An Introduction to Atmospheric Modeling [Colo. State Univ. Course, AT604]

362 Pages·2004·2.48 MB·english

by  D. Randall

An Introduction to
Atmospheric Modeling
Instructor: D. Randall
AT604
Department of Atmospheric Science
Colorado State University
Fall, 2004

i
Announcements
Subject: A practical introduction to numerical modeling of the atmo-
sphere.
Text: Class notes, available at the class website: http://kiwi.at-
mos.colostate.edu/group/dave/at604.html
Course grade: 1/4 on homework, 1/4 on each of two midterms (closed book,
in class), and 1/4 on final (closed book, in class) The final will
emphasize the latter part of the course, and will be held during
finals week.
Access to instructor: As you may know, I have posted office hours, but students in
this class are welcome to come to me with questions any time,
provided only that I am not actually busy with someone else.
Teaching assistant: We are fortunate to have Jonathan Vigh as a TA for this
course. He will grade the homework and will be available to
answer questions on a schedule which he will make known to
you. He may also organized other activities, which will be an-
nounced separately.
Computing: Some of the homework will involve writing computer pro-
grams, plotting results, etc.You can use any computing lan-
guage or plotting software you want. Although you are
certainly encouraged to ask questions about the homework,
neither I nor the TA will help with debugging your programs.
Auditing: Auditing is permitted, provided that you audit officially by fill-
ing out the appropriate form. Auditors are required to attend
class but are not required to hand in homeworks or take exams.
Keep in mind, however, that, like skiing or swimming or bicy-
cling, numerical modeling is learned largely by doing.
Schedule: Classes will be missed occasionally. A calendar will be distrib-
uted.
An Introduction to Atmospheric Modeling
ii
General References
Arakawa, A., 1988: Finite-difference methods in climate modeling. Physically-based
modelling and simulation of climate and climatic change - Part I, M. E. Schlesing-
er (ed.), 79-168.
Arfken, G., 1985: Mathematical methods for physicists. Academic Press, 985 pp.
Chang, J., 1977: General circulation models of the atmosphere. Meth. Comp. Phys., 17,
Academic Press, 337 pp.
Durran, D. R., 1999: Numerical methods for wave equations in geophysical fluid dynam-
ics. Springer, 465 pp.
Haltiner, G. J., and R. T. Williams, 1980: Numerical prediction and dynamic meteorology.
J. Wiley and Sons, 477 pp.
Kalnay, E., 2003: Atmospheric modeling, data assimilation, and predictability. Cam-
bridge Univ. Press, 341 pp.
Manabe, S., ed., 1985: Issues in atmospheric and oceanic modeling, Part A: Climate dy-
namics. Adv. in Geophys., 28, 591 pp.
Manabe, S., ed., 1985: Issues in atmospheric and oceanic modeling, Part B: Weather dy-
namics. Adv. in Geophys., 28, 432 pp.
Mesinger, F., and A. Arakawa, 1976: Numerical methods used in atmospheric models.
GARP Publ. Ser. No. 17, 64 pp.
Randall, D. A., Ed., 2000: General Circulation Model Development. Past, Present, and
Future. Academic Press, 807 pp.
Richtmeyer, R. D., and K. W. Morton, 1967: Difference methods for initial value
problems. Wiley Interscience Publishers, New York, 405 pp.
Washington, W. M., and C. L. Parkinson, 1986: An introduction to three-dimensional
climate modeling. University Science Books, Mill Valley, New York, 422 pp.
An Introduction to Atmospheric Modeling
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Preface
The purpose of this course is to provide an introduction to the methods used in
numerical modeling of the atmosphere. The ideas presented are relevant to both large-
scale and small-scale models.
Numerical modeling is one of several approaches to the study of the atmosphere.
The others are observational studies of the real atmosphere through field measurements
and remote sensing, laboratory studies, and theoretical studies. Each of these four
approaches has both strengths and weaknesses. In particular, both numerical modeling and
theory involve approximations. In theoretical work, the approximations often involve
extreme idealizations, e.g. a dry atmosphere on a beta plane, but on the other hand
solutions can sometimes be obtained in closed form with a pencil and paper. In numerical
modeling, less idealization is needed, but in most cases no closed form solution is
possible. Both theoreticians and numerical modelers make mistakes, from time to time, so
both types of work are subject to errors in the old-fashioned human sense.
Perhaps the most serious weakness of numerical modeling, as a research approach,
is that it is possible to run a numerical model built by someone else without having the
foggiest idea how the model works or what its limitations are. Unfortunately, this kind of
thing happens all the time, and the problem is becoming more serious in this era of
“community” models with large user groups. One of the purposes of this course is to make
it less likely that you, the students, will use a model without having any understanding of
it.
This introductory survey of numerical methods in the atmospheric sciences is
designed to be a practical, “how to” course, which also conveys sufficient understanding
so that after completing the course students are able to design numerical schemes with
useful properties, and to understand the properties of schemes that they may encounter out
there in the world.
The first version of these notes, put together in 1991, was heavily based on the
class notes developed by Prof. A. Arakawa at UCLA, as they existed in the early 1970s,
and this influence is still apparent in the current version, particularly in Chapters 2 and 3.
A lot of additional material has been incorporated, mainly reflecting developments in the
field since the 1970s. The explanations and problems have also been considerably revised
and updated.
The teaching assistants for this course have made major improvements in the
material and its presentation, in addition to their help with the homework and with
questions outside of class.
I have learned a lot by extending and refining these notes, and also through
questions and feedback from the students. The course has certainly benefitted
An Introduction to Atmospheric Modeling
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considerably from such student input.
Finally, Michelle McDaniel has spent countless hours patiently assisting in the
production of these notes. She created the formatting that you see, and organized the
notes into a “book.”
An Introduction to Atmospheric Modeling
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Preliminaries i
CHAPTER 1 Introduction 1
What is a model? .........................................................................................................1
Fundamental physics, mathematical methods, and physical parameterizations .........3
Numerical experimentation .........................................................................................5
CHAPTER 2 Basic Concepts 7
Finite-difference quotients ..........................................................................................7
Difference quotients of higher accuracy ...................................................................11
Extension to two dimensions ....................................................................................18
An example of a finite difference-approximation to a differential equation ............21
Accuracy and truncation error of a finite-difference scheme. ..................................24
Discretization error and convergence .......................................................................25
Interpolation and extrapolation .................................................................................28
Stability .....................................................................................................................29
The effects of increasing the number of grid points .................................................38
Summary ...................................................................................................................39
Problems ..............................................................................................................42
CHAPTER 3 A Survey of Time-Differencing
Schemes for the Oscillation and
Decay Equations 43
Introduction ...............................................................................................................43
Non-iterative schemes. ..............................................................................................43
Explicit schemes ( ).........................................................................................47
Implicit schemes..............................................................................................49
Iterative schemes .......................................................................................................51
Finite-difference schemes applied to the oscillation equation ..................................52
Non-iterative two-level schemes for the oscillation equation ...............................54
Iterative two-level schemes for the oscillation equation ......................................57
The leapfrog scheme for the oscillation equation ...............................................58
The second-order Adams Bashforth Scheme
(m=0, l=1) for the oscillation equation ..............................................................67
A survey of time differencing schemes for the oscillation equation ......................68
Finite-difference schemes for the decay equation ....................................................69
Damped oscillations ..................................................................................................72
Nonlinear damping ...................................................................................................72
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Summary ...................................................................................................................77
A Proof that the Fourth-Order Runge-Kutta Scheme
has Fourth-Order Accuracy........................................................................ 78
Problems ..............................................................................................................83
CHAPTER 4 A closer look at the advection
equation 85
Introduction ...............................................................................................................85
Conservative finite-difference methods ....................................................................88
Examples of schemes with centered space differencing ...........................................93
Computational dispersion .......................................................................................100
The effect s of fourth-order space differencing on the phase speed .......................107
Space-uncentered schemes ......................................................................................108
Hole filling ..............................................................................................................112
Flux-corrected transport ..........................................................................................113
Lagrangian schemes ................................................................................................116
Semi-Lagrangian schemes ......................................................................................118
Two-dimensional advection ....................................................................................120
Summary .................................................................................................................123
Problems ............................................................................................................123
CHAPTER 5 Boundary-value problems 127
Introduction .............................................................................................................127
Solution of one-dimensional boundary-value problems .........................................128
Jacobi relaxation .....................................................................................................130
Gauss-Seidel relaxation ..........................................................................................133
Over-relaxation .......................................................................................................134
The alternating-direction implicit method ..............................................................135
Multigrid methods ...................................................................................................135
Summary .................................................................................................................136
CHAPTER 6 Diffusion 141
Introduction .............................................................................................................141
A simple explicit scheme ........................................................................................143
An implicit scheme .................................................................................................144
The DuFort-Frankel scheme ...................................................................................146
Summary .................................................................................................................147
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Problems ............................................................................................................148
CHAPTER 7 Making Waves 149
The shallow-water equations ..................................................................................149
The wave equation ..................................................................................................150
Staggered grids ........................................................................................................152
Numerical simulation of geostrophic adjustment. as a guide to grid design ..........154
Time-differencing schemes for the shallow-water equations .................................160
Summary and conclusions ......................................................................................167
Problems ............................................................................................................168
CHAPTER 8 Schemes for the one-dimensional
nonlinear shallow-water equations 169
Properties of the continuous equations ...................................................................169
Space differeencing .................................................................................................171
Summary .................................................................................................................178
Problems ............................................................................................................180
CHAPTER 9 Vertical Differencing for Quasi-Static
Models 183
Introduction .............................................................................................................183
Choice of equation set .............................................................................................183
General vertical coordinate .....................................................................................184
The equation of motion and the HPGF.................................................................. 188
Vertical mass flux for a family of vertical coordinates ......................................189
Discussion of particular vertical coordinate systems ..............................................191
Height ...........................................................................................................192
Pressure ........................................................................................................196
Log-pressure .................................................................................................197
The σ-coordinate ..........................................................................................197
More on the HPGF in σ-coordinates .....................................................................200
Hybrid sigma-pressure coordinates .................................................................201
The η-coordinate ....................................................................................................202
Potential temperature .....................................................................................203
Entropy .........................................................................................................206
Hybrid σ-θ coordinates ................................................................................206
Summary of vertical coordinate systems ..........................................................206
Vertical staggering ..................................................................................................208
Conservation properties of vertically discrete models using -coordinates .............210
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Summary and conclusions ......................................................................................221
CHAPTER 10 Aliasing instability 223
Aliasing error ..........................................................................................................223
Advection by a variable, non-divergent current .....................................................227
Fjortoft’s Theorem ..................................................................................................236
Kinetic energy and enstrophy conservation in two-dimensional
non-divergent flow ............................................................................................241
Angular momentum conservation........................................................................... 251
Conservative schemes for the two-dimensional shallow water equations
with rotation ......................................................................................................252
The effects of time differencing on energy conservation .......................................257
Summary .................................................................................................................259
Problems ............................................................................................................260
CHAPTER 11 Finite Differences on the Sphere 261
Introduction .............................................................................................................261
Coordinate systems and map projections ................................................................262
Latitude-longitude grids and the “pole problem” ...................................................267
Kurihara’s grid ........................................................................................................273
The Wandering Electron Grid .................................................................................274
Spherical geodesic grids .........................................................................................274
Summary .................................................................................................................280
CHAPTER 12 Spectral Methods 281
Introduction .............................................................................................................281
Spectral methods on the sphere ...............................................................................289
The “equivalent grid resolution” of spectral models ..............................................294
Semi-implicit time differencing ..............................................................................295
Conservation properties and computational stability ..............................................296
Moisture advection ..................................................................................................296
Physical parameterizations ......................................................................................297
Summary .................................................................................................................297
Problems ............................................................................................................299
An Introduction to the General Circulation of the Atmosphere