Dynamics of crowd disasters: An empirical study

Many observations of the dynamics of pedestrian crowds, including various self-organization phenomena, have been successfully described by simple many-particle models. For ethical reasons, however, there is a serious lack of experimental data regarding crowd panic. Therefore, we have analyzed video recordings of the crowd disaster in Mina/Makkah during the Hajj in 1426H on 12 January 2006. They reveal two subsequent, sudden transitions from laminar to stop-and-go and “turbulent” flows, which question many previous simulation models. While the transition from laminar to stop-and-go flows supports a recent model of bottleneck flows [D. Helbing et al., Phys. Rev. Lett. 97, 168001 (2006)], the subsequent transition to turbulent flow is not yet well understood. It is responsible for sudden eruptions of pressure release comparable to earthquakes, which cause sudden displacements and the falling and trampling of people. The insights of this study into the reasons for critical crowd conditions are important for the organization of safer mass events. In particular, they allow one to understand where and when crowd accidents tend to occur. They have also led to organizational changes, which have ensured a safe Hajj in 1427H.

Figure 1

(Color) (a) Average of the local speeds $V(\vec{r},t)=\parallel \vec{V}(\vec{r},t)\parallel$ as a function of the local density $\rho (\vec{r},t)$. Our own data points are for $R=1\mathrm{m}$ and are shown as red stars. Other symbols correspond to data by Fruin [39] (black triangles), Mori and Tsukaguchi [29] (green circles), Polus et al. [30] (purple squares), and Seyfried et al. [15] (blue dots), obtained with other measurement methods. The solid fit curve is from Weidmann [31]. Scaling the density with a factor of 0.7 (and the Mori data with a factor of 0.6), our data become compatible with Weidmann’s curve (see inset); i.e., the different average projected body areas of people in different countries are very important to consider [40]. Note, however, that the average local speed does not become zero at extreme densities. (b) Average of the local flows $Q(\vec{r},t)=\rho (\vec{r},t)V(\vec{r},t)$ as a function of the local density $\rho (\vec{r},t)$. We have used the same symbols as in (a). Note the second flow peak—i.e., the local maximum at $9\mathrm{persons}∕{\mathrm{m}}^2$. (c) Distribution of local densities $\rho$ for a given average density $\varrho$ (red, $1.6\mathrm{persons}∕{\mathrm{m}}^2$; blue, $3.0\mathrm{persons}∕{\mathrm{m}}^2$; black, $5.0\mathrm{persons}∕{\mathrm{m}}^2$). The $\gamma$ distribution fits the histograms with 50 bins well (solid lines). (d) Distribution of local speeds for the same average densities $\varrho$ as in (c) (same colors for same densities). The distributions deviate from the expected normal distributions, as many pilgrims are parts of large groups including people of high age. Solid lines are smoothed fit curves serving as guides for the eye. Note that, at low densities, a small percentage of pilgrims returns against the main flow direction.

Figure 2

(Color) (a) After a laminar phase, the density of pilgrims shows a sudden transition to stop-and-go waves around $11:53\mathrm{am}$. The densities shown were determined by Gaussian smoothing in space and time. Blue color corresponds to low densities; yellow and orange colors reflect high values. (b) Average flow as a function of time. Note that the drop of the average pedestrian flow below a value of 0.8 persons per meter and second coincides with the occurrence of stop-and-go waves; see Fig. 2a. (c) Location-dependent velocity field $\vec{U}\left(\vec{r}\right)=⟨\vec{V}(\vec{r},t){⟩}_t$ of pilgrim motion (where the brackets indicate an average over the index variable—i.e., over time). Arrows represent pedestrian speeds, averaged over the period from $11:45\mathrm{am}$ to $12:30\mathrm{am}$ on 12 January 2006. To avoid boundary effects, the evaluation focused on the $20\mathrm{m}\times 12\mathrm{m}$ central area of our video recordings. The $x$ coordinate denotes the distance to the on-ramp of the Jamarat Bridge and points opposite to the direction of its entrance. One can clearly see the merging of pedestrians coming from different directions, which caused a bottleneck effect. The contour plot below the arrows represents the “pressure” $P\left(\vec{r}\right)=\rho \left(\vec{r}\right){\mathrm{Var}}_{\vec{r}}\left(\vec{V}\right)$, which we have defined as the average pedestrian density $\rho \left(\vec{r}\right)$ times the velocity variance ${\mathrm{Var}}_{\vec{r}}\left(\vec{V}\right)=⟨\left[V\right(\vec{r},t)-\vec{U}(\vec{r}){]}^2{⟩}_t$ around the average velocity $U\left(\vec{r}\right)$ [51]. The dark red area represents the highest values of the “pressure,” where the most violent dynamics occurred (see the crowd video in the supplementary material [41]). This is also the area where people stumbled and where the accident began (marked by an ellipse).

Figure 3

(Color) (a) Representative trajectories in laminar flow, stop-and-go motion, and “turbulent” flow. Each trajectory extends over an $x$ range of $8\mathrm{m}$, while the time required for this stretch was scaled to 1. To indicate the different speeds, symbols were included in the curves every $5\mathrm{s}$. While the laminar flow (red line) is fast and smooth, motion is temporarily interrupted in stop-and-go flow (black line) and backward motion can occur in “turbulent” flows (blue line). (b) Example of the temporal evolution of the velocity components $V_y\left(t\right)$ in the $y$ direction and $V_x\left(t\right)$ in the $x$ direction during “turbulent” crowd dynamics. A symbol is shown every second. One can clearly see the irregular motion into all possible directions. (c) “Pressure” $P\left(t\right)=\rho \left(t\right){\mathrm{Var}}_t\left(\vec{V}\right)$ as a function of time $t$, where $\rho \left(t\right)$ is the spatial average of the density in the central recorded area and ${\mathrm{Var}}_t\left(\vec{V}\right)=⟨\left[V\right(\vec{r},t)-⟨V{⟩}_{\vec{r}}{]}^2{⟩}_{\vec{r}}$ is the velocity variance. [For the spatial dependence see Fig. 2c, for the spatiotemporal evolution, see the video animation in the supplement [41].] The transition to “turbulent” crowd dynamics (see the crowd video in the supplement [41]) starts at 12:19—i.e., when the “pressure” exceeds the value $0.02∕{\mathrm{s}}^2$. The crowd accident began when the “pressure” reached its peak. (d) Probability density functions of the velocity increment $V_x^{\tau }=V_x(\vec{r},t+\tau )-V_x(\vec{r},t)$ in the laminar and turbulent regimes, determined over many locations $\vec{r}$ for $R=\sqrt{10∕\varrho }$ (see Fig. 1) and $\tau =0.1\mathrm{s}$ (red curves), $\tau =1\mathrm{s}$ (green curves), $\tau =10\mathrm{s}$ (blue curves), and $\tau =100\mathrm{s}$ (black curves). For clarity of presentation, the curves are shifted in vertical direction. Note the nonparabolic, peaked curve for small values of $\tau$, which distinguishes turbulent from laminar flows. (e) Distribution of displacements [i.e., location changes between subsequent stops, defined by $\parallel \vec{V}(\vec{r},t)\parallel <0.1\mathrm{m}∕\mathrm{s}$]. The double-logarithmic representation reveals a power law reminiscent of a Gutenberg-Richter law for earthquake amplitudes. Here, the slope is $2.01\pm 0.15$. (f) Double-logarithmic representation of the structure function $⟨\parallel \vec{V}(\vec{r}+\mathrm{\Delta }\vec{r},t)-\vec{V}\left(\vec{r}\right){\parallel }^2{⟩}_{\vec{r},t}$ of “turbulent” crowd motion, measuring the dependence of the relative speed on the distance $\mathrm{\Delta }\vec{r}$. As in fluids, the slope at small distances is 2, but the slope of $0.18\pm 0.02$ at large separations (in the so-called “inertial regime”) differs from turbulent fluids, probably due to the increased propulsion forces during “crowd panics.”