Mean Meridional Circulations Induced around a 'Vertical' Critical Surface for a Planetary Wave

Abstract

The Eulerian- and Lagrangian-mean flow fields induced around a ‘vertical’ critical surface for a planetary wave propagating latitudinally are discussed by using a simplified model of horizontal wave guide on a beta-plane.
It is shown that the Eulerian-mean meridional stream function xE (not zonal mean part of geostrophic stream function ψ) is forced by a delta-function-like divergence of the Reynolds stress. This xE-equation is solved on the basis of the result of classical hydrodynamics for the critical layer problem (e.g., Lin, 1967; Lindzen and Tung, 1978), as done also by Matsuno and Nakamura (1979) for the problem of vertical propagation.
It is shown that the Eulerian-mean flow fields are confined in the extent given by the Rossby radius of deformation divided by 2π. Particularly, it is noted that the mean zonal acceleration consists of two parts; one is the easterly acceleration concentrated on the critical surface and is associated with no meridional circulation, and the other is the acceleration due to the Coriolis force acting on the wave-induced mean meridional circulation. The latter acceleration is directed toward west in the mid-level and toward east in the upper and the lower layers. It is also shown that the mean vertical flow is discontinuous at the critical surface and has a four sector structure with upward (downward) branch in the southern (northern) side of the surface in the upper layer, and the flow is reversed in the lower layer. The mean meridional circulation consists of two circulations, opposite to each other, in the upper and the lower layers.
Finally, it is shown that the Lagrangian-mean flow fields have the same structure as the Eulerian-mean ones. This is because the second order fields are induced by the divergence of Reynolds stress only, different from the case of vertical propagation (cf. Matsuno and Nakamura, 1979).