School of Geography

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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
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Hillslope Process-Response Models Based on the
Continuity Equation
M.J. Kirkby, 1969
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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
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(A) Continuity equation
(1)
M rate of mechanical lowering D rate of
chemical lowering y elevation t time elapsed
What IS a continuity equation?
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Relationship (B)
mechanical lowering mechanical transport
(2)
(3)
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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
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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
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Does Kirkby address the issue of appropriate
spatial scales for these process response models?
Does grid scale matter?
What about the appropriate timescales?
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Relationship (D) actual transport and
transporting capacity
Transport Limited
Weathering (supply) Limited
S actual sediment transport rate C
transporting capacity of the process
(7)
Removal condition CgtgtS potential rate of
weathering gt rate of transport - sparse soil
inhibits transport from reaching full capacity
Removal condition CS potential rate of
weathering gt rate of transport - soil
accumulates to supply full transport capacity
(6)
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a area drained per unit contour length f(a)
function of a n constant (influence of ?
gradient) 0? a constant gt (0gtagt90)
(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 a
(13)
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Examples showing relevance of the Transport
(process) Laws
soil creep (eq12) where f(a) constant and
n1
(12) gt stable slope a
rivers q discharge C sediment load per unit
width of flow always at full capacity
Scree rock slopes slope of gradient gt a f(a)
constant n1
appropriate estimate of a?
(13) stable slope angle a
All of the above equations can be described by
the form CK(a)m(slope)n
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Characteristic Forms
solution to the continuity equation what are the
assumptions?
(14)
(15)
(16)
13
(17)
(18)
(19)
(20)
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Kirkby is attempting to fit these
process-response models into a unified theory.
What are potential and real benefits and
drawbacks of this approach?
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"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?

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