Modelling agents of change in
the fluvial system
Rob Thomas
Model approaches
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Model approach governed by variety of factors:
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what we model (e.g. behaviour of today’s landscape vs.
landscape change)
scale of prediction (single event vs. evolution of a drainage
network or a mountain range)
why we model (exploratory or explanatory modelling
versus specific prediction), and
to an unknown extent, backgrounds and tastes of
individual modellers
In a river:
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Spatially and/or temporally averaged properties? e.g.
Reynolds-averaging, 2-D or even 1-D flow modelling, bulk
sediment transport
Or movement of individual particles or fluid units
Somewhere in the middle?
Model approaches

Four options for large scale modelling (Haff 1996):
integrate observable, verifiable formulas for erosion and
sediment transport over large times and distances
 define
mid-scale
spatially
integrated
relations,
local
physics
represents
details
of landscape,
butor define
modifiedinformation
transport laws
that hold over
largeroftime
and
immense
requirements
limit scale
prediction
space scales
 find emergentrelations
behaviour
definescale
new predict
relations
process-based
onand
a coarse
consistentatwith
scale
larger
features
and behaviour
landscapes
very
largeoftime
and
space scales,
making of
emergent landforms
interpretation
of causal mechanisms difficult
 definerules
empirical
at necessary
scale properties
general
focusrelations
on summarising
landscape
erosion
transport common
laws capable
of independent
with goaland
of exploring
elements
rather than suite
parameterisation
andproduce
can be any
applied
at landscape
scale,
of mechanisms that
particular
landscape
such that cause and effect can be determined
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Model Approaches
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Range from:
‘Reductionist’- reveal emergent behaviour by ‘scaling up’ smallscale dynamics
through:
‘Synthesist’- ignore collective dynamics to characterise emergent
behaviour of complex systems
Emergent behaviour sometimes cannot be predicted
using smaller-scale physics, but possible using relations
defined at the scale of the emergent behaviour
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Value rests on ability to represent complex interactions
in very simple ways, allowing quick exploration of
to:
possible
solution
spaces
Zero process dynamicsstatic
model,
study long-term evolution to
equilibrium conditions,
but ignore
to equilibrium
(e.g. Optimal
Difficulty
lies in path
choosing
rules to accommodate
Channel Networks (OCN)
[Rodriguez- Iturbe et al., 1992])
situations for which there are no observations or
Common principle =experience
mass or energy conservation
Major differences = equations for erosion and sediment transport
Issues of scale- physical models
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We generally model in the range of scales dealt with by
Newtonian physics and continuum mechanics- both
have bounds of application
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Small-scale phenomena can be modelled from physical
theory. However as scale increases physics becomes
increasingly difficult to apply because:
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both slow- and fast-acting geomorphological processes
different mechanisms dominate change at different
scales
unanticipated processes
unknown initial and forcing conditions
governing relations generally non-linear
Simulating river hydraulics: flow
structures
Flow in rivers is turbulent, with strongly three-dimensional flow
structures that drive deposition and erosion
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eddy sizes range from extremely small up to the order of the channel
dimensions
bedforms also occur at variety of scales
Flow structures in rivers are extremely complex
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Examples from 3D simulations of flow over gravels (Lane et al., in review)
downstream
velocity
cross-stream vectors
120
110
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90
80
70
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40
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Governing equations
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Any fluid flow is described by the Navier-Stokes
equations expressing
the temporal change in momentum
+ the spatial change in momentum
= pressure gradient force
+ change in momentum due to friction
in the downstream, cross stream and vertical directions
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And the Continuity equation expressing
the sum of change in mass (or volume) in the downstream, cross
stream and vertical directions = 0
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However, these equations are too complex to solve
analytically (non-linear, four independent variables)
To obtain approximate numerical solutions we need
simplifications and approximations
General issues..
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Presently unfeasible to solve equations over
spatial or temporal scale large enough to be useful
for modelling rivers
Reducing dimensionality promotes reduction in
effects (e.g. turbulence and secondary circulation)
explicitly dealt with, and reduction in simulation
time.
Additional terms, representing turbulence and
secondary circulation, introduced into 2dimensional form of equations
Potential flow representations
EULERIAN approach
Domain split into small cells fixed in space
Flow described by velocity,
for which we can obtain algebraic equations acceleration and density at points
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Convenient when velocity is constant at the fixed points
Handles flow-field distortion effectively
Material interfaces not easily handled
Difficult to resolve sub-grid scale features
Not suited to unsteady flow
‘Artificial diffusion’ as contaminants apportioned to adjacent
cells
Potential flow representations
LAGRANGIAN approach
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Numerical approximation of equations requires reference
to points that move continuously with respect to each other
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Fluid again separated into finite
zones, characterising individual fluid
elements.
Flow parameters represented with
respect to the fluid itself, providing a
moving frame of reference
Difficulty circumvented by utilising many fluid elements and
recalculating positions and velocities at each timestep
Problems arise when:
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Fluid becomes strongly distorted or large slippages occur
Cavitation occurs or when material interfaces collide with one
another
Potential flow representations
MARKER-IN-CELL approach
Can be considered to combine some
of the best features of both Eulerian
and Lagrangian approaches
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Developed as “Particle-in-Cell” at Los Alamos in the late
1950s for use in particle physics
Adapted to fluvial hydraulics during 1980s by John
Harbaugh and graduate students at Stanford
Eulerian mesh used for characterising the field variables
(e.g. depth and bed elevation)
Lagrangian elements used to characterise the fluid itself
(e.g. velocity and sediment load)
Why hasn’t Marker-in-cell been used
more extensively?
Advantages
Disadvantages
Handles flow-field distortion
Computationally demanding
Handles material interfaces
Flows in which stagnations occur
w.r.t Eulerian mesh are difficult to
resolve
Fluid elements help resolve sub-grid
scale features
Within a large region of flow there
must be no small region for which
detailed resolution is required
Relatively simple to deal with 2+
dimensions (fluid flow in 2-D, plus
depth, logarithmic velocity profile
and deposits in vertical)
It must be not be necessary to know
in detail the fluid variables at
Eulerian grid boundaries at any
given instant of time
Potential flow rules
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Flow Rules:
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Based on 2-D momentum equationVelocity = f (depth, energy slope, g, roughness)
See later for sediment transport
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Murray-Paola (1994, 1997)Velocity = f (bed slope) [to some power]
Sediment transport = f (bedslope · discharge)
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Thomas and Nicholas (2002)Velocity = f (bed slope, depth, g, roughness) [all raised to some
power]
No sediment transport
Sediment transport
Commonly, we
distinguish three
main transport
modes:
1. dissolved load
(wash load)
2. layer spanning
most of water
column where
particles are in
suspension
3. particles that
roll, slide, or
saltate, and are
transported as
bed load
Sediment transport modelling
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Model approaches range from simulating:
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individual particles
multiple size classes (split mixture into a number of
classes, each with different behaviour)
single size class
to ignoring it! (Sediment transport is very difficult- ask
Einstein!)
Transport formulation:
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suspended load
bed load
total load
Governing Equations
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Sediment transport (in 2D) is described by the
following mathematical relationship:
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Mass conservation of sediment by size fraction
 hCk  UhCk  VhCk   
C k   
Ck  

  Ek  Dk  q sk


 e s h
  e s h
t
x
y
x 
x  y 
y 
C is depth-averaged concentration, D is particle deposition rate,
E is particle entrainment rate, h is depth, t is time, U is
downstream velocity, V is cross stream velocity, x is
downstream distance, y is cross stream distance, es is sediment
eddy diffusivity, qs is sediment inflow, and the subscript k
denotes the kth size class
Sediment transport modelling
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Modelling of suspended load via agent-based methods
largely unexplored (but wait..)
Modelling bed load more reasonable. In this case,
agents would be individual particles, rolling, sliding or
saltating
Rules based on:
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empirical functions (e.g. Laursen, Yang, Meyer-Peter and
Mueller)- some excess quantity
stochastic formulations (e.g. Einstein, Shen)
physical properties of particles (e.g. coefficient of restitution,
conservation of mass, momentum and energy)
For example, Schmeeckle and Nelson 2003
Example- bed load transport
Visualization of direct numerical simulation of mixed grain size bedload
transport in response to turbulent sweep event that occurs near the middle
of the animation
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Equations of motion of all particles integrated simultaneously
Distance from left to right is 20 cm, width is 5 cm. Median grain size is
5 mm and s is 2.5 mm
SEDTRA- sediment transport capacity
predictor (Garbrecht et al. 1996)
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Total sediment transport by
size fraction for fourteen
predefined size classes with
suitable transport equation
for each size fraction:
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Wash load (particles less
than 8 mm): after
entrainment, not deposited
Silts and fine sands from 8
mm to 0.25 mm: Laursen
(1958)
Sands from 0.25 mm to 2.0
mm: Yang (1973), and
Gravels from 2.0 mm to 64.0
mm: Meyer-Peter and
Mueller (1948).
Sediment transport modelling
To model sediment
transport processes,
three or more layers can
be distinguished:
•two layers through the
water column, and
•one or more layers
covering the streambed
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Particles exchange between the bed- and
suspended-load layers and the bed
Two options:
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compute sediment transport rates in each layer
combine suspended and bed load layers into single
total load layer
Summary
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Range of model approaches, must be
governed by application
Agents in fluvial systems- I suspect we’re
already doing it, just independently
Language issues- fortran vs java
Others??
Outline
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Model approaches
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Scale issues
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Fluvial hydraulics
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Governing equations
Difficulties
Flow representations
Sediment transport
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Governing equations
Simulation options
Issues of scale- physical models
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Continuum mechanics
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Newtonian physics
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breaks down at elementary particle scale
Quantum physics- “probability functions”
describing particle behaviour
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only works well at certain scales
breaks down when nonlinearity promotes localization
and “shocks,” as in breaking waves, hydraulic jumps,
river channels, caves, etc.
breaks down at even smaller scales
String theory or some yet to be invented theory
Models- What? Why?
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In essence, a model is:
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an idealised representation of reality
a description or analogy used to aid visualisation and
understanding
a system of postulates, data and inferences presented as
a mathematical description of an entity or state
Purposes:
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organise scientific thought
explore controls on landscape form and dynamics
perform experiments beyond the spatial and temporal
range of observations
develop understanding- “thought experiments” (cf. Kirkby)