An Introduction to Atmospheric Modeling, David Randall (2004)

An Introduction to Atmospheric Modeling, David Randall (2004)

(Parte 1 de 5)

An Introduction to Atmospheric Modeling

Instructor: D. Randall

AT604

Department of Atmospheric Science Colorado State University

Fall, 2004

i

An Introduction to Atmospheric Modeling Announcements

Subject :A practical introduction to numerical modeling of the atmo- sphere.

Text:Class notes, available at the class website: http://kiwi.atmos.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 announced separately.

Computing:Some of the homework will involve writing computer programs, plotting results, etc.You can use any computing language 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 filling 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 bicycling, numerical modeling is learned largely by doing.

Schedule:Classes will be missed occasionally. A calendar will be distributed.

An Introduction to Atmospheric Modeling

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 p.

Chang, J., 1977: General circulation models of the atmosphere. Meth. Comp. Phys., 17, Academic Press, 337 p.

Durran, D. R., 1999: Numerical methods for wave equations in geophysical fluid dynamics. Springer, 465 p.

Haltiner, G. J., and R. T. Williams, 1980: Numerical prediction and dynamic meteorology. J. Wiley and Sons, 477 p.

Kalnay, E., 2003: Atmospheric modeling, data assimilation, and predictability. Cambridge Univ. Press, 341 p.

Manabe, S., ed., 1985: Issues in atmospheric and oceanic modeling, Part A: Climate dynamics. Adv. in Geophys., 28, 591 p.

Manabe, S., ed., 1985: Issues in atmospheric and oceanic modeling, Part B: Weather dynamics. Adv. in Geophys., 28, 432 p.

Mesinger, F., and A. Arakawa, 1976: Numerical methods used in atmospheric models. GARP Publ. Ser. No. 17, 64 p.

Randall, D. A., Ed., 2000: General Circulation Model Development. Past, Present, and Future. Academic Press, 807 p.

Richtmeyer, R. D., and K. W. Morton, 1967: Difference methods for initial value problems. Wiley Interscience Publishers, New York, 405 p.

Washington, W. M., and C. L. Parkinson, 1986: An introduction to three-dimensional climate modeling. University Science Books, Mill Valley, New York, 422 p.

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An Introduction to Atmospheric Modeling

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 largescale 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 iv

An Introduction to Atmospheric Modeling 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.”

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An Introduction to the General Circulation of the Atmosphere

Preliminaries i

CHAPTER 1 Introduction 1

What is a model?1
Fundamental physics, mathematical methods, and physical parameterizations3
Numerical experimentation5

CHAPTER 2 Basic Concepts7

Finite-difference quotients7
Difference quotients of higher accuracy1
Extension to two dimensions18
An example of a finite difference-approximation to a differential equation21
Accuracy and truncation error of a finite-difference scheme24
Discretization error and convergence25
Interpolation and extrapolation28
Stability29
The effects of increasing the number of grid points38
Summary39
Problems42
A Survey of Time-Differencing
Schemes for the Oscillation and

CHAPTER 3 Decay Equations43

Introduction43
Non-iterative schemes43
47

Explicit schemes ( )

49
Iterative schemes51
Finite-difference schemes applied to the oscillation equation52

Implicit schemes

54

Non-iterative two-level schemes for the oscillation equation

57

Iterative two-level schemes for the oscillation equation

58
The second-order Adams Bashforth Scheme

The leapfrog scheme for the oscillation equation

67

(m=0, l=1) for the oscillation equation

68
Finite-difference schemes for the decay equation69
Damped oscillations72

An Introduction to the General Circulation of the Atmosphere

Summary7

vi

has Fourth-Order Accuracy78
Problems83

A Proof that the Fourth-Order Runge-Kutta Scheme

A closer look at the advection

CHAPTER 4 equation 85

Introduction85
Conservative finite-difference methods8
Examples of schemes with centered space differencing93
Computational dispersion100
The effect s of fourth-order space differencing on the phase speed107
Space-uncentered schemes108
Hole filling112
Flux-corrected transport113
Lagrangian schemes116
Semi-Lagrangian schemes118
Two-dimensional advection120
Summary123
Problems123

CHAPTER 5 Boundary-value problems 127

Introduction127
Solution of one-dimensional boundary-value problems128
Jacobi relaxation130
Gauss-Seidel relaxation133
Over-relaxation134
The alternating-direction implicit method135
Multigrid methods135
Summary136

CHAPTER 6 Diffusion 141

Introduction141
A simple explicit scheme143
An implicit scheme144
The DuFort-Frankel scheme146

Summary ................................................................................................................. 147

An Introduction to the General Circulation of the Atmosphere

Problems148

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CHAPTER 7 Making Waves149

The shallow-water equations149
The wave equation150
Staggered grids152
Numerical simulation of geostrophic adjustment. as a guide to grid design154
Time-differencing schemes for the shallow-water equations160
Summary and conclusions167
Problems168
Schemes for the one-dimensional

CHAPTER 8 nonlinear shallow-water equations169

Properties of the continuous equations169
Space differeencing171
Summary178
Problems180
Vertical Differencing for Quasi-Static

CHAPTER 9 Models 183

Introduction183
Choice of equation set183
General vertical coordinate184
The equation of motion and the HPGF188
189
Discussion of particular vertical coordinate systems191

Vertical mass flux for a family of vertical coordinates

192

Height

196

Pressure

197

Log-pressure

197
More on the HPGF in -coordinates200

The -coordinate

201
The -coordinate202

Hybrid sigma-pressure coordinates

203

Potential temperature

206

Entropy

206

Hybrid - coordinates

206
Vertical staggering208
Conservation properties of vertically discrete models using -coordinates210

Summary of vertical coordinate systems

An Introduction to the General Circulation of the Atmosphere

Summary and conclusions221

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CHAPTER 10 Aliasing instability223

Aliasing error223
Advection by a variable, non-divergent current227
Fjortoft’s Theorem236
Kinetic energy and enstrophy conservation in two-dimensional
non-divergent flow241
Angular momentum conservation251
Conservative schemes for the two-dimensional shallow water equations
with rotation252
The effects of time differencing on energy conservation257
Summary259
Problems260

CHAPTER 1 Finite Differences on t he Sphere261

Introduction261
Coordinate systems and map projections262
Latitude-longitude grids and the “pole problem”267
Kurihara’s grid273
The Wandering Electron Grid274
Spherical geodesic grids274
Summary280

CHAPTER 12 Spectral Methods281

Introduction281
Spectral methods on the sphere289
The “equivalent grid resolution” of spectral models294
Semi-implicit time differencing295
Conservation properties and computational stability296
Moisture advection296
Physical parameterizations297
Summary297

Problems ............................................................................................................ 299

An Introduction to the General Circulation of the Atmosphere ix

CHAPTER 13 Boundary conditions and nested grids301

Introduction301
Inflow Boundaries301
Outflow boundaries308
Advection on nested grids314
Analysis of boundary conditions for the advection equation using
the energy method320
Physical and computational reflection of gravity waves at a wall323
Boundary conditions for the gravity wave equations with an advection term326
The energy method as a guide in choosing boundary conditions
for gravity waves327
Summary330
Problem330

References and Bibliography 341

An Introduction to the General Circulation of the Atmosphere x

An Introduction to Atmospheric Modeling 1

Copyright 2004 David A. Randall CHAPTER 1 Introduction

1.1What is a model?

The atmospheric science community includes a large and energetic group of researchers who devise and carry out measurements in the atmosphere. This work involves instrument development, algorithm development, data collection, data reduction, and data analysis.

The data by themselves are just numbers. In order to make physical sense of the data, some sort of model is needed. This might be a qualitative conceptual model, or it might be an analytical theory, or it might take the form of a computer program.

Accordingly, a community of modelers is hard at work developing models, performing calculations, and analyzing the results by comparison with data. The models by themselves are just “stories” about the atmosphere. In making up these stories, however, modelers must strive to satisfy a very special and rather daunting requirement: The stories must be true, as far as we can tell; in other words, the models must be consistent with all of the relevant measurements.

A model essentially embodies a theory. A model (or a theory) provides a basis for making predictions about the outcomes of measurements. The disciplines of fluid dynamics, radiative transfer, atmospheric chemistry, and cloud microphysics all make use of models that are essentially direct applications of basic physical principles to phenomena that occur in the atmosphere. Many of these “elementary” models were developed under the banners of physics and chemistry, but some-- enough that we can be proud -- are products of the atmospheric science community. Elementary models tend to deal with microscale phenomena, (e.g. the evolution of individual cloud droplets suspended in or falling through the air, or the optical properties of ice crystals) so that their direct application to practical atmospheric problems is usually thwarted by the sheer size and complexity of the atmosphere.

A model that predicts the deterministic evolution of the atmosphere or some macroscopic portion of it can be called a “forecast model.” A forecast model could be, as the name suggests, a model that is used to conduct weather prediction, but there are other possibilities, e.g. it could be used to predict the deterministic evolution of an individual turbulent eddy. Forecast models can be tested against real data, documenting for example the observed development of a synoptic-scale system or the observed growth of an individual convective cloud, assuming of course that the requisite data can be collected.

We are often interested in computing the statistics of an atmospheric phenomenon, e.g. the statistics of the general circulation. It is now widely known that there are fundamental

Revised April 2, 2004 4:41 pm

2 Introduction

An Introduction to Atmospheric Modeling limits on the deterministic predictability of the atmosphere, due to sensitive dependence on initial conditions (e.g. Lorenz, 1969). For the global-scale circulation of the atmosphere, the limit of predictability is thought to be on the order of a few weeks, but for a cumulus-scale circulation it is on the order of a few minutes. For time scales longer than the deterministic limit of predictability for the system in question, only the statistics of the system can be predicted. These statistics can be generated by brute-force simulation, using a forecast model but pushing the forecast beyond the deterministic limit, and then computing statistics from the results. The obvious and most familiar example is simulation of the atmospheric general circulation (e.g. Smagorinski 1963). Additional examples are large eddy simulations of atmospheric turbulence (e.g. Moeng 1984), and simulations of the evolution of an ensemble of clouds using a space and time domains much larger than the space and time scales of individual clouds (e.g. Krueger 1988).

Forecast models are now also being used to make predictions of the time evolution of the statistics of the weather, far beyond the limit of deterministic predictability for individual weather systems. Examples are seasonal weather forecasts, which deal with the statistics of the weather rather than day-to-day variations of the weather and are now being produced by several operational centers; and climate change forecasts, which deal with the evolution of the climate over the coming decades and longer. In the case of seasonal forecasting, the predictability of the statistics of the atmospheric circulation beyond the two-week deterministic limit arises primarily from the predictability of the sea surface temperature, which has a much longer memory of its initial conditions than does the atmosphere.

In the case of climate change predictions, the time evolution of the statistics of the climate system are predictable to the extent that they are driven by predictable changes in some external forcing. For example, projected increases in greenhouse gas concentrations represent a time-varying external forcing whose effects on the time evolution of the statistics of the climate system may be predictable. Over the next several decades measurements will make it very clear to what extent these predictions are right or wrong. A more mundane example is the seasonal cycle of the atmospheric circulation, which represents the response of the statistics of the atmospheric general circulation to the movement of the Earth in its orbit; because the seasonal forcing is predictable many years in advance, the seasonal cycle of the statistics of the atmospheric circulation is also highly predictable, far beyond the twoweek limit of deterministic predictability for individual weather systems.

Some models predict statistics directly; the dependent variables are the statistics themselves, and there is no need to average the model results to generate statistics after the fact. For example, radiative transfer models describe the statistical behavior of extremely large numbers of photons. “Higher-order closure models” have been developed to simulate directly the statistics of small-scale atmospheric turbulence (e.g., Mellor and Yamada, 1974). Analogous models for direct simulation of the statistics of the large-scale circulation of the atmosphere may be possible (e.g., Green, 1970).

Finally, we also build highly idealized models that are not intended to provide quantitatively accurate or physically complete descriptions of natural phenomena, but rather to encapsulate our physical understanding of a complex phenomenon in the simplest and most compact possible form, as a kind of modeler’s haiku. For example, North (1975) discusses the application of this approach to climate modeling. Toy models are intended primarily as educational tools; the results that they produce can be compared with measurements only in qualitative or semi-quantitative ways.

This course deals with numerical methods that can be used with any of the model “types” discussed above, but for the most part we will be thinking of “forecast models.”

31.2Fundamental physics, mathematical methods, and physical parameterizations

An Introduction to Atmospheric Modeling

1.2Fundamental physics, mathematical methods, and physical parameterizations

(Parte 1 de 5)

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