The shallow water equations do not necessarily have to describe the flow of water. They can describe the behaviour of other fluids under certain situations. For example we can think of the atmosphere as a fluid. The equations governing its behaviour are the Navier-Stokes equations; however, these are notoriously difficult to solve. The shallow water equations can be thought of as an approximation to the Navier-Stokes equations and are solved more readily. There are four points we need to consider (click on the info tabs below for details)
Equation of continuity...
Key laws used in fluid mechanics are continuity equations. For a constant density fluid the continuity equation specifies that there can be no divergence within the fluid. If there were then the fluid could not maintain constant density because mass would either accumulate or be removed from different parts of the fluid. Hence, we set the divergence to zero:
\begin{eqnarray}
\nabla\cdot \vec{v}&=&0\\
\frac{\partial u}{\partial x} +\frac{\partial v}{\partial y} +\frac{\partial w}{\partial z} &=&0
\end{eqnarray}
In this shallow water model we are only considering 2-d flow; however, there is a third dimension, which is the height of the fluid. We assume that vertical motions are the same throughout the depth of the fluid; hence, we can integrate the contunuity equation over
z between
H and
h+H:
\int _H ^{h+H} \frac{\partial u}{\partial x} dz+
\int _H ^{h+H} \frac{\partial v}{\partial y} dz+
\int _H ^{h+H} \frac{\partial w}{\partial z} dz=0
For the first two terms we apply a technique called
differentiation under the integral sign to find, e.g.:
\int _H^{H+h(x,y)} \frac{\partial u}{\partial x} dz = \frac{\partial }{\partial x}\left(\int _H^{H+h(x,y)} u dz \right)-u(x,y)\frac{\partial h}{\partial x}
We also note that the third term \int _H ^{h+H} \frac{\partial w}{\partial z} dz evaluates as w(H+h)-w(H) and since w(H)=0 this is equal to the total derivative of the top of the fluid wrt time, =\frac{\partial h}{\partial t}+u\frac{\partial h}{\partial x}+v\frac{\partial h}{\partial y}; hence:
\frac{\partial h}{\partial t}+\frac{\partial \left(hu\right)}{\partial x}+\frac{\partial \left(hv\right)}{\partial y}=0
This is the equation of continuity solved in the shallow water model; however, more equations need to be solved.
Equation for pressure...
We assume that the fluid is in hydrostatic balance. That is:
\frac{\partial P}{\partial z}=-\rho g
with
\rho equal to a constant.
The hydrostatic equation with constant density implies that we can describe the pressure at height
z as:
P\left(z\right)=-\rho g\left(z-H\right)+P_s\left(x,y,t \right)
where
H is the height of the orography. With the boundary condition that
P\left(z=h+H\right)=0 we may determine
P_s and find that:
P\left(z\right)=-\rho g\left(z-\left(H+h\right)\right)
Non-conservative Horizontal Momentum Equations...
The non-conservative horizontal momentum equations are (i.e. the Navier-Stokes equations in 2-d):
\begin{eqnarray}
\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y} &=& fv -\frac{1}{\rho}\frac{\partial P}{\partial x}\\
\frac{\partial v}{\partial t}+u\frac{\partial v}{\partial x}+v\frac{\partial v}{\partial y} &=& -fu -\frac{1}{\rho}\frac{\partial P}{\partial y}
\end{eqnarray}
Recall from above we found that the pressure field:
P\left(z\right)=-\rho g\left(z-\left(H+h\right)\right)
Now we use this equation to redefine the pressure derivatives on the right hand side of the momentum equations. We find that:
\begin{eqnarray}
-\frac{1}{\rho}\frac{\partial P}{\partial x} &=& -g\frac{\partial \left( h+H\right)}{\partial x}\\
-\frac{1}{\rho}\frac{\partial P}{\partial y} &=& -g\frac{\partial \left( h+H\right)}{\partial y}
\end{eqnarray}
Conservative Horizontal Momentum Equations...
In order to derive the remaining two equations we use the Non-conservative Horizontal Momentum Equations with the Equation of continuity to derive the conservative Horizontal Momentum Equations.
First we multiply the non-conservative horizontal momentum equations by h
\begin{eqnarray}
h\frac{\partial u}{\partial t}+hu\frac{\partial u}{\partial x}+hv\frac{\partial u}{\partial y} &=& fhv -hg\frac{\partial \left(h+H\right)}{\partial x}\\
h\frac{\partial v}{\partial t}+hu\frac{\partial v}{\partial x}+hv\frac{\partial v}{\partial y} &=& -fhu -hg\frac{\partial \left(h+H\right)}{\partial y}
\end{eqnarray}
We then add the product of
u and the continuity equation to the
u momentum equation (above) and the product of
v and continuity equation to the
v momentum equation (above). The
u momentum equation then becomes:
\begin{eqnarray}
h\frac{\partial u}{\partial t}+u\frac{\partial h}{\partial t}+u\frac{\partial \left(hu\right)}{\partial x}+u\frac{\partial \left(hv\right)}{\partial y}+ hu\frac{\partial u}{\partial x}+hv\frac{\partial u}{\partial y} &=& fhv -hg\frac{\partial \left(h+H\right)}{\partial x}
\end{eqnarray}
and there is a corresponding
v momentum equation.
We may gather some of the terms together by making use of the product rule for differentiation. The final result is the shallow water conservative momentum equations:
\begin{eqnarray}
\frac{\partial hu}{\partial t}+\frac{\partial \left(hu^2+\frac{gh^2}{2}\right)}{\partial x}+\frac{\partial uvh}{\partial y} &=& fhv -hg\frac{\partial \left(H\right)}{\partial x}\\
\frac{\partial hv}{\partial t}+\frac{\partial uvh}{\partial x}+\frac{\partial \left(hv^2+\frac{gh^2}{2}\right)}{\partial y} &=& -fhu -hg\frac{\partial \left(H\right)}{\partial y}
\end{eqnarray}
