= Scalar-gradient inclination in a steady planar strain
{title2=$\alpha\to |D|/\sqrt{a^2+b^2}$}
For horizontal strain matrix $S=\bigl(\begin{smallmatrix}a&b\\b&-a\end{smallmatrix}\bigr)$ and zero vertical velocity, horizontal scalar gradients obey $\dot k_h=-Sk_h$. Since $S^2=c^2I$ with $c=\sqrt{a^2+b^2}$, the growing projection is $(I-S/c)k_h(0)/2$. If nonzero it equals $k_0(\cos\phi,\sin\phi)$ and grows as $e^{ct}$. Fixed vertical shears give $\dot k_3=-\sigma_{13}k_1-\sigma_{23}k_2$, so $k_3\sim-k_0D e^{ct}/c$ with $D=\sigma_{13}\cos\phi+\sigma_{23}\sin\phi$. 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.
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