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Über dieses Buch

This textbook introduces step by step the basic numerical methods to solve the equations governing the motion of the atmosphere and ocean, and describes how to develop a set of corresponding instructions for the computer as part of a code. Today's computers are powerful enough to allow 7-day forecasts within hours, and modern teaching of the subject requires a combination of theoretical and computational approaches.

The presentation is aimed at beginning graduate students intending to become forecasters or researchers, that is, users of existing models or model developers. However, model developers must be well versed in the underlying physics as well as in numerical methods. Thus, while some of the topics discussed in the modeling of the atmosphere and ocean are more advanced, the book ensures that the gap between those scientists who analyze results from model simulations and observations and those who work with the inner works of the model does not widen further.

In this spirit, the course presents methods whereby important balance equations in oceanography and meteorology, namely the advection-diffusion equation and the shallow water equations on a rotating Earth, can be solved by numerical means with little prior knowledge. The numerical focus is on the finite-difference (FD) methods, and although more powerful methods exist, the simplicity of FD makes it ideal as a pedagogical introduction to the subject. The book also includes suitable exercises and computer problems.



Chapter 1. Governing Equations and Approximations

The purpose of this chapter is to familiarize the reader with the equations governing the motion of atmospheres and the oceans. It introduces the well-known Boussinesq and hydrostatic approximations, and outlines the way the shallow water equations and the quasi-geostrophic equations follow from these approximations. The importance of boundary conditions is also emphasized. Readers who are familiar with these equations, approximations, and conditions may skip this chapter without loss of continuity.

Lars Petter Røed

Chapter 2. Preliminaries

In this chapter, we present a general analysis of partial differential equations (henceforth, PDEs). These are of importance here because the governing equations of the atmosphere and the ocean, including the hierarchy of simpler equations outlined in Chap. 1 , belong to this class. We shall see that a PDE has a different character depending on the physics it describes. As we shall show in Chaps. 4 – 6 , this has to be taken into account when deciding which numerical method should be used to solve the PDE.

Lars Petter Røed

Chapter 3. Time Marching Problems

The purpose of this chapter is to present some properties inherent in the governing equations listed in Sect. 1.1 . Most problems in the geophysical sciences, including atmospheres, oceans, seas, and lakes, involve solving so-called time marching problems. Typically, the state of the fluid in question is known at one specific time. As postulated by Bjerknes (Meteor Z 21:1–7, 1904) in his comments on weather forecasting (see quote in the preface), the aim is then to predict the state of the fluid at a later time. This is done by solving the governing equations listed in Sect. 1.1 . Such problems are known in mathematics as initial value problems.

Lars Petter Røed

Chapter 4. Diffusion Problem

In this chapter, we discuss the fundamentals of how to cast a PDE into finite difference form. More specifically, the reader will learn how to discretize the diffusion equation, and learn why some discretizations work and some do not. This will be the opportunity to introduce concepts such as numerical stability, convergence, and consistency. It will be explained how to check whether a discretization is stable and consistent, and the reader will learn about explicit and implicit schemes, the rudiments of elliptic solvers, and the concept of numerical dissipation or artificial damping inherent in our discretizations.

Lars Petter Røed

Chapter 5. Advection Problem

The purpose of this chapter is to study potentially useful schemes and discretizations of the linear advection equation. We discuss various stable and consistent schemes such as the leap-frog scheme, the upstream scheme (or upwind scheme), the Lax–Wendroff scheme, and the semi-Lagrangian scheme. The conditions under which they are stable are also discussed, along with ways to avoid the initial problem in centered-in-time schemes. We consider problems like numerical dispersion, numerical diffusion, and computational modes, including ways to minimize their effects, and we discuss flux corrective schemes, which improve the ubiquitous numerical diffusion inherent in low-order schemes like the upstream scheme. Finally, we describe the Courant–Friedrich–Levy (CFL) condition and its physical interpretation.

Lars Petter Røed

Chapter 6. Shallow Water Problem

The purpose of this chapter is to learn how to solve a simple subset of the momentum equations ( 1.1 ) numerically. The focus is on the shallow water equations, and in particular their depth integrated versions ( 1.33 ) and ( 1.34 ). Despite their simplicity, the shallow water equations include the essence of the momentum equations. For instance, we retain the possibility of a geostrophic balance and the impact of nonlinear terms on the dynamics.

Lars Petter Røed

Chapter 7. Open Boundary Conditions and Nesting Techniques

The aim of this chapter is to discuss open boundaries and some of the techniques used to deal with them. An open boundary is defined as a computational boundary at which disturbances originating in the interior of the computational domain are allowed to leave without disturbing or deteriorating the interior solution (Røed and Cooper, Advanced physical oceanographic numerical modelling. D. Reidel Publishing Co, Dordrecht, 1986). Even though the governing equations are still valid at these boundaries, they nonetheless constitute a boundary in a numerical sense. Hence, we focus on how to construct conditions, or open boundary conditions (OBCs), in such a way that disturbances originating in the interior of the computational domain are indeed allowed to leave without disturbing or deteriorating the interior solution.

Lars Petter Røed

Chapter 8. Generalized Vertical Coordinates

So far, we have only used the Cartesian geopotential coordinate system consisting of three orthogonal spatial coordinates x, y, z. Relaxing the orthogonality between the vertical coordinate z and the two horizontal coordinates x, y can make it much easier to analyze phenomena in atmospheres and oceans, and devise compelling models of them. The development of such non-orthogonal coordinate systems remains at the forefront of research in numerical modeling. The purpose of this chapter is, therefore, to present the salient issues relating to these generalized vertical coordinates.

Lars Petter Røed

Chapter 9. Two-Dimensional Problems

This chapter investigates the effect of including more than one dimension in space. In particular, it discusses the impact on numerical stability and the stability criterion. Extension to three dimensions is then straightforward.

Lars Petter Røed

Chapter 10. Advanced Topics

The purpose of this chapter is to use the knowledge acquired in the previous chapters to learn about some slightly more advanced topics. For instance, we sketch ways to construct schemes of higher order accuracy, and ways to solve problems when advection and diffusion are equally important. Furthermore, we consider ways to treat nonlinearities numerically, and ask whether they harbor implications for instability. Since two-way nesting is becoming more and more popular, we also say a few words about smoothing and filtering, and give a detailed presentation of two-way nesting itself. Since the spectral method mentioned in the preface is rather common in global atmospheric models, the chapter ends with a brief description of a one-dimensional application of this method.

Lars Petter Røed

Chapter 11. Quality Assurance Procedures

The aim here is to summarize a set of sound procedures for establishing what is referred to below as a “good” model. The text is based on earlier reports by the author on the subject, in particular McClimans et al. (1992) and Røed (1993). For more extensive reading on the subject, the reader is referred to the in-depth analysis documented in the GESAMP report (GESAMP 1991), or the review Lynch and Davies 1995.

Lars Petter Røed


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