The Pistol Shrimp and the Physics of Cavitation

Biology lens: weapon, signal – and a claw that can switch sides

For the shrimp, the snapping claw is a multi-purpose tool. The shock wave stuns or kills small prey 1; the same snap serves for defence and communication 3. Even without a sound, the claw carries information: in experiments by Hughes, male snapping shrimp reacted aggressively to isolated claws fixed open in the display posture – and how strongly they reacted depended on the size of the claw relative to their own 5. The open claw works as a visual signal of body size, which lets rivals assess each other before a fight escalates.

The claw is also remarkably plastic. In snapping shrimp of the genus Alpheus, the large snapping claw and the small pincer claw can swap roles: after the snapper is lost, the pincer on the opposite side is rebuilt into a new snapping claw over the following moults. Mellon and Stephens showed in 1978 that the form and function of a claw can be transformed even by cutting a nerve on the opposite side of the body – the nervous system controls which claw becomes the weapon 6.

Snapping shrimp shape entire habitats acoustically. They are among the main sources of biological noise in shallow bays, harbours and coastal waters. For the species Synalpheus parneomeris, Au and Banks measured peak-to-peak source levels of 183 to 189 dB re 1 µPa at 1 m, with energy spread from a few kilohertz up to 200 kHz. This crackling can limit human sonar and may interfere with the sounds of dolphins and whales 7. Note that underwater decibels use a reference pressure of 1 µPa instead of 20 µPa in air, so they cannot be compared directly with sound levels in air.

Physics lens: why fast water boils

We usually make water boil by heating it. But boiling depends on pressure just as much as on temperature: at the top of Mount Everest, water boils at around 70 °C. Lower the pressure far enough, and water boils even at room temperature. The pistol shrimp lowers the pressure – with speed.

Bernoulli’s principle says that along a streamline, the static pressure p plus the dynamic pressure ½ρv² stays constant. The faster the water flows, the lower its static pressure:

Versluis and colleagues estimated the speed of the shrimp’s jet from high-speed video of the closing claw: about 25 m/s 1 8. With p₀ = 101 kPa (atmospheric pressure) and ρ = 998 kg/m³ for water, the dynamic pressure at 25 m/s is ½ · 998 · 25² ≈ 312 kPa – three times the atmospheric pressure. Long before that point, at about 14 m/s, the static pressure reaches the vapour pressure of water (2.3 kPa at 20 °C). The water can no longer stay liquid: it cavitates. Simulations suggest the details are more complex: the low pressure arises in the core of a vortex ring that rolls up around the jet, so the cavity is ring-shaped rather than spherical 8.

Engineers capture this with the cavitation number σ = (p_∞ − p_v) / (½ρv²). The smaller σ, the more likely cavitation. For a jet of 25 m/s, σ ≈ 0.3. The mechanical replica of the claw built by Tang and Staack produced jets of about 26 to 37 m/s, with σ = 0.14 to 0.30 – comparable to the real shrimp 3.

The dramatic part comes next. When the jet has passed, the full ambient pressure acts on the bubble, and it collapses faster and faster. The little gas left inside is compressed so violently that it heats up to at least 5,000 K and glows for an instant – shrimpoluminescence, to the authors’ knowledge the first observation of this kind of light production in any animal 2. At the same moment, a shock wave races outwards at the speed of sound in water, about 1,500 m/s 3.

The same physics that the shrimp uses as a weapon is a nuisance for engineers: cavitation bubbles collapsing on ship propellers and pump impellers slowly erode even hardened steel – a process known as cavitation erosion.

Mathematics lens: how long does the collapse take?

In 1917, Lord Rayleigh calculated how an empty spherical cavity in a liquid collapses under a constant pressure difference 4. His result for the collapse time is remarkably simple:

Let us assume a bubble with a maximum radius of R_max = 3 mm – a bubble a few millimetres across. With ρ = 998 kg/m³ and p_∞ − p_v = 101,325 Pa − 2,339 Pa ≈ 99,000 Pa:

Two things follow directly from the formula. The collapse time grows in proportion to the bubble size: a bubble twice as large takes twice as long to collapse. And it is inversely proportional to the square root of the pressure difference: in deeper water, where the ambient pressure is higher, bubbles collapse faster.

Why does the collapse get so hot? If the gas inside is compressed quickly, it has no time to give off heat – the compression is adiabatic. For an ideal gas, the temperature then rises as

where γ is the adiabatic index, the ratio of the gas’s heat capacities c_p/c_V (1.4 for air). A bubble that shrinks to one tenth of its radius already reaches 293 K · 10^1.2 ≈ 4,600 K – the same order of magnitude as the at least 5,000 K derived from the flash 2. The energy spread over the whole bubble is focused into a volume 10³ = 1,000 times smaller. Real bubbles contain water vapour and lose some heat, so this estimate is a rough guide, not a precise prediction.

Computer science lens: simulating a bubble

Rayleigh’s formula assumes an empty bubble. To model a real one – with gas inside, surface tension and viscosity – physicists use the Rayleigh–Plesset equation, an ordinary differential equation for the bubble radius R(t). Versluis and colleagues used a model of this type to reproduce both the bubble radius over time and the sound emitted by the shrimp 1:

Here p_B is the pressure in the liquid at the bubble wall: gas pressure plus vapour pressure, minus the effects of surface tension and viscosity.

One number in this model is unknown: how much gas the shrimp’s bubble contains. Nobody has measured it. So the program below does what scientists do with an unknown input – it tries several values. It simulates the collapse with 10, 100 and 1,000 Pa of gas pressure (at the largest radius), prints collapse time, smallest radius, top wall speed and the temperature estimate from the physics lens, and draws the radius over time:

import numpy as np
import matplotlib.pyplot as plt
from scipy.integrate import solve_ivp

# Water at 20 °C
RHO, P_INF, P_V = 998.0, 101_325.0, 2_339.0   # density (kg/m³), ambient and vapour pressure (Pa)
SIGMA, MU = 0.0728, 1.0e-3                    # surface tension (N/m), viscosity (Pa·s)
T0 = 293.0                                    # temperature of the water (K)

R_MAX = 3.0e-3        # bubble radius at its largest (m)
GAMMA = 1.4           # polytropic exponent of the gas
C_WATER = 1482.0      # speed of sound in water (m/s)

def collapse(p_gas):
    """Simulate the collapse from R_MAX; p_gas = gas pressure in the bubble at R_MAX (Pa)."""
    def rayleigh_plesset(t, y):
        R, dR = y
        p_wall = p_gas * (R_MAX / R) ** (3 * GAMMA) + P_V - 2 * SIGMA / R - 4 * MU * dR / R
        ddR = ((p_wall - P_INF) / RHO - 1.5 * dR**2) / R
        return [dR, ddR]

    def turnaround(t, y):    # event: the bubble wall stops and turns around
        return y[1]
    turnaround.terminal, turnaround.direction = True, 1

    return solve_ivp(rayleigh_plesset, (0, 2e-3), [R_MAX, 0.0], method="LSODA",
                     events=turnaround, rtol=1e-10, atol=1e-13, max_step=1e-6)

rayleigh = 0.915 * R_MAX * np.sqrt(RHO / (P_INF - P_V))
print(f"Rayleigh's formula (empty bubble): {rayleigh * 1e6:.0f} µs\n")
print("gas (Pa)   collapse (µs)   R_min (µm)   max. speed (km/s)   T_max (1000 K)")

plt.rcParams["font.size"] = 13      # stacked and with larger type, so the chart stays readable on a phone
fig, (whole, end) = plt.subplots(2, 1, figsize=(6.4, 8.4), layout="constrained")
for p_gas in [10, 100, 1000]:     # nobody has measured how much gas the shrimp's bubble holds
    sol = collapse(p_gas)
    r_min, speed = sol.y[0, -1], np.abs(sol.y[1]).max()
    t_max = T0 * (R_MAX / r_min) ** (3 * (GAMMA - 1))    # adiabatic heating (physics lens)
    print(f"{p_gas:8}   {sol.t[-1] * 1e6:13.0f}   {r_min * 1e6:10.1f}   {speed / 1e3:17.1f}   {t_max / 1e3:14,.1f}")
    t_us, r_um = sol.t * 1e6, sol.y[0] * 1e6
    fast = np.abs(sol.y[1]) > C_WATER         # wall faster than sound: the model no longer holds
    i = np.argmax(fast) if fast.any() else len(fast)
    for ax in (whole, end):
        line, = ax.plot(t_us[:i + 1], r_um[:i + 1])
        ax.plot(t_us[i:], r_um[i:], ":", color=line.get_color())
    end.annotate(f"{p_gas} Pa", (t_us[-1], r_um[-1]), xytext=(5, 0), textcoords="offset points",
                 va="center", backgroundcolor="white")

for ax in (whole, end):
    ax.axvline(rayleigh * 1e6, color="grey", linestyle="--", label="Rayleigh's formula")
    ax.set_xlabel("time (µs)")
whole.plot([], [], ":", color="grey", label="wall faster than sound: model not valid")
whole.set(ylabel="bubble radius (µm)", title="The whole collapse: the curves overlap")
whole.legend(loc="lower left")
end.set(ylabel="bubble radius (µm, log scale)", xlim=(rayleigh * 1e6 - 4, rayleigh * 1e6 + 5), yscale="log",
        title="Last microseconds: the model reaches its limit")
end.yaxis.set_major_formatter("{x:g}")
plt.show()

• Open in Colab
• Download .py
• How to run this code

Output · what the code prints, re-run at every build

Rayleigh's formula (empty bubble): 276 µs

gas (Pa)   collapse (µs)   R_min (µm)   max. speed (km/s)   T_max (1000 K)
      10             275          3.0                89.7          1,160.6
     100             276         20.5                 5.1            116.3
    1000             279        137.0                 0.3             11.9

What the result teaches:

• The collapse time is robust. However much gas is inside, the bubble collapses after 275–279 µs, within about 1 % of Rayleigh’s formula (1.1 % for the most gas). A prediction that barely depends on an unknown input is one you can trust.
• The end point is not. The smallest radius ranges from 137 µm down to 3 µm – a factor of about 45. Because the temperature grows with (R_max/R_min)^{3(γ − 1)}, the estimate swings from about 12,000 K to over a million kelvin. The model cannot pin down the temperature.
• The model shows its own limits. With 10 or 100 Pa of gas, once the bubble is smaller than about 90 µm (dotted in Fig. 4), the wall would move at 5 to 90 km/s – faster than sound travels in water (about 1.5 km/s). The Rayleigh–Plesset equation treats water as incompressible and ignores heat loss, so in this last phase its numbers are no longer physical. Real bubbles are cushioned by water vapour, heat conduction and the compressibility of water; the flash measured for the shrimp points to at least 5,000 K 2.

A few programming ideas are worth noticing, too:

• Stiffness. Near the collapse, the radius changes extremely fast. An adaptive solver such as LSODA shrinks its time step automatically where needed.
• Events. Instead of guessing how long to simulate, the code stops exactly when the bubble wall turns around – the moment of maximum compression.
• Validation. With almost no gas inside, the simulation must agree with Rayleigh’s analytical result – and it does, to within a microsecond. Comparing numerics with a known limit is a habit worth keeping.

Try it yourself – each change takes one line:

• Set R_MAX = 6.0e-3: the collapse time doubles to 551–557 µs and every radius doubles, but speeds and temperatures stay the same – only the ratio R_max/R_min matters.
• Set P_INF = 201_325.0 (10 m of water depth): the bubble collapses after only 194–195 µs – and harder, so all speeds and temperatures rise.
• Add 10_000 to the list of gas pressures: the gas cushions the collapse, the bubble stops at about 800 µm after 308 µs and heats up to only about 1,400 K.

Rayleigh's formula (empty bubble): 551 µs

gas (Pa)   collapse (µs)   R_min (µm)   max. speed (km/s)   T_max (1000 K)
      10             551          6.0                89.7          1,160.5
     100             552         41.0                 5.1            116.3
    1000             557        274.0                 0.3             11.9

Rayleigh's formula (empty bubble): 194 µs

gas (Pa)   collapse (µs)   R_min (µm)   max. speed (km/s)   T_max (1000 K)
      10             194          1.7               304.3          2,332.3
     100             194         11.5                17.1            233.5
    1000             195         77.4                 1.0             23.6

Rayleigh's formula (empty bubble): 276 µs

gas (Pa)   collapse (µs)   R_min (µm)   max. speed (km/s)   T_max (1000 K)
      10             275          3.0                89.7          1,160.6
     100             276         20.5                 5.1            116.3
    1000             279        137.0                 0.3             11.9
   10000             308        799.8                 0.0              1.4