  In this paper, high order semi-implicit well-balanced and asymptotic
preserving finite difference WENO schemes are proposed for the shallow water
equations with a non-flat bottom topography. We consider the Froude number
ranging from O(1) to 0, which in the zero Froude limit becomes the "lake
equations" for balanced flow without gravity waves. We apply a well-balanced
finite difference WENO reconstruction, coupled with a stiffly accurate
implicit-explicit (IMEX) Runge-Kutta time discretization. The resulting
semi-implicit scheme can be shown to be well-balanced, asymptotic preserving
(AP) and asymptotically accurate (AA) at the same time. Both one- and
two-dimensional numerical results are provided to demonstrate the high order
accuracy, AP property and good performance of the proposed methods in capturing
small perturbations of steady state solutions.
