Tectonic regimes with magmatism (2-D)

How a planet loses its heat depends on how its lid behaves. Lourenço, Rozel, Ballmer & Tackley (2020) showed that two numbers go a long way in deciding it for an Earth-sized planet with magmatism: the yield stress of the lithosphere, and the eruption efficiency, the fraction of the melt that reaches the surface rather than freezing at depth as intrusions. Their regime diagram has four corners: a mobile lid, an episodic lid that breaks and reforms, a plutonic-squishy lid kept thin and weak by intrusions, and a stagnant lid. This example runs one case from each corner in the spherical annulus over 4.5 Gyr.

Temperature (700 to 4500 K), viscosity (10¹⁸ to 10²⁵ Pa s, log scale) and basalt fraction (0 to 1) of the mobile, episodic, plutonic-squishy and stagnant cases (left to right), 0 to 4.48 Gyr, Nr = 64 (64 × 512 cells).

Set-up

shell	$r$ = 3481 to 6371 km, free slip, net rotation removed
walls	300 K at the surface, a cooling core
reference state	anelastic liquid, Perple_X pyrolite table, initial potential temperature 1917 K
viscosity	diffusion creep with a yield stress (below), 10 times higher below 660 km, bounded by 10¹⁸ and 10²⁵ Pa s
heating	radiogenic, 18.77 × 10⁻¹² W kg⁻¹ at the start, half-life 2.43 Gyr
melting	melt from above 300 km is erupted (fraction $E$) or intruded at 60 to 120 km (fraction $1-E$)
basalt	160 kg m⁻³ lighter than the mantle above 60 km, 190 kg m⁻³ denser below (eclogite)
run	4.5 Gyr, Nr = 64, 512 lateral cells as in the paper

The viscosity combines diffusion creep and a pressure-dependent yield stress,

$$\eta =(\frac{1}{{\eta }_{\mathrm{diff}}}+\frac{2{\dot{\epsilon }}_{II}}{{\sigma }_y}{)}^{-1},{\sigma }_y={\sigma }_{\mathrm{duct}}+0.01p,$$

with ${\eta }_0={10}^{21}$ Pa s at 1600 K. The four cases differ only in the surface yield stress ${\sigma }_{\mathrm{duct}}$ and the eruption efficiency $E$, taken from the corners of the paper's Fig. 1:

regime	${\sigma }_{\mathrm{duct}}$	$E$	status
mobile lid	20 MPa	10 %	complete
episodic lid	100 MPa	70 %	complete
plutonic-squishy lid	300 MPa	10 %	complete
stagnant lid	300 MPa	100 %	complete

Mobile lid

Temperature (top), viscosity (middle) and basalt fraction (bottom) at 0.5, 1.5, 2.5, 3.5 and 4.48 Gyr.

The run starts with the lid in place. For the first 430 Myr the surface hardly moves, the interior warms from about 2420 to 2510 K, and small cold drips form under the lid, the beads around the top of the 0.5 Gyr snapshot. Most of the melt is intruded at depth here, and the basalt stays in the upper mantle above a hot, basalt-free lower mantle.

At about 430 Myr the weak lithosphere yields and the lid starts to move. It keeps moving for the rest of the run. The surface travels at 5 to 12 cm per year during the first two billion years, faster than the mantle below it, and the surface heat flow jumps to about 200 mW m⁻². The old lid and its basalt sink as cold sheets, and by 2.5 Gyr the recycled basalt is stirred through the whole mantle in thin, stretched streaks, at a fraction of about 0.2 nearly everywhere. No thick crust and no basal basalt layer survive.

Losing heat this efficiently, the mantle cools for the whole run, from about 2510 K at 0.5 Gyr to 2040 K at 4.5 Gyr, and the core from 3970 to 3780 K. The surface heat flow falls with it, to about 100 mW m⁻² by 2.5 Gyr and 70 to 90 mW m⁻² at the end, and the surface slows to about 2 cm per year. The viscosity strip shows the cost of the cooling: the creep viscosity rises with pressure, and in the colder mantle more and more of the lower mantle reaches the upper bound of 10²⁵ Pa s, about 44 % of the shell by 4.5 Gyr. The flow then concentrates in narrow, weaker sheets between large stiff regions, and the rms velocity of the mantle drops from about 3 to below 1 cm per year. The bound makes these regions softer than the creep law alone would, not stiffer.

Episodic lid

Temperature (top), viscosity (middle) and basalt fraction (bottom) at 0.54, 1.5, 2.5, 3.5 and 4.46 Gyr.

A strong lithosphere keeps the lid in place for the first billion years. The surface hardly moves, the interior warms under the lid from about 2430 to 2610 K, and the surface heat flow settles near 36 mW m⁻². Meanwhile a basaltic crust builds up from the erupted melt, reaching about 140 km by 1 Gyr.

At about 1.0 Gyr the lid fails. It stays mobile for roughly 370 Myr, until 1.4 Gyr. During this episode the surface moves at tens of centimetres per year, the surface heat flow rises to about 120 mW m⁻², and the old lid and its crust sink as cold sheets to the base of the mantle, as the 1.5 Gyr snapshot shows. The mantle loses about 100 K in the process.

The lid then locks again and stays stagnant for the remaining 3 Gyr. The interior warms once more, to about 2760 K at the end, and the surface heat flow drops back to about 33 mW m⁻². The overturn leaves a lasting mark in the basalt: the recycled crust gathers in piles and streaks just above the core, forming a basal layer about 150 km thick by 4.5 Gyr, while a new crust grows at the surface under the reformed lid. The viscosity strip shows the deep mantle starting near 10²⁴ Pa s, stiff from the pressure dependence of creep, and softening as it warms, so that by the end only about 2 % of the shell sits at the upper bound of 10²⁵ Pa s.

Plutonic-squishy corner

Temperature (top), viscosity (middle) and basalt fraction (bottom) at 0.54, 1.5, 2.5, 3.5 and 4.46 Gyr.

This case has the strong lithosphere of the stagnant case, but nine tenths of the melt is intruded at 60 to 120 km instead of erupted. Lourenço et al. (2020) find a plutonic-squishy lid here: a lid kept thin and warm by the intrusions, broken into strong plates separated by weak zones, with slow surface motion. This run takes a different path, and ends as a mobile lid.

For the first 500 Myr the lid holds. The surface hardly moves, the interior warms from about 2430 to 2580 K, and the intruded basalt stays in a thin layer near the surface, the bright rim of the first basalt snapshot. The warm intrusions thin the lid from about 90 km to below 50 km by 1.25 Gyr. Between about 0.5 and 1 Gyr the surface starts to move, slowly at first, and by 1 Gyr the lid is mobile, with the surface moving at about 9 to 10 cm per year and the surface heat flow rising from about 40 to 100 mW m⁻².

The lid stays mobile for the rest of the run. At 1.5 Gyr cold sheets carry the lid and its basalt down to the core in a regular pattern, and from then on the recycled basalt is stirred through the mantle, at a fraction of about 0.2 to 0.4 and slightly higher near the core, with no basal layer. Unlike the mobile case the interior does not cool much: it stays between about 2655 and 2690 K, since most of the heat of the melt is released at depth, while the core cools from 3990 to 3900 K. In the second half of the run a large, cold and stiff region forms in the upper part of the shell, visible in the viscosity strip from 3.5 Gyr, and the surface slows to 4 to 5 cm per year with a surface heat flow near 47 mW m⁻².

Stagnant lid

Temperature (top), viscosity (middle) and basalt fraction (bottom) at 0.54, 1.5, 2.5, 3.5 and 4.48 Gyr.

With the strongest lithosphere and all of the melt erupted, the lid never breaks. It thickens to about 105 to 110 km within the first billion years and then stays at that thickness, and the surface moves at a few thousandths of a centimetre per year against about 2 cm per year in the mantle below.

Because the lid lets little heat through, the interior warms for the whole run, from about 2420 K at the start to 2710 K at 4.5 Gyr, while the surface heat flow stays near 30 to 32 mW m⁻² throughout. The core cools only slowly, from 4000 to 3880 K, as the hot mantle above it keeps the heat flow out of the core low, 5 to 9 mW m⁻² for most of the run. Thin plumes rise from the core-mantle boundary, best seen in the 1.5 and 2.5 Gyr snapshots.

The erupted basalt forms a crust at the surface, the bright rim of the basalt strip. Basalt that leaves the base of the crust, where it becomes denser than the mantle, sinks and is stirred into the mantle below. By 4.5 Gyr the mantle holds basalt at a fraction of about 0.1 to 0.2 nearly everywhere, with no basal layer. The viscosity strip shows the lid as the stiff outer rim above a weaker upper mantle, and a lower mantle that softens as it warms.

Measuring the regime

Following Lourenço et al. (2020), the regime is read from the lid mobility, the rms surface velocity over the rms velocity of the whole mantle, averaged over the run, together with the plateness of the surface deformation. They call a run mobile when the mean mobility exceeds 0.5 and stagnant when it stays below 0.005, with the episodic and plutonic-squishy regimes in between. Both measures describe how the lid moves, not what it is made of: a lid counts as mobile when its surface moves fast compared with the mantle and deforms in narrow zones between rigid parts, as plates do, whether or not it carries a resolved crust. docs/scripts/lid_metrics.jl applies their rules to every VTK frame:

	mobile lid	episodic lid	plutonic-squishy corner	stagnant lid
mobility, 0 to 4.5 Gyr	2.67	0.22	1.94	0.077
mobility, lid rotation removed	2.64	0.17	1.90	0.006
plateness, 0 to 4.5 Gyr	0.72	0.22	0.78	0.15
quiescent plateness	0.01	0.15	0.09	0.15
regime by their rules	mobile	episodic	mobile	episodic

The mobile and episodic cases land in the regime the paper finds for the same parameters, and the criteria of Lyu et al. (2025) agree for the mobile case, over the whole run and over its last third.

The plutonic-squishy corner does not reproduce the paper's regime. The rules of Lourenço et al. give a mobile lid over the whole run. The criteria of Lyu et al. give episodic over the whole run, because the lid is stagnant for its first billion years, and mobile over the last third. The likely reasons are among the differences listed below: the intrusions here are placed at a fixed 60 to 120 km rather than at the base of the crust, and melting follows a different scheme, so the warm, intruded layer that keeps the paper's lid squishy rather than mobile is not reproduced in the same way. The case is kept as it is, as a record of what this set-up gives at that corner.

The stagnant case does not match by the mobility rule, and the reason is worth knowing. Its lid never breaks, but from time to time the whole lid slides around the mantle as one rigid shell, at up to about 0.2 cm per year. The mobility of Lourenço et al. counts that motion, and its mean of 0.077 lies above their stagnant threshold of 0.005. With the rigid rotation of the lid removed, as in the lid mobility MILEVA writes to diagnostics.csv, the mean is 0.006, at the threshold, and the criteria of Lyu et al. (2025), which also use the plateness of the surface deformation, give a stagnant lid in every frame. So the lid behaves as a stagnant lid, while its rotation as a whole puts it on the episodic side of a rule that reads velocity alone. Crustal thickness is not compared with the paper: with 45 km cells at the surface, a thin crust such as that of a mobile lid cannot be resolved, so a thickness derived from this grid would not be meaningful for every case.

There are also some points worth keeping in mind. The episodic case has a single mobile episode in 4.5 Gyr, so its time-mean mobility rests on one event. Lyu et al. (2025) classify regimes over 6 to 10 Gyr, which these runs do not reach. Over the whole run their criteria also give episodic, but over the last third, after the lid has reformed, they give stagnant, so a longer run would be needed to see whether the lid fails again. The set-up also departs from the paper in a few places. The pressure dependence of the yield stress is not given in the paper, and 0.01 is used here. Melting follows MILEVA's peridotite scheme rather than the paper's solidus. Intrusions are placed at a fixed 60 to 120 km rather than at the base of the crust, and there is no enhanced conductivity for highly molten regions. The header of the driver script lists all differences.

Running it

julia --project=. -t 2 examples/annulus_lid_regimes_2d.jl 20 10 1e21 64     # mobile
julia --project=. -t 2 examples/annulus_lid_regimes_2d.jl 100 70 1e21 64    # episodic
julia --project=. -t 2 examples/annulus_lid_regimes_2d.jl 300 10 1e21 64    # plutonic-squishy
julia --project=. -t 2 examples/annulus_lid_regimes_2d.jl 300 100 1e21 64   # stagnant
julia --project=. docs/scripts/lid_metrics.jl output/examples/lid_regime_s20_E10_eta1e+21_N64

The arguments are the surface yield stress in MPa, the eruption efficiency in per cent, the reference viscosity and Nr. On 2 threads, with three other runs sharing the machine, the episodic case took about 31 hours, the stagnant case about 43 hours, the mobile case about two and a half days and the plutonic-squishy case about two and three quarter days, since a fast surface keeps the time step short.

References

Lourenço, D. L., Rozel, A. B., Ballmer, M. D. & Tackley, P. J. (2020), Plutonic-squishy lid: a new global tectonic regime generated by intrusive magmatism on Earth-like planets, Geochemistry, Geophysics, Geosystems 21, e2019GC008756, doi:10.1029/2019GC008756.

Lourenço, D. L., Rozel, A. & Tackley, P. J. (2016), Melting-induced crustal production helps plate tectonics on Earth-like planets, Earth and Planetary Science Letters 439, 18-28, doi:10.1016/j.epsl.2016.01.024.

Lyu, T. et al. (2025), Nature Communications 16, 10037.