TWO-DIMENSIONAL ANALYTIC MODEL FOR OROGENIC COLLAPSE

The decay of orogenic plateaus is generally attributed to erosion and/or gravitational collapse. In this paper we present a relatively simple analytical approach to the quantification of the two mechanisms and their relative importance. Erosion requires differential topography. We treat the erosion problem using the Culling (diffusion equation) approach. The basic assumption is that material transport is proportional to the slope with the equivalent coefficient of diffusion, D, the constant of proportionality. Gravitational collapse is the result of the excess potential energy associated with elevated topography. It is applicable whether the topography is compensated or not. The collapse problem is treated using the thin viscous sheet approximation. The crust with a mean thickness, h_o, is assumed to deform as a linear viscous medium with a viscosity, η, which is large compared with the equivalent viscosity of the underlying mantle. The equations for conservation of mass and for the force balance are required. Both equations are nonlinear and we linearize them assuming shallow topography. Assuming this topography is harmonic with a wavelength, λ, we find an exponential decay of the topography in time. The relative role of erosion versus gravitational collapse for compensated topography is controlled by the value of the nondimensional collapse number C = gh_oλ²ρ_c/16Dηπ². If C is large compared with unity gravitational collapse dominates, if C is small compared with unity erosion dominates. Assuming λ = 500 km, initial topographic amplitude w₁ = 2 km, and an initial velocity of erosion v_e = 1 mm/yr, we find that D = 3x10³ m²/yr. Taking C = 1 with h_o = 30 km, ρ_c = 2,800 kg/m³, and ρ_m = 3,300 km/m³ with other values above we find η = 10²² Pa s. If the equivalent crustal viscosity is greater than this, erosion is expected to dominate over gravitational collapse. For smaller viscosities gravitational collapse will dominate. The value of η = 10²² Pa s is typical of values associated with the deformation of the Tibetan Plateau so that we can not conclude definitively that one mechanism is dominant over the other.