The Shallow Water Equations
Figure 10-2 shows a typical configuration for the flow of fluid in a shallow layer. The upper boundary is referred to as the free surface and the lower boundary of the water layer is referred to as the bottom. The bottom height, hb, is measured with respect to a reference xy-plane placed at z = 0 and it is assumed to be constant in time. The total height of the free surface is denoted H. The height of the water layer is typically measured in relation to the bottom height and denoted h. In dry regions, h = 0 and H = hb.
The equations of continuity and momentum of incompressible inviscid flow subjected to gravity in the vertical direction () are
(10-1)
(10-2)
(10-3)
(10-4)
Figure 10-2: An example illustrating a typical configuration for shallow water equations. A water layer of thickness h flows over a nonflat bottom with topography represented by hb. The total height of the free surface over the reference xy plane at z = 0 is represented by H. The gravity points downward (−z direction) and gives rise to a hydrostatic pressure ranging from 0 atm at the free surface to a value of at the bottom.
Assuming that the water depth h is much smaller than the lateral dimensions of the geometry, the vertical accelerations can be neglected, , and Equation 10-4 reduces to the hydrostatic pressure relation
(10-5)
Both h(x, y) and hb(x, y) are independent of z, implying that . Equation 10-2 and Equation 10-3 then reduces to
(10-6)
(10-7)
which is the momentum equation of the 2D shallow water equations in nonconservative form. Note that u and v represent the horizontal velocity components averaged over h.
The continuity equation is recovered from Equation 10-1 when assuming no penetration on both the free surface and bottom boundaries:
(10-8)
See Chapter 2 in Ref. 1 for more details. Equation 10-6, Equation 10-7, and Equation 10-8 can be combined to express the shallow water equations in conservative form
(10-9)
(10-10)
qx and qy are the components of the water flux in the x and y directions, respectively. The vector can be obtained from the horizontally averaged velocity as
(10-11)
Two different sets of variables are used in the context of the shallow water equations: the conservative variables h and q, and the primitive variables h and u.