When Phillip Thompson began to write the first widely read textbook on numerical weather prediction (NWP), the subject was in its infancy, even though an earlier book, Weather Prediction by Numerical Process by L. F. Richardson (1922), presaged what was to come later in the century after the advent of electronic computers. The availability of computers increased greatly in the 1960s, and universities began to offer courses in atmospheric modeling, but most modelers had to also be model developers because the untested codes had many errors, the numerical schemes for solving the equations and the physical-process representations were not well tested and understood, lateralboundary conditions for limited-area models produced noisy solutions, and codes for defining the initial conditions needed to be further developed. These early practitioners learned the basics of atmospheric modeling from each other, through journal articles, in seminars and conferences, and from early courses on the subject. During the last 30 years of the twentieth century, graduate-level courses in atmospheric modeling flourished at many universities. And because computer modeling of the atmosphere was increasingly becoming an important tool in research and operational weather prediction, these courses were typically filled. Nevertheless, atmospheric modeling was still somewhat of a specialty, and models were not very accessible beyond national centers and a few research universities. Smagorinsky (1983), Thompson (1983), Shuman (1989), Persson (2005), Lynch (2007), and Harper (2008) should be consulted for additional history on atmospheric modeling.

This textbook provides a general introduction to atmospheric modeling for those using models for either operational forecasting or research. It is motivated by the fact that all those who use such models should be aware of their strengths and limitations. Unlike the many other books that specialize in particular aspects of atmospheric modeling, the aim here is to offer a general treatment of the subject that can be used for self study or in conjunction with a course on the subject. Even though there is considerable space devoted here to numerical methods, this is not intended to be the major focus. As the reader will see, there are many other subjects associated with the modeling process that must be understood well in order for models to be used effectively for research or operations. For those who need more information on particular topics, each chapter includes references to specialized resources. It is assumed that the reader has a Bachelors Degree in atmospheric sciences, with mathematics through differential equations.

Abbreviations or acronyms, as well as symbols, will be defined in the text the first time that they appear, and for future reference they are also defined in the lists that appear before Chapter 1. Even though the student should focus on concepts rather than jargon, a technical vocabulary is still necessary in order to discuss these subjects. Thus, commonly used, important terms will appear in italics the first time, in order to identify them as worth remembering.

Backround

The term atmospheric predictability may be defined as the time required for solutions from two models that are initialized with slightly different initial conditions to diverge to the point where the objective (e.g., RMS) difference is the same as that between two randomly chosen observed states of the atmosphere. In the practical context of a forecast, the no-skill limit that defines the predictability may be the forecast lead time when the model-simulated state has no greater resemblance to the observed state of the atmosphere than does a reference forecast based on persistence or climatology. Many of the other chapters in this text address the various components of the modeling process that limit predictability, from dataassimilation systems to numerical methods to physical-process parameterizations, as well as metrics for quantifying it. This chapter will review the general concept of theoretical and practical limits to forecasting skill.

Model error and initial-condition error

As shown in the previous chapter, error that limits predictability originates in both the model and the initial conditions. Refer to Section 7.3 for more information, especially about the various sources of error associated with the model. Often the concept of predictability is discussed in the context of the system's response to infinitesimally small perturbations in the model initial conditions. This predictability is an inherent property of the fluid system and not of the model. Indeed, it is sometimes assumed in this hypothetical discussion that the model is perfect.

Many NWP courses involve the coding by students of one- or two-dimensional shallow-fluid models, and the use of these models in experiments to evaluate the influence of different numerical methods on model solutions (described in Chapter 3). This allows students to become familiar with the structural components of models, to gain experience in debugging model code, and to conduct experiments to confirm concepts discussed in the text.

This appendix suggests an overall framework for coding the shallow-fluid equations that are described in Section 2.3.3, as well as some experiments that can be part of a laboratory component of an NWP course. Because the specific programming language used will determine the details of the model code, only a high-level outline will be provided here. The best approach is to start with the development of a one-dimensional model. Figure A.1 shows a schematic of the procedure for solving such a system, using an advection equation as a simple example. The abscissa is the space dimension and the ordinate is time. A predictive equation would of course be required for u, unless a constant mean speed is employed.

The parameterization problem involves algorithmically or statistically relating the effects of physical processes that cannot be represented directly in a model to variables that are included. Physical processes are parameterized for a few reasons.

• The small scales involved make it too computationally expensive to represent a process directly.

• The complexity of a process makes it too computationally expensive to represent directly.

• There is insufficient knowledge about how a process works to explicitly represent it mathematically.

The representation of atmospheric processes in models takes place within the dynamical core as well as through the so-called model “physics”. The dynamic processes include the propagation of various types of waves (e.g., advective, Rossby, inertia–gravity). Even though the physics processes are parameterized to a large degree, their correct rendering by a model is nevertheless essential for the prediction of virtually all of the dependent variables. The parameterized processes that are discussed in this chapter include cumulus convection, cloud microphysics, turbulence, and radiation. Land-surface processes are also parameterized because they occur on too small a scale to be represented directly, but they are discussed separately in Chapter 5.

Even though parameterizations are typically developed and discussed independently from each other, and from the dynamical core, this is artificial and should be avoided. This is because parameterizations do interact, and the realism of this interaction determines the accuracy of the model. For example, the parameterized spectral solar radiation represents an energy flux at the land surface, and the land-surface parameterization partitions some of it to the sensible heating of the ground.

As we have seen in previous chapters, there is a variety of generally unavoidable sources of model error, including

• initial conditions,

• lateral-boundary conditions for LAMs,

• land/water-surface conditions,

• numerical approximations used in the dynamical core, and

• parameterizations of physical processes.

Each of these input data sets or modeling approaches introduces some error in the modeling process, and ensemble prediction involves performing parallel forecasts or simulations using different arbitrary choices for the above imperfect data or methods. The objective of defining the different conditions for each model integration is to sample the uncertainty space associated with the modeling process in order to define how this uncertainty projects onto the uncertainty in the forecasts. As a preliminary example of the sensitivity of model forecasts to the above factors, Fig. 7.1 illustrates an ensemble of 5-day track predictions for hurricane Katrina in 2005. The forecasts are based on the ECMWF ensemble prediction system. The tracks are strongly dependent on the specific errors in the input observations as well as the model configurations employed.

An ensemble of forecasts is more useful than an individual, deterministic forecast for the following reasons.

• The mean of the ensemble of forecasts is generally more accurate than the forecast from an individual ensemble member, when the statistics are computed over a number of forecasts.

• The difference (spread, variance) among the ensemble members can be an indication of the flow-dependent quantitative uncertainty in the ensemble-mean forecast, given a proper calibration.

• […]

Background

The expression Computational Fluid Dynamics (CFD) modeling comes from engineering, and refers to methods that can be used for the simulation of very-fine scales of motion. The terminology is confusing in the context that weather and climate modeling also involves the use of computational methods to solve the dynamic equations for a fluid. When the term CFD modeling is used in its conventional way in the atmospheric sciences, it refers to the simulation of motions that can synonymously be referred to as occurring on the sub-mesogamma scale, the microscale, or the turbulence scale.

Because we are revisiting the concept of the scales of motion that are represented by a model solution, a reminder of the pertinent discussions in Chapter 3 is appropriate. There is a tendency to think of the 2Δx length scale as the resolution limit of a model, although it has been shown by Skamarock (2004) (e.g., Fig. 3.36) and others that spatial filters associated with the finite-differencing scheme and the explicit diffusion in a model can cause the effective resolution to be quite different from this limit. Motions unresolved by the model can generally be referred to as the subfilter-scale (SFS).

Types of CFD models

There are three general categories of CFD models, although there are myriad methods for solving the equations, just as with larger-scale models.

Background

This chapter describes methods for (1) the graphical display and interpretation of model output, and observations; (2) the calculation of derived variables from model output, which can help in the analysis of processes; and (3) the mathematical processing of model output, which can reveal properties and patterns that are not apparent from the dependent variables themselves. The comparison of the model output with observations is a type of analysis of course, but Chapter 9 on model verification is devoted to this subject. Also, the application of post-processing algorithms, for example to remove systematic error, is a special type of mathematical processing of the output, and this subject is treated in Chapter 13.

Graphical methods for displaying and interpreting model output and observations

Much of the material in this section is covered in courses on meteorological analysis; however, it is provided here because many students of NWP have not had such a course available to them. More in-depth material can be found in texts such as Saucier (1955) and Bluestein (1992a,b).

There have been so many creative ways of displaying model output, and comparing it with observations, that it is impossible to present a thorough treatment here. Nevertheless, some examples will be provided and the student is encouraged to review the literature and become familiar with typical techniques (see chapter Problems 1 and 3). This subject is important because successfully publishing research, whether it is model-based or not, depends on displaying the results in an easily and quickly understood format.