For 2D turbulence, a commonly-observed phenomenon is the formation of large structure vortices, such as the Great Red Spot in the upper atmosphere of Jupiter, where the flow is approximately two dimensional. Lars Onsager proposed such phenomenon "suggests an explanation on statistical grounds" (1949). He tried to use statistical mechanics to analyze 2D point vortices system, and came up with the idea of "negative temperature" to describe the clustering of vortices. This thesis focuses on this novel "negative temperature" concept, and uses both numerical simulation and theoretical analysis to explain it. Here, we develop a numerical method to simulate the evolution of point vortices in a periodic square. Our simulation shows that point vortices system could capture the phenomenon of the coalescence of same sign vortices. However, it requires specific set-up of the distribution of initial vortices. We observe that such distribution has high energy in the Hamiltonian sense, and usually already has some small structures of clustering of vortices. By analyzing its phase space, we find that the entropy of such distribution will decrease as energy grows, which is consistent with the concept of "negative temperature". We also discover other distributions of point vortices that have different properties from the "negative temperature" case, which we call "infinity temperature" and "positive temperature". Our simulation shows this "positive temperature" and "negative temperature" can neutralize each other. By following the "temperature" idea, we can regard the coalescence of same sign vortices as the balancing of temperature. In the last chapter, we analyze the structure of the phase space of the point vortices system, and proposed a new idea of "energy shell thickness" to explain the coalescence of same sign vortices.
University of Minnesota M.S. thesis. December 2020. Major: Mathematics. Advisor: Hao Jia. 1 computer file (PDF); vi, 81 pages.
Simulation of point vortex system and Onsager’s negative temperature" theory of 2D turbulence".
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