Magnetohydrodynamics

The equations of classical magnetohydrodynamics (MHD) are used to model the dynamics of an electrically ideally conducting fluid with comparable hydrodynamic and electromagnetic forces. When modelling astrophysical objects with strong gravitational fields, e.g. neutron stars, it becomes necessary to model the background space–time as well. We use the standard 3 + 1 split to decompose the four dimensional space–time manifold into 3D hyper-surfaces parameterised by a time coordinate \( t \). The background space–time is introduced into the equations in the form of a non-conservative product.

We have

\begin{eqnarray*} \frac{\partial}{\partial t} \begin{pmatrix} \sqrt{\gamma}D \\ \sqrt{\gamma} S_j \\ \sqrt{\gamma} \tau \\ \sqrt{\gamma}B^j \\ \varphi \\ \alpha_j \\ \beta \\ \gamma_m \end{pmatrix} + \nabla \cdot \begin{pmatrix} \alpha v^i D - \beta^i D \\ \alpha T_j^i - \beta^i S_j \\ \alpha(S^i - v^i D) - \beta^i \tau \\ (\alpha v^i - \beta^i)B^j - (\alpha v^j - \beta^j)B^i 0 \\ 0 \\ 0 \\ 0 \end{pmatrix} + \begin{pmatrix} \sqrt{\gamma}(\tau \partial_j\alpha - \frac{1}{2} T^{ik} \partial_j \gamma_{ik} - T_i^j \partial_j \beta^i) \\ \sqrt{\gamma}(S^j\partial_j\alpha - \frac{1}{2} T^{ik} \beta^j \partial_j \gamma_{ik} - T_i^j \partial_j \beta^j)\\ -\beta^j \partial_i(\sqrt{\gamma}B^i) + \alpha \sqrt{\gamma}\gamma^{ji}\partial_i \varphi \\ \sqrt{\gamma}\alpha c_h^2 \partial_j(\gamma B^i) - \beta^j \partial^j \varphi \\ 0 \\ 0 \\ 0 \end{pmatrix} = 0 \end{eqnarray*}

where \( i,j = 1, 2, 3\) and \( m = 1, ..., 6\)

The curved space–time is parameterised by several hypersurface variables: lapse \( \alpha \), spatial metric tensor \( \gamma \) shift vector \( \beta \) and extrinsic curvature \( K \) . The spatial metric tensor is given as a vector of its six independent components \( \gamma = (\gamma_{11}, \gamma_{12}, \gamma_{13},\gamma_{22}, \gamma_{23}, \gamma_{33}) \) and has the determinant \( \sqrt{\gamma} := \det \gamma \). Further, \( D = W \rho \) is the conserved density, which is related to the rest mass density \( \rho \) by the Lorentz factor \( W \) , \( v^i \) is the fluid velocity, \( T \) is the Maxwell 3-energy momentum tensor, \( S \) is the conserved momentum, \( B \) is the magnetic field and \( \tau \) is the conserved energy density. Finally, \( \varphi \) is an artificial scalar introduced to ensure a divergence-free magnetic field, and ch is the characteristic velocity of the divergence cleaning.