Question about conditions for Force Free Fields in plasma

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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?
 
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9,206
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.
 
38
3
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.
Ok, i see! Many thanks for the help :D
 

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