Nonlinear dynamics of steady oblique rolls in rotating magnetoconvection: Pattern transition, flow multiplicity and hysteresis

Abstract

We report results of the numerical investigation carried out to discern the instabilities and pattern transitions near the onset of rotating magnetoconvection (RMC) using the plane layer Rayleigh–Bénard geometry when both rotation and magnetic field are comparable and nonparallel. A parametric study has been conducted for this purpose by varying the Taylor number (Ta, strength of rotation), the Chandrasekhar number (Q, strength of the magnetic field), and the Prandtl number (Pr) in the ranges of 2.5×103≤Ta≤3×104, 0<Q≤100, and 0.38≤Pr≤0.7, respectively. Our analyses reveal the presence of two structurally distinct oblique rolls at the onset of convection, namely, positive oblique roll (SOR+) and negative oblique roll (SOR−) that lie at angles ±γ with the magnetic field. The appearance of these two oblique rolls is found to divide the (Q, Ta) plane into three regions where SOR+, SOR−, and double-roll (both SOR+ and SOR−) emerge as the primary states. With the increasing Rayleigh number (Ra), the SOR− goes through subsequent transitions to produce a plethora of flow patterns in the form of secondary and higher order states. On the contrary, for all (Ta, Q), the SOR+ does not go through any bifurcation when it appears as the primary state and remains stable in the entire range of Ra considered in this study. We find that the Nusselt number (Nu) corresponding to both SOR+ and SOR− scales as Nu∼1+rα, where r=Ra/Rac is the reduced Rayleigh number with Rac being the critical Rayleigh number for the onset of convection. However, while the value of α is found to vary non-monotonically with Q for SOR−, it remains constant (α=0.9) for SOR+. At certain parameter values, we find the emergence of steady or time-dependent finite amplitude flow states in the form of transverse rolls (TR), parallel rolls (PR), and bifurcating states of SOR+. The appearance of these finite amplitude states leads to the phenomena of flow multiplicity, accompanied by the hysteresis in certain parameter regimes where two or more states coexist depending on the history of the preceding states. Finally, we uncover the effect of Pr on the oblique roll instability at the onset of convection. We find that at low Pr, the onset of convection can be subcritical depending on Ta and Q; finite amplitude steady oblique roll persists there. However, as either of Ta, Q, and Pr increases, the subcritical convection inhibits and supercritical convection takes place.