Small scale dynamics in incompressible fluid flows

1. Nonlinear enhanced dissipation for the 2D Euler equation with horizontal viscosity

Consider the 2D Euler equation with horizontal viscosity on $\mathbb{T}\times [0,1]$: $$\partial_t \omega + u \cdot \nabla \omega = \nu \partial_{xx} \omega,$$ $$u=\nabla^\perp\Delta^{-1}\omega.$$ How large can $\nu \int_\Omega |\partial_x \omega|^2$ get as $\nu\to 0$ and $t\rightarrow\infty$?