Lecture Notes on Hillslope Evolution

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. 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] h = z cosα ∆s = ∆x cos α Volume of sediment (V); Volume flux of soil (qv) [x-component]; Depth of soil (h); Elevation soil surface (z); ∆s; ∆x; ∆t 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. Volume of sediment in box (unit width) Change in volume of sediment in box: V = h∆s = z cos α ∆x = z∆x cos α ∆V = ∆h∆s = ∆z∆x = q vin ∆t − q vout ∆t q v − q vin qv − q vin ∂q ∂z ∆h ∆z =− v = − out ; = − out ; ∆t ∆s ∆t ∆x ∂t ∂x E. Soil Mantled Slopes: Steady State Forms Generic Transport Relationship Kirkby (1971): q v = kq m ' S n = k ′x m S n 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 steadystate 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: ⎛ ∂z ⎞ qv = K c S = K c ⎜ − ⎟ ⎝ ∂x ⎠ Kc = f(rainfall, windiness (tree throw), freeze-thaw cycles, soil texture, clay mineralogy, etc) [L2/T] Continuity: ∂q ∂z =− v ∂t ∂x Substitute: Diffusion Eqn ∂z ∂ ⎛ ∂z ⎞ ∂ 2z = − ⎜ − Kc ⎟ = Kc 2 ∂t ∂x ⎝ ∂x ⎠ ∂x Steady State Condition: ∂z ∂ 2z = −ε&= K c 2 ∂t ∂x Integrate (w/ respect to x): − ε& x = Kc ∂z + C1 ∂x B.C.: no flux across ridge: S=0 at x=0 C1 = 0 ; Separate variables, integrate 1 2 − ε& x = K c z + C2 2 B.C.: z=zo at ridge top, x=0 C2 = −K c zo Steady-state solution (parabola) 1 2 − ε& x = K c z − K c zo 2 z = zo − − ε& x = Kc ∂z ∂x x2 ε& 2K c 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. ⎡ x ⎤ z ( x, t ) = aerf ⎢ ⎥ + bx ⎣ 2 Kt ⎦ Where a is scarp amplitude, b is far-field slope, and erf is the error function: ⎡ x ⎤ 2 erf ⎢ ⎥= π ⎣ 2 Kt ⎦ ( ) x 2 Kt −η 2 ∫e dη 0 Maximum scarp angle (θs) occurs at x = 0 and has the convenient formulation: a ⎡ dz ⎤ ⎢ dx ⎥ = πKt + b ⎣ ⎦ x =0 Another 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: 2 X 84 = 2 Kt t= or X 84 4K Scarp studies (southwestern US: semi-arid; granular soils; sparse vegetation) generally find: 10-3 < Kc < 10-2 m2/yr. Dietrich et al. in humid/temperate N. California and Oregon coastal mountains (moist winter conditions, dense vegetation) find: Kc ~ 5 x 10-3 m2/yr. Small et al. on frost-dominated, unvegetated summit flats eroding at rates of ~15 µm/yr estimate Kc ~ 1.7 x 10-2 m2/yr. Non-linear Creep As hillslope gradient approaches the angle of repose (or critical threshold for landsliding) can expect transport rates to increase rapidly: increased probability of mass wasting events. Observations on fault scarps suggest non-linear transport processes or a scarp-height dependence on the coefficient of diffusion. Pierce and Colman (GSA Bull, 1986, 97, p. 869885) find dependence of “diffusivity” on scarp height and scarp orientation (micro-climate). Several non-linear diffusion models have been proposed to capture this effect. Only that due to Roering et al is based on data. Andrews and Hanks, late 1980’s: q v = KS n ; n ~ 2-3 Note: must recalibrate K for different values of n – to get same transport rate at low slopes. Anderson and Humphrey, 1989: ⎛ S ⎞ q v = KS ⎜⎜1 + ⎟⎟ ⎝ Sc ⎠ Howard, 1994: a ; a is a “high power” ⎛ ⎞ 1 ⎟ − 1 qv = K c S + K m ⎜ ⎜ 1 − ( S S )a ⎟ c ⎝ ⎠ ; a~3 Roering et al., 1999: ⎛ ⎞ 1 ⎟ q v = KS ⎜ ⎜ 1 − ( S S )2 ⎟ c ⎝ ⎠ Rainsplash-Sheetwash Environmental Conditions: Thin or impermeable soils, sparse vegetation, Ri > Ic. Near the ridge crest (x < xc; v. shallow HOF) rainsplash dominates, away from ridge crest (x > xc) sheetwash (unchanneled) dominates. Transition is due to: (1) sheetwash attains depth at which shear stress exceeds critical value to entrain sediment; (2) deepening sheetwash protects bed from raindrop impacts. Transport Relations Rainsplash (empirical: field/lab) ⎛ ∂z ⎞ qs = k r S = kr ⎜ − ⎟ ⎝ ∂x ⎠ kr = f(veg. cover, rainfall intensity, raindrop size, soil texture) Sheetwash (empirical/theoretical) q s = k w ( x − xc ) S = k w ( x − xc ) 2 2 2 ⎛ ∂z ⎞ ⎜− ⎟ ⎝ ∂x ⎠ kw = f(rainfall intensity, infiltration capacity, soil texture, veg cover, etc) Why m ~ 2, n ~ 2 for sheetwash? Definition sketch: thickening sheet of overland flow on hillslope Conservation of Mass (water) Pe = Ri − I c q w (x ) = Pe x = uh( x ) 2 Conservation of Momentum u = Cf −1 2 τ b = ρghS ; ghS Find h(x) 12 P x C f Pe x h( x ) = e = u ghS h( x ) = C f 13 g −1 3 Pe x 2 3 S −1 3 23 Find τb(x) τ b (x ) = ρC f 1 3 g 2 3 Pe 2 3 x 2 3 S 2 3 Fluvial Transport Relation (for sheetwash: abrupt transition from no motion to suspension common) q s = α sτ b a ; a ~ 2.5 – 3 (take a = 3, for example) q s = α sτ b ( x ) = α s ρ 3 C f g 2 Pe x 2 S 2 = k w x 2 S 2 3 2 k w = α s ρ 3 C f g 2 Pe 2 Note xc represents position x at which τb = τcr q s = k w (x − xc ) S = k w (x − xc ) 2 2 2 ⎛ ∂z ⎞ ⎜− ⎟ ⎝ ∂x ⎠ 2 Hillslope Profiles Rainsplash dominates for x < xc, in Horton’s (1945) “belt of no erosion”. Diffusive transport, so steady state solution is same as for creep. 0 < x < xc z = zo − x2 ε& 2K r Note IF rainsplash measured in terms of volume transport of sediment (not soil) then Kr = kr/(1-λp); else Kr = kr. for x > xc k w ∂ ⎛⎜ ∂z (x − xc )2 ⎛⎜ − ∂z ⎞⎟ =− ⎜ (1 − λ p ) ∂x ⎝ ∂t ⎝ ∂x ⎠ 2 ⎞ ⎟ ⎟ ⎠ Steady state condition: ∂z ∂ ⎛ ∂z ⎞ 2⎛ = −ε&= − K w ⎜ ( x − x c ) ⎜ − ⎟ ⎜ ∂t ∂x ⎝ ⎝ ∂x ⎠ Integrate (w/ respect x) ε& x + C1 = K w ( x − x c ) For C1, evaluate at x = xc − ε& xc = C1 2 2 ⎛ ∂z ⎞ ⎜− ⎟ ⎝ ∂x ⎠ ⎞ ⎟ ⎟ ⎠ 2 ⎛ ∂z ⎞ ⎟ ⎝ ∂x ⎠ ε&(x − xc ) = K w ( x − xc )2 ⎜ − 2 ∂z ε& =− K w ( x − xc ) ∂x Separate variables and integrate z = −2 Use z at x = xc (rainsplash) for C2 z = zo − ε&( x − xc ) Kw 2 xc ε& 2K r + C2 = C2 Steady State Solution: (x < xc): z = zo − (x > xc): z = zo − x2 ε& 2K r 2 xc ε& 2K r −2 ε&(x − xc ) Kw Convexo-Concave Profile Form. Generic Controls on Profile Form (Smith and Bretherton, 1972) Generic hillslope transport relation: qv = kq w S n = k ′x m S n m' Conservation of mass (water) for HOF (implies m’ = m) and sediment: q w = Pe x ; q v ( x, S ) = ε& x differentiate qv with respect to x: ε&= ∂q v ∂q v ∂S + ∂x ∂S ∂x multiply both sides by either qw or Pex: ε& Pe x = q w ∂q v ∂q ∂S + Pe x v = Pe qv ∂x ∂S ∂x q w ∂qv ∂q ∂S =x v Pe ∂x ∂S ∂x LHS x > 0, ∂q v ∂S > 0 , thus ∂S ∂x < 0 (Æ concave-up profile) if and only if RHS is < 0: qv − q w ∂q v Pe ∂x True for any function qv(qw,S). m' For specific case of qv = kq w S n = k ′x m S n this condition is satisfied by the following conditions: qv < 1< m < n+1 Æ concave all m,n > 1 Æ concave n>1; 0> soil grain size). [Sketches] Examples: Kirkby (1969) Creep Rates from Be10 Budgets Be10 rapidly adsorbed onto soil particles (soil traps all of atmospheric flux delivered by rainfall, except overland flow loses). If no Be10 in bedrock (meteoric Be10 ~ 5 orders magnitude > in situ), simple Be10 mass budget can be used to estimate creep rates, and thus test slope dependence, depth dependence, soil moisture, etc. McKean et al (1993, Geology) describe simple balance assuming plug flow (constant velocity with depth -- flux should be depth dependent! -- but only 20-25% difference to Be10 budget). Mass balance at position x from the divide for steady-state soil and Be10 budget can be written: x ∫P Be h (x )dx = ∫ ρ(x, z)Vs (x, z )ε Be (x , z)dz 0 0 where Vs is the velocity of soil movement, ρ is soil bulk density, and εBe is the concentration of Be10. Note that mass flux of soil is qs = ρVsh, and mass flux of Be10 is ρVsh CBe, where CBe is the vertically averaged concentration. Assumptions, integrating. qs = PBe x ρC Be (x ) G. Weathering and Soil Production Transport-limited conditions require non-zero steady-state soil thickness. Soil Production is often modeled as decaying exponentially with soil thickness: thicker soils insulate the soil-rock interface and inhibit weathering and mechanical soil production (including biogenic mixing): W (h ) = Ws e −Wa h Wa is the rate of soil production decline with depth. At steady state, soil thickness is unchanging (dh/dt = 0), therefore the rate of soil production (rate of lowering of the soil-rock interface) must equal the rate of soil surface lowering: W (hss ) = ∂z = −ε& ∂t Thus faster erosion rates lead to thinner soils. Since at steady state the surface lowering rate is everywhere the same, this implies that steady state soil thickness is constant along the hillslope! Where Ws < ε& , the soil is stripped and bare bedrock weathering-limited slopes result. If erosion rate is high, bedrock-involved landsliding is the dominant transport process. New data from Arjun Heimsath indicates maximum values of Ws are on the order of 0.5 mm/yr. Thus the rate of soil production is a fundamental control on landscape form, erosion process and hillslope hydrology. Soil mantled Æ SSF; Bare bedrock Æ HOF. Keep in mind: hillslope erosion rate is set by the incision rate of bordering channels. In tectonically active landscapes, channel incision tends to adjust to match the rock uplift rate and many such landscapes are landslide-dominated (either Ws < ε& or S > tanφ [the angle of repose]).