MaxwellBoussinesq

MaxwellBoussinesq is a framework that combines Maxwell's equations with the Boussinesq approximation to model the interaction of buoyancy-driven fluid flow and magnetic fields in electrically conducting fluids. It provides a reduced, tractable set of equations suitable for studying magnetoconvection and dynamo action where density variations are small except in buoyancy forces.

The model assumes incompressible flow, negligible displacement currents, and small density changes. Density variations enter only in the buoyancy term, through a linear relation such as ρ ≈ ρ0[1 − α(T − T0)]. The governing equations typically include: the momentum equation, which balances inertia, pressure, viscous diffusion, buoyancy proportional to temperature deviation, and the Lorentz force from the magnetic field; the incompressibility condition, div u = 0; the induction equation for the magnetic field, which describes its evolution via advection, stretching, and diffusion; the solenoidal condition, div B = 0; and an advection-diffusion equation for the temperature (or another scalar) driving buoyancy.

MaxwellBoussinesq is widely used in geophysical and astrophysical contexts, such as modeling convection in planetary cores and stellar interiors, as well as in laboratory experiments with liquid metals. It provides insights into how thermal driving and magnetic fields interact to influence flow patterns, stability, and the onset of dynamo action. Limitations include the neglect of compressibility and non-Boussinesq density variations, which can be important at large temperature differences or in strongly stratified systems.