Magnetohydrodynamics on Unstructured Grids (MUG): 3D linear/nonlinear extended MHD simulation package

Table of Contents

• MUG Examples
• MHD System

MUG solves the linearized or full nonlinear MHD equations in three dimensions using finite elements on unstructured grids of tetrahedra or hexahedra.

MUG has two different implementations:

• The primary implementation (xmhd) using H(Curl) and Lagrange basis functions
• An alternative implementation (xmhd_lag) using only Lagrange basis functions and divergence cleaning

Warning

Not all capability is implemented in the Lagrange only variant.

MUG Examples

The following examples illustrate usage of MUG to run time-dependent resistive and Hall MHD simulations.

• MUG Example: Spheromak Tilt
• MUG Example: Spheromak Heating
• MUG Example: Slab Reconnection

MHD System

Below is the complete system of equations supported by the primary implementation. Most features are also supported by the Lagrange only variant.

\[ \frac{\partial n}{\partial t} + \nabla \cdot \left( n u \right) = D \nabla^2 n \]

\[ \rho \left[ \frac{\partial u}{\partial t} + u \cdot \nabla u \right] = J \times B - \nabla nk(T_e + T_i) - \nabla \cdot \Pi \]

\[ \frac{\partial B}{\partial t} = - \nabla \times \left( -u \times B + \eta J + \frac{1}{ne} \left( J \times B - \nabla nkT_e \right) + \frac{m_e}{ne^2} \frac{\partial J}{\partial t} \right) \]

\[ \frac{n}{\gamma-1} \left[ \frac{\partial T_i}{\partial t} + u \cdot \nabla T_i \right] = -nkT_i \nabla \cdot u - \nabla \cdot q_i + Q_i \]

For two temperature

\[ \frac{n}{\gamma-1} \left[ \frac{\partial T_e}{\partial t} + u_e \cdot \nabla T_e \right] = -nkT_e \nabla \cdot u_e - \nabla \cdot q_e + Q_e \]

\[ u_e = u - \frac{J}{ne} \]

Thermal transport

\[ q_s = - n \left[ \chi_{\parallel,s} \hat{b} \hat{b} + \chi_{\perp,s} \left( I - \hat{b} \hat{b} \right) \right] \cdot \nabla T_s \]

With xmhd_brag=T and single temperature ( \( T_e = T_i \))

\[ \chi_{\parallel,i} = \chi_{\parallel,e}, \chi_{\perp,i} = min(\chi_{\perp,i},\chi_{\parallel,i}) \]

Heat sources

For single temperature ( \( T_e = T_i \))

\[ Q_i = \left[ \eta J^2 - \left( \nabla u \right)^T : \Pi \right] / 2 \]

For two temperature

\[ Q_i = - \left( \nabla u \right)^T : \Pi, Q_e = \eta J^2 \]

Viscosity

\[ W = \left( \nabla u + (\nabla u)^T - \frac{2}{3} I \nabla \cdot u \right) \]

With ‘visc_type='kin’`

\[ \Pi = - \nu \nabla u \]

With ‘visc_type='iso’`

\[ \Pi = - \nu W \]

With ‘visc_type='ani’`

\[ \Pi = - \left[ \nu_{\parallel} \hat{b} \hat{b} + \nu_{\perp} \left( I - \hat{b} \hat{b} \right) \right] \cdot W \]

where \( \mathcal{P} \), \( \mathcal{S} \), and \( \mathcal{C} \) are axisymmetric domains corresponding to the plasma, passive conducting structures (eg. vacuum vessels), coils respectively.