Modelling agents of change in the fluvial system Rob Thomas Model approaches Model approach governed by variety of factors: 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: 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 Model Approaches 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 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 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 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 velocity cross-stream vectors 120 110 100 90 80 70 60 50 40 30 20 10 10 20 30 40 50 60 70 80 90 100 110 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 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 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 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 Numerical approximation of equations requires reference to points that move continuously with respect to each other 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: 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 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 Flow Rules: Based on 2-D momentum equationVelocity = 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 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 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 Governing Equations Sediment transport (in 2D) is described by the following mathematical relationship: 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 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 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 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). 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 Summary 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 Model approaches Scale issues Fluvial hydraulics Governing equations Difficulties Flow representations Sediment transport Governing equations Simulation options Issues of scale- physical models Continuum mechanics Newtonian physics breaks down at elementary particle scale Quantum physics- “probability functions” describing particle behaviour 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? 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)