A model for Rayleigh-Bénard magnetoconvection

Abstract

A model for three-dimensional Rayleigh-Bénard convection in low-Prandtl-number fluids near onset with rigid horizontal boundaries in the presence of a uniform vertical magnetic field is constructed and analyzed in detail. The kinetic energy K, the convective entropy Φ and the convective heat flux (N u − 1) show scaling behaviour with ǫ = r − 1 near onset of convection, where r is the reduced Rayleigh number. The model is also used to investigate various magneto-convective structures close to the onset. Straight rolls, which appear at the primary instability, become unstable with increase in r and bifurcate to three-dimensional structures. The straight rolls become periodically varying wavy rolls or quasiperiodically varying structures in time with increase in r depending on the values of Prandtl number P r. They become irregular in time, with increase in r. These standing wave solutions bifurcate first to periodic and then quasiperiodic traveling wave solutions, as r is raised further. The variations of the critical Rayleigh number Raos and the frequency ωos at the onset of the secondary instability with P r are also studied for different values of Chandrasekhar's number Q.

Magnetoconvection transient dynamics by numerical simulation

“…Spectral simulations have also been performed to characterize chaotic structures for a large range of ǫ, up to ∼ 500 [13]. Recently Basak et al [14,15] showed that the energy was proportional to ǫ for small values of ǫ < 1 and Ha < 10 using DNS in a square box. We did not find numerical studies of pattern motion and of dynamics for intermediate values of (Ha, Ra) and 0 < ǫ < 10 (Ha ∼ 50 and Ra ∼ 10 5 ).…”

Energy-conserving model of hexagonal pattern in Rayleigh-Bénard convection

“…The curve is asymmetric for steady sausage pattern (figure 7(b)), but symmetric for rhombus and hexagonal patterns. In our model, the nature of θ rms shows both unimodal (stationary roll pattern) and bimodal (stationary sausage and rhombus patterns) behaviour for stationary convection whereas it is only unimodal in RB convection in the presence of external magnetic field (Basak and Kumar 2015). θ rms is bi-modal for time-dependent convection with or without the external magnetic field.…”

Transitions near the onset of rotating magnetoconvection

“…Interesting to note here that for low Prandtl number fluids, due to the dominance of the velocity nonlinearity in the momentum equation over the nonlinearity in the energy balance equation, vertical vorticity is generated very close to the onset of convection (Busse 1972) and as a result, rich bifurcation structures are observed there (Schlüter et al 1965;Willis & Deardorff 1970;Maurer & Libchaber 1980;Coste & Peyraud 1981;Lyubimov & Zaks 1983;Jenkins & Proctor 1984;Meneguzzi et al 1987;Chiffaudel et al 1987;Fauve et al 1987;Clever & Busse 1990;Busse et al 1992;Hof et al 2004;Mishra et al 2010;Pal et al 2013;Dan et al 2014Dan et al , 2015Nandukumar & Pal 2016). Also RBC of electrically conducting low Prandtl number fluids has been investigated experimentally as well as theoretically in presence of horizontal (Libchaber et al 1982;Busse & Clever 1983;Fauve et al 1984a,b;Burr et al 2002;Yanagisawa et al 2010;Pal & Kumar 2012; and vertical (Nakagawa 1955;Busse & Clever 1982;Clever & Busse 1989;Cattaneo et al 2003;Basak et al 2014;Basak & Kumar 2015 external magnetic fields. It has been revealed that horizontal magnetic field does not influence the onset of instability but vertical magnetic field inhibits the onset of convection (Nakagawa 1955).…”