Smooth and filamental structures in chaotically advected chemical fields

This paper studies the spatial structure of decaying chemical fields generated by a chaotic-advection flow and maintained by a spatially smooth chemical source. Previous work showed that in a regime where diffusion can be neglected (large Péclet number), the structures are filamental or smooth depending on the relative strength of the chemical dynamics and the stirring induced by the flow. The scaling exponent, ${\gamma }_q$, of the $q\text{th}$-order structure function depends, at leading order, linearly on the ratio of the rate of decay of the chemical processes, $\alpha$, and the average rate of divergence of neighboring fluid parcel trajectories (Lyapunov exponent), $\overline{h}$. Under a homogeneous stretching approximation, ${\gamma }_q/q=\text{max}\left\{\alpha /\overline{h},1\right\}$ which implies that a well-defined filamental-smooth transition occurs at $\alpha =\overline{h}$. This approximation has been improved by using the distribution of finite-time Lyapunov exponents to characterize the inhomogeneous stretching of the flow. However, previous work focused more on the behavior of the exponents as $q$ varies and less on the effects of $\alpha$ and hence the implications for the filamental-smooth transition. Here we set out the precise relation between the stretching rate statistics and the scaling exponents and emphasize that the latter are determined by the distribution of the finite-size (rather than finite-time) Lyapunov exponents. We clarify the relation between the two distributions. We show that the corrected exponents, ${\tilde{\gamma }}_q$, depend nonlinearly on $\alpha$ with ${\tilde{\gamma }}_q<{\gamma }_q$ for ${\tilde{\gamma }}_q<q$. The magnitude of the correction to the homogeneous stretching approximation, ${\tilde{\gamma }}_q-{\gamma }_q$, grows as $\alpha$ increases, reaching a maximum when the leading-order transition is reached $\left(\alpha =\overline{h}\right)$. The implication of these results is that there is no well-defined bulk filamental-smooth transition. Instead it is the case that the chemical field is unambiguously smooth for $\alpha >h_{\text{max}}$, where $h_{\text{max}}$ denotes the maximum finite-time Lyapunov exponent and unambiguously filamental for $\alpha <\overline{h}$, with an intermediate character for $\alpha$ between these two values. Theoretical predictions are confirmed by numerical results obtained for a linearly decaying chemistry coupled to a renewing type of flow together with careful calculations of the Crámer function.

Figure 1

(a) The function $\Lambda \left(\sigma \right)$ corresponding to the alternating sine flow [Eq. (36)] for $A=\pi /2$, $\pi$, and $2\pi$ (where $A=\pi U\tau$, $\tau =1$ and $U$ varies). (b) The corresponding derivative of $\Lambda \left(\sigma \right)$.

Figure 2

The Crámer function $G\left(h\right)$ corresponding to the alternating sine flow [Eq. (36)] for $A=\pi /2$, $\pi$ and $2\pi$ $\left(\tau =1\right)$.

Figure 3

The theoretical prediction for ${\tilde{\gamma }}_1$ [see Eq. (26)] is calculated using Eq. (34) and plotted against $\alpha$ for different values of $A$ (thick solid line) for $\tau =1$. Also plotted ${\gamma }_1$ (dashed line), the homogeneous-stretching approximation [Eq. (1)] as well as ${\tilde{\gamma }}_1^{\prime }$ (thin, solid line), the quadratic approximation [Eq. (35b)]. The theoretical results are in close agreement with the numerical results obtained from a set of simulations whereby an ensemble of $2000\times 2000$ fluid parcels satisfying Eq. (7) are advected backward in time by Eq. (36) $\left(T=50\tau \right)$. $\mathcal{F}$ is given by Eq. (3) and ${\mathcal{F}}_0\left(\mathit{x}\right)$ by Eq. (40). The scaling exponents are determined by calculating $S_1\left(\delta x\right)$ along a single intersection $\left(y=0\right)$ and for two intervals: ${10}^{-5}<\delta x<{10}^{-2}$ (squares) and ${10}^{-5}<\delta x<{10}^{-4}$ (circles).

Figure 4

Same as Fig. 3 but this time the focus is on the second-order scaling exponent (${\tilde{\gamma }}_2$ (thick solid line), ${\gamma }_2$ (dashed line), and ${\tilde{\gamma }}_2^{\prime }$ (thin solid line).

Figure 5

(Color online) Same as Fig. 3 where this time the scaling exponents are plotted as function of $q$ for $\alpha =0.25\overline{h}$ so that ${\gamma }_q=0.25q$ (dashed black line) [see Eq. (1)]. Also plotted ${\tilde{\gamma }}_q$ [see Eq. (26)—thick solid black line] and ${\tilde{\gamma }}_q^{\prime }$ [see Eq. (35b)—thin solid black line]. A line in solid light gray (red) of slope equal to $\alpha q/h_{\text{max}}$ is plotted for large $q$ [see also Eq. (32)].

Figure 6

(Color online) Same as Fig. 5 but this time $\alpha =\overline{h}$ so that ${\gamma }_q=q$ (dashed black line) [see Eq. (1)]. A line in solid light gray (red) of slope equal to $\alpha q/h_{\text{max}}$ is plotted for large $q$ [see also Eq. (32)].

Figure 7

(Color online) Snapshots of the reactive field at statistical equilibrium $\left(T=50\tau \right)$ for various values of $\alpha$ $\left(A=\pi /2\right)$. Case (b) corresponds to the value of $\alpha$ for which the homogeneous-stretching approximation predicts a bulk filamental-smooth transition. The color bars on the right of each graph give the concentration values associated with the different colors.