The depth-averaged shallow-water vorticity equation is often separated to vorticity-tarnsport, -source, and -sink terms. If one would like to know the total effects of the topography on the equilibrium described by the steady vorticity equation, the variable should be used is the stream function of the depth-integrated velocity field. We deduce the stream-function - topography form of the steady depth-averaged vorticity-advection and -diffusion equation with illustrative simple solutions on the topography effects. We also analyze a wind-induced flow model on shallow-lake scale where an evolving underwater hill results in a bifurcation of the steady-states.