Semigeostrophic Flow over Orography in a Stratified Rotating Atmosphere. Part I: Steady Three-Dimensional Solutions over Finite Ridges

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  • 1 Department of Astrophysical, Planetary, and Atmospheric Sciences, University of Colorado. Boulder, CO 80309
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Abstract

Three-dimensional, steady and inviscid flow over orography is examined by means of a semigeostrophic model. A constant Coriolis parameter f, uniform potential vorticity and a uniform basic flow characterize this model, first used by Merkine and Kálnay-Rivas. It is demonstrated that the neglect of ageostrophic accelerations, which characterize a semigeostrophic model, essentially requires low Rosssby number flow Ro≲0.3, and relatively small values of the nondimensional mountain height ε/D ≲ 0.5, where D ∼3 × 103 m is the deformation depth. In this parameter range the disturbance potential is a solution of Laplace's equation; the atmosphere is semi-infinite and the finite-amplitude lower boundary is an isentropic surface.

The basic solution, expressed in prolate spheroidal coordinates, provides disturbance flows over an isolated mountain, finite ridges, and the limiting flow over a two-dimensional ridge. A comparison between the quasi-geostrophic and semigeostrophic responses to flow over the isolated mountain shows that (i) a rotational gradient wind correction to quasi-geostrophic flow that is independent of the basic current is introduced; (ii) the ageostrophic response associated with the basic current is an irrotational flow; and (iii) transformation of the semigeostrophic solution from geostrophic coordinate space to physical space displaces the circulation features on level, surfaces radially outward from the vertical axis, and the displacement is proportional to the mountain height.

An anticyclonic bound vortex, representing the sum of the quasi-geostrophic solution and the rotational wind correction, occurs over an isolated circular mountain (mountain parameter a = 0) and over long ridges (a > 0). However, these circulations are not typically Taylor columns, since they are not two-dimensional: vertical motions occur for 0 < a < . The existence of a closed circulation when a basic current is considered is dependent upon Ro, ε/D and a. In general, relatively larger values of Ro (or smaller ε/D) inhibit the formation of a closed circulation, but there is slight dependence on a. In agreement with Merkine and Kálnay-Rivas, it is shown that a closed circulation will occur around a mountain at a larger value of Ro when the ridge line is aligned with the basic current than when the ridge fine is normal to the current. The difference is relatively small, and probably unobservable in real flows.

A passive scalar disturbance is advected over ridges, characterized by different values of the mountain parameter a and orientation relative to the basic flow. This scalar field is advected anticyclonically around the lip of the ridge north of the current axis, but is retarded at the southern extremity of the ridge where the anticyclonic mountain circulation opposes the basic current. These features of frontal deformation are in qualitative agreement with synoptic analyses in geographic regions of orographic influence. Computed frontogenetical characteristics vary along selected trajectories. The distinctive features of frontogenesis are interpreted and compared with recent results obtained for flew over an infinite ridge.

Abstract

Three-dimensional, steady and inviscid flow over orography is examined by means of a semigeostrophic model. A constant Coriolis parameter f, uniform potential vorticity and a uniform basic flow characterize this model, first used by Merkine and Kálnay-Rivas. It is demonstrated that the neglect of ageostrophic accelerations, which characterize a semigeostrophic model, essentially requires low Rosssby number flow Ro≲0.3, and relatively small values of the nondimensional mountain height ε/D ≲ 0.5, where D ∼3 × 103 m is the deformation depth. In this parameter range the disturbance potential is a solution of Laplace's equation; the atmosphere is semi-infinite and the finite-amplitude lower boundary is an isentropic surface.

The basic solution, expressed in prolate spheroidal coordinates, provides disturbance flows over an isolated mountain, finite ridges, and the limiting flow over a two-dimensional ridge. A comparison between the quasi-geostrophic and semigeostrophic responses to flow over the isolated mountain shows that (i) a rotational gradient wind correction to quasi-geostrophic flow that is independent of the basic current is introduced; (ii) the ageostrophic response associated with the basic current is an irrotational flow; and (iii) transformation of the semigeostrophic solution from geostrophic coordinate space to physical space displaces the circulation features on level, surfaces radially outward from the vertical axis, and the displacement is proportional to the mountain height.

An anticyclonic bound vortex, representing the sum of the quasi-geostrophic solution and the rotational wind correction, occurs over an isolated circular mountain (mountain parameter a = 0) and over long ridges (a > 0). However, these circulations are not typically Taylor columns, since they are not two-dimensional: vertical motions occur for 0 < a < . The existence of a closed circulation when a basic current is considered is dependent upon Ro, ε/D and a. In general, relatively larger values of Ro (or smaller ε/D) inhibit the formation of a closed circulation, but there is slight dependence on a. In agreement with Merkine and Kálnay-Rivas, it is shown that a closed circulation will occur around a mountain at a larger value of Ro when the ridge line is aligned with the basic current than when the ridge fine is normal to the current. The difference is relatively small, and probably unobservable in real flows.

A passive scalar disturbance is advected over ridges, characterized by different values of the mountain parameter a and orientation relative to the basic flow. This scalar field is advected anticyclonically around the lip of the ridge north of the current axis, but is retarded at the southern extremity of the ridge where the anticyclonic mountain circulation opposes the basic current. These features of frontal deformation are in qualitative agreement with synoptic analyses in geographic regions of orographic influence. Computed frontogenetical characteristics vary along selected trajectories. The distinctive features of frontogenesis are interpreted and compared with recent results obtained for flew over an infinite ridge.

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