Uniform Potential Vorticity Flow: Part I. Theory of Wave Interactions and Two-Dimensional Turbulence

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  • 1 Department of Astro-Geophysics, University of Colorado, Boulder 80309
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Abstract

Uniform potential vorticity flows are examined. In the quasi-geostrophic system, conservation of total energy and conservation of available potential energy on plane rigid horizontal boundaries imply a restriction on energy exchanges as a result of scale interactions. It is shown that for the Eady problem instability is always associated with energy transfer both up and down the vertical wavenumber spectrum although energy transfer from small to large three-dimensional wavenumbers may occur over a finite range of the spectrum.

An inertial theory of two-dimensional turbulence is also presented. The formal analysis, based on Leith's diffusion approximation, predicts two inertial subranges: −5/3 and −1 power dependences on the horizontal wavenumber for available potential energy on horizontal boundaries. In the former range, available potential energy on horizontal boundaries cascades at a constant rate toward higher wavenumbers; in the latter range, the depth-integrated total energy cascades at a constant rate toward lower wavenumbers.

Analysis of the semi-geostrophic equations, in the form presented by Hoskins, shows that a formal analogy exists between energy exchanges in this system and energy exchanges in the quasi-geostrophic system. The transformation back to physical space reveals that the mean strain rate, due to vertical wind shear, affects the complete spectrum of interacting waves. This latter result brings the concept of a local inertial energy transfer theory of turbulence for synoptic- and subsynoptic-scale motions into question, although it is concluded that further analysis and observational evidence would be required to resolve the problem.

Abstract

Uniform potential vorticity flows are examined. In the quasi-geostrophic system, conservation of total energy and conservation of available potential energy on plane rigid horizontal boundaries imply a restriction on energy exchanges as a result of scale interactions. It is shown that for the Eady problem instability is always associated with energy transfer both up and down the vertical wavenumber spectrum although energy transfer from small to large three-dimensional wavenumbers may occur over a finite range of the spectrum.

An inertial theory of two-dimensional turbulence is also presented. The formal analysis, based on Leith's diffusion approximation, predicts two inertial subranges: −5/3 and −1 power dependences on the horizontal wavenumber for available potential energy on horizontal boundaries. In the former range, available potential energy on horizontal boundaries cascades at a constant rate toward higher wavenumbers; in the latter range, the depth-integrated total energy cascades at a constant rate toward lower wavenumbers.

Analysis of the semi-geostrophic equations, in the form presented by Hoskins, shows that a formal analogy exists between energy exchanges in this system and energy exchanges in the quasi-geostrophic system. The transformation back to physical space reveals that the mean strain rate, due to vertical wind shear, affects the complete spectrum of interacting waves. This latter result brings the concept of a local inertial energy transfer theory of turbulence for synoptic- and subsynoptic-scale motions into question, although it is concluded that further analysis and observational evidence would be required to resolve the problem.

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