We present and discuss new shallow-water equations that model
the long-time effects
of slowly varying bottom topography and weak hydrostatic imbalance on the
vertically
averaged horizontal velocity of an incompressible fluid
possessing a free surface and
moving under the force of gravity. We consider the regime where the Froude
number
ε is much smaller than the aspect ratio δ of the
shallow domain. The new equations are obtained from the
ε→0 limit of the Euler equations (the rigid-lid approximation)
at the first order of an asymptotic expansion in δ2.
These equations possess local
conservation laws of energy and vorticity which reflect exact layer mean
conservation
laws of the three-dimensional Euler equations. In addition, they convect
potential
vorticity and have a Hamilton's principle formulation.
We contrast them with the Green–Naghdi equations.
Zeroth‐order conservation laws of two‐dimensional shallow water equations with variable bottom topography
