Anelastic Semigeostrophic Flow over a Mountain Ridge

Peter R. Bannon Department of Meteorology, The Pennsylvania State University, University Park, Pennsylvania

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Pe-Cheng Chu Department of Ocoanography, Naval Postgraduate School, Monterey, California

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Abstract

Scale analysis indicates that five nondimensional parameters (R02 ε, μ λ and kλ) characterize the disturbance generated by the steady flow of a uniform wind (U0, V0) incident on a mountain ridge of width a in an isothermal, uniformly rotating, uniformly stratified, vertically semi-infinite atmosphere. Here μ = h0/HR is the ratio of the mountain height h0 to the deformation depth HR = fa/N where f is the Coriolis parameter and N is the static buoyancy frequency. The parameters λ = HR/H and kλ are the ratios of HR to the density scale height H and the potential temperature scale height H/k respectively. There are two Rossby numbers: One based on the incident flow that is parallel to the mountain. ε = V0/fa, and one normal to the mountain, R0 = U0/fa. If R02 ≪1, then the mountain-parallel flow is in approximate geostrophic balance and the flow is semigeostrophic.

The semigeostrophic case reduces to the quasi-geostrophic one in the limit as μ and ε tend to zero. If the flow is Boussinesq (λ = 0), then the semigeostrophic solutions expressed in a streamfunction coordinate can be derived from the quasi-geostrophic solutions in a geometric height coordinate.

If the flow is anelastic (λ ≈ 1), no direct correspondence between the two approximations was found. However the anelastic effects are qualitatively similar for the two and lead to: (i) an increase in the strength of the mountain anticyclone, (ii) a reduction in the extent (and possible elimination) of the zone of blocked, cyclonic flow, (iii) a permanent turning of the flow proportional to the mass of air displaced by the mountain, and (iv) an increase in the ageostrophic cross-mountain flow. The last result implies an earlier breakdown of semigeostrophic theory for anelastic flow over topography.

Apart from a strengthening of the cold potential temperature anomaly over the mountain, the presence of a finite potential temperature scale height (i.e., k nonzero) does not significantly alter the flow solution.

Abstract

Scale analysis indicates that five nondimensional parameters (R02 ε, μ λ and kλ) characterize the disturbance generated by the steady flow of a uniform wind (U0, V0) incident on a mountain ridge of width a in an isothermal, uniformly rotating, uniformly stratified, vertically semi-infinite atmosphere. Here μ = h0/HR is the ratio of the mountain height h0 to the deformation depth HR = fa/N where f is the Coriolis parameter and N is the static buoyancy frequency. The parameters λ = HR/H and kλ are the ratios of HR to the density scale height H and the potential temperature scale height H/k respectively. There are two Rossby numbers: One based on the incident flow that is parallel to the mountain. ε = V0/fa, and one normal to the mountain, R0 = U0/fa. If R02 ≪1, then the mountain-parallel flow is in approximate geostrophic balance and the flow is semigeostrophic.

The semigeostrophic case reduces to the quasi-geostrophic one in the limit as μ and ε tend to zero. If the flow is Boussinesq (λ = 0), then the semigeostrophic solutions expressed in a streamfunction coordinate can be derived from the quasi-geostrophic solutions in a geometric height coordinate.

If the flow is anelastic (λ ≈ 1), no direct correspondence between the two approximations was found. However the anelastic effects are qualitatively similar for the two and lead to: (i) an increase in the strength of the mountain anticyclone, (ii) a reduction in the extent (and possible elimination) of the zone of blocked, cyclonic flow, (iii) a permanent turning of the flow proportional to the mass of air displaced by the mountain, and (iv) an increase in the ageostrophic cross-mountain flow. The last result implies an earlier breakdown of semigeostrophic theory for anelastic flow over topography.

Apart from a strengthening of the cold potential temperature anomaly over the mountain, the presence of a finite potential temperature scale height (i.e., k nonzero) does not significantly alter the flow solution.

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