Numerical Weather Prediction and Data Assimilation [6, 1 ed.] 1786301415, 9781786301413

Table of contents :
Cover
Numerical Weather Prediction
and Data Assimilation, Volume 6
Copyright Page
Contents
Preface
Introduction
1. The Primitive Equations
1.1. Wind forecast equations (conservation of momentum)
1.1.1. Real forces
1.1.2. Apparent forces
1.1.3. Equation of motion
1.2. Continuity equation (conservation of mass)
1.3. Temperature forecast equation (conservation of energy)
1.4. Moisture forecast equation (conservation of water vapor)
1.5. Synopsis of equations
2. Solving Methods in NWP Models
2.1. Decomposition of variables – the perturbation method
2.1.1. Synoptic scale equations
2.1.2. Mesoscale equations
2.2. Numerical solutions of partial differential equations
2.2.1. Computations by finite difference schemes
2.2.2. Derivation of finite difference representations
2.2.3. Methods for solving the advection–diffusion equation
2.3. Time splitting
3. Domain Structures and Boundary Conditions
3.1. Horizontal grid structure and resolution
3.2. The vertical coordinate system
3.2.1. Terrain-following coordinate system
3.3. Boundary conditions
3.3.1. Lateral boundary conditions
3.3.2. Upper (top) boundary conditions
3.3.3. Lower (bottom) boundary conditions
3.4. Design of a simulation
4. Introduction to Data Assimilation
4.1. Successive correction methods
4.2. Least square method
4.3. Variational approach
4.4. Generalization of the methods
5. Desert Dust Modeling
5.1. Dust uptake mechanisms formulation
5.2. Dust advection and deposition
5.3. Parameterization of the dust feedbacks on climate
6. Simulations of Extreme Weather and Dust Events
6.1. Case study 1: numerical simulation of a Mediterranean cyclone and its sensitivity on lower boundary conditions
6.1.1. Description of the synoptic conditions
6.1.2. Design of the simulations
6.1.3. Analysis of the numerical simulations
6.2. Case study 2: nowcasting an extreme precipitation event
6.2.1. Synoptic analysis of the event
6.2.2. Nowcasting methodology and results
6.3. Case study 3: seasonal predictability of a large-scale heat wave
6.3.1. Description of the synoptic conditions
6.3.2. Model description and methodology
6.3.3. Predictability assessment
6.4. Numerical study of a severe desert dust storm over Crete
Appendices
Appendix 1
Appendix 2
A2.1. Turbulent diffusion parameterizations
A2.2. Planetary boundary layer (PBL) parameterization
A2.2.1. Surface layer parameterization
A2.2.2. Viscous sublayer parameterization
A2.2.3. Transition layer parameterization
A2.3. Transforming the vertical coordinate
References
Index
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Citation preview

Numerical Weather Prediction and Data Assimilation

Engineering, Energy and Architecture Set coordinated by Lazaros E. Mavromatidis

Volume 6

Numerical Weather Prediction and Data Assimilation

Petros Katsafados Elias Mavromatidis Christos Spyrou

First published 2020 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc.

Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address: ISTE Ltd 27-37 St George’s Road London SW19 4EU UK

John Wiley & Sons, Inc. 111 River Street Hoboken, NJ 07030 USA

www.iste.co.uk

www.wiley.com

© ISTE Ltd 2020 The rights of Petros Katsafados, Elias Mavromatidis and Christos Spyrou to be identified as the authors of this work have been asserted by them in accordance with the Copyright, Designs and Patents Act 1988. Library of Congress Control Number: 2020931739 British Library Cataloguing-in-Publication Data A CIP record for this book is available from the British Library ISBN 978-1-78630-141-3

Contents

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

vii

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

ix

Chapter 1. The Primitive Equations . . . . . . . . . . . . . . . . . . . . . . . . . .

1

1.1. Wind forecast equations (conservation of momentum) . . . 1.1.1. Real forces . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.2. Apparent forces . . . . . . . . . . . . . . . . . . . . . . 1.1.3. Equation of motion . . . . . . . . . . . . . . . . . . . . 1.2. Continuity equation (conservation of mass) . . . . . . . . . 1.3. Temperature forecast equation (conservation of energy) . . 1.4. Moisture forecast equation (conservation of water vapor) . 1.5. Synopsis of equations . . . . . . . . . . . . . . . . . . . . .

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1 6 14 17 19 21 26 26

Chapter 2. Solving Methods in NWP Models . . . . . . . . . . . . . . . . . . . .

31

2.1. Decomposition of variables – the perturbation method . . . 2.1.1. Synoptic scale equations . . . . . . . . . . . . . . . . . 2.1.2. Mesoscale equations . . . . . . . . . . . . . . . . . . . 2.2. Numerical solutions of partial differential equations . . . . 2.2.1. Computations by finite difference schemes . . . . . . . 2.2.2. Derivation of finite difference representations . . . . . 2.2.3. Methods for solving the advection–diffusion equation 2.3. Time splitting . . . . . . . . . . . . . . . . . . . . . . . . .

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31 39 40 43 44 46 50 63

Chapter 3. Domain Structures and Boundary Conditions . . . . . . . . . . .

67

3.1. Horizontal grid structure and resolution . . . . . . . . . . . . . . . . . . . . . . 3.2. The vertical coordinate system . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1. Terrain-following coordinate system . . . . . . . . . . . . . . . . . . . . .

68 71 72

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3.3. Boundary conditions. . . . . . . . . . . . . 3.3.1. Lateral boundary conditions . . . . . . 3.3.2. Upper (top) boundary conditions . . . 3.3.3. Lower (bottom) boundary conditions . 3.4. Design of a simulation. . . . . . . . . . . .

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86 88 90 90 99

Chapter 4. Introduction to Data Assimilation . . . . . . . . . . . . . . . . . . .

101

4.1. Successive correction methods . 4.2. Least square method . . . . . . . 4.3. Variational approach . . . . . . 4.4. Generalization of the methods .

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Chapter 5. Desert Dust Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . .

119

5.1. Dust uptake mechanisms formulation . . . . . . . . . . . . . . . . . . . . . . . 5.2. Dust advection and deposition . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3. Parameterization of the dust feedbacks on climate . . . . . . . . . . . . . . . .

121 127 132

Chapter 6. Simulations of Extreme Weather and Dust Events . . . . . . . .

143

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102 103 108 114

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6.1. Case study 1: numerical simulation of a Mediterranean cyclone and its sensitivity on lower boundary conditions . . . . . . . . . . . . . . . . . 6.1.1. Description of the synoptic conditions . . . . . . . . . . . . . . . 6.1.2. Design of the simulations . . . . . . . . . . . . . . . . . . . . . . 6.1.3. Analysis of the numerical simulations . . . . . . . . . . . . . . . 6.2. Case study 2: nowcasting an extreme precipitation event . . . . . . . 6.2.1. Synoptic analysis of the event . . . . . . . . . . . . . . . . . . . . 6.2.2. Nowcasting methodology and results . . . . . . . . . . . . . . . . 6.3. Case study 3: seasonal predictability of a large-scale heat wave . . . 6.3.1. Description of the synoptic conditions . . . . . . . . . . . . . . . 6.3.2. Model description and methodology . . . . . . . . . . . . . . . . 6.3.3. Predictability assessment . . . . . . . . . . . . . . . . . . . . . . . 6.4. Numerical study of a severe desert dust storm over Crete . . . . . . .

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143 143 145 147 151 151 153 157 158 159 161 163

Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

169

Appendix 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

171

Appendix 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

179

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

197

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

207

Preface

This book reflects the need to provide fundamental knowledge about theoretical and applied numerical weather prediction (NWP) and data assimilation (DA). It has been written to support undergraduates and graduates in atmospheric or Earth sciences and introduce students to elementary atmospheric dynamics and modeling, finite difference methods, numerical parameterizations, and optimization methods. This book has a more introductory and applied approach to the methods and techniques for NWP. It also includes essential materials for meteorological DA and modeling of the desert dust cycle in the atmospheric environment. Emission patterns, advection methods and deposition processes are all deployed in the chapter describing the desert dust cycle. The final chapter consists of real case studies simulating extreme weather events and a dust outbreak in hindcasting, nowcasting and forecasting modes. This book is divided into six main chapters and two appendices. Apart from a brief introduction to NWP and DA in the Introduction, the theoretical material is covered in Chapters 1–4. The primitive equations governing the main atmospheric motions are analytically presented in Chapter 1, and the methods of solutions and finite difference schemes are included in Chapter 2. The implementation of the primitive equations on grid structures with boundary condition treatment is presented in Chapter 3. DA including successive correction methods and the variational approach with simple examples are introduced in Chapter 4. Chapter 5 is devoted to the analysis and modeling of desert dust processes, as well as a review of the parameterizations of dust feedbacks on climate. Finally, the simulations of three extreme weather events and a desert dust outbreak are presented in Chapter 6. The cases have been chosen as paradigms of phenomena on different time and spatial scales and to explore how such processes are eventually resolved in an atmospheric simulation. The appendices are presented at the end of this book, which include the basics of vector analysis and transformations into a rotating coordinate system, as well as turbulent diffusion and planetary boundary layer parameterizations.

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Acknowledgments: We are indebted to a number of colleagues and PhD candidates for their contributions during the preparation of this book. Dr. George Varlas, Ms. E. Papadopoulou, Ms. V. M. Nomikou and Ms. A. Pappa are all acknowledged for their contributions to performing a part of the embedded simulations and results analysis. The European Centre for Medium Range Weather Forecasts (ECMWF), the National Center for Environmental Predictions (NCEP) and the National Oceanic and Atmospheric Administration (NOAA) are acknowledged for providing gridded analyses and climatologies, as well as surface observational data. The National Center for Atmospheric Research (NCAR) and the University Corporation for Atmospheric Research (UCAR) are also acknowledged for making available to us the Community Atmosphere Model version 3 (CAM3) and the Weather Research and Forecasting (WRF) model. Finally, we are grateful to the Hellenic National Meteorological Service (HNMS) and the National Observatory of Athens (NOA) for providing the precipitation measurements used in the case study of nowcasting in Chapter 6. Petros KATSAFADOS Elias MAVROMATIDIS Christos SPYROU February 2020

Introduction

Numerical weather prediction (NWP) is the state-of-the-art method for supporting atmospheric modeling and weather forecasting that combines a set of differential equations, describing grid scale motions, with parameterizations of the non-physically resolved processes usually deployed in the sub-grid scale. All of these are applied to a geographical domain with specific resolution and integrated on the basis of initial and domain boundary conditions. The set of differential equations govern changes in the motion and thermodynamics of the atmosphere, which are derived from conservation laws of mass, momentum, energy and moisture. They are written in the Eulerian framework, in which values and their partial derivatives (changes in the variable over time, for example, ∂T/∂t, or space ∂T/∂x) are considered at fixed locations on Earth. The atmospheric variables of the equations (e.g. temperature, humidity, wind components, pressure and many others) have independent variables in space, longitude (x), latitude (y), height (z) and time (t). The partial derivatives of the atmospheric variables are extremely complex, hence they cannot be solved analytically. Therefore, only approximate solutions are obtained through advanced numerical methods. Since these equations govern how the variables change in space and time, knowledge of the initial condition of the atmosphere is essential to solve the equations and estimate new values of these variables. Thus, NWP is considered as an initial value problem. Various types of weather observations can serve as input to produce initial conditions of the differential equations through a process called data assimilation (DA). It is a method of combining observations with model outputs in order to reduce the errors of the latter. This method is based on the optimal fitting of the model state to the observations for a given time to produce analysis fields which correspond to the best estimation of the atmospheric variables.

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A notable pioneer of meteorology, Vilhelm Bjerknes, initially approached the NWP concept in the beginning of the 20th Century. He postulated that governing equations of fluid dynamics could be solved forward in time to predict the future state of the atmosphere given its current state. The fundamental problem raised in this hypothesis is that the complex set of governing equations has approximate solutions instead of analytical solutions. In addition, the accurate measurement of the current atmospheric state was almost impossible at that period of time. It was not until almost 20 years later that Lewis Fry Richardson attempted to solve the partial differential equations of fluid dynamics by hand. He initiated his calculations based on the recorded atmospheric observations on May 20, 1910 at 07:00 to estimate the air pressure over Western Europe 6 hours later from the initial date. It took him almost 6 weeks to solve the set of equations, and he predicted a rather unrealistic rise of air pressure of 145 hPa. Despite his errors, Richardson was the first to attempt weather prediction almost 30 years before the first atmospheric simulation carried out in the Electronic Numerical Integrator and Computer (ENIAC). The aim of the project deployed in the ENIAC was to predict the weather by simulating the dynamics of the atmosphere. In April 1950, using the ENIAC, Jule Charney and John von Neumann performed the first atmospheric simulation by solving the barotropic vorticity equation over a domain covering Northern America. Since largescale atmospheric motions are assumed to be predominantly barotropic, this was the first step towards predicting the weather. The ENIAC required more than 1 day to perform a 24-hour weather forecast, and therefore the calculation process lasted longer than the actual weather to occur. In the following decades, the continuous progress of computing power made NWP more robust and reliable. In 1961, Edward Lorenz, an American mathematician and meteorologist, proposed chaos theory for weather prediction. Lorenz realized that errors had been introduced into the model, which were impossible to prevent, propagate in the computational domain and would eventually attract the forecast into chaos. Thus, infinitesimal discrepancy on initial and boundary conditions would lead to completely different deterministic forecasts. The range of these differences would depend on the accuracy of the initial and boundary conditions. This idea was troubling, because it meant that there was a limited time frame within weather forecasts to be reliable. Despite this restriction, the predictability of deterministic forecasts has been increasing by almost 1 day per decade. Nowadays, operational NWP offers reliable products in a forecast window of up to 10 days thanks to the dramatic advances in high-performance computing (HPC), the atmospheric modeling and the optimization of the simulation codes thereof.

1 The Primitive Equations

The primitive equations describing changes in the atmospheric motion and thermodynamics are based on the mathematical expressions of a complete set of conservation principles. These equations are applied to individual air parcels and, due to their accuracy, allow continued progress in NWP models. There are three fundamental physical principles that govern the atmospheric motions: the conservation of momentum (provides equations of the three-dimensional motion), the conservation of the dry air mass (this equation is known as the continuity equation) and, finally, the conservation of heat (based on the first law of thermodynamics). The set is completed by formulas related to the conservation of water (all phases: solid–liquid–gas), as well as with the equation of state for perfect gases. Spectral and grid point models use the same coupled set of governing equations to describe changes occurring at discrete locations within the forecast domain. In order to extract the mathematical equations expressing the aforementioned principles, the system is usually analyzed using a differential control volume. In fluid dynamics, there are two types of control volume in use. The first type consists of a parallelepiped in fixed location on the Earth relative to the coordinate system (Eulerian framework), where the variables (e.g. B) and their derivatives (variable changes over time or space, for example ∂B/∂t or ∂B/∂x) are estimated. According to the second type, the control volume follows the fluid motion assuming that it always contains the same part of the fluid which comprises marked fluid particles (Lagrangian framework). 1.1. Wind forecast equations (conservation of momentum) The conservation of momentum in the atmosphere comes from Newton’s second law of motion. According to this law, a body (e.g. air parcel) is accelerated when a

Numerical Weather Prediction and Data Assimilation, First Edition. Petros Katsafados, Elias Mavromatidis and Christos Spyrou. © ISTE Ltd 2020. Published by ISTE Ltd and John Wiley & Sons, Inc.

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force or a sum of forces is applied to it. The acceleration of the body is proportional to the applied force, inversely proportional to the body’s mass, while its direction is the same as the direction of the force. The acceleration can be written in vector form as:  ΣF α= m 

[1.1]

 Above, ΣF is the sum of forces acting on the body (in Newton), m its mass  (in kg) and α is the total or inertial acceleration (in ms−2). The inertial acceleration is the rate of velocity’s change of the body in relation to an inertial coordinate system. An inertial coordinate system is a system that moves with a constant velocity (v = constant without excluding zero), while a non-inertial reference frame is the one that accelerates or rotates.

 On an inertial reference frame, the absolute acceleration of an air parcel ( α ) can be written as: 

α=

 d aVa dt

[1.2]

Above, the subscript α refers to the non-accelerating coordinate system, while   the acceleration (α ) is the change of velocity d aVa with respect to time ( dt )

(

)

following the parcel of air during its movement. Nevertheless, atmospheric motions are related to the rotating Earth. Hence, on a rotating coordinate system centered at the center of the planet with constant   angular velocity Ω (Ω = 7.29 × 10−5 rad/sec), the absolute velocity Va of an air parcel can be written as the sum of two velocities: a) the velocity relative to the Earth and b) the velocity due to the rotation of the Earth (see Appendix 1, equation [A1.5] or [A1.7]):     Va = V + Ω × R

[1.3]

At the equation above, the position vector of an object standing on the rotating  Earth (e.g. parcel of air) is indicated by R , as it is measured from the Earth’s center. Accordingly, the total time differential operator in an inertial system dα/dt can be represented by the sum of the total time derivative d/dt, which is the derivative

The Primitive Equations

3

in relation to the surface of the Earth, adding the changes resulted from the rotation rate of the planet (Pielke 2002). According to the above, this operator is given by:

da d = + Ω× dt dt

[1.4]

 For any vector A , equation [1.4] gives:

   da A  d    d a A dA   =  + Ω × A  = +Ω×A dt dt dt  dt 

[1.5]

  For A = Va , the above formula becomes:

  d aVa dVa   = + Ω × Va dt dt

[1.6]

 Finally, substituting the vector Va (from equation [1.3]) and considering that   V = dR dt (Pielke 2002), equation [1.6] yields (see Appendix 1, equation [A1.8]):

            d aVa d (V + Ω × R )  dV d (Ω × R )    = + Ω × (V + Ω × R ) = + + Ω × V + Ω × (Ω × R )  dt dt dt dt        dV  dR    dV      + Ω× + Ω × V + Ω × (Ω × R ) = + Ω × V + Ω × V + Ω × (Ω × R )  α= dt dt dt 

      dV + 2(Ω × V ) + Ω × (Ω × R ) α= dt 

[1.7] The first term on the right-hand side of equation [1.7] is the local acceleration   ( alocal = dV dt ) . Local acceleration depicts the rate of velocity’s variation of a moving object (e.g. parcel of air) in relation to a rotating coordinate system fixed on the Earth (relative coordinate system). The second term is related to the Coriolis    acceleration [ aCo = −2(Ω × V )] . Coriolis acceleration is the rate of velocity’s change due to the rotation of the Earth and operates only with the existence of motion. The     third term indicates the centripetal acceleration [acp = Ω × (Ω × R)] . Specifically, the centripetal acceleration is the rate of change of the object’s velocity variation on

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account of its motion around the Earth’s axis of rotation. This rate of change, which is displayed by δV in Figure 1.1, is directed to the rotation axis (Holton 2004). On the right-hand side of equation [1.1], terms of real forces are included. These terms, which affect the local acceleration of an air parcel, are the gravitational force   ( Fg * ), the pressure gradient force ( Fp ) arising from spatial pressure gradients and  the viscous force ( Fv ) due to the air molecules exchanging momentum.

Figure 1.1. The direction of centripetal acceleration. The distance between the air parcel and the axis of rotation is denoted with r and is perpendicular to this axis

Expanding the right-hand side of equation [1.1] in combination with [1.7] gives:          dV 1   ΣF a=  + 2(Ω × V ) + Ω × (Ω × R ) = ( Fg * + Fp + Fv ) m Eq 1.7 dt m

[1.8]

The above equation indicates that on a rotating coordinate system, there are also in action two apparent forces per unit mass. The first is the Coriolis force (for m = 1 kg), namely the Coriolis acceleration (mentioned above), which is a fundamental force for air motion in meso- and large-scale (far from the equator):        FCo    FCo = aCo  FCo = −2(Ω × V ) = −2Ω × V aCo = m = 1 kg m

[1.9]

Furthermore, the abovementioned centripetal acceleration can also be considered as a force per unit mass (centripetal force for m = 1 kg). The centripetal force continuously acts on a parcel of air as it turns around the Earth and is given by (see Appendix 1, equation [A1.14]):            acp = Fcp m  Fcp = acp = Ω × (Ω × R ) = Ω × (Ω × r ) = −Ω 2 r m =1kg

[1.10a]

The Primitive Equations

5

In the above equation, r is the distance of the air parcel that is perpendicular to the axis of rotation. Instead of centripetal, we can use the centrifugal force per unit mass. The centrifugal force is an additional apparent force, opposite of the centripetal force:

       macf = Fcf = − Fcp = −m Ω × (Ω × r ) = −m(−Ω 2 r )     macf = m Ω 2 r = Ω 2 r

[1.10b]

m =1kg

The application of the centrifugal force is necessary for an observer, who stands on the rotating coordinate system and has to apply Newton’s second law in order to describe the motion in relation to the rotating and non-inertial coordinates. For any such observer, who participates in the motion, the centrifugal force can be considered as a real force. This occurs due to the fact that the air body is stationary at the rotating coordinate system even though the centripetal force still acts upon it. Therefore, the centrifugal force is introduced in order to balance the centripetal one. To summarize, if the rotating air parcel is observed from a coordinate system fixed in the space, it undergoes a centripetal acceleration. If the parcel of air is observed from a rotating system along with it, the air parcel can be considered as stationary and the centripetal force is balanced by the centrifugal force. The expressions of local acceleration are commonly used in atmospheric models. Hence, the equation of motion can be written in a more convenient reference frame fixed to the rotated system. To achieve this objective, the combination of   equation [1.8] up to [1.10] gives for alocal = dV dt :

    FCo Fcp 1   +  ( F * + Fp + Fv ) = alocal − m g m m     1   alocal = ( Fp − Fcp + Fg* + Fv + FCo )  m     1   alocal = ( Fp + Fcf + Fg* + Fv + FCo ) m

[1.11]

It should be mentioned that the forces acting on a parcel of air can also be classified into the following two types in dependence to their point of action and their relation to the body mass. a) Body forces: they are proportional to mass and act on the center of mass and b) surface forces: they are independent of mass and act on the surface of parcel.

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Numerical Weather Prediction and Data Assimilation

1.1.1. Real forces Pressure gradient force (PGF). The PGF acts on an air parcel and is classified as surface force. It is referred to as pressure gradient because the net force is not related to the pressure itself but is proportional to the gradient of the pressure field. It should be noted that the pressure gradient of atmosphere is equal to the variation in atmospheric pressure directed perpendicular to the isobaric curves. As discussed previously, it is common in meteorology to use the force per unit mass (acceleration), instead of the force itself. Therefore, what we refer to as PGF is actually the part of the acceleration due to the PGF. To facilitate the study, we often visualize the air parcel as a cube-like shaped differential (infinitesimal) element of air with the following characteristics: the parcel is centered at the point (x0, y0, z0), its dimensions are δx-δy-δz, and its volume is equal to δ V = δ xδ yδ z (Figure 1.2). After the above explanations, the PGF in the x-direction is derived as follows: considering that the pressure at the center (x0) of air parcel is equal to p0, the pressure on a wall (e.g. A in Figure 1.2) can be expressed (Holton 2004) in a one-dimensional Taylor series as:

PA = p0 +

1 ∂p δ x + (C + .......... + Z ) 2 ∂x

[1.12]

where (C+……..+Z) are higher order terms.

Figure 1.2. A schematic representation of the pressure gradient force (x-component) acting on a differential fluid parcel. The parcel’s volume is δV = δxδyδz

The Primitive Equations

7

Ignoring the higher order terms in equation [1.12], the force acting on the wall A of the volume element can be written as: 1 ∂p   FA = PA S A = −  p0 + δ x  δ yδ z 2 ∂x  

[1.13]

where the multiplication δ yδ z represents the surface area (SA) of the wall A, while the direction to the right is considered as the positive one. Similarly, the force that acts on the wall B of the volume element is expressed by: 1 ∂p   FB = PB S B = +  p0 − δ x  δ yδ z 2 ∂x  

[1.14]

Consequently, the net component of the force acting on the fluid element in x-axis is: FPx = FA + FB  FPx = − p0 δ yδ z −

FPx = −

1 ∂p 1 ∂p δ x δ yδ z + p0 δ yδ z − δ x δ yδ z  2 ∂x 2 ∂x

[1.15]

∂p δ x δ yδ z ∂x

The mass m of the element can be calculated by multiplying the density ρ by the volume, namely m = ρδ xδ yδ z , while the x component of the PGF per unit mass is given by the equation: FPx 1 ∂p =− m ρ ∂x

[1.16]

In the same way, the other two components of the PGF per unit mass (y and z components) are given by: FPy

1 =− m ρ FPz 1 =− m ρ

∂p ∂y ∂p ∂z

[1.17]

8

Numerical Weather Prediction and Data Assimilation

    FP = FPx i + FPy j + FPz k

∂p  ∂p  ∂p   i+ j + k = ∇p (Appendix 1, ∂x ∂y ∂z equations [A1.9] and [A1.10]), the total PGF per unit mass is given by:

Since

and

 FP FPx  FPy  FPz  1 ∂p  1 ∂p  1 ∂p  i+ j+ k =− i− j− k = m m m m ρ ∂x ρ ∂y ρ ∂z

[1.18]  1   FP = a p = − ∇p m =1kg

ρ

The following properties related to PGF arise from equation [1.18]: – the PGF is always directed to the opposite direction of the pressure gradient  ∇p ;

( )

– the stronger the pressure gradient, the stronger the PGF. It should also be noted that, if the distance between the adjacent isobars is known (from isobaric maps), the PGF can be estimated using the following approximation Δp ∇p ≅ , where Δp is the contour interval of isobars and Δn is the horizontal Δn distance between the isobars.  Gravitational force ( Fg * ). The gravitational force is the second real external

force. According to Newton’s gravitational law, two hypothetical point masses somewhere in the universe attract each other by a force. This force has two key features: a) it is proportional to the point masses (M and m) and b) it is inversely proportional to the square of the distance (r) among them. For finite bodies (non-point masses), the length (r) is equal to the distance between the mass centers of the bodies. Thus, assuming that Mp is the mass of the Earth and m is the mass of  an atmospheric element, the force ( g * ) per unit mass due to the gravitational attraction of the Earth is given by:  Fg*  *  GM p m  GM p  * Fg* ≡ mg = − r0  ≡ g = − 2 r0 2 m r r

[1.19]

Above, G is the universal gravitational constant (6.673 × 10−11 N m2 kg−2) and  r0 is the unit vector on the axis that crosses the two centers (Figure 1.3).

The Primitive Equations

9

It is common in meteorology to use, as a vertical coordinate, the height above mean sea level (MSL). If the mean radius of the Earth is indicated by R (6,378 km) and the height above MSL is indicated by z, it is valid that r = R + z. Therefore, equation [1.19] can be rewritten as:

GM p  GM P / R 2  GM P / R 2   g* = − r = − r = − r0  0 0 ( R + z )2 ( R + z)2 / R2 ( R / R + z / R) 2 GM P / R 2  GM P  1 1   g* = − r =− r0 = g0* 2 0 2 2 2 (1 + z / R) (1 + z / R) R (1 + z / R)

[1.20]

Figure 1.3. Two spherical masses (the Earth and an atmospheric element) whose centers are separated by a distance r

GM    In the above equation, g 0* ( g 0* = − 2 P r0 ) is the gravitational force (per unit R mass) at MSL (Holton 2004). The height z is considered to be much lower than R in meteorological applications (z>acf , namely g*>>Ω2r, while  the small variations of the modified gravity ( g ), appearing due to the topography of the area (different heights above sea level) or due to the geographical location  of the area, are usually ignored. For these reasons, g can be considered constant and equal to 9.8067 ms−2.    Coriolis force ( FCo = − 2Ω × V ). The Coriolis force is the second apparent force that arises from a rotating reference frame. A body moving on a plane normal to the Earth’s axis of rotation undergoes this force, called Coriolis in honor of the French scientist Gustav-Gaspart Coriolis, who described it mathematically (1835). The Coriolis force is due to the rotation of the Earth and acts only in the direction of motion. The acceleration due to the Coriolis force is equal to the Coriolis force per unit mass and is given by the expression:     aCo = FCo = −2Ω× V . The Coriolis force gives horizontal acceleration normal m =1kg

to the direction of motion depending on the direction of rotation. This means that if the system rotates counter-clockwise (anticlockwise), similar to the rotation of  the Earth, the force will be directed to the right-hand side of velocity V and vice versa. Thus, if an object is moving eastwards (u > 0), then it is deflected southward (towards the equator), whereas an object moving towards the West (u < 0) will be deflected poleward. In a similar way, it can be derived that an object will be deflected eastward due to the Coriolis, if it is moving towards the North (υ > 0). In any case, Coriolis tends to deflect the bodies in the Northern Hemisphere to the right and to the left of the motion in the Southern Hemisphere.   In relation to the rotating reference frame at the surface of the Earth, Ω and V can        be written in component form as Ω = Ω cos ϕ j ′ + Ω sin ϕ k ′ and V = ui ′ + υ j ′ + wk ′ (see Appendix 1, Figures A1.2a and A1.2b). Therefore, the Coriolis acceleration has the components presented below (see Appendix 1, equation [A1.15]): 







α Co = (2Ωυ sin ϕ − 2Ω w cos ϕ )i ′ − 2Ωu sin ϕ j ′ + 2Ω u cos ϕ k ′

[1.31]

According to the above equation, the Coriolis force increases from the equator (zero value) to the poles, where it is maximized and becomes equal to

16

Numerical Weather Prediction and Data Assimilation

2ΩV (Figure 1.7). The angular velocity of the Earth given by Ω = 2π rad/day is equal to 7.292 × 10−5 s−1. Note that the duration of the day is indicated by the stellar day, which lasts 23 hours and 56 minutes. Furthermore, the influence of a counterclockwise rotating sphere (Earth) to the rectilinear motion of an object is presented in Figure 1.8.

Figure 1.7. Variation of the Coriolis force (ms−2) in accordance with the latitude and wind speed

Figure 1.8. The influence of a counterclockwise rotating sphere (Earth) to the rectilinear motion of an object. Objects deflect to the right in the Northern Hemisphere and to the left in the Southern Hemisphere. For a color version of the figure, see www.iste.co.uk/KMS/numerical.zip

The Primitive Equations

17

The Coriolis force is substantial for long time scale phenomena such as the synoptic scale systems, while it is negligible for motions in very short time scales. In a spherical coordinate system, the horizontal component of the Coriolis force is given in vector form (see Appendix 1, equation [A1.17]):      FCo = −2Ω × V = − fk ′ × V

[1.32]

 Above, with f is denoted the coriolis parameter [ f = 2Ω sin ϕ ], k ′ corresponds to the unit vector that is normal to the horizontal surface of motion and is parallel to  the rotation axis only near the Poles, V is the horizontal velocity and φ is the latitude (Holton 2004). The Coriolis deflection character becomes evident by the    fact that the vector −k ′ × V rotates 90° to the right of velocity V . Thus, it is obvious that the Coriolis force is able to change only the direction rather than the speed of the motion.

1.1.3. Equation of motion

The conservation of momentum equation in the non-inertial coordinate system (equation [1.11]) can be expressed as:

        Fp Fg Fv FCo dV 1   = ( Fp + Fg + Fv + FCo ) = + + +  alocal = dt m m m m m      dV 1  = − ∇p − gk +ν ∇ 2V − 2Ω × V ρ dt

[1.33]

Since the velocity is a function of time and location at a specific time,   V = V [ x(t ), y (t), z(t), t] , the left-hand side of equation [1.33] can be written (see

{

}

Appendix 1, equation [A1.12]) as:

         ∂V dV ∂V dx ∂V dy ∂V dz ∂V = + + + = (V ⋅∇)V + dt ∂x dt ∂y dt ∂z dt ∂t ∂t

[1.34]

The left-hand side of equation [1.33] or [1.34] can also be analyzed in Cartesian coordinates to:     dV d (ui + υ j + wk ) du  dυ  dw  = = i+ j+ k dt dt dt dt dt

[1.35]

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Numerical Weather Prediction and Data Assimilation

The analysis of the right-hand side of equation [1.34] also gives:     ∂  ∂  ∂      ∂V ∂V   (V ⋅∇)V + j + k V  = + (ui + υ j + wk ) ⋅  i + ∂t ∂t ∂y ∂z   ∂x      ∂V  ∂  ∂  ∂ ∂   (V ⋅∇)V + =  +  u + υ + w   (ui + υ j + wk )  ∂t  ∂t  ∂x ∂y ∂z       ∂V  ∂u ∂u ∂u ∂u   (V ⋅∇)V + =  + u +υ + w i ∂t  ∂t ∂x ∂y ∂z   ∂υ ∂υ ∂υ ∂υ   + +u +υ +w  j ∂x ∂y ∂z   ∂t

[1.36]

 ∂w ∂w ∂w ∂w   + +u +υ + w k ∂x ∂y ∂z   ∂t Therefore, according to equation [1.34], equation [1.33] can be rewritten in the standard form of the conservation of momentum that is called the equation of motion:      1     ∂V [1.37] = −(V ⋅ ∇)V − ∇p − gk + ν ∇ 2V − 2Ω × V ∂t ρ Furthermore, the gradient squared term can be analyzed in Cartesian coordinates to: 

  

 ∂2 ∂2 ∂2 + 2+ 2 2 ∂y ∂z  ∂x

ν ∇ 2V = ν (∇ ⋅∇)V = ν  

     (ui + υ j + wk )  

 ∂ 2 u ∂ 2 u ∂ 2 u    ∂ 2υ ∂ 2υ ∂ 2υ   + 2 + 2 i + v 2 + 2 + 2  j + 2 ∂y ∂z  ∂y ∂z   ∂x  ∂x

ν ∇ 2V = v 

[1.38]

 ∂2 w ∂2 w ∂2 w   v 2 + 2 + 2 k ∂y ∂z   ∂x

If the model’s horizontal grid size is greater than 10 km, the vertical component of the equation of motion is replaced by its hydrostatic approximation. In the hydrostatic approximation, the vertical acceleration is considered negligible, in comparison to the buoyancy (gravitational acceleration). Thus, it is a convenient way to use the atmospheric pressure as a vertical coordinate, instead of height.

The Primitive Equations

19

1.2. Continuity equation (conservation of mass) The conservation of mass is the second fundamental physical principle, whose mathematical expression is called the continuity equation. Therefore, the mass has neither sinks nor sources within the atmosphere. Consequently, the net rate at which the mass is inserted into a differential box (inflow minus outflow of the box) equals the rate of mass change within the air volume. Considering a differential volume element with dimensions fixed in a Cartesian coordinate system denoted as δx, δy, δz (Eulerian derivation sketched in Figure 1.9), the rate of inflow of the air mass per unit area through the left face is given by 1 ∂( ρ u ) ρu − δ x , whereas the corresponding outflow through the right face can be 2 ∂x 1 ∂( ρ u) δx. written as ρ u + 2 ∂x

Figure 1.9. Schematic representation of a fixed differential volume element and mass inflow into it due to Eulerian motion in parallel direction to the x-axis of the Cartesian coordinate system

Since the area of each face is equal to δyδz, the net flow rate within the differential volume due to the x velocity component is given by (Holton 2004): 1 ∂( ρ u )  1 ∂( ρu)  ∂( ρu)    ρ u − 2 ∂x δ x  δ yδ z −  ρ u + 2 ∂x δ x  δ yδ z = − ∂x δ xδ yδ z    

20

Numerical Weather Prediction and Data Assimilation

Similarly, the net flows for the y- and z-directions are expressed by:

  1 ∂( ρυ )  1 ∂( ρυ )  ∂( ρυ )  ρυ − 2 ∂y δ x  δ yδ z −  ρυ + 2 ∂y δ x  δ yδ z = − ∂y δ xδ yδ z     1 ∂ ( ρ w)  1 ∂ ( ρ w)  ∂ ( ρ w)    ρ w − 2 ∂z δ x  δ yδ z −  ρ w + 2 ∂z δ x  δ yδ z = − ∂z δ xδ yδ z    

Finally, the net rate of mass inflow is:

−

∂( ρ u ) ∂( ρυ ) ∂( ρ w) δ xδ yδ z − δ xδ yδ z − δ xδ yδ z = ∂x ∂y ∂z [1.39]

 ∂ ( ρ u ) ∂ ( ρυ ) ∂ ( ρ w)  δ xδ yδ z − + + ∂y ∂z   ∂x From the above equation, the mass inflow per unit volume is:

   ∂( ρ u ) ∂ ( ρυ ) ∂( ρ w)  = − ∇ ⋅ ρV − + +  ∂y ∂z  ( A.11)  ∂x

( )

[1.40]

The mass inflow per unit volume given by equation [1.40] must be equal to the rate of mass increase per unit volume, which, in turn, is equal to the local density change ∂ρ ∂t . Hence, the mass divergence form of the continuity equation is given by equation [1.41]:

( )

   ∂ρ ∂ρ  = −∇ ⋅ ρV  + ∇ ⋅ ρV = 0 ∂t ∂t

( )

( )

[1.41]

An alternative form of the continuity equation can be obtained by applying in equation [1.41] the two equations, which are given subsequently. Since       d ∂   ≡ + V ⋅∇ (equation [A1.12]), the continuity ∇ ⋅ ρV ≡ ρ∇ ⋅ V + V ⋅ ∇ρ and dt ∂t equation is transformed into:

( )

   ∂ρ  ∂ρ   + ∇ ⋅ ρV = 0  + V ⋅ ∇ρ + ρ∇ ⋅ V = 0  A1.12 ∂t ∂t     dρ 1 dρ + ρ∇ ⋅ V = 0  + ∇ ⋅V = 0 ρ dt dt

( )

[1.42]

The Primitive Equations

21

Equation [1.42] denotes that the rate of density, if following the motion of an air parcel, is opposite of the velocity divergence and can be considered to be the expression of the divergence form of continuity equation. Accordingly, equation [1.41] denotes that the local rate of density’s change is opposite of the mass divergence. However, for an incompressible fluid, the density remains constant if following the motion. In this case, the velocity divergence vanishes due to the fact that d ρ dt = 0 and equation [1.42] gives:   1 dρ   + ∇ ⋅V = 0  ∇ ⋅V = 0 ρ dt    ∂  ∂  ∂    i + j + k  ⋅ ui + υ j + wk ∂y ∂z   ∂x

(



)

[1.43]

∂u ∂υ ∂w + + =0 ∂x ∂y ∂z

The above approximation applies for horizontal flow. In this case, the atmosphere behaves as an incompressible fluid. However, the presence of vertical motion requires to be taken into account the dependence of density to height (in other words, the compressibility of the atmospheric air). 1.3. Temperature forecast equation (conservation of energy)

The third fundamental conservation principle is the conservation of energy (heat), which is applied to a moving air parcel through the atmosphere. The atmosphere on the mesoscale behaves, in most of cases, as an ideal gas being in local thermodynamic equilibrium. This means that if a system is at rest, it remains at rest despite the fact that it has exchanged heat with its surrounding area and has produced work on the surrounding area. The thermodynamic processes that control the evolution of weather systems are represented by the first law of thermodynamics, which is an expression of the principle of conservation of energy, that is, the fact that in an isolated system, the energy is neither destroyed nor created by zero but is transformed from one form to another. The first law is a prognostic equation related to the rate of temperature change of the moving air parcel. Indeed, the differential amount of heat (dQ) absorbing or eliminating a thermodynamic system is equal to the algebraic sum of the differential change in internal energy (dU) and the differential work (dW) that the system produces or expends, as represented by the equation below (Jacobson 2005): dQ = dU + dW

[1.44]

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Numerical Weather Prediction and Data Assimilation

Concerning the differential dW, it is obtained with a positive sign when the work is produced by the system under study, while it is marked with a negative sign when it is produced by the external forces that affect the system. The differential dQ is considered to be positive when the system absorbs this amount of heat, while it is obtained with a negative sign when released from the system into the environment. If equation [1.44] is applied in a box (Pielke 2002) with volume δxδyδz (as shown in Figure 1.10), since the term δyδzdx represents a change in the volume of the box equal to dV, a differential increase of work in the x-direction caused by a force F is given as:

dW = Fdx

=

F = pδ yδ z

pδ yδ zdx  dW = pdV

[1.45]

For a substance of unit mass, the volume V can be replaced by the specific volume α (volume per unit mass), and the work that is done after a differential increase of specific volume by dα is given as follows (Wallace and Hobbs 2006): dW = pda

[1.46]

Figure 1.10. A schematic representation of a box of gas as it changes in size, due to a force F exerted over the surface δyδz

The combination of equations [1.44] and [1.46] gives an alternative expression of the first law of thermodynamics: dQ = dU + pda

[1.47]

Since the atmosphere approximates an ideal gas, as referred to above, the pressure in equation [1.47] is exerted in a uniform manner on all sides of

The Primitive Equations

23

the gas box. Assuming that a quantity of heat (dQ) is given to a unit mass of a material in the gas phase and the temperature of the material increases (temperature increment is equal to dT) without any change in phase, the ratio dQ/dT is the specific heat of the material. The specific heat depends on how the gas receives the heat. If the gas volume remains constant during the process, the specific heat is defined as the specific heat at constant volume (cv): cv = ( dQ dT )v = const

[1.48]

If the volume of the material is constant (dα = 0), then, according to equation [1.47], it is valid that dQ = dU and cv = ( dU dT )v = const . Nevertheless, for an ideal gas and according to Joule’s law, U depends only on temperature. For this reason, it is generally valid to write: cv = ( dU dT )  dU = cv dT

[1.49]

From equations [1.47] and [1.49], the first law of thermodynamics can be written as: dQ = cv dT + pda

[1.50]

Proportionally, a specific heat at a constant pressure is defined as:

c p = ( dQ dT ) p =const

[1.51]

In this case, the gas, as heat added to it, expands at constant pressure, while at the same time, its temperature rises. A certain amount of the heat absorbed by the system is spent to produce work as the system expands against the constant pressure of the environment. This means that, for the same temperature increase, larger quantity of heat must be added to the system in the constant pressure case than that in the constant volume case. For an ideal gas, it is valid that:

c p = cv + R

[1.52]

The specific heat at constant volume for dry air is equal to 717 JK−1kg−1, while the specific heat at constant pressure is equal to 1,004 JK−1kg−1. The difference between them equals the gas constant for dry air (R) that is equal to 287 JK−1kg−1. Equation [1.52] is proved as follows: [1.50]  dQ = cv dT + d ( pa) − adp

[1.53]

24

Numerical Weather Prediction and Data Assimilation

From the ideal gas equation: pa = RT  d ( pa) = d ( RT )  d ( pa ) = RdT

[1.54]

From equations [1.53] and [1.54]: dQ = cv dT + d ( pa ) − adp  dQ = cv dT + RdT − adp 

[1.55] dQ = ( cv + R ) dT − adp

At constant pressure, the term adp is equal to zero. Therefore: dQ = ( cv + R ) dT − 0  dQ = ( cv + R ) dT

[1.56]

Nevertheless, from equation [1.51]

[1.51]  dQ = c p dT

[1.57]

The combination of equations [1.56] and [1.57] gives equation [1.52], while from equation [1.55], an alternate form of the first law of thermodynamics is obtained:

dQ = c p dT − adp

[1.58]

If q represents the diabatic heating rate (Jkg−1s−1) and dt is a differential time interval, then the quantity of heat per unit mass applied to the parcel can be written as dQ = qdt , and the first law of thermodynamics becomes:

qdt = c p dT − adp  c p

dT dp dT a dp q = a +q = + dt dt dt c p dt c p

[1.59]

Replacing the specific volume a by its equal RT p (from the equation of state), equation [1.59] gives the thermodynamic energy equation considering that R c p = κ = 0.286 :

dT a dp q RT q dT κ T q ω+  ω+ = + = = dt c p dt c p pc p cp dt p cp

[1.60]

The Primitive Equations

25

Above, the vertical wind component in the (x, y, p) coordinate system is ω = dp dt . The first term on the right-hand side of equation [1.60] represents the rate of temperature’s change due to adiabatic changes (expansion or compression), while the second term includes the effects of diabatic heat sources and sinks (e.g. absorption of solar radiation, absorption or emission of longwave radiation, latent heat release). Using the expansion for total derivative (see Appendix 1, equation A1.12), the local time rate of temperature’s change can be obtained by:   dT κ T q ∂T κT q = ω+  + (V⋅∇)T = ω+  dt p cp ∂t p cp   ∂T κT q = −(V⋅∇)T + ω+  ∂t p cp

[1.61]

∂T ∂T ∂T ∂T κ T q = −u −υ −w + ω+ ∂t ∂x ∂y ∂z p cp

In equation [1.61], w indicates the vertical component of the wind in the (x,y,z) coordinate system, while the two vertical velocities (ω and w) are related as follows:

ω≡

dp ∂p   ∂p = + (V⋅∇) p + w dt ∂t ∂z

[1.62]

The above equation can be simplified as ω = w (∂p ∂z )  w = ω (∂z ∂p ) , due to the fact that the local time rates of pressure change (the first term on the right-hand side of equation [1.62]) are small in extratropical weather systems, while the second term tends to be even lower than the first one because of the quasi-geostrophic character of atmospheric motions on a large scale. After definitions above, equation [1.61] becomes: ∂T ∂T ∂T ∂T κ T q = −u −υ −w + ω+  ∂t ∂x ∂y ∂z p cp ∂T ∂T ∂T ∂z ∂T κ T q = −u −υ −ω + ω+  ∂t ∂x ∂y ∂p ∂z p cp  ∂T ∂T ∂T   κ T ∂T  q = −u +υ − + ω + ∂t ∂y   p ∂p  cp  ∂x

[1.63]

26

Numerical Weather Prediction and Data Assimilation

In equation [1.63], the first term on the right-hand side is the horizontal advection term, while the second term is the combined effect of adiabatic compression and vertical advection. 1.4. Moisture forecast equation (conservation of water vapor)

The moisture forecast equation indicates that in a parcel of air, the total amount of water is conserved as the parcel moves around. Exceptions to this rule are the existence of sources (evaporation of water E) and sinks (condensation of water C):

dq = E −C dt

[1.64]

dq ∂q   = + V ⋅∇q dt ∂t and add the continuity equation [1.41] multiplied by q, we can write the conservation of water in a flux form (see Appendix, equation [A1.18] up to [A1.20]): If we multiply equation [1.64] by ρ, expand the total derivative

  ∂( ρ q) = −∇ ⋅ ( ρ qV ) + ρ ( E − C ) ∂t

[1.65]

1.5. Synopsis of equations

Eventually, there are seven equations for seven unknown variables: V(u, v, w), T, p, ρ, and q. In order to simplify the above equations, it is assumed that the lower atmosphere is incompressible. This eliminates the dependence of density on the continuity equation, which is a reasonable approach for the lower layer of the atmosphere (0–4 km), as shown by Holton (2004). Subsequently, by introducing a new non-dimensional pressure variable π (called Exner function):

π ≡ ( p ps )

R

cp

[1.66]

and, defining the potential temperature as: Θ ≡ ( ps p )

R

cp

[1.67]

The Primitive Equations

27

it can be easily proved that: Θ=T π

[1.68]

Above, ps is a reference pressure usually 1,000 hPa. According to the above   definitions, the term (1 ρ ) ∇p can be replaced by cp ⋅ Θ ⋅ ∇ π (see Appendix 1, equation [A1.21]), and the governing equations are transformed as follows: Equation of motion (three equations, one for each component)       dV = −c p Θ∇π − gk + ν ∇ 2V − 2Ω × V  dt          ∂V = −(V ⋅ ∇)V − c p Θ∇π − gk + ν ∇ 2V − 2Ω × V ∂t

The horizontal components of the above equation can be written as:

∂u ∂u ∂u ∂u ∂π = −u − υ − w − cpΘ ∂t ∂x ∂y ∂z ∂x  ∂ 2u ∂ 2u ∂ 2u  + 2υΩ sin ϕ − 2wΩ cos ϕ + v  2 + 2 + 2  ∂y ∂z   ∂x

[1.69]

and

∂υ ∂υ ∂υ ∂υ ∂π = −u −υ −w − cpΘ ∂t ∂x ∂y ∂z ∂y  ∂ 2υ ∂ 2υ ∂ 2υ  − 2uΩ sin ϕ + v  2 + 2 + 2  ∂y ∂z   ∂x

[1.70]

The term 2Ωw cos ϕ in equation [1.69] can be removed, due to the fact that vertical scalar velocities are much smaller in comparison to horizontal velocities. According to the above, the vertical component of wind can be written as:

∂w ∂w ∂w ∂w ∂π = −u −υ −w − cpΘ −g ∂t ∂x ∂y ∂z ∂z  ∂2 w ∂2 w ∂2 w  + 2uΩ cos ϕ + v  2 + 2 + 2  ∂y ∂z   ∂x

[1.71]

28

Numerical Weather Prediction and Data Assimilation

The vertical component of the Coriolis force is also smaller than other vertical components in the equation of motion. For this reason, the term 2Ωu cos ϕ can be neglected. Another simplification can be made based on the extent of molecular viscosity. The molecular viscosity is too small in the atmosphere except in a thin layer within a few centimeters of the Earth’s surface. This is the reason that the last terms in the above three equations are negligible and can be removed. If the horizontal grid size of the model is greater than 10 km (large compared to the vertical scale), the vertical component of the equation of motion is replaced by its hydrostatic approximation. Assuming the hydrostatic approximation, the gravitational force is balanced by the vertical component of the atmospheric PGF. Eventually, in an atmosphere, which is in hydrostatic equilibrium, the equilibrium of the forces on the vertical axis is given by the equation:

∂p ∂p gp = −g ρ  =− , or p ∂z ∂z RT ρ= RT

∂p ∂π ∂π 1 ∂π g = ρcpΘ = −g ρ  =− gρ  =− ∂z ∂z ∂z ρcpΘ ∂z cpΘ

[1.72]

Microscale processes such as turbulent motions near the surface, whose horizontal and vertical dimensions are comparable, cannot be approached hydrostatically, and the non-hydrostatic approach should be followed, which, however, increases the computational time of the simulation. Continuity equation 1 dρ   + ∇ ⋅ V = 0 , or ρ dt

 ∂ρ  ∂ρ ∂ ( ρ u ) ∂ ( ρυ ) ∂ ( ρ w) + ∇ ⋅ ρV = 0  =− − − ∂t ∂t ∂x ∂y ∂z

( )

[1.73a]

For an incompressible fluid (dρ/dt = 0), equation [1.73a] can also be written as:   ∂u ∂υ ∂w ∇ ⋅V = 0  + + =0 ∂x ∂y ∂z

[1.73b]

The Primitive Equations

29

Equation of state of ideal gas pa = RT

[1.74]

Temperature forecast equation  ∂T dT q a dp q ∂T ∂T   κ T ∂T  = +  = −u +υ − + ω + dt c p c p dt cp ∂t ∂y   p ∂p   ∂x

dT q a dp = +  dt c p c p dt

, or

[1.75]

If the thermodynamic energy equation is written in terms of potential temperature, the last term in equation [1.75] can be eliminated (Jacobson 2005) and is transformed as (see Appendix 1, equations [A1.22]–[A1.23]):  ∂Θ q ∂Θ ∂Θ ∂Θ  = −u +υ +w + ∂t ∂y ∂z  π c p  ∂x

[1.76]

Equation of moisture

dq = E − C , or dt   ∂( ρ q) = −∇ ⋅ ( ρ qV ) + ρ ( E − C )  ∂t ∂( ρ q) ∂ ( ρ qu ) ∂ ( ρ qυ ) ∂ ( ρ qw) =− − − + ρ (E − C) ∂t ∂x ∂y ∂z

[1.77]

2 Solving Methods in NWP Models

To describe various atmospheric phenomena through an atmospheric model, the spatial domain is divided into finite-sized mesh (grid) cells. Time is also divided into finite-sized time steps. It is also a common practice to decompose each corresponding variable according to different horizontal scale classifications. Finally, the system of equations is simplified by eliminating certain terms through scale analysis. The atmospheric phenomena can be classified into four types with respect to the spatial scale: (1) microscale phenomena (ranging up to 1 km), (2) mesoscale phenomena (ranging from 1 up to 102 km), (3) synoptic scale phenomena (ranging from 102 up to 104 km) and iv) global scale phenomena that range over 104 km. For some real atmospheric motions (e.g. fluctuations due to small turbulent eddies), the temporal scales (ranging up to 1 or 2 seconds) are, in general, lower than the model time steps (reduced to up to 5 seconds for a typical mesoscale model). For instance, if turbulent flow is evident within a fluid, the value of a measured variable fluctuates rapidly over time, as eddies of various scales pass through the point where the measurement is performed. This is why in many models not resolving eddies, where it is desirable to take into account the sub-grid scale disturbances, a process called Reynolds averaging is used (Reynolds-averaged models). By this process, the flow is averaged over a time interval long enough to moderate sub-grid scale fluctuations but small enough to maintain the trends for the large-scale eddy field. In order to achieve this, it is assumed that the variables can be divided into mean fields changing slowly or turbulent components changing rapidly. 2.1. Decomposition of variables – the perturbation method Each variable in a real atmosphere contains scales that are resolved within the grid of a numerical model, while it may contain smaller sub-grid scales. In the

Numerical Weather Prediction and Data Assimilation, First Edition. Petros Katsafados, Elias Mavromatidis and Christos Spyrou. © ISTE Ltd 2020. Published by ISTE Ltd and John Wiley & Sons, Inc.

32

Numerical Weather Prediction and Data Assimilation

perturbation method, each field variable can be separated into two components: (1) a basic state part and (2) a perturbation part, which reflects the local field deviation from the basic state. According to the comments above, each quantity can be written as a sum of its components:

Α* = Α + Α′

[2.1]

ˆ is the basic state portion and In the above equation, the over bar Α = Α + Α ˆ represents the spatial average over a grid ( Α is the component of the variable in synoptic scale and A is the component in mesoscale), while Α′ is the sub-grid scale perturbation (microscale component), which, in practice, is the deviation from the average. It is convenient to assume that the averaged variables change spatiotemporal considerably less in comparison to the deviations from the average. This implies that the averaged variables are approximately constant while the perturbations fluctuate significantly within the grid shell Δx–Δy-Δz and time interval Δt. This is why the microscale components cannot be adequately resolved by a mesoscale grid and might be parameterized through the so-called gradient-flux approximation. On the other hand, the synoptic scale variables can be considered as not changing significantly over time. Therefore, the equations in synoptic scale can be isolated from the general equations. It should be noted that the averaging period in this case would be large in comparison to the time scale of fluctuations in the mesoscale. This situation removes any energy exchange in the model between the mesoscale and synoptic scales. According to the above assumptions, each independent variable can be broken into the following components and takes the form:

u* ( x, y, z, t ) = uˆ ( z, t ) + u( x, y, z, t ) + u′( x, y, z, t )

[2.2]

υ * ( x, y, z, t ) = υˆ ( z, t ) + υ ( x, y, z, t ) + υ ′( x, y, z, t )

[2.3]

w* ( x, y, z, t ) = wˆ ( z, t ) + w( x, y, z , t ) + w′( x, y, z , t )  −> 0

w ( x, y, z, t ) = w( x, y, z, t ) + w′( x, y, z, t ) *

[2.4]

π * ( x, y, z, t ) = πˆ ( x, y, z ) + π ( x, y, z, t ) + π ′( x, y, z, t )

[2.5]

ˆ ( z, t ) + Θ( x, y, z, t ) + Θ′( x, y, z, t ) Θ* ( x, y, z, t ) = Θ

[2.6]

q* ( x, y, z, t ) = qˆ ( z, t ) + q( x, y, z, t ) + q′( x, y, z, t )

[2.7]

Solving Methods in NWP Models

33

Noteworthy is the absence of the w component for the synoptic scale in equation [2.4]. It has been deleted due to the fact that it can be considered too small in relation to the synoptic scale horizontal wind components. All the other synoptic variables are independent in the horizontal space (x- and y-directions). Substituting the equations from [2.2] to [2.7] into the system of equations of Chapter 1 (Equations [1.69] up to [1.77]), considering all the approximations made and averaging over a time scale compatible with the time scale of micrometeorological turbulent eddies, the following set of equations is obtained (for further details, see Appendix A2 – Turbulent diffusion parameterizations). Note that the set of brackets < > is used to denote the quantities averaged over time, while the system of the equations below is simplified eliminating the terms, which are several orders of magnitude lower than the remaining ones. The only term that is by default equal to zero is the term: ∂u ′ ∂x + ∂υ ′ ∂y + ∂w′ ∂z = 0 in the equation of vertical component of wind (see Appendix A2). Horizontal x-momentum equation

Defining that Fx = v ( ∂ 2 u ∂x 2 + ∂ 2 u ∂y 2 + ∂ 2 u ∂z 2 ) and averaging over time, equation [1.69] can be transformed as: ∂u* ∂u* ∂u* ∂u* ∂π * = −u* −υ* − w* − c p Θ* ∂t ∂x ∂y ∂z ∂x  ∂ 2 u* ∂ 2 u* ∂ 2u*  + 2υ *Ω sin ϕ − 2w*Ω cos ϕ + v  2 + 2 + 2   ∂y ∂z   ∂x ∂(u ′ + u ) ∂(u ′ + u ) ∂(u ′ + u ) = − (u ′ + u ) − (υ ′ + υ ) ∂t ∂x ∂y − ( w′ + w)

∂ (π ′ + π ) ∂ (u ′ + u ) − c p ( Θ′ + Θ ) ∂z ∂x

+ 2(υ ′ + υ )Ω sin ϕ − 2( w′ + w)Ω cos ϕ + Fx′ + Fx  ∂u ∂u ′ ∂u ∂u ′ ∂u ∂u ′ ∂π ∂(u ) = −u − u′ −υ − υ′ − w − w′ − cpΘ ∂t ∂x ∂x ∂y ∂y ∂z ∂z ∂x − c p Θ′

∂π ′ + 2Ω sin ϕ υ ′ + 2υΩ sin ϕ − 2Ω cos ϕ w′ ∂x −−−−−−−−> 0 −−−−−−−−−−> 0

−−−−−−−−> 0

− 2wΩ cos ϕ + Fx′ + Fx −−> 0

−−> 0

[a]

34

Numerical Weather Prediction and Data Assimilation

From [a]: ∂u ∂u ∂ u ′u ′ ∂u ′ ∂u ∂ υ ′u ′ ∂υ ′ ∂u = −u − + u′ −υ − + u′ −w ∂t ∂x ∂x ∂x ∂y ∂y ∂y ∂z

−

∂ w′u ′ ∂z

+ u′

∂w′ ∂π + 2υΩ sin ϕ − 2wΩ cos ϕ − c p Θ  ∂z ∂x

∂u ∂u ∂u ∂u ∂u ′ ∂υ ′ ∂w′ = −u −υ −w + u′ ( + + ) + 2υΩ sin ϕ ∂t ∂x ∂y ∂z ∂x ∂y ∂z −−−−−−−−−−−> 0

[b]

∂π ∂ u ′u ′ ∂ υ ′u ′ ∂ w′u ′ − − −  ∂x ∂x ∂y ∂z ∂u ∂u ∂u ∂u ∂π = −u −υ −w + 2υΩ sin ϕ − 2 wΩ cos ϕ − c p Θ ∂t ∂x ∂y ∂z ∂x − 2 wΩ cos ϕ − c p Θ

+

∂  m ∂u  ∂  m ∂u  ∂  m ∂u  + + Kx Ky Kz ∂x  ∂x  ∂y  ∂y  ∂z  ∂z 

Further decomposition of [b] in synoptic and mesoscale scales yields to [c] and, as a consequence, to equation [2.8]: ∂uˆ ∂u ∂ (uˆ + u ) ∂ (uˆ + u ) ∂ (uˆ + u ) + = −(uˆ + u ) − (υˆ + υ ) − ( wˆ + w) ∂t ∂t ∂x ∂y ∂z ˆ π π ( ) ∂ + ˆ + Θ) + f (υˆ + υ ) − fˆ ( wˆ + w) − c p (Θ ∂x 2 2 (uˆ + u )  u u ∂ ∂ ∂ ∂  + K xm 2 + K ym 2 + ( Kˆ zm + K zm )  ∂z  ∂z  ∂x ∂y ∂uˆ ∂u ∂uˆ ∂u ∂uˆ ∂u + = −(uˆ + u ) − (uˆ + u ) − (υˆ + υ ) − (υˆ + υ ) ∂t ∂t ∂x ∂x ∂y ∂y −−> 0

−−> 0

∂ (uˆ + u ) − ( wˆ + w) + f (υˆ + υ ) − fˆ ( wˆ + w) −−> 0 −−> 0 ∂z 2 2 ˆ ˆ + Θ) ∂ (π + π ) + K m ∂ u + K m ∂ u − c p (Θ x y ∂x ∂x 2 ∂y 2 +

∂  ˆm ∂ (uˆ + u )  ( K z + K zm )  ∂z  ∂z 

[c]

Solving Methods in NWP Models

[c ] 

∂uˆ ∂u ∂u ∂u ∂ (uˆ + u ) + = −(uˆ + u ) − (υˆ + υ ) − w ∂t ∂t ∂x ∂y ∂z ˆ π ( ∂ +π) ˆ − c (Θ ˆ + Θ) + f (υˆ + υ ) − fw p ∂x 2 2 ∂u ∂u ∂  ∂ (uˆ + u )  + K x 2 + K y 2 + ( Kˆ zm + K zm ) ∂x ∂y ∂z  ∂z 

35

[2.8]

Horizontal y-momentum equation  ∂ 2υ ∂ 2υ ∂ 2υ  Working in the same way and defining that Fy = v  2 + 2 + 2  , ∂y ∂z   ∂x equation [1.70] gives:

∂υˆ ∂υ ∂υ ∂υ ∂ (υˆ + υ ) + = −(uˆ + u ) − (υˆ + υ ) −w − f (uˆ + u ) ∂t ∂t ∂x ∂y ∂z 2 2 ˆ + Θ) ∂ (πˆ + π ) + K m ∂ υ + K m ∂ υ − c p (Θ x y ∂y ∂x 2 ∂y 2

+

[2.9]

∂  ˆm ∂ (υˆ + υ )  ( K z + K zm )  ∂z  ∂z 

The coefficient K zm (equations [2.8] and [2.9]) is the vertical eddy exchange coefficient of momentum (m2s−1) and represents the parameterization of eddy transport processes in the planetary boundary layer, Kˆ zm participates respectively in eddy friction terms for momentum on the synoptic scale, K xm and K ym are the horizontal eddy exchange coefficients of momentum, f is the Coriolis parameter (=2Ωsinφ) and fˆ is equal to 2Ωcosφ (see Appendix A2 – Turbulent diffusion parameterizations). The horizontal counterparts of eddy exchange coefficients ( K xm and K ym ) are convenient to be invariant within the horizontal space (x- and y-directions), and, unlike the vertical coefficients, it has been chosen here not to be decomposed into synoptic and mesoscale components. Vertical z-momentum equation

The vertical momentum equation is reduced to the hydrostatic equation. Averaging  ∂2 w ∂2 w ∂2 w  over time, equation [1.71] and defining that Fz = v  2 + 2 + 2  yields: ∂y ∂z   ∂x

36

Numerical Weather Prediction and Data Assimilation

∂ ( w′ + w) ∂ ( w′ + w) ∂ ( w′ + w) ∂ ( w′ + w) = − (u ′ + u ) − (υ ′ + υ ) − ( w′ + w) ∂t ∂x ∂y ∂z − c p ( Θ′ + Θ ) ∂ ( wˆ + w) −−> 0

∂t

= −u

∂ (π ′ + π ) ∂z

+ 2(u ′ + u )Ω cos ϕ − g + Fz



−−> 0

∂w ∂w′ ∂w ∂w′ ∂w ∂w′ − u′ −υ − υ′ −w − w′ + 2u Ω cos ϕ ∂x ∂x ∂y ∂y ∂z ∂z

∂π −g  ∂z ∂w ∂w ∂w ∂π ∂u ′ = −u −υ −w + 2u Ω cos ϕ − c p Θ − g + w′ ∂x ∂y ∂z ∂z ∂x − cpΘ

∂w ∂t

−−> 0

−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−> 0

+ w′

0

∂υ ′ ∂y

= −c p Θ

+ w′

w′

∂w′ ∂ ∂ ∂ u ′w′ − υ ′w′ − w′w′  − z x y ∂ ∂ ∂ ∂z

  ∂π ∂  ∂w  ∂  m ∂w  ∂  m ∂w  Ky Kz − g +  K xm + + ∂z ∂x  ∂x  ∂y  ∂y  ∂z  ∂z  −−> 0  −−> 0    −−> 0   ∂π ∂u ′ ∂υ ′ ∂w′  −c p Θ −g =0 + + ∂z ∂x ∂y ∂x −− −−−−−−−−−−−> 0

ˆ + Θ) ∂ (πˆ + π ) − g = 0 − c p (Θ ∂z Equation of Hydrostatic Equilibrium

[2.10] Continuity equation

By averaging over time, the equation for incompressible flow [1.73b] yields: ∂u * ∂υ * ∂w* ∂u ∂υ ∂w ∂u ′ ∂υ ′ ∂w′ + + ≡0 + + + + + ≡0 ∂x ∂y ∂z ∂x ∂y ∂z ∂x ∂y ∂z

[a]

Nevertheless, due to the fact that the fluctuations are assumed to be random quantities, their average is, by definition, zero ∂u ′ ∂υ ′ ∂w′ ∂ < u ′ > ∂ < υ ′ > ∂ < w′ > + + = + + =0 ∂x ∂y ∂z ∂x ∂y ∂z

[b]

Solving Methods in NWP Models

37

Substituting [b] into [a], it is obtained: ∂u ∂υ ∂w ∂ (uˆ + u ) ∂ (υˆ + υ ) ∂ ( wˆ + w) + + =0 + + =0 ∂x ∂y ∂z ∂x ∂y ∂z ∂uˆ ∂u ∂υˆ ∂υ ∂wˆ ∂w ∂u ∂υ ∂w + + + + + =0 + + =0 ∂x ∂x ∂y ∂y ∂z ∂z ∂x ∂y ∂z

−−> 0

[2.11]

−−> 0

−−> 0

∂u ∂υ ∂w + + =0 ∂x ∂y ∂z

The remaining prognostic equations are similar in form, and the equivalent averaged forms can be written as: Thermodynamic energy equation

By averaging over time equation [1.76] and considering that the motions are adiabatic (q = 0) yields: ˆ ∂Θ ˆ + Θ) ∂Θ ∂Θ ∂Θ ∂ (Θ ∂2Θ + = −(uˆ + u ) − (υˆ + υ ) −w + K xθ 2 ∂t ∂t ∂x ∂y ∂z ∂x [2.12] ˆ  ∂ Θ ∂  ˆθ θ ∂ (Θ + Θ) ( ) + K θy + + K K   z z 2 ∂z  ∂z ∂y  2

In the above equation, similar to equations [2.8] and [2.9], K xθ and K xθ are the horizontal eddy exchange coefficients of heat, Kˆ θ represents the vertical exchange z

θ

coefficient of heat on the synoptic scale, K z is the vertical mesoscale exchange coefficient of heat and all the coefficients are measured in m2s−1. Similar to the exchange coefficients of momentum, K xθ and K θy are not decomposed into synoptic and mesoscale components, as well as they were chosen to be invariant within the spatial x and y. Moisture equation

Working in the same way and considering, as above, that the variables K xq and

K yq are the horizontal eddy diffusivity coefficients of moisture, K zq is the vertical mesoscale exchange coefficient of moisture, Kˆ zq represents the vertical exchange

38

Numerical Weather Prediction and Data Assimilation

coefficient of moisture on the synoptic scale and all the coefficients are measured in m2s−1, the moisture forecast equation is given by: ∂qˆ ∂q ∂q ∂q ∂ (qˆ + q ) + = −(uˆ + u ) − (υˆ + υ ) − w ∂t ∂t ∂x ∂y ∂z + K xq

∂2q ∂2q ∂  ∂ (qˆ + q )  + K yq 2 + ( Kˆ zq + K zq ) + (E − C) 2 ∂z  ∂z  ∂x ∂y

[2.13]

However, it is necessary to further manipulate equations [2.8]–[2.13] because the synoptic and the mesoscale variables are still mixed. In order to achieve this target, we have to define the geostrophic wind set of equations and introduce some additional assumptions. The geostrophic wind components are defined in association with the synoptic pressure gradient through the following equations: ug = −

ˆ ∂πˆ cpΘ f

∂y

, υg = +

ˆ ∂πˆ cpΘ f

∂x

[2.14]

It is further assumed that the synoptic state is in hydrostatic equilibrium, expressed as: ˆ ∂πˆ  ∂πˆ = − g g = −c p Θ ˆ ∂z ∂z cpΘ

[2.15]

It is also considered that the fluctuations of the potential temperature on the mesoscale are very low in magnitude comparative to their synoptic counterparts. This situation is expressed by: Θ

ˆ

∂πˆ ∂πˆ f υ g ∂πˆ fc p Θυ g = f υg  =  cpΘ =  ˆ ˆ ∂x ∂x c p Θ ∂x cpΘ

∂πˆ Θ cpΘ = f υg ˆ ∂x Θ

[b]

--------------------■------------------Substituting [b] into [a] yields: ∂u ∂u ∂u ∂ (uˆ + u ) = −(uˆ + u ) − (υˆ + υ ) − w ∂t ∂x ∂y ∂z + f (υ − υ g +

2 2 Θ ˆ −c Θ ˆ ∂π + K m ∂ u + K m ∂ u ) − fw p x y ˆ ∂x ∂x 2 ∂y 2 Θ

[2.23]

∂  ˆm ∂u  ∂ ∂uˆ ( K z + K zm )  + ( K zm )  ∂z  ∂z  ∂z ∂z

∂π is deleted due to the fact that is much ∂x ˆ ∂π [according to equation [2.16], Θ

+

∂z * ∂zG ( z * − s ) ∂zG ∂υ ( z * − s ) ∂zG ∂u 1 υ + + ( s − zG ) ∂z * ∂y ( s − zG ) ∂y ∂z * ( s − zG ) ∂x ∂z *

=

∂zG ( z * − s ) ∂zG ∂u s ∂w u + + ( s − zG ) ∂z * ( s − zG ) ∂x ( s − zG ) ∂x ∂z *

+

∂zG ( z − s ) ∂zG ∂υ υ +  ( s − zG ) ∂y ( s − zG ) ∂y ∂z * *

∂zG ( z * − s ) ∂zG ∂u s u ∂w ∂w* = − − ( s − zG ) ∂z * ∂z * ( s − zG ) ∂x ( s − zG ) ∂x ∂z * −

υ

∂zG ( z * − s ) ∂zG ∂υ + ( s − zG ) ∂y ( s − zG ) ∂y ∂z *

[d]

Domain Structures and Boundary Conditions

85

Substituting [b], [c] and [d] into [a] and rearranging yield: ∂u ∂x

+ Z*

∂υ ∂y

+ Z*

∂z 1  ∂zG ∂w* − +υ G u * ∂z ∂y ( s − zG )  ∂x

 ∂u ( z − s) ∂zG ∂u ( z * − s) ∂zG  − *  *   ∂z ( s − zG ) ∂x ∂z ( s − zG ) ∂x   ∂υ ( z * − s) ∂zG ∂υ ( z * − s ) ∂zG + * − *  ∂z ( s − zG ) ∂y ∂z ( s − zG ) ∂y

 + 

*

∂u ∂x

+ Z*

∂υ ∂y

+ Z*

 =0 

∂z ∂w* 1  ∂zG − +υ G u * ∂z ∂y ( s − zG )  ∂x

 =0 

[3.22]

Synoptic scale variables

For the synoptic scale, the same procedure gives:

∂uˆ ∂t ∂uˆ ∂t

= f (υˆ − υ g )+ Z

∂ ˆ m ∂uˆ (Kz ) ∂z ∂z 2

Z*

( s − z* ) ∂zG  s  ∂ ˆ m ∂uˆ = f (υˆ − υ g )− g + (Kz * )  Z* s ∂x  s − zG  ∂z * ∂z

[3.23]

--------------------■------------------∂υˆ ∂t ∂υˆ ∂t

= f (u g − uˆ ) + Z

∂ ˆ m ∂υˆ (K z ) ∂z ∂z 2

= f (u g Z*

Z*

( s − z * ) ∂zG  s  ∂ ˆ m ∂υˆ − uˆ ) − g + (Kz * )  ∂y  s − zG  ∂z * s ∂z

[3.24]

--------------------■------------------2 ˆ ˆ ˆ ˆ  s  ∂ ˆ θ ∂Θ ∂Θ ∂ ∂Θ ∂Θ = ( Kˆ zθ = ) (K z * )  * ∂t Z ∂z ∂z ∂t Z *  s − zG  ∂z ∂z

--------------------■-------------------

[3.25]

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∂qˆ ∂t

Z

∂ ∂qˆ ∂qˆ = ( Kˆ zq )  ∂z ∂z ∂t

2

Z*

 s  ∂ ˆ q ∂qˆ (K z * ) =  * ∂z  s − zG  ∂z

[3.26]

3.3. Boundary conditions

As discussed previously, the integration domain, which is identical to the area covered by the numerical model, is limited and artificially enclosed. It is therefore necessary to give the values of the dependent variables at all grid points on the perimeter of the integration domain. Such values are known as boundary conditions. All types of models (e.g. global or hemispheric, mesoscale and nested models) integrate over time the approximate forms of primitive equations by means of boundary conditions. For mesoscale models, lateral and vertical (top and bottom) boundary conditions are needed, while for global models, only vertical boundary conditions are required. Boundary conditions are necessary for time integration of the conservation laws. This means that the governing equations may result in a solution only when the boundary and initial conditions are specified. The form of the boundary conditions required by any partial differential equation depends on the equation itself and the way that it has been discretized. Note that the selection of convenient boundary conditions can also reduce the computational time. The boundary conditions, the accuracy of which significantly affects the estimation of prognostic variables, are usually derived from low-resolution models (with integration area wider than the area of interest). They may come from various sources, such as: – data assimilation systems; – values coming from a current or previous forecasting cycle of a model with a wider integration area; – climatic or constant values for specific surface characteristics such as soil moisture, surface sea temperature and vegetation type. Common boundary conditions are classified either in terms of the numerical values that have to be set or in terms of the physical type of the boundary condition. For example, there are three common numerical boundary conditions that can be specified for steady-state fluid problems: Dirichlet boundary condition: According to the Dirichlet boundary condition, the values of the variable F on the boundary are known constants f(x,y,z). This requires a simple input to be made, in order to fix the value of the variable at the boundary:

Domain Structures and Boundary Conditions

F = f ( x, y , z )

87

[3.27]

Neumann boundary condition: According to the Neumann (or flux) boundary condition, the derivatives of the variable F on the boundary are known constants f(x,y,z): ∂F = f ( x, y , z ) ∂n

[3.28]

Above, n is equal to x, y or z.

Mixed-type boundary condition: aF + b

∂F = f ( x, y , z ) ∂n

[3.29]

Above, n is equal to x, y or z. On the other hand, in fluid problems (such as the atmospheric motions), the commonly observed boundary conditions are the physical boundary conditions, which, in general, are as follows:

Solid walls (either moving or stationary): In this case, if the flow is laminar, then the velocity components can be set to be equal to the wall’s velocity, while the situation is more complex for turbulent flow. Inlets: The fluid enters the domain. In this case, we need to know either its velocity or the pressure or the flow rate of mass. Certain fluid characteristics (e.g. turbulence) need also to be defined. Symmetry boundaries: For symmetrical flow, where there is not any flow through the boundary and the variable derivatives normal to the boundary are zero. Cyclic/periodic boundaries: Appear in pairs and the flow variables are same at equivalent positions on both boundaries. Periodic boundary conditions are used as the west–east boundary conditions in global and hemispheric models or in the case of choice of spectral or pseudospectral methods for the horizontal part of the system. In this case, the westernmost is simply adjacent to the easternmost grid cell and its western edge is identical to the eastern edge of the aforementioned eastern grid. Consequently, if xN = x0 + ( NX − 1) Δx and y N = y0 + ( NY − 1) Δy , for any dependent variable G, the boundary conditions can be summarized as follows: (a) G ( x0 , y , z , t ) = G ( xN , y , z , t ) , (b) G ( x, y0 , z , t ) = G ( x, y N , z , t ) .

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Pressure boundary conditions: In general, pressure boundary conditions cannot be used at boundaries where velocities are also known and specified, due to the fact that velocities are influenced by pressure gradients. Outflow boundary conditions: In many simulations, it is needed to have fluid flow out of one or more boundaries of the computational region. The simplest and most commonly used outflow condition is that for all quantities the normal derivatives at the boundary are equal to zero. This condition is known as “continuative” boundary. Open–closed boundary conditions: Open boundary conditions are used if the flow of the fluid comes across the surface of the boundary in all directions. In this case, the total amount of fluid flows outside the domain, into the domain or a combination of the above two. As an example, the perturbations in the mesoscale can pass through or out of the model mesh. If the perturbations are not permitted to enter or exit, then the boundary conditions are characterized as closed. Mesoscale atmospheric models require special handling in introducing initial and boundary conditions for solving the differential equations system. Their implementation should address the problem of an adequate assessment of the initial boundary conditions (initial-value problem). In addition to the above, the solution of the differential equations in a limited-area mode requires the frequent updating of the boundary conditions at the grid points located on the borders of the domain. It is also common to consider for lateral, top and bottom borders separately. It should be noted that the top and lateral boundaries are not physical boundaries and therefore have no physical meaning. Their existence is only due to the finite domain of these models, which must be limited horizontally and vertically because of the constraints on computational power. On the contrary, a real boundary is the bottom border. Hence, the transfer of physical properties across this interface (e.g. heat, moisture or momentum) is very important for a series of mesoscale meteorological circulations. 3.3.1. Lateral boundary conditions

Since the boundary conditions are usually derived from low-resolution models involving significant errors and difficulties of finding physically accurate and appropriate values to specify, the area of interest should be far enough beyond the boundaries of the integration field. This practice, which is the boundary removal as far as possible from the region of interest, is a common practice in limited-area models. The extent of the domain of interest compared to the total integration area should be about four times lower (1:4). The lateral boundary conditions usually come from previous forecasting cycles of models with larger integration domains. For example, a 6- to 54-hour forecast from a global model with initialization at

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89

06 UTC can feed a lateral boundary condition to a limited-area model for a 48-hour forecast starting at 12 UTC. Using this horizontal grid expansion mechanism, it is feasible to minimize the result of the lateral boundary on the region of interest. According to Pielke (2002), there are several types of open or closed lateral boundary conditions in use, some of which are as follows:

Lateral conditions of constant inflow and gradient outflow: During the air inflow at the model domain, the dependent variables are assumed that remain unchanged at inflow boundaries (in this case, we have a closed boundary). However, during the air outflow from the model domain, it is assumed that the variables have instantly the same values with those at a grid point before the boundary. At this point, it is necessary to clarify that the terms inflow and outflow are defined relative to wind direction at the boundaries.

Radiative boundary conditions: According to this procedure, the values of any prognostic-dependent variable at the boundaries are modified in order to minimize the reflection of disturbances back to the model domain. In order to implement radiative conditions, any one of the variables can be evaluated at the boundary using an equation of the following form: ∂ϕ ∂t = −c ∂ϕ ∂x . In the method of Orlanski −∂u ∂u ( 0 ≤ c ≤ Δx Δt ), while, according to Klemp and Lilly (1978) (1976), c = ∂t ∂x or Klemp and Wilhelmson (1978), the parameter c represents the dominant phase velocity of an internal gravity wave, is constant and equal to c = H  f BV π , where H is the height of the model domain and f BV is the Brunt–Väisälä frequency. Sponge boundary conditions: Sponge boundary conditions are a type of radiative boundary conditions discussed previously. Filters are added close to the lateral boundaries in order to damp the wave and advective disturbances during their moving towards the margins of the model domain. An important difference, which increases the computational cost of the model simulation, is that by this method, additional grid points are needed close to the boundaries. The additional grid points are added to permit the increase of smoothing gradually. The prognostic equation of any dependent variable can be written in the ∂ϕ ∂ϕ following form: = −u − r (ϕ − ϕ0 ) , where r is the relaxation coefficient ∂t ∂x (Davis 1983) that reaches its maximum at the boundary and ϕ0 is the desirable value of ϕ at the boundary.

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Periodic boundary conditions: As discussed above, the dependent variables at one end of the model domain are assumed to be identical to the values at the other boundary. 3.3.2. Upper (top) boundary conditions

The vertical model conditions include top and bottom boundary conditions. Similarly to the lateral boundaries, the top of a mesoscale model should be removed as far as possible from the area of significant mesoscale disturbance. Using the height of the stable thermodynamic stratification layer as the criterion, modelers have placed the tops of their domains deep inside the stratosphere, at the tropopause or into the stable tropospheric layer. Another important parameter is the type of the top. Modelers have used rigid tops (u = υ = w = Θ = π = 0), impervious material surfaces, porous lids and absorbing layers. Most models apply the fixed ceiling (cap) as the top boundary condition. A fixed ceiling at the model top cuts off any upward movement through this layer. However, because of this condition, gravity waves can be reflected backwards. Such waves are created in areas characterized by strong vertical convergence (e.g. storms) or in areas with intense relief. The problem is usually resolved by applying absorption or damping of unwanted waves near the upper model’s boundary, which usually extends between 50 and 1 hPa. According to Pielke (2002), based on a series of studies of boundary conditions, the following conclusions applicable to a mesoscale model can be drawn: (a) if the vertical propagation of the internal energy of gravity waves is equal to or greater than the advective properties, an absorbing layer is necessary. Otherwise, it is sufficient to add a plane as a top in parallel with an isentropic surface, which should be avoided in case of significant vertical advection. (b) The use of a rigid top is appropriate only in the case of the prevalence of advective effects. The depth of the model must also be significantly greater than the area where mesoscale disturbances occur. The exact form of the top (e.g. selection of a material surface area, an absorbing layer or rigid top) is not important, as the disturbances for any of the dependent variables reaching that level will be insignificant. 3.3.3. Lower (bottom) boundary conditions

As mentioned in section 3.3, the bottom, which is a real boundary, is also important from a physical point of view. This is because the gradients of the dependent variables over the lower interface produce and have a significant impact on many mesoscale circulations and mesoscale flows entering or leaving the

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atmosphere. For example, energy and moisture fluxes are exchanged between the surface of the Earth (bare soil, vegetation, water) and the atmosphere. Aerosol particles and gases are also removed from the lower tropospheric layer through dry deposition, wet deposition and sedimentation fluxes, or instead they are emitted into the atmosphere. According to the above, the fluxes of heat, moisture and momentum can be included as lower boundary conditions. Attention should be paid to the fact that the use of insufficient data such as data of topography, soil and vegetation type, surface albedo, surface temperature or availability of water may be sources of significant errors that can directly affect the forecasting meteorological parameters near the surface (wind field, surface temperature, humidity). It should be noted that any change over time in the lower boundary that may come from anthropogenic or animal activity (e.g. deforestation, overgrazing) can cause significant climatic changes. Due to the essential significance in mesoscale atmospheric systems, this component has to be represented with a very high accuracy. 3.3.3.1. Water bodies

For a proper representation of water surfaces (lakes, bays and oceans) in a mesoscale atmospheric model, the dynamic/thermodynamic interactions between air and water might be taken into account. For example, small-scale boundary layer interactions (e.g. gas exchanges between the air–water interface) and large-scale heat transfers by currents driven by wind should be involved. In general, these interactions include complex nonlinear processes. A very effective practice is the application of an oceanographic model in order to properly simulate the interactions as well as to provide suitable lower boundary conditions over water. In recent years, several researchers have coupled atmospheric models with ocean models or, in general, water body models, where the lower boundary of the total model is the topographical bottom of the water body. By applying such modeling systems, they have documented various impacts of the wind on ocean dynamics. For example, Avissar and Pan (2000), for the first time, developed a coupled modeling system between atmosphere and lake, using the Regional Atmospheric Modeling System (RAMS). They simulated hydrological and meteorological processes at Kinneret Lake (Israel) during the summer period. Costa et al. (2001) have also coupled RAMS with the Princeton Ocean Model (POM). His purpose was to study deep cumulus cloud–ocean interactions over the region of western tropical Pacific Ocean. In other past works, any spatiotemporal change in fields of atmospheredependent variables over water surfaces can cause variations in upper water levels (on a daily basis) in the vertical gradients of salinity, temperature and other aerosol

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and gaseous materials. According to Pielke (1991), for proper simulation of the ocean uptake of carbon dioxide, the mesoscale resolution of ocean upwelling must be known. The temperature of the coastal waters may be also affected by the advection of aerosol. In general, the wind at low levels influences vertical turbulent mixing within the water body, while the surface temperature of water and its spatiotemporal variability primarily influence the atmospheric circulation. Recently, Katsafados et al. (2016) have introduced the Workstation Eta nonhydrostatic regional model fully coupled with the ocean wave model WAM. The socalled WEW is a two-way fully coupled atmosphere–ocean wave system, designed to explicitly resolve air–sea interactions supporting research and operational activities over the Mediterranean Sea. The system was built in the Multiple Program Multiple Data (MPMD) environment where the atmospheric and ocean wave components are handled as parallel tasks on different processors. Interactions considered in WEW were mainly driven by the momentum exchanges within the ocean wind–wave system and included the effects of the resolved wave spectrum on the drag coefficient and its feedback on the momentum flux. WEW offered a more realistic representation of the aerodynamic drag over rough sea surfaces. The performance of the system was tested in a high-impact atmospheric and sea-state case study of an explosive cyclogenesis in the Mediterranean Sea showing an overall improvement of the root-mean-square error (RMSE) up to 11%. The evolution of WEW is the Chemical Hydrological Atmospheric Ocean System (CHAOS) consisting of three components: the Weather Research and Forecasting Model (WRF-ARW) coupled with the Nucleus for European Modeling of the Ocean (NEMO) ocean circulation model coupled with the ocean wave (WAM) model. All the components are online connected under the OASIS3-MCT coupler environment, leading to the development of a three-way fully coupled system (Varlas et al., 2018). Physical interactions considered in CHAOS or in similar coupled systems are mainly driven by the sensible and latent heat exchanges at the air–sea interface and include the effects of the explicitly resolved ocean circulation on the estimated sea surface temperature (SST) and its feedback on the thermodynamic flux. 3.3.3.2. Land bodies

When the bottom surface is represented by land, it is suitable to separate bare soil from vegetated one. This is because it is easier to simulate bare soil. Besides, the mesoscale models have become quite sophisticated to represent such surfaces. On the contrary, the vegetation effects are very complex. For this reason, several field campaigns have been performed in order to develop a better understanding of interactions between soil and atmosphere, while Avissar (1995) gives an overview of land–atmosphere interactions. Nowadays, all atmospheric models are combined

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93

with a single- or multilayer soil model. In such cases, the bottom boundary of the overall model over soil is the lowest ground level. The surface fluxes needed as lower boundary conditions are usually calculated via a bulk parameterization based on the Monin–Obukhov similarity theory (1954). Based on this theory, the wind and temperature profiles in the turbulent surface layer can be represented by a set of equations that depends only on a few parameters (e.g. the surface roughness length z0; see Appendix A2, Eq. A2.1 up to Eq. A2.45b). Based on many observational studies, the similarity theory indicates that the momentum and heat fluxes are almost constant with height within the surface layer (10–100 m deep), which is much thinner than the planetary boundary layer. 3.3.3.3. Surface energy balance

In the first mesoscale models, the surface temperature was considered as a function of periodic heating. Later, more general periodic forms were used. According to this process, temperature measurements were fitted to a series of periodic functions. Although such formulations were simple to apply, they implied an infinite reservoir of heat and did not allow feedback between the ground surface temperature and the mesoscale circulation within the atmosphere. The first mesoscale model, developed by Physick (1976), allowed feedback between the ground surface temperature and the atmospheric circulation. He applied a heat budget technique according to which the surface of the ground had zero heat storage. Similar techniques have been subsequently implemented by Mahrer and Pielke (1977) and Estoque and Gross (1981) among others. According to the heat budget method, all the amounts of heat reaching the ground surface are balanced, which results in an equilibrium surface temperature. Formally: −QG + QC + QR = 0

[3.30a]

Here, QG, QC and QR are respectively the conductive, convective and radiative components of heat flux to the surface of the ground. An example of the balance of heat fluxes is given in the following balance equation (Mahrer and Pielke, 1977):

Rs + RL + ρ Lu* q* + ρ c p u*θ* − ρ s cs K s

∂T ∂z

− σ TG4 = 0

[3.30b]

G

In the above equation, RS is the incoming solar radiation (shortwave), RL is the incoming long-wave radiation, L is the latent heat, cp is the specific heat of air at constant pressure, ρs is the soil density, cs is the specific heat of soil and Ks is the soil heat diffusivity. The third, fourth and fifth terms are respectively the latent, sensible

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and soil heat fluxes, while the sixth term is the outgoing long-wave radiation from the Earth’s surface (σ is the Stefan–Boltzmann constant). The surface friction velocity ( u* ), surface friction temperature ( θ* ) and surface friction specific humidity ( q* ) are computed using equation (A2.41), (A2.43) or (A2.45), in accordance with the stability conditions applicable on a case basis. Shortwave radiation

Shortwave radiation consists of two components: direct radiation and diffuse radiation. The direct radiation reaches an area without any absorption or scattering as it passes through the atmosphere. On the contrary, the diffuse component of radiation reaches the ground after being scattered. The diurnal variation of the direct downward solar radiative flux on a horizontal surface located at the top of the atmosphere can be computed as follows:   a2  S0  2  r  S =  0 

  cos Z 

 for Z < 900     0  for Z ≥ 90  

[3.31]

The quantity S0 = 1376 W m−2 is the solar constant, while the ratio (α2/r2), related to the Earth–Sun distance, can be calculated, according to Paltridge and Platt (1976), by the formula: a2 = 1.000110 + 0.034221cos d 0 + 0.00128sin d 0 r2 + 0.000719 cos 2d 0 + 0.000077 sin 2d 0

[3.32]

The parameter d0 is equal to (2π m 365) in equation [3.32], where the day number m starts as 0 on January 1 and ends as 365 on December 31. Furthermore, the distance between the Sun and Earth ranges between r = 0.98324a (early January) and r = 1.01671a (in early July), while the parameter a is the distance between the Sun and the semi-major axis of the Earth’s elliptical orbit. The variable Z in equation [3.31] is the zenith angle, which is equal to 90° when the disk of the Sun crosses the horizon. The zenith angle is equal to 0° when the disk of the Sun is overhead. Sunrise and sunset occur when Z = ±90°.

Domain Structures and Boundary Conditions

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In detail, the zenith angle is given by: cos Z = sin ϕ sin δ + cos ϕ cos δ cosψ

[3.33]

Here, φ is the latitude, δ is the solar declination (range: from +23.5° on June 21 to -23.5° on December 22) and ψ is the solar hour angle (at noon ψ = 0), given by:

ψ = arccos ( − tan δ tan ϕ )

[3.34]

On the surface of the Earth, the solar irradiance can be calculated using empirical formulations, which account for absorptivity and transmissivity of shortwave radiation (Mahrer and Pielke, 1977). In a clear atmosphere, water vapor is the main source of heating by absorbing shortwave radiation within the troposphere (List, 1971; Paltridge & Platt, 1976), while the heating resulted by shortwave absorption is substantial within the boundary layer, as shown by Moores (1982). An empirical formula, which accounts for the absorptivity of water vapor, has been expressed by McCumber (1980). This formulation was obtained by McDonald (1960) and is given by:  r( z)  aw = 0.077    cos Z 

0.3

[3.35]

where r(z) is the optical path length of water vapor above the layer z up to the top of the atmosphere, given as: top

r( z) =

 ρ qdz

[3.36]

z

Here, ρ is the water density (1 g/cm3) and q is the specific humidity (g/kg). An expression for fractional transmissivity of shortwave radiation reaching the ground has been modified and applied by Atwater & Brown (1974). This formulation, proposed by Kondratyef (1969), includes absorption and molecular scattering caused by oxygen, ozone and carbon dioxide and is given by:   0.000949 ∗ P + 0.051  G = 0.485 + 0.515 1.041 − 0.16   cos Z   

12

  

[3.37]

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Here, P is the pressure in hPa. Finally, the net shortwave radiative flux at the surface is:   a2  S0  2 RS =   r 0 

  cos Z (1 − A)(G − aw ) 

 for Z < 900   for Z ≥ 900 

[3.38]

Above, A is denoted the albedo. Long-wave radiation

To calculate the long-wave radiation, the optical path lengths of its emitters, as well as its emissivities, are necessary. With CO2 and water vapor considered as emitters, the path lengths can be computed for each model layer from the surface to the top of the model by the following formulas (Mahrer and Pielke, 1977): For water vapor: Δr j = −

( Pj +1 − Pj ) g

[3.39a]

qj

For carbon dioxide: Δc j = −0.4148239( Pj +1 − Pj )

[3.39b]

The total path lengths are given by adding the individual path lengths from the first up to the nth model level: n

n

j =1

j =1

rn =  Δrj , cn =  Δc j

[3.40]

The emissivity function for carbon dioxide is given in a form proposed by Kondratyev (1969):

(

ε co (i, j ) = 0.185 1 − exp −0.3919 ci − c j 2



)

0.4

 

[3.41]

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97

Accordingly, the emissivity for water vapor is given in the form proposed by Jacobs et al. (1974), which was derived from the data of Kuhn (1963):

(

0.11288log10 1 + 12.63 ri − rj    0.104 log10 ri − rj + 0.440   0.121log r − r + 0.491 j 10 i   ε r (i, j ) =   0.146 log10 ri − rj + 0.527   0.161log10 ri − rj + 0.542    0.136 log10 ri − rj + 0.542

)

for log10 ri − rj ≤ −4     for log10 ri − rj ≤ −3    for log10 ri − rj ≤ −1.5    [3.42]  for log10 ri − rj ≤ −1    for log10 ri − rj ≤ 0     for log10 ri − rj > 0  

Here, ri − rj is the optical path length (in grams per centimeters squared) for use in equation [3.42] from the ith up to the jth level. Finally, the emissivity at each level is given by:

ε (i, j ) = ε co (i, j ) + ε r (i, j ) 2

[3.43]

Using the emissivity functions defined above, the downward and upward fluxes at any level N can be calculated by the following equations: Rdown ( N ) =

top −1

σ

 2 (T j=N

4 j +1

+ T j4 ) ε ( N , j + 1) − ε ( N , j ) 

4 + σ Ttop 1 − ε ( N , top )  N −1

Rup ( N ) =  j =1

σ 2

(T

4 j +1

+T

4 j

) ε ( N , j ) − ε ( N , j + 1)

+ σ TG4 1 − ε ( N , 0 ) 

[3.44]

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The radiative cooling at each layer N can be computed from: 1  Rup ( N + 1) − Rup ( N ) + Rdown ( N ) − Rdown ( N + 1)   ∂T    = z ( N + 1) − z ( N )  ∂t  N ρ c p

[3.45]

The terrain effects on radiation

The solar radiation on a slant surface is calculated by the Kondratyev (1969) expression given below: S sl = S cos ϕi

[3.46]

Here, ϕi is the angle of incidence of solar rays on the inclined surface. It is valid that:

cos ϕi = cos a cos Z + sin a sin Z cos( β − n)

[3.47]

Here, Z is the zenith angle given by equation [3.33], while α is the slope angle, and β−n are the solar and slope azimuths which can be calculated using the following expressions:  ∂z  2  ∂z  2  a = tan  G  +  G    ∂x   ∂y   −1

 cos δ sinψ  sin Z

β = sin −1 

 ∂z n = tan −1  G  ∂y

  

[3.48]

∂zG  π − ∂x  2

In the above equation, ZG is the ground elevation, while for a slant surface, the solar and infrared radiations are modified to: RS

sl

RL

sl

cos ϕi cos Z = RL cos a = RS

[3.49]

Vegetation

The correct representation of boundary conditions at the bottom becomes more difficult when the vegetation is taken into account than for bare soil. This is due to the fact that the information on fluxes of heat, humidity, momentum and other

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materials (gases and aerosols) inside and outside the vegetation remains limited even today. Nevertheless, since the vegetation dynamically changes over time, it is necessary to be included in a mesoscale model, as an important component of the ground characteristics. Even for arid areas, the importance of sparse vegetation in thermal flux has been demonstrated. For example, McCumber (1980), using the synoptic meteorological data for south Florida (as model input for July 17, 1973), showed substantial differences between profiles of potential temperature developed over a forested area and over a bare soil (during the afternoon or morning). Both regions were of the same soil type (sandy loam). The differences in temperature reached up to 3°C, while the depth of the mixed layer reached a difference of more than 300 meters during the day. In the same study, large variations of foliage, canopy and ground temperatures over four different types of vegetation combinations have also been illustrated. Due to the fact that the surface temperature prevails the forcing of various types of mesoscale systems, the different types of soil or vegetation are expected to play an important role in the atmospheric phenomena and therefore need to be included in model simulations. According to the above, incorporating vegetation into a mesoscale model can be very important. 3.4. Design of a simulation

Numerical limited-area atmospheric models allow the study of phenomena of different spatiotemporal scales. They can be used as a tool for weather forecasting as well as for designing studies to investigate the effect of various factors on the development and evolution of a variety of phenomena, such as the effect of surface sea temperature on the intensity of a storm. In order to exploit the potential of a numerical model, some key steps should be taken, such as: – Identification of the problem: it needs to be clarified what exactly it is intended to study with the implementation of the model. It is essential to know the phenomena that can play an important role in shaping a particular atmospheric situation. – Determination of the spatiotemporal scale of the problem: once the atmospheric state, which we are trying to describe by the model, is determined, it is crucial to know the spatiotemporal scale of the phenomena involved, in order to make their description as accurate as possible. – Determination of the extent of the model integration area: applying a limitedarea arithmetic model, it is assumed that the boundaries of the integration area should be placed far enough from the area of interest. In this way, it is intended to ensure that the described atmospheric flow will not be able to approach and will incorrectly modify the atmospheric situation in the area of interest due to incomplete

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or even incorrect information at the boundaries. Particular attention should also be paid to the fact that at the lateral boundaries of the integration area, there do not exist or there are as few mountainous volumes as possible with a steep slope. Thus, flows are transferred smoothly into and out of the integration area without being artificially blocked by mountainous volumes. – The choice of physical, dynamic (and chemical) processes to be simulated: depending on the nature of the problem, physics laws are numerically solved, adopting assumptions and simplifications. – The choice of the computing machine in which the model algorithms will be encoded and calculated: as is mentioned, the implementation of numerical models is intertwined with the evolution of computer technology. The computational power, the precision associated with the cutoff error in the calculations and the capabilities offered by the software companies are necessary tools to be taken into account when designing the implementation of a numerical model. – Determination of spatial discretization and time step of the model integration: the criteria that will lead to their definition are a combination of the necessity of a detailed description of the atmospheric situation, the available computational power and the time required to make the model outputs ready for use. – Definition of initial and lateral boundary conditions: applying a numerical model depends directly on the quality of the input data to be used. In order to calculate the initial boundary conditions and to update its lateral boundaries, it is necessary to provide as much as possible information about the initial dynamic state and the evolution of the atmosphere–ground–sea system. – Ensuring a sufficient number of measurements, which will be used for evaluating its predictive efficiency. – Development of the software, which will be used for availability and visualization of the model outputs. – Finally, it is necessary to improve the model algorithms so that the outputs are as close as possible to real atmospheric conditions.

4 Introduction to Data Assimilation

Numerical weather prediction (NWP) is assumed to be an initial/boundary value problem. Given an estimate of the atmosphere’s present state (initial conditions) and appropriate surface and lateral boundary conditions, the model simulates the evolution of the atmosphere. The more accurate the estimate of the initial conditions, the better the quality of the predictions. Meteorological data assimilation (DA) is the entire procedure for constructing initial conditions by combining gridded fields (a priori state) with in situ or remote-sensed observations. It is flexible enough to allow a range of many different types of atmospheric observations to be digested within a framework of a numerical model. In this way, outputs from NWP forecasts are enriched with measurements provided by a variety of sources such as surface meteorological stations, radiosondes, weather and satellite radars, and other metering platforms, as shown in Figure 4.1. The obtained fields are characterized as analyses, meaning that they are three-dimensional gridded fields able to reproduce the real atmospheric conditions for a certain reference time. Analyses are usually used as initial conditions for NWP models. DA uses statistical methods to merge observations and short-term forecasts. Therefore, the purpose of this process is to combine past knowledge of a system, in the form of an NWP model, and new information about the system, in the form of observations (Kalnay 2003). Instrument errors are discarded, systematic deviations are removed and each observation is compared with its neighbors. In 1980s and 1990s, the early DA methods used were successive corrections, polynomial interpolation, nudging and optimal interpolation. Then, two fundamental methods were mostly applied in DA models. Variational analysis is based on optimal control theory, which aims to minimize a given cost function that measures the model-to-data misfit. It consists of:

Numerical Weather Prediction and Data Assimilation, First Edition. Petros Katsafados, Elias Mavromatidis and Christos Spyrou. © ISTE Ltd 2020. Published by ISTE Ltd and John Wiley & Sons, Inc.

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–least square method; –optimal interpolation (OI); –three-dimensional variational assimilation (3DVAR); –four-dimensional variational assimilation (4DVAR). Kalman filters and relevant approaches (extended Kalman filter and ensemble Kalman filter) are included in sequential methods in which the observations are assimilated as soon as they become available. Successive corrections and least square methods are briefly presented in this chapter. An introduction to the theoretical background of variational analysis with practical exercises, applications and case studies is also included.

Figure 4.1. Schematic of the current global atmospheric observing system. For a color version of the figure, see www.iste.co.uk/KMS/numerical.zip

4.1. Successive correction methods The successive correction model was initially presented by Bergthorsson and Doos (1955). In this method, an earlier first guess or a background field xb(i) gives the first estimate of the gridded field at the ith grid point using the weighted sum of differences between the background field and observations (y(j)-xb(j)) surrounding the ith grid point. The value xa(i) is the obtained analysis at the grid point i using observations, which is given by:

Introduction to Data Assimilation

xa ( i ) = xb ( i )

 +

pj j =1

wi , j  y ( j ) − xb ( j ) 

 j =1wi, j pj

103

[4.1]

The weights are defined by Cressman (1959) as:  R 2 − di2, j for di2, j ≤ R 2  wi , j =  R 2 + di2, j  0 for d 2 ≥ R 2 i, j 

[4.2]

where di , j measures the distance between the grid point i and the observation point j, and R is the radius of influence beyond which the observations are not taken into account. This method is empirical, simple and economical, which provides reasonable analyses but is limited because it does not guarantee that the chosen empirical formulas are appropriate. Barnes (1964) introduced an empirical method of successive corrections that was used when the background field or the first guess is not available. In this case, the error covariance is considered to be very large and the weights are given by: − rij2 2

w = e Rn n ij

[4.3]

where rij is the distance from the observation point to the grid point. The radius of influence R is changed by a constant factor at each iteration: Rn2+1 = γ Rn2

[4.4]

For γ=1, large scales are only captured, and for γn), then the obtained analysis vector is:

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xa = ( H T H ) H T yo −1

[4.40]

If the number of observations is less than the number of state x (m 60 μm are lifted from the ground and, after a small parabolic trajectory, return to the surface. Due to the impact, smaller particles are created, which in turn are lifted and transported at larger distances (Alfaro and Gomes 2001; Zender et al. 2003); (3) disaggregation, where the energy applied during impact can lead to their disaggregation and release of finer dust particles (Chappell et al. 2008).

Figure 5.2. Processes that initiate desert dust production (source: Shao et al. 2015)

The most frequent process is the saltation bombardment, which is initialized when the wind exceeds a critical friction velocity that is needed to lift the large sand particles. This critical friction velocity can be parameterized as in Zender et al. (2003):

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  ρ p gDsand   0.129 ⋅     ρ air  0.03 < Rn < 10  (1.928 Rn0.092 − 1)0.5  U *T =   0.12 ⋅  ρ p gDsand  1 − 0.0858e −0.0617( Rn −10)   ρ air   

(

[5.1]

)

Rn > 10

where Dsand is the optimal sand size for starting the process, ρp is the particle density, ρair is the air density and Rn is the Reynolds number that can be derived from a formula given by Marticorena et al. (1997):

Rn = 1331⋅ D1.56 zand + 0.38

[5.2]

There are two main factors that affect the efficiency of the sandblasting process. These are the drag partitioning and the soil moisture (Zender et al. 2003). The drag is affected mainly by the presence of non-erodible elements in an arid soil, like small rocks, which affect the threshold friction velocity by protecting the erodible parts, while draining momentum from the wind (Bergametti et al. 2007). Therefore, a formulation is needed to parameterize the above process. How efficient is the partitioning between the parts of the soil that can be eroded and the non-erodible soils can be expressed as suggested by Marticorena and Bergametti (1995):     Zo     ln  S     Z  0      DrgP = 1 −   0.8     0.1         ln  0.35  Z S      0        

−1

[5.3]

where Z0 is the roughness length and Z 0s is the roughness length of a bed of potentially erodible particles without any non-erodible elements (Marticorena and Bergametti 1995). Using equation [5.3], the threshold friction velocity is modified as:

U*T = U*T * DrgP

[5.4a]

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123

Additionally, one of the most important factors for dust production is how dry or wet the soil is. A moist soil hinders the sandblasting process and therefore the number of dust particles that can be injected into the atmosphere. To factor this in equation [5.1], the threshold friction velocity becomes: U *T WT ≤ WTt  U *T =  0.68 T U * ⋅ 1 + 1.21(WT − WTt )

WT > WTt

[5.4b]

where WT is the soil moisture and WTt is the maximum amount of absorbed water in the soil. The later can be parameterized according to Fecan et al. (1999): WTt = 0.014(ClayF ) 2 + 0.17(%ClayF )

[5.5]

where ClayF is the clay fraction (%) for each soil type. The clay fraction plays an important role in dust production schemes; therefore, the clay content can be defined through global datasets such as the FAO soil dataset from the UN Food and Agriculture Organization (FAO 1991) with a 5 min resolution (Figure 5.3). Having defined the threshold over which the saltation processes begin, the horizontal flux of the large sand particles (that initiate the bombardment process) can be derived following White (1979): H FLX =

C ρ airU *3 g

 U *T 1 + u* 

   U *T   1 −     u*

  

2

   

[5.6]

where C = 2.61 is a constant that was determined from the wind tunnel experiments performed (White 1979), ρair is the air density, U* is the friction velocity and g = 9.81 ms-1 is the acceleration due to gravity. Knowing the flux of the large sand particles, we can define the vertical flux of small dust aerosols into the atmosphere. In order to do this, the Marticorena and Bergametti (1995) formulation can be used:

VFLX = H FLX * Seff

[5.7]

where Seff is the sandblasting mass efficiency:

(

)

 13.4 M clay − 6 ln10  

Seff = 100e 

[5.8]

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Figure 5.3. Clay content (in %) derived from the global FAO dataset. For a color version of the figure, see www.iste.co.uk/KMS/numerical.zip

The above equation [5.8] is derived from wind tunnel experiments and works for clay fractions up to 20% (Marticorena et al. 1997). In creating a complete numerical formulation, several other factors need to be taken into account. Dust uptake is dependent on the soil characteristics and the type of vegetation, as well as the clay fraction. Several global datasets exist that provide such information. Typically, the 30-second global land use database of the 24-category U.S. Geological Survey (USGS) is used (Anderson et al. 1976). This global database was synthesized from the 1-km Advanced Very High Resolution Radiometer (AVHRR) dataset. This highresolution global land use dataset provides a very detailed spatial distribution of vegetation, as well as the separation between water bodies and land surface for highresolution applications. The dominant vegetation type in each grid box can be used to represent the grid level vegetation characteristics (Spyrou et al. 2010). Knowing the vegetation types, a desert mask fraction Adesert can be calculated following Nickovic et al. (2001). Depending on the ratio of the horizontal grid distribution and the resolution of the available dataset (Figure 5.4), if each model grid box contains at least one point from the global vegetation grid, then:

Desert Dust Modeling

Adesert =

number of dust points total number of vegetation points

125

[5.9]

If the grid boxes do not contain any point from the global vegetation grid, then the desert mask fraction can be defined from interpolating the four nearest global points: 4

Adesert =

W

n

[5.10]

* MSK n

n =1

where Wn is the bilinear interpolation weighting factor and MSKn is the global dataset desert mask.

(a)

(b)

Figure 5.4. Specification of the desert fraction. The bold-line and the dashed-line grids denote the vegetation/desert and the model data respectively. (a) Example with eight vegetation/desert points inside the model grid box, where fraction is calculated according to equation [5.9] and (b) example with no vegetation/desert points inside the box, where fraction is calculated according to equation [5.10] (source: Nickovic et al. 2001)

Alternatively, Solomos et al. (2019) used the MODIS Normalized Difference Vegetation Index (NDVI) to identify areas that can act as dust sources. The 500-m 16-day averaged NDVI product from MODIS (Didan 2015) has been implemented to work in the NMME-DREAM v1.0 model instead of the more commonly used USGS dataset. The NDVI is a normalized transform of the near-infrared to red reflectance ratio, designed to provide a standard for vegetation and takes values between -1 and +1. Since it is expressed as a ratio, it has the benefit of minimizing certain types of band-correlated noise (positively correlated) and errors caused by variations in irradiance, clouds, atmospheric attenuation and other parameters

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(Solano et al. 2010). Areas with NDVI < 0.1 can be classified as active dust sources and inserted into the model using equations [5.9] and [5.10]. An example is given in Figure 5.5.

Figure 5.5. Dust source strength as defined by the August 1–16, 2016 mean NDVI (source: Solomos et al. 2019). For a color version of the figure, see www.iste.co.uk/KMS/numerical.zip

It is assumed that the vertical flux of dust in the source areas is distributed according to the size of the particles following a trimodal lognormal probability density function (Zender et al. 2003). This size distribution of background dust is set into discrete bins, as proposed by DʼAlmeida (1987), which are called source modes. The characteristics of this distribution are presented in Table 5.1, where number median diameter,

~ Dn is the

~ Dν is the mass median diameter, σ is the standard

deviation and M frac is the mass fraction in each bin. When dust leaves the ground, it changes distribution and can be transported into a number of discreet size bins. The greater the number of bins, the better the representation of dust particles in the atmosphere. However, this number is usually chosen to balance accuracy and computational needs. For example, the Dust Entrainment and Deposition (DEAD) model uses 4 size bins (Zender et al. 2003), the Dust Regional Atmospheric Modeling System (DREAM) uses 8 size bins, (Perez et al. 2006) and the Global Ozone Chemistry Aerosol Radiation and Transport (GOCART) model (Ginoux et al. 2004) uses 5 size bins, all with diameters ranging from 0.1 to 10 μm. This range of particles are those that are small enough to be transported away from the sources and affect areas beyond the deserts (Spyrou et al. 2010).

Desert Dust Modeling

~ Dn

~ Dν

σ

M frac

0.16

0.832

2.1

0.036

3.19

4.82

1.9

0.957

10.0

19.38

1.6

0.007

127

Table 5.1. Trimodal size distribution of sources (source: DʼAlmeida 1987; Zender et al. 2003)

The mass overlap between the source (I) and transport modes (J) can be calculated based on Schulz et al. (1998) as:

  1 I M J = erf 2  

  rmax, J    ln     Dν , I     − erf  2 ln σ I     

  rmin, J     ln      Dν , I       2 ln σ I       

[5.11]

where rmax and rmin are respectively the maximum and minimum radius of each size bin, Dn is the number median diameter, Dv is the mass median diameter, σ is the standard deviation and erf is the standard error function. Taking into account all of the above, the vertical dust flux from the surface equation [5.7] can be expressed as: NPS

VFLX = T ⋅ H FLX ⋅ Seff ⋅ Adesert ⋅

M

I ,J

[5.12]

I =1

where NPS is the number of transport size bins and T = 7 *10−4 is a global tuning factor chosen to give reasonable results (Zender et al. 2003). 5.2. Dust advection and deposition

Once airborne, dust particles can be transported away from the source areas and affect a wide region. The movement of particles into the atmosphere is controlled by the prevailing wind patterns and can be expressed using an Euler-type continuity equation:

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∂CJ ∂C ∂C ∂C J ∂CJ ∂ = −u J − v J − W − ∇ ( K H ∇C J ) − ( K z )+ ∂t ∂x ∂y ∂z ∂z ∂z [5.13] (

∂CJ ∂C ) SOURCE − ( J ) SINK ∂t ∂t

where CJ is the concentration of particles per size J and x, y, z is the three∂CJ ∂C − v J is ∂x ∂y the horizontal advection, with u, v the horizontal wind speed components, while the ∂C J corresponds to the vertical advection. For this term, the Van Leer term −W ∂z (1977) scheme is used and the vertical advection becomes:

dimensional coordinate system. The term on the right-hand side −u

∂CJ ∂ ∂W  ∂C J  = −W = − (C J W ) + C J  ∂t  ∂z ∂z ∂z  vadv

[5.14]

where W = w - vg is the relative vertical wind speed, where w is the vertical wind speed and vg the gravitational settling velocity of each particle bin. The term −∇( K H ∇C J ) denotes the lateral diffusion, with KH being the lateral diffusion ∂C J ∂ (Kz ) is the vertical diffusion of dust, with KZ the ∂z ∂z ∂C turbulence exchange coefficient. The term ( J ) SOURCE corresponds to increases in ∂t ∂C dust concentration over time due to production from sources and ( J ) SINK ∂t corresponds to decreases in dust concentration over time due to dry and wet deposition removal processes.

coefficient. Accordingly, −

In creating a dust transport scheme, the viscous sublayer model proposed by Janjic (1994) can also be used. In this parameterization, it is assumed that fluids cannot penetrate a small surface boundary, which leads to a small gradient of the wind speed close to the ground (Figure 5.6; Schindler and Ackerman 2010).

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Figure 5.6. Boundary layer vertical structure. The red line denotes the change of wind speed with height (source: Schindler and Ackerman 2010). For a color version of the figure, see www.iste.co.uk/KMS/numerical.zip

According to this, in the case of dust mixing just above the desert surface, a thin sublayer is created under smooth flow conditions. The transition towards the rough flow happens gradually as the turbulent mixing increases. When the surface turbulence is fully developed, this sublayer vanishes and the mobilization of dust particles and the saltation processes can begin. Therefore, the concentration above the viscous sublayer CAVS,J for each size bin can be defined as:

z C AVS , J = CJ +  S ω

  FS 

[5.15]

where CJ is the surface dust concentration as derived from [5.13], ω is the viscous diffusivity, FS is the surface turbulent flux above the viscous sublayer and ZS is the depth of the sublayer, defined as: zS = ζ M

ω Rn1/ 4 Sc1/ 2 U∗

[5.16]

where ζ = 0.35, M is a constant depending on the flow regime (30 for smooth flow and 10 for rough flow; Liu et al. 1979), ω is the viscous diffusivity, Rn is the Reynolds number, U* is the friction velocity and Sc is the Schmidt number defined as: Sc =

ν ω

[5.17]

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where v = 0.000015 m 2 s −1 is the momentum molecular diffusivity. Removal of the dust particles from the atmosphere is described by the term ∂C ( J ) SINK in equation [5.13], which contains the two mechanisms of dry and wet ∂t deposition. Very small dust particles (with diameters less than 0.1 μm) are removed via coagulation with the other aerosols present in the atmosphere and Brownian motion. Above 0.1 μm, particles are heavy enough to start falling to the ground due to gravitational forces, with larger particles falling faster than smaller ones (Figure 5.7).

Figure 5.7. Settling velocity of dust, as a function of the particle size (source: Gualtieri and Mihailovic 2012)

The falling speed (settling velocity) of the particles (per size bin) can be parameterized according to Kumar et al. (1996) as:

Vd , J = Vsed +

1 ra + rb + ra rbVsed

[5.18]

where ra and rb are the aerodynamic resistance and the boundary (ground) resistance respectively and Vsed is the threshold deposition velocity as calculated by Stokes law. The aerodynamic and boundary resistances can be calculated as: ra =

1 kU *

  1    ln   − φh    z0  

[5.19]

Desert Dust Modeling

rb =

1 U* ( Sc−2/ 3

131

[5.20]

+ 10−3/ St )

where k is the von Kármán constant, z0 is the roughness length, φh is a stability correction term, which can be calculated using the Monin–Obukhov length L as proposed by Abbasi et al. (2018):

(

 2 ln 0.5 1 + 1 − 16 L 0.5  )    (  −5 L

φh = 

)

L0

where St is the Stokes number and Sc is the Schmidt number, as defined in [5.17]. As far as the wet deposition is concerned, dust particles can be removed by precipitation scavenging, cloud interception and fog and snow removal. Wet deposition is more prominent on aerosols smaller than 5 μm (Miller et al. 2006), but in general, it affects the entire dust size spectrum. For parameterizing the wet deposition, we can use a simple washout parameter and the model grid precipitation proposed by Nickovic et al. (2001) as: ∂  ∂P   ∂C J  = −WP  C J  ∂t  ∂ z ∂t    WET

[5.22]

where WP = 5⋅105 is the washout parameter and P is the total precipitation (rain, snow, graupel, etc.). However, in creating a wet deposition scheme, more developed parameterizations have been designed. The most commonly used is the parameterization proposed by Seinfeld and Pandis (1998), which includes both incloud and below-cloud scavenging processes. According to this, changes in the atmospheric concentration of dust particles are assumed to depend on a scavenging coefficient Λ:  ∂C J  = −ΛCJ  ∂t   WET

[5.23]

For dust inside the cloud, the scavenging coefficient can be written as: Λ incloud = 4.2 × 10−7

EP dcloud

[5.24]

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where E is the collection efficiency (assumed constant 0.9), P is the total precipitation and dcloud is the cloud mean droplet diameter. Clouds in a numerical model can be identified by the cloud water mixing ratio and relative humidity (Tsarpalis et al. 2018). Below for the cloud bottom, the same equation [5.24] is applied, with two major differences: first the collection efficiency in not constant, but depends on the dust particle size, and is derived following Seinfeld and Pandis (1998):

(

)

4   1/ 2 1/ 3 1/ 2 1/ 2 +  E (d J ) = RnS 1 + 0.4 Rn Sc + 0.16 Rn Sc s           3/ 2 *  St − S    μ  1/ 2   4ϕ  μ + ϕ (1 + Rn )  +  S − S * + 2 / 3     t    w 

[5.25]

where dj is the particle size diameter, Rn is the Reynolds number, μ is the kinematic viscosity of air (1.8 10-5kg m-1 s-1), μw is the kinematic viscosity of water (10-3kg m-1 s-1), ϕ = d rain / d J is the ratio of dust particle size to raindrop size, Sc is the Schmidt number, St is the Stokes number and S* is calculated as: S* =

1.2 + ln(1 + Rn) /12 1 + ln(1 + Rn)

[5.26]

5.3. Parameterization of the dust feedbacks on climate

As discussed in the introductory section of this chapter, desert dust particles have a number of feedbacks on weather and climate. Broadly, these are categorized into direct effects (dust particles modifying radiative transfer in the atmosphere) and indirect effects (dust particles modifying cloud properties and precipitation). The Earth climate is controlled by the energy balance among the incoming solar radiation, the outgoing long-wave flux and the internal atmospheric processes. When these fluxes are balanced, the Earth’s climate is stable, but any deviation will lead to a hotter or colder climate until the balance is re-established (Pandis and Seinfeld 1993). In contrast to other greenhouse gases (like CO2), desert dust has the ability to affect both sides of the energy spectrum: on the shortwave, particles scatter and absorb the incoming solar radiation, thus shading the planet surface, while on the long wave, they emit thermal radiation towards the ground and the top of the atmosphere (Tegen 2003; Helmert 2007; Spyrou et al. 2013; Boucher et al. 2013).

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133

This way they can cause both warming and cooling of the atmospheric column (Kauffman et al. 2005). When a beam of light hits a dust particle, the amount scattered depends on the intensity of the incident radiation: Fscat = Cscat F0

[5.27]

where Cscat is the single-particle scattering cross section. Accordingly, for absorption: Fabs = Cabs F0

[5.28]

where Cabs is the single-particle absorption cross section. The conservation of energy dictates that the energy that is removed from the beam is equal to the energy that is scattered (in all directions) and absorbed. The combined effect of scattering and absorption is called extinction, and therefore the single-particle extinction cross section is defined as: Cext = Cscat + Cabs

[5.29]

The scattering efficiency Qscat is defined as: Qscat =

Cscat A

[5.30]

where A is the cross-sectional surface. Accordingly, Qabs and Qext can be calculated. The ratio of Qscat to Qext is called single-scattering albedo: SSA =

Qscat Cscat = Qext Cext

[5.31]

Therefore, the portion of the energy scattered by a particle is SSA and the portion absorbed is 1 - SSA. Depending on the wavelength of the incoming radiation with respect to the particle size, there are different types of scattering that can take place. However, if we consider the size of dust particles that are injected into the atmosphere and transported over large areas, then the scattering of the incoming solar radiation can be described by the Rayleigh and Mie scattering theory (also known as Lorenz–Mie–Debye; Kerker 1969; Bohren and Huffman 1983).

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The main parameters that dominate extinction from a particle are: (1) the wavelength of the incoming radiation λ, (2) the size of the particle and (3) the complex refractive index. The size of the dust particles can be expressed by the dimensionless size parameter a as:

a =π

Dp

[5.32]

λ

where Dp is the particle diameter and π = 3.14. Depending on the value of the size parameter, the scattering is divided into Rayleigh (for small particles), Mie (for particles comparable in size with the wavelength) or geometric scattering (for large particles), as illustrated in Figure 5.8 (Alkholidi 2014).

Figure 5.8. Patterns of Rayleigh, Mie and geometric scattering (source: Alkholidi 2014)

The complex refractive index is expressed as: N = n(λ ) + ik (λ )

[5.33]

where the real part represents the extinction and the imaginary part represents the attenuation of radiation. Both parts are a function of the wavelength. Usually, the refractive index is normalized using the complex refractive index of the air N0 (Seinfeld and Pandis 1998). Some examples of the refractive index are given in Table 5.2. In most cases of a particle scattering light, the distribution of scattered light around the particle (one or many) is azimuthally symmetrical with respect to the incident beam direction. This can be described using a scattering angle θ, and the angular distribution of the scattered energy can be expressed via the phase function (Kokhanovsky 2008):

Desert Dust Modeling

P (θ ) =

F (θ )

π

135

[5.34]

 F (θ ) sin θ dθ 0

where F(θ) is the intensity of the beam. The most useful parameter that can be derived from the phase function is the asymmetry parameter ASP defined as: π

1 ASP = 2

 cosθ F (θ )sin θ dθ 0

π

π

=

 F (θ ) sin θ dθ

1 cos θ P (θ ) sin θ dθ 2



[5.35]

0

0

Atmospheric element

N = n + ik n

k

Water

1.333

1.96 x 10-9

Ice

1.309

1.96 x 10-9

NaCl

1.544

0

H2SO4

1.426

0

NH4NSO4

1.473

0

(NH4)2NSO4

1.521

0

SiO2

1.55

0 (λ = 550 nm)

Mineral dust

1.51

0.0055 (λ = 550 nm)

Table 5.2. Complex refractive index for several atmospheric elements at wavelength 598 nm (Seinfeld and Pandis 1998)

For a particle that scatters the incoming radiation isotropically in every direction, the asymmetry parameter is ASP = 0. Positive values mean that the particle scatters more radiation towards the direction of the incoming beam, while negative values mean that the particle scatters more in the opposite direction. The examination of the scattering and absorption of a group of dust particles is a demanding and time-consuming process. However, if the mean distance between the particles is larger than the size of the particles, then we can assume that the total scattered (and absorbed) energy is the sum of the individual scattered (and absorbed)

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energies of each dust particle, and a unified parameterization can be assumed (Seinfeld and Pandis 1998). Therefore, for an atmospheric layer with thickness dz, which contains dust particles, the modification of the incoming radiation can be expressed as: dF = −bext Fdz

[5.36]

where bext is the extinction coefficient, which can be written as: bext = Cext N

[5.37]

where N is the total number concentration of dust particles. By assuming that the intensity of the incoming radiation at the top of the atmosphere is F0, the intensity at height z is given by the Beer–Lambert law (Ingle and Crouch 1988): F = exp(−bext z ) F0

[5.38]

The unidimensional parameter τ = bextz is called the aerosol optical depth (AOD) and is directly related to the number of dust particles in the atmosphere. The extinction coefficient is related to the extinction efficiency:

bext =

π d J2 4

N J Qext

[5.39]

where dj is the particle size diameter and Nj is the number concentration of each size bin. The extinction coefficient can be expressed as the sum of the scattering coefficient and absorption coefficient. Similarly, the extinction efficiency can be expressed as:  bext = bscat + babs     Qext = Qscat + Qabs 

[5.40]

Once the optical depth and the extinction efficiencies have been defined, the direct radiative feedback of desert dust particles can be calculated and added in a numerical system. Such a work was performed by Spyrou et al. (2013), who found for a 6-year period that during spring, especially in May, the direct radiative feedback reaches a maximum of approximately –55 Wm−2 for the shortwave and +30 Wm−2 for the long wave on a daily average, especially in the source areas (Figure 5.9(a) and (b)).

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137

(a)

(b)

Figure 5.9. Dust radiative feedback in spring in the incoming shortwave (a) and long wave (b) radiation at the surface, as averaged for a 6-year period (source: Spyrou et al. 2013). For a color version of the figure, see www.iste.co.uk/KMS/numerical.zip

Another important impact of dust aerosols is their ability to act as cloud condensation nuclei (CCN), which changes the cloud physical, optical and chemical properties and modifies precipitation (indirect effect; Solomos et al. 2011; Boucher et al. 2013). Calculation of the aforementioned cloud properties, due to the presence of dust aerosols, often relies on empirical formulations and correlations, since

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explicitly resolving all the processes that affect cloud droplet numbers (like activation of aerosol into cloud droplets and evaporation) is time-consuming and impractical (or even unfeasible) for numerical models. Therefore, the quantification of the aerosol indirect effect must rely on parameterizations on cloud and aerosol interactions (Fountoukis and Nenes 2005). One of the most advanced parameterizations and comprehensive works remains the study of Nenes and Seinfeld (2003), which was later updated in Fountoukis and Nenes (2005). This work is based on a sectional parameterization of the aerosol size (which can vary in time) and composition (internally or externally mixed). The methodology can be separated into two stages following Fountoukis and Nenes (2005): first, the number concentration of the CCN is calculated as a function of supersaturation, following Köhler theory (Seinfeld and Pandis 1998). Then, this CCN spectrum is introduced into an adiabatic air mass with a constant vertical wind speed and the maximum supersaturation achieved during the vertical movement of particles, and the number of CCN that will form cloud droplets is calculated. To represent the entire range of CCN, a multiple lognormal form is assumed:

nmod

n( D p ) =

 J =1

 2 Dp  ln Dmean exp   2 ln 2 σ J 2π ln(σ J )   NC J

     

[5.41]

where Dp is the particle diameter, NCj is the aerosol number concentration, Dmean is the geometric mean diameter of each mode (bin) and σj is the standard deviation of each mode. The particle number concentration can be expressed in terms of the critical supersaturation distribution as: n s ( s) =

where

d ln( D p ) dNC dN . =− ds d ln( D p ) ds

dN = d ln( D p )

nmod

 J =1

 2 Dp  ln NC J Dmean exp   2π ln(σ J ) 2 ln 2 σ J  

[5.42]    derived from [5.41].   

According to Seinfeld and Pandis (1998), the critical supersaturation of a particle with diameter Dp can be calculated as:

Desert Dust Modeling

s=

2  A  B  3D p

3/ 2

[5.43]

νρ s M w , with ρs and Ms respectively the density and ρw ρw M s molecular weight of the dissolved substance, ρw and Mw respectively the density and molecular weight of water and v the number of ions resulting from the dissociation of one solute molecule. Using the above equations, the critical supersaturation distribution in [5.42] becomes: where A =

4σ M w

   

139

and B =

2/3   2  sg , J   ln    nmod s    3NCJ  s exp  n (s) =  2  2 ln σ J  J =1 3s 2π ln(σ J )    



[5.44]

where sg,J is the critical supersaturation of a particle with diameter Dmean. Using equation [5.44], the full spectrum of CCN can be calculated as the concentration of particles whose supersaturation is greater than the critical supersaturation value:

nmod

F ( s) =

 J =1

  sg , J    2ln   NCJ  s  erf  2 3 2 ln(σ J )     

[5.45]

If the maximum supersaturation is known, then the activated droplet number can be calculated from equation [5.45]: N d = F ( smax ) . In order to examine the sensitivity of this scheme to aerosol properties, Solomos et al. (2011) performed a set of idealized simulations for a convective cloud system over flat terrain. In the two scenarios considered, one was representative of a remote area with a relatively clean atmosphere and a total particle concentration of 100 cm−3 (named pristine case), and in the other, the authors assume a polluted atmosphere with a total aerosol concentration of 1500 cm−3 (named hazy case), which is a typical value for urban and industrial zones, or in cases of severe desert dust episodes. The particles were assumed to follow a bimodal lognormal distribution with fixed sizes, a geometric standard deviation of 2 and a number median diameter of 0.2 and 2 μm for each mode. The chemical composition of the particles was assumed to be ammonium

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sulfate (soluble fraction). The two simulations behaved in a similar manner after 80 min (runtime) and developed two distinct convective areas (Figure 5.10(a)). After the initial development of the convection, significant variations between the pristine and hazy scenarios were observed. In the pristine case, the cloud droplets number concentration remained low and fewer nuclei were completing for the available water. Therefore, large cloud and raindrops developed, and 100 min after initialization, there was intense precipitation (Figure 5.10(b)).

−1

Figure 5.10. Total condensates mixing ratio (g kg ) for the pristine (left column) and the hazy (right column) scenarios, after (a) 80 min, (b) 100 min and (c) 170 min simulation time (source: Solomos et al. 2011). For a color version of the figure, see www.iste.co.uk/KMS/numerical.zip

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141

In contrast, when the aerosol count was increased (hazy scenario), initiation of rain was delayed. In the first two hours of the cloud formation, the concentration of cloud droplets was very high. Due to this delay, the conversion of cloud to rain droplets was very low, inhibiting precipitation. The two clouds were both present after almost 3 hours of simulation time for the pristine conditions. During the hazy case scenario, the two cloud systems had merged into one, forming an anvil at their upper layers (Figure 5.10(c)). It must be noted that the indirect aerosol feedback is a highly nonlinear phenomenon, and the responses to precipitation, in regard to aerosol properties (composition, size distribution, hydroscopicity and others), indicate that a large portion of uncertainty unfortunately remains unresolved and unanswered (Solomos et al. 2011).

6 Simulations of Extreme Weather and Dust Events

6.1. Case study 1: numerical simulation of a Mediterranean cyclone and its sensitivity on lower boundary conditions This particular case study has been selected to demonstrate the impact of the lower boundary conditions on the structure of a deep cyclonic system developed over the Eastern Mediterranean Sea on January 21, 2004. Various comparative numerical simulations have been performed in order to determine the sensitivity of the cyclone to lower boundary conditions in the form of different sea surface temperature (SST) forcings. It is worth mentioning that, according to the MEDiteranean Experiment database (MEDEX), this cyclonic system has been classified as one of the deepest Mediterranean cyclones found in the last 40 years. Three comparative numerical simulations were performed based on a non-hydrostatic regional model. Gridded analysis, satellite-measured and climatological SST were used as model lower boundary conditions, while the rest of the atmospheric initial and boundary conditions of the simulations remained identical. In more detail, the three different sets of SST applied as lower boundary conditions are: a) gridded SST analyses from the European Centre for MediumRange Weather Forecasts (ECMWF), b) satellite-derived SST from the Advanced Very High Resolution Radiometer (AVHRR) and c) 30-year monthly climatological SST from the National Oceanic and Atmospheric Administration (NOAA). 6.1.1. Description of the synoptic conditions In order to investigate the atmospheric synoptic conditions that prevailed in the phases of cyclone formation and evolution, gridded analyses from ECMWF are

Numerical Weather Prediction and Data Assimilation, First Edition. Petros Katsafados, Elias Mavromatidis and Christos Spyrou. © ISTE Ltd 2020. Published by ISTE Ltd and John Wiley & Sons, Inc.

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used. The precursor to the surface cyclone was a cutoff low over Libya, while the surface cyclogenesis began at about noon (12:00 UTC) January 21, 2004, in the Gulf of Sidra, with 998.9 hPa central pressure value (Figure 6.1a). At that time, the maximum vorticity at 500 hPa reached 30 × 10−5 s−1, and it was located over the western part of the surface cyclone (Figure 6.1b). This is a strong indication of a rapid baroclinic development. As mentioned in the literature, during the explosive phase of the cyclone, a vorticity maximum (at the 500 hPa pressure level) appears at about 500 km westerly of its center on the surface (McDonald and Reiter 1988).

Figure 6.1. ECMWF analyses of (a) MSLP (hPa) and (b) absolute vorticity (s−1, in color) with geopotential height (gpm, in solid lines) at 500 hPa, valid at 12:00 UTC, January 21, 2004, (c) MSLP (hPa) and (d) absolute vorticity (s−1, in color) with geopotential height (gpm, in solid lines) at 500 hPa, for January 22, 2004, 00:00 1 UTC, (e) MSLP (hPa) and (f) absolute vorticity (s− , in color) with geopotential height (gpm, in solid lines) at 500 hPa, for January 22, 2004, 12:00 UTC. For a color version of the figure, see www.iste.co.uk/KMS/numerical.zip

The system was further deepened over the next 12 hours and moved over the Greek Peninsula. The mean sea level pressure (MSLP) at the center of the cyclone reached a value of 985.7 hPa, as shown in Figure 6.1c, while a two-trough system dominated at the 500 hPa pressure level over the Sidra Gulf and Ionian Sea. The absolute vorticity got its maximum of 40 × 10−5 s−1 over north-eastern Libya (Figure

Simulations of Extreme Weather and Dust Events

145

6.1d). The cyclone arrived over the coastline of Turkey and the Eastern Aegean Sea at 12 UTC on January 22 (Figure 6.1e) having been deepened to 976 hPa (according to ECMWF analysis), and it was very close to the minimum value of 977 hPa recorded at that time on the nearby surface meteorological station at Samos Island. At the isobaric level of 500 hPa, the two troughs merged into one, while the maximum absolute vorticity of 40 × 10−5 s−1 was located over Cyprus (Figure 6.1f). The moving of the cyclone at a latitude of about 36 degrees north as well as the deepening of its central MSLP during the last 24 hours equal to 22.5 hPa (equivalent to 1.42 bergerons) fulfilled the Sanders and Gyakum criterion (Sanders and Gyakum 1980) for upgrading the cyclone to a meteorological bomb. 6.1.2. Design of the simulations The sensitivity of the cyclone development on different lower boundary conditions is realized through the application of Weather Research and Forecasting limited-area model with the embedded dynamical core of Non-hydrostatic Mesoscale Model (WRF-NMM). The numerical simulations are performed on a single 305 × 273 domain with 0.09° × 0.09° grid spacing and 38 vertical levels, while the domain structure is following the Arakawa E-staggered grid, and it is centered over 39.50° N and 14.95° E. ECMWF analyses on 0.50° × 0.50° horizontal grid increment and 11 isobaric levels are used for model initial and boundary conditions. The entire set of simulations initiated on January 21, 2004, at 00:00 UTC with a forecast window of 48 hours up to January 23 at 00:00 UTC. The initial and boundary conditions are identical and based on ECMWF analyses except for the SST used as lower boundary conditions. The simulations initiated 12 hours before cyclone formation or 36 hours before it reached its minimum MSLP. Three experiments are conducted in order to examine the mesoscale and synoptic atmospheric response to various SST forcings as lower boundary conditions. The SST sources for the three simulations referred to in this case study are tabulated in Table 6.1. Simulation

Source of SST

Horizontal resolution (degrees/km)

Temporal increment

1. ECMWF-SST

Operational gridded analysis, source: ECMWF

0.50 × 0.50 ~(46 × 56)

Daily average

0.25 × 0.25 ~(23 × 28)

Daily average

1.0 × 1.0 ~(91 × 111)

Monthly average

2. AVHRR-SST Infrared Satellite, source: AVHRR 3. ClimOI-SST

Optimum Interpolation V2, source: NOAA

Table 6.1. List of the simulations and their SST characteristics

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The first simulation that is considered as the control experiment consisted of ECMWF analysis SST (Figure 6.2) on 0.50° × 0.50° horizontal resolution (namely ECMWF-SST simulation). As shown in Figure 6.2, a temperature gradient of +4°C is noticeable over the Ionian Sea (at a longitude of about 18° east) and extends up to the coastal areas of Cyprus and the Middle East.

Figure 6.2. Horizontal distribution of ECMWF SST analysis (°C) valid for January 21, 2004, in the model domain. For a color version of the figure, see www.iste.co.uk/KMS/numerical.zip

For the second experiment, the infrared satellite AVHRR SST data (namely, AVHRR-SST simulation) is used on 0.25° × 0.25° horizontal resolution and on daily temporal increment. The 30-year (1971–2000) climatological SST product derived from NOAA is applied as a lower boundary condition in the third experiment. This product known as Optimum Interpolation V2 (hereafter ClimOI-SST) has horizontal resolution on 1° × 1° (Reynolds et al. 2002). Regarding the comparison of ECMWF and AVHRR SST (Figure 6.3a), the SST difference is small but locally noticeable, with a maximum temperature difference of +1.5°C along the axis starting from the Gulf of Sidra towards the southern Crete. It should be noted that local maxima are also displayed at: a) the central Aegean Sea and b) in the vicinity of the north-western coast of Cyprus. The area with the largest SST differences (close to +2.0°C) appeared locally at the area between the Peloponnese and the northern coastline of Crete (over the Aegean Sea). The SST difference in the ECMWF versus ClimOI comparison is smaller (Figure 6.3b) with most values ranging between −0.5°C and +0.5°C. The greater differences are located around the Greek Peninsula with larger values (up to +1.5°C) at the eastern Aegean Sea. Larger differences (positive/negative) appeared only locally, and they are located close to the coastline. This was probably due to the fact that the higherresolution SST (ECMWF) was able to analyze local effects more accurately than the coarser climatic one.

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147

Figure 6.3. Comparative horizontal plots of the differences (in °C) between (a) ECMWF and AVHRR and (b) ECMWF and ClimOI, valid on January 21, 2004, 00:00 UTC. Positive values point out areas of warmer SST in the preceding dataset. For a color version of the figure, see www.iste.co.uk/KMS/numerical.zip

6.1.3. Analysis of the numerical simulations MSLP is relatively insensitive to different sources of SST as shown in Figure 6.4, valid for 18:00 UTC January 21, 2004. Simulated central pressure has been also extracted every three hours for the period January 21, 2004 at 12:00 UTC up to January 22, 2004 at 12:00 UTC (24 hours). Significant differences are not detected in cyclone central pressure, as is shown in Table 6.2. ECMWF operational analysis shows a central pressure of 985.7 hPa (last rows in Table 6.2) on January 22, at 00:00 UTC, while all three simulations overpredicted the intensity estimating a central pressure of around 987 hPa. This is an indication that the simulations predicted a slightly less intensive cyclone than in reality. Additionally, a surface pressure of 976 hPa was recorded from a local meteorological station at Samos Island (eastern Aegean Sea) on January 22, at 12:00 UTC, while the predicted MSLP nearby Samos ranged from 974.1 to 974.7 hPa. Cyclone deepening rates for

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all three simulations also ranged from 1.39 to 1.43 bergerons that are close to 1.42 appearing in the ECMWF analyses.

Figure 6.4. Comparisons of the MSLP (at 4 hPa intervals) for the entire simulations, valid on January 21, 2004, at 18:00 UTC. The blue, black and red contours correspond to the ECMWF-SST, AVHRR-SST and CLIM-SST simulations respectively. The minimum contoured values of MSLP are 995.8, 995.6 and 995.7 hPa respectively. For a color version of the figure, see www.iste.co.uk/KMS /numerical.zip

Experiment/deepening rate (bergerons)

January 21, 2004 12:00

15:00

18:00

21:00

1. ECMWF SST/1.43

996.1

996.8

995.8

991.2

2. AVHRR SST/1.40

996.2

996.8

995.6

991.0

3. CLIM-OI SST/1.39

996.1

996.8

995.7

991.2

ECMWF Analysis/1.42

998.9

-

993.2

-

January 22, 2004 00:00

03:00

06:00

09:00

12:00

1. ECMWF SST/1.43

987.0

983.4

979.6

975.5

974.1

2. AVHRR SST/1.40

986.9

983.5

979.7

975.8

974.6

3. CLIM-OI SST/1.39

987.1

983.4

980.0

975.9

974.4

ECMWF Analysis/1.42

985.7

-

982.1

-

976.4

Table 6.2. Cyclone central MSLP (hPa) as estimated in the numerical simulations. Last rows represent minimum MSLP from operational ECMWF analyses on 6 hours temporal increment

Simulations of Extreme Weather and Dust Events

149

Simulated cyclone trajectories that resulted from different lower boundary condition forcings are illustrated in Figure 6.5. The representation was based on 3hourly MSLP, starting on January 21, 18:00 UTC up to January 22, 12:00 UTC, which corresponds to 7 time spots. The red solid line corresponds to the first experiment’s track (ECMWF-SST), the blue line represents the cyclone track of the second experiment (AVHRR-SST) and the green dashed line represents the third experiment’s track (ClimOI-SST). The same colors are used for the labels referring to the MSLP estimated from the three simulations respectively (Figure 6.5). As it is shown, there is no difference between ECMWF-SST and ClimOI-SST simulations, while small track deviations are evidenced in the AVHRR-SST. It is noticeable that integer rounded MSLP remains the same in all three numerical simulations. Such similarities suggest that different SST forcings pose a negligible impact on the location and strength of the cyclone.

Figure 6.5. Simulated cyclone trajectories, valid from January 21, 2004, at 18:00 UTC to January 22, 2004, at 12:00 UTC on a time increment of 3 hours (7 time spots). The labels indicate the central MSLP of the cyclone for each another of the experiments. For a color version of the figure, see www.iste.co.uk/KMS/numerical.zip

The response of the simulated rainfall patterns on the different sources of SST has been also accessed. To this end, the 12 hours of accumulated precipitation from January 21, 2004, 18:00 UTC up to January 22, 2004, 06:00 UTC, as well as their differences in all three simulations are depicted in Figure 6.6. Despite the fact that

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precipitation maxima remained almost identical, the different SST forcings influenced the spatial distribution of the rainbands to a significant extent. Thus, the rainband appears to move faster in ECMWF-SST simulation against the AVHRRSST (Figure 6.6a). Additionally, the rainband in ClimOI-SST simulation is moving faster against AVHRR-SST (Figure 6.6b). The increase in the transition speed of the rainbands in the simulations with the warmer SST is mainly attributed to the enhanced surface fluxes that impose a stronger destabilization in the boundary layer of the cyclone and finally alter the speed of the simulated precipitation patterns.

Figure 6.6. Differences of 12-hourly accumulated precipitation (mm), as simulated for the time period from January 21, 2004 (18:00 UTC), up to January 22, 2004 (06:00 UTC), between: (a) ECMWF-SST and AVHRR-SST, and (b) AVHRR-SST and ClimOI-SST. Values in the range of ±2 mm are covered in white. For a color version of the figure, see www.iste.co.uk/KMS/numerical.zip

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151

6.2. Case study 2: nowcasting an extreme precipitation event The second case study is related to the implementation of a data assimilation (DA) model in the nowcasting mode to predict in the short term an extreme precipitation event that occurred on November 22, 2013, on the island of Rhodes, Greece. The theoretical background for this case study is presented in Chapter 4. On November 22, 2013, a warm front passed over the southeastern Aegean Sea and triggered torrential rainfall of nearly 100 mm within 4 hours mainly affecting the northern part of Rhodes. Four casualties were caused by the resulted flash flood, making it the deadliest event ever in the region. In general, DA systems involve the NWP model outputs with a wide range of observational data (METAR, SYNOP, RAOB, satellite retrievals, etc.) to present the state of the atmosphere and to derive 3D analysis schemes. As reported by Mass (2012), the massive increase of satellite and radar observations, coupled with new approaches of DA systems (e.g. 3DVAR or 4DVAR), has led to significant improvement in synoptic scale forecasts and analyses. Nowcasting is a form of very short-range weather forecast, up to 3–6 hours, initiating from observational data merged with predictions from conventional atmospheric models. The nowcasting procedure applied here is based on the implementation of the mesoscale DA model local analysis and prediction system (LAPS) that is able to combine regional surface and upper air observations with remote-sensed data and coarse grid model predictions to produce objective analyses. Therefore, DA can improve our knowledge regarding the state of the atmosphere and might serve as the initialization procedure of NWP models, or it can be used in very short range forecasting (nowcasting) weather events (Albers et al. 1996) to provide early warnings in cases of high-impact weather events such as storm surges or floodings. In the context of this study, the LAPS model has been applied in nowcasting mode to predict the extreme precipitation of November 22, 2013, on Rhodes. Additional sensitivity tests of LAPS have been also performed, wherein highresolution remote-sensed quantitative precipitation estimates (QPE) had been ingested into the nowcasting procedure. The results are evaluated through available observations obtained from independent networks of surface meteorological stations. 6.2.1. Synoptic analysis of the event The meridional circulation that prevailed over western Europe on November 20, 2013, led to a rapidly intensified trough that covered northwestern Europe, transferring polar continental air masses towards the central Mediterranean Sea. Next days, the trough tilted eastward and formed an upper air cutoff low (Figure 6.7a). These synoptic conditions resulted in the formation of surface

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depressions accompanied by cold fronts over Italy and Greece. The passage of a cold front on November 22 triggered torrential precipitation and gale force winds over western Greece. At that time, a warm front was extended from the Balkan Peninsula to the eastern Aegean Sea (Figure 6.7b). Intense warm air advection caused heavy rainfall and flooding over the southeastern Aegean Sea islands and mainly over the northern areas of Rhodes.

(a)

(b)

Figure 6.7. (a) Geopotential height (solid black lines in gpdam) and MSLP (solid white lines in hPa) for November 22, 2013, 00:00 UTC, based on the Global Forecasting System (GFS) analysis data. (b) The UK Meteorological Office surface analysis chart for November 22, 2013, 12:00 UTC (source: archive of http://www.wetter3.de). For a color version of the figure, see www.iste.co.uk/KMS/ numerical.zip

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153

6.2.2. Nowcasting methodology and results For this case study, LAPS was configured over a domain that covered the eastern Mediterranean on a 3 km horizontal grid increment and 41 vertical levels. Near-toanalysis forecasts obtained from the Global Forecasting System (GFS) are applied as LAPS background fields, using the 21-11-2013, 12 UTC and 22-11-2013, 00 UTC forecast cycles, in order to estimate the impact of each forecast cycle on the nowcasting efficiency. Surface and upper air observations (METAR, SYNOP and RAOBS) obtained from the Global Telecommunication System (GTS) were assimilated in the aforementioned control experiments (CTRL12 and CTRL00, hereafter) to produce hourly objective analyses of various atmospheric parameters and the accumulated precipitation as well. Additional experiments were conducted to the control ones employing high-resolution satellite QPE released by EUMETSAT and the National Oceanic and Atmospheric Administration (NOAA). The assimilating method of QPE is following the Barnes multivariate analysis scheme presented in Chapter 4. This scheme is a method of successive corrections, wherein each LAPS grid point is given a first guess value, which is iteratively corrected on the basis of the amount of error computed at the satellite data grid points. Each satellite data point is finally weighted according to its distance to the LAPS grid point. The experiments (see Table 6.3) have been evaluated against independent precipitation observations obtained from two conventional surface meteorological stations located in the vicinity of the flooding area and operated by the Hellenic National Meteorological Service (HNMS) and the National Observatory of Athens (NOA).

Simulation abbreviation CTRL12

Forecast cycle 21-11-2013 12 UTC

Forecast cycle 22-11-2013 00 UTC

X

CTRL00

METAR SYNOP RAOB

MPE

QMORPH

X X

X

CTRL12MPE

X

X

CTRL12QMORPH

X

X

CTRL00MPE

X

X

CTRL00QMORPH

X

X

X X X

Table 6.3. List of LAPS simulations characteristics including background fields and the ingested observations

X

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The Multi-Sensor Precipitation Estimate (MPE) is an EUMETSAT product that consists of the near-real-time rain rates. Each image is in the original pixel resolution of about 4 km in the equator, time increment of 15 minutes and the algorithm of precipitation estimation is based on the combination of polar orbit microwave measurements and images in the Meteosat infrared (IR) channel by the so-called blending technique. NOAA QMORPH satellite product is based on the Climate Prediction Center MORPHing technique (CMORPH) and produces global precipitation analyses at a spatiotemporal resolution of 8 km over the equator and 30 minutes respectively. This technique uses precipitation estimates derived from low orbiter satellite microwave observations, whose properties are advected via spatial propagation information that is obtained from geostationary satellite IR data (Joyce et al. 2004). The combination of the MPE and QMORPH products with CTRL12 and CTRL00 simulations yielded a total of six LAPS simulations that are presented in Table 6.3. The spatial distribution of the estimated hourly accumulated precipitation is found to depend strongly on the experiment and the ingested information (Figure 6.8). In more detail, CTRL12 and CTRL00 simulations ingesting only surface and upper air observations failed to provide an accurate estimation of precipitation analysis. Both simulations underestimated the maximum precipitation putting it almost 3 hours before the peak of the event, which was at 14–15 UTC (Figures 6.8a and 7.8b). This is a strong indication that the background fields play a key role in the quality of the produced analysis fields, and only a large number of assimilated point-wise observations could substantially alter the resulted objective analysis (Katsafados et al. 2012). The assimilation of satellite retrievals massively improved the spatiotemporal distribution of the precipitation. The experiments CTRL12MPE and CTRL00MPE (Figure 6.8c and d) provided the most accurate estimation of the precipitation in terms of space-time distribution and the total amount of water. The simulations based on the QMORPH retrievals showed a northeastern shift of the main precipitation core (Figure 6.8e and f), which could be mainly attributed to the coarser native spatiotemporal resolution of QMORPH products comparing against the MPE ones. The impact of the satellite retrievals assimilation on the precipitation analysis is evaluated using a direct comparison of the measured precipitation against the relevant LAPS estimates. Therefore, independent observational records from two surface meteorological stations operated by HNMS and NOA in the vicinity of the flooding area are applied to verify LAPS precipitation estimates. The timeplots in Figure 6.9 indicate that the assimilation of MPE retrievals significantly improves LAPS objective analyses. CTRL12QMORPH and CTRL00QMORPH simulations also provide relevant improvement, but both still underestimate the total amount of precipitation, especially at the location of the HNMS station (Figure 6.9b and d).

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(a)

(b)

(c)

(d)

(e)

(f)

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Figure 6.8. Hourly accumulated precipitation (color shaded in mm) and MSLP (solid line in hPa), valid for November 22, 2013, at 15:00 UTC for (a) CTRL12, (b) CTRL00, (c) CTRL12MPE, (d) CTRL00MPE, (e) CTRL12QMORPH and (f) CTRL00QMORPH simulations. For a color version of the figure, see www.iste.co.uk/KMS/numerical.zip

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(a)

(c)

(b)

(d)

Figure 6.9. Timeplots of the measured accumulated precipitation (histograms in mm 1 1 h− for NOA observational records and mm 3 h− for HNMS observational records) in November 22, 2013, with the relevant LAPS precipitation analyses (in dashed lines) for (a) NOA observations with the CTRL12, CTRL12MPE, CTRL12QMORPH simulations, (b) HNMS observations with the CTRL12, CTRL12MPE, CTRL12QMORPH simulations, (c) NOA observations with the CTRL00, CTRL00MPE, CTRL00QMORPH experiments and (d) HNMS observations with the CTRL00, CTRL00MPE, CTRL00QMORPH simulations. For a color version of the figure, see www.iste.co.uk/KMS/numerical.zip

Eventually, the CTRL00MPE analysis, which is considered as the most accurate, in conjunction with the first guess fields of LAPS, is implemented to provide nowcasts of the entire atmospheric variables in a forecast window of 60 minutes. Thus, a simple space-time extrapolation scheme has been applied in LAPS analyses for the estimation of each field tendency (Browning and Collier 1989). The hourly accumulated precipitation nowcast at 15 UTC based on November 22 at 14 UTC LAPS analysis (Figure 6.10a) is depicted in Figure 6.8b.

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(b)

Figure 6.10. (a) CTRL00MPE simulation analysis of hourly accumulated precipitation (color-shaded in mm) and MSLP (solid line in hPa) for November 22, 2013, at 14 UTC and (b) nowcast at 15 UTC of the hourly accumulated precipitation (colorshaded in mm) and the MSLP (solid line in hPa) based on the 14 UTC analysis. For a color version of the figure, see www.iste.co.uk/KMS/numerical.zip

6.3. Case study 3: seasonal predictability of a large-scale heat wave The applied methodology to perform long-term dynamical simulations and their capabilities to accurately simulate a large-scale heat wave are discussed in this case study. Such simulations rely on comprehensive atmospheric global circulation models (AGCM) covering the entire globe to avoid lateral boundary conditions forcing and usually coupled with land surface and hydrodynamic circulation models to sustain balanced lower boundaries for long simulation periods. It is known from Chapter 3 that AGCM, as any other NWP model, are sensitive to the initial conditions, which imposes a finite limit, ranging from several days to few weeks, of the atmospheric predictability. Practically, this sensitivity can be reduced by performing stochastic simulations instead of the trivial deterministic ones. Therefore, the ensemble method is introduced in the context of NWP and it is assumed as a feasible technique to integrate a deterministic simulation with an estimate of the probability distribution of possible atmospheric states (Buizza 1997). Ensemble provides an average forecast instead of an individual one, which is considered as more accurate because the most uncertain forecasted components are tended to be averaged out. It also provides an estimation of the forecast reliability, which can vary from simulation to simulation and sets the quantitative basis for probabilistic forecasting (Kalnay 2003). An ensemble procedure consisting of 31 members (simulations) is applied to predict a strong heat wave event that occurred during the end of July and the beginning of August 2010 over eastern Europe and western Russia. Maximum

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temperatures exceeded 40°C in early August, characterizing this event as a mega heat wave and causing over 55,000 deaths. The prevailed drought in Russia led to ignition of severe wildfires, amplifying in this manner the disaster in the area. The Russian heat wave resulted from an unusually strong blocking anticyclone, which formed an omega block over eastern Europe. The system transferred warm air from Africa and the Arabic peninsula to western Russia, leading to abnormal temperatures. An opposite northerly cold airflow was developed on the easterly side of the anticyclone. The cold air masses interacted with humid and warm air of low atmospheric levels and initiated a heavy rainfall across the Gangetic Plains, which are located between the Bay of Bengal and northern Pakistan (Webster et al. 2011). 6.3.1. Description of the synoptic conditions The upper air synoptic flow in July 2010 over western Russia is characterized by a persistent omega blocking pattern, which increased the meridional component of the anomalous flow at the middle troposphere transferring warm air masses to the area (Figure 6.11). This atmospheric pattern that prevailed over Russia for almost 3 weeks resulted in extremely high surface temperatures that mainly occurred around the center of the block due to the combination of northward displaced subtropical air and descending air motions.

Figure 6.11. Geopotential height at 500 hPa (solid lines in gpm) and temperature at 850 hPa (color-shaded in K), valid on July 29, 06:00 UTC based on ECMWF analysis. For a color version of the figure, see www.iste.co.uk/KMS/numerical.zip

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The mean monthly temperature anomalies for July 2010 with respect to the 1971–2000 base period released from NOAA indicate a surface temperature positive anomaly of 5°C over eastern Europe and Russia (Figure 6.12). The unprecedented surface temperature of 42.2°C was recorded in Jaskul at southern Russia, which was almost 10 degrees above its historical maximum (Table 6.4).

Figure 6.12. Mean temperature anomalies (°C) for July 2010 with respect to the 1971–2000 base period (source: National Climatic Data Center, NESDIS/NOAA). For a color version of the figure, see www.iste.co.uk/KMS/numerical.zip

Meteorological station

Date

Max. temperatures (°C)

Jaskul (Russia)

8/8/2010

42.2

Moscow (Russia)

30/7/2010

39.0

Gomel (Belarus)

7/8/2010

38.9

Joensuu (Finland)

29/7/2010

37.2

Table 6.4. Records of near-surface maximum temperatures at four meteorological stations around the affected area (source: ECMWF)

6.3.2. Model description and methodology The seasonal simulations are carried out with an advanced AGCM, the Community Atmosphere Model version 3 (CAM3) of the National Center for

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Atmospheric Research (NCAR), which is the atmospheric component of the Community Climate System Model (CCSM). CAM3 is a spectral model using the spectral technique to solve the entire set of the primitive equations presented in Chapter 1 instead of the finite difference method. In spectral techniques, the dependent variables (temperature, pressure, wind components, etc.) are transformed into their respective spectral coefficients via Fourier transformation. The spectral coefficients are then estimated instead of directly predicting the dependent variables. An inverse transformation back into physical space is finally applied to reconstruct the dependent variables. In this case study, CAM3 is configured on 85-wave triangular truncation (T85L26) and 26 levels for the vertical discretization. The specific Eulerian truncation corresponds to a zonal resolution of 1.41° × 1.41°. CAM3 includes also the community land model (CLM) to simulate the land surface energy exchanges. In the framework of ensemble forecasting, a modified version of the lagged average forecast (LAF) method is used (Hoffman and Kalnay 1983) in which the seasonal-scale simulations performed by CAM3 are initialized from consecutive daily analyses. Time-variant climatological SST dataset is also applied for the definition of the sea surface boundary condition. No lateral boundary conditions are used due to global domain coverage. In more detail, CAM3 is initialized by the global forecasting system (GFS) global analyses on 00:00 UTC for each of the 31 consecutive days in January 2010. These consecutive simulations consist of the ensemble members, and each of them has been integrated up to 8 months ahead starting all from January. In this way, 31 members are selected within a lead time of 5–7 months for the period June, July, August (JJA). Each member is initialized by the GFS analyses for each consecutive day of January at 00:00 UTC having a common end on September 1, 2010, at 00:00 UTC. Therefore, the first member (simulation) is initialized on January 1 at 00:00 UTC, and the simulation period is 243 days (8 months). The second member starts on January 2 at 00:00 UTC for a 242-day simulation. Finally, 31 simulations have been carried out, and the entire ensemble structure is shown in Figure 6.13.

Figure 6.13. The ensemble structure

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Given the ensemble simulations, the estimated temperature anomaly is extracted as the difference of the monthly averaged model results and the NCEP/NCAR longterm monthly means valid for the period of 1971–2000 (Kalnay et al. 1996). Additionally, spaghetti plots of the temperature at 850 hPa have been also produced to evaluate each member deviation from the ensemble and climatological means. The spaghetti plots depict a single isothermal line for each member to be directly compared with relevant climatological values. They are also useful to reveal regions of high predictability, if the contours have a recognizable pattern agreement, or to detect where there exists large uncertainty if the contours are eventually spreading away. 6.3.3. Predictability assessment Temperature anomaly at 850 hPa indicates the model predictability in comparison to the long-term monthly means from NCEP/NCAR. The temperature at this level is mostly influenced by synoptic scale motions because this level is usually located above the atmospheric boundary layer, but, at the same time, it affects the surface conditions.

(a)

(b)

Figure 6.14. Mean monthly temperature anomaly (K) at 850 hPa for (a) July and (b) August 2010 based on the ensemble members initialized in January 2010. Shaded areas indicate the 95% exceedance confidence level. For a color version of the figure, see www.iste.co.uk/KMS/numerical.zip

Figure 6.14 depicts the mean monthly temperature anomalies during the period that the heat wave occurred in July and August 2010. A prevailing temperature anomaly of almost +6°C is noticeable over the Middle East and northern Saudi Arabia (Figure 6.14a). Secondary maxima of temperature anomaly up to +2°C are

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located over southern Russia and the Balkan peninsula, but they are not considered as statistically significant in 95% confidence level. The mean August temperature anomaly shows a similar pattern as well (Figure 6.14b). These results indicate preliminary signals of the upcoming heat wave despite the overall low predictability to accurately estimate the strength and the location of the heat wave in a 5–7 month lead period. For the definition of the spaghetti diagrams, the mean monthly isotherm of 283 K for each of the 31 members is directly compared to the NCAR/NCEP long-term mean monthly contours of 283 K (10°C) and 278 K (5°C) respectively. The spaghetti plots reveal whether the estimated temperatures will exceed the relevant climatological values for the period and the area under consideration. According to Figure 6.15a, most ensemble members exceed the long-term mean monthly temperature at 283 K for July 2010, indicating the occurrence of the increased possibility of higher-than-normal temperatures. The few members that also lie close to the 278 K suggest that the predicted temperature anomaly will be lower than 5 K. Similarly, in August 2010, all members exceeded the climatological threshold of 283 K and even half of them exceeded the contour of 278 K (Figure 6.15b). This is a strong indication of the persistence of higher-than-normal temperatures in the entire area.

(a)

(b)

Figure 6.15. Spaghetti diagrams (red solid lines) of the mean monthly temperature at 850 hPa isotherms of 283 K (10°C) for the entire ensemble members, valid in (a) July and (b) August 2010. NCEP long-term mean monthly isotherms at 850 hPa of 283 K (10°C) and 278 K (5°C) are displayed in blue and cyan solid lines respectively. For a color version of the figure, see www.iste.co.uk/KMS/numerical.zip

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6.4. Numerical study of a severe desert dust storm over Crete On March 22, 2018, a severe Saharan desert dust transport affected Greece and, more specifically, the island of Crete (the largest island of the country). The event was captured by in situ and remote sensing measurements, namely the Finokalia atmospheric monitoring station, operated by the Environmental Chemical Processes Laboratory (ECPL) of the Department of Chemistry of the University of Crete. The concentration of dust particles at the PM station increased abruptly during the afternoon, and at 17:20 UTC a new record value for Greece was recorded at a staggering amount of 6,340 μg.m−3 (Solomos et al. 2018). The episode was also captured by MSG-SEVIRI RGB remote sensing images, as shown in Figure 6.16.

Figure 6.16. Desert dust identification from the MSG-SEVIRI dust RGB images on the March 22, 2018, at 15:00 UTC (source: Solomos et al. 2018). For a color version of the figure, see www.iste.co.uk/KMS/numerical.zip

A twofold mechanism initiated this event. First, a low-pressure system was formed on March 21, 2018, at 18 UTC over the Gulf of Sirte (Figure 6.17). The depression intensified and moved towards the Ionian Sea, reaching a minimum pressure of 990 hPa at the Gulf of Taranto on March 22, 2018, at 12 UTC. Additionally, there exists a warm conveyor belt ahead of the cold front, which is the main driving pathway for the dust advection from Libya. The low-level convergence in the warm sector of the depression helps to trap desert dust particles ahead of the cold front and transport them towards Crete (Solomos et al. 2018).

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Figure 6.17. Surface analysis chart for March 21, 2018, at 18:00 UTC (source: UK Meteorological Office)

The severity of the dust transport event and the underlying physical mechanisms makes it ideal for testing the theoretical procedure for desert dust modeling. To this end, the Advanced Weather Research Forecasting model (WRF-ARW) coupled to Chemistry (WRF-Chem) version 4 is used (Grell et al. 2005; Skamarock et al. 2008; Powers et al. 2017). The modular nature of the WRF-ARW allows the user to use a desert dust description sub-model, which describes the life cycle of particles from production to transport and eventually deposition of dust. First and foremost, in order to examine this severe phenomenon, the proper domain configuration must be selected. This was a transport episode which was initiated and transported in mesoscale, while the effects were apparent in a localized area. Therefore, the chosen domain configuration must contain the appropriate nesting to fully resolve the various scales. An example of such a configuration is given in Figure 6.18a. The outer domain is set up at a 20 km resolution, the intermediate at 5 km, covering the Mediterranean area and finally the fine domain at 1 km is localized over Crete, the area of interest. This configuration will allow for the dynamic downscaling of atmospheric processes, which in turn will properly simulate the phenomenon. The outer domain contains all the potential dust sources, including the Saharan desert and the Arabian Peninsula. As mentioned, the Saharan area is of great interest in this case; therefore, we can use vegetation cover to define these areas that can produce dust particles. This can be achieved using the in-built vegetation categorization of WRF-ARW (Skamarock et al. 2008), as seen in Figure 6.18b. As already discussed in Chapter 5,

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the production of desert dust is initiated mostly by the sandblast bombardment process, where large sand particles are lifted by the wind, follow a small trajectory and fall again to the ground, breaking into smaller particles, which are then released into the atmosphere. Therefore, the sand fraction shown in Figure 6.19a is essential for accurately describing the saltation process. As expected, the main source areas of the Saharan desert have a high percentage of sand in the soil. Additionally, the clay content in arid soils (shown in Figure 6.19b) dictates the strength of the dust flux as proposed by Marticorena and Bergametti (1995).

(a)

(b)

Figure 6.18. a) WRF-ARW domain configuration, where the three nests are shown. The intermediate domain covering the Mediterranean area is denoted as d02 and Crete is encompassed in d03, b) WRF-ARW 24 vegetation categorization covering entire simulation area. For a color version of the figure, see www.iste.co.uk/KMS/ numerical.zip

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(a)

(b)

Figure 6.19. Spatial distribution of WRF-ARW for a) sand fraction percentage and b) clay fraction percentage. For a color version of the figure, see www.iste.co.uk/KMS/numerical.zip

Having defined source areas, the host meteorological model will provide the necessary atmospheric condition field that rules production of dust (e.g. soil moisture, wind speed, temperature) and the transport of particles over the Mediterranean Sea. The prevailing synoptic conditions favor dust transport from its coastal sources over Libya towards Crete, increasing dust concentration to an unprecedented level. The flow pattern associated with this is known as the Khamsin winds in Libya and Egypt, and it mostly occurs during spring (Edgell 2006). In Figure 6.20, the MSLP for March 22, 2018, shows the position of the low system on

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the coast of Libya. The flow pattern at 850 hPa caused the dust instruction in the area of interest (Figure 6.21a) and set the favorable conditions for the produced dust particles to be transported over the Mediterranean, towards eastern Europe and Greece. In Figure 6.21b, the peak of the vertically integrated mass of dust (or dust load) is located over Crete but mostly affects eastern Mediterranean and southern Greece.

Figure 6.20. Mean sea level pressure valid on March 22, 2018, 00:00 UTC, as simulated by the WRF-ARW model. For a color version of the figure, see www.iste.co.uk/KMS/numerical.zip

(a)

(b)

Figure 6.21. a) Wind speed (in ms−1) at 850 hPa, as simulated by the WRF-ARW 2 model and b) dust load (in mgr⋅m− ), as simulated by the Chem component of WRF-ARW, both valid on March 22, 2018, at 12:00 UTC. For a color version of the figure, see www.iste.co.uk/KMS/numerical.zip

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It is apparent that the model successfully simulated the pathway of Saharan dust as it is advected from synoptic and mesoscale circulations. The complex effects of topography on the surface wind speed over Crete seem to play a key role in increasing dust concentration (Figure 6.22a). Also the desert dust transport pattern is interrupted by the topography of the island causing it to change direction and tunnel between the mountains (Figure 6.22b). The simulated maximum daily dust concentration at the Finokalia monitoring station reached approximately 5,500 μg.m−3, very close to the observed one at 6,340 μg.m−3. Therefore, the model domain configuration chosen here gave us the ability to accurately describe this intense transport phenomenon on March 22, 2018. We demonstrated how this problem was approached through modeling simulations and how the dust cycle is handled in general. Additionally, we have shown that the horizontal resolution effects are of significant importance depending on the type and scale of the physical phenomenon we want to study.

(a)

(b)

Figure 6.22. a) Wind speed at 10 m (in ms−1) and b) surface dust 3 concentration (in μgr⋅m− ), over the finest model domain, valid on March 22, 2018, at 12:00 UTC. For a color version of the figure, see www.iste.co.uk/KMS/numerical.zip

Appendices

Numerical Weather Prediction and Data Assimilation, First Edition. Petros Katsafados, Elias Mavromatidis and Christos Spyrou. © ISTE Ltd 2020. Published by ISTE Ltd and John Wiley & Sons, Inc.

Appendix 1

Inertial and rotating reference frame: Suppose that we have at our disposal two reference frames, one inertial and one rotating, having common origin and common z-axis, while it is possible for a vector to be represented in component form in either frame. The rotating frame moves around the z-axis with an angular velocity of Ω (rad/s). This case is represented below.

    Consider a random vector function A and let us call i , j and k the unit    vectors of the absolute inertial coordinate system, while i ′ , j ′ and k ′ are the unit vectors of a relative coordinate system that rotates in connection to Earth. At any time, it is valid:

       A = Α x i + Α y j + Α z k = Α′x i′ + Α′y j ′ + Α′z k ′

[A1.1]

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 The right-hand side of equation [A1.1] represents the variable A analyzed in the relative coordinate system, while the left-hand side represents the variable analyzed in the absolute system. According to the above equation, the total time derivative of  A at the inertial system is given by:  d a A dAx  dAy  dAz  = i+ j+ k = dt dt dt dt ( A.1)    dAx′  dAy′  dAz′  di ′ dj ′ dk ′ i′ + j′ + k ′ + Ax′ + Ay′ + Az′ dt dt dt dt dt dt

[A1.2]

Note that the unit vectors of the relative coordinate system vary in terms of their direction. Furthermore, the equations below are always valid, regardless of the direction of the unit vectors:    di ′   dj ′   dk ′   = Ω × i′ , = Ω × j ′ and = Ω × k′ dt dt dt

[A1.3]

Eventually, placing equations [A1.3] into [A1.2]:        d a A dAx′  dAy′  dAz′  = i′ + j′ + k ′ + Ax′ (Ω × i ′) + Ay′ (Ω × j ′) + Az′ (Ω × k ′) = dt dt dt dt       dAx′  dAy′  dAz′  i′ + j′ + k ′ + (Ω × Ax′ i ′) + (Ω × Ay′ j ′) + (Ω × Az′k ′) = dt dt dt    dAx′  dAy′  dAz′   i′ + j′ + k ′ + Ω × ( Ax′ i ′ + Ay′ j ′ + Az′k ′)  dt dt dt   d a A dA   = + Ω× A dt dt

[A1.4]

 As mentioned above, the vector A is a random function. By setting in place of   A the position vector R , equation [A1.4] results in [A1.5]:       d a R dR   = + Ω × R  Va = V + Ω × R dt dt

[A1.5]  which means that the absolute velocity Va is equal to the relative velocity plus the velocity due to the motion of the relative coordinate system. The previous equation can be derived by another way. Let us consider that a body moves in relation to a coordinate system fixed in the space (inertial reference frame), but it is stationary in relation to a relative coordinate system (e.g. a fixed body at a certain point M on the

Appendix 1

173

surface of the Earth). In this case, it is valid that r = R cos ϕ , where R is the radius of the Earth at the location of M, r is the distance of M that is perpendicular to the axis of rotation and φ is the latitude of M (Figure A1.1).

Then, it is true that:        V = 0, Va = VΩ = Ω × R = Ω × r

[A1.6]

 However, if the body is moving in relation to the Earth (i.e. V ≠ 0 ), it will travel  on the relative coordinate system at a space interval equal to Vdt (within a time interval dt), while on the inertial coordinate system, the body will travel at a space  interval equal to Va dt . The point under consideration will also move due to the    rotation of the Earth at a space interval equal to VΩ dt = (Ω × R) dt .

Obviously, as shown in Figure A1.1, it is valid that:

       Va dt = Vdt + VΩ dt  Va dt = Vdt + (Ω × R)dt          Va = V + Ω × R  Va = V + Ω × r

Figure A1.1. Relation between the absolute and the relative velocity at the rotating coordinate system fixed to the Earth

[A1.7]

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Accordingly, using the expressions [A1.4], [A1.5] and [A1.6], equation [A1.8] can be obtained:          d aVa dVa   d aVa d (V + Ω × R )  = + Ω × Va  = + Ω × (V + Ω × R ) dt dt dt dt         dV d (Ω × R) = + + Ω × V + Ω × (Ω × R ) dt dt    dV dR      = + Ω× + Ω × V + Ω × (Ω × R ) dt dt  dV        = + Ω × V + Ω × V + Ω × (Ω × R )  dt        d aVa dV = + 2(Ω × V ) + Ω × (Ω × R) dt dt       dV = + 2(Ω × V ) + Ω × (Ω × r ) dt

[A1.8]

 Del operator ( ∇ ): The del operator is a vector differential operator. In Cartesian coordinates, it is written as:

 ∂  ∂  ∂  ∇= i + j+ k ∂x ∂y ∂z

[A1.9]

 Gradient of a scalar ( ∇S ): By definition, the gradient of a scalar (S) is a vector given by:

 ∂S  ∂S  ∂S  gradS = ∇S = i+ j+ k ∂x ∂y ∂z   Divergence of a vector ( ∇ ⋅ A ):     ( A = Ax i + Ay j + Az k ) is scalar given by:

   ∂A ∂Ay ∂Az divA = ∇ ⋅ A = x + + ∂x ∂y ∂z

[A1.10] The

divergence

of

a

vector

[A1.11]

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175

Total differentiation: For a variable which is a function of time and location at a specific time {e.g. A = A[ x (t ), y (t), z(t), t] }, the total derivative can be expanded in Cartesian coordinates as follows: dA ∂A dt ∂A dx ∂A dy ∂A dz = + + +  dt ∂t dt ∂x dt ∂y dt ∂z dt   ∂A ∂A ∂A dA ∂A ∂A = + + (V ⋅∇) A u+ v+ w= ∂y ∂z ∂t dt ∂t ∂x

[A1.12]

where the total derivative dA dt of a variable A is its rate of change following the motion, and ∂A ∂t is the time rate of the variable’s variation at a fixed point (local derivative).

      Curl of a vector ( ∇ × A ): The curl of a vector ( A = Ax i + Ay j + Az k ) is similarly a vector that is given by:  i

  j k

  ∂ ∂ ∂ ∇× A = ∂x ∂y ∂z Ax Ay Az

[A1.13]

    Evidence of the expression ( −Ω × (Ω × R ) = Ω 2 r ): The angular velocity vector (in rad/s) and the radius vector of the Earth (in m) in spherical–altitude coordinates    (Figure A1.2a) are defined as: Ω = Ω cos ϕ j ′ + Ω sin ϕ k ′ (Figure A1.2b) and   R = Rk ′ .

(a)

(b)

(c)

Figure A1.2. a) Symbols of the spherical coordinate   system. R is the radius of the Earth, φ is the latitude, λ is the longitude, and i ′, j ′, k ′ are the west–east, south– north and vertical unit vectors respectively, b) components of the angular velocity vector of the Earth and c) centrifugal acceleration components

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  The term Ω × R is the rate of variation of the body’s position due to the rotation of the Earth.  i′

  Ω× R = 0 0

 j′ Ω cos ϕ

 k′

 Ω sin ϕ = ΩR cos ϕ i ′

0

R

=

Fig . A1.2 a

   Ωri ′ = Ω × r

and finally:    i′ j′ k′    −Ω × (Ω × R) = − 0 Ω cos ϕ Ω sin ϕ = Ωr 0 0   −(Ω 2 r sin ϕ j ′ − Ω 2 r cos ϕ k ′) =     Ω 2 (r cos ϕ k ′ − r sin ϕ j ′) = Ω 2 rr0 = Ω 2 r

[A1.14]

Fig . A1.2 c

  Coriolis force: Ω and V can be written in component form in spherical– altitude coordinates (Figure A1.2a) as:        Ω = Ω cos ϕ j ′ + Ω sin ϕ k ′ and V = ui ′ + υ j ′ + wk ′

 i′

    aCo = FCo = −2Ω × V = −2 0 Ω cos ϕ m =1kg

u

 k′

 j′

Ω sin ϕ 

υ

w

[A1.15]

     aCo = FCo = 2Ω(υ sin ϕ − w cos ϕ )i ′ − 2Ωu sin ϕ j ′ + 2Ωu cos ϕ k ′ m =1kg

 In the above equation, the term 2Ωw cos ϕ i ′ can be removed due to the fact that vertical scalar velocities are smaller than horizontal ones. The vertical component of the Coriolis force is also smaller than other vertical components in the momentum  equation. For this reason, the term 2Ωu cos ϕ k ′ can be removed. After these simplifications, the equation for the Coriolis force per unit mass in spherical–altitude coordinates becomes:

        aCo = FCo = −2Ω × V = 2Ωυ sin ϕ i ′ − 2Ωu sin ϕ j ′ = f υ i ′ − fuj ′ m =1kg

where f = 2Ω sin ϕ is the Coriolis parameter.

[A1.16]

Appendix 1

177

Equation [A1.16] gives another form of the Coriolis force (since w=0, only the horizontal wind speed will be considered):  i′

  j′ k ′

      aCo = FCo = −2Ω × V = f υ i ′ − fuj ′ = − f 0 0 m =1kg

u υ

  1 = − fk ′ × V

[A1.17]

0

Equation of conservation of water:   ∂q   ∂q = − ρV ⋅∇q + ρ ( E − C ) + V ⋅ ∇q = E − C  ρ ∂t ∂t

[A1.18]

      ∂ρ ∂ρ = −∇ ⋅ ρV  = − ρ∇ ⋅ V − V ⋅∇ρ  ∂t ∂t

( )

[A1.19]     ∂ρ q = − ρ q∇ ⋅ V − qV ⋅∇ρ ∂t

By adding the above equations ([A1.18] and [A1.19]):       ∂q ∂ρ +q = − ρV ⋅∇q − ρ q∇ ⋅ V − qV ⋅∇ρ + ρ ( E − C )  ∂t ∂t   ∂( ρ q) = −∇ ⋅ ( ρ qV ) + ρ ( E − C ) ∂t

ρ

Proof of equation (

[A1.20]

 1  ∇p = c p ⋅ Θ ⋅ ∇π ):

ρ

     Θ ⋅ π = T  ∇(Θ ⋅ π ) = ∇T  Θ∇π + π ∇Θ = ∇T        Θ∇π = −π ∇Θ + ∇T  c p Θ∇π = −c pπ ∇Θ + c p ∇T  R

 R 

R

R

 

−1    p  c p  ps  c p   p  c p R  ps  c p   ps   c p Θ∇π = −c p     ∇T − c p   T    − 2  ∇p + c p ∇T cp  p   p   ps   p   ps   R   −1  

 p R  p  c p = − c p ∇T − c p T   ps c p  ps   RT  1  c p Θ∇π = ∇p = ∇p p ρ

 R   −1  

 ps  c p    p

  ps    − 2  ∇p + c p ∇T   p 

[A1.21]

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Numerical Weather Prediction and Data Assimilation

Thermodynamic energy equation in terms of potential temperature: From [1.75] 

dT q a dp = + dt c p c p dt

( )

 p d T s p  dΘ =  dt dt

=

( p) ps

R

cp

R

cp

  

( )

dT R ps ( R c p −1)  ps  dp −T  − 2  p dt cp  p  dt

[A1.22]

d Θ 1 dT RΘ dp = + dt π dt c p p dt

According to Jacobson (2005), the last term in equations [1.75] and equation [A1.22] can be eliminated (dp/dt=0). Therefore:

d Θ 1 dT dΘ 1 q ∂Θ ∂Θ ∂Θ ∂Θ q =  =  +u +υ +w =  dt π dt dt π c p ∂t ∂x ∂y ∂z π c p [A1.23] ∂Θ ∂Θ ∂Θ ∂Θ q = −u −υ −w + ∂t ∂x ∂y ∂z π c p

Appendix 2

A2.1. Turbulent diffusion parameterizations In order to account for turbulent effects in the governing equations, describing the changes in the atmosphere (e.g. momentum or thermodynamic energy equations), it is necessary to apply a kind of parameterization. It is assumed that any flow quantity (e.g. velocity u) can be decomposed into a mean flow component u and a microscale component u ′ that designates at any time the deviation from the mean value. The mean quantity can be interpreted as an average over a time interval sufficiently large to filter out small-scale fluctuations (time scales are very large in comparison to the time scale of the turbulent fluctuations), but small enough to allow important large-scale temporal variations to be conserved. For the chosen time interval Δt, the time averaged quantity is given by the relation:

u=

1 Δt

t0 + Δt



udt

[A2.1]

t0

Similar expressions can be written for the other variables in the governing equations such as: pressure π*, potential temperature Θ*, moisture q* and friction terms (represented by F). All these quantities are defined as the sum of a mean and perturbation quantities:

u* = u + u ′,υ* = υ + υ ′, w* = w + w′,

π * = π + π ′, Θ* = Θ + Θ′ and q* = q + q′

[A2.2]

Due to the fact that the fluctuations are assumed to be random quantities, their average is by definition zero. Then, it is valid that:     ∇ ⋅U = ∇ ⋅U ′ = 0 [A2.3]

Numerical Weather Prediction and Data Assimilation, First Edition. Petros Katsafados, Elias Mavromatidis and Christos Spyrou. © ISTE Ltd 2020. Published by ISTE Ltd and John Wiley & Sons, Inc.

180

Numerical Weather Prediction and Data Assimilation

This can be proven by obtaining an average of the continuity equation:

∂u * ∂υ * ∂w* ∂u ∂υ ∂w ∂u ′ ∂υ ′ ∂w′ + + ≡0 + + + + + ≡0 ∂x ∂y ∂z ∂x ∂y ∂z ∂x ∂y ∂z

[a]

By defining that: (a) ∂u ′ ∂υ ′ ∂w′ ∂ < u ′ > ∂ < υ ′ > ∂ < w′ > + + = + + = 0 ∂x ∂y ∂z ∂x ∂y ∂z

  ∂u ∂υ ∂w + + = ∇ ⋅U = 0 ∂x ∂y ∂z

[A2.4]

A similar procedure can be applied in all the governing equations (motion, energy, moisture). For simplicity, one equation (in our case, the x-momentum equation) will be treated to provide a general methodology, which can be applied to all other equations. u*

∂u ∂u ′ + ∂x ∂x ∂u ∂u ′ ∂u ∂u ′ = u + u + u′ + u′ ∂x ∂x ∂x ∂x

∂u * = u + u′ ∂x

−−−> 0

[A2.5]

−−−> 0

∂u ∂u ′ =u + u′ ∂x ∂x

In the above equation, it is assumed that the average of a mean flow quantity is the quantity itself, while the average of a microscale quantity is by definition equal to zero. Similarly:

υ*

∂u * = υ +υ′ ∂y = υ

∂u ∂u ′ + ∂y ∂y

∂u ∂u ′ ∂u ∂u ′ + υ + υ′ + υ′ ∂y ∂y ∂y ∂y −−−> 0

=υ

∂u ∂u ′ + υ′ ∂y ∂y

−−−> 0

[A2.6]

Appendix 2

w*

∂u ∂u ′ + ∂z ∂z ∂u ∂u ′ ∂u ∂u ′ = w + w + w′ + w′ ∂z ∂z ∂z ∂z

181

∂u * = w + w′ ∂z

−−−−> 0

[A2.7]

−−−−> 0

∂u ∂u ′ =w + w′ ∂z ∂z

The last term in equations [A2.5]–[A2.7] can be transformed as: u′

∂ u ′u ′ ∂u ′ ∂u ′ = − u′ ∂x ∂x ∂x

[A2.8]

υ′

∂ υ ′u ′ ∂u ′ ∂υ ′ = − u′ ∂y ∂y ∂y

[A2.9]

w′

∂ w′u ′ ∂u ′ ∂w′ = − u′ ∂z ∂z ∂z

[A2.10]

The following expressions are also valid: ∂u * ∂ (u + u ′) ∂u ∂u ′ ∂u = = + = ∂t ∂t ∂t ∂t ∂t

[A2.11]

−−−> 0

cpΘ *

∂π * ∂ (π + π ′) ∂π ∂π ′ = c p (Θ + Θ′) = cpΘ + cpΘ ∂x ∂x ∂x ∂x −−−−−> 0

+ c p Θ′

∂π ∂π ′ ∂π * ∂π ∂π + c p Θ′  cpΘ * = cpΘ = cpΘ ∂x ∂x ∂x ∂x ∂x

−−−−−−−> 0

[A2.12]

−−−−−−> 0

2υ * Ω sin ϕ = f (υ + υ ′) = f υ + f υ ′ = f υ

[A2.13]

−−> 0

2 wΩ cos ϕ = fˆ ( w + w′ ) = fˆ w = fˆ w −−> 0

 ∂ 2u ∂ 2u ∂ 2u  v  2 + 2 + 2  = Fx ∂y ∂z   ∂x FRICTION TERM

[A2.14a]

[A2.14b]

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Numerical Weather Prediction and Data Assimilation

Then, the x-momentum equation can be written as: ∂u ∂u ∂u ∂u ∂π = −u −υ −w + f υ − fˆ w − c p Θ ∂t ∂x ∂y ∂z ∂x ∂ u ′u ′

−

∂x

−

∂ υ ′u ′ ∂y

−

∂ w′u ′ ∂z

∂u ′ ∂υ ′ ∂w′ + + + Fx + Fx′  ∂x ∂y ∂z −−> 0

+ u′

[A2.15]

−−−−−−−−−−> 0

∂u ∂u ∂u ∂u ∂π = −u −υ −w + f υ − fˆ w − c p Θ + Fx ∂t ∂x ∂y ∂z ∂x −

∂ u ′u ′ ∂x

−

∂ υ ′u ′ ∂y

−

∂ w′u ′ ∂z

The turbulence flux divergence terms can be approximated by:

− − −

∂ u ′u ′ ∂x ∂ υ ′u ′ ∂y ∂ w′u ′ ∂z

=

∂  m ∂u   Kx  ∂x  ∂x 

=

∂  m ∂u   Ky  ∂y  ∂y 

=

∂  m ∂u   Kz  ∂z  ∂z 

[A2.16]

where K zm is the turbulent or eddy exchange coefficient of momentum on the vertical. By analogy, the exchange coefficients of momentum on the horizontal are represented by K xm , K ym . Substituting [A2.16] into [A2.15], the x-momentum equation can be written as:

∂u ∂u ∂u ∂u ∂π = −u −υ −w + f υ − fˆ w − c p Θ + Fx ∂t ∂x ∂y ∂z ∂x [A2.17]

+

∂  m ∂u  ∂  m ∂u  ∂  m ∂u  Kx + Ky + Kz ∂x  ∂x  ∂y  ∂y  ∂z  ∂z 

Appendix 2

183

Due to the fact that the effect of turbulent eddies is considerably higher than the friction, the friction term can be neglected and [A2.17] takes the form:

∂u ∂u ∂u ∂u ∂π = −u −υ − w + f υ − fˆ w − c p Θ ∂t ∂x ∂y ∂z ∂x ∂  ∂u  ∂  m ∂u  ∂  m ∂u  Ky Kz +  K xm + + ∂x  ∂x  ∂y  ∂y  ∂z  ∂z 

[A2.18]

Similarly, the y-momentum equation is given by: ∂υ ∂υ ∂υ ∂υ ∂π = −u −υ −w − f u − cpΘ + Fy ∂t ∂x ∂y ∂z ∂y ∂ u ′υ ′ ∂ υ ′υ ′ ∂ w′υ ′ − − − ∂x ∂y ∂z

[A2.19]

The turbulence flux divergence terms can be approximated by:

− − −

∂ u ′υ ′ ∂x ∂ υ ′υ ′ ∂y ∂ w′υ ′ ∂z

=

∂  m ∂υ   Kx  ∂x  ∂x 

=

∂  m ∂υ   Ky  ∂y  ∂y 

=

∂  m ∂υ   Kz  ∂z  ∂z 

[A2.20]

Substituting [A2.20] into [A2.19], the y-momentum equation can be written as:

∂υ ∂υ ∂υ ∂υ ∂π = −u −υ −w − f u − cpΘ ∂t ∂x ∂y ∂z ∂y ∂  ∂υ  ∂  m ∂υ  ∂  m ∂υ  Ky Kz +  K xm + + ∂x  ∂x  ∂y  ∂y  ∂z  ∂z 

[A2.21]

Respectively, the thermodynamic energy and the moisture equations are given by the following relations:

∂Θ ∂Θ ∂Θ ∂Θ = −u −υ −w ∂t ∂x ∂y ∂z ∂  ∂Θ  ∂  θ ∂Θ  ∂  θ ∂Θ  +  K xθ  + K y  + Kz  ∂x  ∂x  ∂y  ∂y  ∂z  ∂z 

[A2.22]

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Numerical Weather Prediction and Data Assimilation

∂q ∂q ∂q ∂q = −u −υ −w ∂t ∂x ∂y ∂z

[A2.23]

∂  ∂q  ∂  q ∂q  ∂  q ∂q  Ky Kz +  K xq + + ∂x  ∂x  ∂y  ∂y  ∂z  ∂z  A2.2. Planetary boundary layer (PBL) parameterization

The planetary boundary layer (PBL) is the lowest part of the atmosphere, which is directly influenced by contact with the surface of the planet. Due to the fact that the model grid resolution is considerably large for explicitly resolving the small-scale fluxes prevailing in the PBL, the representation of this layer in mesoscale models is mainly manipulated through the subgrid-scale correlation terms. A common approach to mesoscale models is to resolve the PBL into a number of discrete sections: the viscous sublayer, the surface and the transition layer. The turbulent fluxes of momentum, heat and moisture are analogous to molecular diffusion, which can be expressed as: ∂u ∂υ , − w′υ ′ = K zm , ∂z ∂z ∂Θ ∂q − w′Θ′ = K zθ , − w′q ′ = K zq ∂z ∂z − w′u ′ = K zm

[A2.24]

From equation [A2.24]: w′u ′ = − K zm

∂u = −u*2 cos μ ∂z

[A2.25] w′υ ′ = − K zm

∂υ = −u*2 sin μ ∂z

( u ) , while the parameter u

In the above equation, μ = arctan υ friction velocity. By defining V =

(u

2

*

is the surface

+ υ 2 ) as the magnitude of the grid-averaged

flow, equation [A2.25] can be written as

K zm

∂V = u*2 ∂z

[A2.26]

Appendix 2

185

Integrating [A2.26] between the level defined as zo (where V = 0 ) and an arbitrary random height z above the ground level, the following equation can be obtained: Z

V (z) =

Z

u u ∂V dz =  * dz = * ∂ z kz k Z0 Z0



Z



Z0

dz u* z = ln z k z0

[A2.27]

The above equation is called the logarithmic wind profile, where k = 0.35 is the von Kármán constant and zo is the aerodynamic roughness length (in meters). The value of zo depends on the surface characteristics, ranging from 0.001 cm over ice to 10 m over large buildings. Roughness length can be a function of the friction velocity above some surfaces (e.g. water surfaces). For example, over water, Garratt (1992) suggested the form as represented by [A2.28a], while Sellers et al. (1996) suggested an expression for the surface roughness length over a vegetation canopy (equation [A2.28b]):

z0 = ( 0.01625 ± 0.00225 )

(

z0 = hc 1 − 0.91e −0.0075 LT

u*2 g

)

[A2.28a]

[A2.28b]

where hc is the canopy height and LT is the leaf area index, which is equal to square meters of leaf surfaces per square meter of underlying ground. If there are no neutral conditions within the atmospheric layer near the ground, equation [A2.26] must be generalized to include buoyancy effects. Thus, according to Prandtl’s mixing length theory, the following formula is also valid:

kz ∂V ∂V u* z z = Φm    = Φm   ∂z kz u* ∂z L L

[A2.29]

where Φm(z/L) is an experimental non-dimensional wind profile, u* is the surface friction velocity ( u*2 = − w′u ′ ) and L is the Monin–Obukhov length. Similar expressions are also valid for the distribution of the potential temperature, as well as for moisture:

kz ∂Θ  z  ∂Θ θ* z = Φh    = Φh   ∂z kz θ* ∂z L L

[A2.30a]

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Numerical Weather Prediction and Data Assimilation

kz ∂q ∂q q* z z q z = Φq    = Φq   ≈ * Φh   ∂z kz q* ∂z L  L  kz L

[A2.30b]

where Φh(z/L) is an experimental dimensionless temperature gradient, θ* is a potential temperature scale (in Kelvin), which can be considered as the flux temperature ( θ* = − w′Θ′ / u* ), Φq(z/L) ≈ Φh(z/L) is a dimensionless specific humidity gradient and q* is the water vapor scale (kg/kg). The Monin–Obukhov length is a length scale (in meters) proportional to the height above the surface on which the buoyant production of turbulence dominates rather than the mechanical production of turbulence. It is a parameter that designates the atmospheric stability within the surface layer. It cannot be measured directly, and in general the following assumptions are valid: for stable conditions, L>0; for unstable conditions, L0)

z z z z Φ m   = 1 + 4.7 and Φ h   = 0.74 + 4.7 L L L L K zm ( z ) =

kzu*

1 + 4.7

z L

and K zθ ( z ) = K zq ( z ) =

[A2.35]

kzu*

0.74 + 4.7

[A2.36]

z L

For unstable conditions (z/L< 0)

z z  Φ m   = 1 − 15  L L    

−1/ 4

z z  and Φ h   = 0.74 1 − 9  L L     1/ 4

z  K zm ( z ) = u* kz 1 − 15  L 

−1/ 2

[A2.37] 1/ 2

z  and K zθ ( z ) = K zq ( z ) = 1.35u* kz 1 − 9  L 

[A2.38]

The above set of equations are valid in the case of mechanical turbulence. If there is turbulence due to convection (z/L < −5), the following expressions can be used: 1/3

1/3

 0.078 z   2.162 z  θ q K zm ( z ) = u* z  −  , K z ( z ) = K z ( z ) = u* z  −  k L k L  

[A2.39]

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Numerical Weather Prediction and Data Assimilation

The surface friction velocity is obtained by vertically integrating equation [A2.29] from a height zo very close to the ground up to a height z (usually the anemometer height). It is assumed that V ( z0 ) = 0: z

z

Φ ( z / L) ∂V kz ∂V z dz = u*  m dz  = Φm     k u* ∂z z  L  z0 ∂z z0 z

z

Φ m ( z / L) dz  z z0

k  dV = u*  z0

z

[A2.40a]

Φ ( z / L)   k V ( z ) − V ( z0 )  = u*  m dz  u* = z =0   z0

kV ( z ) Φ ( z / L) z m z dz 0 z

Similarly, by integrating equations [A2.30], we get: k Θ( z ) − Θ( z0 )  Φ h ( z / L) dz  θ* = z z Φ ( z / L) z0 z h z dz 0 z

k Θ( z ) − Θ( z0 )  = θ* 

k [ q ( z ) − q ( z0 ) ] Φ ( z / L) k [ q ( z ) − q ( z0 ) ] = q*  h dz  q* = z z Φ ( z / L) z0 z h z dz 0 z

[A2.40b]

Under neutral conditions, it is valid u* =

kV ( z ) z



z0

dz z

 u* =

u  z  kV ( z )  V ( z ) = * ln   k  z0   z  ln   z  0

k Θ( z ) − Θ( z0 )  k Θ( z ) − Θ( z0 )  θ* =  z  θ* =    z  0.74 ln    z0 

dz z 0.74 z 0

Θ( z ) = Θ( z0 ) + 0.74

θ* k

 z  ln    z0 

[A2.41a]

Appendix 2

q* =

k [ q ( z ) − q ( z0 ) ] z

dz z 0.74 z 0

q ( z ) = q ( z0 ) + 0.74

 q* =

k [ q ( z ) − q ( z0 ) ]  z  0.74 ln    z0 

189



[A2.41b]

q*  z  ln   k  z0 

Under stable conditions, it is valid z

z

Φ ( z / L) z z dz =  Φ m   = 1 + 4.7   m L z L z0 z0

z z (1 + 4.7 ) z z 4.7 L dz = dz + L dz  z z z z z0 0

[a] = ln( z / z0 ) +

4.7 ( z − z0 ) L

Applying [a] as denominator in [A2.40a], the following equations can be obtained: u* =

kuV ( z ) z 4.7 ln + ( z − z0 ) z0 L

[A2.42]

 u  z 4.7 V ( z ) = *  ln + ( z − z0 )  k  z0 L 

Using the same methodology, similar expressions for friction temperature and moisture can be derived:

θ* =

q* =

k[Θ( z ) − Θ( z0 )] z 4.7 0.74 ln + ( z − z0 ) z0 L k[q ( z ) − q ( z0 )] z 4.7 0.74 ln + ( z − z0 ) z0 L

 z 4.7 ( z − z0 )   0.74 ln + k  z0 L    q z 4.7 q ( z ) = q ( z0 ) + * 0.74 ln + ( z − z0 )  k  z0 L 

Θ ( z ) = Θ ( z0 ) +

θ* 

[A2.43]

190

Numerical Weather Prediction and Data Assimilation

Under unstable conditions, it is valid z   dz Φ m ( z / L) dz = z z z  1 4 1 − 15( z / L)  z 0 0 z

[a]

By setting: F = 4 1 − 15( z / L)  F 4 = 1 − 15( z / L)  15( z / L) = 1 − F 4  z =

L (1 − F 4 )  15

[b]

L 4 F 3 dF L dz dz 4 F 2 dF 3 15 dz = − 4 F dF  =−  = 4 L 15 z z ( F − 1) F (1 − F 4 ) 15

Substituting [b] into [a]:  1  dz F 4 F 3 dF F 4 F 2   4 1 − 15( z / L)  z = F ( F 4 − 1) F = F ( F 4 − 1)dF z0  0 0 z

F

=

4F 2  2 + 1)( F + 1)( F − 1)dF F0 ( F F

=



  ( F

F0

[c]

A B C  + + dF + 1) ( F + 1) ( F − 1) 

2

A B C + + = ( F + 1) ( F + 1) ( F − 1) 2

A( F + 1)( F − 1) + B( F 2 + 1)( F − 1) + C( F 2 + 1)( F + 1) = ( F 2 + 1)( F + 1)( F − 1) =

( A + B) F 3 + (C − A + B) F 2 + ( A + B) F + ( A − B + C ) ( F 2 + 1)( F + 1)( F − 1)

[d]

Appendix 2

191

From [b] and [d], we obtain that i) A + B = 0 => A = −B, ii) C − A + B = 4 and iii) A − B + C = 0 => 2A + C = 0 => C = −2A. Finally, A = −1, B = 1, C = 2 and equation [c] implies that: F  A 4F 2 B C  = dF F ( F 2 + 1)( F + 1)( F − 1) F  ( F 2 + 1) + ( F + 1) + ( F − 1) dF = 0 0 F

F

F

F

1 1 1 dF +  dF + 2  2 dF = + − + 1) F F F ( 1) ( 1) ( F0 F0 F0

− F

F

F

1 1 1 d ( F + 1) +  d ( F − 1) + 2  2 dF = F + 1) F − 1) F0 ( F0 ( F0 ( F + 1)

−

−[ln( F + 1) − ln( F0 + 1)] + [ln( F − 1) − ln( F0 − 1)] + 2[arctan(F) − arctan(F0 )]  z

Φ m ( z / L) dz = [ln( F0 + 1) − ln( F + 1)] − z z0



[ln( F0 − 1) − ln( F − 1)] + 2[arctan(F) − arctan(F0 )] = ln

( F0 + 1) ( F − 1) − ln 0 + 2[arctan(F) − arctan(F0 )] = ( F + 1) ( F − 1)

 ( F + 1)  ln  0   ( F + 1) 

 ( F0 − 1)   ( F − 1)  + 2[arctan(F) − arctan(F0 )]   

z

Φ m ( z / L) ( F + 1)( F − 1) dz = ln 0 + 2[arctan(F) − arctan(F0 )] z ( F + 1)( F0 − 1) z0



In

the

above

equation,

F0 = 4 1 − 15( z0 / L ) = [1 − 15( z0 / L )]

[e]

1

4

and

1

F = [1 − 15( z / L)] 4 . Finally, applying [e] as the denominator in [A2.40a], the following equation for u* is obtained: u* =

kV ( z ) kV ( z ) = ( 1)( 1) F F + − Φ m ( z / L) 0 z z dz ln ( F + 1)( F0 − 1) + 2[arctan(F) − arctan(F0 )] 0 z

[A2.44] V ( z) =

 u*  ( F0 + 1)( F − 1) + 2[arctan(F) − arctan(F0 )] ln k  ( F + 1)( F0 − 1) 

192

Numerical Weather Prediction and Data Assimilation

Accordingly:

θ* =

k[Θ( z ) − Θ( z0 )] z

Φ h ( z / L) dz z z0



=

k[Θ( z ) − Θ( z0 )] (G + 1)(G − 1) 0.74 ln 0 (G + 1)(G 0 − 1)

[A2.45a] q* =

k[q ( z ) − q ( z0 )] k[q ( z ) − q ( z0 )] = z (G 0 + 1)(G − 1) Φ h ( z / L) z z dz 0.74 ln (G + 1)(G 0 − 1) 0

Θ( z ) = Θ( z 0 ) +

θ* 

(G 0 + 1)(G − 1)   0.74 ln  k  (G + 1)(G 0 − 1) 

[A2.45b] q ( z ) = q ( z0 ) +

q* k

 (G 0 + 1)(G − 1)   0.74 ln  (G + 1)(G 0 − 1)  

In the above equations: G0 = 1 − 9( z0 / L) = [1 − 9( z0 / L )]

1

2

and

G = [1 − 9( z / L )]

1

2

Another useful parameter is the convective vertical velocity scale w* which is equal to zero under stable conditions, while under unstable conditions (or free convection) it can be expressed by the following equation suggested by Deardorff (1974):

 gu θ z  w* =  − * * e  Θ0  

1

3

[A2.46]

where Θ0 is the potential temperature above the surface layer and Ze is the top of the PBL. A2.2.2. Viscous sublayer parameterization

The viscous sublayer is the level near the ground (z < z0). Zilitinkevich (1970) and Deardorff (1974) suggest parameterizations for temperature and

Appendix 2

193

specific humidity above the layer in relation to the corresponding surface values as follows: θ  u z  Θ( z0 ) = ΘG + 0.0962  *   * 0   k  υ 

0.45

[A2.47]  q  u z  q ( z0 ) = qG + 0.0962  *   * 0   k  υ 

0.45

In the above expressions, υ is the kinematic viscosity of air (approximately equal to 1.5 × 10−5 m2s−1), while within this layer u = υ = w = 0 . A2.2.3. Transition layer parameterization

The functional form of the eddy exchange coefficients K zm and K zθ above the surface layer is determined using an interpolating polynomial suggested by O’Brien (1970), taking into account that the diffusivity coefficients and their vertical gradient are continuous with height in the PBL:  K zm ( z s ) − K zm ( z PBL ) +    m  ∂K z ( z s )  2  + (z − z)    K zm ( z ) = K zm ( z PBL ) + PBL ∂z    ( zPBL − zs ) ( z − zs )  K m ( z ) − K m ( z )   ( z s z PBL )  2   z z − ( )    PBL s  

[A2.48]

In the above equation, z is the height at which the coefficient K zm must be determined, zPBL is the top of the PBL and zs is the top of the surface layer. Note that this equation is valid for z PBL ≥ z ≥ zs . A similar expression can be written for

K zθ = K zq :  K zθ ( zs ) − K zθ ( z PBL ) +    z z − ( )   ∂K θ ( z ) K zθ ( zs ) − K zθ ( zPBL ) )   K zθ ( z ) = K zθ ( zPBL ) + PBL (  z s  +2 ( zPBL − zs ) ( z − zs )  ( zPBL − zs )  ∂z     [A2.49] 2

194

Numerical Weather Prediction and Data Assimilation

The height of the PBL ze can be calculated under neutral conditions, as suggested by Blackadar and Tennekes (1968): ze =

λ u*

[A2.50]

f

For stable and unstable conditions, the above formula is corrected by Zilitinkevich (1972): For stable conditions, ze = λ ( Lu* f

)

(

1

[A2.51]

2

1

For unstable conditions, ze = λ u* fL 3

)

3/ 2

[A2.52]

where f is the Coriolis parameter and λ is a constant equal to 0.35, as first proposed by Zilitinkevich (1972). An expression, which describes the diurnal variations of ze, was suggested by Deardorff (1974):

∂ze ∂z ∂z ∂2 z ∂2 z = −uZe e − υZe e + wZe + K x 2e + K y 2e + Ge ∂t ∂x ∂y ∂x ∂y

[A2.53]

where under unstable conditions (L