Shallow-Water Modelling
- The linearized 2D shallow water equations (for $f = 0$) are $$ \frac{\partial u}{\partial t} = - \frac{\partial h}{\partial x}, \quad \frac{\partial v}{\partial t} = - \frac{\partial h}{\partial y}, \quad \frac{\partial h}{\partial t} = -c^2 \left( \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} \right) $$
- Discrete version for staggered grid $$ \frac{d u_{j,k}}{dt} = - \delta_x^+ h_{j,k}, \quad \frac{d v_{j,k}}{dt} = - \delta_y^+ h_{j,k}, \quad \frac{d h_{j,k}}{dt} = -c^2 \left( \delta_x^- u_{j,k} + \delta_y^- v_{j,k} \right) $$
- With the finite differencing operators $$ \delta_x^+ h_{j,k} = \frac{h_{j+1,k} - h_{j,k}}{\Delta x}, \quad \delta_x^- h_{j,k} = \frac{h_{j,k} - h_{j-1,k}}{\Delta x} $$ $$ \delta_y^+ h_{j,k} = \frac{h_{j,k+1} - h_{j,k}}{\Delta y}, \quad \delta_y^- h_{j,k} = \frac{h_{j,k} - h_{j,k-1}}{\Delta y} $$
A simple shallow-water demo that captures only linear wave dynamics with basic forcing and boundaries
A full shallow-water system including nonlinear terms, variable Coriolis, and friction, capable of simulating realistic jets, eddies, and instabilities
1Markus Jochum and Carsten Eden. Bornö Summerschool 2024.