So as showed by Frisch *et al.* (a), the 2D Euler equation $$v_{t}+ v\cdot \nabla v=\mu \Delta v$$ can be derived by the Hexagonal-placed automaton (for low velocity).

I am curious about the existence of similar derivations for the following equations:

the vorticity form with $\omega=\nabla \times v$: $$\omega_{t}+ v\cdot \nabla \omega=\mu \Delta \omega$$

The vorticity form with white noise: $$\omega_{t}+ v\cdot \nabla \omega=\mu \Delta \omega+\phi$$

Did I miss any obvious references that contain these derivations? Thank you.

*Remark:* Once we have a derivation for $\omega_{t}+ v\cdot \nabla \omega=\mu \Delta \omega$, then the force can be represented by adding a momentum at each site (see Rothman (b) for details). So (2) should follow from (1).

*Update*: I found this paper from 2011 "A lattice Boltzmann model for the eddy–stream equations in two-dimensional incompressible flows", which seems to tackle this problem. But I still would like to see any other more detailed treatments or extensions because I don't want to rely on a single paper.

**References**

(a) Uriel Frisch, Brosl Hasslacher and Yves Pomeau. *Lattice-gas automata for the Navier-Stokes equation.* Physical Review Letters **56**, 14 (1986) 1505.

(b) Daniel H. Rothman and Stiphane Zaleski. *Lattice-gas cellular automata: simple models of complex hydrodynamics.* Vol. 5. Cambridge University Press, 2004.

(c)Yan, Bo, et al. "A lattice Boltzmann model for the eddy–stream equations in two-dimensional incompressible flows." Applied Mathematical Modelling 35.5 (2011): 2358-2365.