The GRP algorithm was developed in Part I and tested against a variety of analytical examples. The next challenge is to adapt it to the needs of “Scientific Computing,” that is, to implement it in the simulation of problems of physical significance. This is a formidable task for any numerical algorithm. The problems encountered in applications usually are multidimensional and involve complex geometries and additional physical phenomena. In this chapter we first describe in more detail some of these problems, as a general background for the following chapters. We then proceed to the main goal of the chapter, namely, the upgrading of the GRP algorithm (which is essentially one dimensional) to a two-dimensional, second-order scheme. To this end we recall (Section 7.2) Strang's method of “operator splitting” and then (Section 7.3) apply it to the (planar) fluid-dynamical case. Although the method can be further extended to the three-dimensional case, we confine our presentation to the two-dimensional setup, which serves in our numerical examples (Chapters 8 and 10).

General Discussion

In Part I we studied the mathematical basis of the GRP method. We also considered a variety of numerical examples for which analytic (or at least asymptotic) solutions were available. This provided us with some measure of the accuracy of the computational results, and helped identifiy potential difficulties (such as the resolution of contact discontinuities).

For the analysis and depiction of meteorological data it is useful and customary to map the surface of the earth onto a plane. Therefore, it is advisable for purposes of numerical weather prediction to formulate and evaluate the atmospheric equations in stereographic coordinates. Such a map projection should represent the spherical surface as accurately as possible, but obviously some features will be lost. It is extremely important to preserve the angle between intersecting curves such as the right angle between latitude circles and meridians. Maps possessing this desirable and valuable property are called conformal. If distances were preserved by mapping the sphere onto the projection plane, the map would be called isometric. Mapping from the sphere to the stereographic plane is conformal but not isometric. In order to remove this deficiency to a tolerable level, a scale factor m, also called the image scale, will be introduced.

The stereographic projection

We will now describe the sphere-to-plane mapping by introducing a projection plane that is parallel to the equatorial plane. On the projection plane we may construct a Cartesian (x, y, z)-coordinate system with a square rectangular grid. The stereographic Cartesian coordinate system differs from the regular Cartesian coordinates since the metric fundamental quantities are not gij = δij. In fact, they still contain the Gaussian curvature 1/r of the spherical coordinate system, showing that in reality we have not left the sphere.

The GRP method was developed (Chapter 5) for compressible, unsteady flow in a duct of varying cross section. In the case of a planar two-dimensional duct, the quasi-1-D formulation is taken to be a reasonable approximation of the actual (2-D) flow. In this chapter we study a duct flow where an incident wave interacts with a short converging segment, producing interesting wave structures. An illustrative case is that of a rarefaction wave propagating through a “converging corridor,” producing (at later times) a complex “reflected” wave pattern. Such a case is studied (numerically) in this chapter, using (a) the quasi-1-D approach of Chapter 5 and (b) the full two-dimensional computation as described in Section 8.3 (with the duct contour taken as a stationary boundary). The comparison between the two computations reveals some bounds of validity of the quasi-1-D approximation. We conclude the chapter by listing (Remark 10.1) several articles describing the application of the GRP to diverse fluid-dynamical problems, including well-known test cases, shock wave reflection phenomena compared to experimental observation, and even a case where a “moving boundary” experiment is favorably compared to the corresponding GRP solution.

Consider a centered rarefaction wave that propagates in a planar duct comprising two long segments of uniform cross-sectional area joined by a smooth converging nozzle.

In the previous chapter we introduced the vertical coordinate η to handle orographic effects in mesoscale models. In the synoptic-scale models we are going to replace the height coordinate z which extends to infinity by a generalized vertical coordinate ξ. The introduction of ξ is motivated by the fact that we cannot integrate the predictive equations using z as a vertical coordinate to infinitely large heights. Replacing z by the atmospheric pressure p, for example, results in a finite range of the vertical coordinate. We will see that another advantage of the (x, y, p)-coordinate system is that the continuity equation is time-independent. There are other specific coordinate systems that we are going to discuss. Therefore, it seems of advantage to first set up the atmospheric equations in terms of the unspecified generalized vertical coordinate ξ. Later we will specify ξ as desired. We wish to point out that the introduction of the generalized coordinate is of advantage only if the hydrostatic equation is a part of the atmospheric system.

We will briefly state the consequences of the transformation from the stereographic (x, y, z)-coordinate system to the stereographic (x, y, ξ)-coordinate system, which henceforth will be called the ξ system.

Introduction

So far we have treated the so-called primitive equations of baroclinic systems which, in addition to typical meteorological effects, automatically include horizontally propagating sound waves as well as external and internal gravity waves. These waves produce high-frequency oscillations in the numerical solutions of the baroclinic systems, which are of no interest to the meteorologist. Thus, the tendencies of the various field variables are representative only of small time intervals of the order of minutes while the predicted weather tendencies should be representative of much longer time intervals.

In order to obtain meteorologically significant tendencies we are going to eliminate the meteorological noise from the primitive equations b y modifying the predictive system so that a longer time step in the numerical solution becomes possible. We recall that the vertically propagating sound waves are no longer a part of the solution since they are removed by the hydrostatic approximation. The noise filtering is accomplished by a diagnostic coupling of the horizontal wind field and the mass field while in reality at a given time these fields are independent of each other. The simplest coupling of the wind and mass field is the geostrophic wind relation. The mathematical systems resulting from the artificial inclusion of filter conditions are called quasi-geostrophic systems or, more generally, filtered systems. For such systems at the initial time t0 = 0 only one variable, usually the geopotential, is specified without any restriction.