For the $2d$ KPZ eqn:
$$h_{t}=\mu\Delta h+\frac{1}{2}(\nabla h)^{2}+\eta$$
there are various growth discrete models that converge to it; for example TASEP and lozenge tilings.
By taking the curl of
$$v=-\nabla h$$
we obtain the $2d$ Burgers eqn
$$v_{t}+v\cdot \nabla v=\mu \Delta v+\nabla \eta$$ and $\nabla \times v=0$. So $v$ is interpreted as the slope of the growth pyramids.
Any suggested analogous discrete growth models for 2d Euler: $v_{t}+v\cdot \nabla v=\mu \Delta v+\nabla \eta$ and $\omega=\nabla \times v=-\Delta \psi$ is the vorticity.
In other words, is there a corresponding "KPZ" and discrete growth model for 2d Euler as we have for 2d Burgers?
