Modelling agents of change in the fluvial system

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Modelling agents of change in the fluvial system

• Rob Thomas

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Model approaches

• Model approach governed by variety of factors
• what we model (e.g. behaviour of todays
  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
• 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?

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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,
  or define modified transport laws that hold over
  larger time and space scales
• find emergent behaviour and define new relations
  consistent with scale of larger features and
  behaviour of emergent landforms
• define empirical relations at necessary scale

local physics represents details of landscape,
but immense information requirements limit scale
of prediction
process-based relations on a coarse scale predict
landscapes at very large time and space scales,
making interpretation of causal mechanisms
difficult
general rules focus on summarising landscape
properties with goal of exploring common elements
rather than suite of mechanisms that produce any
particular landscape
erosion and transport laws capable of independent
parameterisation and can be applied at landscape
scale, such that cause and effect can be
determined
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Model Approaches

• Range from
• Reductionist- reveal emergent behaviour by
  scaling up small-scale dynamics
• through
• Synthesist- ignore collective dynamics to
  characterise emergent behaviour of complex
  systems
• to
• Zero process dynamics- static model, study
  long-term evolution to equilibrium conditions,
  but ignore path to equilibrium (e.g. Optimal
  Channel Networks (OCN) Rodriguez- Iturbe et al.,
  1992)
• Common principle mass or energy conservation
• Major differences equations for erosion and
  sediment transport

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Issues of scale- physical models

• We generally model in the range of scales dealt
  with by Newtonian physics and continuum
  mechanics- both have bounds of application
• Small-scale phenomena can be modelled from
  physical theory. However as scale increases
  physics becomes increasingly difficult to apply
  because
• 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

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Simulating river hydraulics flow structures

• Flow in rivers is turbulent, with strongly
  three-dimensional flow structures that drive
  deposition and erosion
• 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
• Examples from 3D simulations of flow over gravels
  (Lane et al., in review)
• downstream cross-stream vectors
• velocity

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Governing equations

• 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
• And the Continuity equation expressing
• the sum of change in mass (or volume) in the
  downstream, cross stream and vertical directions
  0
• However, these equations are too complex to solve
  analytically (non-linear, four independent
  variables)
• To obtain approximate numerical solutions we need
  simplifications and approximations

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General issues..

• 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
  2-dimensional form of equations

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Potential flow representations
EULERIAN approach
Domain split into small cells fixed in space for
which we can obtain algebraic equations
Flow described by velocity, acceleration and
density at points

• 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

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Potential flow representations
LAGRANGIAN approach
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

• Numerical approximation of equations requires
  reference to points that move continuously with
  respect to each other
• Difficulty circumvented by utilising many fluid
  elements and recalculating positions and
  velocities at each timestep
• Problems arise when
• Fluid becomes strongly distorted or large
  slippages occur
• Cavitation occurs or when material interfaces
  collide with one another

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Potential flow representations
MARKER-IN-CELL approach
Can be considered to combine some of the best
features of both Eulerian and Lagrangian
approaches

• 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)

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Why hasnt Marker-in-cell been used more
extensively?
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Potential flow rules

• Flow Rules
• Based on 2-D momentum equation-
• Velocity f (depth, energy slope, g, roughness)
• See later for sediment transport
• Murray-Paola (1994, 1997)-
• Velocity f (bed slope) to some power
• Sediment transport f (bedslope discharge)
• Thomas and Nicholas (2002)-
• Velocity f (bed slope, depth, g, roughness)
  all raised to some power
• No sediment transport

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Sediment transport

• Commonly, we distinguish three main transport
  modes
• dissolved load (wash load)
• layer spanning most of water column where
  particles are in suspension
• particles that roll, slide, or saltate, and are
  transported as bed load

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Sediment transport modelling

• Model approaches range from simulating
• 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
• suspended load
• bed load
• total load

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Governing Equations

• Sediment transport (in 2D) is described by the
  following mathematical relationship
• Mass conservation of sediment by size fraction
• 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

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Sediment transport modelling

• 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
• 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

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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

• 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

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SEDTRA- sediment transport capacity predictor
(Garbrecht et al. 1996)

• Total sediment transport by size fraction for
  fourteen predefined size classes with suitable
  transport equation for each size fraction
• 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).

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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

• Particles exchange between the bed- and
  suspended-load layers and the bed
• Two options
• compute sediment transport rates in each layer
• combine suspended and bed load layers into single
  total load layer

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Summary

• Range of model approaches, must be governed by
  application
• Agents in fluvial systems- I suspect were
  already doing it, just independently
• Language issues- fortran vs java
• Others??

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Outline

• Model approaches
• Scale issues
• Fluvial hydraulics
• Governing equations
• Difficulties
• Flow representations
• Sediment transport
• Governing equations
• Simulation options

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Issues of scale- physical models

• Continuum mechanics
• only works well at certain scales
• breaks down when nonlinearity promotes
  localization and shocks, as in breaking waves,
  hydraulic jumps, river channels, caves, etc.
• Newtonian physics
• breaks down at elementary particle scale
• Quantum physics- probability functions
  describing particle behaviour
• breaks down at even smaller scales
• String theory or some yet to be invented theory

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Models- What? Why?

• In essence, a model is
• 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
• 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)