Magnetic stabilization of the buoyant convection between infinite horizontal walls with a horizontal temperature gradient

Abstract

Buoyant convection induced between infinite horizontal walls by a horizontal temperature gradient is characterized by simple monodimensional parallel flows. In a layer of low-Prandtl-number fluid, these flows can involve two types of instabilities: two-dimensional stationary transverse instabilities and three-dimensional oscillatory longitudinal instabilities. The stabilization of such flows by a constant magnetic field (vertical, or horizontal with a direction transverse or longitudinal to the flow) is investigated in this paper through a linear stability analysis and energy considerations. The vertical magnetic field stabilizes the instabilities more quickly than the horizontal fields, but the stabilization is only obtained up to moderate values of Hartmann number $Ha$ (before disappearance of the instabilities). Characteristic laws, given by the critical Grashof number $\Gr_c$ as a function of $Ha$ (proportional to the intensity of the magnetic field), have been found for the initial stabilization at small $Ha$. They are $\Gr_c \sim \Gr_{c_0} \exp(Ha^2)$ for the two-dimensional instabilities and $\Gr_c - \Gr_{c_0} \sim Ha^2$ for the three-dimensional instabilities (where $\Gr_{c_0}$ is the critical Grashof number at $Ha=0$), indicating that the three-dimensional instabilities, less stabilized, will prevail in a vertical magnetic field. It has been shown by an energy analysis that the strong stabilization of the two-dimensional instabilities is connected to the strong diminution of the destabilizing shear energy term when the velocity profiles are modified by the vertical magnetic field, and affected little by the Lorentz energy term. For the horizontal magnetic fields, the stabilization is very weak at small $Ha$, but then reaches an asymptotic behaviour corresponding to $\Gr_c \sim Ha$. This asymptotic stabilization is connected to the decrease of the destabilizing shear energy term due to the increase of the marginal cell length in the horizontal magnetic field. In fact, this stabilization only concerns the two-dimensional modes in the longitudinal field and the three-dimensional modes in the transverse field.

Directional effect of a magnetic field on oscillatory low-Prandtl-number convection

“…The corresponding spatial fields e͑x , y , z͒ can in turn be analyzed to locate the production regions. Note that, as shown by Kaddeche et al, 16 E visc and E magn are stabilizing by nature and thus negative terms.…”

“…III B to extend the analysis of the more academic case of a fluid layer of infinite lateral extent, confined between horizontal plates and subject to a horizontal temperature gradient, which is strongly stabilized in the presence of a vertical magnetic field. 16 The spatial distribution of the fluctuating kinetic energy budget is subsequently analyzed for the magnetic-fielddelayed transition in the three-dimensional cavity in Sec. III C, based on the methods introduced in Sec.…”

“…Thus, a detailed examination of the spatial distribution of the shear energy is necessary to gain insight into the damping action of the magnetic field. Our first approach is to consider the simpler problem of magnetohydrodynamic damping in an extended fluid layer confined between rigid, horizontal walls and subject to a horizontal temperature gradient, 16 which presents important similarities with our three-dimensional problem and offers the advantage of an analytical basic flow solution. Indeed, the linear stability analysis of this basic flow yields a strong increase of the threshold for the two-dimensional steady instability, scaling as Gr c / Gr c ͑Ha= 0͒ϳexp͑Ha 2 ͒.…”

“…An advantage of the analysis performed by Kaddeche et al 16 is that the basic flow u͑Gr, Ha͒ is directly proportional to Gr, u͑Gr, Ha͒ =Gr u G ͑Ha͒. Thus, Gr can be factored out of the shear energy term, so that E shear Ј =GrE shear Љ .…”

“…15 Most previous theoretical work on the magnetohydrodynamic damping of oscillatory convection has been focused on the linear stability analysis of convective flows in infinitely extended layers subject to a horizontal temperature gradient. [16][17][18][19] When the horizontal confining plates are rigid, 16 the vertical field is most effective at stabilizing the flow, suppressing both two-dimensional steady instability modes ͓Gr c / Gr c ͑Ha= 0͒ϳexp͑Ha 2 ͔͒ and three-dimensional oscillatory modes ͓Gr c / Gr c ͑Ha= 0͒ −1ϳ Ha 2 ͔. The strong stabilization of the two-dimensional modes correlates with a similar reduction in the shear energy normalized by Gr c .…”