Momentum fluxes due to three‐dimensional gravity‐waves: Implications for measurements and numerical modelling

A three‐dimensional wave‐action equation for linear internal gravity‐waves is derived. When the basic‐state flow does not turn with height, this equation can be used to derive the three‐dimensional version of a theorem of Eliassen and Palm. This states that, for steady flows in the absence of dissipation, the vertical flux of both horizontal components of momentum is independent of height. When the basic‐state flow exhibits turning with height, analysis of the wave‐action equation indicates that, in general, the wave field cannot be regarded as steady and the three‐dimensional Eliassen and Palm theorem does not apply. It is also shown that the wave‐action equation has serious implications for the interpretation of atmospheric measurements of the momentum flux: in order to make accurate measurements of the momentum flux arising from three‐dimensional orographically forced gravitywaves, it is necessary to average ρmu'w' (where ρm is the basic‐state density and u' and w' are the horizontal and vertical wave‐induced velocity‐perturbations respectively) on horizontal planes downstream of the orography, rather than along lines in the downstream direction (as has previously been done with aircraft measurements). Using a three‐dimensional linear numerical model, we demonstrate the importance of this in even the most simple idealized cases.