Controls on trench topography from dynamic models of subducted slabs

Transcription

1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 99, NO. B8, PAGES 15,683-15,695, AUGUST 1, 1994 Controls on trenh topography from dynami models of subduted slabs Shijie Zhong Department of Geologial Sienes, University of Mihigan, Ann Arbor Mihael Gurnis Seismologial Laboratory, California Institute of Tehnology, Pasadena Abstrat. A finite element method with onstrained elements and Lagrange multipliers is used to study tetoni faults in a visous medium. A fault, representing the interfae between overriding and subduting plates, has been inorporated into a visous flow model of a subdution zone in whih both thermal buoyany and the buoyany assoiated with the phase hange from olivine to spinel are inluded. The fault auses stress to onentrate in its viinity, giving rise to a weak plate margin and a mobile plate if a power law rheology is used. Surfae dynami topography with either a Newtonian or a power law rheology and with typial subdution zone parameters is haraterized by a narrow and deep trenh and a broadly depressed bak ar basin. This suggests that oeani trenhes and bak ar basins over subdution zones are dynamially ompensated by visous flow. Our models show that trenh depth inreases with fault dip angle, slab dip angle, slab length, and age of oeani lithosphere just prior to subdution. The influene of fault dip angle and age of lithosphere on trenh depth is greater than the influene of slab dip angle and slab length. These relationships of trenh depth versus subdution zone parameters explain well the statistis of observed trenh depths. For those subdution zones with oeani lithosphere on both sides of the trenh, observed trenh depths have been orreted for fault and slab dip angles, based on the relationships from the dynami models. After orretion to a ommon set of parameters, trenh depth orrelates linearly with age of lithosphere prior to subdution with a slope whih is lose to what models having high visosities within the transition zone and lower mantle predit. Comparison between the trenh depths, orreted for fault and slab dip angles, and model trenh depths suggests that the resisting tangential stress on faults in subdution zones may range from 15 MPa to 3 MPa, depending on model details. Introdution Topography on the Earth's surfae originates from rustal thikness variations and stresses indued by super-rustal and sub-rustal loads. The topography ontrast between oeans and ontinents and the topographi variations in orogeni belts are generally due to rustal thikness variations. Topography due to rustal thikness variations is usually isostatially ompensated and not assoiated with geoid anomalies at long wavelengths. The response of the Earth's surfae to stress indued by buoyany (like subduted slabs) within the mantle results in dynami topography. Dynami topography onstrains the thermal and rheologial struture of the mantle and lithosphere. Sine topography is signifiantly affeted by rustal thikness variations, it is usually diffiult to distinguish dynami topography from isostatially ompensated topography. Long-wavelength dynami topography may be dedued from residual seafloor topography [Cazenave et al., 1989; Davies and Priba, 1993; Priba, 1991], anomalously depressed ontinental hypsometry [Gurnis, 1993], and from studies of the geoid and mantle tomography [Hager et al., 1985). Copyright 1994 by the Amerian Geophysial Union. Paper number 94JB /94/94JB-89$5. Outer rise-trenh-island ar-bak ar basin topography is a diagnosti feature of subdution zone dynamis. These topographi features may be related to subdution proesses in a systemati way. While island ar and bak ar basin topography are influened by volanism and bak ar spreading [Karig, 1971], trenh topography probably has a dynami origin. Outer rises are usually onsidered as a seondary topographi feature aompanying trenhes. Trenhes as dynami topography signals may offer us important insight into subdution dynamis. A number of haratelistis of onverging margins orrelate with trenh topography_ Grellet and Dubois [1982] reported that trenh depths are positively orrelated with subdution rates_ Hilde and Uyeda [ 1983] showed that trenh depths inrea$e with the age of lithosphere just prior to subdution. Hilde and Uyeda [1983] suggested that the positive orrelation between trenh depths and subdution rates, reported by Grellet and Dubois [1982}, may be due to the positive orrelation between age and subdution rates. In a statistial analysis of trenh depth for subdution zones, Jarrard [1986] showed that orrelation oeffiients of trenh depth with age of subduting lithosphere and subdution rate were.57 and.45, respetively. Jarrard [1986] argued that the orrelation with subdution rate is degraded, if the orrelation between subdution rate and age prior to subdution is exluded, as suggested by Hilde and Uyeda [1983]. Moreover, intermediate 15,683

2 15,684 ZHONG AND GURNIS: TRENCHES FROM SUBDUCTED SLAB MODELS subdution dip angles measured from the surfae to 1 km in depth are orrelated with trenh depths with a orrelation oeffiient of.52, and the overall orrelation oeffiient to age of subduting lithosphere and intermediate subduting dip angles is.77 [Jarrard, 1986]. However, dip angles of slabs at depths greater than 1 km and Benioff zone length are apparently not orrelated with trenh depths [Jarrard, 1986]. The generally deep bathymetry of bak ar basins may also be related to the dynamis of slabs. Slater [1972] and Slater et al. [1976] pointed out that bak ar basins in the western Paifi are anomalously deep for their age. Reent studies on the Philippine Sea bak ar basins indiate that basement depths of those bak ar basins are about.8 km deeper than those of major oean floor of the same age [Park et al., 199]. Residual topography, the topography orreted for rustal thikness variations, sediment loading, and lithospheri age, is anomalously deep in the bak ar basin regions of the western Paifi [Priba, 1991]. Anomalously deep bak ar basin topography may be related to slabs. Systemati studies on the relationship between the harateristis of slabs and residual bak ar basin topography have not yet been undertaken. Three kinds of models have been proposed to explain the origin of outer rise, trenh, island ar, and bak ar basin topography of subdution zones. First, models of a thin elasti plate with a bending moment ating on trenhes provide plausible explanations for outer rises and trenhes [e.g., Watts and Talwani, 1974]. De Bremaeker [1977] suggested that the outer rise and trenh topography ould also be explained with a thin visous plate loaded at trenhes. However, these models not only fail to address how the subdution proess gives rise to the fore whih bends plates but also annot explain the residual bak ar basin topography. Seond, trenh and island ar topography has been asribed to the subdution of a plate. MKenzie [1969] showed, with a simple visous orner flow, that subdution of a plate into the mantle indues trenh-like topography over subdution zones but that the deepest part was on the overriding plate. Using a similar onept but a visoelasti finite element method, Melosh and Raefsky [198] found that not only trenhes but also island ar topography an be explained by the pressure differenes generated within the slab. The later numerial models make a diret physial onnetion between the origin of trenhes and the subdution proess. Finally, muh of the trenh-island ar-bak ar basin topography may be diretly attributed to the negative buoyany of slabs. The negative buoyany not only represents the primary driving fore for mantle onvetion but also dynamially depresses the Earth's surfae in the viinity of subdution zones. To explain Aleutian subdution zone topography, Sleep [1975] formulated a visous flow model with a heterogeneous rheology and realisti density struture. Davies [1981] realized the importane of faults at plate boundaries, and he proposed a fratured elasti plate model in whih trenhes and bak ar basins were partially ompensated by negatively buoyant slabs. Although Davies' models an explain topography, free air, and geoid anomalies over subdution zones, the surfae topography was obtained by a simple fore projetion rather than the solution to the momentum equation. By introduing a fault representing the plane separating the overriding from the subduting plate, Zhong and Gumis [1992] showed, with a visous flow model, that a trenh with a depth of 3 km and a width of 1 km and a broad bak ar basin with a depth of about 1 km an be generated with a slab of realisti thermal buoyany and a resisting shear stress of - 5 MPa aross the fault. Model trenh depth is primarily determined by dip angle of a fault and age of a slab [Zhong and Gumis, 1992], qualitatively onsistent with the statistis of observed trenh depths. The magnitude of shear stress on large faults has long been a ontroversial subjet [Kanamori, 198]. With a fritional heating model, shear stress on the San Andreas fault is dedued from heat flow measurements to be smaller than 1 MPa [e.g., Lahenbruh and Sass, 1988); in subdution zones shear stress on faults ranges from 14 to 4 MPa, depending on model details [Tihelaar and Ruff, 1993]. With a simple mehanial model, Davies [198] found that shear stress on fault planes in subdution zones had to be over 1 MPa in order to balane internal buoyany. The magnitude of shear stress on faults in subdution zones may be onstrained by observed trenh depths: a larger shear stress on a fault implies a stronger oupling between overriding plate and subduting plate, and hene slab, and for a given buoyany fore, a larger shear stress would give rise to a shallower trenh. Faults are essential to understanding lithospheri deformation in subdution zones. Mehanial models of tetoni faults have been onsidered in relation to earthquakes [Jungels and Frazier, 1973; Melosh and Reafsky, 1981; Lyzenga et al., 1991] and graben formation [Melosh and Williams, 1989]. Sine previous studies have dealt with problems at relatively small temporal and spatial sales, the media surrounding faults are assumed to be visqelasti. For problems at large temporal and spatial sale, suh as subdution zone dynamis, a visous rheology is more appropriate. In Zhong and Gurnis' [1992] visous flow models of subdution zones, fault planes are treated as rigid boundaries; normal veloities on the faults vanish, and tangential veloities are disontinuous aross the faults. However, rigid faults may result in disontinuous normal stresses on faults, and this requires a physially ad ho assumption about the faults, that is, faults have to be able to support the disontinuous normal stresses without deformation. Faults may be more properly treated as deformable in the normal diretion (e.g., a vertial fault of this type has been given by Barr and Houseman [1992]). For the deformable faults, both normal veloity and normal stress on faults are ontinuous, while the tangential veloity may be disontinuous. Despite the differene between these two types of fault models, faults are represented as a disontinuity in the tangential veloity aross faults. In this paper, we will study deformable faults using finite element methods with onstrained elements and Lagrange multipliers. We will primarily fous on the flow field and surfae dynami topography indued by both a thermal buoyany assoiated with subduting lithosphere and a buoyany due to the phase hange from olivine to spinel at a depth of 41 km; the latter buoyany was not onsidered by Zhong and Gumis [1992] and may be important in determining the stress on faults [Davies, 198]. Our omputations show that for either a Newtonian or power law rheology, dynami topography is haraterized by a narrow and deep trenh and a broad bak ar basin, similar to those reported by Zhong and Gumis [1992]. Stresses onentrate in the viinity of a fault, and when a power law rheology is used, the high stresses give rise to a weak plate margin and a mobile plate. Our models

3 ZHONG AND GURNIS: TRENCHES FROM SUBDUCTED SLAB MODELS 15,685 indiate that trenh depth is primarily ontrolled by fault dip angle and age of lithosphere, onsistent with the observed trend of trenh depths [Jarrard, 1986]. On the basis of our models, we have orreted observed trenh depths for both fault dip angle and slab dip angle for subdution zones in whih the overriding plates are oeani. After orretion to a ommon set of parameters, the previously sattered trenh depth to age of lithosphere relation beomes nearly linear. We suggest, from the orreted trenh depth to age relation that the shear stress aross faults may range from 15 MPa to 3 MPa. Physial Models and Method We have developed a two-dimensional visous flow model of a subdution zone with a fault using finite elements. Tetoni faults are ompliated, and simplifiations are neessary based on our urrent understanding. For the deformable fault, both normal veloity and normal stress are ontinuous aross the fault interfae, but the tangential veloity may be disontinuous aross the fault plane. The flow is oupled on either side of the fault interfae by an imposed resisting shear stress. A more advaned possibility would be to assume an effetive oeffiient of frition aross the fault. Visous flow models have suessfully been used to predit slab dip angles [Hager and O'Connell, 1978] and the depth dependene of seismiity [Vassiliou and Hager, 1988]. Our visous flow model with a fratured lithosphere (Figure 1) inludes a negative thermal buoyany assoiated with subduting slabs and the negative buoyany due to the phase hange from olivine to spinel at a depth of 41 Ian. The fault plane is assumed to extend down to the base of a highvisosity lithosphere. Fault dip angles, 9/, are impliitly the subdution dip angles within the lithosphere and are usually muh smaller than slab dip angles, e,, representing the Benioff zone dip angles at depths greater than 1 Ian [Jarrard, 1986]. Sine trenh depth strongly orrelates with the 9/ but not with the 9s [Jarrard, 1986], we independently varied these two dip angles. In all models presented here, depths of the bottoms of lithosphere, the upper mantle, and the transition zone are assumed to be 1 km, 41 km, and 67 km, respetively; the thikness of slabs is 1 Jan; the height and length of the model box are 15 km and 45 km, respetively; free slip boundary onditions are applied on the top and bottom boundaries, and refleting boundary onditions are used for the two vertial boundaries. We have found that the size of model box is large enough so that the bottom and two vertial boundaries do not greatly affet the model results near the onverging margin. There are two kind of buoyany fores within subduted lithosphere: thermal and that due to defletion of phase boundaries. Subduted lithosphere may experiene three major phase hanges: basalt to elogite at a depth of 6 to 8 km, olivine to spinel at a depth of 41 km, and spinel to post spinel at a depth of 67 km [Ringwood, 1975]. Although phase hange from basalt to elogite is loalized within a layer of about 6-7 km thik, the negative buoyany indued by this phase hange may be omparable to loal thermal buoyany, due to the large density hange of this phase hange. However, this negative buoyany should be offset by the positive buoyany of the basalt-depleted mantle underlying the oeani rust [O.xburgh and Parmentier, 1977]. Therefore the buoyany aused by this phase hange is not node al node ar a) r fault b) Figure 1. A visous flow model of subdution zones with a fault. Coeffiient of thermal expansion a, gravitational aeleration g, density of the mantle p, and referene temperature T, are 2 x w- 5 K- 1, 1m s x 1 3 kg m 3 and 15 K, respetively. inluded in this study. The phase hange al 67 km is not inluded beause slabs in most of the ases studied here are not long enough to enounter the 67-km phase boundary and beause, as we will show, this phase hange boundary is too deep to greatly affet the surfae dynami topography. Therefore only thermal buoyany and buoyany due to phase hange from olivine to spinel at 41 km are taken into aount. In our models, the thermal buoyany within the slab is assumed to be uniform and is equated to the buoyany within the oeani plate just prior to subdution [e.g., Hager and O'Connell, 1981]. While the thermal buoyany is relatively well onstrained, the phase hange buoyany is not well known. If the thermal struture of the slab is known, then the negative buoyany assoiated with the phase hange from olivine to spinel may be estimated. Shubert et al. [1975] found that the olivine and spinel phase boundary was displaed upward by a maximum 115 Ian within a subduting slab; this is equivalent to a total buoyany of about 1.6 x 1 13 N m 1 per unit length of trenh [Davies, 198]. Shubert et al. [1975] impliitly assumed that phase hange reations instantaneously go to ompletion; the negative buoyany would be redued if phase hange kinetis are taken into aount [Sung and Bums; 1976]. Sine we do not solve for the thermal struture, the phase hange buoyany is simplified to a 6% density inrease (2 kg m 3, Turotte and Shubert [1982]) over a length Lpfor a 1-km-thik slab at 41 Jan depth. Two different phase hange buoyany models are tested. (1) The length of the segment over whih the phase hange ours is assumed to be linearly proportional to the age of a slab. The older a slab is, the older it is, and the larger the length of the segment is. For a 1 m.y. old slab, the length of segment is assumed to be 6 Ian, giving a negative buoyany lose to that of Davies [198]. (2) Regardless of age

4 15,686 ZHONG AND GURNIS: TRENCHES FROM SUI3DUCTED SLAB MODELS of subduted lithosphere, the length of segment Lp is set to 6 km. Within the Earth's mantle the inertial fores are negligible, and the momentum and ontinuity equations, using the Boussinesq approximation, beome and (1) U;,; =, (2) where u;, /;. and O;j are the flow veloity, the body fore, and the stress tensor, respetively. Throughout the paper, repeated indies denote summation. The body fore and the stress tensor may be expressed in terms of a state equation and a onstitutive equation as and where Tis the temperature, Po and T are the referene density and temperature, respetively, g is the gravitational aeleration, : is the oeffiient of thermal expansion, and oij is kroneker delta, p is the pressure, and fl is the dynami visosity. The equations are solved with a finite element method. In order to enfore inompressibility, a penalty formulation in whih -'Au;,; is used to approximate the pressure P and seletive and redued integration are used [Hughes, 1987; King et al., 199}. Speial finite element tehniques are needed in order to solve the visous flow with a fault plane. Those nodes (Figure 1) having the same oordinates but on different sides of the fault are assigned different degrees of freedom (dof), as done in interfae problems in engineering [Cook, 1981]. With suh dofs the flow an be deoupled or weakly oupled on either side of the fault, depending on the onstraints and presribed shear stress. Constraints on the fault may be realized with onstrained element tehniques for those elements adjaent to the fault. For a system like ours with few onstraints, a Lagrange multiplier method is very effetive [Cook, 1981]. The onstraints for realizing the faults an be written for deformable faults as and (3) (4) ar a1 O Un -un =, (5) where u' and u! 1 are normal veloities for nodes ar and al on the fault whih have the same oordinates but are on different sides of the fault (i.e., nodes ar and al are on the right and left sides of the fault, respetively; e.g., Figure 1a); On is a normal stress on the fault; element indies er and el are for those elements that have one side on the fault plane (e.g., Figure la). The normal stress an be expressed as on = sin 2 (9 1 )o xx +os 2 (9 1 )Oyy +sin(29 1 )o xy. (7) The onstraints (5) and (6) an be rewritten, respetively, as u;' sin(b 1 )+ u;' os(b 1 )-u; 1 sin(b 1 )-uf os(b 1 ) =, (8) (6) and n,,.!a a 2a a """!a a 2a a LUerUx+ferUy)- LUeJUx+feJ Uy)=O, (9) a=! a=! where n,, is the number of nodes per element and equal to four for the quadrilateral elements used here; / 4 and /ia are defined for element e (e = er or el) as (1) where N a,x and N a,y are the derivatives of shape funtions of node a with respet to x andy oordinates, respetively. Equations (6) to (9) are essentially a projetion of elemental to nodal stresses. Suh a projetion is neessary, beause stress is defined at redued gaussian quadrature points [Hughes, 1987] rather than diretly on the nodes of elements in our finite element solution of inompressible visous flow. Any onstraint, like these made in equations (8) and (9), an be written in the form of a matrix equation: [ C]{D} = {Q}, (12) where { D} is a vetor ontaining veloity. To solve for the flow with a onstraint, equation ( 12), is equivalent to minimizing a potential [Cook, 1981] 11"'.!_(D} 1 [K](D} -{D} 1 {R} + {u}([ C](D}- (Q}), ( 13) 2 where the first two terms represent the regular potential for the fluid flow and the third term is from the onstraint, equation (12); entries in { o:} are alled Lagrange multipliers; [K] and (R} are the ordinary stiffness matrix and fore vetor, respetively. Besides body and boundary fores, {R} also inludes ontributions from the resisting shear stress on the faults, through whih fore flow is oupled between two sides of the faults. Minimizing 11 with respet to both {D} and {o:} yields a matrix equation [ ]{}={} (14) Solving equation (14) gives the flow inorporating the onstraint, equation (12). Equation (14) must be arefully solved sine it is not positive definite. For a system with a few onstrained elements, the Lagrange multiplier method is very effiient and easily implemented. Both Newtonian and non-newtonian rheologies have been used in this study. For Newtonian models, visosities of the lithosphere, the upper mantle, transition zone, and the lower mantle are assumed to be 1.5 x 1 22, 3 x ufw, 6. x 1 21, 3 x 1 22 Pa-s, respetively (Figure 2); visosity of slab within the upper mantle and the transition zone are 4.5 x 1 21 and 6. x Pa s, respetively. This radial visosity struture for the lithosphere and mantle is generally onsistent with that derived from long-wavelength geoid studies [Hager, 199]. Beause of the temperature dependent rheology of mantle materials, slabs may have a higher visosity than the ambient mantle. Non-Newtonian rheology, or strain rate dependent visosity, is essential to making the plate mobile by weakening plate margins [Christensen, 1983; King and Hager,

5 ZHONG AND GURNIS: TRENCHES FROM SUBDUCTED SLAB MODELS 15, ]. Following King (1991], for non-newtonian flow, the strain rate is related to stress as (15) The average effetive visosity of the lithosphere within trenhes is about 1 21 Pa s in our non-newtonian models; therefore the assumption that the model trenh and bak ar topography reahes equilibrium is valid. where E. and a are the seond invariant of the strairi rate and Results and Disussion stress tensors, respetively; a is a referene stress; n is the exponent and is equal to 3 in this study; A represents the Our omputations for a wide range of model parameters, temperature, pressure, and ompositional dependene of inluding Newtonian and power law rheologies, onsistently visosity. The effetive visosity is defined as show that when a deformable fault is introdued into a visous flow model of subdution zones, dynami topography over a (16) slab is haraterized by a narrow and deep trenh and a broad The non-newtonian rheology auses the equation of motion,. equation (1), to be nonlinear; the nonlinear equation is solved iteratively [King, 1991]. The. lithosphere, the upper mantle, the transition zone, and the lower mantle have different A (equation (15)); A is hosen suh that the final average effetive visosity for a region is lose to that used in Newtonian models. The surfae has zero normal veloity as a boundary ondition, but normal stress indued by flow an be nonzero. Dynami topography is determined from surfae normal stress by assuming that the surfae will deform with the normal stress suh that no net tration ats on the surfae; dynami topography is (J h = ll, pg (17) where t.p is the density ontrast between lithosphere and the media (e.g., water) overriding lithosphere. A onsistent boundary flux method [Zhong et aj., 1993] is used to ompute normal stresses on the surfae after obtaining flow velocity from equation (14). Beause the normal stress is disontinuous at the intersetion of the fault with the surfae, the normal stress is assigned by the average value of the two adjaent surfae nodes. In general, the normal veloity on the model fault is nonzero, and this implies that the fault may migrate horizontally and hange its dip angle, depending on the distribution of buoyany and geometry of the fault plane and slab. The nonstationary fault is onsistent with observations of trenh migration whih presumably means that faults migrate, In this study, we do not study the time dependene of subduted slabs and faults, whih requires a oupled solution to the energy equation. Here we attempt to investigate the flow and topographi struture aused by a given buoyany artd fault geometry. We assume that the buoyany struture and fault geometry have been maintained for suh a long time that the alulated dynami topography is established. It is the buoyany fores that drive both inantle flow and motion of the fault, and fault geometry annot hange signifiantly unless the buoyany distribution hanges signifiantly. This suggests that the timesale for motion of the fault annot be larger than the timesale for the redistribution of the buoyany. The harateristi time for the redistribution of the buoyany assoiated with slabs may be 1 m.y. or larger, and the.1 to.1 m.y. is the time it takes dynami topography to reah equilibrium at a wavelength of about 1 to 1 3 km given a visosity of 121 Pa s [Rihards and Hager, 1984]. bak ar basin. A fault auses stress to onentrate in its viinity, whih, when a power law rheology is used, gives rise to a weak plate margin and the plate beomes mobile. Trenh depth varies with fault dip angles a! slab dip angles es, slab age 't 5, slab length L 5, and the resisting shear stress on the fault. However, for a given resisting shear stress, fault dip angle and slab age are the two most important parameters influening trenh depth. The observed trenh depths. after being orreted for slab dip and fault dip based on our models, show a lear linear orrelation to slab age and with a slope whih is lose to what our models predit. From the orreted trenh depth relationship to slab age, we suggest that shear stress on onvergent boundaries ranges from 15 MPa to 3 MPa. In what follows we will first show general harateristis of dynaini topography and flows from our fault models with different rheologies; seond, hek our model resolution and disuss the possible effets of lithosphere flexure and volani ars on dynami topography; third, systematially determine the relation between trenh depth and fault dip, slab dip, age and length of slab, and resisting shear stress on faults; and, finally, interpret observed trenh depths from the perspetive of model results. Dynami Topography and Flow: Effets of Faults Two ases with a deformable fault are omputed for Newtonian and non-newtonian rheology, respetively; the other parameters are zero resisting shear stress, a fault dip ef of 3, a slab dip es. of 6, a length of 7 km, and an age of 8 m.y. For the non-newtonian ase, the preexponents of the power law rheology (i.e., A in equation (15)) for the lithosphere, the upper mantle, the transition zone, the lower mantle, and the slab are hosen suh that the horizontally averaged effetive visosity profile is similar to that used in Newtonian models (Figure 2). For both ases, dynami topography is haraterized by a narrow and deep trenh-like and a broad depression or bak ar basin-like topography (Figure 3a). For the non-newtonian ase, the trenh is about 1 km wide and 5. km deep; the maximum depression of 3.6 km behind the trenh is about 15 km from the trenh; at a distane of about 3 km behind the trenh, the magnitude of depression dereases rapidly to about.6 km, and from there the depression of the bak ar region gradually dereases to zero over about 17 km (Figure 3a). We also observe an outer r1se topography of about.3 km for the non-newtonian ase (Figure 3a). For the Newtonian ase, the outer rise topography beomes less evident, and the trenh depth and the maximum depression behind the trenh tend to be larger than those from the non-newtonian models (Figure 3a).

6 15,688 ZHONG AND GURNIS: TRENCHES FROM SUBDUCTED SLAB MODELS ]' 5 "-" "I 'I } - "' " ' \.,) {\ -.. \ ---. Newtonian '. \.. \ Non-Newtonian with fault. ' \ - -- Non-Newtonian with no fault.., Visosity (Pa s). ' 'I Figure 2. Visosity profiles for Newtonian and non Newtonian deformable fault models and a non-newtonian model with no fault. Fault Plane a 2 >... l)i) Ci 18 s 4 E z l)i) -2 E-< g > "' l)i) u >. Ci Newtonian Non-Newtonian With no fault Distane (km) a) b) ---- Without a rust With a rust Distane (km) ) Without a elasti lid ---- With a elasti lid -- With a elasti lid & a rust Distane (km) Figure 3. Dynami topography profiles (a) for fault models with Newtonian rheology and non-newtonian rheology, and a non-newtonian model with no fault, (b) for models with and without a rust, and () for models with and without a thin elasti plate. Figure 4. Streamlines with the same ontour values for (a) Newtonian and (b) non-newtonian deformable fault models and () non-newtonian model with no fault. The maximum and minimum values of stream fu ntion for Figures 4a, 4b, and 4 are 6. X 1 5 m2 s l and -5.6 X 1 5 m2 s l, 5.3 X 1Q 4 m2 s l and -5.2 X 1 4 m 2 s l, and 5.3 X 1 4 m 2 s l and -6.4 X 1-4 m2 s 1, respetively. The dynami topography from the ases with deformable faults (Figure 3a) is similar to that from a rigid fau lt model [Zhong and Gurnis, 1992], although the magnitude of the maximum depression behind the trenh from the deformable faults is substantially larger. The rigid fa ult of Zhong and Gurnis [1992] may support muh of the topography that would be otherwise transmitted onto the surfae and ause a greater depression behind the trenh if normal deformation ori faults is allowed. Nevertheless, the similarities between dynami topography from these two different kinds of faull models suggest that the disontinuity in tangential veloity on faults, a ommon feature of these two faults, is essential to produing trenh topography. The flows differ signifiantly between the Newtonian and non-newtonian ases. For the Newtonian ase, the high visosity of the slab auses the flow to lok up (Figure 4a); the maximum veloities of lithosphere just prior to subdution and of the slab within the mantle are 1.9 m/yr and 3.8 m/yr, respetively. If the slab visosity is dereased to the values of its ambient mantle, then the maximum veloity of slab inreases to about 7.6 m/yr, while the lithosphere remains immobile with a maximum veloity of 1. em/yr. The lithosphere veloity prior to subdution anno t be signifiantly less than slab veloity; otherwise, the slab would detah from the lithosphere [Sleep, 1975]. When the

7 ZHONG AND GURNIS: TRENCHES FROM SUBDUC1ED SLAB MODELS 15,689 non-newtonian rheology is used, the subduting lithosphere has a muh larger veloity than the overriding lithosphere does, indiating asymmetri subdution; the subdution veloity is lose to the slab veloity (Figure 4b plotted with the same ontour values as those in Figure 4a). The maximum veloity of the lithosphere just prior to subdution is 13.6 m/yr; the slab veloity is about 22. m/yr. The mobile plate results from the weakening of plate margins due to the stress dependent visosity [e.g., Christensen, 1983). When a fault is present, stress tends to onentrate near the fault, espeially within the subduting plate, whih is evident from the surfae dynami topography (e.g., Figure 3a); the high stress near the fault weakens plate margins and gives rise to a mobile subduting plate (Figure 4b). For omparison, a ase, whih does not inlude any fault, but otherwise is idential to the non-newtonian fault ase, is presented (visosity profile in Figure 2, topography in Figure 3a, and flow field in Figure 4). It is evident that stresses near the fault within the subduting plate beome larger when a fault is present (Figure 3a for topography or surfae normal stress), whih weakens the subduting plate more effiiently. As a result, the lithosphere visosity beomes smaller ompared to the no fault ase (Figure 2), and subdution veloity and asymmetry of subdution are enhaned (Figures 4b and 4). Asymmetri subdution and a omparable veloity between subduting lithosphere and slab [Sleep, 1975] are two important harateristis of subdution zones. We have found that both harateristis an be ahieved in the models shown in Figures 4a, 4b, and 4 with a high-visosity slab and lithosphere, although the magnitudes of veloities for the Newtonian model are muh smaller. When the slab visosity is assumed to be the same as ambient mantle, veloity of the subduting lithosphere beomes omparable to veloity of overriding lithosphere but muh smaller than slab veloity. This suggests that both the asymmetri subdution and a omparable veloity between subduting lithosphere and slab are probably due to the oupling between the high-visosity slab and lithosphere. For either Newtonian or non-newtonian ases, when the resisting shear stress on the fault is zero, the tangential veloity is disontinuous aross the fault, and the tangential veloity on the subduting lithosphere side of the fault is muh larger than that on the ovniding lithosphere side. For the non-newtonian ase (Figure 4b), the average tangential veloities on the subduting lithosphere side and the overriding lithosphere side are 16.7 m/yr and 2.8 m/yr, respetively. The differene between these two tangential veloities beomes small when the resisting shear stress or the fritional fore on the fault is inreased. Another important differene between Newtonian and non Newtonian models is the nature of normal stress near faults. Sine model trenh topography (i.e., the normal stress) hanges greatly near faults (Figure 3a), detailed studies inluding resolution study on the topography near faults are neessary. For the ases presented in Figure 3a, a mesh refetted to as mesh 1 is refined near the fault with 1 km horizontal spaing between elements in the viinity of the fault. When the mesh is doubled near the fault (referred to as mesh 2), for both Newtonian and non-newtonian ases with faults (Figure 3a), the maximum hanges in flow veloity due to the hanges in meshes are found to be only about.1 %. However, dynami topographies from the Newtonian and the non-newtonian > <.) e:::::l,..._ ] 2.._ CD >. Q 21 s Mesh h. - - Mesh Distane (krn) Distane (km) --e--mesh h. - - Mesh Figure S. Resolution studies of dynami topography for (a) Newtonian and (b) non-newtonian rheology. ases show different responses to the hanges in meshes. For the Newtonian ase, while the doubled mesh (mesh 2 in Figure Sa) yields almost the same topography as mesh 1 does (mesh 1 in Figure 5a) at most nodes, the topography at the node on the fault and the node next to the fault node on the left side of the fault are deeper than those from mesh 1 (Figure 5a). The deepest trenh depth for the Newtonian models inreases as the mesh is refined; this is probably due to the singularity in normal stress indued by the faults. For the non-newtonian rheology, the singularity apparently vanishes, and both mesh 1 and mesh 2 produe almost idential dynami topography (Figure 5b). The stress-dependent rheology is ruial not only for produing mobile plates but also for yielding physially reasonable normal stresses. For this sake, we will use non Newtonian rheology in all the following models. The morphology of the model outer rise, trenh, and bak ar depression (3 km behind the trenh) from the non Newtonian ase (Figure 3a) are lose to those for observed outer rises, trenhes, and bak ar basins, assuming that island ar topography is due to isostati ompensation. The model outer rise, same as the trenh, is a purely visous phenomenon. The elastiity of lithosphere is not neessary to be invoked to explain the outer rise [De B remaeker, 1977]. The model bak ar depression whih in general is about.3 km over a broad bak ar region may be slightly smaller than observed bak ar basin depth. Other proesses whih are not onsidered in our models inluding bak ar spreading may also ontribute to the observed bak ar depression. It should be pointed out that the maximum depression of 15 km behind the model trenh (Figure 3a) is usually where topographially high volani island ars are loated. As we mentioned earlier, island ar topography is primarily due to the isostati ompensation of volani roks [Sleep, 1975]. Island ars primarily onsist of basalts; the lower-density basalts ould be loally ompensated and give rise to topographi highs. In order to illustrate this proess, a ase whih only differs from the non-newtonian ase in having a rust of volani roks

8 15,69 ZHONG AND GURNIS: 1RENCIIES FROM SUDDUC1ED SLAB MODELS has been omputed. The density of the rust is taken to be 3 kg m 3 smaller than that of ordinary lithosphere. The rust is loated right above the fault; the width of the rust is about 3 km on the surfae, and its width tapers to about 7 km at 45 km depth. The density and shape of the rust are very lose to those given by Sleep [ 1975]. The resulting dynami topography (Figure 3b) explains well the observed subdution zone topography inluding trenh, island ar, and bak ar basin. More importantly, the model trenh topography is insensitive to the inlusion of the rust, and the trenh depth dereases by only about.25 km, ompared with the ase without the rust (Figure 3b). Therefore in all the subsequent models of trenh depth, no rust of volani roks is inluded. The dynami topography determined from purely visous stresses may also be affeted by the elasti rigidity of the lithosphere. This effet is investigated by a thin elasti plate with a end free [Zhong and Gurnis, 1992]. The elasti lithosphere thikness inreases with lithospheri age, but this thikness beomes onstant at about 25 km when lithospheri age is greater than 35 m.y. [Watts et al., 198]. This orresponds to a flexural rigidity of 1.4 x 1 23 N m assuming a Young's modulus of 1. x 1 11 N m 2 and a Poisson's ratio of.25 [Watts et al., 19&]. Sine the ages of lithosphere in our models of trenh depth are all greater than 4 m.y., we study only the effet of a thin elasti plate with a flexural rigidity of 1.4x 1 23 N m. After loading the thin elasti plate by the normal stress from the non-newtonian ase (Figure 3a), the general harateristis of the topography remain unhanged (Figure 3}, but the trenh depth dereases by about 22% (i.e., 1.1 km} and the island ar topography is slightly dereased and smoothed (Figure 3). It should be pointed out that for a given flexural rigidity and distribution of normal stress ating on the elasti plate, the deformation (i.e., topography) of the plate is linear to the magnitude of normal stress [Hetenyi, 1946]. Relationships of Trenh Depth to 9,, 9 fo L and 'ts We now systematially study the effets of the four primary parameters of subdution zones, i.e., dip angles of faults, e 1. dip angles of slabs, 9s, length of slabs, L., and age of lithosphere prior to subdution, 'ts, on trenh depth. When omputing these ovariations, we fix three of the four parameters, and then ompute trenh depth as a funtion of the fourth parameter. Eah set of subdution zone parameters represents a distint model of subdution zones; an evolutionary relation is not assumed between any two subdution zones. Our omputations show that the trenh depth in general inreases with slab dip angle (Figure 6a), fault dip angle (Figure 6b), slab length (Figure 6) and age of lithosphere (Figure 6d). For these models of trenh depth, resisting shear stresses on faults are either zero or 1 MPa, and the effets of elastiity are not inluded. From these relationships of trenh we an observe that trenh depth depth to e. 9/, L 9, and 't 9, does not uniformly vary with these four parameters; fault dip angles 9/, age of lithosphere 't 9, and slab length Ls have more important effets on trenh depth than do slab dip angles e. For zero resisting shear stress ases, trenh depth inreases by about 3.9 km, 3.6 km and 2.7 km, respetively, as e 1, 't 9, and Ls are varied from 2 to 45 (Figure 6b), from 4 m.y. to Q -B. <I> g Q g 4..Q 'ts= 8 m.y. Ls = 7km b) Slab Dip Angle (degree) Fault Dip Angle (degree) _... 8!!! 6-5..Q ) s. 4 g OMPa Length of Slab (km) d) tl.l r. n :::. :s,cy -' 15 < 4... ()" h'' 9 OMPa.-... ' - - e - - MPa with elastiity n 1!. Subdution veloity '< Figure 6. Dependenies of trenh depth on (a) slab dip angle, (b) fault dip angle, () length of slab, and (d) age of lithosphere. Trenh depth with a thin elasti lid and subdution veloity relations to age of lithosphere are shown (Figure 6d). The other three parameters and the resisting shear stress for eah ase are also shown.

9 ZHONG AND GURNIS: TRENCHES FROM SUBDUCTED SLAB MODELS 15,691 m.y. (Figure 6d), and from 2 km to 8 km (Figure 6); trenh depth inreases by only about 1. km when es hanges from 3 to 85 (Figure 6a), Trenh depth inreases rapidly with slab length when slab length inreases up to about 4 km, but then hanges only slightly as slab length inreases beyond 45 km (Figure 6). The rapid hange in trenh depth when Ls is about 4 km results from the rapidly inreasing buoyany whih is indued by the phase hange at 41 km. Further inreases of slab length from 45 km do not greatly hange trenh depth. This suggests that trenh depth is not sensitive to farfield buoyany; this justifies our exlusion of the buoyany assoiated with the phase hange from spinel to pervoskite at 67 km. Trenh depth dereases as the resisting shear stress on faults inreases to 1 MPa, but the slopes hange only slightly (Figures 6a and 6b ). When a thin lasti plate with a flexural rigidity of 1.4 x 1 23 N m is loaded by the normal stresses resulting from the above models, trenh depths are redued by approximately 22%, but so ar the slopes of the trenh depth to the four subdution zone parameters (e.g., Figure 6d). The redutions in trenh depth are similar beause the distributions of the normal stress ating on the elasti plates are all quite similar, and as disussed earlier, the deformation of the plates is linear to the magnitude of normal stress. The inlusion of a thin elasti lid does not hange the onlusion of whih parameters influene trenh depth most. These relationships allow us to qualitatively explain observed statistis of trenh depth [Jarrard, 1986]. Parameters for subdution zones are generally within the ranges shown in Figure 6. Age of lithosphere " varies from about 15 m.y. in the southwest Mexian subdution zone to about 155 m.y. in Marianas subdution zone [Jarrard, 1986]; Othanges from about 15 in Chile subdution zone to 44 in New Hebrides subdution zone [Jarrard, 1986]; dip angle of slabs varies between 3 and 85 ; and Ls; if we equate it with the length of Benioff zone, varies from 1 to over 1 3 km. Among the four subdution zone parameters, slab length may be the most unertain. Seismi studies suggest that the length of Benioff zones poorly represents the length of the slabs. For example, the Benioff zone in the Aleutians extends to only 2 km depth, but a detetable high seismi veloity anomaly may exist as deep as 4 km [Zhao et al., 1993]. If the length of thermal slabs is Indeed large enough that slabs experiene the 41-km phase hange, our model results (Figure 6) suggest that trenh depth has a high orrelation with ts and 9/ and little or no orrelation with L 5 and 8 5, onsistent with the observed statistis of trenh depth [Jarrard, 1986). The orrelation between trenh depth and subdution veloity [Grellet and Dubois, 1982; Hilde and Uyeda, 1983] is also a diret onsequene of the physis in our models. Clearly, an older lithosphere means a larger amount of negative buoyany (assuming that other parameters are idential) and therefore results in a larger subdution veloity (Figure 6d). It should be pointed out that in our non-newtonian (stress dependent) alulations, the effetive visosity struture may be different from ease to ase, beause any hange in those four subdution parameters will ause hange in stress, i.e., the effetive visosity. The preexponnts, A, in equation (15) for lithosphere, the lower mantle, and slab are larger than those for the upper mantle and transition zone in order. to obtain higher visosities within lithosphere, the lower mantle, and the slab. For models with slabs of a given age, a set of preexponents, A, has been determined suh that the horizontally averaged visosities are similar to those used in Newtonian models (e.g., Figure 2). Suh determined A have been used for models with slabs of that given age. This is partly justified sine A represents the temperature dependene of visosity and should be dependent on slab age. However, we have found that the use of a ommon set of prexponents for different slab age auses only a slight hange in the topography; the relative differene in trenh depth between 4 and 16 m.y. old slabs hanges by only.15 km. However, using a ommon set of A results in subdution veloities that are muh too large for old slabs ompared to young slabs: old slabs have more buoyany and result in smaller effetive visosities. Dependeny of Observed TreJth Depth on Age Although the dependene of observed trenh depth on age of lithosphere is sattered, a positive orrelation is evident. Hilde and Uyeda [1983] suggested that trenhes with anomalous depths were assoiated with subdution of marginal basin lithosphere, but Jarrard [1986] argued that no suh relationship ould be identified from his statistial analyses of trenh depth. Sine we have obtained theoretial relationships between trenh depth and four important parameters of subdution zones and sine resisting shear stress affets only the absolute magnitude of trenh depth and not the slopes (Figure 6), we may use these relationships to orret the observed trenh depth for fault dip, slab dip, and slab length to a ommon set of parameters. If shear stresses on fault planes do not differ signifiantly among subdution zones, we should expet the orreted trenh depth to be linearly proportional to age. We will first present the orreted trenh depths derived from those relationships exluding an elasti lid, and then show results inluding one. Ten subdution zones in whih 81, 'ts, Lsz (Lsz here is Benioff zone length), and 8 5 are known have been hosen from Jarrard [1986] (Table 1 ). We have avoided ontinental overriding plates to avoid possible ompliations by rheologial and density struture. The trenh depth to age relation for these subdution zones is not learly linear (Figure 7a). The Marianas trenh, with a 155 m.y. lithosphere, is 4.95 km deep (Figure 7a). The Solomon trenh is nearly as deep, 4.42 km, but the subduting lithosphere is only 5 m.y. old. For the orretions, we have hosen a standard model: e 1 = 3, 8 5 = 6, and Ls = 7 km. Corretion for 9 5 bas only a slight effet on the trenh depth to age relation, beause trenh depth is not very sensitive to 9 5 (Figure 6a). The largest hange aompanies the orretion for Marianas trenh: trenh depth dereases by.4 km after orreting 8 s from 81 to 6. The further orretion for 8/dramatially hanges the trenh depth to age relation (Figure 7b). Beause of large 8 1 in Solomon, South Sandwih, New Britain and New Hebrides subdution zones, these trenhes derease in depth substantially after orretion, and the Marianas and Lesser Antilles trenhes inrease in depth beause of their relatively small 8/" The resulting trenh depth to age relation (Figure 7b) displays a muh better linear trend than does the observed (Figure 7a). However, further orretion for LBz results in a signifiant inrease in trenh depth for young subdution zones but almost no hange for old subdution zones (Figure 7). This is beause Benioff zone length for young subduting lithosphere is muh shorter than that for old subduting lithosphere and the 7 km standard.

10 15,692 ZHONG AND GURNIS: TRENCHES FROM SUBDUCTED SLAB MODELS Table 1. Subdution Zone Parameters [after Jarrard, 1986] Symbol Segment Name ef' es, 't,, LBz, Trenh Depth, deg deg m.y. km km Ker Kermade Ton Tonga Nhb New Hebrides Sol Solomon Nbr New Btitain Mar Marianas Izu Izu-Bonin Alu Central Aleutians Ant Lesser Antilles So South Sandwih Note that 9/, Bs and trenh depth are Dipl, DipD, and differential trenh depth, lld, respetively, in Jarrard [1986] notation. L8z are omputed from the vertial and horizontal slab extent given by Jarrard [1986] Linear regressions for orreted trenh depths show that the slopes of regression lines are 32m m.y.- 1 and 11 m m.y.- 1 and that the oeffiients of orrelation are.83 for trenh depths orrected for 9 8 and e 1 (Figure 7b) but only.42 for trenh depths orreted fore,, e 1 and Lsz (Figure 7). Trenh depth to age relations for the standard model with zero shear stress and 3 MPa shear stress on the fault have a slope of 3 m m.y.- 1 and 33m m.y: 1, respetively (figure 8b), lose to that with orretion for as and Bt The orretion for Benioff zone length not only signifiantly degrades the orrelation between trenh depth and age but it also leads to a muh smaller slope ompared to the dynami model. This suggests that Benioff zone length may not represent the length of thermal slabs, onsistent with travel time residual sphere analysis [e.g., Creager and Jordan, 1984} and seismi tomography [e.g., Zhao et al., 1993]. To use those trenh depth relations to 9 9, e 1, and L 8 z with a thin elasti lid inluded to do the orretions does not signifiantly affet our results. This is beause orretions involve only rehitive variations of trenh depth and the elasti lid uniformly redues the relative depths by about 22% for the rigidity used. The slope of the linear regression for the trenh :3.. Q 4...d. 4 a),.--.. g Sol Ton D Ker So 4. Nbr Izu -= () o Nhb : D Ant Alu Observed values Age of lithosphere (m.y.) D D Corret for 9 5 and ef :3.. Q 4... l. 4 ) D D -:3.. Q.. () Lithosphere Age (m.y.) d) Corret for 6s and ef with elastiity 8 12 Figure 7. (a) Observed trenh depths and (b) those after orretion for slab and fault dip angles, and () after orretion for slab and fault dip angles and slab length. (d) Trenh depths orreted for slab and fault dip angles with a thin elasti lid. The solid lines in Figures 7b and 7d are the linear regression lines. See Table 1 for abbreviations of segments. 16

11 ZHONG AND GURNIS: TRENCHES FROM SUBDUCTED SLAB MODELS 15,693! :[ 2 a) j... o -2 & -4.;.! i -6 = Distane (km) 8 8 Without elastiity '""' ':3 Q 4 OMPa g..d Q, Q, Q = IOMPa =..., 4 ---OMPa MPa With elastiity..:: u 2 lompa MPa dil MPa MPa With no PHCB d) e) Constant PHCB ""' o,.-,. A g s..:: E.. -- g..:: u 4 Q..d u = = Figure 8. (a) Effets of resisting shear stress aross faults on dynami topography. Trenh depth relations to age of lithosphere for the standard models with different resisting shear stresses on faults (b) exluding and () inluding a thin elasti lid. Effets of (d) no phase hange buoyany and (e) onstant phase hange buoyany on the estimate of resisting shear stress on faults are also shown. PHCB stands for phase hange buoyany. The solid lines are the linear regression lines for orreted trenh depths for eah ases ) depths orreted for 9s and a 1 terases to 27 m m.y: 1 and the oeffiient of orrelation also slightly redues to.78 (Figure 7d). This slope is also lose to those for the standard model when the effets of elastiity are inluded, whih are about 23 and 24 m m.y:l for zero and 2 MPa shear stresses, respetively (Figure 8). Impliations to Resisting Shear Stress on Convergent Plate Boundaries Inreasing resisting shear stress on the fault plane (i.e., inreasing the oupling between subduting and overriding plates) will ause trenh depth to derease [Zhong and Gurnis, 1992]. This suggests that we may estimate the magnitude of resisting shear stress by omparing model trenh depth with observed trenh depth. Although the oupling between subduting and overriding plates (i.e., the resisting shear stress) may differ among subdution zones in terms of maximum sizes of earthquakes [Uyeda and Kanamori, 1979], we have not attempted to study the variability of the resisting shear stress between different subdution zones; but rather, we have assumed a onstant shear stress for all subdution zones. The estimated resisting shear stress from models exluding the elastiity will be presented first and then the effets of elastiity and different buoyany distribution will be disussed. For the standard model (i.e., Bt== 3, Bs:. 6, and Ls = 7 km), a 2-MPa resisting shear stress, applied uniformly on the upper 5 km of the fault in depth, auses trenh depth to derease by.7 km ompared to that with a zero resisting shear stress (Figure 8a). The shear stress results in less depression of the bak ar region (Figure 8a). The tangential veloities averaged over the segment of fa ult on whih the 2- MPa shear stress is applied are 11. m/yr on the subduting lithosphere side and 6. m/yr on the overriding lithosphere

12 15,694 ZHONG AND GURNIS: TiffiNCI IES FROM SUDDUCfED SLAB MODELS side, in ontrast to 16.7 m/yr and 2.8 m/yr for the ase with zero shear stress. Trenh depth to age relations are omputed for the standard model with different shear stresses on the fault. Although trenh depth hanges with the shear stress. the slopes of trenh depth versus age for diffe rent shear stresses are nearly idential (Pigure!!b). Plotted on Figure Hb ar the trenh depths orreted for a/ and es and their linear regression (i.e., Figure 7b); the effets of elastiity were not onsidered in these orreted trenh depths. We observe that the 3-MPa shear stress line is lose to the linear regression line for the orreted trenh depth. This leads us to onlude that shear stress on faults in subdution zones may be about 3 MPa. When the effets of elastiity are inluded, omparison between the orreted trenh depths and those from standard models suggests a 15-MPa resisting shear stress (Figure 8). It should be pointed out that a few orreted trenh depths are either far below the 3-MPa line in Figure 8b or above the -MPa line in Figure 8. However, we find that the resisting shear stress annot exeed 4 MPa; otherwise, for subdution zones younger than 6 m.y., the tangential veloity on the fault on the overriding lithosphere side beomes larger than that on the subduting lithosphere side, whih ontradits the original sign of the resisting shear stress. This implies that our models annot-explain some orreted trenh depths. Two possibilities may resolve this disrepany: (1) some raw trenh depths inlude signifiant fration of nondynami topography inluding sediments, as disussed by Hilde and Uyeda [1983], making trenhes appear shallower, and (2) unertainties in the measured fault and slab dips and unertainties in model parameters inluding negative buoyany and mantle visosity, as disussed below. Tbe magnitude of the resisting shear stress determined from our models depends on total buoyany within the slab and visosity struture. A smaller phase hange buoyany or a higher mantle visosity would yield a smaller resisting shear stress. Phase hange kinetis whih were exluded in previous alulations of phase hange buoyany tend to redue the negative buoyany [Sung and Burns, 1976], and thus the estimated resisting shear stress. However, we find that even after the phase hange buoyany is exluded, a 2-MPa resisting shear stress is still needed in order to math the trenh depths orreted for fault and slab dips without taking into aount elastiity (Figure 8d). We have assumed a linear relationship between the amount of phase hange buoyany and age of lithosphere in omputing trenh depth to age relation (e.g., Figure 8b). To assume a onstant phase hange buoyany for different ages of subduting slabs auses the slopes of trenh depth to age relations to derease by about 5 m m.y.- 1 (Figure 8e without the elastiity). This does not greatly hange our estimate o f resisting shear stress (Figure 8e). When alulating those orreted trenh depths in Figures 8d and 8e, trenh depth to fault dip and slab dip relations for models with orresponding phase hange buoyany strutures were used, respetively. A higher-visosity mantle provides larger visous resisting fore to balane the negative buoyany assoiated with slabs and auses Jess deformation on the surfae, onsequently, smaller resisting shear stress on fault plane is required in order to math observed trenh depths. Our visosity struture with high visosities for the lower mantle and transition zone is basially the same as Hager's (199]; this visosity struture suggests a resisting shear stress of about 15-3 MPa o n onvergent plate boundaries, whih is smaller than the 1 MPa of Davies [198]. but is in general onsistent with the 14-4 MPa inferred from fritional heating models of subdution zones [TihellJllr and Ruff, 1993]. Conlusions We have developed a finite element method with onstrained elements and Lagrange multipliers to study tetoni faults in a visous medium. With this method, a fault whih represents the interfae between an overriding and a subduting plate has been inorporated into a visous flow model of subdution zones. Surfae dynami topography from the model with typial subdution zone parameters is haraterized by a narrow and deep trenh and a broadly depressed bak ar basin. This suggests that trenhes and bak ar basins may originate from the ompensation of subduting slabs. Our models indiate that trenh depth in general inreases with fault dip, slab dip, slab length, and age of lithos{>here prior to subdution. However, the fault dip angle and the age o f lithosphere are more important in ontrolling trenh depth than the slab dip angle. The length of thermal slabs does not greatly effet trenh depth, as long as they are long enough to experiene the phase hange from olivine to spinel at 41 km depth. These trenh depth to subdution zone parameter relations explain well the statistis of observed trenh depths [Jarrard, 1986]. Observed trenh depths of subdution zo nes with oeani lithosphere on both sides of trenhes have been orreted for fault dip angles, slab dip angles, and length of Benioff zone, based on the relationships from o ur models. Trenh depth orreted only for fault dip and slab dip displays a lear orrelation with age of lithosphere prior to subdution with a slope of 32 m m.y.-l (27 m m.y.- 1 when elastiity is onsidered), whih is onsistent with that predited by our models and is lose to the seafloor subsidene rate of 36 m m.y.- 1. Further orretion of trenh depth for length of Benioff zones not only degrades the orrelation between trenh depth and age but also gives rise to muh smaller slope than predited by our models; this s uggests that the length of Benioff zones used for the orretion does not represent well the length of the thermal signal of slabs. Comparison between the trenh depths orreted for dip angles of faults and slabs and those from models with different shear stresses on faults suggests that shear stress on faults in subdution zones may range from 15 MPa to 3 MPa, depending on model details. Akaowledp..as. We thank L. Ruff and G. Houseman for onstrutive disussions and M. Parmentier, S. 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