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)
next slide
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
next slide
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
next slide
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
next slide
Return of the Blob
R=1/125 R=1/10000
Porosity
Pressure
LowPressure
next slide
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
next slide
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
next slide
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)
next slide
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