Variational analysis of energy-enstrophy theories on the sphere

Chjan C. Lim; Da Zhu

doi:10.3934/proc.2005.2005.611

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2005, Volume 2005: 611-620. Doi: 10.3934/proc.2005.2005.611 open access

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Variational analysis of energy-enstrophy theories on the sphere

Chjan C. Lim^1, and
Da Zhu¹

1.
Department of Mathematical Sciences, Rensselaer Polytechnic Institute, Troy, NY 12180

Received: August 31, 2004

Revised: March 31, 2005

Published: August 2005

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Abstract

Kraichnan's energy-enstrophy theory for 2D inviscid flows on the sphere is discussed within a variational framework. We will give necessary and sufficient conditions for the existence and uniqueness for the extremals of the energy with zero circulation under different values of the temperature parameter $\beta$. The unboundedness of the augmented energy functional in this model when $\beta$ is located in the certain intervals will be shown and related to energy catastrophe of the energy-enstrophy model.

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Mathematics Subject Classification: Primary: 76B47, 82B20; Secondary: 65C05.

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