In this paper, we study the nonlinear asymptotic stability of the 2D Boussinesq equations near the Couette flow in \(\mathbb{T}\times \mathbb{R}\) under the condition that the Richardson number satisfies \(\gamma^{2} > {1\over 4}\) . We allow for different viscosity ν and thermal diffusivity μ and establish stability in the regime \(\mu^{3}\leq \nu \leq\mu^{ {1\over 3}}\) . We show that if the initial perturbation is sufficiently small in HN+1 × HN+2 (with N ⩾ 6) in the sense that, for some sufficiently small α > 0 (0 < α ≪ 1), \(\parallel v_{\text{in}}-(y,0)\parallel_{H^{N+1}}+\parallel\rho_{\text{in}}+\gamma^{2}y-1\parallel_{H^{N+2}}\leq\varepsilon_{0}\min\{\nu,\mu\}^{{1\over 2}+{4\over 3}\alpha},\) then the Couette flow is asymptotically stable. The recent result of Zhai and Zhao (2023) represents an important contribution to this problem, and our work constitutes a substantial improvement.
