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On the Statistical Theory of Two-Dimensional Topographic Turbulence

J. R. HerringNational Center for Atmospheric Research, Boulder, Colo. 80307

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Abstract

Two-dimensional rotating turbulent flow above a random topography is investigated using the direct interaction approximation and an extension of the test field model, which includes equations for the lagged covariance spectra. For topographic dominated flows (at large scales) the flow predicted is strongly locked to topography. If inertial effects dominate (at smaller scales), three enstrophy-inertial subranges of progressively smaller scales are suggested: a k−1 energy range, followed by two physically distinguishable k−3 ranges. We discuss these inertial ranges by a heuristic theory based on the test field model similar to that proposed by Leith (1968). The origins of these inertial subranges are explained by considering the dominant vorticity distortion (or transfer) process at a given scale, and the coherence time (the length of time the distorting process lasts) at that scale. If topography determines both distortion and the time scale, a k−1 range results; the first k−3 range is an inertial distortion and topographic Rossby wave time-scale regime, and the second k−3 range is the usual two-dimensional inertial range. We examine in some detail the predictions of the theory for stationary turbulence maintained by random stirring at large scales. The theory predicts that the lagged covariance of the vorticity field has a static component which is strongly correlated with topography. The relative magnitude of this static component is determined in terms of a nondimensional measure of topography.

Abstract

Two-dimensional rotating turbulent flow above a random topography is investigated using the direct interaction approximation and an extension of the test field model, which includes equations for the lagged covariance spectra. For topographic dominated flows (at large scales) the flow predicted is strongly locked to topography. If inertial effects dominate (at smaller scales), three enstrophy-inertial subranges of progressively smaller scales are suggested: a k−1 energy range, followed by two physically distinguishable k−3 ranges. We discuss these inertial ranges by a heuristic theory based on the test field model similar to that proposed by Leith (1968). The origins of these inertial subranges are explained by considering the dominant vorticity distortion (or transfer) process at a given scale, and the coherence time (the length of time the distorting process lasts) at that scale. If topography determines both distortion and the time scale, a k−1 range results; the first k−3 range is an inertial distortion and topographic Rossby wave time-scale regime, and the second k−3 range is the usual two-dimensional inertial range. We examine in some detail the predictions of the theory for stationary turbulence maintained by random stirring at large scales. The theory predicts that the lagged covariance of the vorticity field has a static component which is strongly correlated with topography. The relative magnitude of this static component is determined in terms of a nondimensional measure of topography.

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