Exercise 9

Learning goals

This exercise has learning goals:

  1. Get a feeling for the differences between the Shallow Water Equations and non-hydrostatic models.
  2. Understand the derivation of non-hydrostatic models
Reference

See the excellent book of Castro-Orgaz and Hager for more information.

Depth-averaged non-hydrostatic free surface flow

Given the depth-averaged mass and momentum balance in 1D

\[ \begin{aligned} &\frac{\partial}{\partial t} \int_{b}^{s} u \, dz + \frac{\partial}{\partial x} \int_{b}^{s} u^2 \, dz &= -\frac{1}{\rho} \left[ \frac{\partial}{\partial x} \int_{b}^{s} \sigma_{xx} \, dz + (\sigma_{xx})_b \frac{\partial b}{\partial x} - (\sigma_{xz})_b \right] \end{aligned} \tag{1}\]

with

\[ \begin{aligned} \sigma_{zz}(z) &= \sigma_{zz}(s) + \rho g(s - z) - \rho w^2 \\ &+ \rho \frac{\partial}{\partial t} \int_{z}^{s} w \, dz' + \rho \frac{\partial}{\partial x} \int_{z}^{s} wu \, dz' + \int_{z}^{s} \frac{\partial \sigma_{zx}}{\partial x} \, dz' \end{aligned} \tag{2}\]

and

\[ \begin{aligned} w(z) = & = \frac{\partial b}{\partial t} - \frac{\partial }{\partial x} \left(\int_b^z u \right) \, dz' \end{aligned} \tag{3}\]

Boussinesq-type model

Water waves over horizontal topography can be derived from Equation 1 - Equation 3 using the assumptions:

  • horizontal channel: \(b=0\)
  • \(\sigma_{xx} = \sigma_{zz} = p\) is the water pressure
  • the stress \(\sigma_{zx}\) can be neglected
  • a stress-free free surface
  • \(u(t,x,z) = U(t,x)\) constant in z.
  1. Derive an expression for \(\dfrac{p}{\rho}\)

The mass and momentum balance can be written in vector notation as

\[ \dfrac{\partial \mathbf{Q}}{\partial t} + \dfrac{\partial \mathbf{F}}{\partial x} = \mathbf{0} \]

where

\[ \mathbf{Q} = \begin{bmatrix} h\\ hU \end{bmatrix}, \;\; \mathbf{F} = \begin{bmatrix} hU\\ M \end{bmatrix} \]

  1. Derive an expression for \(M\), the momentum flux.

  1. \[ \dfrac{p}{\rho} = g (h - z) + (U_{x}^2 - U_{xt} - UU_{xx}) \dfrac{(h^2 - z^2)}{2} \]

  2. \[ M = g \dfrac{h^2}{2} + U^2 h + (U_{x}^2 - U_{xt} - UU_{xx}) \dfrac{h^3}{3} \]

Disperson Relation of the Boussinesq-type model

  1. Compute the dispersion relation for the Boussinesq-type equations derived before

Steps:

  • Linearize \[ h = h_0 + \tilde{h} (x, t) \, , \, u = U_0 + \tilde{u} (x, t) \] around the lake at rest solution \[ h_0 = H = \text{const} \quad U_0 = 0 \quad .\]

  • Plug in the plane wave ansatz \[ \begin{aligned} \tilde{h} &= \hat{h} e^{i (kx-\omega t)}\\ \tilde{u} &= \hat{u} e^{i (kx-\omega t)} \end{aligned} \]

  • Solve the resulting system to compute \(\omega(k)\)

  1. How does the phase velocity \(v_{phase} = \frac{\omega}{k}\) behave compared to the phase velocity of the shallow water equations?
  1. \[ w(k) = \pm k \sqrt{\dfrac{g h_0}{1 - \frac{1}{3} h^2_0 k^2}} \]
  2. \[ w(k)^{SWE} = \pm k \sqrt{g h_0} \]

The phase velocity of the SWE are constant in contrast to the phase velocity of the Boussinesq type model. The latter is therefore dispersive.