The dynamics of wind-driven coastal upwelling and downwelling are studied using a simplified dynamical model. Exact solutions are examined as a function of time and over a family of sloping topographies. Assumptions in the two-dimensional model include a frictionless ocean interior below the surface Ekman layer and no along-slope dependence of the variables; however, dependence in the cross-slope and vertical directions is retained. Density and the along-slope component of momentum are advected by the cross-slope velocity, with thermal wind balance maintained at all times. The time-dependent initial value problem is solved with constant initial stratification and no initial along-slope flow. Previously, this model has been used to study upwelling over flat-bottomed ocean, but the novel features in this work are the study of exact solutions for a family of sloping topographic profiles, for both upwelling and downwelling. The exact solutions are compared to numerical solutions from a primitive equation ocean model configured in a similar twodimensional geometry. The exact solutions predict that deep undercurrents will develop only over steep topographic slopes and the cross-slope flow in the deep frictionless interior will be increasingly surface intensified as the topographic slope increases.