Summary

Most flows encountered in the coastal oceans are turbulent. Furthermore, these flows are three-dimensional. Fluid mechanics deal with the flow of fluids.

  • The set of equations solved has long been known. Details can be found in Blumberg and Mellor [1987], amongst others.
  • The hypotheses under which these equations (so called primitive equations) are obtained are classic approximations:
    • the Boussinesq assumption : density in the medium slightly deviates from a reference density and therefore can be replaced by a reference density except within the gravity term,
    • the hydrostatic approximation resulting from scaling the equations: the horizontal movement scale is assumed to be an order of magnitude larger that the vertical one.
  • These equations are transformed within a sigma framework to make free surface processing easier.
    • \(\sigma=\frac{z+H}{\xi+H}\)
      • \(\sigma\) : vertical coordinate
      • \(H(x,y)\) : absolute value of bottom position
      • \(\xi(x,y)\) : sea surface elevation
  • The horizontal turbulent diffusion operators are not written in a fully discretized form, but simplified according to Mellor [1985]. This approximation is relatively acceptable, as long as the discretized vertical layers cross the isopycns smoothly, i.e. in the case of a smooth bathymetry gradient. In most of application, the two components of horizontal eddy viscosity are set equal to a constant.
  • The most often used closure scheme in coastal ocean models at present are the two-equation closure model that follows the k–kl theory assumptions and the k-epsilon ones.
  • The main original aspect of MARS model is the coupling between barotropic and baroclinic modes especially.
  • The mode splitting approach leads to build a specific barotropic model and considering kinematic boundary conditions. The time stepping used to solve the barotropic mode is an Alternate Direction Implicit (hereafter noted ADI) scheme. The ADI method is only implicit with respect to the direction of the computation. Thus, the free surface elevation is calculated every half time step, whereas mean courrents are calculated alternatively. Computation of mean u and \(\xi\) are performed in a row-wise manner whereas mean v and \(\xi\) computations are performed in column-wise
  • The use of a spatially centered second order scheme and the staggered Arakawa C grid lead to a tridiagonal linear system being solved using LU factorization. Special attention was paid to the discretization of the advection operators which was based on the efficient schemes given by Leonard [1979,1991].