Modeling Earth Surface Modeling Earth Surface Dynamics from Source to Dynamics from Source to Sink Sink Matthew Wolinsky NCED Videoconference April 25, 2006
Dec 16, 2015
Modeling Earth Surface Modeling Earth Surface Dynamics from Source to SinkDynamics from Source to Sink
Matthew Wolinsky
NCED VideoconferenceApril 25, 2006
IntroductionIntroduction
• NCED studies the diverse channel systems that serve as the arterial network of Earth's “Critical Zone”
– mountain streams, alluvial fans, river floodplains and deltas, submarine canyons and fans, …
• Over geologic time channel systems sculpt erosional landscapes and deposit sedimentary records of the past
• On continental scales linked channel systems transport sediment from high mountain source areas to deep marine sink areas
Processes, Environments, and BoundariesProcesses, Environments, and Boundaries
Processes, Environments, and BoundariesProcesses, Environments, and Boundaries
Bedrock-Alluvial Transition
Shoreline
Shelf Edge
Sedimentary Processes and Boundary CouplingSedimentary Processes and Boundary Coupling
Boundary coupling is as
important as sedimentary
processes in determining
surface dynamics and
stratigraphy!
Long Profile ModelLong Profile Model
A unified framework to explore …
• Large-scale evolution of source to sink system
– Coupled landscape, seascape, and stratigraphic evolution– signatures of paleo-environments and processes
• Medium-scale system evolution of sub-systems
– migration of boundaries between sedimentary environments– propagation of “signals” upstream/downstream within environments
• Large-scale consequences of alternative process models and hypotheses developed by NCED research
Outline of TalkOutline of Talk
1) Overview of modeling framework
– Conservation of mass and momentum between flow and sediment– Generalized morphodynamic evolution equation
2) Prototype model
– Simplified bedrock-alluvial-marine model– Response to sea level cycles
3) Gravel-sand transitions
– Sharp vs diffuse transitions– Physics of equilibrium transitions– Transient modeling– MATLAB Dynamic Stratigraphy Toolbox
Conservation of Sediment MassConservation of Sediment Mass
(Exner Equation) qxt
surface elevation , q sediment flux
subsidence
surface elevation =
sediment thickness + basement elevation
surface change =
deposition/erosion + uplift/subsidence
Sediment Flux LawsSediment Flux Laws
Non-Equilibrium Flux Laws
bedrock incision
passive settling
Sqx ~
qqx ~
Sqfqx ,,
“Fast” equilibration between sediment flux and bed/flow conditions
“Slow” equilibration between sediment flux and bed/flow conditions
Equilibrium Flux Laws
bedload diffusion Sq ~
Sfq ,
• Exner + Flux Law Advection-Diffusion-Reaction Equation
• Nonlinear coefficients (Velocity, Diffusivity, Source)
• Conservation of (grain-size specific) sediment flux
xxxt V
qSqSqVV ,,,,,,,
itixq
Morphodynamic Evolution EquationMorphodynamic Evolution Equation
Simplified Bedrock-Alluvial-Marine ModelSimplified Bedrock-Alluvial-Marine Model
• Single grain size• Lump marine processes into “diffusion”• Linear coefficients + moving boundaries Nonlinear system
( Humphrey and Heller,1995; Jordan and Flemings, 1991 )
Environment V Threshold
Bedrock V0 x h , q qa
Alluvial a >zSL
Marine m zSL
Simplified Bedrock-Alluvial-Marine ModelSimplified Bedrock-Alluvial-Marine Model
Simplified Bedrock-Alluvial-Marine ModelSimplified Bedrock-Alluvial-Marine Model
Simplified Bedrock-Alluvial-Marine ModelSimplified Bedrock-Alluvial-Marine Model
Simplified Bedrock-Alluvial-Marine ModelSimplified Bedrock-Alluvial-Marine Model
• Over long timescales source area corresponds to region of uplift, sink area corresponds to region of subsidence
• “Equilibrium” bedrock-alluvial transition = tectonic transition
• Sea level forcing causes shoreline migration, which forces migration of bedrock-alluvial transition
• Boundary migration triggers upstream waves of deposition and erosion, as seen by …
Depositional History and Moving BoundariesDepositional History and Moving Boundaries
Summary of Bedrock-Alluvial-Marine ModelSummary of Bedrock-Alluvial-Marine Model
• Large change (discontinuity) in deposition across shoreline, with strong localization of deposition (erosion) near shoreline
• Transgression triggers upstream waves of deposition in coastal plain (“upward tilt” to deposition contours)
• Sea level changes cause perturbations in relative uplift, preventing equilibrium bedrock channel profiles
• Perturbations in bedrock erosion rates “passively” advected upstream without decay (steady tectonics)
Gravel-Sand TransitionsGravel-Sand Transitions
• Typically downstream fining is relatively continuous
within gravel-bed rivers and within sand-bed rivers
• However there is typically a rapid downstream transition
in bed grain size and slope between these two river types
• Modeling formation and dynamics of gravel-sand transitions an essential component of the long profile model
• Two classes of models: explicit interface vs self-organized
– Explicit interface models assume a sharp transition– Self-organized models allow for diffuse transitions
Explicit-Interface Gravel-Sand ModelsExplicit-Interface Gravel-Sand Models
(Marr et al., 2000)Two grain sizes: gravel + sand
Aggregated diffusive flux law
gs qqq
0, V
Environment Threshold Deposition
Gravel qg g ∂x qg
Sand qg s ∂x qs
Self-Organized Gravel-Sand ModelsSelf-Organized Gravel-Sand Models
Conservation of grain-size fractions (Hirano, 1971):
sxtsssta PqFPFH
Ps sand fraction in transport
Fs sand fraction in active layer
Ha active layer thickness
Grain-size specific (diffusive) flux laws
Quasi-static hydrodynamic momentum balance
Effective diffusivity averaged over bed composition
ggssi FF
2/31 iii WFRgq
Mixed-Grain Bedload FluxMixed-Grain Bedload Flux
SqgC wD23
Preferential Transport of SandPreferential Transport of Sand
(Ferguson, 2003)
Sand more mobile than gravel
preferential sand transport
1s
ss F
FP
sg WW
L
qF tst
0,0
Steady State Gravel-Sand TransitionSteady State Gravel-Sand Transition
Steady aggrading profile (e.g. downstream dam) Steady bed composition, constant aggradation
xL
FPP
L
xqq ss
sx
,10
,,
2/3
2/3
ss
x
ggss
sss
sssx
FPy
yfyM
WFWFRgq
WFqRgPxL
FPP
Steady State Gravel-Sand TransitionSteady State Gravel-Sand Transition
• Steady Hirano + Flux laws system of equations for unknown sand fractions and bed shear stress
• Degenerate 3rd order ODE system (index-1 DAE)
• Integrate downstream to solve
Transport coefficients a function of relative shear stress …
• Grain-size specific
reference shear stress
• Transport “maxes out”
at large relative stress
Transport Coefficients: Similarity CollapseTransport Coefficients: Similarity Collapse
iri WW
,
72.0,8,14 A
(Wilcock and Crowe, 2003)
Reference stress depends on mean grain size …
ss Fg
Fsmmrmmr DDDRgD 1*
,, ,
Reference Shear Stress: Hiding EffectsReference Shear Stress: Hiding Effects
(Parker and Klingeman, 1982; Wilcock and Crowe, 2003)
030.0,40.0, *,
,
,
mr
b
m
i
mr
ir bD
D
But also on constituent grain sizes … (hiding effects)– Larger grains harder to move, but protrude higher into flow
– Smaller grains easier to move, but “hidden” among larger grains
Steady Gravel-Sand Transition: Bed CompositionSteady Gravel-Sand Transition: Bed Composition
Sharp transition at low sediment influx
Diffuse transition at higher sediment influx
Steady Gravel-Sand Transition: Bed ProfileSteady Gravel-Sand Transition: Bed Profile
Sharp transition has abrupt slope break
Diffuse transition has smooth slope
Sharpness of transition depends on preferential transport …
… So need large contrast in transport coefficients Wi
Depends on ratio of constituent grain sizes (potential contrast)
But also on stress level (i.e. sediment influx)
Scaling and Dimensional AnalysisScaling and Dimensional Analysis
23*,
*0** ,
mr
g
b
s
g qQ
D
DD
Large D* + Small Q* Abrupt transition
Transient Gravel-Sand ModelingTransient Gravel-Sand Modeling
• Changes in external forcing transient response
• Must solve transient Hirano with possibility of erosion– Must keep track of bed composition (i.e. stratigraphy)– Rarely able to solve transient Hirano analytically
• Stratigraphy is typically very dynamic – columns grow/shrink due to deposition/erosion – A difficult computational problem
• Use discrete data structure to efficiently store, access, and update stratigraphy … while hiding details from user
% SedLayer data structure
SedLayer = { dz,phi,tdep,F[Ngrain] }
% Matlab interface routines StratPtr = InitStrat(eta0,Hactive,Ngrain) SrfLyr = GetSrfLyr(StratPtr,col) EroLyr = ColUpdate(StratPtr,col,deta) Strat = GetStrat(StratPtr) FreeStrat(StratPtr)
% Matlab display routinesZcont = ContourTime(Strat,Tcont) ShadeStrat(Strat,fname)
MATLAB Dynamic Stratigraphy ToolboxMATLAB Dynamic Stratigraphy Toolbox
sxtsssta PqFPFH
Transient Gravel-Sand Transition: Steady ForcingTransient Gravel-Sand Transition: Steady Forcing
Time to reach equilibrium appears to depend on two factors:
1) gravel diffusion time and 2) basin filling time
Transient Gravel-Sand Transition: Cyclic Forcing (Transient Gravel-Sand Transition: Cyclic Forcing (PPss00))
Sharp transition simulation has small fluctuations in interface
position, but large fluctuations in slope (unconformities)
StratPtr=InitStrat(eta0,Hactive,Ngrain); for i=1:nx-1
% compute diffusivities SrfLyr=GetSrfLyr(StratPtr,i); K=(SrfLyr.F).*W;
% compute potential erosion/deposition dQ=Q-K*S; deta=dQ*dt/dx;
% compute actual erosion/deposition deta=ColUpdate(StratPtr,i,deta); dQ=deta*dx/dt;
% update surface and fluxes eta(i)=eta(i)+sum(deta); Q=Q-dQ;
end Strat=GetStrat(StratPtr); FreeStrat(StratPtr);
Example MATLAB RoutineExample MATLAB Routine
Future WorkFuture Work
• Complete alluvial component of long-profile model– gravel-sand model, floodplain evolution (mud)
• Refine and implement marine process models– turbidity currents, sand-mud dynamics
• Input from NCED community– What problems/questions are of interest?– What capabilities are needed to address these?– What sedimentary processes/models are appropriate?– MATLAB Dynamic Stratigraphy Toolbox (anybody interested?)
AcknowledgementsAcknowledgements
Vaughan Voller
Chris Paola
Participants in the
2nd NCED Workshop on
Geomorphic Interfaces