  In this article we discuss flows in shallow, stratified horizontal layers of
two immiscible fluids. The top layer is an electrolyte which is
electromagnetically driven and the bottom layer is a dielectric fluid. Using a
quasi-two-dimensional approximation, we derive the depth-averaged
two-dimensional (2D) vorticity equation which includes a prefactor to the
advection term, previously unaccounted for. In addition, we study how the
horizontal components of velocity vary in the vertical direction. For a
Kolmogorov-like flow, we evaluate analytical expressions for the coefficients
in the generalized 2D vorticity equation, uncovering their dependence on
experimental parameters. To test the accuracy of these estimates, we
experimentally measure the horizontal velocity fields at the free-surface and
at the electrolyte-dielectric interface using particle image velocimetry (PIV).
We show that there is excellent agreement between the analytical predictions
and the experimental measurements. Our analysis shows that by increasing the
viscosity of the electrolyte relative to that of the dielectric, one may
significantly improve the uniformity of the flow along the vertical direction.
