1. Modelling of Marine Systems. Shallow waters Equations

2. Conservation Principle “The rate of accumulation inside a Control volume balances the fluxes across its boundaries plus the Sources minus the Sinks”! Or, in tensorial notation:

3. Mass conservation Principle Mass is conserved. This implies that the rate of accumulation inside a volume balances the input and output budgets:

4. Em incompressível

5. Boussinesq Approximation Density can be considered as being constant unless if multiplied by the gravity acceleration (much bigger than flow acceleration).

6. Momentum conservation One must evaluate the forces per unit of volume. =>

7. Pressure force resultant

8. Force resultant (including friction and weight)

9. Summing up This is the Momentum transport equation into its differential form. It states that the fluid acceleration is the result of the 3 forces (pressure, friction and gravity).

10. The Navier-Stokes Equations

11. In case of Geophysics Boussinesq approximation Vertical velocity is small Vertical acceleration negligible and consequently Pressure is Hydrostatic

12. Equations to solve

13. Pressure Force Pressure force has baroclinic and barotropic components.

14. Reference Free Surface H=  +h z xi Hydrographic Reference h  c

15. Generic Property Free Surface

16. Mass Conservation If the fluid is incompressible density is constant and the continuity equation becomes: In cartesian coordinnates :

17. 2D depth Integrated Model Free Surface

18. 2D Case (Small vertical gradients)

19. 2D Model

20. Computational Grid Free Surface