$\boldsymbol{\nabla} \boldsymbol{\cdot} \mathbf{B}$, outer boundary, and Biot-Savart in magnetosphere MHD simulations

We examine the size of $\boldsymbol{\nabla} \boldsymbol{\cdot} \mathbf{B}$ and outer surface boundary integrals in estimating the surface magnetic field from magnetohydrodynamic (MHD) simulations. Maxwell's equations tell us $\boldsymbol{\nabla} \boldsymbol{\cdot} \mathbf{B}$ = 0, which may be violated due to numerical error. MHD models, such as the Space Weather Modeling Framework (SWMF) and the Open Geospace General Circulation Model (OpenGGCM), use different techniques to limit $\boldsymbol{\nabla} \boldsymbol{\cdot} \mathbf{B}$. Analyses of MHD simulations typically assume $\boldsymbol{\nabla} \boldsymbol{\cdot} \mathbf{B}$ errors are small. Similarly, analyses commonly use the Biot-Savart Law and magnetospheric current density estimates from MHD simulations to determine the magnetic field at a specific point on Earth. This calculation frequently omits the surface integral over the outer boundary of the simulation volume that the Helmholtz decomposition theorem requires. This paper uses SWMF and OpenGGCM simulations to estimate the magnitudes of the $\boldsymbol{\nabla} \boldsymbol{\cdot} \mathbf{B}$ and outer boundary integrals compared to Biot-Savart estimates of the magnetic field on Earth. In the simulations considered, the $\boldsymbol{\nabla} \boldsymbol{\cdot} \mathbf{B}$ and outer surface integrals are up to 30 percent of Biot-Savart estimates when the Biot-Savart estimates are large. We conclude rather than using the Biot--Savart Law to estimate the magnetic field from the magnetosphere, it is better and computationally more efficient to use the integral over the inner boundary of the magnetosphere. The conclusions are the same for a simulation involving a simple change in the interplanetary magnetic field and a more complex superstorm simulation.