Abstract

In the long-term numerical integration of the equations of motion required for medium-range weather forecasting, the study of the atmospheric general circulation, and other applications, the finite difference formulation of the nonlinear terms may give rise to a special type of instability. This difficulty was first noted in the meteorological literature by Phillips [4]. Phillips pointed out that a unique feature of this instability is that it cannot be suppressed by using shorter time steps. Arakawa [l] has made a very valuable contribution in showing that a numerical scheme which retains certain integral properties of the continuous equations eliminates nonlinear instability.* It must be pointed out that the formulation proposed by Arakawa does not guarantee accuracy. This is assured only if all the significant energy of the flow is in scales of motion that are large enough to ‘be adequately resolved by the numerical grid. A system free of nonlinear instability has the merit, however, that relatively minor truncation errors in grid-scale motions do not lead to large spurious increases in energy. The work of Arakawa [I] has been extended by Lilly [ 2 ] . Lilly has devised a method of numerically integrating the primitive equations which, except for time truncation, exactly conserves finite difference expressions for the kinetic energy of both the divergent and nondivergent components of the flow. This method is currently being used in an extension of investigations of the atmospheric general circulation initiated by Smagorinsky [ 5 ] . The present note is concerned with a generalization of the ideas of Arakawa [I] and Lilly [ 2 ] . 111 many applications of the techniques of numerical weather forecasting to other geophysical problems it may be necessary to use grids with irregularly spaced points. For example, it may be important t o join two different types of nets together, or the peculiar geometry of the region under consideration requires an irregular arrangement of points. Consider the following equation