Statistical mechanics of 2-D fluids
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"Two dimensional fluid flows are realized in situations where one dimension is much smaller than the other two or if symmetry allows the neglect of one of the dimensions. Other factors, too, can contribute to result in effectively two dimensional (henceforth 2-D) flows. An example is the planetary atmospheric and oceanic flow where rotation of the fluid about the planet's axis locks it into 2-D motion (Taylor-Proudman theorem). See Greenspan (1968) for the dynamics of rotating fluids. A constant magnetic field perpendicular to a layer of plasma has a similar locking effect (Kraichnan and Montgomery, 1980) which is why there is some interest in 2-D magnetohydrodynamics. One of the interesting features of 2-D flows is the formation of coherent structures, an example of which is the Great Red Spot of Jupiter. An experiment by Sommeria, Meyers and Swinney (1988) demonstrated the formation of coherent structures in 2-D turbulent shear by creating a "Great Red Spot" in a rotating tank of fluid. The problem of 2-D flows is also interesting purely from the point of view of studying the dynamics, hence the wish to study 2-D fluid flows. Various approaches to study the turbulent relaxation of 2-D flows are discussed in this thesis. We begin by setting up the basic equations and by reviewing the Hamiltonian formulation of the dynamics in Chapter 1. In Chapter 2 we discuss statistical approaches to solve the problem of turbulent relaxation. These approaches include the point vortex approximation and the method of maximizing entropy. While the fluid is approximated by a collection of point vortices in the former case, the latter approach allows us to deal with continuous vorticity distributions, although, in practice, one discretizes the vorticity in order to compute the results numerically. Chapter 3 is devoted to the study of a selective decay hypothesis based on arguments of cascade of energy and enstrophy to different scales. After having discussed these three different approaches, we then compare their predictions to the observations in an experiment on electrons in a magnetized column in Chapter 4. The possibility of a simple monotonic restacking of the vorticity is also discussed in Chapter 4"--Introduction.