Layered Canyon Explorer: Stream-Power Incision, Knickpoint Retreat and Caprock Hoodoos

Presets

Live readouts

Graphs

River long profile: strata bands, plateau, bed now (blue)

Valley cross-section at the chosen row

Knickpoint retreat: model vs uniform-K closed form (km from outlet)

Canyon depth (m) and rim width (m/10) over time

Stratigraphic column: erodibility K and critical slope of each layer

Numerical checks

running...

How to use

What to try: press Drop base level and watch a knickpoint (waterfall) climb up the river. Harden the caprock and see the cliff stall; soften the shale and watch benches form. Use Mesa country for hoodoos. Change the slope exponent to n = 2 and the step smooths out. Move the cross-section slider to compare reaches. Hover or tap the terrain to probe the rock layer, its erodibility and the local slope.

The science

The river sets the pace. Where bedrock is eroded by flowing water, the stream-power law says the bed lowers at a rate proportional to the power the water spends on it:

dz/dt = U - K A^m S^n

z bed elevation, A upstream drainage area, S = -dz/dx local bed slope, K erodibility of the rock the river is cutting into (it changes layer by layer), U rock uplift. Drainage area follows Hack's law, A = k_a x^h with h = 1.67, measured from the divide.

Knickpoints. For n = 1 the equation is a pure advection equation: a step in the bed moves upstream at c = K A^m, x_k(t)^(1-hm) = x_0^(1-hm) - (1-hm) K k_a^m t The step does not smooth out; it travels intact and leaves a graded reach behind it. For n > 1 the step diffuses. The page solves the profile with an implicit upwind scheme (unconditionally stable) and checks it against this closed form.

Steady state. Setting dz/dt = 0 gives S = (U/K)^(1/n) A^(-m/n) a power-law profile with concavity theta = m/n. The page integrates a profile to steady state and regresses log S on log A to recover m/n.

Layers. Each layer has its own K, its own weathering rate and a critical slope. Hard caprock and cliff-forming limestone have low K and steep critical slopes (cliffs). Shales have high K and gentle slopes. When a soft layer under a hard one wears away faster, the hard layer is undermined and retreats in blocks: this is differential erosion, and where fractures isolate a block you get buttes and hoodoos.

Valley walls. The heightmap is evolved by exchanging material between neighbouring cells. For each pair the higher cell gives a flux made of three terms: weathering w s/s_c, linear diffusion D s/dx and, when the slope s exceeds the layer's critical slope s_c, a slope-failure term proportional to the excess. Material reaching the channel is carried away by the river (the volume ledger is shown as a check).

Honest simplifications. The river is one channel with a fixed sinuous path; the layers are horizontal (a small regional tilt follows the plateau); there is no sediment-cover feedback, no lateral river migration, no overhangs (a heightmap cannot represent them) and no rockfall mechanics. K, w, D and critical slopes are plausible illustrative values for a dry plateau, not measurements of a real canyon; time is in model years and the time-lapse runs roughly a million years in ten seconds. The base-level drop is imposed instantly at the outlet; layer names are generic.

Live checks. (1) the numerical knickpoint position against the closed-form retreat integral, (2) the steady-state slope against (U/K)A^(-m), (3) the regressed concavity against m/n, (4) the live volume ledger: change of rock volume equals minus the exported sediment to 1e-6.