Atmospheric gravity wave spectra' A stochastic approach

The evolution of the spectral distribution of atmospheric gravity wave energy, pseudomomentum, and action with altitude is calculated. We apply the eikonal, or ray, formalism to the description of the off-resonant interaction between scale-separated waves where the wave action is projected on elementary wave packets distributed in a position-wave number phase space. We then integrate the ray equations in random vertical shears of horizontal winds. Using stochastic theory, we derive the universal form of the energy-spectrum dependence on the vertical wave number which results from the interactions and possesses the main features of observed spectra or their related empirical descriptions. An asymptotic limit at large wave number (tail) is formed for arbitrary sources, scaling as N^2m^-3 (where N is the Brunt-Väisälä frequency and m is the vertical wave number). A knee in the spectrum appears at a central wave number m* as the wave field propagates upward. Whereas the tail limit remains approximately constant, m* decreases with altitude to account for the increase of the wave energy. If, as a proxy for dissipation, we impose a complete loss of action for waves whose wave numbers are larger than a given threshold m*, there is still a large spectral range where the asymptotic limit N^2m^-3 remains unchanged. With such a proxy, energy and pseudomomentum deposition can be estimated from the particle statistics. This provides the first steps toward a new parameterization of the wave-induced drag on the mean flow that is required in general circulation models.