GLOBAL ATMOSPHERIC RESEARCH PROGRAMME (GARP) WMO- ICSU Joint Organizing Committee
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NUMERICAL METHODS USED IN ATMOSPHERIC MODELS
By F. Mesinger and A. AJakawa
VOLUME I
GARP PUBLICATIONS SERIES No. 17
August 1976 TABLE OF CONTENTS
Page
FOREWORD v
SUMMARY . VI f. INTRODUCTION ; GENERAL REMARKS ABOUT GRID POINT METHODS. I. Historical introduction ...... 2. Methods for the numerical solution of the equations of motion . 3. Basic elements of the grid point method 2 4. Finite difference schemes. 3 5. Convergence 5 6. Stability 6
II. TIME DIFFERENC ING SCHEMES .. . . 9 I. Definitio ns of some schemes ...... 9 2. Properties of schemes applied to the oscillation equation 11 3. Properties of schemes applied to the friction equation 19 4. A combination of schemes ...... 21
Ill. THE ADVECTION EQUATION • . . • . • ...... 22 I. Schemes with centered second-order space differencing 22 2. Computational dispersion ...... 26 3. Schemes with uncentercd space differencing . . . . . 30 4. Schemes with centered fourth-order space differencing 33 5. The two-dimensional advection equation...... 34 6. Aliasing error and nonlinear instability ...... 35 7. Suppression and prevention of nonlinear instability . 37
IV . THE GRAVITY AND GRAVITY-INERTIA WAVE EQUATION •.•.• 43 I. One-dimensional gravity waves: centered space differencing 43 2. Two-dimensional gravity waves ...... 44 3. Gravity-inertia waves and space distribution of variables . 46 4. Time differencing; the leapfrog scheme and the Eliassen grid 50 5. Economical explicit schemes . . . 53 6. Implicit and semi-implicit schemes 55 7. The splitting or Marchuk method 58 8. Two-grid-interval noise 59 9. Time noise and time filtering 61 10. Dissipation in numerical schemes 62 SUMMARY
This publication is discussing methods that are used oscillation (or frequency) equation, and friction equation. for the solution of hydrodynamic governing equations in Discussion of the leapfrog scheme includes a more numerical models of the atmosphere. The number of detailed analysis of the computational mode problem. methods in use in these models is, one might find, sur Chapter Ill deals with the numerical solution of those prisingly great; thus, in addition to analysis of problems forms of the advection equation which describe advection involved and techniques used for investigation of proper of one dependent variable. Schemes are analysed first ties of various schemes, a discussion is included only of considering the simplest one-dimensional linear advec schemes which are more widely used, or which are tion equation, with special emphasis given to the prob expected by the authors to become more widely used in lems of phase speed errors and computational dispersion, the near future. and group velocity errors. Then a brief account is The present volume is restricted to grid point finite included of the extension to two space dimensions. difference methods, and, furthermore, to problems and Finally, nonlinear advection equation is considered. methods used for time and horizontal space differencing. Aliasing error and nonlinear instability is discussed, and One remaining topic of the horizontal space differencing, a review is given of methods used to suppress or prevent that of the numerical solution of the advection equation nonlinear instability in atmospheric numerical models, with two dependent variables (advection terms of the including a detailed exposition of the principle of the two-dimensional primitive equations) will be included in Arakawa method. the Volume II of the publication. In Chapter IV schemes and problems related to the numerical solution of the gravity and gravity-inertia wave In Chapter I of this volume, following a short histori equations are considered. First, a discussion is given of cal introduction on the development and use of numerical the effects of space differencing on the numerical solution methods in atmospheric models, avilable methods for numerical solution of the differential equations governing of the gravity wave equations. Having now two or the atmosphere are briefly reviewed. Then, basic ele three dependent variables, the problem of the space ments of the finite difference method for solving these distribution of variables becomes of interest. Con sidering gravity-inertia wave equations, five different equations are introduced. Finally, the concept of space distributions are analysed with respect to their stability of finite difference equations, and methods for effect on the geostrophic adjustment process. Then, a testing the stability of these equations, are considered at some length. review is given of schemes and methods used to ac complish an economical use of computer time and/or Chapter II presents a discussion of time differencing elimination of computational modes in handling the schemes which are elementary enough so that they can gravity wave terms - the Eliassen grid, economical be defined using a simple ordinary differential equation, explicit schemes, the semi-implicit scheme, and the with one dependent variable. After defining a number splitting method. Finally, as the gravity waves can of such schemes. behaviour of numerical solutions is generate a false space and/or time noise in the calcula investigated which are obtained when these schemes are tions, some techniques available for prevention or sup· used for two specific ordinary differential equations: pression of such space (two-grid-interval) and time noise.
T hi~ volume is haset.l <>n a rcvi~et.l tran~lation of a part of the by A . Arakawa for his course 212A, Numerical Methods in Dynamic text hook "" <.Jynamk meteo rology, written by the fi rst of the present Meteorology, that he has been teaching at the Department of author~. for senior year stut.lents o r the D.:partment of Meteorology, Meteorology, University of California at Los Angeles. The revised University of Bclgrat.le . In writing the translatet.l part of that translation was read and further revised by A. Arakawa and, textbook, however, extensive use was made of lecture notes written finally, edited by M. J . P. Cullen of the Meteorological Office. TABLE OF CONTENTS
Page
FOREWORD v
SUMMARY • VI
I. I NTRODUCTION; GENERAL REMARKS ABOUT GRID POINT METHODS . I. Historical introduction ...... 2. Methods for the numerical solution of the equations of motion . 3. Basic elements of the grid point method 2 4. Finite difference schemes. 3 5. Convergence 5 6. Stability . . 6
II. TIME DIFFERENCING SCHEMES ...... 9 I. Definitions of some schemes ...... 9 2. Properties of schemes applied to the oscillation equation II 3. Properties of schemes applied to the friction equation 19 4. A combination of schemes ...... 21
Ill. THE ADVECTION EQUATION ...... • . . 22 I. Schemes with centered second-order space differencing 22 2. Computational d ispersion ...... 26 3. Schemes with uncentered space differencing . . . . . 30 4. Schemes with centered fourth-order space differencing 33 5. The two-dimensional advection equation...... 34 6. Aliasing error and nonlinear instability ...... 35 7. Suppression and prevention of nonlinear instability . 37
IV. THE GRAVITY AND GRAVITY-INERTIA WAVE EQUATION ...•. 43 I. One-di mensional gravity waves: centered space djfferencing 43 2. Two-dimensional gravity waves ...... 44 3. Gravity-inertia waves and space distribution of variables . 46 4. Time differencing; the leapfrog scheme and the Eliassen grid 50 5. Economical explicit schemes . . . 53 6. Implicit and semi-implicit schemes 55 7. The splitting or Marchuk method 58 8. T wo-grid-interval noise . . . . . 59 9. Time noise and time filtering 61 JO . Dissipation in numerical schemes 62 FOREWORD
Meteorology was one of the very first fields of physical indicated a need for a means to rapidly assimilate the science that had the opportunity to exploit high speed accumulated experience of meteorology. The first attempt computers for the solution of multi-dimensional time was at the hands of two able mathematicians, H. Kreiss depcndent non-linear problems. The authors of this and J. Oliger, who contributed a much needed sense of monograph trace the precedents from Bjerknes to von mathematical unity in their monograph " Methods for Neumann. The numerical techniques first employed were the Approximate Solution of Time-Dependent Problems" based on a small existing body of methodology, much (G.P.S. No. 10, 1973). This present volume, more of which was drawn from engineering practice, such as specifically reflecting experience with atmospheric models, the application of relaxation methods to the solution of has been written by two outstanding workers in the field, Poisson's equation. The working repertoire of numerical Prof. F. Mesinger and Prof. A. Arakawa, with Dr. A. methods rapidly expanded as the physical problems grew Robert as general editor. An additional volume will be in complexity and as practical experience accrued. The published containing chapters on the subjects: spectral growth was almost exclusively the result of the innova methods, global mapping problems, and finite element tions of the " using" physical scientists themselves. As a methods. consequence these advances often lacked the rigour and proof that might have been expected from applied mathematicians. The results of this evolution are to be found scattered throughout the meteorological literature of the past 25 years and it became apparent that there was a growing need for a systematic account of the rationale and development of technique. The JOC felt that GARP's needs, as reflected by the rapid influx of new scientists into numerical modelling, would be well served by the availability of a si ngle definitive source. J. SMAGORINSKY Other related disciplines such as oceanography have also Chairman, Joint Organizing Committee SUMMARY
This publication is discussing methods that are used oscillation (or frequency) equation, and friction equation. for the solution of hydrodynamic governing equations in Discussion of the leapfrog scheme includes a more numerical models of the atmosphere. The number of detailed analysis of the computational mode problem. methods in use in these models is, one might find, sur Chapter Ill deals with the numerical solution of those prisingly great; thus, in addition to analysis of problems forms of the advection equation which describe advection involved and techniques used for investigation of proper of one dependent variable. Schemes are analysed first ties of various schemes, a discussion is included only of considering the simplest one-dimensional linear advec schemes which are more widely used, or which are tion equation, with special emphasis given to the prob expected by the authors to become more widely used in lems of phase speed errors and computational dispersion, the near future. and group velocity errors. Then a brief account is The present volume is restricted to grid point finite included of the extension to two space dimensions. difference methods, and, furthermore, to problems and Finally, nonlinear advection equation is considered. methods used for time and horizontal space differencing. Aliasing error and nonlinear instability is discussed, and One remaining topic of the horizontal space differencing, a review is given of methods used to suppress or prevent that of the numerical solution of the advection equation nonlinear instability in atmospheric numerical models, with two dependent variables (advection terms of the including a detailed exposition of the principle of the two-dimensional primitive equations) will be included in Arakawa method. the Volume II of the publication. In Chapter JV schemes and problems related to the numerical solution of the gravity and gravity-inertia wave In Chapter I of this volume, following a short histori equations are considered. First, a discussion is given of cal introduction on the development and use of numerical the effects of space differencing on the numerical solution methods in atmospheric models, avilable methods for of the gravity wave equations. Having now two or numerical solution of the differential equations governing three dependent variables, the problem of the space the atmosphere are briefly reviewed . Then, basic ele distribution of variables becomes of interest. Con ments of the finite differe·nce method for solving these sidering gravity-inertia wave equations, five different equations are introduced. Finally, the concept of space distributions are analysed with respect to their stability of finite difference equations, and methods for effect on the geostrophic adjustment process. Then, a testing the stability of these equations, are considered review is given of schemes and methods used to ac at some length. complish an economical use of computer time and/or Chapter II presents a discussion of time differencing elimination of computational modes in handling the schemes which are elementary enough so that they can gravity wave terms - the Eliassen grid, economical be defined using a simple ordinary differential equation, explicit schemes, the semi-implicit scheme , and the with one dependent variable. After defining a number splitting method. Finally, as the gravity waves can of such schemes. behaviour of numerical solutions is generate a false space and/or time noise in the calcula investigated which are obtained when these schemes are tions, some techniques available for prevention or sup used for two specific ordinary differential equations: pression of such space (two-grid-interval) and time noise.
This volume is b.ascd on a rcvi5cd 1ransla1ion of a parl of lhc by A. Arakawa for his course 212A. Numerical Methods in Dynamic lcxlbook ,rn dyn.imk melc INTRODUCTION; GENERAL REMARKS ABOUT GRID POINT METHODS In this chapter. following a short historical introduction was constructed. The absolute vort1c1 ty conservation on the development and use of numerical methods in equation, and this first electronic computer, were used atmospheric models, methods available for numerical by Charney, Fjertoft and von Neumann in the late I 940's solution of the differential equations governing the for the first successful numerical forecast (Charney et al .. atmosphere will be briefly reviewed. Then, basic elements 1950). of the finite difference method for solving these equations Much faster computers, and improved understanding will be introduced. Finally, the concept of stability of of computational proble ms, now also enable long-term finite difference equations, a nd methods for testing the integrations of the basic primitive equations. It is gener stability of such equations, will be discussed at some ally considered that integration of the primitive equations length. enables easier incorporation of various physical processes than the integration of modified equations, that is, inte I. Historical introduction gration of the divergence and vorticity equations. Thus, It is considered that Wilhelm Bjerknes ( 1904) was the it is mostly the primitive equations that are used today for first to point o ut that the future state of the atmosphere practical numerical forecasting by meteorological services. can in principle be obtained by an integration of differen Charts obtained by numerical forecasting are used by tial equations which govern the behaviour of the atmo synopticians in these services as the principal basis for sphere, using as initial values fields describing an observed decisions on forecasts issued for public use. state of the atmosphere. Such an integration performed A number of research groups have been actively usi ng numerica l methods is called numerical weather engaged for more than a decade in development of models prediction. When, however, a numerical integration is for the numerical si mulation of the general circulation performed starting from fictitious initial fields, it is called of the atmosphere. In such simulations starting from a numerical simulation . fictitious initial state, e.g. an isothermal and motionless A first practical attempt at a numerical weather predic atmosphere, is often considered to be an advantage for tion was made by Richardson. After ve ry tedious and the experiments. It enables a test of the ability of th::: time-consuming computations, carried out mostly during computational and physical schemes of the model to the First World War, Richardson obtained a totally simulate an atmosphere with statistical properties similar unacceptable result. Despite this, he described his to those of the real atmosphere, with no, or not much, method and results in a book (Richardson, 1922), and prior information on these properties. this is today o ne of the most famous in meteorology. Numerical models a re also very frequently developed The wrong result obtained by Richardson. and his for studies of some smaller-scale atmospheric phenomena. estimate that 64.000 men are necessary to advance the Foremost among these are studies of the cumulus convec calculations us fast as the weather itself is advancing. tion problem, a nd simulation of processes within the left some douht as to whether the method would be of planetary boundary layer. In this text , however. we practical use. A number of developmenls that followed, shall primarily have in mind the application of numerical however. improved the situation. Courant. Friedrichs methods to predictio n and simulation of large-scale and Lewy ( 192~) found that space and time increments atmospheric motions. in integral ions of this type have to meet a certain stability criterion. Mainly due to the work of Rossby in the late 1930's, it hecame. understood that even a rather simple 2. Methods for the numerical solution of the equations of motion equation, that describing the conservation of absolute vorticity following the motion of air particles. suffices for Numerical solution of the equations of motion today an approximate description of large-scale motions of the in most cases is performed using the grid point method. a tmosphere. Finally, in 1945, the first electro nic computer In this method a set of points is introduced in the region ENIAC (Electronic Numerical Integrator and Computer) of interest and dependent variables are initially defined 2 INTRODUCTION; GENERAL REMARKS ABOUT GRID POINT MF.TIIOOS and subsequently computed at these points. This set of using differences of dependent variables over finite points is called the grid. The words mesh or lattice arc space and time intervals ; for that reason this approach also used. It is necessary to have the grid points at fixed is called the finite dif]<'ren,·t• method. The approximations locations in the horizontal. This means that, according for derivatives arc then used to construct a system of to the Eulerian system of equations, space and time algebraic equations that approximates the governing coordinates are chosen as independent variables. partial differential equations. This algebraic system is A number of attempts have been made to develop considered valid at each of the interior grid points of atmospheric models usi ng an approach which is at least the computation region. For the initial time and at partly Lagrangian. Serious difficulties are encountered space boundary points, additional constraints or equa when a straightforward numerical integration of the tions are defined that approximate the initial and bound Lagrangian system of equations is undertaken. However, ary conditions as required by the physics of the problem. it is possible to construct methods with some Lagrangian The set of algebraic equations obtained in this way is properties; for example, to have some or all of the compu then solved, usually usi ng an electronic computer, by a tation points moving with the fluid. In hydrodynamics suitable step-wise procedure. a number of such methods have proved to be very useful, We shall now consider some basic elements of the especially for some problems which are not amenable finite difference method. For simplicity, we start by to treatment by a strictly Eulerian technique (e.g. Harlow considering a function of one independent variable and-Amsden, 1971). However, in meteorology the per u = u (x). formance of Lagrangian or semi-Lagrangian models that have so far been developed has not been quite satis The function u is a solution to a differential equation factory. A discussion of one way of constructing a that we are interested in. We want to find an approxima Lagrangian model, and a review of earlier attempt!>, can tion to this solution in a bounded region R of the inde be found in a paper by Mesinger (1971). pendent variable, having a length L. The simplest way Another possible approach is to express the spatial of introducing a set of grid points is to require that they dependence of the variables in terms of a series of ortho divide the region R into an integer number of intervals gonal functions, and then substitute this into the governing of equal length Ax. This length Ax is called the grid equations. In this way the equations reduce to a set of interval, or grid length. Let us denote the number of ordinary differential equations, so that the coefficients grid intervals by J. It is convenient to locate the origin of the series can be computed as functions of time. This of the x axis at the left-hand end of the region R. Thus, is the spectral method of solving the governing equations. we are looking for approximations to u (x) at discrete Until relatively recently it was considered that in effi points x = jAx, wherej takes integer values 0, 1,2, ... , J. ciency the spectral method could not be competitive These approximate values we shall denote by with the grid point method. But the use of the fast u = u (jAx). Fourier transform has completely changed the situation 1 1 and investigation of spectral methods is now the subject Thus, we are interested in finding J + l values u • of intensive research. 1 Knowledge of a discrete set of values u1, even if the In the following we shall consider the technique of approximations were perfect, offers, obviously, less using the grid point method, and the problems associated information than knowledge of the function u (x). Let with it, using grid of computation points fixed in space. us briefly consider the situation in that respect. We shall This is the most direct way of solving the equations of very often find it convenient to think of the function motion numerically. Furthermore, knowledge of this u (x) as being formed by a sum of its Fourier compo method is necessary for the investigation and under nents, that is standing of the relative merits of other alternatives mentioned in this section. u(x)= ~ + ~ (an cos 21tn {: + bn sin 21tn [). n>I 3. Basic clements of the grid point method With the grid point method. the most common way Now, the available J + I values u1 do not enable the tlf ~olving the governing equations is to find approximate computation of all of the coefficients a,., b,.; rather, expressiuns for derivatives appc:iring in the equations. they can be used to compute only J + I different These approximate expressions arc defined using only coefficients. A natural choice is to assume that the values of the dependent va riables at the grid points. J + I values u1 define the near value a0 and as many as and at discrete time intervals. Thus, they are formed possible of the coefficients of the Fourier components at CHAPTER I 3 the lo ng wave length end of the series, that is, coefficients The difference between this expression and the derivative du) be' · d · h' for n = 1,2,3, ... , ~. Of these components, the one with ( - mg approximate ; in t 1s case the shortest wavelength wiJI have n = J /2, with the wave dx J t = J_ (g) ,:Jx + .!. (d3~) (L1x)2 + .... length L 2L 2L 2 dx i 6 dx· i n = 7 = L / L1x = i t1x. is called the truncation error of the approximation to the Having made that choice, we can say that with values u 1 deri vative. These are terms that were "truncated off" at discrete points x = jL1x it is not possible to resolve to form the approxi mation. The truncation error gives waves wi th wave length shorter than 2L1x. a measure of how accurately the difference quotient 'Now let us consider the differences between values u, approximates the derivative for small values of Llx. that will be used to construct approximations to deriva The usual measure of this is the order of accuracy of an tives. These differences a re called.finite differences. They approximation. This is the lowest power of L1 x that can be calculated over one or more of the intervals L1x. appears in the truncation error. Thus, approximation Depending on the relation of the points from which the (3 . I) is of the first order of accuracy. We can write values are taken to the point where the derivative is required, they can be centered or uncentered. An un £ = 0 ('1x). centered difference is, for example, the forward difference For an approximation to the derivative to be consistent it must, obviously, be at least first order accurate. L1u1 = u1+i - u1• More often centered (or central ) differences are used, such as 4. Finite difference sche.mes In a centered difference the difference is between values The a lgebraic equation obtained when derivatives symmetrical about the point where the difference is in a differential equation are replaced by appropriate being calculated. finite difference approximations is called a finite difference One way to construct an approximation to a differen approximation to that differential equation, or a finite tial equation is to simply replace the derivatives by appro difference scheme. In this section we shall introduce the priate finite difference quotients. For example, for the concepts of consistency, truncation error, and accuracy, first derivative one can use the approximation for a finite difference scheme. As an example, we shall use the linear advection (3. 1) equation iJu iJ u ( ) . . d t +COX = OJ U = U X , I . C = a positive const.(4 . 1) T he finite difference quotient here is. of course, only one of many possible approximations to the first derivat ive ft describes advection of the variable u at a constant at point j. velocity c in the direction of the x axis. The solution If a finite difference quotient. or a more complex to this simple equation can, of course, also be obtained expression, is to be used as a n approximation to a deri by a n analytic method . It will be useful to obtain the vative. it is required, above all, that this approximation analytic solution first, in order to investigate properties be consistent. This means that the a pproximat ion should of numerical solutions by comparing them against approach the derivative when the grid interval approaches known properties of the true solution. zero. The quotient (3 . 1), obviously. has that property. It is convenient to this end to change from variables Important information is obatined when the true x, t to variables ~. t with the substitution ~ = x-ct. solution u (jL1x) is substituted into an approximation Using the notation u(x, t )= U(~.t) to the derivative in place of the grid point values u1, and u (jL1x) is expanded in a Taylor series about the central we obtain point. For the quotient (3. 1) this procedure gives au au a~ au a, au au a, = a{ a, + a, a, == -c a~ + a, · 3 Uj+ 1 - U · (du) I (d~u) I (d u) 2 L1x J ._ dx . + 2 W L1x + 6 -;f;! . (Ax) + .... au au a~ au a, au J J .I rJx= ~ iJx + a, ax= ~ - 4 INTRODUCTION ; GENERAL REMARKS ABOUT GRID POINT METHODS Substitution of these expressions into (4. I) gives i) a, u <1;. ,) = o. (n + l)At u" nAt i Thus, it is seen that U cannot be a function of t, but can be an arbitrary function of I;. A solution of (4.1) is, (n-1) At therefore, u = f(x-ct), (4.2) where f is an arbitrary function. This, we see, is the x general solution of the advection equation (4. I), since {j- 1) Ax jAx (j + I) Ax it can satisfy an arbitrary initial condition Figure 4.2 A firute difference grid for finding an approximate solution of (4 . I). u (x,O) = F(x). (4.3) Thus. by a forward difference quotient, and the space derivative u = F(x-ct), {4.4) by a backward difference quotient. In this way we obtain the scheme is the solution of (4. I) satisfying the initial condition n+I n n n (4 .3). uj - uj + c uj - uj-1= 0. (4 .5) For a physical interpretation, it is often convenient to At Ax consider the solution in the x, t plane. In the present This scheme could be called a forward and upstream case, we see that the solution takes constant values along scheme, the latter word indicating the position of the the straight lines point j - 1 relative to the advection velocity. lt is, of x - ct = const. course, only one of many possible consistent finite difference schemes for the differential equation. There These lines are the characteristics of the advection equa are many schemes which approach the differential equa tion; one of them is shown in Fig. 4 . 1. We can say tion when the increments Ax, At approach zero. that the solution propagates along the characteristics. Since for small values of Ax, At a finite difference equation approximates the corresponding differential x - ct = const = x0 equation, we can expect that its solution will be an approx imation to the solution of that equation. We shall call u = const = u (x0 , 0) solutions given by finite difference schemes numerical solutions. There are, of course, both approximate and numerical solutions obtained by other methods which will not be considered in this publication. It is most convenient to study the properties of numerical solutions when they can be compared with known solutions of the original differential equation, which we shall refer to as Xo true solutions. The difference between the numerical and the true solution Figure 4 . 1 One of the characteristics of the linear advection equation (4.1). uj -u (jAx, nAt) (4. 6) is the error of the numerical solution. Let us now construct a scheme for finding an approxi mate solution to (4. I) using the grid point method. For obvious reasons, we cannot often expect to know the error of the numerical solution. However, we can We are now looking only for an approximate solution always find a measure of the accuracy of the scheme by at the discrete points in the (x,t) plane formed by the substituting the true solution u (jAx, nAt) of the equation grid shown in Fig. 4. 2. The approximate solution at a into the numerical scheme. Since the true solution will point ( j .'1x, 11:1 t) is denoted by u'j. not satisfy the numerical equations exactly, we will have The hchaviour of the true solution, which propagates to add an additional term to keep the equation valid. al u (jtJx, (n + 1) /Jt) - u (jtJx, ntJt) (b) What is the behaviour of the erroru,'-u (jtJx. nAt) tJt + when, for fixed values of Ax, At, the number of time (4. 7) steps n increases? + c u (jtJx, ntJt) - u {{j-1) tJx, ntJt) = £. tJx The answer to the first of these questions depends on the convergence of the numerical solution : if the error The term £ we shall call the truncation error of the finite approaches zero as the grid is refined (as .dx, tJ t ... 0) difference scheme. It shows how closely the true solution the solution is called convergent. If a scheme gives a satisfies the equation of the scheme, and, thus, gives a convergent solution for any initial conditions, then the measure of the accuracy of the scheme. scheme also is called convergent. We can obtain a more useful form for the expression for Consistency of a scheme does not guarantee conver the truncation error by performing a Taylor series gence ; we shall illustrate this by a simple example. expansion of the true solution about the central space We still consider the scheme (4. 5) ; its truncation error a nd time point. Using the original differential equation (4 .8) approaches zero as the grid is refined, and, there to eliminate the leading term we obtain the truncation fore, this is a consistent scheme. But consider the error (4. 7) as numerical solution, when the grid lines and character 2 istics are as shown in Fig. 5 . I . The characteristic passing I u u I iJ3u 2 £ 3:1 /Jt + - ~ (tJt) + ... - = -2 at 6 dt through the grid point taken as the origin in this example passes through another grid point, A, denoted by a (4.8) I iJ2u I a3 u 2 ) square. Thus, the true solution at A, is equal to the - c - 3':l tJ:x - - T.:] (tJx) + ··· . ( 2 ax 6 ax initial value at the origin. However the numerical solution given by (4. 5) A is. computed using the values As before. these are the terms that were "truncated at points denoted by circles. The shaded domain, off" to make the differential equation reduce to our including all of these points, is called the domain of finite difference scheme. dependence of the numerical scheme. The grid point at In the same way as for an approximatio n to the deri the origin is outside that domain, and, thus, cannot vative. the order of accuracy of a finite difference scheme affect the numerical solution at A 0 • Therefore, the error is the lowest power of tJx and /Jt that appears in the can be arbitrarily great. If the space and time steps truncation error. T hus, scheme (4. 5) is first order were reduced by the same relative amount, say to one accurate. We can write half of their values in the figu re, the domain of dependence would still remain the same, and this situation would not & = 0 (/Jt) + 0 (/Jx), change. T hus. as long as the ratio of the steps tJx and /Jt or remains the same, refinement of the grid cannot bring & = 0 (/Jx, tJt). a bout a reduction in the error of the numerical solution. It is useful to make a distinction between orders of accuracy in space and in time, especially when the lowest x-ct = const powers of tJx and /Jt are not the same. As before, a necessary condition for consistency of a scheme is that it be at least of the fi rst order of accuracy. 5. Convergence The truncation error of a consistent scheme can be made arbitrarily small by a sufficient reduction of the increments Ax and At. Unfortunately, we cannot be sure Figure S. I A possible relative position or a characteristic and or that this will also result in a reduction of the error of a domain or dependence. the numerical solution. For that reason, we return to consideration of the error uj - u (jtJx, nAt). A necessary condition for convergence of a scheme is. obviously. that the characteristic defining the true solu Following Richtmyer and Morton (1 967) we ask two tion at a grid point is inside the domain of dependence questions: of the numerical solution at that point. In our example, (a) What is the behaviour of the erroru1- u (jtJx, nAt) this wi ll happen when the slope of the characteristics when. for a fixed total time ntJt, the increments tJx, At is greater than the slope of the dashed line bounding the approach zero? domain of dependence. that is, when 6 INTRODUCTION ; GENERAL REMARKS ABOUT G RID POINT METHODS cL1t ~ L1x (5 . I) This proves the boundedness of the numerical solution. Hence, I - µ ~ 0 is seen to be a sufficient condition for Thus, this 1s a necessary condition for convergence stability of (6 . I). of (4 . 5). This direct testing of the stability is simple. Unfortun ately, as might be anticipated from the argument, it is successful only for a rather limited number of schemes. 6. Stability Energy method. This method is of a much wider appli The answer to the second question raised at the begin cability, and can be used even for nonlinear equations. ning of the section 5 depends on the stability of the numer If we know that the true solution is bounded, we test ical solution. A rigorous definition of stability employs whether (uj)2 is also bounded. If it is, then every the concepts of functional analysis, and refers to the 4 boundedness of the numerical solution only (e .g. Richt value uj must be bounded, and the stability of the myer and Morton, 1967). The difficulties in defining scheme has been proved. The method is called the energy stability are caused by the fact that the true solution, method since in physical applications u2 is often propor in general, does not have to be bounded. However, tional to some form of energy. Of course, there are when we know that the true solution is bounded, as in examples when this is not so . the equations we are interested in here, we can use a Squaring (6. I) and summing over j we obtain definition referring to the boundedness of the error u7 - u (jL1x, nL1t) . We say that a solution u7 is stable if 1 2 2 2 this error remains bounded as n increases, for fixed ~ (u;+ ) = ~[(1-µ) (u;) + 2µ(1 -µ)u; ui~• + values of L1x. L1t . As before, we say that a finite difference 1 1 (6.3) $Cherne is stable if it gives a stable solution for any initial + µ2 (ui~/J. conditions. Stability of a scheme is a property of great practical We shall assume a cyclic boundary condition, for example significance. There are consistent schemes, of a high order of accuracy, that still give solutions diverging unacceptably fast from the true solution. Thus, conditions then for stability, if any, should be known. There are three 2 (u~_ / (u/) • (6 .4) methods that can be used to investigate the stability I: 1 = ~ of a scheme, and we shall give an example of each of J J these methods. We shall do this by considering again Now, using Schwarz's inequality the forward and upstream scheme (4.5) . Direct method. Since we know that the true solution is bounded. it suffices to test the boundedness of the numer and (6. 4), we can write ical solution. The scheme (4 .5) can be written as 2 1 u;+ = ( I - µ) u7 + µ ui-1, l6, I ) ~u/ u1~1 .; ~ ~=I: l u7l+µlui- 1I· (6.2> or 1 1 We can apply this at the point where at time level n + I ~ (u/ +1 12< ~ ( ut)2. 1 I u;+' I is a maximum, Max We will usually not be able to use it to test the stability This gives of nonlinear equations, and will have to resort to the analysis of their linearized versions. A solution to a linear equation, however, can be expressed in form of a For each harmonic solution t 6 . 8) to be stable it is required Fourier series, where each harmonic component is also that a solution. Thus, we can test the stability of a si ngle I U (n) I = I AI (n) I U (0) I < 8 , harmonic solution ; stability of all admissible harmonics will then be a necessary condition for stability of the where Bis a finite number. This gives scheme. For an illustration of this method, it is useful first to n In I A I < In (B/1 u<0>I)= B' , obtain an analytic solution of the equation (4. 1) where B' is a new constant. Since n = 1/At, the necessary condition for stability becomes B' In IAI < At. (6. II) in the form of a single harmonic 1 Now, suppose that we require boundedness of the solu u (x, I)= R~ (1) e ikxJ. (6.6) [u tion for a finite time t. Conditio n (6.11) can then be Here U (1) is the wave amplitude, and k the wave number. written as Substituting this into the preceding equation we obtain dU + ikcU= O. If we now define di IAI =I+ 6, Thus, the problem of solving a partial differential equation we see, in view of the power series expansion of In ( I 6), has been reduced to that of solving this ordinary differen + tial equation. Its solution is that the stability condition obtained is equivalent to U (1) = U (0) e -ttct, 6 ~ 0 (At). or where U (O) is the initial value of the amplitude. Hence, (6.12) the desired harmonic solution is u (x, I ) = Re (U(O) e•t An equation such as (6. 15) gives further information the graphs of the I i..1 2 curves as shown in Fig. 6. I. In about the behaviour of the numerical solution. This can general, as the wave length L increases, that is, as k be obtained by studying the variation of I A I with µ for approaches zero, the amplification factor approaches various fixed values of kLJx . To this end we plot the unity for any value of the parameter µ . I I i.. I curves : (6 . 15) shows that in the present case all The figure shows that within the stable region the scheme of these curves are parabolas. Furthermore, recall that is damping for all values µ < I. The damping increases the minimum resolvable wave length is 2L1x. Thus, the as the wave length decreases. Since the true solution maximum value that wave number k can take is 1t/L1x. We has a constant amplitude, this damping reveals an error thus plot the I i.. I 2 curves for this maximum value due to finite differencing. We see that this error increases k = 1t/L1x (or wave length L = 2L1x), and for half this as the wave length decreases. At the shortest resolvable value, k = n/2L1x (L = 4L1x}, and a quarter of this wave length, L = 2L1x, the error may be very great value, k = n /4L1x (L = 8L1x). The first derivative unless At is extremely small. It is even possible for 2 this wave to be completely removed after only a single d I i.. 1 -- = -2 (1-2µ) (1-cos kLJx) time step ! The dependence of the error on wave length, dµ . as seen here, might have been anticipated by considering representation of harmonics of various wave lengths by shows that all the I i.. I 2 curves have minima at µ = I /2. the finite difference grid. The shortest resolviible wave, This information, in addition to calculation of the ordi with only two data points per wave length, is very poorly nates of (6. 15) at µ = 0, I /2 and I, suffices for sketching represented ; as the wave length increases, the representa tion by a finite difference grid improves, and approaches I I 'I!, L = 4L1x the continuous representation as the wave length tends , ,.,• L = 8.dx to infinity. -··------·------·-- --- ,, ...... There exists a wealth of more precise definitions of ,,~'" ,: stability and convergence, as well as stability criteria . ••• -····· I ------· I For a further discussion of these subjects, and of the I relation between the properties of stability and conver I I gence, the interested reader is referred to the book by I Richtmyer and Morton (1967) and to the publication I 0.5 I by Kreiss and Oliger (1973). However, for application I of numerical methods to atmospheric models, it is more I important to discuss other problems than to refine the I I stability and convergence concepts beyond the outline I L = 2.dx given here. These numerical problems, such as phase. I I I speed errors and computational dispersion, nonlinear / / instability, effect of the space-time grid on the properties 0 / of the numerical solution, and, also the ideas behind 0 0 .5 µ = cL1t/L1x and properties of the great variety of schemes that are Figure 6. I Sketches of I >.12 curves, for the scheme (6 .1), and currently being used in atmospheric models, will be for various values of l. discussed in the remaining chapt«s of this publication. CHAPTER II TIME DIFFERENCING SCHEMES In this chapter we consider ordinary differential equa an ordinary differential equation, the equation and a tions with om• dependent and one independent variable. single initial condition is all that is required for an exact Although atmospheric models are essentially always solution. Thus, the error of the numerical solution models for solving a complex set of partial differential is entirely due to the inadequacy of the scheme. With a equations, in some formulations the numerical solution of partial differential equation, the error of the numerical ordinary differential equations forms an important solution is brought about both by the inadequacy of the part of the computational proc-edure. For instance in scheme and by insufficient information about the initial spectral models the governing partial differential equa conditions, since they are known only at discrete space tions reduce to a set of ordinary differential equations for points. Thus, an increase in the accuracy of the scheme the expansion coefficients as dependent variables. A set improves only one of these two components, and the of ordinary differential equations will also be obtained results arc not too impressive. if a Lagrangian method is used, in which the computa Another reason for not requiring a scheme of high tional points move with the flu id. But , most of all, accuracy for approximations to the time derivative terms schemes for solving ordinary differential equations are is that, in order to meet a stability requirement of the of interest here since they are often used without modi type discussed in the preceding chapter, it is usually fication to construct approximations to the time deri necessary to choose a time step significantly smaller than vative terms in the governing partial differential equa that required for adequate accuracy. With the time step tions. Knowledge of the properties of schemes for solving usually chosen, other errors, for example in the space ordinary differential equations will then be used in inves differencing, arc much greater than those due to the time tigating the properties of more complex schemes for differencing. Thus, computational effort is better spent solving the partial differential equations. in reducing these other errors, and not in increasing the With that in mind. we shall here fi rst define some of the accuracy of the time differencing. This, of course, does schemes that will be interesting to analyze. Then we not mean that it is not necessary to consider carefully the shall investigate the behaviour of numerical solutions properties of various possible time differencing schemes. obtained when these schemes are used for two specific Accuracy, is only one important consideration in choosing ordinary differential equations : the oscillation (or a scheme. frequency) equation, and the friction equation. These To define some schemes, we consider the equation equations will serve as prototypes for later extension of the results to advection, gravity-inertia wave, and diffusion processes within the atmospheric primitive ~~=f(U,1) 1 V=U(t). (I.I) equations. The independent variable t is here called time. We I . Definitions of some schemes divide the time axis into segments of equal length '11. Schemes used for the time derivative terms within the We shall denote by u (n+ I) 41 the value of U obtained for time level n + I is then used U(n+I) = U(n)+ /(U,t)dt. (I. 2) for an approximation to J<" + 11 • and this approximate n41 J value J<" +1> • is used to make a backward step. Thus, We shall first list several schemes which do not use an ij(HH) • . UM I 4t·f'">. iterative procedure·. lj ( Jt + I) LJlH) + At·!'" I II•. (I. 6) A I. Euler (or forward) scheme. This is the scheme where u1n +o = Ul"l + tdt·Jt">, where (1.3) This is an explicit scheme, of the first order of accuracy. AS. Heun scheme. Here, in much the same way, an The truncation error of this scheme is O (tdt). Thus, this approximation is constructed to the trapezoidal scheme. is a first order accurate scheme. For the integrand in Thus, (I . 2) we have here taken a constant value equal to that ij(n+ I)• = ij("I + .di ,J(n), at the lower boundary of the time interval. Thus, fin (1. 7) (I . 3) is not centered in time, and the scheme is said to be uncentered. J n general, uncentered schemes will be found to be of the first order of accuracy, and simple centered schemes to be of the second order of accuracy. Thus, this is also an explicit scheme. It is of the second order of accuracy. A2. Backward scheme. We can also take a constant B. Three level schemes. Except at the first step, value off equal to that at the upper boundary of the one can store the value U If, as here, a value off depending on u1n+ IJ appears (11+I) 41 in the difference equation, the scheme is called implicit. u< 11 + o = u This also is a second order accurate scheme. As another simple example we can consider the accel There are many other rather obvious possibiljties. eration a nd Coriolis terms of the horizontal component For example, one can approximate the integral in (I . 8) of the equation of motion of the atmosphere, that is using Simpson's rule, that is. by fitting a parabola to the 11 du dv valuesJ<" - ,J<"> and pu 11 • The implicit scheme obtained -=fv, - =-Ju. in this way is called the Milne-Simpson scheme. To dt dt illustrate the wealth of possible alternatives we note that If we define in a paper by Young (1968) properties of 13 different U = u + iv, schemes have been studied. Furthermore, when we are solving a more complicated partial differential equation, we can write these two equations as or a system of such equations, time (or space-time) differencing schemes can be constructed which are more = -i/U. complex than those which can be defined using the !~ simple equation (I. I). Such schemes are widely used in atmospheric models, and some of them will be described This again reduces to (2. 1), this time if we substitute in later chapters of this publication. (I) = - f. Since there are many more important types of wave 2. Properties of schemes applied to the oscillation equation motion, we can hope that results obtained by a study of (2 . l) will be much more general. It can, indeed, be The stability and other important properties of the shown (e.g. Young, 1968) that the equation (2. 1) can time differencing schemes defined in section ( I) depend be obtained from a rather general linearized system of on the form of the function f(U,t). Thus, in order to governing equations, describing a number of types of discuss these properties we have to prescribe this function. wave motion in the atmosphere. For applications in atmospheric models it is of particular interest to consider the case The genera l solution of (2.1) is 1 f = iroU, U (t) = U (0) ecai , that is, the equation or, for discrete values t = ntJt, U (ntJt) = U (0) e'"m,1 1 (2.2) ~~ = iwU1 U = U(t). (2. I) Thus, considering the solution in a complex plane, Equation (2. I) we shall call the oscillation equation. The its argument rotates by rotJt in each time step At, and word frequency equation is also used. We allow U to be there is no change in amplitude. complex; then (2 . 1) can be thought of as representing a system of two equations. The parameter ro is real, and The pro perties of various schemes when applied to is called the frequency. (2. I) are conveniently analyzed usi ng the von Neumann method. This method, as we have seen, involves defining It is easy to give some justification for our interest in the a variable ).. by eq!,Jation (2.1). As an example, recall that the harmonic component u au au Thus, the numerical solution can formally be written as Qt + C ax = 0 I C = COnSt., U <"> =I). I" u In accordance with this and (2. 5), we shall say that a Therefore, scheme is unstable 1,.= l+iap. (2.10) neutral if l-il3p damping or, (or dissipative) It will also be instructive to compare the phase change 2 of the numerical solution per time step, 9, with that of the "' = I+ ~2 p2 (l-aj3p + ip). true solution, o.u1t. The ratio of these changes, 9/0>'11, is the relative phase change of the numerical solution. Substituting for a and 13 allows us to investigate the Obviously, we can say that a scheme is effect of particular schemes. For the Euler scheme we have accelerating "'=I + ip, (2.11) of no effect on 9 > if -= l. phase speed O>iJt < for the backward scheme decelerating . For accuracy, therefore, it is desirable to have both the "' = I ) p2 (1 + ip), (2.12) amplification factor and the relative phase speed close to unity. Exceptions to this arc so-called "computa and, for the trapezoidal scheme, tional modes", which, as we shall see later, can appear as false solutions superposed on the physical solution. 1,.=+. (1- !p2+ ip)· (2. 13) These are solutions that do not approach the true solution I +4P2 4 as the space and time steps approach zero. If such solutions exist they will each have their own value of To test for stability we need to know I A I· Since the amplifica~ion factor. Since they arc not an approxima the modulus of a ratio of two complex numbers is equaf tion to the true solution, it is desirable to have their to the ratio of their moduli, we can obtain the values ol amplitudes as small as possible, that is, to have their I A I directly from (2. 10). For the Euler scheme we have amplification factors less than unity. (2 . 14) We shall now discuss the properties of the schemes that have been defined in the preceding section. The Euler scheme is, thus, always unstable. It is interest Two level schemes. The three non-iterative two level ing to note that, if tJt is chosen so as to make p relatively schemes can be described by a single finite difference small, we have equation u<-+ 1>= U<"> + tJt (aj<•> + l3j<-+ 1>), (2 .6) l1,.l = I + ! p2 + .... (2 .15) 2 with a consistency requirement 2 This shows that I A I = 1 + 0 [('1t) ], that is, I A I - 1 is a+l3=l. an order of magnitude less than the maximum allowed by the von Neumann necessary condition for stability. Obviously, a = I , P = 0 for the Euler scheme, a = 0, However, experience shows that an indiscriminate use 13 = 1 for the backward scheme, and a= 1/2, 13 = 1/2 for of the Euler scheme for solution of the atmospheric the trapezoidal scheme. equations leads to amplification at a quite unacceptable Applied to the oscillation equation (2. 6) gives rate. For the backward scheme we obtain U<"+ 1>= U'"' + i0>tJt (aU'"> + 13 u In order to evaluate A we must solve this equation for (2 . 16) U <"+1>. Denoting, for brevity, The backward scheme is, thus, stable no matter what p = O>iJt, (2 .8) value of tJt is chosen. Thus, it is an unconditionally we obtain stable scheme. We can, furthermore, notice that it is u<"+ I) = I+ iap u<"> (2.9) damping, and that the amount of damping increases as l-ll3p . the frequency 0> increases. This is often considered to CHAPTER 2 13 be a desirable property of a scheme . For instance, we can and for the Heun scheme think of a system in which a number of frequencies are present at the same time : for example, solving a system of equations of the type (2. 1). This situation is similar to (2. 22) that existing in the real atmosphere. It would appear to be necessary to maintain the amplitudes of motions To test for stability we evaluate I A I, For the Matsuno of different frequencies in the correct ratio. However, scheme we obtain in numerical integrations, high frequency motions are often excited to unrealistically large amplitudes through I A I = (l-p2 + p')Y·. (2.23) errors in the initial data. It may then be desirable to reduce the amplitudes of high frequency motions by a Thus, the Matsuno scheme is stable if selective damping in the time differencing scheme. In other words, a scheme with frequency dependent damping IPI ~ I. properties can be used to filter out undesirable high frequency motions. In other words, to achjeve stability we have to choose At For the trapezoidal scheme we find sufficiently small so that (2.24) (2.17) The Matsuno scheme, thus, is conditionally stable. The The trapezoidal scheme is, thus, always neutral. The higher the frequency, the more restrictive is the stability amplitude of the numerical solution remains constant, condition. just as does that of the true solution. It is useful to note that both the implicit schemes considered here were Differentiating (2.23) we find that stable no matter how large a value of At was chosen. The iterative two level schemes can also be described d I AI _ p (i_ 2 ..2) by a single equation in the same way as (2.6). Thus, we dp - (l-p2 +p•)Y, P • write u<01>• = u<"> + .dt·J<">, Hence, the amplification factor of the Matsuno scheme 1 LJ<..+ 1> = U<"> + At(aJ<"> + l3J<"+ >•), (2. 18) has a minimum for p = I /V2. Therefore, as pointed a+l3=1. out by Matsuno (1966a) when dealing with a system with a number of frequencies we can choose a time step Now a = 0, 13 = I for the Matsuno scheme, and a = 1/2, so as to have O < p < I /y2 for all the frequencies 13 = 1/2 for the Heun scheme. present, and then, in the same way as backward implicit scheme, this scheme will reduce the relative amplitudes Applied to the oscillation equation (2 . 18) gives of high frequencies . This technique has recently become very popular for initialization of atmospheric models, where it is used to damp the spurious high frequency noise generated by the assimilation of the observed data. As shown by Matsuno (1966b) higher order accuracy schemes with similar filtering characteristics can be con Eliminating u Thus, IA.I= I+ 4I p4 )'" (2 . 25) A = I -j3p2 + ip (2.20) ( Substituting the appropriate values of 13 we now obtain This is always greater than unity. Thus, the Heun the values of A. for the two schemes. Hence, for the scheme is always unstable, like the Euler scheme. Matsuno scheme However, instead of (2. 15), for small p we now have I 4 A= 1-p• + ip, (2 .21) IAI =I + 8 p + ··· . (2.26) 14 TIME DI FFERENCING SCHEMES that is, I >.. I = I + 0 [(.dt)4). This instability is quite Since the right-hand side is always less than unity, we weak. Experience shows that it can be tolerated when we can see that these two schemes are decelerating. For can choose a relatively small value of .dt . (Note that, p = I we have 0/p = 1t/4. whenever the amplification rate is less than that allowed In other cases the effect may not be so obvious. For by the von Neumann necessary condition, the total the Matsuno scheme, for example, (2.21) gives amplification in a given time is reduced as the time step is reduced.) 0 I p (2.31) Fig. 2 . I summarizes the results obtained for the five ,;= ;arc tan l-p2 • schemes considered so far. For all of these schemes the amplification factors were found to be even functions It is not obvious whether the right-hand side here is of p. so the amplification factor curves are shown only greater or less than unity. However , the behaviour for p ~ 0. of (2.31) for all p is of no practical interest, since we , Matsuno scheme already know that p must be chosen less than unity in I Euler scheme order to ensure stability, a nd rather small for frequencies )'' for which we want the integration errors to be small. ,• I _,/ I Thus, we need only consider (2 .3 1) for small p; we .• I •• I Heun scheme obtain ,,.··· I ,, I ,,,,,,,,•• ~ I 0 2 .. -= l+ -pi+ .. · . ... p 3 ...--·"' The Matsuno scheme, therefore, is seen to be accelerating . . For the special value p = I this can be seen directly from Backward scheme (2. 13), since then 0/p = 1t/2. Analysis of phase errors of schemes applied to the oscillation equation is not so important as analysis of the amplification factor. Phase errors do not affect stability ; and when these schemes are used to solve the partial 0 l/y2 p differential equations of motion additional phase errors due to space differencing will appear. We will then be Figure 2 . 1 The amplification factor as a function of p .., c.,;,JJt for interested only in the total phase error , and it will be five two level sche.mes and for the true solution. found that the error due to space differencing is usually dominant. It is a lso of interest to consider the phase change per time step, 0, and the relative phase change per time step, Three level schemes and computational modes. We 0/p. Using the notation consider first the leapfrog scheme (I . 9) . Applied to the oscillation equation it gives (2.27) ij(ll+l) = ucn- 1) + i2ro.dt u ).. . A problem with all three o r more level schemes includ 0 = arc tan J1!J • (2.28) ing this is that they require more than one initial condition 'A.,r to star~ the computation. From a physical standpoint a single initial condition U 10> should have been sufficient. or 0 I ')... . However, in addition to the physical initial condition. - = - a rc tan 'Jl!J, (2 .29) three level schemes require a computational initial condi P p Au tion u< 1>. This value cannot be calculated by a three level scheme, and, therefore, it will usually have to be F or the Euler and the backward schemes, using obtained using one of the two level schemes. (2.11) and (2. 12) we obtain According to (2. 3) we also have 0 I - = - a re tan p. (2.30) p p (2. 33) CHAPTER 2 15 When these relations are substituted into (2.32) we Thus, we obtain a numerical solution that is equal to the obtain true solution (2. 36), and consists of the physical mode only. 8 . Suppose calculating u 0 > gives v u> = -u<0>. a second degree equation for A. It has solutions Then we obtain, for all n, A1=~ + ip, or (2.34) u The leapfrog scheme, applied to (2. 35), gives Therefore, the amplitudes of the physical and of the computational modes are seen to be proportional to, lj(n + I) = lj(n - 11. (2.37) respectively, f-or a given physical initial condition u<0>, we consider I lj(ll_ ).2 LJ(O) I and I LJlll _ )., u 1 0 02 = rr. - 01. u< > = i..1 u<> so as to eliminate the computational mode. Numerical methods are used in practice to solve Specifically, as p -• 0, 9 · • p. while 02 • rr. --p. Thus, equations that cannot be solved by analytical methods, 1 for small LJt the physical mode is seen to approximate and arc more complex than the simple oscillation equa the true solution, while the behaviour of the computa tion (2. 1). In these cases we will not know the exact tional mode is quite different. For the case p < 0, we values ofi.. and At· Thus, 1> is usually computed using 1 u< obtain in the same way one of the two level schemes. The simplest method is to use the Euler scheme, or, a more refined procedure could be used, for example the Hcun scheme. Using (2.39) it can be shown that the latter alternative will give Thus, for p ~ 0, a smaller amplitude of the computational mode. et = ± rr. - 01. (2.42) We also note that even if we did know the exact value of i..1, this would still not allow the computational mode rr. to be eliminated in a practical numerical calculation. ,,. The numerical solution which we calculate is not an ..,,.,.,'7t exact solution of the finite difference equations, since the __ ,,,. ,,.,,,. -2 arithmetical operations arc performed in practice only ------to a finite number of significant digits. The error produced in this way is called round off error, though in electronic computers results of arithmetic operations are sometimes truncated to a given number of digits, instead of being rounded off. With round off errors present, permanent elimination of the computational mode is not possible Figure 2.2 Phase change of the physical and of the computational in principle, since the computational mode would appear mode for the leapfrog scheme. in the course of integration in any case even if it were absent initially. However, it is usually found that round off errors are of little importance in atmospheric For accuracy of the physical mode, 01 should closely models, and in solving partial differential equations in approximate the phase change of the true solution, p. general. For small p (2 .41)1 gives Proceeding now to the stability analysis, in view of I (2 . 38) and our inability to eliminate the computational a, = p + - i3 +···. 6 mode completely, we will have to require for stability that neither of the two amplification factors is greater Thus, the leapfrog scheme is accelerating; the accelera than unity. It is convenient to consider three special tion, though, is four times less than that of the Matsuno cases. scheme. Jt is instructive to note that schemes of different I. Ip I < I. In (2. 34) t - p1 is positive, and we obtain orders of accuracy can stilt have the same order of leading term in power series expansions of either the amplification I i..d = I i..t I = 1. (2.40) factors or the phase changes. Thus, in this case both modes arc stable and neutral. Differentiating (2.41)i we find For the phase change. using (2. 28) ~ - I dp - v1-p2 2 01 = arc tan (p/V l - p ), (2.41) The phase error, thus, is seen to increase sharply asp - I, 03 = arc tan (- pl~). when 01/p _. 1t/2. It is instructive to consider the behaviour of 0, as a It may be useful to illustrate the behaviour of the two function of p. especially asp · • 0. We consider first the modes obtained case p '> 0. Since for both modes "'''" = I i.. I sin 0 = p we have O < 0 < 1t. Considering the signs of i..., we find that O --... 01 < 1t/2, and rr./2 < 02 < rr.. To illustrate these results, the phase changes (2. 41) arc plotted in in the complex plane. For simplicity, we consider the Fig. 2 .2. We sec that, for alt p, case 01 = 1t/8 and assume that the imaginary part of CHAPTER 2 17 the solution is equal to zero at the initial moment. The A detailed knowledge of the behaviour of the computa physical mode, as seen in (2.43), rotates in the positive tional mode may be helpful in recognizing its excessive sense by an angle 01 in each time step .tlt, while at the presence in an integration. Thus, we plot the real and same time the computational mode, in the case p > 0, imaginary parts of the computational mode as functions rotates by an angle n - 01 • Therefore, the two modes can of time. This can be done by using an alternative form be represented graphically as in Fig. 2. 3. of (2.43)z 1 1 u<;> = ( - 1)" U ~ (cosn01 - isinn01), I or directly from Fig. 2. 3. We obtain diagrams as shown 4 in Fig. 2 .4. Because of the factor ( - 1)", both real and 5 ) imaginary parts oscillate between time steps. u•• n n = O Physical mode n 3 5 Figure 2 .4 Real and imaginary parts of the computational mode, for the leapfrog ~heme, for the same case as shown io the preceding figure, as functions of time. II . Ip I = I. This is a limiting case of the solutions considered for Ip I < I. (2. 34) shows that the values of ). are now equal, A.1 = A.2 = ip. 2 Therefore (2.44) 4 Computational mode Thus, both modes are still neutral. Since neither of them has a real pan, we obtain, for p = ± I , Figure 2. 3 The physical and computational modes, for the leap frog ~heme, when 81 = 11/8 and when the imaginary parts are zero at the initial moment, for a number of (2.45) values of n. 18 TIME DIFFERENCING SCHEMES Therefore, the two modes can be written in the form check for the presence of growing oscillations of that period. (2.46) In a complex plane, they rotate by an angle of ± rc/2 in each time step, while the true solution rotates by an angle of ± I only. T he phase error, thus. is large. 111 . Ip I > I : Both values of >.. in (2 . 34) still have imaginary parts only. so that >.., = i(p + v9=°i). n 2 A.2 = i (p - Vp - I), where the expressions in parentheses are real. Therefore, I>.., I = IP + v9=°i I, (2.47) 2 I A.2 I= IP - v'P - 11. Figure 2. 5 The real part of the unstable mode, for the leapfrog scheme, in case when Il l = 1. 1 and when the imaginary part is zero at the initial moment, as a Thus, for p > I we have I >..1 1 > I, a nd for p < -I function of time. I >..2 I > I. Therefore, for Ip I > I the leapfrog scheme is unstable. The instability increases sharply as IPI increases beyond I ; we can sec this, for example for To sum up, advantages of the leapfrog scheme are p > I , because that it is a very simple scheme, of second order accuracy, and neutral within the stability range I c1.u1t I ~ I. A d isadvantage of the leapfrog scheme is the presence of a ~~= I + p dp ~ neutral computational mode. With nonlinear equations there is a tendency for a slow amplification of the compu which is unbounded as p -+ I . tational mode. An example of this growth can be seen, for example, in one figure of a paper by Lilly (1965). Since the two values of >.. still have no rear parts, we The usual method used for suppressing this instability again have is the occasional insertion of a step made by a two level scheme, which eliminates the computational mode. (2 .48) A multi-level scheme that damps the computational mode could also be used for this purpose. When solving the system of gravity wave equations, The two modes for p ~ I, can thus be written as as wi ll be shown in Chapter JV , it is possible to construct grids and/or finite difference schemes which have essen u<~> =IP+ vp2 - l I" u <~>e *'"i. (2. 49) tially the same properties as the leapfrog scheme, but in which the computational mode is absent. These u<;> = IP - 2 - I I" u <~>e*' ";. vp methods calculate the physical mode only, and at the In the complex plane, both modes again rotate by an same time require only one half of the computation angle of I n/2 in each time step. However, this time time needed for the regular leapfrog scheme as described the amplitude of one of the modes increases. and that here. of the other decreases with time. The real part of the We consider, finally. stability and other properties unstable mode can. for instance. be represented as a of the Adams-Bashforth scheme (1.10). Applied to function of time by a graph like that in Fig. 2.5. Because the oscillation equation it gives of (2.48) the period of the unstable oscillation is always 4.dt. This can be used to diagnose the instability: if the results appear unsatisfactory, it is a good idea to CHAPTER 2 19 Substituting the relations (2. 33) we find I IAd= I + 4 p4 + ···. (2. 53) IA21=1p+···. Expressions (2. 52) and/or (2. 53) show that the physical mode of the Adams-Bashforth scheme is always unstable. We have, of course. again obtained a second degree However, as for the Heun scheme, the amplification is equation for A. It has the solutions only by a fourth order term, and it can be tolerated when a sufficiently small value of Llt is chosen. Note that the amplification given by (2. 53) is twice that given by (2. 26) A1 + i p + p2 + ip). 1 =I (1 % Ii-~ for the Heun scheme. Since the amplification is propor (2.51) tional to (Llt )4. however, a small reduction in time step would compensate for that difference. Thus, the Adams A I + i p - p2 + ip)· 2 = t ( % vi - ~ Bashforth scheme, with only one evaluation of the right hand side per time step, can still be considered Thus. as p -• 0 1-1 - I, while Ai 0. We see that the much more economical. It has been fairly frequently solution associated with A1 again represents a physical used in meteorological numerical studies. For example, it mode, and that associated with A2 a computational is being used by Deardorff in his numerical simulations mode. However, while for the leapfrog scheme the of the planetary boundary layer (e.g. Deardorff, 1974). computational mode was found to be neutral, here it is Analyses of the properties of some other schemes, seen to be damped. This is a very useful property of the applied to the oscillation equation, can be found in papers Adams-Bashforth scheme as the computational mode by Lilly (1965), Kurihara (1965) and Young (1968). cannot cause inconveniences. In practice the choice of a scheme will depend not only The exact analysis of the amplification factors here is on the properties considered here, but a lso on some more difficult because of the presence of square roots in practical considerations. For example, we might expect (2 . 51 ). However. since for reasons of accuracy we that the three level schemes, since they use more informa have to choose a relatively small value of p in any case. tion, would generally give better results than the two it will suffice to consider amplification factors for small level schemes. Our findings agree with that conjecture ; values of p only. T he power series expansion of (2.51) for example, for second order accuracy the explicit then gives three level schemes required only one evaluation of the right hand side per time step, while the two level schemes ).. = I + ip - J.. p2 + ; .!_ pl - l p4 + ... required two evaluations. As another example, if we I 2 4 8 • want to damp high frequency motions with three level . I I . I I schemes we can linearly extrapolate the derivative beyond A2 := 1 - p + - p2 - 1 - pl + - p4 _ ... 2 2 4 8 . the centre of the interval (nLJt, (n + I) Llt), and thus Now. after rearranging the terms these series can be obtain a scheme that will perform such a damping in a written as more selective and more economical way than the Mat suno scheme (Mesinger, 1971). However. three level schemes generally require more core storage space in the computer than two level schemes and this may affect our our decision. 3. Properties of schemes applied to the friction equation which can be used to obtuin the amplification factors We shall now consider the properties of schemes when applied to the equation dU (2. 52) -=-te. U, U =U(t). te. > O. (3. I) di We shall call this equation the friction equation. The higher order terms have been omitted. A final Again it is easy to justify our interest in this equation. expansion gives For example, if we define U =u + iv, it describes the 20 TIMI! DIFFERENCING SCHEMES effect of friction proportional to the velocity vector, as For the trapezoidal scheme a = 1/2 and 13 = 1/2 ; the is often assumed for motions near the ground. As scheme is again always stable for K > 0. The solution another example, note that when seeking a solution of the does not oscillate if K < 2. heat transfer, or Fick's diffusion equation Considering the iterative two level scheme (2. 18) we au a2u obtain Q( = (J a;'! I (J > 0, U in the form of a single harmonic component Therefore, both the Matsuno and the Heun scheme are stable for sufficiently small values of K . u (x, t) = Re [(U {t} e•h], It is instructive to consider in some detail the behaviour of the numerical solution obtained using the leapfrog we obtain scheme. Applied to (3 . I) it gives dU 2 u 1 ( . ) giving the solutions U (t) = U (0) e-• • 3 2 A1 - K Thus, both the real and the imaginary part decrease = + Vt+K2, (3 . 9) exponentially with time. 2 )..2 = - K - VI + K • The properties of schemes applied to (3. I) will again be analyzed using the von Neumann method. As in the As K .... 0 )...... I, while ~ .... - I ; thus, the solution previous section, we consider first the non-iterative two 1 level scheme (2 .6). Applied to the friction equation. associated with )..1 again represents the physical mode, (2 .6) gives and that associated with )..2 the computational mode. For K > 0, that is, for the normal case of a forward lj(fli o = U"-IC.dt(aU1") + pu u,. 4. A combination of schemes A A natural question to ask at this point is what can we do if, for example, the equation contains both the oscilla tion and the friction term, that is dU . U U dt = 1(1) - IC • (4 . 1) o~~...... ~~..__~_._~~.L-~-'-~4--.J....l.~-'-~n I Herc we might like to use the leapfrog scheme because of the oscillation term iroU, but we know that it cannot Figure J. I An example illustrating the instability of the leapfrog be used for the friction term - IC U. In this and similar scheme applied to the friction equation. AC is con structed to be parallel to the tangent T. situations we can use different schemes for the different terms ; for example, we might use the leapfrog scheme Finally, for the Adams-Bashforth scheme we obtain for the oscillation term and the fo rward scheme for the friction term. We then obtain The Adams-Bashforth scheme, thus, is stable for suffi ciently small values of K. The computational mode is damped. Other combinations, of course , are also possible. CHAPTER III THE ADVECTION EQUATION We now consider differential equations with one in the preceding chapter. For example, when the time dependent and two or three independent vaTiables, that derivative is approximated using the leapfrog scheme, is, partial differential equations. More specifically, we we obtain shall consider various simplified forms of the advection n+I n-1 n n equation, describing advection of a dependent variable. Uj -UJ =-CUJ+1-UJ-I, (I .4) This is considered in practice to be the most important 2At 2Ax part of the atmospheric governing equations. as one of many possible consistent schemes for the We have already discussed the one-dimensional linear numerical solution of (I. I). advection equation to some extent in the introductory chapter. We shall organize the analysis here so as first The properties of schemes constructed in this way can to continue considering problems associated with the be inferred from the known properties of time differencing simplest one-dimensional linear form of the advection schemes applied to the oscillation equation. To see this, equation. and then to proceed to problems introduced we substitute into (I. 3) a tentative solution in form of by more complex forms of the advection equation. the single harmonic component (1. 5) I . Schemes with centered second--0rder space differencing After some rearrangement, this gives We shall first consider the equation au i)u dU . ( c ) i)r + c i)x = 0, c = const. (I. I) dt = 1 - Ax sin kAx U. (1.6) Here u = 11 (x ,t) is a function of two independent varia If we denote bles : the independent variable x will represent a space variable, and, thus, (I . I) will be called the one-dimen co sin kAx (I. 7) =- Ax.!.... ' sional linear advection equation. As seen earlier, its general solution is this is equivalent to the oscillation equation of the previous u = J(x- ct), (I. 2) chapter. Now, if we approximate (I .6) using one of the where f is an arbitrary function. The name "advection time differencing schemes studied in that chapter, the same equation" was suggested by Phillips (1960). finite difference equation is obtained as when we apply One of the finite difference schemes for ( I . I) is the that scheme to (I. 3) and then substitute the wave solution forward and upstream scheme, which has been shown ( 1. 5). Hence, properties of finite difference schemes excessively damping in Chapter I. derived from (I . 3) can be inferred from the results of Section 2 of Chapter II, where the frequency co is If the space derivative in (I . I) is approximated by a given by (I. 7). centered finite difference quotieni using values at the two nearest points, we obtain for the time derivative As an example, if (I . 6) is approximated using the lc•apfrog scheme, we obtain du;_ u ,+1 - Uj-1 __._ - -c (I. 3) a, 2Ax U(n+I) = U(n- t) + 2i (-c :; Sin kLlx) U(n). (1.8) The subscript here, as before, denotes the distance from the origin in space increments; that is. x = jAx. A Using the notation of chapter II , we write number of schemes for the numerical solution of (I. I) can now be constructed because we can approximate the Ar . k (I. 9) time derivative in (1.3) by one of the methods studied P=-CAxsin Ax. CHAPTER 3 23 We obtain the same finite difference equation {1.8) by For the computational mode, on the other hand, we fi rst applying the leapfrog scheme to {I. 3) giving {I .4) and obtain then substituting ( I . 5) into ( I .4). Thus, properties of {I . 4) can be inferred from {I. 7) and from the known 1141 u; Re [ (-1)" {jdx - ) (1.14) properties of the leapfrog scheme applied to the oscillation = u!o) /k k!, J. equation. Let us look at some of conclusions obtained in These expressions can be compared with the true solution this way. For stability of the leapfrog scheme it was of {I. I) in the form of a single harmonic component, required that the condition Ip I :.,; I be satisfied for all as given in chapter I, values of co occuring. Thus, we have to satisfy u (x, t) = Re [U (0) c'·t<.• - <11 ] . (I. 15) We find that the phase speed of the physical mode, c1 , is equaJ to - 0/kLJt, and the phase speed of the computa tional mode. c2 , considering the even time steps only, is for any admissible k. Since I sin kL1x I does reach the equal to 0/kLJt. Eq. (1. 12)2 shows that as Lit -+ 0 0 - p, maximum value of unity in the admissible range of k, a nd {1.9) shows that as Ax -+ 0 p~ -ckL1t. Thus, as we obtain the stability condition Ax, Lit .... 0 c1 -+ c, that is, the phase speed of the physical mode approaches the phase speed of the true solution, while at the same time c • -c. In addition, the compu I. (1.10) 2 lei:: < tational mode changes sign at all grid points from time step to time step, because of the factor ( - 1)" in (I. 14). This criterion, obtained already in Chapter I, shows Now let us use a nother scheme from chapter 11 to that stability cannot simply be achieved by reduction approximate the time derivative in {1 . 3), the Matsuno of the time and space increments. Rather, it is necessary scheme. First t he approximate values u,<'a I) • are com to reduce the ratio of these increments L1 t /L1x to obtain puted using the forward scheme, that is, stability. The condition (1 . 10) was first found by Courant, Friedrichs and Lewy {1928), and, therefore, is usually ( 11 + 1>° II fl II Uj - Uj = - C Uj+ I - Uj-1 . (1. 16) referred to as the Courant-Friedrichs-Lewy, or CFL, Lit 2L1x stability criterion. It is instructive to note that the maximum value of Then, these approximate values are used in the backward IP I, that is, the minimum stability, is associated with scheme, that is 0 0 the wave with kL1x = rt /2. This is the component with n+I II (11+1) (11 + 1) Uj - Uj wave length 4L1x , twice the shortest resolvable wave = _ C Uj + I - Uj-1 {I . 17) length 2L1x. L1 t 2L1x We can also use other results of the previous analysis. It is instructive to eliminate the approximate values There a re two solutions for u 11 + 1 " " " A1 .ind )..:! are given here by Eqs. (2. 34) of Chapter II. Uj -Uj Uj+t - UJ-1 2L1x In the stable case, we have. for p ~ O. Lit = - c + (I. 18) II II II 2 Uj + 2 - 2uj + Uj-2 + c Lit (2.dx)2 ( I . 12) Without the last term. this represents the finite difference equation obtained when the forward scheme is used for Using ( I . 5), it is seen that the approximation uj also the time derivative in ( I . 3). The third term approaches hns a physical an and it has a damping effect. This damping effect, how Adding this to the relation gives ever, is dependent on the wave length. As the third term is calculated over an interval of 4£1x. the maximum damping occurs for a wave with wave length of 4£1x. There is no damping of the shortest resolvable wave with wave length 2£fx. Even if a damping effect were desir· able when solving the advection equation, we would not Substituting (1. 21), and eliminating P using P = I-a, want this particular dependence on wave length. Thus, we obtain, after some rearrangement the Matsuno scheme does not appear suitable for solving the advection equation. 11 11 ~L;"i I [("'1 +1)2 - ("i )2] = (a-2 I)~ ~ ("i 11+1 -ui11)2 .(1.24) It is convenient to include here one more example of J J the use of the energy method for testing stability. In addition to being applicable also to nonlinear equations, Thus, if a > I /2 we have an unstable sche me; if a = I /2 it can be used to study the effect of boundary conditions a stable and neutral scheme, and, if a < I /2 a stable and on stability. We will use the energy method here to test damping scheme, which makes the total "energy" the stability of a group of schemes that can be used for solving (1.3). L ~ (uj)z monotonically decrease with time. A fairly wide class of schemes for solving ( I . 3) can , be written as Finally in this section, we shall analyze a scheme that (1.19) was proposed by Lax and Wendroff (1960) and is thus called the Lax-Wendroff scheme, or, more specifically, the two-step Lax-Wendroff scheme. In contrast with the where µ = C.1t/.1X, (1.20) schemes discussed so far, the Lax-Wendroff scheme cannot be constructed by an independent choice of finite and u; is a linear function of a number of values u;. difference approximations to the space and to the time derivative of the advection equation. To describe the For example, to obtain the non-iterative two level schemes procedure, we use the stencil shown in Fig. I . I. First, we write • " A t1+l u1 = a u1 + .,u, . (1.21) I For the iterative two level schemes write n+I l • II p ( II II ) Uj: Uj - - µ Uj+I - Uj-1 • (1. 22) 2 I n+ . 2 Finally, for the Adams-Bashforth scheme (I. 23) n x . I . I I j-1 1-- j J + - j+ Here we shall analyze the stability of the non-iterative 2 2 two level schemes. It is convenient first to multiply Figure I . I The space-time stencil used for the construction of the (I . 19) by u; and sum for all j; we obtain Lax-Wendroff scheme. ~ • ( 11+1 ") I )' • ( • • ) '-j u1 u1 -ui =-2 µ 'r u1 "i+ 1- u1•1 • provisional values are calculated at the centres of the two rectangular meshes of this stencil, for points denoted The right hand side vanishes if cyclic boundary conditions by x signs. This is done using centered space and for ure assumed : we then have ward time differencing, taking for u; •. ,, and u;_,1, arith metic averages of the values u; at the two nearest grid points. Therefore, CHAPTER 3 25 Uj+'I,n+Y, - °!( Uj"+I + Uj") _ Uj+I" - Uj" Therefore - -(' I ,, ,1x ' ). = I µ2 (cos k,1x- I) - iµ sin k,1x. (I . 30) -at + 2 (1. 25) Since " +Y, ,t( " " ) " " Uj -'11 - Uj - Uj-1 _ Uj - Ul-1 . 2 k,1x I - -c ,1x . cos kt1x - I = - 2 sin 2 , - ,1t 2 . . kt1x k,1x Stn k,1 X = 2 Stn l COS l , Using these provisional values another step is made, centered in both space and time ; thus we finally obtain n+I " 11+% n+'I, Uj - Uj = - C Uj+Y, - Uj-Y, 2 2 4 k,1x]Y, (I . 26) I i..l = I - 4 µ h - µ ) sin . (I . 31) ,1t t1x [ 2 Substitution of the provisional values from (I . 25) into The expression within the bracket is a sum of two squares this equation gives and never negative . Thus, the Lax-Wendroff scheme 2 n+I II II II is stable for I - µ ;;;;,: 0, or , for Uj - Uj = - c· Uj+I - UJ- 1 + (1. 27) '11 2 ,1x ,1t It II II lcl ,1x < I. I 2 Uj+1 - 2Uj + Uj- 1 + 2 c '11 (,1x)2 This is again the Courant-Friedrichs-Lewy stability It is interesting to note that this finite difference equa criterion (I . JO) . The scheme is damping for I c I t1t/ tion is very similar to (I . 18) , that is, to the scheme / ,1x < I. obtained using simple centered space differencing and the It is instructive to analyze in some detail the dependence Matsuno time differencing. The only difference is in the of the damping wave length and on µ. For the shortest last term. This again approaches zero as ,1x, ,1t -+ 0. resolvable wave length of 2,1x we have k,1 x = n, and, However, if t1x-+ 0 with t1t fixed, it now approaches therefore, Y1c2 ,1t ( d2u/ux2). Thus, we see that it is again equivalent in form to a fin ite difference approximation to the F ickian diffusion term, but with a coefficient of half the size given by (1 . 18). Furthermore , this term is now For waves of twice the wave length, 4'1x, k.,1x = n/2, calculated over an interval of 2,1x, and its damping effect a nd will be a maximum at that wave length. This sort of (1 - µ2 µ4)'' ·· ( I . 33) dependence on wave length is often considered desirable 11.l = + for damping in a finite difference scheme. This is because, as we will see later, there are serious problems with In general, since finite difference calculations for small wave lengths, 2) . 4 kt1x d l 1.I 4µ(1-2µ sm - especially a round 2'1x. It is often possible to alleviate --=------~"--~ dµ 2 2) . 4 kt1x these problems by using a dissipative scheme, which da mps I - 4 µ ( I - µ sm - - the two-grid-interval waves preferentially. 2 While (1 . 18) was of the first order of accuracy in time, all the I k I curves have minima at µ = I /vi Substitut (I . 27) has truncation error O [('1x)2] + 0 ((,1t)2) ; thus, ing this value ofµ into (I . 31) we find that these minimum it is of second order accuracy in both space and time. values of the amplification factor are equal to • 4 kt1x)Y, To test the stability of the Lax-Wendroff scheme, we 1- s,n - (1 . 34) substitute ( 2 u; = Re [U The amplification factors for wave lengths of 2.dx negligible compared with the physical dissipation, and a~d 4.dx, as calculated in ( I . 32) and ( I . 33), are shown it can also be useful for controlling the shortest waves. in Fig. I. 2. The amount of damping is seen to be gener If the physical dissipation is very small or non-existent, ally quite large for shorter wave lengths, especially for the it is better to use a neutral scheme. wave length 2.dx. The amount of damping also depends One should point out here that we can change the time on the time step and the advcction velocity. This is differencing scheme intermittedly during an integration, a disadvantage of the Lax-Wendroff scheme because so as to get the required amount of a particular effect. there is no reason why the amount of damping should For example, in the general circulation model developed depend on these quantities and it is either impractical at the National Center for Atmospheric Research, or impossible to control the damping by changing them. Boulder, Colo., the Lax-Wendroff scheme is used because For example, for small values ofµ expansion of (1.31) of its damping effect on the shortest waves. However, gives to keep the amount of damping small, it is used only once 2 . 4 k.dx in every hundred steps, the rest are made using a neutral I). I = I - 2µ sin + .... 2 leapfrog scheme (Kasahara, 1969). On the other hand, showing that for a given amount of time (a fixed value of in the British operational model (e.g. Gadd, 1974b) n.dt) the total damping will be approximately propor the Lax-Wendroff scheme is used every time step, with the tional to .di. However, we wish to choose .di to give authors of the model making no mention of any excessive the best accuracy and stability properties, not to give damping due to the scheme for the purposes of that the optimal amount of damping. model. 2. Computational dispersion As we have seen, the linear advection equation iJu iJu ...... ______iJt + c iJx = 0 1 c = const, (2. I) has a solution in the form of a single harmonic component u (x, t) = Re [U (t) e'h]. (2. 2) 0.5 provided that L = 2.dx dU + ikcU = 0. (2.3) dt 0 µ In this oscillation equation kc is equal to the frequency v, and c = v/k is the phase speed of the waves. It is seen 0 0.5 that waves of all wave lengths are propagated with the Figure I .2 Amplification factor of the Lax-Wendrotf scheme, as same phase speed, that is, the function u (x, t) is advected a function ofµ ~ c.dt/.dx, for the wave lengths 2.dx with no change in shape at a constant velocity c along the x and 4.dx. axis. There is no dispersion. Now consider the equation The Lax-Wendroff scheme has been fai rly widely used in atmospheric models, due to a recommendation du · u ·+1 - u · 1 by Richtmycr (1963). and its reasonably good behaviour. ::.::J. + c 1 ,- = 0. (2.4) It is second order nccur.ue, explicit, not unconditionally a, 2.dx unstable. and since it is a two levels scheme there is no computational mode. None of the schemes obtained by that is obtained by approximating the space derivative combining centered space differencing with one of the in (2. I) by a centered difference quotient. The equation seven time differencing schemes studied in chapter II (2.4) is neither a differential nor a difference equation, has all of these advantages. The dissipation of the Lax but a hybrid of these two. An equation of this type can Wendroff scheme will not be harmful if its total effect is be called a differential-difference equation, or a semi- CHAPTER 3 27 discrete equation. The finite difference equations obtained u, when the time derivative in (2.4) is approximated using a consistent time differencing scheme will approach (2 .4) as the time step approaches zero. Thus, for small .dt (2. 4) represents a n approximation to these finite difference equations. Since the time derivative has retained its differential form, a ny error in (2.4) is due to the space d ifferencing. For this reason, equations of this type a re used to study the effect of particular sp~ce difference approximations Figure 2.1 A plot of the " two-grid-interval" wave, with a wave on the properties of the numerical solution. length of 2..4x. Recall that (2. 4) has a solution in the form of a single harmonic component If the pattern that is being advected represents a super position of more than one wave, this false dispersion u (t) = Re [ U (t) e1t J4 1 "'J, (2.5) wi ll result in a deformation of that pattern. It is especially provided that small-scale patterns in the atmosphere, e.g. fronts, shear lines, etc., that represent a superposition of many waves, including a significant proportion of the shortest !~ + ik ( c si ~.dk:x) U = O. (2.6) waves. For this reason. in numerical forecasting, such patterns, if present in the initial fields, are fairly rapidly We have now written this so that it can conveniently be deformed. until they aquire a form which is less sharp compared with (2.3). Instead of the constant phase than in the beginning. Since such small-scale features speed c, we see that waves now propagate with the phase are of particular importance in weather processes, speed the effect of computational dispersion d eserves very careful consideration. • sin k.dx We now turn our attention to the group velocity. In the c c k.dx . (2 . 7) = case of the linear equation (2. I) we obtain fo r the group velocity c, T his phase speed is a fu nction of the wave number k. Thus. the finite differencing in space causes a dispersion of the waves ; we shall call this effect computational (2.8) dispersion . As k.d x increases from zero, the phase speed c• monotonically decreases from c, and becomes zero for Thus, the group velocity is constant a nd equal to the the shortest resolvable wave length 2.dx, when k.dx = 1t. phase speed c. With the differential-difference equation Thus, all waves propagate at a speed that is less than the (2.4), however, (2 . 7) gives for the group velocity c; true phase speed c. with this decelerating effect increasing as the wave length decreases. The two-grid-interval • d (kc") Cg dk = C COS k.dx. (2. 9) wave is stationary. = Thus. as k.dx. increases from zero. the group velocity c;. T he reason for the two-grid-interval wave being sta decreases monotonically from cv, and becomes equal to tionary is obvious when we look at the plot of that wave , - c, for the shortest resolvable wave lengt h of 2.dx. shown in Fig. 2. I. For this wave u ,. 1 • ~ u _, at all grid 1 1 These results are summarized in Fig. 2. 2. For the tlu1 points. and (2. 4) gives a zero value for · J- . exact advect ion equation (2. I) both individual waves ' I and wave packets, that is. places where superposition of waves results in a m.iximum amplitude of a group of We have encountered two effects here. Firstly. the neighbouring wave numbers, propagate at the same con ad vcction speed is less than the true advcction speed. stant velocity c = cv· Introduction of the centered space The consequence of this error is a general retardation of fini te difference quotient in (2.4) both makes the phase the advection process. Secondly. the advection speed speed and the group velocity decrease as the wave number changes with wave number ; this false dispersion is increases. The error is particularly great for the shortest particularly serious for the shortest waves. resolvable wave lengths ; waves with wave lengths less 28 THE ADVECTION EQUATION Phase This situation can be illustrated by a simple example. or Let us define Y (x) as a function that is slowly varying group in space ; for example, it can be a sine function of a large velocity wave length. Furthermore, we define c = c, c u, = (- J)I Y,, (2. 10) ...... , ' as shown in Fig. 2 . 3. Thus, the function ± Y (x) is the envelope of the function u • Suppose that we calculate ' 1 ' ' the advection of u, using (2 . 4), then substituting (2. I 0) ' into (2 .4) we obtain ' \ \ \ ktJx ~ - c Y1+ 1 - YJ-1 = O. 0 a, 2'1x 1t/2 \ 1t \ \ Thus, the advection of Y1 is governed by the same \ equation as the advectio n of u1, except that the advection velocity appears with an opposite sign ! Therefore, as the ' ,.c individual short waves of the function u1 slowly propagate ' ' .... __ in the direction of the positive x axis, the envelope -c ' ± Y (x), which has a long wave length, propagates rela tively fast in the opposite direction. When it is a sine function, so that it consists of a single harmonic, it 20 10 5 4 3 2'1x propagates with no change in shape. Because of (2 . 10), Wave length u1 must a lso be advected with no change in shape; from Figure 2 .2 Phase speed and group velocity, in the case of the this we conclude that u1 also consists of a single harmonic linear advection equation, c and c,, and in the case component. If, on the other hand, the function Y (x) of the corresponding diff'erential·diff'erence equation consisted of a number of harmonic components. with second-order centered space diff'erencing, c• the shapes of both Y (x) and u would change during and c; (Matsuno, 1966c). 1 the advection process, as a result of the computational dispersion of these components. It is possible to obtain an analytic solution of (2 .4), than 4'1x even have a negative group velocity. This which can be used to analyze its behaviour for some means that wave packets made up of these waves propa given initial conditions of interest. To this end it is gate in the direction opposite to the advection velocity convenient to define a non-dimensional time variable and opposite to the direction of propagation of individual waves. 't = ct/ tJx, (2. 11) ~I\/\ I\~\ I\ I\ I) x - Y (x) Figure 2.3 Function~ 11, and ± Y (x), illustrating propaiation of an cnvclupe of short waves when their advection is calculated using second· order centered space differencing. CHAPTER 3 29 and, after dividing equation (2.4) by c/2~x. to write It is instructive to look in some detail at the solution it in the form satisfying the initial conditions 1 forj=O (2.12) rdO) ={ (2.16) J O for j~O. This can be recognized as the recurrence formula of the the simplest solution of the form (2. 13), for different Bessel function of the first kind of order j, 11 ('r), (e.g. values of the no n-dimensional time. At the initial Courant and Hilbert, 1953, p. 488). In other words, moment the function u1 consists of a single pulse-like disturbance, centered at the point j = 0, as shown in the u, (t) = 1, (t) (2.13) upper diagram of Fig. 2 . 5. We note that, because of (2.12), du /dt is then equal to zero at all points except at is a solution of (2 . 12). Several of these functions, of the 1 j - 1 and j I , where it is equal to - I /2 and I /2, lowest order. are shown in Fig. 2.4. The figure illustrates = = respectively. more of these functions than indicated, since, for any j , Thus, at the initial moment the disturbance propagates 1_, = ( -1)11,. at the same rate in the directions of both the positive and the negative x axis. Further propagation of the dis Note, furthermore, that in (2. 12) the subscript j can turbance according to (2.13) can be followed using take any integer value, since the locatio n of the grid Fig. 2 .4, or, more accurately, using some tables of Bessel point for which we choose j = 0 is arbitrary. Thus, a functions. Solutions obtained in this way for t = 5 and solution that is more general than (2. 13) is t = 10 arc shown in the middle and lower diagrams of Fig. 2 . 5, respectively. u, (t) = 1,-p (t), t = O Figure 2.4 The Bessel functions Jo (t), J 1 (t) and J s (t). where p is an arbitrary integer. Since we arc solving a linear equation. a still more general solutio n is a linear combination of all these solutions. that is II} ( The three diagrams present an example of the computa physical reasons to have one of these points as the tional dispersion of the second-order centered space central point and the second located on the side from d ifferencing. We note t hat if we expand a single pulse which the fluid is being advected toward the centre. like disturbance as a cosine Fourier integral Therefore we approximate (2 . I) by u (;) =J ~a (k) cos kx dk, ~ + UJ - Uj-l = 0 for c > 0, (3. la) at c ,:Jx • 00 a (k) = ~1u (x) cos kx dx, It O ~ + c UJ+~;u1 =0. for c < 0. (3 . I b) and a (k) is calculated numerically using the grid point values only. we obtain a constant value T hese equations are again differential-difference equa 2 a (k) = -,:Jx. tions. (3 . la) employs the backward and (3. lb) the for 1t ward difference quotient for the approximation to the Therefore, all the harmonic components have equal space derivative. However, in both cases the differences amplitude. By analogy with the 1ight spectrum, such a are calculated on the side from which the advection function is called white noise if it does not appear for velocity reaches the centre; hence, these differences are physical reasons. Its Fourier components are advected called upstream differences. Calculated on the opposite with different phase speeds, as summarized in Fig. 2. 2, side the differences would be called downstream differ bringing about a dispersion of the disturbance. With the ences. non-dimensional time chosen here we see from (2. 12) that the physical advection velocity should keep the pulse Eqs. (3. I) can be used to construct schemes for the located at the pointj = •. Because of the space difference advection equation, by approximating the time derivative approximation, however, all the phase speeds are less by one of the many possible consistent methods. The than the physical advection velocity. The main distur resulting schemes will only be of the first order of accuracy. bance, as seen in Fig. 2. 5, is advected at a speed only However, they have a particular advantage over centered slightly less than the physical one; obviously it mostly schemes in space when applied to the advection of a consists of the longer wave components, which have an disturbance similar to the one considered in the preceding advection speed not much different from the physical section. T his is that, with upstream differences, a dis advection velocity. However, it is seen to be diffusing turbance cannot propagate in the direction opposite to away with t ime, which is again a result of the dispersion. the physical advection. Thus, no parasitic waves will We also observe propagation of a group of short waves contaminate the numerical solution. in the direction opposite to that of the physical advection. Since the appearance of these waves contradicts the If, specifically, a forward difference is used for the time physical properties of the advection equation, such waves derivative in (3.1), we obtain, for c > 0, are called parasitic waves. n+I n n n The solution of the differential-difference equation is, Uj - Uj + C Uj - Uj-1 = O. (3 . 2a) obviously. quite unsatisfactory as an approximation to '11 ,:Jx the true solution. However, this example, with the initial disturhance located at one point only. is completely unsuitahle for a good solution by a difference approxima This is the scheme that was used for the examples of the tion. This is exactly the reason why it provides an in introductory chapter. It was found that this scheme structive illustration of the difficulties involved. was damping, with the amount of damping depending on the wave length, with a maximum for the shortest resolv Analytic solutions for a more general case. when a able wave length of 2L1x. The analytic solution of the centered difference approximation is made to the time difference equation (3. 2a) has been discussed by Wurtele derivative also have hecn considered by Egger (1971). (1961). 3. Schemes with uncentered space differencing The advantage that is accomplished, at least in prin ciple, by using upstream differencing as compared with The space derivative in (2. I) can also be approximated centered or downstream differencing, can be illustrated using uncentered differencing. Still using values at by considering the domain of influence of a grid point two points for this approximation it is attractive for in different schemes. We still consider the case c > O. CHAPTER 3 JI x - ct = const x Figure 3.1 Domnins of influence of a grid point. for the scheme (3.2a) with upstream differencing, for corresponding schemes with centered and intersect downstream differencing, and for the true solution. In the true solution, a grid point value can be said to from the difference schemes. As in Section 2 we introduce propagate along the characteristic x - ct = const. the non-dimensional time T = ct/Ax. Eq. {3 . la) can Fig. 3 . I shows a grid point ma rked by a circle with the then be written as associated characteristic passing through it. With upstream differencing as in (3. 2a), the value at that grid point will influence the values at points within the domain shaded by vertical lines. The figure also shows the domains of influence with centred and downstream A solution of this equation is the Poisson frequency differencing. Of the three domains of influence, that function given by upstream differencing is clearly the best approxi -t j-p e 't mation to the characteristic line representing the domain for j:>p. of influence in the true solution. uj ('t) = (j~p) ! (3. 4) This discussion suggests constructing a scheme for I for j < p. (2. I) by tracing a characteristic back from the point as can easily be checked by substitution. Here pis again (jAx, (n + I) At) to intersect the previous time level an arbitrary integer, that is, we have al ready taken into t = nAt and calculating the val ue u• at the point of inter account the fact that the location of the point j = 0 is section by interpolation. We then set u; + 1 = u• . Choosing a linear interpolation procedure, that employs values at two neighbouring points at the time nAt. we obtain x - ct = const n n n+ I n Uj -UJ- 1 ( ) n I I u1 = ui-I + dx .dx - cAt . This can he idc:ntical to the scheme (3 . 2a). with up stream differendng. If, on the other hand. a quadratic interpolation procedure is chosen, using three neighbour ing points, one obtains the Lax-Wendroff scheme, as the n ,\' reader can readily verify. j-1 j j -1 I For further insight into the properties of schemes that can be obtained from (3. I a) we consider the analytic Figure 3.2 Sketch for construction of schemes by calculation of a solution of this differential-difference equation. For small previous value on a characteristic passing through values of At this will approximate the solution obtained the point (}'1.x, (n + I) '11) . 32 THE ADVECTION EQUATION As in section 2 we can form a solution more general than (3 . 4), as a linear combination of all possible solu tions (3 . 4), that is J -t J-p ()-~ae t , (3. 6) "1 t - LJ p ( ·- ) ! P=- oo J p where ar, are arbitrary constants. Substituting t = 0 into (3. 6) we obtain u1 (0) = a1. (3. 7) 0 2 3 4 5 6 7 8 9 10 11 j - p Thus, the constants ar, can again be chosen so as to satisfy arbitrary initial conditions u1 = u1 (0), and so Figure 3. 3 The Poisson frequency function (3. 4), for the case (3. 6) represents the general solution of (3 . 3), or (3. I a). t = 4. Considering the behaviour of the simple solution (3. 4), and the summation limits in (3 .6), we see that in general the value u1 (t) at a point j can be considered as a result arbitrary. An example of the Poisson frequency function of superposition of the effect of the initial values at that is shown in Fig. 3 . 3 ; the graph in the figure represents point and of the initial values at all the points located the shape of this function fort = 4. There is no need to upstream of it. include a vertical scale. since the area enclosed by the An example of the solutions (2. 14), for centered graph of a frequency function has to be equal to unity, so differencing, a nd (3. 6), for upstream differencing, for e-t /-P an initial disturbance of a somewhat larger space scale . )' = I. (3. 5) 1 for j = -1, 0, 1 j-~p :O (J-p . "J (0) = Thus, fort = 0, when, as shown by (3.4), the histogram O for j ;c -1, 0, I consists of a single rectangle, its ordinate is equal to l unity. . is shown in Fig. 3. 4. If the grid distance is of the o rder 1 Consider now the change in shape of the histogram of 300 km, and c is about 15 msec- we can see that (3.4) as the non-dimensional time t increases from zero. The initial shape of this histogram, that of a parallelogram having a base t1x and erected at a single 1.0 t = 5 . \ l grid point, is, of course. equivalent to the shape of the pulse-like d isturbance (2. 16) used for the example of 0.5 (\ , the previous section. As t increases beyond zero (3. 4) . 11 # • 0 ...... _-~ . ' transforms into a skewed bell-shaped histogram of the - 15 - 10 .- 5.' . 0: 5 10 15 20 25 type as shown in the figu re, with its mean posi tion on .'• ' 1.0 the x axis t = 10 ' \ . 0.5 ' • 00 -· J-p ' ~ (j- p) e( ._: )! = t, ,'. f~ I - AM~' I \ ,.fr;() J p 0 ~~ 0 ,5 • • • I- ' 15 20 \ ,,' moving at u constant speed. Thus. the mean position 1.0 t = 15 propa~atcs with a speed equal to the physical advcction velocity. The maximum point of the histogram, however, 0.5 lags behind as is shown by the skewed shape of the histo .. ,•' . grum. Physically unjustified negative values of u 0 ~ - ; 1 \,·O \ :s , 10 15 20 25 j never occur and no parasitic waves appear on the ..,, opposite side of zero from the direction of the physical Figure 3.4 Analytic solutions of the exact advection equation a 5 units of non-dimensional time approximately corres Llx away from the centre. A simple calculation shows pond to the physical time of one day. Thus. the damping that the approximation (4 . 3) is obtained by extrapolation effect of the upstream differencing is seen to be quite for the value d = 2.LJx/3. Of course, there is no reason severe. The figure also illustrates the properties of the to expect that the accuracy (lf such an approximation two methods described, but to a lesser extent than the should decrease monotonically as ti decreases. examples with the initial disturbance limited to a single We now want to look at the effect on the phase speed grid point only. Thus we can hardly claim that the use of using the approximation (4. 3) for the space derivative of upstream differencing instead of centered second in the advection equation. Replacing the space derivative order differencing has, generally speaking, improved the in (2.1) by (4 .3) we obtain the differential-difference solution. equation auj + C (~ UJ+ ( - UJ-1 - .!_ UJ+2-U}-2) = 0. (4.4) 4. Schemes with centered fourth-order space differencing a, 3 2.dx 3 4.dx Most of the difficulties that have been discussed in As in Section 2, we investigate the behaviour of a this chapter. in particular the phase speed error and the tentative solution in form of a harmonic component computational dispersion, have been due to the approxi mations used for space differencing. Thus, consideration u (t) = Re [U (t) e'kJdz]. should be given to other possibilities ; one is to employ 1 approximations of a higher order of accuracy. We shall · With second-order space differencing, we obtain the phase first construct such an approximation. speed When the approximate value u1 a re expanded into Taylor series about the central point, and substituted • sin k.dx c = c • into the finite difference quotient, we obtain kLlx J "L+t- u/-l_au I au( )2 [( )4I Now, in the same way, with fourth-order differencing we .dx - ax + 3! ax) .dx + O tlx . (4. 1) 2 find the phase speed Thus, this quotient is of the second order of accuracy. .. _ (4 .sin k.dx I sin 2kL1x) (4 . S) It is formed by taking differences of values of u1 at points c - c 3 k.dx - 3 2k.dx · one grid distance away from the central point. Similarly a quotient can be formed by taking differences of values We shall compare these two results. For second order two grid distance away. We then obtain. replacing tlx differencing, we obtain by series expansion for small in (4.1) by 2t1x, values of k 3 "iti - "l - 2 - au 4 a u ( )2 [ 41 44x - ax + 3! ax) tlx + 0 (tlx) . (4.2) This quotient is still second order accurate, but the On the other hand, with fourth order differencing we have coefficients are larger. Other consistent approximations to ;Ju/dx can be formed as linear combinations of the 0 4 quotients (4. 1) and (4.2). The combination for the c = c (1- i (kLJx) + ···). second order terms in the truncation errors of (4. 1) a nd (4. 2) cancel is particularly important. This is Thus, even though the decelerating effect is still present, the phase speed error has been much reduced for small -4 "i+I - "·-1-- 1 - -I u,+ 2-"i -2 - -au + O [( Llx )41 . (4 .3) values of k. 3 2J:< J 4.dx ax These phase speeds are shown in Fig. 4. I as functions of kLlx, for all admissible values of k. The figure illus and represents a fourth-order accurate approximation to ilu/Jx. trates the very significant increase in accuracy of the phase speed for large-scale and medium-scale waves. We can also think of the approximation (4. 3) as repre However, as the wave length approaches its minimum senting a linear extrapolation of the quotients (4. 2) and value of 2Llx, the increase in phase speed obtained (4 . I) so as to simulate an approximation corresponding to by fourth order differencing diminishes, until, finally, dilferences taken between points at a distance d less than the wave with wave length 2.dx is again stationary. 34 nrn ADVECTION EQUATION Phase For small scale disturbances, of a scale close to two speed grid intervals in space, no finite difference method is really satisfactory. Additional ways of constructing difference schemes for the advection equation are discussed in the paper by Anderson and Fattahi ( 1974) where further references are given. If it is felt that, in a particular situation, the improvement of the advection of scales close to two grid intervals is necessary, the obvious method is to find a way of making the computation with a reduced grid size. As can be inferred from Fig. 4 . 1., halving the grid size with simple second-order differencing 1t makes the stationary two-grid-interval wave move at a speed of almost 2/3 of its physical advection speed. But this, of course, is not easily attainable : in two Figure 4.1 Phase speed for the linear adve<:tion equation, c, and dimensional problems, halving the grid size increases the for the corresponding differential-difference equations 0 with second order (c ) and with fourth order (c00) computational time requirements by a factor of four, centered space differencing. and, with the usual time difference schemes by an addi tional factor of two in order to maintain computational stability. Still, a steady increase in the capabilities of Moreover, for short waves the slope of the phase speed commercially available computers enables constant curve is greater than with second order differencing, and, improvements of this kind, so that it is expected that in a therefore. the computational dispersion of these waves few years the resolution of atmospheric models may be is greater. Thus, although a short wave-length disturb such that advection errors will not be a major problem. ance will now be advected at a somewhat greater speed At present it is estimated that the horizontal truncation its false deformation. due to computational dispersion, errors in the advection terms are the largest single source will be faster. of errors in short range numerical forecasting, accounting for almost 40 per cent of the total error (Robert, 1974). Because of the decrease in the phase speed error of the longer waves, the use of fourth order schemes for advec Another way of improving the advection of small tion has brought about significant improvements in scale systems may be to develop a computational method operational numerical forecasting in both the U.S.A. more in spirit of the Lagrangian system of equations. and Japan, in barotropic and quasi-geostrophic baroclinic As yet, such methods have not been very much explored models. With primitive equation models, now in use in meteorology. in all advanced forecasting centres, fourth order advection schemes are not yet quite so widespread. Still, it is gener ally believed that fourth-order advection schemes should 5. The twHimensional ad,ection equation be used in operational forecasting. However, the use of advection schemes of a high-order of accuracy in general We now consider the two-dimensional linear advection circulation models may not be so important. The choice equation is hetween increasing the accuracy of space differencing au au au or spending an equivalent amount of extra computation at + c.,.. ax + Cy ay = 0, Cz, C11 = const (5. J) time in reducing the grid size of the model. A straight· where u = u (x, y, t) is a function of two space variables, forward calculation (e.g. Thompson, 1961. p. 157) and c~,c.,, are the components of the advection velocity. shows that the first alternative should be advantageous. Thus, the advection speed is given by The use of the additional grid points needed for higher order differencing docs create some side difficulties. c = vc: + c:. (5. 2) In much the same way as using more than two levels for time differencing resulted in the appearance of computa We shall test the stability of schemes for the numerical tional modes in time, so the use of additional grid points solution of (5. I) by the procedure of Section I . Thus, for space differencing results in the appearance of compu space derivatives are approximated by standard second tational modes in space. Furthermore, formulation of order difference quotients, giving boundary conditions becomes more complicated. Simply formulated boundary conditions may be a source of -ua . =-c U'· + a·I . - U,-,1-c, I . U'·I, · +I - u1,1-1 , (5 .3) serious problems. a, ,., .,.. 2,fr ')' iJy CHAPTER 3 35 Here, as is usual for two-dimensional problems, we have , r,:: Llt r L. c .dx I. (5. 7) changed the choice of subscript denoting the grid points < a long x axis, so that the coordinates of the grid points arc now x = iLlx, y = jLly, and approximate values Therefore, in the two-dimensional case we have to choose u (iLlx, jLly) are denoted by u,. ,. As a tentative solution a time step that is V2 times less than that permitted in of (5. 3) we substitute the one-dimensional case. 1 111 We note that the minimum stability is associated with u,. 1 = Re [U(t) e If the leapfrog scheme is used for the time derivative, we obtain as the stability criterion is, however, greater by a factor ofV2 than wave numbers along the axes, and its wave length is therefore shorter I(;; sin kLlx + i sin /Lly) Llt I< I. (5.6) by the same factor. This applies to all waves with k = I. This has to be satisfied for all admissible values of the wave numbers k, I. For simplicity, we shall consider only the cases where 6. Aliasing error and nonlinear instability Llx = Lly ; we denote this grid size by d•. In the wave number plane, that is, a diagram with co-ordinates k ,I, Another generalization of the simple one-dimensional the admissible wave numbers are contained within the linear advection equation is to consider the nonlinear square region shown in Fig. 5. I. Inside that region the advection equation maximum value of the left-hand side of (5. 6) is obtained iJu iJu at the centre of the square, marked by a circle. The wave -+ u -= 0, (6.1) represented by that point has wave lengths 4d• in both a, ax the x and y directions so that sin kLlx = sin /Lly = I. We have returned to dimension, so that u = u (x. r). td• Shuman (1974) calls (6.1) the shock equation. Its general solution (e.g. Platzman, 1964) is 7t u = J(x- ut), as can readily be verified. Here/is an arbitrary function. Here we consider only the effect of the multiplication in (6 . I). When performed in finite differences, it results in an error related to the inability of the discrete grid to resolve wave lengths shorter than 2Llx, that is, wave numbers greater than kmu = n/Llx. Thus, consider a 0 function u (x) which can be represented by values at 0 7t grid points, for example Figure ~. I Admis~ible region of wave numbers for a square two. dimensional grid. with grid length &x = &y = u•. u = sin kx, (6.2) where k < krnu . However, substituting (6. 2) into the For a given value of the advection speed the left-hand nonlinear term of (6. I) gives side of (5. 6) has a maximum value at this point if the iJu k 'k 1 k · k advection velocity makes an angle of n /4 with the x axis. U iJx = Sin X COS kx = 2 Stn 2 X. .int ht.s case c~ = c = vi- - c. Th us we o b tam. t h c stab 'tltty. 11 2 Hence, if the wave number in (6.2) is in the interval criterion '12krnu < k ~ kmu, the nonlinear term will give a wave 36 TIIE Al)VECTION F.QUATION number that is beyond the range that can be resolved by . k . 2n ( 2n k) the grid. It cannot, therefore, be properly reproduced sm x = sm Ax x cos Ax - x - in a finite difference calculation. To gain some insight into what happens in such a - cos -2n x s.in ( -2n - k ) x. situation, consider a wave for which k > Krn,x· For Ax Ax example, let L = 4Ax/3. A wave of that wave length is shown by the full line in Fig. 6 . 1. Knowing only the However, at the grid points x = jAx, and values at grid points we will not be able to distinguish . 2n .,, 2n ,., I sin - JaX = 0 , COS - jaX = . this wave from the one shown by the dashed line. Thus, Ax Ax with the convention adopted earlier which assumes that the longest waves are present. we will make an error. Therefore, we find This is called aliasing error. sin kjAx = - sin (2krnax - k)jAx. (6.3) In this way, we see that, knowing only the grid point values, we cannot distinguish the wave numbers k from 2km ..- k. Thus, if k > krnax , using the convention x mentioned earlier. we can say that the wave number k is misrepresented as the wave number (6.4) Hence, as shown in Fig. 6 .2, the resulting wave has a Figure 6. I A wave of wave length 4'1x/ 3, misrepresented by the wave number k* which is less than krnax by an amount finite difference grid as a wave of wave length 4.dx. equal to that by which - k was greater than krnax. We can think of the wave number k* as being an image obtained by the reflection of k across the value krnax into the admissible range of wave numbers. In a more general case. suppose that the function u consists of a number of harmonic components u = ,., u •. k ....J n 0 2k,nax The nonlinear term will then contain products of harmo Figure 6 . 2 Misrepresentation of a wave number k > kmn, in nics of different wave lengths, such as accordance wi1h (6. 4). sin k 1x sin k 2x. As an example consider the case L = 4Ax/3, illustrated However, in Fig. 6 . 1. Then k = 3n/2Ax, and (6.4) gives k* = = n/ 2Ax as the wave number "seen" by the finite differ ence grid. This. of course, is the same wave. of wave length 4Ax, as the one found graphically and shown Thus. even if a finite difference calculation is started with by the dashed line. waves which all have k ·' kn,.x• very soon through this Now consider the consequences of al iasing errors in a process of nonlinear interaction waves will be formed numerical integration. An atmospheric variable, as a wi th k · k 11.,.. . and a miHepre:.cntation of waves will function of space co-ordinates, can be thought of as occur. consisting of a series of harmonic components. It is In general we can write useful to consider the "energy'' of these components, that is. their contribution to the mean square value of the variable considered as a function of wave number. sill k .,· -- sin [2kn,a• (2knmx ·-k)J x. This is the spectrum of the "energy". For example, Substituting here km,x n/ /Jx and using the formula if the variables are velocity components, this function for the sine of a d ifference, we obtain is the kinetic energy spectrum. This spectrum describes CHAPTER 3 37 the relative importance of features of different scales in experiment, but after every two hours of simulated time the field of the variable. Now, experience shows that he performed a ha rmonic analysis of the vorticity fields. the spectrum of atmospheric variables does not change and eliminated a ll components with k ..... Yikmu· If much with time. On synoptic maps we do not have there are no components with k - · Yikmu the advection situations where small scale features a re dominant on one term cannot produce waves with k > kmu.· We expect day, and absent on the next. Accordingly, spectra of that it will be some time before the amplitudes of the atmospheric variables also do not change much in their eliminated waves are built up again to an appreciable general shape. The energy of a particular component can, extent. This filtering procedure eliminated the a ppear of course, change , but the characteristic shape of the spec· a nce of the spurious increase in energy, thereby confirm trum as a whole is fairly constant. For example, a zonal ing this explanation of the instability. spectrum of the eastward velocity component in middle latitudes typically has a maximum for wave numbers 4 to 7, that is, 4 to 7 wave lengths along a latitude circle, 7. Suppression and preYe.ntlon of nonlinear instability with the energy tapering off rather rapidly as the wave number increases beyond about 10. Thus, there is ve ry If an integration is to be performed for an extended little energy in wave numbers of the order of the maximum period of time, it is necessary to suppress or prevent wave numbers that can be resolved by finite d ifference nonlinear instability. For short range integrations it is grids used in atmospheric models. not necessary to do this, though such a procedure might still have a beneficial effect on the model. In a finite difference integration, in additio n to these relatively small physical changes, the shape of a spectrum It has been pointed out by Orszag ( 1971 ) that to is subject to changes due to aliasing errors. If we have a eliminate aliasing errors it is not necessary to fi lter the spectrum of the shape just described, and consider the top half of the admissible wave numbers. It is sufficient to eliminate the top one-third, because, if waves with representation of various combinations k 1 + k 2 that 2 2 a re greater than kmu, we see that most of the energy of k > /akmu. a re filte red out, all the aliases satisfy k > / 3 km&X such combinations will belong to components with wave and will thus be eliminated. numbers not much g reater than kmax· T hus, due to If, however, we consider that such a suppression of aliasing errors a spurious energy inflow is expected at the shortest waves is a satisfactory method of dealing wave numbers that arc not much less than kmu , and, in with the problem, it would be simpler to use a differencing time, the energy of these components can be expected to scheme that has a built-in damping of the sho rtest grow beyond physically acceptable lim its. Experience waves. This idea is due to Richtmyer (1963), who sug shows that, if no precautionary measures are taken, this gested use of the Lax-Wendroff scheme for this purpose. can indeed happen, and even cause a catastrophic end to It was found by experience that such a practice does the integration. The phenomenon is due to the nonlinear suppress the nonlinear instability, and that to do this terms of the equations, and, therefore, is called the it is sufficient to use an intermittent Lax- Wendroff step nonlinear instability. Nonlinear instability was first en at quite long intervals (Kasahara, 1969). Kreiss and countered by Norman Phillips ( 1956) in his famous work Oliger ( 1973), on the other hand, recommend adding a that laid the foundation for the numerical modelling of the dissipative term to a scheme which is not dissipative, so atmospheric general circulation. Starting from an atmo that the amount of dissipation can be contro lled in a sphere at rest, he integrated the vorticity equation for a more practical way. simulated time of the order of30days. The calculation then Another way of avoiding nonlinear instability is to came to an end due to an explosive increase in the total use a Lagrangian formulation of the advection terms energy of the system, associated with a n appearance of instead ofa Euleri an formulation. We calculate t he position elongated shapes in the vorticity field . Phillips initially of the parcel that should be advected to the grid point believed th.it the breakdown was due to excessive trunca considered in step Jt. A value of the dependent variable tion errors. and he later repeated the experiment using can be found corresponding to that position by interpola space :ind time steps both reduced to about ha lf of their tion in space. T he change due to advection is set equal previous va lues. T his must have greatly reduced the trun to the difference between the value obtained by interpola cmion errors, but the catastrophic increase in total energy tion and that a t the grid point. ln some cases. these sche still happened at about the same t ime. mes turn out to be identical to schemes obtained using the In a later paper Phillips ( 1959) gave the interpretation Eulerian formulation, but other schemes can also be of nonlinear instability similar to what has been presented obtained. A procedure of this type was first used by here, but for the nondivergent vorticity equation. For Leith (1965) ; an example of its use more recently is a test of this explanation Phillips again repeated his given in the paper by Krishnamurti et al. (1973). 38 THE ADVECTION EQUATION A conceptually elegant approach for dealing with the large scale atmospheric processes. Thus, for more than nonlinear instability problem has been suggested and a decade it has been used as a basic prognostic equation developed by Arakawa (1966, 1972). H is idea is that for the numerical weather prediction, eventually supple it is better, if possible, to use schemes for the advection mented by some additional terms of a smaller order of terms that are not only free of the nonlinear computa magnitude. tional instability but also free of the spurious inflow of energy to these short waves, instead ofartificially suppress To illustrate the Arakawa procedure for the vorticity ing their amplitudes. The amplitudes of the shortest equation (7. 3), we need some knowledge of its integral waves in atmospheric models are small initially. and they properties in wave number space. We want to study the will rt:main small if a false generation of these short energy exchanges between different harmonics that are waves is avoided. Arakawa has shown that it is possible permitted by that equation. to construct such schemes, and that they are obtained Consider first the kinetic energy spectrum when the when care is taken to conserve in the finite difference form velocity is two-dimensional and nondivergent, so that some integral properties of the original differential equa it can be given by (7 .2). We can almost always assume tions. that in the region considered A , the stream function can When the Arakawa conservation schemes are used be expressed as a series of orthogonal functions there is no need for an artificial dissipation in the advec tion process. This enables the statistical properties of the 'I' = I:'l'n, (e.g. Courant and Hilbert, 1953, p . 369) (7 .4) schemes to be maintained under advection, a feature n especially useful in general circulation studies. where the functions 'l'n are eigenfunctions of the Helm- holtz equation It has sometimes been argued that, because the phase (1 .5) error of the short waves is very large, they should be eliminated before they erroneously affect the longer The parameters An arc known as the generalized wave waves through nonlinear interactions .. This argument does numbers of the components 'l'n· not take account of several factors. If the phase speeds of the short waves are wrong, the situation will not necessar As an example, let A be a rectangular region with ily improve if their amplitudes are also made wrong. • sides L.,, L11 For boundary conditions assume that the They may still be performing a useful function of a stream function is periodic in x with period L., and is statistical nature. Also damping or elimination of the zero along the lower and upper boundary. Then we can shortest waves will also remove some energy from the write the stream function longer waves that we are interested in. If we wish to dissipate energy, it is obviously better to do so for 2 cos ;:· x physical and not for computational reasons. "'= I: (a,,,,,, + We shall introduce the procedure of Arakawa by " •· "· (7 .6) . 2nn1 ) • nn2 considering the vorticity equation + b,, n sin --x sm -L y. I ' L'I( 'Y 2 ~ + , . rt = 0 I t = p ljl, (7, )) Differentiating this we obtain 2 _ 2nn nn 1 2 ( 2)2 ] where the velocity , is assumed to be nondivcrgent, p '¥,, - -[ ( Lx ) + T:; 'I',,, that is , = k x pljl. (7 . 2) that is, Sub~tituting this into (7 . I) we obtain a 2 2 Ji p ljl = J (p ljl, ...,). (7 . 3) If the region A had different geometry, another set of This equation gives the local change in vort1c1ty as a orthogonal functions would satisfy (7. 5) and the bound result of advcction by a two-dimensional nondivergent ary conditions, and could be used for the expansion (7 . 4). velocity. It is also a nonlinear advection equation. These functions will be solutions of the Helmholtz However, in contrast with the one-dimensional equation equation (7. 5) . (6. I), (7. 3) gives a good approximate description of Define the average of a variable a by CHAPTER 3 39 the coefficients of these components ; in (7. 4), these - I a= A I/dA. coefficients have been absorbed into the functions 'Vn · Since the values of An are already known, for commonly We are interested in the average value of the kinetic used geometries, we can calculate the kinetic energy energy per un it mass spectra using (7. 9). Such a calculation, as well a s the calculation of the spectra of other variables, has been - I I 2 2) ( K = u + v = 17'1' • 17'1'· (7. 7) used for numerous studies of the behaviour and structure 2 2 of both the observed and numerically simulated fields of atmospheric variables. Substituting (7 . 4), and assuming that this series can be differentiated and integrated term by term, we obtain The mean square vorticity K =~ Ji'~"'". Ji'~"''" = ~ ~Ji'ljl II. ~ Ji'ljf'" = can be expressed as a sum of contributions of different VIII · VW. . =24,,Jn~ )' ~ m n harmonics in a similar way. Substituting (7.4), using We note that (7. 5), and the orthogonality of the functions 'l'n, we obtain (7. 10) Assume that no mass transport occurs through the bound Substituting the expression (7. 9) for the kinetic energy of aries of A, that is, a component 'l'n ; we find for the average value of the enstrophy half the vorticity squared, 17 · ('I'm Ji'ljln) = 0. (7 . 11 ) Using (7. 5), we then obtain Comparing this with (7. 8) we see that the average wave number is related to average values of enstrophy and kinetic energy. Define the average wave number as Since the functions ljl" are orthogonal, that is, (7 . l 2) 'I'm ljln = 0 for m # n, Substituting (7. 11) and (7 . 8) we find the double sum reduces to a sum over only a single subscript, namely. >- =ff¥. (7 . 13) K =i ~A~;;:. We have therefore expressed the average kinetic energy Thus, when the velocity is two-dimensional and nondiver in the region A as a .)'Um of contributions of different gent , the average wave number is determined by the harmonics ratio of the average values of enstrophy and kinetic energy. K= )' K (7 .8) 4,-J n ' We originally wished to study the time dependence of where the energy of spectral components permitted by the vorticity equation (7. 3). It will suffice to look at the time ( 7. 9) dependence of (7. 13). (7. 3) gives The contributions A·,.. considered as a function of n, (7 . 14) represent the kinetic energy spcc.:trum . A:. scl!n from (7 . 9). they arc nl!ver ncg;1tive. Wl1l!n the stream func tion 'V is known, thl! func.:tions 'l'n can he computed by Again assuming no mass transport through the bound i.tandard series expansion methods. In fact . we calculate aries of A. we find 40 T HE ADVECTION EQUATION a- a , --2 a - K = - - (PII') = P'I' · ;;- P'I' = , , iJt dt 2 ut (1 . 15) ":,______.1_-y!--- - ,-!-----f---- = - "' :, l 11' = - "' J (t. "'). ••• However, for any two scalar quantities p, q, we have Figure 7. I A mechanical analogy of the interchange of energy J (p, q) = k . p x (ppq) = - k . p x (qpp). between harmonic components. Using Stokes' theorem, we see that We now return to the numerical solution of (7 . 3) and J(p, q) = 0, (7 . 16) the associated nonlinear instability problem. Obviously, if a finite difference scheme could be constructed so as to if either p or q is constant along the boundary of A. conserve the average values of cnstrophy and the kinetic Under the same conditions, we have energy, the average wave number would not change, and, therefore, a systematic transport of ener~ toward the pJ (p, q) = 0, qJ (p, q) = 0. {7 . 17) highest wave numbers would not be poss1~le. ~ rakawa has pointed this out, and showed that. fintte d1ff~ ren~e Therefore, if we assume that II' is constant along the approximations can be constructe~ that, m_deed, mamtatn boundary of A , (7. 14) and (7. 15) give the properties (7. 17) of the analytic Jacobian. Therefore, I - 2 - average enstrophy and ki netic en~rgy are conserved - ~ = const, and K = const. (7 . 18) 2 within the advection terms, and so ts the average wave number. Nonlinear instability is therefore prevented. In this way, we find that the average wave number does His approximations, in addition, maintain the proF.rtY not change with time with two-dimensional nondivergent (7. 16), and thus also conserve the average. v~rt1c1_ty. flow . In other words, a systematic energy cascade Thus the gross characteristics of the frequency d1stnbut1on toward higher wave numbers is not possible. Furthermore, of the vorticity field are also conserved. The true non since to obtain the cnstrophy the contributions K" are divergent vorticity equation conserves all moments of the multiplied by the wave number squared, the fraction of frequency distribution of the vorticity since the area the energy that can flow to high wave numbers is clearly and the vorticity of individual fluid parcels are both limited, and the higher the wave number, the more it is conserved. Maintaining properties {7. 16) and (7. 17)1 limited (Fj0rtoft, 1953). in a finite difference calculation will guarantee the conser As pointed out by Charney (1966) this situation can vation of the first two moments of this distribution. be illustrated by a simple mechanical analogy. T he We illustrate Arakawa's method by considering how foregoing relations show that to satisfy {7. 17)1 • In o ur finite difference calculation it 2 takes the form KA = ~ K,, A!= const, On the left hand side here each of the two factors is c J .. We shall consider only approximations of the second This approximation is also of the second order of accur order of accuracy. With the simplest centered space acy. differencing, we require values of p, q from a box of nine adjacent grid points to evaluate (7 .20), as shown When evaluating the sum in (7. 19) using (7. 22) we in Fig. 7 . 2. Write d for the grid size, and p,, , q,, for the obtain 24 terms from each grid point in the computa values of p, q at the point denoted by k. We then obtain tional region. All of these terms will be of the form the followi ng approximations to the expressions (7 . 20) const · ~" ~1 'I'm · By choosing the constants d, 13, y appropriately we can make all of these terms cancel out in the summation process, thereby fulfilling (7. 19). For example, the point O will contribute terms to (7. 19) of 6 2 5 the form ("'- I ~o l o (~ .111) d (o.~~ 111 23 more terms). = 4 2 1 2 + 3 0 ' A term containing ~o ~1 '1'2 will also appear in the expres sion for ~1 J 1 (~. 'I'). Because of the form of the Jacobian approximations (7. 21) it will have to come from the 7 4 8 product p3 q8 • Thus, one contribution from the point I - will be Figure 7.2 Stencil used 10 define approximations 10 Jacobian. These two terms wifl cancel if a = J3 . Arakawa has shown that, when (7.2la) not only do all the terms in the sum (7. 19) cancel. but also all the terms in the expression for the conservation of x+ I ( the average kinetic energy, and the average vorticity J (p. q) = -:--'.'! x q (p - p ) - 4d 2 s 6 (Arakawa, 1966: Lilly, 1965). Thus, the approximation (7 .2lb) - q4 (p8 - P7) - q, (ps - p8) + q3 (p6 - P1)J. JA =} (J++ + Jx+ + J +x) (7. 23) will conserve average vorticity. enstrophy and kinetic energy when used for the numerical solution of (7. 3). (7 .2lc) This is more than sufficient for the prevention of nonlinear instability. The approximation (7. 23) is usually called - p (q - q) - p (q - q) + p (q - q >J. .16 7 256 4l!7 the Arakawa Jacobian. Arakawa has also shown how to construct an approximation of fourth order accuracy to the Jacobian. conserving these three quantities. The superscripts I· and x denote the pos1t1ons of the It has recently heen demonstrated (Jespersen. 1974) pllints frnm which values of p and q. respectively, are used that the Arakawa Jacohian can be derived as a special to form the approximation . Each of the approximations case of the Sll called "finite clement method ... a relatively (7. 2) is consistent and of the second order of accuracy. new and promising development in the field of the A more general approximation can now be formed as numerical solution of partial differential equations. a linear combination of these three. that is Instead of approximating the space derivatives by fin ite differences, the finite clement method consists of usi ng J (p,q) ~ 0. Ji I + J3 J"' I· y J '". (7 . 22) an interpolation procedure to convert a set of values with the consistency requirement given at grid points into a field given everywhere. This is done using a variational formulation. minimizing the a + J3 + y = l. error of the approximation (e.g. Cullen, 1974). 42 nm Al)VF.C'TION EQUATION Arakawc1 has constructed an analogue of the scheme Thus, it is not only a conservation of energy. as has (7. 23) for the vorticity equation to approximate the sometimes incorrectly been implied . For example. an advection terms in the primitive equations in the case approximation can easily be constructed for the non· when the wind is nondivergent. This scheme is then linear term of the one-dimensional advection equation generalized to allow for divergence (Arakawa, 1972 ; (6. I) which would conserve the kinetic energy. Using Arakawa and Lamb, 1976). such an approximation, however, would not prevent nonlinears instability in the way that the Arakawa In conclusion, we stress that the essence of the Arakawa scheme does. The Arakawa procedure does not have a method is to control the computational energy cascade, one-dimensional analogue, as the nondivergent vorticity hy conservation of the average wave number within the equation (7. 3) is not nonlinear when applied to a one advection terms due to the nondivergent part of the flow. dimensional problem. CHAPTER IV THE GRAVITY AND GRAVITY-INERTIA WAVE EQUATIONS In this chapter we consider the equations describing exclusively in atmospheric models today. T hey are the horizontal propagation of gravity and gravity-inertia generally considered superior for both research and waves. Mathematically, this means that we will be dealing operational applications (e.g. Sawyer, 1972). The speed with a system of two or three partial differential equa of propagation of the gravity and gravity-inertia waves, tions of the first order. Thus, we will now have two or and their sensitivity to various numerical errors mean that three dependent variables. The system of equations will their treatment requires especially careful consideration. always be equivalent to a single differential equation of a higher order. This equation can be obtained from the system by elimination of dependent variables. I . One-dimensional gravity waves: centered space We first put this problem in perspective. Arakawa, differencing (Arakawa, 1970), has stated that there are two main problems in fin ite difference integrations of the atmo We shall first consider the simplest case of gravity spheric governing equations. One is a proper simulation waves where the dependent variables are functions of one of the geostrophic adjustment process. Through this space variable. They a re governed by the linearized process the atmosphere establishes a characteristic quasi equations nondivergent state, mostly as a result of the dispersion of the gravity-inertia waves. The associated computa au rlh uh = _ H au a, = - g ax . a, ax I g. H = const. (I . I) tional considerations will be discussed in this chapter. The second problem is the prediction or simulation of the Thus, we have a system with two dependent and two large-scale quasi-nondivergent flow after it has been established. Here the horizontal advection is the dominat independent variables. ing mechanism. The associated computational consi We seek wave solutions of (I. I) in the form I\ derations were discussed in the preceding chapter. u(x,t) = Re[Del(k.r•vl)]. h(x,t) = Re[hel(k.r• YI) ],(1.2) Extensive study of the problems in integrations of the gravity-inertia wave equations began in atmospheric and obtain the homogeneous system modelling much later than studies of the advection I\ I\ I\ " problem. After Richardson's (1922) first unsuccessful vu = gkh, vh = Hku , numerical integration of the complete primitive equa tions. the successful result of Charney, Fj0rtoft and von giving the frequency equation Neumann (1950) was largely due to the exclusion of 2 v2 = gHk • (I. 3) gr.ivity-inertia waves from their equations by using the gellStmphic approximation in the vorticity equation. Thus, The glwcrning equations with the gravity-inertia waves excluded. arc customarily called the filtered equation.,·. c = = ± viii. ( I .4) They hypass thl· gcostrnphic adjustment problem. T he i liltcrcd equations were used almost exclusively in the lirst dl'l·adc of' numerical forecasting rese.irch. showing that the grnvity waves can propagate along the x axis in both directions at a speed ~H. This Elfclrts Ill improve the performance of numeric.ii v' models led to a desire to include the non-geostrophic speed is not a function of wave number so that there is no effects. This is very dillicult to do within the modified dispersion of the waves. system of equations. Thus, starting with the first success Consider now the differential-difference equations ful experiments by Hi nkelmann (1959), modellers came back to using the primitive equations. Except for special 0!1=- hi+1-h1-1. ~= - H Uj+,·Ui -1. (1.5) purposes, the primitive equations are used almost Ut g 2.dx iJ I 2.dx 44 TIIE GRAVITY AND GRAVITY-INERTIA WAVE EQUATIONS that we obtain when the space derivatives in (I . I) are in the figure contains two elementary " subgrids", with approximated by centered finite difference quotients the solution on one of these subgrids being completely usi ng values at the two nearest points. The solutions decoupled from the other. Thus. it would be better to (I . 2) now take the form calculate only one of these solutions. that is. to use a grid as shown in Fig. I . 2. Such a grid, with variables A A 1 4 4 carried at alternate points in space, is called a staggered u1 (t ) = Re [u e 2 2 = H (sin kt1x) (I. 7) Figure 1.2 A grid with two dependent variables that are carried v g AX . at alternate grid points. Thus, instead of a constant phase speed, the gravity waves now propagate with the phase speed group velocities. Thus, when using such a staggered grid, the phase speed and group velocity diagram shown ,r;-; sin kAx c• = ± rgH kLJx (1. 8) in Fig. 2. 2 of the preceding chapter is reduced to its left half, covering waves with wave lengths of up to 4L1x only. T his is a tremendous improvement. or If we wish to have waves with wave lengths between si n kAx 4Ax and 2Ax in our calculation we can reduce the grid c• c --- ( I . 9) length by a factor of two and perform a much more = kAx accurate integration, using the same amount of compu This phase speed is a function of wave number, and, thus, tation time than with a grid that is not staggered. we see that the space differencing again results in compu tational dispersion. The formula ( I . 9) is the same as 2. Two-dimensional gravity waves the one obtained in the preceding chapter when consider ing the advection equation. Therefore, both the phase We now consider two-dimensional gravity waves. T hus, speed and the group velocity depend on the wave number we consider the system of linearized equations as shown in Fig. 2 . 2 of the preceding chapter. T he du iJh av iJh phase speed decreases as the wave length decreases, and -=-g-· -=-g- · the wave with wave length 2Ax is stationary. di ax a, oy (2. I) There ii.. however. an important difference between iJh =- H fi'· 1> . this problem and the advection problem because we <7 r now have two dependent variables. We have assumed that they arc buth carried at every grid point, as shown Substituting the wave solutions in Fig. 1. 1. u = Re [0e' Figure I. I A grid wilh two dependent variables that are both we now obtain carried at every grid point. (2 . 3) Thus, in the two-dimensional case the gravity waves As far as the system (I . 5) is concerned, however, the underlined variables in the figure depend only on propagate with the same constant phase speed v'g H. other underlined variables. The same statement holds Because of the results obtained in the preceding for the variables that arc not underlined. Thus, the grid section, we fi rst consider the spatial distribution of the CHAPTER 4 45 variables. With two dimensions and three dependent of the area of the admissible region of the wave number variables, a large number of spatial arrangements of the plane. variables arc possible. For the present we consider the The same standard finite difference approximations three rectangular arrangements shown in Fig. 2. I. T he can be used for the space derivatives in (2. l) for all thr~c identifying 1::tters (A), (E) and (C) are chosen so as to lattices. We write these approximations using the conform with the symbols used by WinninghofT and difference operators 6,. and 011 , defined by, fo r example, (A) (E) (C) Figure 2 . I Three types of lattice considered for the finite difference solution of 2. I. Arakawa (Arakawa, 1972). We shall denote the shortest Sxh ~.rh(x+d·.yl- h (x-d•.y)J. distance between the grid points by d*. With the same = 2 value of d* the lattice (E) will have twice, and the lattice {C) four times less variables per unit area than the Thus, (2. l) can be approximated by lattice {A) , as in Fig. 2. I. The lattice (E) can be obtained by a superposition of two (C) lattices, and the lattice (A) au av -i)r = -gox h • -iJt =- go y h • by a superposition of two (E) lattices, or of four (C) (2. 4) lattices. iJh ~ = - H (ox u + oy v). The admissible regions of wave numbers in the wave iJ I number plane can be found by considering the shortest resolvable wave lengths. Note that with lattice (E) the Substituting wave solutions analogous as in (2. 2), we lines joining the nearest points with the same variable obtain make an angle of it/4 with the grid lines while with the H sin~ kd• sin2 Id• 2 + (2 . S) other two lattices these lines arc along the grid lines. v = R t1 • Fig. 2.2 shows the admissible wave numbers. A halving we define of the number of variables is associated with a halvi ng (A) (E) (C) 1t Id* 0 () 1t 0 1t r'igure 2.2 Admissible region~ of w:ive numbers for the three types of lattice shown in Figure 2.1 . 46 THE GRAVITY AND GRAVITY-INERTIA WAVE EQUAT IONS the ratio of the phase speed given by (2.5), c• , to the are constant, they will be a stationary solution on that true phase speed y gH, can be written as lattice, no matter what the values of the variables on the other (C) lattice are. Two stationary solutions, with 2 2 different constant values on each of these complementary C* -vsin X + sin Y (2. 6) lattices, will give a stationary wave represented by the right v'if1 - x2 + y2 · hand corner of the triangular region in Fig. 2.3, with a This formula reduces to the previous formula, (1. 8) or zero phase speed. This wave is usually referred to as the (I . 9), when applied to the one-dimensional case. two-grid-interval wave. In the same way, the (A) lattice admits four independent stationary solutions, with The values of the relative phase speed (2.6) on the different constant values on each of its four type (C) wave number region admitted by lattice (E) are shown subgrids. in Fig. 2 . 3. By symmetry about the line l = k only half of the region needs to be shown. Fig. (2. 2) shows that The two-grid-interval wave can easily be generated lattice (C) admits only the left half of the triangular region when boundary conditions are artificially described, and, with more complete equations, in cases when gravity waves are generated inside the computational region. 0.6 These can be caused by heating, for example through the release of latent heat, and by the influence of mountains. When gravity waves are excited involving variables of '+- ,1 I ,~ I 0.4 one of the (C) subgrids only, for example by forcing at Jd• / I / • individual grid points or lines of points, the gravity wave '1 l •' / I will propagate through the variables of this subgrid only. J l I ( l The variables of the other (C) subgrid will be influenced 0.0 only through the Coriolis and advection terms on a much larger time-scale. Thus physical effects which may 0 rt/2 1t excite relatively long waves in the atmosphere may excite kd* spurious waves with wave lengths of approximately Figure 2. 3 Relative phase speed of gravity waves when the space two grid intervals in a computation. When these reach derivatives in (2 . 1) arc approximated by straightfor an excessive amplitude, some remedial measures have to ward space-centered finite difference analogues. be taken. These will be discussed in a later section. shown in the diagram. Clearly lattice (C) gives a more accurate phase speed for gravity waves than the other 3. Gravity-inertia waves and space distribution of variables lattices considered here. Unfortunately, because it does not carry the two velocity components at the same points, In this section we discuss the effect of centered space there is some difficulty with the Coriolis force term. differencing on gravity-inertia waves. Thus, we consider Of the other lattices, the staggered lattice (E) is much the system of linearized equations superior to the non-staggered lattice (A). A result with the same truncation error can be achieved in about half au ah av ah of the computation time, (exactly half if the equations -dt =-g-ax + Jv . a, = - g dy - Ju. are linear), and a sizable fraction of wave numbers that (3. I) i}h are associated with large phase speed errors and compu -= - HP · 11 . tational dispersion are eliminated. The additional time a, needed for a calculation on an (A) lattice is spent on waves that can hardly be expected to improve the inte These equations differ from those of section 2 in the gration. appearance of the two Coriolis terms. The Coriolis terms contain no derivatives. However, they are difficult As we can see from the phase speed diagram, lattice to calculate on the (C) lattice, which was ideal for pure (E) is not free of computational problems. As with the gravity waves. non-staggered one-dimensional grid discussed in the preceding section , the solutions of (2. 4) on each of the Thus, we reconsider the problem of the distribution of two type (C) subgrids forming the (E) lattice are inde the variables. pendent and can diverge from each other. This can be a It is not obvious how we should analyse various arran source of serious problems. For example, if the values gements of variables. Our primary concern here is to of the dependent variables on one of these (C) lattices consider (3. I) as part of the complete system of primitive CHAPTER 4 47 equations. We are interested in large-scale motions, derivatives, leaving the time derivatives in their differen otherwise we would not be including the Coriolis terms. tial form. On the large sca:lc, the primitive equations admit two The discussion is based on that by Winninghoff and district types of motion : low-frequency, quasi-geostrophic Arakawa, as presented by Arakawa (Arakawa, 1972; and quasi-nondivergent flow ; and high-frequency gravity Arakawa et al. 1974). inertia waves. Gravity-inertia waves are continually. We consider five ways of distributing the dependent excited in the atmosphere ; however, as they are dispersive, variables in space, shown in Fig. 3. I . We denote by d the a local accumulation of wave energy disperses with time. shortest distance between neighbouring points carrying This process is known as geostrophic adjustment ; the the same dependent variable. In the figure dis the same remaining motion is in approximate geostrophic balance for each of the five lattices ; thus, all the lattices have the and changes only slowly in time. In this chapter we are same number of dependent variables per unit area. The concerned with the correct simulation of this process, computation time needed for an integration on each which is essentially governed by the gravity-inertia wave of the lattices will be about the same ; the properties of equations (3. 1). the solution obtained, though, will differ because of the We are interested both in waves caused by physical effect of the space arrangement of variables. effects, and in those caused by inadequacies of the Using the subscripts shown in the figure, we define initial data and of the numerical procedures. the centered space differencing operator by However, the details of the adjustment process do not I matter as much as the correctness of the resulting quasi (Ofa) . = -d, \a,· +.! . - a,--,'.,·) , geostrophic flow. . 1 •) : · J We shall therefore investigate the effect of the space this rotation is applicable to all the lattices. Here d' is the distribution of dependent variables on the dispersive distance between the points between which the finite properties of the gravity-inertia waves. This will be done difference is taken. Thus, for lattices (A) through (D) d' using the simplest centered approximations for the space is equal to the grid size d, and for the lattice (E) it is (A) (B) 1. + I __u, v, ______h u, v, h _, u, v, h 1. + I h h h ,. u, v u, v • • j u, v, h u, 11, h u, v, h h h j h u, v U, V • • (E) . I u, v, h u, v, h u, v, h . I h h h . I u, v h u, v l - i- l i i + I l- i- 1 i i + I ; + 2 ...... - ---- +-d-+ ~d-+ ~) (~ u h u h v h v h j + I h v v v u u u h u h u v h v h j j h v v u u u u v 11 j - 1 h h u h j - 1 h h i- 1 i i + I i- "1 i i + I ~d---+ +-d---+ Fiaure 3. I Five types of lattice considered for the finite difference solution of (3 . I). 48 THE GRAVITY AND GRAVITY-INERTIA WAVE EQUATIONS equal to V2d. We also define an average taken over the Substituting the wave solutions {1 .2), we obtain the same two points by frequency equation which can be written as litl, . = -1 la,+.!. .+ a,_.!. ·) . •J 2 ,·J l •J (3 .4) (6.,a),. 1 and (a.II),, 1 arc defined in the same way, but with respect to they axis. Finally, Thus, as the radius of deformation - .. A = Vgll/f, (ciltll),., = (ciZ),,,. is never equal to zero, the frequency of the gravity For each of the five lattices we use the simplest centered inertia waves is a monotonically increasing function of k. approximations for the space derivatives and Coriolis Therefore, the group velocity iJv/iJk is never equal to terms (3 . 1). We obtain the difference systems zero. This is very important for the geostrophic adjust iJu -x av -y ment process, as it precludes a local accumulation of g 6xh fv, a, g 6yh - fu. a, = - + = - wave energy. (3.2)A i:Jh ....--:x -.YI We now look at the effect of the finite differencing in a, = - H 1OxU + Oy v I space in this case. As the variables arc assumed not to depend on y, the systems (3. 2) reduce to au - Y av -X - =-g6 h +fv a, = - g 6yh - fu. av a, x • a,= - fu, (3.2)B (3 . S)A ah=_ H ,C,f a, x ' iJu ., h -xy av ., h -XY iJt : - g Ux + fV ' at = - g Uy - fU ' av a, = - fu . ah (3. 2)c a, = - H l&xu + 6y v) , (3 , S)B au -xy -xy av -xy -xy a, = - g 5xh + fv • a, = - g 5yh - fu • (3 .2) au ., -x i)v - x 0 - =-g u h +fv fu . ah H 1r-uxy + r-:-vxy ) a, x • a, = - at = - Ox1' Oy• I (3 .5)c a v a, = - g 6yh - f u . (3 , 2)E av -x i)t = - fu ' We sha ll first analyze a one-dimensional case, that in (3 . S)o ah ~... which the variables u, v and h do not vary with y. Thus, a, = - H OxU; we have u, v,h = u, v, h (x, t). The system (3 . I) then reduces to au a h av av -a, =-g-ax + fv . a, = -fu. a,= - fu . (3 .3) (3 .5)g iJha, -__ Hauax. I N ,: / ) "'- . CHAPTER 4 49 Substitution of wave solutions into these systems gives associated with a zero group velocity. For A/d = I /2 the the frequency equations group velocity is equal to zero for all k. Lattice (D) : The frequency reaches a maximum at 2 (3 . 6)A (A/d) 2 cos kd = l /4 . The two-grid-interval wave at (}}2 = t + (~r sin kd, kd = 1t is stationary. Lattice (E) : The frequency reaches a maximum at kd = 1t/V2. The shortest resolvable wave with kd = V21t v)2 ()..)2 . 2 kd (3. 6)s (l = I + 4 d sin 2 , behaves like a pure inertia oscillation, and its group velocity is again zero. A summary of these results is shown in Fig. 3 . 2. It v)2 2 kd ()..)2 2 kd shows the functions Iv llf, in the case y/d = 2. (J = cos 2 + 4 d sin 2 . (3.6)c 2 (lvr = cos2 2kd + (A.)2d, ~in kd, (3.6)o 6 4 2 ~ (lvr = l + 2 (dAr. sin. V2kd . (3.6)a f 2 The non-dimensional frequency vif is now seen to depend on two parameters, kd and )../d. We shall analyze the dispersion properties revealed by these expressions for each of the five lattices. The wave length of the shortest resolvable wave along the x kd/ 1t axis is 2d for lattices (A) through (D), and y2d for the lattice (E). Thus, we have to consider the range O < kd ~ 1t Figure 3.2 The functions Iv l/f given by (3 .4) and (3.6), with for lattices (A) through (D), and the range O < kd ~ ')..Jd= 2. ~ v'2x for the lattice (E). Lattice (A) : The frequency reaches a maximum at kd = x/2. Thus, the group velocity is zero for k equal The figure vividly illustrates the inade~uacy.of the latti~es to x/(2 d). If gravity-inertia waves of approximately that (D) and (A). The phase speed and d1spers1on properties wave number are excited near a point inside the computa of the remaining three lattices are much better : however. tional region, for example by nonlinear effects or forcing zero group velocities occur with every ~attice. Thus, wit_h through heating or ground topography, the wave energy any lattice there will be difficulties m the geostroph1c stays near that point. Beyond this maximum value, adjustment process. for x /2 < kd <-. 1t, the frequency decreases as the wave The difference between the results for lattices (B) and number increases. Thus, for these waves the group velocity has the wrong sign. Finally, the two-grid-interval (E) is interesting because these two lattices can be obtained from one another by a rotation through an angle of x/4. wave with kd = 1t behaves like a pure inertia oscillation, llnd its group velocity is again zero. If we consider the one-dimensional case in which the dependent variables are constant along t~e lines Lattice (B): The frequency increases monotonically y = x + c, we obtain results for these two lattices that throughout the range O < kd < 1t . However, it reaches are exactly opposite to those in Fig. 3.2: In general, a maximum at the end of the range, so that the group we define the coordinate system x', y' by rotating the system velocity is zero for the two-grid-interval wave with x,y in the positive direction through an angle of x/4, kd = 1t. and then, using the relations Lattice (C) : The frequency increases monotonically with kd if )../d > l /2 and decreases monotonically with kd if V2 u· == V2 (u+v) . v· = - (-u + v). :A./d < I / 2. It again reaches an extreme value at kd = x, 2 2 50 THE GRAVITY AND GRAV ITY·INERTIA WAVE EQUATIONS change from variables u, v, la to new dependent variables The (B) or (E) lattices have a problem with fal se low u', v' ,la. We find that this transforms the system (3. 2)B frequencies of the shortest waves. The two-grid-interval into (3 . 2)s, and, conversely, (3. 2)n into (3. 2)8 . Thus, wave, that was stationary as a pure gravity wave, now the dispersion properties of the lattices ( B) and ( £ ) can behaves like a pure inertia oscillation. The difficulty be considered equivalent. A gravity-inertia wave in one arises from decoupling of the gravity wave solutions on of these lattices has phase speed and dispersion properties the two complementary (C) type subgrids. Methods of identical to those of the same wave with its front rotated dealing with this will be discussed later. through an angle of rt/4 in the other lattice. Obviously, we should also consider the two-dimensional case. The values of I v I// that are obtained in the two· 4 . Time differencing; the leapfrog scheme and the dimensional case for the true solution and those using EUasseo &rid lattices (B) and (C) are shown in Fig. 3. 3 with '>. /d' = 2. Properties of time differencing schemes applied to the The diagram for lattice (E) can be obtained by a counter· gravity wave equations can be deduced from the analysis clockwise rotation of the (B) lattice diagram. of Chapter 2, as was done for the advection equation. (A) (B) (C) 1.0 1.0 1.0 ·,4 ~ ' ) 0.8 0.8 '· \ \ }s' 0.8 3.5 \ 2 ...___- \ 2.5 ____ 0.6 0,.6 \ ~0.6 ~ ~ -'.:!:! --~ ·, -~ 0.4 0.4 ' ·,....-- 0.4 3.5 ' \ 0.2 0.2 0.2 4\ \ 0 0.2 0.4 0.6 0.8 1.0 0 0.2 0.4 0.6 0.8 I .0 0 0.2 0.4 0.6 0.8 1.0 kd /rt kd /rt kd /rt Fiaure 3.3 The functions Jv J//, for the true solution and for solutions of systems (3.2)Band (3. 2}c, with 'J../d= 2. The diagram for lattice (C) in the two-dimensional case We shall demonstrate this for the one-dimensional is seen to be a much better approximation to the exact equations solution than the (B) or (E) lattice diagram. In the (B) lattice diagram the dot-dashed line shows the maximum au + c au + g ah = O. I v I/Jfor a given ratio 1/k; note that there is no such line at ax ax in the (C) lattice diagram and the exact solution. Such (4.1) a maximum occurs at only two corner points of the (C) ah + c an + H au = O. lattice diagram. Thus, with the (C) lattice, no waves at ax ax have a group velocity with the wrong sign . The situation, though, does depend on the parameter '>. /d. With a stra· tified atmosphere the radius of deformation ). depends We first multiply the second of these equations by an arbitrary parameter )., and add the result to the first on the stability ; if the stability is so weak as to make '>. /d of the order of l or less, the (C) lattice loses the advantages equation. We obtain shown in Fig. (3 . 3). However, for typical grid sizes used in atmospheric models this is not the case and therefore a ~ ~ a, (u +).la)+ (c + '>.H) ax + ( g+ '>.c) ax = 0. (4.2) Arakawa (Arakawa and Lamb, 1976) concludes that the lattice (C) is the best lattice to simulate the geostrophic adjustment process. Accordingly, it is at present being used in the general circulation model at the University We wish to choose ). so that of California at Los Angeles, and also in the British g + ).c = ). (4.3) operational model. c+'A.H ' CHAPTER 4 51 to obtain an equation with only one dependent variable, u + Ah. The two solutions of (4 . 3) are V(f)2 + yRT, where 'Y = cp/c.. If we neglect the first term and recall (4.4) that the scale height of an isothermal atmosphere is H• = RT/g, we see that the phase speed of the Lamb waves is of the Substituting these into (4. 2) we obtain same order of magnitude as that of the external gravity waves. Thus, in view of the relation between stability f;, + (c + ~) JxJ(u + V,f h) = O, and the phase speed , we see that (4. 7) should also repre sent an approximately correct stability requirement in (4. 5) the three-dimensional case. With the highest phase r:i + (c _ ~) :x (u _ wh) = 0. speeds of the order of 300 m sec-•, an\'.I a grid size of 1 about 100 km, this requirement does not permit time This is the normal form of the system (4 . 1). It shows that steps longer than about 5 minutes. This time will be smal (4 . I) is equivalent to a system of two advection equations. ler by a factor of V2 with two horizontal coordinates. The quantity u + v'g / Hh is seen to be advected at a The CFL stability condition thus means that a large amount of computer time is required for integration of velocity c in the direction of the x axis, while, + VgH the primitive equations, especially when the grid size is at the same time, the quantity u-v'g / Hh is advected in small to reduce errors in space differencing. F or this the same direction at a velocity c-VgH. reason some investigators prefer using implicit time dif Suppose now we choose a grid that carries both u and h ferencing schemes, so that the choice of time step can be at every grid point. The systems obtained by using based solely on accuracy and not on stability. centered space differencing in (4 . 1) and (4.5) are then We can also study the stability and other properties of equivalent. We can therefore use the same procedure time differencing methods applied to the gravity wave as in Section I of Chapter 3 to analyse time differencing equations by direct substitution of wave solutions. For schemes. We obtain the same results as before, except example, consider the leapfrog scheme with centered that in ~ce of the advection velocity c we now have space differencing applied to the two-dimensional system c + v'gll. Thus, if the leapfrog scheme is used for the au ah av+ i)h = O time differencing. and c is considered positive, we obtain - + g-= 0 ar gay · the Courant-Friedrichs-Lewy stability criterion for this a, ax (4.8) case as air a,+HP·11=0. '11 (c + v'if/) '1x < I. (4 .6) Using one of the lattices of Fig. 2. I as well as the notation of Section 2, we obtain The advection velocity in the atmosphere is normally 1 1 1 about an order of magnitude less than the phase speed u"+ = u"- -2g'1t6%h". vn+I = v"" -2gLJt611 h", of external gravity waves. Accordingly. in the foreg~ (4.9) criterion c is often neglected compared with v'g H, giving the stability requirement Substituting the wave solutions (4. 7) (4. 10) When using the three-dim<•nsional primitive equations, external gravity waves arc normally eliminated by per· we obtain the homogeneous system mitting no vertical velocity at the upper boundary. The highest phase speed admitted by the system is then that (i,..2 -1) t + i/1.2 v'2 gµ sin X h = 0 of the Lamb waves, which for an isothermal atmosphere is (i,..2- l)~ + iA2V2gµsinYh=0, (4. 11) 52 THE GRAVITY AND GRAVITY·INERTIA WAVE EQUATIONS i')..2 v'2 Hµ (sin X ~ +sin Y t) + ('}..2 -1) h = 0. 2VgHµ ~ I, (4. 15) Here X and Y are defined as in Section 2, while or (4. 16) that is, µ = '1t/d when the lattice (E) is chosen. The properties of the numerical solution can now be This is in agreement with the previous results. The studied by analyzing (4. 11). The requirement that its nondimensional constant on the left side is sometimes determinant be equal to zero gives six solutions for ')... called the Courant number. Two of these are With solutions like (4.10), the frequency v is given by ')..=I (4. 12) ').. = I').. I e-1,41. and Thus, the expressions obtained for >.. can be used to calcu ').. = - 1. (4.13) late the relative phase speed c• /Vgii using the relation The remaining four are given by c• I ->... vfi1 =.dt i4H(k 2 + /2) arc tan ,..:~. (4.17) >..2 = l-4A ± 2 V2A (2A - 1) (4. 14) If we are given ')..2 rather than>.., as here, we can express where 2 the relative phase speed as a function of (>.. ) 1,,. and A = gHµ2 (sin1 X + sin2 Y). ().a),.. Thus, using (4. 14), we find for 2A ~ I : We can now analyze the solutions (4.10) associated c• I with the values found .for ""· The first of these values, vf'i1 = 2µ V2gH (X2 + Y2) x (4. 12), gives a neutral and stationary solution. If either (4. 18) sin X or sin Y is non-zero in this neutral and stationary x ( 2 v'2A (I - 2A )) /\ arc tan ~ _ A . case then, according to (4.11), we have h = 0, and the 1 4 solution represents a physically acceptable translatory motion . If, however, sin X and si n Y are both equal to zero, the amplitudes of all three dependent variables can This expression, of course, approaches (2 . 6) as '1 t take arbitrary values. In addition to the physically approaches zero. acceptable solution where all the dependent variables For a more explicit illustration of the effect of time are constant (k = I = 0), there is a solution with one or differencing, we can perform a series expansion of (4. 18). both of the wave numbers k and I equal to rc /d•. This One obtains, for V2A < 1/V2, is the two-grid-interval wave, discussed already in Section 2 2. It again appears as a false computational solution ; c• = ysin X + sin2 Y (t + ! A + _! A 2 + ···) since it is stationary. it is not affected by the introduction viR x 1 + Y 2 3 10 · of time differencing. The second value, ').. = - I, represents a false compu· The factor multiplying the series in parenthesis describes tational mode in time, with a period of 2'1t. This com· the decelerating effect of space differencing. as given putational mode results from using a three time level by (2. 6). The acceleration resulting from the leapfrog scheme. time differencing is beneficial, as it reduces the phase error due to space differencing. To prove stability of the scheme the behaviour of the remaining solutions given by (4. 14) has to be investigated. The values of the relative phase speed ( 4 . 18) are shown They will all also be neutral for 2A < I. To obtain the in Fig. 4 . I, for the physical mode with 2v'gHµ = 0.5. condition in the form B'1t ~ I we write The wave number region shown here is the same as in Fig. 2. 3, where the effect of space differencing alone was considered. Comparison of these figures shows little difference between the two families of isolines. The Since this has to be satisfied for all the admissible waves, relative acceleration due to the time differencing has a we find that the CFL criterion in the two-dimensional maximum at the upper corner of the diagram, but the case is now relative phase speed here is sti ll poor. CHAPTER 4 53 grid the R ichardson grid. A single Richardson grid is considered as a time-staggered version of the (C) lattice and suffices for the solution of the pure gravity wave system (4.9); thus, on an Eliassen grid the system (4 .9) has two independent solutions. Using the difference Id* system considered above to approximate the differential system {4 . 19), these solutions are coupled only through the two Coriolis terms. 0 rc/2 ft kd* Figure 4 .1 Relative phase speed of gravity waves, with centered time and space differencing, and Courant number equal to O.S. Finally, we point out that time differencing suggests new ways of distributing the variables, as the grid can no w also be time-staggered. A good example is given by the linearized system du <1h av ah --a, + g -ax - Jv = 0 . a, + g ay + f u = o. (4. 19) i)h + H(au + iJv) = O. iJt iJx iJ y x approximated using lattice (E), the leapfrog scheme and centered space d ifferencing. If all the variables were Figure 4.2 A space-time grid- staggered both in space and time, convenient for the leapfrog scheme associated with calculated at every time level, there would be two inde centered space differencing. pendent solutions. The solution involving the variables of the space-time grid shown in Fig. 4. 2 would be indepen dent of that involving the variables that are left out in this 5. Economical explicit schemes figure. The second _$rid can be obtained by shifting the The fact that we are now solving two equations, the grid a distance V2d* along the line y = x. Thus, as equation of motion and the continuity equation, sug with the space grids discussed in Section 2, the space gests new ways of constructing time differencing sche time grid formed by using the {E) lattice at every time mes. Some of these have recently attracted the attention level can be considered as a superposition of two element of atmospheric modellers. ary subgrids of the type shown in Fig. 4 . 2. Solving the As seen in the section 4, an inconvenient feature of system ( 4 . 19) on only o ne of these saves half the computa gravity waves is the high computer time required for tion time, with no change in the truncation error. In a solution using explicit schemes for the time differencing. addition the computational mode in time, given by The t ime step imposed by the CFL stability criterion is (4 . 13), is eliminated , as the variables at alternate time generally considered to be much less than that required levels arc missing. Thus, with a more complete system for an accurate integration of the slower quasi-geostro of equations, the gradual separation of solutions at phic motions. With these steps, the errors due to space alternate time levels is not possible. The advantages of differencing are much greater than those due to time the space-time grid shown in the figure were pointed out differencing. Robert (1974), for example, estimates by Eliassen (1956) at an early stage in the study of the that the typical errors due to space differencing in present primitive equations, and it is called the Eliassen grid. atmospheric models amount to nearly 40 per cent, and However, as pointed out by Platzman (1958 ; 1963) tho~ due to time differencing only to about I per cent the grid in Fig. 4 . 2 can again be considered as formed by a of the total error. Thus, any economy that can be made superposition of two subgrids, where in each of these in time differencing is welcome , as the time that is saved subgrids only the height is kept at one time level and the can usefully be used to increase the accuracy of the space velocity components at the next. Platzman calls t his sub- differencing. 54 THE GRAVITY AND GRAVITY-INERTIA WAVE EQUATIONS Two explicit schemes that are more c_conomical than backward scheme, with (4.14), for the leapfrog scheme. the standard leapfrog scheme will be given here. They The right-hand side of (5 . 3), with A replaced by 4A, both achieve economy by using a different integration is equal to the right-hand side of (4.14). Because of the procedure for the height gradient terms of the equation definition of A , this means that A for the forward-back of motion and for the divergence term of the continuity ward scheme is identical to A2 for the leapfrog scheme equation. For brevity, we call these terms the gravity when time steps are used for the forward-backward wave terms of the governing equations. scheme twice as long as those for the leapfrog scheme ! We shall discuss the properties of one of these "econo Thus, the forward-backward scheme gives the same mical" schemes in some detail. It is ~btained by first result using only half the computation time needed for integrating the gravity wave terms of either the equation the leapfrog scheme. In addition, as a two level scheme, of motion or of the continuity equation forward, and it has no computational mode in time. then those of the other equation backward in time. Thus, To understand this advantage of the forward-back this scheme could be called the forward-backward scheme. ward over the leapfrog scheme we compare the finite With centered space differencing, (4. 8) is approximated difference analogues that these two schemes give for the by wave equation, since the system of gravity wave equa tions is equivalent to a single wave equation. Consider u"+l = u" - g.dt6J,", v"+i = v" - g.dt611h", the one-dimensional version of this system : (5 . I) hn+I = h" - H.dt (6su + 611vr+ 1, au+ iJh _ O a, g ax - . (5 .5) or by an analogous system in which the order of integra tion is reversed. Eliminating one of the variables u, h we obtain a wave Substituting the wave solutions (4. JO) we find three solutions for A. One of these, equation A = I , (5.2) (5. 6) gives again a neutral and stationary solution. The remaining two are We can perform the same elimination for each of the finite difference schemes. A = I - A ± VA(A-2), (5.3) The forward-backward and space-centered approxi· where the quantity A is defined as in the preceding mation to (5. 5) is section. Solutions (5 . 2) and (5 . 3) are obtained for both u~+ 1 -u~ I I h/'+ 1 - hj-1= O versions of the scheme, that is, no matter which of the .dt + g 2.dx two equations - the equation of motion or the co~ti (5. 7) nuity equation - is first integrated forward. n+ I h" n+I n+t hI - j + HUj+I - UJ - 1 =O Examination of the amplification fac~ors given by .dt 2.dx ' (5 . 3) shows that the scheme is stable and neutral for A ~ 2, that is, for We now substract from the second of these equations an analogous equation for time level n-1 instead of n, divide the resulting equation by .d t, and, finally, eliminate To satisfy this for all the admissible waves, we must have all u values from it using the first of Eqs. (5. 7), written for space points j + I and j-I instead of j. We obtain 2-VgHµ ~ 2. (5.4) 11+1 II 11-I n II II hJ - 2hJ + hJ _ H hj+2-2h1 + h1 - 2 = O (5 8) Thus, the forward-backward scheme is stable and neutral ( .dt )2 g (2.dx )2 • · with time steps twice those allowed by the CFL criterion for the leapfrog scheme (Ames, 1969). This is a finite difference analogue of. the wave equation The amplification factors of the forward-backward and (5. 6). Note that although each of the two equations of the leapfrog scheme are equal within their regions of (5 . 7) is only of the first order of accuracy in time, the stability. We now compare their effect on the phase wave equation analogue equivalent to (5 . 7) is seen to be speed by comparing the expression (5.3), for the forward· of the second order of accuracy. CHAPTER 4 55 If we use a leapfrog and space-centered approximation in meteorology, by Gadd (1974) in experiments with the to (5 . 5), and follow an elimination procedure like that British operational model. used in deriving (5. 8), we obtain Another way of constructing an economical explicit scheme was pointed out by Shuman, Brown and Campana n+I ih"-l+Jr"-3 lri - j j (1974) , and it is now used in an operational model at (2.dt)2 the National Meteorological Center. For the shallow n -1 n-1 n-1 (5.9) water equations with this scheme, the height values at _ HlrJ+i-21rJ +lr1-2 _ time level n + 1 are first calculated using the leapfrog 2 0 g (2.dx) - · scheme, and then the equation of motion is integrated using the height field averaged over the time interval This also is an analogue to the wave equation (5.6) of 2.dt by the bi-trapezoidal rule : second-order accuracy. However, in (5. 8) the second time derivative was approximated using values at three ~1r""'+ 11r"+ i1r"+ '. consecutive time levels ; in (5. 9) it is approximated by values at every second time level only, that is, at time Substitution of wave solutions into the equations of this intervals 2.dt. Thus, while the time step required for scheme gives the value {5. 3) for k, in addition to the linear stability with the leapfrog scheme was half that neutral values. Thus, the stability criterion and the pro with the forward-backward scheme, (5. 9) shows that perties of the physical solution are the same as with the we can omit the variables at every second time step, and forward-backward scheme. Even though this Shuman thus achieve the same computation time as using the Brown-Campana (SBC) scheme is a three level scheme, forward-backward scheme with double the time step. time staggering of the grid is not possible because of the This method was discussed in the previous section for averaging of the height values. Thus, the economy the two-dimensional case, it is the Eliassen grid. Thus, accomplished by the SBC scheme is again equivalent comparing (5 . 8) and (5. 9) shows that the economy to that accomplished with leapfrog time differencing by accomplished by tire forward-backward scheme is equivalent the Eliassen grid. The SBC scheme has somewhat larger to that accomplished with leapfrog time differencing by tire storage requirements than the forward-backward scheme. Eliassen grid. Both of these methods avoid calculating However, it does permit the use of the leapfrog scheme the false time computational mode, and thus save half for the advection terms. of the computation time with no effect on the physical mode of the solution. 6. Implicit and semi-implicit schemes Comparing these two methods, the forward-backward The time step permitted by the economical explicit scheme has some advantages. With the forward-back schemes, twice that prescribed by the CFL criterion, is ward scheme all the variables are defined at all grid still considerably shorter than that required for accurate points at every time step ; this facilitates the programming integration of the quasi-geostrophic motions. Even with work. In addition, the forward-backward scheme can these schemes the time differencing error is still much less be modified to allow propagation of gravity waves than the space differencing error for typical current between all points of the grid preventing two-grid-interval atmospheric models. Thus, we consider implicit schemes noise. This modification will be described in Section 8. which are stable for any choice of time step. We shall No analogous method, however, has so far been proposed consider here only the simplest of the implicit schemes, for the leapfrog scheme with the Eliassen grid. the trapezoidal rule. For brevity it will simply be called A disadvantage of the forward-backward scheme is tire implicit scheme. that it is not possible to use the leapfrog scheme for the We shall first discuss the properties of the implicit advcction terms. However, the second-order accurate scheme applied to the system (4.8) in some detail, that Adams-Bashforth scheme can be used for these terms. is, the case of pure gravity waves. Thus, we consider the Its weak instability should cause no trouble because of finite difference system the relatively slow speed of the advcction processes. 1 11 For example, in experiments of Mcsingcr and Janjic 11" + = u"- g .dt.!.2 (6x lr"+6x 1r +•) • (1974), where a multi-level model was used for simulation of the growth of a baroclinic wave, a forward scheme 11 11 11 v"+ 1 = v -g.dt1(6ylr +6ylr +•). (6.1) was used for the advection terms, and no signs of insta bility were noticed for about a two week period. The 11 forward-backward scheme has been used for the storm lr"+ I = Ir "-H.dt H (6xu + 6yv)" + (6xu + c5y1,) + ']. surge problem by Fischer (1959) and Siclccki (1968), and, 56 THE GRAVITY AND GRAVITY·INERTIA WAVE EQUATIONS Substituting the wave solutions (4.10) we find three With an ordinary oscillation equation, (2. 7) in Chap solutions for l. One of these, ter II , this can be done very simply. For the system (6 . 1) 01 01 it is more complex. The quantities 6zu and 611v can l = I, (6.2) be eliminated from the third equation by applying operators 6., and 6 to the first and second of these equa· is again that associated with a neutral and stationary solu 11 tions and substituting the results into the third equation. tion. The remaining two are This gives an equation for the height which can be solved using a number of standard methods : the most popular l=+.(1-!A ±V-2A) . (6 .3) of these is the relaxation method which is discussed l+-A 2 2 later in this section. Examination of (6. 3) shows that it always gives ampli Two methods are used to deal with the advcction, fication factors satisfying Coriolis, and other terms of the governing equations, in atmospheric models. One of these, the splitting method, I l I= t, (6.4) will be discussed in the next section. The other is the semi-implicit method. There is no advantage in using and so the scheme is unconditionally stable and neutral. an implicit method for these additional terms of the Using (6 .3) and (4.17), we find for the relative phase governing equations. They are associated with slower speed of the nonstationary solutions, phase speeds, and should not require excessively smaJI time steps for linear stability when calculated explicitly. c• 1 ( Thus, they can be calculated by an explicit scheme. Since (X2 y2j arc tan =f 1-VU-)A (6. 5) 'vj11 = µ V2gH + ! the trapezoidal implicit scheme is a two level scheme like the forward-backward scheme, it is convenient to The numerical values given by (6. 5) for the physical mode use the Adams-Bashforth scheme for this purpose. with 2,VgHµ = Sare shown in Fig; 6.1. The time step Robert (1969) in a spectral model, and subsequently is chosen to be of the same order of magnitude as the Kwizalc and Robert (1971) in a grid point model, chose, time steps that are currently used with implicit ~hemes however, to use the leapfrog scheme. We then need variables at the middle of the time step used for the implicit differencing, and, therefore, it has to be performed over a time interval of 2.dt. However, the scheme is now less economical for gravity waves since these steps have to be made separately for each of the two time levels stored in the leapfrog scheme. In return, we have a differencing for the advection and other additional terms that is neutral and more accurate than Adams Bashforth's. Kwizak and Robert caJI this combined scheme the semi-implicit scheme. It has been used for a 0 1t/2 number of years in the Canadian operational model, kd• and is. becoming increasingly popular in some other operational numerical prediction centres. Fiaure 6.1 R.elatiYe phue speed or gravity waves, with implicit time and cente,ed apace difrerenc.lng, and Courant The usual procedure used for solving the semi-implicit number equal to S. difference system for variables at time level n + I will be illustrate(\ for the shallow water equations. These in atmospheric models. The wave number region in equations can be written in a compact form the figure is the same as in the earlier diagrams, Figs. 2 . 3 and 4. 1. Comparing the isolines of the present figure au ah a... a1r with those of Fig. 2 . 3, where the effect of space differenc at = - g ax + A., . at = - g ay + Ar• ing alone was considered, shows that the effect of time (6.6) differencing on phase speed is now not negligible. Implicit ah = - HP·,+ A time differencing is seen to result in a considerable retarda ot 4 · tion of gravity waves of the same order of magnitude as that due to centered space differencing. where A., A. and AA denote the terms that were omitted To apply an implicit method it is necessary to solve in the system (4.8) describing the propagation of pure the difference system for variables at level n + 1. gravity waves. When we use leapfrog differencing for CHAPTER 4 57 these additional terms, and implicit differencing over a y time interval 2L1t for the gravity wave terms and centered space differencing, (6. 6) is replaced by u"+i = u"- 1 - gt1t (6.,Ji"-1 + 6.,Jin+ 1) + 2L1t A:, 1 yn+I = v"- -gt1t(611 Jin-l + 611 Jin+I) + 2L1t A:, (6.7) 01 1 x h = h"- - HLJt [(6.,u + 611vr-1 + (6.,u + 6.,v)01] + + 2L1t A:. We now apply the operator 6.z to the first, and 611 to the second of these equations, respectively, and add the results. We introduce the notation Figure 6.2 Stencil used to calculate the approximation , 21i. We obtain (6.zu + 611 v)"+I = (6.,u + 611 v)"-1 - - gt1t [(6u + 61111) Jin-I + {6u + 61111) Jin+l] + + 2L1t (6z A,. + 611 A.,)". Substituting the right-hand side into the third of Eqs. Thus, (6.8) is a finite difference approximation to an (6. 7), and defining the "finite difference Laplacian" by elliptic equation p 1 h + ah+ b(x,y) = 0. To solve such an equation, it is necessary to know the we find values of h (x, y) at the boundaries of the computation region. For a numerical solution we write (6. 8) at each 1 Jin+I = Ji•H-2HL1t (6zu + 611v)"- + of the interior grid points where the variable h is carried. 1 1 1 2 + gH(L11) (,- Ji"- + ,. h"+1) + In this way we obtain a system with one equation for 11 each interior grid point. There is one unknown for each + 2L1t[A,.-HL1t(6zA,. + 6 A.,)] • 11 grid point. In each of the equations, except the equations for points adjacent to the boundary, there are five of Using, in addition, the definitions these unknowns. There are no difficulties in principle in solving such a system of linear equations, but, since 1 F'H =h"- -2HL1t(6zU + 611v)" - 1 + gH(L1t)2 ,-2 h"- 1 , the number of equations is normally exceedingly large, of the order of 1000 or more, it is not obvious how to G" =2L1t [A11- HL1t(6zAN 6 A.,)]", + 11 set about it. this can be written as The method usually used is the relaxation method. This consists of the followiTtg steps. a) An arbitrary guess i., made for the field h" +1. Usually the field of the preceding time step, h", is taken The terms httve been ammged to show that at time level n as this first guess. the right-hand side is known at all space grid points. b) At each of the grid points the value Jin+ 1 is changed Once this equation has been solved for the values hn+ 1 , so as to satisfy the difference equation, in our case (6 .8). 1 u..+ und v" + 1 can be obtained directly from the first and These changes can be made simultaneously at all grid second of Eqs. (6. 7). We now consider ways of solving points (simultaneous or Richardson relaxation), or sequen (6.8). tially, point by point (sequential or Liebmann relaxation). The quantity ,.,h on the left side of (6. 8) is an approxi c) The preceding step is repeated as many times as mation to p'h. Using the notation of Fig. 6. 2, it can needed to make the change at every point less than some be written as preassigned small value. 58 THE GRAVITY AND GRAVITY-INERTIA WAVE EQUATIONS The relaxation method always converges. Experience forecasts made with explicit differencing and time steps shows that the convergence is faster for sequential relaxa of IO min are almost identical to those made with semi tion, and also if the changes calculated to satisfy the implicit differencing and time steps of 60 min. Later equation exactly arc multiplied by a factor having a value Robert et al. (1972) have calculated the differences be between I and 2 (overrelaxation factor) before being added tween 5-day forecasts obtained using 30 and 60 min time on. For a particular problem the optimum value of this steps for a baroclinic 5-level semi-implicit model. These ovcrrelaxation factor can easily be found by numerical differences were found to be insignificant compared to experiments, in which the number of iterations required other sources of error normally present in numerical is plotted as a function of the value of the overrelaxation models. However, the model used for these experiments factor. This optimum value can be shown to be not much did not include topography, surface friction, and other less than 2. More details on the relaxation method can physical processes ; one might expect the deceleration be found in textbooks by Thompson (1961) and by of gravity waves to have a more noticeable effect when Haltincr (1971). these physical processes (e.g. the release of latent heat) are present, since then the gravity waves should be more The algebraic system given by equations of the type significant. On the other hand, the computation time (6.8) can also be solved by direct methods (e.g. Kreiss saved by the implicit differencing can be used to reduce and Oliger, I 973, p . 54). Direct method can be more the grid size on the computation. This would decrease efficient than the relaxation procedure : thus, they are the phase speed error for all the waves, including the typically used when relaxation requires very large compu gravity waves. tation time, as may happen, for example, in convection studies. When implicit schemes are used for simulation or prediction of large scale atmospheric motions, the time needed fo r relaxation is several times less than the 7. The splitting or Marchuk method time needed for other steps of the integration procedure, The complexity of the system of hydrodynamic equa so that only a small fraction of the total computer time tions, that is, the simultaneous presence of a number of can be saved by using a faster direct method. For that physical factors, may cause some difficulties. One reason the use of direct methods, requiring a larger difficulty was mentioned in the preceding section : if we programming effort, is not popular in these models. wanted to approximate (6. 6) using a fully implicit scheme Generalization to the three-dimensional case of the we would obtain a system for the variables at level n + I procedure for solving the semi-implicit system for that is practically impossible to solve. Also since different variables at level n + I outlined here is not quite trivial. physical factors are present in this system we will nor The reader is referred to the paper by Robert et al. (1972). mally wish to use different schemes for terms associated with them. Thus, considering the linearized system with Implicit schemes were first used extensively in atmo advection and gravity wave terms, spheric models by Marchuk (Map'iyK, 1957). With the semi-implicit scheme it is also possible to construct au + au ah an economical grid analogous to the Eliassen grid for a, c ax + g ax = o. the leapfrog scheme; the appropriate space-time stagger (7. I) ing of the variables was pointed out by Gerrity and ah ah au McPherson (1971 ). A semi-implicit scheme somewhat -a, + c-ax +H-ax = O . different from the one outlined here has been developed by Burridge. and is now used in the British operational we might wish to use one scheme for the advection terms, model (Burridge and Hayes, 1974). Implicit and scmi and another for the gravity wave terms - in much the implicit schemes arc undoubtedly the most efficient same way as was done within the semi-implicit scheme. schemes used in atmospheric models. To achieve this In such a situation, even though both of the schemes economy we have to put additional effort into solving to be used arc stable considered one at a time, we cannot an elliptic equation. Furthermore they arc associated be certain that the scheme obtained as a combination of with an appreciable deceleration of gravity waves. Thus, the two will also be stable. An example where it is not the implicit schemes do not seem suitable for the study was given by Kasahara (1965). of details of the geostrophic adjustcmcnt process. How These problems can be avoided by using the splitting ever, this deceleration docs not appear particularly method. The idea of this method is to construct schemes harmful for the simulation and prediction of the largc for a complex system of equations so that within each scalc quasi-gcostrophic motions. For example, Kwizak time step this system is split into a number of simpler and Robert (1971) have found that barotropic 5-day subsystems, which are then solved consecutively one at CHAPTER 4 59 a time. In the case of (7. I), within a given time step, we splitting method : we can choose a relatively long time could first solve the system of advection equations step for the subsystem governing a slow process, advection in the present example, and then use a number of smaller au + c-=du steps to calculate the faster process. Since the advection a, ax o. process is the most expensive in computation time (7 .2) within the primitive equations, significant economies can i)h iJh + c-= 0. be accomplished in this way. A disadvantage of the a, ax method is that calculation of the effects of different physical factors one at a time usually leads to an increase in Denote the provisional values u" +1, h" +1 obtained in this way by u•, h*. Use these values at the beginning of the the truncation error. For example, Burridge and Hayes time step for solving the remaining subsystem (1974) suggest that the technique of splitting the governing equations into advection and adjustment stages does not allow time steps longer than 12 to 15 min if the time truncation is not to become significant. (7 .3) The splitting method was first used in atmospheric models by Marchuk (Map'fyK, 1967) ; thus, in mete orology it is also known as the Marchuk method. It would appear that the splitting method is used for most 1 1 The values u"+ , h"+ , obtained after solving also this atmospheric models in the Soviet Union. The splitting other subsystem, are now taken as actual approximate technique is used also in the British operational model values of these variables at the level n + I. The procedure (Burridge and Hayes, 1974), and in a limited area model is repeated in each following time step. by Lepas and his collaborators (Lepas et al., 1974). A solution obtained by the splitting method will repre· sent a consistent approximation to the true solution. This can be proved easily for a particular choice of schemes for solving the subsystems. The approximate 8. Two-grid-intenal noise values of the dependent variables then have to approach the true values as the time step approaches zero. Unless we are using the lattice (C) shown in Fig. 2. I To study the stability of schemes constructed by the we will always have a problem with two-grid-interval splitting method, we consider the example above. Denote waves. These are false stationary waves appearing as neutral solutions of the difference equations for gravity by A4 and Xb the values of X of the schemes chosen for the numerical solution of subsystems (7. 2) and (7. 3), waves. When the Coriolis terms arc also present, as respectively. Then, we have seen in Section 3, the two-grid-interval waves appear with false low frequencies as pure inertia waves, or, with u• = R e(XaX"t e'"). lattice (D), as stationary waves. A number of methods have been used to cope with this. and In many models dissipative schemes are used to give maximum damping for the two-grid-interval wave, or lateral diffusion is added with relatively large diffusion Therefore, we find, coefficients. The appearance of excessive two-grid interval noise is thereby suppressed. However, instead of attacking the consequences of inadequacies in a simula and tion of a physical process, it is generally better to look for a method that would achieve a physically correct simulation of that process, and thus eliminate the cause Thus, if both of the schemes chosen for the solution of of the difficulty, One method this kind for dealing with subsystems (7 . 2) and (7. 3) arc stable, the combined the two-grid-interval wave problem has been suggested scheme constructed by the splitting method will also be and used by Arakawa (1972). It consists of an intermittent ~table. This conclusion cun be generalized for an arbi· use of uncentered space differencing within the gravity trary system of equations and number of subsystems. wave terms. performed alternately on opposite sides of When applying the splitting method, we do not neces the central point. sarily have to use equal time steps for each of the subsys Mesinger (1973) showed how two-grid-interval wave tems. This may well be the main advantage of the noise could be prevented in some cases even by using cen- 60 THE GRAVITY AND GRAVITY-INERTIA WAVE EQUATIONS tered differencing; this method will be outlined briefly y here. We consider the system of linearized gravity wave equations iJu + i)h = iJv + i)h = 0 a, g ax o. iJt g iJy . (8 .. 1) x iJh iJt + HP· 11 = 0. Consider any two neighbouring height points for example within the lattice (E). A height perturbation at one of these points cannot affect the other point because there Figure 8. I Stencil used to denote the height and velocity grid is no velocity point in between ; this velocity is needed point values surrounding a height point. to cause a height change at the other point through the divergence term in the continuity equation. To circum vent this difficulty we can introduce auxiliary velocity points midway between the height points. Velocity space-averaging, that is components at these auxiliary points can be assumed equal to an average of velocities at the two neighbouring velocity u'" = Y2 Ii/'+ ;Y')". v"=-, y'2 (- -x'u +v-X ')". 2 points at the beginning of a time step, and the acceleratjon 2 contributions can then be evaluated and added to these An overbar denotes a two-point average taken along the initial values to obtain components at the middle or at direction indicated following the bar sign. Acceleration the end of the time step. Only the velocity components contributions are added to these initial values to obtain and accelerations along directions joining the two height points are needed, and these accelerations can be calcu values at the end of the time step, lated using the height values at the two points. The resulting velocity components can then be used for a more accurate calculation of the divergence term in the conti nuity equation. In this way schemes are obtained in · I ( h" 6 h"+I) which a height perturbation at a single grid point is v'" + I = V" - gt1t 2 6y • + y • • propagated by gravity waves to all the other height grid points. Therefore there can be no grid-splitting and two The velocity divergence in the continuity equation can grid-interval noise in the height field. Since a velocity now be approximated by perturbation can propagate as a gravity wave only by exciting height perturbations, the procedure will prevent false two-grid-interval noise in all the variables. We shall illustrate this procedure using the implicit giving equal weight to all eight directions of the lattice. scheme. (6.1). The velocity components at regular In this way the implicit approximation to the continuity velocity points are computed in the same way as before, so equation may be obtained as the first two equations of that system remain unchanged. To calculate the velocity divergence in the continuity n+l " '6 6 )" equation we define auxiliary velocity points midway h = h - H'1t I xu + yv + between the neighbouring height points, as shown by the (8.2) circled numbers 5, 6, 7 and 8 in Fig. 8 . I. Using the 1 system x', y' shown in this figure, components u' are +!gH/'1t)2 ( P~h"+ P~h"+ ) . 4 needed at points 5 and 7, and components v' at points 6 and 8. At the beginning of the time step '1t these compo Here the velocity components at level n + I have already nents are obtained by been eliminated using the first two of Eqs. (6. 1), and CHAPTER 4 61 9. Time noise and time filtering (8.3) ln addition to the appearance of spurious short-wave noise in space, spurious short-wave noise in time, that is, high frequency noise can appear in numerical models. This is again a finite difference approximation to p 2h, but One mechanism causing this when the leapfrog scheme now it is calculated using the height values of nine neigh is used for nonlinear equations is the separation of bouring height points. solutions at alternate time steps, generating two-grid Comparing this scheme with the standard implicit interval noise in time. Such separation is illustrated scheme of Section 6, the only modification is that this in a paper by Lilly (1965 , p. 23). nine-point Laplacian has replaced the five-point Lapla High frequency noise appears in atmospheric models cian (6. 9) in the continuity equation. This allows also as a result of difficulties in observing initial condi propagation of gravity waves between all height points tions representative of the large scale atmospheric of the grid, thus admitting no false space noise in the motions. The observed initial conditions contain instru height field. A more detailed analysis of the properties mental errors, are influenced by meso and small scale of the scheme can be found in Mesinger (1973). The motions, are not known at grid points of the model, and, modification, for example, has no effect on the uncondi tional stability of the implicit scheme; however, instead finally, are completely absent over relatively large areas of being neutral for all waves, the scheme now damps of the globe. As a result of all of these factors, if initial shorter waves to some extent. The modified scheme grid point values are interpolated directly from the has a smaller truncation error than the unmodified observed data the numerical forecasts will contain scheme . spurious gravity waves of unrealistically large amplitudes. Analogous modifications of some other schemes have In the early successful integrations of the primitive been discussed in papers by Janjic (1974) and Mesinger equations these problems were partially by-passed by (1974). All of these papers show that the modified obtaining the initial winds from the initial geopotential schemes are strikingly superior in the case of a stationary fields as a solution of the balance equation - the equation circular vortex, forced at a single height grid point. A obtained by assuming the initial velocity divergence and note by Mesinger and Janjic (1974) provides, furt1'er its time derivative to be equal to zero. Initial conditions more, a dramatic illustration of the advantages of the prepared in this way (e.g. Haltiner, 1971) prevent exces proposed method in the case of a limited area model, sive high-frequency gravity wave noise. requiring lateral boundary conditions to be prescribed. In a 5-level model using the unmodified forward-back It is now generally accepted that this is not the best ward scheme, intense short-wave noise was generated way of preparing the initial conditions. First, the wind at the boundaries of the region, a problem noticed also data are not used when solving the balance equation, and by earlier investigators (e.g. Miller et al., 1972 ; Krishna some information is lost. It has also been shown (e.g. murti et al., 1973). With the scheme modified along Phillips, 1960b ; Winninghoff, 1968) that the presence these lines, however, there were no difficulties due to the of a realistic initial divergent wind field should have a prescribed boundary conditions, even with no lateral beneficial effect on the forecast. Finally, an increasing diffusion in the model. fraction of the observations are now continuous, and in time not obtained at specific times. The methods being It is imponant to be aware t hat this method is not used to extract the maximum information from this attempting to improve the calculation of short gravity type of data rely more on running a prediction model waves of wave lengths close to two grid intervals. At to adjust the data in space and time (e.g. Bengtsson, this scale the finite difference representation is very poor, 1975). In such an integration relatively intense high and significant improvements in accuracy can hardly frequency noise is generated. be expected. The problem is that gravity waves, with longer wave lengths can propagate independently on The first of the mechanisms mentioned here, separation individual (C) type subgrids, and thus erroneously of solutions at alternate time steps, has to be suppressed appear to have wave lengths close to two grid intervals. in some way - otherwise it may lead to a complete Thus, we are confronted with a kind of aliasing error. breakdown of the integration. One method that is used The proposed method enables these waves to appear for this purpose is an intermittent step made with a with wave lengths close to their physical value instead two level scheme. A weakness of such a procedure is in the noise region with wave lengths close to two grid that the choice of the solution that is eliminated is arbi intervals. trary. 62 TIIE GRAVITY AND GRAVITY-INERTIA WAVE F.Q UATIONS Experience shows that the noise generated by assi been changed prior to changing U (t). It is then appro- milation of the observed data typically dies out to an priate to consider the filter acceptable level in about 24 hours of simulated time I u(t >= u The situation is different when we arc now considering -· aod llilbcrt, D ., l'>SJ. Mt·tluuls of Mu1h,·111e11icul l'llysi,·s, the gravity-inertia wave terms, governing the gcoMrophic Vol. I. New York, lntcrscicm:c. 562 pp. adjustment process. This process is a resul t of the dis Cullen, M . J . P. , 1974. lntegrutiun u/ thl' primitive eq11atiuns on a persion of high frequency waves. Use of a frequency sphere using the finite ell'ml'nt method. Quart. J. Roy. Meteor. selective dissipative scheme will make these high fre Soc., 100, 555-562. quency waves damp out at a faster rate, and thus acce Deardorff, J , W., 1974. Three-dimensional numerical study of the lerate the adjustment process, although the actual phy height and mean structure of a heated planetary boundary layer. sical process is dispersive rather than dissipative. This Boundary-Layer Meteor.• 7, 81-l06. gives an effect much the same as that of time filtering. Egger, J ., 1971. Mindestgriisse vun Gebirgen und Konvektions Therefore, if we are only interested in the final result of the gebietl'n, die in dm Modl'Jll'n du numuischen Vurhersage beriick geostrophic adjustment process, dissipation in the sichtigt werden kiinnen . Beitr. Phys. At':'1os., 44, 245-271. gravity-inertia wave terms may be helpful, especially Eliassen, A., 1956. 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The von Neumann condition allows an exponential, Each wave component is, thus, advected at a constant but no faster, growth of the solution. This, of course, velocity c along the x axis with no change in is needed to analyze cases when the true solution grows amplitude. exponentially. However, when we know that the true Returning to the von Neumann method, we now look solution does not grow, as in our example (6 . 7), it is for an analogous solution of the finite difference equation customary to replace ( 6. 12} by a sufficient condition (6. 1). Into this equation we substitute a solution of the form (6.13) (6.8) This condition is much less genero us than that required by the original defin ition of stability. Returning to our Here U