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Vorticity and Potential Vorticity in Mountain Wakes

R. RotunnoNational Center for Atmospheric Research,Boulder, Colorado

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V. GrubišićNational Center for Atmospheric Research,Boulder, Colorado

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P. K. SmolarkiewiczNational Center for Atmospheric Research,Boulder, Colorado

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Abstract

A wake is traditionally defined as the region of nearly stagnant flow downstream of a body in a uniform stream. In a stratified fluid, the motions and density surfaces downstream of an obstacle become primarily horizontal; the vertical component of the vorticity associated with the wake, coexisting with the stable vertical density stratification, implies that there is potential vorticity (PV) in the wake. Recent work has demonstrated that dissipation aloft, associated with a breaking mountain wave over an isolated peak, produces a dipole in PV downstream; the dipolar vertical vorticity of the wake is associated with the PV dipole. Although one may infer the existence of vorticity downstream, the PV argument is silent on the question, Where does the wake vorticity come from? To answer this question, a weakly nonlinear model for PV production and wake formation in the case of a small-amplitude mountain has been analyzed, and numerical simulations pertaining to the strongly nonlinear large-amplitude case have been carried out. The simple model indicates that even with dissipation in the system, the vertical vorticity of the wake arises through the tilting of baroclinically generated horizontal vorticity by the dissipating mountain wave. This analysis shows that there need not be any direct effect of friction in the vorticity equation to produce the vorticity of the wake; dissipation (due to friction and/or heating) enters indirectly through its effect on the tilting term. Analysis of numerical simulations of the large-amplitude case shows that the conclusions from the weakly nonlinear model regarding the source of wake vorticity continue to hold in the strongly nonlinear regime.

Corresponding author address: Dr. Richard Rotunno, NCAR/MMM, P.O. Box 3000, Boulder, CO 80307-3000.

Email: rotunno@ucar.edu

Abstract

A wake is traditionally defined as the region of nearly stagnant flow downstream of a body in a uniform stream. In a stratified fluid, the motions and density surfaces downstream of an obstacle become primarily horizontal; the vertical component of the vorticity associated with the wake, coexisting with the stable vertical density stratification, implies that there is potential vorticity (PV) in the wake. Recent work has demonstrated that dissipation aloft, associated with a breaking mountain wave over an isolated peak, produces a dipole in PV downstream; the dipolar vertical vorticity of the wake is associated with the PV dipole. Although one may infer the existence of vorticity downstream, the PV argument is silent on the question, Where does the wake vorticity come from? To answer this question, a weakly nonlinear model for PV production and wake formation in the case of a small-amplitude mountain has been analyzed, and numerical simulations pertaining to the strongly nonlinear large-amplitude case have been carried out. The simple model indicates that even with dissipation in the system, the vertical vorticity of the wake arises through the tilting of baroclinically generated horizontal vorticity by the dissipating mountain wave. This analysis shows that there need not be any direct effect of friction in the vorticity equation to produce the vorticity of the wake; dissipation (due to friction and/or heating) enters indirectly through its effect on the tilting term. Analysis of numerical simulations of the large-amplitude case shows that the conclusions from the weakly nonlinear model regarding the source of wake vorticity continue to hold in the strongly nonlinear regime.

Corresponding author address: Dr. Richard Rotunno, NCAR/MMM, P.O. Box 3000, Boulder, CO 80307-3000.

Email: rotunno@ucar.edu

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