Scaling relations in quasi-static magnetoconvection with a strong vertical magnetic field

Abstract

The scaling law for the horizontal length scale $\ell$ relative to the domain height $L$ , originating from the linear theory of quasi-static magnetoconvection, $\ell /L \sim Q^{-1/6}$ , has been verified through two-dimensional (2-D) direct numerical simulation (DNS), particularly at high values of the Chandrasekhar number ( $Q$ ). This relationship remains valid within a specific flow regime characterized by columnar structures aligned with the magnetic field. Expanding upon the $Q$ -dependence of the horizontal length scale, we have derived scaling laws for the Nusselt number ( $Nu$ ) and the Reynolds number ( $Re$ ) as functions of the driving forces (Rayleigh number ( $Ra$ ) and $Q$ ) in quasi-static magnetoconvection influenced by a strong magnetic field. These scaling relations, $Nu \sim Ra/Q$ and $Re \sim Ra Q^{-5/6}$ , have been successfully validated using 2-D DNS data spanning a wide range of five decades in $Q$ , ranging from $10^5$ to $10^9$ . The successful validation of the relations at large $Q$ values, combined with our theoretical analysis of dissipation rates and the incorporation of the horizontal length scale's influence on scaling behaviour, presents a valid approach for deriving scaling laws under various conditions.