  In this paper, we focus on the two-dimensional surface quasi-geostrophic
equation with fractional horizontal dissipation and vertical thermal diffusion
which represents a general case of the classical surface quasi-geostrophic
equation. On the one hand, we will show the local existence and uniqueness of
the solution in Sobolev space $H^{2-2\alpha}(\mathbb{R}^2)\cap
H^{2-2\beta}(\mathbb{R}^2)$, which is the critical space in the classical case.
Furthermore, we will demonstrate that the solution is global even when the
initial data is very small. Finally, we will study the asymptotic
representation of our global solution in infinity.
