Constraints on Solutions of Long's Equation for Steady, Two-Dimensional, Hydrostatic Flow over a Ridge

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  • 1 Department of Astrophysical, Planetary and Atmospheric Sciences, University of Colorado, Boulder, Colorado
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Abstract

Two-dimensional, stratified shear flow over a ridge is considered. The finite-amplitude disturbances are steady and hydrostatic, and solutions are derived from the Boussinesq from the Long's equation. Two limiting solutions are examined; viz., 1) the case of marginal or neutral static stability and 2) the case of infinite static stability either at or above the lower boundary. The former case is associated with a critical point for the horizontal flow velocity, u=0; an infinite value of u accompanies the latter case. The conditions for neutral static stability that have been derived for uniform upstream flow conditions are shown to apply to the case when both the upstream static stability N¯(z¯) and the horizontal velocity u¯(z¯) are nonuniform in the vertical direction z¯. Upstream variations of N¯(z¯) and u¯(z¯) cannot be specified arbitrarily if the relative vorticity vanishes at some point either at the ridge or in the airstream above. An unbounded solution, u = ∞, of Long's equation will occur unless the condition [N¯−2(u¯−2/2)z¯]z¯ < 1 is satisfied. The physical interpretation of this constraint on the upstream flow is provided. It is also noted that the same condition has been derived by Abarbanel et al. as a sufficient condition for the nonlinear stability of a stratified shear flow to three-dimensional distrurbances. However, the physical relationship between these two model results has not been established.

Abstract

Two-dimensional, stratified shear flow over a ridge is considered. The finite-amplitude disturbances are steady and hydrostatic, and solutions are derived from the Boussinesq from the Long's equation. Two limiting solutions are examined; viz., 1) the case of marginal or neutral static stability and 2) the case of infinite static stability either at or above the lower boundary. The former case is associated with a critical point for the horizontal flow velocity, u=0; an infinite value of u accompanies the latter case. The conditions for neutral static stability that have been derived for uniform upstream flow conditions are shown to apply to the case when both the upstream static stability N¯(z¯) and the horizontal velocity u¯(z¯) are nonuniform in the vertical direction z¯. Upstream variations of N¯(z¯) and u¯(z¯) cannot be specified arbitrarily if the relative vorticity vanishes at some point either at the ridge or in the airstream above. An unbounded solution, u = ∞, of Long's equation will occur unless the condition [N¯−2(u¯−2/2)z¯]z¯ < 1 is satisfied. The physical interpretation of this constraint on the upstream flow is provided. It is also noted that the same condition has been derived by Abarbanel et al. as a sufficient condition for the nonlinear stability of a stratified shear flow to three-dimensional distrurbances. However, the physical relationship between these two model results has not been established.

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