Flow around a river bend sets up a spiral secondary (helical) current: fast surface water is thrown to the outer bank, sinks, and returns along the bed toward the inner bank. That skews the fastest near-bank flow toward the outer bank, so the outer (concave) bank erodes while the inner (convex) bank accretes into a point bar from the sediment the flow can no longer carry. This is the Ikeda–Parker–Sawyer excess-velocity mechanism:
∂ζ/∂t = E₀ · U · κ̃(s) (bank migration rate)
κ̃(s) = ∫₀ˢ κ(s′) e^{-(s-s′)/λ} ds′/λ (upstream-lag-filtered curvature)
ζ is the bank's lateral position, κ(s) the channel centerline curvature, U the mean velocity, and λ a lag length set by width-to-depth ratio and friction — curvature drives migration, but with a downstream phase lag, so the fastest-migrating point sits slightly downstream of the sharpest bend. Because migration is proportional to curvature, bends are a positive-feedback instability: any curvature grows itself over time, which is exactly why straight channels evolve into meanders and meanders keep tightening.
- Migration rate E₀ — the curvature→migration coupling constant; higher values grow loops faster.
- Curvature lag — the filter length λ in the formula above; larger lag smooths and shifts the migration lobe further downstream of the bend apex.
- Sediment deposition rate — how fast eroded sediment rebuilds the point bar on the inner bank (visualized as sand-colored accretion and instanced grains).
When a growing loop's neck narrows until the channel meets itself, the river cuts off: it takes the shorter new path, and the abandoned loop is stranded as a crescent-shaped oxbow lake — the same process visible in satellite imagery of the Mississippi, Amazon, or any low-gradient alluvial river.