SUPPLEMENTARY INFORMATION

Transcription

1 SUPPLEMENTARY INFORMATION Linking mantle upwelling with the lithosphere decent and the Japan Sea evolution: a hypothesis Alik Ismail-Zadeh 1, 2, 3 *, Satoru Honda 4, Igor Tsepelev Institut für Angewandte Geowissenschaften, Karlsruher Institut für Technologie, Karlsruhe, GERMANY Institut de Physique du Globe de Paris, Paris, FRANCE Institute of Earthquake Prediction Theory and Mathematical Geophysics, Russian Academy of Sciences, Moscow, RUSSIA Earthquake Research Institute, University of Tokyo, Tokyo, JAPAN Institute of Mathematics and Mechanics, Russian Academy of Sciences, Yekaterinburg, RUSSIA * Correspondence to Alik.Ismail-Zadeh@kit.edu or aiz@ipgp.fr Here we describe the detail of the mathematical model including the governing equations, the rheological law, the background temperature, and the boundary conditions. Numerical methods and solvers are then presented. The performance of the sensitivity analysis is illustrated by several case studies related to (i) variations in mantle viscosity, (ii) model depth variation, and (iii) the presence of phase changes in the mantle. Also here we compare qualitatively three seismic tomography models.

2 MODEL DESCRIPTION Mathematical statement In the three-dimensional (3-D) rectangular domain [0, x1 l1] [0, x2 l2] [0, x3 h] and for time interval t [0, ] we solve the regularized Stokes, the incompressibility, and the backward heat balance equations using the quasi-reversibility method 42 and the extended Boussinesq approximation 10,1 : T P ( [ u ( u) ]) ( E ) RaT a La ( ) a La ( ) e, (1) u 0, (2) * 1 2 * 2 ( E ) T u T A B Di Ra u3 T A T Di ( eij ) (3) t i, j 1 with appropriate boundary and initial conditions (see below). Here * A 1 a a Di La T 0, w1 w2 La 2 2 B 1 a a Ra w w , 1 1 tanh i i, i zi x3 i( T Ti), i = 1, 2, 2 w i x = (x 1, x 2, x 3 ), u = (u 1, u 2, u 3 ), t, T, P, and are the dimensionless Cartesian coordinates, velocity, time, temperature, pressure, and viscosity, respectively; is the present time; e ( u ) { u / x u / x} is the strain rate tensor; e (0,0,1) is the unit vector; is the ij i j j i gradient operator; and E is the unit operator. With regard to the phase changes around 410 km and 0 km, respectively, 1 and 2 are the dimensionless excess pressures; and 1 and 2 are the phase functions describing the relative fraction of the heavier phase, respectively, and varying between 0 and 1 as a function of depth and temperature. The Rayleigh (Ra), Laplace (La), and 2

3 modified dissipation (Di * ) dimensionless numbers are defined as * * 3 * 1 Ra g T h ( ), La gh ( ), and * 3 * 1 Di ( ch T ) * * * 2 * 1, respectively. The operator ( ) 2 1 E is applied to the right-hand side of the Stokes equations (1) to smooth temperature jumps at the phase boundaries and to enhance the stability of our computations. The higher dissipation term, whose magnitude is controlled by the small parameter, is added in order to regularize the heat balance equation (3). Length, temperature, and time are normalized by h, respectively. The physical parameters used in this study are listed in Table S1. T, and h 2 1, We consider a temperature- and depth-dependent Newtonian rheology ( T( x ), x3) : E gxv RT, exp a * 3 a 0 where 0 is determined so that it will give Pa s at the depth of 290 km and temperature of 198 K. Other parameters of the rheological law are listed in Table S1. We set the upper limit of the viscosity at ~10 22 Pa s, which results in the viscosity increase from the upper to the lower mantle by about two orders of magnitude. Background temperature The initial conditions for temperature are determined from the seismic tomography model as described in the main text. The background temperature used for deriving the temperature from seismic velocity anomalies is presented in Fig. S1. Note that the solidus for dry rocks (black curve) is used in this derivation. 3

4 Fig. S1. Background temperature used in the model (the light blue curve for the temperature determined from the cooling half-space model 18, and the dark blue curve for the temperature from the geothermal adiabat 19 ). The liquidus (brown curve 2 ) and solidus for dry (black curve 18 ) and wet (red and orange curves 3 ) rocks are also presented. Boundary conditions At the upper surface of the model boundary we prescribe the velocity W (, tx1, x2, x 3 0) and temperature T = T u. The velocity W is constructed as described in the main text. Due to complexities and significant uncertainties in trench migration (oceanward versus trenchward) patterns observed on the wide Pacific and Philippine Sea subduction zone 4, we do not include the trench migration in the present model. Also we do not introduce a weak (artificial) zone at the plate interfaces as it is used in forward modelling to reduce stresses at the interfaces and to promote subduction. Such a weak zone is partly accommodated in backward modelling, because in the backward (time-reverse) modeling the hotter material from below moves upward toward the divergent plate interfaces reducing the viscosity there. At the lower surface of the model boundary we set the velocity u 0 (no-slip) and fixed temperature T = T l. To allow for the flow to pass through the lateral boundaries, we prescribe at the lateral sides of the boundary the following conditions: u T 0, 0. 4

5 P Also we prescribe 0 at the model boundary. We tested several other conditions at the lateral sides and at the bottom of the model domain (see Table S2) to study how the boundary conditions influence the restoration results. We found that the major features related to the present conclusions are reserved among the above different cases. NUMERICAL METHODS AND SOLVERS The governing equations with the prescribed boundary and initial conditions are solved numerically by the finite-volume method 10 using open source computational fluid dynamics software package OpenFoam ( We use finite volumes (rectangular hexahedrons), and hence a horizontal resolution of the model is 20 km 20 km. The model domain is divided into five horizontal layers: Layer 1 (from the surface to the depth of 400 km), layer 2 ( km), layer 3 ( km), layer 4 (50-70 km), and layer 5 (70 to 800 km). Within each layer 0, 40, 35, 40, and 15 grid points are used. Therefore, a vertical resolution of the model varies from 0.5 to 8.7 km. The accuracy of the numerical solutions has been verified by several tests including grid changes, volume preservation, and principle of the maximum 10,5. Several approaches have been developed to reconstruct the past thermal state and flow in the crust and mantle: backward advection method -9, sequential filtering method 70, variational/ adjoint method 71-73, and quasi-reversibility (QRV) method 43. Among these methods, the QRV method is less susceptible to a noise (small temperature perturbations) in restoration models 74. The QRV method for data assimilation introduces a new term in the heat balance equation (the first term in Eq. 3) to regularize the equation when solving it backward in time. The additional term describes heat flux relaxation 43,75. Velocity u and pressure P are found from the equations (1) and (2) using the SIMPLE method 10,7. The regularized heat balance equation (3) is approximated by the Euler method using the implicit approximation of the advective term and the explicit approximation of the conductive term: 5

6 n 1 n 2 T T n 1 n n ( E D) CT DT f ( u, T ) 0, dt where the discrete operators T C= C and T D=D approximate the advective and conductive terms, respectively. To solve the numerical scheme we use the splitting method 77 introducing the convection/anti-diffusion and regularization parts as n 1/2 n n ( E dtc) T ( E dtd) T dt f ( u, T ), (4) 2 n 1 n 1/2 ( E D ) T T. (5) The system of the discrete equations (4) is solved by the BiConjugate Gradient method 78 using the incomplete LU-factorization as a pre-conditioner 79. The system (5) is solved by the conjugate gradient method 10. NUMERICAL TESTS Table S3 lists the case studies performed to analyse the effects of (i) the variation in mantle viscosity (case studies ), (ii) the variation in the depth of the model domain (case studies 2.1 and 2.2.), and (iii) the phase changes (case studies ). In the following results shown in Figs. S2 S8, the figures present a 3-D (south to north) view of the temperature anomaly and viscosity in the region ( 4000 km km depth ). The region presents a central part of the model domain truncated equally from the north and from the south. A transparent mode of visualization with transparency coefficient 0.5 is used. Variations in mantle viscosity To analyse a sensitivity of the model to variations in mantle viscosity, we performed three case studies varying the dimensionless viscosity from 45 to 591 by changing the activation volume V a : m 3 mol -1 (case study 1.1, Fig. S2), m 3 mol -1 (case study 1.2, Fig. S3), and m 3 mol -1 (case study 1.3, Fig. S4). We found that small-scale upwellings in the model show persistency despite the realistic variations in viscosity of the mantle and the slab. Particularly, the upwelling beneath the northern Japan Sea linking the hotter material in the subslab mantle with the mantle wedge region is clearly recognized in Figs. S3 and S4 (see areas marked by green ovals in the figures).

7 Fig. S2. Evolution of the temperature anomaly and viscosity in case study 1.1 Fig. S3. Evolution of the temperature anomaly and viscosity in case study 1.2 7

8 Fig. S4. Evolution of the temperature anomaly and viscosity in case study 1.3 Change of the depth of the model domain To analyse a sensitivity of model results to the conditions imposed on the lower boundary of the model domain (no-slip conditions), we performed two case studies by increasing the depth of the model domain to 1000 km (case study 2.1) and 2000 km (case study 2.2). Figure S5 compares temperature anomalies and viscosities in case study 1.2 with those in the case studies 2.1 and 2.2. We see that the results are rather robust to the depth of the domain. And again there exists a link between the hotter rocks in sub-slab mantle with those in the mantle wedge (see areas marked by green ovals in Fig. S5). Influence of phase changes To analyse the effect of phase changes on the data assimilation, three case studies have been performed. Figures S-S8 present the results (temperature anomalies and viscosities) of the case studies. The upwellings connecting the sub-slab mantle with the mantle wedge are observed in the models despite changing conditions at the interfaces of 410 km and 0 km depths (see areas marked by green ovals in the figures). 8

9 Fig. S5. Evolution of the temperature anomaly and viscosity for different depths of the model domain 9

10 Fig. S. Case study 3.1: As case study 1.2, but with no phase change introduced Fig. S7. Case study 3.2: As case study 1.2, but only with the 410-km phase change interface 10

11 Fig. S8. Case study 3.3: As case study 1.2, but only with the 0-km phase change interface COMPARISON OF SEISMIC TOMOGRAPHY IMAGES Figure S9 illustrates three images of P-wave seismic tomography models. The resolution of the models is as follows: WEPP2 5 : 128 (latitude) 25 (longitude) 32 (depth), GAP_P2 15 : 288 (latitude) 57 (longitude) 29 (depth), and MIT 58 : 25 (latitude) 512 (longitude) 4 (depth). The image of the lithospheric slab and the sub-slab hot anomaly in the MIT model are less pronounced although its resolution power is comparable with the resolution power of the GAP_P2. If the MIT model is considered, the fading slab image in the northern part of the studied region (see cross section CS3 in Fig. S9c) may be much favourable for the potential link between the sub-slab mantle and the mantle wedge, than the tomography models used in the study. 11

12 Fig. S9. Three seismic tomography images: WEPP2 5, MIT 58, and GAP_P

13 Movie Linkage between the hot sub-slab mantle and mantle wedge The movie (Linkage_BandC.mp4) presents the evolution of two iso-surfaces of hot 4% anomaly (red) and cold 4% anomaly (blue). It illustrates the linkage between the hot anomaly in the subslab (region c in Fig. 3) and the hot anomaly in the mantle wedge (region B in Fig. 3) in 38.9 Ma. The movie s show starts at 38.9 Ma and finishes at present. The movie can be downloaded from References 1. Christensen, U. R. & Yuen, D. A. Layered convection induced by phase transitions. J. Geophys. Res. 90, (1985). 2. McKenzie, D. & Bickle, M. J. The volume and composition of melt generated by extension of the lithosphere, J. Petrology 29, (1988). 3. Kawamoto, T. Hydrous phase stability and partial melt chemistry in H 2 O-saturated KLB-1 peridotite up to the uppermost lower mantle conditions. Phys. Earth Planet. Int., , (2004). 4. Schellart, W. P., Stegman, D. R. & Freeman, J. Global trench migration velocities and slab migration induced upper mantle volume fluxes: Constraints to find an Earth reference frame based on minimizing viscous dissipation. Earth-Science Reviews 88, (2008). 5. Samarskii, A. A. Theory of Finite Difference Schemes (in Russian). Nauka, Moscow (1977).. Steinberger, B. & O Connell, R. J. Advection of plumes in mantle flow: implications for hotspot motion, mantle viscosity and plume distribution. Geophys. J. Int. 132, (1998). 7. Ismail-Zadeh, A. T., Talbot, C. J. & Volozh, Y. A. Dynamic restoration of profiles across diapiric salt structures: numerical approach and its applications. Tectonophysics 337, 21-3 (2001). 8. Conrad, C. P. & Gurnis, M. Seismic tomography, surface uplift, and the breakup of Gondwanaland: Integrating mantle convection backwards in time. Geochem. Geophys. Geosys. 4, 1031 (2003). 9. Ismail-Zadeh, A. T., Tsepelev, I. A., Talbot, C. J. & Volozh. Y. A. Three-dimensional forward and backward modelling of diapirism: Numerical approach and its applicability to the evolution of salt structures in the Pricaspian basin. Tectonophysics 387, (2004). 13

14 70. Bunge, H.-P., Richards, M. A. & Baumgardner, J. R. Mantle circulation models with sequential data-assimilation: Inferring present-day mantle structure from plate motion histories. Phil. Trans. Royal Soc. A 30, (2002). 71. Bunge, H.-P., Hagelberg, C. R. & Travis, B. J. Mantle circulation models with variational data assimilation: Inferring past mantle flow and structure from plate motion histories and seismic tomography. Geophys. J. Int. 152, (2003). 72. Ismail-Zadeh, A. T., Korotkii, A. I. & Tsepelev, I. A. Numerical approach to solving problems of slow viscous flow backwards in time. In: Computational Fluid and Solid Mechanics, Bathe, K. J. (ed.), Elsevier Science, Amsterdam, pp (2003). 73. Ismail-Zadeh, A., Schubert, G., Tsepelev, I. & Korotkii, A. Inverse problem of thermal convection: Numerical approach and application to mantle plume restoration. Phys. Earth Planet. Inter. 145, (2004). 74. Ismail-Zadeh, A., Schubert, G., Tsepelev, I. & Korotkii, A. Three-dimensional forward and backward numerical modeling of mantle plume evolution: Effects of thermal diffusion. J. Geophys. Res. 111, B0401 (200). 75. Yu, N., Imatani, S. & Inoue, T. Characteristics of temperature field due to pulsed heat input calculated by non-fourier heat conduction hypothesis. JSME Int. J., Series A 47, (2004). 7. Patankar, S. V. Numerical Heat Transfer and Fluid Flow. McGraw-Hill, New York (1980). 77. Samarskii, A. A. & Vabischevich, P.N. Computational Heat Transfer. Vol. 1. Mathematical Modeling. John Wiley & Sons, New York (1995). 78. Van der Vorst, H. A. Bi-CGSTAB: A fast and smoothly converging variant of Bi-CG for the solution of nonsymmetric linear systems. SIAM J. Sci. Stat. Comput. 13, (1992). 79. Saad, Y. Iterative Methods for Sparse Linear Systems, PWS, Boston (199). 14

15 TABLES Table S1. Model parameters Parameter Symbol Value Dimensionless density jump at the 410-km phase boundary a Dimensionless density jump at the 0-km phase boundary a Thermal conductivity c 1250 W m -1 K -1 Activation energy E a J mol -1 Acceleration due to gravity g m s Depth h 800 km Length (in x-direction) l km Length (in y-direction) l km Universal gas constant R J mol -1 K -1 Difference between the temperatures at the lower (T l ) T * 1594 K and upper (T u ) model boundaries Dimensionless temperature at the upper model boundary T u 290 / T * Dimensionless temperature at the lower model boundary T l 1884 / T * Dimensionless temperature at the 410-km phase boundary T / T * Dimensionless temperature at the 0-km phase boundary T / T * Activation volume V a 4 10 m 3 mol -1 Dimensionless width of the 410- km phase transition w 1 10 km / h Dimensionless width of the 0- km phase transition w 2 10 km / h Dimensionless depth of the 410- km phase boundary z km / h Dimensionless depth of the 0- km phase boundary z km / h Thermal expansivity K -1 QRV regularization parameter Dimensionless Clapeyron (pressure-temperature) slope at the 410-km phase boundary Dimensionless Clapeyron slope at the 0-km phase boundary 1 2 Pa K -1 * * T ( gh) 2 10 Pa K -1 * * 1 T ( gh) 15

16 Reference viscosity * Pa s Thermal diffusivity m s Reference density * 3400 kg m -3 Phase regularization parameter Table S2. Conditions (on velocity and pressure) at the lateral sides of the model boundary Lateral sides u P 0 0 u P 0 0 Lower surface u P 0 0 u 0 P 0 u P 0 P 0 u 0 0 P u u* 0 0 u 0 P 0 Table S3. A list of case studies No. case study Activation volume, V a, m 3 mol -1 Model depth, km Phase change introduced km and 0 km km and 0 km km and 0 km km and 0 km km and 0 km No phase changes km km 1