American Institute of Mathematical Sciences

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

Pattern formation and dynamic transition for magnetohydrodynamic convection

Taylan Sengul 1, and  Shouhong Wang 2,

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.
Keywords: pattern formation, hexagonal pattern, metastability., dynamic transition, Magnetohydrodynamic convection.
Mathematics Subject Classification: Primary: 76W05, 35Q35, 35B3.
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

show all references

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

[1]

Tian Ma, Shouhong Wang. Dynamic transition and pattern formation for chemotactic systems. Discrete & Continuous Dynamical Systems - B, 2014, 19 (9) : 2809-2835. doi: 10.3934/dcdsb.2014.19.2809
[2]

Julien Cividini. Pattern formation in 2D traffic flows. Discrete & Continuous Dynamical Systems - S, 2014, 7 (3) : 395-409. doi: 10.3934/dcdss.2014.7.395
[3]

Yuan Lou, Wei-Ming Ni, Shoji Yotsutani. Pattern formation in a cross-diffusion system. Discrete & Continuous Dynamical Systems - A, 2015, 35 (4) : 1589-1607. doi: 10.3934/dcds.2015.35.1589
[4]

Peter Rashkov. Remarks on pattern formation in a model for hair follicle spacing. Discrete & Continuous Dynamical Systems - B, 2015, 20 (5) : 1555-1572. doi: 10.3934/dcdsb.2015.20.1555
[5]

Rui Peng, Fengqi Yi. On spatiotemporal pattern formation in a diffusive bimolecular model. Discrete & Continuous Dynamical Systems - B, 2011, 15 (1) : 217-230. doi: 10.3934/dcdsb.2011.15.217
[6]

Julien Barré, Pierre Degond, Diane Peurichard, Ewelina Zatorska. Modelling pattern formation through differential repulsion. Networks & Heterogeneous Media, 2020, 15 (3) : 307-352. doi: 10.3934/nhm.2020021
[7]

Shanshan Chen, Junping Shi, Guohong Zhang. Spatial pattern formation in activator-inhibitor models with nonlocal dispersal. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020042
[8]

Martin Baurmann, Wolfgang Ebenhöh, Ulrike Feudel. Turing instabilities and pattern formation in a benthic nutrient-microorganism system. Mathematical Biosciences & Engineering, 2004, 1 (1) : 111-130. doi: 10.3934/mbe.2004.1.111
[9]

Ping Liu, Junping Shi, Zhi-An Wang. Pattern formation of the attraction-repulsion Keller-Segel system. Discrete & Continuous Dynamical Systems - B, 2013, 18 (10) : 2597-2625. doi: 10.3934/dcdsb.2013.18.2597
[10]

Evan C. Haskell, Jonathan Bell. Pattern formation in a predator-mediated coexistence model with prey-taxis. Discrete & Continuous Dynamical Systems - B, 2020, 25 (8) : 2895-2921. doi: 10.3934/dcdsb.2020045
[11]

Fengqi Yi, Eamonn A. Gaffney, Sungrim Seirin-Lee. The bifurcation analysis of turing pattern formation induced by delay and diffusion in the Schnakenberg system. Discrete & Continuous Dynamical Systems - B, 2017, 22 (2) : 647-668. doi: 10.3934/dcdsb.2017031
[12]

Alexandra Köthe, Anna Marciniak-Czochra, Izumi Takagi. Hysteresis-driven pattern formation in reaction-diffusion-ODE systems. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3595-3627. doi: 10.3934/dcds.2020170
[13]

Hyung Ju Hwang, Thomas P. Witelski. Short-time pattern formation in thin film equations. Discrete & Continuous Dynamical Systems - A, 2009, 23 (3) : 867-885. doi: 10.3934/dcds.2009.23.867
[14]

R.A. Satnoianu, Philip K. Maini, F.S. Garduno, J.P. Armitage. Travelling waves in a nonlinear degenerate diffusion model for bacterial pattern formation. Discrete & Continuous Dynamical Systems - B, 2001, 1 (3) : 339-362. doi: 10.3934/dcdsb.2001.1.339
[15]

H. Malchow, F.M. Hilker, S.V. Petrovskii. Noise and productivity dependence of spatiotemporal pattern formation in a prey-predator system. Discrete & Continuous Dynamical Systems - B, 2004, 4 (3) : 705-711. doi: 10.3934/dcdsb.2004.4.705
[16]

Guanqi Liu, Yuwen Wang. Pattern formation of a coupled two-cell Schnakenberg model. Discrete & Continuous Dynamical Systems - S, 2017, 10 (5) : 1051-1062. doi: 10.3934/dcdss.2017056
[17]

Qingyan Shi, Junping Shi, Yongli Song. Hopf bifurcation and pattern formation in a delayed diffusive logistic model with spatial heterogeneity. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 467-486. doi: 10.3934/dcdsb.2018182
[18]

Kolade M. Owolabi. Numerical analysis and pattern formation process for space-fractional superdiffusive systems. Discrete & Continuous Dynamical Systems - S, 2019, 12 (3) : 543-566. doi: 10.3934/dcdss.2019036
[19]

Christian Kuehn, Pasha Tkachov. Pattern formation in the doubly-nonlocal Fisher-KPP equation. Discrete & Continuous Dynamical Systems - A, 2019, 39 (4) : 2077-2100. doi: 10.3934/dcds.2019087
[20]

Evelyn Sander, Thomas Wanner. Equilibrium validation in models for pattern formation based on Sobolev embeddings. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020260

2019 Impact Factor: 1.105

Tools

Metrics

  • PDF downloads (30)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences