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