  In this paper, we consider the stability threshold for the shear flows of the
Boussinesq system in a domain $\mathbb{T} \times \mathbb{R}$. The main goal is
to prove the nonlinear stability of the shear flow $(U^S,\Theta^S)=((e^{\nu
t\partial_{yy}}U(y),0)^{\top},\alpha y)$ with $U(y)$ close to $y$ and
$\alpha\geq0$. We separate two cases: one is $\alpha\geq 0$ small scaling with
the viscosity coefficients and the case without smallness of $\alpha$ and fixed
heat diffusion coefficient. The novelty here is that we don't require $\mu=\nu$
and only need to assume that $\mu$ is scaled with $\nu$ or fixed, where $\mu$
is the inverse of the Reynolds number and $\nu $ is the heat diffusion
coefficient.
