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# 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

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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 parameterizations | 3 |

Numerical experimentation | 5 |

CHAPTER 2 Basic Concepts7

Finite-difference quotients | 7 |

Difference quotients of higher accuracy | 1 |

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 |

A Survey of Time-Differencing | |

Schemes for the Oscillation and |

CHAPTER 3 Decay Equations43

Introduction | 43 |

Non-iterative schemes | 43 |

47 |

Explicit schemes ( )

49 | |

Iterative schemes | 51 |

Finite-difference schemes applied to the oscillation equation | 52 |

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 equation | 69 |

Damped oscillations | 72 |

An Introduction to the General Circulation of the Atmosphere

Summary | 7 |

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has Fourth-Order Accuracy | 78 |

Problems | 83 |

A Proof that the Fourth-Order Runge-Kutta Scheme

A closer look at the advection |

CHAPTER 4 equation 85

Introduction | 85 |

Conservative finite-difference methods | 8 |

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

An Introduction to the General Circulation of the Atmosphere

Problems | 148 |

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CHAPTER 7 Making Waves149

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 |

Schemes for the one-dimensional |

CHAPTER 8 nonlinear shallow-water equations169

Properties of the continuous equations | 169 |

Space differeencing | 171 |

Summary | 178 |

Problems | 180 |

Vertical Differencing for Quasi-Static |

CHAPTER 9 Models 183

Introduction | 183 |

Choice of equation set | 183 |

General vertical coordinate | 184 |

The equation of motion and the HPGF | 188 |

189 | |

Discussion of particular vertical coordinate systems | 191 |

Vertical mass flux for a family of vertical coordinates

192 |

Height

196 |

Pressure

197 |

Log-pressure

197 | |

More on the HPGF in -coordinates | 200 |

The -coordinate

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The -coordinate | 202 |

Hybrid sigma-pressure coordinates

203 |

Potential temperature

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Entropy

206 |

Hybrid - coordinates

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Vertical staggering | 208 |

Conservation properties of vertically discrete models using -coordinates | 210 |

Summary of vertical coordinate systems

An Introduction to the General Circulation of the Atmosphere

Summary and conclusions | 221 |

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CHAPTER 10 Aliasing instability223

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 1 Finite Differences on t he Sphere261

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 Methods281

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 ix

CHAPTER 13 Boundary conditions and nested grids301

Introduction | 301 |

Inflow Boundaries | 301 |

Outflow boundaries | 308 |

Advection on nested grids | 314 |

Analysis of boundary conditions for the advection equation using | |

the energy method | 320 |

Physical and computational reflection of gravity waves at a wall | 323 |

Boundary conditions for the gravity wave equations with an advection term | 326 |

The energy method as a guide in choosing boundary conditions | |

for gravity waves | 327 |

Summary | 330 |

Problem | 330 |

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.”

31.2 | Fundamental physics, mathematical methods, and physical parameterizations |

An Introduction to Atmospheric Modeling

1.2Fundamental physics, mathematical methods, and physical parameterizations

(Parte **1** de 5)