Global regularity for the 2d non-diffusive anisotropic critical dissipation

. This paper is concerned with the global regularity of the twodimensional generalized Boussinesq equations. When the dissipation mechanism is only fractional vertical dissipation in the horizontal velocity and horizontal dissipation in the vertical velocity, it is examined that the system has a unique global smooth solution. The finding not only settles a problem remarked in (J. Differential Equations 268: 910-944, 2020), but also relaxes the restriction on the initial data.