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In this paper, we consider the initial boundary value problem in a cylindrical domain to the three-dimensional primitive equations with full eddy viscosity in momentum equations but with only horizontal eddy diffusivity in the temperature equation. Global well-posedness of a z-weak solution is established for any such initial datum such that itself and its vertical derivative belong to L2. This not only extends the results in the work of Cao, Li, and Titi [Physica D 412, 132606 (2020)] from the spatially periodic case to general cylindrical domains but also weakens regularity assumptions on the initial data, which are required to be H2 there.

In this paper, we provide rigorous justification of the hydrostatic approximation and the derivation of primitive equations as the small aspect ratio limit of the incompressible three-dimensional Navier-Stokes equations in the anisotropic horizontal viscosity regime. Setting ε > 0 to be the small aspect ratio of the vertical to the horizontal scales of the domain, we investigate the case when the horizontal and vertical viscosities in the incompressible three-dimensional Navier-Stokes equations are of orders O(1) and O(ε α ), respectively, with α > 2, for which the limiting system is the primitive equations with only horizontal viscosity as ε tends to zero. In particular we show that for "well prepared" initial data the solutions of the scaled incompressible threedimensional Navier-Stokes equations converge strongly, in any finite interval of time, to the corresponding solutions of the anisotropic primitive equations with only horizontal viscosities, as ε tends to zero, and that the convergence rate is of order O ε β 2 , where β = min{α − 2, 2}. Note that this result is different from the case α = 2 studied in [Li, J.; Titi, E.S.: The primitive equations as the small aspect ratio limit of the Navier-Stokes equations: Rigorous justification of the hydrostatic approximation, J. Math. Pures Appl., 124 (2019), 30-58], where the limiting system is the primitive equations with full viscosities and the convergence is globally in time and its rate of order O (ε).

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