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

TL;DR

This study reframes magnetospheric field estimation at Earth by applying the Helmholtz decomposition to separate Biot-Savart contributions from $\nabla\cdot\mathbf{B}$ and outer-boundary effects. Comparing SWMF/BATS-R-US and OpenGGCM under identical solar-wind forcing, the authors show that the inner-boundary integral $\mathbf{B}_{\text{in}}$ largely captures the magnetospheric field at Earth, with $\mathbf{B}_{\text{out}}$ and $\mathbf{B}_{\text{div}}$ contributing up to $\sim$20–30% in many cases and more when the Biot-Savart estimate is small. They demonstrate that deviations in gap-region currents are divergence-free and that non-smooth MHD fields (shocks) can break the Helmholtz-based relations, highlighting a need for careful modeling of domain boundaries. The results advocate using $\mathbf{B}_{\text{in}}$ rather than a full Biot-Savart calculation for efficiency and consistency, with implications for interpreting magnetospheric contributions to ground magnetic measurements across simple and extreme solar events (including superstorms). These insights are robust across model differences and event types, offering practical guidance for space weather analysis and ground-field predictions.

Abstract

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.

Figures (14)

• Figure 1: Schematic representation of MHD simulation volume. The blue rectangle is the magnetosphere. The pale red circle encloses the gap region. The dashed black circle is the ionosphere. The small gray circle is Earth. The outer and inner magnetosphere boundaries are identified.
• Figure 2: $\mathbf{B}_{\text{BS}}$, $\mathbf{B}_{\text{in}}$, $\mathbf{B}_{\text{div}}$, and $\mathbf{B}_{\text{out}}$ versus time derived from BATS--R--US simulation. Plots show $\mathbf{B}$ at Colaba, India. $\mathbf{B}_{\text{in}}$ (blue line) explains most of the variation in $\mathbf{B}_{\text{BS}}$; with $\mathbf{B}_{\text{BS}}$ (black line) equal to the sum of $\mathbf{B}_{\text{in}}$, $\mathbf{B}_{\text{out}}$, and $\mathbf{B}_{\text{div}}$ (red line).
• Figure 4: The $x$, $y$, and $z$ components (GSM) of the BATS--R--US and OpenGGCM $\mathbf{B}$ measured along a line parallel to the $x$ axis. $y$ and $z$ values shown in titles. The top graph is at 03:00 (UTC), the bottom at 10:00 (UTC). OpenGGCM has narrow spikes near $x=-20$ in the top graph and near $x=0$ in the bottom graph.
• Figure 5: $\mathbf{B}_{\text{BS}}$, $\mathbf{B}_{\text{in}}$, $\mathbf{B}_{\text{div}}$, and $\mathbf{B}_{\text{out}}$ versus time derived from BATS--R--US simulation grid assuming infinite line current outside the simulation domain. Plots show $\mathbf{B}$ at Colaba, India. Since $\nabla \cdot \mathbf{B} = 0$ and $\nabla \times \mathbf{B} = 0$, we expect $\mathbf{B}_{\text{BS}} = 0$ and $\mathbf{B}_{\text{div}} = 0$. In addition, the Helmholtz decomposition theorem requires that $\mathbf{B}_{\text{in}} = - \mathbf{B}_{\text{out}}$.
• Figure 7: $\mathbf{B}_{\text{BS}}$, $\mathbf{B}_{\text{in}}$, $\mathbf{B}_{\text{div}}$, and $\mathbf{B}_{\text{out}}$ versus time derived from BATS--R--US simulation grid assuming a $B_Z^\text{\tiny{IMF}}$ flip within the simulation domain. Plots show $\mathbf{B}$ at Colaba, India. Since the magnetic field is discontinuous at $y=\pi$, we expect the Helmholtz Decomposition Theorem to be violated. Consistent with this, the red and black lines do not overlap.