Onset of oscillatory Rayleigh-Bénard magnetoconvection with rigid horizontal boundaries

Abstract

We present the results of linear stability analysis of oscillatory Rayleigh-Bénard magnetoconvection with rigid and thermally conducting boundaries. We have investigated two types of horizontal surfaces: (i) electrically conducting and (ii) boundaries which do not allow any outward current normal to the surface (magnetic vacuum conditions). For the case of electrically conducting boundaries, the critical Rayleigh number Rao(Q,Pr,Pm), the critical wave number ko(Q,Pr,Pm), and the frequency at the instability onset ω(Q,Pr,Pm) increase as the Chandrasekhar number Q is raised for fixed non-zero values of thermal Prandtl Pr and magnetic Prandtl number Pm. For small values of Pr, the frequency of oscillation ω at the primary instability shows a rapid increase with Pm for very small values of Pm followed by a decrease at relatively larger values of Pm. In the limit of Pr→0, Rao and ko are found to be independent of Q. However, the frequency ω increases with Q, but decreases with Pm in this limit. The oscillatory instability is possible at the onset of magnetoconvection if and only if Chandrasekhar's criterion is valid (i.e., Pm > Pr) and Q is raised above a critical value Qc(Pr,Pm) such that the product Pm*Qc≈91 for large Pm. For the stellar interior of an astrophysical body (Pm≈10−4 and Pr≈10−8), the value of this product Pm*Qc≈230. The boundary conditions for magnetic vacuum change the critical values of Rayleigh number, wave number, and frequency of oscillation at the onset. The oscillatory magnetoconvection occurs in this case, if Q>Qc, where Pm*Qc≈42 for large Pm. For steller interior, this value is approximately 64. A low-dimensional model is also constructed to study various patterns near the onset of oscillatory convection for rigid, thermally and electrically conducting boundaries. The model shows standing and drifting fluid patterns in addition to flow reversal close to the onset of magnetoconvection.