Teleconnections in the Source-to-Sink System

John Swenson

Department of Geological SciencesUniversity of Minnesota Duluth

THANKS : Chris Paola, Tetsuji Muto, and Lincoln Pratson

Teleconnections: Context

La Nina anomalous SL pressure

Strong statistical relationship between ‘weather’ in different

parts of the globe

Information propagates through the atmosphere

Long-distance propagation of allogenic forcing (e.g. sea level change) through the transport system via erosion and deposition on geologic time scales

Road map:

(1) Downstream (eustatic) forcing

(1a) Response to steady Rsl fall

(1b) Response to periodic perturbations (filtering)

(2) Upstream forcing: Propagation of sediment-supply signals

(3) Wave energy and mesoscale suppression of fluvial aggradation

Talk = theory & experiments

Theory developed for geologic time scales, where forcing data are poorly constrained / non-existent

average over many ‘events’

implicitly involves the ‘upscaling’ problem

Teleconnections in S2S fundamentally involve coupling of environments

Cannot overemphasize the need to treat morphodynamics of the transport environments and the coupling of environments with equivalent levels of sophistication*

*Requires considerable simplification of transport relations…

A few important points (and a plea for forgiveness…)

(1a) Downstream forcing

Fluviodeltaic response to steady sea-level fall

Classic teleconnection problem: How do alluvial rivers respond to sea level and what is the upstream (‘stratigraphic’) limit of sea-level change?

Sequence stratigraphy (e.g. Posamentier & Allen, 1999): Fall in relative sea level (Rsl) at shoreline = degradation & sequence-boundary development

Recent models & experiments (e.g. Cant, 1991; Leeder and Stewart, 1996; Van Heijst

and Postma, 2001): Rivers can remain aggradational during Rsl fall

Let’s analyze the response to a steady rate of fall…

Note: Rsl is falling everywhere

Investigate how allogenic forcing (sediment & water supply, fall rate) and basin geometry control aggradation?

Morphodynamics:

€

∂ ′ η ∂t

= υ∂2 ′ η

∂x2

€

′ η =η+σt

Problem is not closed…

Diffusive fluvial morphodynamics (Paola, 2000):

Shoreline BC:

Alluvial-basementtransition BC:

€

′ η s,t( ) = Rsl t( )

€

′ η r,t( ) = −φr

€

Rsl t( ) = Zsl t( ) + σt

Swenson & Muto (2006), Swenson (2005)

€

υ∝ qwo = Ifqwf

Absorb subsidence:

€

qs = −υ∂ ′ η

∂x

+hydrodynamics& stress closure

Closure (moving boundaries):

€

qs s,t( ) = −υ∂ ′ η

∂x s,t

=1

β− φ( )φs +Rsl( ) β

ds

dt+

dRsl

dt

⎛

⎝ ⎜

⎞

⎠ ⎟

€

qs r,t( ) = υ∂ ′ η

∂x r,t

= qso

…need additional pair of equations to locate shoreline and alluvial-basement transition and close problem

Shoreline:

Alluvial-basement transition:

€

τ =1

υ

qso2

dRsl dt2

Scaling

dRsl/dt ~ 1 mm/a(late-Quaternary systems )

Non-dimensionalization:

€

ηH

= Fx

Λ,t

τ;qso

υφ,φ

β

⎛

⎝ ⎜

⎞

⎠ ⎟

Response time:

Invent one:

Dimensionless numbers for morphodynamics

Elevation scale:

€

Λ =qso

dRsl dt

€

H =qso

υ

qso

dRsl dt

€

qso υφ < 1

€

φ β <<1

No imposed length scale…

Three-phase evolution: Widespread aggradation degradation

widespread aggradation

(onlap)

widespreaddegradation

(offlap)

‘mixed’

timelines

modified Wheeler diagram

Focus on timing of offlap (toff):

Gross measure ofaggradational interval

Good experimental observable

Scaling arguments:

What controls the duration of aggradation?

€

toff τ = f φ β,qso υφ( )

€

qs

qw

~ κ Sα ~ 3.6; ~ 1.95υ = qw

Supporting flume experiments (Tetsuji Muto, Nagasaki University)

Theory and experiments similarly‘sophisticated’

Require similarity in qso/(υ) & /β

Scale issue: Sediment flux varies non-linearly with slope; resort to blatant empiricism

€

∂ ′ η ∂t

= α κqw( )∂ ′ η

∂x

α−1

⋅∂2 ′ η

∂x2Gives non-linear morphodynamics

Representative experimental results

Sensitivity study: variations in qso/(υ)

Shoreline

Source of ‘noise’ = Stick-slip on the delta foreset

Swenson & Muto (2006, Sedimentology)

(1b) Downstream forcing

Fluviodeltaic response to periodic eustatic forcing: Frequency dependence of teleconnection between shoreline and alluvial-basement transition

Perturbation theory with two moving boundaries…wiggle sea level

€

Zsl = −iεeiωt€

s = so + iεαei ωt+ψ( )

€

r = ro + iεβei ωt+φ( )

€

η=ηo − iεη1ei ωt+θ( )

forcing

Imposed response

Operate on ‘imposed’ response with governing PDE and BCs

Determine amplitude and phase of shoreline and alluvial-basement transition

Gives frequency dependence (‘filtering’)

Note change in basin response time (diffusive timescale):

qs = steady

€

Λ

€

τb = Λ2 υ

€

eiψ = em

c1 + ic2

c3

€

βeiφ =eb

1− eb

c4 + ic5

H2

€

c1 = 1+1

1− so*

T

4πτb

em 1− so*2

( ) +1[ ] Re G[ ] + Im G[ ]( ) + em

T

2πτb

1+ so*( )1− so*( )

G2

€

c2 =1

1− so*

T

4πτb

em 1− so*2

( ) −1[ ] Re G[ ] − Im G[ ]( )

€

c3 = 1+ 2em 1+ so*( )T

4πτb

Re G[ ] + Im G[ ]( ) + em2 T

2πτb

1+ so*( )2

G2

€

c4 = Re H[ ] ⋅ 1+ so*2 −1( )em

c1

c3

⎡

⎣ ⎢

⎤

⎦ ⎥+ Im H[ ] ⋅ so*

2 −1( )em

c2

c3

€

c5 = Re H[ ] ⋅ so*2 −1( )em

c2

c3

− Im H[ ] ⋅ 1+ so*2 −1( )em

c1

c3

⎡

⎣ ⎢

⎤

⎦ ⎥

€

G = tanhπT

τb

1+ i( ) ⋅so*

⎡

⎣ ⎢

⎤

⎦ ⎥=

sinh so* πT τb( ) + i ⋅sin so* πT τb( )

cosh so* πT τb( ) + cos so* πT τb( )€

H = coshπT

τb

1+ i( ) ⋅so*

⎡

⎣ ⎢

⎤

⎦ ⎥= cos so*

πT

τb

⎛

⎝ ⎜

⎞

⎠ ⎟cosh so*

πT

τb

⎛

⎝ ⎜

⎞

⎠ ⎟+ i ⋅sin so*

πT

τb

⎛

⎝ ⎜

⎞

⎠ ⎟sinh so*

πT

τb

⎛

⎝ ⎜

⎞

⎠ ⎟

Shoreline:Alluvial-basementtransition:

Puff…

Teleconnections: fluviodeltaic systems as filters to eustasy

Swenson (2005, JGR)

Why? ‘Skin’ depth:

€

λ = υT

€

λ* = T τb

Experimental ‘test’ (XES Facility, SAFL group, C. Paola, W. Kim)

Kim et al. (2006, JSR)

(2) Upstream forcing

Shoreline response to fluctuations in sediment supply:Frequency dependence

€

qs = qso − iεeiωt

€

s = so − iεαei ωt+ψ( )

forcing

response

Determine amplitude and phase of shoreline

More perturbation theory…wiggle upstream BC

€

η=ηo − iεη1ei ωt+θ( )

Teleconnections: fluviodeltaic systems as filters to qs

(3) Mesoscale teleconnections between shallow-marine and fluvial systems

Suppression of avulsion via increases in wave energy

Avulsion frequency is dominant control on ‘mesoscale’ stratigraphic architecture…

Avulsion frequency: Quick overview

Previous studies ignore potential role of nearshore processes

Figure by Paul Heller

Avulsion appears to be driven by superelevation of channel…

Rivers are one part of a linked depositional system…

Avulsion frequency = F(sedimentation rate)

Past studies (field, experimental, theoretical) have focused on this relationship, using sediment supply, subsidence, or changes in relative sea level as proxies for sedimentation rate

Deltaic (distributary) systems: The larger problem…

Deltaic systems have two fundamental degrees of freedom:

(1) Adjust number of channels (N)

(2) Adjust channel residence time (τ) or avulsion frequency (1 / τ)

Nile (N = 4)

Lena(N > 100)

Images courtesy of James Syvitski ( INSTAAR)

Problem statement:

To what extent does wave energy affect avulsion frequency?

Can the tail wag the dog?

Today… force N =1; analyze τ

Basic hypothesis: Wave energy suppresses avulsion

Mechanism du jour = Wave energy & alongshore sand transport

Observation: Fluviodeltaic systems prograde as approximately self-similar waveforms (clinoforms)

Conceptual model: Idealized ‘highstand’ deltaic system

Steady sand / water supply & wave climate

Sand channel belt & shoreface

Channel belt = fixed width

Shoreface = fixed geometry

Mud floodplain and pro-delta

No tides, subsidence, or sea-level change

Basic assumptions:

Channel-belt (fluvial) morphodynamics:

‘Cheat’ and assume diffusive morphodynamics (Paola, 2000) works:

€

qs = −υ∂η

∂x

Upstream condition:

€

∂η∂t

= υ∂2η

∂x2

€

∂η∂x x=0

= −qso

υ€

η scb,t( ) = 0Shoreline condition:

€

qso = Ifqsf

Shoreface morphodynamics (map view):

Simplify and use generalized CERC relationship (Komar, 1988; CERC, 1984):

€

qls = −κ∂s

∂y

Channel-belt condition:

€

∂s

∂t= κ

∂2s

∂y2

€

limy→±∞ s y,t( ) = 0Far-field condition:€

∝ IsHb( )5 2

Diffusivity:

Longshore transport on long timescales is poorly understood (Cooper & Pilkey, 2004)

€

s 0,t( ) = scb t( )

Coupling channel-belt & shoreface morphodynamics:

Problem is closed mathematically

Channel-belt progradation:

€

−∂s

∂yy=0

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟D =

1

2qswW

Longshore flux and wave extraction from channel-belt:

€

−υ∂η∂x x=scb t( )

− qsw = Ddscb

dt

Avulsion criterion

Avulsion set-up:superelevation ~ channel depth

€

he = αh

€

0.6 < α < 1.1

Geometric argument:

€

scrit ~ h So

Diffusion gives:

€

So ~ qso υ

€

h∝1 So

(Mohrig et al., 2000)

Shoreface (map view) Channel belt (cross section)

Shoreface / channel-belt evolution

€

∂s

∂t= κ

∂2s

∂y2

€

∂η∂t

= υ∂2η

∂x2Solve and subject to boundary & initial conditions…

Tim

e

€

qsoW( )τ ~h

So

WD +1

2

h

So

hW +h

So

Dπ

2κ ⋅ τ

Solution:

Lazy approach…use simple scaling arguments

Supply over‘lifespan’ (τ)

Channel beltprogradation

Channel beltaggradation

(superelevation)

Cuspate ‘wings’(‘smearing’)

€

ττo

= f ξ( )τo = Avulsion time scale (zero-energy limit)

= Dimensionless parameter groupingthat embodies interplay of river and waves

Relative importance of fluvial input and wave energy ()

€

=1

Wqso

π

2Dhκυ

€

∝qwf

If1 2qsf

3 2 ⋅ IsHb( )5 4

qwf = flood water flux

qsf = flood sediment flux

If = flood intermittency

Hb = storm breaker height

Is = storm intermittency

Expanding…

€

→ 0

Analytical solution (approximate):

€

ττo

=ξ

2⋅ 1+ 1+

4

ξ2

⎡

⎣ ⎢

⎤

⎦ ⎥

€

ττo

→ 1

€

ττo

~ ξ2€

→ 0

€

>>1

General solution:

River-dominated limit:

Wave-dominated systems:

‘Smearing’ length >> channel-belt width

€

qsoτ ~h

So

D 1+1

2

h

D+

π 2( )κ ⋅ τ

W

⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥

Sand budget (from before):

<<1 (generally)

Teleconnection: wave-driven suppression of avulsion

after Swenson (2005, GRL)

Morphodynamic models hint at long-term teleconnections in the S2S system:

Alluvial aggradation during Rsl fall can be long lived

Eustasy can affect the entire alluvial system

Fluviodeltaic systems behave as low-pass filters to both upstream and downstream forcing

Wave energy can effectively suppress avulsion on appropriate spatiotemporal scales

How do we test predictions in natural systems?

Conclusions

€

∝qwf

If1 2qsf

3 2 ⋅ IsHb( )5 4

€

τb =1

υ

qso2

dRsl dt2

€

λ* = T τb ~ 1

€

τ τo ~ ξ2