School of GeographyUniversity of Leeds

http://www.geog.leeds.ac.uk/people/m.kirkby/

2004: Fellow American Geophysical Union 2002-: Emeritus Professor, University of Leeds1999: Royal Geographical Society / Institute of British Geographers: Founder's Medal1989: Leverhulme Research Fellowship1989: British Geomorphological Research Group: David Linton Award.1976: Royal Geographical Society: Gill Memorial Award.1973-: Professor of Physical Geography, University of LeedsHead of Department 1978-81, 1984-87, 1992-95.1967-73: Lecturer in Geography, University of Bristol1965-7: NERC Research Fellow, University of Cambridge (Department of Geography)1965: Research Collaborator, The Smithsonian Institution, Washington, DC1964-5: Research Associate, The Isaiah Bowman Dept of Geography, The Johns Hopkins University1963-4: Research Associate, US Geological Survey, Washington DC (with Dr. L.B. Leopold)1963: PhD (University of Cambridge): Geomorphology (supervised by Prof R.J. Chorley)1960: BA (University of Cambridge): Mathematics (Part II) and Geography (Part II) (Trinity College): Philip Lake Prize

Mike Kirkby

M.J. Kirkby, 1969

Hillslope Process-Response Models Based on the Continuity Equation

Objective: examine a series of process-response models of slope developmentbased on field measurement (empirical) rather than theory

“…attempt to formalize process-response models of hillslopes into a single theory”

Approach: - defines a general equation (continuity equation) for soil and sediment flux- process based models are developed from continuity equation*what major assumptions are inherent in these models? base level conditions…?

Section 1 (A – G): Continuity Equation and Transport LawsEquations 1-13 are setting up the methodology for the describingcharacteristic forms

Section 2: Characteristic FormsEquations 14-27 use the continuity equation to derive equations for characteristic forms

p. 15. Why is he considering a system in ‘cycle time’;

Kirkby wanted to develop models "based on field measurement rather than theory" (p. 15). Why?

(A) Continuity equation

(1)

M = rate of mechanical loweringD = rate of chemical loweringy = elevationt = time elapsed

What IS a continuity equation?

(2)

mechanical lowering & mechanical transport

M = rate of mechanical lowering = vector divergenceS = actual sediment transport rate

Relationship (B):

(3)

(4)

rate of lowering & soil thickness

z = soil deptht = time elapsedy = elevationW = rate of lowering of the soil-bedrock interface

Relationship (C):

(5)

rate of lowering & soil thickness

Relationship (C):

*degree of soil development is related to the relative magnitudes of mechanical and chemical removal

- soil thickness is considered constant- land surface and soil-bedrock interface are lowered at same rate

M = rate of mechanical loweringD = rate of chemical loweringW = rate of lowering of the soil-bedrock interfaceμ = extent of weathering at the surface

Does Kirkby address the issue of appropriate spatial scales for these process response models?

Does grid scale matter?

What about the appropriate timescales?

Transport Limited Weathering (supply) Limited

Removal condition: C=S

potential rate of weathering > rate of transport

- soil accumulates to supply full transport capacity

Removal condition: C>>S

potential rate of weathering > rate of transport

- sparse soil; inhibits transport from reaching full capacity

S = actual sediment transport rateC = transporting capacity of the process

(6) (7)

Erosion LimitedIntermediate condition:- unconsolidated material- where the transporting process is operating at variable depths (river bed-load)- erosion rate (-∂y/∂t) is proportional to ‘surplus’ (surplus = C > volume available for removal - k = erosion constant

t

y

kSC

1

SCk

SCk

0(8)

Relationship (D): actual transport and transporting capacity

(12)

(G) Transport (process) Law:

Simpler-slow mass movements, surface wash, stream transport

a = area drained per unit contour lengthf(a) = + function of an = constant (influence of ↑ gradient) 0→+α = constant > (0>α>90°)

More Complex- landslides, talus movement (stable slope angle)- rate of transport ↑ w/gradient above critical angle α

(13)

QUESTION

What “special case” of equation 13 does eq. 12 represent?

Examples showing relevance of the Transport (process) Laws:

soil creep:= (eq12) where f(a) = constant and n=1

rivers:q = dischargeC = sediment loadper unit width of flow*always at full capacity

Scree & rock slopes:slope of gradient > αf(a) = constantn=1

(13)

sta

ble

slop

e an

gle

α(1

2) >

sta

ble

slo

pe α

appropriate estimate of α?

All of the above equations can be described by the form:

C=K*(a)m(slope)n

(14)

(15)

(16)

Characteristic Forms:

solution to the continuity equationwhat are the assumptions?

(17)

(18)

(19)

(20)

Kirkby is attempting to fit these process-response models into a unified theory.

What are potential and real benefits and drawbacks of this approach?

"As many factors as possible have been left in the equations at each stage, to retain maximum flexibility in the solutions...... . At many points, however, it has been convenient to make simplifying assumptions..." (p. 27). What are examples of these simplifying assumptions?

-links between form and process

-conservation of mass + empirical process laws to calculate approx. slope forms towards which hillslopes will develop (obliterating initial form)

-From a characteristic form (plus assumptions) one can deduce the information about the processes that formed it

- how do we identify a characteristic form in a landscape?

Conclusions: