Analytical and numerical methods for wave propagation in fluid media

Analytical and Numerical Methods for Wave Propagation in Fluid Media

SERIES ON STABILITY, VIBRATION AND CONTROL OF SYSTEMS

Founder and Editor: Ardeshir Guran Co-Editors: A. Belyaev, H. Bremer, C. Christov, G. Stavroulakis & W. B. Zimmerman

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Vol. 7

Analytical and Numerical Methods for Wave Propagation in Fluid Media Author: K. Murawski

SERIES ON STABILITY, VIBRATION AND CONTROL OF SYSTEMS Series A

Volume?

Founder and Editor: Ardeshir Gliran Co-Editors: A. Belyaev, H. Bremer, C. Christov, G. Stavroulakis & W. B. Zimmerman

A n f l l v t f r j t l n u l l lVTiiftiprirsil IWivMltJW.3 M i l

wwstwv i

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in Uiiiil Miedifi

K. Murawski Uniwersytet Marii Curie-Skfodowskiej, Lublin, Poland

1111% World Scientific

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This book is dedicated to my wife, Ewa, and son, Kamil, for their love and support.

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Preface

Much of the matter filling the Universe is in a state of fluid which comprises liquid, gas, and plasma. The mathematical description of fluid motion makes use of partial differential equations which propagate the fluid variables in time. Many phenomena that occur in the fluid, treated as a continuous medium, can be studied in the frame of hydrodynamics (HD) or magnetohydrodynamics (MHD), which is a relevant and simple tool to describe the behaviour of the fluid. Among many extraordinary phenomena found in fluid such as, for instance, convective cells (see the Figure on the book cover for the convective cell seen in the Bieszczady Mountains in Poland on June 22, 2000), waves are the most pervasive and can be observed everywhere in the Universe (molecular clouds, extragalactic jets, accretion disks, Sun, solar wind, magnetospheres, magnetotail, cometary tails). We will point out the ubiquity of wave phenomena in fluids by discussing few selectively chosen examples such as acoustic waves, emission of air-pollutions, magnetohydrodynamic waves in the solar corona, solar wind interaction with Venus, and ion-acoustic waves. This book is primarily concerned with the analytical and numerical solutions to hyperbolic system of wave equations. Mathematically the most interesting feature of such systems is that they admit a shock solution which is a discontinuity that can form as a consequence of nonlinearity even from small initial condition after sufficiently long time. No attempt has been made in this book to study in any detail the physics of fluids and waves, or to explore the physical significance of the problems we solve using the presented analytical and computational techniques. Although the reader is expected to be conversant in the basic concepts of fluid

vu

Vlll

Preface

dynamics, as expounded, for example, in Landau and Lifshitz (1986) this book consists an overview of main ideas and approaches with a hope that they will serve as a roadmap to students or researches new to this area. This book is not intended to be complete. Some of the results are presented without derivation and it is left to the reader to consult the references provided for a complete presentation. Very often a considerable effort is required to add the details that have been omitted. However, our intention is that the reader will find this overview a useful introduction to the analytical and numerical methods for solving nonlinear equations which describe wave propagation in fluid media. While this book is in no sense a treatise on the whole subject of analytical and numerical techniques for fluid dynamics, it spans a relatively wide range of topics. The emphasis is on methods in use by the author, although brief descriptions of other techniques are included as background material too. We describe those techniques which are of greatest performance and are most widely used. Likewise, the references are for the most part to most-known papers but there are many approaches and interesting works which are not cited or surveyed here. This book would never appear without my teachers and coworkers. I wish to express my profound gratitude to all of them but in particular to Profs. Wieslaw A. Kamiriski and Eryk Infeld for their unfailing support and the latitude they gave me in choosing my sometimes meandering research direction, and to Prof. Bernard Roberts for his introduction to solar physics. Thanks are also due to many coworkers (Jose Luis Ballester, Rick DeVore, Sasha Kosovichev, Randy LeVeque, Valery Nakariakov, Luigi Nocera, Ramon Oliver, Efim Pelinovsky, Tomasz Plewa, Mike Ruderman, Rich Steinolfson, Oskar Steiner, Robin Storer, Takashi Tanaka, Rolf Walder, Ivan Zhelyazkov, and others) who have proven to be excellent guides through the jungle of problems some of which are undertaken in this book. Notification of errors, criticism, and suggestions for improvement are welcome.

K. Murawski Lublin 2002

Contents

Preface

vii

Chapter 1 Introduction 1.1 Limitations of analytical and experimental methods 1.2 Numerical simulations — a bridge between analysis and experiment 1.3 Advantages of computer simulations 1.4 Scope of the book Chapter 2 Mathematical description of 2.1 Classification of differential equations 2.1.1 Advection equation 2.1.2 Characteristics of the advection equation 2.2 A conservation equation 2.3 The Navier-Stokes equations 2.4 The one-dimensional Euler equations 2.5 Plasma and the Maxwell's equations 2.6 Kinetic plasma 2.7 Quasi-particle approximation 2.8 Magnetohydrodynamic approximation 2.8.1 Lagrangian picture 2.8.2 Incompressible limit 2.8.3 Cold plasma limit Chapter 3 Linear waves 3.1 Waves in homogeneous fluids

fluids

1 1 2 3 4 7 7 8 9 10 12 13 14 15 18 19 22 22 23 25 26

ix

x

Contents

3.1.1

3.2

MHD waves 3.1.1.1 The Alfven wave 3.1.1.2 Magnetosonic waves 3.1.2 Ion-acoustic waves Waves in inhomogeneous fluids 3.2.1 Acoustic and internal gravity waves in gravitationally stratified medium 3.2.2 Sound waves in random fields 3.2.2.1 Waves in random mass density field

Chapter 4 Model equations for weakly nonlinear waves 4.1 Inviscid Burgers equations for fast MHD waves 4.1.1 Nonlinear interactions 4.1.2 Modified inviscid Burgers equation 4.1.3 Inviscid Burgers equation 4.2 The Burgers equation for acoustic waves in viscous fluid . . . . 4.3 The Korteweg-de Vries equation for long waves in a cylinder . . 4.4 Few modifications of the KdV equation 4.5 The Zakharov-Kuznetsov equation for strongly magnetized ionacoustic waves 4.6 The nonlinear Schrodinger equation for modulational waves in a cylinder 4.6.1 Few remarks on the NS equation 4.7 Few other model wave equations 4.8 Remarks on multi-dimensional wave equations

33 36 37 45 45 46 47 49 50 52 56 58 58 60 61 62

Chapter 5 5.1 5.2 5.3 5.4

5.5

Analytical methods for solving the classical model wave equations Analytical solution of the inviscid Burgers equation The direct method for the Burgers equation Backlund transformation for the Korteweg-de Vries equation . 5.3.1 Solitons Inverse scattering method 5.4.1 Lax criterion 5.4.2 Inverse method Stationary wave solutions of the nonlinear Schrodinger equation

26 28 29 30 33

65 65 66 69 70 71 71 72 74

Contents

Numerical methods for a scalar hyperbolic equations 6.1 Finite-difference approximations 6.2 Simple finite-difference schemes 6.2.1 Lax-Wendroff and Lax-Fredrichs schemes 6.2.2 The Beam-Warming method 6.2.3 The MacCormack scheme 6.2.4 A stable scheme 6.2.5 Hermitian compact scheme 6.2.6 Upwind differencing 6.2.7 Method of lines discretization 6.3 Temporal discretization 6.3.1 Runge-Kutta methods 6.3.2 Multigrid methods 6.3.3 Linear multi-step methods 6.4 Finite-volume methods 6.5 Von Neumann stability analysis 6.6 Explicit and implicit time integrations 6.6.1 An implicit scheme 6.6.2 Semi-implicit method 6.7 Numerical errors 6.7.1 Spurious modes 6.7.2 Overshoots and undershoots 6.7.3 Monotonicity, positivity, and causality 6.8 Problems with source terms 6.8.1 A fractional step method 6.9 Open boundary conditions 6.10 Shock-capturing schemes 6.10.1 Algebraic schemes 6.10.2 Geometric schemes 6.10.3 Godunov method 6.10.4 Riemann problem 6.10.5 The MUSCL scheme 6.10.6 Higher-order schemes 6.10.7 Kurganov schemes 6.11 Flux-corrected transport method 6.11.1 Convection 6.11.2 Diffusion 6.11.3 Anti-diffusion

xi

Chapter 6

79 79 81 83 84 85 85 85 86 88 89 89 90 90 91 93 94 95 95 97 100 100 101 104 104 105 107 107 108 108 108 109 Ill 112 113 114 115 115

xii

Contents

6.12 Spectral methods 6.12.1 The Fourier transform method 6.12.2 The Chebyshev expansion method 6.13 Finite-element method 6.14 The locally one-dimensional method

Chapter 7

Review of numerical methods for model wave equations

116 118 119 121 122

123

Chapter 8

Numerical schemes for a system of one-dimensional hyperbolic equations 127 8.1 Linear system of one-dimensional equations 127 8.1.1 Characteristic variables 128 8.1.2 Riemann problem for the linear equations 129 8.1.3 The wave propagation method 130 8.2 Nonlinear system of one-dimensional equations 131 8.2.1 Flux-difference splitting scheme 131 8.2.2 Euler equations 133 8.3 The shock tube problem 135 8.4 Rankine-Hugoniot jump condition 136 8.5 The Riemann problem for the Euler equations 137 8.5.1 The HLL Riemann solver 139 8.5.2 The Roe approximate Riemann solver 139 8.5.3 A relaxation Riemann solver 141 8.5.4 Extension of the Roe scheme for a general equation of state 142 8.6 Deficiences of Godunov-type schemes 144 8.6.1 Entropy fix 144 Chapter 9

A hyberbolic system of two-dimensional equations 9.1 Operator splitting schemes 9.2 Operator unsplit methods 9.3 Grid generation 9.3.1 Structured and unstructured grids 9.3.2 Other grid generation methods

149 150 151 152 152 154

Contents

9.4 9.5 9.6

Adaptive mesh refinement method 9.4.1 AMR codes Implicit hydrodynamic schemes 9.5.1 Barely implicit scheme for the Euler equations Few specific examples of hydrodynamic schemes

xiii

. . . .

156 157 158 161 163

Chapter 10 Numerical methods for the M H D equations 165 10.1 Problems with the MHD equations 166 10.2 Conservative form of the MHD equations 167 10.3 Non-conservative equations 168 10.4 Eigenvalues and eigenvectors 170 10.5 Singularities 171 10.6 Problems with MHD Riemann solver 173 10.7 Divergence cleaning schemes 173 10.8 A scheme for a strong magnetic field 176 10.9 Few specific examples of explicit MHD schemes 178 10.9.1 9-th wave Riemann solver for two-component MHD equations 179 10.10 Implicit MHD schemes 184 Chapter 11 Numerical experiments 11.1 Numerical solution of the inviscid Burgers equations 11.2 The effect of random mass density fields on acoustic waves . . 11.2.1 Seeding time-dependent random field 11.2.2 Numerical results 11.2.3 Summary 11.3 Numerical simulations of air-pollutions 11.3.1 Numerical model 11.3.2 Numerical results and discussion 11.4 Driven MHD waves in the solar corona 11.4.1 Physical model 11.4.2 Numerical solution of MHD equations for the solar coronal plasma 11.4.3 Numerical results 11.4.4 Summary 11.5 Solar wind interaction with Venus 11.5.1 Numerical model

187 188 191 192 193 194 194 195 196 199 200 200 201 208 209 212

xiv

Contents

11.5.2 Numerical results and discussion 11.5.3 Concluding remarks 11.6 Ion-acoustic waves and solitary waves

214 216 217

Chapter 12

221

Summary of the book

Bibliography

223

Index

235

Chapter 1

Introduction

1.1

Limitations of analytical and experimental methods

Traditionally the scientific methods involve a mutual interplay between experiment and analysis. The former tries to collect information by repeated events. The latter attempts to order the accumulated knowledge. Analysis and experiment interact with each other via mutual stimulation and feed back. However, the traditional methods of investigating nature have their limitations. Often the complexity of the physical phenomenon and the simultaneous interaction of various effects make a complete analysis impossible. On the experimental side, one is limited to measurements of only a small fraction of the quantities of interest and even they can be sampled only at a few times and spatial locations and with a limited degree of accuracy. Consequently, one is then faced with the task of interpreting limited observations with theories that are incomplete. For example for astrophysical plasma, it is usually impossible to repeat experiments and the available observational data is sparse and sporadic. The observational data is mainly collected by satellite observations, space active experiments and ground based active or passive remote diagnostics. Consequently, the accumulation of such data is an extremely expensive and time consuming task. Additionally, the data very often depends on time and space. The sheer volume of exploration in space is so large that the essentially point observations by spacecraft alone often leads to misinterpretation of the data. Computer simulations can reproduce the global phenomena which can never be measured simultaneously even by a number of spacecrafts. The present computers can be used to model a macroscopically neu-

1

2

Introduction

tral medium containing free electrons and ions. This medium constitutes a plasma when the long-range Coulomb forces produce collective behavior. The need for time-dependent calculations of plasma is stimulated by observations and theory which show the existence of time-dependent behavior. The development of supercomputers makes detailed and accurate calculations with relatively short calculation times feasible.

1.2

Numerical simulations — a bridge between analysis and experiment

Numerical simulations - that is, the use of computers to solve problems by simulating theoretical models - is part of new methodology that has taken its place alongside pure theory and experiment during the last few decades. Numerical simulations permit one to solve problems that may be inaccessible to direct experimental study or too complex for theoretical analysis. Computer simulations can bridge the gap between analysis and experiment. Very often the simplest physical phenomena are described by complicated mathematical equations which cannot be solved analytically and require numerical treatment. The basic idea of computer experiments is to simulate the physical behavior of complicated natural systems by solving an appropriate set of mathematical equations that are built on the basis of a physical model. A typical way for computer simulation is to develop a mathematical model, perhaps in a series of differential or integral equations and then to transform them to a discrete form that can be numerically treated. By this way, numerical simulations attempt to initiate the dynamic behavior of a system and to predict or calculate subsequent events. Numerical simulations have emerged as a new branch in physics complementing both experiments and theory. A simulation can sometimes replace a physical experiment, although most often a simulation and an experiment are complementary. Results of scientific experiments are often explained by simulations, and simulations are often calibrated by experiments. The experiments provide input for the simulations which are viewed as experimenting with theoretical models. The feedback of numerical results into theoretical modeling and the continues interaction with laboratory experiments and analytical theory make computing an indispensable tool for science. Therefore, the increase in computing power in both speed and

Advantages

of computer

simulations

3

storage has given computational physics its significance. Improved computer capacity and the solution algorithms themselves, have a large effect on the quality of solutions obtained. Numerical simulations can be used to study the dynamics of complex physical systems. Although the variety of complex flows that computational fluid dynamics can analyze continues to increase, the solutions to much more complex flows are desired. A numerical model can be used to interpret measurements and observations, extend existing analytical models into new parameter regimes and quantitatively test existing theories. That can be done by comparing model predictions to experimental data. Modern computers are fast and do not complain of boredom when repeating the same procedure millions of times. Analytical methods, on the other hand, have been plagued with this problem. With the use of computer, one can often test theoretical predictions and approximations. The numerical models are simpler and more idealized than the actual physical system. However, they are far more complete and realistic than we can handle analytically.

1.3

Advantages of computer simulations

Computer simulations contain many advantages over conventional experiments. Simulations can evaluate the importance of a physical effect by turning this effect on or off, changing its strength, or changing its functional form. This way of isolating effects is an important advantage that a simulation has over an experiment. The main advantage of computer experiments is that complicated physical system involving nonlinearity and inhomogeneity can be treated without difficulty as easily as much simpler linear and homogeneous systems are dealt with. As a consequence of that, nonlinearity and inhomogeneity is no longer an obstacle in exploitation of physical systems. The computer simulations reproduce both linear and nonlinear behavior of a physical system. One can compare the results of such calculations with the behavior of real physical systems and with theory. These results can then be used to test theoretical predictions. Both laboratory measurements and computer simulations are never exact. Theoretical researcher, by contrast, is blessed with the possibility of generating exact solutions. The computer does not treat the analytical for-

4

Introduction

mulae in a way a theoretical physicist is fond of manipulating. Instead, many bits of numbers are crunched by the computer. Consequently, one can get only a single event instead of a physical law out of the computer. To learn the general behavior and laws of nature, we have to interpret and analyze the computer results. Both simulations and laboratory experiments benefit greatly from focusing on specific mechanisms of complex phenomena. Therefore, much can be learned about physical phenomena by idealizing and simplifying the problem. As a consequence of that, an experiment is not always a better probe of a physical system than a simulation. Simulations can be used to test the range of validity of theoretical approximations. For example, when a linear theory breaks down, the reason of breakdown can be studied by simulations. So, the simulations can be used to test and extend analytical results. The reverse case is also true as theory can be used to validate a numerical model. Numerical simulations, analysis and experiment cover mutual weakness of both pure experiment and pure theory. These simulations will remain a third dimension in fluid dynamics, of equal status and importance to experiment and analysis. It has taken a permanent place in all aspects of fluid dynamics, from basic research to engineering design. The computer experiment is a new and potentially powerful tool. By combining conventional theory, experiment and computer simulation, one can discover new and unsolved aspects of natural processes. These aspects could often neither have been understood nor revealed by analysis or experiments alone.

1.4

Scope of the book

The purpose of this book is to present few analytical and numerical methods of solving wave propagation problems. The book is organized along the following guidelines. Chapter 2 is devoted to mathematical descriptions of fluids. Here several types of plasma approximations are presented. They include equations for hydrodynamics and magnetohydrodynamics. In Chapter 3 the linear dispersion relations for waves in homogeneous and inhomogeneous media are reviewed with a description of the problems which appear when inhomogeneous media, such as stratified by gravity or contained random fields, are taken into account. The following Chapter presents several model equations

Scope of the book

5

for weakly nonlinear waves and solitons. A few analytical methods of solving these equations are described in Chapter 5. Chapters 6-10 introduce various numerical methods, pointing out their strengths and weaknesses. Some of these methods are used to solve the inviscid Burgers equations and in the simulations of random acoustic waves, air-pollutions, waves in the solar corona, solar wind interaction with non-magnetic bodies such as Venus, and ion-acoustic waves. These numerical experiments are performed in Chapter 11. This book is completed by Summary and the list of references as well as by the subject index.

Chapter 2

Mathematical description of fluids

2.1

Classification of differential equations

We consider the following second-order differential equation: autXX + butXt + cu,tt + dutX + eu,t + gu = s,

(2.1)

where s is a source term, u = u(x, t) and the comma with indices denote the partial differentiation, e. g. U xx

'

_d2u ~ Ox2'

This notation for partial derivatives will be used throughout this book. Classification of differential Eq. (2.1) is based on the sign of the discriminant A = b2 - 4ac.

(2.2)

For A < 0, A = 0, and A > 0 we have respectively the elliptic, parabolic, and hyperbolic equations. These equations model different sorts of physical phenomena in fluid dynamics and require different analytical and numerical methods for their solution. Typical examples are Poisson equation, u,xx + u,tt = s,

(2.3)

for an elliptic equation, the diffusion equation, u,t = (£>«,X),B, 7

(2.4)

8

Mathematical

description

of fluids

for a parabolic equation, and the wave equation, ujt ~ c2utXX = 0,

(2.5)

for a hyperbolic equation. Here D{x) is the diffusion coefficient and c is the wave speed. For c = const this equation can be rewritten as a set of two first-order equations vtt + cw,x

=

0,

(2.6)

w,t + cv,x

=

0,

(2.7)

where: v = — cuiX,

w = utf

(2.8)

For s = 0, Eq. (2.3) is called Laplace'a equation. When a, b, c, d, e, g or s are functions of u or space and time, Eq. (2.1) is nonlinear and the class of an equation may vary according to the sign of A in particular time and location in space. 2.1.1

Advection

equation

This book is primarily concerned with first-order hyperbolic system of n equations of the following form: u t + Au

x

+ Bu,y + C u

z

= 0,

(2.9)

where u(x, t) 6 Rn and A, B, C are nxn matrices which have real eigenvalues, and are diagonalizable, i. e., have complete sets of linearly independent eigenvectors. Equation (2.9) is equivalent to Eqs. (2.6) and (2.7) if

and B = C = 0

(2.11)

or d/dy = d/dz = 0. In fluid dynamics u represents conserved quantities such as the densities of mass, momentum, and energy.

Classification

of differential

9

equations

The simplest example of a hyperbolic equation is the scalar advection equation uit + cu,x=0

(2.12)

for which the initial condition is given by u(x,0)=uo(x).

(2.13)

Here, c = const can be thought as a convective velocity for the generalized density u. Equation (2.12) can be easily solved by rewriting it as

i + «5)- a

t = cu,x+utt

= 0.

(2.21)

So, the solution u(t) is constant along each characteristic. It will be shown in Sees. 6.10 and 8.1 that characteristics are important for developing accurate numerical schemes for hyperbolic equations.

2.2

A conservation equation

Many physical processes are governed by fundamental laws such as a conservation of mass, energy, momentum, and charge. These processes can often be described by the conservation equation which is a generalization of Eq. (2.16), viz. Q,t + V • (gv) = git + v • Vg + pV • v = 0.

(2.22)

Here g is a generalized density and v is a flow speed. The term v • Vg describes convection and the expression gV • v corresponds to compression or to the reverse effect, rarefaction. We will derive now the equation for conservation of mass in a onedimensional gas dynamic system. A more general derivation can be found, for example, in Landau and Lifshitz (1986) and Wendt (1992). Let x and g(x,t) represent the distance and the mass density at the point x, respectively. If we assume that there are nor sinks nor sources (mass is neither created nor destroyed) then the mass in section < x\, x-i > can be changed only because of fluid flowing across the end-points x\ and

A conservation equation

11

xi of this section (Fig. 2.1). We can express that by the following formula: d d~t

rX2 nX2

I

g{x,t)dx = g(xi,t)v{x1,t)-Q(x2,t)v(x2,t),

(2.23)

J X\

where v(x,t) is a fluid velocity and the product g(x,t) and v(x,t) is equal to the flux of fluid. From this equation we get / J X\

Qttdx = - /

(gv)iXdx

(2.24)

JX\

or

[g,t + (gv),x]dx = o.

(2.25)

Since this equality holds for any interval < xi, x2 > the integrand must be zero, i. e. (2.26)

g,t + (QV),X = 0.

This is the one-dimensional counterpart of the differential form of the mass continuity equation. In the case of constant velocity v in Eq. (2.26) we obtain the simple advection equation, (2.27)

g,t + vgyX = 0.

*>

X

X

Fig. 2.1 The mass is changed in the interval < xi, X2 > as a result of inflow from the left and outflow at x = X2Equation (2.26) can also be written in Lagrangian form as: dg g,t + vg,x = -£=

~ev,x,

(2.28)

where the two terms on the left hand side are the Eulerian time derivative and the advective space derivative. Together these two terms comprise the

Mathematical description of fluids

12

Lagrangian derivative dg/dt which denotes the change of Q in a reference frame, moving with the speed v.

2.3

The Navier-Stokes equations

The Navier-Stokes equations were obtained in the first half of the nineteenth century (in 1845) independently by the Frenchman M. Navier and the Englishman G. Stokes. In Cartesian coordinates, the Navier-Stokes equations for two-dimensional flows (with d/dz = 0) can be written as the set of conservation equations of Eq. (2.22), viz. (

\

\

QVX T

QVX

QK ~ xx

+

QVy

QVxVy

V EJ

\

+

7~xy

(E + p)vx - VXTXX - vyTxy + qx J

\

QVy

V

Qvl - r, y -vv (E+p)vy- VXTyX

. VyTyy

~\~ Qy

Here S is the source term, the velocity v = [vx,vy,0], energy density such that the pressure p is given by, P = (7 ~ 1)

= s.

(2.29)

/

and E is the total

E-jQtvl+tf)

(2.30)

The specific heats ratio 7 = cp/cv is such that 7 = (m + 2)/m, where m is the number of internal degrees of freedom of the fluid molecules, cp and cy are the specific heats at constant pressure and volume, respectively. For a monoatomic fluid m — 3, while for diatomic molecules m = 5. In the late seventeenth century Isaac Newton claimed that shear-stress T in a fluid is proportional to velocity gradients. Such fluids are called Newtonian fluids in opposite to non-Newtonian fluids such as blood for which this dependence does not exist. For Newtonian fluids Stokes showed that the normal (TXX, ryy) and shear (TXV, ryx) stresses are TXx

-

T~xy

=

AV -v + 2fivXtX, ^\Vx,y

T Vy x),

Tyy = W -y + 2fxvyy, TyX =

TXy,

(2.31)

The one-dimensional

Euler

equations

13

where fj, is the molecular viscosity coefficient and A is the bulk viscosity coefficient. The normal stresses are related to the time rate of change of volume of the fluid element, whereas the shear stresses are associated with the time rate of change of the shearing deformation of the fluid element. Stokes made the hypothesis that

This relation is frequently used but is not definitely confirmed. For gases, the viscosity coefficient fj, is assumed to vary in accordance with Sutherland's law (Quirk 1991), / T \ 3 / 2 7 o + 110

^°UJ

r+iio'

(2 32)

"

where free stream conditions are denoted by the subscript 0. Heat transfer by thermal conduction is proportional to the local temperature gradient qx —

qy =

-KT,X,

~KT 0). The other sources in these equations are neglected. Note that while the mass density g and the momentum gv are convected with the speed v, the effective speed at which energy is transported is veff =

14

Mathematical

description

of fluids

v(E + p)/E. This fact has direct consequences in developing numerical methods for the Euler equations.

2.5

Plasma and the Maxwell's equations

The term plasma was first coined by Tonks and Langmuir (1929) in there studies of oscillations in electric discharges. Plasma can be generated thermally. As heat is added to a solid such as for instance ice, it undergoes a phase transition to a new state which is called liquid. If heat is further added to a liquid, it becomes the gaseous state. An additional energy input results in the ionization of some of the atoms. At a sufficiently high temperature (that usually exceeds 105 K) most matter exists in an ionized state which is called plasma. A plasma state can exist at temperatures lower than 105 K if there is a mechanism for ionizing the gas and if the mass density is low enough so that recombination is not rapid. Unlike conventional gases but like most liquids, plasma is capable of conducting an electric current. Since this plasma state behaves quite differently from the other states, it is often called the fourth state of matter. This term was introduced by Crookes (1879) to describe the ionized medium generated in a gas discharge. Although there is little in the way of natural plasma on the Earth's magnetosphere as the low temperature and high density of the Earth's near atmosphere preclude the existence of plasma, it exists in a variety of places, some as familiar as a fluorescent light or perhaps less well-known locations such as the leading edge of high speed space-shuttles. Plasma exists in the upper atmosphere (ionosphere) too where it is created by photoionization of the tenuous atmosphere. Father out from the Earth, plasma streams from the Sun in the form of the solar wind which consists mainly of protons, and fills many regions of interstellar space. It is estimated that as much as 99.9% of the universe is comprised of plasma. Many phenomena in plasma can be described by the Maxwell's equations (e. g., Dendy 1990) which can be written as follows: V B

=

0,

(2.37)

V-E

=

-,

(2.38)

-VxB

=

£E,t-rj,

(2.39)

Kinetic plasma

VxE

=

-Bt.

15 (2.40)

In these equations the relations B = /iH and D = eE have been used to eliminate the magnetic field strength H and electric displacement D . Here E is the electric field, q is the charge density, fj, and e are respectively the magnetic permeability and permittivity. For vacuum these values are H = 4?r • 1(T 7 H/m and e ~ 8.854 • 1CT12 F/m. The symbol j denotes the current density that is given by Ohm's law j = a(E + v x B ) ,

(2.41)

where a is the electric conductivity. The Maxwell's equations have a simple physical interpretation. Equation (2.37) implies that there are no magnetic monopoles, whereas Eq. (2.38) and Faraday law (2.40) assume that either electric charges or timevarying magnetic fields are able to generate electric fields. Ampere law (2.39) shows that either currents or time-varying electric fields can give rise to magnetic field.

2.6

Kinetic p l a s m a

There are situations in geophysics (e. g., Shizgal, Hubert 1989) and in space physics (Shizgal, Blackmore 1986) where the mean free path of particles, which is the average distance traveled between particle collisions, is comparable to, or higher, that a local spatial scale. For these rarefied gas dynamical situations the Euler equations are no longer valid and a kinetic theory treatment is required. We consider i-th species of the plasma, represented by particles with electric charge e*, mass rrii, number density rij, velocity v, and temperature T{. Kinetic processes are characterized by time-scales which are associated with the plasma frequency,

«>i = J^-,

(2-42)

and the cyclotron frequency in the magnetic field B,

fl, = ! * £ . rn.i

(2.43)

16

Mathematical description of fluids

Length-scales involve these frequencies and relevant velocities. They are: a) the Debye length,

XDi = J ^ 4 t

(2-44)

with the thermal speed

c.=

/MI V mi

Here ks is Boltzmann's constant; b) the thermal gyroradius

m=

Ci

(2.45)

c) the direct gyroradius rdi

=

Ai

=

Vi

(2.46)

d) the inertial length

c

(2.47)

A kinetic (fluid) plasma description is required when these scales are higher than or comparable to (smaller than) the scales of physical phenomena. There is also an intermediate regime, for example, when kinetic effects along an ambient magnetic field are important but they are negligible in the perpendicular direction. Alternatively, the various plasma species can be treated differently. In a hybrid approach one species is treated kinetically and the other species are considered as a fluid. The gaseous state can be described by the distribution function for the different species. For sufficiently rarefied gases, the single distribution function, / ( v , r, t), is sufficient for this purpose, assuming that there are no two particle correlations. The distribution function is defined as the quantity f(v,r,t)dvdr is equal to number of particles with velocity in the interval < v, v + d v > and position < r, r+dr > at time t. This distribution function depends on seven dependents: three components of velocity, three position variables and the time. A reduction of the number of dependents is required to make the problem tractable. This reduction is achieved by expressing the distribution function in terms of moments.

Kinetic plasma

17

The first moment is the number density, n(r,t) = J / ( v , r , t ) d v .

(2.48)

Hence the mass density g is Q(r,t) = mn(r,t),

(2.49)

where m is the mass of a particle. Now, we can refer to a microscopic description in terms of / ( v , r, t) and to a macroscopic description with the use of g{r,t). Kinetic phenomena are described by the Vlasov equation (e. g., Dendy 1990, Nocera, Mangeney 1999), fi,t + v • V/i + -^-[E + (v x B)] • / i i V = 0,

(2.50)

77l{

where fi(r, v) is the distribution function of the i-th species. The electric E and magnetic B fields are described by the Maxwell's equations which contain the source terms coming from the moments of all plasma species VxE

=

-B,t,

(2.51)

-VxB V V B

=

^ e * f fivdv + eE,t, J i 0,

(2.52) (2.53)

Y^eiffidv.

(2.54)

eV-E

=

In evaluations of Eqs. (2.51)-(2.53) it is important to decide what kind of electromagnetic field is needed. For strong magnetic field we can neglect magnetic field perturbations at all frequencies. In this case, electric charge density fluctuations are important and the electrostatic field is evaluated from Eq. (2.54). Both ions and electrons are often represented in this case kinetically with the relevant spatial and temporal scales equal to the electron Debye length and the inverse of the electron plasma frequency, respectively. For frequencies lower than the ion plasma frequency, electrons are treated adiabatically with the electron number density, ne = neo exp (j^f-)

-

(2-55)

Mathematical description of fluids

18

where <j> is the electrostatic potential such that E — —V, and ne0 is the ambient electron density. Perturbations which satisfy this constraint are called electrostatic. For finite magnetic field, there are various regimes which can be classified in dependence on the wave frequency in comparison to the electron cyclotron frequency Qe. For frequencies much higher than the electron cyclotron frequency, both electrostatic and electromagnetic oscillations are present but they are generally weakly coupled. For a description of the electrostatic oscillations the electrostatic approximation can be used. If, on the other hand, electromagnetic waves are of interest, the Maxwell's equations have to be solved instead. For frequencies close to the electron cyclotron frequency Cle of Eq. (2.43), dynamics of electrons have to be taken into account. However, ions can be treated as an immobile fluid for appropriate time-scales. For fij0

(3-26)

or — iV or where fcy and k± are the components of the wave vector k parallel and perpendicular to E, respectively. In the magnetic-field-free case Q, = 0 and we recover Eq. (3.23). So that for small k, ion-acoustic waves are low-frequency, almost dispersionless waves, similar to sound waves in air. The same dispersion relation characterizes the ion-acoustic waves which propagate parallelly to the magnetic field (k± = 0 ) . As the perpendicular waves with k\\ = 0 exhibit the dispersion relation

"2 = ^ + iffci

(3 27)

"

the ion-acoustic waves are anisotropic. The presence of fi 2 in Eq. (3.27) results in a cut-off frequency. Free oscillations are possible for OJ2 > fi 2 . Below the cut-off frequency the oscillations are evanescent. We will see in the forthcoming part of this

Waves in inhomogeneous fluids

33

monograph that similar properties possess acoustic and internal gravity waves in gravity field.

3.2 3.2.1

Waves in inhomogeneous fluids Acoustic and internal stratified medium

gravity waves in

gravitationally

We consider gravitationally permeated fluid, ignoring magnetic effects. In the presence of constant gravity, momentum equation attains its form Qv,t +

• V)v = - V p + gg.

Q(V

(3.28)

In the case of gravity pointing in the negative z-direction, g = — gz, and for the static (v = 0) equilibrium this equation gives ~90o-

Po„

(3.29)

So, the inclusion of gravity introduces an additional force and a preferred direction. As a consequence of that wave motions are anisotropic and they are driven by a buoyancy force. Gravity imposes a length-scale in the fluid, which is called the density scale-height, H, Qo

H =

(3.30)

0o.. There is also a length-scale introduced by pressure variations, the pressure scale-height A such that A=

Po

^s

P0,z

19

(3.31)

Higher values of H and A correspond to weaker stratification. In particular, constant Qo{z) and po{z) profiles imply H, A —> oo. These spatial scales impose time-scales, defined as the time taken for a wave to pass the distance H and back again, viz. 2H/cs. The inverse of it is called the acoustic cut-off frequency 0J„

2H'

(3.32)

34

Linear waves

This frequency has a simple physical interpretation as the waves are propagating for their frequencies which are higher than w > ua. For lower frequencies, these waves are evanescent; they decay with z. Additionally, we can define the buoyancy (or Brunt-Vaisala.) frequency ujg such that 2 _ U) s

1

1,

9'

9

**-E>~4-li-

(3 33)

-

If u)g < 0 the equilibrium is unstable and convection sets in. This condition is known as the Schwarzschild criterion for instability. From this equation we get that the condition w^ < 0 is equivalent to 7A > H or