Semigeostrophic Disturbances in a Stratified Shear Flow over a Finite-Amplitude Ridge

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Abstract
Steady-state, two-dimensional disturbances forced by flow over a finite-amplitude ridge are considered. The model represents an extension of the one presented by Robinson (1960). This study is based on the semigeostrophic system of equations for uniform potential vorticity flow. The model equations satisfy the Cauchy-Riemann conditions, and solutions for uniform flow over various shaped ridges may be obtained in terms of a complex potential. The novel result is the determination of solutions for disturbances in a zonal current with linear shear. The boundaries are tilled in the cross-stream direction to coincide with basic state potential temperature surfaces. This simplification, which provides isentropic boundaries, permits the solutions for disturbances in a shear flow to be obtained directly from solutions forced by uniform flow over the same ridge. Physical properties of the solutions are presented in terms of three parameters: &epsi/D, r and δ. The amplitude of the ridge is &epsi& and D is the deformation depth, based on the characteristic width of the ridge L r represents the ratio of &epsi& to the channel depth and δ is the constant shear of the basic current. Solutions corresponding to uniform flow, δ = 0, in an unbounded fluid, r = 0, represent a limit that is compared with a previous study (Pierehumbert 1985). The present results confirm Pierrehumbert's conclusion that upstream deceleration is not significant, and that the characteristic vertical depth, over which the disturbances decay, is D. Confinement of the flow by a rigid lid (r ≠ 0) and consideration of a shear (δ ≠ 0) do not affect that flow deceleration, nor do these features affect the characteristic decay of the ageostrophic velocity components. However, the presence of a lid causes the geostrophic velocity component to become relatively independent of depth. It is also shown that an increase in the static stability (&epsi&/D increasing) enhances the ageostrophic circulation in a manner that is similar to the effect of increasing the shear δ from negative to positive values. Moreover, a linear lower boundary condition may be used in some circumstances because the velocity components on the ridge are relatively insensitive to changes in r when 0 > r ≳ 0.3 and δ0 However, linearization of the boundary condition cannot be supported when the basic flow changes with height, δ≠ 0. The geostrophic momentum approximation is shown to be valid over most of the domain, but may be violated along the windward slope unless &epsi&/D ≤ 0.6, with δ ≤ 0.5. Other considerations that need to be addressed to apply semige-ostrophic theory to mountain flows include a stability analysis of the present solutions and the use of nonisentropic boundary surfaces.

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