Background:
A thin filament of fluid mixes into a flow field in accord with the so-called "compression-diffusion equation" (CDE), which is essentially an advection-diffusion equation with a particular space-time dependent velocity field $u(x,t) = \dot{s}(t)x/s(t)$. The variable $s(t)$ is the time-dependent thickness of the filament, and the quantity $\dot{s}/s$ is the compression rate of the filament induced by the flow. This compression rate depends on the specifics of the flow field in question, see this paper.
The CDE has the following form: $$ \partial_t c(x,t) = - \frac{\dot{s}(t)x}{s(t)}\partial_x c(x,t) + D \partial_x^2 c(x,t). $$ One can make the so-called ``Ranz transformation" to variables $$ \tau(t) = D \int_0^t \frac{dt'}{s(t')^2}$$ $$ \xi(x,t) = \frac{x}{s(t)},$$ and this provides the canonical diffusion equation: $$ \partial_\tau c(\xi,\tau) = \partial_\xi^2 c(\xi,\tau).$$ Therefore, the solution to the compression-diffusion equation is $$ c(x,t) = \frac{c_0}{\sqrt{1+4\tau(t)}}\exp\left(- \frac{x^2}{s(t)^2(1+4\tau(t))}\right),$$ for an arbitrary $s(t)$ and initial condition $c(x,0)=c_0e^{-x^2/s_0^2}$. Clearly, the Ranz transformation greatly simplifies a challenging PDE to provide an exact solution.
Question:
I would like to better interpret the variable $\tau$ which enacts this transformation. I understand that $\tau$ encodes a trade-off between the compression and diffusion terms, but I am not sure how to show this. I am interested to find a derivation of the variable $\tau$, perhaps from a dominant balance argument. Further, I would be interested for any physical explanation of why this is the timescale in which the CDE becomes canonical.
Attempt:
At short times $\tau(t)\ll 1$, the compression term dominates the CDE, while at long times $\tau(t)\gg 1$, the diffusion term dominates. This trade-off of compression-dominated to diffusion-dominated behavior originates from the fact that the compression term amplifies the concentration gradients ($\partial_x c$) which in turn fuel the diffusive flux ($J=-D\partial_x c$). The limit $\tau=1$ describes equal influence of the compression and diffusion terms, and the resulting timescale $t_B$, defined by $$ 1 = D\int_0^{t_B} \frac{dt'}{s(t')^2},$$ is known as the Batchelor time (in reference to George Batchelor).
I thought it may be possible to find the advection and diffusion-dominated solutions, then merge the two to find $\tau(t)$ and thereby gain insight into its meaning. In the advection-dominated regime, we have $$ \partial_t c = -( \dot{s}x/s ) \partial_x c, $$ for which it is challenging for me to find a solution (which could be called $c_<(x,t)$, while the diffusive regime we have $$ \partial_t c = D\partial_x^2 c, $$ which has solution $c_>(x,t) = \frac{c_0}{\sqrt{4Dt}}\exp(-x^2/(4Dt)).$ Perhaps it is possible to solve the advection-dominated problem and merge the solution via matched asymptotic expansions to discover the Batchelor time divides the two?
Note on tags This problem is equivalent to Brownian motion in an inhomogeneous environment. It also relates to boundary layers in the sense of matched asymptotic expansions. I am attempting to cast a wide net.