  We study the asymptotic dynamics of the lake equations in the following two
cases, an island shrinking to a point and an emerging island. For both cases,
we derive an asymptotic lake-type equation. In the former case, the asymptotic
dynamics includes an additional Dirac mass in the vorticity. The main
mathematical difficulty is that the equations are singular when the water depth
vanishes. We provide new uniform estimates in weighted spaces for the related
stream functions which will imply the compactness result.
