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In this paper we study the equation of an anisotropic rotating fluid evolving in a three-dimensional torus. The anisotropy of the fluid concerns the vertical viscosity which can be considered to be null. We prove that the equation is well posed on arbitrarily long time intervals, if the Rossby number is small enough. We also prove that the “filtered” solutions converge to the solution of the limit system. Those results hold for all tori (without imposing the non-resonant condition on the domain), in a functional frame very close to the critical regularity level. When the initial data is of null horizontal average, in the case of non-resonant tori as well as for certain resonant tori, we obtain the global existence and uniqueness of the solution. [Copyright &y& Elsevier]