Rheology and melt migration

Magma Dynamics I
Sash Hier-Majumder
University of Maryland
Also:
Jodi Gaeman
Jesse Wimert
Matt Abbott
Questions Related to Magma
Dynamics
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Evolution of the early Earth and planets (Solomatov, 2007
(review), Samuel et al (2009), Labrosse et al (2007), Gaeman et
al. (2010), Mosenfelder et al. (2007))
Volcanism at plate boundaries (Spieg-talk, Cagnioncle (2008),
Iwamori (2000), Richard et al. (2007))
Storage, transport efficiency, and localization of magma in the
mantle (Katz et al 2005, Stevenson, 1986, Hier-Majumder et al.
2006, Hernlund and Jellinek, 2010)
Melting and nonequilibrium effects (Sramek et al. (2007), Hewitt
and Fowler (2009), Rudge et al (under review), Katz, (2009))
Volcanology (Dufek and Manga (2008), Michaut and Bercovici
(2010) )
Length scales
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Micro – scale of a few grains and melt units
Meso – A statistically significant number of
grain and melt units
Macro – Mantle/planetary length scale
Micro
Variables in melt geometry
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Contiguity
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Dihedral angle
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Melt fraction
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Deformation
Contiguity
Agg
=
A g g A g m
Grain boundaries: Twist
Grain boundaries: Tilt
McKena and Schluger, (2008)
Low angle grain boundaries
Hiraga and Kohlstedt (2004)
Dihedral angle
Wray's (1976) classification
=70.53
≥70.53
70.53≥≤60
≤60
Melt fraction
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Grains are contiguous
below disaggregation
fraction
Suspension at high
enough melt fraction
Dihedral angle and melt fraction
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Von Bargen and Waff (1986)
Larger degree of
melting needed to
interconnect at high
dihedral angle
Dihedral angle
mimics high melting
Areas in partially molten rocks
Von Bargen and Waff (1986)
Experimental measurement
Yoshino et al. (2005)
Deformation
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Tubules and pockets
in undeformed state
Melt wets grain
boundary under
stress
Grain boundary wetting
Hier-Majumder and Kohlstedt (2006)
Hier-Majumder et al. (2006)
Melt segregation
Holtzman et al. (2003)
Katz et al. (2006)
Deformation and contiguity
Takei (2005)
Meso
Elasticity
Upper and lower bounds
Hard
Soft
Voigt or upper bound
Hard
Soft
Reuss or lower bound
Bounds and average (on board)
Microstructure?
?
The equilibrium geometry model
Bulk and shear moduli of the
skeleton
contiguity
contiguity
Surface tension
Disaggregation and surface
tension
=  1−2 cos 2   2 cos  if s


1− if ≥s
4
Hier-Majumder et al, [2006]
Viscosity
Rheology of partially molten
rocks
 
n

̇= A m e
d


e
H
−
 
RT
=g e− 
Scott and Kohlstedt, [2006].
Rheology of partially molten
rocks
=melt 1.35 −0.35−2.5
− 
=g e
Scott and Kohlstedt, [2006].
Contiguity and viscosity
Takei and Holtzman (2009a)
Models of contiguity
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Melt fraction
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Dihedral angle
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Deformation
Takei and Holtzman (2009a)
Yoshino et al. (2005)
Governing equations
∇⋅u=0
Mass conservation
∇⋅T =0
Momentum conservation
¿
0= u
k
No-slip
k

 T ⋅n =  ∇⋅n  n − ∇ 
k
k
k
k
k
k
Jump condition
∂F
k
k
0=
u ⋅∇ F
∂t
Kinematic equation
Laplace vs Marangoni on the
board
Marangoni effect
Hosoi and Bush (2001)
Suminagashi (Marbling) using
Marangoni effect
Basic medium
=
Tap
water(unthickened)
Colour(Soft film) =
Same as classic
suminagashi
Colour(Hard film) =
Add metaric compound
to dye or colour
liquid(see "Method")
Use of Marangoni
effect= Expansion and
compression direction
http://www5e.biglobe.ne.jp/~kuroda/room-3e.htm
Microstructural models
German (1985)
Hopper (1990)
Air bubbles in corn syrup
Manga and Stone (1993)
Models of microstructure
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Geometric models (Wray (1976), von Bargen and Waff
(1986), German (1985))
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Prescribed dihedral angle
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Steady-state (no forces)
Dynamic models (Kuiken, Hopper)
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Models viscous flow within grains
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Limited to one or two grains, does not reproduce dihedral angles
Combined model
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Multiparticle interaction (numerical)
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Single particle analysis (analytical)
Analytical solution
Macro
Two-phase flow
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Coupled viscous flow of
the melt and the matrix
Surface tension balance
pressure, viscous stress,
and body forces
Incorporates both melt
geometry and
disaggregation.
Governing equations
∂ ∂

=  1−w −
∂t ∂ y
m
2
 

Mass conservation
2

∂  ∂  4 ∂ 1− ∂ w
4 w
1−

− R 1−−
=0
2
2
3
∂ ∂ y 3 ∂ y  ∂ y
Dm 
∂
=− PB
∂
Dt
Action-reaction equation
Phenomenological relation
Bercovici et al. (2001a)
UltraLow Velocity Zones
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Topography varies
between 5-40 km
Highly variable shear
to P wave velocity
drop
Garnero (2004)
Density increase of
10% on average
Williams and Garnero (1996)
Compaction layer (on board)
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Profile of a dense melt-rich layer
Neutrally buoyant layer
Retention
Depth
Self separation
Melt fraction
Hier-Majumder et al. [2006]
Homogenization
n mul oc f ot hgi e H
Solitary waves
Melt fraction
Hier-Majumder et al. [2006]
surface tension
=
buoyancy
Surface tension and chemistry
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Strong gradients of
surface tension may
impose gradient of
chemical potential
Mass transport due to
Gibbs-Thompson
effect
Different from
capillary tension
Annealing melt bands
After deformation
After 50 hr annealing
King et al. (in preparation)
Length scales
Takei and Hier-Majumder (2009)
Worked example
Dense, partially molten layer
Mineral Physics model
Potential temperature 1600 K, basalt fraction 18%
Xu et al. (2008)
Thermal vs chemical anomaly
Thermal vs chemical anomaly
Mineral Physics model
Potential temperature 1600 K, basalt fraction 18%
Xu et al. (2008)