The multi-layer shallow water model with free surface
We consider n immiscible, homogeneous, inviscid and incompressible fluids, with no surface tension. Derived from the Euler equations with hydrostatic assumption (the pressure is no more an unknown of the problem) and long-wave hypothesis, this model is a simplified two-dimensional description of three-dimensional processes with large horizontal length scales. The simplest case is with one layer and was introduced by A.B. de Saint-Venant, in Comptes rendus hebdomadaires des séances de l'académie des sciences , 1871. As this case is entirely known, scientists are studying the case n ≥ 2 for a few decades.
Configuration of the multi-layer shallow wataer model with free surface
In order to study the local well-posedness of this model, I study the eigenstructure of the advection matrices associated to. The issue is to find conditions on density, velocity and height, of each layer, such that all the eigenvalues are real (and distinct would be even better!). This is not obvious, even in the two layer model. And for any number of layers, it is quite difficult.
In order to provide the underwater forecast, SHOM uses HYCOM (HYbrid Coordinate Ocean Model), a code based on the multi-layer shallow water model with free surface. Below, 3 examples of outputs made by HYCOM.
Direction and strength of currents at the surface (Bay of Biscay, Apr. 04-06, 2016)
Height of water referenced to mean sea level (Bay of Biscay, Apr. 04-06, 2016)
Temperature at the surface (Bay of Biscay, Apr. 04-06, 2016)
