Trailing Behind the Bandwagon:

Transition from Pervasive to Segregated Melt Flow in Ductile Rocks

James Connolly and Yuri Podladchikov

Sowaddahamigonnadoaboutit?

Flog a dead hypothesis: reexamine mechanical flow instabilities in light of a rheological model for plastic decompaction

•Review steady flow instabilities in viscous matrix

•Consider the influence of plastic decompaction

•General analysis of the compaction equations for disaggregation conditions

Review of the Blob, an Old MovieD

epth

5 km

5 km

Initial condition Birth of the “B lob”

Porosity, t=0/0~10

t=3.3 /0~50

5

5

-45

-40

-35

-30

-25

porosity/0

y/

1.5

2

2.5

-10 0 10

-45

-40

-35

-30

-25

-peffective

/(g)

x/

y/

-1

0

1

0 20 402

4

6

time/

Amplitude (blue); Velocity (red)

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A differential compaction model: Death of the Blob?

What’s wrong with the Blob?

Compaction and decompaction are asymmetric processes

pe > 0com paction

pe < 0 decompaction

viscousviscous

Bulk viscosity ( ): c c

d c

d c

R

R

plastic

d

0

Channelized flow, characteristic spacing ~ c

Domains carry more than the excess flux?

Flow channeling instability in a matrix with differential yielding

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Numerical Problem

A traveling wave with gradients on drastically different spatial scales

A variable resolution grid that propagates with the

center of mass

Intrinisic flow instability in viscoplastic media

Waves nucleate spontaneously from vanishingly small heterogeneities and grow by drawing melt from the matrix

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c cd d

c cP r e s s u r e

P r e s s u r e

P o r o s i t y

P o r o s i t y

a ) C o n s t a n t v i s c o s i t y , s t e a d y s t a t e w a v e .

b ) D i f f e r e n t i a l y i e l d i n g , t r a n s i e n t w a v e .

Constant Viscosity vs. Differential Yielding

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Return of the Blob

R=1/125 R=1/10000

Porosity

Pressure

LowPressure

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0 200 400 600 800

200

400

600

800

1000

1200

time/(cR3/8)

ampl

itude

200 400 600 800 1000 1200

0

10

20

30

amplitude

velo

city

/vd

200 400 600 800 1000 12000

5

10

15

20

amplitude

wav

elen

gth/

c

200 400 600 800 1000 12000

5

10

15

x 104

volu

me/

(c2 R

1/2 )

amplitude

Scaling?

1D analyticR = 1/156R = 1/625R = 1/2500R = 1/10,000R = 1/40,000R = 1/160,000

3 8

c

tA

R

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10-1

100

101

102

10-2

100

102

104

time/c

Dis

sipa

tion

Is there a dominant instability?

R = 1/156R = 1/625R = 1/2500R = 1/10,000R = 1/40,000R = 1/160,000

D t R.

at =10m ax3/2 4

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Wave growth rate ~R3/8/tc*

For R ~ 10-3 an instability grows from = 10-3 to disaggregation in ~103 y with v ~ 10-500 m/y over

a distance of 30 km

Yes and Maybe

Yes, the mechanism is capable of segregating lower asthenospheric melts on a plausible time

scale

If the waves survive the transition to the more voluminous melting regime of the upper

asthenosphere, total transport times of ~1 ky are feasible.

Alternatively, waves could be the agent for scavenging Actinide excesses that are

transported by a different mechanism, e.g., RII or dikes.

So does it work for the McKenzie MORB Actinide Hypothesis?

dry meltingequilibrium 2 30 2 38Th/ U

damp/carbonate melting, < 0.1 %disequilibrium

f230 23 8Th/ U, 226Ra

100-

150

km

Mid-Ocean Ridge(Hirth & Kohlstedt 1995, Dasgupta et al. 2004)

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Conclusions I

Pipe-like waves are the ultimate in porosity-wave fashion:

nucleate from essentially nothingsuck melt out of the matrix

grow inexorably toward disaggregation

Growth/dissipation rate considerations suggest R~10-4, mechanistic arguments would relate R to the viscosity of the suspension

Toward a Complete Classification of Melt Flow Regimes

Transition from Darcyian (pervasive) to Stokes (segregated “magmatic”) regime

• Darcy’s law with k = fk()

• Viscous bulk rheology with e

s

qpv

1

s R 1s

1

d

, 1 eg

, McKenzie/Barcilon

qQT q

q

f kq A

f f

• Neglect perturbations to the solid pressure field (small porosity limit)

• 1D stationary states traveling with phase velocity

Balancing ball

gv h

t x

v p

x

t z

0 ,p

fz

xv

t

1( )q

s

pf

z

v g h

x v x

sq

p H

p

0h

vdv g dxx

0 q

sH

p dp d

2

2

vE hg

1

q

sp

U Hq

sg

Porosity Wave Balancing Ball

Wave Solutions as a Function of Flux

Phase diagram

/x

Sensitivity to Constituitive Relationships

Conclusions II

Lithosphere

Partia lly (3 vol % ) m oltenasthenosphere

Basalt d ikes

Basalt s ills

M assive D unites

R eplacive D unites

R eplacive D unites = reactive transport channeling instability?

Basalt d ikes = se lf propagating cracks?

Basalt s ills = segregation caused by m agica l perm eability barriers?

M assive D unites = rem obilized replacive dunite?

M id-O cean R idge

Objectives

• Review steady flow instabilities => birth of the blob

• Consider the influence of differential yielding => return of the blob

• Analysis of the compaction equations for dissagregation conditions

So dike-like waves are the ultimate in porosity-wave fashion:

They nucleate out of essentially nothing They suck melt out of the matrix

They seem to grow inexorably toward disaggregation

But

Do they really grow inexorably, what about 1?

Can we predict the conditions (fluxes) for disaggregation?

Simple 1D analysis

Wave growth rate ~R3/8/tc*

For R ~ 10-4 (10-8) an instability grows from = 10-3 to disaggregation in ~104 y with v ~ 1-50 m/y

over a distance of 30 (1) km

Adequate to preserve actinide secular disequilibria?

Excuses:

McKenzie/Barcilon assumptions give higher velocities and might be justified at large porosity

The waves are dike precursors?

So does it work for MORB transport?

100

-150

km

M id -O cea n R idge

Conclusions I

Pipe-like waves are the ultimate in porosity-wave fashion:

nucleate from essentially nothingsuck melt out of the matrix

grow inexorably toward disaggregation

Growth/dissipation rate considerations suggest R~10-4, mechanistic arguments would relate R to the viscosity of the suspension

Velocities are too low to explain MORB actinide signatures, but the waves could be precursors to a more efficient mechanism

Problem: Geochemical constraints suggest a variety of melting processes produce minute quantities of melt, yet that this melt segregates and is transported to the surface on

extraordinarily short time scales

Hypotheses: dikes (Nicolas ‘89, Rubin ‘98), reactive transport (Daines & Kohlstedt ‘94, Aharanov et al. ‘95) and shear-induced instability (Holtzman et al. ‘03, Spiegelman ‘03)

partial explanations

Flog a dead hypothesis: reexamine mechanical flow instabilities in light of a rheological model for plastic decompaction

•Review steady flow instabilities => birth of the blob

•Consider the influence of differential yielding => return of the blob

•Analysis of the compaction equations for disaggregation conditions

Sowaddahamigonnadoaboutit?

A Pet Peeve:Use and Abuse of the Viscous Compaction Length, Part II

• Is bulk viscosity

>> shear viscosity ?

• All formulations have effectively the same definition

for 1k k

q

• The compaction length is not stress dependent for power law rheologies

1

1

g

q

qk

Good News for Blob Fans

• Soliton-like behavior allows propagation over large distances

Bad News for Blob Fans

• Stringent nucleation conditions

• Soliton-like behavior prevents melt accumulation

• Small amplification, low velocities

• Dissipative transient effects

Is there a dominant instability?

10-1

100

101

102

100

105

time/c

Dis

sipa

tion

10-2

10-1

100

101

102

103

100

105

volume/c2

Dis

sipa

tion R = 1/156

R = 1/625R = 1/2500R = 1/10,000R = 1/40,000R = 1/160,000

D t R.

at =10m ax3/2 4

SS stage 1

SS stage 2

transient

Conclusions I

Pipe-like waves are the ultimate in porosity-wave fashion:

nucleate from essentially nothingsuck melt out of the matrix

grow inexorably toward disaggregation

Growth/dissipation rate considerations suggest R~10-4, mechanistic arguments would relate R to the viscosity of the suspension

Velocities are too low to explain MORB actinide signatures, but the waves could be precursors to a more efficient mechanism