Semi-geostrophic flow of a stratified atmosphere over elongated isentropic valleys and ridges

A study is undertaken of semi-geostrophic flow of a stratified inviscid atmosphere over various forms of infinite (and some forms of finite) length isentropic orographic features on an f-plane. Finite-amplitude, explicit solutions are derived under the stipulation that the velocity (U) and the Brunt Väisälä frequency (N) of the upstream flow are both uniform.

In the case of the infinite ridge configuration, it is shown that, in contrast to the case of orographically forced buoyancy waves where breakdown is associated with the waves attaining a vertical tilt, the breakdown of semi-geostrophic flow is accompanied by a local fusion of the isentropes at the surface with an associated infinite across-ridge flow. The criteria for, and the spatial location of, the breakdown are demonstrated to be sensitively dependent upon the dimensionless Rossby radius of deformation (NH/fL) and the “orographic shape”. (Here H and L refer to the characteristic height and width of the ridge.) In particular, the separate influence upon the flow of the curvature and the asymmetry of the orography is explored. It is shown that large convex curvature enhances the likelihood of flow breakdown and that an asymmetric valley-ridge profile is accompanied by a non-zero mean along-ridge flow component but with no attendant net Coriolis lift.

Three-dimensional solutions are derived for flow over some special, “zero net volume” forms of orography. In addition to the breakdown of semi-geostrophy, the flow response can also exhibit a Taylor cone structure. The criterion for the cone’s existence is related to the value of a Froude number (defined in terms of the amplitude of the cross-ridge flow component of the incident flow).

Attention is also drawn to the formal limitation of semi-geostrophy and some consideration is given to the implications of its violation.