Diffusive time scale ($t_D\sim L^2/\kappa$)

A concentration varying over length $L$ has Laplacian of order its amplitude divided by $L^2$. Balancing the time derivative with $\kappa \Delta \chi$ in the diffusion equation gives the displayed relaxation time. A multiple-scale expansion with spatial scale $\epsilon x$ therefore uses slow time ${\epsilon }^{2}t$ when the effective diffusivity is of order one.