Formulation

The non-dimensionalized equations for incompressible magnetohydrodynamics can be expressed as:

where ${\bf v}$ denotes the velocity of the fluid with components ${\bf v} =$ [ u(x,y,t) , v(x,y,t)]^T in the x and y directions; p(x,y,t) is the pressure; and ${\bf B}$ denotes the magnetic field of the fluid with components ${\bf B} =$ [ B_x(x,y,t) , B_y(x,y,t)]^T in the x and y directions. The non-dimensional parameters are shown in table 6.1.

 

Variable Description

$$\eta$$ Magnetic resistivity

$$\mu$$ Viscosity

$$S_v= \frac{\rho_0 V_A L_0}{\mu}$$ Viscous Lundquist number

$$S_r= \frac{V_A L_0}{\eta}$$ Resistive Lundquist number

$$V^2_A = \frac{{\bf B}\cdot{\bf B}}{\rho}$$ Alfven wave speed

$$A = \sqrt{\frac{V_A^2}{V_0^2}}$$ Alfven Number

L₀ Characteristic length scale

V₀ Characteristic velocity

$$\rho_0$$ Characteristic density

In two-dimensions we used a magnetic streamfunction $\phi$ to represent the magnetic field:

$$\bf B= \nabla\times(\phi {\bf k})$$

This will automatically satisfy the magnetic divergence condition. The two magnetic fields will now be evolved by the equation:

$$\frac{\partial \phi}{\partial t} = {\bf v}\times{\bf B} + \frac{1}{S_r} \nabla^2 \phi$$