School of Geography University of Leeds http://www.geog.leeds.ac.uk/people/m.kirkby/

1. School of Geography University of Leeds http://www.geog.leeds.ac.uk/people/m.kirkby/ Mike Kirkby 2004: Fellow American Geophysical Union 2002-: Emeritus Professor, University of Leeds 1999: Royal Geographical Society / Institute of British Geographers: Founder's Medal 1989: Leverhulme Research Fellowship 1989: 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 Bristol 1965-7: NERC Research Fellow, University of Cambridge (Department of Geography) 1965: Research Collaborator, The Smithsonian Institution, Washington, DC 1964-5: Research Associate, The Isaiah Bowman Dept of Geography, The Johns Hopkins University 1963-4: 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

2. Hillslope Process-Response Models Based on the Continuity Equation M.J. Kirkby, 1969

3. Objective: examine a series of process-response models of slope development based 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 Laws Equations 1-13 are setting up the methodology for the describing characteristic forms Section 2: Characteristic Forms Equations 14-27 use the continuity equation to derive equations for characteristic forms Kirkby wanted to develop models "based on field measurement rather than theory" (p. 15). Why? p. 15. Why is he considering a system in ‘cycle time’;

4. (A) Continuity equation (1) M= rate of mechanical lowering D= rate of chemical lowering y= elevation t= time elapsed What IS a continuity equation?

5. Relationship (B): M= rate of mechanical lowering = vector divergence S= actual sediment transport rate mechanical lowering & mechanical transport (2) (3)

6. Relationship (C): rate of lowering & soil thickness (4) z = soil depth t= time elapsed y= elevation W= rate of lowering of the soil-bedrock interface

7. Relationship (C): rate of lowering & soil thickness (5) M= rate of mechanical lowering D= rate of chemical lowering W= rate of lowering of the soil-bedrock interface μ= extent of weathering at the surface - soil thickness is considered constant - land surface and soil-bedrock interface are lowered at same rate *degree of soil development is related to the relative magnitudes of mechanical and chemical removal

8. Does Kirkby address the issue of appropriate spatial scales for these process response models? Does grid scale matter? What about the appropriate timescales?

9. Relationship (D): actual transport and transporting capacity Erosion Limited Intermediate 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 (8) Transport Limited Weathering (supply) Limited S= actual sediment transport rate C= transporting capacity of the process (7) Removal condition: C>>S potential rate of weathering > rate of transport - sparse soil; inhibits transport from reaching full capacity Removal condition: C=S potential rate of weathering > rate of transport - soil accumulates to supply full transport capacity (6)

10. a = area drained per unit contour length f(a) = + function of a n = constant (influence of ↑ gradient) 0→+ α= constant > (0>α>90°) (G) Transport (process) Law: Simpler -slow mass movements, surface wash, stream transport QUESTION What “special case” of equation 13 does eq. 12 represent? (12) More Complex - landslides, talus movement (stable slope angle) - rate of transport ↑ w/gradient above critical angle α (13)

11. Examples showing relevance of the Transport (process) Laws: soil creep: = (eq12) where f(a) = constant and n=1 (12) > stable slope α rivers: q = discharge C = sediment load per unit width of flow *always at full capacity Scree & rock slopes: slope of gradient > α f(a) = constant n=1 appropriate estimate of α? (13) stable slope angle α All of the above equations can be described by the form: C=K*(a)m(slope)n

12. Characteristic Forms: solution to the continuity equation what are the assumptions? (14) (15) (16)

13. (17) (18) (19) (20)

15. Kirkby is attempting to fit these process-response models into a unified theory. What are potential and real benefits and drawbacks of this approach?

16. "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? Conclusions: • 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?