Wilson and Head’s magma ascent in dikes (1981)

Lionel Wilson and James Head followed magma up dikes from its source to the surface on the Earth and the Moon. Buoyancy drives laminar flow between the walls at a speed that grows as the square of the width, and the magma’s dissolved water comes out of solution as the pressure falls. Basalt in a dike a metre wide rises at metres per second; a viscous rhyolite barely moves.

import matplotlib.pyplot as plt
import numpy as np

from earthsciencekit.constants import GRAVITY
from earthsciencekit.volcanology import dike_ascent_velocity, water_solubility

width = np.logspace(-1, 1, 100)
magmas = {"basalt, 100 Pa s": 100.0, "andesite, 1e5 Pa s": 1e5, "rhyolite, 1e8 Pa s": 1e8}
fig, (left, right) = plt.subplots(1, 2, figsize=(10, 4))
for name, viscosity in magmas.items():
    left.loglog(width, dike_ascent_velocity(width, 300, viscosity, GRAVITY), label=name)
speed = dike_ascent_velocity(1.0, 300, 100, GRAVITY)
assert 2700 * speed * 1.0 / 100 < 2000  # laminar
left.set(xlabel="dike width [m]", ylabel="ascent speed [m/s]", title="Speed grows as width squared")
left.legend(fontsize=8)

depth = np.linspace(0, 8000, 400)
pressure = 2400 * GRAVITY * depth
dissolved = water_solubility(pressure, 4.1e-6)
saturation = (0.05 / 4.1e-6) ** 2 / (2400 * GRAVITY)
assert np.isclose(water_solubility(2400 * GRAVITY * saturation, 4.1e-6), 0.05)
right.plot(100 * dissolved, depth / 1e3, label="solubility")
right.axvline(5, color="0.5", ls="--", label="magma with 5 wt% water")
right.axhline(saturation / 1e3, color="C3", lw=1, label=f"bubbles from {saturation / 1e3:.1f} km")
right.invert_yaxis()
right.set(xlabel="dissolved water [wt%]", ylabel="depth [km]", title="Water exsolves on the way up")
right.legend(fontsize=8)
fig.tight_layout()

plt.show()