A THEORY OF LARGE-SCALE DISTURBANCES IN NON-GEOSTROPHIC FLOW

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Abstract

The primary aim of this article is to derive a complete system of modified hydrodynamical equations which govern a large class of “meteorological” motions of the atmosphere, but which cannot generate solutions corresponding to sound and gravity waves. The latter have no significant effect on the behavior of the large-scale disturbances; they do, however, create serious computational problems (through the mere possibility of their mathematical existence), and cannot be resolved by measurements at widely separated points.

By a detailed analysis of the motions of a linear system, it has been found that the omission of the total derivative of the horizontal divergence from the divergence equation is necessary to exclude gravity-wave solutions. In effect, this modification diverts all of the transformed internal and potential energy into the wave energy of the large-scale disturbances. In slightly stronger form, this type of modification is also sufficient to exclude gravity-wave solutions from the general non-linear equations, and is essentially a generalization of the “filtering approximations” introduced by Charney (1948).

The scheme proposed for integrating the modified non-linear equations is an iterative method, whereby a conditionally convergent sequence of approximate solutions can be generated from a known initial distribution of pressure. This method is applied to the solution of a system of equations which is one order of approximation higher than the equations of quasi-geostrophic motion, and provides the basis for a practicable method of numerical forecasting.

Abstract

The primary aim of this article is to derive a complete system of modified hydrodynamical equations which govern a large class of “meteorological” motions of the atmosphere, but which cannot generate solutions corresponding to sound and gravity waves. The latter have no significant effect on the behavior of the large-scale disturbances; they do, however, create serious computational problems (through the mere possibility of their mathematical existence), and cannot be resolved by measurements at widely separated points.

By a detailed analysis of the motions of a linear system, it has been found that the omission of the total derivative of the horizontal divergence from the divergence equation is necessary to exclude gravity-wave solutions. In effect, this modification diverts all of the transformed internal and potential energy into the wave energy of the large-scale disturbances. In slightly stronger form, this type of modification is also sufficient to exclude gravity-wave solutions from the general non-linear equations, and is essentially a generalization of the “filtering approximations” introduced by Charney (1948).

The scheme proposed for integrating the modified non-linear equations is an iterative method, whereby a conditionally convergent sequence of approximate solutions can be generated from a known initial distribution of pressure. This method is applied to the solution of a system of equations which is one order of approximation higher than the equations of quasi-geostrophic motion, and provides the basis for a practicable method of numerical forecasting.

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