How can "Geysers" on Europa reach heights of 100km?

Question

After seeing today's Washington Post's article NASA just saw Europa’s geysers erupting into space. Again., and especially the phrase "geysers erupting into space" I checked and Europa has a surface gravity of about 1.3 $m/s^2$, which is substantial - almost the same as the Moon's at about 1.6 $m/s^2$!

The BBC's website article Europa moon 'spewing water jets' says

"Further evidence has been obtained to show that Jupiter's icy moon Europa throws jets of water out into space."

and

"The suggestion is that the jets reach several hundred kilometres in height before then falling back on to Europa."

On earth, pressure can come from geological hydrostatic forces and by steam produced by geothermal heating, but they rarely rise beyond a few dozen meters. That would be perhaps 100 meters in Europa's gravity.

The images shown of the water above Europa show heights of 100 kilometers and more.

Unlike Earth rock being heavier than water, Europa's ice crust floats. What could be sources of hydrostatic pressure or pressurized steam near the surface that can create such "geysers" or "jets" that can project water to such heights in this gravity?

above: The geyser Strokkur in Iceland, Earth, from here.

Answer 1

Technically those aren't "geysers" on Europa, they're cryovolcanos. Though that definition may be a bit imprecise as well, but the 100 km eruptions on Europa probably have more in common with volcanic eruptions on Earth than geysers. Europa undergoes significant tidal flexing which, in combination to it's solid icy crust, could be compared to explosive volcanic eruptions on Earth and not the cyclic geysers, which are far smaller.

Volcanoes on Earth can shoot material high into the atmosphere on Earth but volcanic ash isn't a good example because it's not so much "shot" into the air as it rides rising hot air to altitudes 30 and 40 KM high.

Magma fountains on Earth can reach a few hundred meters and Tephra (anything larger than Ash, from pebbles to large rocks/boulders) can be found as far as a mile from a volcano and can exit the volcano at several hundred meters per second. From the Tephra article:

Blocks and bombs as large as 8-30 tons have fallen as far away as 1 km from their source (Bryant, 1991). Small blocks and bombs have been known to travel as far away as 20-80 km (Scott, 1989)! Some of these blocks and bombs can have velocities of 75-200 m/s (Bryant, 1991)

Europa's Tidal-flexing can cause significant pressures within it's icy crust. See here. and here.

So, if we draw an Earth-Europa comparison, and assuming my math is right, using 1/2 at^2, all you'd need for material to reach 100 km on Europa is an exit velocity from the ground of a bit over 500 meters per second. Or, some 20%-30% faster than a 22 caliber bullet. That's faster than material usually leaves an Earth Volcano, but not all that much faster. When frozen, Europa's outer icy crust is probably very hard and brittle. In pressure spots where the pressure exceeds the structural integrity of the icy crust, it's not all that surprising that the results are pretty explosive, leaving the surface at 500 meters per second.

Ice is very strong and brittle and when cold enough, where the Earth's crust, while rocks are quite brittle and strong, the crust overall can also be quite malleable, especially a few miles down in hotter temperatures under an erupting volcano.

Very cold ice is resistant to cracking, but under enough pressure, when it finally does crack it can release explosive energy. Ice cracks in water due to differential expansion, (Silicate rock doesn't expand with temperature nearly as much as frozen ice does), and while the cracking of an ice cube dropped into water might not seem like much, a big enough ice cube dealing with enough tidal heating can have explosive results.

While this fun little video might seem unrelated (turn down your volume before watching), it demonstrates the explosive force that rapid warming can have on a hard block of ice. This isn't steam pressure, as the steam is all escaping. This is the inside of the block expanding faster than the outside can and the result is impressive. https://www.youtube.com/watch?v=epkRd-w3TGw So, with that in mind, explosive energy from a hard, cold 10 mile thick icy surface under the influence of tidal flexing shouldn't be all that surprising.

Answer 2

I wanted to throw my hat into the mix to flesh out another contributing factor. One reason these "geysers" are so capable of achieving great heights is a lack of atmosphere on Europa to slow them down (and to a lesser extent, weaker gravity). I wrote a basic "physics simulation" in Python 3 which illustrates this purpose, the code for which is below.

import numpy as np

# Define properties of Europa
G = 6.67408E-11     # Gravitational constant, m^3 kg^-1 s^-2
M = 4.7998E22       # Mass of Europa, kg
R = 1560000         # Radius of Europa, m
rho0 = 0.1          # Density of air at surface, kg m^-3
H = 100             # Scale Height of Atmosphere, m

# Define properties of ejected water droplet
C_D = 0.5           # Drag coefficient, unitless
A = np.pi*0.01**2   # Cross-sectional area, m^2
m = 0.001           # Mass of droplet, kg
x = 0               # Initial height above surface, m
v = 500             # Initial velocity, m/s
t = 0               # Initial time, s
dT = 1E-3           # Timestep in "simulation", s

# Run "Simulation"
while v > 0:

    a = -(G*M*m/(R+x)**2 + 0.5*(rho0*np.exp(-x/H))*v**2*C_D*A) / m
    v = v + a*dT
    x = x + v*dT

    t += dT

# Print final results
print('Time:',round(t,4),'(s)\nHeight:',round(x/1000,4),'(km)\nVelocity:',round(v,5),'(m/s)')

This starts off by defining a droplet with the specified properties (e.g., mass, cross-sectional surface area) and ejects it up with some initial speed from the surface of Europa. As it travels up, the acceleration on it due to gravity and atmospheric drag is constantly calculated and the velocity and subsequently height are updated at every time step. The simulation stops when the droplet reaches the peak of it's upwards journey, indicated by the velocity becoming negative (i.e., the particle has begun falling down again). The final time taken to reach that height, the final height, and the velocity at that height (which should be very nearly zero) are then printed out. The air drag model I used was the basic drag equation. Feel free to substitute something more complex and potentially more realistic. The atmosphere itself, is modeled with a simple exponential model as described here.

Now to analyze a few results:

No atmosphere

In this case just set rho0 = 0. This will mean only gravity is acting to slow down the droplet. Without an atmosphere, I get the output:

Time: 413.093 (s)
Height: 101.1158 (km)
Velocity: -0.00046 (m/s)

To be fair, I chose an ejection velocity which would give me a height of 100 km (which I stole from userLTK's answer). Play around with other ejection velocities as you wish.

Thin Atmosphere

There are two factors here to produce a thin atmosphere. First, setting the density of air at the surface. Without spending the time to dig into more realistic numbers, I'll use rho0 = 0.1 ($kg/m^3$). For reference, on Earth it is $1.225\:kg/m^3$ at the surface. The other factor is the scale height. This number controls how rapidly the atmosphere disappears as you go up. A smaller number means a thinner atmosphere that ends more quickly. A larger number means a persistently thicker atmosphere that extends much farther. Again, I didn't look up what a proper scale height should be, but for reference on Earth the scale height is about $8\:km$. Here I used H = 100 ($\mathrm{meters}$). In this case, I get the output:

Time: 175.804 (s)
Height: 19.9797 (km)
Velocity: -0.00028 (m/s)

Even the addition of this thin, weak atmosphere dropped the height achieved by an order of magnitude.

Thick Atmosphere

Let's up the density of the air and make the atmosphere extend more by increasing the scale height. Now rho0 = 0.5 ($kg/m^3$) and H = 1000 ($\mathrm{meters}$). The output is:

Time: 7.158 (s)
Height: 0.1182 (km)
Velocity: -0.00127 (m/s)

We've dropped the height by three orders of magnitude compared to the no atmosphere model!

Conclusion

While the physics model is basic, I don't think it is unrealistic. I'm sure with more investigation, more accurate numbers and models could be used, but I believe the results would be more or less the same. On Earth, it is not gravity which primarily limits the height of our geysers, it the strong atmosphere drag. If I run this simulation on Earth, I find a similar water droplet would reach a height of $40\:m$ in our atmosphere, but without that atmosphere, it would get up to $13\:km$, more than 300 times higher. The reason these geysers can reach such large heights is primarily because of the lack of atmosphere drag, compared to what we see on Earth. The weaker gravity does help in the geysers reaching higher heights, but to a lesser extent.

Answer 3

I've drawn information from the helpful links below this question and this partial answer

I don't think astronomers or planetary scientists expect a geyser of liquid water to be squiring 100 to 200 kilometers above Europa's surface. There are now however at least two published detection of plumes of water vapor that reach that high.

So after doing some reading, asking around, and with helpful conversations in the comments of this questions and the answers here, I think it's pretty clear that there are not geysers on Europa and the term was used incorrectly by the Washington Post. However, Based on the sub-surface jet model, it might not be incorrect to call them jets, as in the BBC article.

above: NASA hubblesite images from here and here respectively.

The image on the left is reminiscent of a water fountain and probably gives the wrong impression. While there is probably some directionality, it might not be a coherent, tight column like that.

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Possibility #1 Squirting Water from an Over-Pressured Ocean:

While it's easy to imagine that stresses and pressure in the thick ice is compressing the liquid below so much that it would squirt out of a hole like a small hole in a water balloon, probably not. Ice floats. The density of ice is about 0.92 to 0.93 (when very cold) compared to cold water at about 1.00. That means if you dug a hole in the ice and reached the ocean, it would rise, but probably not to the surface. For example, if everything is in equilibrium and the ice were 10km deep, it might end up something like 800 meters below the average surface. There could be short term non-equilibrium, but ice deforms and cracks and refreezes, and over time the system would not build up and maintain huge hydrostatic pressure beyond that caused by the weight of the ice. The mechanics and dynamics of the ice a whole topic in itself.

above: Image of ice fishing from here

cf. "I've been waiting all morning - where is the geyser?"

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Possibility #2 Squirting Water from an Over-Pressured Trapped Lake:

This one is a bit easier to imagine. A volume of water trapped within the ice and subjected to huge tectonic forces, a bit like a surface spring of subsurface water on Earth. However a volume of trapped gas would really be helpful here. Since water is nearly incompressible, the loss of even a tiny volume will rapidly reduce the pressure.

above: "Europa's ice-trapped lake sits above the ocean in an illustration, Illustration courtesy Britney Schmidt and Dead Pixel FX, University of Texas at Austin" from National Geographic's "Great Lakes" Discovered on Jupiter Moon? (cropped).

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Possibility #3 Sub-surface Vent:

above: Figure 3 from Jared James Berg's thesis Simulating water vapor plumes on Europa.

Berg's thesis presents Direct Simulation Monte Carlo (DSMC) calculations for a scenario where a reservoir of trapped liquid water below the surface of Europa feeds a volume of water vapor which continuously vents to the surface. If the cross section of the passage has a constriction or narrowing followed by and expansion area, this forms a sort of nozzle which converts some of the random thermal motion of the atoms into a directed flow. The velocity will be somewhat focused upwards, and the magnitude will be of the same order as the thermal velocity.

This model does not rely on high pressures in the crust, but instead "focuses" the thermal motion upwards.

The simulation includes the dynamics small "grains" of water ice in addition to water vapor molecules. At some point early in the trajectory, the mean free path becomes so large that further condensation stops and molecules are traveling on nearly ballistic trajectories under the influence of gravity.

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Comments on Geysers and Atmospheres:

The image of the geyser in the question has initiated some discussion on the effects of atmospheric drag. Geysers on earth reach anywhere from sub 1 meter to as much as 20 meters typically. How much higher would it go if there were no atmosphere?

Here's a calculation (python script below) of isolated droplets launched into various density atmospheres with an initial velocity of 25 m/s just as a ballpark estimate of a tall geyser. In each plot the heigh vs time is calculated for five radii geometrically spaced between 0.1 and 10.0 millimeters. It can be a factor of 10 or 20 effect for the smallest droplets, and maybe a factor of 2 or 3 for the largest, but it doesn't get you to 100 kilometers if there's no air.

Yes if Europa had Earths atmosphere, the geysers wouldn't be there, but no, there aren't geysers on Europa because there's no atmosphere.

But this kind of calculation is an overestimate of the impact of drag. The effect in a real geyser would probably be much smaller than this kind of calculation suggests. Take a look at an actual geyser! The mass of the water probably dominates the column, and rapidly transfers momentum to the air in the column by the same drag force. The force ($dp/dt$) is equal and opposite, so actually drag may speed up the air much more than it slows down the water. And of course, this is hot air and steam which are accelerated by buoyancy forces as well.

The source is the following video - it really gets going around 02:00, and if you keep watching you can see a double rainbow as a bonus! (original)

https://www.youtube.com/watch?v=y8gLhHzPY5M

def Fdrag(x, v):

    if hscale:
        rho = rho0 * np.exp(-x/hscale)
    else:
        rho = rho0

    return -0.5 * rho * v * abs(v) * CD * area

def deriv(X, t):

    x, v     = X
    acc_drag = Fdrag(x, v)/mass

    xdot     = v
    vdot     = -acc_grav + acc_drag

    return np.hstack((xdot, vdot))

class Droplet(object):
    def __init__(self, r):

        self.r = r

class Atmosphere(object):
    def __init__(self, rho0, hscale=None):

        self.rho0   = rho0
        self.hscale = hscale

import numpy as np
from scipy.integrate import odeint as ODEint
import matplotlib.pyplot as plt
import copy

CD_sphere    = 0.47
H2O_density  = 1E+03        # kg/m^3

hscale_Earth = 1E+04        # meters
g_Earth      = 9.8          # m/s^2
rho0_Earth   = 1.3          # kg/m^3

# make some droplets

v0_all = 25.    # m/s

radii = np.logspace(-2, -4, 5)  # meters

droplets = [Droplet(r) for r in radii]    # instantiate

for drop in droplets:     # I love python objects :)
    drop.area   = np.pi * drop.r**2
    drop.volume = (4./3.) * np.pi * drop.r**3
    drop.mass   = H2O_density * drop.volume
    drop.CD     = CD_sphere
    drop.v0     = v0_all
    drop.x0     = 0.0

# make some atmospheres

fractions   = np.array([1E-04, 1E-02, 1])
names       = ['1E-04', '1E-02', '1.0']

atmospheres = []

for frac, name in zip(fractions, names):

    rho0 = rho0_Earth * frac

    atmosphere = Atmosphere(rho0, hscale=hscale_Earth)

    atmosphere.name = name + " of Earth"
    atmosphere.droplets = copy.deepcopy(droplets)  # DEEP copy!
    atmosphere.acc_grav = g_Earth

    atmospheres.append(atmosphere)

tol  = 1E-09
time = np.linspace(0, 6, 200)

xpts = [2.6, 1.8, 1.2, 0.5, 0.3]
ypts = [19, 11.8, 6.3, 3.1, 1.2]
labs = ['0.1', '0.32', '1.0', '3.2', '10'][::-1]
labs = [n + 'mm' for n in labs]

labzip = zip(xpts, ypts, labs)

for atmosphere in atmospheres:

    for drop in atmosphere.droplets:

        X0       = np.hstack((drop.x0, drop.v0))
        mass     = drop.mass
        area     = drop.area
        CD       = drop.CD
        rho0     = atmosphere.rho0
        hscale   = atmosphere.hscale
        acc_grav = atmosphere.acc_grav

        answer, info = ODEint(deriv, X0, time,
                              rtol=tol, atol=tol,
                              full_output=True )
        drop.answer = answer

plt.figure(figsize=[14,5])

for i, atmosphere in enumerate(atmospheres):

    plt.subplot(1,len(atmospheres),i+1)

    for drop in atmosphere.droplets:
        x, v = drop.answer.T
        plt.plot(time, x)

    plt.ylim(0, 34)
    plt.xlim(0, 5.2)

    title = atmosphere.name + ", v0=" + str(v0_all) + " m/s"
    plt.title(title, fontsize=14)

    if i == 2:
        for x, y, s in labzip:
            plt.text(x, y, s, fontsize=12)

    plt.xlabel('time (sec)', fontsize=14)
    plt.ylabel('height (m)', fontsize=14)

plt.suptitle('Single, Isolated droplets in air', fontsize=18)

plt.show()