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Suppose now we consider the shallow water model of section (8.3).

p=p
s
z=h z
g
H ρ = constant

h
b
x

Figure 10.2.1 The shallow water model.

The principal dynamical simplification of such a model is that the vertical velocity is

constrained by the geometry to be so small that the vertical acceleration term in the
vertical equation o f motion is negligible and this allows the pressure to be calculated

using the hydrostatic approximation as discussed in chapter 9. Thus

!p
= "#g (10.2.6)
!z

and for a fluid of constant density this can be integrated immediately,
p = !"gz + C(x,y,t) (10.2.7)

To determine the “constant” C we apply the condition that the pressure match the

atmospheric pressure p (x,y,t) on the surface z=h(x,y,t)or
s

p = !g(h " z) + p (10.2.8)
s

For steady flow the momentum equations in the horizontal direction are,

Chapter 10 8