Gravity waves generated by sheared three-dimensional potential vorticity anomalies

The gravity waves (GWs) produced by three-dimensional potential vorticity (PV) anomalies are examined under the assumption of constant vertical shear, constant stratication, and unbounded domain. As in the two-dimensional case analyzed in an earlier paper, the disturbance near the PV anomaly is well modeled by quasi-geostrophic theory. At larger distances the nature of the disturbance changes across the two inertial layers that are located above and below the anomaly, and it takes the form of a vertically propagating GW beyond these. For a horizontally monochromatic PV anomaly of innitesimal depth, the disturbance is described ana-lytically using both an exact solution and a WKB approximation; the latter includes an exponentially small term that captures the change of the solution near the PV anomaly induced by the radiation boundary condition in the far eld. The analytical results reveal a strong sensitivity of the emission to the Richardson number and to the orientation of the horizontal wavenumber: the absorptive properties of the inertial layers are such that the emission is maximized in the Northern Hemisphere for wavenumbers at negative angles to the shear. For localized PV anomalies, numerical computations give the temporal evolution of the GW eld. Ana-lytical and numerical results are also used to establish an explicit formfor the Eliassen-Palmux that could be used to parameterize GW sources in GCMs. The properties of the Eliassen-Palm ux vector imply that in a westerly shear, the GWs exert a drag in a southwest direction in the upper inertial layer, and in a northwest direction at the altitudes where the GWs dissipate aloft.