On the equations of the large-scale ocean.

*(English)*Zbl 0766.35039This contribution is part of a series of papers of the authors in which they analyze equations that are relevant in climatology. The work under review describes the ocean as a (slightly) compressible fluid; the unknowns are velocity, pressure, density, temperature and salinity.

As the general equations are too complicated various approximations are derived. The assumption that density variations are relevant only in the buoyancy force and in the thermodynamic equation leads to the Boussinesq equations. Due to the fact that the ocean’s depth is small compared to its horizontal extension the pressure gradient and the gravitational force are the dominant terms in the equation of motion for the vertical component of the velocity. This leads to the “primitive equations” of the ocean. The assumption of different viscosities in the horizontal and the vertical equations finally leads to the “primitive equation with vertical viscosity” which are introduced by the authors.

These equations are then analyzed. Appropriate function spaces are introduced which allow a formulation of weak solutions and which take care of a nonlocal side condition that replaces \(\text{div }\underline{v}=0\) in the Navier-Stokes equations. Then existence of solutions is shown, uniqueness under suitable smallness assumptions, and analyticity in time.

Finally attractors of the equations are investigated, and estimates for their dimensions are given.

As the general equations are too complicated various approximations are derived. The assumption that density variations are relevant only in the buoyancy force and in the thermodynamic equation leads to the Boussinesq equations. Due to the fact that the ocean’s depth is small compared to its horizontal extension the pressure gradient and the gravitational force are the dominant terms in the equation of motion for the vertical component of the velocity. This leads to the “primitive equations” of the ocean. The assumption of different viscosities in the horizontal and the vertical equations finally leads to the “primitive equation with vertical viscosity” which are introduced by the authors.

These equations are then analyzed. Appropriate function spaces are introduced which allow a formulation of weak solutions and which take care of a nonlocal side condition that replaces \(\text{div }\underline{v}=0\) in the Navier-Stokes equations. Then existence of solutions is shown, uniqueness under suitable smallness assumptions, and analyticity in time.

Finally attractors of the equations are investigated, and estimates for their dimensions are given.

Reviewer: J.Bemelmans (Aachen)

##### MSC:

35Q35 | PDEs in connection with fluid mechanics |

86A05 | Hydrology, hydrography, oceanography |

76D05 | Navier-Stokes equations for incompressible viscous fluids |