# Why can't wave models model storm surge?

I found this nice module here on MetEd that gives an overview of the WAVEWATCH wave model. In it, in the Introduction section, it is mentioned that WAVEWATCH cannot model storm surge or tsunamis. Is this true of all 3rd generation models? Can source terms be added to the energy balance equation to model these phenomena?

Yes. Spectral wave models cannot model storm surge because the wave energy balance equation that they integrate does not describe the physical processes associated with storm surge.

Wave models solve the wave energy balance equation:

$$\dfrac{\partial E}{\partial t} + \dfrac{\partial (c_gE)}{\partial x} + \dfrac{\partial (\dot{k}E)}{\partial k} + \dfrac{\partial (\dot{\theta} E)}{\partial \theta} = \rho_w g\sum S_i(x,k,\theta)$$ where the terms on the LHS represent local change in time and advection of energy in geographical ($x$), wavenumber ($k$), and directional ($\theta$) space, respectively. The source terms $S_i$ on the RHS determine the growth and decay of wave energy due to wind input, wave dissipation, and non-linear wave-wave interactions.

While mean water depth, wind forcing, and background Eulerian currents all affect the evolution of wave energy, the wave energy balance equation only solves for the statistical distribution of wave energy and its change in time. It does not provide information about the change of mean water level or Eulerian currents. The simplest equation set necessary to explicitly model surge are the single-layer (2-D) shallow water equations:

$$\dfrac{d\mathbf{u}}{dt}=-g\nabla\eta + \dfrac{1}{\rho_w} \dfrac{\partial \boldsymbol{\tau}}{\partial z}$$

$$\dfrac{\partial \eta}{\partial t} = -\nabla \cdot (\mathbf{u}(H+\eta))$$ where $\mathbf{u}$ is the Eulerian velocity and $\eta$ is the displacement from the mean water level $H$. $\nabla$ is the horizontal gradient operator. $\dfrac{\partial \boldsymbol{\tau}}{\partial z}$ is the vertical stress gradient that forces the flow - essentially the difference between surface and bottom stresses. The momentum equation describes how the flow velocity changes due to horizontal pressure gradient and surface forcing (stress). The mass continuity equation describes how the water elevation changes due to divergence of the flow. For spatial scales larger than 10 km or so, Earth's rotation becomes important and the RHS of the momentum equation must include the Coriolis acceleration.

Another requirement for the model to simulate storm surge is to allow for wetting and drying. What this means is that the land cells, which are originally above mean water level, and thus not part of the model solution, can become wet once the water level reaches their height and "submerges" them. These grid cells then become part of the model solution. While this may sound obvious and physically intuitive, implementing a wetting-and-drying scheme in a model is a non-trivial challenge regarding software design.

When the wave and surge models are coupled, the Eulerian velocity field $\mathbf{u}$ is mainly forced by wave dissipation, and less by wind directly. The surface elevation changes in response, and together with Eulerian velocity feeds back to the wave model.

• Great answer. Just wanted to mention that the $u$ in the shallow water equations is the depth-averaged value of the horizontal component of the Eulerian velocity (maybe the $\nabla$ should be $\nabla_H$ to imply it is the horizontal divergence). Although all this becomes obvious once we realize that $\partial \eta/\partial t$ is just the difference between the value of the vertical component of the Eulerian velocity at the surface and at the bottom. – Isopycnal Oscillation Aug 17 '15 at 21:21
• @IsopycnalOscillation Good point, made that more clear in the text. – milancurcic Aug 18 '15 at 1:56