  We address the persistence of regularity for the 2D $\alpha$-fractional
Boussinesq equations with positive viscosity and zero diffusivity in general
Sobolev spaces, i.e., for $(u_{0}, \rho_{0}) \in W^{s,q}(\mathbb R^2) \times
W^{s,q}(\mathbb R^2)$, where $s> 1$ and $q \in (2, \infty)$. We prove that the
solution $(u(t), \rho(t))$ exists and belongs to $W^{s,q}(\mathbb R^2) \times
W^{s,q}(\mathbb R^2)$ for all positive time $t$ for $q>2$, where
$\alpha\in(1,2)$ is arbitrary.
