## Analysis of some oceanography physics problems by the Galerkin’s method.(English)Zbl 0891.76067

Díaz, Jesús Ildefonso (ed.), The mathematics of models for climatology and environment. Proceedings of the NATO Advanced Study Institute, Puerto de la Cruz, Tenerife, Spain, January 11–21, 1995. Berlin: Springer. NATO ASI Ser., Ser. I, Glob. Environ. Chang. 48, 135-180 (1997).
The aim is to determine the velocity fields of marine streams with a view of solving biologic models and pollution transport. The studied models concern phenomena specific to the seas where the main forcing is the wind. We present the study of two models: a three-dimensional model, and a shallow water model in depth-mean velocity formulation.
First of all, we present an existence result for the shallow water problem. The equations of this model differ from classical equations by the boundary conditions that are $$u\cdot n$$ and $$\text{curl }u$$ fixed on the boundary. This difference allow us to solve the equations by the Galerkin’s method using the eigenbasis of the Laplace operator $$\Delta$$ with $$u\cdot n= 0$$ and $$\text{curl }u= 0$$ on the boundary. These boundary conditions endow this basis with particular properties that allow to smoothly solve the equations of the shallow water model. This basis is also used to solve the three-dimensional model.
For the entire collection see [Zbl 0861.00022].

### MSC:

 76M25 Other numerical methods (fluid mechanics) (MSC2010) 86A05 Hydrology, hydrography, oceanography