Self-organized criticality in cumulus clouds

The shape of clouds has proven to be essential for classifying them. Our analysis of images from fair weather cumulus clouds reveals that, in addition to turbulence, they are driven by self-organized criticality. Our observations yield exponents that support the fact the clouds, when projected to two dimensions, exhibit conformal symmetry compatible with $c=-2$ conformal field theory. By using a combination of the Navier-Stokes equation, diffusion equations, and a coupled map lattice, we successfully simulated cloud formation, and obtained the same exponents.

Figure 1

Three samples of (a) LERW and (b) the boundaries of cumulus clouds.

Figure 2

(a) Illustration of the winding angle statistics of random curves. (b) The box-counting method defining the quantities $l$ and $L$. (c) The $l-r$ scaling method for extracting the closed loops.

Figure 3

The fractal properties of clouds from observational data. (a) Log-log plot of trace lengths $l$ in terms of $L$ (the box linear size) for BoCLs. The dashed line is a linear fit with slope $D_f=1.248\pm 0.006$. The inset is the end-to-end distance $\overline{R}$ in terms of $N$ with the exponent $\nu =0.81\pm 0.01$. (b) Ensemble average of ${log}_{10}l$ in terms of ${log}_{10}r$ ($〈〉$ means the ensemble average) with slope $D_f=1.22\pm 0.02$ for BuCLs. The log-log plot of the distribution functions of $l$ is shown in the inset, with the exponent ${\tau }_l=2.38\pm 0.02$. (c) Semilog plot of the LGF in terms of $\tilde{r}$. The log-log plot of the distribution functions of $l$ is shown in the inset, with exponents ${\tau }_r=2.12\pm 0.03$. (d) Semilog plot of the distribution of the winding angle $p\left(\theta \right)$ in terms of $\theta$ for three BoCLs. (e) $〈{\theta }^2〉$ and $〈\theta 〉$ for both BoCLs and BuCLs. The inset shows the semilog plot of the variance in terms of $l$ for both cases with slopes $0.42\pm 0.01$ and $0.36\pm 0.02$ for boundary and bulk curves, respectively.

Figure 4

The schematic representation of the process of generating a curve starting and ending on the horizontal (real) axis.

Figure 5

The results for the SLE analysis for the BoCLs. The main panel shows the variance of the driving function in terms of “time,” with the slope $\kappa =2.2\pm 0.2$. The upper inset shows that the mean value of the driving function is almost zero, and the lower one shows the distribution function of $\frac{{\zeta }_t}{\sqrt{2}}$ at time $t={10}^4$, with a Gaussian fit corresponding to that time.

Figure 6

(a) 2D image of cloud simulation and (b) its contours.

Figure 7

The results of simulations of clouds using the CML method. (a) $〈{log}_{2}l〉$ in terms of $〈{log}_{2}r〉$ with the slope $D_f=1.247\pm 0.016$. The inset shows the log-log plot of the distribution functions of $l$ with exponent ${\tau }_l=2.14\pm 0.05$. (b) Semilog plot of the LGF in terms of $\tilde{r}$. The inset shows the log-log plot of the distribution functions of $r$ with exponent ${\tau }_r=2.35\pm 0.06$. (c) Variance of $\theta$ and its average, where the inset shows the semilogarithmic graph with slope $0.506\pm 0.015$.

Figure 8

RGB and gray maps of an image that was taken from cumulus clouds. The color bars show the intensity of light of each color.

Figure 9

(a)–(d) are four sample pictures from cumulus clouds. (e) and (f) are the boundaries of two samples of cumulus clouds in a gray map.

Figure 10

(a) and (b) are inner contour lines of cumulus clouds.