Chaotic advection and linear damping

If the advected field is reactive, e.g. in the case of concentration field of a chemical product, the resulting structure of the concentration field is determined by the competition between the mixing process and the chemical reaction. We are interested in the simplest case of reaction, that is an exponential decay of the concentration field originating form its finite lifetime, because of its analogy with the friction term in the equation for vorticity.

In this case the two processes in competition are the direct cascade of scalar fluctuation produced by the flow, and the exponential decay of the same fluctuation due to the reaction term. The stretching exerted by the incompressible smooth velocity generates small scale features at exponential rate (Lyapunov):

$$r = L e^{\lambda t}$$ (2.11)

At the same time fluctuations initially generated by the external forcing at the large scale $L$ decay at exponential rate given by the inverse of finite lifetime $\alpha = 1/\tau$:

$$\delta \theta({\mbox{\boldmath$r$}} ,t) = \delta \theta(L ,0) e^{\alpha t}$$ (2.12)

Combining the two exponential behavior one has:

$$\delta \theta({\mbox{\boldmath$r$}} ,t) = \delta \theta(L ,0) \left( {r \over L} \right)^{\alpha / \lambda}$$ (2.13)

and thus, in stationary conditions one gets the following scaling law for the structure functions:

$$S^{\theta}_p (r) \equiv \langle (\delta_r \theta)^p \rangle \sim r^{\zeta^{\theta}_p}$$ (2.14)

with the scaling exponents:

$$\zeta^{\theta}_p = \min \left[ p {\alpha \over \lambda} , p \right]$$ (2.15)

When the omogeneization produced by the linear damping is stronger than the stirring $\alpha > \lambda$ the resulting field is smooth with Hölder exponent $h = 1$, otherwise one obtains a filamental rough field with $h = \alpha / \lambda$. This phenomenon is known as smooth-filamental transition [36].

In the scaling range the prediction for the power spectrum of passive scalar fluctuations reads:

$$E_{\theta}= 2\pi k \langle \vert \hat{\theta}({\mbox{\boldmath$k$}}) \vert^2 \rangle \sim k^{-1-2 \alpha / \lambda }$$ (2.16)

The spectral slope shows an explicit dependence on the linear damping coefficient $\alpha = 1/\tau$, and in the limit of infinite lifetime $\alpha \to 0$ the Batchelor's $k^{-1}$ prediction is recovered.