Simulation and Visualization of Magnetohydrodynamics

Toroidal flux ropes set up that looks like the inside of a fusion reactor.

Magnetohydrodynamics describe the fluid mechanics of plasma. Plasma fluids are so hot that their electrons move freely inside of them and thus induce a magnetic field which then again manipulates the fluid flow. Regular fluid dynamics are typically described using the incompressible Navier-Stokes equations:

    \begin{align*} \frac{\partial V}{\partial t} + (V\cdot\nabla)V &= \nu\nabla^2V-\nabla p + F \\ \frac{\partial\rho}{\partial t}+\nabla\cdot(\rho V) &= 0 \end{align}

where V describes the velocity field, t the time, \rho the density, p the pressure, F the external forces and \nu the kinematic viscosity. In Magnetohydrodynaimcs we add several new terms and equations. In addition to the above variables we also have the magnetic field B, the current density J, the electric field E and the constant value for ratio of specific heats \gamma:=\tfrac{5}{3}. The raw equations are the following:

Continuity Equation:

    \begin{equation*} \frac{\partial\rho}{\partial t}+\nabla\cdot(\rho V) = 0 \end{equation}

Momentum Equation:

    \begin{equation*} \rho\left(\frac{\partial}{\partial t}+V\cdot\nabla\right)V=J\times B-\nabla\rho \end{equation}

Ampere’s law:

    \begin{equation*} \mu_0J-\nabla\times B \end{equation}

Faraday’s law:

    \begin{equation*} \frac{\partial B}{\partial t} = -\nabla\times E \end{equation}

Ideal Ohm’s law:

    \begin{equation*} E+V\times B = 0 \end{equation}

Divergence constraint:

    \begin{equation*} \nabla\cdot B = 0 \end{equation}

Adiabatic Energy Equation

    \begin{equation*} \frac{d}{dt}\left(\frac{p}{\rho^{\gamma}}\right)=0 \end{equation}

The goal is now to reduce these above equations effectively while still capturing the most important features. For that we want to make use of the flux ropes model as they have been used to describe solar eruptions. The goal is to explore their potential for fast simulations of Magnetohydrodynamic phenomena and animations.

Flux ropes as in a coronal loop.