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Time and earthquakes
Soil transport
Initial scarp and view
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The science
What a fault scarp is. When an earthquake ruptures the surface, one side of the fault steps up relative to the other and leaves a sharp step in the ground, the scarp. From that moment, soil creep (tree-throw, burrowing animals, freeze-thaw, wetting and drying) moves soil downhill, the crest lowers, the base fills with colluvium and the step rounds off. The older the scarp, the smoother its profile, so geomorphologists can estimate its age from the shape.
Linear diffusion. If the soil flux is proportional to slope, q = −D S with S = dz/dx, then conservation of mass gives the diffusion equation:
∂z/∂t = D ∂2z/∂x2 D in m2/kyr (typical 1 to 20)For an initial step of height H at x = 0 the exact solution is
z(x, t) = (H / 2) [ 1 + erf( x / (2 √(D t)) ) ] steepest slope Smax = H / √(4 π D t)The product kt = D t (units m2) is the "diffusion age". Fitting the erf curve to a surveyed profile gives kt; if D has been calibrated on scarps of known age, the age is kt / D. That is how earthquake scarps in the western United States, New Zealand and Turkey have been dated.
Nonlinear creep. Soil cannot hold slopes steeper than a critical slope Sc = tan(angle of repose), so a nonlinear flux is often used (Roering et al. 1999):
q = −D S / (1 − (S / Sc)2)The flux grows without limit as S approaches Sc, so a fresh, nearly vertical scarp collapses within the first few steps into a free face at the critical slope and then degrades more slowly. The model caps S/Sc at 0.98 in the denominator, an effective diffusivity boost of about 25 times. Switch the nonlinear option off to get pure linear diffusion. A linear fit to a nonlinear scarp tends to over-estimate the age, which the fourth graph shows.
Repeated earthquakes. The recurrence slider adds a new offset every so many kyr (or press "New earthquake"). The result is a compound scarp whose young step sits on an older rounded one; a single erf then fits poorly, a reminder that kt dating assumes one event.
Numerical checks run on this page. (1) A separate linear run to D t = 120 m2 is compared with the exact erf solution. (2) The same run is compared with Smax = H / √(4 π D t), and the kt recovered by the least-squares erf fit is compared with the true D t. (3) The live profile obeys mass conservation: the volume now equals the volume at the start plus the volume added by earthquakes; the flux-form scheme moves soil but cannot create it.
Numerics. Finite volumes on 401 nodes (0.5 m spacing, 200 m wide), explicit time stepping with an adaptive step proportional to dx2 / (D times the diffusivity boost), no-flux ends (flat far field on both sides).
Honest simplifications. One-dimensional profile, uniform soil, a single D for the whole slope, no gully or stream erosion, no deposition from the upslope side, no spatial changes in climate, no fault-zone damage, and an initial step with a sharp vertical offset (real scarps already have a colluvial wedge and a free face). The 3D landscape extrudes the 1D profile along the fault; the vegetation, boulders, survey rods and total station are visual props, not part of the calculation. The vertical exaggeration slider only stretches the picture, and kt-age estimates from real scarps depend on a calibrated D that varies with climate and soil.