View the Slides - Tyler Drombosky

DI22A-08
Dynamics of ULVZ-mantle interaction
using fast multipole boundary element
method NSF EAR 0911094 and EAR 1215800
Tyler Drombosky - drombosk@math.umd.edu
University of Maryland College Park
Applied Mathematics and Scientific Computation
Saswata Hier-Majumder - saswata@umd.edu
University of Maryland College Park
Department of Geology
Center for Computational Sciences and Mathematical Modeling
of Earth’s Lower Mantle
density elevated by a few percent relative to
surrounding mantle; if higher, structures w
Edward J. Garnero* and Allen K. McNamara
flatten out along the CMB, and if lower, they w
be easily entrained in upwellings. If the resu
Processes within the lowest several hundred kilometers of Earth’s rocky mantle play a critical role
density is less than that of the surrounding ma
in the evolution of the planet. Understanding Earth’s lower mantle requires putting recent seismic and
the material is buoyant and forms large, dom
mineral physics discoveries into a self-consistent, geodynamically feasible context. Two nearly
structures that actively rise through the mantle
antipodal large low-shear-velocity provinces in the deep mantle likely represent chemically distinct
2C). Alternatively, these thermochemical su
and denser material. High-resolution seismological studies have revealed laterally varying seismic
plumes may heat up and rise because of ex
velocity discontinuities in the deepest few hundred kilometers, consistent with a phase transition from
thermal buoyancy, then cool and sink becau
perovskite to post-perovskite. In the deepest tens of kilometers of the mantle, isolated pockets of
decreased thermal buoyancy (11). Smaller plu
ultralow seismic velocities may denote Earth’s deepest magma chamber.
that entrain some of the denser material can for
2
A.K. McNamara et al. / Earth and Planetary Science Letters 299 (2010) 1–9
arth’s most profound internal boundary plate boundaries overlie regions of D″ with higher the tops of these structures. Assuming this
lies roughly halfway to its center, at a depth than average velocities, and that hot-spot volcanoes havior for Earth implies that LLSVPs are su
of nearly 2900 km, where the solid mantle overlie regions with lower than average velocities. plumes in various stages of ascent or fall (11)
If LLSVP thermal and compositional buo
meets the fluid outer core. Emerging research Such spatial correlations, combined with evidence
characterizes structure and processes on the mantle for high P- and S-wave velocities mimicking slab cy are roughly balanced, then near-neutra
side of this boundary that influence chemistry shapes extending from beneath some subduction negative buoyancy can yield long-lived s
and convection throughout the mantle, heat loss zones well into the lower mantle, constitute one structures (9, 10, 12, 13). Piles are passively s
from the core, and Earth’s thermal structure and argumentinfavorofwhole-mantleconvection(2,3). and shaped by mantle convection, preferen
Seismic data suggest that two broad regions accumulating into ridge-like structures ben
evolution. The fields of seismology, mineral physics, geodynamics, and geochemistry have been key with lowered shear-wave speeds and higher than dominant upwelling centers (the Pacific
in providing new information. To better understand average density lie beneath the Pacific and Africa Africa). Plumes may originate from pile
Earth’s lowermost mantle, and hence whole-mantle (4, 5). The African anomaly appears to extend particularly at peaks or ridges (Fig. 2B), and en
processes, we summarize several recent observa- upward from the CMB about 1000 km (6), where- nominal amounts of pile material. If the
tions and examine them in a geodynamical context. as the height of the Pacific anomaly is less certain modulus of pile material is anomalously high
Historically, the lowermost few hundred but probably at least 400 to 500 km. Each anom- consistent with subducted basaltic crust), struc
kilometers of the mantle was noted as having aly is about 15,000 km across, and together they intermediate to piles and superplumes can f
a reduced seismic velocity gradient with depth, cover nearly 50% of the CMB. The boundaries which have steeper sides than the other cases
Lower-mantle chemical heterogeneity m
interpreted as being caused by a lowermost- between these large low-shear-velocity provinces
mantle thermal boundary layer above the hot (LLSVPs) and normal mantle are sharp, as implied also have existed since Earth’s early differentia
1. Global distribution of ULVZ based on seismic studies. Blue areas in the foreground indicate probed areas lacking evidence for ULVZ structure, while red patches in th
seismic
waves that graze
LLSVP
edges (6–8). Seismic images of present-day structure w
core. This view was modified inFig.
the
early 1980s, by
foreground mark regions with detected
ULVZs. Background colorset
showal.
lowermost
mantle seismic shear wave velocities from the tomographic study by Ritsema et al., 2004; sca
[McNarama
2010]
bar
at
the
bottom
is
for
the
background
tomography
model.
Details
on
studies
summarized
here can be found in the Supplemental material.
when seismologists observed a first-order discontinuous increase in velocity between 250 and
350 km above the core-mantle boundary (CMB)
Upper
mantle density material can accumulate in these regions, and several importan
(1). This discontinuous jump is chemical
typicallyheterogeneity
referred (e.g., Hofmann, 1997). Chemically distinct
questions remain:
of ancient oceanic
to as the D″ discontinuity. In theLLSVPs
past may
few have
years,formed by the accumulation
Seismic
crust that has long since subducted (e.g., Brandenburg
and van Keken, Lower
reflections
observations, modeling, and predictions
1. Can the vigor of deep mantle convection dynamically suppo
2007a, 2007b; have
Brandenburg et al., 2008; Christensen and Hofmann, mantle
LLSVP
topography on material of such high density (as inferred from
1994;
Davies,
2008;
Hirose
et
al.,
1999;
Hofmann
and
White,
1981;
shown that the deepest mantle is complex (Fig. 1)
High T,
Sharp
seismic observations of small scale ULVZs juxtaposed with mant
Nakagawa and Buffett, 2005; Nakagawa et al., 2009; Xie and
Tackley,
high ,
and much more anomalous than the rest of the
side
2004a, 2004b), or may 0be primordial remnants of dVs
a mantle
lacking ULVZ structure), or would high density ULVZ material flatte
lower mantle. The term D″ is used
to refer toprocess
the that occurred much earlier in Earth's history
outD"
into a ubiquitous layer along the CMB?
differentiation
general depth shell of the lowermost
2.
On
the
otherPv
hand, can such small volumes of dense ULVZ materi
(e.g., Boyet several
and Carlson, 2005;
Kellogg
et
al.,
1999;
Labrosse
et
al.,
0
STZ
66
survive
intact
for geologic times, or would they be complete
2007;
Lee
et
al.,
2010;
Solomatov
and
Stevenson,
1993;
Tolstikhin
hundred kilometers of the mantle, and does not
ULVZ
pPv
entrained
and
mixed
with the surrounding mantle?
and Hofmann, 2005). If denser than
the
background
mantle,
LLSVP
0
50
denote any specific structural characteristic.
CMBconvecting
High T,
Core
3. Can
changing subduction patterns cause large composition
material can survive wholesale1 entrainment into the
The UltraLow Velocity Zones
E
3:1 s-wave to p-wave
loss in velocity
50-100 km wide
10-40 km thick
1-2 orders of magnitude
less viscous
10% more dense
Dynamically coupled to
surrounding mantle
reactions +dVs
reservoirs (LLSVPs) to change shape and location over time, and
mantle, forming long-lived compositional reservoirs (e.g.,Bull et
al.,
0
0
pPv
Long-Wavelength Heterogeneity
so, how would accumulations
of dense ULVZ material respond t
2009; Davaille, 1999; Deschamps and25Tackley,
2008; 2009; Garnero
Low
T,
0
these changes?
and McNamara, 2008; Jellinek and Manga,902002; Kellogg et al., 1999;
and Implications
+dVs
2
Lassak et al., 2007, 2010; McNamara and Zhong, 2004a, 2004, 2005;
Earth’s internal structure is predominantly
imaged
these questions,
we performed
very in
high
resolutio
Nakagawa and
Tackley, Fig.
2008;1.
Olson
and Kincaid, 1991;
Tackley,
Tomographically
derived
(43) high To
andexplore
low seismic
shear velocity
variations
Earth’s
m
whole mantle, geodynamical convection calculations of a thre
2000,inversions
2002; Tan and
2007, 2005; Youngs and Houseman,
by seismic methods. Tomographic
of Gurnis,
(blue
and
red,
respectively)
are
shown
in
an
equatorial
cross
section
(right)
viewed
from
the
south,
a
component thermochemical system: background mantle, large compo
2009; Zhong et
al., 2008). This would result in a thermochemical style
seismic data yield maps of seismic-wave
speed
withisan
enlarged
paneland
(left)
depicting several
seismic
findings
in the
D″ region.
A large low-shear-vel
sitional
reservoirs
intended
to represent
seismically-observed
LLSVP
of mantle convection which
driven
by thermal
compositional
[Garnero and McNarama 2008]
Experimental Model
Forcing terms
Buoyancy
Mantle
ULVZ
ULVZ
Imposed velocity field
(flow within LLSVP)
Modulating terms
Viscosity ratio varies
patches desire to
deform
Patch interaction
Core
Core
Experimental Model
Forcing terms
Buoyancy
Mantle
ULVZ
ULVZ
Imposed velocity field
(flow within LLSVP)
Modulating terms
Viscosity ratio varies
patches desire to
deform
Patch interaction
Core
Core
What is the drainage
rate of the ULVZ?
Governing Equation
The Stokes Boundary Integral Equation (BIE) for
multiple domains [Pozrikidis 2001]
1+
2
q
uj (x)
=u1
j (x0 )
Z
P
X
Rp
p=1
+
P
X
1
p=1
p
4⇡
4⇡
Z
PV
(p)
fi (x)Uij (x, x0 ) d m (x)
p
ui (x)Tijk (x, x0 )n̂k (x) d
p (x)
p
The interfacial surface term is buoyancy driven
2
µ
p
(⇢
⇢
)gx
m
p
c
(p)
p =
R =
µm
uc µm
Non-dimensionalizion yields the viscosity ratio and
compositional Rayleigh number [Leal 2007]
Governing Equation
The Stokes Boundary Integral Equation (BIE) for
multiple domains [Pozrikidis 2001]
1+
2
q
uj (x)
=u1
j (x0 )
Z
P
X
Rp
p=1
+
P
X
1
p=1
p
4⇡
4⇡
Z
PV
(p)
fi (x)Uij (x, x0 ) d m (x)
p
ui (x)Tijk (x, x0 )n̂k (x) d
p (x)
p
The interfacial surface term is buoyancy driven
2
µ
p
(⇢
⇢
)gx
m
p
c
(p)
p =
R =
µm
uc µm
Non-dimensionalizion yields the viscosity ratio and
compositional Rayleigh number [Leal 2007]
Governing Equation
The Stokes Boundary Integral Equation (BIE) for
multiple domains [Pozrikidis 2001]
1+
2
q
uj (x)
=u1
j (x0 )
Z
P
X
Rp
p=1
+
P
X
1
p=1
p
4⇡
4⇡
Z
PV
(p)
fi (x)Uij (x, x0 ) d m (x)
p
ui (x)Tijk (x, x0 )n̂k (x) d
p (x)
p
The interfacial surface term is buoyancy driven
2
µ
p
(⇢
⇢
)gx
m
p
c
(p)
p =
R =
µm
uc µm
Non-dimensionalizion yields the viscosity ratio and
compositional Rayleigh number [Leal 2007]
Governing Equation
The Stokes Boundary Integral Equation (BIE) for
multiple domains [Pozrikidis 2001]
1+
2
q
uj (x)
=u1
j (x0 )
Z
P
X
Rp
p=1
+
P
X
1
p=1
p
4⇡
4⇡
Z
PV
(p)
fi (x)Uij (x, x0 ) d m (x)
p
ui (x)Tijk (x, x0 )n̂k (x) d
p (x)
p
The interfacial surface term is buoyancy driven
2
µ
p
(⇢
⇢
)gx
m
p
c
(p)
p =
R =
µm
uc µm
Non-dimensionalizion yields the viscosity ratio and
compositional Rayleigh number [Leal 2007]
Governing Equation
The Stokes Boundary Integral Equation (BIE) for
multiple domains [Pozrikidis 2001]
1+
2
q
uj (x)
=u1
j (x0 )
Z
P
X
Rp
p=1
+
P
X
1
p=1
p
4⇡
4⇡
Z
PV
(p)
fi (x)Uij (x, x0 ) d m (x)
p
ui (x)Tijk (x, x0 )n̂k (x) d
p (x)
p
The interfacial surface term is buoyancy driven
2
µ
p
(⇢
⇢
)gx
m
p
c
(p)
p =
R =
µm
uc µm
Non-dimensionalizion yields the viscosity ratio and
compositional Rayleigh number [Leal 2007]
Governing Equation
The Stokes Boundary Integral Equation (BIE) for
multiple domains [Pozrikidis 2001]
1+
2
q
uj (x)
=u1
j (x0 )
Z
P
X
Rp
p=1
+
P
X
1
p=1
p
4⇡
4⇡
Z
PV
(p)
fi (x)Uij (x, x0 ) d m (x)
p
ui (x)Tijk (x, x0 )n̂k (x) d
p (x)
p
The interfacial surface term is buoyancy driven
2
µ
p
(⇢
⇢
)gx
m
p
c
(p)
p =
R =
µm
uc µm
Non-dimensionalizion yields the viscosity ratio and
compositional Rayleigh number [Leal 2007]
Numerical Method
20
40
60
Collocation
80
100
Cubic spline interpolation for
140
geometry
160
120
180
Gaussian quadrature or
200
RIM2D (singular) integration
Generally dense and
asymmetric
50
100
150
200
Fast Multipole Method
Approximate the integral over a boundary element
Move series expansions up and down a tree
Translate
Combine
Distant interactions
represented by a series
Nearby interactions are
computed directly
Direct Method
Tree structure results in O(N log N ) iterative method
Fast Multipole Method
Approximate the integral over a boundary element
Move series expansions up and down a tree
Translate
Combine
Distant interactions
represented by a series
Nearby interactions are
computed directly
Fast Method
Tree structure results in O(N log N ) iterative method
ULVZ Simulation #1
Buoyancy Driven System
No Imposed Velocity
ULVZ Dynamics (5 Myrs)
Left: High patch viscosity, low buoyancy force
Right: Low patch viscosity, high buoyancy force
Patch Settling
−0.5
−1
Drainage Rate (cm/year)
−1.5
−2
−2.5
−3
−3.5
−4
R=−7.88e+00
R=−1.84e+01
R=−2.89e+01
R=−3.94e+01
−4.5
−5
−5.5
0
0.2
0.4
0.6
Viscosity Ratio ( )
p
µp
=
µm
0.8
1
R
(p)
=
(⇢m
⇢p )gx2c
uc µm
Patch Settling
−0.5
−1
Drainage Rate (cm/year)
−1.5
−2
−2.5
−3
−3.5
−4
=1.00e+00
=6.70e−01
=3.40e−01
=1.00e−02
−4.5
−5
−5.5
−40
−35
−30
−25
−20
−15
−10
Compositional Rayleigh Number (R)
p
µp
=
µm
R
−5
(p)
=
(⇢m
⇢p )gx2c
uc µm
Patch Settling
−0.5
−1
Drainage Rate (cm/year)
−1.5
−2
−2.5
−3
−3.5
−4
=1.00e+00
=6.70e−01
=3.40e−01
=1.00e−02
−4.5
−5
−5.5
−40
−35
−30
−25
−20
−15
−10
Compositional Rayleigh Number (R)
p
µp
=
µm
R
−5
(p)
=
(⇢m
⇢p )gx2c
uc µm
ULVZ Simulation #2
Upwell Driven Imposed Flow
Upwell Dynamics (10 Myrs)
Left: High patch viscosity
Right: Low patch viscosity
Summary
Numerically experimented via FMBEM the ULVZ patch
drainage in LLSVP for various densities and viscosities
Inverse linear relationship between viscosity and
drainage rate
Linear relationship between compositional Rayleigh
number and drainage rate
Interaction of patches leads to asymmetric spreading
Evidence of core deformation for ULVZ-like parameters
References
X. Gao, “Numerical evaluation of two-dimensional singular boundary integrals--Theory and
Fortran code”, Journal of Computational and Applied Mathematics, Volume 118, Pages 44-64,
2006
E.J. Garnero and A.K. McNamara, “Structure and Dynamics of Earth's Lower Mantle”, Science,
Volume 320, Issue 5876, Pages 626-628, 2008
S. Hier-Majumder and J. Revenaugh, “Relationship between the viscosity and topography of
the ultralow-velocity zone near the core–mantle boundary”, Earth and Planetary Science
Letters, Volume 299, Pages 382-386, 2010
D.M. Koch and D.L. Koch, “Numerical and theoretical solutions for a drop spreading below a
free fluid surface”, Journal of fluid mechanics, Volume 287, Issue 1, Pages 251-278, 1994
L.G. Leal, Advanced Transport Phenomena: Fluid Mechanics and Convective Transport
Processes, Cambridge Series in Chemical Engineering, 2007
T.M. Lassak et al., “Core-mantle boundary topography as a possible constraint on lower mantle
chemistry and dynamics”, Earth and Planetary Science Letters, Volue 289, Pagrs 232-241,
2010
Y.J. Liu and N. Nishimura, “The fast multipole boundary element method for potential problems:
a tutorial”, Engineering Analysis with Boundary Elements, Volume 30, Issue 2, Pages 371-381,
2006
References
Y.J. Liu, “A new fast multipole boundary element method for solving 2-D Stokes flow problems
based on a dual BIE formulation”, Engineering Analysis with Boundary Elements, Volume 32,
Issue 2, Pages 139-151, 2008
M. Manga and H.A. Stone, “Buoyancy-Driven Interactions Between 2 Deformable Viscous
Drops”, Journal of fluid mechanics, Volume 256, Pages 647-683, 1993
A.K. McNamara, E.J. Garnero, and S. Rost , “Tracking deep mantle reservoirs with ultra-low
velocity zones”, Earth and Planetary Science Letters, Volume 299, Pages 1-9, 2010
C. Pozrikidis, Boundary integral and singularity methods for linearized viscous flow, Cambridge
Texts in Applied Mathematics, 1992
C. Pozrikidis, “Interfacial Dynamics for Stokes Flow”, Journal of Computational Physics, Volume
169, Issue 2, Pages 250-301, 2001
S. Rost et al., “Seismological constraints on a possible plume root at the core–mantle
boundary”, Nature, Volume 435, Issue 7042, Pages 666-669, 2005
M.S. Warren and J.K. Salmon, “A Parallel Hashed Oct-Tree N-Body Algorithm”, Proceedings of
the 1993 ACM/IEEE conference on Supercomputing, AMC, Pages 12-21, 1993
DI22A-08
Dynamics of ULVZ-mantle interaction
using fast multipole boundary element
method NSF EAR 0911094 and EAR 1215800
Tyler Drombosky - drombosk@math.umd.edu
University of Maryland College Park
Applied Mathematics and Scientific Computation
Saswata Hier-Majumder - saswata@umd.edu
University of Maryland College Park
Department of Geology
Center for Computational Sciences and Mathematical Modeling
Extra Slides
FMM Expansions
Validity of Expansion
Expansions are valid based on the L2 distance
between boundary elements and Greens functions
2
O(N
) time and space to
Distance matrices cost
compute
x=0.01010101110
Approximate distances using
y=0.10111010100
L1 norm to determine areas of
01100111011001
validity
A Morton number scheme is used to greatly accelerate
the process [Warren 1993]
Extra Slides
CMB 4x4 Simulation
Settling Interaction
Two ULVZ patches are placed above a core
The viscosity ratio and density contrast of the
patches with respect to the mantle is varied
The viscosity ratio parameter ranges from 1e-2 to 1
The density of the patches are varied 2-8% more
dense than the mantle
Simulations are run for 5 million years
Compositional Rayleigh Number (R)
−3.942000e+01
−2.890800e+01
−1.839600e+01
−7.884000e+00
1.000000e−02 3.400000e−01 6.700000e−01 1.000000e+00
Viscosity Ratio ( )
Extra Slides
Imposed Flow Simulation
Rising Interaction
Two patches are placed in an upswell imposed velocity
similar to a plume.
The viscosity ratio and density contrast of the
patches with respect to the mantle is varied
The viscosity ratio parameter ranges from 1e-2 to 1
The density of the patches is varied between 102%
and 110% of mantle density
Simulations are run for 5 million years
Rising Analysis
For low compositional Rayleigh number, solution is
dominated by the imposed velocity field
For moderately size compositional Rayleigh numbers
the upswell is able to support some or all of the patch
With large compositional Rayleigh numbers the density
contrast of the patch overcomes the imposed velocity
For all compositional Rayleigh numbers low viscosity
ratios allows the patch to deform easily in the direction
of greatest velocity
Compositional Rayleigh Number (R)
−3.94e+01
−2.89e+01
−1.84e+01
−7.88e+00
1.00e−02
3.40e−01 6.70e−01
Viscosity Ratio ( )
1.00e+00
Patch Interaction
Velocity is able to uplift part
of the particles
In low velocity regions,
gravitational force
overcomes
As particles approach,
pressure builds and
velocities repel
No spreading due to lack of
lower boundary
Extra Slides
Evidence Of Core Deformation
[Lassak et al. 2010]
Numerical model of core mantle boundary shows
deformation in areas of detected ULVZs.