- #1
Clear Mind
- 38
- 3
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?
$$\vec{\nabla} P = \frac{1}{c} \vec{J} \times \vec{B}$$
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?