If $v : \mathbb{R}^3 \to \mathbb{R}^3$ is smooth, periodic ,divergence-free and of unit norm, can we say anything about $(\nabla \times v) \times v$?

Question

The question is as in the title.

Let $v : \mathbb{R}^3 \to \mathbb{R}^3$ be a smooth vector field that is periodic on the $3-$dimensional torus $\mathbb{T}^3$.

Further assume that $\lVert v(x) \rVert=1$ for all $x \in \mathbb{R}^3$ and $\nabla \cdot v=0$.

Then, can we say anything about $(\nabla \times v) \times v$? I found out that it is equal to $(v \cdot \nabla)v$ but cannot do anything further.

I wish it were just zero, but cannot find a way to prove or disprove..

Could anyone please help me?