Eath Planets Space, 62, 665 673, 2010 Constuction of semi-dynamic model of subduction zone with given plate kinematics in 3D sphee M. Moishige 1, S. Honda 1, and P. J. Tackley 2 1 Eathquake Reseach Institute, the Univesity of Tokyo, 1-1-1 Yayoi, Bunkyo-ku, Tokyo 113-0032, Japan 2 Institute of Geophysics, NO H9.1 ETH Zuich CH-8092, Zuich, Switzeland (Received May 31, 2010; Revised Septembe 17, 2010; Accepted Septembe 28, 2010; Online published Decembe 13, 2010) We pesent a semi-dynamic subduction zone model in a thee-dimensional spheical shell. In this model, velocity is imposed on the top suface and in a small thee-dimensional egion aound the shallow plate bounday while below this egion, the slab is able to subduct unde its own weight. Suface plate velocities ae given by Eule s theoem of igid plate otation on a sphee. The velocity imposed in the egion aound the plate bounday is detemined so that mass consevation inside the egion is satisfied. A kinematic tench migation can be easily incopoated in this model. As an application of this model, mantle flow aound slab edges is consideed, and we find that the effect of Eath cuvatue is small by compaing ou model with a simila one in a ectangula box, at least fo the paametes used in this study. As a second application of the model, mantle flow aound a plate junction is studied, and we find the existence of mantle etun flow pependicula to the plate bounday. Since this model can natually incopoate the spheical geomety and plate movement on the sphee, it is useful fo studying a specific subduction zone whee the plate kinematics is well constained. Key wods: Semi-dynamic model, subduction zone, 3D sphee, mantle flow. 1. Intoduction Roughly speaking, thee ae two types of appoaches to modeling subduction zones. One is to impose the velocity and the geomety of the plate in advance, an appoach that has been used fo a long time (e.g., McKenzie, 1969; Knelle and van Keken, 2007, 2008). The othe is to calculate slab movement moe o less dynamically (e.g., Billen et al., 2003; Billen and Hith, 2005; Schellat et al., 2007; Moa et al., 2009). Both appoaches have advantages and disadvantages. Conceptually, fully dynamic models ae undoubtedly pefeable but this appoach is justified only when we know the heology of plate well enough. In addition, even if we know such a heology, it is often difficult to apply this model to the specific subduction zone, since we cannot contol the velocity and geomety of subduction zones. Howeve, it is a useful technique to study the geneal chaacteistics of subduction zones. On the othe hand, using the kinematic appoach, we can constuct models that incopoate the pesent velocity and geomety of specific subduction zones to compae the models with obsevations. The demeit is that this kind of model may miss some impotant dynamics of subduction, such as the change in plate boundaies and tench cuvatue with time (Schellat et al., 2007). Thus, we believe that these two types of appoaches ae complementay to each othe and that an undestanding of both type of appoaches will lead to futhe undestanding of subduction zones. Copyight c The Society of Geomagnetism and Eath, Planetay and Space Sciences (SGEPSS); The Seismological Society of Japan; The Volcanological Society of Japan; The Geodetic Society of Japan; The Japanese Society fo Planetay Sciences; TERRAPUB. doi:10.5047/eps.2010.09.002 The philosophy of imposing velocity on some pat is not as novel as mentioned above. But, in eality, it is not so easy to constuct models based on it because it is difficult to find an appopiate velocity distibution to impose. Because of simpleness, this type of model has been constucted in the two-dimensional (2D) case since the ealy days of plate tectonics, but 3D modeling is a ecent development (e.g., Knelle and van Keken, 2007, 2008; Honda, 2009). Knelle and van Keken (2007, 2008) imposed the velocity of the whole subducting plate in a ectangula box to study the 3D flow of the cuved subduction zone. Honda (2008, 2009) constucted a model somewhat between kinematic and dynamic models in a ectangula box. The velocity on the suface and the small egion suounding a shallow plate bounday ae given a pioi. The velocity imposed in the plate bounday egion is detemined so that mass consevation inside the egion is satisfied unde the assumption that the elative velocity of oceanic subegion with espect to continental subegion is paallel to the plate bounday. Deepe flow is affected by the slab density. The model can include the tench migation kinematically. In this pape, we show that we can extend the same concept of Honda (2008, 2009) to moe ealistic geomety (i.e., cylindical and spheical geomety) and show some examples of the applications fo the case of spheical geomety. 2. Geneal Model Desciptions 2.1 Basic equations The basic equations used in this study ae common in the field of mantle convection studies. Unde the assumptions of an incompessible viscous fluid, negligible inetia and the Boussinesq appoximation, the basic equations to be 665
666 M. MORISHIGE et al.: CONSTRUCTION OF SEMI-DYNAMIC SUBDUCTION ZONE MODEL Table 1. The symbol used in this study and its meaning and value. Symbol Meaning Value v v i τ(τ ij ) Velocity i-component of velocity Deviatic stess tenso p Pessue ρ Density g Gavitational acceleation 10 m/s 2 e i Unit vecto in i-diection η Viscosity ρ 0 Refeence density 4000 kg/m 3 α The coefficient of themal expansion 3 10 5 /K T Tempeatue κ Themal diffusivity 10 6 m 2 /s ω A Angula velocity of plate A See text. ω B Angula velocity of plate B See text. V AS Speed of plate A at θ = 90 in spheical geomety See text. V BS Speed of plate B at θ = 90 in spheical geomety See text. V C Velocity component on the continental subegion See text. V O Velocity component on the oceanic subegion See text. 0 Minimum of the bounday egion See text. 1 Maximum of the bounday egion See text. θ 0 Minimum θ of the bounday egion See text. θ 1 Maximum θ of the bounday egion See text. φ 0 Minimum φ of the bounday egion See text. φ 1 Maximum φ of the bounday egion See text. Radial distance of a finite volume See text. θ Inclination of a finite volume See text. φ Azimuth of a finite volume See text. φ tench The position of tench See text. η 0 Refeence viscosity See text. E Activation enegy 210 kj/mol R Gas constant 8.314 J/K/mol solved ae given by (the meanings of symbols and physical popeties ae summaized in Table 1), v = 0 (Mass consevation) (1) τ p = ρge z (Equation of motion), (2) τ = η { ( v) + ( v) T } (Constitutive elation), (3) ρ = ρ 0 (1 αt ) (4) (Equation of state: Boussinesq appoximation), T + v T = κ 2 T (5) t (Enegy equation with constant themal conductivity and no intenal heating). These equations ae discetized using finite volumes and they ae solved by the pogam StagYY (Tackley, 2008) which is modified to include plate-like featues as descibed below. Othe details ae descibed in the section of examples. 2.2 Implementation of subduction-like featues In the following discussion, the noth of the spheical coodinate system coesponds to the axis of otation and the convegence of each plate is pependicula to longitude, that is, the axes of both plate otations ae the same. In ode fo the slab to subduct at a given angle in the (adius)-φ (longitude) plane, we impose the velocity inside a small egion aound shallow plate bounday, which we heeafte call the bounday egion (Fig. 1). Figue 1 shows a θ (colatitude) slice of the bounday egion and its suoundings. The top suface is divided into two plates (plates A and B), and each plate velocity is constained by the igid otation given by V A = 1 ω A sin θe φ = V AS sin θe φ, (6) V B = 1 ω B sin θe φ = V BS sin θe φ whee 1 is the adius of the Eath, θ is the colatitude measued fom the noth pole, ω A and ω B ae the angula velocities of plates A and B, espectively. V AS (= 1 ω A ) and V BS (= 1 ω B ) ae the suface speeds at θ = 90 of the plates A and B, espectively. Equation (6) means that we can natually incopoate the vaiation of plate velocity along the tench by consideing spheical geomety. The velocity inside the bounday egion is detemined as follows. The bounday egion can be divided into two subegions: the continental subegion (uppe ight egion of Fig. 1) and the oceanic subegion (lowe left egion of Fig. 1). Defining V C and V O as the velocity inside each subegion, we assume that thei hoizontal components ae
M. MORISHIGE et al.: CONSTRUCTION OF SEMI-DYNAMIC SUBDUCTION ZONE MODEL 667 Fig. 1. Schematic view of the bounday egion aound the tench. The egion extends fom 0 to 1 (-diection), fom θ 0 to θ 1 (θ-diection), and fom φ 0 to φ 1 (φ-diection). The egion is divided into two subegions (oceanic subegion and continental subegion) and a diffeent velocity is imposed on each subegion (V O fo oceanic subegion and V C fo continental subegion). A finite volume whose size is (-diection) θ (θ-diection) φ (φ-diection) and whose cente is on the bounday ( PB,θ PB,φ PB ) is also shown. those of the igid body otation associated with each plate movement ω A sin θ (continental subegion) and ω B sin θ (oceanic subegion) whee is the adius. It may be a good fist ode appoximation to assume that the adial (vetical) component of V C is zeo, that is, the continental plate cannot subduct. Thus, V C and V O ae expessed as V C = ω A sin θe φ, (7) V O = V e + ω B sin θe φ whee V is -component of V O. In ode to detemine V = V (,θ,φ), mass consevation is consideed. Note that mass consevation in the continental subegion is aleady satisfied by the above equation. The plate bounday is placed so that it passes the vetex as PB = 0 1 φ 1 φ 0 (φ PB φ 1 ) + 0, (8) whee ( i,φ i )(i = 0, 1) is the adial and longitudinal coodinate that chaacteizes the size of the bounday egion, and the subscipt PB implies the plate bounday. Fist, we conside mass consevation in the finite volume along the plate bounday. The plate bounday is equied to intesect all the finite volumes along the plate bounday at thei vetices, which means that the numbe of finite volumes in the bounday egion is the same in the - and φ-diections. This means that the shape of the finite volumes in the bounday egion detemines the subduction angle imposed. Conside one of those finite volumes as shown in Fig. 1. Consideing that the velocity is unifom on each plane of the volume, mass consevation in this volume, whose size is (adius) θ (colatitude) φ (longitude), is given by ( PB ω A sin θ PB ) PB θ = ( PB ω B sin θ PB ) PB θ +V PB ( PB /2,θ PB,φ PB )( PB /2) 2 2 sin θ PB sin( θ/2) φ, (9) which gives V PB ( PB /2,θ PB,φ PB ) PB 2 θ = ( PB /2) 2 φ 2 sin( θ/2) (ω A ω B ). (10) Below this plate bounday shown by the shadowed aea in Fig. 1, we conside only the mass balance in the adial diection since the hoizontal component of the velocity does not change with φ. This is given by o V (,θ PB,φ PB ) 2 = V PB ( PB /2,θ PB,φ PB )( PB /2) 2 (11) ( ) PB /2 2 V (,θ PB,φ PB ) = V PB ( PB /2,θ PB,φ PB ). (12) This is because the aea pependicula to the adial diection is popotional to 2. Finally, fom Eqs. (8), (10) and (12), we have 0 1 (φ φ 1 ) + 0 φ V (,θ,φ)= 1 φ 0 θ φ 2 sin( θ/2) (ω A ω B ). (13) Theefoe, the velocity imposed in the bounday egion 2
668 M. MORISHIGE et al.: CONSTRUCTION OF SEMI-DYNAMIC SUBDUCTION ZONE MODEL is given by V C = ω A sin θe φ 0 1 (φ φ 1 ) + 0 φ V O = 1 φ 0 θ φ 2 sin( θ/2) (ω A ω B )e +ω B sin θe φ 2. (14) In the limit of infinitesimally small, θ and φ, this educes to V C = ω A sin θe φ 0 1 (φ φ 1 ) + 0 φ V O = 1 φ 0 1 0 φ 1 φ 0 (ω A ω B )e + ω B sin θe φ 2. (15) Note that Eq. (15) also means that, at the plate bounday, the elative velocity of the oceanic subegion viewed fom the continental plate is paallel to the diection of the plate bounday. Based on the same concept, we can also obtain the velocity in the bounday egion in cylindical geomety as V C = ω A e φ V O = 0 1 φ 1 φ 0 (φ φ 1 ) + 0 1 0 φ 1 φ 0 (ω A ω B )e + ω B e φ, (16) whee the velocity of plate A and B ae expessed as V A = 1 ω A e φ. (17) V B = 1 ω B e φ Thus, this method is consideed to be a natual extension of that used by Honda (2008, 2009) in a ectangula geomety. The position of the tench, defined by the suface expession of plate bounday and expessed as an abitay function of θ, φ tench (θ, t), moves in the φ-diection as φ tench (θ, t) = φ tench (θ, 0) + ω A t. (18) 3. Examples 3.1 Flow aound slab edges: convegent-tansfom fault bounday 3.1.1 Model As a fist example of application of the model, flow aound slab edges, that is, a convegenttansfom fault bounday as studied by Honda (2009) fo a ectangula box, is consideed. A schematic view of the model is shown in Fig. 2(a). The model is tuned to be simila to that of Honda (2009), and it coves the egion 5370 (km) 6370, 85.5 θ( ) 94.5, 13.5 φ( ) 13.5. Thus a depth extent of 1000 km and a hoizontal extent of about 3000 km (φ-diection) 1000 km (θ-diection) is consideed. Initially, the tench is located at φ = 0 and 90 θ( ) 94.5, and the tansfom fault is located at 13.5 φ( ) 0 and θ = 90. The size of the bounday egion is 100 km (-diection) 4.5 (θ-diection) 0.9 (φ-diection), and this means that the angle of subduction nea the suface is 45. The top bounday condition fo tempeatue is T = 0. The bottom bounday conditions fo velocity and tempeatue ae pemeable (v θ = v φ = 0 and τ = 0) and no vetical conductive heat flux ( T/ = 0), espectively. The bounday condition on the walls pependicula to the θ-diection is impemeable fee slip fo velocity and no hoizontal conductive heat flux fo tempeatue (v θ = τ θφ = τ θ = 0, T/ θ = 0). The bounday condition on the walls pependicula to the φ diection is v = v θ = p/ φ = 0 fo velocity and pessue, and no hoizontal conductive heat flux ( T/ φ = 0) fo tempeatue. These bounday conditions ae equivalent to those of Honda (2009) fo a ectangula box. The Newtonian viscosity η is given by whee ( η = η 0 h() exp E R(T + 273) 1 (5960 < (km) 6370) h() = 10 (5710 < (km) 5960) 100 (5370 (km) 5710) ), (19) (20) and η 0 is set such that the viscosity at T = 1300 C is 10 20 Pa s with h = 1 (Honda, 2009). The maximum and minimum viscosity ae set to 10 17 Pa s and 10 23 Pa s, espectively. The initial tempeatue is given by a half-space cooling model, that is, ( ) z T = T m ef 2, (21) κt age whee z is the depth and T m = 1300 C. t age is set to 25 My (unde the plate A) and 120 My (unde the plate B). The numbe of finite volumes used is 128 (-diection) 128 (θ-diection) 384 (φ-diection). Resolution tests have been pefomed fo a 2D model. It is desiable to check the effect of esolution in thee dimensions. Howeve, since a 3D calculation with highe esolution equies an impactical amount of compute time, we made tests fo a 2D model. Since the geomety of ou model is complex, we used a simple successive oveelaxation method to solve the equations of motion. Figue 3 shows the effect of esolution on the esults. The numbe of finite volumes used in the calculation shown in Fig. 3(b) is doubled in both the - and φ-diections (256 768) compaed to that shown in Fig. 3(a) (128 384). Results show that although mino diffeences can be seen, the oveall behavio is simila between these cases. Thus, this suppots the adequacy of the esolution used in this study.
M. MORISHIGE et al.: CONSTRUCTION OF SEMI-DYNAMIC SUBDUCTION ZONE MODEL 669 Fig. 2. (a) Schematic view of the model of a convegent-tansfom fault bounday. The dimension of the egion is 1000 km (-diection) 9 (θ-diection) 27 (φ-diection). The velocities V A and V B ae imposed on the plates A and B, espectively. A detailed view of the bounday egion whee velocity is imposed is shown in Fig. 1. The dimension of the bounday egion is 100 km (-diection) 4.5 (θ-diection) 0.9 (φ-diection). (b) Schematic view of the model aound a plate junction. The dimension of the egion is 1000 km (-diection) 10.8 (θ-diection) 18 (φ-diection). Fig. 3. The effect of the numbe of finite volumes on esults fo a 2D case. Tempeatue and velocity field ae shown. The numbe of finite volumes used is (a) 128 (-diection) 384 (φ-diection) and (b) 256 (-diection) 768 (φ-diection). The time elapsed fom the stat of calculations is shown on the bottom-left cone of each figue. 3.1.2 Results Figue 4 shows the esults of thee cases with a convegent-tansfom fault bounday fo spheical shell geomety. (V AS, V BS ) used in the calculations in Fig. 4(a, b, c) ae ( 7.5 cm/y, 0 cm/y), i.e., eteating tench, (0 cm/y, 7.5 cm/y), i.e., fixed tench, and (7.5 cm/y, 15.0 cm/y), i.e., advancing tench, espectively. We can see fom this figue that the subduction angle becomes gentle as the tench eteats, and it becomes steepe as the tench advances. Figue 5 shows coss-sectional views at depths of 200 and 400 km fo the cases of eteating tench (Fig. 5(a)), fixed tench (Fig. 5(b)), and advancing tench (Fig. 5(c)). Only the egions aound the subducting slab ae shown. We can see that the thickness of slab in the coss section diffes in these thee cases because of the diffeence in subduction angle, as seen in Fig. 4. The slab edge flow shown in Fig. 5 is inteesting in tems of the existence/non-existence of along-ac flow in the subslab mantle (e.g., Long and Silve, 2008). Honda (2009) fist epoted the esults of this type of model and showed that significant along-ac flow in the sub-slab mantle does not exist unless the tench eteat is lage. Compaing the esults shown in Fig. 5(a, c) with the coesponding case in a ectangula box (see figues 2 and 3 in Honda (2009)), we can see that the hoizontal flow aound slab edges in spheical geomety is simila to that in a ectangula box,
670 M. MORISHIGE et al.: CONSTRUCTION OF SEMI-DYNAMIC SUBDUCTION ZONE MODEL (a) 3.84My 7.39My (b) Reteating tench 3.73My 7.62My (c) Fixed tench 3.12My 5.79My Advancing tench Fig. 4. Tempeatue stuctue aound slab edges. (a) The case with a eteating tench (V AS = 7.5 cm/y, V BS = 0 cm/y). (b) The case with a fixed tench (V AS = 0 cm/y, V BS = 7.5 cm/y). (c) The case with an advancing tench (V AS = 7.5 cm/y, V BS = 15.0 cm/y). The cyan plane shows an isothemal suface of 1000 C. Yellow lines shows the isothems of 400, 800, and 1200 C on the plane with θ = 94.5. Time shown in the bottom-left cone of each figue is the time elapsed fom the stat of each calculation. which means the effect of Eath cuvatue on hoizontal flow is small, at least fo the paametes consideed in this study. 3.2 Flow aound a plate junction 3.2.1 Model As a second example of the application of the model, we conside the flow aound a plate junction with a fixed tench in a spheical shell. A schematic view of the model is shown in Fig. 2(b). Only the model egion and tench position ae diffeent fom the pevious example, and othe physical popeties, such as viscosity, bounday, and initial conditions, ae the same. The dimension of the model egion is 5370 (km) 6370, 84.6 θ( ) 95.4, 6.3 φ( ) 11.7, and the position of tench in φ- diection φ tench (θ) is given by φ tench (θ) 0 at 84.6 θ( ) 89.1 =, (22) (89.1 θ) /1.2 at 89.1 θ( ) 95.4 which means that the angle between the stike of the tench and the diection of plate velocity at 89.1 θ( ) 95.4 is nealy 50. (Note that it is not necessay fo the position of the tench to be a linea function of θ; it could be abitay.) The size of bounday egion is 100 km (-diection) 0.9 (φ-diection) fo each θ. The numbe of finite volumes
M. MORISHIGE et al.: CONSTRUCTION OF SEMI-DYNAMIC SUBDUCTION ZONE MODEL 671 Fig. 5. Hoizontal view at depths of 200 and 400 km of flow aound the slab edges fo the cases of (a) eteating tench, (b) fixed tench and (c) advancing tench. Tempeatue and velocity fields ae shown in the aea nea the subducting slab. An equidistant cylindical pojection is used. Note that the scale of the velocity vectos is diffeent in each figue. The time elapsed fom the stat of calculations is (a) 7.39 My, (b) 7.62 My, and (c) 5.79 My, espectively.
672 M. MORISHIGE et al.: CONSTRUCTION OF SEMI-DYNAMIC SUBDUCTION ZONE MODEL Fig. 6. Flow and tempeatue aound a plate junction. (a) 3D image of the tempeatue field. The cyan plane shows an isothemal suface of 1000 C. Yellow lines shows isothems of 400, 800, and 1200 C on the plane with θ = 95.4. (b) Hoizontal view at a depth of 100 km of (a). An equidistant cylindical pojection is used. Tempeatue and velocity fields ae shown. Flow pependicula to the plate bounday exists in the aea enclosed by the white dotted line. The time elapsed fom the stat of calculation is 7.02 My. used is 128 (-diection) 128 (θ-diection) 256 (φdiection). 3.2.2 Results Figue 6 shows that the esulting tempeatue and velocity stuctue. (V AS, V BS ) used in this study is (0 cm/y, 7.5 cm/y). Figue 6(a) shows the 3D tempeatue field and Fig. 6(b) shows a hoizontal view at a depth of 100 km. Inteestingly, we see that thee is a flow pependicula to the plate bounday in the aea enclosed by white dashed line, and the magnitude of this velocity component is much lage than that of adial velocity component. Simila flow is also epoted by Knelle and van Keken (2008) in 3D ectangula geomety. Nakajima et al. (2006) examined the fast diections of shea-wave splitting nea the plate junction of the southwesten pat of the Kuile ac and the notheasten Japan ac. Fom thei analysis, they suggested that mantle etun flow occus sub-paallel to the local maximum dip of the slab. The esult shown in Fig. 6 may suppot thei infeence. Howeve, futhe systematic study, such as the estimate of seismic anisotopy (e.g., Knelle and van Keken, 2007, 2008) is necessay. 4. Discussion and Conclusion In this pape, we have constucted a semi-dynamic subduction zone model in a 3D spheical shell and shown some applications. Ou model enables the slab to subduct by imposing velocities in a small bounday egion so that the numbe of finite volumes needed is smalle than that equied fo a complex slab heology to achieve subductionlike featues, such as the naow low viscosity shea zone (Billen and Hith, 2005). Additionally, the model can easily incopoate obsevations such as the geomety of the shallow pat of the slab and plate bounday. The model can also include the ove-iding plate, which is not consideed
M. MORISHIGE et al.: CONSTRUCTION OF SEMI-DYNAMIC SUBDUCTION ZONE MODEL 673 in the fee subduction model (e.g., Schellat et al., 2007; Moa et al., 2009) but is sometimes impotant (Yamato et al., 2009). Howeve, the model also has a numbe of limitations, such as an inability to handle moe than two plates. In this study, we assume that the pole of both plate motions coincides, although this is not tue in geneal. In theoy, such a case could be handled by splitting the plate motion into the motion elative to the efeence plate and the absolute motion of the efeence plate. In pactice, howeve, this will make the calculations difficult because of the complex bounday conditions. Despite these difficulties, ou model is useful fo undestanding the chaacte of subduction zones and can be applied fo a paticula pai of plates, such as the Pacific plate and slowly moving Euasia plate. It is obvious that ou model is bette on the point that this uses a 3D sphee, which is the Eath s geomety, and the effects of Eath cuvatue can be coectly taken into account. In ou examples, we do not see a significant diffeence between the esults in the 3D ectangula box and spheical shell geomety. If, howeve, we modeled a boade aea o deepe pocesses, such as the defomation of subducted slabs that include the stagnation and the tea in the tansition zone and the buckling at the CMB (e.g., Fukao et al., 2001; Loubet et al., 2009; Obayashi et al., 2009), the diffeence is expected to become moe significant, and ou model might give a bette undestanding of such phenomena. Acknowledgments. A pat of this wok was done, while P. J. T. was a visiting eseache of Eathquake Reseach Institute, the Univesity of Tokyo. This wok was suppoted by Gant-in-Aid fo JSPS Fellows (21-8038) and fo Scientific Reseach (19104011). The Geneic Mapping Tools (Wessel and Smith, 1998) wee used to daw figues in this study. Fo this study, we have used the compute systems of the Eathquake Infomation Cente of the Eathquake Reseach Institute, the Univesity of Tokyo. We thank Masanoi Kameyama fo his useful comments. Refeences Billen, M. I. and G. Hith, Newtonian vesus non-newtonian uppe mantle viscosity: Implications fo subduction initiation, Geophys. Res. Lett., 32, L19304, doi:10.1029/2005gl023457, 2005. Billen, M. I., M. Gunis, and M. Simons, Multiscale dynamics of the Tonga-Kemadec subduction zone, Geophys. J. Int., 153, 359 388, 2003. Fukao, Y., S. Widiyantoo, and M. Obayashi, Stagnant slab in the uppe and lowe mantle tansition egion, Rev. Geophys., 39, 291 323, 2001. Honda, S., A simple semi-dynamic model of the subduction zone: effects of a moving plate bounday on the small-scale convection unde the island ac, Geophys. J. Int., 173, 1095 1105, 2008. Honda, S., Numeical simulations of mantle flow aound slab edges, Eath Planet. Sci. Lett., 277, 112 122, 2009. Knelle, E. A. and P. E. van Keken, Tench-paallel flow and seismic anisotopy in the Maiana and Andean subduction systems, Natue, 450, 1222 1226, 2007. Knelle, E. A. and P. E. van Keken, Effect of thee-dimensional slab geomety on defomation in the mantle wedge: Implications fo shea wave anisotopy, Geochem. Geophys. Geosyst., 9, Q01003, doi:10. 1029/2007GC001677, 2008. Long, M. D. and P. G. Silve, The subduction zone flow field fom seismic anisotopy: A global view, Science, 319, 315 318, 2008. Loubet, N., N. M. Ribe, and Y. Gamblin, Defomation modes of subducted lithosphee at the coe-mantle bounday: An expeimental investigation, Geochem. Geophys. Geosyst., 10, Q10004, doi:10. 1029/2009GC002492, 2009. McKenzie, D. P., Speculations on the consequences and causes of plate motions, Geophys. J. R. Aston. Soc., 18, 1 32, 1969. Moa, G., P. Chatelain, P. J. Tackley, and P. Koumoutsakos, Eath cuvatue effects on subduction mophology: Modeling subduction in a spheical setting, Acta Geotech., 4, 95 105, 2009. Nakajima, J., J. Shimizu, S. Hoi, and A. Hasegawa, Shea-wave splitting beneath the southwesten Kuile ac and notheasten Japan ac: A new insight into mantle etun flow, Geophys. Res. Lett., 33, L05305, doi:10.1029/2005gl025053, 2006. Obayashi, M., J. Yoshimitsu, and Y. Fukao, Teaing of stagnant slab, Science, 324, 1173 1175, 2009. Schellat, W. P., J. Feeman, D. R. Stegman, L. Moesi, and D. May, Evolution and divesity of subduction zones contolled by slab width, Natue, 446, 308 311, 2007. Tackley, P. J., Modeling compessible mantle convection with lage viscosity contasts in a thee-dimensioal spheical shell using the yin-yang gid, Phys. Eath Planet. Inte., 171, 7 18, 2008. Wessel, P. and W. H. F. Smith, New impoved vesion of the Geneic Mapping Tools eleased, Eos Tans. AGU, 79, 579, 1998. Yamato, P., L. Husson, J. Baun, C. Loiselet, and C. Thieulot, Influence of suounding plates on 3D subduction dynamics, Geophys. Res. Lett., 36, L07303, doi:10.1029/2008gl036942, 2009. M. Moishige (e-mail: stellvia@ei.u-tokyo.ac.jp), S. Honda, and P. J. Tackley