The Solution of Nonlinear Meteorological Problems by the Method of Characteristics

Abstract

The Method of Characteristics. The method of characteristics is only one means of solving hyperbolic differential equations or systems of differential equations with real characteristics. Problems of this kind are also solved by related methods, such as simple wave theory, the Riemann method of integration, and the marching method. In addition, in a nonlinear system, shocks or jumps occur which cannot be studied by means of systems of equations with real characteristics, but must be studied by other methods. Loosely speaking, all methods mentioned above can be grouped under the single heading—the method of characteristics. If this definition is used, the method of characteristics is not new to meteorology. Richardson used the marching method correctly in the initial example in his monumental work on numerical weather prediction [15, pp. 6–10]. Rossby [16] used the Riemann method of integration to discuss the effect of a line source of planetary waves in a barotropic atmosphere. In all of its phases, however, the method of characteristics has been applied most often and most thoroughly to problems in gas dynamics and the flow of a shallow layer of fluid with a free surface. Riemann devised his method of integration to solve the problem of one-dimensional unsteady flow of a gas in a pipe, and most other methods have been developed with such problems in mind. Courant and Friedrichs [5] give a bibliography of this work.