# American Institute of Mathematical Sciences

2005, 2005(Special): 611-620. doi: 10.3934/proc.2005.2005.611

## Variational analysis of energy-enstrophy theories on the sphere

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

Received  September 2004 Revised  April 2005 Published  September 2005

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.
Citation: Chjan C. Lim, Da Zhu. Variational analysis of energy-enstrophy theories on the sphere. Conference Publications, 2005, 2005 (Special) : 611-620. doi: 10.3934/proc.2005.2005.611
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