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 shallow water equations. 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 vertical shear in velocity (\(\frac{\partial u}{\partial z}=\frac{\partial v}{\partial z}=0\)), and the hydrostatic equation (Eq. (2.6)):

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

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 start by integrating the Hydrostatic balance Eq. (2.6) from a chosen level \(z\) to the free surface:

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

We can now differentiate (2.14)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 (\(p_0\)).

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

By replacing the pressure in Eq. (2.4) with the expressions in Eq. (2.15), we are done with the Navier-Stokes derivations.

We can now 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.

(2.16)\[\begin{split}\begin{aligned} \int_{-h}^\eta\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}+\frac{\partial w}{\partial z}dz=0\\ w(\eta)-w(-h)+\int_{-h}^\eta\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}dz=0\\ \frac{\partial\eta}{\partial t}+\int_{-h}^\eta\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}dz=0\\ \frac{\partial\eta}{\partial t}+\frac{\partial}{\partial x}\int_{-h}^\eta u\, dz+\frac{\partial}{\partial y}\int_{-h}^\eta v\, dz=0\\ \frac{\partial\eta}{\partial t}+\frac{\partial}{\partial x}u(\eta+h)+\frac{\partial}{\partial y}v(\eta+h)=0 \end{aligned}\end{split}\]

We now arrive at the shallow-water equation:

The Shallow Water Equations

(2.17)\[ \begin{align}\begin{aligned}\begin{aligned}\\\begin{split}\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}-fv=-g\frac{\partial \eta}{\partial x}\\\end{split}\\\begin{split}\frac{\partial v}{\partial t}+u\frac{\partial v}{\partial x}+v\frac{\partial v}{\partial y}+fu=-g\frac{\partial \eta}{\partial y}\\\end{split}\\\frac{\partial\eta}{\partial t}+\frac{\partial}{\partial x}u(\eta+h)+\frac{\partial}{\partial y}v(\eta+h)=0\\\end{aligned}\end{aligned}\end{align} \]

If we assume that the wave height \(\eta<<h\), we can neglect \(\eta\) in the quadratic terms. We could also assume that the variations in \(h\) is small, leaving us with a constant water depth H that can be moved outside the differentiation. The simplified shallow water waves now becomes:

(2.18)\[ \begin{align}\begin{aligned}\begin{aligned}\\\begin{split}\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}-fv=-g\frac{\partial \eta}{\partial x}\\\end{split}\\\begin{split}\frac{\partial v}{\partial t}+u\frac{\partial v}{\partial x}+v\frac{\partial v}{\partial y}+fu=-g\frac{\partial \eta}{\partial y}\\\end{split}\\\frac{\partial\eta}{\partial t}+H(\frac{\partial}{\partial x}u+\frac{\partial}{\partial y}v)=0\\\end{aligned}\end{aligned}\end{align} \]

We can continue simplifying the equations by assuming some of the terms to be small/negligible. We could, e.g., assume geostropic balance and remove the advection terms. We could also remove the coriolis terms if we are looking at a smaller area where the effect of Earth rotation is small compared to the other terms.

Shallow Water gravity waves

Here we assume Earth rotation and advection terms are negligible:

(2.19)\[ \begin{align}\begin{aligned}\begin{aligned}\\\begin{split}\frac{\partial u}{\partial t}=-g\frac{\partial \eta}{\partial x}\\\end{split}\\\begin{split}\frac{\partial v}{\partial t}=-g\frac{\partial \eta}{\partial y}\\\end{split}\\\frac{\partial\eta}{\partial t}+H(\frac{\partial}{\partial x}u+\frac{\partial}{\partial y}v)=0\\\end{aligned}\end{aligned}\end{align} \]

Kelvin waves

If we assume a shallow water wave travelling meridionally (north/south), and no advection terms, we can assume that the eastward velocity is zero and arrive at the Kelvin wave equations:

(2.20)\[ \begin{align}\begin{aligned}\begin{aligned}\\\begin{split}-fv=-g\frac{\partial \eta}{\partial x}\\\end{split}\\\begin{split}\frac{\partial v}{\partial t}=-g\frac{\partial \eta}{\partial y}\\\end{split}\\\frac{\partial\eta}{\partial t}+H\frac{\partial}{\partial y}v=0\\\end{aligned}\end{aligned}\end{align} \]

A Kelvin wave always travel with a coast to “lean on”. On the northen hemisphere, the coast is to the right of the kelvin wave propagation (looking in the direction the wave is moving). On the southern hemisphere, the coast is to the left of the propagation direction. A special case of Kelvin waves can also occur at the equator. Here, you can have one Kelvin wave on either side of the equator, leaning against each other. Whivh direction do you think these equator Kelvin waves will propagate? East or West?