A Divergence-Form Wave-Induced Pressure Inherent in the Extension of the Eliassen–Palm Theory to a Three-Dimensional Framework for All Waves at All Latitudes

Classical theory concerning theEliassen–Palmrelation is extended in this study to allowfor a unified treatment of midlatitude inertia–gravity waves (MIGWs), midlatitude Rossby waves (MRWs), and equatorial waves (EQWs). A conservation equation for what the authors call the impulse-bolus (IB) pseudomomentum is useful, because it is applicable to ageostrophic waves, and the associated three-dimensional flux is parallel to the direction of the group velocity of MRWs. The equation has previously been derived in an isentropic coordinate system or a shallow-water model. The authors make an explicit comparison of prognostic equations for the IB pseudomomentum vector and the classical energy-based (CE) pseudomomentum vector, assuming inviscid linear waves in a sufficiently weak mean flow, to provide a basis for the former quantity to be used in an Eulerian time-mean (EM) framework. The authors investigate what makes the three-dimensional fluxes in the IB and CE pseudomomentum equations look in different directions. It is found that the two fluxes are linked by a gauge transformation, previously unmentioned, associated with a divergence-form wave-induced pressure L. The quantity L vanishes for MIGWs and becomes nonzero for MRWs and EQWs, and it may be estimated using the virial theorem. Concerning the effect of waves on the mean flow, L represents an additional effect in the pressure gradient term of both (the three-dimensional versions of) the transformed EM momentum equations and the merged form of the EMmomentumequations, the latter of which is associated with the nonacceleration theorem