Hopf bifurcations and global nonlinear $L^2$-energy stability in thermal MHD

Abstract

The transfer of heat and mass by convection in a fluid horizontal layer $\text{L}$, heated from below – in the past as nowadays – has attracted the attention of many scientists since it can be driven by different factors and has a very relevant influence on the behaviour of many phenomena of the real world concerning meteorology, sun and stars physics, oceanography, heat insulation, air and water pollution, ... (see [1–12, 25–29] and references therein). When $\text{L}$ is filled by a plasma and is embedded in a transverse constant magnetic field, in the non-relativistic scheme of magneto-hydrodynamic (MHD), in the early of 1950 ([1–2]), the Nobel Prize laureate (1983) S. Chandrasekhar – in linear thermal MHD – obtained a relevant inhibition of convection by a magnetic field, verified experimentally [9] and appeared in 1961 in the celebrated monograph [3]. Successively, during the years, many efforts have been done in order to recover this relevant stabilizing effect in the nonlinear thermal MHD theory (see [6, 10, 11, 13, 16]). This goal has been reached partially in 1988 in [13] and totally in [16] but only under very severe restrictions on the initial data (of the order $<10^{-6}$). Recently in [17], via a non standard approach, assuming the verticality of the gradient pressure perturbations, it has been totally recovered in the nonlinear thermal MHD theory, the linear inhibition of convection by magnetic field for any admissible initial data (Linearization Principle). In the present paper, we return to the problem and show that, in thermal MHD, the linear asymptotic stability implies the global exponential nonlinear $L^2-$energy stability, without requiring the verticality of the perturbations to the pressure gradient. As concerns the onset of instabilities in the free-free case, since $P_r\ge P_m$ (with $P_r,P_m$ Prandtl and Prandlt magnetic numbers), implies the onset of steady bifurcation for any value of the Chandrasekhar number $Q^2$, we analyze the case $P_r<P_m$ via the introduction of two critical numbers $R_{C_2},R_{C_3}$ coming from the application of the Hurwitz criterion to the spectral equation of the nth-component of the perturbations to the thermal conduction $m_0$ and obtain that if and only if $R_{C_2}<R_{C_3}$ the Hopf bifurcation occurs. Named Hopf bifurcation number the threshold $Q_c$ that the Chandrasekhar number has to cross for the occurring of Hopf bifurcations, we obtain that $Q_c=\frac{1+P_r}{P_m-P_r}{\pi }^2$. This formula – new in the existing literature – removes the difficulties (mentioned in page 184 of [3]) on finding a "simpler formula which gives $Q_c$ as function of $P_r$, $P_m$".