Another Chapter in

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Another Chapter in THE SEARCH FOR THE HOLY
GRAIL A MECHANISTIC BASIS FOR HYDRAULIC
RELATIONS FOR BANKFULL FLOW IN GRAVEL-BED RIVERS
Gary Parker With help from François Metivier
and John Pitlick
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What is the physical basis relations for bankfull
geometry of gravel-bed streams?
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Where do the following relations come from?

• Bankfull Depth Hbf (Qbf)0.4
• Bankfull Width Bbf (Qbf)0.5
• Bed Slope S (Qbf)-0.3
• where Qbf bankfull discharge

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THE GOAL A Mechanistic Description of the Rules
Governing Hydraulic Relations at Bankfull Flow in
Alluvial Gravel-bed Rivers
The Parameters Qbf bankfull discharge
(m3/s) QbT,bf volume bedload transport rate at
bankfull discharge (m3/s) Bbf bankfull
width (m) Hbf bankfull depth (m) S bed slope
(1) D surface geometric mean or median grain
size (m) g gravitational acceleration
(m/s2) R submerged specific gravity of sediment
1.65 (1) The Forms Sought
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DATA SETS

• Alberta streams, Canada1
• Britain streams (mostly Wales)2
• Idaho streams, USA3
• Colorado River, USA (reach averages)
• 1 Kellerhals, R., Neill, C. R. and Bray, D. I.,
  1972, Hydraulic and
• geomorphic characteristics of rivers in Alberta,
  River Engineering
• and Surface Hydrology Report, Research Council of
  Alberta, Canada,
• No. 72-1.
• 2 Charlton, F. G., Brown, P. M. and Benson, R.
  W., 1978, The
• hydraulic geometry of some gravel rivers in
  Britain, Report INT 180,
• Hydraulics Research Station, Wallingford,
  England, 48 p.
• 3 Parker, G., Toro-Escobar, C. M., Ramey, M. and
  Beck S., 2003,
• The effect of floodwater extraction on the
  morphology
• of mountain streams, Journal of Hydraulic
  Engineering, 129(11),
• 2003.
• 4 Pitlick, J. and Cress, R., 2002, Downstream
  changes in the channel of a
• large gravel bed river, Water Resources Research
  38(10), 1216,
• doi10.1029/2001WR000898, 2002.

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NON-DIMENSIONALIZATION
These forms supersede two previous forms,
namely which appear in reference 3 of the
previous slide. Note
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WHAT THE DATA SAY
The four independent sets of data form a coherent
set!
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REGRESSION RELATIONS BASED ON THE DATA
To a high degree of approximation,
Remarkable, no?
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WHAT DOES THIS MEAN?
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THE PHYSICAL RELATIONS NECESSARY TO CHARACTERIZE
THE PROBLEM

• Required four relations in the four unknowns
• Hbf, Bbf, S, QbT,bf.
• Resistance relation (Manning-Strickler)
• Gravel bedload transport relation (Parker 1979
  approximation of Einstein 1950)
• Relation for channel-forming Shields number ?bf
  (Parker 1978) and
• Relation for gravel yield from basin (not
  determined solely by channel mechanics).

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RESISTANCE RELATION
Manning-Strickler form where Ubf Qbf/(Bbf Hbf)
denotes bankfull flow velocity,
Here we leave ?r and nr as parameters to be
evaluated.
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BEDLOAD TRANSPORT RELATION
Use Parker (1979) approximation of Einstein
(1950) relation applied to bankfull flow
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RELATION FOR CHANNEL-FORMING SHIELDS NUMBER
Base the form of the relation on Parker (1978)
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RELATION FOR GRAVEL YIELD FROM BASIN AT BANKFULL
FLOW
This relations is external to the channel itself,
and instead characterizes how the channels in a
watershed interact with the unchannelized
hillslopes. The necessary relation should be a
dimensionless version of the form where nbT
must be evaluated.
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WORKING BACKWARD
Rather than working forward from the basic
physical relations to the hydraulic relations,
lets work backward and find out what the form
the physical relations must be to get the
observed hydraulic relations.
Recall that
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RESISTANCE RELATION
The desired form is
Now using the definition of Cz, the
non-dimensionalizations and the relations it is
found that But
so that
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RELATION FOR BANKFULL SHIELDS NUMBER
By definition Using the relations it is
found that This can be rewritten as
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RELATION FOR GRAVEL TRANSPORT AT BANKFULL FLOW
Recall that Now from the last relation of the
previous slide, Using the previously-introduce
d non-dimensionalizations, Thus
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EVALUATION OF THE CONSTANTS
From the regression relations, In addition, for
natural sediment it is reasonable to assume In
the Parker approximation of the Einstein
relation, The data of the four sets indicate
an average value of ?bf of 0.04870, or thus
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THE RESULTING RELATIONS
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TEST OF RELATION FOR Cz using all four data sets
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TEST OF RELATION FOR ?bf using all four data seta
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FINAL RESULTS
If we assume mechanistic relations of the
following form
resistance
bedload transport
channel-forming Shields number
sediment yield relation
then we obtain the results
The first three of these correspond precisely to
the data!
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Test against the original data set
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Test against the original data set
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Test against the original data set
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Test against the original data set
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Test against four new data sets
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Test against four new data sets
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Test against four new data sets
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Test against three new data sets
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BRITAIN II STREAMS ROLE OF BANK STRENGTH Class 1
has least vegetation, Class 4 has most vegetation
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RELATION BETWEEN VEGETATION DENSITY AND BANK
STRENGTH, BRITAIN II STREAMS
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HOW WOULD VARIED BANK STRENGTH (r), SEDIMENT
SUPPLY (?Y) AND RESISTANCE (?r) AFFECT HYDRAULIC
GEOMETRY?
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VARIATION IN r (BANK STRENGTH)
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VARIATION IN ?y (GRAVEL SUPPLY)
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VARIATION IN ?r (CHANNEL RESISTANCE)
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QUESTIONS?