Structure Functions and Intermittency for Coarsening Systems

Abstract

In studies of turbulence, there has been extensive use of physical quantities such as {\it energy transfers} and {\it structure functions}. We examine whether these quantities can be useful in understanding problems of domain growth or coarsening, as modeled by the {\it time-dependent Ginzburg-Landau} (TDGL) equation and the {\it Cahn-Hilliard} (CH) equation. This paper has two major themes. First, we review our recent papers on energy transfers in domain growth. Second, we study structure functions and intermittency for coarsening systems. As a consequence of sharp interfaces, the structure functions scale as $S_q \sim r^{ζ_q}$, where $r$ is the distance between two points. For the TDGL and CH models, $ζ_q = 1$, indicating {\it anomalous scaling}

Figures (8)

• Figure 1: Plot of the order parameter $\psi(x,t)$ vs. $x$, depicting a typical order parameter profile of the TDGL equation. The locations of kinks and anti-kinks are denoted by $\sigma_n$.
• Figure 2: Schematic of domain of size $R(t)$ at time $t$, with an interface of width $\xi$, in a system of linear size $L$. The $+$ region, where $\psi > 0$, is surrounded by the $-$ region, where $\psi < 0$. The circumference of the $+$ domain is denoted by $l_c$.
• Figure 3: Plots showing the evolution of $\psi({\bf x},t)$: (a) For 1D TDGL equation, $\psi(x,t)$ vs. $x$ at $t=0, 20, 500$. (b)-(c) For 2D TDGL equation, density plots of $\psi({\bf x},t)$ at $t=5$ and 50. Points with $\psi > 0$ are marked yellow, and points with $\psi < 0$ are unmarked. (d) For 1D CH equation, $\psi(x,t)$ vs. $x$ at $t=10, 20, 10^7$. (e)-(f) For 2D CH equation, density plots of $\psi({\bf x},t)$ at $t=100$ and 1500.
• Figure 4: 1D TDGL equation: (a) Profile $\psi(x,t=500)$ vs. $x$. (b) For the profile in (a), structure function $S_q(r,t)$ vs. $r$ for $q =2$ to $6$. The inset shows $\zeta_q$ vs. $q$ for $r <\xi$. (c) For the data in (b), normalized structure function $L S_q(r,t)/(2^q N_0 r) \simeq 1$ with $N_0=2$. The symbols used are the same as those in (b). (d) Hardened $\psi(x,t=500)$ vs. $x$. (e)-(f) $S_q(r,t)$ and normalized $S_q(r,t)$ for the hardened profile in (d). The symbols used are the same as those in (b).
• Figure 5: 2D TDGL equation: (a) Density plot of the hardened field $\psi({\bf x},t=100)$. (b) Structure function $S_q(r,t)$ vs. $r$ for $q=2$ to $6$. The inset shows $\zeta_q$ vs. $q$ for $r <\xi$. The deviation from $\zeta_q = q$ is due to under-resolution of the interface region. (c) Structure function $S_q(r,t)$ vs. $r$ for $q=2$ to $6$ computed from the hardened data shown in (a). (d) For the data in (c), the normalized structure function $L^2 S_q(r,t)/(2^q l_c r) \simeq 1$. The symbols used are the same as those in (c).