SpatioTemporal Scaling of Channels in Braided Streams

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Spatio-Temporal Scaling of Channels in Braided
Streams

• A.G. Hunt
• Dept. of Physics and
• Department of Geology
• Wright State University

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What is a braided stream?
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What is a braided stream?

• There are many channels
• The flow pattern shifts
• Channels are born and die
• Usually considered examples of complex dynamics
  in sediment choked streams with high gradients
  (steep slopes)
• I dont argue with picture I try to explain it
  with reductionist technique

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Fig. 1 Braided lahar runout channel,
Pasig-Potrero River, Philippines. Note standing
wave patterns in each of the braids, indicating
near-critical flow conditions (Tinkler, 1997ab).
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What is critical flow?
Froude number is a dimensionless quantity based
on the ratio of kinetic to gravitational energy.
Froude number 1 is critical flow.
(1)
For Frlt1, flow tends to be deeper and slower,
while for Frgt1, flow is shallow and fast. Change
from one to the other along the bed of a stream
is called hydraulic jump (mountain
streams). Fr1 is a typical condition for
streams with high gradients and large sediment
capacity.
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What are the spatio-temporal scaling properties?
(2)
? channel length
? channel lifetime
Not diffusion-related!
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Check units of two equalities

• Length divided by time squared is acceleration
• Implies importance of forces

• Constancy of ratio of flow conditions to g
  constrains flow
• In context of previous example relates forces to
  flow conditions

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Can one generate a link between microscopic
transport theory and macroscopic observation?

• I believe yes, but only with new stochastic
  theory of sediment transport

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Replace traditional Newtonian mechanics with
probabilistic formulation
We still have logarithmic vertical velocity
profile
(3)
u velocity at height y, y0 is a reference height
known as surface roughness, u is quantity that
appeared in Fr.
But reinterpret quadratic bed stress ? as energy
density (near bed)
(4)
?w density of water
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Traditional Interpretations of sediment
entrainment (picking up sediment)
U
Fupd2G
Note that these particles are all of identical
size
dparticle diameter
Fdown(m-mw)g(?-?w)gd3
Combination of gravity and buoyancy
Entrainment occurs when upward force exceeds
downward force.
Set two forces equal for largest particle
entrained and result is
(5)
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What about energy interpretation?
Journal of Geophysical Research, 1998
Entrainment occurs most of the time when,
just
Kinetic gt Potential
Analogy to Boltzmann factor.
Probability P goes rapidly to zero when the
numerator exceeds the denominator
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Ad hoc use of a Boltzmann factor has
uncertainties

• 1 What is the theoretical basis for the
  assumption? (analogy to statistical mechanics and
  large number of available states in the turbulent
  stream)
• 2 Practical
• Is the argument of the function verified? (Yes,
  entrainment condition above)
• Is the functional form justified? (Perhaps
  surface roughness calculations total flow)

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This ansatz implies physics

• That there is a localized source of random
  kinetic energy
• That this energy can be transferred to the
  particle in a short time

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Effectively a single-collision (one- eddy)
scattering ansatz

• Eddy the size of particle is absorbed completely
  inelastically (like a phonon).
• Kinetic energy fluctuations available are
  proportional to mean kinetic energy.
• Any excess kinetic energy absorbed goes to
  raising particle conservation of energy gives
  maximum height of entrainment (raise the particle
  higher off the bed).

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Take several threads from above and develop
individually, Then weave them back together
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Fundamental Rate Equation Set settling and
entrainment rates equal for steady-state
Settling rate
Entrainment rate
(6)
(7)
Inverse of a bursting period attempt frequency
(8)
For particle entrained to maximum height ym
Find vy from force balance
We will use eqn(6) later, but eqn(7) and eqn(8)
yield an effective clast velocity
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Eqns(6-8) have a number of applications

• Derive implied surface roughness and compare with
  phenomenology.
• Consider total flow and similarity to result from
  minimization of specific energy.
• Compare implied downstream effective velocity of
  bedload with experiment and field observations.
• Analyze potential implications for
  spatio-temporal scaling

The fourth involves the first and the third as
well, but 2, which derives from 1, can be
ignored here
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Surface Roughness I Large Fr and Large
Entrainment Rates
In this case AltltB can be used to approximate
solution to eqn(6)
(9)
and
gives an estimate for surface roughness
Note that this expression yields a very similar
expression for total flow as does minimizing the
specific energy
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Surface Roughness II Small Fr and Small
Entrainment Rates
(10)
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Large u
Empirical
Small u
y0
Fr
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Effective Downstream Velocities
tA-1
(11)
equals no. times per second that particle is
entrained times distance traveled per
entrainment. Multiply by a time (duration of
flood) to find a distance per flood, L.
NOTE these calculations apply to bedload, not
suspended load
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(12)
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Range of particle sizes carried as part of bedload
This gives maximum particle size since ln(1) 0
Minimum particle size is obtained from eqn(12)
for when Lut if result implies that the
particle moves downstream as fast as the water
near the bed, then the particle is part of the
suspended load. Both dmax and dmin turn out to
be proportional to h, the stream depth.
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Sedimentation catastrophe for Frlt1 Property of
interaction between bedload and surface roughness
(13)
(14)
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Fr near 1 implies that
with x and t the distance and time of fluid
motion near the bed.
Depth h is now known to be proportional to
transport distance L (for all particle
sizes), If L ?? (channel length), then minimum
transport time t?? (channel lifetime) If the
maximum value of L is ? then x?
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Alternate Interpretation Relates to the Influence
of Large Channels on Smaller Channels. Is There a
Connection?

• Likely yes Small flow fluctuations in larger
  channels will always produce (in a relative
  sense) larger flow fluctuations in smaller
  channels (mass conservation).

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Conclusions

• A stochastic formulation of sediment dynamics is
  clearly required.
• The present formulation may or may not turn out
  to be rigorously correct, but is certainly a good
  start because it can
• Predict the maximum particle entrained on a bed
  of uniform particle sizes,
• Predict the relevance of Froude number 1 to
  braided streams,
• Predict the distance of downstream transport as a
  function of particle size,
• Explain an origin of spatio-temporal scaling of
  braided streams.