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Energy and Enstrophy Spectra of Geostrophic Turbulent Flows Derived from a Maximum Entropy Principle

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  • 1 Royal Netherlands Meteorological Institute (KNMI), De Bilt, Netherlands
  • | 2 School of Mathematical Sciences, University College Dublin, Belfield, Dublin, Ireland
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Abstract

The principle of maximum entropy is used to obtain energy and enstrophy spectra as well as average relative vorticity fields in the context of geostrophic turbulence on a rotating sphere. In the unforced-undamped (inviscid) case, the maximization of entropy is constrained by the constant energy and enstrophy of the system, leading to the familiar results of absolute statistical equilibrium. In the damped (freely decaying) and forced-damped case, the maximization of entropy is constrained by either the decay rates of energy and enstrophy or by the energy and enstrophy in combination with their decay rates. Integrations with a numerical spectral model are used to check the theoretical results for the different cases. Maximizing the entropy, constrained by the energy and enstrophy, gives a good description of the energy and enstrophy spectra in the inviscid case, in accordance with known results. In the freely decaying case, not too long after the damping has set in, good descriptions of the energy and enstrophy spectra are obtained if the entropy is maximized, constrained by the energy and enstrophy in combination with their decay rates. Maximizing the entropy, constrained by the energy and enstrophy in combination with their (zero) decay rates, gives a reasonable description of the spectra in the forced-damped case, although discrepancies remain here.

Corresponding author address: Dr. W. T. M. Verkley, Royal Netherlands Meteorological Institute (KNMI), P.O. Box 201, 3730 AE De Bilt, Netherlands. Email: verkley@knmi.nl

Abstract

The principle of maximum entropy is used to obtain energy and enstrophy spectra as well as average relative vorticity fields in the context of geostrophic turbulence on a rotating sphere. In the unforced-undamped (inviscid) case, the maximization of entropy is constrained by the constant energy and enstrophy of the system, leading to the familiar results of absolute statistical equilibrium. In the damped (freely decaying) and forced-damped case, the maximization of entropy is constrained by either the decay rates of energy and enstrophy or by the energy and enstrophy in combination with their decay rates. Integrations with a numerical spectral model are used to check the theoretical results for the different cases. Maximizing the entropy, constrained by the energy and enstrophy, gives a good description of the energy and enstrophy spectra in the inviscid case, in accordance with known results. In the freely decaying case, not too long after the damping has set in, good descriptions of the energy and enstrophy spectra are obtained if the entropy is maximized, constrained by the energy and enstrophy in combination with their decay rates. Maximizing the entropy, constrained by the energy and enstrophy in combination with their (zero) decay rates, gives a reasonable description of the spectra in the forced-damped case, although discrepancies remain here.

Corresponding author address: Dr. W. T. M. Verkley, Royal Netherlands Meteorological Institute (KNMI), P.O. Box 201, 3730 AE De Bilt, Netherlands. Email: verkley@knmi.nl

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