  In this paper, we study the 3D regularized Boussinesq equations. The velocity
equation is regularized \`a la Leray through a smoothing kernel of order
$\alpha$ in the nonlinear term and a $\beta$-fractional Laplacian; we consider
the critical case $\alpha+\beta=\frac{5}{4}$ and we assume $\frac 12
<\beta<\frac 54$. The temperature equation is a pure transport equation, where
the transport velocity is regularized through the same smoothing kernel of
order $\alpha$. We prove global well posedness when the initial velocity is in
$H^r$ and the initial temperature is in $H^{r-\beta}$ for
$r>\max(2\beta,\beta+1)$. This regularity is enough to prove uniqueness of
solutions. We also prove a continuous dependence of the solutions on the
initial conditions.
