Ideal MHD equilibrium equations

[vibe3]

I have a question about the ideal equilibrium MHD equation:

$$\vec{J} \times \vec{B} = \nabla p$$

where $\vec{J}$ is the current, $\vec{B}$ is the magnetic field, and $p$ is the plasma pressure.

Does the magnetic field here represent the total magnetic field (any applied external fields plus fields due to the plasma currents?) Or does it represent only the plasma current fields?

I'm asking because one consequence of this equation is:

$$\vec{B} \cdot \nabla p = 0$$

But if you consider a system with an arbitrarily strong applied external field, and a low plasma density, this condition will certainly not be true. (ie: the small plasma currents will not be strong enough to change the external field enough to make this condition true)

 

[Valerie Park]

The magnetic field in the ideal equilibrium MHD equation represents only the magnetic field due to the plasma currents. It does not include any applied external fields. As a result, the equation \vec{B} \cdot \nabla p = 0 will be true for any system with no external magnetic field present.