MHD Equations

In MHD, the magnetic induction ${\bf B}$ and the plasma velocity ${\bf v}$ are determined by the equation of motion

$$\rho {D{\bf v}\over Dt} = \rho {\partial {\bf v}\over \part... ...){\bf v} = -\nabla p + {\bf J}\times {\bf B} + \rho {\bf g},$$ (4.1)

and the ideal induction equation

$${\partial {\bf B}\over \partial t} = \nabla \times ({\bf v}\times {\bf B}),$$ (4.2)

In addition, there is the equation of mass continuity

$${\partial \rho \over \partial t} + \nabla \cdot (\rho {\bf v}) = 0,$$ (4.3)

and the remainder of Maxwell's equations

$$\nabla \cdot {\bf B} = 0,$$ (4.4)

$${\bf J} = \nabla \times {\bf B}/\mu_{0},$$ (4.5)

plus

$$p = \rho R T,$$ (4.6)

and an energy equation. This may, for example, be taken as an isothermal plasma.

Note,

1. In the equation of motion we have an extra magnetic force, ${\bf J}\times {\bf B}$ (the Lorentz force), that is normal to both the magnetic field and the current. On using (4.5), this force may be written in the form
   

   $${\nabla \times {\bf B}\over \mu_{0}}\times {\bf B} = {1\ove... ...ght ){\bf B} - \nabla \left ({B^{2}\over 2\mu_{0}}\right ).$$ (4.7)

   
   
   The second term on the right hand side represents a magnetic pressure force, $B^{2}/2\mu_{0}$, acting from regions of high magnetic pressure to low magnetic pressure. The first term is interpreted as a magnetic tension force which acts when the field lines are curved. Magnetic tension is similar to the force exerted by a stretched elastic band. Interpreting the Lorentz force in terms of these two basic quantities is important since it is possible to see how the plasma will respond to a given magnetic field configuration.

2. In the Maxwell equation,
   
   $${\nabla \times {\bf B}\over \mu_{0}} = {\bf J} + {\partial \over \partial t}\left (\epsilon_{0}{\bf E}\right ),$$
   
   
   which is called Ampère's Law, the last term is negligible, except for extremely rapid time variations. Dropping this term gives (4.5). In order of magnitude, the Maxwell equation,
   
   $$\nabla \times {\bf E} = -{\partial {\bf B}\over \partial t},$$
   
   
   which is called Faraday's Induction Law, can be approximated by $E/L = B/t$, where $t$ and $L$ are typical time and length variations of the plasma. Comparing the left hand side with the last term (the displacement current) in Ampère's law, we find that
   
   $${B\over L} >> \mu_{0}\epsilon_{0}{E\over t}.$$
   
   
   Thus,
   
   $$c^{2} >> {L^{2}\over t^{2}} = V^{2},$$
   
   
   where we have used
   
   $$c^{2} = 1/\mu_{0}\epsilon_{0}$$
   
   
   for the speed of light and the typical plasma velocity is $V = L/t$. Hence, (4.5) is valid provided the plasma flows are small compared with the speed of light.

3. A plasma moving with a velocity ${\bf v}$ experiences an electric field ${\bf v}\times {\bf B}$ in addition to ${\bf E}$ so that Ohm's Law is
   

   $${\bf J} = \sigma\left ({\bf E} + {\bf v}\times {\bf B}\right ).$$ (4.8)

   
   
   Normally, the left hand side is negligible so that (4.8) reduces to
   

   $$0 ={\bf E} + {\bf v}\times {\bf B}.$$ (4.9)

   
   
   Using (4.9) to eliminate ${\bf E}$ the Maxwell equation
   
   $$\nabla \times {\bf E} = -{\partial {\bf B}\over \partial t},$$
   
   
   reduces to (4.2).