Energy Conservation : New Research [1 ed.] 9781608767786, 9781606922316

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Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

ENERGY CONSERVATION: NEW RESEARCH

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

No part of this digital document may be reproduced, stored in a retrieval system or transmitted in any form or by any means. The publisher has taken reasonable care in the preparation of this digital document, but makes no expressed or implied warranty of any kind and assumes no responsibility for any errors or omissions. No liability is assumed for incidental or consequential damages in connection with or arising out of information contained herein. This digital document is sold with the clear understanding that the publisher is not engaged in rendering legal, medical or any other professional services.

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ENERGY CONSERVATION: NEW RESEARCH

GIACOMO SPADONI

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

EDITOR

Nova Science Publishers, Inc. New York

Copyright © 2009 by Nova Science Publishers, Inc. All rights reserved. No part of this book may be reproduced, stored in a retrieval system or transmitted in any form or by any means: electronic, electrostatic, magnetic, tape, mechanical photocopying, recording or otherwise without the written permission of the Publisher. For permission to use material from this book please contact us: Telephone 631-231-7269; Fax 631-231-8175 Web Site: http://www.novapublishers.com NOTICE TO THE READER The Publisher has taken reasonable care in the preparation of this book, but makes no expressed or implied warranty of any kind and assumes no responsibility for any errors or omissions. No liability is assumed for incidental or consequential damages in connection with or arising out of information contained in this book. The Publisher shall not be liable for any special, consequential, or exemplary damages resulting, in whole or in part, from the readers’ use of, or reliance upon, this material. Any parts of this book based on government reports are so indicated and copyright is claimed for those parts to the extent applicable to compilations of such works. Independent verification should be sought for any data, advice or recommendations contained in this book. In addition, no responsibility is assumed by the publisher for any injury and/or damage to persons or property arising from any methods, products, instructions, ideas or otherwise contained in this publication. This publication is designed to provide accurate and authoritative information with regard to the subject matter covered herein. It is sold with the clear understanding that the Publisher is not engaged in rendering legal or any other professional services. If legal or any other expert assistance is required, the services of a competent person should be sought. FROM A DECLARATION OF PARTICIPANTS JOINTLY ADOPTED BY A COMMITTEE OF THE AMERICAN BAR ASSOCIATION AND A COMMITTEE OF PUBLISHERS.

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LIBRARY OF CONGRESS CATALOGING-IN-PUBLICATION DATA Energy conservation : new research / editor, Giacomo Spadoni. p. cm. Includes bibliographical references and index. ISBN 978-1-60876-778-6 (E-Book) 1. Energy conservation--Research. I. Spadoni, Giacomo. TJ163.3.E398 2009 333.791'6--dc22 2008047492

Published by Nova Science Publishers, Inc.

New York

CONTENTS Preface

vii

Short Communications On Energy Conservation in Infinite Systems Jon Perez Laraudogoitia Numerical Analysis of Double Diffusive Natural Convection in Trapezoidal Enclosure Filled with Porous Medium Under Magnetic Field R. Younsi

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Research and Review Studies

1 3 13

33

Chapter 1

Identifying Energy Conservation Behaviors of 6, 7, and 8th Grade Elementary School Children: A Pilot Study with Turkish Children Arzu Şener and Seval Güven

35

Chapter 2

Energy Conservation Measures as Investments Joshua M. Pearce, David Denkenberger and Heather Zielonka

67

Chapter 3

Towards Sustainable Energy Consumption Patterns: Economic Modeling of Energy Conservation in Greece Eleni Sardianou

87

Chapter 4

Energy Indeterminacies and Inequalities During Tunneling Theodosios G Douvropoulos

103

Chapter 5

Energy Conservation in Hawking Radiation as Tunneling Fujun Wang and Yuanxing Gui

123

Chapter 6

Spray Combustion Simulation for Low-NOx Emissions Hirotatsu Watanabe, Yohsuke Matsushita, Hideyuki Aoki and Takatoshi Miura

129

Chapter 7

Mass and Energy Conserving Fully Discrete Schemes for the Shallow-Water Equations Yuri N. Skiba and Denis M. Filatov

155

vi Chapter 8

Examination of Two Turbulent Kinetic Energy Parameterizations in a Convective Boundary Layer: Comparison of Two Models on Case Studies Mehrez Samaali, Dominique Courault, Albert Olioso, Michael Bruse, René Occelli and Pierre Lacarrère

199

Chapter 9

Energy Conservation in Nanoelectronics J. Hoekstra

215

Index

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Contents

247

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PREFACE This new book concentrates on energy conservation which is the practice of decreasing the quantity of energy used. It may be achieved through efficient energy use, in which case energy use is decreased while achieving a similar outcome, or by reduced consumption of energy services. Energy conservation may result in increase of financial capital, environmental value, national security, personal security, and human comfort. Individuals and organizations that are direct consumers of energy may want to conserve energy in order to reduce energy costs and promote economic security. Industrial and commercial users may want to increase efficiency and thus maximize profit. Energy conservation is an important element of energy policy. Energy conservation reduces the energy consumption and energy demand per capita, and thus offsets the growth in energy supply needed to keep up with population growth. This reduces the rise in energy costs, and can reduce the need for new power plants, and energy imports. The reduced energy demand can provide more flexibility in choosing the most preferred methods of energy production. By reducing emissions, energy conservation is an important part of lessening climate change. Energy conservation facilitates the replacement of non-renewable resources with renewable energy. Energy conservation is often the most economical solution to energy shortages, and is a more environmentally benign alternative to increased energy production. One recurrent debate in the technical and philosophical literature in the last third of the 20th century has touched on indeterministic processes composed exclusively of deterministic sub-processes, a type of physical process with some conceptually interesting consequences. From their definition, such processes typically (although not necessarily) involve systems with an infinite number of component subsystems (which I shall call infinite systems) and in this first Short Communication, I shall be limiting my attention to this latter type. The reason for this restriction is that in a very general class of processes that take place in infinite systems, the law of the conservation of energy is violated in an unexpected way. This very general class of processes to which I refer are called indeterministic supertasks and what is unexpected in the violation of energy conservation that they entail lies in the fact that the violation does not take place in any of their component sub-processes. Furthermore, we are dealing with a-causal processes in the precise sense that physical theory does not provide any causal factor that accounts for such sudden changes in the total energy of the systems. All examples of indeterministic supertasks proposed to date correspond to processes in which

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viii

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energy conservation is violated. The question is: are there indeterministic supertasks in which the energy is conserved? The central purpose of this paper is to answer that there are. The possibility of indeterministic supertasks in which the energy is conserved arises as a consequence of a new variety of indeterminism in the supertask sphere that does not depend on the presence of self-excitation processes. A model is constructed that facilitates detailed criticism of some of the interpretations appearing in the recent philosophical literature in respect of supposed varieties of indeterminism unacceptable in classical dynamics. One interesting defence of this criticism is that, unlike the excessively specific character of several results concerning dynamic supertasks, in my argument I do not need to assume that the physical systems involved are composed of point particles or rigid solids. This last is particularly significant: in the case of deformable bodies, although the energy may be distributed in some extremely complicated ways in their internal degrees of freedom, beyond the analytical difficulties one may rigorously prove that there are indeterministic supertasks in which the energy is conserved. Double-diffusive convection flow is of a great of interest in several industrial applications. Convection plays a dominant role in crystal growth in which it affects the fluidphase composition and temperature at the phase interface. It is the foundation in modern electronics industry to produce pure and perfect crystals to make transistors, microwave devices, infrared detectors and memory devices. Natural convection adversely affects local growth conditions and enhances the overall transport rate. The problem considered in this paper is a two-dimensional natural convection flow in a trapezoidal porous cavity filled with a binary fluid. The physical model for the momentum conservation equation makes use of the Darcy–Brinkman equations, which allows the no-slip boundary condition on a solid wall to be satisfied. Laminar regime is considered under steady state condition. Different types of boundary conditions have been employed. Dirichlet conditions are prescribed along the bottom and inclined surfaces for temperature and concentration. On top surface, Neuman, i.e., zero gradient conditions are assigned to temperature and concentration. A uniform magnetic field was applied in horizontal direction. The two-dimensional flow equations, expressed here in a velocity–pressure (UVP) formulation, along with the energy and concentration equations. The computational fluid dynamics (CFD) simulation was carried out using FEMLAB, which is application software from MATLAB. This second Short Communication was done for constant Buoyancy ratio N=1, Lewis number Le = 2, and Prandtl number Pr = 0.149. An extensive series of numerical simulations is conducted in the range of 104 ≤ Ra ≤ 107, 1 ≤ Ha ≤ 4.102 , 10-9 ≤ Da ≤ 1, where Ra, Ha and Da are Rayleigh number, Hartman number and Darcy number, respectively. Streamlines, isotherms and iso-concentrations are produced to illustrate the flow structure transition from. In addition, the predicted results for the average Nusselt and Sherwood numbers are presented and discussed for various parametric conditions. It was observed that heat and mass transfers were decreased with increasing Hartmann number and decreasing value of Darcy number. The importance of the issue of energy is felt with its gradually increasing dimensions in our country as well as in the world. With the unsustainable consumption models of the developed technology in our country and in the world, increasing population and production of the world produce an increasing pressure on the air, ground, energy and other necessary resources. As the needs increase, the scarce energy resources are consumed fast and thus, the insufficiency of energy comes in front of us as one of the biggest problems of 2000’s.

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Preface

ix

Meeting the needs and expectations of the human being is the prime objective of the development. Sustainable development is to search the ways of meeting the today’s needs and expectations without compromising from the future needs and expectations. To make the living standard higher than the minimum fundamental level sustainable, the consumption standards in every field should have reached to the long-term sustainability. Contrary to this, most of us live beyond the ecologic means of the world. For example, the data on electric consumption shows clearly that this is the case. The energy consumption was first of most valid indicators to measure the development level of the countries until the recent times. As much as higher energy consumption per capita occurred in a country, it was deemed that the country was most developed. Today, rather than energy consumption amount, its efficiency is discussed. For now, it is important how the produced or acquired energy is used efficiently, not how much energy is produced. This drives the people search the ways to use the energy resources more efficiently and economically. In our country where is not rich in respect of energy resources, the abroad dependence already at 62% in this field will increase by the time, as the population will increase. Especially, the number of children and young people within the increasing population ratio enhances their importance on energy consumption as being in every field. There are 1.7 billion children who are younger than 15 years-old in the world. More than 32.0% of the world population is formed by this population group. In the developing countries, the proportion of the children to the total population is higher than in the developed countries. According to the recent census, 12.6% of the population consists of age group of 0-4 yearsold in our country’s population, 11.47% 5-9 years-old, 10.56% 10-14 years-old. When one considers those proportions, it is necessary to increase the sensitivity of the child and young population who will be the future’s adult to the energy consumption. The study in Chapter 1, which was executed on 150 students, is a pilot study in order to determine weather the students, who attend to the 6, 7, 8 grades in the primary school, use the limited energy resources sufficiently productively and rationally without wasting them in more rational manner. Many energy conservation measures (ECMs) that utilize energy more efficiently than standard devices or practices will save money in the long term; however, decision makers often fail to deploy ECMs because of what is viewed as a prohibitive initial investment and long payback times. In order to improve the probability of implementation of both the quantity and size of ECMs, the economic justification must be clarified for decision makers at all levels. In this chapter, a method is reviewed that overcomes common mistakes that encourage poor economic decisions to be made in favor of less efficient technologies or processes. By treating ECMs as standard investments, rational decisions can be made by comparing ECMs to other more traditional investments. Chapter 2 provides a graphical tool to simplify the calculations for the return on investment (ROI) of any ECM from only the simple payback and device lifetime. For example, an individual might have an investment option of 4% ROI after taxes and inflation in the bond market and is considering an energy efficient lighting upgrade. For a device lifetime of 20 years (such as an electronic ballast fluorescent lighting system), as long as the payback is 13 years or less, the ECM (and in this case the more efficient ballast) is the superior investment. Utilizing this method will encourage the increased deployment of energy efficiency and renewable energy technologies, while

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improving the economic performance of the companies and individuals that utilize the method. Chapter 3 presents insights into the profile of consumers’ who are willing to restrict their final energy consumption as a reaction to higher energy prices employing cross section data for Greece. Probit regression analysis is used to investigate disparities in consumer’s choice to conserve energy from heating and mobility among different socio-economic groups. The empirical results support that income is the strongest predictor variable of conserving energy from both heating and mobility. It is estimated that individuals' characteristics may explain differences among consumers’ decision to conserve energy in case of an increase of energy prices. The role of quantity d κ / dE , which is the derivative of the action in a classically forbidden region of motion with respect to energy, is discussed in detail. As a result the concept of the energy indeterminacies as well as the mechanism of the broadened turning points, are introduced. These indeterminacies produce interesting inequalities that arise from the tunneling process, and relate the above mentioned mechanism to the uncertainty principle. Energy is conserved during tunneling through the energy indeterminacies that the entry and exit turning points create, and in addition a time scale for tunneling is generated due to the fact that the classically forbidden action is a decreasing function of the energy. The analysis shows that during the tunneling process, position becomes a function of the potential and this is the reason for the derivative dy / dU to appear. This derivative seems to play a crucial role, since it determines the entry and exit turning points. The previous mentioned indeterminacies are discussed in Chapter 4, for two different types of generic potentials, namely the single and double well potential. In the first case they produce the real energy shift that characterizes the resonance states, while in the second case produces the energy indeterminacies during tunneling. The law of energy conservation, which was one of the most important discovery in nineteen century, plays a critical role in physics. With the development of modern physics, energy conservation law also makes great contributions to black holes. Recently a direct semi-classical derivation of black hole radiation as a tunneling process was provided to avoid ”information loss paradox”. The tunneling picture possesses a dynamic background since the energy conservation law is valid. When the outgoing particle tunnels out the horizon, the black hole will shrink, so the tunneling barrier is defined by the tunneling particle itself. In Chapter 5, we investigate Hawking radiation as massless particles tunneling across the event horizon of the Schwarzschild black holes. The conclusion shows that the black hole spectrum is not a precisely thermal emission spectrum, but is consistent with an underlying unitary theory. The ingoing particle also has influence on tunneling probability as well as outgoing paricle in the viewpoint of energy conservation. Chapter 6 describes a validation of numerical models for spray combustion simulation and a numerical investigation of spray combustion for low NOx emission. In a combustor used in this study, kerosene spray was injected with a pressure atomizer, and high-speed preheated air was introduced towards the spray flow through 6 air inlet nozzles to improve the mixing of the spray and the air. In the numerical simulation, the conservative equations of mass, momentum and energy in the turbulent flow field were solved in conjunction with the k-ε two-equation model. The soot distribution in the combustor was calculated by solving a transport equation for the soot mass fraction using simple expressions for the soot formation

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Preface

xi

and oxidation rates. A post-processing NO formation model was used to predict the rate of thermal and prompt NO formation. The validity of the numerical models was presented by comparing with the experimental data. The exhaust NO mole fraction with considering soot showed a good agreement with the experimental data. Then, the effect of the diameter of air inlet nozzles on combustion behavior and NO emission was numerically investigated. When the diameter of air inlet nozzles decreased from 8 to 4 mm, the calculated exhaust NO mole fraction was drastically decreased by about 80%. The increase in the inlet velocity resulted in the improvement of the mixing of the spray and the air, and hence the high temperature region where thermal NO was formed became narrow. As a result, exhaust NO mole fraction decreased. Furthermore, the decrease in NO mole fraction was explained from the decrease in the residence time in the high temperature region above 1800 K. Consequently, the numerical results indicated that the mixing process of the fuel and the air had an important effect on low NOx combustion. A numerical simulation of spray combustion was also performed for the case that a baffle plate was equipped in the spray combustor because it does not require drastic changes to the combustor shape. When a suitable baffle plate was applied to the combustor, NO emission decreased by about 40%. It was numerically shown that the baffle plate was not only a flame holder but also a tool for reducing NO. In Chapter 7, a new method for constructing finite difference schemes for the shallowwater model (SWM) is suggested. The model equations can be considered in a limited area, in a doubly periodic domain, in a periodic channel on the plane, on a whole sphere, and in a periodic channel in the longitudinal direction on the sphere. An essential advantage of the method is that it produces fully discrete (both in time and in space) shallow-water schemes that exactly conserve the mass and the total energy and whose numerical implementation is computationally inexpensive. Our approach is based on splitting of the SWM operator by coordinates and by physical processes. Thereby the solution of the original system of 2D partial differential equations reduces to the solution of three simple problems containing either 1D partial differential equations or ordinary differential equations. In fact, an infinite family of such conservative schemes is proposed, which are either linear or nonlinear depending on the choice of certain scheme parameters. On a doubly periodic manifold, the method allows constructing conservative finite difference schemes of arbitrary approximation orders in the spatial variables. Moreover, if the SWM is considered on the entire sphere (which is not a doubly periodic manifold) then the method makes it possible to use the same numerical schemes (of arbitrary approximation order in space) and algorithms as for a doubly periodic manifold. The numerical SWM algorithms are computationally cheap, because each scheme is easily realised by fast direct methods of linear algebra. The skill of the finite difference schemes is illustrated by numerical results. Modeling of energy and mass interactions within the atmospheric boundary layer at meso and global scales is essential for weather forecast and climate analysis. Interactions near the surface are commonly described in Soil-Vegetation-Atmospheric models which involve energy budget calculations for the soil and the vegetation and require temperature, humidity and wind descriptions for heat and latent flux calculations. However, turbulence at the upper layers of the atmosphere is described with turbulence closure models which account for momentum and heat processes. Higher closure model performances have been demonstrated in describing turbulent mixing for various atmospheric boundary layer states (i.e. stable/unstable, clear/cloudy, etc). In Chapter 8, two higher Turbulent Kinematic Energy (TKE or E) models are compared in a convective boundary layer. The first model is based on

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a local turbulence closure approach using a modified version of the standard Launder and Spalding (1974) TKE model which calculates two prognostic equations for TKE and its dissipation. This closure model is implemented in a simple (compared to other ABL models) model named ENVIronmental METeorological (ENVIMET). The second one corresponds to a non-local turbulence closure and uses the Bougeault and Lacarrère’s (1989) parameterization which consists of solving a prognostic equation for TKE and using mixing lengths to describe turbulence. Although simulations are compared on two case studies, a comparison to relevant results published in the literature is also outlined. Conservation of energy is a general principle in nature, however, some modern theories on nanoelectronics seem to violate this principle. For example, the orthodox theory of singleelectronics predicts tunneling of electrons through a junction in nanoelectronic structures when the free-energy of the system is lowered as a result of the tunnel event. But the energy loss cannot always be modeled as dissipation by heat or by radiation in this theory. Especially, when the tunnel event of a single electron is considered the dissipation cannot be modeled by a resistance. The impossibility to model radiation is due to the description of the system in terms of (electronic) circuit theory. Due to the fact that the tunnel junction in Coulomb blockade is modeled as a capacitor, energy conservation in these nanoelectronic circuits can be investigated by studying a classical problem in circuit theory: the switched two-capacitor network. Subsequently, in Chapter 9, energy conservation in the two-capacitor network is discussed in case of both bounded and unbounded currents. Because nanoelectronic tunneling of single electrons can be modeled with an unbounded current, it is shown that many common features exist.

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SHORT COMMUNICATIONS

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In: Energy Conservation: New Research Editor: Giacomo Spadoni, pp. 3-12

ISBN: 978-1-60692-231-6 © 2009 Nova Science Publishers, Inc.

ON ENERGY CONSERVATION IN INFINITE SYSTEMS Jon Perez Laraudogoitia Dept. de Logica y Filosofia dela Ciencia, Univ. del Pais Vasco, Victoria-Gasteiz, Spain

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Abstract One recurrent debate in the technical and philosophical literature in the last third of the 20th century has touched on indeterministic processes composed exclusively of deterministic sub-processes, a type of physical process with some conceptually interesting consequences. From their definition, such processes typically (although not necessarily) involve systems with an infinite number of component subsystems (which I shall call infinite systems) and I shall be limiting my attention to this latter type here. The reason for this restriction is that in a very general class of processes that take place in infinite systems, the law of the conservation of energy is violated in an unexpected way. This very general class of processes to which I refer are called indeterministic supertasks and what is unexpected in the violation of energy conservation that they entail lies in the fact that the violation does not take place in any of their component sub-processes. Furthermore, we are dealing with a-causal processes in the precise sense that physical theory does not provide any causal factor that accounts for such sudden changes in the total energy of the systems. All examples of indeterministic supertasks proposed to date correspond to processes in which energy conservation is violated. The question is: are there indeterministic supertasks in which the energy is conserved? The central purpose of this paper is to answer that there are. The possibility of indeterministic supertasks in which the energy is conserved arises as a consequence of a new variety of indeterminism in the supertask sphere that does not depend on the presence of self-excitation processes. A model is constructed that facilitates detailed criticism of some of the interpretations appearing in the recent philosophical literature in respect of supposed varieties of indeterminism unacceptable in classical dynamics. One interesting defence of this criticism is that, unlike the excessively specific character of several results concerning dynamic supertasks, in my argument I do not need to assume that the physical systems involved are composed of point particles or rigid solids. This last is particularly significant: in the case of deformable bodies, although the energy may be distributed in some extremely complicated ways in their internal degrees of freedom, beyond the analytical difficulties one

4

Jon Perez Laraudogoitia may rigorously prove that there are indeterministic supertasks in which the energy is conserved.

Introduction In its most general form, the principle of energy conservation establishes that the variation in kinetic energy plus internal energy, per unit of time, is equal to the sum of the work plus any other energy introduced or extracted, also per unit of time. Energies introduced may include thermal energy, chemical energy, electromagnetic energy and so on. If we only consider the introduction or extraction of thermal energy, the principle of energy conservation takes the form of the first law of thermodynamics. And if we only consider mechanical quantities, then it becomes a directly demonstrable theorem based on the equations of motion. However this demonstration presupposes that the physical system involved consists solely of a finite number of parts or subsystems. When that is not the case, the energy (which is mechanical energy) may not be conserved in some processes. In fact, going by the results obtained to date in the literature, in the processes called indeterministic supertasks at first glance it seems that the energy is never conserved (assuming of course that it is well-defined, and in particular that it is not infinite). In Pérez Laraudogoitia (2008) I have however shown that there is no intrinsic connection between the indeterministic nature of a supertask and the non-conservation of energy. Here I take a further step in this direction by showing that there are indeterministic supertasks in which the energy is conserved. To do so requires the use of an interesting new model of indeterministic behaviour.

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1. Indeterminism through Deterministic Subprocesses Indeterminism is not unconnected to dynamics (whether classical or relativistic). The simultaneous collision between at least three particles in one-dimensional motion already leads to a final state that the laws of conservation are not capable of determining uniquivocally. In this case, the multiple interaction that takes place is not capable of determining the final state of the three-particle system because it is not capable of determining the final state of any of the particles (and it would be sufficient if it were incapable of determining the final state of at least one of them). In general, if the interaction to which at least one part of a system is subjected does not permit the future evolution of that part of the system being determined uniquivocally then the process of evolution of the complete system will be indeterministic and we will say that in such a case indeterminism appears through at least one indeterministic subprocess. But it is a notable fact that some times things can become subtler: indeterminism may appear despite the fact that the interactions to which each part of a system is subjected permit the evolution to be uniquivocally determined, and we will say in such a case that indeterminism appears through deterministic subprocesses. A striking example of this is the temporal inversion of Xia’s cosmic game of tennis, admirably described by Stewart (1996). In the direct process, a set of five point particles subjected exclusively to their mutual gravitational interaction evolves in such a way that each one escapes to the infinite

On Energy Conservation in Infinite Systems

5

in a finite time (the same for all). The temporal inversion of this process leads to the sudden appearance (from infinity) of a set of five particles that, under their mutual gravitational interaction, evolve in a perfectly deterministic way. What is not determined in all this is the instant (if any) at which Xia’s particles will erupt in the space from infinity. However, in this article (given my stated object) I shall concentrate more on situations in which the indeterminism arising from deterministic subprocesses is of a more standard type: they correspond to processes that involve the indeterministic temporal evolution of a given dynamic system and, unlike Xia’s example, require the consideration of infinite systems in the precise sense that they consist of an infinite number of components. This leads directly to the need to consider supertasks.

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2. Indeterministic Supertasks In principle, a supertask may be understood as an infinite sequence of actions or processes that take place in a finite total time. As we are interested in dynamic systems—ones that are furthermore free from external actions—from now on we shall think of a supertask in terms of an infinite sequence of processes that involve particles in interaction. We shall also suppose, for simplicity’s sake, that the interactions between particles take place by means of collisions considered elastic (inelasticity is impossible if we are dealing with point particles or rigid bodies and, when it is possible, we are in fact outside of the realm of pure mechanics by allowing the dissipation of mechanical energy in the form of heat). Even with the restrictions we have just assumed, the concept of supertask introduced is too ample because it applies to any infinite sequence of elastic collisions between particles (in a finite time), however mutually independent these collisions are. To narrow things down acceptably, global independence is an adequate concept. Let us suppose then that the particles of the set P evolve (during a finite period of time) exclusively by means of an infinite number of elastic collisions between them following the process PR from the initial condition C given at the instant t0. For each q∈P, let Pq be the set of particles of P causally connected with q (obviously, through elastic collisions) during the process PR from the initial condition C. We shall say that the collisions involved in PR (from C) are globally independent when, for each q∈P, a) Pq is finite and b) if there is more than one evolution possible for q from C this is exclusively due to the existence of indeterministic collisions at PR. Now we will understand a supertask as being a process that lasts a finite time and develops exclusively by means of an infinite number of elastic collisions not globally independent. Should the process in question be indeterministic we shall properly say the supertask is indeterministic. However, this is a concept of an indeterministic supertask that we might call “weak”. The reason for this weakness lies in the fact that the source of the indeterminism may be in the rather trivial circumstance that at least one of the collisions involved is indeterministic. Subtler and more interesting is the concept of an indeterministic supertask in a “strong sense”. An indeterministic supertask in a strong sense is an indeterministic process that lasts a finite time and develops exclusively by means of an infinite number of elastic deterministic collisions not globally independent. By requiring all the collisions to be deterministic, the condition of global independence is reduced to the requirement that, for each q∈P, a) Pq is finite and b) only one single evolution is possible for q from C. Therefore, the negation of global independence

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supposes the existence of at least one q∈P such that a) Pq is not finite or (a non-exclusive disjunction) b) there is more than one possible evolution for q from C. The notion of indeterministic supertask in a strong sense encompasses processes in which indeterminism is possible in such a subtle way that, in contrast to what might have been suspected until now, it coexists with the conservation of energy. Let us look first at what this subtlety consists of.

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3. A New Route to Indeterminism The conjunction of the two central notions introduced in sections 1 and 2 leads to the idea of indeterministic process through deterministic subprocesses, performed by means of an indeterministic supertask in a strong sense. This is the type of process we shall be concentrating on. As noted above, it involves physical systems that consist of an infinite number of components, in our case particles, and having some measure of its complexity is pertinent. For our purposes, the complexity of an infinite system of particles is suitably characterized by an ordered triple of ordinal types. Considering a suitable orthonormal system of spatial axes, the first component of the triple is the ordinal type of the set of points resulting from the projection of the points occupied by the particles (or by their centres of masses, if they are not point particles) on the first coordinate axis, the second component of the triple is the ordinal type of the set of points resulting from the projection of the points occupied by the particles on the second coordinate axis and analogously for the third component of the triple. For example, if an infinite set consists of particles occupying all the points of the space of coordinates (h,0,0) (0,m,0) and (0,0,n), with h,m,n any positive integers, then their complexity is described by the triple (ω,ω,ω) where ω is the ordinal type of the positive integers in their natural ordering. If we consider onedimensional configurations of particles only (as is, with clear preference, the case in what has been done up to now in the study of dynamic supertasks) then it is enough to provide a sole ordinal type to measure their complexity: the one corresponding to the particles in the order in which they appear in the configuration. The lowest level of complexity for an infinite one-dimensional configuration clearly corresponds to the ordinal type ω and this is, by a long way, the case studied in most detail in the literature. It is common knowledge that indeterministic supertasks in a strong sense can be developed in an infinite one-dimensional configuration of particles of complexity ω. However this is only possible in a simple way by means of its spontaneous self-excitation and this clearly implies a violation of energy conservation. Indeed, as an indeterministic supertask in a strong sense will consist exclusively of determinist collisions, the only source of indeterminism can lie in whether there occurs or not a certain possible infinite sequence of collisions (determinist, of course) in which there is no first collision (if there is a first collision in a possible sequence of determinist collisions then it is automatically determined that that sequence will occur). Given the ordinal type ω of the set of particles, an infinite sequence of collisions in which there is no first collision implies, in the simplest case, that for any positive integer n > 1, the particle n–1th collides with the nth, which has previously collided with n+1th. As this infinite sequence of collisions constitutes an indeterministic supertask it should take place in a finite time. It thus corresponds to an excitation of the system of particles and, given

On Energy Conservation in Infinite Systems

7

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that it may take place spontaneously, it in fact corresponds to a spontaneous self-excitation. Obviously the conservation of energy is violated to the extent necessary for the spontaneous process to actually take place. New forms of indeterminism are obtained by “augmenting” the ordinal type (i.e. the complexity) of the one-dimensional configuration of infinite particles. For instance, for a complexity ω+1 (essentially identical in all relevant aspects to a complexity ω+n) the possibility of indeterminism by global interaction appears (Pérez Laraudogoitia, 2005). We may also consider complexities that aren’t ordinal numbers, although the simplest of them all, namely ω* (the ordinal type of negative integers) and n+ω*, are merely mirror images of ω and ω+n, which means they are of no interest here. It is interesting to consider the “lowest” complexities that are not ordinal numbers or mirror images of them. ω+ω* is one of the two simplest ordinal types that differ non-trivially (i.e. not through a mere transformation of symmetry) from an ordinal number (the other is ω*+ω). Let us see how a configuration of complexity ω+ω* provides in a simple manner a new route to indeterminism that is radically different from the above, a route that will reveal the possibility of indeterministic supertasks in a strong sense in which energy is conserved. The idea is simple. Let us number the particles of the subconfiguration ω with the positive entire numbers in their natural order and the particles of the subconfiguration ω* with the entire negative numbers also in their natural order. We know from before that in an indeterministic supertask in a strong sense the root of indeterminism must lie in whether a certain possible infinite sequence of determinist collisions, in which there is no first collision, takes place or not. The difference is that now a lot of different, infinite, possible sequences of deterministic collisions in which there is no first collision (without this implying the need for a self-excitation) are intuitively evident, at least in principle. For example, for any positive integer n, particle n could collide with particle –n (and be scattered as a consequence of the collision, “opening the door” to other collisions). Or, again for any positive integer n, particle n might collide with particle –n–1, particle –1 being the only one not to undergo any collision whatsoever. By this path clearly there are in principle an infinite number of different possible sequences of deterministic collisions in which none is the first. Let us see in particular how these possibilities may be updated in the exact sphere of dynamics to prove the existence of indeterministic supertasks in a strong sense in which energy is conserved.

4. An Indeterministic Supertask in a Strong Sense in which Energy Is Conserved As a preliminary exercise, let us consider the collision of two rigid spheres that initially move at parallel velocities but in opposite directions. If the radii of the spheres are r* and r we shall suppose that their parallel velocity vectors before the collision are not on the same line but on lines separated at a distance (r+r*)/√2. This guarantees that, when they come into contact, the tangent common to both will form an angle of π/4 with the horizontal (we shall call this type of collision semi-frontal). The situation is described in detail in the figure below: the common tangent HJ forms an angle of π/4 with the horizontal FG:

8

Jon Perez Laraudogoitia

L V1

V’1 J

A

F

N E K

B

C

G

M

V2 D

H

V’2

After the collision, the spheres scatter with velocity vectors V1’ and V2’, which we shall determine. For simplicity’s sake, we will suppose that the initial velocities of both (moving in opposite directions) have value one. With A and D the geometric centres of the spheres (of mass m* and m and radii r* and r respectively) and C their point of contact at the instant of collision, the conservation of linear momentum leads to the pair of equations m* – m = m* V1’ Cos ^(AKB) + m V2’ Cos ^(DME)

(1)

0 = m* V1’ Sin^(AKB) – m V2’ Sin^(DME)

(2)

and the conservation of energy to

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m* + m = m* V1’2 + m V2’2

(3)

Finally let us posit the equation of conservation of the angular momentum taken with regard to point C. The perpendicular to the line FG from A goes through B and AB = r*/√2. The perpendicular to FG from D goes through E and DE = r/√2. The angular momentum before the collision is therefore (m* r* /√2) + (m r /√2). As ^(ACJ) = π/2, ^(JCE) = π/4 and ^(BCA) + ^(ACJ) + ^(JCE) = π, clearly ^(BCA) = π/4 = ^(KCA). Therefore ^(KAC) = π – (π/4) – ^(AKC)) = (3π/4) – ^(AKC) and so ^(LAC) = ^(AKC) + π/4. But the perpendicular to the support line of velocity vector V1’ from C goes through L, which means that LC = AC Sin ^(LAC) = r* Sin(^(AKC) + π/4) = (r*/√2) (Sin ^(AKC) + Cos ^(AKC)). Also ^(CMD) = π – ^(DME) and as ^(DCM) = π/4, it happens that ^(CDM) = ^(DME) – π/4. But the perpendicular to the support line of velocity vector V2’ from C goes through N, which means that CN = CD Sin ^(CDM) = r Sin(^(DME) –π/4) = (r/√2)(Sin ^(DME) – Cos ^(DME)). The angular momentum after the collision is therefore (m* r*/√2)(Sin ^(AKC) + Cos ^(AKC)) V1’ + (m r/√2)(Sin ^(DME) – Cos ^(DME)) V2’ and the equation of the conservation of angular momentum:

On Energy Conservation in Infinite Systems

9

(m* r*/√2) + (m r/√2) = (m* r*/√2)(Sin ^(AKC) + (4) + Cos ^(AKC)) V1’ + (m r/√2)(Sin ^(DME) – Cos ^(DME)) V2’

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The system of transcendental equations (1) – (4) becomes a system of algebraic equations V2’ Cos with the change of variables V1’ Cos ^(AKC)) = ax , V1’ Sin ^(AKC)) = ay , ^(DME)) = bx and V2’ Sin ^(DME)) = by . By resolving the latter and undoing the change we find V1’ = (√(m*2 + 5m2 – 2mm*))/(m + m*)

(5)

V2’ = (√(m2 + 5m*2 – 2mm*))/(m* + m)

(6)

^(AKC) = arctg (2m/(m* – m))

(7)

^(DME) = arctg (2m*/(m* – m))

(8)

Thus the directions and velocities of the spheres after the collision are determined. In particular, if their masses are equal (m* = m), then ^(AKC) = ^(DME) = arctg ∞ = π/2 and both spheres are scattered at final (opposing) velocities perpendicular to their initial velocities. If m* = 2m then ^(AKC) = arctg 2 ≈ 63.43º and ^(DME) = arctg 4 ≈ 75.96º so that the more massive particle is scattered in a direction that forms an angle of 63.43º above the horizontal and the less massive is scattered forming an angle of 75.96º below the horizontal. Our next step will be to construct in detail a certain configuration of rigid spheres of complexity ω+ω* following the indications given at the end of the previous section. Let us consider an infinite set of rigid spheres en (n a positive or negative integer, not null) of masses mn = 1/2 | n | and radii rn = 1/10 | n |. Initially each en with n > 0 moves horizontally to the right at unit velocity and each en with < 0 moves horizontally to the left also at velocity of value one. In the initial instant, each en with n > 0 has its geometric centre above point xn = -1/2 | n | of a horizontal axis Ho and at a distance of rn/√2 above it. Analogously, each en with n < 0 has its geometric centre below the point xn = 1/2 | n | of the Ho axis and at a distance rn/√2 beneath it. Clearly, if the only spheres present were, for instance, the two ea (a > 0) and eb (b < 0), they would collide semi-frontally and the analysis made at the beginning of this section would be valid for them. Let us see now how, from this initial configuration, Ci, of complexity ω+ω*, an indeterministic supertask in a strong sense in which the energy is conserved may develop. Certainly one possible evolution (which I shall call I) from Ci is the one where, ∀n ≠ 0, en collides with (and only with) e–n, all of them being scattered in a direction perpendicular to Ho. Total energy conservation is immediate here because each en only collides with e–n and in each of these collisions the energy is conserved. Furthermore, the infinite sequence of collisions occurs in an interval of time of finite duration ( 0, en collides with (and only with) e–n–1, all of the en (n > 0) being scattered in a direction that forms an angle of 63.43º above Ho and all the e–n (–n < –1) in a direction forming an angle of 75.96º beneath Ho. The possibility of process II (in which, as in I, all the collisions are deterministic) shows two things. One, that the collisions in process I are not globally independent, which makes process I a supertask (in

10

Jon Perez Laraudogoitia

which the energy is conserved). The other, that process I is an indeterministic supertask. In view of the fact that, as stated, all the collisions involved in process I are deterministic, we deduce finally that said process is an indeterministic supertask in a strong sense in which the energy is conserved. This is the principal conclusion of the present article, as I announced at the outset.

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5. A Criticism From the initial condition Ci we have found, ∀n > 0, a possible form of evolution (I) where en collides with e–n without doing so with e–n–1 and another (II) where en collides with e–n–1 without doing so with e–n. The type of indeterminacy to which this gives rise was baptized by Angel (2005) as even/odd indeterminacy. He considered a bi-dimensional initial configuration of infinite identical particles Pn (n ≥ 1) with unbounded velocities and showed that there were at least two possibilities of evolution: a) P2n+1 hits P2n without P2n hitting P2n–1, b) P2n hits P2n–1 without P2n+1 hitting P2n. Arguing that “we can’t avoid making choices between these possible outcome paths” (p.186) he concluded that the even/odd indeterminacies seem ontologically unappealing. As I have already said, Angel’s model depends on the presence of unbounded velocities, and it was by rejecting this latter that he proposed to avoid the possibility of even/odd indeterminacies. The supertask presented in the previous section shows that the possibility of even/odd indeterminacy persists even with bounded velocities (it was this that enabled us to talk meaningfully of energy conservation, through the energy not being “infinite”). But why should this form of indeterminacy be ultimately unacceptable in dynamics? Indeterminism in a system of particles appears when it is not possible to determine uniquivocally which force will act on each particle in each instant of time. This may be due to the fact that the interaction to which at least one particle of the system is subjected in at least one instant of time does not permit the force acting on it at that instant of time to be uniquivocally determined (as in the triple collision mentioned in section 1, where the force is not a function but a distribution), but it might also be due to the fact that it is not possible to determine the interaction to which at least a particle in the system will be subjected in each instant of time (even when more than one of such possible interactions, if it actually occurs, enables the evolution of the particles involved to be determined uniquivocally).Indeed, this latter is what characterizes indeterminism through deterministic subprocesses in general, indeterministic supertasks in a strong sense in particular and, in short, Angel’s even/odd indeterminacies.

6. Extensions My proof of the existence of indeterministic supertasks in a strong sense in which the energy is conserved depends on assuming beforehand the existence of solid bodies. Suppositions of this sort are typical of the majority of the literature on dynamic supertasks and many results in this area depend crucially on them (point particles are another habitual supposition). Fortunately, this is not the case here. To see why we need to remember that an homogenous isotropic elastic solid is characterized by three numerical parameters, the longitudinal modulus of elasticity (E), the transverse modulus of elasticity (G) and the Poisson’s rate (μ),

On Energy Conservation in Infinite Systems

11

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which enable us to relate the deformations with the tensions by means of generalized Hooke’s laws εx = (1/E)(σnx – μ (σny + σnz))

(9)

εy = (1/E)(σny – μ (σnx + σnz))

(10)

εz = (1/E)(σnz – μ (σnx + σny))

(11)

γxy = τxy/G

(12)

γyz = τyz/G

(13)

γxz = τxz/G

(14)

and which satisfy the relation G = E/(2(1 + μ)). The limit of rigid solid is reached with E,G →∞ (maintaining μ constant for instance) which means that the result of the collision between two elastic spheres will differ arbitrarily little from the case where the spheres are rigid taking E and G sufficiently large (this can be easily verified, for instance, for the case of the interval of time that a collision lasts. See Landau & Lifshitz (1986), chap. 1, sec.9). Let us call this difference the error. If, in an infinite sequence of (causally connected) binary collisions between rigid spheres, we substitute the latter by elastic spheres, the errors propagate and magnify with time as the sequence of collisions becomes longer and it is not clear in principle that there is any way of making them all arbitrarily small. However, in the indeterministic supertasks in section 4, the binary collisions of the infinite sequence of collisions are not causally connected (something that did not stop them from not being globally independent) meaning that, as the errors do not propagate, we may bound all of them uniformly by choosing E and G sufficiently large and initial velocities sufficiently small. What this means is that those indeterministic supertasks in which the energy is conserved remain valid even when we use elastic spheres (with E and G sufficiently large) instead of rigid spheres. This is definitely of interest. No exact analytical solution for the general problem of the collision between two elastic spheres exists. It is not just a question of a part of the kinetic energy of translation being converted in potential energy of deformation during the collision but that yet another fraction produces elastic waves whose propagation in the elastic medium is complex. In the case of section 4’s semi-frontal binary collision the angles of dispersal would no longer be 63.43º and 75.96º (and it may perhaps be impossible to calculate them exactly) but, even so, we know they will differ only slightly from those values (with suitable E and G) so that, even with elastic (deformable) bodies, there are indeterministic supertasks in a strong sense in which the energy is conserved.

Acknowledgement Many thanks to Prof. David Atkinson for reading and commenting on a previous draft of this paper.

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Jon Perez Laraudogoitia

References

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Angel, L. (2005). Even and Odds in Newtonian Collision Mechanics. The British Journal for the Philosophy of Science. 56, pp. 179-188 Landau, L.D. & E.M. Lifshitz (1986). Theory of elasticity. Pergamon Press. Oxford Pérez Laraudogoitia, J. (2005). An Interesting Fallacy Concerning Dynamical Supertasks. The British Journal for the Philosophy of Science. 56, pp. 321-334 Pérez Laraudogoitia, J. (2008). Energy Conservation and Supertasks. Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics. 39, pp. 364-379 Stewart, I. (1996). From here to infinity. A Guide to Today’s Mathematics. Oxford University Press. Oxford

In: Energy Conservation: New Research Editor: Giacomo Spadoni, pp. 13-31

ISBN: 978-1-60692-231-6 © 2009 Nova Science Publishers, Inc.

NUMERICAL ANALYSIS OF DOUBLE DIFFUSIVE NATURAL CONVECTION IN TRAPEZOIDAL ENCLOSURE FILLED WITH POROUS MEDIUM UNDER MAGNETIC FIELD R. Younsi∗ Applied Sciences Department, University of Quebec at Chicoutimi, Chicoutimi (Qc), Canada

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Abstract Double-diffusive convection flow is of a great of interest in several industrial applications. Convection plays a dominant role in crystal growth in which it affects the fluidphase composition and temperature at the phase interface. It is the foundation in modern electronics industry to produce pure and perfect crystals to make transistors, microwave devices, infrared detectors and memory devices. Natural convection adversely affects local growth conditions and enhances the overall transport rate. The problem considered in this paper is a two-dimensional natural convection flow in a trapezoidal porous cavity filled with a binary fluid. The physical model for the momentum conservation equation makes use of the Darcy–Brinkman equations, which allows the no-slip boundary condition on a solid wall to be satisfied. Laminar regime is considered under steady state condition. Different types of boundary conditions have been employed. Dirichlet conditions are prescribed along the bottom and inclined surfaces for temperature and concentration. On top surface, Neuman, i.e., zero gradient conditions are assigned to temperature and concentration. A uniform magnetic field was applied in horizontal direction. The two-dimensional flow equations, expressed here in a velocity–pressure (UVP) formulation, along with the energy and concentration equations. The computational fluid dynamics (CFD) simulation was carried out using FEMLAB, which is application software from MATLAB. This study was done for constant Buoyancy ratio N=1, Lewis number Le = 2, and Prandtl number Pr = 0.149. An extensive series of numerical simulations is conducted in the range of ∗

E-mail address: [email protected]. Tel:+1418-545-5011; Fax: +1418-545-5012; Author information: Applied Sciences Department, University of Quebec at Chicoutimi, 555 Boul. De l’Université G7H 2B1 Chicoutimi (Qc), Canada;

14

R. Younsi 104 ≤ Ra ≤ 107, 1 ≤ Ha ≤ 4.102 , 10-9 ≤ Da ≤ 1, where Ra, Ha and Da are Rayleigh number, Hartman number and Darcy number, respectively. Streamlines, isotherms and isoconcentrations are produced to illustrate the flow structure transition from. In addition, the predicted results for the average Nusselt and Sherwood numbers are presented and discussed for various parametric conditions. It was observed that heat and mass transfers were decreased with increasing Hartmann number and decreasing value of Darcy number.

Keywords: Double-diffusive flow; Heat and mass transfer; Magnetic field; Numerical solution, Porous medium, Trapezoidal cavity.

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I. Introduction Magnetohydrodynamic (MHD) natural convection heat transfer flow is of considerable interest in the technical field due to its frequent occurrence in industrial technology and geothermal application, high-temperature plasmas applicable to nuclear fusion energy conversion, liquid metal fluids, and (MHD) power generation systems. Research on convective instability in a porous medium under the influence of an imposed magnetic field has gained momentum only recently due to its various applications in engineering and technology, particularly in manufacturing of the magnetic fluids. One important example of double-diffusive convection can be found in material solidify processes. Since solidification of alloys and crystals necessary involves the simultaneous flows of momentum, heat and solute. The appearance of thermal and concentration gradients near the solid-liquid interface can causes a uniform density distribution and convectiondiffusion motion there, which may have a profound effects on the solid structure as it is crystallized from the liquid state. Electromagnetic field has been used in the metal industry to control microstructures solidification and to reduce or eliminate natural convection in the melt. In crystal growth process, the objective is to adjust the process and characteristics of the magnetic filed in order to eliminate the deleterious unsteadiness in the melt motion and to achieve a steady melt motion which produces uniform and controllable dopant and contaminant concentrations in the crystal. A major advantage of a magnetic field is that it can be tailored to achieve different field strengths and orientations at different positions in the melt and at different stages during the growth of a crystal. The linear stability theory of thermal convection in a thin horizontal layer of an electrically conducting fluid, heated from below, subject to vertical magnetic field of uniform strength (hydromagnetic Benard problem) was studied earlier by Chandrasekhar [1] and Platten and Rass [2]. The combined heat and mass transfer in porous media is limited, because of complexities involved in double-diffusive natural convection. Most of previous studies in this topic use Darcy’s law for solving flow within the porous medium. Natural convection of heat and mass transfer in a square porous cavity subjected to constant temperature and concentration has been investigated by Trevisian and Bejan [3]. The authors used the Darcy’s model for modeling the flow in porous medium. The numerical study has been carried out for a given range of Darcy-Rayleigh number, Lewis number and buoyancy ratio. Lage [4] studied the effect of the convective inertia term on Benard convection in a porous medium. The author shows that inertia term included in the general momentum equation has no effect on the overall heat transfer. Mahmud and Fraser [5] investigated the magnetohydrodynamic free

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Numerical Analysis of Double Diffusive Natural Convection…

15

convection and entropy generation for a square cavity for low Hartmann numbers. They observed that increasing the value of Hartmann number (i.e. magnetic force) have a tendency to retard the fluid motion in the cavity. Agrawal et al. [6] discussed thermal and mass diffusion on hydromagnetic viscoelastic natural convection past an impulsively started infinite plate in the presence of a transverse magnetic filed. Helmy [7] studied the unsteady laminar free convection flow of an electrically conducting fluid through a porous medium bounded by an infinite vertical plane surface of constant temperature. Shanker and Kishan [8] presented the effect of mass transfer on the MHD flow past an impulsively started infinite vertical plate. Ram et al.[9] studied the MHD free convection flow past an impulsive started vertical infinite plate when a strong magnetic field of uniform strength was applied transversely to the direction of flow. The first work in MHD using the state space approach was done by Ezzat [10-12], where the heated vertical plate problem was solved using a numerical inverse Laplace transform. Ezzat et al. [11] formulated the state space approach for the one dimensional problem of viscoelastic magnetohydrodynamic unsteady free convection flow with the effects on a viscoelastic boundary layer flow with one relaxation time. Bian et al. [13] considered the interaction of an external magnetic field with convection currents in a porous medium. The porous medium was modeled according to Darcy’s model. It is found that the application of a magnetic filed, modifies the temperature and flow fields significantly. Sparrow and Cess [14] studied the effect of magnetic field on the natural convection heat transfer. Hady et al. [15] analyzed the MHD free convection flow along a vertical wavy surface with heat generation or absorption. Romig, [16] studied the effect of electric and magnetic fields on the heat transfer to electrically conducting fluids. Ece and Buyuk [17] illustrated the natural convection flow under a magnetic field in an inclined rectangular enclosure heated and cooled on adjacent walls using differential quadratic (DQ) method. They demonstrated that circulation inside the enclosure and therefore the convection become stronger as the Grashof number increases while the magnetic field suppresses the convective flow and the heat transfer rate. Riley [18] analyzed the (MHD) free convection heat transfer. Garandet et al. [19] analyzed the buoyancy driven convection in a rectangular enclosure with a transverse magnetic field. Chamkha [20] made a numerical study for hydromagnetic combined convection flow in a vertical lid-driven cavity. They took into account the internal heat generation or internal heat absorption in their analysis. Sacheti et al. [21] found an exact solution for the transient (MHD) free convection flow with constant surface heat flux. Recently Saravanan and Yamaguchi [22] have investigated the centrifugal convection in a magnetic-fluid-saturated porous medium under zero gravity environment. The purpose of the present paper is to study the double-diffusive natural convection flow behaviour and its effects on heat and mass transfer in a trapezoidal porous cavity submitted to transverse magnetic filed. The flow is modeled using the generalized model of DarcyBrinkman.

II. Problem Definition and Governing Equations The physical domain under investigation is a two-dimensional fluid-saturated Darcy– Brinkman porous enclosure (see Figure 1). The trapezoidal enclosure filled with a binary fluid is of width L and height H, and the Cartesian coordinates (x,y), with the corresponding velocity components (u, v), are indicated herein. It is assumed that the third dimension of the

16

R. Younsi

enclosure is large enough so that the fluid, heat and mass transports are two-dimensional. At the bottom wall a heating and salting is subjected to higher and constant temperature and concentration, T1 and C1, while the top one is assumed thermally insulated and impermeable. The inclined walls are maintained at lower temperature and concentration constants, T0 and C0. Gravity acts in the negative y-direction. Both velocity components are equal to zero on boundaries. A magnetic field with uniform strength B0 is applied in the horizontal direction. The porous matrix is assumed to be uniform and in local thermal and compositional equilibrium with the saturating fluid. Thermophysical properties are supposed constant. The flow is assumed to be laminar and incompressible. Viscous dissipation and porous medium inertia are not considered, and the Soret and Dufour effects are neglected. Joule heating, and viscous dissipation are all assumed negligible. Density of the saturated fluid mixture is assumed to be uniform over all the enclosure, exception made to the buoyancy term, in which it is taken as a function of both the temperature T and concentration C through the Boussinesq approximation:

ρ = ρ [1 − βT (T − T0 ) − βC (C − C0 ) ]

(1)

Where ρ0 is the fluid density at temperature T0 and concentration C0, and βT and βC are the thermal and concentration expansion coefficients, respectively. Subscript 0 refers to the condition over the inclined vertical walls of the enclosure. U=V=0 ∂Θ ∂Φ = =0 ∂X ∂X

G B U=V=0 Θ=Φ=0

G g Porous medium U=V=0

U=V=0 Θ=Φ=0

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Y, V X, U

Figure 1. Schematic representation of the geometry and the mesh.

The governing equations for convection flow using conservation of mass, momentum, energy, species and electric charge can be written as: Continuity equation

∇.V = 0

(2)

Numerical Analysis of Double Diffusive Natural Convection…

17

Momentum equations

G ∂V 1 μ J + (V .∇ )V = − ∇ p + ν∇ 2V + ⊗ B − g βT (T − T0 ) − g β C (C − C0 ) − V ∂t ρ0 ρ0 ρ0 K

(3)

Energy equation

∂T + (V.∇)T = α∇ 2T ∂t

(4)

∂C + (V.∇ )c = D∇ 2c ∂t

(5)

Species equation

Electric charge

∇.J = 0

J = σ(−∇φ + V ⊗ B)

(6)

where t stands for time; ∇ is the gradient operator; ∇2 is the Laplacian operator, g, B, P, T, C, and V represent the gravitational acceleration vector, magnetic induction vector, pressure, temperature, concentration and velocity vector, respectively; J is the current density vector; ∇φ is the associated electric field; σ is the fluid electrical conductivity; β is the volumetric expansion coefficient; K is the permeability of the porous medium; and ν, μ, ρ0, and cp are the effective kinematic viscosity, effective dynamic viscosity, density, and specific heat, respectively. D is the mass diffusivity. As discussed by Garandet et al. [10], for a two dimensional situation, Eq. (6) for the electric potential reduces to∇2φ=0. The unique solution is ∇φ=0 since there is always an electrically insulating boundary around the enclosure on which ∂φ/∂n=0. It follows that the electric field vanishes everywhere. In the absence of an electric field, the small magnetic Reynolds number assumption uncouples Maxwell equations from the momentum equation. The Lorentz force then reduces to a systematically damping factor.

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Using the following transformation of variables: ( X ,Y ) =

P=

( x, y ) , L

p ρ 0 g β ΔTL

(U ,V ) =

t g β ΔTL , (u , v) , τ= L g β ΔTL

Θ=

T − T0 , Φ = C − C0 , T1 − T0 C1 − C0 (7)

The definition of all variables is given in nomenclature. By employing the aforementioned assumptions into the macroscopic conservation equations of mass, momentum, energy and species, a set of dimensionless governing equations is obtained as:

18

R. Younsi Continuity equation

∂U ∂V + =0 ∂X ∂Y

(8)

X-momentum equation

Pr ⎛ ∂ 2U ∂ 2U ∂U ∂U ∂U ∂P +U +V =− + + ⎜ Ra ⎝ ∂X 2 ∂Y 2 ∂τ ∂X ∂Y ∂X

⎞ Pr U ⎟− Ra Da ⎠

(9)

Y-momentum equation Pr ⎛ ∂ 2V ∂ 2V ⎞ Pr V Pr ∂V ∂V ∂V ∂P V +U +V =− + + + (Θ + N Φ ) − Ha 2 ⎜ ⎟− Ra ⎝ ∂X 2 ∂Y 2 ⎠ Ra Da Ra ∂τ ∂X ∂Y ∂Y

(10)

Energy equation

∂Θ ∂Θ ∂Θ +U +V = ∂τ ∂X ∂Y

1 ⎛ ∂ 2Θ ∂ 2Θ ⎞ ⎜ 2+ 2⎟ ∂Y ⎠ Ra Pr ⎝ ∂X

(11)

Species equation

⎛ ∂ 2Φ ∂ 2Φ ⎞ 1 ∂Φ ∂Φ ∂Φ +U +V = + ⎜ ⎟ ∂τ ∂X ∂Y Le Ra Pr ⎝ ∂X 2 ∂Y 2 ⎠

(12)

The initial and boundary conditions for the dimensionless equations are as follows:

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Initial condition (at τ = 0)

Θ = Θ 0 = 0, ⎫ ⎪ Φ = Φ 0 = 0,⎬ U = V = 0 ⎪⎭

(13)

Boundary conditions:

Θ = Φ = 1,U = V = 0 Θ = Φ = 0,U = V = 0

on on

∂Θ ∂Φ = = 0, U = V = 0 ∂X ∂X

on

⎫ ⎪ AB ⎪ AD, BC ⎬ ⎪ ⎪ DC ⎭

(14)

Numerical Analysis of Double Diffusive Natural Convection…

19

The general transport equations can be written in a canonical form as:

∂Ω ∂ + ∂τ ∂X

∂Ω ⎞ ∂ ⎛ ∂Ω ⎞ ⎛ ⎜ U Ω − ΓΩ ⎟+ ⎜V Ω − ΓΩ ⎟ = SΩ ∂X ⎠ ∂Y ⎝ ∂Y ⎠ ⎝

(15)

Where Ω stands for U, V, Θ and Φ, ΓΩ is the diffusion coefficient and SΩ is the source term. Eqs. (8)–(12) indicate that the present problem is governed by the following dimensionless parameters, namely, Prandtl number Pr, Darcy number Da, the thermal Rayleigh number Ra, Hartman number Ha, the solutal to thermal buoyancy ratio N and Lewis number Le defined as:

Pr =

⎫ β ΔC gβ ΔTL3 K ν α , Da = 2 , Le = , Ra = T , N= C , Ha = B0 L σ ρ0 ν ⎬ α να β T ΔT L D ⎭ (16)

The overall heat and mass transfer rates across the system are important in engineering application. It is appropriate at this stage to define the Sherwood and Nusselt numbers on the surface of mass and heat sinks can be written respectively as: 1

∂Φ Sh 0 = ∫ − ∂Y 0

1

dX

,

Y =0

Nu 0 = ∫ − 0

∂Θ dX ∂Y Y =0

(17)

Streamfunction and streamlines are routinely the best way to visualize the convective flow in the enclosure. The dimensionless streamfunction ψ is defined such:

U=

∂Ψ ∂Y

,

V=−

∂Ψ ∂X

(18)

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III. Solution Methodology Eqs. (8)–(14) are solved numerically using commercial finite element method based software FEMLAB [23]. The computational scheme was to simultaneously solve the highly non-linear equations. An unstructured mesh consisting of Lagrange quadratic elements was created. To solve temperature and velocity values at the interfaces of spherical objects and surrounding media accurately, a relatively fine mesh was generated near the interfaces. A FEM solution was considered converging when the difference in maximum temperature and concentration between successive calculations was less than 0.1% for a doubling of the number of elements. In the convergence studies, simulation results were found to be meshing independent and the optimum mesh generation yielded 14750 elements. The ‘GMRES’ differential equation solver in FEMLAB was used to achieve convergence. All computer simulations were performed on

20

R. Younsi

a Dell Pentium 4 with 3.2 GHz processors and 1 GB RAM running a Windows XP 32-bit operating system.

IV. Results and Discussion In this section the numerical results are discussed in order to study the effects of the presence of the magnetic field, porous medium, buoyancy ration and intensity of convection in a trapezoidal cavity. Computations are carried out for Ra ranging from 104 to 107, Ha is ranging from 0 to 400, Da is ranging from 10-9 to 1, Lewis number and buoyancy ratio from 0 to 10. The PrandtI number is fixed in all calculations to Pr =0.149 which corresponds to a titanium based alloy. Various numerical visualizations of the flow and temperature fields are depicted graphically.

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IV.1. Influence of Hartman Number Figure 2 illustrates the effect of Hartmann number Ha, on the flows patterns, isotherms and isoconcentrations contours. To highlight on the effect of Ha, the Rayleigh number is kept constant Ra = 106, Pr = 0.149 Le = 2, N = 1 and Da=10-4.The influence of a magnetic field on flow and temperature and concentration distribution is apparent from this figure. In the absence of a magnetic field (Ha=0), there is a circulating flow in the enclosure, fluid rise up from middle portion of the bottom wall and flow down along the two vertical walls forming two strong symmetric rolls with clockwise and anti-clockwise rotations.Also it is seen that both isothermals and isoconcentrations are stratified in vertical direction. As the magnetic field is imposed Ha =200 and 400, the circulation is observed to be weaker in these cases as compared to the corresponding previous case where Ha=0, since the magnetic field considerably retards the convection currents and the convective motion inside the cavity is almost completely damped (Ψmax =8.10-3 for Ha=400). The isotherms and isoconcentrations resemble to the heat and mass transfer by pure double diffusion. Indeed, when the magnetic field is applied along the X-axis, the magnetohydrodynamic body force acts along the Y-axis and works to retard the fluid particles carried upward. Another view of the effect of Hartman on heat and mass transfer is found in Figure 3, where Nusselt and Sherwood numbers are plotted as a function of Ha. The analysis of this figure indicates that for small values of Ha, the boundary layer regime prevails. As the Hartman number increases, the electromagnetic body force increases which suppresses progressively the strength of the convective motion, and thus boundary layer regime is followed by the double diffusive regime for which Nusselt and Sherwood numbers tend to one. One can conclude that the main contribution of the magnetic effect is the suppression of the overall heat and mass transfer in the enclosure

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Ha=0

Ha=200

Ha=400 Streamlines

Velocity-vectors

Isotherms

Isoconcentrations 6

Figure 2. Streamlines, Velocity-vectors, isotherms and isoconcentrations versus Hartman number: Pr=0.149, Ra=10 , Da=10-4, N=1, Le=2.

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R. Younsi

3,0

Nu

0

2,5

2,0

1,5

1,0

0

70

140

210

280

350

420

280

350

420

Ha 3,5

3,0

Sh

0

2,5

2,0

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1,5

1,0

0

70

140

210

Ha Figure 3. Effect of Hartman number on the Nusselt and Sherwood numbers: Pr=0.149, Ra=106, Da=104, N=1, Le=2.

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Ra=1E4

Ra=1E6

Ra=1E7 Streamlines

Velocity-vectors

Isotherms

Isoconcentrations

Figure 4. Streamlines, Velocity-vectors, isotherms and isoconcentrations versus Rayleigh number: Pr=0.149, Ha=0, Da=10-4, N=1, Le=2.

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R. Younsi

IV.2. Influence of Rayleigh Number Figure 4 presents the effect of thermal Rayleigh number on the streamlines velocity vectors, isotherms and isoconcentrations contours for Ha = 0, Pr = 0.149, Le = 2, N = 1 and Da=10-4. In this figure the effect of both thermal buoyancy force and solutal buoyancy force are equal. Therefore, the double diffusive flow is applicable. Due to the cold vertical walls, fluids rise up from middle portion of the bottom wall and flow down along the two vertical walls forming two symmetric rolls with clockwise and anti-clockwise rotations. For low Rayleigh number Ra = 104, the pure double diffusion regime is dominant. The flow consists of very weak clockwise and anti-clockwise cells with maximum strength Ψmax =0.012 and the temperature and solutal profiles are almost invariant. As the Rayleigh number increases the buoyancy driven circulation inside the cavity also increases (Ψmax =0.12 for Ra=106 and Ψmax =0.3 for Ra=107), the distortion of the isotherms and concentrations increases gradually and the advection takes the command, becoming the dominant mode of heat transfer. The circulations are greater near the center and least at the wall due to no-slip boundary conditions. For higher Rayleigh number (intense convection) the concentration and temperature gradients in the vicinity of the bottom and inclined walls increases and develop thermal and solutal boundary layers. The streamlines are crowded near the cavity wall and the cavity core is empty as well as both isothermals and isoconcentrations are stratified in vertical direction except near the insulated surfaces of the enclosure (Ra=107). The influence of Rayleigh number on the Nusselt and Sherwood numbers becomes more significant at higher Rayleigh numbers (see Figure 5).

5

4

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Nu

0

3

2

1

0 2,0x106

4,0x106

6,0x106

Ra

8,0x106

Figure 5. Continued on next page.

1,0x107

1,2x107

Numerical Analysis of Double Diffusive Natural Convection…

25

6 5

Sh

0

4 3 2 1 0 2,0x106

4,0x106

6,0x106

Ra

8,0x106

1,0x107

1,2x107

Figure 5: Effect of Rayleigh number on the Nusselt and Sherwood numbers: Pr=0.149, Da=10-4, Ha=0, N=1, Le=2.

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IV.3. Influence of Darcy Number The effect of Da on the flow patterns and isotherms inside the cavity in the absence of the magnetic field at Ra =106 is shown in Figure 6. In general, the fluid circulation is strongly dependent on Darcy number. For low values Da, the flow is seen to be very weak as observed from stream function contours. Therefore, the temperature and concentration distributions are similar to that with stationary fluid and the heat transfer and mass transfers are due to purely double diffusion. The convective heat and mass transfer decreases because of the retardation of the buoyancy force. In the limit of Da=0, the permeability of the medium approaches zero, causing the flow to eventually cease (Ψmax = 2.5 10-4). As Da increases, the strength of flow is increased and the flow patterns will be that of the corresponding free convection problem with no porous medium. The stronger circulation causes the temperature and concentration contours to be concentrated near the side walls and near the edges of bottom wall which may result in greater heat and mass transfer rate due to convection. The effect of varying the Darcy number on the heat and mass transfer, Nu and Sh are illustrated in Figure 7. As the permeability of the porous medium Da is increased, the boundary frictional resistance becomes gradually less important and the fluid circulation within the cavity is progressively enhanced. Indeed, increasing the Brinkman term implies that the balance between the Darcy term and the buoyancy force in the boundary layer is progressively replaced by the balance between a viscous force and the buoyancy term. The viscous force enhances the velocity at high Darcy numbers. From this figure it appears that when Da is large enough, Nu and Sh tend asymptotically toward constant values. Nu and Sh are observed to decrease considerably with decreasing Da toward the pure double diffusion limit. This is expected since the Brinkman model reduces to Darcy Law when Da tends to 0. As the Darcy number is decreased, the boundary frictional resistance increases progressively and adds to the bulk frictional drag induced by the solid matrix to slow the convection motion.

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Da=1E-7

Da=1E-4

Da=1 Streamlines

Velocity-vectors

Isotherms

Isoconcentrations -6

Figure 6. Streamlines, Velocity-vectors, isotherms and isoconcentrations versus Darcy number: Pr=0.149, Ra=10 , Da=10-4, N=1, Le=2.

Numerical Analysis of Double Diffusive Natural Convection…

27

4

Nu

0

3

2

1 1E-9

1E-6

Da

1E-3

1

1E-6

Da

1E-3

1

4

Sh

0

3

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2

1 1E-9

Figure 7. Effect of Darcy number on the Nusselt and Sherwood numbers: Pr=0.149, Ra=106, Ha=0, N=1, Le=2.

IV.4. Influence of Lewis and N The buoyancy ratio effects on Nu and Sh, are shown in Figure 8 for Ra=106, Da=10-4 and Ha=0 and typical values of Le. It can be seen from the figures that the general tendency is a monotonic increase of heat and mass transfer with N .These figures show also that the heat

28

R. Younsi

and mass transfer is more important at lower and greater values of Lewis numbers, respectively. In fact, the increase of Le induces a decrease of the solutal boundary layer thickness (which increases Sh), but it also engenders a decrease of the flow intensity (as this reduces the concentration gradient in the core region), which leads to a decrease of Nu. The increase of Sh with N is more pronounced at larger Le. In the case of Nu, more important improvements are obtained at small Le. 7 6

Nu

0

5 4 3

Le=0.1 Le=1 Le=5 Le=10

2 1

0

2

4

6

8

10

12

8

10

12

N 15 Le=0.1 Le=1 Le=5 Le=10

9

Sh

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0

12

6

3

0

2

4

6

N Figure 8. Effect of Buoyancy ration on the Nusselt and Sherwood numbers: Pr=0.149, Ra=106, Da=104, Ha=0.

Numerical Analysis of Double Diffusive Natural Convection…

29

V. Conclusion Double-diffusive natural convection in trapezoidal porous cavity in the presence of magnetic field has been studied numerically. The Brinkman extended Darcy model has been used through this study. A parametric study was carried out. It was found that the heat and mass transfer mechanisms and the flow characteristics inside the enclosure depended strongly on the strength of the magnetic field and Darcy number. Significant suppression of the convective currents was obtained by the application of a strong magnetic field and/or presence of a porous medium. Also, the effects of the magnetic field and the porous medium were found to reduce the heat and mass transfers and fluid circulation within the cavity. The present analysis is focused on the influence of a limited number of dimensionless parameters. As an extension of this work, it is particularly relevant to take into account the aspect ratio (L/H), the Prandtl number (Pr) and correlate heat and mass transfer.

Nomenclature G B C cp D Da

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G g

H Ha k K L Le N Nu0 P Pr Ra Sh0 T t U(V) V (X,Y)

Θ Φ ΔT ΔC

:magnetic field :concentration[ ] :specific heat at constant Pressure,[ Jkg-1K] :mass diffusivity, [m2s-1] :Darcy number (Eq.16) :acceleration due to gravity,[m.s-2] :Cavity high ,[m] :Hartman number (Eq.16) :thermal conductivity,[Wm-1 K] :permeability, [m2] :cavity width, [m] :Lewis number (Eq.16) :buoyancy ratio (Eq.16) :overall Nusselt number :pressure, [Pa] :Prandtl number (Eq.16) :Rayleigh number (Eq.16) :overall Sherwood number :temperature[C] :time,[s] :dimensionless velocity in X (Y) direction [ms-1] :velocity vector : dimensionless Cartesian coordinate [m] :dimensionless temperature (Eq.7) :dimensionless concentration (Eq.7) : temperature difference between plates (=T1-T0) : concentration difference between plates (=C1-C0)

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R. Younsi

ψ

: stream function

Subscripts 1 0

: heated surface : cooled surface

Greek Symbols α βT βC ρ ν τ

:thermal diffusivity = [m2 s-1] : isobaric coefficient of thermal expansion fluid. : isobaric coefficient of solutal expansion fluid. : density, [Kg.m-3] :kinematic viscosity, [m2s-1] : dimensionless time

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References [1] Chandrasekhar, S. Hydrodynamic and hydromagnetic stability; University press, Oxford 1961, 1313-1460. [2] Platten, J. K., Rasse, D. Entropie. 1972, 45, 7-16. [3] Trevisan, O.; Bejan, A. Int. J. Heat and Mass Transfer. 1985, 28, 1597-1611. [4] Lage, J. L. Num. Heat Transfer. 1992, 22(4), 469-485. [5] Mahmud, S.; Fraser, R.A. Int. J. Heat Mass Transfer. 2004, 47, 3245–3256. [6] Agrawal, A.K.; Samria, N.K.; Gupta, S.N. J. En. Heat Mass Tran. 1998, 20, 35-42. [7] Helmy, K.A. ZAMM., 1998, 78, 255-270. [8] Shanker, B.; Kishan, N. J. En. Heat Mass Tran. 1987, 19, 273-278. [9] Ram, P.C.; Takhar, H.S. Fluid Dyn. Res. 1993, 11(3), 99-105. [10] Ezzat, M.A. Can. J. Phys. 1994, 1, 311-317. [11] Ezzat, M.A.; Zakaria, M.; Shaker, O.; Barakat, F. Acta Mech. 1996, 119, 147-164. [12] Ezzat, M.A.; Abd-Elaal, M. J. Franklin Inst. 1997, 334B, 685-709. [13] Bian, W.; Vasseur, P.; Bilgen, E.; Meng, F. J. Heat Transfer and Fluid Flow. 1996, 17(1), 36-44. [14] Sparrow, E.M.; Cess, R.D. Int. J. Heat Mass Transfer. 1961, 3, 267–274. [15] Hady, F.M.; Mohamed, R.A.; Mahdy, A. Int. Commun. Heat Mass Transfer. 2006, 33, 1253–1263. [16] Romig, M. Adv. Heat Transfer. 1964, 1, 268–352. [17] Ece, M.C.; Buyuk, E. Fluid Dyn. Res. 2006, 38, 564–590. [18] Riley. N. J Fluid Mech. 1964, 18, 577–586. [19] Garandet, J.P.; Alboussiere, T. Int. J. Heat Mass Transfer. 1992, 35, 741–749. [20] Chamkha, A.J. Numer Heat Transfer A. 2002, 41, 529–546.

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[21] Sacheti, N.C.; Chamdran. P.; Singh, A.K. Int. Commun. Heat Mass Transfer. 1994, 21, 131–142. [22] Saravanan, S.; Yamaguchi, H. Phys. Fluids. 2005, 17, 1–9. [23] Femlab comsol AB version 2.0, reference manual, 2000.

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RESEARCH AND REVIEW STUDIES

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In: Energy Conservation: New Research Editor: Giacomo Spadoni, pp. 35-66

ISBN: 978-1-60692-231-6 © 2009 Nova Science Publishers, Inc.

Chapter 1

IDENTIFYING ENERGY CONSERVATION BEHAVIORS TH OF 6, 7, AND 8 GRADE ELEMENTARY SCHOOL CHILDREN: A PILOT STUDY WITH TURKISH CHILDREN Arzu Şener∗ and Seval Güven♣ Hacettepe University Department of Family and Consumer Sciences 06100 Samanpazarı/Ankara-Turkey

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Abstract The importance of the issue of energy is felt with its gradually increasing dimensions in our country as well as in the world. With the unsustainable consumption models of the developed technology in our country and in the world, increasing population and production of the world produce an increasing pressure on the air, ground, energy and other necessary resources. As the needs increase, the scarce energy resources are consumed fast and thus, the insufficiency of energy comes in front of us as one of the biggest problems of 2000’s. Meeting the needs and expectations of the human being is the prime objective of the development. Sustainable development is to search the ways of meeting the today’s needs and expectations without compromising from the future needs and expectations. To make the living standard higher than the minimum fundamental level sustainable, the consumption standards in every field should have reached to the long-term sustainability. Contrary to this, most of us live beyond the ecologic means of the world. For example, the data on electric consumption shows clearly that this is the case. The energy consumption was first of most valid indicators to measure the development level of the countries until the recent times. As much as higher energy consumption per capita occurred in a country, it was deemed that the country was most developed. Today, rather than energy consumption amount, its efficiency is discussed. For now, it is important how the produced or acquired energy is used efficiently, not how much energy is produced. This drives the people search the ways to use the energy resources more efficiently and economically.

∗ ♣

E-mail address: [email protected] E-mail address: [email protected]

36

Arzu Şener and Seval Güven In our country where is not rich in respect of energy resources, the abroad dependence already at 62% in this field will increase by the time, as the population will increase. Especially, the number of children and young people within the increasing population ratio enhances their importance on energy consumption as being in every field. There are 1.7 billion children who are younger than 15 years-old in the world. More than 32.0% of the world population is formed by this population group. In the developing countries, the proportion of the children to the total population is higher than in the developed countries. According to the recent census, 12.6% of the population consists of age group of 0-4 years-old in our country’s population, 11.47% 5-9 years-old, 10.56% 10-14 years-old. When one considers those proportions, it is necessary to increase the sensitivity of the child and young population who will be the future’s adult to the energy consumption. This study, which was executed on 150 students, is a pilot study in order to determine weather the students, who attend to the 6, 7, 8 grades in the primary school, use the limited energy resources sufficiently productively and rationally without wasting them in more rational manner.

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Introduction Environment is the habitat in which humans and other living organisms interact through their life span [Anon, 1998]. Through their history, humans have made astounding progress in science and technology. Yet, they have failed to show the same success in preservation of the environment which has thus become an alarming concern of the global community [Çevik, 1999]. The degree and extent of pollution and shortage in resources have led the world population to feel increased need for altering its existing relationship with the environment. Rapid increase in population has led to the boosted the demand for land, water and air while industrial and urban waste conversely pollutes the existing sources of water, land and air [Kinnear, Taylor & Ahmed, 1974]. During the last 350 years, the world population has increased by nearly 5 billion. With this explosive increase in the population, the consumption of resources and energy has also increased rapidly. For example, during the years 1945–1975, the world economy and primary energy consumption showed a spanning fourfold growth. Recently, environmental problems resulting from energy production, conversion, and utilization have caused increased public awareness in all sectors (public, industry, and government) and in developed as well as developing countries. The idea that consumers should share responsibility for pollution and its cost has been receiving increased acceptance. In some jurisdictions, the prices of several energy resources have increased over the last 1–2 decades, to at least partially account for environmental costs [Dinçer, 2001]. World population is expected to double by the middle of the 21st century. Similar progress and change will continue to occur in the economic development. Both global demands for energy services and primary-energy demands are expected to increase. Simultaneously, concern about energy-related environmental problems such as acid precipitation, stratospheric ozone depletion, and global climate change will also continue to grow [Dinçer, 2001; Dinçer, 2003]. Furthermore, wasteful use has exhausted resources [Erten, 2001]. Pollution and destruction to the environment has adverse effects on mental and physical wellbeing both at the individual and societal levels [Kinnear, Taylor & Ahmed, 1974].

IdentifyingEnergy Conservation Behaviors...

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Humans have a right to attaining economic growth and wealth. However, in today’s world there is greater need for a mindful attitude toward the environment while striving toward attainment growth and wealth [Çevik, 1999]. Energy is one of the indispensable needs for humans in a variety of areas of life (i.e., in industry, transportation, heating, lighting, cooking etc.). How energy sources are handled can have positive and negative impacts on quality of living. Energy is an important determinant of economic, social and political affairs and legislation. It is also an essential input to the social and economic growth of today’s countries. The contemporary concept of sustainability is most relevant to the energy sector. Sustainable handling of energy involves policies, technologies and practices that enable continual delivery of clean, affordable and reliable energy to all spheres of the society [Anon, 2008]. Thus, energy wastage can be reduced by; •

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•

Industry; developing methods and processes of production that allow effective use of energy and eliminates energy wastage, Consumers; mindful use of energy by the ordinary person.

Energy wastage is a concern of every nation in today’s world. It is imperative to foster national ethics about elimination of energy wastage. A lack of such ethics and ignorance about energy use leads to shortage in energy, increased prices, exhaustion of energy resources and destruction of the environment [Yavuz, 1985]. A genuine concern for leaving a livable environment to the next generations should guide energy production and consumption [DPT, 2000]. Turkey depends on other countries for its energy needs. It imports 70% of its energy. As a country with a rapidly growing population, dependence on other countries will continue to grow correspondingly. Increases in population brings about growing numbers of youth whose attitude and behaviors about energy use is essential in how the country will handle energy matters in the future [Büyükmıhçı, 2004]. The number of individuals younger than 15 years of age is 1.7 billions worldwide. In other words, this age group constitutes over 32% of the world population. Compared to developed countries, young persons make up bigger portions of the general population in the developing world. In order to answer the question of “why energy?” one must take a close look at the relationship between energy and environment. [Erten, 2001]. In the last census held in Turkey, 12.6% of the population was between ages of 0-4; 11.47% between 5-9; and 10.56% was between ages of 10 and 14 [DİE, 2003]. Given the considerable number of young population of the country, the attitudes and behaviors of these individuals will have significant implications on the future of energy use in the country. Attaining insight into young persons’ attitudes and behaviors about energy and effective use of energy will guide interventions with the young population.

Energy Saving And Productivity Energy and Environment Energy is considered a prime agent in the generation of wealth and a significant factor in economic development. The importance of energy in economic development has been

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Arzu Şener and Seval Güven

recognized universally; the historical data attest to a strong relationship between the availability of energy and economic activity. During the past two decades, the risk and reality of environmental destruction have become more apparent. Environmental issues have become more visible as a result of the increase in the world population, consumption, industrial activity, etc. [Kaygusuz, 2002 a, DPT, 2000]. Energy is one of the most essential inputs into production. If not the starter of the damage to the environment, it is a considerable contributor. It is present in almost all human activities in varying ways and degrees (industry, heating, transportation, lighting, cleaning etc.). Its use in a highly populated and industrialized world often does irreversible damages to the land, air and water resources (Figure 1) [Pala, 1998]. Issue yIncreased energy

Outcomes yIncreased prices yIncreased air and water temperature yPoisonous gases and environmental hazards

Solutions yProduction of natural energy (water, energy,wind) yIncreased CO2 and climate disasters yDeveloping environment free technologies yEnergy saving

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Figure 1. Energy and Environmental Issues (Cepel, 1992).

The chief reason for increased destruction of the environment is energy consumption of the industrialized countries that lack a sustainable approach [Keating, 1993]. In other words, these countries are mainly responsible for air pollution, ozone depletion, and carbon emissions. Yet, compared to these countries, developing countries have great room for improvement in their use of energy. Major changes in the production and use of energy are urgently needed. Some significant steps are taken nationally and internationally toward promotion of energy transition, increase of energy efficiency in terms of conservation, promotion of renewable energy technologies, and promotion of sustainable transport systems [Dinçer, 1998]. The world has increasingly been experiencing floats, drought, severe windstorms and extremely hot summers which cause tremendous degree of material and non-material harms. Although the world community seems surprised at the occurrence of these catastrophes, scientists have insistently warned the world community about the alarming environmental changes. However, these warnings were often dismissed, at least in part, because of the ambition for attainment of wealth. Continuing this indifference is an invitation for more disasters to come. At the heart of the indifference toward the environment lies the human selfcentered use of energy resources [Erten, 2001]. Energy is mostly used in the industry, transportation, residences and services. Heat is used in cleaning, personal hygiene, heating, lighting, cooking and production whereas power is used for transportation and machinery use etc [TC Çevre Bakanlığı, 1997, Akalın, 1990]. Currently, fossil-based non-renewable energy sources such as coal, gas, petroleum, nuclear energy etc. and non-fossil-renewable sources such as geothermal energy, solar energy, wind and water are being used [TC Çevre Bakanlığı, 1997; Yılmaz & Can, 2005]. Future development depends on the degree to which renewable energy sources are used in environment friendly manners. Achieving solutions to environmental problems faced today requires long-term actions for sustainable development. Utilizing renewable energy resources

IdentifyingEnergy Conservation Behaviors...

39

can at least partially be an ample solution. Use of these resources can make sustainable development possible [Kaygusuz, 2002 a]. On the other hand, renewable energy is not the answer for all our energy problems. “There is a fallacy in believing that energy conservation and solar energy alone can save our energy future. If world population and per capita use of energy continues growing at current rates or higher, our demand for energy is likely to grow faster than our ability to supply it from renewable sources” [McCluney, 2003]. The world has not figured out any sources of energy for future needs. Human actions have brought energy sources close to an end which comprises the linkage between energy issues and environmental concerns of today’s world. Future of the world’s energy needs is thus the chief concern of the contemporary world which has alarmed scientists and policy makers all around the globe [Dünya Çevre ve Kalkınma Komisyonu, 199; Göka & Görmez, 1993]. The existing non-sustainable ways of consumption and energy supplies need reexamination. Likewise, along with discovering new energy sources, today’s world must find ways in which wasteful use of existing resources and pollution can be alleviated [Keating, 1993; Göka & Görmez, 1993]. Today’s world faces the challenge of figuring out the future of energy issues [Dünya Çevre ve Kalkınma Komisyonu, 1991]. This requires a sustainable development approach aiming at minimizing costs and hazards to the environment while supporting economic and social development [Gürbüz, 2005].

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Figure 2. Sustainable Life Triangle [Gürbüz, 2005, Schmitt, Schöpfer, Björnsen, 2005].

Energy Consumption in Turkey Energy is an indispensable element in human life. Fossil based sources such as coal, petroleum and gas constitute a great portion of the energy used by today’s world (85-90%). These energy sources are limited and will come to an end. In fact, as indicated by numerous authors, if the existing pace of consumption continues, the world will run out of its petroleum reserves in 40-45 years, gas in 60-70 years and coal in 240-250 years [TMMOB, 2008]. Similar with the rest of the world, Turkey relies on fossil based resources for its energy needs. Turkey pays about 30 million dollars each year to import these fuels. In addition to this high cost, fluctuations in foreign currencies have considerable adverse impacts on the country’s economy as well as on family budgets [TMMOB, 2008].

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Arzu Şener and Seval Güven

Some consider energy consumption per person as a criterion for a given country’s degree of prosperity (Figure 3). In Turkey, energy consumption per person is 30% lower than the world average. Turkey’s existing population of 70 millions increases about 1.3 million each year. As a country depending on importing of energy, Turkey will continue to face ample challenges in meeting energy needs of its increasing population while also having to consider environmental damages consumption of these energy sources cause (Figure 3) [Göka & Görmez, 1993].

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Figure 3. Energy Consumption per Person [Büyükmıhçı, 2006].

Turkey needs to strive toward bringing its energy consumption level to those of the developed countries. Thus, the existing energy consumption patterns call for improvement [Büyükmıhçı, 2006]. Despite its rapid growth and its striving toward membership to the European Union, Turkey’s energy consumption is no where near to that of the developed world [Ural et al., 2008]. “Turkey has dynamic economic development and rapid population growth. It also has macro-economic, and especially monetary, instability. The net effect of these factors is that Turkey's energy demand has grown rapidly almost every year and is expected to continue growing, but the investment necessary to cover the growing demand has not been forthcoming at the desired pace. On the other hand, meeting energy demand is of high importance in Turkey” [Kaygusuz, 2002b] . Turkey has to import energy because its energy need exceeds its energy production [Kaygusuz, 2002 b]. In fact, imported energy constitutes 70% of the total energy used in the country. Drastic changes in the country’s energy policies are not feasible in the near future [Ural et al., 2008].

IdentifyingEnergy Conservation Behaviors...

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Turkey, according to 2000 data, produces 27.67 Mtoe (million tons of oil equivalent) per year from its own primary sources and consumes 79.46 Mtoe a year (see Table 1). It is projected that by 2020, the primary energy production will be 85 Mtoe, while primary energy consumption will be 318 Mtoe. Given that Turkey’s fossil fuel reserves are about a total of 2454 Mtoe, the country will have to import energy in increasing proportions in the future [Kaygusuz, 2002 b]. Table 1. Primary Energy Production and Consumption of Turkey During 1996-2000 [ktoe] [MENR [2001] Energy Production 1997

1998

Energy consumption

1999

2000

1997

1998

Hard coal 1347 1678 2729 Lignite 11759 12514 12685 Oil 3630 3230 3056 Natural gas 230 684 662 Total fossil 16966 18106 19132 Hydropower 3424 3632 2982 Geothermal 179 256 274 Solar 80 98 114 Wood 5512 5512 5293 Waste and dung 1512 1492 1510 Total renewable 10707 10878 10650

1769 12830 2925 631 18155 2656 286 120 5081 1376 9519

8495 12280 30515 9165 60455 3424 179 80 5512 1512 10707

8160 12414 32083 10635 63292 3632 256 98 5512 1492 10878

1999

2000

11286 12984 32916 12902 70088 2982 274 114 5293 1510 10650

8149 12830 34893 14071 69943 2656 286 120 5081 1376 9519

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Turkey does not posses a wealth of energy resources. Its total energy potential which consists of coal geothermal and hydro electrical power, equals to only 1% of the world’s total energy potential. Turkey has considerably limited petroleum and gas reserves [Ural et al., 2008].

Figure 4. Turkey’s Energy Consumption per Sector (2003) [Gürbüz, 2005].

2006 data from the Turkish Ministry of Energy and Natural Resources indicates that the country consumes energy equivalent to approximately 93 million tons of petroleum annually and this consumption increases about 5% each year. Most of the consumption takes place in three sectors: industry, buildings and transportation (Figure 4). Buildings constitute 30% of this consumption. Consumption in buildings involves heating, lighting and use of electrical

42

Arzu Şener and Seval Güven

appliances. Studies show that at least 30% of energy used in buildings can be saved [TMMOB, 2008]. Turkey’s use of non-renewable energy sources such as coal, petroleum, gas etc. increases each year. This does not only contribute exhaustion of these limited energy sources but also adds to the environmental pollution. In other words, pollution to the environmental is worsened in the process of production and consumption of energy [TMMOB, 2008]. Given the interrelations between environment, energy, industrialization and changes in population, if today’s world fails to make drastic improvements in efficient use and conservation of energy and if the production electrical power, continues to rely heavily on non-renewable fossil based fuels, it will have to pay profound costs perhaps sooner than the next century [Pala, 1998].

Efficient Use of Energy

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In order to establishing sustainable handling of energy, efficiency in energy use needs to be improved (Figure 5). Energy intensity, which is used as the chief criterion in determining the degree to which a country uses energy, refers the rate between the amount of energy consumed and the country’s gross national product per capita. Turkey has an intensity rate of .38 while the average of the OECD countries is 0.19. In other words, Turkey’s energy consumption is twice as inefficient as the average of OECD countries [Çağlar, 2005].

Figure 5. Efficient Use of Energy [Büyükmıhçı, 2006].

IdentifyingEnergy Conservation Behaviors...

43

Efficient use of energy; • • • • •

a clean environment, more prosperity, more affordable energy , reduced expenditure on foreign currency, sustainable energy which refers to more production with the same amount of energy consumption [EİEİGM, 2007].

Efficient use of energy is closely linked to a country’s prosperity and well-being. Turkey does not own its own energy resources for sustainable development. It has not only been purchasing a great deal of energy from other countries but also been using this expensive energy inefficiently. Therefore, there is urgent need for taking firm steps toward efficient use of energy [Çağlar, 2005].

Energy Conservation

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The energy crisis of 1973-1974 alarmed scientists and policy makers of the developed world for the first time. The search for new solutions brought about several ideas of which two stood out the most: finding new (and renewable) energy sources and reducing the number of motor vehicles in use. More importantly, it was concluded that conserving the existing energy sources was even more vital. Energy conservation was to be accomplished at various levels. It requires new technologies, action on the part of institutions and governments as well as effort by each individual. Hence, energy conservation has received more attention than ever, since early 1970s [Erten, 2001]. Energy conservation does not mean minimizing or limiting the need for energy. It refers to effective and efficient use of energy which involves minimization of the amount of energy spent in the delivery of a service or production of a good without comprising the need and demand for energy [EİEİGM, 2007]. In short, energy conservation is any behavior that results in the use of less energy. Energy conservation is about; • • •

reducing wasteful use, doing more with the same amount of energy or doing the same thing with less energy [Anon, 2005].

In short energy conservation involves, • • •

right production, right planning and right use [TMMOB, 2008].

Efforts toward energy conservation should strive toward rational use of energy without compromising economic or social development [Zuiches, Morrrison & Gladhart, 1976]. The

44

Arzu Şener and Seval Güven

community of individuals concerned about energy conservation is in agreement that the cheapest energy is the energy obtained by conservative and efficient use of energy [EİEİGM, 2007]. Energy conservation can be viewed at two levels: • •

First level involves less energy use by each person without compromising from life standards. The second level involves some practices on a larger scale that can contribute to conservation of energy. Some of these practices involve industrial processes, isolation of buildings and developing ways in which leakage of heat can be eliminated. This category of practices has to do with the sort of conservation that leads to production which contributes to states’ gross income and strengthens individuals’ habits that support economic and social development. The purpose of these actions is to use less energy while doing the same things and by doing so eliminating the risk for future catastrophes [Zuiches, Morrison & Gladhart, 1976].

These practices involve;

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• • • •

Pollution to the air and soil, Dependence on foreign petroleum, Minimize acid precipitation and Conserve resources for the use of future generations while promoting development [Prokop, 1994].

Humans are a part of the environment and are also capable of altering it and reconstructing it. They are also users of the environment. No person in any given society can possibly escape from consuming. There is an undeniable link between individuals’ behaviors as consumers and reduction or increase in environmental pollution or energy consumption [Gül & Güven, 2000]. “Despite decades of research in energy conservation and other pro-environmental behaviors, considerable uncertainty remains about what motivates people to behave in environmentally responsible ways. Studies have investigated, for example, the contributions and links between environmental knowledge [e.g., Ostman & Parker, 1987], environmental values [e.g., Kempton, Boster, and Hartley, 1995; attitudes [e.g., Becker, Seligman, Fazio, and Darley, 1981; Geller, 1995], personal characteristics [e.g., Allen & Ferrand, 1999], and behaviors [e.g., Harrigan, 1991; Katzev & Johnson, 1987]. Numerous theoretical approaches have been developed to integrate various combinations of these factors.” However, similar with other human behaviors, energy-conservation behavior is multifaceted and complex, thus challenge attempts to explain and predict it. Both effective use and conservation of energy involve human behaviors. Individuals’ and families’ conduct reflect their attitudes on energy conservation. Human behaviors are part of creation of the energy problems and can be part of solutions to these problems. Studies show that individuals’ lack of knowledge or misinformation leads to ineffective use of energy [Cook & Berrenberg, 1981]. Therefore, energy conservation depends heavily on

IdentifyingEnergy Conservation Behaviors...

45

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understanding the fact that there is a limited of supply of energy resources and that inefficient use of this supply will soon bring it to an end. Thus, energy conservation is a social responsibility of every individual [Ruffin & Weinstein, 1979]. Unless behaviors and habits of individual consumers change to a significant extent, drastic improvements in energy conservation cannot take place [Gül & Güven, 2000]. For example, leaving television or lights on, continual use of electrical appliances etc. lead to increased consumption of energy every year. Producing more energy leads to green house effect which causes more warming in the atmosphere and more thermal energy plants are built to allow more use of energy. Reckless and wasteful use of energy resources corresponds to wasteful use of individual and family budgets [Güven, 1999]. Individuals who have no idea as to how to conserve energy cannot effectively do so [Rudd, 1978]. The most significant means that can promote consciousness about energy conservation is education, which can through formal and informal means inform individuals not only about the issue but also about alternative ways in which it can be handled [Ruffin & Weinstein, 1979]. Formal and informal education of new generations should start in early ages in order to effectively foster awareness about sustainability of energy and development (Figure 6).

Figure 6. Education and Sustainable Development.

46

Arzu Şener and Seval Güven The purpose of this education should be; •

•

Encouraging young individuals toward attainment of knowledge, skills and values and responsible behaviors about energy and environmental issues [Ergün, 1993]. Transmission of the culture and promotion of cultivated new generations [Göka & Görmez, 1993].

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It is important to start education on environment and energy issues in early ages so as to foster attitudes, values and behaviors at developmentally critical stages. Success of such education does not only depend on what is done at schools but rather on the collaboration between schools, families and communities (Figure 7)[Güven, 1999].

Figure 7. Environmental Education: Framework and Process [Ayhan, 1999].

Children’s first observations and experiences about use of resources take place in the family environment [Paolucci & Hogan, 1973]. For instance, in order to reduce the amount of electricity consumed, families can use lights only in areas of the house that are in active use or to reduce energy consumed for heating they can pay attention to the isolation of their buildings. Families can also reduce the amount of energy they use by taking short showers and paying close attention to not running water that is not in active use. Likewise, they can separate and collect materials made of glass, paper, plastic and metals for recycling which will not only have enormous contributions to the protection and efficient use of resources but also set examples for education of children [Göka & Görmez, 1993]. The general climate of

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IdentifyingEnergy Conservation Behaviors...

47

the family, relationships among family members, ways in which goods and services are purchased and utilized-all have influences on behaviors and attitudes of children [Güven, 1999]. Household use constitutes one third of total energy consumption. Improvement in conservation of energy can have positive implications for the family and for the protection of the environment [İnan, 2005]. Families can have remarkable impact on new generations’ energy consumption habits given that families consumes around 80% all goods and services of a country and that individuals spend developmentally critical years in families [Güven, 1999]. Energy and environmental issues are of vital importance for the future of the planet, the economy and human societies. Schools can take significant roles in coping with these issues. “Given national and international commitments to Agenda 21 following the Rio de Janeiro Earth Summit Conference, it is surely essential to view energy as a vital ingredient of school education world-wide”. Although education is vital, it cannot be the sole remedy to energy issues. State laws, policies on development or even governments’ decisions can also have significant implications. Disciplines such as law, politics, economy, biology, sociology, psychology, medicine all need to be actively involved in environmental issues because an interdisciplinary approach to these issues can lead to more useful insight. Improved collaboration between industries and government is needed for acquiring safety and sustainability in handling of resources and energy [Ergün, 1993]. The most important issue is how today’s world goes about fostering sustainability of resources and development. This can be possible through education of the youth who will grow into adulthood and be actively responsible in handling future of energy and resources [Dünya Çevre ve Kalkınma Komisyonu, 1991]. Given that energy is an indispensable part of the daily life, sufficient steps should be taken to ensure that obtaining and using energy today do not compromise living habitats and circumstances of proceeding generations [Gürbüz, 2005]. “Efficiency and conservation are key components of energy sustainability-the concept that every generation should meet its energy needs without compromising the energy needs of future generations. Energy sustainability focuses on long-term energy strategies and policies that ensure adequate energy to meet today’s needs, as well as tomorrow’s. Sustainability also includes investing in research and development of advanced technologies for producing conventional energy sources, promoting the use of alternative energy sources, and encouraging sound environmental policies” [NEED, 2007]. In addition, the need for more natural resources that is due to non-sustainable consumption need to be reevaluated. Likewise, alternative ways that can slow down pollution and exhaustion of resources should be discovered soon [Keating, 1993]. Given the rapid increase of population in Turkey and the considerable number of young population, fostering consciousness about environmental issues can greatly contribute to continual development of the country [Ergün, 1993]. Well informed young individuals can have positive impact on their peers as well as on future policy making. Improved consciousness about environmental issues will contribute to protection of the environment [Nazlıoğlu, 1991].

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Arzu Şener and Seval Güven

Method Work on effective use and conservation of energy has revealed that 30% of energy consumption takes place in residences. Therefore, individuals’ and families’ handling of energy has significant implications. Children have considerable influence on the amount of energy consumed in households because of the impact they have on families’ consuming behaviors and because of their own activities and consumption. Therefore, the purpose of this study was to identify behaviors and attitudes of 6th, 7th, and 8th graders on energy conservation. Part of the purpose of the study was to motivate future researchers to work on various aspects of energy issues with various samples.

Sample and Data Analysis Participants of this study were 6, 7 and 8th grade students attending to “İnönü Elementary School” located in Polatlı, in the district of Ankara. A total of 150 (62 females and 88 males) volunteering students answered the survey packet. Surveys were given to students in face-toface contact with the students. Chi Square procedure was used in for data analysis [Sümbüloğlu & Sümbüloğlu, 1994].

Findings Descriptive Statistics Students’ distribution to gender was as follows: 41.3% and 58.7%. Majority of the female participants were 6th graders (58%) and majority of males were (74%) were 7th graders. Students whose mothers had “elementary education or less” were 59.4%. Those whose fathers had “elementary education or less” were 56.7%. Majority of the students (53.4%) were from families made of 4-5 individuals (58.0% of 6th graders; 52.0% of 7th graders; and 50.0% of 8th graders).

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The Need for Energy Saving in Turkey As a result of its efforts toward industrialization, attainment of new technologies, and improvement of living standards and its increasing population Turkey’s need for energy increases each year. The country wastes 3 million dollars each year due to inefficient use of energy. In other words, by lacking efficient use of energy, Turkey wastes an amount of money with which two dams capable of producing 6 billion 600 million kilowatt/hour can be built. The rate of energy waste in Turkey is significantly higher than the overall average of the world (20%). Turkey obtains 70% of energy from other countries. Thus, energy prices in Turkey are extremely high. Therefore, efficient use of energy is urgent and a civic duty of every citizen [TMMOB, 2008]. Eighty-two percent of students responded “yes” to the question of “Do you think energy saving is necessary in Turkey?” while 18% answered “no.” The number of male students (85.2%) who answered “yes” to the question was higher than that of female students (77.4%)

IdentifyingEnergy Conservation Behaviors...

49

(p>0.05). Students’ likelihood to believe in the necessity of energy saving increased with grade level (6th grade: 80.0%, 7th grade: 82.0%, 8th grade: 84.0%) (p>0.05) (Figure 8). This finding indicates that students are aware of the necessity for energy saving.

90,0 80,0 70,0 60,0 50,0 40,0 30,0 20,0 10,0 0,0

85,2

77,4

22,6

Female

20,0

14,8

Male

6th grade Yes

84,0

82,0

80,0

16,0

18,0

7th grade

8th grade

No

Figure 8. Students’ Beliefs about Energy Saving.

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Switching off Lights in Unused Areas Nineteen percent of energy is used for lighting worldwide. Electricity is highly expensive in Turkey. Yet, 25% of the total electrical energy consumption is used for lighting. Twenty percent of electrical energy consumption in residences is used for lighting [Anon, 2005]. Much of this energy is wasted due to in inefficient use [NEED, 2007]. For example, one-hour use of a 100-watt light bulb on a daily basis amounts to 36.5 kilowatt/hour in a year which results in approximately 20 kg of CO2. Thus, considerable amount of energy is used for lighting at homes, schools and workplaces [Anon, 2005]. Seventy percent of the participants of this study reported switching off the lights in unused areas while 22% indicated leaving the lights on. The number of students who turn off the lights was significantly higher than those who do not (Figure 9). The percentage of students who believe in the necessity of energy saving and those who turn off the lights when not needed was considerably high. However, future work is needed to determine if this attitudes are due to economic difficulties or to an awareness about energy saving. As noted earlier, electrical energy is extremely expensive in Turkey. Income lower than 2000 TL (TL: Turkish Lira) constitutes poverty limit of a 4-person family and 772 TL is limit for starvation limit for the same size family. Individuals living under the poverty limit make up about 20% of the approximately 70 million population of the country. Considering these circumstances and the fact that the minimum wage is 435 TL in Turkey, efficient energy use is not only important for the country but for each family alike. Thus, students’ attitudes about energy saving could be due to the limited family income more so than an awareness about environmental issues.

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Arzu Şener and Seval Güven

Percentage of female students who reported turning the light off while leaving a room was significantly higher than that of male students (82.3% and 75.0%) (p>0.05). Students in higher grades were more likely to turn off the light while leaving a room (6th grade: 66.0%, 7th grade: 78.0% and 8th grade: 90.0%) (p0.05). The likelihood of not unplugging household appliances after use increased with grade level (6th grade: 84.0%, 7th grade: 88.0%; and 8th grade: 96.0%) (p>0.05) (Figure 10).

IdentifyingEnergy Conservation Behaviors...

100,0 90,0 80,0 70,0 60,0 50,0 40,0 30,0 20,0 10,0 0,0

95,2

85,2

14,8

96,0

88,0

84,0

16,0

51

12,0 4,0

4,8 Female

Male

6th grade Yes

7th grade

8th grade

No

Figure 10. Students’ Likelihood of Turning off and Unplugging Household Equipment After Use.

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Turning the Tap off while Busy or not Actively Using the Water Although water is essential for life, it is often taken for granted. Individuals are satisfied as long as an unlimited supply of good quality water is available. However, water supplies in many countries are no longer unlimited. Escalating water development costs have become a serious obstacle to expanding water supplies [Sharpe & Theodore 1989]. Thus, the amount of water used while taking a shower, washing hands, shaving, brushing teeth, washing vegetables or fruits is worth consideration. After leaving the tap, clean becomes sewage water. The cost of obtaining water and that of purifying sewage is considerably high. Although two third of the planet is made of water, only 0.3% of it can used as drinking water. The amount of available water for each person in Turkey is 1430 m3 annually. Considering the amount of each person consumes daily is 111 litters, the need for careful use of water is obvious. Nicholas Stern, a chief economist of the World Bank notes that the most significant environmental and economic threat is warming which leads to drought in the Middle East, Mediterranean countries and Turkey (Cited in, Özmumcu, 2007]. Stern also cautions that drought has dangerous impact on water resources, which if continued to be wasted as they are, will become even more valuable than petroleum and will perhaps become a factor leading to international conflicts and wars [Özmumcu, 2007]. Indeed, Turkey had an abundant amount of 45.000 m3 available water per person in the 1960s while this amount is 1.430 m3 at the present [Güven, 2007]. Given that the world’s food supplies also rely on availability of water, mindful use of existing sources of water is of vital importance [Keating, 1993]. Results of this study showed that almost all the students (98.7) reported turning the water off when they are not using it. While looked at in terms of gender; 98.4% of female and 98.9% of male participants reported turning the water off while not using it. Likewise, all the 6th and 7th graders (100%) reported doing so. This rate was 96% among the 8th graders (X²= has not been applied) (Figure 11). Those who reported turning the water off while brushing teeth were 87.7% of the participants. Female students who do not turn the water off while brushing teeth were 16.1% and male students were 25%. (p>0.05). None of the students in 6th

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Arzu Şener and Seval Güven

and 7th grades indicated not turning the water off while only about 4% of those 8th grade indicated doing so. These rates changed regarding tooth brushing: 14% of 6th graders; 18% of 7th graders; and 32% of 8th graders reported that they not turn off the water while brushing teeth. In other words, as their grade level increased students were more likely to not turn the water off when not actively using it. (p>0.05) (Figure 12). This finding illustrates that students might lack awareness about saving water while brushing their teeth. Individuals can save between 15 to 35 litters of water if they simply turn off the water during teeth brushing. 98,9

98,4 100,0 90,0 80,0 70,0 60,0 50,0 40,0 30,0 20,0 10,0 0,0

1,6

Female

100,0

6th grade Yes

96,0

1,3

0,0

0,0

1,1

Male

100,0

7th grade

8th grade

No

Figure 11. Students’ Distribution in terms of Whether or not They Turn the Water off while not Actively Using It.

90,0 80,0

83,9

86,0

75,0

82,0 68,0

70,0 60,0 Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

50,0 40,0 30,0

32,0

25,0 16,1

14,0

18,0

20,0 10,0 0,0

Female

Male

6th grade Yes

7th grade

8th grade

No

Figure 12. Students’ Distribution in terms of Whether or not They Turn the Water off while Brushing Teeth.

IdentifyingEnergy Conservation Behaviors...

53

Thus, changing traditional taps to new-automatic ones could eliminate 25% of water consumption [Güven, 2007].

Showering Preferences Water heating is the third largest energy expense of households. It typically accounts for about 14 percent of utility bills (NEED, 2007]. At national levels, pumping water from one place to another constitutes a significant portion of countries’ water budget. At individual level, individuals typically pay more for heating water than for all the water they use. Saving water will save energy, which, in turn, will save money on water and heating bills, as well as on municipal energy costs. As water users, children can have a great influence families’ water consumption (Sharpe & Theodore, 1989]. Heated water is used for showers, baths, laundry, dishwashing and general cleaning. There various ways to cut water heating bills. One way to conserve hot water is taking showers instead of baths and taking shorter showers [NEED, 2007]. Such showers can save up to 8% of total water used. However, findings of this study showed that over the half of participants did not take short showers (53%). Only 37% indicated taking short showers. While 41.9% of female students took short showers, 56.8% of males took long showers (p>0.05). Sixty-eight percent of sixth graders, 40% of 7th graders and 52% of 8th graders reported taking short showers. Thus, 6th graders were significantly more likely to take short showers than 7th and 8th graders (X²=has not been applied) (Figure 13).

68

70,0

56,8

60,0 50,0

41,9

40,0 34,1

40,0

52

50,0

48,4

32,0

30,0

30,0 20,0

16,0 9,1

9,7

2,0K

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

10,0 0,0

10,0

Female Taking a bath

Male

6th grade

Taking a Short Shower

7th grade

8th grade

Taking a Long Shower

Figure 13. Distribution of Students’ Showering Preferences.

Use of Blow-Dryers Hair drying is can best be done with a towel without using a drying machine because the use of a blow-dryer for 10 minutes costs as much energy that of a 60-watt light bulb running for three hours [TMMOB, 2008].

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Majority of the participants indicated not using a blow-dryer (68.7%). Those who used a blow-dryer were 31.3%. Female students were more likely to use (37.1%) than male students (27.3%) (p>0.05). Students’ use of the hair dryer decreased with grade 40%, 28% and 26% for 6th, 7th and 8th graders respectively (p>0.05) (Figure 14). In short, prevalence of blowdryer use was low. Erten [2001] also found that students did not report over-use of electrical household appliances.

80,0

72,7

62,9

70,0

60,0

74,0

72,0

60,0 50,0

40,0

37,1

40,0

27,3

28,0

26,0

7th grade

8th grade

30,0 20,0 10,0 0,0

Female

Male

6th grade Yes

No

Figure 14. Distribution of Students’ Blow-Dryer Use.

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Purchasing Recyclable Products “Using recycled materials almost always consumes less energy than using new materials. Recycling reduces energy needs for mining, refining, and many other manufacturing processes”. Individual consumers can have an impact on industrial energy use through the product choices they make and through what we do with packaging and products they no longer use. “The most effective way for consumers to help reduce the amount of energy consumed by industry is to decrease the number of unnecessary products produced and to reuse items wherever possible”. Likewise, purchasing only products that are necessary, and reusing and recycling then can reduce the overall energy use. Hence, reduction in waste will save money, energy and natural resources and thus help product of the environment [NEED, 2007]. “For example, recycling a pound of steel saves enough energy to light a 60-watt light bulb for 26 hours. Recycling a ton of glass saves the equivalent of nine gallons of fuel oil. Recycling aluminum cans saves 95% of the energy required to produce aluminum from bauxite. Recycling paper cuts energy usage in half” [NEED, 2007]. Majority of the current sample (70%) indicated that they did not pay particular attention to purchasing recyclable products (Figure 15). Percentage of males preferring recyclable goods was significantly higher (39.8%) than of females was 16.1%. (p 1.0 [43]. Therefore, Thermal NO dominates in exhaust NO. Thermal NO increases exponentially above about 1800 K. The residence time of gas in the each computational cell is calculated from the cell volume dividing the volume flow rate of combustion gas in the cell. Therefore, the residence time in high temperature region, th, is expressed as the sum of the residence time in the cell above 1800K. th is evaluated as follows:

1 1 1 ⎞ ⎛ 1 . = ∑⎜ + + ⎟ th Δr / V rΔθ / W ⎠ t >1800 K q ⎝ Δx / U

(26)

Figure 11 shows the effect of inlet Reynolds number on exhaust NO mole fraction. Increasing the inlet Reynolds number steadily decreases the exhaust NO mole fraction. This result shows that the mixing process of the fuel and the air has an important effect on lowNOx combustion. 400

Exhaust NO mole fraction (at 0%O2) [ppm]

(d = 8 mm, n = 6) (d = 10 mm, n = 6) 300

(d = 7.3 mm, n = 4) 200 (d = 5.2 mm, n = 8)

100

0 15000

(d = 4 mm, n = 6)

(d = 8 mm, n = 6)

20000

25000

30000

35000

40000

45000

50000

Inlet air nozzle Reynolds number [-]

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Figure 11. The effect of inlet Reynolds number on exhaust NO mole fraction [14].

Figure 12 shows the effect of residence time in high temperature region on exhaust NO mole fraction. Exhaust NO mole fraction increases with an increase of the residence time in the high temperature region. When the number of the air inlet nozzles decreases, the residence time in high temperature region increases because the region where the mixing of the spray and the air deteriorates due to nonuniform air supply. Therefore, exhaust NO mole fraction increases with a decrease of the number of the air inlet nozzles. When the residence time in high temperature region is higher than 0.66 (at d = 8 mm, n = 6), that does not greatly affect the exhaust NO mole fraction. This is because NO formation reaction is almost terminated in such long residence time.

Spray Combustion Simulation for Low-NOx Emissions

147

Exhaust NO mole fraction (at 0%O2) [ppm]

450 (d = 8 mm, n = 6)

400 350

(d = 10 mm, n = 6)

300 250 (d = 5.2 mm, n = 8)

200

(d = 7.3 mm, n = 4) 150

(d = 6 mm, n = 6)

100 50 0

(d = 4 mm, n = 6) 0

0.1

0.2

0.3 0.4 0.5 0.6 0.7 Residence time in high temperature region [sec]

0.8

0.9

Figure 12. The effect of residence time in high temperature region on exhaust NO mole fraction [14].

3.3.4. The Effect of Baffle Plate on Combustion Behavior The difference in the predicted temperature distribution between Figure 6(a)(air ratio 3) and Figure 6(b)(λ = 2) are greatly noticeable. At λ = 3, a high temperature region near the atomizer is small and combustion reaction is almost terminated in the upstream. However, at λ = 2, the high temperature region extends to the downstream because the mixing of spray and air deteriorates. As a result, exhaust NO mole fraction at λ = 2 is larger than that at λ = 3.

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ൺ 15 ms

-1

Baffle plate

(a) Without baffle plate

(b) With baffle plate A

(c) With baffle p late B

(d) With baffle plate C

Figure 13. Predicted velocity vectors near injector (λ = 2) [13].

148

Hirotatsu Watanabe, Yohsuke Matsushita, Hideyuki Aoki et al.

Droplet

Injector

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Figure 14. Predicted droplets trajectories near injector (λ = 2).

To shorten the flame length, a baffle plate is applied to the combustion chamber as an easy and reasonable addition to the jet burner. Moreover, it does not require drastic changes to the burner shape. In this section, the effect of the baffle plate on combustion behavior is numerically investigated. Figure 13 shows the time-mean velocity vectors of the combustion gas flow near the baffle plate. The zero-axial velocity isoline is also plotted in Figure 13. The size and position of the baffle plate have a strong influence on the gas flow. Figure 14 shows predicted droplet trajectories and the total amount of adhesion of droplets to the baffle plate in accordance with the total amount of injected fuel when baffle plate B or baffle plate C is used. The size and position of the baffle plate have a influence on the total amount of adhesion of droplets. The effects of baffle plate on combustion characteristics are investigated from the point of view of the gas flow and the amount of adhesion of droplets to the baffle plate.Figure 15 shows the predicted temperature contours and the ratio of the predicted adhesion of droplets to the baffle plate in accordance with the total amount of injected fuel. Figure 16 shows the predicted NO mole fraction distribution and exhaust NO mole fraction at 0%O2 concentration. When baffle plate A is set in the chamber, the maximum high temperature region is extended downstream as shown in Figure 6(b) and Figure 15(a). This is because the distance between the atomizer and the baffle plate is too short to provide air sufficiently near the atomizer as shown in Figure 13(b). When baffle plate B, whose inner diameter is larger than that of baffle plate A, is set in the chamber, the high temperature region becomes narrow and shifts upstream as shown in Figure 15(a) and Figure 15(b). As a result, the NO mole fraction and exhaust NO mole fraction in Figure 16(b) is lower than that in Figure 16(a). This is because the mixing of spray and air is improved and air is sufficiently supplied to the vicinity of the injector by increasing the inner diameter of baffle plate as shown in Figure 13(c). When baffle plate C, in which the distance between the spray atomizer and the baffle plate is larger than that in baffle plate B, is set in the chamber, the temperature and NO mole fraction distribution show a little difference, as shown in Figure 15(b), (c), Figure 16(b), (c). However, the adhesion of droplets to the baffle plate increases greatly.

Spray Combustion Simulation for Low-NOx Emissions

149

Exhaust NO fraction decreases using baffle plate, especially using baffle plate B or C. However, the more adhesion of droplets to the baffle plate increases, the more difficult it is to carry out stabilized continuous operation, due to carbon deposits. Consequently, the jet burner using the baffle plate B is found to provide better combustion behavior than the existing jet burner without a baffle plate. Inlet air

Baffle plate(Adhesion 2.7 wt%)

(a) With baffle plate A(Tmax=2281 K) Baffle plate(Adhesion 0.49 wt%)

(b) With baffle plate B(Tmax=2355 K) Baffle plate(Adhesion 6.5 wt%)

(c) With baffle plate C(Tmax=2385 K) 593 K

2400 K

Figure 15. Predicted temperature contour (λ = 2) [13]

Inlet air

Baffle plate

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(a) With baffle plate A(Exhasut NO( at 0%O2) = 337 ppm)

(b) With baffle plate B(Exhasut NO( at 0%O2) = 248 ppm)

(c) With baffle plate C(Exhasut NO( at 0%O2) = 211 ppm) 0 ppm

550 ppm

Figure 16. Predicted NO mole fraction distribution (λ = 2) [13].

150

Hirotatsu Watanabe, Yohsuke Matsushita, Hideyuki Aoki et al.

4. Conclusion The present paper describes the basics of spray combustion simulation and numerical investigation of spray combustion for low NOx emission. A spray combustion simulation was carried out in the jet-mixing type combustor. The validity of the numerical model was investigated. Then, the effect of diameter of air inlet nozzle on combustion behavior and NO emission was numerically discussed, In addition, the effect of a baffle plate on the combustion behavior was also investigated numerically. When the diameter of air inlet nozzle decreased from 8 to 4 mm, calculated NO concentration in exhaust gas was drastically decreased by about 80%. The increase in the inlet velocity causes the improvement of mixing of spray and air, and hence high temperature region where thermal NO is formed become narrow. Consequently, numerical results indicated that the mixing process of the fuel and the air had an important effect on low NOx combustion. It is noted that the exhaust NO mole fraction correlate with the residence time in high temperature region. When using a baffle plate in the combustor, the distance between the injector and the baffle plate is an important parameter. In this case, NO concentration decreased by about 40% using preferable baffle plate.

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Nomenclature A C Cμ, Ck, Cε1, Cε2 d E I k Le lb mi n p q R Re Ri r Sij Sφ Sdφ T t U, V, W w

pre-exponential factor of rate constant concentration constants in k-ε two equation model diameter activation energy radiative intensity turbulent kinetic energy length scale of turbulent eddy the distance between the atomizer and the baffle plate mass fraction for chemical species i the number of air inlet nozzles pressure the number of total computational grids universal gas constant Reynolds number chemical reaction rate stoichiometric oxygen weight required to burn 1kg of fuel strain rate source term for φ source term evaluated using a PSI-Cell model temperature time velocity component weight function

[-] [molm-3] [-] [m] [Jmol-1] [Wm-2sr] [m2s-1] [m] [m] [-] [-] [Pa] [-] [m] [-] [molm-3s-1] [-] [s-1]

[K] [s] [ms-1] [-]

Spray Combustion Simulation for Low-NOx Emissions x, r, θ

151

cylindrical coordinate

Greek Symbols Γφ Δ ε κ λ μ μ, ζ, η ρ φ Ω

diffusion coefficient for φ filter width eddy dissipation rate absorption coefficient air ratio viscosity direction cosine density dependent variable direction of radiation

[m] [m2s-3] [m-1] [-] [Pa・s] [-] [kgm-3] [sr]

Subscripts arr eddy eff fu g h ox t

Arrhenius eddy diffusion effective fuel gas high temperature region oxidant turbulence

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Superscripts m, m’ + -

direction number of radiation negative direction positive direction space average

References [1] Arai, M., Hiroyasu, H. Nakamori, K. and Nakaso, S. Nonluminous spray combustion in a jet-mixing-type combustor. Transactions of the Japan Society of Mechanical Engineers (B) (in Japanese), 56(530), 332-338 (1990)

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[2] Furuhata, T., Amano, K., Yotoriyama, K. and Arai M. Development of Can-type Low NOx Combustor for Micro Gas Turbine (Fundamental Characteristics in a Primary Combustion Zone with Upward Swirl), Fuel, 86, 2463-2474 (2007) [3] Grandinger T. B., Inauen, A., Bombach, R., Kappeli, B., Hubschmid, W. and Boulouchous, K. Liquid-Fuel/Air Premixing in Gas Turbine Combustors: Experiment and Numerical Simulation. Combustion and Flame, 124, 422-443 (2001) [4] Abu-Zaid, M. Performance of Single Cylinder, Injection Diesel Engine Using Water Fuel Emulsions. Energy Conversion and Management, 45, 697-705 (2004) [5] Lin C.Y. and Wang K. H. Diesel Engine Performance and Emission Characteristics Using Three-phase Emulsions as Fuel. Fuel, 83, 537-545 (2004) [6] Tsue, T, Kadota, T., and Segawa, D. Statistical Analysis Onset of Microexplosion for an Emulsion Droplet. Proceedings of the Combustion Institute, 26, 1629-1635 (1996) [7] Mikami, M, Yagi T. and Kojima, N. Occurrence Probability of Microexplosion in Droplet Combustion of Miscible Binary Fuels, Proceeding of the Combustion Institute, 27, 1933-1941 (1998) [8] Harada, T., Watanabe, H., Matsushita, Y., Tanno, S., Aoki, H. and Miura, T. The effect of Water/n-dodecane Emulsified Fuel Droplet Temperature and Initial Diameter on Secondary Atomization. Kagaku Kogaku Ronbunshu (in Japanese), 34, 161-167 (2008) [9] di Mare, F., Jones, W. P. and Menzies, K. R. Large eddy simulation of a model gas turbine combustor. Combustion and Flame, 137, 278-294 (2004) [10] Sharma, N. Y. and Som, S. K. Influence of fuel volatility and spray parameters on combustion characteristics and NOx emission in a gas turbine combustor. Applied Thermal Engineering, 24, 885-903 (2004) [11] Aoki, H., Tanno, S., Miura, T. and Ohnishi, S. Three dimensional spray combustion simulation in a practical boiler. JSME Int. J. series B, 35, 428-434 (1992) [12] Furuhata, T., Tanno, S., Miura, T., Ikeda, Y. and Nakajima, T. Performance of numerical spray combustion simulation. Energy Conversion and Management, 38, 11111122 (1997) [13] Watanabe, H., Suwa, Y., Matsushita, Y., Morozumi, Y., Aoki, H,, Tanno, S. and Miura, T. Spray Combustion Simulation Including Soot and NO Formation. Energy Conversion and Management, 48, 2077-2089 (2007), with permission from Elsevier [14] Watanabe, H., Suwa, Y., Matsushita, Y., Morozumi, Y., Aoki, H., Tanno, S., Miura, T. Numerical investigation of spray combustion in jet mixing type combustor for low NOx emission. Energy Conversion and Management, 49, 1530–1537 (2008), with permission from Elsevier [15] Launder, B. E. The prediction of laminarization with a two-equation model of turbulence. Int. J. Heat and Mass Transfer, 15, 301-314 (1972) [16] Furuhata, T, Tanno, S, Miura, T. Prediction of NO concentration in a liquid-fueled gas turbine combustor. In:Chan S. H. (editor). Transport phenomena in combustion. Taylor & Francis, 2, 1271-1282 (1996) [17] Launder, B. E. Progress in the development of a Reynolds-stress turbulence closure. J. Fluid Mech., 63, 537-566 (1975) [18] Carvalho, M. G. and Farias, T. L. Modeling of heat transfer in radiating and combusting systems. Trans. Inst. Chem. Eng., A 76, 175-184 (1998) [19] Germano, M., Piomelli, U., Moin, P. and Cabot, W. H. A dynamic subgrid-scale eddy viscosity model. Phys Fluids, A3, 1760-1765 (1991)

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[20] Meneveau, C., Lund, T. S., and Cabot, W. H. A Lagrangian Dynamic Subgrid-Scale Model of Turbulence, J Fluid Mech., 319, 353-385 (1996) [21] Pope, S. B. PDF methods for turbulent reactive flows. Progress in Energy and Combustion Science, 11, 119-192 (1985) [22] Magnussen, B. F. and Hjertager, B. H. On mathematiacl modeling of turbulent combustion with special emphasis on soot formation and combustion. 16th Symp. (Int.) on Combustion, 719-729 (1976) [23] Gran, I. R. and Magnussen, B. F. A numerical study of a bluff-body stabilized diffusion flame. part 2. influence of combustion modeling and finite rate chemistry. Combust. Sci. and Tech., 119, 191-217 (1996) [24] Viskanta, R. and Mengüç, M. P. Radiative heat transfer in combustion systems. Prog. Energy Combust. Sci., 13, 97–160 (1987) [25] Siddall, R. G. and Selçuk, N. Evaluation of a new six-flux model for radiative transfer in rectangular enclosures. Trans. Inst. Chem. Eng., 57, 163–169 (1979) [26] Fiveland, W. A. Discrete-ordinates solutions of the radiative transport equation for rectangular enclosures. Trans. ASME J. Heat Transfer, 106, 699–706 (1984) [27] Thynell, S. T. Discrete-ordinates method in radiative heat transfer. Int. J. Eng. Sci. 36, 1651–1675 (1998) [28] Mengüç, M. P. Viskanta, R. Radiative transfer in axisymmetric, finite cylindrical enclosures. Trans. ASME J. Heat Transfer, 108, 271–276 (1986) [29] Lockwood, F. G. and Shah, M. G. A new radiation solution method for incorporation in general combustion prediction procedures. 18th Symp (Int) on Combustion, 1405–1414 (1981) [30] Raithby, G. D. and Chui, E. H. A finite-volume method for predicting a radiant heat transfer in enclosures with participating media. Trans. ASME J. Heat Transfer, 112, 415–423 (1990) [31] Hottel, H. C. and Cohen, E. C. Radiant heat exchange in a gas-filled enclosure: Allowance for nonuniformity of gas temperature. AIChE J., 4, 3–14 (1958) [32] Hayasaka, H., Kudo, K., Taniguchi, H., Nakamachi, I., Omori, T., Katayama T. Radiative heat transfer analysis by the radiative heat ray method (Analysis in a twodimensional model). Heat Transfer Jpn Res, 17, 35–47 (1988) [33] Maruyama, S. and Aihara, T. Radiation heat transfer of arbitrary three dimensional absorbing, emitting and scattering media and specular and diffuse surfaces. Trans ASME J. Heat Transfer, 119, 129–136 (1997) [34] Gosman, A. D. and Lockwood, F. C. Incorporation of a flux model for radiation into a finite-difference procedure for furnace calculations. 14th Symp (Int) on Combustion, 661–671 (1973) [35] Crowe C. T., Sharma M. P. and Stock D. E. The particle-source in cell (PSI-Cell) model for gas-droplet flows. Transaction of ASME Journal of Fluids Engineering, 99, 325-332 (1977) [36] Mostafa A. A. and Mongia H. C. On the interaction of particles and turbulent fluid flow. Int. J. Heat Mass Transfer, 31, 2063–2075 (1988) [37] Shuen J. S., Solomon, A. S. P., Zhang, Q. F. and Faeth, G. M. Structure of particle-laden jets: Measurements and predictions, AIAA J., 23, 396–404 (1985) [38] Berlemont, A., Desjonqueres, P. and Gouesbet, G. Particle Lagrangian simulation in turbulent flow. Int. J. Multiphase Flow, 16, 19–34 (1990)

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[39] Zeldvich, Y. B. (1946). The oxidation of nitrogen in combustion and explosions. J. Acta Physicochim, 21, 577 (1946) [40] Haynes, B. S., Iversch, D. and Kirov, N. Y., The behavior of nitrogen species in fuel rich hydrocarbon flames, 16th Symp. (Int.) on Combustion, 1103-1112 (1976) [41] Smith, P. J. Hill, S. C. and Smoot, L. D., Theory for NO formation in turbulent coal flames, 19th Symp. (Int.) on Combustion, 1103-1112 (1982) [42] Saario, A. Rebola, A., Coelho, P. J., Costa, M. and Oksanen, A. Heavy fuel oil combustion in a cylindrical laboratory furnace: measurements and modeling, Fuel, 84, 359-369 (2005) [43] Fenimore, C. P. Formation of nitric oxide in premixed hydrocarbon flames. 13th Symp. (Int.) on Combustion, 373-380 (1971) [44] Takagi, T., Ogasawara, M., Daizo, M. and Tatsumi, T. NOx formation from nitrogen in fuel and air during turbulent diffusion combustion. 16th Symp. (Int.) on Combustion, 181-187 (1976) [45] Meunier, Ph., Costa, M. and Carvalho, M. G. Fuel, 77, 1705-1714 (1998) [46] De soete. Overall reaction rates of NO and N2 formation from fuel nitrogen. 15th Symp. (Int.) on Combustion, 1093-1102 (1974) [47] Hampartsoumian E., Nimmo, W., Pourkashanian M. and Williams, A. Combust. Sci. and Tech., 93, 153-172 (1993) [48] Li, Z. Q., Wei, F. and Jin, Y., Numerical simulation of pulverized coal combustion and NO formation, Chemical Engineering Science, 58, 5161-5171 (2003) [49] Boardman, R. D., Eatough, C. N., Germane, G. J. and Smoot, L. D., Comparison of Measurements and Predictions of Flame Structure and Thermal NOx, in a Swirling, Natural Gas Diffusion Flame, Combust. Sci. and Tech., 93, 193-210 (1993) [50] Beer J. M., Heat Transfer in Flames. John Wiley & Sons, 1974 [51] Faeth, G. M. Evaporation and Combustion of Sprays. Progress Energy Combustion and Science, 9, 1-76 (1983)

In: Energy Conservation: New Research Editor: Giacomo Spadoni, pp. 155-197

ISBN 978-1-60692-231-6 c 2009 Nova Science Publishers, Inc.

Chapter 7

M ASS AND E NERGY C ONSERVING F ULLY D ISCRETE S CHEMES FOR THE S HALLOW-WATER E QUATIONS Yuri N. Skiba1,∗ and Denis M. Filatov2,† 1 Centro de Ciencas de la Atm´osfera (CCA), Universidad Nacional Aut´onoma de M´exico (UNAM), Cd. Universitaria, C.P. 04510, M´exico D. F., M´exico 2 Centro de Investigaci´on en Computaci´on (CIC), Instituto Polit´ecnico Nacional (IPN), C. P. 07738, M´exico D. F., M´exico

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Abstract A new method for constructing finite difference schemes for the shallow-water model (SWM) is suggested. The model equations can be considered in a limited area, in a doubly periodic domain, in a periodic channel on the plane, on a whole sphere, and in a periodic channel in the longitudinal direction on the sphere. An essential advantage of the method is that it produces fully discrete (both in time and in space) shallow-water schemes that exactly conserve the mass and the total energy and whose numerical implementation is computationally inexpensive. Our approach is based on splitting of the SWM operator by coordinates and by physical processes. Thereby the solution of the original system of 2D partial differential equations reduces to the solution of three simple problems containing either 1D partial differential equations or ordinary differential equations. In fact, an infinite family of such conservative schemes is proposed, which are either linear or nonlinear depending on the choice of certain scheme parameters. On a doubly periodic manifold, the method allows constructing conservative finite difference schemes of arbitrary approximation orders in the spatial variables. Moreover, if the SWM is considered on the entire sphere (which is not a doubly periodic manifold) then the method makes it possible to use the same numerical schemes (of arbitrary approximation order in space) and algorithms as for a doubly periodic manifold. The numerical SWM algorithms are computationally cheap, because each scheme is easily realised by fast direct methods of linear algebra. The skill of the finite difference schemes is illustrated by numerical results. ∗ †

E-mail address: [email protected] E-mail address: [email protected]

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Yuri N. Skiba and Denis M. Filatov

Keywords: Shallow-water equations, splitting method, conservative finite difference schemes, nonlinear interaction, modelling of solitary waves.

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1.

Introduction

It is known that a solution to the shallow-water model (SWM) for an ideal and unforced fluid satisfies various local and global (integral) conservation laws [18]. Among the integral invariants of motion the most important ones are the mass, the total (kinetic plus potential) energy and the potential enstrophy (see Section 3). However, in a fully discrete (i.e. discrete both in space and in time) SWM the total energy and the potential enstrophy usually stop being invariant [33]. This leads to such an undesired effect as the distribution of energy over a spectrum of movements of different scales, which can produce the nonlinear instability [17, 28]. As a consequence, the numerical results, especially at long-term calculations, can be far from the exact ones in some norms. In order to reduce the approximation and instability errors, conservative finite difference schemes have to be used [23, 24]. In the last forty years there have been suggested several methods of discretising the SWM in space. The resulting semidiscrete models (discrete in space but still continuous in time) conserve some or other integral invariants of motion of the SWM equations [1–4, 7, 9, 13, 16, 20–22, 32, 34]. However, any explicit discretisation in time of such semidiscrete models leads to fully discrete schemes which conserve the mass but not the total energy and the potential enstrophy. If the Crank-Nicolson scheme is then used to conserve the total energy [5], the resulting methods turn out to be computationally time-consuming. For a doubly periodic domain and a periodic channel on the plane a family of fully discrete linear and nonlinear SWM schemes conserving the mass and the total energy was suggested in [25]. For shallow-water flows on a sphere fully discrete schemes were constructed in [26, 27, 29, 30]. In this work we describe in detail a new method for the numerical simulation of shallowwater flows. The method permits to conserve the mass and the total energy of a fully discrete shallow-water system, is simple and fast in realisation, and applicable both in the Cartesian and spherical geometries. Our approach is based on splitting of the SWM operator by coordinates and by physical processes [14, 25, 37]. Thereby the solution of the original system of 2D partial differential equations reduces to the solution of three simple split problems containing either 1D partial differential equations or ordinary differential equations. Due to specially chosen spatial approximations each split system conserves the mass and the total energy. In fact, an infinite family of such conservative schemes is proposed, which are either linear or nonlinear depending on the choice of certain scheme parameters. On a doubly periodic manifold the method allows constructing finite difference schemes of arbitrary approximation orders in space. Moreover, if the SWM is considered on the entire sphere (which is, evidently, not a doubly periodic domain) then the method makes it possible to use the same numerical schemes and algorithms as for a doubly periodic region. Thus the schemes of arbitrary approximation order in the spatial variables can be used for the whole sphere as well. The numerical SWM algorithms are computationally cheap, because each scheme can easily be realised by fast direct methods of linear algebra. The material is organised as follows. In Section 2 we, basing on the theory of shallowwater flows, derive the shallow-water equations (SWE) in the Cartesian geometry. In Sec-

Mass and Energy Conserving Fully Discrete Schemes...

157

tion 3 we consider the principal local and global conservation laws underlying the SWM. In Section 4 we introduce a divergent form of the SWE and split the original two-dimensional system in time and in space. In Section 5 we show how the previously obtained divergent form can be used for constructing a second-order conservative finite difference scheme of the SWM. In Section 6 we present conservative SWE finite difference schemes of the first four orders in space, as well as give a general formula for deriving arbitrary-order mass and total energy conservative SWM schemes. Section 7 is devoted to the shallow-water equations on a sphere. There we demonstrate how the method of splitting can efficiently be used for adapting the developed Cartesian finite difference schemes to the spherical geometry. In Section 8 we test the developed approach numerically considering three different shallow-water problems. Conclusions are given in Section 9.

2.

Shallow-Water Equations for a Rotating Unforced Ideal Fluid

Let us consider a rotating unforced ideal fluid with a constant density (ρ = Const) in the gravitational field. Let h(x, y, t) be the height of the free surface of the fluid (Fig. 1). Suppose that the fluid is rotating around the z-axis with an angular velocity Ω, that is ~ Ω = Ω~k, where ~k = (0, 0, 1) is the unit vector parallel to the z-axis, f = 2Ω is the Coriolis parameter, g is the gravitational acceleration, hT (x, y) is the bottom topography, and H(x, y, t) = h(x, y, t) − hT (x, y)

(1)

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is the total depth of the fluid at a point (x, y) at a time moment t.

Figure 1. Shallow-water model. Let us denote by D the mean depth of the fluid layer, by L the characteristic scale of horizontal motions, and by U and W the characteristic values of the horizontal and vertical velocities, respectively. The fundamental SWE parameter is [18] δ=

D ≪ 1. L

158

Yuri N. Skiba and Denis M. Filatov From the continuity equation ∇ · ~u ≡

∂u ∂v ∂w + + =0 ∂x ∂y ∂z

(2)

∂u ∂v we have UL ∼ W D , and hence W ∼ δU . However, in some situations the terms ∂x and ∂y ∂v δU 2 compensate each other and ∂u ∂x + ∂y ∼ L . Thus W ∼ δ U , and in the Euler equations the terms containing the vertical velocity component w can be ignored [18], and the hydrostatic approximation can be used:

∂u ∂u ∂u 1 ∂p +u +v − fv = − , ∂t ∂x ∂y ρ ∂x ∂v ∂v ∂v 1 ∂p +u +v + fu = − , ∂t ∂x ∂y ρ ∂y ∂p . −ρg = ∂z

(3) (4) (5)

The condition p(x, y, z)|z=h = p0 = Const at the free surface implies that p(x, y, z, t) = ρg (h(x, y, t) − z) + p0 .

(6)

Thus, at the point (x, y, z) the pressure p exceeds the value p0 by the weight of the unit column of the fluid situated above the point. We should note that the pressure derivatives ∂p ∂h = ρg , ∂x ∂x

∂p ∂h = ρg ∂y ∂y

(7)

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are independent of z, and from (3)-(4) it follows that the velocities are horizontal (independent of z) if they are horizontal at the initial time moment. This result is known as the Taylor-Proudman theorem [18]. Suppose now that the conditions of the Taylor-Proudman theorem are fulfilled. Then equations (3)-(4) take the form ∂u ∂u ∂u ∂h +u +v − f v = −g , ∂t ∂x ∂y ∂x ∂v ∂v ∂h ∂v +u +v + f u = −g . ∂t ∂x ∂y ∂y

(8) (9)

Using the independence of u and v from z and integrating the continuity equation (2) in z, we obtain   ∂u ∂v + + w(x, ˜ y, t). (10) w(x, y, z, t) = −z ∂x ∂y Since the bottom condition at z = hT requires w(x, y, hT , t) ≡

∂hT ∂hT dhT =u +v , dt dx dy

Mass and Energy Conserving Fully Discrete Schemes...

159

we get ∂hT ∂hT +v + hT w(x, ˜ y, t) = u dx dy



∂u ∂v + ∂x ∂y



,

and hence, equation (10) is reduced to w(x, y, z, t) = (hT − z)



∂u ∂v + ∂x ∂y



+u

∂hT ∂hT +v . dx dy

(11)

At the free surface z = h(x, y, t) the corresponding kinematic condition is w(x, y, h, t) ≡

dh ∂h ∂h ∂h = +u +v . dt ∂t ∂x ∂y

(12)

The combination of (11) and (12) results in ∂ ∂ ∂h + [(h − hT ) u] + [(h − hT ) v] = 0. ∂t ∂x ∂y

(13)

In terms of the total depth (1) the equation of mass conservation (13) is written as ∂h ∂(uH) ∂(vH) + + = 0, ∂t ∂x ∂y

(14)



(15)

or as dH = −H dt

∂u ∂v + ∂x ∂y



.

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Thus, the shallow-water model consists of equations (8)-(9) and equation (14) or (15). Therefore, the condition δ = D L ≪ 1 has permitted to reduce the number of independent variables and the number of the Euler equations by eliminating the variable z and the vertical velocity component w.

3.

Local and Global Conservation Laws of the SWM

It is well known that the shallow-water model as a closed physical system (without sources and sinks of energy) possesses various conservation laws, both local (when a physical characteristic is conserved at each point of the fluid) and global (when an integral physical characteristic is conserved) ones. In particular, it conserves such important characteristics of motion as the potential vorticity in every fluid particle (a local law) and the mass, total energy and potential enstrophy (global laws) [33]. Each conservation law establishes a specific limitation on the SWM solution, or, in other words, on the motion of the fluid. In this section we shall briefly examine the basic local and global conservation laws for the SWM.

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3.1.

Yuri N. Skiba and Denis M. Filatov

Conservation of the Mass in a Thin Fluid Column

Let us apply (14) to a vertical thin fluid column localised at a point (x, y). If the horizontal divergence ∇ · (~uH) ≡

∂(uH) ∂(vH) + ∂x ∂y

at this point is positive (or negative) then the expansion (compression) of the column must be balanced by the decrease (increase) of the column height (Fig. 2). Because the area A of the transversal section of the column is changed according to 1 dA ∂u ∂v = + , A dt ∂x ∂y

(16)

1 dA 1 dH + = 0, H dt A dt

(17)

d (HA) = 0. dt

(18)

one can obtain that

or

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In other words, the total volume HA (and hence, the mass) of the fluid column is conserved.

Figure 2. Conservation of the mass in a thin fluid column.

3.2.

Conservation of the Relative Vertical Position of a Fluid Particle

Again consider a vertical column of the fluid. Using (15), we eliminate the divergence of velocity in (13) and obtain w=

z − hT dH ∂hT ∂hT dz = +u +v , dt H dt ∂x ∂y

(19)

Mass and Energy Conserving Fully Discrete Schemes...

161

or z − hT dH d(z − hT ) = . dt H dt The last equation implies the conservation of the relative vertical position particle:   d z − hT = 0. dt H Note that 0 ≤

3.3.

z−hT H

z−hT H

of each

(20)

≤ 1.

Conservation of the Potential Vorticity in a Thin Fluid Column

For the components of the relative vorticity vector ω ~ ≡ rot ~u = (ωx , ωy , ωz ) we have the estimates ∂w ∂v ∂w W U − = ∼ ∼δ , ∂y ∂z ∂y L L ∂u ∂w ∂w W U ωy = − =− ∼ ∼δ , ∂z ∂x ∂x L L ∂v ∂u U ωz = − ∼ . ∂x ∂y L ωx =

(21)

From (21) it follows that the horizontal components of the vorticity ω ~ are insignificant in the SWM, and ω ~ has, primarily, the vertical direction: ω ~ = (0, 0, ωz ). Differentiating (8) and (9) with respect to y and x, respectively, and subtracting the results, we obtain the vorticity equation for ζ ≡ ωz :   ∂ζ ∂ζ ∂ζ ∂u ∂v dζ ≡ +u +v = −(ζ + f ) + . (22) dt ∂t ∂x ∂y ∂x ∂y

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Using (15), equation (22) can be written as dζ ζ + f dH = . dt H dt

(23)

Thus, if dH dt > 0, that is the fluid column is compressing and rising (Fig. 2), then the intensification of the relative vorticity ζ is proportional to the product of the growth rate ζ+f dH dt and the potential vorticity q ≡ H . Since f is a constant, statement (23) yields the conservation of the potential vorticity   d ζ +f ∂q ∂q ∂q dq ≡ ≡ +u +v = 0. (24) dt dt H ∂t ∂x ∂y Hence, the relative vorticity ζ in the SWM must increase (or decrease) simultaneously with the total depth H in order to keep the potential vorticity q invariant. In other words, the relative vorticity is produced due to stretching of a fluid column. Besides, if initially the

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Yuri N. Skiba and Denis M. Filatov

relative vorticity of a fluid column is zero (ζ = 0) then it can remain zero only if the total depth H of the column does not change. In addition to the local conservation laws, shallow-water flows have various integral invariants of motion — global conservation laws. Consider the SWE (8)-(9), (14) in a two-dimensional domain D that can be one of the following regions: 1. A doubly periodic domain D = {(x, y) : 0 < x < X, 0 < y < Y } with the boundary conditions {u, v, h, H}x=X = {u, v, h, H}x=0 ,

{u, v, h, H}y=Y = {u, v, h, H}y=0 ; (25)

2. A channel D = {(x, y) : 0 < x < X, 0 < y < Y } with the periodic conditions in the x-direction {u, v, h, H}x=X = {u, v, h, H}x=0

(26)

v(x, 0, t) = v(x, Y, t) = 0

(27)

and the slip condition

at the lateral boundaries of the channel; 3. An arbitrary region with the slip condition at its boundary S un |S = 0,

(28)

where ~n is the unit outward normal to S, and un is the normal component of the velocity ~u = (u, v).

3.4.

Conservation of the Total Mass

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If we integrate equation (14) over the domain D and use the boundary conditions (25), (26)-(27) or (28) then we obtain the conservation of the total mass Z ∂ hdD = 0. (29) ∂t D Indeed, for a doubly periodic domain (case No. 1 above) the integral  Z  ∂(uH) ∂(vH) + dD ∂x ∂y D

(30)

is equal to zero due to condition (25):  Z X Z  Z Y ∂(uH) ∂(vH) X (vH)Yy=0 dx = 0. (uH)x=0 dy + + dD = ∂x ∂y 0 D 0 In the third case (condition (28)) the integral (30) is equal to zero due to the Gauss theorem [11]:  Z  Z Z Z ∂(uH) ∂(vH) ∇ · (~uH)dD = (~uH) · ~ndS = un HdS = 0. + dD = ∂x ∂y D D S S The same result can be obtained for the mixed conditions (26)-(27) (case No. 2).

Mass and Energy Conserving Fully Discrete Schemes...

3.5.

163

Conservation of the Potential Enstrophy

Another important characteristic of the SWM solution related with the potential vorticity of the fluid particles is the potential enstrophy   Z Z 1 ζ +f 2 1 J≡ H dD = Hq 2 dD. (31) 2 D H 2 D The potential enstrophy is a measure of rotation of the fluid particles. We shall now demonstrate that the potential enstrophy is an invariant of the shallow-water motion, i.e. dJ = 0. dt

(32)

Indeed, note that due to (1) equation (14) can be written as ∂H ∂(uH) ∂(vH) + + = 0. ∂t ∂x ∂y Multiplying (33) by

q2 2 ,

(33)

as well as (24) by Hq, we obtain q 2 ∂H q 2 ∂(uH) q 2 ∂(vH) + + =0 2 ∂t 2 ∂x 2 ∂y

(34)

and Hq

∂q ∂q ∂q + uHq + vHq = 0. ∂t ∂x ∂y

Rewriting the last equation as       ∂ q2 ∂ q2 ∂ q2 H + uH + vH =0 ∂t 2 ∂x 2 ∂y 2 and summing it with (34), we get ∂Π ∂Π ∂(uΠ) ∂(vΠ) + + = + ∇ · (~uΠ) = 0, ∂t ∂x ∂y ∂t

(35)

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2

where Π = H q2 . Integrating (35) over D and using the Gauss theorem, we obtain (32).

3.6.

Conservation of the Total Energy

Multiply equation (8) by uH, equation (9) by vH and equation (14) by gh. Then, summing the three resulting equations, we obtain   2 

u + v2 1∂  2 2 2 2 u + v H + g h − hT + ∇ · ~uH + gh = 0. (36) 2 ∂t 2 Integrating (36) over D and taking into account that due to the Gauss theorem   2  Z u + v2 ∇ · ~uH + gh dD = 0, 2 D

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we obtain the total energy conservation law Z  2

 1 ∂ u + v 2 H + g h2 − h2T dD = 0. 2 ∂t D

(37)

Here the first summand stands for the kinetic energy Z
1 u2 + v 2 HdD, EK ≡ 2 D while the second summand represents the potential energy [29] Z
1 g h2 − h2T dD. EP ≡ 2 D

4.

Divergent Form and Splitting of the SWM Equations

For constructing a fully discrete SWM (that is discrete both in time and in space) and finite difference schemes that exactly conserve the mass and the total energy we need to write the original SWM in a divergent form. To this end, we introduce new variables [24, 25] √ (38) z ≡ H, U ≡ uz, V ≡ vz and rewrite the SWE (8)-(9), (14) in the divergent form     ∂U 1 ∂(uU ) ∂U 1 ∂(vU ) ∂U ∂h + +u + +v −f V = −gz , ∂t 2 ∂x ∂x 2 ∂y ∂y ∂x | {z }

(39)

divergent terms

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∂V 1 + ∂t 2 |



∂V ∂(uV ) +u ∂x ∂x



1 + 2 {z



∂V ∂(vV ) +v ∂y ∂y



+f U = −gz

∂h , ∂y

(40)

}

divergent terms

∂h ∂(zU ) ∂(zV ) + + = 0. ∂t ∂x ∂y | {z }

(41)

divergent terms

The validity of (41) can be shown in a trivial way, so we shall only clarify the derivation of equation (39), since the validity of (40) can be established in the same manner. Multiplying (8) by z, we get   ∂z ∂u ∂u ∂h ∂U + −u +U +V − f V = −gz . (42) ∂t ∂t ∂x ∂y ∂x Thus, (39) is valid if it holds −u

∂z ∂u ∂u 1 +U +V = ∂t ∂x ∂y 2



∂(uU ) ∂U +u ∂x ∂x



+

1 2



∂(vU ) ∂U +v ∂y ∂y



.

(43)

Mass and Energy Conserving Fully Discrete Schemes...

165

∂z ∂t

using the mass conservation equation (14) as follows   ∂z 1 ∂z 2 1 ∂H 1 ∂h 1 ∂U z ∂V z = = = =− + ∂t 2z ∂t 2z ∂t 2z ∂t 2z ∂x ∂y     1 ∂U 1 ∂z ∂V ∂z =− − u . + +v 2 ∂x ∂y 2 ∂x ∂y

To prove (43), we calculate

Multiplying the last equation by u, we obtain   ∂z 1 ∂U ∂V 1 ∂z 1 ∂z −u = u +u + u2 + uv . ∂t 2 ∂x ∂y 2 ∂x 2 ∂y | {z } | {z } | {z } B

A

(44)

C

The terms A, B and C from (44) can be written as follows       1 ∂U ∂V 1 ∂(uU ) ∂(uV ) 1 ∂u ∂u A≡ u +u = + − U +V 2 ∂x ∂y 2 ∂x ∂y 2 ∂x ∂y     ∂u 1 ∂(uU ) ∂(vU ) 1 ∂u + +V = − U , 2 ∂x ∂y 2 ∂x ∂y   1 ∂z 1 ∂u 1 ∂z 1 ∂U = u u − U , B ≡ u2 = u 2 ∂x 2 ∂x 2 ∂x 2 ∂x   1 ∂z 1 ∂z 1 ∂u 1 ∂U C ≡ uv = v u − V . = v 2 ∂y 2 ∂y 2 ∂y 2 ∂y Substitution of these expressions into (44) yields     ∂z 1 ∂(uU ) ∂U ∂U ∂u 1 ∂(vU ) ∂u −u =A+B+C = +u +v −V . + −U ∂t 2 ∂x ∂x 2 ∂y ∂y ∂x ∂y This proves statement (43). The SWM can be written in the operator form

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~ ∂ψ ~ = 0, + A(ψ) ∂t

(45)

~ is the SWM nonlinear operator, ψ ~ ≡ (ψ1 , ψ2 , ψ3 ) = (U, V, h√g) is the soluwhere A(ψ) tion. It is easy to see that the operator of system (39)-(41) is antisymmetric, i.e. D E ~ ψ ~ = 0 ∀ψ ~ 6= 0 A(ψ), (46) in the inner product defined as D

3 E Z X ~ ~ φi ψi dD. φ, ψ = D i=1

Indeed, due to (45) D

E

~ ψ ~ =− A(ψ),

*

~ ∂ψ ~ ,ψ ∂t

+

1∂

~ 2 =−

ψ , 2 ∂t

(47)

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Yuri N. Skiba and Denis M. Filatov

and hence, defining the L2 -norm of the solution as 1/2

D E1/2 Z 1


~ 2 2 2 ~ ~ U + V + gh dD , =

ψ ≡ ψ, ψ D 2

(48)

~ = Const in time [25]. For this we multiply equations (39)it is sufficient to show that kψk (41) by U , V and gh, respectively, and integrate the sum of the resulting equations over D. Then, using the Gauss theorem, we immediately obtain Z


1∂

~ 2 1 ∂ U 2 + V 2 + gh2 dD

ψ ≡ 2 ∂t 2 ∂t D    2 Z U +V2 + gHh dD = 0. (49) =− ∇ · ~u 2 D Since the bottom topography hT (x, y) is time-independent, square of the solution’s norm ~ 2 ) coincides with the total energy of the SWM (cf. (37) and (48)). It should (i.e. kψk be noted that unlike the total energy, square of the potential enstrophy (31) is not even a seminorm of the potential vorticity q, and all the more not a seminorm of the SWM solution ~ ψ. Divergent forms of hydrodynamic closed systems play a key role in the development of conservative finite difference schemes [23, 24]. This property will be used in the next section in order to construct finite difference schemes which conserve the mass and the total energy for the SWM. ~ as the sum of three simpler operators Now represent the operator A(ψ) ~ = A1 (ψ) ~ + A2 (ψ) ~ + A3 ψ, ~ A(ψ)

(50)

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and let [tn , tn+1 ] be a sufficiently small time interval (tn+1 = tn +τ ). Here the operators A1 and A2 are nonlinear, whereas the operator A3 is linear (see systems (52)-(54) below). We apply the splitting method to approximate the SWM (45) in [tn , tn+1 ] by the three simple problems ~1 ∂ψ ~1 ) = 0, + A1 (ψ ∂t ~2 ∂ψ ~2 ) = 0, + A2 (ψ ∂t ~3 ∂ψ ~3 = 0, + A3 ψ ∂t

(51)

~ n ) from the previous which are solved one after another, and besides the solution to (45) ψ(t ~1 (tn ) = ψ(t ~ n ), time interval [tn−1 , tn ] is the initial condition for the first problem in (51): ψ ~ ~ ~ ~ and further ψ2 (tn ) = ψ1 (tn+1 ) and ψ3 (tn ) = ψ2 (tn+1 ). Finally, the SWM solution at the ~3 (tn+1 ), and the process is repeated for the moment tn+1 is approximated by the solution ψ next time interval [tn+1 , tn+2 ] [15, 25, 26, 37]. Note that the first two problems in (51) are one-dimensional and nonlinear, while the third problem is linear and represents the system of two simple ordinary differential equations describing the fluid rotation. So, we have:

Mass and Energy Conserving Fully Discrete Schemes...

167

Problem 1. The first problem in (51) gets solved with respect to the variable x for all fixed y’s:   1 ∂(uU ) ∂U ∂h ∂U + +u = −gz , ∂t 2 ∂x ∂x ∂x   1 ∂(uV ) ∂V ∂V + +u = 0, (52) ∂t 2 ∂x ∂x ∂h ∂(zU ) + = 0. ∂t ∂x Problem 2. The second problem in (51) gets solved with respect to the variable y for all fixed x’s:   ∂U 1 ∂(vU ) ∂U + +v = 0, ∂t 2 ∂y ∂y   ∂V 1 ∂(vV ) ∂V ∂h + +v = −gz , (53) ∂t 2 ∂y ∂y ∂y ∂h ∂(zV ) + = 0. ∂t ∂y Problem 3. The third problem in (51) describes the fluid rotation (Coriolis term): ∂U − f V = 0, ∂t ∂V + f U = 0. ∂t

(54)

The accuracy of approximation of the solution to (39)-(41) by the solution to (52)-(54) can be improved with the decrease of the mesh size.

5.

A Conservative Finite Difference Scheme for the SWM

Now we shall construct conservative finite difference schemes of the second approximation order for systems (52)-(54) [25]. Let us introduce the following notations: τ = tn+1 − tn ,

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Rn +Rn+1

n = R(x , y , t ), R = kl kl ∆x = xk+1 − xk , ∆y = yl+1 − yl , Rkl , where R reprek l n kl 2 sents any of the functions u, v, z, U , V , H and h. Then system (52) can be approximated as   Ukn+1 − Ukn 1 uk+1 Uk+1 − uk−1 Uk−1 Uk+1 − Uk−1 + + uk τ 2 2∆x 2∆x hk+1 − hk−1 = −gz k , (55) 2∆x   Vkn+1 − Vkn 1 uk+1 Vk+1 − uk−1 Vk−1 Vk+1 − Vk−1 + + uk = 0, (56) τ 2 2∆x 2∆x

Hkn+1 − Hkn z k+1 Uk+1 − z k−1 Uk−1 + = 0. τ 2∆x

(57)

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Yuri N. Skiba and Denis M. Filatov

(Here the invariable subindex l is omitted in order to simplify the denotations). Besides, system (53) is discretised as Uln+1 − Uln 1 + τ 2



Vln+1 − Vln τ

 v l+1 Ul+1 − v l−1 Ul−1 Ul+1 − Ul−1 = 0, + vl 2∆y 2∆y   1 v l+1 Vl+1 − v l−1 Vl−1 Vl+1 − Vl−1 + + vl 2 2∆y 2∆y hl+1 − hl−1 = −gz l , 2∆y Hln+1 − Hln z l+1 Vl+1 − z l−1 Vl−1 + =0 τ 2∆y

(58)

(59) (60)

(similarly, the invariable subindex k is omitted). Finally, system (54) is approximated as n+1 n − Ukl Ukl − fl Vkl = 0, τ Vkln+1 − Vkln + fl Ukl = 0. τ

(61) (62)

It is easy to show that the first split system (55)-(57) possesses two conservation laws. Indeed, multiplying (57) by ∆x∆yτ , summing over all internal points of the domain, and using the boundary conditions, we obtain X X n+1 n M n+1 ≡ ∆x∆y Hkl = ∆x∆y Hkl ≡ M n, (63) k,l

k,l

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that is the mass of the split discrete system is constant in time. Also, multiplying (55) by ∆x∆yτ Uk , (56) by ∆x∆yτ Vk and (57) by ∆x∆yτ ghk , summing the resulting equations, integrating over all internal points of the domain, we get E n+1 ≡ ∆x∆y

X1

n+1 2 2 ([Ukl ] + [Vkln+1 ]2 + g[hn+1 kl ] )

= ∆x∆y

X1

n 2 ([Ukl ] + [Vkln ]2 + g[hnkl ]2 ) ≡ E n ,

k,l

k,l

2 2

(64)

and hence, the total energy is conserved in the discrete split system too. In a similar way it can be shown that the mass and the total energy are invariants of systems (58)-(60) and (61)(62). As a result, the whole scheme (55)-(62) also conserves these two important characteristics of motion [25]. Note that the mass and the total energy are conserved independently of the choice of the parameters ukl , v kl and z kl . Therefore, equations (55)-(62) represent, in fact, a family of finite difference schemes that approximate the original nonlinear systems (52)-(54). In particular, scheme (55)-(62) is linear if ukl = unkl ,

n , v kl = vkl

n ; z kl = zkl

in case of ukl = ukl , v kl = vkl and z kl = zkl the scheme turns out to be nonlinear.

(65)

Mass and Energy Conserving Fully Discrete Schemes...

169

Therefore, due to the splitting the solution of the original system of 2D partial differential equations reduces to the solution of three simple problems containing either 1D partial differential equations or ordinary differential equations. The split problems are easily resolved one after another by rapid methods of linear algebra [19,26,29]. Besides, in the case of linear schemes the use of direct methods keeps the conservation of mass and total energy during the calculations. In this connection we note that the use of the Crack-Nicolson scheme for approximating semidiscrete problems obtained in [20,22] generally requires the application of iterative methods, which will violate the conservation laws.

6.

SWM Schemes of Arbitrary Approximation Order in Space

The finite difference schemes considered in Section 5 are of the second approximation order in the spatial variables. In this section we give SWM finite difference schemes of the first four approximation orders in space, as well as suggest a general formula for constructing arbitrary-order conservative schemes for the SWM. We shall only write down finite difference schemes for Problem 1 (system (52)). Formulas for Problem 2 (system (53)) can be derived in the same way. As for Problem 3 (system (54)), that is a system of ordinary differential equations in time, and its solution is straightforward (see (61)-(62)).

6.1.

First-Order Approximation Schemes

Denote a = {u, v}, B = {U, V }. Let us discretise (52) in the form   Ukn+1 − Ukn 1 uk Uk+1 − uk−1 Uk−1 hk+1 − hk + , = −gz k τ 2 ∆x ∆x   Vkn+1 − Vkn 1 uk Vk+1 − uk−1 Vk−1 + = 0, τ 2 ∆x

(66)

Hkn+1 − Hkn z k Uk − z k−1 Uk−1 + = 0. τ ∆x

(68)

(67)

Here we used the first-order approximation

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∂uB ∂B uk Bk − uk−1 Bk−1 Bk+1 − Bk uk Bk+1 − uk−1 Bk−1 +u ≈ + uk = . ∂x ∂x ∆x ∆x ∆x Another first-order scheme can be obtained if we take ∂B uk+1 Bk+1 − uk Bk Bk − Bk−1 uk+1 Bk+1 − uk Bk−1 ∂uB +u ≈ + uk = ; ∂x ∂x ∆x ∆x ∆x then Ukn+1 − Ukn 1 + τ 2



 hk − hk−1 uk+1 Uk+1 − uk Uk−1 = −gz k , ∆x ∆x   Vkn+1 − Vkn 1 uk+1 Vk+1 − uk Vk−1 + = 0, τ 2 ∆x Hkn+1 − Hkn z k+1 Uk+1 − z k Uk + = 0. τ ∆x

(69) (70) (71)

170

6.2.

Yuri N. Skiba and Denis M. Filatov

Second-Order Approximation Scheme

If we employ the central second-order finite difference stencil then we shall have the scheme (repeated from Section 5)   Ukn+1 − Ukn 1 uk+1 Uk+1 − uk−1 Uk−1 Uk+1 − Uk−1 = + + uk τ 2 2∆x 2∆x hk+1 − hk−1 = −gz k , (72) 2∆x   Vkn+1 − Vkn 1 uk+1 Vk+1 − uk−1 Vk−1 Vk+1 − Vk−1 = 0, (73) + + uk τ 2 2∆x 2∆x Hkn+1 − Hkn z k+1 Uk+1 − z k−1 Uk−1 + = 0. τ 2∆x

6.3.

(74)

Third-Order Approximation Schemes

Increasing the approximation order to three, we can write either ∂B ∂aB +a ≈ ∂x ∂x 11ak Bk − 18ak−1 Bk−1 + 9ak−2 Bk−2 − 2ak−3 Bk−3 ≈ + 6∆x 2Bk+3 − 9Bk+2 + 18Bk+1 − 11Bk +ak = 6∆x 2Bk+3 − 9Bk+2 + 18Bk+1 = ak + 6∆x −18ak−1 Bk−1 + 9ak−2 Bk−2 − 2ak−3 Bk−3 (3) =: A−+ (a, B) + 6∆x

(75)

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or ∂aB ∂B +a ≈ ∂x ∂x 2ak+3 Bk+3 − 9ak+2 Bk+2 + 18ak+1 Bk+1 − 11ak Bk ≈ + 6∆x 11Bk − 18Bk−1 + 9Bk−2 − 2Bk−3 +ak = 6∆x 18ak+1 Bk+1 − 9ak+2 Bk+2 + 2ak+3 Bk+3 + = 6∆x −18Bk−1 + 9Bk−2 − 2Bk−3 (3) +ak =: A+− (a, B), 6∆x which will provide either with Ukn+1 − Ukn 1 (3) 2hk+3 − 9hk+2 + 18hk+1 − 11hk + A−+ (u, U ) = −gz k , τ 2 6∆x Vkn+1 − Vkn 1 (3) + A−+ (u, V ) = 0, τ 2 Hkn+1 − Hkn 11z k Uk − 18z k−1 Uk−1 + 9z k−2 Uk−2 − 2z k−3 Uk−3 + =0 τ 6∆x

(76)

(77) (78) (79)

Mass and Energy Conserving Fully Discrete Schemes...

171

or with Ukn+1 − Ukn 1 (3) 11hk − 18hk−1 + 9hk−2 − 2hk−3 + A+− (u, U ) = −gz k , τ 2 6∆x Vkn+1 − Vkn 1 (3) + A+− (u, V ) = 0, τ 2 n+1 n Hk − Hk 2z k+3 Uk+3 − 9z k+2 Uk+2 + 18z k+1 Uk+1 − 11z k Uk + = 0, τ 6∆x

(80) (81) (82)

respectively.

6.4.

Fourth-Order Approximation Scheme

With the central fourth-order stencil we shall obtain the following finite difference scheme of approximation order O(τ + ∆x4 ): Ukn+1 − Ukn 1 + τ 2

−uk+2 Uk+2 + 8uk+1 Uk+1 − 8uk−1 Uk−1 + uk−2 Uk−2 + 12∆x  −Uk+2 + 8Uk+1 − 8Uk−1 + Uk−2 +uk = 12∆x −hk+2 + 8hk+1 − 8hk−1 + hk−2 = −gz k , 12∆x  1 −uk+2 Vk+2 + 8uk+1 Vk+1 − 8uk−1 Vk−1 + uk−2 Vk−2 + + 2 12∆x  −Vk+2 + 8Vk+1 − 8Vk−1 + Vk−2 +uk = 0, 12∆x

(83)

Hkn+1 − Hkn −z k+2 Uk+2 + 8z k+1 Uk+1 − 8z k−1 Uk−1 + z k−2 Uk−2 + = 0. τ 12∆x

(85)

Vkn+1 − Vkn τ

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6.5.



(84)

Arbitrary-Order Approximation Schemes

Generally, an arbitrary p-order noncentral finite difference scheme can be obtained either with (backward-forward approximation) p p ∂B 1 X 1 X ∂aB +a ≈ ck−i (aB)k−i − ak ck−i Bk+i , ∂r ∂r ∆r ∆r i=0

z

∂h 1 ≈ −zk ∂r ∆r

i=0 p X

(86)

ck−i hk+i ,

(87)

p ∂zB 1 X ≈ ck−i (zB)k−i ∂r ∆r

(88)

i=0

i=0

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Yuri N. Skiba and Denis M. Filatov

or with (forward-backward approximation) p p ∂aB ∂B 1 X 1 X +a ≈− ck−i (aB)k+i + ak ck−i Bk−i , ∂r ∂r ∆r ∆r i=0

z

∂h 1 ≈ zk ∂r ∆r

i=0 p X

(89)

ck−i hk−i ,

(90)

p 1 X ∂zB ≈− ck−i (zB)k+i . ∂r ∆r

(91)

i=0

i=0

An arbitrary q-even-order central finite difference scheme comes from q

q

2 2 ∂aB ∂B 1 X 1 X +a ≈ cˆk+i (aB)k+i + ak cˆk+i Bk+i , ∂r ∂r ∆r ∆r q q

i=− 2

(92)

i=− 2 q

2 1 X ∂h ≈ zk cˆk+i hk+i , z ∂r ∆r q

(93)

i=− 2

q 2

∂zB 1 X ≈ cˆk+i (zB)k+i . ∂r ∆r q

(94)

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i=− 2

The coefficients {ci } and {ˆ ci } determine the corresponding finite difference stencils and can be found, e.g., in [10]. For example, for scheme (72) we have q = 2 and cˆk+1 = −ˆ ck−1 = 21 , cˆk = 0. Remark. From (86)-(94) it is seen that the corresponding finite difference schemes produce matrices with either 2p + 1 or q + 1 nonzero diagonals. Therefore, for an even q ≥ 4 and an odd p = q − 1 it holds 2p + 1 = 2q − 1 > q + 1, that is a lower-order scheme is implemented on matrices with a greater number of nonzero diagonals than a higher-order scheme is (cf., e.g., (75)-(82) vs. (83)-(85)). Nonetheless, the presented formulas (86)-(94) provide a general algorithm for straightforward constructing arbitrary-order finite difference schemes for the SWM. Obviously, finite difference approximations distinct from (86)-(94) can also be suggested. Several examples can be found in [6, 31]. One can make sure that all the presented schemes conserve the mass and the total energy. As it was pointed out in Section 4, the keypoint of that is the divergent form of the SWE [26, 27]. Analogously to the second-order schemes presented in Section 5, all other of the developed schemes can be resolved by a direct (i.e. non-iterative) method, and therefore the conservation laws are not violated. Indeed, consider, for example, equation (66). For its right-hand side, taking into account the definition of Rkl (see Section 5), we can write −gz k

hn+1 + hnk+1 − hn+1 − hnk hk+1 − hk k = −gz k k+1 , ∆x 2∆x

and hence, expressing from (68) the function Hkn+1 and then substituting hn+1 = Hkn+1 + k

Mass and Energy Conserving Fully Discrete Schemes...

173

hT k into (66), we shall obtain Ukn+1 − Ukn + P1 + P2 = P3 + P4 , τ

(95)

where P1 = P3 = −

1 S1 (n), 4∆x

1 gz k τ S1 (n + 1), P2 = − S2 (n + 1), 4∆x 4∆x2  gz k  n τ P4 = − 2(hk+1 − hnk ) − S2 (n) 2∆x 2∆x

and n n S1 (n) = uk Uk+1 − uk−1 Uk−1 ,

n n − 2z k Ukn + z k−1 Uk−1 . S2 (n) = z k+1 Uk+1

Therefore, if (65) holds then scheme (95), (67)-(68) can be solved by a direct linear algebra method. Of course, if (65) holds then all the other schemes in x and in y also reduce to systems of linear algebraic equations. Moreover, due to the use of splitting all these systems will have band matrices, and thus fast band linear solvers can be employed for their solution [19]. As we noted in Section 5, the functions ukl , v kl and z kl can be defined in an arbitrary way. So, apart from (65) we can write wkl =

X

(w)

n , αk+i,l+j wk+i,l+j

i,j

w = {u, v, z},

(96)

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(w)

where αk+i,l+j are weight coefficients for u, v and z, while i, j vary in some ranges over neighbouring nodes. Such approximations may be useful for reducing computational modes in the solutions if one employs an even-order central finite difference scheme. The issue of reducing undesired computational modes will also be discussed in Section 8. Independently of the accuracy of the spatial approximations the developed schemes are only of the first order in time. To improve the temporal accuracy of the solution the method of dicyclic splitting can additionally be employed [14]: instead of solving the shallow-water system as   n+ 13 n+ 31 n n ~ ~ ~ ~ ψ1 −ψ ψ +ψ  + A1  1 = 0, τ 2   n+ 32 n+ 13 n+ 31 n+ 23 ~ ~ ~ ~ ψ2 − ψ1 + ψ1  ψ + A2  2 = 0, τ 2 2

(97)

(98)

2

~ n+ 3 ~ n+1 − ψ ~ n+1 + ψ ~ n+ 3 ψ ψ 3 2 2 + A3 3 = 0, τ 2

(99)

174

Yuri N. Skiba and Denis M. Filatov

it should be solved as 1



 n+ 51 n ~ ~ ψ +ψ  + A1  1 = 0, 2   n+ 25 n+ 51 ~ ~ ψ + ψ1  + A2  2 = 0, 2

~ n+ 5 − ψ ~n ψ 1 τ 2

2

1

~ n+ 5 − ψ ~ n+ 5 ψ 2 1 τ 2

3

2

3

(100)

(101)

2

~ n+ 5 − ψ ~ n+ 5 ~ n+ 5 + ψ ~ n+ 5 ψ ψ 3 2 2 + A3 3 = 0, τ 2   4 4 3 3 ~ n+ 5 + ψ ~ n+ 5 − ψ ~ n+ 5 ~ n+ 5 ψ ψ 2 3 2 3  = 0, + A2  τ 2 2   4 4 ~ n+1 − ψ ~ n+ 5 ~ n+1 + ψ ~ n+ 5 ψ ψ 1 2 2  = 0. + A1  1 τ 2 2

(102) (103)

(104)

In [14, 15] it was demostrated that such a procedure yields solutions of the second order ~ and A2 (ψ). ~ As to the in time even for systems with noncommutative operators A1 (ψ) Coriolis operator A3 , its discretisation does not affect the scheme’s spatial approximation order.

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7.

Shallow-Water Model on a Sphere

The sphere is a very important geometric manifold for many meteorological applications of the SWE. A great advantage of the splitting-based approach is that it allows using the developed ‘planar’ algorithm for studying shallow-water flows on a rotating sphere. Consider the shallow-water equations in the spherical coordinates (λ, ϕ)      ∂U 1 1 ∂uU ∂U 1 ∂vU cos ϕ ∂U + +u + + v cos ϕ − ∂t R cos ϕ 2 ∂λ ∂λ 2 ∂ϕ ∂ϕ   u gz ∂h − f + tan ϕ V = − , (105) R R cos ϕ ∂λ      1 ∂V ∂V ∂V 1 ∂vV cos ϕ 1 ∂uV + +u + v cos ϕ + + ∂t R cos ϕ 2 ∂λ ∂λ 2 ∂ϕ ∂ϕ   gz ∂h u , (106) + f + tan ϕ U = − R R ∂ϕ   ∂H 1 ∂zV cos ϕ ∂zU + + = 0. (107) ∂t R cos ϕ ∂λ ∂ϕ Here λ is the longitude (positive eastward), ϕ is the latitude (positive northward), R is the sphere’s radius, f = 2Ω sin ϕ, and Ω is the angular rotation rate. Problem (105)-(107) is being studied on a sphere S. Let us note that the same sphere can be represented by various coordinate maps defined in different ways. For example, let λ vary from 0 to 2π, while ϕ be within the interval

Mass and Energy Conserving Fully Discrete Schemes...

175

[− π2 , π2 ]. Then we shall obtain the standard coordinate map for S shown in Fig. 3 by dashdot lines. On the other hand, if we define λ to be from 0 to π, whereas ϕ to be from 0 to 2π, then we shall have another coordinate map for S, which is shown in Fig. 4 (dash-dot lines).

Figure 3. The first coordinate map (dash-dot lines) and the corresponding grid covering (solid lines) for the sphere S. The point of using the operator splitting in the spherical geometry is that although the sphere is not a doubly periodic manifold, our algorithm developed for the Cartesian geometry can be applied to the sphere without any essential modifications. In fact, the metric term 1 R cos ϕ has merely to be taken into account. Indeed, for the first coordinate map we define the corresponding grid covering as follows n (1) S∆λ,∆ϕ = (λk , ϕl ) :

∆λ 2

≤ λk < 2π +

∆λ π 2 ,−2

+

∆ϕ 2

≤ ϕl ≤

π 2

−

L−1/2

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

In (108) we moved the grid to a half step in the direction ϕ: {ϕl }1/2 1 R cos ϕ

∆ϕ 2

o

.

= {− π2 +

(108) ∆ϕ 2

+

l∆ϕ, l = 0, L − 1}, because due to the metric term equations (105)-(107) are not valid in the poles. Therefore we do cover the whole sphere and at the same time do not have to find the solution at the pole nodes [35]. This grid is used for computing the solution in the λ-direction. For computing the solution in the ϕ-direction we can use the second coordinate map, on which the grid is defined as n (2) S∆λ,∆ϕ = (λk , ϕl ) :

∆λ 2

≤ λk ≤ π −

∆λ ∆ϕ 2 , 2

≤ ϕl < 2π +

∆ϕ 2

o

.

(109)

One can see that the grid covering (109) has the same nodes as in (108) (Figs. 3-4, solid lines). Thus, after making slight modifications in the above-given formulas (86)-(94) (as 1 we noted, only the metric term R cos ϕ has to be added) they can be applied for constructing

176

Yuri N. Skiba and Denis M. Filatov

Figure 4. The second coordinate map (dash-dot lines) and the corresponding grid covering (solid lines) for the sphere S.

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conservative finite difference schemes for system (105)-(107). For instance, the secondorder finite difference scheme in λ has the form

Ukn+1 − Ukn 1 + τ 2cl

Vkn+1 − Vkn τ



 uk+1 Uk+1 − uk−1 Uk−1 Uk+1 − Uk−1 + uk 2∆λ 2∆λ gz k hk+1 − hk−1 =− , cl 2∆λ   1 uk+1 Vk+1 − uk−1 Vk−1 Vk+1 − Vk−1 + + uk = 0, 2cl 2∆λ 2∆λ

(110)

Hkn+1 − Hkn 1 z k+1 Uk+1 − z k−1 Uk−1 + = 0, τ cl 2∆λ

(112)

(111)

Mass and Energy Conserving Fully Discrete Schemes...

177

while in ϕ we shall get Uln+1 − Uln 1 + τ 2cl

Vln+1 − Vln τ



v l+1 Ul+1 cos ϕl+1 − v l−1 Ul−1 cos ϕl−1 2∆ϕ  Ul+1 − Ul−1 +v l cos ϕl = 0, 2∆ϕ  1 v l+1 Vl+1 cos ϕl+1 − v l−1 Vl−1 cos ϕl−1 + 2cl 2∆ϕ  Vl+1 − Vl−1 gz l hl+1 − hl−1 +v l cos ϕl =− , 2∆ϕ R 2∆ϕ

Hln+1 − Hln 1 z l+1 Vl+1 cos ϕl+1 − z l−1 Vl−1 cos ϕl−1 + = 0. τ cl 2∆ϕ

(113)

(114) (115)

Here cl = R cos ϕl . Analogously, schemes of other orders can easily be written. The rotation problem is   n+1 n Ukl − Ukl ukl − fl + tan ϕl Vkl = 0, τ R   Vkln+1 − Vkln ukl + fl + tan ϕl Ukl = 0. τ R

(116) (117)

It is noteworthy that the use of two different maps (108)-(109) is possible exclusively due to the splitting. Therefore, if (65) holds, we keep the simple band structure of the matrices, and hence the solution appears to be cheap from the computational standpoint. It is to remark in this connection that if one solves the complete 2D shallow-water system without splitting then the problem of complicating the matrix structure evidently arises.

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8.

Numerical Results

In this section we present some results of numerical experiments with the developed SWM finite difference schemes. All the experiments are divided into two groups. The main purpose of the first group of experiments described in subsections 8.1 and 8.2 was to test the schemes of different approximation orders in space and to compare their skill on grids of different resolution. In particular, we tested the schemes using a Rossby-Haurwitz wave solution (subsection 8.1) and randomly distributed initial data that did not satisfy the geostrophic relations (subsection 8.2). The primary attention was given to the study of time-space structure of the numerical solutions. In the second group of experiments (subsection 8.3) the schemes of different approximation orders are used to study the dynamics of soliton-like solutions generated by a topography (two mountains). We stress again that each finite difference scheme exactly conserves the mass and the total energy, but not the potential enstrophy. Since the potential enstrophy is one of the basic invariants of the shallow-water motion, temporary behaviour of the potential enstrophy is considered in all the experiments as an important integral characteristic of the schemes’ quality.

178

8.1.

Yuri N. Skiba and Denis M. Filatov

Rossby-Haurwitz Waves

In these tests we verified the developed schemes on the wavenumber-4 Rossby-Haurwitz (RH4) wave with the maximum velocity 20 m/s and the mean height H(λ, ϕ, 0) = 12, 000 m [8, 36]. The grid steps ∆λ, ∆ϕ and τ were chosen such that the finite different schemes were accurate and the matrices corresponding to each split system were diagonally dominant. We had the sequence of spatial grids 12◦ × 12◦ , 6◦ × 6◦ , 3◦ × 3◦ and 1.5◦ × 1.5◦ . As it is known, odd-order finite difference schemes used for solving SWM systems produce either physical or purely computational modes in the solution, depending on whether the advective terms are approximated correctly or not from the physical point of view. On the other hand, even-order schemes simultaneously generate both physical and computational modes [19]. So, in accordance with the theory, the purely odd-order schemes provided physically unreliable solutions, just because neither up-wind-like nor down-wind-like schemes are suitable for simulating such a complicated flow as the RH4 wave. Unlike that, the even-order schemes properly approximated the solution and provided reliable results. Because the mass and the total energy were proved to be constant, we took the potential enstrophy J(t) as an important integral characteristic for estimating the quality of the numerical solution. The corresponding results for maximum variation of the potential enstrophy J(t) are shown in Table 1. Table 1. RH4 wave. Maximum variation (in %) of the potential enstrophy for schemes of different approximation order Grid × 12◦ 6◦ × 6◦ 3◦ × 3◦ 1.5◦ × 1.5◦

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12◦

1st > 10 > 10 > 10 > 10

2nd 2.899 0.710 0.331 0.169

1st − 2nd 3.777 1.067 0.306 0.151

3th > 10 > 10 > 10 > 10

4th 1.734 0.497 0.286 0.157

3rd − 4th 1.504 0.490 0.292 0.161

In order to reduce the effect of computational modes generated by even-order schemes, we used ‘conditional’ computing. Namely, when computing in the direction λ, at each fixed ϕl we employed an odd-order up-wind scheme if all the uk ’s were positive, an odd-order down-wind scheme if all the uk ’s were negative, or the even-order scheme if there were the uk ’s of both signs. Similarly we did in the direction ϕ checking each time the signs of the v l ’s. Therefore we slightly improved the solution (compare columns ‘1st − 2nd ’ vs. ‘2nd ’ and ‘3rd − 4th ’ vs. ‘4th ’, respectively). Because of complexity of the flow the odd-order schemes were applied rather seldom, and so the improvement was not substantial. However, the more directional is the flow the more essential accuracy improvement can be achieved while using odd-order schemes [26, 29]. In Fig. 5 we show graphs of the potential enstrophy in time. As one may see, the J(t) variations are within narrow bands and very small — the quantity δJ(t) ≡ max J(t)−min · min J(t) ◦ ◦ 100% does not exceed a third of a percent on fine (3 × 3 or better) grids (Table 1). In Figs. 6-8.1. we show the depth and the velocity fields for several time moments. These results nicely match a real RH4 wave solution [33, 36].

Mass and Energy Conserving Fully Discrete Schemes... 4

4

Potential Enstrophy

x 10

2.97

2.965

2.965

2.96

2.96

J(t)

J(t)

2.97

2.955

2.95

2.945

2.945

0

0.5

1

1.5

2

2.5 Time (days)

3

3.5

4

4.5

5

Potential Enstrophy

x 10

2.955

2.95

2.94

179

2.94

0

0.5

1

1.5

2

2.5 Time (days)

3

3.5

4

4.5

5

Figure 5. RH4 wave. Behaviour of the potential enstrophy J(t) in time on the grid 6◦ × 6◦ , τ = 7.2 min, second-order (left) and fourth-order (right) schemes.

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8.2.

Unbalanced Solutions

In these experiments we tested the developed schemes on highly unbalanced initial data. Namely, we considered the SWE with randomly distributed initial conditions whose mean values satisfied the geostrophic balance with H(λ, ϕ, 0) = 1000 m, while maximum variation of the depth field was 80 m and the maximum velocity variation was about 2 m/s. We found out that on the grid 3◦ × 3◦ maximum variation of the potential enstrophy was about 0.0116% for the second-order scheme and 0.0111% for the fourth-order one (for comparison, in [20] for similar initial conditions it was 0.05%). In Fig. 8 we plot the graphs of log10 J(t)−J(0) for both schemes. It can be seen that the higher order scheme produces J(0) a solution which is moving to a J(t)-steady state faster than the solution provided by the lower order scheme is. In Fig. 9 there are power spectra (in a logarithmic scale) of the total energy and the potential enstrophy at t = 0 days and t = 30 days corresponding to the second-order scheme. The results are averaged in frequencies ω1 , ω2 . In Fig. 10 we show analogous results corresponding to the fourth-order scheme. It can be seen that no apparent energy cascades are observed for both schemes, nor evident enstrophy buildups take place at any scale — the curves at t = 30 days are of a chaotic behaviour and resembling to those at t = 0 days, while the latters, being consistent with the randomly distributed inital data, demonstrate the white noise.

8.3.

Soliton-Like Solutions Generated by a Model Topography

For these experiments we had a simple set of initial conditions H(λ, ϕ, 0) = 3000 m, u(λ, ϕ, 0) = 0, v(λ, ϕ, 0) = 0. We introduced a relief given by two mountains, whose heights did not exceed 1000 m (Fig. 11). The larger mountain covers the area from −50◦ to +50◦ meridionally, locating at 250◦ − 280◦ eastwards, which roughly corresponds to the American Cordillera including the mountains of North, Central and South America. The smaller mountain is located near 30◦ − 50◦ northwards and 80◦ − 100◦ eastwards

180

Yuri N. Skiba and Denis M. Filatov Depth at t = 0.2 days 80

60

60

40

40

20

20

0

0

ϕ

ϕ

Depth at t = 0.1 days 80

−20

−20

−40

−40

−60

−60

−80

−80 50

1.02

100

1.04

1.06

150

1.08

λ 1.1

200

1.12

250

1.14

1.16

300

1.18

350

50

1.2

1.02

100

1.04

1.06

150

1.08

λ 1.1

200

1.12

250

1.14

1.16

300

1.18

350

1.2

4

4

x 10

x 10 Depth at t = 0.4 days

80

80

60

60

40

40

20

20

0

0

ϕ

ϕ

Depth at t = 0.3 days

−20

−20

−40

−40

−60

−60

−80

−80 50

1.02

100

1.04

1.06

150

1.08

λ 1.1

200

1.12

250

1.14

1.16

300

1.18

350

50

1.2

1.02

100

1.04

1.06

150

1.08

λ 1.1

200

1.12

250

1.14

1.16

300

1.18

350

1.2

4

4

x 10

x 10 Depth at t = 0.6 days

80

80

60

60

40

40

20

20

0

0

ϕ

ϕ

Depth at t = 0.5 days

−20

−20

−40

−40

−60

−60

−80

−80

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

50

1.02

100

1.04

1.06

150

1.08

λ 1.1

200

1.12

250

1.14

1.16

300

1.18

350

1.2

50

1.02

100

1.04

1.06

150

1.08

λ 1.1

200

1.12

250

1.14

4

x 10

1.16

300

1.18

350

1.2 4

x 10

Figure 6. RH4 wave. Depth field for several time moments. representing the Himalaya. Note that we did not keep the height ratio between the real mountains and the model ones. In Table 2 we give the maximum variation of the potential enstrophy, analogously to Table 1. Again, we see that the purely odd-order schemes failed, whereas the even-order schemes approximated the solution correctly. Besides, computational modes in the solution can effectively be avoided by employing ‘conditional’ schemes.

Mass and Energy Conserving Fully Discrete Schemes...

181

Velocity at t = 0.1 days 80 60 40

ϕ

20 0 −20 −40 −60 −80 50

100

150

λ

200

250

300

350

250

300

350

250

300

350

Velocity at t = 0.2 days 80 60 40

ϕ

20 0 −20 −40 −60 −80 50

100

150

λ

200

Velocity at t = 0.3 days 80 60

20 ϕ

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40

0 −20 −40 −60 −80 50

100

150

λ

200

Figure 7. Continued on next page.

182

Yuri N. Skiba and Denis M. Filatov Velocity at t = 0.4 days 80 60 40

ϕ

20 0 −20 −40 −60 −80 50

100

150

λ

200

250

300

350

250

300

350

250

300

350

Velocity at t = 0.5 days 80 60 40

ϕ

20 0 −20 −40 −60 −80 50

100

150

λ

200

Velocity at t = 0.6 days 80 60

20 ϕ

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40

0 −20 −40 −60 −80 50

100

150

λ

200

Figure 7. RH4 wave. Velocity field for several time moments.

Mass and Energy Conserving Fully Discrete Schemes... Scheme 2nd

Scheme 4th

−4

−4

−5

−5 log10 (J(t) − J(0)) / J(0)

−3

log10 (J(t) − J(0)) / J(0)

−3

−6

−6

−7

−7

−8

−8

−9

0

5

10

15 Time (days)

20

25

−9

30

0

5

J(t)−J(0) J(0)

Figure 8. Shock-like solution. Graph of log10 fourth-order (right) schemes.

10

−12.6

−12.7

−12.7

−12.8

−12.8

Log E(t) PS

Log E(t) PS

30

Log Energy Power Spectrum

−12.9

−12.9

−13

−13

−13.1

−13.1

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

−13.2

1.8

0

0.5

1

log w1

log w2

Log Potential Enstrophy Power Spectrum

Log Potential Enstrophy Power Spectrum

14.7

1.5

14.7 t = 0 days t = 30 days

t = 0 days t = 30 days

14.6

14.6

14.5

14.5

14.4

14.4

Log J(t) PS

Log J(t) PS

25

t = 0 days t = 30 days

−12.6

14.3

14.3

14.2

14.2

14.1

14.1

14

20

−12.5 t = 0 days t = 30 days

−13.2

15 Time (days)

for the second-order (left) and

Log Energy Power Spectrum −12.5

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183

0

0.2

0.4

0.6

0.8

1 log w1

1.2

1.4

1.6

1.8

14

0

0.5

1

1.5

log w2

Figure 9. Shock-like solution. Power spectra of the total energy (top) and the potential enstrophy (bottom) at t = 0 and t = 30 days, second-order scheme. The results are averaged in ω2 (left) and ω1 (right).

184

Yuri N. Skiba and Denis M. Filatov Log Energy Power Spectrum

Log Energy Power Spectrum

−12.5

−12.5 t = 0 days t = 30 days

−12.6

−12.6

−12.7

−12.7

−12.8

−12.8

Log E(t) PS

Log E(t) PS

t = 0 days t = 30 days

−12.9

−12.9

−13

−13

−13.1

−13.1

−13.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

−13.2

1.8

0

0.5

log w1

Log Potential Enstrophy Power Spectrum 14.7 t = 0 days t = 30 days

t = 0 days t = 30 days

14.6

14.6

14.5

14.5

14.4

14.4

Log J(t) PS

Log J(t) PS

1.5

Log Potential Enstrophy Power Spectrum

14.7

14.3

14.3

14.2

14.2

14.1

14.1

14

1 log w2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

14

0

0.5

log w1

1

1.5

log w2

Figure 10. Shock-like solution. Power spectra of the total energy (top) and the potential enstrophy (bottom) at t = 0 and t = 30 days, fourth-order scheme. The results are averaged in ω2 (left) and ω1 (right).

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

Table 2. Soliton-like solutions. Maximum variation (in %) of the potential enstrophy for schemes of different approximation order Grid × 12◦ 6◦ × 6◦ 3◦ × 3◦ 1.5◦ × 1.5◦ 12◦

1st 1.129 1.661 > 10 > 10

2nd 0.396 0.130 0.047 0.039

1st − 2nd 0.326 0.102 0.041 0.030

3th 1.091 1.287 > 10 > 10

4th 0.212 0.102 0.051 0.046

3rd − 4th 0.151 0.081 0.046 0.038

In Fig. 12 we plot graphs of the potential enstrophy in time. Here the variations are also within limited bands, and on finer grids the quantity δJ(t) does not exceed 0.051%. We found out that the taken initial conditions generate a stable λ-periodic soliton-like solution ~η = (u, v, H)T , whose depth field has two observable peaks. In Fig. 13 we show the function H(λ, ϕ, t) for several time moments. It can be seen that just in the beginning the larger mountain (‘Cordillera’) generates two solitary peaks (located at λ ≈ 210◦ −

Mass and Energy Conserving Fully Discrete Schemes...

185

Relief Field 80

60

40

ϕ

20

0

−20

−40

−60

−80 0

50

100

100

200

150

300

λ

400

200

250

500

300

600

700

350

800

Figure 11. Soliton-like solutions. Relief used in the numerical experiments (maximum height of the mountains is about 900 m). 5

1.024

1.0238

1.0238

1.0236

1.0236

1.0234

1.0234

1.0232

1.0232

1.023

1.0228

1.0226

1.0226

1.0224

1.0224

1.0222

1.0222

0

0.5

1

1.5

2

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transports the first peak and a turbulent convergent-divergent vortex (well-observed at λ = 220◦ − 250◦ at t = 0.2) which moves the second, weaker peak. Note that the ‘Himalaya’ mountain also produces wave solutions, which, however, have much smaller amplitudes compared to those generated by the larger mountain (nicely observed at t = 0.2). Depth at t = 1.1 days 80

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Figure 13. Soliton-like solutions. Depth field. Generation and subsequent interaction of two solitary peaks propagating in opposite directions. From t = 1.1 to t = 1.5 we observe an interaction of the two peaks at λ ≈ 100◦ −110◦ . Again, the first peak remains almost unperturbed, whereas the second one changes its shape and loses in the amplitude (Fig. 13, t = 1.5). In Fig. 8.3. the same peaks interaction is shown as it is modelled by the velocity field. In Figs. 15-8.3. we show the propagation of the peaks over the ‘Cordillera’ mountain. What is interesting here is that we can observe two different manners of propagation of the solitary waves over an obstacle. Indeed, the stronger peak goes over the ‘Cordillera’ without considerable changes both in the depth and the velocity fields (Figs. 15-8.3., t = 2.0 − 2.4). On the other hand, the weaker solitary wave, being located at t = 2.6 between the ‘Cordillera’ and the stronger peak, changes its velocity field essentially when passing the mountain at t = 2.8 (Fig. 8.3.). In Figs. 17-8.3. we show the solution for further time moments. While propagating, the

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peaks meet one another and interfere at t = 3.7. After the nonlinear interaction the waves continue propagating east- and westwards, which is observed clearer in Fig. 8.3. than in Fig. 17. Comparing the solutions at t = 1.3 and t = 3.7, we find the full period of revolution over the sphere to be ∆T = 3.7−1.3 = 2.4 days. At the same time, the gravitational waves 6 m 2πR revolution is known to be ∆Tgrav = √ ≈ 40·10 171 m/s = 2.7 days [18, 33], and, following gH the theory of nonlinear waves propagation [12], the time ∆T should be a little bit greater than ∆Tgrav . The explanation for why ∆T happened to be less than ∆Tgrav is the effect of nonlinear wave interaction. Indeed, say, in Fig. 13 we see that the stronger peak, when interacting with the weaker one, passes from λ ≈ 80◦ (t = 1.1) to λ ≈ 150◦ (t = 1.5) for t = 1.5 − 1.1 = 0.4 days. Without the nonlinear interaction such a rapid translation would be impossible, since it requires the solitary wave velocity to be close to 225 m/s. The same is observed in Fig. 15: here the stronger peak, due to the nonlinear interaction

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Figure 14. FiSoliton-like solutions. Velocity field. Generation and subsequent interaction of two solitary peaks propagating in opposite directions.

with the ‘Cordillera’ mountain, passes from λ ≈ 230◦ (t = 2) to λ ≈ 290◦ (t = 2.4) for the same 0.4 days, which translation would only be possible at the peak’s velocity about 193 m/s.

9.

Conclusion

A new method for constructing finite difference schemes for the shallow-water model was suggested. The model equations were considered in a limited area, in a doubly periodic domain, in a periodic channel on the plane, as well as on a whole sphere and in a periodic

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processes. Therefore the solution of the original system of 2D partial differential equations

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Figure 17. Soliton-like solutions. Depth field. Stability and periodicity of the solution.

reduces to the solution of three simple problems containing either 1D partial differential equations or ordinary differential equations. In fact, an infinite family of such conservative schemes is proposed, which are either linear or nonlinear depending on the choice of certain scheme parameters. On a doubly periodic manifold, the method allows constructing conservative finite difference schemes of arbitrary approximation orders in space. Moreover, if the SWM is considered on the entire sphere (which is not a doubly periodic manifold) then the method makes it possible to use the same numerical schemes (of arbitrary approximation order in space) and algorithms as for a doubly periodic manifold. The numerical SWM algorithms are computationally cheap, because each scheme is easily realised by fast direct methods of linear algebra. The skill of the finite difference schemes was illustrated by numerical results. The numerical experiments were divided into two groups. The first group of experiments represented various tests to verify the schemes of different approximation orders in space and to compare their skill on grids of different resolution. In particular, we tested the schemes using a Rossby-Haurwitz wave solution and randomly distributed initial data that did not satisfy the geostrophic relations. The primary attention was given to the study of time-

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space structure of the numerical solutions. In the second group of experiments the schemes of different approximation orders were used to study the dynamics of solitary waves generated by a model topography. Each finite difference scheme exactly conserves the mass and the total energy, but not the potential enstrophy. Because the potential enstrophy is one of the basic invariants of the shallow-water motion, temporary behaviour of the potential enstrophy was considered in all the experiments as an important integral characteristic of the schemes’ quality. The last experiments are of physical interest as they allowed to study the nonlinear dynamics of soliton-like solutions, in particular the interaction of solitary waves with topography.

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Figure 18. Soliton-like solutions. Velocity field. Stability and periodicity of the solution.

Acknowledgements This research was partially supported by the grants No. 14539 y No. 26073 (Sistema Nacional de Investigadores, CONACyT, Mexico), and by the three projects: PAPIIT-UNAM IN105608 (Mexico), 46265 A-1 (CONACYT, Mexico) and FOSEMARNAT-CONACyT 2004-01-160 (Mexico).

References [1] A. Arakawa and V.R. Lamb, A Potential Enstrophy and Energy Conserving Scheme for the Shallow-Water Equation, Mon. Wea. Rev. 109 (1981).

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[2] E. Audusse, F. Bouchut, M.-O. Bristeau, R. Klein and B. Perthame, A Fast and Stable Well-Balanced Scheme with Hydrostatic Reconstruction for Shallow-Water Flows, SIAM J. Sci. Comp. 25 (2004) 2050-2065. [3] M. Berger, C. Helzel and R. LeVeque, h-Box Methods for the Approximation of Hyperbolic Conservation Laws on Irregular Grids, SIAM J. Numer. Anal. 41 (2003) 893918. [4] F. Bouchut, J. Le Sommer and V. Zeitlin, Frontal Geostrophic Adjustment and Nonlinear Wave Phenomena in One-dimensional Rotating Shallow Water. Part 2: HighResolution Numerical Simulations, J. Fluid Mech. 514 (2004) 35-63. [5] J. Crank and P. Nicolson, A Practical Method for Numerical Evaluation of Solutions of Partial Differential Equations of the Heat Conduction Type. Proc. Cambridge Philos. Soc. 43 (1947) 50-67. [6] F. Gibou and R. Fedkiw, A Fourth Order Accurate Discretization for the Laplace and Heat Equations on Arbitrary Domains, with Applications to the Stefan Problem, J. Comput. Phys. 202 (2005) 577-601. [7] R. Heikes and D.A. Randall, Numerical Integration of the Shallow-Water Equations on a Twisted Icosahedral Grid. Part I: Basic Design and Results of Tests, Mon. Wea. Rev. 123 (1995) 1862-1880. [8] R. Jakob-Chien, J.J. Hack and D.L. Williamson, Spectral Transform Solutions to the Shallow Water Test Set, J. Comput. Phys. 119 (1995) 164-187. [9] V.F. Kim, Numerical Analysis of Some Conservative Schemes for Barotropic Atmosphere, Rus. Meteor. Hydr. 6 (1984) 251-261.

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[10] G.A. Korn and T.M. Korn, Mathematical Handbook for Scientists and Engineers. McGraw-Hill, New York, 1968. [11] A.N. Kolmogorov and S.V. Fomin, Elements of the Theory of Functions and Functional Analysis. Dover Publications, 1999. [12] P.K. Kundu, Fluid Mechanics. Academic Press, 1990. [13] R.J. LeVeque and D.L. George, High-Resolution Finite Volume Methods for the Shallow-Water Equations with Bathymetry and Dry States. In: Advanced Numerical Models for Simulating Tsunami Waves and Runup, H. Yeh, P.L. Liu, C.E. Synolakis (eds.), Advances in Coastal and Ocean Engineering, vol. 10. World Scientific Publishing Co., Singapore, 2008. [14] G.I. Marchuk, Methods of Numerical Mathematics. Springer-Verlag, Berlin, 1982. [15] G.I. Marchuk, Splitting Methods. Moscow, Nauka, 1988 (in Russian).

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[16] Y. Masuda and H. Ohnishi, An Integration Scheme of the Primitive Equations Model with an Icosahedral-Hexagonal Grid System and Its Application to the Shallow-Water Equations. In: T. Matsuno (ed.), Short- and Medium-Range Numerical Weather Prediction, Universal Academy Press, 1987, 317-326. [17] K. Morton and D. Mayers, Numerical Solution of Partial Differential Equations. Cambridge University Press, Cambridge, 1994. [18] J. Pedlosky, Geophysical Fluid Dynamics. New York, Springer, 1979. [19] W.H. Press, S.A. Teukolsky, W.T. Vetterling and B.P. Flannerty, Numerical Recipes in C: The Art of Scientific Computing. Cambridge University Press, New York, 1992. [20] T.D. Ringler and D.A. Randall, A Potential Enstrophy and Energy Conserving Numerical Scheme for Solution of the Shallow-Water Equations on a Geodesic Grid, Mon. Wea. Rev. 130 (2002) 1397-1410. [21] R. Sadourny, The Dynamics of Finite-Difference Models of the Shallow-Water Equations, J. Atmosph. Sci. 32 (1975) 680-689. [22] R. Salmon, Poisson-Bracket Approach to the Construction of Energy- and PotentialEnstrophy-Conserving Algorithms for the Shallow-Water Equations, J. Atmosph. Sci. 61 (2004) 2016-2036. [23] A.A. Samarskii and Yu.P. Popov, Completely Conservative Difference Schemes, Zh. Vychisl. Mat. Mat. Fiz. 9 (1969) 953-958 (in Russian). [24] Yu.I. Shokin, Completely Conservative Difference Schemes. In: Computational Fluid Dynamics, G. de Vahl Devis and C. Fletcher (eds.), Amsterdam, Elsevier, 1988, 135155.

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[25] Yu.N. Skiba, Finite-Difference Mass and Total Energy Conserving Schemes for Shallow-Water Equations, Russ. Meteor. Hydr. 2 (1995) 35-43. [26] Yu.N. Skiba and D.M. Filatov, On Splitting-based Mass and Total Energy Conserving Shallow-Water Schemes, Int. Conf. Mathematical and Numerical Aspects of Waves, SIAM, Boulder, USA, 2005, 285-287. [27] Yu.N. Skiba and D.M. Filatov, Conservative Splitting-based Schemes for Numerical Simulation of Vortices in the Atmosphere, Interscience 31 (2006) 16-21. [28] Yu.N. Skiba and D. Parra Guevara, Efectos negativos de la aproximacion y representacion falsa de ondas sobre una malla, Misc. Mat. 43 (2006) 133-151. [29] Yu.N. Skiba and D.M. Filatov, On Splitting-based Mass and Total Energy Conserving Arbitrary Order Shallow-Water Schemes, Numer. Meth. PDE 23 (2007) 534-552. [30] Yu.N. Skiba and D.M. Filatov, Conservative Arbitrary Order Finite Difference Schemes for Shallow-Water Flows, J. Comp. Appl. Math. 218 (2008) 579-591.

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[31] Yu.N. Skiba and D.M. Filatov, On an Efficient Splitting-based Method for Solving the Diffusion Equation on a Sphere, Numer. Meth. PDE submitted. [32] K. Takano and M.G. Wurtele, A Four-Order Energy and Potential Enstrophy Conserving Difference Scheme. The US Air Force Geophysics Laboratory, Tech. Rep. AFGL-TR-82-0205, 1982, 85 pp. [33] C.B. Vreugdenhil, Numerical Methods for Shallow-Water Flow. Kluwer Academic, Dordrecht, 1994. [34] A.A. White, A View of the Equations of Meteorological Dynamics and Various Approximations. In: Large-Scale Atmospheric Dynamics, vol. I. Analytical Methods and Numerical Models, J. Norbury and I. Roulstone (eds.), Cambridge University Press, Cambridge, 2002, pp. 1-100. [35] D.L. Williamson, Difference Approximations of Flow Equations on the Sphere. In: Numerical Methods Used in Atmospherical Models, vol. II, GARP Publ. Series, 1979, No 17. [36] D.L. Williamson, J.B. Drake, J.J. Hack, R. Jakob and P.S. Swarztrauber, A Standard Test Set for Numerical Approximations to the Shallow-Water Equations in Spherical Geometry, J. Comput. Phys. 102 (1992) 211-224. [37] N.N. Yanenko, The Method of Fractional Steps. Springer-Verlag, Berlin, 1971.

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Reviewed by Prof. Dr. Mikhail Alexandrov, Autonomous University of Barcelona, Bellaterra (Barcelona), 08193, Spain Tel: (34) 93 581 1410 Email: [email protected]

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In: Energy Conservation: New Research Editor: Giacomo Spadoni, pp. 199-214

ISBN: 978-1-60692-231-6 © 2009 Nova Science Publishers, Inc.

Chapter 8

EXAMINATION OF TWO TURBULENT KINETIC ENERGY PARAMETERIZATIONS IN A CONVECTIVE BOUNDARY LAYER: COMPARISON OF TWO MODELS ON CASE STUDIES Mehrez Samaali1, Dominique Courault1, Albert Olioso1, Michael Bruse2, René Occelli3 and Pierre Lacarrère4 Institut National de la Recherche Agronomique, Unité Climat Sol et Environnement, Domaine Saint Paul, Site Agroparc, 84 914 Avignon Cedex 9 (France). 2 Institute of Geography, Building NA 5/173, University of Bochum, Universitätsstrasse 150, 44 780 Bochum (Germany). 3 Polytech’Marseille (Département de Mécanique Energétique) - Technopôle de Château Gombert. 5, rue Enrico Fermi - 13453 Marseille Cedex 13 (France). 4 CNRM, météo-France, 42 avenue Gaspard Coriolis 31057, Toulouse (France) 1

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Abstract Modeling of energy and mass interactions within the atmospheric boundary layer at meso and global scales is essential for weather forecast and climate analysis. Interactions near the surface are commonly described in Soil-Vegetation-Atmospheric models which involve energy budget calculations for the soil and the vegetation and require temperature, humidity and wind descriptions for heat and latent flux calculations. However, turbulence at the upper layers of the atmosphere is described with turbulence closure models which account for momentum and heat processes. Higher closure model performances have been demonstrated in describing turbulent mixing for various atmospheric boundary layer states (i.e. stable/unstable, clear/cloudy, etc). In this paper, two higher Turbulent Kinematic Energy (TKE or E) models are compared in a convective boundary layer. The first model is based on a local turbulence closure approach using a modified version of the standard Launder and Spalding (1974) TKE model which calculates two prognostic equations for TKE and its dissipation. This closure model is implemented in a simple (compared to other ABL models) model named ENVIronmental METeorological (ENVIMET). The second one corresponds to a non-local turbulence closure and uses the Bougeault and Lacarrère’s (1989) parameterization which consists of solving a prognostic equation for TKE and using mixing

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Keywords: convective boundary layer, ENVIMET, Meso-NH, turbulent kinetic energy budget.

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Introduction It is commonly known that the land surface acts on the atmosphere’s vertical structure by exchanging energy and mass. Energy and mass exchanges depend on land surface characteristics involving Leaf Area Index (LAI), the vegetation and soil surface radiative optical properties (albedo, emissivity), soil moisture conditions, and dynamic and optical properties of air masses. Interactions near the surface are classically described within SoilVegetation-Atmospheric models (e.g. Olioso, 1992; Tuzet et al., 1993) which are based on an energy budget concept (i.e. the available net radiative energy is dissipated in transpiration by leaves, evaporation, heat and photosynthesis) applied for the soil and the vegetation, and requires temperature, humidity and wind description for flux calculations. However, mass and energy exchanges in the atmospheric layers are described by the so-called Reynolds average equations which involve turbulent flux calculations which need, in turn, to calculate exchange coefficients prognostically. For that, a wide range of turbulence schemas ranging from simple (e.g. Blackadar, 1979) to very complex (e.g. Yamada and Moller, 1975; Wilson, 1988; André et al., 1978), were proposed in recent decades. However, they are basically classified into two categories: non local and local methods (Stull, 1988). In non-local methods, the mixing due to larger turbulent eddies is considered in the calculation of momentum, heat and humidity fluxes terms of the Reynolds equations. Some examples of these closures were discussed in the literature (e.g. Bougeault and Lacarrère, 1989; Stull, 1984; Stull, 1993; Zhang and Stull, 1992) and showed good performance in simulating mixed boundary layer and also compared well with measurements. However, their implementation in ABL models has the disadvantage of increasing considerably the computing time making them less popular than local turbulence closures (Bélair et al., 1999). In local methods, mixing due to large turbulent eddies is not directly considered and unknown quantities (i.e. turbulence fluxes of momentum, heat and humidity) at one grid point are determined from mean atmospheric variables and/or their gradient at that point using turbulent closure models (Stull, 1988). Within this category of turbulent closure methods there are many approaches applied in ABL models: first order (e.g. Pielke and Mahrer, 1975), one-and-half (e.g. Launder and Spalding, 1974; Huang and Raman, 1991; Xu and Taylor, 1997), and higher order (e.g. Yamada and Moller, 1975; Wilson, 1988; André et al, 1978) closures. Note that the order of these schemas refers to the highest momentum to be calculated when solving the Reynolds equations. The higher order schemas have particularly demonstrated to be consistent in describing the evolution of daytime and nocturnal boundary layers (Yamada and Moller, 1975; Wilson, 1988; André et al., 1978). However, they have the disadvantage of being very time demanding: for example if compared to order one-and-half they are 100 times slower (André et al., 1980). These limits make the one-and-half closures more popular and commonly used to describe turbulence in atmospheric boundary layer models.

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The most used one-and-half closure scheme is the E − ε (named also k − ε ) model. It is based on two prognostic equations describing the turbulent kinematic energy and its dissipation rate, respectively. The first standard version of this model was proposed by Launder and Spalding (1974) and has been devoted for engineering applications where in general good performance has been observed. However, when this model was applied within ABL models, many problems related to the mixing representation were identified. In particular, the depth of the planetary boundary layers was overestimated compared to observations in different (neutral and stable) conditions (Mellor and Yamada, 1982; Detering and Etling, 1985; Duynkerke, 1988; Aspley and Castro, 1997; Xu and Taylor, 1997). Hence, many solutions have been proposed to avoid these problems: adjusting the empirical coefficients used in the standard model (Detering and Etling, 1985), adjusting some of these coefficients and modifying the parameterization of the production of ε equation (Duynkerke, 1985), keeping the standard coefficients unchanged, but introducing a maximum mixing length ( l max ) in the parameterization of the ε production term (Aspely and Castro, 1997), adjusting one of the standard coefficients and introducing a new parameterization for mixing lengths involved in the ε production term and the exchange coefficient calculation (Xu and Taylor, 1997). In some other investigations, a simple formulation called the Kolmogorov relation (Kolmogorov, 1941) is used to ensure balance between the production and dissipation terms in the TKE equation. For example, Huang and Raman (1991) applied this formulation and used a new formulation for ε calculation. However, Wright et al. (1998) followed the same method with the exception of introducing a new equation to a parameter involved in the ε production term calculation. In this work, we pursue the E − ε closure issues within the comparison of a local turbulence closure model based on an improved version of the standard E − ε closure against a non-local formulation using the Bougeault and Lacarrère (1989) schema. This improved E − ε standard model uses a modified dissipation and production representation including the Kolmogorov relation. It is implemented in a 3D atmospheric boundary layer model called ENVIronmental METeorological (ENVIMET) developed at the University of Bochum in Germany (Bruse and Fleer, 1998; Bruse, 1998). However, the Bougeault and Lacarrère (1989) scheme consists of a mixing length closure implemented in Meso-NH which is the French research community model, mainly developed by the Aerology Laboratory and the CNRM (Centre National de Recherches Météorologiques). Despite that the model includes many closures suitable for scales ranging from large (synoptic) to small (large eddy) scales and many configurations (1-2 or 3D), this closure is mainly intended to 1D simulations. Thus, this study uses exclusively 1D versions of both models and Meso-NH simulations will be considered as a base case. In particular, the vertical thermal structure of the atmosphere as well as vertical profiles of TKE and its budget components simulated with ENVImet will be compared to those of Meso-NH on case studies . Comparison to anterior TKE results such as those of Stull (1988) will be also discussed. The following is organized in 4 sections. In sections 1 and 2, the turbulence parameterizations of ENVImet and Meso-NH will be presented. Then, model settings and the case studies will be briefly described in section 3. Finally, section 4 will outline the results of the comparison between the two models and section 5 will draw the sum up of this study.

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The Envimet Model

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Brief Description ENVImet is a 3D non-hysdrostatic meteorological local-scale model created at the Univrsity of Bochum within the Climate Research Group (CRG) (Bruse and Fleer, 1998; Bruse, 1998). Basically, ENVImet is designed to study energy and mass interactions between the surface and the atmosphere in urban areas at small scales (lower than 10 km²). It includes a complete description of radiation, heat, and water vapor near the surface as well as in the upper layers of the atmosphere. The model uses a 1D version which runs in parallel to the main model (i.e. the 3D model) and provides inflow lateral and upper boundary conditions. However, the main model feeds the 1D model with surface boundary conditions. The model was originally devoted to study the influence of building and trees on the atmosphere structure in urban areas. However, it has been applied on an agricultural field with special care to surface energy fluxes (Samaali, 2002; Samaali et al., 2007). In particular, the surface schema has been analyzed in detail and improved by comparing it to realistic radiative transfer models (Verhoef, 1984, 1985; Prévot, 1985) as well as surface radiation measurements acquired on a soybean field experiment (Samaali et al., 2007; Samaali, 2002). From a surface flux modeling point of view, the soil surface is treated independently from the vegetation, which is split into many layers for which the number depends on a user configuration. Thus, radiative (shortand long-wave) and energy budgets are solved for the soil surface as well as each vegetation layer. One main particularity is that the short- and long-wave radiative measured data (total and diffuse short-wave and atmospheric long-wave radiation data) are used to drive the model at each time step. The first model grid box is systematically split into five layers in order to better represent energy and mass exchanges near the surface. The soil model solves the 1D exchange of water and heat in fourteen layers and in two meters of depth with fine nodes near the surface and coarse nodes in the deep layers. Soil hydrological input parameters are prescribed from Clapp and Hornberger (1978) and soil surface turbulent fluxes are calculated based on the Monin-Obukhov (1954) similarity approach which includes the description of atmospheric stability. The initialization of temperature and humidity in the atmosphere is based on radiosonde measurements used for the upper layers whereas soybean experiment measurements are used for the lower layers of the atmosphere. For the initialization of the wind speed, ENVImet uses horizontal wind speed magnitude at ten meters above the surface and derives an initial profile based on a logarithmic formulation. The initialization of soil humidity and temperature is also based on soybean experiment data. Given the complexity of this model, providing all the information related to its concepts, parameterizations and hypothesis in this chapter would be out of scope. However, further details can be found in Bruse and Fleer (1998), Bruse (1998), Samaali et al. (2007) and Samaali (2002).

Turbulence Closure Schema As mentioned above, this work aims to analyze the 1D version of ENVImet as being an essential component for the 3D model since it provides the boundary conditions. The 1D turbulence scheme is identical to the standard E − ε model (Launder and Spalding, 1974) that we repeat here for completeness:

Examination of Two Turbulent Kinetic Energy Parameterizations…

203

∂E = − Tr + Pr + Th − Di ∂t

(1)

∂ε ∂t I

= − Trε + A. Pr + B.Th − C I2

I3

I4

(2)

I5

and ε are turbulent kinematic energy and its dissipation rate, respectively. Assuming horizontal homogeneity, we can obviously note that the advection term due to the 1D configuration does not appear in equations 1 and 2. Terms Tr and Trε correspond to vertical

E

transport of turbulent kinematic energy and its dissipation rate, respectively. They are described as follows:

Tr =

∂( w' E )

(3)

∂z

Trε =

∂( w' ε ) ∂z

(4)

w' is the fluctuation of the vertical wind speed component. Terms Pr and Th represent the production of energy and the buoyancy effect due to vertical wind speed and temperature gradients, respectively. They are defined as follows: 2 ⎧⎛ ∂u ⎞ 2 ⎛ ∂ v ⎞ ⎫⎪ ⎪⎜ ⎟ ⎟ ⎜ Pr = K m ⎨⎜ ⎟ + ⎜ ⎟ ⎬ ⎝ ∂z ⎠ ⎪⎭ ⎪⎩⎝ ∂z ⎠

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Th =

g

θ

Kh

(5)

∂θ ∂z

(6)

(u, v) are the horizontal wind speed components calculated by solving wind speed prognostic equations. g is the vertical acceleration due to gravity, K m and K h are momentum and heat exchange coefficients, respectively ( K m = K h = C μ E / ε , with C μ 2

an empirical constant taken to be 0.09) and θ is the average potential temperature calculated by solving a prognostic equation. The terms A , B and C which arise in equation 2 depend on E and ε , and involve empirical constants of the E − ε standard model ( c1 = 1.44 ; c2 = 1.92 ; c3 = 1.44 ) :

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Mehrez Samaali, Dominique Courault, Albert Olioso et al.

A = c1

B = c3

C = c2

ε

(7)

E

ε

(8)

E

ε2

(9)

E

As mentioned in the introduction, the application of the E − ε model within the ABL has shown problems resulting in mixing overestimation which affects exchange coefficients and the TKE calculation. Thus, many solutions have been investigated to avoid these issues. Some authors have modified the original ε equation by adding source terms, using diffusion processes and adding energy production (Mellor and Yamada, 1982; Duynkerke, 1988), setting a maximum for boundary layer length scale to ovoid TKE overestimation (Aspley and Castro, 1997), setting a maximum for more than one length scale (Xu and Taylor, 1997) or using the Kolmogorov relation (Kolmogorov, 1941) with a specific parameterization for a parameter involved in the ε calculation (Huang and Raman, 1991). Here the Kolmogorov relation is used to calculate the dissipation without including any other parameterization for the sake of simplicity:

εk =

(11)

κ is the Von Karman constant (0.4), C μ is a constant set to 0.09 and lε is the

dissipation mixing length. However, we consider the maximum value between

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(10)

lε

lε = κ .z

with, where

Cμ E 3 / 2

εk

calculated

in eq.10 (Kolmogorov relation) and ε in eq.2 (prognostic ε equation) to avoid possible overestimation of mixing and TKE due to the non-parameterisation of relevant parameters and to ensure model stability:

Di = max(ε , ε k )

(12)

The Meso-NH Model Brief Description Meso-NH is a 3D non-hydrostatic meso-scale meteorological model which has been developed in France by the Laboratoire d'Aérologie (UMR 5560 UPS/CNRS) and CNRMGAME (URA 1357 CNRS/Météo-France). It is used for research as well as various other applications. It should be mentioned that Meso-NH is more complex than ENVImet since it

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205

includes topography effects, cloud and radiation interactions as well as rain processes. Unlike ENVImet, no connection between the 1D and 3D Meso-NH versions is needed and they do not have to run in parallel. Each of them can be run separately for a given case study. Despite its name, which indicates meso-scale and non hydrostatic applications, it has been also used for local scale and hydrostatic simulations. The description of the surface interactions is based on the ISBA scheme (Interactions Soil-Biosphere Atmosphere) which solves radiation and energy budgets of the soil surface and vegetation (both in one layer) using the force restore method (Deardorff, 1978). ISBA has been tested and evaluated for different soil and vegetation types (Noilhan and Planton, 1989; Noilhan and Mahfouf, 1996; Jacquemin and Noilhan, 1990). Soil and atmospheric model initialization can be based on measured or simulated profiles. However, many methods are used to define the boundary conditions. Those that are most used consist of nested techniques. The atmospheric system has been analyzed in many previous studies: for example, the implementation of a convective schema for regional and global applications based on Bechtold et al. (2001); the use of an adequate turbulence schema for meso-scale and large eddy simulations (Cuxart et al., 2000) and the introduction of a module that calculate the radiation in relation to cloud effects (Morcrette, 1991). A complete description of Meso-NH can be found at http://mesonh.aero.obsmip.fr/mesonh/. For the purpose of the comparison to ENVImet, only the Meso-NH 1D version will be used.

Turbulence Closure Schema Unlike ENVImet which involves two prognostic equations, Meso-NH uses an E − l model based on only one prognostic equation for TKE (the same as eq. 1). A mixing length formulation is introduced to account for turbulence. It is based on the non-local Bougeault and Lacarrère (1989) formulation which is recommended for large horizontal grid sizes where vertical transport is dominant compared to horizontal transport (i.e. 1D configurations). The main originality of this approach consists of using mixing lengths for TKE ( l k ) and dissipation ( lε ), and including non-local features in the turbulence description.

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At each level of the atmosphere, these two mixing lengths take into account the potential vertical upward and downward displacements that parcels with a mean TKE at this level could achieve before being stopped by buoyancy effects. Thus, two mixing lengths l up and

l down are introduced to represent maximum up and down vertical distances that parcels could reach starting from their original equilibrium level. The following formulations are applied to calculate K m , l k and lε , respectively :

K m = l k .E 1 / 2 .

(13)

l k = min(lup ,l down )

(14)

lε = (lup l down )1 / 2

(15)

206

Mehrez Samaali, Dominique Courault, Albert Olioso et al. The dissipation is therefore calculated as follows:

ε = Cε lε E 3 / 2

(16)

Despite the simple parameterization of these mixing lengths, this turbulence model was proven to reproduce very well what happens in the boundary layer and capture very well TKE behaviour (Bougeault and Lacarrère, 1989). In addition, some studies (Cuxart et al., 1994; Alapaty et al., 1997) outlined a comparable performance of this approach to more sophisticated methods such as transilient turbulent theory, particularly for convective boundary layer cases. However, this approach seems to have limitations due to the overestimation of mixing length scales in the stable boundary layer (Bélair et al., 1999; Mailhot and Loch, 2004), hence the focus on convective boundary layer cases in this chapter.

Models Settings and Case Studies

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In order to assess as best as possible the comparison of both turbulence closures, common sources of differences between ENVImet and Meso-NH, such as initialization, and some design features were set similarly: 1) Since the soil initialization in ENVIMET and Meso-NH are different (values for three layers in ENVImet and two in Meso-NH) with non-matching layer spacing, constant and identical vertical profiles are used to set the initial conditions for soil temperature and humidity (table 1). 2) The vertical grid mesh of Meso-NH can be defined in model inputs. However, that of ENVImet is hard-coded. To avoid possible effects of the vertical grid size on the comparison results, ENVImet was modified to allow the definition of the number of vertical layers as well as their respective thicknesses as input by the user. A vertical grid mesh using thin layers (1 m) near the surface and thicker layers at the upper levels (up to 400 m) was used in both models. 3) In order to initialise both models with realistic temperature and humidity in the atmosphere, measured profiles acquired in the Alpilles experiment available at http://avi1.avignon.inra.fr/reseda (Olioso et al, 2002, Courault et al, 2003) were used. However, given the different methods used to initialize the wind speed (Meso-NH uses radiosonde input profiles, but ENVImet uses a value at 10 m above the surface and calculates an initial logarithmic profile), the profiles issued from the ENVImet logarithmic calculation was used as input in Meso-NH. An initial horizontal wind speed value at 10 m equal to 3m s-1, corresponding to moderate dynamic conditions, was used. 4) Meso-NH calculates itself the solar (total and diffuse) flux radiations. However, ENVImet uses external values (measurements, eventually) or calculates it based on Teasler and Andreson (1984) formulas. In order to minimize the effect of radiative transfer parametrization near the surface, both models were driven by similar solar radiation flux values with a 20 min time step. These values are

Examination of Two Turbulent Kinetic Energy Parameterizations…

207

provided by a reference run corresponding to clear sky conditions performed using Meso-NH. 5) ENVImet 1D version is devoted to provide lateral boundary conditions for the main model and conceptually does not take into account the vegetation interactions with the soil and the atmosphere despite that all dynamic processes are represented. Thus, simple cases of wet and dry bare soils were considered in this comparison. Main characteristics of these cases are given in table 1. Both models were run for a period of 24 hours starting at midnight. Only midday profiles are analysed here. Table 1. Main test characteristics of ENVImet and Meso-NH comparison. Dry soil

Wet soil

Initial constant soil temperature profile (soil surface

276

276

included) Initial constant soil humidity profile (soil surface included)

25

65

3

3

Total solar radiation at midday ( Rg ), ( w.m )

605

605

Soil surface roughness ( z 0 ), ( m)

0.01

0.01

Wind speed at 10 m above the soil surface −2

3

Note that all TKE budget terms (i.e. Tr, Pr, Th and ε) are normalized by ( w*) / z i , where w * is called convective velocity scale and given by:

[

w* = ( g .zi / θ ).( w'θ ' ) s

]

1/ 3

(17)

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g is the gravitational acceleration, z i is the height of the convective boundary layer, ( w 'θ ' ) s is the surface heat flux and θ is the average potential temperature. This normalization uses buoyancy flux ( g / θ .( w θ ) s ) at the surface and z i since they are key '

'

variables for a free convection description (Stull, 1988). The normalization by the convective velocity scale also allows a comparison with relevant published results such as those outlined by Stull (1988).

Results and Discussion The vertical profiles of potential temperature show that under the influence of surface heating the convective boundary layer reaches a height of more than 500 meters in the dry soil surface

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Mehrez Samaali, Dominique Courault, Albert Olioso et al.

case and more than 150 meters in the wet case (figs. 1a and 1b). Both models simulate well the temperature inversion where the temperature gradient change is obvious. ENVImet seems to predict the inversion height (around 325 m) in agreement with Meso-NH in the wet case, but an over-prediction could be observed in the dry soil case (250 m difference, approximately). Despite that the influence of the soil surface is well captured by both models, significant differences in potential temperature are noticed near the surface: around 5°C in the dry case and approximately 4°C in the wet one. This is due to the over-prediction of midday surface turbulent heat flux for the dry case (ENVImet predicted ( w θ ) s = 0.15m / s.K '

'

versus Meso-NH predicted ( w θ ) s = 0.12m / s.K ) and the wet case (ENVImet simulated '

'

( w 'θ ' ) s = 0.08m / s.K versus Meso-NH simulated ( w 'θ ' ) s = 0.05m / s.K ). However, it

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is not obvious to only attribute these differences to turbulence closure since both models also have different surface parameterizations which acts differently on model results depending on the description of short and long-wave radiation interactions with the soil surface, the number of prognostic equations and parameters in the soil, the atmospheric stability functions, etc. Note that ENVImet is based on a multi-layer land surface model (Bruse et al., 1998; Samaali et al., 2007) whereas Meso-NH is based on ISBA (Noilhan and Planton, 1989). The midday TKE profiles of both models are in good agreement for the two simulated cases (figs. 2a and 2b). The typical shape of TKE due to daytime unstable conditions is well simulated. For both cases, TKE increases from the surface reaching a maximum at about z / z i = 0.3 . This increase is due to mixing under dominant convective conditions. Above this maximum TKE decreases rapidly with height since the effect of the convection become much less stronger than near the surface. Such results have been outlined by Therry and Lacarrère (1983) during a typical convective day of Wangara experiment. TKE values in the dry soil case are higher (more than 0.6 m2/s2) than in the wet soil case (maximum of 0.53 m2/s2) at the convective boundary layer. The development of the TKE profile with height is more significant (more than 500 m, fig. 2a) in the dry soil case than in the wet one (less than 500 m, fig. 2b). This is mainly due to the surface heat flux which is higher in the dry soil case than in the wet one (differences of 30 w/m2 are observed). Despite the TKE profile shape was well captured by both models, slight differences arise near maximum values: around 0.2 m2/s2 for the dry case and 0.1 m2/s2 for the wet one. Using the Meso-NH turbulence closure (i.e. Bougeault and Lacarrere’s mixing length approach) as a reference, the modified standard E − ε used here tends to slightly over-predict mixing which leads to an over-prediction of TKE despite the application of the limitation for ε calculation expressed in eq. 12. However, even with this over-prediction, ENVImet is able to reasonably predict TKE values below the maxima (i.e. very close to the soil surface) which could give more confidence in the prediction of surface energy budget fluxes and particularly the portioning between heat and latent flux under dry or wet conditions. This analysis can be completed by focusing on TKE budget terms normalized by the convective mixing scale (figs. 3a and 3b). The buoyancy term (Th) decreases linearly with height. Near the ground, it is large and positive meaning a large generation of turbulence since the surface is warmer than the atmosphere. This is a typical behaviour characterised by perfect clear sky days over land (Stull, 1988). Both model buoyancy terms match well and are in a reasonable agreement with the Stull (1988) results (figs. 3, 4 versus fig. 5). Regarding the

Examination of Two Turbulent Kinetic Energy Parameterizations…

209

dissipation term (-Di), ENVImet shows an under-prediction compared to Meso-NH with significant discrepancies very close to the surface (figs. 3 and 4). These discrepancies are more significant for the dry case simulation (fig. 3) than for the wet case simulation (fig. 4) (4.5 m2/s3 and 1.9 m2/s3 at the first level above the ground, respectively). However, both models show similar dissipation values at the upper layers of the atmosphere. The ENVImet underestimation of dissipation near the ground could be explained by the dissipation formulation which seems not to dissipate energy enough. Although Meso-NH dissipation profiles are in the range of values given by Stull (1988) (figs. 3b, 4b versus fig. 5), they could be over-predicted since the Bougeault and Lacarrère (1989) non-local parameterization seems to over-predict the dissipation mixing length ( lε ) near the upper and the lower parts of the atmospheric boundary layer (Belair et al., 1999). This could lead to an over-prediction of dissipation (eq. 16). The shear (or mechanical) production term (Pr) is maximum at the surface for both simulations with both models (figs. 3 and 4). This is not surprising since the greatest wind shear occurs at the surface. However, this term varies little with height since the shear is very low (near zero), hence the low production term values obtained. The transport term (-Tr) shows negative values near the surface meaning a loss of TKE. It increases with height in both cases and changes to a production of TKE (i.e. the values become positive). This change occurs at z / z i = 0.2 for the dry case for both models (fig 3). However, for the wet case ENVImet and Meso-NH show production of TKE at z / z i = 0.16 and

z / z i = 0.3 , respectively (fig 4). Note that for both cases the magnitude of this term is not

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significant compared to the -Di and Th terms particularly at a height above z / z i = 0.1 . However, ENVImet Tr term becomes significant very close to the surface and agrees well with the Stull (1988) range of values (fig. 5). Meso-NH Tr term near the ground is underestimated compared to the Stull (1988) transport range of values and shows a different profile shape (figs. 3 and 4 versus fig. 5). Perhaps, one explanation could be the fact that the Stull (1988) range of values probably does not cover all possible cases of convective boundary layers even if it includes results issued from many studies (i.e. Deardorff, 1974; André et al. 1978; Therry and Lacarrère, 1983; Lenschow, 1974; Pennell and LeMone, 1974; Zhou et al., 1985; Chou et al., 1986). The limitation of the non-local parameterization of the Bougeault and Lacarrère (1989) near the surface outlined by Bélair (1999) could also have an impact on vertical transport processes. Despite the discrepancies observed with some of the TKE budget terms simulated by both models, results are overall comparable to the results outlined by Stull (1988). Not only the typical shape of the budget terms is well reproduced but also most of the predicted values are in the range of values reported by Stull (1988). Overall, results from both models fit well at heights above z / z i = 0.1 , but show discrepancies below this height. Near the ground, the dissipation (-Di), the buoyancy (Th) and the production (Pr) terms are the most significant contributors in the TKE budget. However, at the upper layers of the atmosphere only dissipation and buoyancy become dominant for both of the dry and wet cases.

Mehrez Samaali, Dominique Courault, Albert Olioso et al. (a) 1500

H e ig h t (m )

1500

0

500

(b)

ENVImet-12h00 Meso-NH-12h00

0

H e ig h t (m )

ENVImet-12h00 Meso-NH-12h00

500

210

5

10

15

20

0

5

10

T-air (°C)

15

20

T-air (°C)

Figure 1. Vertical potential temperature profiles simulated with ENVImet and Meso-NH at 12:00 UTC: dry (a) and (b) wet soil cases.

(a) 1500 0

0

500

H e ig h t (m )

1500

(b)

ENVImet-12h00 Meso-NH-12h00

500

H e ig h t (m )

ENVImet-12h00 Meso-NH-12h00

0.0

0.2

0.4

0.6

0.8

1.0

0.0

0.2

0.4

E (m2/s2)

0.6

0.8

1.0

E (m2/s2)

(a) ENVImet

(b) Meso-NH

1.0

-Di Pr Th -Tr

0.0

0.5

z/zi 0.5

z/zi

1.0

-Di Pr Th -Tr

0.0

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1.5

1.5

Figure 2. Vertical turbulent kinetic energy profiles simulated with ENVImet and Meso-NH at 12:00 UTC: dry (a) and (b) wet soil cases.

-8

-6

-4

-2

0

2

4

Terms of the TKE budget (m2/s)

-8

-6

-4

-2

0

2

4

Terms of the TKE budget (m2/s)

Figure 3. Normalized terms of the TKE budget equation obtained from ENVImet (a) and Meso-NH (b) at 12:00 UTC for the dry soil case. All terms are divided by

w*3 / z i .

1.4

1.4

Examination of Two Turbulent Kinetic Energy Parameterizations…

(b) Meso-NH

1.2 1.0 z /z i

0.8

1.0 0.8 0.6

0.0

0.0

0.2

0.2

0.4

0.4

z /z i

-Di Pr Th -Tr

0.6

1.2

(a) ENVImet -Di Pr Th -Tr

211

-5

-4

-3

-2

-1

0

1

2

-5

-4

-3

Terms of TKE budget (m2/s)

-2

-1

0

1

2

Terms of TKE budget (m2/s)

Figure 4. Normalized terms of the TKE budget equation obtained from ENVImet (a) and Meso-NH (b) at 12:00 UTC for the wet soil case. All terms are divided by

Th

w*3 / z i .

Convective mixed layer

-Tr -Di

-Di

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Pr

Figure 5. Normalized terms of the TKE budget equation (Stull, 1988; pp 155; with kind permission of Springer Science and Business Media). These results are based on data and models from Deardorff (1974), André et al., (1978), Therry et Lacarrère (1983), Lenschow (1974), Pennell et LeMone (1974), Zhou et al., (1985) and Chou et al., (1986). Possible ranges of values are represented by the shaded areas. All terms are divided by

w*3 / z i .

Conclusion This chapter dealt with the comparison of local and non-local higher order TKE closure models tested in a convective boundary layer for dry and wet soil cases. The analysis of temperature as a key variable in TKE behaviour in the mixed layer showed the models’s

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Mehrez Samaali, Dominique Courault, Albert Olioso et al.

ability of capturing the temperature inversion for both cases despite the significant discrepancies (more than 4°C) obtained near the surface. In agreement with temperature overprediction near the surface, ENVImet heat surface fluxes were found to be higher than those of Meso-NH with differences attributed to land surface parameterization: particularly the radiative transfer calculation and the number of equations and surface parameters. Both ENVImet (local closure) and Meso-NH (non-local closure) were also able to reproduce quite well the shape of TKE profiles as well as values near the surface and above z / z i = 0.4 . Maximum discrepancies are below z / z i = 0.4 and are in the order of 0.2 m2/m2 and 0.1

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m2/m2 for the dry and wet cases, respectively. These discrepancies were attributed to the local closure technique used in ENVImet which may not dissipate energy enough and consequently over-predicts mixing in the convective layer. The comparison of TKE budget terms normalized by the convective velocity scale showed a reasonable agreement of buoyancy terms predicted by both the local and non-local closures. However, the dissipation term (-Di) showed significant discrepancies particularly near the ground which are higher in the dry case (4.5 m2/m3) than in the wet one (1.9 m2/m3). Despite that the dissipations for both models are in the range of values given by Stull (1988), these discrepancies were attributed to the method applied to calculate the dissipation within ENVImet, which consists of using the maximum between the Kolmogorov-based dissipation and the prognostic one and does not allow enough dissipation. Meso-NH dissipation values near the surface could be overpredicted since the Bougeault and Lacarrère (1989) non-local formulation has been proven to over-predict mixing near the surface and the atmosphere top (Bélair et al., 1999). The production (Pr) term vertical profile shape is well simulated by both models. Although, ENVImet values are slightly over-predicted near the ground, values of both models are comparable to the range values of Stull (1988). The transport term (-Tr) profile shape is well captured by ENVImet and agrees well with results of Stull (1988), particularly near the ground. However, Meso-NH seems to under-predict this term with values out of the Stull (1989) range of values. This could be due to the fact that the results of Stull (1989) may not cover all possible cases of a convective boundary layer driven by different ground heat flux despite that a reasonable number of studies were considered to construct Figure 5. Overall, both of the local and non-local closures reproduce quite well the profile shapes of the TKE budget terms and most predicted values are in agreement with the results shown in the literature (Stull, 1988). Profile values of both closures agree well at heights above z / z i = 0.1 , whereas discrepancies remain below this height. These discrepancies are related to the hypotheses assumed in each of these closures.

References Alapaty, K.; Pleim, J. E.; Raman, S.; Niyogi, D. S.; Byun, D. W.; J. Appl. Meteor. 1997: 36, 214–233. Andre, J. C.; DeMoor G.; Lacarrere, P.; Therry, G.; DuVachat, R. J. Atmos. Sci. 1978, 35, 1861-1883. André, J. La Météorologie, 1980, 22, 6-49. Aspley D.; Castro I. Bound. -Layer Meteor. 1997, 83, 75-98.

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Examination of Two Turbulent Kinetic Energy Parameterizations…

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Bechtold, P.; Bazile, E.; Guichard, F.; Mascart, P.; Richard, E. Quart. J. Roy. Meteor. Soc. 2001, 127, 869-886. Bélair, S.; Mailhot, J.; Strapp, J.W; MacPherson, J.I. J. Appl. Meteor. 1999, 38, 1499-1518 Blackadar, A.. K. Advances in Environmental and Scientific Engineering, J. R. Pfafflin and E. N. Ziegler, Eds., Gordon and Breach, 1979, I, 55-85. Mon. Wea. Rev. 1989, 117, 1872-1890. Bruse M. - Die Auswirkungen kleinskaliger umweltgestaltuung auf das mikroklima. Entwicklung des prognostischen numerichen modells ENVI-met zur simulation der wind, temperatur und feutchteverteilung in städtischen strukturen. Ph.D. thesis. University of Bochum (Germany), 1998, 1-186. Bruse, M.; Fleer H. Environ. Model. & Soft. 1998, 13, 373-384. Chou, S. H.; Atlas, D.; Yeh, E. N. J. Atmos. Sci. 1986, 43, 547-564. Clapp, R.B.; Hornberger, M. Water Resour. Res. 1978, 14, 601-604 Courault, D,; Lacarrere, P,; Clastre, P,; Lecharpentier, P,; Jacob, F,; Marloie, O,; Prévot, L.; Olioso A, Can. J. Remote. Sens. 2003, 29, 741-754. Cuxart, J.; Bougeault, P.; Lacarrère, P.; Noilhan, J.; Soler, M. R. Bound. -Layer Meteor. 1994, 67, 251–276. Cuxart, J.; Bougeault, Ph.; Redelsperger, J.L. Quart. J. Roy. Meteor. Soc. 2000, 126, 1-30. Deardorff, J. J. Geophys. Res. 1978, 83, 1889-1903. Deardorff, J. W. Boundary Layer Meteorol. 1974, 1, 81-106. Detering, H.; Etling, D. Bound. -Layer Meteor. 1985, vol 33, 113-133. Duynkerke, P. J. Atmos. Sci. 1988, 45, 865-880. Duynkerke, P. J. Atmos. Sci., 1988, 45, 865-880. Huang C.; Raman S. Bound. -Layer Meteor. 1991, 55, 381-407. Jacquemin, B.; Noilhan, J. Bound. -Layer Meteor. 1990, 42, 93-134. Kolmogorov, A, N. Doklady Akad. Nauk SSSR 32,. 1941, 538-541 Launder, B.; Spalding, D. Computational Methods in Applied Mechanical Engineering, 1974, 3, 269-289. Lenschow D. H. J. Atmos. Sci., 1974, 31, 465–474. Mailhot, J. Lock, A. 16th Symposium on Boundary Layers and Turbulence, AMS meetings, Portland, ME, 9-13 August 2004, 1-4. Monin, A..S.; Obukhov, A.M. Akad. Nauk SSSR Geofiz. Inst. Tr, 1954, 151, pp 163-187. Morcrette, J. J., J. Geophys. Res. 1991, 96, 9121-9132. Noilhan, J.; Mahfouf, J. Global & planet. Chang.. 1996, 13, 145-159. Noilhan, J.; Planton, S. Amer. Meteor. Soc., 1989, 117, 536-549. Olioso, A. . et al. Agronomie, 2002, 22, 597-610. Olioso, A.. Simulation des échanges d'énergie et de masse d'un couvert végétal, dans le but de relier la transpiration et la photosynthèse aux mesures de réflectance et de température de surface. Ph.D. Université de Montpellier II, Montpellier (France), 1992, 254 p. Pennell, W. T.; LeMone, M. A. J. Atmos. Sci., 1974, 31, 1308-1323. Pielke, R.A.; Mahrer, Y. J. Atmos. Sci. 1975, 32, 2288-2308. Prevot, L. Modélisation des échanges radiatifs au sein des couverts végétaux - application à la télédétection, validation sur un couvert de maïs. Ph. D. thesis University of Paris VI, 1985, 178 p.

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Samaali, M. Evaluation d’un modèle de couche limite atsmospherique dans un cas homogène: Application à une parcelle agricole de soja. Ph.D. thesis. Ecole polytechnique universitaire de Marseille (France), 2002, 209 p. Samaali, M.; Courault,D.; Bruse, M.; Olioso, A..; Occelli R. Atmos. Res. 2007, 85, pp 183198. Stull, R. An introduction to Boundary Layer Meteorology; ISBN 90-277-2768-6; Kluwer Academic Publishers, Netherlands, 1988; pp 1-250. Stull, R. Bound. -Layer Meteor. 1993, 62, 21-96. Therry G., Lacarrère P. Bound. -Layer Meteor. 1983, 25, 247-266. Tuzet, A.; Perrier, A.; Aissa, A. Agri. & Forest Meteor. 1993, 65, 63-89. Verhoef, W. Remote Sens. of Envi. 1985, 17, 165-178. Verhoef, W. Remote Sens. of Envi., 1984, 16, 125-141. Wilson, J. Bound. -Layer Meteor. 1988, 42, 371-392. Wright, S. D.; Elliott, L.; Ingham, D. B.; Hewson, M. J. C. Bound. -Layer Meteor. 1998, 89, 175-195. Xu D.; Taylor P. Bound. -Layer Meteor. 1997, 84, 247-266. Yamada, T.; Mellor, G. J. Atmos. Sci. 1975, 32, 2309-2329. Zhang, Q.; Stull, R. J. Atmos. Sci. 1992, 49, 2267-2281. Zhou, M.Y.; Lenschow, D. H.; Stankov, B. B.; Kaimal, J. C., Gaynor, J. E. J. Atmos. Sci., 1985, 42, 47-57.

In: Energy Conservation: New Research Editor: Giacomo Spadoni, pp. 215-245

ISBN 978-1-60692-231-6 c 2009 Nova Science Publishers, Inc.

Chapter 9

E NERGY C ONSERVATION IN N ANOELECTRONICS J. Hoekstra∗ Delft University of Technology, The Netherlands

Abstract

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Conservation of energy is a general principle in nature, however, some modern theories on nanoelectronics seem to violate this principle. For example, the orthodox theory of single-electronics predicts tunneling of electrons through a junction in nanoelectronic structures when the free-energy of the system is lowered as a result of the tunnel event. But the energy loss cannot always be modeled as dissipation by heat or by radiation in this theory. Especially, when the tunnel event of a single electron is considered the dissipation cannot be modeled by a resistance. The impossibility to model radiation is due to the description of the system in terms of (electronic) circuit theory. Due to the fact that the tunnel junction in Coulomb blockade is modeled as a capacitor, energy conservation in these nanoelectronic circuits can be investigated by studying a classical problem in circuit theory: the switched two-capacitor network. Subsequently, energy conservation in the two-capacitor network is discussed in case of both bounded and unbounded currents. Because nanoelectronic tunneling of single electrons can be modeled with an unbounded current, it is shown that many common features exist.

1.

Introduction

Energy conservation is a consequence of symmetry of nature under translation in time. In quantum mechanics, laws are symmetrical under translation in time [1]. In classical physics, conservation of energy is an axiom; however, no exceptions are known until now. Whenever energy is lowered in a system, dissipation in the form of heat or radiation has to be present. In the theories describing electronic circuits such as today’s microelectronic integrated circuits (ICs) or computer chips, conservation of energy holds too— although the circuit theories are only a rough approximation of the physical reality described by Maxwell’s equations. The conservation of energy in electronic circuit theory is a consequence of a theorem known as Tellegen’s theorem. Knowing the above, it is remarkable ∗

E-mail address: [email protected]

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J. Hoekstra

that some almost generally accepted models for modern nanoelectronic circuitry seem to violate energy conservation, because in those models both dissipation by heath and by radiation cannot be modeled. As an example of nanoelectronic circuitry, the nanoelectronic metallic single-electron tunneling junctions (metallic SET juctions) are considered. Schematically, the metallic tunnel junction consists of two metal conductors separated by a very thin insulator (typically: 5 nm < d < 10 nm), see Figure 1. Due to the extreme small insulator thickness quantum mechanical tunneling of electrons through this insulator becomes possible.

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Figure 1. (a) Metallic single-electron tunneling (SET) junction schematically: two metal leads separated by a thin layer of insulating material; (b) symbol for the SET junction circuit element. Until 2007 almost generally accepted, the circuit theory for these junctions named orthodox theory of single-electronics ( [2] and refs. therein) allows only quantum mechanical tunneling when the (electrostatic) free-energy of the system lowers during tunneling, that is, the free energy of the circuit after tunneling is lower than the free-energy before. If the energy cannot be lowered, the circuit is said to be in Coulomb blockade. Energy must be lost during a single tunnel event, however, this loss cannot be modeled in the circuits as dissipation in a resistor—because the tunneling time is assumed to be zero in the orthodox theory. In addition, Coulomb blockade in nanoelectronic circuits including resistors cannot be modeled unambiguously; the orthodox theory distinguishes different tunnel conditions for low- or high ohmic environments. As an example consider the nanoelectronic single-electron box excited by an ideal voltage source, see Figure 2. The electron box consist of a tunnel junction (TJ) and a capacitor. The junction will tunnel when the voltage across the junction exceeds the so-called critical voltage V cr . In a low ohmic environment tunneling is possible when the (electrostatic) free energy G = G(W, t) in the circuit is lowered. In the circuit context, P the change in free energy ∆G is defined by the change in energy stored on i capacitors i ∆wsei just before and just after a tunnel event and the energy delivered or absorbed by sources during the tunnel event. If Qsk is the charge transported through the k-th voltage source Vsk then ∆G =

X i

∆wsei −

X k

Qsk Vsk .

(1)

Energy Conservation in Nanoelectronics

217

Figure 2. Nanoelectronic electron-box system and its electrical circuit equivalent according to the orthodox theory of single-electronics, before the actual tunneling the tunnel junction is modeled with a capacitor. According to the orthodox theory tunneling is possible if ∆G < 0.

(2)

By modeling the tunnel junction as a capacitor, however, the system falls within the domain of circuit theory. In circuit theory, energy is conserved in every circuit—based on Tellegen’s theorem. The theorem also holds for circuits consisting of (ideal) voltage sources and capacitors. The scientific question this chapter is investigating is whether ∆G could be unequal to zero in a nanoelectronic circuit theory, in other words, that always holds that ∆G = 0

(3)

for any valid nanoelectronic circuit—that is, a nanoelectronic circuit for which the Kirchhoff laws hold. It is shown that the latter is true.

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2.

Energy Conservation in Classical Electromagnetism

This section reviews briefly energy conservation in classical electrodynamics. The purpose for this review is to formulate the equations that form the starting point for the approximations that are used in the description of (nano)electronic circuits. In the description it is assumed that electrons in the circuits move at speeds much lower than the speed of light. The Maxwell equations form a suitable starting point, combined with the expression of the Lorentz force F = F(r, t) on the electrons. Experiments show that: F is proportional to the electron’s fundamental charge e; F is proportional to its velocity v measured by an observer in the direction perpendicular to v; F has a term that is independent of v. The result is that F = eE + ev × µ0 H,

(4)

in which µ0 is the permeability of the vacuum. The electromagnetic field vectors E = E(r, t) and H = H(r, t) are called the electric field strength and the magnetic field strength

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J. Hoekstra

at the location of the electron, respectively, and both are produced by all the other electrons that are present.

2.1.

Maxwell’s Equations for Vacuum in the Presence of Electrons and Currents

We start with the description of the properties of the electromagnetic field vectors in an area in which electrons or currents are present. This has to be done carefully, because the electromagnetic field vectors can not be measured directly in such area. A suitable starting point for the description of the electromagnetic field vectors is Gauss’s law. Therefore a closed surface S is chosen such that S is in vacuum and all charges are inside. Based on experiments, Gauss’s law may then be stated: the net electric flux through a closed surface S equals the total net charge inside Q the surface divided by the permittivity of the vacuum ǫ0 I Q (5) n · E dA = , ǫ 0 S If ρ = ρ(r, t) is the charge density inside the S, we write Z Q= ρ dV V

Assuming that, in these cases, E is continuously differentiable Gauss’s theorem may be applied, relating the flux of E out of a surface with divergence of E. This results in: Z Z −1 div E dV = ǫ0 ρ dV, (6) V

V

or div E =

ρ . ǫ0

(7)

This is Gauss’s law of the electric field. Experiments measuring H on the surface S reveal: I n · H dA = 0,

(8)

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S

or div H = 0.

(9)

This is Gauss’s law for magnetism, expressing that magnetic monopoles do not exist. Experiments show that moving electrons and currents generate a magnetic field. The correct relation between the current density and the magnetic field strength was found by Maxwell. If J = J(r, t) is the current density: curl H = J + ǫ0

∂E , ∂t

(10)

which holds for all currents. Also holds curl E = −µ0

∂H , ∂t

(11)

Energy Conservation in Nanoelectronics

219

known as Faraday’s law of induction. Equations 7, 9, 10, and 11 are the Maxwell equations for vacuum in the presence of charges and currents as put forward by Lorentz in 1892 in his electron theory (in modern notation); the charges or currents are always described inside a closed surface, while the electromagnetic field vectors are measured along or outside the closed surface. We conclude this subsection by making two further points regarding the Maxwell equations. First, inside conductors—or more general inside matter—electromagnetic field vectors also exist. Because they cannot be measured directly, they are introduced axiomatically with the vectors: P the electrical polarization, M the magnetization, D the electric flux density or displacement, and B the magnetic flux density or magnetic induction. Second, for the accurate description of currents the Maxwell equations are completed with the continuity equation, expressing the conservation of charge: ∂ρ + div J = 0. ∂t

2.2.

(12)

Conservation of Energy and Poynting’s Theorem

The electromagnetic field will exert a Lorentz force given by Equation 4 on a free electron. The work done by the applied force on the electron, when it moves through a displacement, ∆W is defined to be ∆W = F · ∆r and the rate at which the work is done is the power P

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P =

dW = F · v = eE · v, dt

(13)

because v · (v × B) = 0; the magnetic field does no work on the charged particle. If we interpret v as the average velocity of the electrons in a unit volume and using J = ρv we define dw = E · J, (14) dt in which w is the work per unit volume. The expression obtained is for the energy associated with the motion of the electrons. What we would like to have is an expression for the energy that applies to a general distribution of electrons and currents (in ρ and J). This expression can be found by using the Maxwell equations in vacuum in the presence of currents. Multiplying Equation 10 by E and Equation 11 by H and subtracting one from the other, we get using div(E × H) = −E · curl H + H · curl E: E · J = −ǫ0 E ·

∂E ∂H − µ0 H · − div (E × H). ∂t ∂t

(15)

Using E · ∂E/∂t = 1/2∂(E · E)/∂t and H · ∂H/∂t = 1/2∂(H · H)/∂t one finds E·J+

∂ 1 1 ( ǫ0 E · E + µ0 H · H) + div (E × H) = 0. ∂t 2 2

(16)

The standard interpretation of this equation is found after integration of Equation 16 over a closed area V1 in which a source area — an area within which electromagnetic

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J. Hoekstra

source Vsource Ssource

n

V1

n

V2

S

Figure 3. Exchange of electromagnetic energy. energy is generated — is excluded, see Figure 3. The result of the integration is Z Z I I ∂ 1 1 E · J dV + ( ǫ0 E · E + µ0 H · H)dV + n · (E × H)dA = n · (E × H)dA, ∂t V1 2 2 V1 S Ssource (17) where Z I I (18) div (E × H)dV = n · (E × H)dA − n · (E × H)dA V1

S

Ssource

was used; Equation 17 is known as Poynting’s theorem. The physical interpretation of Equation 17 is the following. First note that all of the terms (integrals) in the expression have units of energy per unit time [J/s] or power [W]; therefore, this relation is customarily viewed as a statement of conservation of energy. We examine all the terms in the theorem. The first term represents the rate at which energy is exchanged between the electromagnetic field and the mechanical motion of the charge within the volume V1 , see Equation 14. The second term represent the electromagnetic energy stored in the electric field and the electromagnetic energy stored in the magnetic field in the volume V1 . The third term I def P == n · (E × H)dA (19) Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

S

is interpreted as the instantaneous power transported by the electromagnetic field from the volume V1 to the volume V2 through the surface S. In the same way the last term I n · (E × H)dA (20) Psource def == Ssource

can be interpreted as the instantaneous power provided by the energy source through Ssource to the volume V1 . Equation 17 can, now, be written as: dWmech dWem + + P = Psource dt dt explicitly showing the conservation of energy.

(21)

Energy Conservation in Nanoelectronics

3.

221

Current and Energy Conservation in Circuits

In a classical context, currents in electronic circuits are introduced by the so-called steadystate currents approximations of the continuity equation and the Maxwell equations. The steady-state currents approximation refers to a special kind of dynamic situation with currents consisting of large numbers of conduction electrons in motion that can be approximated by a steady flow of electrons. In this approximation the charge density ρ(x, t) does not change with time, that is, ∂ρ/∂t is zero. Consequently, ∂E/∂t will be zero too.

3.1.

Steady-State Currents in General

Applying ∂ρ/∂t = 0 to the continuity equation and to Maxwell’s equation for the magnetic field, we obtain for steady-state currents: div J = 0

(22)

curl H = J.

(23)

and Equation 22, div J = 0, for steady-state currents has some important consequences. In particular, it leads to Kirchhoff’s current law. Consider therefore the integral form of Equation 22: I n · J dA = 0. (24) S

Suppose we have a node in which several currents come together through N electrodes. Choose S to be the surface of the node. On the isolated parts of S holds: n · J = 0.

(25)

Through the N electrodes current can flow towards and from the node. Applying Equation 24 on these electrodes immediately leads to: N X

In = 0.

(26)

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n=1

Equation 26 is known as Kirchhoff’s current law (KCL). Any steady-state current produces a magnetic field, as was discovered by Oersted in 1820. curl H = J. This H field can be calculated with the help of the so-called vector potential A. Gauss’s law for magnetism div H = 0 is automatically satisfied by setting H = µ−1 0 curl A.

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J. Hoekstra

A is called the vector potential and is arbitrary to within a gradient. The derivation of A can be found in standard textbooks and is given here for reference. In the Coulomb gauge the expression for the vector potential is Z Jp′ dVp′ µ0 Ap = . (27) 4π rpp′

3.2.

Constant Currents

Ohm’s law for resistive circuit elements can be found if we consider constant steady-state currents, in which case the magnetic field will be static too. In general, steady-state currents are define by the requirement that all fields are time independent, that is, ∂ρ/∂t def == 0, def 0; in fact, the steady-state currents describe constant currents ∂E/∂t def 0, and ∂H/∂t == == in circuits. In this approach, the Maxwell equations will become curl H = J,

div H = 0

(28)

and

ρ . (29) ǫ0 The expression curl E = 0 shows that in case of steady-state currents one of the Maxwell equations reduces to the same expression as the equation expressing energy conservation in a conservative field: I I I E·ds = e E·ds = F·ds = 0. curl E = 0,

div E =

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It shows no net work can neither be gained nor lost when a charged particle traverses a closed path in the electric field in case of steady-state currents. An alternative expression for this is E = −grad V, (30) that is, a potential difference V = u1 − u2 can be defined between two points in the electric field (u1 and u2 are called the potential at point 1 and point 2, respectively), which value does not depend on the path taken. Now, the resistance can be introduced by using the experimental fact that when a current flows in a material, the following phenomenological relation exists between the vectors J and E. If σ is the conductivity of the material used then J = σE.

(31)

The above equation is called Ohm’s law, and holds for those values of the electric field for which the relation between J and E is linear. What happens in a homogeneous conductor, having a resistance, when it is carrying a current? We know that the electromagnetic energy to maintain the current is supplied by the source (source area). And, since the conductor has resistance, there is an electric field in the conductor causing the current; without this field the electrons will not move other than in arbitrary directions due to temperature. There is also an electric field just outside the resistive conductor—the field we can measure.

Energy Conservation in Nanoelectronics

223

For a homogeneous conductor, σ is independent of position, constant cross-sectional area A and length l using Ohm’s law we obtain u1 − u2 = l|E|

(32)

I = A|J|

(33)

u1 − u2 = V = RI

(34)

and so that where R = l/(σA). R is the resistance and depends upon the physical parameters of the conductor. Equation 34 is also valid for homogeneous conductors of arbitrary shape [3] and is called Ohm’s law for the resistor. For the last consequence of a constant current treated here, consider energy conservation in case of a circuit consisting of an energy source having a constant voltage and a conductor with a resistance R. In steady-state we obtain from Equation 21 that |Pdiss | = |Psource |, if we do not allow energy to leave the circuit (remember that the fields are static so radiation cannot be included), that is, the electromagnetic energy delivered by the source is converted totally into heat in the conductor. By considering the work per unit time done by the force on the electrons, F = eE, the heat developed per unit volume per unit time is, see Equation 14, Q=J·E

(35)

and is called Joule heat. The total heat produced per second in a homogeneous conductor of length l and cross-sectional area A is Q = J · EAl = I(u1 − u2 ) = I 2 R = Pdiss .

(36)

On the other hand the power delivered by the source area: I Psource = n · (E × H)dA S

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can be found to be equal to [4] Psource =

Z

V

E · JdV,

(37)

using −div (E × H) = −Hcurl E + Ecurl H, curl E = 0, and curl H = J. Consequently, Psource = −Vs I, with V the potential difference across the source. This can be seen by noting that Z Z grad V · JdV. E · JdV = − V

V

Using div(V J) = V div J + J · grad V

(38)

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J. Hoekstra

one finds =−

Z

div (V J)dV +

V

Z

V div JdV.

V

The last term is zero and V is constant (steady state). Using Gauss’s theorem the first term becomes I −V n · JdA = −V I, S

where I is the total current delivered by the source. Equating the power absorbed in the resistor and the power delivered by the source for equal currents (charge conservation) we find: I(u1 − u2 ) = −Vs I and Vs + (u1 − u2 ) = 0,

(39)

the sum of voltages must be zero. We recognize what is known as Kirchhoff’s voltage law in circuit theory. Now, we are going to consider non-stationary currents.

3.3.

Time-Dependent Current Flow

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Switching a current on also requires energy. A constant current causes a magnetic field; a in time changing current will cause a in time changing magnetic field. This effect could not be taken into account in the case of a steady-state current. The description of changing currents relies on a quasi-static fields description, that is, the fields only change a little during the time required for light to traverse a distance equal to the maximum dimension of the system under consideration. This is called the quasistationary approach. The quasi-stationary approach takes into account a nonzero term ∂H/∂t, but ∂E/∂t and ∂ρ/∂t are still considered to be zero, as is done in the steady-state approach. As a consequence, we can still use Kirchhoff’s current law and a changing magnetic field can be taken into account to model time-dependent current flow in circuits. Starting point is Faraday’s law of induction Equation 11, curl E = −µ0

∂H . ∂t

(40)

The vector potential is used to express the amount of energy necessary to build up the magnetic field. Starting with Z 1 W = µ0 H2 dV (41) 2 we find W =

1 2

Z

H· curl A dV.

After partial integration we obtain Z Z 1 1 A·curl H dV = A·J dV. W = 2 2

(42)

(43)

Energy Conservation in Nanoelectronics

225

Using Equation 27 we find µ0 W = 8π

Z

J p J p′ rpp′

dVp dVp′ .

(44)

This equation can be written as

1 (45) W = LI 2 , 2 with I the time varying current through the circuit that causes the magnetic field. The quantity L, the coefficient of self-inductance is defined such that: Z J p J p′ µ0 2 LI = dVp dVp′ , (46) 4π rpp′ L only depends upon the geometry of the circuit. Again consider a circuit with a conductor with resistance R and satisfying Ohm’s law J = σE. From Faraday’s law and the definition of the vector potential it follows that curl (E +

∂A ) = 0. ∂t

Defining a new potential V leads to a generalization of the electrostatic relation E=−

∂A − grad V . ∂t

(47)

The heat produced in the conductor is again I 2 R. The source delivering Z Z Z ∂A dV. E · JdV = − grad V · JdV − J· ∂t V V V Using the same kind of manipulations as in the description of the energy conservation in case of a constant current and filling in Equation 27 for the vector potential, one obtains: Z I dI E · JdV = −V n · JdA − LI , dt V S Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

or

dI dt Thus, energy conservation leads to the important differential equation: I 2 R = −IV − LI

L

dI + RI = −V dt

(48)

where I is the total current delivered by the source. We see that the magnetic field is modeled by a self inductance and again the sum of the voltages is zero if we associate V with the voltage of an energy source. In case of a circuit with a capacitor an additional term is required. In circuits in which a conductor is used to charge a capacitor, the current is time-dependent and we can use Equation 48 with a term relating the voltage across the capacitor, as a consequence of the

226

J. Hoekstra

electric field that is build up by the charge loaded on the capacitor, and the amount of charge Q stored on the capacitor with capacitance C. We obtain Q dI + RI + = −V, dt C

(49)

d2 I dI I dV +R + =− . dt2 dt C dt

(50)

L or, differentiated, L

4.

Energy Conservation in Lumped Circuits

As discussed in the introduction, the effects of tunneling and Coulomb blockade are modeled by lumped circuits. In lumped circuits, the elements have no extension, thus are point like, and the voltage across any element and the current through any element are always well-defined. As discussed in the previous section, from a classical physics and electromagnetic theory point of view using Maxwell’s equations, the lumped circuit is only an approximation: it assumes instantaneous propagation of electromagnetic waves through the circuit, steady-state currents or a quasi-static approach. Energy is conserved and if applied to circuits that cannot radiate, Kirchhoff current and voltage laws could be obtained. In the theory describing lumped circuits, from now on called (electronic) circuit theory, the Kirchhoff’s laws enter axiomaticly. The topology of the a circuit specifies the location of nodes and elements or more precise: nodes and branches; a branch may represent any two-terminal element, and the terminals are called nodes. In discussing circuits the topology is described with nodes n, branch (element) currents in , and branch (element) voltages vn . Kirchhoff’s current law (KCL) and Kirchhoff’s voltage law (KVL) are the two fundamental postulates of circuit theory. They hold irrespective of the nature of the elements constituting the circuit. Both reflect topological properties of the circuit. Kirchhoff’s current law for nodes states that, for all times t, the algebraic sum of the currents i1 , i2 , ..., in leaving any node is equal to zero: X in (t) = 0. (51)

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n

Given any connected lumped circuit having n nodes, we may choose arbitrarily one of these nodes as a datum node, i.e., as a reference for measuring (electric) potentials u1 , u2 , ..., un−1 . Let vi denote the voltage (potential) difference across branch (element) i connected between node k and j. Kirchhoff’s voltage law (KVL) states that, for all choices of the datum node, for all times t, and for all pairs of nodes k and j: vi (t) = uk (t) − uj (t).

(52)

If we now consider a closed node sequence, thus starting end ending at the same node, the voltage difference must be zero. Now, Kirchhoff’s voltage law states that, for all closed node sequences (loops), for all times t, the algebraic sum of all node-to-node voltages (voltage differences) around the chosen closed node sequence is equal to zero: X vn (t) = 0. (53) n

Energy Conservation in Nanoelectronics

227

The KCL and KVL always lead to homogeneous linear algebraic equations with constant real coefficients. Modelled by lumped elements, nanoelectronic devices will be characterized by equivalent circuits. Since the devices are essentially physical devices, the properties we are interested in are closely related to the physics of the device; in particular, we are interested in energy considerations to make a transition possible from a (quantum) physics formulation of nanoelectronics to a circuit theory. In quantum mechanics, the electron tunneling phenomenon is described by the time-independent Schr¨odinger equation: if EP (x) is the potential energy of the electron and E is the total energy, m is the mass, and ψ the wave function of the electron, then (for one-dimensional problems) −

¯2 d 2 ψ h + EP (x)ψ = Eψ, 2m dx2

(54)

where ¯h is Planck’s constant divided by 2π. However, as soon as circuits including tunneling junctions are considered, these circuits are studied using the fundamental quantities voltage and current. We have to find a link between the energies in Equation 54 and the circuit quantities. The formulation of the overall equations involves the combination of the elements with topological properties of the network, and the reduction of all of this information to a set of differential equations in some convenient form. Also the question must be addressed whether it is possible to include tunneling devices—or any other new nanoelectronic device—in electrical circuit theory, and what kind of restrictions there are. From the energy formulation of quantum mechanics, using the Hamiltonian, it is clear that energy in lumped circuit may play an important role. This is the case. Circuit models for the nanoelectronic single-electron tunneling devices and the associated Coulomb blockade effect are described in terms of energy, whether it is energy minimization or just energy conservation. Ultimately, it will be experimental data that must show that quantum mechanical devices can be modeled accurately by circuit models.

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4.1.

Energy Calculations in Some Simple Circuits and Networks

In circuit theory, a circuit is characterized by one or more sources interconnected with one or more receivers, or sinks of electrical energy. In fact, in the circuit energy is supplied by sources, transferred from one place in the circuit to another, and temporarily stored or dissipated in circuit elements. As a first step, energy in some simple circuits is considered. A reasonable lumped circuit to model the charging of a linear capacitor is shown in Figure 4, if we assume that the resistance of the wire is substantial enough and the current is not changing very fast. Supposing that the capacitor is discharged totally during t < 0 s, it will start charging at t = 0 s. The capacitor will be charged with an amount q until the voltage across the capacitor equals Vs (steady-state). Kirchhoff’s voltage law demands vs (t) = i(t)R + q(t)/C. In the interval dt the voltage source delivers the energy dw: dw = Vs idt = Vs dq = i2 Rdt +

q dq. C

(55)

228

J. Hoekstra

i(t) + vs(t)

vs R

Vs

C

0

t

Figure 4. Lumped circuit model for charging a capacitor. The first term is the energy dissipation in the resistor, the second the amount of energy stored in the capacitor. In steady-state the voltage across the capacitor is Vs = q0 /C and the current is zero. The total amount of energy delivered by the source is: Z ∞ Z 1 q0 ws = i2 Rdt + qdq = C 0 0 1 2 1 q 0 = wR + q 0 V s . (56) 2C 2 During charging the source provided an amount of energy of ws = q0 Vs , while the capacitor stored an amount of energy equal to wC = (1/2)CVs2 = (1/2)q0 Vs ; we see that precisely half of the energy is dissipated in the resistor. This result holds for any non-zero value of the resistor, even for very small values. Notice that the source supplies additional energy to charge the capacitor and that energy is conserved in the circuit, because this additional energy is dissipated in the resistance. Equation 55 seems to indicate that in case of R = 0 Ω the resistor term disappears and the source is providing twice as much energy as is stored on the capacitor, giving rise to the speculation that the other half of the energy is released as radiation in the circuit. To investigate this claim, consider the current in the circuit. By differentiating the expression obtained by the Kirchhoff’s voltage law and making use of the constitutional equation of the capacitor iC (t) = CdvC (t)/dt = i(t) we obtain

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wR +

dvs (t) di(t) i(t) =R + . dt dt C

(57)

Solving this differential equation and applying the initial condition at t = 0 s, the solution is found as Vs t i(t) = exp(− ). (58) R RC Taking the limit R → 0 the current in the circuit will become infinite. As the total charge stored on the capacitor is independent of R, the current can be expressed with the (Dirac) delta pulse: i(t) = CVs δ(t). (The definition used for the Dirac delta pulse, here, is δ(t) = lim ae−at ; a→∞

other definitions also exist.)

(59)

Energy Conservation in Nanoelectronics

229

An infinite current, however, is incompatible with the concept of resistance, in which charges must have a finite speed. In addition, the energy dissipation wR (t) = i2 (t)R in a resistor cannot be defined for infinite currents, because the square of a delta function is not defined in the theory of generalized functions. To clarify energy calculations in capacitor networks without resistors we consider the network of Figure 5. Note that this network describes really the charging of a capacitor without a resistor (the description of a circuit with even a very small resistor would fall within the previous description); circuits that are less realistic for modeling reality will be called networks from now on.

i(t)

vs

+

q(t < 0) = 0

C

vs(t)

Vs 0

∆t

t

Figure 5. Network to describe the charging of a capacitor with a finite current (bounded current) without a resistor. To analyze energy in the network of Figure 5, we determine v(t), i(t), the energy stored on the capacitor wse , and the total energy delivered by the (voltage)source ws . The capacitor is defined by the relation V = C −1 q. Knowing the voltage as a function of time  for t < 0  0 (Vs /∆t)t for 0 ≤ t ≤ ∆t v(t) =  Vs for t > ∆t we can easily find the current in the circuit using i(t) = C

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:

  0 CVs /∆t i(t) =  0

dv(t) dt for for for

(60) t ∆t.

We can calculate the energy stored on the capacitor after t > ∆t to be wse = 1/2(CVs2 ). The energy delivered by the source in the interval 0 < t < ∆t can be found as: ws =

Z

∆t 0

CVs2 ivdt = ∆t2

Z

∆t

tdt = 0

1 CVs2 2 1 ∆t = CVs2 . 2 ∆t2 2 We notice that energy is conserved in this network. =

230

J. Hoekstra

i(t)

vs

+ C

vs(t)

q(t < 0 ) = 0

Vs t

0

Figure 6. Network to describe the charging a capacitor with an infinite current (unbounded current) without a resistor. To go back to our original challenge, the energy description in the capacitor circuit of Figure 4 in the limit R = 0 Ω, consider the limit ∆t ↓ 0, see Figure 6. We can easily perform the calculations and find that also in this case the amount of energy delivered by the source is ws = 1/2(CVs2 ). That is, energy is conserved and there is no need for the introduction of energy loss by radiation. Also in this case the current can be described by a (Dirac) delta pulse, as is seen in Figure 7. While ∆t approaches zero, the current grows to infinity but the area in the it-diagram stays the same, namely CVs . Following the mathematical definitions of the step and delta pulse, see also the appendix: mathematical preliminaries, one finds for the current i(t), i(t) = Cdv/dt = CVs d(ǫ(t))/dt = CVs δ(t). The delta current pulse CVs δ(t) that transports in zero time an amount of charge Q = CVs

vs

vs Vs

Vs 0 i

∆t

t

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t

CVs

CVs/∆t 0

0 i

t

0

t

Figure 7. Voltages and currents in circuits of Figures 5 and 6. is, generally, called an unbounded current. I want to point out that we can also calculate directly the energy provided by the source by noting that while the source steps from 0 V to Vs the current through the source is the delta pulse CVs δ(t). And, the energy provided by the source ws is Z ∞ Z ∞ 1 (61) |ws | = ivdt = CVs2 δ(t)ǫ(t)dt = CVs2 . 2 −∞ −∞

Energy Conservation in Nanoelectronics

231

Where we used that the integral can be calculated by partial integration and realizing that the delta pulse is defined as the derivative of the step function: Z ∞ 1 1 ∞ δ(t)ǫ(t)dt = [ǫ2 ]−∞ = . (62) 2 2 −∞ The important difference with the circuit described in Figure 4 is the absence of the resistor. Also based on energy arguments, it shows that an impulsive current is not compatible with the presence of a resistor in the circuit. Finally, we notice that the network of Figure 6 represents a switched network, as shown in Figure 8.

i(t)

t=0

+ Vs

C

q(t < 0) = 0

Figure 8. Switched capacitor network with an unbounded current when the switch is closed.

4.2.

Tellegen’s Theorem

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We saw in the previous subsection that energy was conserved in all the circuits and networks that were considered. This is no coincidence, it is a general property of lumped circuits and based on a theorem called Tellegen’s theorem. Tellegen’s theorem is general in that it depends solely upon Kirchhoff’s laws and the topology of the network. It applies to all (sub)circuits and elements, whether they be linear or nonlinear, time-invariant or time variant, passive or active, hysteretic or nonhysteretic. The excitation might be arbitrary; the initial conditions may also be arbitrary. It describes elements as ”black boxes” only assuming a circuit topology and the validity of Kirchhoff’s laws.

Figure 9. ”Black box” description of an element as a one port. Tellegen’s theorem (1952) can be formulated as follows. Suppose we have two circuits with the same topologies, and all N elements are represented by a one-port as in Figure 9. For each element k in one circuit (the unprimed circuit), there is a corresponding element k′ in the other circuit (the primed circuit). If we now take the voltage across element k, vk , and multiply it with the current through the corresponding element k′ , i′k , and sum this for

232

J. Hoekstra

all N elements then:

N X

vk i′k = 0.

(63)

k=1

To fully understand this theorem consider the next example [5], see Figure 10.

Figure 10. Example of the Tellegen’s Theorem We have a unprimed circuit and a primed circuit consisting of three elements a, b, and c. Using Kirchhoff’s voltage law the voltages v across the elements are expressed in potentials u with respect to the datum (ground). Now we can easily check that: va i′a + vb i′b + vc i′c = u1 i′a + (u1 − u2 )i′b + u2 i′c =

u1 (i′a + i′b ) + u2 (−i′b + i′c ) = u1 0 + u2 0 = 0,

1 and 2 sum up to zero. because according to Kirchhoff’s current law the currents at node

4.3.

Proof of Tellegen’s Theorem

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Consider a general element in a given circuit—the unprimed circuit—and a corresponding element in another circuit—the primed circuit—having exactly the same topology. Because α and β in the unthe topologies are the same, there is for any element between nodes primed circuit a corresponding element between the same nodes in the primed circuit. We assume that there are N elements and nt nodes and investigate the sum N X

vk i′k = 0.

k=1

Let us assume, for simplicity, that the circuit has no branches in parallel; i.e., there exists only one branch between any two nodes. The proof can be easily extended to the general case. (If branches are in parallel, replace them by a single branch whose current is the sum of the branch currents. If there are several parts, the proof shows that Tellegen’s theorem holds for each of them. Hence it holds also when the sum ranges over all branches of the graph.) 1 . Thus, We first pick an arbitrary node as a reference node, and we label it node u1 = 0. Let uα and uβ be potentials of the αth node and the βth node, respectively, with

Energy Conservation in Nanoelectronics

233

α and node β Figure 11. An arbitrary branch, k , connects node

respect to the reference node. It is important to note—in both the unprimed circuit and the primed circuit—that once the branch voltages (v1 , v2 , ..., vN ) are chosen, then by KVL the node potentials u1 , u2 , ..., uα , ..., uβ , ... are uniquely specified. Let us assume that branch k α and node β as shown in Figure 11, and let us denote by i′αβ the current connects node α to node β in the primed circuit. The voltage vk is the flowing in branch k from node voltage across element k in the unprimed circuit, then vk i′k = (uα − uβ )i′αβ

(64)

β to node α Obviously, vk i′k can also be written in terms of i′βα , the current from node in the primed circuit , as vk i′k = (uβ − uα )i′βα

Adding the above two equations, we obtain

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vk i′k =

 1 (uα − uβ )i′αβ + (uβ − uα )i′βα 2

(65)

Now, if we sum the left-hand side of Equation 65 for all the branches in the graph we obtain N X vk i′k k=1

The corresponding sum of the right-hand side of Equation 65 becomes, using i′βα = −i′αβ , n

n

t X t 1X (uα − uβ )i′αβ 2

α=1 β=1

where the double summation has indices α and β carried over all the nodes in the graph. The sum leads to the following equation: N X k=1

n

vk i′k =

n

t X t 1X (uα − uβ )i′αβ . 2

α=1 β=1

(66)

234

J. Hoekstra

α to node β we set i′αβ = i′βα = 0. Now Note that if there is no branch joining node that Equation 66 has been established, the right-hand side of Equation 66 is split as follows: N X k=1

vk i′k

=

nt 1X

2

α=1



uα 

nt X

β=1



i′αβ 

n

t 1X − uβ 2

β=1

nt X

i′αβ

!

.

(67)

α=1

Pt β For each fixed α, nβ=1 i′αβ is the sum of all branch currents entering node . By KCL, each one of these sums is zero, hence, N X

vk i′k = 0.

k=1

Thus we have shown that given any set of branch voltages subject to KVL only and any set of branch currents subject to KCL only, the sum of the products vk i′k is zero. This concludes the proof.

4.4.

Conservation of Energy in Lumped Circuits

For a single arbitrary circuit we can take the same circuit for both the primed circuit and the unprimed circuit. Tellegen’s theorem now states the conservation of energy in the circuit. We have, with the notations of Tellegen’s theorem, N X

vk (t)ik (t) = 0

for all t.

k=1

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Since vk (t)ik (t) is the power delivered at time t by the circuit to branch k, the theorem may be interpreted as follows: at any time t the sum of the power delivered to each branch of the circuit is zero. Suppose the circuit has several independent sources; separating in the sum the sources from the other branches, we conclude that the sum of the power delivered by the independent sources to the circuit is equal to the sum of the power absorbed by all the other elements of the circuit. This means that as far as lumped circuits are concerned, the validity of KVL and KCL together imply conservation of energy.

5.

Bouded and Unbouded Currents

In the previous section it was mentioned that the delta current pulse i(t) = CVs δ(t) transporting in zero time an amount of charge Q = CVs , is an unbounded current. The concepts bounded and unbounded currents come from some considerations in network theory related to circuits including capacitors and switches (or, more general, stepping voltages). As such they are closely related to intriguing networks, such as the switched capacitor network of Figure 12 that seems to violate energy conservation. They are also related to the description of energy in networks that have reduction in “free-energy” and thus to nanoelectronics. Figure 12 shows two capacitors with equal capacitance values. Suppose that before the

Energy Conservation in Nanoelectronics

235

t=0

+ v1(t)

+ C

C

v2(t)

_

_

Figure 12. Switched capacitor network example. Assume an initial voltage across one of the capacitors and then close the switch; the energy stored on the capacitors will be halved. Note that the voltages on both capacitors step when the switch closes. switch is closed one of the capacitors is uncharged, while the other is charged with Q0 . The energy stored in the circuit before the switch is closed is wse|before =

(Q0 )2 (Q0 )2 +0= . 2C 2C

After the switch is closed the energy stored in the circuit is wse|after =

(Q0 /2)2 (Q0 /2)2 (Q0 )2 + = . 2C 2C 4C

We see that at t = 0 s the charge is redistributed in the network. This suggests that there was an unbounded current at t = 0 s. Before examining this network in detail, first the continuity property of bounded capacitor currents and the properties of capacitor voltages in networks with unbounded currents are reviewed.

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5.1.

Continuity Property in Linear Networks

It is possible to predict whether a capacitor voltage vC (t) in a network (circuit) containing switches is continuous, that is if v(0+ ) = v(0− ) for this capacitor. In the two capacitors network (Figure 12), for example, we see that the capacitor voltages are discontinuous, if v1 (0− ) 6= v2 (0− ). The following theorem predicts whether capacitor voltages are continuous in networks. Theorem: [6] In a linear network consisting of resistors, inductors, capacitors, switches operating simultaneously, d.c. voltage sources and current sources, one is sure that: the capacitor voltages are continuous except in those capacitors which, together with at least one make contact, form a loop in the network derived from the original by reducing all source intensities to zero.

5.2.

Continuity Property of Bounded Capacitor Currents

For convenience, only linear time-invariant capacitors are assumed. In general, using the constitutional relation v = C −1 q and the definition of the current i = dq/dt the voltage

236

J. Hoekstra

across a capacitor is given by: v(t) = C

−1

q(t0 ) + C

−1

Z

t

i(τ )dτ t0

t ≥ t0 .

(68)

More specific, if q(0− ) is the charge on a capacitor with capacitance C just before closing the switch the voltage across a capacitor is: v(t) = C −1 q(0− ) + C −1

Z

t

i(τ )dτ

0−

t ≥ 0− .

(69)

Clearly, to know the voltage across the capacitor we need to know the entire past history. To examine the continuity of the capacitor voltage we focus at t = 0, the moment of switching. Bounded currents can now be defined [7]. If the current iC (t) in a linear time-invariant capacitor is bounded in a closed interval [0− , 0+ ], then the voltage vC (t) across the capacitor is a continuous function in the open interval (0− , 0+ ). In particular, for time t = 0 satisfying 0− < 0 < 0+ holds: vC (0− ) = vC (0) = vC (0+ ) . Proof: Substituting t = t0 +∆t in Equation 68, where ta < t0 < tb and ta < t0 +∆t ≤ tb , we get: vC (t0 + ∆t) − vC (t0 ) = C

−1

Z

t0 +∆t

iC (τ )dτ t0

t0 + ∆t ≥ t0 .

(70)

Since iC (t) is bounded in [ta , tb ], there is a finite constant M such that |iC (t)| < M for all t in [ta , tb ]. It follows that the area under the curve iC (t) from t0 to t0 + ∆t is at most M ∆t (in absolute value), which tends to zero as ∆t → 0. Hence Equation 70 implies that vC (t0 + ∆t) → vC (t0 ) as ∆t → 0. This means that the voltage vC (t) is continuous at t = t0 . Filling in ta = 0− and tb = 0+ the continuity for bounded currents at t = 0 is proved. (A continuity property also holds for inductor currents if the voltage across the inductor is bounded.)

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5.3.

Unbounded Currents

In this context an unbounded current is an impulsive current; an unbounded current may appear in capacitor networks with stepping ideal voltage sources in the absence of resistive elements, such as the example of Figure 5. In this example an initially uncharged capacitor was excited by an ideal voltage source. We found that when the voltage source waveform is a step function stepping at t = 0 s from 0 V to Vs volt, the current through the circuit at t = 0 s was calculated to be i(t) = Cpv(t) = Cp[(Vs ǫ(t)] = CVs δ(t),

(71)

where the p-operator is used as an operator representing differentiation (see also the appendix), and CVs is the charge transferred during the current pulse in this circuit.

Energy Conservation in Nanoelectronics

5.4.

237

Voltages in Circuits with Unbounded Currents

A definition of the voltage across a capacitor in the unbounded case, can be found using Equation 69. If ∆q is the charge transferred during the delta pulse then for t > 0 s holds: Z t −1 − −1 v(t) = C q(0 ) + C ∆q δ(τ )dτ (72) 0−

= C

−1

−

q(0 ) + C

−1

∆q.

(73)

∆q is a positive quantity if positive charge is entering a capacitor and negative if positive charge is leaving a capacitor.

6.

Solutions of the Two Capacitor Problem

For the energy problem of the capacitor network of Figure 12 there exists two solutions, the first considers only bounded currents, the second only unbounded currents.

6.1.

Solution A: Bounded Currents

First, we have the standard textbook solution in which a resistor is placed between the two capacitors. By inserting the resistor, the current in the circuit is bounded, the voltages across the capacitors will only change gradually, continuity of the capacitor currents is guaranteed, and the resistor absorbs the missing energy. Again, assuming v2 (0− ) = 0 V we can easily find the bounded current in the circuit (remember that both capacitances are the same and equal to C):   −2t v1 (0− ) exp i(t) = t>0 (74) R RC

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and for the energy dissipated in the resistor wR :     −4t C[v1 (0− )]2 exp −1 wR (t) = − 4 RC

t > 0.

(75)

We see that in steady-state the amount of energy dissipated in the resistor is Cv1 (0− )2 /4, which is equal to the amount of energy stored on both capacitors after the switch is closed and half of the energy stored on capacitor 1 before the switch was closed. We see that energy is conserved at all times.

6.2.

Solution B: Unbounded Currents

Second, there exists a solution in which the current is unbounded. This solution starts by representing charged capacitors by their equivalent initial condition models, in which the charged capacitor is modeled as an uncharged capacitor in series with a voltage source representing the charge on the capacitor. Because the change of charge on the capacitors will now be modeled as a step function in the initial conditions (from v(0− ) to v(0+ )) the resulting current will be an impulse function—the current through the capacitors being

238

J. Hoekstra

proportional to the derivative of the voltage across them. For example, Davis [8] gives an excellent treatment. For our circuit with equal capacitances we find that the initial charges are distributed over both capacitances in such a way that the Kirchhoff voltage law holds: the voltages across both capacitances must be the same. Consequently, the charge on both capacitors will be q1 (0− ) + q2 (0− ) . 2 The delta current, thus, moved an amount charge of q1 (0− ) −

q1 (0− ) + q2 (0− ) q1 (0− ) − q2 (0− ) = , 2 2

and leads directly to the current 1 i(t) = C[v1 (0− ) − v2 (0− )]δ(t). 2

(76)

This current is the current redistributing the charge at t = 0 s. Again we can consider the case that the initial charge on the second capacitor is zero. We see from Equation 76 that half of the initial charge on the first capacitor is transported to the second with an impulsive current i(t) = (1/2)Cv1 (0− )δ(t). Now, just as in the first solution we are going to consider the energy in this circuit: what is dissipating half of the energy in the absence of a resistor?

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6.3.

Energy Generation and Absorbtion in Circuits with Unbounded Currents

The last example of subsection 4.1. showed the energy generation and the energy balance in the simplest network involving an impulsive current, namely, a single capacitor charged by an ideal voltage source. The network charges an initially uncharged capacitor with a charge q. The ideal voltage source, therefore, steps from 0 V to C −1 q at t = 0 s. An impulsive current will transfer an amount of q Coulomb towards the capacitor—and, of course, at the same time carry off an amount of (-q) from the negative side of the capacitor plates. The energy stored the capacitor is raised from 0 J to q 2 /(2C). We calculated the energy delivered by the source ws : Z Z q2 −1 2 ws = vidt = C q . ε(t)δ(t)dt = 2C We noticed that, of course, energy stored on the capacitor is provided exactly by the source; energy is conserved in this circuit. Note that we are now able to calculate the energy generated by a stepping voltage source at t = 0 s through which a delta current pulse flows at t = 0 s. If the step had been down the source would have been absorbing energy; this should not be a surprise, because the source is just a non-linear resistor being able to be both passive or active. The result can be generalized by considering a source stepping from vbefore to vafter through which a delta current pulse i(t) = qδ(t) is flowing. The stepping source can be represented by: v(t) = vbefore (t) + (vafter (t) − vbefore (t))ε(t) (77)

Energy Conservation in Nanoelectronics So, the energy delivered by the source in this case is Z Z ws = vidt = q [vbefore (t) + (vafter (t) − vbefore (t))ε(t)] δ(t)dt

239

(78)

using that all the voltages are constant just before and after the event leads to Z q ws = qvbefore + q(vafter − vbefore ) ε(t)δ(t)dt = qvbefore + (vafter − vbefore ) 2 giving the important result

6.4.

q ws = (vbefore + vafter ). 2

(79)

Solution B: Energy Conservation

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Now, we can attack the energy problem in case we have an unbounded current. The energy balance consists of the energy storage before the switch is closed (that is, the energy at t = 0− ), the energy storage after the switch is closed (that is, the energy at t = 0+ ), and the energy generated and/or dissipated during closing of the switch at t = 0 s. Again we consider for simplicity the charge on the first capacitor to be q and the charge on second capacitor to be zero. We find: the energy stored before closing is q 2 /(2C); the energy stored after closing is 2(q/2)2 /(2C) = q 2 /(4C). For the energy generated and absorbed at t = 0 we realize that during closing a current i(t) = (q/2)δ(t) flew through the circuit while the voltage sources representing the initial conditions—in this case, the values at t = 0− and t = 0+ —were both stepping: the source of the first capacitor stepped down from qC −1 to (q/2)C −1 , and the source of the second capacitor stepped up from 0 to (q/2)C −1 . Using Equation 79 we calculate that the source of the first capacitor while stepping down absorbed an amount of energy equal to (3/8)(q 2 /C), while the source of the second capacitor while stepping up generated an amount of energy equal to (1/8)(q 2 /C). Consequently, the net result is that during switching the sources absorbed q 2 /(4C)—that is, in total they dissipated the missing energy at t = 0 s!

6.5.

Unbounded or Bounded Currents through Circuits

We have seen that the energy delivered by a stepping voltage source for charging a capacitor with an unbounded current is: ws =

q2 2C

unbounded current.

(80)

This is fundamentally different from the case a bounded current is used. To obtain a bounded current in this circuit a resistor must be added, in series with the capacitor, to limit the current. The energy delivered by a stepping voltage source in the latter case is Z Z −1 −1 ws (t) = i(t)C qε(t)dt = C q i(t)dt t > 0. (81)

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J. Hoekstra R

In steady state i(t)dt will be the total amount of transported charge q, so that we obtain in steady state: q2 bounded current. (82) ws = C The resistor will dissipate the difference between the amount of energy delivered by the source and the amount of energy stored by the capacitor. In fact the above described difference proofs that an unbounded current cannot flow (other than strictly local—we will find this in the description of the tunnel current in the next section) through a circuit in which resistors are present. The proof is thus based on energy arguments. Also the reverse is true: in a circuit including resistors an unbounded current cannot flow through the circuit.

7.

Energy Conservation in Some Nanoelectronic Circuits

In nanoelectronics quantum-classical equivalent circuits exist for tunneling in metalinsulator-metal diodes and single-electron tunneling transistors [9]. The dominant one in a circuit synthesis context is the impulse model [10, 11], it models the current representing a single tunneling electron by a impulsive current i(t) = eδ(t − t0 )—with e the elementary electron charge— in parallel with the junction capacitance CTJ , and the critical voltage above which tunneling starts is modeled by a hot-electron scheme: if τ is the transit time, that is, the total time for an electron to tunnel and to restore the thermal equilibrium distribution, then t0 ∈ [0, τ ] and cr {vTJ = vTJ (t) : vTJ (t + τ ) = −vTJ (t)} .

(83)

Now, we have enough tools to describe nanoelectronic circuits that include metallic singleelectron tunneling junctions. I describe a couple of them.

7.1.

Tunnel Junction Excited by an Ideal Current Source

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As a first example the SET junction excited by an ideal current source is considered; the source is switched on at t = 0 s and has a constant value Is , see Figure 13.

i(t) is Is

is(t) 0

TJ

t (a)

(b)

Figure 13. Single-electron tunneling junction excited by an ideal current source: (a) source waveform and (b) circuit.

Energy Conservation in Nanoelectronics

241

Although probably unrealistic, consider a zero tunneling time. This choice makes it possible to obtain a description that can be compared with the orthodox theory of singleelectronics that assumes a zero-tunneling time. The behavior of the circuit in time shows so-called Coulomb oscillations, see Figure 14. The explanation of this behavior is the following. Due to the discreteness of charge only a whole electron charge, e, will be transfered during tunneling of an electron.

vs v cr t

0 -v cr

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

Figure 14. Coulomb oscillation in circuit in which the single-electron tunneling junction is excited by an ideal current source. The figure shows the oscillations when the tunneling time is assumed to be zero. At time t = 0 s, the current source will start charging the junction capacitance. If it would be possible to tunnel as soon as some charge appears on the tunnel junction, the phenomenon Coulomb oscillations would not be visible. The reason for not tunneling directly can be attributed to violation of energy conservation in the circuit. During tunneling the total charge of the electron is moved from the negative side of the metal to the positive side and can be represented by a tunnel current i(t) = eδ(t) because the tunneling time is assumed zero. This representation can also be proved rigorously [12]. We know that during tunneling the current source cannot generate any energy, because the tunneling time is zero. As a consequence the only allowed tunnel transition must satisfy that the energy (dissipation) involved must be zero too; in this way energy conservation in the circuit is guaranteed. During tunneling the voltage across the tunneling junction steps, because of the immediate transfer of the charge e. The energy involved in this process of a stepping potential difference as a consequence of the delta tunnel current is given by Equation 79. To obtain a elastic transition (energy conserving) wTJ must be zero, that is e (84) wTJ = (vbefore + vafter ) = 0. 2 This immediately leads to the desired result that the voltage across the tunnel junction immediately before tunneling vTJ|before plus the voltage across the junction immediately after tunneling vTJ|after must be zero, that is vafter = −vbefore .

(85)

This relation together with the requirement that a whole electron charge has to be transferred during tunneling through the junction leads to the result: e cr = vTJ . (86) 2CTJ

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J. Hoekstra

As is clear from the above argument, this is the only transition that does not violate energy conservation.

7.2.

Tunnel Junction Excited by an Ideal Voltage Source

Second, we consider a metal-insulator-metal tunnel junction excited by a constant voltage source. For large capacitances the response is well-known and shows the vi-characteristic of a linear resistor at low source voltages. We assume it can be modeled with by a tunnel junction exited by an ideal constant voltage source, in addition we consider electrons to tunnel through the tunneling junction one-by-one. Figure 15 shows the circuit and the energy-band diagram if we follow a single electron. Due to the fact that our model circuit

EF

eδ(t)

eVs EF

+ Vs _

CTJ

tunneling electron

_

+ Vs

Figure 15. Single-electron tunneling junction excited by an ideal voltage source above the critical voltage and energy-band diagram showing the tunneling of a single electron from the Fermi level.

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

only consists of an ideal voltage source and a capacitor representing the junction’s capacitance, but that resistors are absent, the unbounded impulsive tunnel current can propagate through the whole circuit. Analyzing this circuit we find: KVL:

vTJ = Vs

(87)

KCL:

i(t) = eδ(t)

(88)

Also the energy balance in the model circuit is well described. It is clear from the energy-band diagram that the tunneling electron, in this case, dissipates an amount of energy equal to eVs during the transition from the Fermi level at the negative side of the source to the Fermi level at the positive side of the source. The energy provided by the source is Z Z Vs eδ(t)dt = eVs δ(t)dt = eVs (89) So, the source provides the dissipated energy at the time of tunneling. What remains, now, is to prove that the equivalent circuit for the tunneling junction also absorbs an amount of energy of eVs .

Energy Conservation in Nanoelectronics

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Again we consider the equivalent circuit, representing tunneling, to be a capacitor in parallel with a impulsive current source i(t) = eδ(t) and find the energy dissipated by this equivalent circuit  e − v(0 ) + v(0+ ) , wTJ = 2 where the voltage across the tunneling junction is constant. Filling in that v(0− ) = v(0+ ) = Vs , we easily obtain the energy absorbed by the impulsive source to be eVs too. Energy is again conserved.

7.3.

Tunnel Junction Excited by a Non-ideal Voltage Source

As might be clear from the previous discussions the modeling of the tunneling circuit with a real source, that is, with the inclusion of a resistor will prohibit the impulsive current to propagate through the equivalent circuit. As a consequence Coulomb oscillations will appear, like in a circuit in which the tunneling junction is excited by a current source; also Coulomb blockade is feasible. As in this circuit all currents and voltages are bounded the validity of the Kirchhoff laws as well as energy conservation is always guaranteed. In this case the inclusion of a tunneling time will result in an uncertainty in the voltage across the tunnel junction, however, a state in which the voltage step and associated impulsive current exactly absorb the amount of energy to obtain balance in energy can always be found [10]. To find the critical voltage, the voltage before and after the tunnel event must be equal but opposite and the energy absorbed will be zero; because of the change in polarity the source will also not deliver any net energy.

7.4.

Remarks

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Two remarks must be made. First, as a consequence of the definition of the critical voltage and the conservation of energy, the inclusion of a tunneling time will result in a description in which the impulsive current will take place in the interval [0, τ ]. Second, to obtain a continuous current from the tunneling of single electrons, a Poisson distribution of the tunneling events has to be assumed.

8.

Conclusion

In this chapter, bounded and unbounded currents were considered in the context of nanoelectronic single-electron tunneling circuits and switched networks. This was necessary because some nanoelectronic theories claim reduction of free energy in (electronic) networks, and thus violation of energy conservation. Shown was that, first, energy is always conserved in these circuits—as is mandated by Tellegen’s theorem. Second, that if energy is lost or minimized in the final or steady state it just means that the missing energy was absorbed/dissipated during an unbounded current at t = 0 s. Some single-electron tunneling circuits have been analyzed to illustrate the theory, in contrast it showed that based on energy conservation correct behavior can be explained and predicted.

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J. Hoekstra

Appendix: Mathematical Preliminaries To describe circuits that include switches we assume that the switch opens (or closes) at t = 0 s, we will distinguish t = 0− just before opening (closing) the switch, and t = 0+ immediately after opening (closing) the switch. Defined are: f (0+ )

def == lim f (t) t↓0

(90)

f (0− )

def == lim f (t) t↑0

(91)

and

In most textbooks on circuit or system theory models for switches are described using either operational calculus or Laplace transforms. For our purposes, the models are best described using (Heaviside’s) operational calculus, making it possible to describe the models in the time-domain. Used are the following definitions for the operators p and 1/p: p and

1 p

def ==

def ==

Z

d dt

(92)

t

()dα.

(93)

−∞

For a capacitor with a constant capacitance C holds: i(t) = Cpv(t)

(94)

1 i(t) Cp

(95)

and v(t) =

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Furthermore the following notation is used: -the unit step function ε(t) is defined by the relation  1 t>0 def ε(t) == 0 t0  0R ∞ def δ(τ )dτ = 1 t = 0 δ(t) ==  −∞ 0 t