Earth

Hotspot island chain

Move an ocean plate over a mantle plume and read the chain it leaves: distance over age difference is the plate’s speed, the islands subside with the cooling lithosphere and erode once above the sea until the oldest are seamounts, and a change of course at 40 Myr bends the chain by exactly the change of azimuth without touching the old islands. Let the plume drift and the chain shifts with it.

By Mykhailo Makarov

Actual application image

What happens

An oceanic plate moves over a mantle plume at a set speed and azimuth (1 cm/yr is 10 km/Myr). Every Myr a new island is seeded over the plume and every island rides with the plate, gathering volcanic volume at the plume’s Gaussian output while it is within reach; it stands as a cone of the set slope on a 4 km sea floor, sinks with the cooling lithosphere as k·(√(t + 1) − 1) and loses height to erosion while it is above the sea, the eroded volume booked so that volume plus eroded equals erupted. The plate turns to a new azimuth at the turn time; the plume may drift. Chain length over the oldest age is the plate speed and the bend is the change of azimuth.

Try this

Watch the chain grow and the oldest islands sink; at 40 Myr the course changes and the chain bends by the azimuth change. Give the plume a drift and see the chain shift.

The model and its limits

A fixed-output plume with a Gaussian reach, cones of one slope, a square-root subsidence law and erosion linear in height; no flexure, no reef growth, no real chain reproduced.

Equations used in the module

Equations
dx/dt = v,  v = 10·u (km/Myr per cm/yr)
dV/dt = Q·e^(−d²/r_p²)
H = (3·V·tan²φ/π)^(1/3)
z = H − D − k·(√(t + 1) − 1) − Δh

Related experiments

Back to catalogue

Earth

Sedimentary basin

Watch a 100 km basin fill over ten million years: the basement sinks faster the farther from the hinge, sediment arrives at the hinge and spreads downslope by diffusion, faster on land than under water, and every quarter million years a marker bed is laid. Sand stays near the source and mud travels on; porosity squeezes out with depth so the compacted pile stands taller than the solid it holds. Raise the sea and the shoreline retreats; triple the supply and the basin overfills and spills past its far edge. Supplied, exported and held sediment balance to rounding, and a diffusing mound spreads its variance by exactly 2 K t.

Earth

Folding of strata

Squeeze a 100 m sandstone bed a hundred times stiffer than its shale along its length: every wavelength seeded in it grows, but the one near Biot’s 2 pi h (eta_l / 6 eta_m)^(1/3) grows fastest and the bed buckles into regular folds, their amplitude the exponential of the growth integrated over the shortening history. Double the thickness and the folds double in wavelength; raise the contrast and they lengthen as the cube root; take the contrast away and the bed only thickens with its matrix. The viscous power of the shortening is read in watts per metre of fold.

Earth

Metamorphism

Bury a shale at a kilometre per million years down a 25 K/km geotherm, hold it, and lift it back: its point moves across the pressure-temperature grid of facies, chlorite gives way to biotite, garnet and staurolite as the equilibrium shares shift, and because the reaction rate falls a thousandfold between 600 and 400 degrees the peak assemblage is carried to the surface almost untouched, a little more retrograded when the exhumation is slow. Steepen the burial under a cold gradient and the path runs into blueschist; let a pluton heat a shallow rock and it bakes to hornfels; hold a sample isothermal and its shares relax on the closed-form exponential.