Stability for the Boussinesq Equations with Horizontal Dissipation near the Hydrostatic Balance on $\mathbb{R}^2$

The hydrostatic balance is a fundamental equilibrium state in stratified fluids and plays a central role in geophysical fluid dynamics. Understanding its stability under incomplete dissipation is a longstanding challenge, since anisotropic diffusion alone is generally insufficient to control the nonlinear evolution and no robust stabilizing mechanism is known for the corresponding anisotropically dissipative Navier--Stokes equations. In this paper, we investigate the two-dimensional Boussinesq equations on $\mathbb{R}^2$ with only horizontal dissipation near the hydrostatic equilibrium $(U,\Theta)=(0,x_2)$. We show that the velocity--temperature coupling generates internal gravity waves whose dispersive decay, together with the horizontal dissipation, provides an effective stabilizing mechanism that compensates for the complete absence of vertical dissipation. This identifies a mechanism by which dispersive wave propagation restores stability in an incompletely dissipative fluid system. For sufficiently small initial perturbations in $H^k(\mathbb{R}^2)\cap W^{3,1}(\mathbb{R}^2)$ with $k\ge14$, we establish the global existence and uniqueness of classical solutions together with explicit anisotropic, componentwise large-time decay rates for the velocity and temperature, including faster decay of the vertical velocity.