Discontinuity, Nonlinearity, and Complexity
On Selective Decay States of 2D Magnetohydrodynamic Flows
Discontinuity, Nonlinearity, and Complexity 4(2) (2016) 209218  DOI:10.5890/DNC.2016.06.008
MeiQin Zhan
Department of Mathematics and Statistics, University of North Florida, Jacksonville, FL32224, USA
Download Full Text PDF
Abstract
The selective decay phenomena has been observed by physicists for many dynamic flows such as NavierStoke flows, barotropic geophysical flows, and magnetohydrodynamic (MHD) flows in either actual physical experiments or numerical simulations. Rigorous mathematical works have been carried out for both NavierStoke and barotropic geophysical flows. In our previous work, we have rigorously showed the existence of selective states for 2D MHD flows. In this paper, we present a partial result on instability of the selective states.
References

[1]  Biskamp, D. ( 2003), Magnetohydrodynamic Turbulence, Cambridge University Press. 

[2]  Brown, M.R. (1997), Experimental evidence of rapid relaxation to largescale structures in turbulent fluids: selective decay and maximal entropy , Plasma Physics, 57, 203227. 

[3]  Constantin, P. and Foias, C. (1988), NavierStokes Equations, Chicago University Press, Chicago. 

[4]  Embid, O. and Majda, A., (1998), Low Froude number limiting dynamics for stably stratified flow with small or finite Rossby numbers, Geophysical & Astrophysical Fluid Dynamics, 87, 150. 

[5]  Foias, C., Manley, O.P., Rosa R., and Temam, R. (2001), Cascade of energy in turbulent flows , C.R. Acad. Sci. Paris Sér., 332, 16 

[6]  Foias, C. and Saut, (1984), Asymptotic behaviour, as t→ꝏ of solutions of NavierStokes equations and nonlinear spectral manifolds, Indiana University Mathematics Journal, 33, 459477. 

[7]  Foias, C., and Temam, R., (1989), Gevrey class regularity for the solutions of the NavierStokes equations, Journal of Functional Analysis, 87, 359369. 

[8]  Kraichnan, R.H. (1959), The structure of isotropic turbulence at very high Reynolds numberss,Journal of Fluid Mechanics, 5, 497543. 

[9]  Kraichnan, R.H. (1965), Lagrangianhistory closure approximation for turbulence, Physics of Fluids, 8, 575598. 

[10]  Kraichnan, R.H. (1967), Inertial ranges in twodimentional turbulence, Physics of Fluids, 10, 14171423. 

[11]  Longcope, D.W. and Strauss, H.R. (1993), The coalescence instability and the development of current sheets in twodimensional magnetohydrodynamics,Physics of Fluids, 135, 18582869. 

[12]  Majda, A., and Holen, M. (1998), Dissipation, topography, and statistical theories for large scale coherent structure, CPAM, L, 11831234. 

[13]  Majda, A., and Wang, X. (2001), The selective decay principle for barotropic geophysical flows, MAA, textbf 8, 579 594. 

[14]  Majda, A., and Wang, X. (2006), Nonlinear Dynamics and Statistical Theories for Basic Geophysical Flows, Cambridge University Press. 

[15]  Matthaeus, W.H., and Montgomery, D. (1980), Selective decay hypothesis at high mechanical and magnetic Reynolds numbers, Annals New York Academy of Sciences, 203222. 

[16]  Matthaeus, W.H., Stribling, W.T., Martinez, D., Oughton, S., and Montgomery, D. (1991), Decaying twodimensional NavierStokes turbulence at very long times, Physica D, 51, 531538. 

[17]  Montgomery, D., Shan, X., Matthaeus,W.H. (1993), NavierStokes relaxation to sinhPoisson states at finite Reynolds numbers, Physics of Fluids A, 5(9), 22072216. 

[18]  Singh, M., Khosla, H. K., andMalik, J.S.K. (1998), Nonlinear dispersive instabilities in Kelvin Helmholtz MHD flows Plasma Physics, textbf59, 2737. 

[19]  Temam, R. (1997), Infinite Dimensional Dynamical Systems in Mechanics and Physics, 2nd Edition , SpringerVerlag, New York. 

[20]  Zhan, M.(2010), Selective Decay Principle For 2D Magnetohydrodynamic Flow, Asymptotic Analysis, 67, 125146. 