On the vertical scale of gravity waves excited by localized thermal forcing

Gravity waves generated by convective heating are widely believed to have vertical wavelengths approximately twice the depth of the heating. The frequency, horizontal, and vertical wavelengths of gravity waves are, however, mutually related through the gravity wave dispersion relationship. For forcing of a given frequency, waves of vertical wavelength of twice the depth of the heating will be efficiently excited only if the horizontal forcing projects significantly onto horizontal scales compatible with the vertical-to-horizontal wavenumber ratio given by the dispersion relationship. The preferred vertical wavelength depends on a nondimensional parameter relating the frequency, horizontal, and vertical scales of the forcing. For the high-frequency waves that dominate the momentum flux in the upper stratosphere and mesosphere, the maximum vertical flux of horizontal momentum generally occurs for waves with vertical wavelengths much greater than twice the depth of the heating.