On the propagation of gravity waves in a hydrostatic, compressible fluid with vertical wind shear

Abstract

The speed of propagation of the vertical modes of gravity waves is found in a non-rotating hydrostatic and compressible fluid with a vertical variation of mean temperature and wind. The analysis is based on a set of linearized equations, and it is assumed that the basic flow as well as the perturbations are hydrostatic. We are therefore concerned with one of the possible wave-type solutions to the primitive equations as they are being used in studies of the general circulation of the atmosphere and in other dynamical studies of the atmosphere. In the case of a constant static stability it is found that the speed of propagation of internal gravity waves is determined by the Richardson number. If the (positive) Richardson number is smaller than 1/4, the waves will move with a speed determined by the basic flow at the bottom of the fluid. For Richardson numbers larger than 1/4 several vertical modes moving with different speed will exist. The speed of propagation is evaluated as a function of the vertical wave number and the vertical windshear for given values of the static stability. The speed of the external gravity waves is evaluated numerically. It is found that the main difference is the existence of a mode which has a numerically large phase speed and which has a maximum amplitude of the vertical velocity at the boundary. The modes of the internal waves move with almost the same speed as before. The amplitudes of vertical velocity and geopotential are computed as functions of pressure for the internal and external gravity waves. The results of this study which apply to a hydrostatic and compressible atmosphere are compared in the last section with certain general results, obtained in earlier investigations, for a non-hydrostatic, incompressible fluid. A sufficient condition for stability has been found and an upper and lower limit on the magnitude of the complex root has been determined.