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Statistical stability arguments for maximum kinetic energy dissipation

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Mathematics for the Fluid Earth

The hypothesis that stationary turbulent flows have maximal mean-flow kinetic energy dissipation (Max-D) is intriguing because mean-flow properties can be predicted without modelling the turbulent component of the flow. Our knowledge of Max-D is largely restricted to relatively simple laboratory flows. Measured Poiseuille flow profiles match Max-D predictions closely and, under these simplified conditions, Malkus’s statistical stability argument provides some theoretical justification for Max-D [1]. However, it is not clear whether Max-D is applicable to more complicated fluid systems, like Earth’s atmosphere [2]. Recent global climate model simulations have found that the calibrated values of important tunable parameters are indeed consistent with Max-D [3]. Furthermore, the maximum entropy framework [4], which naturally gives a Max-D principle in the case of simple laboratory flows, can be readily applied to more complicated systems. I will discuss attempts to gener alise the Malkus statistical stability argument and how this connects with maximum entropy arguments. In doing so I hope to compare the physical insights of statistical stability, which emphasises dynamical resilience to perturbations, with maximum entropy considerations, which ignore system dynamics.

[1] W. V. R. Malkus. Borders of disorder: In turbulent channel ow. Journal of Fluid Mechanics, 489:185{198, 2003. [2] Richard Goody. Maximum entropy production in climate theory. Journal of the atmospheric sciences, 64(7):2735-2739, 2007. [3] Salvatore Pascale, Jonathan M. Gregory, Maarten H.P. Ambaum, and Remi Tailleux. A parametric sensitivity study of entropy production and kinetic energy dissipation using the FAMOUS AOGCM . Climate Dynamics, 38(5-6):1211-1227, 2012. [4] Dewar R and Maritan A. A theoretical basis for maximum entropy production. 2013. In Beyond the Second Law: Entropy Production and Non-equilibrium Systems (eds. R Dewar, C Lineweaver, R Niven, K Regenauer-Lieb), Springer, In Press

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