Scalar-gradient inclination in a steady planar strain ($\alpha \to |D|/\sqrt{a^2+b^2}$)

For horizontal strain matrix $S=\left(\begin{array}{cc}a& b\\ b& -a\end{array}\right)$ and zero vertical velocity, horizontal scalar gradients obey ${\dot{k}}_h=-S{k}_h$. Since $S^2=c^{2}I$ with $c=\sqrt{a^2+b^2}$, the growing projection is $(I-S/c)k_h(0)/2$. If nonzero it equals $k_0(\cos \varphi ,\sin \varphi )$ and grows as $e^{ct}$. Fixed vertical shears give ${\dot{k}}_3=-{\sigma }_{13}{k}_1-{\sigma }_{23}{k}_2$, so $k_3\sim -k_{0}D{e}^{ct}/c$ with $D={\sigma }_{13}\cos \varphi +{\sigma }_{23}\sin \varphi$. The ratio $\alpha =|k_3|/|k_h|$ tends to the displayed value. It measures scalar-isosurface orientation; zero growing projection or $D=0$ must not be described as generic exponential vertical-gradient growth.

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.

Let $c=\sqrt{a^2+b^2}$ with independent $a,b\sim N(0,{\Gamma }^2)$, and let $D\sim N(0,{\Lambda }^2)$ be independent of them. Then $c$ has Rayleigh distribution and $\alpha =|D|/c$ has the displayed cumulative distribution function for $A\ge 0$ and density ${\Lambda }^2\Gamma ({\Gamma }^2{A}^2+{\Lambda }^2{)}^{-3/2}$. Indeed, conditioning on $D$ gives $\mathbb{P}(c\ge |D|/A)=\exp [-D^2/(2{\Gamma }^2{A}^2)]$, and a Gaussian integral completes the calculation. Independence is essential: choosing $D=(\Lambda /\Gamma )a$ has the same Gaussian marginal but makes the ratio bounded by $\Lambda /\Gamma$.