Efficient and fast numerical methods to compute fluid flows in the geophysical β plane

We consider a fluid flow in a rotating sphere with an unit radius. The flow is incompressible and inviscid, and covers the sphere with a constant density. This kind of flow is called a geophysical flow, since it is one of the simplest models of atmospheric flows in the earth. In practical study of geophysical flows, we are sometimes interested in a local flow in the neighborhood of a certain point in the sphere. In that case, we consider flows in a plane which is tangent to the point as an approximation model. The plane is called the geophysical β plane. In the present article, we introduce an equation which describes a motion in the β plane. And a numerical procedure to compute the equation is formulated. Furthermore, we suggest an efficient technique to compute it fast and accurately by using a fast algorithm and a parallelization based on the idea of Domain Decomposition. As an example of its application, we compute a two-dimensional flow problem in the β plane and investigate the effectiveness of the fast method and the effect of rotation on the evolution numerically. Numerical computations of the geophysical flows play an important role in the atmospheric research, such as the weather forecast and the investigation of the environmental issues. In spite of its importance, it is not easy to obtain useful and practical results since it costs too much to compute these problems for sufficiently fine resolutions. That is why a fast and accurate numerical method is required. The purpose of the study is to give an efficient numerical method to compute such flows and to show its effectiveness by applying it to some fluid problem. In the next section, we suggest the fast numerical method: We consider the equation of the flows in the β plane, whose detailed definition and formulation is explained. Our numerical method called the point potential vortex method is introduced. Then, some techniques to compute it fast and accurately are appearing. In the third section, we show some results of the numerical computation of a fluid flow in the β plane: (1) effectiveness of the fast method and (2) investigation of the influence of rotation on the evolution of the flow. The last section is conclusions.