GLOBAL ATMOSPHERIC RESEARCH PROGRAMME (GARP) WMO- ICSU Joint Organizing Committee ~.?·· '), .,, : . II • -:- i" ; ; . ~ . -· •.\. 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