# CMS-Flow:Hydro Eqs

## Continuity and Momentum Equations

On the basis of the definitions Variable Definitions, and assuming depth-uniform currents, the general depth-integrated and wave-averaged continuity and momentum equations may be written as (Phillips 1977; Mei 1989; Svendsen 2006)

 $\frac{\partial h}{\partial t} + \frac {\partial(hV_j)} {\partial x_j} = S^M$ (1)
 $\frac{\partial(hV_i)}{\partial t} + \frac {\partial(hV_jV_i)}{\partial x_j} - \varepsilon_{ij}f_chV_j = -gh\frac{\partial \bar{\eta}}{\partial x_i} - \frac{h}{\rho} \frac{\partial p_{a}}{\partial x_i} + \frac {\partial}{\partial x_j} {\left(v_{t}h \frac {\partial V_i} {\partial x_j} \right)} - \frac{1}{\rho} \frac{\partial} {\partial x_j} \left(S_{ij} + R_{ij} - \rho h U_{wi}U_{wj} \right) + \frac{\tau_{si}}{\rho} - m_{b}\frac{\tau_{bi}}{\rho}$ (2)

where

t = time[s]
$x_j$ = Cartesian coordinate in the $j^{th}$ direction [m], j = 1,2 or x, y
$S^m =$ source term due to precipitation, evaporation and structures (e.g. culverts) [m/s]
$f_c = 2\Omega sin \phi =$Coriolis parameter [rad/s] in which $\Omega = 7.29 \ x \ 10^{-5}$ rad/s is the Earth’s angular velocity of rotation and $\phi$ is the latitude in degrees
$g =$ gravitational constant (~9.81 m/s2)
$p_a$ = atmospheric pressure [Pa]
$\rho =$ water density (~1025 kg/m3)
$v_t =$ turbulent eddy viscosity [m2/s]
$\tau_{si} =$ wind surface stress [Pa]
$S_{ij}$ = wave radiation stress [Pa]
$R_{ij}$ = surface roller stress [Pa]
$m_b$ = bed slope coefficient [-]
$\tau_{bi}$ = combined wave and current mean bed shear stress [Pa].

The above 2DH equations are similar to those derived by Svendsen (2006), except for the inclusion of the water source/sink term in the continuity equation and the atmospheric pressure and surface roller terms and the bed slope coefficient in the momentum equation. It’s also noted that the horizontal mixing term is formulated slightly differently as a function of the total flux velocity, similar to the Generalized Lagrangian Mean (GLM) approach (Andrews and McIntyre 1978; Walstra et al. 2000). This approach is arguably more physically meaningful and also simplifies the discretization in the case where the total flux velocity is used as the model prognostic variable.

# References

• Andrews, D. G., and M. E. McIntyre. 1978. An exact theory of nonlinear waves on a Lagrangian mean flow. Journal of Fluid Mechanics (89):609–646.
• Mei, C. 1989. The applied dynamics of ocean surface waves. New York: John Wiley.
• Phillips, O. M. 1977. Dynamics of the upper ocean, Cambridge University Press.
• Svendsen, I. A. 2006. Introduction to nearshore hydrodynamics, Advanced Series on Ocean Engineering, 124, World Scientific Publishing, 722 p.
• Walstra, D. J. R., J. A. Roelvink, and J. Groeneweg. 2000. Calculation of wave-driven currents in a 3D mean flow model. In Proceedings, 27th International Conference on Coastal Engineering, 1050-1063. Sydney, Australia.