MHD approximations

The following approximations are made to produce a tractable set of equations:

1.

The displacement current term in Ampere's law () is omitted. This can be justified by comparing the LHS of () with the displacement current term:

$$\vert\mbox{\boldmath$\nabla \times B$ }\vert \approx \frac{B}... ...E}} {\partial t} \right\vert \approx \frac{E}{c^{2} \tau} \, ,$$

where L and $\tau$ are the characteristic MHD length and time scales. Thus,

$$\frac{\left\vert\frac{\partial {\bf E}} {\partial t} \right\v... ...x \left( \frac{L}{\tau} \right)^{2} \frac{1}{c^{2}} \ll 1 \, ,$$

where $E/B \approx L/\tau$ from Faraday's law (). Hence () reduces to

$${\mbox{\boldmath$\nabla \times B$ }} = \mu_{0} {\bf J} \, .$$ (3.41)

2.

Charge neutrality ($\rho = 0$) is typically satisfied in a plasma because the forces associated with any unbalanced charges imply a potential energy per particle that well exceeds the mean thermal energy per particle. The charge conservation equation () then reduces to

$${\mbox{\boldmath$\nabla \cdot$ }} {\bf J} = 0 \, .$$ (3.42)

This also follows by taking the divergence of equation ().

3.

The approximation () or () to Ohm's law is assumed.