Dynamic Transitions of Quasi-Geostrophic Channel Flow

The main aim of this paper is to describe the dynamic transitions in flows described by the two-dimensional, barotropic vorticity equation in a periodic zonal channel. In \cite{CGSW03}, the existence of a Hopf bifurcation in this model as the Reynolds number crosses a critical value was proven. In this paper, we extend the results in \cite{CGSW03} by addressing the stability problem of the bifurcated periodic solutions. Our main result is the explicit expression of a non-dimensional number $\gamma$ which controls the transition behavior. We prove that depending on $\gamma$, the modeled flow exhibits either a continuous (Type I) or catastrophic (Type II) transition. Numerical evaluation of $\gamma$ for a physically realistic region of parameter space suggest that a catastrophic transition is preferred in this flow

Dynamic Transitions and Baroclinic Instability for 3D Continuously Stratified Boussinesq Flows

The main objective of this article is to study the nonlinear stability and dynamic transitions of the basic (zonal) shear flows for the three-dimensional continuously stratified rotating Boussinesq model. The model equations are fundamental equations in geophysical fluid dynamics, and dynamics associated with their basic zonal shear flows play a crucial role in understanding many important geophysical fluid dynamical processes, such as the meridional overturning oceanic circulation and the geophysical baroclinic instability. In this paper, first we derive a threshold for the energy stability of the basic shear flow, and obtain a criteria for nonlinear stability in terms of the critical horizontal wavenumbers and the system parameters such as the Froude number, the Rossby number, the Prandtl number and the strength of the shear flow. Next we demonstrate that the system always undergoes a dynamic transition from the basic shear flow to either a spatiotemporal oscillatory pattern or circle of steady states, as the shear strength $\Lambda$ of the basic flow crosses a critical threshold $\Lambda_c$. Also we show that the dynamic transition can be either continuous or catastrophic, and is dictated by the sign of a transition parameter $A$, fully characterizing the nonlinear interactions of different modes. A systematic numerical method is carried out to explore transition in different flow parameter regimes. We find that the system admits only critical eigenmodes with horizontal wave indices $(0,m_y)$. Such modes, horizontally have the pattern consisting of $m_y$-rolls aligned with the x-axis. Furthermore, numerically we encountered continuous transitions to multiple steady states, continuous and catastrophic transitions to spatiotemporal oscillations.Comment: 20 pages, 7 figure

On the Instabilities and Transitions of the Western Boundary Current

We study the stability and dynamic transitions of the western boundary currents in a rectangular closed basin. By reducing the infinite dynamical system to a finite dimensional one via center manifold reduction, we derive a non-dimensional transition number that determines the types of dynamical transition. We show by careful numerical evaluation of the transition number that both continuous transitions (supercritical Hopf bifurcation) and catastrophic transitions (subcritical Hopf bifurcation) can happen at the critical Reynolds number, depending on the aspect ratio and stratification. The regions separating the continuous and catastrophic transitions are delineated on the parameter plane

Equilibrium Statistical Mechanics of Barotropic Quasi-Geostrophic Equations

We consider equations describing a barotropic inviscid flow in a channel with topography effects and beta-plane approximation of Coriolis force, in which a large-scale mean flow interacts with smaller scales. Gibbsian measures associated to the first integrals energy and enstrophy are Gaussian measures supported by distributional spaces. We define a suitable weak formulation for barotropic equations, and prove existence of a stationary solution preserving Gibbsian measures, thus providing a rigorous infinite-dimensional framework for the equilibrium statistical mechanics of the model.Comment: 18 page

ON THE SPECTRAL INSTABILITY AND BIFURCATION OF 2D-QUASI-GEOSTROPHIC POTENTIAL VORTICITY EQUATION

The analysis on hydrodynamic stability of shear flows is an active research direction in fluid dynamics. In this article, the spectral instability and bifurcation of forced shear flows governed by the 2D quasi-geostrophic equation with a generalized Kolmogorov forcing are investigated. We prove that the corresponding eigenvalue problem can be transferred into a family of algebraic equations with infinity number of variables, and the nontrivial solutions to the algebraic equations are expressed in form of continuous fractions. After obtaining the asymptotic estimate for the ratio of the imaginary parts of eigenvalues to a control parameter R as it approaches to infinity, we show that there exists a critical value Rc above which, the forced shear flows become unstable, where the control parameter R is the product of Reynolds number Re and the intensity of the curl of the forcing. To shed light on the bifurcation involved in the losing stability of the forced shear flows, a natural method used to reduce the quasi-geostrophic equation to ODEs is introduced. Based on numerical experiments on the coefficients in the ODEs, we show that both supercritical and subcritical Hopf bifurcations occur in the forced shear flows, which only depend on the type of generalized Kolmogorov forcing

DYNAMIC TRANSITIONS OF QUASI-GEOSTROPHIC CHANNEL FLOW

The main aim of this paper is to study the dynamic transitions in flows described by the two-dimensional, barotropic vorticity equation in a periodic zonal channel. In [Z.-M. Chen et al., SIAM J. Appl. Math., 64 (2003), pp. 343-368], the existence of a Hopf bifurcation in this model as the Reynolds number crosses a critical value was proven. In this paper, we extend these results by addressing the stability problem of the bifurcated periodic solutions. Our main result is the explicit expression of a nondimensional parameter gamma which controls the transition behavior. We prove that depending on gamma, the modeled flow exhibits either a continuous (Type I) or catastrophic (Type II) transition. Numerical evaluation of gamma for a physically realistic region of parameter space suggests that a catastrophic transition is preferred in this flow, which may lead to chaotic flow regimes