LAUSR

The lake equations $$\begin{aligned} \left\{ \begin{aligned}&\nabla \cdot \big ( b \, {\mathbf {u}}\big ) = 0&\text {on}\ {\mathbb {R}}\times D , \\&\partial _t{\mathbf {u}} + ({\mathbf {u}}\cdot \nabla ){\mathbf {u}} = -\nabla h&\text {on}\ {\mathbb {R}}\times D , \\&{\mathbf {u}} \cdot \varvec{\nu } = 0&\text {on}\ {\mathbb {R}}\times \partial D \end{aligned} \right. \end{aligned}$$ ∇ · ( b u ) = 0 on R × D , ∂ t u + ( u · ∇ ) u = - ∇ h on R × D , u · ν = 0 on R × ∂ D model the vertically averaged horizontal velocity in an inviscid incompressible flow of a fluid in a basin whose variable depth $$b : D \rightarrow [0, + \infty )$$ b : D → [ 0 , + ∞ ) is small in comparison to the size of its two-dimensional projection $$D \subset {\mathbb {R}}^2$$ D ⊂ R 2 . When the depth b is positive everywhere in D and constant on the boundary, we prove that the vorticity and energy of solutions of the lake equations whose initial vorticity concentrates at an interior point behaves asympotically a multiple of a Dirac mass whose motion is governed by the depth function b .