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We establish global existence and uniqueness theorems for the two-dimensionalnon-diffusive Boussinesq system with viscosity only in the horizontaldirection, which arises in Ocean dynamics. This work improves the globalwell-posedness results established recently by R. Danchin and M. Paicu for theBoussinesq system with anisotropic viscosity and zero diffusion. Although wefollow some of their ideas, in proving the uniqueness result, we have used analternative approach by writing the transported temperature (density) as$\theta = \Delta\xi$ and adapting the techniques of V. Yudovich for the 2Dincompressible Euler equations. This new idea allows us to establish uniquenessresults with fewer assumptions on the initial data for the transported quantity$\theta$. Furthermore, this new technique allows us to establish uniquenessresults without having to resort to the paraproduct calculus of J. Bony.  We also propose an inviscid $\alpha$-regularization for the two-dimensionalinviscid, non-diffusive Boussinesq system of equations, which we call theBoussinesq-Voigt equations. Global regularity of this system is established.Moreover, we establish the convergence of solutions of the Boussinesq-Voigtmodel to the corresponding solutions of the two-dimensional Boussinesq systemof equations for inviscid flow without heat (density) diffusion on the intervalof existence of the latter. Furthermore, we derive a criterion for finite-timeblow-up of the solutions to the inviscid, non-diffusive 2D Boussinesq systembased on this inviscid Voigt regularization. Finally, we propose aVoigt-$\alpha$ regularization for the inviscid 3D Boussinesq equations withdiffusion, and prove its global well-posedness. It is worth mentioning that ourresults are also valid in the presence of the $\beta$-plane approximation ofthe Coriolis force.

Global Well-posedness for The 2D Boussinesq System Without Heat  Diffusion and With Either Anisotropic Viscosity or Inviscid Voigt-$\alpha$  Regularization