Convective amplification length (${\ell }_a=c/{\sigma }_r$)

If a disturbance grows in time as $e^{{\sigma }_{r}t}$ while it is transported at positive speed $c$, the distance for one e-fold of growth is ${\ell }_a=c/{\sigma }_r$. Traversing distance $d$ gives exponent ${\sigma }_{r}d/c=d/{\ell }_a$, so $d\ll {\ell }_a$ produces little growth. This estimate compares a temporal growth rate with transport; it does not by itself determine the full spatial dispersion relation or stability of a finite domain with feedback.