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Nonlinear stability and statistical equilibrium of forced and dissipated flow Zou, Jieping


A global analysis for the hydrodynamical system defined for a homogeneous, incompressible layer of fluid on the β-plane is performed in both infinite and finite function space. Its application to global stability has yielded an algorithm for characterizing flows based on the existence of initially growing perturbations as opposed to the normal mode analysis; its application to the search for optimal initial perturbations has led to the least upper bound of energy growth rate; its application to multiple equilibria has given rise to a necessary condition for their existence; its application to the study of the relationship of modal to nonmodal growth rates has uncovered the cause underlying many aspects of the limitation of the modal stability analysis including the failure to predict transient growth of disturbances in stable flows and the underestimation of the intensity of initial development of instability in unstable flows. Numerical illustrations made for some specific flows have strengthened the general results, suggesting that a stability analysis of a hydrodynamical system without a global analysis is likely to be limited in many important aspects. The local analysis of asymptotic behavior of nonmodal disturbances to hyperbolic equilibria of the system have established: a) for any subcritical flow outside of monotonic, global stability regime, there exists a finite neighborhood around the origin of RM such that a disturbance initialized in this neighborhood will ultimately decay to zero after it exhibits Orr's temporal amplification; b) for any supercritical flow, there exists a finite neighborhood adjacent to the origin of RM such that a disturbance initialized in this neighborhood will persist as t→∞; and c) the nature of the persistent disturbances is related to the nature of the nonhyperbolic point in parameter space of interest. The numerical experiments are seen to confirm these predictions. Closure modeling of forced-dissipated statistical equilibrium of perturbed flows arising from initially uniform zonal flows over random topography is done with special regard to the correlation between disturbance and underlying topography and the resulting stress. Such an exercise has led to, on one hand, the numerical results for topographical stress suggesting clearly the significance of this force in overall momentum budget of large scale ocean circulations. On the other hand, it has led to an appreciation that the detailed conservation of energy and potential enstrophy, which holds regardless of the presence of dissipation in the system, provides a means for systematic investigation of nonlinear transfer of the these quantities among interacting triads, an area not accessible to other approaches.

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