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The Shallow Water System

The Shallow Water System

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The first example we will work with are the shallow water equations. These are simplifications of the Navier-Stokes equation when the height of the fluid region is small compared to its length, and they are extremely important as model equations for planetary oceans and atmospheres.

The shallow water equations

We begin with he incompressible Navier-Stokes equations with constant density:

(41)\[\begin{align} & \p_t u + u \p_x u + v \p_y u + w \p_z u = - \frac{1}{\rho} \p_x p\\ & \p_t v + u \p_x v + v \p_y v + w \p_z v = - \frac{1}{\rho} \p_y p\\ & \p_t w + u \p_x u + v \p_y w + w \p_z u = - \frac{1}{\rho} \p_z p - g\\ &\p_x u \p_y v + \p_z w = 0 \end{align}\]

We consider a fluid extending from \(z = 0\) to height \(h(x,y,t)\) (in general we can also work with nontrivial bottom topography, but we will for now consider flat bottoms.) The boundary condiions are

(42)\[\begin{align} w(x,y,0,t) & = 0\\ w \Big|_{z = h} & = \frac{d}{dt} h(x,y,t) = \p_t h + u\p_x h + v \p_x h \end{align}\]

where in the second line we have used he fact that \(w\) a the surface should be the velocity of a fluid parcel

We first convert the incompressibility condition to an equation for the height by integrating the former over the entire vertical water column:

\[ \int_0^h dz {\vec\nabla}\cdot{\vec u} = 0 = \int_0^{h} dz (\p_x u + \p_y v) + w(x,y,\eta,t) \]

where we have used the boundary conditions. Next, we can pull the horizontal deivatives outside of the integral if we subtract their action on the integration limits.

\[ w + \p_x \int_0^{h} dz u + \p_y \int_0^{h} dz v - u \p_x h - v \p_y h = 0 \]

Substituting the top boundary conditions, we have

\[ \p_t h + \p_x \int_0^{h} dz u + \p_y \int_0^{h} dz v = 0 \]

Next we start to make some approximations. Assuming that characteristic vertical scales are of order \(H\), horizonal scales of order \(L\), and time scales of order \(T\), we set \(w \sim H/T\), \(u, v \sim L/T\). Thus \(w\) is considerably smaller than \(u,v\). There is no reason for the pressure term to drop out, so we rplace the momentum equation for \(w\) with the hydrostatic condition (we will justify this self-consistently at the end). We can then set \(\p_z p = - \rho g\), or \(p = p_0 + \rho g (h - z)\). The horizontal momentum equations become, after we drop the terms proportional to u:

(43)\[\begin{align} & \p_t u + u \p_x u + v \p_y u = - g \p_x h\\ & \p_t v + u \p_x v + v \p_y v = - g \p_y h \end{align}\]

Since the right hand side is independent of \(z\), we can start with \(u,v\) constant in \(z\), and it will continue to be so. Then the integrals in our conservation condition just become multupilication by \(h\) and we have

\[ \p_t h + \p_x (h u) + \p_y (h v) = 0 \]

Note that this appears as a conservation equation much like that of the density, in the compressible case. We do not impose incompressibility on the horizontal velocities; a divergence will be compensated by a change in heigh, so that the total mass of the fluid is conserved.

These are the shallow water equations. Now note that if we consider the scaling of the momentum equations, then balancing the advective derivative against the bouyancy forcing we have \(u^2 \sim g H\), and matching this to the first term, a time scale of \(T^{-1} \sim u/L \sim \sqrt{g H}/L\). We then have \(w \sim H/T \sim (H/L) \sqrt{g H} \ll \sqrt{g H} \sim u\), and our approximations are self-consistent.

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