If the curl of a velocity field (Vorticity) is twice the local angular velocity, then what is the curl of the vorticity related to?

Question

The curl of the velocity field $\bar V(\bar x,t )$ is known to be related to the local angular velocity $\bar \omega (\bar x,t) $ by the expression

$$\bar \omega = \frac{1}{2}\nabla \times \bar V $$

Is the curl of the vorticity related to any or all of the local acceleration $a = \partial_{t} \bar V $, the local angular acceleration $\alpha = \partial_{t} \bar \omega$ and/or the local gyroscopic/angular precession $\bar L (\bar x,t)$ ?

If so, is there a general expressions similar to the above to describe such a relationship?

$$\nabla \times \bar \omega = ?$$