Pattern-fluid interpretation of chemical turbulence

The spontaneous formation of heterogeneous patterns is a hallmark of many nonlinear systems, from biological tissue to evolutionary population dynamics. The standard model for pattern formation in general, and for Turing patterns in chemical reaction-diffusion systems in particular, are deterministic nonlinear partial differential equations where an unstable homogeneous solution gives way to a stable heterogeneous pattern. However, these models fail to fully explain the experimental observation of turbulent patterns with spatio-temporal disorder in chemical systems. Here we introduce a pattern-fluid model as a general concept where turbulence is interpreted as a weakly interacting ensemble obtained by random superposition of stationary solutions to the underlying reaction-diffusion system. The transition from turbulent to stationary patterns is then interpreted as a condensation phenomenon, where the nonlinearity forces one single mode to dominate the ensemble. This model leads to better reproduction of the experimental concentration profiles for the “stationary phases” and reproduces the turbulent chemical patterns observed by Q. Ouyang and H. L. Swinney [Chaos 1, 411 (1991)].

Figure 1

The pattern-fluid model. Heterogeneous spatial patterns are interpreted as the superposition of $N+1$ randomly arranged fundamental patterns, of weights $(1-\varphi )/N$ and $\varphi$ according to Eq. (3). The fundamental patterns represent solutions of the Lengyel-Epstein equations. The random orientation and position arises either by translation and rotation or naturally from random differences in the perturbations of the homogeneous state that lead to the patterns. The stationary ordered phases arise when the amplitude $\varphi$, of a single pattern, here called dominant pattern, becomes significantly larger than the weight of the other patterns, i.e., $\varphi \gg (1-\varphi )/N$.

Figure 2

Use of Minkowski functionals to characterize different phases of the Lengyel-Epstein model. Numerical solutions of the LE equations in the (a) hexagonal phase, $\sigma =20,a=12,b=0.38$ and (b) stripe phase, $\sigma =20,a=11.6,b=0.3$. The gray scale corresponds to the value of the concentration profile $u(x,y)$, where black is the maximum and white the minimum concentration. (c) Area $V\left(\rho \right)$, perimeter $S\left(\rho \right),$ and Euler characteristic $\chi \left(\rho \right)$ as a function of the binarization threshold $\rho$ for the hexagonal (black solid) and stripe (blue dashed) patterns from (a) and (b). The functional dependence on $\rho$ is characteristic for each phase and can be used to distinguish the phases quantitatively. See also section 1 of the Supplemental Material [19] for details on the Minkowski functional calculations and interpretations.

Figure 3

Substantial differences between the concentration profiles of stationary hexagonal patterns in the CIMA reaction and the reaction-diffusion model are reconciled by the pattern-fluid model. (a) Experimental hexagonal pattern from [13] at concentrations ${\left[{\mathrm{ClO}}_2^{-}\right]}^A=20\mathrm{mM},{\left[{\mathrm{H}}_2{\mathrm{SO}}_4\right]}^B=100\mathrm{mM},\left[\mathrm{MA}\right]=9\mathrm{mM}$. (b) Numerical pattern-fluid for $\varphi =0.425,N=3$ (c) Minkowski functionals $V\left(\rho \right),S\left(\rho \right)$, and $\chi \left(\rho \right)$ of the hexagonal phase of the LE model from Fig. 1 (dots), experimental CIMA pattern (solid) from (a), and pattern-fluid (dashed) from (b). The Minkowski functionals of the LE model and CIMA pattern are significantly different. A much better agreement is found for the pattern fluid.

Figure 4

Optimal parameters for the pattern-fluid model. Determination of optimal parameter values $\varphi$ and $N$ in the pattern-fluid model by minimizing the mean-squared difference $\Delta$ between Minkowski functionals of simulated and experimental CIMA patterns: $\Delta =\text{max}(\Delta V,\Delta S,\Delta \chi )$ for the numerical pattern fluid (averaged over 10 realizations) as a function of $\varphi ,N$ for a hexagonal fundamental pattern $u_0$ $\left(\sigma =20,a=12,b=0.38\right)$ compared to CIMA patterns from [13]. Left: Hexagonal phase ${\left[{\mathrm{ClO}}_2^{-}\right]}^A=20\mathrm{mM},{\left[{\mathrm{H}}_2{\mathrm{SO}}_4\right]}^B=100\mathrm{mM},\left[\mathrm{MA}\right]=9\mathrm{mM}$, where (a) gives $\Delta$ as a function of $N$ for fixed values of $\varphi =0.1$, 0.4, 1 and (b) is a gray-scale plot of $\Delta$ in the $N,\varphi$ plane. Right: Turbulent phase ${\left[{\mathrm{ClO}}_2^{-}\right]}^A=18\mathrm{mM},{\left[{\mathrm{H}}_2{\mathrm{SO}}_4\right]}^B=30\mathrm{mM},\left[\mathrm{MA}\right]=9\mathrm{mM}$, with (c) $\Delta$ as a function of $N$ and (d) $\Delta$ in the $N,\varphi$ plane. Lower values mark a better agreement.

Figure 5

Reproduction of experimental concentration profiles of the turbulent CIMA phase by the pattern-fluid model. (a) Experimental turbulent pattern from [13, 24] with concentrations ${\left[{\mathrm{ClO}}_2^{-}\right]}^A=18\mathrm{mM},{\left[{\mathrm{H}}_2{\mathrm{SO}}_4\right]}^B=30\mathrm{mM},\left[\mathrm{MA}\right]=9\mathrm{mM}$. See [13] for initial chemical concentrations. (b) Numerical pattern fluid according to Eq. (3) with $\varphi =0.1,N=12$. (c) Area $V\left(\rho \right)/A_0$, perimeter $S\left(\rho \right)\lambda /A_0$, and Euler characteristic $\chi \left(\rho \right){\lambda }^2/A_0$ as a function of $\rho$ for the experimental turbulence (solid) and numerical pattern-fluid patterns (dashed) from (a) and (b).

Figure 6

Patterns and nonlinear interaction strength of the $(a,b)$ parameter space of the reaction-diffusion model. Left: Stationary homogeneous (gray square), hexagonal (orange circle), mixed (orange diamond) and stripe (violet triangle) patterns in the Lengyel-Epstein reaction-diffusion model in the $a,b$ plane with $\sigma =20$. The solid line indicates the Turing bifurcation. Right: Nonlinear interaction strength quantified by $\langle {\left(\frac{\partial \overline{u}}{\partial t}\right)}^2+{\left(\frac{\partial \overline{v}}{\partial t}\right)}^2\rangle +c$ in the $a,b$ plane, where $c$ is a constant chosen to be equal to $b$ for alignment with the plot on the left. For $a$ below the Turing bifurcation, the nonlinear interaction strength is 0 since the fundamental patterns are spatially homogeneous.