White-noise vertical shear gives an Ornstein-Uhlenbeck inclination ($d\beta =-c\beta dt-gdW$)

In the long-time growing horizontal-gradient branch, independent vertical shear noises combine as $\cos \varphi d{W}_1+\sin \varphi d{W}_2=dW$. The signed inclination $\beta =k_3{e}^{-ct}/k_0$ satisfies the displayed Ornstein-Uhlenbeck stochastic differential equation. For $c>0$ its stationary density is $\sqrt{c/(\pi g^2)}{e}^{-c{B}^2/g^2}$, with variance $g^2/(2c)$. The Fokker-Planck probability current vanishes at infinity and gives this Gaussian by one integration. At $c=0$ and nonzero noise there is no normalizable stationary density.