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Equilibrium state of quasi-geostrophic point vortices

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The Nature of High Reynolds Number Turbulence

The statistics of quasi-geostrophic point vortices is investigated theoretically and numerically, in order to understand fundamental aspects of quasi-geostrophic turbulence. Numerical simulations of N-vortex system (N = 2000 − 8000) in an infinite fluid domain are performed using a fast special-purpose computer (MDGRAPE-3) for molecular dynamics simulations.[1] The vortices of the same strength are initially located randomly and uniformly in a cubic box, and we choose the state with the highest multiplicity for fixed angular momentum. The axi-symmetric equilibrium state is obtained after about 20 turn over time. The probability density distribution of the center region resembles that of the purely two-dimensional point vortices. The three-dimensional effect appears near the upper and lower lids in the tighter concentration of vortices around the axis of symmetry (End-effect). The most probable vortex distributions are determined based on the maximum entropy theory.[2] We search for the state of maximum Shanon-entrophy under the constraints of vertical vorticity distribution, angular momentum and energy (mean-field approximation). The theoretical predictions agree quite well with the numerical results.[3] We investigate the influence of energy on the equilibrium state, in some detail. Larger vortex clouds have lower energy, larger entropy and lager angular momentum. Each vertical layer has same contribution to the entropy and the angular momentum, whereas the center region has stronger influence to the energy than the lids. Therefore, the distribution in the center region expands radially for lower energy and shrinks for higher energy. In order to keep the angular momentum unchanged, the distribution near the lids should shrink for lower energy and should expand for higher energy. References [1] Yatsuyanagi Y., Kiwamoto Y., Tomita H., Sano M.M., Yoshida T., Ebisuzaki. T.: Dynamics of Two-Sign Point Vortices in Positive and Negative Temperature States. Phys. Rev. Lett. 94:054502, 2005 [2] Kida S.: Statistics of the System of Line Vortices. J. Phys. Soc. Jpn. 39(5):13951404, 1975 [3] Hoshi S., Miyazaki T.: Statistics of Quasi-geostrophic Point Vortices. FDR (Subbmitting), 2008.

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