Lattice Boltzmann magnetohydrodynamics with current-dependent resistivity

Lattice Boltzmann magnetohydrodynamics is extended to allow the resistivity to be a prescribed function of the local current density. Current-dependent resistivities are used to model the so-called anomalous resistivity caused by unresolved small-scale processes, such as current-driven plasma microturbulence, that are excluded by the magnetohydrodynamics approximation. These models closely resemble the Smagorinsky eddy viscosity model used in large eddy simulations of hydrodynamic turbulence. Lattice Boltzmann implementations of the Smagorinsky model adjust the collision time in proportion to the local rate of strain, as obtained from the non-equilibrium parts of the hydrodynamic distribution functions. This works successfully even with a single relaxation time collision operator. However, the existing lattice Boltzmann magnetohydrodynamic implementation contains a spurious term in the evolution equation for the magnetic field that violates the divergence-free condition when the relaxation time varies in space. A correct implementation requires a matrix collision operator for the magnetic distribution functions. The relaxation time imposed on the antisymmetric component of the electric field tensor is calculated locally from the non-equilibrium part of the magnetic distribution functions, which determine the current, while the symmetric component remains subject to a uniform relaxation time to suppress the spurious term. The resulting numerical solutions are shown to converge to independent spectral solutions of the magnetohydrodynamic equations, and to preserve the divergence-free condition up to floating point round-off error.