Earth's mantle with mineral-physics properties (2-D)

Earth's whole mantle in the spherical annulus, with the reference state taken from a thermodynamic table of pyrolite and the viscosity from the profile of Steinberger & Calderwood (2006), which is constrained by mineral physics, the geoid and surface observations (Material tables).

300 Myr of the driver at Nr = 64 (64 × 512 cells, 45 km radially): temperature (left) and log₁₀ viscosity (right). Six hot plumes, seeded at the core-mantle boundary, rise through a lower mantle several hundred times stiffer than the upper mantle; cold sheets descend from the top boundary layer between them.

The same run at 1, 59, 151 and 294 Myr.

The model

shell	$r$ = 3481 to 6371 km, free slip, net rotation removed
walls	273 K at the surface, 3773 K at the core-mantle boundary
reference state	anelastic liquid (ALA), with mass conservation $\mathrm{\nabla }\cdot ({\rho }_{\mathrm{ref}}\mathbf{v})=0$: adiabat, density and thermal expansivity along the adiabat of the Perple_X pyrolite table from $T_{\mathrm{pot}}$ = 1600 K (cfg.referenceTable); 2503 K and 135.5 GPa at the core-mantle boundary
energy	adiabatic heating and viscous dissipation, together (cfg.useShearHeating)
viscosity	${\eta }_r(z)\exp [(H/nR)(z)(T_{\mathrm{ad}}-T)/(T{T}_{\mathrm{ad}})]$, Steinberger & Calderwood (2006) without boundary layers, lateral factor capped at 10³ either way, clamped to ${10}^{20}$-$5\times {10}^{23}$ Pa s
initial state	the table adiabat with a 100 km cold boundary layer at the top, a 150 km hot one at the bottom, six 50 K warm seeds at the CMB, 5 K noise
run	300 Myr; Nr = 64 by default

With an upper bound of $5\times {10}^{23}$ Pa s the cold top boundary layer is only about a thousand times stiffer than the mantle below it, so it is not a rigid lid: the surface moves at about half the mantle's rms velocity (lid mobility 0.2-0.8 after 100 Myr).

Measured on the run above: rms velocity about 1 to 2 cm/yr, surface heat flow 37-50 mW m⁻² and core heat flow 180-200 mW m⁻² after the start-up, both still changing at 300 Myr. The core heat flow is high because the conductivity at the base of the mantle is ${\rho }_{\mathrm{ref}}{c}_p\kappa$ with the table's density there, about 5600 kg m⁻³, 1.7 times the conductivity a constant density of 3300 kg m⁻³ would give. The energy budget residual of diagnostics.csv stays below 0.09 of the surface power in every 100-step window except one, −0.46 at step 500, when the first plumes reach the top boundary layer and the velocity peaks at 7.5 cm/yr. This is a 300 Myr start-up on a coarse grid, a demonstration of the material model, not a statistically steady Earth.

Running it

export GKSwstype=nul
julia --project=. -t 4 examples/annulus_earth_steinberger_2d.jl 64

The argument is Nr (Nt = 8 Nr). The run above took 58 minutes on 2 threads (2615 steps). The same knobs as every example apply: MILEVA_MAXTIME_MYR, MILEVA_MAXSTEPS, MILEVA_OUTDIR, MILEVA_VTKEVERY. The animations were rendered with docs/scripts/render_annulus_gif.jl output/docs_short/earth temperature ... (and viscosity).

Simplifications

• Two-dimensional: the spherical annulus, not a three-dimensional shell.

• Anelastic liquid approximation with the reference density along the table adiabat; the heat capacity is constant (CP_REF) and gravity is constant, 9.81 m s⁻².

• Phase transitions enter through the reference profile only, not as lateral density changes.

References

Steinberger, B. & Calderwood, A. R. (2006), Geophys. J. Int. 167, 1461-1481, doi:10.1111/j.1365-246X.2006.03131.x. The data sources of the table are on the Material tables page.