Volcanic Island Formation: Growth vs. Wave Erosion (2D)

Volcano Supply

Effusion rate 0.30 km³/kyr

Flank angle (magma viscosity) 10°

Ocean Setting

Wave erosion rate 0.15 km³/kyr

Vent depth below sea level 800 m

Live Readout

Elapsed time

0 kyr

Erupted volume

0.00 km³

Summit elevation

−800 m

Island radius

0.00 km

Status

Vent opens on the sea floor…

How it works

The edifice is modelled as a cone of half-angle θ (steeper for viscous andesite/rhyolite magma, shallow for fluid basalt) standing on the sea floor at depth D. Total erupted volume V sets the cone's base radius and height from simple solid-cone geometry:

V = (π/3)·r²·h,  h = r·tanθ
  ⇒ r = (3V / (π·tanθ))^(1/3),  h = r·tanθ
summit elevation = h − D
radius at sea level = r·(1 − D/h)   (if D < h)

Every step adds effusion-rate volume to V (constructive), while wave erosion removes volume at a rate that peaks exactly at the shoreline and fades to zero more than ~300 m above or below sea level — real wave-base planation only attacks the intertidal zone. When erosion draws down V as fast as effusion adds to it, the whole self-similar cone stalls at a fixed size: a classic drowned guyot. When effusion wins, the summit breaks the surface and keeps climbing, now mostly out of erosion's reach.

Cross-section — the growing cone in profile; blue tint marks the submerged part, sea level is the dashed line.
Volume chart — cumulative erupted volume (tan) vs. cumulative eroded volume (blue); the gap between them is what's actually left standing.
Elevation chart — summit height relative to sea level over time; crossing zero is the moment of island birth.

Subaerial rock Submerged flank Sea level