Chun-Hsiung Hsia

Stratified rotating Boussinesq equations in geophysical fluid dynamics: Dynamic bifurcation and periodic solutions

The main objective of this article is to study the dynamics of the stratified rotating Boussinesq equations, which are a basic model in geophysical fluid dynamics. First, for the case where the Prandtl number is greater than 1, a complete stability and bifurcation analysis near the first critical Rayleigh number is carried out. Second, for the case where the Prandtl number is smaller than 1, the onset of the Hopf bifurcation near the first critical Rayleigh number is established, leading to the existence of nontrivial periodic solutions. The analysis is based on a newly developed bifurcation and stability theory for nonlinear dynamical systems (both finite and infinite dimensional) by two of the authors [T. Ma and S. Wang, Bifurcation Theory and Applications, World Scientific Series on Nonlinear Sciences Vol. 53 (World Scientific, Singapore, 2005)].

Tropical atmospheric circulations with humidity effects.

The main objective of this article is to study the effect of the moisture on the planetary scale atmospheric circulation over the tropics. The modelling we adopt is the Boussinesq equations coupled with a diffusive equation of humidity, and the humidity-dependent heat source is modelled by a linear approximation of the humidity. The rigorous mathematical analysis is carried out using the dynamic transition theory. In particular, we obtain mixed transitions, also known as random transitions, as described in Ma & Wang (2010 Discrete Contin. Dyn. Syst. 26, 1399–1417. (doi:10.3934/dcds.2010.26.1399); 2011 Adv. Atmos. Sci. 28, 612–622. (doi:10.1007/s00376-010-9089-0)). The analysis also indicates the need to include turbulent friction terms in the model to obtain correct convection scales for the large-scale tropical atmospheric circulations, leading in particular to the right critical temperature gradient and the length scale for the Walker circulation. In short, the analysis shows that the effect of moisture lowers the magnitude of the critical thermal Rayleigh number and does not change the essential characteristics of dynamical behaviour of the system.

Time-periodic solutions of the primitive equations of large-scale moist atmosphere: existence and stability

This article is devoted to the study of the asymptotic stability of the three-dimensional viscous primitive equations for large-scale moist atmosphere in the pressure coordinate system. An asymptotic stability criterion for the solutions of such model with time-dependent forcing terms is obtained. In particular, under the assumptions that the associated forces are both time-periodic and small (in a suitable sense), we obtain the existence of the time-periodic solution for the primitive equations of large-scale moist atmosphere. Moreover, this time-periodic solution is asymptotically stable in sense.

Bifurcation of binary systems with the Onsager mobility

The main objective of this article is to study the effect of the (nonlinear) Onsager mobility to the phase separation of the binary system, using rigorous bifurcation analysis. In particular, a nondimensional parameter K, depending on the molar density u0 of the homogeneous state, and the critical temperature is derived; the sign of this parameter dictates the type of transition. Also, the analysis indicates that the type of the transition, the critical temperature Tc, and the strength of the deviation of the transition solutions from the homogeneous state are all independent of the choices of the Onsager mobility.

Attractor Bifurcation of Three-Dimensional Double-Diffusive Convection

In this article, we present a bifurcation analysis on the double-diffusive convection. Two pattern selections, rectangles and squares, are investigated. It is proved that there are two different types of attractor bifurcations depending on the thermal and salinity Rayleigh numbers for each pattern. The analysis is based on a newly developed attractor bifurcation theory, together with eigen-analysis and the center manifold reductions.

Hopf bifurcation for two-dimensional doubly diffusive convection

In this article, we present a stability analysis of the Hopf bifurcation for a doubly diffusive problem. It is proved that there exist both continuous and jump transitions and that these are determined explicitly by the variation of certain physically relevant, non-dimensional parameters.

On the long-time stability of a temporal discretization scheme for the three dimensional viscous primitive equations

In this article, a semi-discretized Euler scheme to solve the three dimensional viscous primitive equations is studied. Based on suitable assumptions on the initial data and forcing terms, the long-time stability of the proposed scheme is proven by showing that the

Dynamical bifurcation of the two dimensional Swift-Hohenberg equation with odd periodic condition

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