Module 2, lesson 3 · 45 minutes

Stop-and-go waves and crowd turbulence

Before the crush on the Jamarat Bridge in 2006, video showed the crowd pass through two sudden changes of state. Each one was visible, and measurable, minutes in advance.

You’ll be able to

describe the sequence from laminar flow to stop-and-go waves to crowd turbulence, and explain why stop-and-go waves appear;
define Helbing’s crowd “pressure”, P = ρ · Var(v), and give its unit;
state the thresholds observed at Mina in 2006, and the caveat that comes with them.

On 12 January 2006, during the Hajj, a crush on the on-ramp of the Jamarat Bridge at Mina killed several hundred pilgrims. Even the toll is uncertain contested: the Saudi Interior Ministry first gave 345 and later 363; Arab News reported “at least 363”. Officials blamed luggage dropped at the eastern entrance.

The ramp was filmed. Helbing, Johansson and Al-Abideen later analysed the video and found a longer story. Over about half an hour the crowd went through “two subsequent, sudden transitions from laminar to stop-and-go and ‘turbulent’ flows”, and the accident began about ten minutes after the second.

That matters enormously in practice. If a crowd disaster were a bolt from the blue, nothing could be done. If it is the end of a sequence of visible states, it can be seen coming. This lesson is about recognising those states and measuring them.

Three regimes on one ramp

The analysis describes three states of the flow:

Laminar flow. People move smoothly in the same direction, like the streamlines of a gentle river.
Stop-and-go waves. From about 11:53, when the outflow “dropped below a value of 0.8 persons per meter and second”, the crowd began to alternate between moving and standing still. The waves persisted for more than 20 minutes.
Crowd turbulence. From 12:19, motion became irregular, in all directions at once. The authors: “we will refer to them as ‘crowd turbulence’”.

Stop-and-go waves are bands of stopped people that travel backwards through a dense moving crowd, so that each person alternately walks and stands still. Crowd turbulence (also called a crowd quake) is irregular, violent motion in an extremely dense crowd, in which people are pushed unintentionally in all directions.

Reading a space–time diagram. Each line is one person; a steeper line is a faster walker, a flat segment is someone standing still. The times are from Helbing, Johansson and Al-Abideen; the trajectories are drawn to show the three regimes and are not data.

The 2007 paper describes the turbulent state as one “sometimes called ‘crowd panic’”. It does not itself conclude “turbulence, not panic”. That explicit conclusion comes in Helbing and Mukerji’s 2012 analysis of the Love Parade: “a mass panic was most likely not the cause… do not provide evidence for a stampede of people, while the dangerous phenomenon of crowd turbulence is clearly visible”.

Why stop-and-go waves appear

Go back to the fundamental diagram from Module 1. Flow rises with density up to a maximum, then falls. On the falling side, more density means less flow. That side is unstable, and here is why.

Suppose one person in a dense, moving crowd hesitates for a moment. Density just behind them rises a little. On the falling side of the curve, higher density means lower flow out of that spot, so people arrive there faster than they leave. Density rises further, flow drops further, and in a moment that little knot has become a band of people standing still. The people at the front of the band walk off it; people at the back join it. So the band moves backwards, upstream, while everyone in it moves forwards.

Motorway drivers know the result as a jam with no visible cause. In a crowd it is the first warning that density has passed the flow maximum and that the outflow can no longer keep up with the inflow. At Mina the outflow had dropped below 0.8 p/m/s, well below the 1.23 p/m/s that Weidmann’s curve allows at its peak. Module 1’s lesson on bottlenecks and queues explains why a congested flow often cannot recover its earlier capacity.

In a space–time diagram of a crowd, a band of flat segments moves down and to the right, from downstream towards upstream as time goes on. What is the crowd doing, and in which direction is each person moving?

It is a stop-and-go wave. The stopped band travels upstream, against the flow, while each person in it is either standing still or moving downstream. People leave the band at its front and join it at its back, so the band moves backwards even though nobody walks backwards.

Crowd turbulence and Helbing’s “pressure”

How do you put a number on turbulence? Helbing and colleagues combined two things a camera can measure: how dense the crowd is at a point, and how much the velocities of the people there vary. They called the product the crowd “pressure”: “the local pedestrian density times the local velocity variance”.

$P(\mathbf{r}, t) = \rho(\mathbf{r}, t)\;\mathrm{Var}_{\mathbf{r},t}(\mathbf{v})$

P(r, t) = ρ(r, t) · Var(v), measured around point r at time t

P: crowd “pressure”, 1/s²
ρ: local density, p/m², measured within a radius R
Var(v): local variance of velocity, m²/s²
R: measurement radius (1 m in the paper)

Check the unit: persons per square metre times square metres per second squared leaves 1/s² (persons are counted, not measured). That is why the paper keeps “pressure” in quotation marks and explicitly distinguishes it from mechanical pressure, which is force per area in N/m². It is a measure of how violently a dense crowd is jostling, not of how hard it is pressing on anyone.

On the ramp, turbulence “starts at 12:19, i.e. when the ‘pressure’ exceeds the value 0.02/s2”.

The shape of the warning. Only the threshold (0.02 /s²) and the time it was crossed (12:19) come from the paper; the curve is schematic.

Worked example

A camera tracks a patch of crowd 1 m in radius. The density there is 8 p/m², and individual speeds scatter around their mean with a standard deviation of 0.05 m/s.

Variance is the square of the standard deviation: 0.05² = 0.0025 m²/s².
P = 8 × 0.0025 = 0.02 /s². That is exactly the value at which turbulence began at Mina.
Now double the scatter to 0.1 m/s. The variance quadruples to 0.01, and P = 0.08 /s². Small increases in jostling raise P quickly.

The authors say so directly: “The critical values depend on the choice of R”. Measure density and variance over a bigger or smaller patch and you get different numbers. Treat 0.02 /s² as what was observed on one ramp, with one method, on one day. What generalises is the idea: dense crowds give warning, and a measure combining density with velocity scatter can detect it.

Why is “pressure” in quotation marks in Helbing’s paper?

Because it is not mechanical pressure. Mechanical pressure is force per area, in N/m². Helbing’s quantity is density times the variance of velocity, with unit 1/s²: a gas-kinetic analogy that measures how irregularly a dense crowd is moving. It says nothing directly about the force on anyone’s chest.

What the video showed

The detailed findings repay careful reading, because each one corrects a common assumption.

Local densities “greater than 10 persons per square meter”, measured with a radius R of 1 m. Yet the average density “rarely exceeds values of 6 persons per square meter”. In the paper’s words, “the maximum local densities are twice as high as the average densities”. An average hides the most dangerous spots.
Flow collapses above 6 p/m². “When a local density of 6 persons per square meter is exceeded, the local flow decreases by a factor of 3 or more.” A ramp that is jammed moves far fewer people, so the queue upstream grows faster.
A crowd keeps moving at extreme density. A follow-up analysis found that the average speed was not zero even at about 10 p/m². A crowd can look as if it is flowing while it is in deadly danger.

The paper describes how turbulence turns into an accident. There were “random, unintended displacements into all possible directions, which pushed people around. With a certain likelihood, this caused them to stumble … fallen individuals were trampled … the area of trampled people spread … as they became obstacles for others”. That is the domino mechanism of the previous lesson, set off by turbulence. The abstract compares the motion to geology: “sudden eruptions of pressure release comparable to earthquakes”.

Fruin described the same state twenty years earlier, from incident reports rather than video: “At occupancies of about 7 persons per square meter the crowd becomes almost a fluid mass. Shock waves can be propagated through the mass sufficient to lift people off of their feet and propel them distances of 3 m (10 ft) or more. People may be literally lifted out of their shoes, and have clothing torn off”.

What to do about it

Johansson, Helbing, Al-Abideen and Al-Bosta drew practical rules from the Mina analysis:

keep crowd density below 3–4 p/m²;
make the outflow capacity at least equal to the inflow;
avoid counterflows;
schedule with a 30% safety margin;
have contingency plans.

The changes made for the next Hajj, as listed in the 2007 paper, follow the same logic: elongated elliptical pillars at the Jamarat, “as suggested by Dr. Keith Still”; no crowd accumulation on the plaza; automated counting from cameras; balancing flows between the ground-floor and first-floor ramps; one-way streets and plaza; optimised scheduling and routing; and removal of squatters. Module 8 follows the story through the rebuilt bridge.

Laminar flow, then stop-and-go waves, then turbulence, then an accident. Each transition is visible on a camera, and the first one came about 35 minutes before the accident at Mina. A control room that recognises stop-and-go waves as a warning, and has the authority to stop inflow, can act long before anyone falls.

“Turbulence means people panicked.” The irregular motion is unintended: people are pushed by forces travelling through the crowd. The 2012 analysis of the Love Parade found no evidence of a stampede, while turbulence was clearly visible.
“0.02 /s² is a universal danger threshold.” The critical value depends on the measurement radius R.
“Stop-and-go is harmless shuffling.” At Mina it was the first of two transitions, and persisted for more than 20 minutes before turbulence began.
“The average density at Mina was over 10 p/m².” Local densities exceeded 10 p/m²; the average rarely exceeded 6. Local peaks were about twice the average.