Formulation

The equations for compressible magnetohydrodynamics can be expressed in conservative form in compact notation as:

Alternatively, in flux form with the explicitly stated fluxes as:

with the variables and parameters defined in table 8.1.

 

Variable Description

$$\rho({\bf x}, t)$$ density

$${\bf v}({\bf x}, t)=(u,v,w)({\bf x}, t)$$ velocity

$${\bf B}({\bf x}, t)=(B_x,B_y,B_z)({\bf x}, t)$$ magnetic fields

$$E = \frac{p}{(\gamma-1)} +\frac{1}{2}(\rho {\bf v}\cdot{\bf v}+{\bf B}\cdot{\bf B})$$ total energy

p Pressure

$$\bar{p} = p + \frac{1}{2} {\bf B}\cdot{\bf B}$$ Pressure plus magnetic pressure

$$T = \frac{p}{R \rho}$$ Temperature

$$Pr = \frac{c_p \mu}{\kappa}$$ Prandtl Number

R Ideal gas constant

$$\eta$$ 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

c_p Specific heat at constant pressure

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

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