Exploring from classical vertical convection to magnetoconvection: statistical properties of dissipation and scaling relations

We investigate the statistical properties of kinetic and thermal dissipation rates in two-dimensional/three-dimensional vertical convection of liquid metal ( $Pr = 0.032$ ) within a square cavity. Two situations are specifically discussed: (i) classical vertical convection with no external forces and (ii) vertical magnetoconvection with a horizontal magnetic field. Through an analysis of dissipation fields and a reasonable approximation of buoyancy potential energy sourced from vertical heat flux, the issue of the ‘non-closure of the dissipation balance relation’, which has hindered the application of the GL theory in vertical convection, is partially resolved. The resulting asymptotic power laws are consistent with existing laminar scaling theories and even show certain advantages in validating simulations with large Prandtl number ( $Pr$ ). Additionally, a full-parameter model and prefactors applicable to low- $Pr$ fluids are provided. The extension to magnetoconvection naturally introduces the approximate expression for total buoyancy potential energy and necessitates adjustments to the contributions of kinetic dissipation in both the bulk and boundary layer. The flow dimensionality and boundary layer thickness are key considerations in this analysis. The comprehension of Joule dissipation has been updated: the Lorentz force generates positive dissipation in the bulk by suppressing convection, while in the Hartmann layer, shaping the exponential boundary layer requires the fluid to perform positive work to accelerate, leading to negative dissipation. Finally, the proposed transport equations for magnetoconvection are supported by current direct numerical simulation (DNS) and literature data, and the applicability of the model is discussed.