View the Slides - Tyler Drombosky
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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 - [email protected] University of Maryland College Park Applied Mathematics and Scientific Computation Saswata Hier-Majumder - [email protected] 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 - [email protected] University of Maryland College Park Applied Mathematics and Scientific Computation Saswata Hier-Majumder - [email protected] 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.