Three-dimensional time-dependent water flows with constant non-vanishing vorticity and depth dependent density

We show that the movement of a time-dependent gravity water flow with constant non-zero vorticity and continuously depth dependent density satisfying the three-dimensional water wave equations is essentially two-dimensional: the velocity field, the pressure and the free surface do not change in the direction orthogonal to the direction of propagation. Our result is true both for the inviscid as well as for the viscous water wave problem.