Equations and Modelisation

Equations and Modelisation

Equations

The Non-linear Shallow Water equations (NSW) are formulated below with conservative variables:

where stands for the water depth, for the horizontal velocity and for the bottom elevation. is the Coriolis force and the friction source term.

Hydraulic simulations

For hydraulic simulations, the friction source term is defined by the Manning-Strickler friction source term:

with depending on the uniform Manning’s roughness coefficient.

Oceanic simulations

For oceanic simulations, the friction source term can be written as:

Numerical Resolution

Hyperbolic part

The explicit finite-volume scheme for cell at time step :

The numerical flux and gravitational potential at edge are:

where is the gravitational potential and .

Mesh notation: cell area, edge length, cell perimeter, outward unit normal from through , neighbor cell sharing edge .

Entropy-dissipating artificial diffusion

The diffusion terms and are designed to ensure entropy (energy) dissipation under a CFL condition:

where . The coefficients and satisfy local stability conditions and may depend on the local Froude number and CFL, giving a family of schemes. The scheme mixes centered and upwind discretizations; a semi-implicit variant is also available.

Friction source term

A time-splitting scheme is used to treat the friction source term, then we focus here on the resolution of:

Oceanic friction source term

Semi-implicit resolution

The semi-implicit time step resolution writes:

which easily gives the final expression of the velocity:

Full implicit resolution

The full implicit time step resolution writes:

Remarking that , the solution is given by a quadratic equation:

with the final solution:

Manning-Strickler source term

Semi-implicit resolution

The semi-implicit time step resolution writes:

which easily gives the final expression of the velocity:

Full implicit resolution

The full implicit time step resolution writes:

Again, remarking that , the solution is given by a quadratic equation:

with the final solution:

Coriolis force

The Coriolis term is treated with a Crank-Nicolson semi-implicit scheme that preserves kinetic energy (rotation neither creates nor destroys energy):

Discretizing at mid-time-step:

which has an analytical solution (exact rotation by angle ):

The Coriolis parameter is either read per cell from a latitude file () or set from a beta-plane approximation ().