  We continue our study of the dynamics of a nearly inviscid periodic surface
quasi-geostrophic equation. Here we consider a slightly diffusive stochastic
SQG equation of the form \begin{equation*} \begin{cases} d\theta_t +
|D|^{2\delta}\theta_t\,dx + (u_t \cdot \nabla)\theta_t\,dx + |D|^{\delta}dW_t =
0 \\ u_t = \nabla^\perp|D|^{-1}\theta_t. \end{cases} \end{equation*} We
construct global energy solutions as introduced by P. Goncalves and M. Jara
(2014) for any $\delta > 0$, so that any small amount of diffusion permits us
to construct solutions. We show moreover that pathwise uniqueness of these
energy solutions holds in the presence of sufficiently high diffusion $\delta >
\frac32$.
