Vorticity equations of incompressible Navier-Stokes equations in 2D We know for incompressible Navier-Stokes equations, we have the vorticity equation: $$\omega_t - \Delta \omega + (u \cdot \nabla)\omega = (\omega \cdot \nabla)u$$
But for two dimensional space, $(u \cdot \nabla)\omega $. I don't see why after I plug in the expression of $\omega$. (Here $\omega = \partial_1 u_2 - \partial_2 u_1$)

A solution to the 2D Navier-Stokes equations can be realised as a special kind of solution to 3D Navier-Stokes, namely one for which
$$ u(x,y,z) = (u_1(x,y),u_2(x,y),0) , \quad w(x,y,z)=\nabla \times u(x,y,z) =(0,0,\omega(x,y)).$$
Here, $\omega = \operatorname{curl}_{\text{2D}}(u_1,u_2). $ See for instance here.
Plugging this into the above vorticity equation, we notice in particular that 
$$ w\cdot \nabla = \begin{pmatrix} 0 \\ 0 \\ \omega  \end{pmatrix}\cdot \begin{pmatrix} \partial_1 \\ \partial_2 \\ \partial_3  \end{pmatrix}  = \omega\partial_3$$
So since $u$ does not depend on $x_3$, the term $(w\cdot\nabla) u $ vanishes.
