  We compute the generation of vorticity from velocity dispersion in the dark
matter fluid. For dark matter at zero temperature Helmholtz's theorem dictates
that no vorticity is generated and we therefore allow the dark matter fluid to
have a non-vanishing velocity dispersion. This implies a modification to the
usual hydrodynamical system (continuity and Euler equations): we have to
consider the Boltzmann hierarchy up to the second moment. This means that the
Euler equation is modified with a source term that describes the effect of
non-zero velocity dispersion. We write an equation for the Eulerian vorticity
in Lagrangian coordinates and show that it has a growing mode already at second
order in perturbation theory. We compute the power spectrum of the vorticity
and the rotational velocity at second order in perturbation theory.
