Energy spectra associated with long's model for steady finite-amplitude flow over orography

Abstract

The process of wave steepening in Long's model of steady, two-dimensional stably stratified flow over orography is examined. Under conditions of the long-wave approximation, and constant values of the background static stability and basic flow, Long's equation is cast into the form of a nonlinear advection equation. Spectral properties of this latter equation, which could be useful for the interpretation of data analyses under mountain wave conditions, are presented. The principal features, that apply at the onset of convective instability (density constant with height), are:

i) a power spectrum for available potential energy that exhibits a minus eight-thirds decay, in terms of the vertical wavenumber k _z ^-;

ii) a rate of energy transfer across the spectrum that is inversely proportional to the wavenumber for large k _z ^-;

iii) an equipartition between the kinetic energy of the horizontal motion and the available potential energy, under the longwave approximation, although all the disturbance energy is kinetic at the point where convective instability is initiated. It is also shown that features i) and ii) apply to more general conditions that are appropriate to Long's model, not just the long-wave approximation. Application to fully turbulent flow or to conditions at the onset of shearing instability are not considered to be warranted, since the development only applies to conditions at the onset of convective instability.