A river feeds sediment into a basin. Water parcels random-walk down the deepest, straightest route; where the jet slows, grains settle, a bar grows, the flow splits around it. Simplified cellular model, 100 m cells, not a calibrated forecast.
New land area (km², cyan) and channel count at the mouth (orange, per 5 km² scale).
Second check: Ferguson-Church settling law vs an independent drag force balance (Schiller-Naumann drag, solved by bisection).
Four headless runs of 6 years each with identical flow. Prediction: the trapped share should rise monotonically with settling velocity.
The science
Water routing. A few hundred water parcels enter at the river inlet. Each step a parcel moves to one of 8 neighbours with probability proportional to hθ · (0.05 + cosα)κ: depth h attracts flow, and the angle α to the previous heading models momentum (jet inertia κ, heading memory). The count of parcels crossing a cell, divided by the number released, is the fraction of discharge through it: u = Q f / (Δx h). This follows the idea of the DeltaRCM family of models, in a much cruder form: no water-surface solver.
Sediment. A parcel carries solids volume Vs = (C/ρs) Vw. Local transport capacity is Ceq = Cin(u/uref)3 (a power law of the Engelund-Hansen type; uref is the inlet velocity, so the feeder river is in equilibrium). Above capacity, a fraction 1 - exp(-ws Δx /(u h)) of the excess settles in the cell (residence time Δx/u, fall height h); below capacity the flow erodes, limited to 3 m of the original bed. The settling speed ws = R g D² / (C1ν + √(0.75 C2 R g D³)) is the Ferguson-Church law (R = 1.65, C1 = 18, C2 = 1).
Morphology. Bed change = deposited volume / (cell area (1 - λ)), porosity λ = 0.4. Wave smoothing is a conservative diffusion of the shallow bed. Sea level rises by the chosen rate and the whole bed subsides by the subsidence rate. Cells higher than sea level are land and carry no flow.
Why a mouth bar forms. Leaving the channel the jet spreads over many cells and gets deeper, so u falls and the cubic capacity collapses: sediment drops out just offshore. The bar shallows its own cells, parcels avoid shallow cells (weight ∝ hθ), and the flow bifurcates around it. Coarser grains stop sooner and build blunt, bar-fronted deltas; finer grains travel offshore and build smooth, muddy fronts.
Simplifications, honestly. No tides, waves only as diffusion, no vegetation feedback on flow, no cohesion, no floodplain overbank deposition, upstream reach of the river is held at equilibrium, one grain size, 2D depth-averaged. The morphological time step (0.04 yr) is accelerated: parameters give plausible orders of magnitude, not a prediction for any real delta. The discharge, load and rates in the presets are illustrative.
Checks. (1) Solids are conserved: supplied = exported offshore + stored in the bed (re-summed from the final heightfield, corrected for subsidence). (2) The settling law is compared with an independent force-balance solution.