7
z
A
- - - -----,. --- _ .
If\ h(x,y,l)
H(x,y)
Figure 2: The flow domain
1.2.1 The shallow water equations
In this section we recall the shallow water equations (see for instance Abbot [1980], Rahman [1988], Stoker [1957]) which are a useful model for the flow of water in a shallow
domain, i.e. a domain with a small depth compared to its other dimensions.
Consider an incompressible fluid in a shallow domain defined by (see figure 2).
where n is the Xl> X2 projection of the domain filled by the fluid, h( Xl, X2, t) is the height
of the fluid layer at point (Xl, X2) at time t, X3 = b(xl> X2) is the equation of the bottom
surface and H( Xl> X2) = A - b( Xl> X2) is the depth from a fixed reference level A. Whithout
loss of generality we can assume H is non-negative. The shallow water equations can
be obtained from the incompressible Navier-Stokes equations by assuming pressure is
hydrostatic and integrating in depth. Taking into account the kinematic and kinetic
conditions both at the free boundary and at the bottom we get
(1.14)