Question about conditions for Force Free Fields in plasma

1. Jan 20, 2016

Clear Mind

My question is about Force Free fields in the study of plasma stability (in MHD regime): Consider an isolated ideal plasma in an equilibrium state (where the effect of selfgravity is negligible), from the Navier-Stokes equation we get that:

$$\vec{\nabla} P = \frac{1}{c} \vec{J} \times \vec{B}$$

Now, if $P=const$ and $\vec{J}$ (in MHD $\vec{J}\propto\vec{\nabla}\times\vec{B}$) is parallel to $\vec{B}$, we get that $(\vec{\nabla} \times \vec{B}) \times \vec{B}=0$. Thus implies that:

$$(\vec{\nabla} \times \vec{B}) = \alpha(r) \vec{B}$$

That is the condition for a Free-Froce fields. So ... the question is, shouldn't be the curl of a vector always be orthogonal to the vector?

2. Jan 20, 2016

Staff: Mentor

The cross-product is orthogonal to the two vectors, but the curl is not a proper cross-product. It can have a component along the direction of the vector. This is easy to see if you add a constant to B: its curl won't change, but you can change the direction of B arbitrarily.

3. Jan 21, 2016

Clear Mind

Ok, i see! Many thanks for the help :D

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