Shallow Water Equations

The Shallow Water equations, SWE in the following, are derived from the Navier Stokes equations. The fluid is assumed to have constant and uniform density. If the water movements are mainly horizontal and the effects of small scale velocity fluctuations are aggregated into shear-stress and viscosity terms, the equations of motion can be simplified in a two-dimensional system(Liu & Leendertse, 1978). An assumption in shallow water theory is that vertical scale H is much smaller than the horizontal scale L: L/H<<1 (Edlund, 1996; Neta, 1992).

A vertical integration (3.1) for the horizontal water speed is introduced to obtain the averaged horizontal velocity (shown in figure 3.1) that appears in the 2D equations, as in Liu & Leendertse (1978).

  

Figure 3.1: Speed profile.

$$\frac{\int_{-H_u}^{\eta} u dz}{H_u+\eta} = \overline{u} \qquad \frac{ \int_{-H_u}^{\eta} v dz}{H_u+\eta} = \overline{v}$$ (6.1)

Where: $\eta(x,y,t)$ is the water height above the reference level $\overline{H}$, H_u represents the depth below that level, $H = H_u + \eta$is the distance from sea bottom to water surface (see figure 3.1).

Shallow-Water equations set are (3.2) for mass balance and (3.3-3.4) for momentum conservation in (x,y)directions.

$$\frac{\partial \eta }{\partial t} + \frac{\partial H \overlin... ...{\partial x} + \frac{\partial H \overline{v}}{\partial y}= 0$$ (6.2)

$$\frac{\partial \overline{u}}{\partial t} + \overline{u}\frac{... ...frac{\tau _{sx}}{\rho H} - \varepsilon \Delta \overline{u}=0$$ (6.3)

$$\frac{\partial \overline{v}}{\partial t} + \overline{u}\frac{... ...\frac{\tau _{sy}}{\rho H} - \varepsilon \Delta \overline{v}=0$$ (6.4)

Where: $\overline{u}, \overline{v}$ are the velocities in the Eulerian coordinate system (x,y), g is the gravity acceleration, $\Omega=2 \omega \sin{\phi}$ is the coefficient giving place to Coriolis force, $\omega$ is earth's angular velocity, $\phi$the latitude, C_h=7.83 ln (0.37 H /Z₀) is bottom stress Chezy coefficient, Z₀ is the bottom roughness, $\rho$ is the sea water density, $\tau _{sx},\tau _{sy}$ (eq. 3.5) are the wind stress components at sea surface and $\varepsilon$ is the coefficient for turbulent viscosity.

$$\tau _{sx}=C_D\rho_aW_x^2 \qquad \tau _{sy}=C_D\rho_aW_y^2$$ (6.5)

W_x, W_y are wind velocity components measured 10 meters above sea level, $\rho_a$ is air density and C_D is the sea surface stress coefficient.