Shallow water model

The ShallowWaterModel simulates the shallow water dynamics for a fluid of constant density but with varying fluid depth $h(x, y, t)$ and whose velocity only varies in the horizontal, $\boldsymbol{u}(x, y, t) = u(x, y, t) \boldsymbol{\hat x} + v(x, y, t) \boldsymbol{\hat y}$.

ShallowWaterModel allows users to prescribe the shallow water dynamics using two different formulations: VectorInvariantFormulation() and ConservativeFormulation().

The VectorInvariantFormulation uses the horizontal velocity $\boldsymbol{u}$ and the total depth of the fluid, $h$, as the dynamical variables. Furthermore, the advective terms are rewritten via the vector identity:

\[\boldsymbol{u} \boldsymbol{\cdot} \boldsymbol{\nabla} \boldsymbol{u} = (\boldsymbol{\nabla} \times \boldsymbol{u}) \times \boldsymbol{u} + \boldsymbol{\nabla} \left( \frac1{2} \boldsymbol{u} \boldsymbol{\cdot} \boldsymbol{u} \right ) \, .\]

Thus, the shallow water dynamics in vector-invariant form become:

\[\begin{align} \partial_t \boldsymbol{u} + (\zeta \boldsymbol{\hat z} + \boldsymbol{f}) \times\boldsymbol{u} & = - \boldsymbol{\nabla} \left [ g (h +b) + \frac12 \boldsymbol{u} \cdot \boldsymbol{u} \right ] \, , \\ \partial_t h + \boldsymbol{\nabla} \boldsymbol{\cdot} (\boldsymbol{u} h) & = 0 \, , \end{align}\]

where $\zeta(x, y, t) = \partial_x v - \partial_y u$ is the vertical component of the relative vorticity.

The elevation of the bottom bathymetry, measured with respect to the free-surface at rest, is $b(x, y)$. The free-surface elevation $\eta$ is then:

\[\eta(x, y, t) = h(x, y, t) + b(x, y) \, .\]

The ConservativeFormulation() uses the volume transport along each direction $\boldsymbol{u} h = (u h, v h)$ and the total depth of the fluid $h$ as the dynamical variables. The shallow water dynamics in conservative form is:

\[\begin{align} \partial_t (\boldsymbol{u} h) + \boldsymbol{\nabla} \boldsymbol{\cdot} \left ( \boldsymbol{u} \boldsymbol{u} h \right ) + \boldsymbol{f} \times (\boldsymbol{u} h) & = - g h \boldsymbol{\nabla} \left ( h + b \right ) \, ,\\ \partial_t h + \boldsymbol{\nabla} \boldsymbol{\cdot} (\boldsymbol{u} h) & = 0 \, , \end{align}\]

where $\boldsymbol{\nabla} \boldsymbol{\cdot} \left ( \boldsymbol{u} \boldsymbol{u} h \right )$ denotes a vector whose components are $[\boldsymbol{\nabla} \boldsymbol{\cdot} (\boldsymbol{u} \boldsymbol{u} h)]_i = \boldsymbol{\nabla} \boldsymbol{\cdot} (u_i \boldsymbol{u} h)$. We can retrieve the flow velocities by dividing the corresponding transport by the fluid's height, e.g., v = vh / h.