A second-order analytic solution for oscillatory wind-induced flow in an idealized shallow lake

We present a second-order analytic solution to the nonlinear depth-integrated shallow water equations for free-surface oscillatory wind-driven flow in an idealized lake. Expressing the solution as an asymptotic expansion in the dimensionless wave amplitude (ζ/h), which is considered to be a small parameter, enables simplification of the governing equations and permits the use of a perturbation approach to solve them.This analytic solution provides a benchmark for testing numerical models. In particular, the main merit of this solution is that it accounts for advective effects, which are typically omitted from analytic solutions of two-dimensional free surface flow. In order to retain these effects in an analytic solution, we restrict our attention to forcing from a monochromatic wind stress, consider a constant depth rectangular lake, and simplify the governing equations by omitting the Coriolis and eddy viscosity terms and using a linearised friction factor. As such, the analytic solution is of limited use for considering real world problems. Due to the complexity of the analytic solution computer code for this solution is available online.Our solution is valid for cases where changes in the water surface level are small compared with the depth of the lake, and the advective terms in the momentum equations are small compared with acceleration terms. We examine the validity of these assumptions for three test cases, and compare the second-order analytic solution to numerical results to verify an existing hydrodynamic model.