that the Bayesian inference procedure is sensitive to key parameters of the SMCM such as the normalization values of CAPE and dryness, CAPE0 , and T0 but it is hardly sensitive to the prior distribution provided that the training times series is long enough. While the results shown in Figure 10.16 were obtained with uniform priors, a rerun conducted with Gaussian priors using the transition time scales of Peters et al. [217] led to almost identical results [36]. Table 10.3 Giga-LES inferred transition time scales associated with the 2 × 2 and 4 × 4 partitions. The last column shows the τ parameters used by Peters et al. [217]. Adopted from [36]. Parameter

τ01 τ10 τ12 τ02 τ23 τ20 τ30

Mean (SD) [hours] 2 × 2 Partition 4 × 4 Partition Peters et al. (Formation of congestus) 27.686 (8.233) 31.789 (4.795) 1 (Decay of congestus) 7.426 (11.155) 1.761 (0.224) 1 (Conversion of congestus to deep) 0.208 (0.006) 0.238 (0.001) 3 (Formation of deep) 17.950 (3.507) 11.821 (0.211) 4 (Conversion of deep to stratiform) 0.359 (0.001) 0.2570 (0.0001) 0.13 (Decay of deep) 10.126 (15.674) 9.551 (13.146) 5 (Decay of stratiform) 1.444 (0.021) 1.021 (0.002) 5

10.6 SMCM with Nearest Neighbour Interactions To include local interactions between convection sites and cloud types directly at the microscopic scale, i.e., in addition to those occurring conditional on the large-scale indicators such as CAPE and mid-tropospheric dryness, the author [110] introduced a multi-type interacting particle Hamiltonian, as a generalization of the Ising model used for CIN in Section 10.2. The representation of such cloud-cloud direct interactions would allow a direct representation of self-organization of cloud clusters at the mesoscale, occurring in nature, for example, over the eastern Pacific ITCZ [43, 124], in coastal regions, and over the tropical forests of Amazonia and Africa, that are induced by unresolved processes such as eddies due to barotropic and baroclinic instability during the ITCZ break down [51, 298], sea-breezes around small Islands [12], and the diurnal cycle of convection [297, 60, 262], respectively. While the details of the SMCM with local interactions including its coarse grained approximation are found in [110], here we provide a brief overview to foster interest in its application to parameterize convective organization at the subgrid scale in circumstances such as those enumerated above.

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Fig. 10.16 Sequence of posterior and prior distributions associated with the sequential Bayesian inference procedure when applied to the Giga-LES data using a 2 × 2 ( top set of panels) and 4 × 4 domain partitions (bottom panels). From [36].

10.6 SMCM with Nearest Neighbour Interactions

223

10.6.1 The Multiple Particle Hamiltonian We begin by introducing the multiple particle Hamiltonian to account for the energy due to local interactions between lattice sites at the microscopic level 3

H(X,U) =

3

∑ ∑ Ek,l (X) + gk (U)X

k=1 l=k N

Ek,l (X) = −

∑

Jk,l (|i − j|)1{X i =k} 1{X j =l} , 1 ≤ k ≤ l ≤ 3,

(10.39) (10.40)

i, j=1,i= j

where Jk,l is the location interaction potential between a particle (or cloud) of type k and a particle of type l, and gk (U) are the external potentials which depend on the large-scale indicators denoted by the generic variable U. For the case of nearest neighbour interactions considered in [110], we have √ Jk,l (r) = 0 ⇐⇒ r = 1 or 0 < r ≤ 2, depending whether the corner neighbours are counted for or not. Khouider [110] introduced the interaction matrix J = [Jk,l ] as a parameter set to be tuned or inferred from data. Then by enforcing “partial” detailed balance on the particle stochastic system (Xt )t>0 in (10.19), i.e., detailed balanced at each site, ignoring transition occurring at all other sites, the transition rates at each fixed site j = 1, 2, · · · , N satisfy the following equations: R02 + R01 =

ρ1 ρ2 ρ3 R10 eH0 −H1 + R20 eH0 −H2 + R30 eH0 −H3 ρ0 ρ0 ρ0

ρ1 (R12 + R10 )eH0 −H1 ρ0 ρ2 ρ1 R02 = (R20 + R23 ) eH0 −H2 − R12 eH0 −H1 ρ0 ρ0 ρ3 R30 = R23 eH2 −H3 , ∀ j = 1, · · · , N, ∀X ∈ Σ , ρ2 R01 =

(10.41)

where Hk is the Hamiltonian associated with the configuration X such that X j = k and Σ = {0, 1, 2, 3}N as the state space, i.e., the set of all possible lattice configurations with N = n2 is the lattice size. Notice that only the energy differences appear in the transition rate equations the absolute value of the Hamiltonian energy is unimportant. As in the Ising model, a transition is favoured (i.e., its probability is increased) when the Hamiltonian decreases afterwards.

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10 Stochastic Birth and Death Models for Clouds

By introducing a background distribution that relates solely to the large-scale dynamics as in (10.27), the new transition rates are obtained by solving the above equations to obtain Rk0 = R˜ k0 , k = 1, 2, 3 R12 = R˜ 12 eH1 −H2 ρ1 R01 = (R˜ 12 + R˜ 10 ) eH0 −H1 = R˜ 01 eH0 −H1 ρ0 ρ3 1 R02 = (ρ2 R˜ 20 − ρ1 R˜ 12 )eH0 −H2 + R˜ 30 eH0 −H3 ρ0 ρ0 ρ3 ˜ H2 −H3 R23 = R30 e = R˜ 23 eH2 −H3 , ρ2

(10.42)

where R˜ kl are the background rates depending solely on the large-scale indicators as in (10.27). In particular, the death rates (i.e., transitions back to clear sky) depend only on the environmental large-scale variables as in (10.25). Straightforward computations show that the Hamiltonian energy jumps are explicitly given by N

∑

(H0 − H1 ) j =

J11 (|i − j|)1{X i =1} + J1,2 (|i − j|)1{X i =2} + J1,3 (|i − j|)1{X i =3}

i=1,i= j N

∑

(H0 − H2 ) j =

J22 (|i − j|)1{X i =2} + J1,2 (|i − j|)1{X i =1} + J2,3 (|i − j|)1{X i =3}

i=1,i= j N

∑

(H0 − H3 ) j =

J33 (|i − j|)1{X i =3} + J1,3 (|i − j|)1{X i =1} + J2,3 (|i − j|)1{X i =2}

i=1,i= j N

∑

(H1 − H2 ) j =

J22 (|i − j|)1{X i =2} − J11 (|i − j|)1{X i =1} + J1,2 (|i − j|)×

i=1,i= j



 1{X i =1} − 1{X i =2} − J1,3 (|i − j|)1{X i =3} + J2,3 (|i − j|)1{X i =3} (10.43)

(H2 − H3 ) j =

N

∑

J33 (|i − j|)1{X i =3} − J22 (|i − j|)1{X i =2} − J1,2 (|i − j|)1{X i =1}

i=1,i= j



+ J1,3 (|i − j|)1{X i =1} + J2,3 (|i − j|) 1{X i =2} − 1{X i =3}

We note that the dependence on the external potentials gk (U) is accounted for in the background rates R˜ kl .

10.6 SMCM with Nearest Neighbour Interactions

225

This completes the definition of the SMCM with local interaction. Though, it is worthwhile noting that the “partial” detailed balance equations guarantee that the stochastic process Xt preserves the Gibbs measure and its equilibrium distribution

μ (Xt ) ∝ e−H(Xt ) .

10.6.2 Coarse Grained Approximation To obtain approximate dynamics for the coarse grained birth-death process yielding the evolution of cloud types area fractions, σc , σd , σs , on the GCM grid cell, directly without going through the microscopic lattice, the coarse grained Hamiltonian is defined as the conditional expectation of the microscopic Hamiltonian given the coarse grained averages of Xt (or area fractions) on the coarse mesh, denoted by X¯t : 3

¯ X) ¯ = E[H(X)|X] ¯ = H(

3

3

M

k=1

i=1

¯ + ∑ gk (U) ∑ Nki , ∑ ∑ E[Ekl (X)/X]

k=1 l=1

where 3

¯ =−∑ E[Ekl (X)/X]

N

3

N

∑∑ ∑

k=1 j=1 l=1 r=1,r= j

 E Jkl (| j − r|)1{X j =k} 1{X j =l} /X¯ .

By assuming that the particles, within each coarse cell, are uniformly redistributed2 on the microscopic level, we arrive at 3 3 M   ¯ = − 1 ∑ ∑ ∑ Jkl0 nb (q − 2)2 + 4(nb − nb )(q − 2) + 4n1 Nli E[Ekl (X)/X] Q2 k=1 l=1 i=1  



+ nb (q − 2) + n2 Nlis + Nlin + Nlie + Nliw + n3 Nlisw + Nlise + Nlinw + Nline Nki ,

where Nkix is the number of particles (or cloud sites) of type k that are contained in the neighbour x of the coarse cell i under-consideration, where e, w, s, n denote east, west, south, and north neighbour, respectively, while se, sw, nw, ne are, respectively, the south-east, south-west, north-west, and north-east corners.

10.6.3 Mean Field Equation As in the interaction free case, mean field equations for the area fractions can be derived from the expected values in the limit when both the number or microscopic and coarse lattice sites go to infinity [110]. We have 2

This is the main approximation used here in order to obtain coarse grained dynamics in closed form. Nonetheless, as demonstrated by numerical tests the coarse grained model preserves the bulk statistics of the microscopic mode.

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10 Stochastic Birth and Death Models for Clouds

∂ σ1 = α01 σ0 − (α10 + α12 )σ1 ∂t ∂ σ2 = α02 σ0 + α12 σ1 − (α20 + α23 )σ2 ∂t ∂ σ3 = α23 σ2 − α30 σ3 , ∂t

(10.44)

where αkl (x,t)σk is the formal limiting value of the infinitesimal rates associated with the transitions k to l. Here σ0 is the clear sky area fraction and σ1 , σ2 , σ3 are, respectively, the area fractions of congestus, deep, and stratiform cloud types. In particular, we have

αk0 = R˜ k0 , k = 1, 2, 3, α01 = R˜ 01 eΓ10 , α12 = R˜ 12 eΓ12 , α23 = R˜ 23 eΓ23 , ρ3 1 α02 = (ρ2 R˜ 20 − ρ1 R˜ 12 )eΓ20 + R˜ 30 eΓ30 , (10.45) ρ0 ρ0 where [84] 3

Γ0k (x) = ∑ Jkl ∗ σl (x) and Γkl (x) = Γ0l (x) − Γ0k (x),

(10.46)

l=1

with f ∗ g(x) =



f (x − y)g(y) dy as the convolution operation.

10.6.4 Numerical Testing The SMCM with local interactions is tested offline (i.e., without coupling it to a large-scale atmospheric model) in [110] and the microscopic (q = 1), the coarse grained, and the mean field limit were compared to each other for both the case when local interactions were enforced and the case when local interactions were ignored. The coupling of the SMCM with local interactions to an atmospheric model remains a future research project. With the parameter values provided in Table 10.4, the different models were run for a long period of time, until statistical equilibrium. We note that with n = 20 and q = 10 we have a lattice of 20 × 20 microscopic sites divided into 4 coarse cells having 10 × 10 microscopic sites each. Figure 10.17 shows the snapshots obtained with the microscopic model with and without local interactions. In addition to the random nature of the two configurations, the case with local interactions shows clear aggregations or congestus sites (light blue) due to the fact that J11 = 0.25 is the largest interaction potential value.

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227

Table 10.4 List of parameter values used in the numerical simulations. See Section 10.3 for physical meanings.

τ10 = 5 hours τ20 = 5 hours τ30 = 5 hours τ12 = 2 hours τ01 = 2 hours τ02 = 2 hours τ23 = 3 hours C = 0.25 Convection potential D = 0.5 Tropospheric dryness h10 = Γ (D)/τ10 h20 = (1 − Γ (C))/τ20 h30 = 1/τ30 h12 = Γ (C)(1 − Γ (D))/τ12 h23 = Γ (C)/τ23 h01 = Γ (D)Γ (C)/τ01 h02 = (1 − Γ (D))Γ (C)/τ02 Γ (x) = 1 − e−x if x > 0 ⎡ 0 otherwise ⎤ 0.25 0 0 J0 Interaction matrix ⎣ 0 0.125 0.05 ⎦ 0 0.05 0.125 n = 20 q = 10 nb = 8

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Fig. 10.17 Snapshot of the multicloud microscopic lattice model. (a) With local interactions and (b) without local interactions. n = 40. From [110].

In Figure 10.18, we show the cloud area fraction time series obtained with the various models. In all cases the deterministic mean field limit converges to a stationary equilibrium around which the stochastic models fluctuate. Also the convergence to equilibrium occurs fairly fast at the same time scales, for both stochastic and deterministic models. More importantly, despite the associated enormous computational gain, the coarse grained model provides similar statistical behaviour as the more expensive microscopic version. Moreover, local interactions can boost the stochastic dynamics by exhibiting more and larger intermittent excursions from the equilibrium as seen, for example, in the congestus (green) time series in panels (a) and (c). Also, the local interactions substantial deviate the mean climatology from the large-scale background as the converged mean field solution separates significantly from its background equilibrium solution (compare panels a) and b) with panel c). There is way more equilibrium cloud fractions (especially congestus, as expected) in the case with local interactions (a and b) then in the case without local interactions (c).

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Fig. 10.18 Time evolution of cloud area fractions for the case of local interactions using (a) the microscopic lattice model and (b) the coarse grained process and (c) for the case without local interactions. Solid: stochastic process, Dashes: mean field equation, Dots: prior distribution. From [110].

Cloud couverage

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228 10 Stochastic Birth and Death Models for Clouds

Chapter 11

Implementation of the SMCM in a Global Climate Model

Here, we discuss the implementation of the stochastic multicloud model (SMCM) presented in the previous chapter in comprehensive climate models. First, we look at the case of an aquaplanet setting using the HOMME atmosphere only, dynamical core used in Chapter 9 and then consider, in Chapter 12, the case of a more involved state-of-the-art ocean-atmosphere coupled model, used in actual-operational climate predictions, namely, the National Centers for Environmental Predictions (NCEP)’s Climate Forecasting System (CFS). As portrayed by the results presented below important improvements in terms of the simulation of the MJO and convective coupled waves as well as the monsoon variability are achieved through the addition of stochasticity to the HOMME-MCM aquaplanet simulations presented in Chapter 9. Moreover, the implementation of the SMCM in CFS resulted in unprecedented improvements in terms of the simulation of the tropical modes of atmospheric variability on synoptic and intra-seasonal scales in coarse resolution GCMs. We thus start by presenting the improved HOMME-SMCM aquaplanet simulations both for equatorial warm pool and for a monsoon-like configuration as done in Chapter 9. In particular, we will discuss the importance of stratiform clouds (in terms of diabatic heating and rainfall) for the large-scale organization of convection. We then discuss the implementation of the SMCM in CFS and present some results to showcase the superiority of the SMCM parameterization in capturing the key tropical modes of variability as observed in nature.

11.1 SMCM-HOMME Aquaplanet To implement the SMCM in HOMME, Deng et al. [37] choose the straightforward route of simply replacing the closure equations for the heating and cooling rates associated with the three cloud types, as presented in Chapter 9, with their stochastic equivalents discussed in the previous chapter (c.f. equation (10.34)). As discussed in Chapter 10, there are many ways to use the SMCM for the parameterization of © Springer Nature Switzerland AG 2019 B. Khouider, Models for Tropical Climate Dynamics, Mathematics of Planet Earth 3, https://doi.org/10.1007/978-3-030-17775-1 11

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11 Implementation of the SMCM in a Global Climate Model

cumulus convection in a climate model and this is perhaps the simplest and more conservative way as it builds on the success of the HOMME-MCM results shown in Chapter 9. Thus, without further due, we discuss the SMCM-HOMME simulations as reported in three main research articles.

11.1.1 Case of a Uniform Forcing The first results of the SMCM-HOMME model were reported in [37]. Specifically, the deterministic aquaplanet simulation discussed in Section 9.3, without doubling the moisture background constants and with αs = 0.5, is repeated to assess the effect of stochasticity on the parameterization of organized convection. The reader is reminded that for this particular parameter regime, the deterministic HOMME-MCM simulation consisted of superposition of convectively coupled Kelvin, Rossby, and westward inertia-gravity waves perpetually circling the globe (c.f. Figure 9.4). To obtain an MJO-like signal the deterministic model required the doubling of the moisture gradient. The associated Hovm¨oller diagram of the Deng et al. simulation is reported in Figure 11.1. The last 500 days of a 200-day simulation are shown. Unlike the deterministic simulation, the stochasticity somehow helps the emergence of intermittent MJO-like disturbances somewhat resembling nature. As we can see from the time series in Figure 11.1, the solution undergoes frequent regime change sometimes displaying single wavenumber 1 MJO disturbances lasting between 50 and 100 days (e.g., between 1600 and 1700 days) to episodes dominated by wavenumber 2 packets of two successive MJOs (e.g., around simulation time 1850– 1900 days). Also, streaks of meso- to synoptic scale westward moving waves are clearly visible within the MJO envelope, particularly in the low-level zonal winds and deep convection contours. This is consistent with the observed multiscale organization of convection suggesting two-way interactions between the planetary and meso- and synoptic scale systems through momentum transport and modulation of thermodynamic variables. Also, another manifestation of the stochasticity appears from the fact that the individual MJO events move at different speeds, varying from 4 to 10 m s−1 , and larger on rare occasions. However, as reported in the histogram in Figure 11.2, the distribution of the MJO phase speeds peaks at roughly 5 m s−1 , which is the typically observed MJO propagation speed. Beside the fact that the MJO-like disturbances possess most of the key observed features of the MJO as was the case for the deterministic simulation in Chapter 9, it is worth saying that some key statistics of the observed rainfall and moisture data are also reproduced. As reported in Figure 11.3, rainfall decorrelates at a faster rate than moisture, a 1–2 hours versus 20–25 hours, consistent with observations [206, 220, 221, 243]. It is interesting to note that the decorrelation of moisture typically increases significantly with length scale (It takes 15 hours to reach 0.6 when the

11.1 SMCM-HOMME Aquaplanet

231

Fig. 11.1 Hovm¨oller diagram for uniform aquaplanet HOMME-SMCM simulation. The plots correspond to meridional averages (between 10°S and 10°N) of low-level and upper-level zonal winds and deep convective heating. The stochastic parameters are as in equation (10.35) with 20 and τgrid = 1 hour. Compared with the result obtained with the deterministic HOMME-MCM in Figure 9.4. Adopted from [37].

original GCM-grid data is used v.s. 20 hours to reach 0.6 when the data is group in 5 × 5 grid box averages), while the decorrelation of rainfall is essentially insensitive to the averaging operation. This is a clear indication that large-scale organization of convection is mainly controlled by moisture and rainfall is essentially a stochastic process responding to smooth moisture variations. Also, there is a clear sensitivity of the decorrelation times to changes in the transition time scales (compare the left v.s. the right panels of Figure 11.3). While the decrease of the rainfall decorrelation time with decreasing transition time scales (τgrid = 4 hours v.s. τgrid = 1 hour) is somewhat expected, the decrease of that of moisture results from the strong two-way interactions between moisture and rainfall. Consistently, Majda and Stechmann [178] have a successful MJO theory based on a predator-prey oscillator-like behaviour between rainfall and low-level moisture via a convective-activity function.

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11 Implementation of the SMCM in a Global Climate Model

Fig. 11.2 Histogram of the MJO propagation speed of the HOMME-SMCM simulation. The red line represents a gamma-like distribution fit. Adopted from [37].

Fig. 11.3 Decorrelation times of moisture and precipitation in the HOMME-SMCM simulations. Top are for original grid data and bottom are for data averaged on 5 × 5 grid box areas. The panels on the right are obtained with τgrid = 1 hours and those on the left correspond to τgrid = 4 hour. Adopted from [37].

11.2 Role of Stratiform Heating on Organized Convection

233

11.2 Role of Stratiform Heating on Organized Convection 11.2.1 MJO and Kelvin Waves Deng et al. [38] considered the case of a warm pool like sea-surface temperature profile in the aquaplanet setting for the SMCM-HOMME climate simulations, as done in [3] and reported at the end of Chapter 9. While searching for the most adequate parameter regime for the realistic simulation of organized convection, they noticed an acute sensitivity of the results to stratiform heating parameters. Here, we illustrate this behaviour and discuss its implication on the parameterization of organized convection. Deng et al. [38] discovered that for the case of a warm pool SST profile, the SMCM transition time scales have to be substantially tuned to obtain satisfying results. More importantly important regime changes were observed with regards to changes in the stratiform death time scale τ30 . In the standard parameter regime, we have

τ01 = 40τgrid , τ10 = τ12 = τgrid , τ02 = 4τgrid , τ23 = τ20 = 3τgird , τ30 = 5τgrid , (11.1) where τgrid = 1 hour in this subsection. We note that the large value used for the birth of congestus cloud, τ01 = 40τgrid , compensates for the use of strong warm forcing leading to large CAPE values that would otherwise lead to an exaggerated congestus heating at the expense of deep convection and large-scale organization. We note that in the standard regime, the stratiform heating fraction is still kept at αs = 0.5. In Figure 11.4, we show the Hovm¨oller diagrams of deep and congestus heating for the three parameter regimes: αs = 0.5, τ30 = 5τgrid , αs = 0.25, τ30 = 5τgrid , and αs = 0.25, τ30 = 10τgrid . As we can see, for the large αs value, the simulation consists of MJO-like disturbances, i.e., a planetary and intra-seasonal scale organization of convection. When αs is reduced but τ30 is kept at the same value, the organization shifts to a synoptic scale regime, with the solution dominated by eastward moving Kelvin waves (as revealed by spectral power plots not shown here) but when at the same time τ30 is increased to τ30 = 10τgrid , the solution reverts back to the MJO regime. Thus, as far as the scale of convective organization is concerned, the parameters αs and τ30 play essentially the same role. This, in fact, could be somewhat understood if we consider the expectation value of stratiform heating when the stochastic cloud fraction is set at its equilibrium value with respect to the slowly varying large scale. For a given deep heating, the stratiform portion increases naturally with increasing αs by construction and with increasing τ30 or equivalently decreasing death rate R30 , through the increase of the equilibrium stratiform area fraction, according to equation (10.27). Thus, the reported experiments clearly demonstrate a key role played by stratiform heating regarding the large-scale organization of convection. While the importance of stratiform heating for organized convection has been

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corroborated by various observational studies [e.g., 183, 235, 260, 255, 152, 139], it is not clear through which physical mechanism(s) this dependence occurs. Is it due to the effect of the titling and top heavy heating of the large-scale dynamics [184, 175, 139] or to the modulation of stratiform rain evaporation-induced downdrafts [174]? The latter effect could influence, for example, strength and size of the cold pool in the boundary layer which are believed to be important for organized convection [26, 49]. To test the second hypothesis, Deng et al. [38] repeated the simulations when the parameter μ fixes the relative strength of stratiform heating in the downdraft mass flux, according to Table 9.1. The results are reported in Figure 11.5. As can be seen from this figure, the MJO-like organization progressively diminishes and reduces to synoptic scale waves as μ is decreased from μ = 0.1 to μ = 0.01. Note that the standard value is μ = 0.25. This is consistent with the results of Majda et al. [174] who showed that the stratiform instability [184, 175] is mainly controlled by the parameter μ , by regulating the regeneration of the boundary θe and convective instability. The parameter μ somehow sets a time scale at which CAPE is consumed and regenerated and thus sets a time scale for convective organization.

11.2.2 Case of Asian Monsoon-Like Warm Pool As discussed in Chapter 10, the Asian monsoon variability is also comprised of multiple scale convective organized systems. On the intra-seasonal and synoptic scales in particular, we can distinguish the east-northward ISO’s and north-westward moving low-pressure systems, respectively. Similarly to the equatorial case, observations revealed that stratiform anvils play a major role in the dynamics of these Asian monsoon convectively coupled disturbances [199, 198, 197, 24, 28, 134]. Ajayamohan et al. [5] investigated the effect of changing the stratiform parameters for the monsoon-like aquaplanet warm pool setting in Figure 9.10(b), using the HOMME-SMCM model. Similar regime shift as in the case of the equatorial warm pool is observed when the parameter αs is decreased. Figure 11.6 displays the Wheeler-Kiladis-Takayabu type spectra for two different warm pool configurations, when the heating centre is at 10° North (top 6 panels) and when the heating centre is at 15° North (bottom panels). Both the east-west and north-south diagrams are shown for the parameter values αs = 0.5, 0.25, 0.125, yielding a total of six experiments. As we can see from these panels, for the large value of αs = 0.5, northward and eastward propagation of intra-seasonal disturbances (with periods around 40 days or more) dominates the wave spectrum but as αs decreases, synoptic scale waves become dominant. Interestingly, for the 10° North case, eastward and northward propagation dominates at the intra-seasonal scale, suggesting MJO-like disturbances co-existing with monsoon ISO signals when αs = 0.5, while at αs = 0.125, we see both eastward and westward moving synoptic scale waves without any significant signal in the north-south spectrum. The latter are likely the analogs of convectively coupled equatorial Kelvin and westward interio-gravity or mixed

11.2 Role of Stratiform Heating on Organized Convection

235

Fig. 11.4 Hovm¨oller diagrams for SMCM-HOMME climate simulations with a warm pool SST profile. Top panels: αs = 0.5, τ30 = 5τgrid , middle: αs = 0.25, τ30 = 5τgrid , bottom: αs = 0.25, τ30 = 10τgrid . Adopted from [38].

Rossby-gravity waves. When the warm pool is at 15°N on the other hand, while the north-eastward movement is still dominant at the large αs = 0.5 value, westward disturbances dominate the synoptic scale variability at αs = 0.125. These are essentially monsoon depressions that are observed to dominate the Indian summer monsoon variability at synoptic scales [e.g., 238, 70, 31, 63, 133, 197] as confirmed by the analysis of their physical structure and by their geographical location [5], reported below.

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11 Implementation of the SMCM in a Global Climate Model

Fig. 11.5 Same as Figure 11.4 but with αs = 0.5, τ30 = 5τgrid but with smaller values of stratiform downdraft fraction. Top panels: μ = 0.1, middle: μ = 0.05, bottom: μ = 0.01. Adopted from [38].

The simulated mean monsoon climatologies associated with the six experiments listed above are displayed in Figures 11.7 and 11.8, for the cases of a warm pool forcing located at 10°N and 15°N, respectively. Consistent with observations and with the deterministic simulation in Figure 9.11, all six experiments exhibit Asian monsoon-like flow structure characterized by south-easterly flow near the warm pool region with a maximum around 10°N and a strong return easterly flow between 20 and 30°N, independently of α and the exact warm location. A patch of positive vorticity defining the monsoon trough dominates the northern flank of the warm pool. The positive vorticity patch decreases in strength and shifts south westwards as αs is decreased. At the same time the Hadley circulation is dominated by a second baroclinic flow structure with air converging and raising in the upper part of the mid-troposphere and diverging and sinking in the lower mid-troposphere within the warm pool region, a signature of the upper stratiform dominated heating (shown in colours). The dominant stratiform flow is attenuated in Exp 3 and 6 because of the small value αs = 0.125 used there and a deeper Hadley circulation is recovered, more in line with Figure 9.11.

Period (days)

1

5 North

10

15

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–1

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5

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–5

40 100

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Fig. 11.6 East-West (a, b, c) and North-South (d, e, f) Wheeler-Kiladis-Takayabu spectra of precipitation averaged between 0 and 20°N, for the Asian monsoon HOMME-SMCM simulations with varying stratiform heating fraction: αs = 0.5 (a and d), αs = 0.25 (b and e), αs = 0.125 (c and f). Top panel for the case when warm pool centre is at 10°N and bottom panels for the case when the warm pool is 15° North. Adopted from [5].

Period (days)

2

11.2 Role of Stratiform Heating on Organized Convection 237

Period (days)

1

e

–10

–5

0

1

5

10 EXP5

EXP5

2

–1

3

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–5

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–1

3

Wavenumber

Wavenumber

40 100 5 –5

4

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North

40 100

–3

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20

4

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EXP4

20

North

10

–5

South

10

d

10

4

20 40 100 15 –15

10

East

20 40 100 15 –15

5

West

20 40 100 –15

0

b

10

2

10

EXP4

10

–5

East

4

–10

West

4

a

4

2

Fig. 11.6 (continued)

Period (days)

f

c

–3

–10

0

1 Wavenumber

–1

South

–5

West

North

5

East

3

10 EXP6

5

15

EXP6

9

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238 11 Implementation of the SMCM in a Global Climate Model

11.2 Role of Stratiform Heating on Organized Convection

239

It is interesting to note that the cases with large αs , in Figures 11.7 and 11.8, characterized with a strong vorticity maximum extending in the northward direction are also characterized by north-eastward propagating intra-seasonal disturbances in 11.6, while the cases with smaller αs values characterized with weaker vorticity maximum extending rather in the east-west direction exhibit east and/or westward moving synoptic scale waves. Compare in particular Exp4 and Exp6. This intriguing feature raises the question of whether the trough structure and the persistence of westward moving synoptic scale low-pressure systems versus north and eastward propagating intra-seasonal precipitation signals have cause and effect relationship or these are two products of the same cause. This issue requires further investigation and could be addressed by simple dynamical models as those presented in Chapter 7. A short first glimpse into this issue is provided by the plots in Figure 11.9 where the mean zonal wind vertical shear and vertical vorticity profile are plotted for the three experiments in Figure 11.8 and for ERA-interim reanalysis data [5, 223]. As we can see, the shear profile starts up westerly in the lower troposphere and easterly above 400 hPa and below 100 hPa when αs = 0.5. It progressively weakens and approaches the observed profile (black line) at αs = 0.125 which is mainly easterly throughout. The vertical vorticity follows a similar path in approaching the observation reference profile though not quantitatively. The apparent resemblance between the observed and the αs = 0.125 vorticity profiles is in the level at which low-level positive vorticity switches to negative vorticity aloft. While this transition occurs between 400 and 200 hPa for the αs = 0.5 and αs = 0.25 cases, consistent with the Hadley cell structures in Figure 11.8, it occurs roughly near 600 hPa in both the observations and the αs = 0.125 case. As these results clearly demonstrate the sensitivity of GCM simulation of tropical climate dynamics to stratiform clouds, they raise the question on how nature actually controls the various cloud fractions since in nature the synoptic systems associated with small stratiform fractions and the intra-seasonal systems associated with the large αs values coexist. Moreover, the mean shear, vorticity, and local Hadley cell obtained with the small αs are more realistic. Thus, in order to obtain a realistic simulation, a better balance between deep and stratiform clouds needs to be simulated. The inability of the aquaplanet model to capture both scales at once is perhaps due to the fact that other types of stratiform clouds are not represented, mainly those that form in situ due to sudden local drop in temperature in the upper troposphere as opposed to stratiform clouds that form in the wake of deep convection. This is somewhat justified by the fact that the comprehensive GCM simulations presented in the next chapter, which happen to represent both forms of stratiform clouds, do capture both intra-seasonal and synoptic scale disturbances simultaneously as in nature.

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11 Implementation of the SMCM in a Global Climate Model

(a)

(b)

(c)

(d)

(e)

(f)

Fig. 11.7 Mean monsoon climatology for the 10°N case. Left panels (a,c,e): Vertical vorticity (colours, 10−6 s−1 ) and horizontal velocity profile (arrows, 10 m s−1 reference arrow on top panel) averaged between 0 and 20°N. Right panels (b,d,f): Local Hadley cell as characterized by v − w velocity arrows (2 m s−1 reference arrow on top panel) and heating and cooling (colours, K day−1 ), averaged between 60 and 180°E. The dotted line marks the centre of warm pool forcing. Top to bottom: αs = 0.5, 0.25, 0.125. Adopted from [5].

11.2 Role of Stratiform Heating on Organized Convection

(a)

(b)

(c)

(d)

(e)

(f)

Fig. 11.8 Same as Figure 11.7 but the 15°N case. Adopted from [5].

241

242

(a)

11 Implementation of the SMCM in a Global Climate Model

Zonal wind

(b)

Vorticity

Fig. 11.9 Mean zonal wind vertical shear (left) and mean vertical vorticity profile, average over the monsoon trough region 10–20°N and 80–180°E for the three experiments in Figure 11.8 and ERA-interim reanalysis data (observations). Adopted from [5].

We conclude this subsection by showing in Figure 11.10 the physical structure of the simulated synoptic scale low-pressure systems corresponding to EXP6 in Figure 10.8. Two separate events are displayed one at roughly 1790 days and the other at roughly 1870 days, as indicated by the red and black ovals drawn on top of the Hovm¨ller diagram of precipitation shown on the right panel, which indeed confirm the dominating westward synoptic systems consistent with Figure 11.6 (panel c) on bottom). In both cases, we see warm temperature overlying cold temperatures at the location of largest potential temperature anomalies. Note also that the 1790 days case exhibit a packet of two waves that move together and the second case has a secondary weaker wave that is following behind. As we can see from the Hovm¨oller diagram, the simulation is characterized by convectively active and convectively suppressed episodes each lasting up to 10 days and the active episodes are characterized by wave packets that propagate next to each other. The meridional wind anomalies are characterized by northerly winds in front of the wave (to the east of the maximum θ ) and southerly winds in the back below roughly 300 hPa and reverse wind aloft. This suggests cyclonic vorticity in the lower troposphere surmounted with negative vorticity aloft with a hint of a strong barotropic component consistent with observations [131].

11.2 Role of Stratiform Heating on Organized Convection Precipitation

EXP 6

243 q and V anomalies

100

(a)

(b)

1880 Pressure (hPa)

200

1860

1840

400 600

1820 Time (days)

850 1000 80E 100

1800

120E

160E

120E

160E

(c)

1780 Pressure (hPa)

200 1760

1740

1720

0 1

60E 2

3

120E 4

5

6

8

600

850 1000 240W 80E

130E 7

400

9

10

–7

–6

–5

–4

–3

–2

–1

0

1

2

3

Fig. 11.10 Vertical structure of westward propagating low-pressure systems and at two separate times, indicated by the red and black circles on the Hovm¨oller diagram (left panel) of precipitation for the case αs = 0.125 and 15°N warm pool. The red and black dashed lines indicated propagation speeds of roughly 19 m s−1 . Adopted from [5].

Chapter 12

SMCM in CFS: Improving the Tropical Modes of Variability

12.1 Introduction The failure of the traditional cumulus parameterizations to adequately represent organized convective systems and tropical climate variability such as the MaddenJulian oscillation, convectively coupled waves, and monsoon weather and climate [e.g. 151, 95, 97] led the climate modelling community to think outside the box [224]. With the increase in computing power in the early 2000s and late 1990s, climate scientists begin to think about possibilities to directly numerically resolve clouds and convection instead of representing them via ad hoc closure formulae based on not fully justified assumptions such as quasi-equilibrium [8]. Two such ideas have been widely popular. The technique of cloud resolving convective parameterization (CRCP) or superparameterization introduced by Grabowski [74] and Khairoutdinov and Randall [107] consists in superimposing a 2D cloud resolving model (CRM) within each GCM grid column in lieu of a cumulus parameterization. As its name suggests, global CRM amounts to simply solving the equations of motion, including the conservation of the various water species (vapour, cloud water, ice, rain, snow, etc.), on the global domain at high enough resolution to actually represent clouds and convection features. It has been successfully tempted by Japanese scientists to take advantage of the availability at their disposal of one of the most powerful computers in the world, at that time, namely the Earth Simulator [234]. Despite the enormous computational burden the global CRM run rather successfully albeit for a short period of 10 days. It was run at 7 km resolution, which seems to be enough to capture cloud clusters that are of central importance for large-scale tropical variability consistent with the simulation presented in Section 8.2. Superparameterization is intermediate, in terms of computational complexity, between the global CRM and conventional coarse resolution GCMs and is more affordable and had more tangible impact on climate modelling research. Though still computationally expensive to compete with “traditional” state-ofthe-art climate models on the operational level, CRCP or superparameterization cli© Springer Nature Switzerland AG 2019 B. Khouider, Models for Tropical Climate Dynamics, Mathematics of Planet Earth 3, https://doi.org/10.1007/978-3-030-17775-1 12

245

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12 SMCM in CFS: Improving the Tropical Modes of Variability

mate models were very successful as research tools. Their superiority in terms of representing the key modes of atmospheric tropical variability resulted in significantly reducing many known climate simulation biases [109, 129, 71, 72, 191]. At the heart of this success, there is the fact that the superparameterization approaches allow some degree of freedom for the sub-GCM-grid convection to build up and die at its own leisure. It is believed that this added variability captures the essence of convection intermittency and the moisture preconditioning that allows a realistic organization of convective systems at the large scale as observed in nature [76]. Instead of fully or partially resolving convection, stochastic parameterization attempts at reinforcing this missing variability at the subgrid scale, without computational overhead, via the use of random parameters [20, 211, 154, 173, 153, 222, 111]. A comprehensive overview of stochastic parameterization has recently been put together by Berner et al. [13], see also [30]. While many stochastic parameterization approaches such the one in Buizza et al. [20], known as the stochastically perturbed parameterization tendency (SPPT), and that of Lin and Neelin [153] rely on introducing some ad hoc random parameter to perturb the existing parameterization outcome, there are few approaches that tried to use a more rigorous method. Plant and Craig [222], for instance, used equilibrium statistical mechanics tools applied to cloud resolving data to come up with distributions of mass flux among various cloud types. However, the distribution of the unresolved variability is fixed in time and does not respond to changes in the large-scale state. Despite their limitations, these early stochastic methods have demonstrated some skill and proved very useful in paving the way forward for improvements and new ideas [33, 277, 276]. By construction the SMCM [111] aims at representing the subgrid variability associated with organized convection via a Markov process whose distribution evolves in time, with the large-scale state. In essence the SMCM emulates a superparameterization model without the burden of a computational overhead. Its success in idealized settings of reduced coarse vertical resolution and aquaplanet global models made the prospect for its use in a full-fledged state-of-the-art climate model become very attractive. As mentioned earlier, there are many ways to use the SMCM as a cumulus parameterization including the possibility of coupling it to an existing cumulus parameterization [42, 218]. However, in Goswami et al. [67, 66, 65] the more straightforward and conservative approach of adopting prescribed heating profiles for the three cloud types that proved to be successful in the aquaplanet setting, for both equatorial convective variability and monsoon climate [37, 38, 5, 4]. Next, we discuss briefly the subtleties associated with the implementation of the multicloud model with prescribed heating profiles in the NCEP’s Climate Forecasting System GCM with a fully-vertically resolved moisture profile and existing shallow convection and boundary layer schemes, as opposite to the aquaplanet case where a one-layer (vertically averaged) moisture and boundary layer θe equations were used as in the idealized models of Chapters 6–8 and Section 10.4. Then, a few key results will be given in comparison to observation and to a control simulation using the default CFS model, which uses the simplified Arakawa-Schubert parameterization [213, 59].

12.2 Implementation of SMCM in CFS

247

12.2 Implementation of SMCM in CFS The implementation of the SMCM in CFS, following Goswami et al. [67, 66, 65], starts with the careful design of the heating profiles, φc (z), φd (z), φs (z), associated with congestus, deep, and stratiform cloud types, respectively. Instead of using the theoretical basis functions associated with the vertical normal mode expansion as done for the aquaplanet HOMME implementation in Chapter 9 and the previous sections, Goswami et al. [e.g. 65] opted for the observed heating profiles [242, 236] shown in Figure 12.1.

Fig. 12.1 Basis functions φc (z), φd (z), φs (z) for, respectively, congestus heating (red), deep convective heating (blue), and stratiform heating (yellow) profiles. Note that the basis function is zero beyond 200 hPa to avoid spurious heating in the upper atmosphere.

The total convective heating is accordingly set to Qtot (z) = Hd φd (z) + Hc φc (z) + Hs φs (z).

(12.1)

Here Hc , Hd , and Hs are, respectively, the associated heating rates, which are parameterized using the stochastic area fractions, σd , σc , and σs , with some substantial modification with respect to what was used in the previous two chapters. In

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12 SMCM in CFS: Improving the Tropical Modes of Variability

particular, here, we have

σd Qd σ¯d σc Hc = αc Qc σ¯c   ∂ Hs 1 σs = αs Hd − Hs , ∂t τs σ¯s Hd =

(12.2)

where σ¯c , σ¯d , and σ¯s are fixed background values of σc , σd , and σs , respectively. The mean area fractions are calculated according to the equilibrium distribution of the SMCM in (10.27), corresponding to the observed global mean-climatological state. As earlier, αc and αs are the congestus and stratiform adjustment coefficients, respectively, while τs is the stratiform convection adjustment time scale. As already mentioned earlier, one major difference between the aquaplanet implementation and the present one in that because of many other physics components such as radiation, shallow convection, and large-scale condensation, to name the most obvious ones, which rely directly on the vertical distribution of moisture it is no longer reasonable to use a one-layer moisture equation as done in (9.6). Instead, the SMCM parameterization needs to be modified to accommodate a full vertical resolution moisture profile. The challenge is to come up with an adequate parameterization for moisture source and sink. Moreover, both in its toy GCM and aquaplanet implementations, the multicloud heating rates and downdrafts are formulated in terms of deviations from a radiative convective equilibrium background. However, in the case of a real-world GCM where the presence of land, topography, and climate nonhomogeneities, such easily computed uniform RCE solution is no longer meaningful as it will undoubtedly be far from the real climatology. Instead, observational data (CFS reanalysis [102] to be more specific) were used to compute climatological background values, denoted here by overbars. More specifically, we have used climate data to compute surrogates for the RCE solution as time and spatial means for a set of boxes centred over different areas of relatively homogeneous climatology. These different areas (boxes) of relatively homogeneous climatology are shown in Figure 12.2 with the long term mean of specific humidity at the surface is shown in the background in order to provide a rationale behind choosing these boxes. Important modifications have also been introduced in the transition rates of the multidimensional birth-death process governing σc , σd , σs . In addition to the mid tropospheric dryness (MTD), which here is defined simply through the relative humidity and the convective available potential energy (CAPE), the transition rates also depend on the convective inhibition (CIN) and vertical velocity (W) to better connect the SMCM deep cumulus scheme with the existing shallow convection scheme. As reported in Table 12.1, the rates for the birth of congestus and deep cloud types are modified so that deep convection is inhibited when there is strong CIN or strong large-scale subsidence. Also unlike the aquaplanet simulation of the previous chapter the transition time scales obtained by the Bayesian inference tech-

12.2 Implementation of SMCM in CFS

249

Fig. 12.2 Boxes for climatological RCE computations. See text for details.

nique of De La Chevroti`ere et al. [36] is used here despite the new modifications in the formulations of the transition rates. Further, the potential for deep, Qd , and congestus, Qc , convection are given by   +  1 Lv  1  qm + θeb − γc θm Qd = Q¯d + τq C p τc 

 +   1 ¯ Qc = Qc + θ − γc θ m (12.3) τc eb while the downdraft is set to  Dc = μ

Hs − Hc Q¯c

+ .

(12.4)

Important modifications worth noting here include the systematic use of adjustment time scales τc and τq in the formulation of Qd and Qc . Here, Q¯d , Q¯c , Q¯s are the background potentials for deep, congestus, and stratiform convection, respectively, computed by projecting the background (i.e., observed climatology from the reanalysis data) convective heating onto the three basis functions in Figure 12.1. The values of the adjustment time scale τq and τc and other new parameters used here are listed in Table 12.2. The parameters τq and τc , which also directly affect the precipitation rate, are pre-tuned using reanalysis data by comparing the implied

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12 SMCM in CFS: Improving the Tropical Modes of Variability

Table 12.1 SMCM transition rules. The transition rates are given in terms of the large-scale predictors CAPE, C = CAPE/CAPE0, low-level CAPE, CL = LCAPE/LCAPE0, dryness, D = (100 − H )/MT D0, where H is the relative humidity (in %), large-scale subsidence, WN = − min(0,W /W 0), and CN = −CIN/CIN0. Here LCAPE is the part of the CAPE integral between LFC and the freezing level. We note that CIN is by definition a negative definite quantity, so that when CIN is large, Γ (CN ) −→ 1. )

(1 − e−x ), if x > 0 Time scale (hours) 0, otherwise

Description

Transition rate, where Γ (x) =

Formation of congestus

R01 =

(1−Γ (WN ))+(1−Γ (CN )) 1 τ01 (Γ (CL )Γ (D) 2

Decay of congestus

R10 =

1 τ10 Γ (D)

Conversion of R12 = congestus to deep Formation of deep

R02 =

Conversion of R23 = deep to stratiform

τ01 =32 τ10 =2

1 τ12 Γ (C)(1 − Γ (D)) (1−Γ (WN ))+(1−Γ (CN )) 1 τ02 (Γ (C)(1 − Γ (D)) 2

τ12 =0.25 τ02 =12 τ23 =0.25

1 τ23

Decay of deep

R20 =

1 τ20 (1 − Γ (C))

Decay of stratiform

R30 =

1 τ30

τ20 =9.5 τ30 =1

precipitation rates with TRMM data [94] when the SMCM is forced with reanalysis    data. The quantities qm , θeb , θm are deviations of the middle troposphere moisture, equivalent potential temperature at the planetary boundary layer (PBL) and middle troposphere potential temperature from their respective background states (observed climatology): 

qm = qm − q¯m 

θeb = θeb − θ¯eb  θm = θm − θ¯m .

(12.5)

The subscript b indicates variables averaged over the boundary layer height defined  as, Xb = 1h 0h X(z) dz. The PBL height, h, is provided by the boundary layer scheme in CFSv2. The height of the stable PBL, h, is estimated iteratively from ground up using bulk Richardson number (Rb ) until a critical value Rbc = 0.25 is reached [261]. It is precisely the height of the mixed layer consistent with the design of the multicloud model [118, 121, 269]. The subscript m indicates values of the corresponding variables at the middle troposphere. Based on the climatological profiles of equivalent potential temperature (θe ) and convective heating (not shown here),

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we have defined the middle and low troposphere value of θe at 500hPa and 700hPa, respectively. Table 12.2 Typical parameter values used in SMCM-CFS simulations. See Equations 12.2–12.5 and Table 12.1 for definitions. Parameter τq τs τc μ γc

αc αs CAPE0 LCAPE0 MT D0 CIN0 W0−

Value 144 hrs 96 hrs 240 hrs 0.0125 0.1

Remarks Moisture adjustment time scale Stratiform convection adjustment time scale Congestus convection adjustment time scale Relative contribution of stratiform evaporative cooling to downdraft Contribution of mid-troposphere potential temperature to the convective potential 0.1 Congestus adjustment coefficient 0.2 Stratiform adjustment coefficient 5000 J/kg Reference value of CAPE 2000 J/kg Reference value of LCAPE 5% Reference value of MTD 5 J/kg Reference value of CIN 0.05 m/s Reference value of vertical velocity

12.2.1 Prescribed Vertical Profiles of Moistening/Drying To cope with the full vertical resolution moisture equation, similarly to the prescribed heating profiles, φc (z), φd (z), φs (z), we introduce vertical profile for moisture source and sink, called for convenience, Q2 (z) and δm (z), respectively. While the profile of Q2 (z) is inspired directly from the observed moisture sink profile of Yanai et al. [295], δm is based on the intuition that the main moistening source comes from the evaporation of stratiform rain and detrainment of shallow and congestus clouds. The latter is set to have a maximum at roughly 850 hPa and then decrease linearly in both directions. The two profiles are shown in Figure 12.3.

Fig. 12.3 Prescribed moistening (right) and drying (left) profiles for the SMCM-CFS model. See text for details.

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To further constrain the moisture convective tendency, we require, in addition, that vertically integrated moist static energy is conserved. This is achieved by setting the total precipitation to the vertical integral of convective heating  H

P= 0

Qtot (z) Q2 (z) dz,

and the total evaporation is balanced by the downdraft. Accordingly, the evaporation rate is set to   Dc E(z) = δm (z)  m θe , H where m X = Xb − Xm , the subscript b and m indicate the PBL and middle troposphere values of X. Here H = 16 km is the average height of the tropical troposphere. The δm function is constructed according to 

) MID | 2 exp −αm |p(z)−p pBOT −pT OP , if z ≥ h δm (z) = 0, if z < h 

where αm is a constant, so that, H1 0H δm (z) = 1, p(z), pMID , pbot , ptop are the pressure values at respectively an arbitrary level z, middle, surface, and top of the troposphere. The temperature and moisture tendencies are given by   ∂ θ (z) = Qtot (z) − Dbθ , (12.6) ∂t SMCM   ∂ q(z) = −P(z) + E(Z) − Dbq , (12.7) ∂t SMCM where, Dbθ and Dbq are, respectively, the rates of cooling and drying due to downdrafts below the PBL. They are given by ) Dc m q if z < h, Dbq (z) = h 0 if z > h ) Dc m θ if z < h, Dbθ (z) = h 0 if z > h

12.3 Results The model was further calibrated in situ through a series of 5-year simulations where key parameters such as those listed in Table 12.2 were systematically varied. The mean climatology and climate variability were then systematically compared with observations and with a control simulation produced by the default CFSv2

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using the simplified Arakawa-Schubert cumulus parameterization [213]. As carefully documented in [65], in terms of the climatology, the CFS-SMCM model is robust to small-to-moderate changes in the SMCM parameters in Table 12.2. The CFS-SMCM climatology remains close to that of the default CFSv2 with a few notable improvements as reported in [66]. Though one parameter of interest is one that controls the transition to deep convection based on the mid-tropospheric humidity, namely MTD0 ; The amount of precipitation and the dry and moist biases in CFS climatology were found to be quite sensitive to this parameter (more on this below). We note that both the control and SMCM CFS simulations were run at T126 horizontal resolution (∼100 km) with 48 vertical levels and a 10 minute time step.

12.3.1 Improved Climatology To illustrate, we show in Figures 12.4 and 12.5 geographical maps of the mean rainfall, zonal mean temperature biases, and rainfall variances including both the total and their partition into intra-seasonal oscillation (ISO) and synoptic scale variances [67], respectively. Both annual and summer climatologies are shown. As can be seen from this figure both simulation overestimate the rainfall amounts especially around the Indian Ocean-Western Pacific warm pool region. Also both models predict a double ITCZ in the central Pacific; we see two bands of maximum rainfall running parallel on both sides of the equator, especially in the annual means, in both simulations while the southern ITCZ in the observations (TRMM) tilts significantly towards the South. This is indeed a recurrent problem in almost all (if not all) coupled climate models and it is not clear whether this is due to the atmospheric models alone or the ocean models are also to blame. A proper representation of air-sea interactions and boundary layer processes may be the key for a faithful simulation of the ITCZ. While the simulated climatologies look globally similar, CFS-SMCM shows a few improvements in some key tropical locations. For example, the dry biases over the Indian summer monsoon (ISM) region, northern Australia, and Amazonia and the wet bias over the western and central North Pacific have substantially reduced. Nonetheless, substantial exaggerations in precipitation amounts by the SMCM model are also produced, particularly west of a few mountain ranges, namely Congo, Mekong, and the Andes. The most important improvement achieved by the SMCM-CFS, in terms of climatology, is the noticeable reduction in the upper atmosphere warm bias (compare the control CFSv2 v.s. the CFS-SMCM plots on the corresponding panels of Figure 12.4). However, the cold bias in the upper troposphere (between 400 and 200 hPa) is not reduced. The improvement of the stratospheric warm bias is probably due to the overall improvement of the simulated tropical modes of variability, associated with organized convection as shown below, since there is documented evidence that convectively coupled waves are linked to the QBO [271, 294, 68] (See Chapter 3). The middle troposphere cold bias is thought to result from the lack of a proper radiative cloud feedback due to stratiform

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clouds. These issues remain ongoing research topics. More importantly, while, from Figure 12.5, the SMCM exaggerates the overall rainfall variance it does put back a significant amount of it into the synoptic scales which is underestimated in the control-CFSv2 simulation. While both the SMCM simulation and TRMM show balanced amounts of synoptic versus intra-seasonal oscillation (ISO) variances, the ISO variance in the CFSv2 simulation dominates by far, especially in the Indian Ocean-Western Pacific warm pool region. This is in fact a signature of the improved representation of the convectively coupled spectrum by CFS-SMCM as reported below.

12.3.2 Improved Tropical Modes of Variability To have a closer look on how good a climate model simulates the climate variability, beyond the distribution of variance, climate scientists look at how good the model represents a few key climate and weather patterns that are known to characterize the climate system. In the tropics, such patterns comprise the spectrum of convectively coupled waves and the MJO as well as the monsoon intra-seasonal oscillations and the associated synoptic scale systems. For convenience, we refer to these wave disturbances here as the tropical modes of variability. There are of course many other disturbances in the tropics that are not counted for here such easterly waves, which originate, for example, over Africa and propagate towards and over the tropical Atlantic or South America and propagate towards the East Pacific, that are notorious for being the precursors of devastating hurricanes and other tropical cycles [45], tropical cyclones, and mesoscale convective systems of various sorts [195]. Thus, as done in the previous chapters, we begin with the Wheeler-Kiladis-Takayabu spectral diagrams [283]. However, this time we will make direct comparisons with observations and with the state-of-the-art control-CFSv2 simulation. Figure 12.6 displays the outgoing long wave radiation (OLR) spectra for the two model simulations compared with satellite data [283]. Once again the observations are characterized by significant spectral peaks associated with equatorial Kelvin waves, Rossby waves, mixed Rossby-gravity (MRG) waves, and westward inertio-gravity (WIG) waves that lie on top of the corresponding dispersion-relation branches and an isolated MJO signal1 . The CFSv2 simulation has some skill in capturing an MJO-like signal though it appears somewhat slower (peaks significantly below the 30 days period mark), a decent Rossby wave peak, and a somewhat weakish Kelvin wave peak. It has to be noted here that CFSv2 is among the most successful models in the world [e.g. 97]. Note that both the MRG and WIG wave peaks are missing in the CFSv2 plots. The CFS-SMCM simulation on the other hand improves on the CFSv2 results significantly. First, we have a much stronger Kelvin

1

Note that the bright blurb on the right boundary of the observation panels is a data processing artefact that should be ignored.

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Fig. 12.4 Mean rainfall, temperature bias (difference between observed-reanalysis temperature profiles for observations (TRMM for rainfall), control CFSv2 and CFS-SMCM simulations. From [67].

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Fig. 12.5 Same as Figure 12.4 but for the total, synoptic, and intra-seasonal (ISO) rainfall variances. From [67].

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wave peak, and a stronger and frequency-wise more accurate MJO peak, a similarly good Rossby wave peak and more importantly we see power on the MRG and WIG branches, all consistent with the observations. This is a breakthrough in terms of the simulation of tropical waves by a coarse resolution GCMs [151, 95, 97]. This is even more so when considering other aspects of the CFS-SMCM simulations such as the dynamical and morphological features of these tropical waves, including the Asian monsoon modes of variability, at both synoptic and intra-seasonal scales, consistent with the aquaplanet simulations of the previous chapters, through now they are all part of the same simulation or parameter regime. To dig deeper into the discrepancies between the two model simulations and confirm the superiority of the CFS-SMCM in terms of simulating these tropical modes of atmospheric variability, we show in Figures 12.7 and 12.8 the variances associated with the MJO and the different convectively coupled waves for a 15 year runs corresponding to the two model versions, in comparison with observations. As we can see in Figure 12.7 while both simulations produce significant MJO variance in the vicinity of the Indian Ocean-Western Pacific warm pool consistent with the observations, the one produced by the CFS-SMCM simulation is more realistic. Also both model have a hard time putting significant MJO variance over the maritime continent, unlike the observation, the CFSv2 features a double peak straddling the equator, instead of the variance being centred at the equator. Nonetheless, the CFSSMCM variance is unrealistically weaker and is attained more to the east, in the Arabian sea. This is probability related to difficulties associated with simulating convection over the maritime continent, which is notoriously hard to almost if not all GCMs. The discrepancies between the simulated CCEWs in Fig 12.8 are more spectacular. While the CFS-SMCM shows a reasonably good skill in putting variance at the right location in each one of the wave-types, CFVS2 has hardly any variability except for the equatorial Rossby waves. This is consistent with Figure 12.5 which showed very little rainfall variance. We recall here that equatorial Rossby waves are low enough frequency to qualify as ISOs.

12.3.3 MJO and Monsoon ISO Propagation The Hovm¨oller diagrams of the MJO-filtered and unfiltered OLR for the two simulations and for the observations are reported in Figure 12.9. Consistent with the peak in the spectrum diagrams in Figure 12.6, we see eastward moving streaks in all of the filtered fields. However, the CFSv2 MJO signal is much weaker and maximizes around 120°W, over the central Pacific. The SMCM MJO filtered signal is more consistent with the observation both in terms of propagation speed and geographical location. When looking at the unfiltered fields, the CFSv2 simulation is much worse as it hardly exhibits a coherent MJO signal over the tropical warm pool unlike the observation and the SMCM. The phase composites of the MJO over its life cycle presented in Figure 12.10 further clarify the superiority of the SMCM

Fig. 12.6 Wheeler-Kiladis-Takayabu spectral diagram for satellite observations (NOAA OLR, left), CFSv2 control simulation, and CFS-SMCM simulation. Top panel shows the equatorial symmetric waves while the lower panels are for the anti-symmetric waves. The black lines represent the dispersion relations of equatorially trapped waves with varying equivalent depths. The vertical lines mark the 30, 6, and 3 days periods. From [67].

258 12 SMCM in CFS: Improving the Tropical Modes of Variability

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Fig. 12.7 MJO filtered OLR variance in the CFS-SMCM and CFSv2 simulations and in observations. From [67].

simulation. The CFSv2 simulation hardly shows a coherent MJO propagation. The spectrum power can be indeed misleading. According to the last two metrics CFSv2 can hardly be qualified as being able to simulate an MJO signal. This is consistent with the fact that the MJO variance does not peak at the equator. Thus what CFSv2 presents as an MJO in terms of spectral power has very little to do with the MJO as observed in natures. The additional diagnostics of MJO physical structure and mechanisms provided in the following subsections further confirm this assertion.

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Fig. 12.8 Same as Figure 12.7 but for the convectively coupled wave signals. From [66].

The ability of the two models to simulate the monsoon intra-seasonal disturbances is tested in Figure 12.10. While on appearance both simulations show comparable skill in terms of monsoon ISO simulation, when compared to the observations. CFSv2 exaggerates the variance in the Western Pacific and has a much slower northward propagation speed. CFS-SMCM on the other hand has little to no variance over the maritime continent and the Bay of Bengal while it slightly exaggerates

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Fig. 12.9 Hovm¨oller diagrams of MJO filtered (top) and unfiltered (bottom) OLR anomalies for the CFS-SMCM (left), observations (middle), and CFSv2 (right). From [67].

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the monsoon ISO variance in the Arabian sea consistent with the MJO variance in Figure 12.7.

Fig. 12.10 MJO filtered OLR distribution at various phases of its lifetime (indicated in days) as it propagates through the Indian Ocean-Western Pacific warm pool in CFS-SMCM (left), observations (middle), and CFSv2 (right). From [66].

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Fig. 12.11 Monsoon ISO variance distribution (top), northward propagation (middle), and flow structure (bottom). From [66].

12.3.4 Further Physical Aspects of the MJO Signal Further dynamical and morphological aspects of the simulated MJO-filtered disturbances are summarized in Figures 12.12 and 12.13 [66]. From the flow structure in Figure 12.12, we see that while the CFS-SMCM simulation captures an MJOlike disturbance which has a lot in common with the observed MJO, the CFSv2’s filtered signal doesn’t present any kind of resemblance to it. The observed MJO is characterized by low-level equatorial divergence capped by upper-level equatorial convergence in phase with the minimum OLR and a quadruple vortex structure straddling the equator, in both the lower and upper troposphere. Except for evident

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discrepancies in terms of location size and strength of the vortices, the CFS-SMCM MJO is a much better representation of the observations. Figure 12.12 already disqualifies the CFSv2 simulation for producing a physically meaningful MJO, and warrants the GCM-community for not rushing to call any eastward spectral signal in the intra-seasonal planetary scale range an MJO. Figure 12.13 provides further evidence that this is indeed the case and confirms that the success of the CFS-SMCM simulation is rooted from its design principle based on the interactions between multiple cloud types. While the CFS-SMCM has all the right physical features, when compared to observations, the CFSv2 simulation has again hardly anything in common. All the features that are known to characterize the MJO are lacking in the CFSv2 simulation [127], starting with the sharp peaking down of the OLR corresponding to the MJO convection peak. The following features are common to both the CFS-SMCM and observations but are badly lacking in the CFSv2 signal: 1. a tilted zonal wind profile characterized by westerly winds in the lower troposphere capped by easterlies aloft following the convection maximum, 2. tilted low-level convergence capped by upper-level divergence in phase with the OLR minimum and with upward vertical velocity and positive heating, 3. positive temperature anomalies in the upper troposphere in phase with the convection centre, 4. a tilted relative humidity peaking in the lower troposphere a few days prior to deep convection. These are all characteristic features of the multicloud model since its inception which are shown here to be resilient and demonstrate once again that the multicloud paradigm is at work at multiple scales as anticipated. Moreover, the same conclusions about the two simulations were reached in [66] when similar diagnostic where applied on the monsoon ISO disturbances in Figure 12.11. While the MJO (and to some extent the monsoon ISO) features represented in Figures 12.12 and 12.13 are universal, it is not easy to draw conclusion on how they contribute to the dynamics, initiation, amplification, maintenance, propagation of the MJO (and monsoon ISO) though many interpretations have been speculated in the literature [e.g., 96, 97, 95]. The interested reader is directed to those research articles and to [66] for more details. Since its discovery in the early 1970s, many theoretical models have been proposed for the MJO with very mixed results. Among the most recent ones, we can list four that all present some features of the MJO at least as a slow-eastward propagating planetary equatorial wave. They have been recently the subject of a workshop on the MJO [161]. These theories could be viewed as complementary or contradictory depending on whether the glass is seen half full or half empty. The MJO-skeleton model of Majda and Stechmann [178] assumes that convective instabilities occur at synoptic scales at lower and their planetary scale envelope

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interacts with low-level moisture in an oscillatory motion to produce a neutrally stable MJO mode in addition to westward moving planetary scale waves. The moisture mode theory [239, 2] suggests a model where the wind field is set to a stationary Gill-model solution in response to moisture perturbations that control the heating anomalies though such steady state simplification is not physically nor mathematically justified [101]. Due to the strong Rossby gyres to the west of the convergence maximum, the moisture anomalies are advected eastward by the Gill-response wind background. Wang and Rui [272] propose a model with strong boundary layer frictional convergence which then acts to produce the desired Gill-type wind response. Finally, Yang and Ingersoll [296] suggest that the MJO is a gravity wave packet owing its eastward propagation to the beta induced asymmetry between eastward and westward moving inertio-gravity waves. Despite their apparent differences all these theories seem to produce an MJO-like disturbance, i.e., a slowly moving eastward wave with some key structure resembling the MJO. Similarly to the fact that all these theories cannot all possibly be valid, it is also possible for GCMs to produce MJO like signals for reasons that have nothing to do with reality as it is obviously the case of CFSv2. Nonetheless, a consistent argument that has been put forward by observational studies [e.g., 126, 100] is one that connects the eastward propagation of the MJO to low-level moistening to its front. The same principle seems to apply for the northward propagation of monsoon ISO [96, 1] as it is also well captured by the SMCM [66]. However, the exact mechanisms of eastward and northward propagation of intra-seasonal convective disturbances, especially the why and how the moisture preconditioning occurs, remain an active research area. The roles and contributions of congestus and shallow cloud detrainment, moisture advection, and moisture convergence remain to be determined [82, 100, 268]. More importantly, the role of the beta-effect, which is arguably the main source of asymmetry for both eastward and northward propagation, remains to be elucidated, even though there are arguments in favour of symmetry breaking due to faster eastward moving Kelvin waves and slower westward moving Rossby waves, which are thought to be the main dynamical components of the MJO, but the phase speed difference exceeds 10 m s−1 while the MJO phase speed is only about 5 m s−1 !

12.3.5 Rainfall Event Distribution It is known from observational records that rainfall events of 200 mm day−1 are typically produced by deep and high clouds, which are characterized by low ORL values while thin cirrus and shallow cumulus and cumulus congestus clouds typically drizzle or produce only light rain events. Also while strong rainfall events are typically more concentrated in the OLR-rainfall plane they also exhibit a fair amount of scattering along the OLR axis as shown on the left panels in Figure 12.14. State-of-the-art GCMs using deterministic parameterizations tend to produce too

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Fig. 12.12 MJO flow structure. Colour shading is for OLR while stream line contours are pictured in grey scale. From [66].

much light rainfall events as the underlying cumulus parameterizations tend to produce rain and form deep clouds too often and too quickly. Thanks to the multicloud model paradigm of inhibiting deep clouds when the atmosphere is too dry to produce proper amounts of rain, the CFS-SMCM produces rainfall-OLR distribution that are more consistent with the observations as shown in Figure 12.14. Moreover, we recall that a key parameter that affects the rainfall distribution is the one that controls the sensitivity of the transition to deep convection due to midlevel moisture, namely MTD0 (see the caption of Table 12.1). During the tuning of this parameter [65], it is found that a large MTD0 tends to produce larger amounts of rain, consistent with the fact that a larger MTD0 value allows transition to deep convection at drier tropospheric conditions. Thus in order to further reduce the dry bias over land without affecting the precipitation climatology over the oceans, Goswami et al. [64] conducted a simulation with the CFS-SMCM model when the parameter MTD0 assumes distinct values over land and over the oceans. More precisely a value MTD0 = 5 is used over the oceans while MTD0 = 25 over land. This modification led to important improvements in the global precipitation climatology without significantly affecting the synoptic and intra-seasonal scale variability [64].

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Fig. 12.13 Zonal and vertical structure of MJO-filtered OLR, zonal wind, zonal convergence, relative humidity, potential temperature, vertical velocity, and heating, perturbations for CFS-SMCM, observations, and CFSv2. From [66].

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Fig. 12.14 Rainfall versus OLR distributions during summer over the Indian Summer Monsoon (ISM) region (top panel) and during the year for the whole tropics (bottom panels) for conservations (TRMM v.s. NOAA ORL, left), CFSv2 simulation (middle), and CFS-SMCM simulation (right). From [67].

Glossary

List of Symbols g =9.8 m s−2 Acceleration of Earth’s gravity. p Atmospheric pressure. v = (u, v) Horizontal flow velocity field with u, v are the zonal (east-west) and meridional (north-south) velocity components. w Vertical velocity component. x, y, z Cartesian coordinates in zonal, meridional, and vertical directions. t Time. ∇ = (∂x , ∂y ) Horizontal gradient operator. div(v) = ∇ · v = ∂x u + ∂y v Horizontal divergence operator.

ρ Air density. ρd Density of dry air. ρv Density of water vapour. q Specific humidity. r Water vapour mixing ratio. T Air temperature.

θ Potential temperature. θe Equivalent potential temperature. θe∗ Saturation equivalent potential temperature.

© Springer Nature Switzerland AG 2019 B. Khouider, Models for Tropical Climate Dynamics, Mathematics of Planet Earth 3, https://doi.org/10.1007/978-3-030-17775-1

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Glossary

Γ0 Dry adiabatic laps rate. Rate at which a vertically lifted an unsaturated parcel of air cools down due to expansion. Γs Moist adiabatic laps rate. Rate at which a vertically lifted parcel of saturated air cools down given that part of its expansive cooling is compensated by latent heat release from condensation in order to keep the parcel at saturation level. N Brunt-V¨ais¨al¨a buoyancy frequency. Frequency at which vertically displaced air parcels oscillate. Lv Latent heat of vapourization/condensation. Cp Heat capacity at constant pressure. Cv Heat capacity at constant volume.

Ω = 2π day−1 Frequency of Earth’s rotation. β = 2.2804 × 10−11 m−1 s−1 Gradient of the Coriolis parameter at the Equator. v0 , v1 , v2 , · · · Barotropic, first and second baroclinic horizontal velocity components, etc. w1 , w2 , · · · First and second baroclinic vertical velocity components, etc.

θ1 , θ2 , · · · Potential temperature components associated, respectively, with first, second, etc., baroclinic modes. λm Vertical structure separation constants. θeb Equivalent potential temperature in the ABL. θem Equivalent potential temperature in the mid-troposphere. c Gravity wave speed.

ω Wave frequency or phase. Called generalized phase when complex. k Zonal wavenumber. wc Updraft vertical velocity. Vertical velocity of convecting parcel.

σc Cloud (or convective) area fraction. Proportion of surface area occupied by convection (cumulus clouds) in a given horizontal grid box. Also refers to relative area occupied by congestus clouds while σd and σs correspond to deep convective and stratiform clouds, respectively. mc = σc wc Convective mass flux. n = −1, 0, 1, 2, · · · Meridional index of equatorially trapped waves. Hc , Hd , Hs Parameterized diabatic heating associated with congestus, deep, and stratiform clouds. QR Radiative cooling rate. P Precipitation rate. Qc Bulk diabatic heating due to parametrized convection.

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D Downdrafts.

αs Stratiform fraction adjustment parameter. αc Congestus fraction adjustment parameter. τs Stratiform adjustment time scale. τc Convective time scale. μ Relative contribution of stratiform rain evaporation to downdrafts. m0 Downdraft mass flux reference scale.

τkl Stochastic transition time scale from cloud state k to cloud state l. Q˜ j , j = 0, 1, 2 Projection of background moisture gradient onto barotropic ( j = 0) and first ( j = 1) and second ( j = 2) baroclinic modes.

λ˜ = Q˜ 2 /Q˜ 1 Relative contribution of second baroclinic convergence due to background moisture. α˜ Contribution of second baroclinic convergence due to moisture perturbation.

Technical Terms Ansatz A prescribed functional form of solution for a differential equation or system of equations often based on an educated guess. Aquaplanet Idealized setting for climate simulations where the land cover and topography are ignored assuming a water covered world. Atmospheric boundary layer. Also sometimes called planetary boundary layer (PBL). The thin layer of air of roughly 500 m to 2 km, between the surface and the stably stratified tropospheric layer typically above the trade wind inversion and the well-mixed layer near the ground. Baroclinic flow Part of the tropospheric flow that depends on height often directly driven by diabatic forcing due to condensational latent heat release, i.e., convection. Baroclinic modes A sequence of tropospheric flow components (modes) with a distinct vertical structure well approximated by a corresponding sequence of sines and cosines. Also known as modes of vertical structure. Barotropic mode Part of the tropospheric flow that doesn’t depend on height. Beta-plane approximation An approximation of the governing equations of atmospheric flow retaining the effect of Earth curvature in the sole Coriolis term. The approximation is thought to be acceptable along an up to 60◦ -wide strip running parallel to the equator.

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Birth-death process A Markov jump process whose infinitesimal generator is a tridiagonal matrix. The chain can only go up or down by one unit if the “distance” between successive states were made uniform and normalized. Convectively available potential energy, CAPE . Total buoyant energy of an adiabatically raising moist parcel integrated from LFC to LNB. Convective inhibition, CIN Energy that an adiabatically lifted parcel need to overcome become it reaches its LFC. Vertically integrated negative buoyancy of an adiabatically raising moist parcel between the surface and LFC. Cumulus congestus Cumulus clouds that extend from the top of the PBL to near the freezing level. Often termed as warm clouds because they do not contain ice crystals. Deep convection Convective flows that expand from the top of the ABL to the upper troposphere, near the tropopause. Process leading to the formation of cumulonimbus clouds. Dispersion relation An explicit or implicit algebraic relation that links the wave frequency to the wavenumber. Downdrafts Flux of mass due to subsiding air under the influence of negative buoyancy or large scale divergence. Eigenvalue Problem A mathematical equation involving a differential or an algebraic operator (a Matrix, when the problem is linear as it is often the case) whose unknowns are pairs each containing a scalar (called the eigenvalue) and a function or vector (called an eigenfunction or eigenvector). Equatorially trapped waves A set of solutions to the linear equatorial beta-plane shallow water equations including the Kelvin waves, inertio-gravity waves, and the mixed-Rossby gravity (MRG) waves, also known as the Yanai wave. The observed analogues are called by the generic name of “convectively coupled waves”. These waves vanish rapidly away from and move along the equatorial (the equatorial acts as a wave-guide). Equilibrium distribution A probability distribution of a stochastic process which is invariant in time. Equivalent height,√ H ∗ A hypothetic fluid column height so that the gravity wave speed given by c = gH ∗ is that of the corresponding baroclinic mode. Ergodic Markov chain A type of Markov chain for which in particular the limiting distribution exists and is given by a unique equilibrium distribution. Galerkin projection A mathematical/numerical method permitting to reduce the dimension of a partial differential equation (or system of equations) by projecting its solution in a set of orthogonal basis functions.

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Group velocity Speed at which wave-energy and wavepackets propagate. Derivative of phase with respect to wavenumber. Growth/Decay rate Rate of wave exponential growth/decay. It is given by the imaginary part of the generalized wave frequency. The corresponding wave/mode is said to be unstable when this quantity is positive and stable otherwise. Hadley cell Mean flow running perpendicular to the equator consisting of a closed circulation cell (or cells) in which air rise over the tropical region (the ITCZ) and descend in the subtropics roughly around 30◦ North and South. Hovm¨oller diagram 2d colour plot in time-longitude domain revealing streaks of propagating wave-like features. Infinitesimal generator The matrix whose off-diagonal entrees are the transition rates of a Markov chain and its rows sum to zero. Kelvin wave The simplest linear wave solution for the equatorial beta-plane shallow water equations which moves eastward and is characterized by vanishing meridional velocity and pressure perturbations that are in phase with the zonal winds. Also bears the name an observed tropical disturbance in winds, temperature, rainfall, and cloudiness which has a zonal structure that resembles that of the shallow water solution. Level of free convection Level at which a moist parcel raised adiabatically from the surface first becomes positively buoyant. Level of neutral buoyancy. Level at which a convectively buoyant parcel first loses it buoyancy without mixing with the environment. Lifted condensation level Level at which a moist parcel raised adiabatically from the surface first becomes saturated. Limiting distribution The probability distribution, when it exists, that a stochastic process approaches when the time index goes to infinity. Low level CAPE, CAPEl . Buoyant energy of an adiabatically raising moist parcel integrated from LFC to freezing level. Madden-Julian oscillation A planetary scale disturbance in wind, temperature, precipitation, and cloudiness that is observed to occur over the warm waters of the Indian Ocean and Western Pacific. It forms roughly over the middle of Indian Ocean and moves eastward at roughly 5 m s−1 and has a variable period ranging between 40 to 60 days. Unlike other known “convectively coupled waves”, the MJO doesn’t seem to have an equatorially trapped waves. Markov Chain A stochastic process whose distribution at given time depends only on the knowledge of its state at one single previous time. Also known as a memoryless process. Markov jump process A continuous time Markov chain with a finite state space.

274

Glossary

Moist convection or simply convection Vertical air motion driven by buoyancy as a result of latent heat release from condensation. Monsoon low pressure systems Westward moving wave-like disturbances occurring within the monsoon trough carrying significant cyclonic vorticity and precipitation perturbation. Monte Carlo Simulation A technique for representing a stochastic process on a computer program. Organized convection Coherent ensemble of convection driven flows forming a distinct wave-like flow pattern with significant velocity, temperature, moisture, and precipitation perturbations. Phase speed Speed of wave propagation given by the (real part of) the wave frequency and the wavenumber. Probability transition matrix The matrix whose entrees are the transition probabilities of a Markov chain. Shallow water equations A set of partial differential equations governing the flow of a thin fluid which is homogeneous in the vertical used to approximate geophysical flows to the first order. Stochastic matrix A square matrix whose entrees are all nonnegative and all its rows sum to one. Stochastic process A sequence of random variables indexed by a discrete or continuous counter often set to increase from zero to infinity and referred to as time so the process is viewed as a dynamical system. Stratiform instability Instability of moist gravity wave due to the positive feedback of stratiform heating on convective activity involving coupled between first and second baroclinic modes. Stratiform clouds Anvil-like shaped clouds that form in the wake of deep convection in the upper troposphere. State space A discrete or continuous set of real numbers on which a stochastic process takes its values. Trade wind inversion A sharp interface in the lower troposphere between the free troposphere and the ground where the stability increases rapidly in a jump-like manner. Transition probability The conditional probability for a Markov chain to switch from two given state in a given time period. Transition rate The transition probability of a Markov chain over an infinitesimally small time period. Updrafts Flux of mass due to raising air as a result of convection.

Glossary

275

Walker cell Mean flow running parallel to the equator consisting of a closed circulation cell (or cells) in which air rise over the Indian Ocean/Western Pacific warm pool and descend over the Eastern Pacific/Central American continent. Wheeler-Kiladis-Takayabu diagram 2d colour plot of spectrum power usually in zonal wavenumber-frequency domain, revealing dominant peaks of coherent wave features in an observed or simulated time series.

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Index

A Adiabatic lapse rates, 34–35 Asian monsoon-like warm pool convectively coupled disturbances, 234 ERA-interim reanalysis data, 239 GCM simulation, 239 ISO signals, 234 mean monsoon climatology, 239–242 mean zonal wind vertical shear, 242 simulated mean monsoon climatologies, 236 SMCM-HOMME, 235 Wheeler-Kiladis-Takayabu spectra, 237–238 Atmospheric circulation, 163 B Barotropic/external mode, 16, 22, 118, 123, 127, 221 Bayesian inference procedure, 215, 217–219, 221, 222, 248 Birth death processes atmospheric convection, 186 CIN and CAPE, 186 cloud types, 188 coarse grained model, 203–206 continuous time Markov chains Kolmogorov forward and backward equations, 65–66 limiting distribution and detailed balance, 66–67 M/M/m process, 68 transition rates, 67 waiting time and transition rates, 64–65 discrete time Markov chains, 61–64 hypothetical parcel, 187 LCL and LFC, 186

physical mechanisms, 185, 187–188 Poisson process, 63–64 precipitation patterns, 185 C Climate modelling, 36, 38, 78, 144, 164, 245 Cloud resolving convective parameterization (CRCP), 245 Clouds area fractions, 201–202 birth and death models (see Birth death processes) MCM (see Multicloud model (MCM)) and precipitating atmosphere, 25–26 rain droplets, 23 saturation and formation, 32–33 in tropics, 43–48 Coarse graining approximation, 225 birth-death stochastic model, 192, 203–206 cell size, 215 Markovian birth-death processes, 189 mesoscopic stochastic model, 192–195 statistical behaviour, 227 time evolution of cloud area, 228 Computations with random numbers acceptance-rejection method, 60 inverse transform method, 59 large numbers law, 58 Monte Carlo integration, 58 Convective inhibition (CIN) and CAPE, 38, 39 coarse grained mesoscopic stochastic model, 192–194 energy barrier, 38

© Springer Nature Switzerland AG 2019 B. Khouider, Models for Tropical Climate Dynamics, Mathematics of Planet Earth 3, https://doi.org/10.1007/978-3-030-17775-1

299

300 Convective inhibition (CIN) (cont.) Gillespie’s exact algorithm, 195–196 microscopic stochastic model, 189–192 negative energy, 186 numerical tests, 196–197 transition probability matrix, 195 Convective instability of the second kind (CISK), 76–79, 81, 87, 107, 164 Convectively coupled equatorial waves (CCEWs), 50–52 baroclinic modes, 52 CFS-SMCM, 257 dry equatorial shallow water waves, 50–51 equations, 118–123 Hovm¨oller diagram, 43 MGWI, 117 OLR and GARP-GATE, 42 physical structure and dynamical features, 51 QBO, 41 spectral peaks, 41 systematic link, 42 tropical clouds, 43–48 uniform background, 123–127 See also Tropical waves Convective momentum transport (CMT) background, 140–143 multiscale model, 151, 153–155 parameterization, 146–151 simulated Kelvin wave, 135–140 stochastic model, 143–146 vertical transport, 133 See also Madden Julian Oscillation (MJO) Cumulus parameterization adjustment-type schemes, 163 CISK/wave-CISK, 164 climate model, 115 GCM, 163–165 global climate and numerical weather, 163 HOMME, 165 MCM, 165–168 mid-tropospheric and boundary layer, 92 multicloud model (see Multicloud model (MCM)) SMCM (see Stochastic multicloud model (SMCM)) subgrid effect, 135 D Dew-point temperature, 32 Dry dynamics air density, 4 Coriolis force, 5–6 global climate, 3

Index hydrostatic primitive equations, 3 normalized quantity, 5 PBL, 5 primitive equations, 6–8 Taylor expansion, 4 E Entropy, 27, 30, 31, 33, 34, 38, 47, 104 Equatorially trapped waves dispersion relation curves, 42 extra-prognostic equations, 127 Hurricane initiation, 42 multicloud convection scheme, 117 shallow water (see Shallow water approximation) synoptic scale, 41 Wheeler-Kiladis-Takayabu spectral diagrams, 178, 258 See also Convectively coupled equatorial waves (CCEWs) G Geostrophic balance relation, 6 Gibbs free energy, 29 Giga-LES dataset, 216, 218–222 Gillespie’s exact algorithm, 195–196, 204–206 Global climate model (GCM) analogy, 78 cumulus parameterization, 163–165 HOMME-SMCM parameterization, 229–230 uniform forcing, 230–232 moist air, 83 H High order methods modelling environment (HOMME) aquaplanet model, 169 atmospheric general circulation model, 164 SMCM-HOMME model, 229–232, 234, 235 I Internal waves, 9 Intertropical convergence zone (ITCZ), 44, 158, 172, 174, 219, 221, 253 Intra-seasonal variability, 151, 156, 163, 164, 175, 177, 229, 233, 234, 239, 253, 254, 264–266 L Level of neutral buoyancy (LNB), 36, 38, 101, 104, 164 Lifted condensation level (LCL), 32, 36 Liquid water enthalpy, 30

Index M Madden Julian Oscillation (MJO) cartoons, 48, 50 vs. CCWs, 133, 168–172 CMT (see Convective momentum transport (CMT)) convectively coupled waves, 178 dynamics and propagation, 49 equatorial waves convectively coupled waves, 155 lag time, 157 low-level westerly shear amplifies, 160–161 multiscale waves, 161 RCE, 156 WWB, 162 zonal wind, 158, 160 Kelvin waves, 43 large-scale disturbance, 48 low-frequency mode, 42 moisture convergence, 49 monsoon ISO propagation, 257–263 physical aspects, 263–265 planetary scale disturbances, 133 synoptic scale waves, 110 two-way interactions, 134 vortices, 49 warm pool (see Warm pool simulations) Markov processes birth death (see Birth death processes) CMT, 144 likelihood function factors, 218 Poisson process, 63–64 random numbers (see Computations with random numbers) subgrid variability, 246 transition probabilities, 204 Mass-flux schemes, 75, 82–86, 164 Meridional index, 13, 14, 18–20, 85, 270 Meridional shear, 117, 120, 123, 127–130 Moist dynamics cloudy and precipitating atmosphere, 25–26 conserved moist thermodynamic variables, 30–31 equation of state, 24–25 first law, 26–27 moisture/water vapour, 23–24 phase change processes, 27–30 saturation and formation of clouds, 32–33 second law, 26–27 stability, 35–38 weather forecasting models, 23 Moist gravity wave instability (MGWI), 109–113, 117

301 Moist gravity waves adjustment scheme, 78–81 CCEWs, 117 convectively coupled tropical waves, 75 low-frequency wave streaks, 111 mass-flux schemes, 82–86 MGWI, 109 parameterization schemes, 75 quasi-equilibrium schemes, 75–76 wave-CISK, 76–78 wave left cycle, 110 WISHE, 82–86 Monte Carlo simulation Markov chains (see Markov processes) mean flow strength, 212 non-zero transition rates, 205 Multicloud model (MCM) aquaplanet global climate (see Global climate model (GCM)) baroclinic modes, 158 CIN, 100 closure equations, 154 convectively coupled equatorial waves (see Convectively coupled waves) convective systems, 52 cumulus parameterization, 165–168 linear stability analysis, 105–110 model formulation, 100–105 multiscale convective systems, 99 nonlinear simulations, 111–115 parameter inference, 66 simulations, aquaplanet, 168–172 warm pool simulations, 172–181 WISHE, 99 Multiscale convective systems, 52–55 N Nearest neighbour interactions coarse grained approximation, 225 ITCZ, 221 mean field equation, 225–226 multiple particle Hamiltonian, 223–225 numerical testing, 226–228 Nonlinear simulations, 92–97, 111–115, 117 O Outgoing long wave radiation (OLR), 42, 44, 53, 54, 254, 257, 258, 261, 263–268 P Parabolic cylinder functions, 13, 14, 18, 19, 121, 122, 126 Planetary boundary layer (PBL), 5, 250, 252

302 Precipitation climatology, 266 convection, 42 coupled wave, 108 decorrelation times, 232 efficiency, 90 and evaporation, 77 Hovm¨oller diagram, 173, 175, 177 intra-seasonal precipitation signals, 178, 239 liquid/solid particles, 25 low-level moisture, 49 mid-tropospheric evaporation, 167 moisture, 88 10N simulation, 178 substantial exaggerations, 253 surface, 45, 136, 140 tropical, 6 vertical integral, 252 Wheeler-Kiladis-Takayabu spectral diagrams, 178 Q Quasi-biannual oscillation (QBO), 41, 253 R Radiative convective equilibrium (RCE), 84, 90, 98, 105–110, 248 Riemann invariants, 18–20 S Saturation and formation of clouds, 32–33 Shallow water approximation eigenfunctions, 8, 9 Kelvin wave, 9–11 Rossby, gravity and mixed Rossby-gravity waves, 11–15 water-like equations, 9 Simulated Kelvin wave, 135–140 Static stability, 5, 7 Stochastic multicloud model (SMCM) building block, 187 CIN model, 199 cloud area fractions, 201–202 coarse grained birth-death stochastic model, 203–206 cumulus parameterization atmospheric circulation model, 210 mass-flux, 209 quasi-equilibrium assumption, 209 Toy-GCM, 210–215 deterministic mean field equations and numerical simulation, 206–208 equilibrium statistics, 201–202 GCM (see Global climate model (GCM))

Index HOMME (see High order methods modelling environment (HOMME)) implementation basis functions, 247 improved climatology, 253–254 MJO and monsoon ISO propagation, 257–263 RCE, 248 transition rates, 249, 250 tropical modes of variability, 254–257 vertical profiles of moistening/drying, 251–253 inference parameters from data Bayesian inference procedure, 217–219 Giga-LES dataset, 219–222 large-scale numerical model, 200 lattice cloud model, 198 organized tropical convection, 186 prescribed values, 201 rainfall event distribution, 265–268 stationary distribution, 201–202 transition probabilities, 199 tropical modes (see Tropical modes) unresolved variability, 198 Stochastic processes, 57, 61, 64, 192, 198–200, 202, 225, 228 Stratiform instability budget analysis, 90 CAPE evolution, 90, 92, 93 convectively coupled tropical waves, 87 direct-deep convective heating, 89 downdrafts, 89 nonlinear simulations, 92–97 organized convection Kelvin waves, 233–234 MJO, 233–234 physical structure of waves, 96 quasi-equilibrium, 87 sensitive parameters, 90 two-baroclinic model, 90 wave-CISK and WISHE theories, 87 WISHE, 94 T Trapped waves, see Equatorially trapped waves Tropical modes of variability CRCP and CRM, 245–246 Markov process, 246 random parameters, 246 SMCM parameterization (see Stochastic multicloud model (SMCM)) Tropical waves GARP-GATE, 42 Hovm¨oller diagram, 42, 43

Index OLR, 42 QBO, 41 systematic use, 41, 42 V Vertical normal modes, 15–17 Vertical shear, 12, 117, 120, 130–132, 151, 153, 239, 242 W Warm pool simulations climatological means, 176 HOMME-MCM model, 175

303 horizontal structure, 178 ITCZ, 172, 173 lag-latitude plots, 177, 178, 181 MJO-like disturbances, 173 time mean vorticity contours, 178, 180 vertical structure, 176 Wave instability, 84, 129, 158 Wet-bulb temperature, 32 Wind-induced surface heat exchange (WISHE), 75, 82–86, 94, 99 Y Yanai waves, 13–16, 126