Why Euler equation in Ideal MHD is not identically zero?

Question

In Ideal MHD, we assume the plasma to be force-free:

$$\vec{E} = - \frac{1}{c}\left(\vec{v}\times \vec{B}\right)$$

But the Euler equation of motion is written as:

$$\rho \frac{d\vec{u}}{dt} = \vec{j}\times\vec{B}$$

assuming a cold plasma, where $d\vec{u}/dt$ is the material derivative. I would expect it to be 0 on the RHS due to the force-free condition, but apparently not. Why?

Answer 1

The term "force free" can be applied in one of two ways:

1. $\mathbf{j}\times\mathbf{B}=0$ (see this Wikipedia article)
2. $\mathbf{F}=\mathbf{E}+\mathbf{v}\times\mathbf{B}=0$ (or $\sigma\to\infty$) (see this Wikipedia article or this Physics.SE Q&A)

When dealing with fluid dynamics (MHD), we typically mean the latter of the two. This allows you to replace all electric fields with $-\mathbf{v}\times\mathbf{B}$ and deal only with the magnetic fields. This also means that, $$\mathbf{j}\sim\nabla\times\mathbf{B}\implies\mathbf{B}\times(\nabla\times\mathbf{B})\sim\nabla\vert\mathbf{B}\vert^2-\left(\mathbf{B}\cdot\nabla\right)\mathbf{B}\neq0$$ as the magnetic field must also feed back into the fluid as a pressure component to interact dynamically (rather than passively) with the fluid.