Interaction between depth variation and turbulent diffusion in depth-averaged vorticity equations

Steady, depth-averaged, shallow water vorticity transport equations including advection, surface and bed shear stresses, and turbulent diffusion effects are written out in vorticity-velocity and stream function formalisms. The Boussinesq approximation is used to represent turbulent stresses in the effective stress tensor. We consider two different forms of the curl of the effective stress tensor: its complete form and the commonly used form neglecting the terms expressing interaction with variable water depth. After deriving the two equations in vorticity-velocity formalism, we recast the equations into stream function formalism, revealing all the internal effects associated with variable water depth. We examine the differences between the models through analytical solutions of the stream function equations for simple but realistic flows. The solutions are validated with CFD simulations.