Beltrami flows have zero parallel gradients?

[TheCanadian]

In the following wiki article for Beltrami flows, it is stated that the nonlinear terms are identically zero. I can easily prove the terms are equivalent and thus cancel one another, to yield the linear equation. But after a bit of algebra, I don't see why the terms are zero themselves? Why would a vorticity vector being parallel with velocity result in the parallel gradients going to 0?

 

[fresh_42]

I think this is a mistake. It should be $\displaystyle (\mathbf {v} \cdot \nabla ){\boldsymbol {\omega }}-({\boldsymbol {\omega }}\cdot \nabla )\mathbf {v} =0$ which is all we need.

Otherwise there should be a physical meaning behind $J_\mathbf{v} \cdot \mathbf{v} =\mathbf{0}$.