# American Institute of Mathematical Sciences

November  2014, 13(6): 2609-2639. doi: 10.3934/cpaa.2014.13.2609

## Pattern formation and dynamic transition for magnetohydrodynamic convection

 1 Department of Mathematics, Purdue University, West Lafayette, IN 47906, United States 2 Department of Mathematics, Indiana University, Bloomington, IN 47405

Received  June 2013 Revised  November 2013 Published  July 2014

The main objective of this paper is to describe the dynamic transition of the incompressible MHD equations in a three dimensional (3D) rectangular domain from a perspective of pattern formation. We aim to classify the formations of roll, rectangle and hexagonal patterns at the first critical Rayleigh number.

When the first eigenvalue of the linearized operator is real and simple, the critical eigenvector has either a roll structure or a rectangle structure. In both cases we find that the transition is continuous or jump depending on a non-dimensional number computed explicitly in terms of system parameters.

When the critical eigenspace has dimension two corresponding to two real eigenvalues, we study the transitions of hexagonal pattern. In this case, we show that all three types of transitions--continuous, jump and mixed--can occur in eight different transition scenarios.

Finally, we study the case where the first eigenvalue is complex, simple and corresponding eigenvector has a roll structure. In this case, we find that both continuous and jump transitions are possible.

We give several bounds on the parameters which separate the parameter space into regions of different transition scenarios.
Citation: Taylan Sengul, Shouhong Wang. Pattern formation and dynamic transition for magnetohydrodynamic convection. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2609-2639. doi: 10.3934/cpaa.2014.13.2609
##### References:
 [1] Subrahmanyan Chandrasekhar, Hydrodynamic and Hydromagnetic Stability,, Clarendon Press, (1961). Google Scholar [2] Michael C. Cross and Pierre C. Hohenberg, Pattern formation outside of equilibrium,, \emph{Reviews of Modern Physics}, 65 (1993), 851. Google Scholar [3] Alexander V. Getling, Rayleigh-Bénard Convection: Structures and Dynamics,, World Scientific, (1998). Google Scholar [4] Ernst L. Koschmieder, Bénard Cells and Taylor Vortices,, Cambridge University Press, (1993). Google Scholar [5] Tian Ma and Shouhong Wang, Bifurcation Theory and Applications,, Hackensack, (2005). doi: 10.1142/9789812701152. Google Scholar [6] Tian Ma and Shouhong Wang, Phase Transition Dynamics,, Springer-Verlag, (2013). Google Scholar [7] M. R. E. Proctor and N. O. Weiss, Magnetoconvection,, \emph{Reports on Progress in Physics}, 45 (1982), 1317. Google Scholar [8] Roger Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics,, 2$^{nd}$ edition, (1997). Google Scholar

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##### References:
 [1] Subrahmanyan Chandrasekhar, Hydrodynamic and Hydromagnetic Stability,, Clarendon Press, (1961). Google Scholar [2] Michael C. Cross and Pierre C. Hohenberg, Pattern formation outside of equilibrium,, \emph{Reviews of Modern Physics}, 65 (1993), 851. Google Scholar [3] Alexander V. Getling, Rayleigh-Bénard Convection: Structures and Dynamics,, World Scientific, (1998). Google Scholar [4] Ernst L. Koschmieder, Bénard Cells and Taylor Vortices,, Cambridge University Press, (1993). Google Scholar [5] Tian Ma and Shouhong Wang, Bifurcation Theory and Applications,, Hackensack, (2005). doi: 10.1142/9789812701152. Google Scholar [6] Tian Ma and Shouhong Wang, Phase Transition Dynamics,, Springer-Verlag, (2013). Google Scholar [7] M. R. E. Proctor and N. O. Weiss, Magnetoconvection,, \emph{Reports on Progress in Physics}, 45 (1982), 1317. Google Scholar [8] Roger Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics,, 2$^{nd}$ edition, (1997). Google Scholar
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