2.5. The Shallow water equations

The shallow water equations describe waves that have long wave lengths compared to the water depth. The name suggests that these waves occur over shallow water. When surface gravity waves arise in the open ocean, they does not follow the shllow water equtions. However, when the waves reach a sloping beach, they will start to behave like shallow water waves when the water column thickness becomes small relative to the wave lengths.

Shallow water waves an also occur in deep waters, when the wave lengths are sufficently long. See Fig. 2.3 for a sketch of the shallow water wave.

Fig. 2.3 Sketch of a shallow water wave. The bottom depth is given by \(h(x,y)\) and the surface elevation is given by \(\eta(x,y,t)\).

To derive the shallow water equations, we start with the continuity equation for incompressible fluids (Eq. (2.3)), the Navier stokes equation (Eq. (2.4)), neglecting friction, and the hydrostatic equation (Eq. (2.6)):

(2.12)\[\begin{split}\begin{aligned} \frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}+\frac{\partial w}{\partial z}&=0\\ \frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}+w\frac{\partial u}{\partial z}-fv&=-\frac{1}{\rho_0}\frac{\partial p}{\partial x}\\ \frac{\partial v}{\partial t}+u\frac{\partial v}{\partial x}+v\frac{\partial v}{\partial y}+w\frac{\partial v}{\partial z}+fu&=-\frac{1}{\rho_0}\frac{\partial p}{\partial y}\\ g\rho&=-\frac{\partial p}{\partial z} \end{aligned}\end{split}\]

The vertical boundary conditions can be expressed as:

(2.13)\[\begin{split}\begin{aligned} \frac{D\eta}{Dt}&&=\frac{\partial \eta}{\partial t}+u\frac{\partial \eta}{\partial x}+v\frac{\partial \eta}{\partial y}=w(z=\eta)\qquad \text{Surface boundary conditions}\\ \bar{u}\cdot\nabla(z+h(x,y))&&=0\qquad \text{at} z=-h(x,y)\qquad\qquad \text{Bottom boundary conditions} \end{aligned}\end{split}\]

We can integrate the continuity equation over the water column depth:

Note

\[ w(\eta)=\frac{\partial \eta}{\partial t} \]

\[ w(-h)=0 \]

For a continuous function, the integral and derivation order can be swapped.

(2.14)\[\begin{split}\begin{aligned} \int_{-h}^\eta\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}+\frac{\partial w}{\partial z}dz\\ w(\eta)-w(-h)+\int_{-h}^\eta\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}dz\\ \frac{\partial\eta}{\partial t}+\int_{-h}^\eta\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}dz\\ \frac{\partial\eta}{\partial t}+\frac{\partial}{\partial x}\int_{-h}^\eta u\, dz+\frac{\partial}{\partial y}\int_{-h}^\eta v\, dz\\ \frac{\partial\eta}{\partial t}+\frac{\partial}{\partial x}u(\eta+h)+\frac{\partial}{\partial y}v(\eta+h) \end{aligned}\end{split}\]

Now, we integrate the Hydrostatic balance Eq. (2.6) from a chosen level \(z\) to the free surface:

(2.15)\[\begin{split}\begin{aligned} \int_z^\eta g\rho\,dz=-\int_z^\eta\frac{\partial p}{\partial z}\,dz\\ p(\eta)-p(z)=-g\rho_0(\eta-z)\\ p(z)=p(\eta)+g\rho_0(\eta-z) \end{aligned}\end{split}\]

We can differentiate (2.15)with respect to \(x\) and \(y\) to find the right hand sides of the horizontal Navier-Stokes equation. Note that \(p(\eta)\) is a constant, typically set as a reference pressure of zero (\(p_0\)).

(2.16)\[\begin{split}\begin{aligned} \frac{\partial p}{\partial x}=\frac{\partial}{\partial x}(p_0-g\rho_0(\eta-z))=-g\rho_0\frac{\partial \eta}{\partial x}\\ \frac{\partial p}{\partial y}=\frac{\partial}{\partial y}(p_0-g\rho_0(\eta-z))=-g\rho_0\frac{\partial \eta}{\partial y} \end{aligned}\end{split}\]

\( \frac{\partial p}{\partial y}=\frac{\partial}{\partial y}(p_0-g\rho_0(\eta-z))=-g\rho_0\frac{\partial \eta}{\partial y} \)

If we further assume no vertical shear in the horizontal velocities (\(\frac{\partial u}{\partial z}=\frac{\partial v}{\partial z}=0\)), the shallow water equations become:

definition:SW)=

The Shallow Water Equations

(2.17)\[ \begin{align}\begin{aligned}\begin{aligned}\\\begin{split}\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}-fv=-g\frac{\partial \eta}{\partial x}\\\end{split}\\\begin{split}\frac{\partial v}{\partial t}+u\frac{\partial v}{\partial x}+v\frac{\partial v}{\partial y}+fu=-g\frac{\partial \eta}{\partial y}\\\end{split}\\\frac{\partial\eta}{\partial t}+\frac{\partial}{\partial x}u(\eta+h)+\frac{\partial}{\partial y}v(\eta+h)\\\end{aligned}\end{aligned}\end{align} \]