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 equat ion can be used to derive the three-dimensional version of a theorem of Eli assen and Palm. This states that, for steady flows in the absence of dissip ation, the vertical flux of both horizontal components of momentum is indep endent of height. When the basic-state flow exhibits turning with height, a nalysis of the wave-action equation indicates that, in general, the wave fi eld cannot be regarded as steady and the three-dimensional Eliassen and Pal m theorem does not apply. It is also shown that the wave-action equation ha s serious implications for the interpretation of atmospheric measurements o f the momentum flux: in order to make accurate measurements of the momentum flux arising from three-dimensional orographically forced gravity-waves, i t is necessary to average rho(m)u'w' (where rho(m) is the basic-state densi ty and u' and w' are the horizontal and vertical wave-induced velocity-pert urbations respectively) on horizontal planes downstream of the orography, r ather than along lines in the downstream direction (as has previously been done with aircraft measurements). Using a three-dimensional linear numerica l model, we demonstrate the importance of this in even the most simple idea lized cases.