12.163/12.463 Surface Processes and Landscape Evolution K. Whipple September, 2004 VIII. HILLSLOPE EVOLUTION A. Definitions: Transport Limited and Weathering Limited Landscapes Transport-limited hillslopes: delivery of sediment to streams is limited by the rate at which soil and rock can be transported (supply >> capacity). Hillslope form dictated by transport processes and their spatial variability (conservation of mass; divergence of sediment flux). Weathering-limited (detachment-limited) hillslopes: delivery of sediment to streams is limited by the rate of sediment production (supply << capacity) by the various mechanisms of chemical weathering, physical weathering, and erosional detachment (overland flow; mass movement). Hillslope form is dictated by weathering and erosional processes, divergence of sediment flux is not relevant. B. Introduction to Hillslope Hydrology Flow Pathways: 1. Horton Overland Flow (HOF). Rainfall intensity exceeds infiltration capacity, overland flow occurs regardless of soil saturation state. Typically arid regions and bare bedrock slopes (small fraction of Earth’s surface today). Sharply peaked hydrographs (minutes to channel): storm flow. HOF may have dominated early Earth history before landplants. 2. Subsurface Storm Flow (SSF) (“Throughflow”). Shallow groundwater flow. Infiltration capacity (rate) exceeds rainfall intensity, downslope flow in saturated zone, usually a thin soil above bedrock or other discontinuity in hydraulic conductivity. Flow rates cm/s -cm/hr, contributes to storm flow and base flow (most flow gets downslope in hrs -10’s hrs) --strong rise in hydrographs. Dominant mechanism in humid/temparate regions (most of Earth’s surface). 3. Return Flow and Saturated Overland Flow (RF, SOF). Variable source area concept: saturated zones at base of hills (concave topography) grow during wet season/storms. When subsurface flow capacity (“transmissivity”) is exceeded, SSF is forced to return to the surface, contributing to overland flow and storm flow component of hydrographs. 112.163/12.463 Surface Processes and Landscape Evolution K. Whipple September, 2004 4. Groundwater flow. Slow vertical percolation of water in soil/rock. Rates from cm/hr to cm/yr. Important to chemical weathering of bedrock, contributes to base flow only (minimal storm response). C. Hillslope Transport Processes Slow/Continual Processes 1. Soil Creep (humid/temperate -SSF) Biogenic Mechanisms (Burrowing, Tree Throw, etc); Frost Heave; Shrink/Swell (clays); Rheologic Creep (slow plastic flow; solufluction -- freeze thaw or wet/dry) 2. Rainsplash/Sheetwash (arid -HOF) Rainsplash -Rainflow -Sheetwash Continuum. Rain drop impacts displace sediment “splash”, net down-slope transport. “Rainflow” is transport caused by disturbance of thin, laminar sheet flow by rain drop impacts. Will consider only unchanneled sheetwash initially. Rapid/Stochastic Processes 1. Masswasting Slumps, Earth flows, Landslides, Debris flows, Rock Fall, Rock Avalanches D. Mathematical Description of Processes Conservation of Mass [Sketch] Volume of sediment (V); Volume flux of soil (qv) [x-component]; Depth of soil (h); Elevation soil surface (z); Δs; Δx; Δt Note for creep processes volume flux of soil includes the pore space (not volume flux of sediment), for sheetwash can define transport rate of soil or transport rate of sediment; the latter requires a porosity correction in mass balance equation. !coszh =!cosxs"="212.163/12.463 Surface Processes and Landscape Evolution K. Whipple September, 2004 Volume of sediment in box (unit width) Change in volume of sediment in box: xzxzshV !=!=!=""coscostqtqxzshVoutinvv!"!=!!=!!=!sqqthinoutvv!""=!!; xqqtzinoutvv!""=!!; xqtzv!!!!"=E. Soil Mantled Slopes: Steady State Forms Generic Transport Relationship Kirkby (1971): nmnmvSxkSkqq!=='Kirkby gives empirical evidence for m,n values for different hillslope processes. We will (later) examine evidence and derive values from theory, develop understanding of geologic, climatic and biotic control of various parameters. First we examine implications of different forms of the transport relationship for equilibrium hillslope form. Lab exercise will pursue numerical implementation to allow investigation of boundary conditions (link to rest of landscape) and hillslope responses to transients (e.g. change in climate or river incision rate) Coupled with continuity equation -can derive relationships for steady-state slope forms developed under different sets of processes (e.g. different climates) Soil Creep Generally Humid/temperate conditions: Ic >> Ri; m=0, n=1 Transport law: Kc = f(rainfall, windiness (tree throw), freeze-thaw cycles, soil texture, clay mineralogy, etc) [L2/T] !"#$%&'==xzKSKqccv((Continuity: xqtzv!!!!"=22xzKxzKxtzcc!!!!!!!!="#$%&'((=Substitute: Diffusion Eqn 312.163/12.463 Surface Processes and Landscape Evolution K. Whipple September, 2004 Steady State Condition: 22xzKtzc!!"!!=#=&1CxzKxc+=!""#&C10=xzKxc!!"=#&xKxzc!""&#=2221CzKxc+=!"&oczKC !=2occzKzKx !=!221"&coKxzz22!&"=Integrate (w/ respect to x): B.C.: no flux across ridge: S=0 at x=0 ; Note this is solution for steady-state slope: Separate variables, integrate B.C.: z=zo at ridge top, x=0 Steady-state solution (parabola) How are boundary conditions reflected in this formulation? Geologic and climatic factors? Use in Fault-Scarp Age Determination Analytical solution for transient behavior: error function solution. Initial profiles assumed to be at angle of repose. Diffusion models “fit” to observed slope profiles -- “fit” only derives estimate of K*t product -- need independent estimate of t to “calibrate” K for field setting. Sketch: Hanks et al, 1984 definition sketch. 412.163/12.463 Surface Processes and Landscape Evolution K. Whipple September, 2004 Initial Condition: Vertical Scarp. bxKtxaerftxz +!"#$%&=2),(Where a is scarp amplitude, b is far-field slope, and erf is the error function: ( )!"=#$%&'(KtxdeKtxerf20222)*)Maximum scarp angle (θm) occurs at x = 0 and has the convenient formulation: bKtadxdzmx+==!"#$%&='(tan0Another convenient relation is found by recognizing that erf(1) = 0.84. Thus if we define X84 as the position at which 84% of scarp offset a is reached, we can write: KtX 284=or KXt4284=tXK4284=Pierce and Colman (1986) – Paper available – present a solution for a similar problem, with these differences: no background slope (b = 0); initial 512.163/12.463 Surface