Comparison of statistical theories for large scale two-dimensional flow / Margaret Joyce Holen.

Thesis (Ph. D.)--Princeton University, 1995.

Includes bibliographical references.

In computation, experiment and observation, nearly steady large scale flows are seen in two-dimensional fluid flows with high Reynolds numbers. Three statistical theories that attempt to explain and predict the appearance of these large scale structures have been proposed: the point vortex statistical theory, the classical statistical theory, and the Robert/Miller theory. Each of the three statistical theories gives a different prediction. The prediction of the Robert/Miller theory is the most complex, as it involves coupling between the equation for the most probable state and an infinite family of constraint equations. This thesis compares the predictions of the statistical theories for a family of quasi-geostrophic systems (QGS's) that model large-scale two-dimensional geophysical flow. Chapter 1 introduces the dynamical equations for this family of QGS's and discusses the conserved quantities for this family. Chapters 2 and 3 review the classical and Robert/Miller statistical theories for this family of QGS's. The point-vortex statistical theory is discussed as a special case of the Robert/Miller theory. In chapter 4, we use formal asymptotics to derive an approximation to the Robert/Miller most probable state equation in the limit that the coupling between the most probable state equation and the infinite family of constraints is small. In chapter 5, we use the small coupling expansion derived in chapter 4 to construct approximate solutions of the Robert/Miller most probable state equation. Chapter 6 is dedicated to proving that the small coupling expansion derived in chapter 4 is indeed a good approximation to the Robert/Miller most probable state. Finally, in chapter 7, we present numerical results that attempt to establish a connection between the predictions of the statistical theories and the dynamics of QGS's with dissipation.