OF PLATE MOTION. J. Phys. Earth, 33, , 1985 THE MAGNITUDE OF DRIVING FORCES

Transcription

1 J. Phys. Earth, 33, , 1985 THE MAGNITUDE OF DRIVING FORCES OF PLATE MOTION Shoji SEKIGUCHI Disaster Prevention Research Institute, Kyoto University, Uji, Kyoto, Japan (Received February 22, 1985; Revised July 25, 1985) The absolute magnitudes of a variety of driving forces that could contribute to the plate motion are evaluated, on the condition that all lithospheric plates are in dynamic equilibrium. The method adopted here is to solve the equations of torque balance of these forces for all plates, after having estimated the magnitudes of the ridge push and slab pull forces from known quantities. The former has been estimated from the age of ocean floors, the depth and thickness of oceanic plates and hence lateral density variations, and the latter from the density contrast between the downgoing slab and the surrounding mantle, and the thickness and length of the slab. The results from the present calculations show that the magnitude of the slab pull forces is about five times larger than that of the ridge push forces, while the North American and South American plates, which have short and shallow slabs but long oceanic ridges, appear to be driven by the ridge push force. The magnitude of the slab pull force exerted on the Pacific plate exceeds to 40 % of the total slab pull forces, and that of the ridge push force working on the Pacific plate is the largest among the ridge push forces exerted on the plates. The high correlation that exists between the mantle drag force and the sum of the slab pull and ridge push forces makes it difficult to evaluate the absolute net driving forces. However, the slab resistances appear to contribute more to cancelling the driving forces than the mantle drag force. From stress estimation, it was found that high stresses are concentrated around the leading edge of the downgoing slab. 1. Introduction Since the plate tectonic hypothesis has gained credibility in explaining a variety of geophysical and geological observations, the problem of the driving mechanism of plate motion has become an important subject, and has been closely investigated by several authors (e.g., FORSYTH and UYEDA, 1975; CHAPPLE and TULLIS, 1977; RICHARDSON et al., 1976; HARPER, 1978; DAVIES, 1978; HAGER and O'CONNELL, 1981; CARLSON et al., 1983). Plate motion may be closely related to the thermal convection in the mantle. Although mantle convection has been investigated with idealized models (e.g., RICHTER, 1973; MCKENZIE et al., 1974; PARMENTIER, 1978; DE BREMAECKER, 1977; CSEREPES, 1982; JARVIS and Mc- KENZIE, 1980), it seems rather difficult, at present, to understand the driving mech- 369

2 370 S. SEKIGUCHI anism of plate motion, in terms of thermal convection in a realistic mantle, because of the uncertainties in the distributions of temperature and viscosity in the mantle. On the other hand, the kinematics of plate motion, i.e. the configuration and the velocity of the plate, is now well understood, on the basis of the spreading rate of mid-oceanic ridges, the strike of transform faults, and the slip vectors of major earthquakes (MINSTER and JORDAN, 1978). FORSYTH and UYEDA (1975) examined the driving mechanism of plate motion, by making use of the kinematics of plates. They estimated the relative magnitudes of plate driving forces, on the assumption that each plate is in dynamical equilibrium, and obtained the following results: 1) the slab pull is an order of magnitude larger than any other force; 2) the slab pull force is nearly balanced with the slab resistance; 3) the mantle drag exerted on the bottom of the plates, which resists plate motion, is much stronger under the continents than under the oceans. However, all the magnitudes of these forces acting on plates have been treated as unknown parameters, so that the absolute magnitudes of these forces could not be directly determined by solving equilibrium equations. Also, in their treatment, slab pull was assumed to be proportional to the length of the trench, but this assumption appears simplistic. CHAPPLE and TULLIS (1977) followed an approach similar to that by FORSYTH and UYEDA (1975). They estimated the slab pull in advance, using the analytical solution of McKENZIE (1969) for temperature and density distributions in a downgoing slab. They also took account of the focal mechanisms of intermediate and deep-focus earthquakes. It is also stated in their conclusions that the slab pull force is nearly balanced with a resistive force working at subduction zones, although it is not clear whether the slab resistance or the colliding resistance cancels the slab pull force. The main purpose of the present study is to estimate the absolute magnitude of various forces that could contribute to plate motion and for better understanding of global tectonics, earthquake mechanisms, as well as the plate driving mechanism itself. The major difference of our study from the previous studies is that we estimate the slab pull and ridge push forces as definitely as possible, before solving the equilibrium equations. Since the thermal structure of oceanic lithospheres has been well explained in some models (e.g., PARKER and OLDENBURG, 1973 ; SCLATER and FRANCHETEAU, 1970), we can calculate the ridge push force, which may be the force arising from density variations in cooling oceanic lithospheres. The slab pull force resulting from the density contrast between the slab going down from the trench and the surrounding mantle is also calculated in advance. Since the slab resistance plays an extremely important role in the net driving force of plate motions, we investigated, in detail, several different types of the slab resistance, such as those exerted along the surface of downgoing slabs and around the edge of the slabs. With these improvements, we shall discuss the results obtained here in comparison with those obtained by FORSYTH and UYEDA (1975) and CHAPPLE and TULLIS (1977).

3 The Magnitude of Driving Forces of Plate Motion Method In order to estimate the magnitude of the driving forces, we shall use the dynamic equilibrium conditions, following the previous studies (FORSYTH and UYEDA, 1975; CHAPPLE and TULLIS, 1977). The equation of torque balance for plate motions can be written in the form, (1) where subscripts i(=0, c, N) and j(=1, 2, 3) indicate an index of a plate and a component, respectively, and repeated subscripts k indicate summation over the set of unknown parameter xk. The equation a3i+j,k includes the geometrical and dynamical factors for the length of arms from the rotational axes of plates, the areas of plates, the length of trenches, the length of the convergence plate boundaries, the thicknesses and the length of the slab, and the relative and absolute velocities of plate motions. The equation b3i+j indicates the ridge push, and c3i+j denotes the slab pull forces. In the present study, the torques to be computed are the following two types. The first type is the torque whose directions and magnitudes will be calculated in advance, which includes the slab pull and the ridge push, and the second is those of which directions are calculated, leaving the magnitudes and signs unknown. If we consider 12 plates, then there are 36 linear algebraic equations for the unknown parameters xk. This set of equations can be solved by least squares (e.g., AKI and RICHARDS, 1980). Next, we formulate the magnitude and type of the forces working on all plates. Here, we consider 8 forces, i.e., the ridge push (FRP), mantle drag (FD,), continental drag (FCD), colliding resistance (FCR), slab pull (FSP), slab surface resistance (FSSR), slab edge resistance (FSER), and suction (FSC). We know that the first three forces are exerted on the area of plates. However, we assume that the last five forces act on the boundaries of plates, although, strictly speaking, these forces do not exactly act on the boundaries. Resistances along transform faults and pushing forces from hot spots have been neglected in this study, because CHAPPLE and TULLIS (1977) have shown that these forces produce such insignificant torques on the plates. Twelve plates are incorporated in this study (Fig. 1). They are African (AF), Antarctic (AN), Arabian (AR), Caribbean (CA), Cocos (CO), Eurasian (EU), Indian (IN), North American (NA), Nazca (NZ), Pacific (PA), Philippine Sea (PH), and South American (SA) plates. The plate boundaries and the edges of the continents were digitized from Plate 1 of SCLATER et al. (1981). The ages of the oceanic plates are also taken from the same data. We refer to the absolute and relative velocities of plate motions from AM1-2 and RM2 of MINSTER and JORDAN (1978, 1979), respectively. Although AM1-2 has been derived from the assumption that hot spots are fixed at the base of the mantle, there could be an alternative approach to estimate absolute plate velocities, which is based on the assumption that no net torque is exerted on the lithosphere as a whole, and also

4 372 S. SEKIGUCHI some assumptions about the forces driving plate motions (SOLOMON and SLEEP, 1974). This approach gives absolute plate velocities that are very similar to those calculated on the assumption that a set of hot spots provides a fixed reference frame (SOLOMON and SLEEP, 1974). For this reason, even if we use this absolute velocity, the results obtained in this paper would not be seriously affected. 2.1 Ridge push It is well known that there are close relations between the depth of the ocean floor, heat flow and the age of oceanic plates (PARSONS and SCLATER, 1977). These relations can be explained either by a plate model (SCLATER and FRANCHE- TEAU, 1970) or by a thickening model (PARKER and OLDENBURG, 1973; YOSHII, 1973; KONO and YOSHII, 1975). Following ARTYUSHKOV (1973), there should exist forces which act in the horizontal direction with considerable magnitudes, due to the lateral density variations. The variations come either from its own lateral variations themselves, or from the thickening of the oceanic plate. In this study, we assume that the forces due to the thickening of the plate are the ridge push forces. LISTER (1975) has calculated the forces, in the case of a thickening oceanic plate, taking into account the thermal structures within the plate. For simplicity, however, we assume that each of the densities of the lithosphere and the asthenosphere is constant, respectively. In the case shown in Fig. 2, where the plate is assumed to be flat, not spherical, the force exerted on the shaded area of the plate per unit length is given by (2) where d2-d2., and l2-l1 are sufficiently smaller than the distance between points 1 and 2, g is the gravitational acceleration (=9.8 m/s2), di and li are the depth to the ocean floor and the thickness of the plate at point i(i=1, 2), respectively, and ĕl and pa are the densities of the oceanic plate and asthenosphere, respectively. Assuming Fig. 1. A schematic map of plate boundaries and continental margins.

5 The Magnitude of Driving Forces of Plate Motion 373 Fig. 2. A schematic diagram showing a thickening lithosphere (ĕl), underlying the ocean (ĕw), and resting on the asthenosphere (ĕa). The horizontal force, frp, acts on the shaded area. isostatic compensation at a certain depth below the plate, the thickness of the plate, (m), is given by l (3) where d is the depth to the ocean floor in meters, ĕw is the density of the water, ĕ a-ĕw and ĕl-ĕw are assumed to be 2.3 (g/cm3) and (g/cm3), respectively, and the depth to the top of a spreading mid-oceanic ridge is assumed to be 2,500 m, Furthermore, following PARSONS and SCLATER (1977), the relation between the depth and the age of the ocean floor is expressed as, (4) Therefore, from Eqs. (2), (3), and (4), we can estimate the magnitude of the ridge push force from the age of the ocean floor. It follows from Eq. (2) that the ridge push force working on an intermediate portion of the plate bounded by two different points is determined from the ages of the plate at these points. For practical calculations, a smoothing and interpolating technique has been applied to the data of ocean floor ages based on SCLATER et al. (1981). We divided all the oceanic plates into approximately 2 ~2 areas, calculated the torques of the ridge push forces exerted on each of the divided elements by applying Eq. (2) in the latitude and longitude directions, and then sumed up the calculated values for each plate. The forces due to lateral density variations in marginal basins and continents have been neglected. This is because the marginal basins occupy only small portions of all oceanic areas and show relatively small variations in their ages. Also, the continents have no weak zones corresponding to oceanic ridges and hence may be rigid enough to sustain the stress.

6 374 S. SEKIGUCHI 2.2 Slab pull When a cooled plate sinks into the uppermost mantle, the density contrast between the subducted plate or slab and the mantle comes from temperature anomalies, and hence it causes slab pull forces. The density distribution within and around the slab can be determined, if the temperature distribution around it is known. The temperature is, however, controlled by many factors (e.g., MINEAR and TOKSOZ, 1970a, b; TOKSOZ et al., 1971, 1973), for example; average temperatures in the earth, as a function of the radius, adiabatic compression, conduction, frictional heating, viscous heating, radioactive heating, phase transition (SCHUBERT et al., 1975; SUNG and BURNS, 1976), thickness of the slab, subduction rate, and so on. In this study, we assume that the slab has a constant density contrast (ƒ ƒï) of g/cm3, that does not depend on the depth and thickness of the slab. In general, the slab corresponds to the deep siesmic zone in the mantle, under island arcs. JORDAN (1977) and CREAGER and JORDAN (1984) suggest the possibility that the slab penetrates below the deepest portion of the zone. It is not known whether the slab actually exists in the seismicity gap between intermediate and deep seismic zones in all subduction zones. However, recent studies of threedimensional seismic velocity structures (e.g., HIRAHARA, 1977; HIRAHARA and MIKUMO, 1980) show that the Pacific plate, with high seismic velocities and a constant thickness, extends continuously down to a depth of 500 km under the northeastern Japan arc. From the above evidence, we assume, in this paper, that the slab with a constant thickness continues down to the deepest part of the seismically active zone. The slab pull force, due to the negative bouyancy per unit length along the strike of trenches (fsp) is given by, where ƒ ƒï is the density contrast between the slab and the mantle, l, L, and ƒæ are the thickness, length, and dip angle of the slab, respectively, and DS is the depth to the leading edge of the slab. The length of the slab is measured perpendicularly to the strike of the trench. The thickness of the slab is estimated from the age of ocean floor at the trench, using Eqs. (3) and (4). The slab pull force is assumed to work in the direction perpendicular to the strike of the trench. 2.3 Mantle drag force and continental drag force We assume that the mantle drag force per unit area (fdf) is expressed as where Va is the absolute velocity of plate motions. Whether the mantle convection drives the plate or not will be determined by solving the equilibrium equation. FORSYTH and UYEDA (1975) and CHAPPLE and TULLIS (1977) show that the mantle resistance under the continents is larger than that under the oceans. For

7 The Magnitude of Driving Forces of Plate Motion 375 this reason, we shall introduce here the continental drag force, which is exerted only on the part of the plate under the continents. The formulation for this force is the same as in the mantle drag force. For continental regions, we consider the sum of these two forces, the mantle drag force and the continental drag force. 2.4 Colliding resistance We assume that, at covergent plate boundaries, colliding resistive forces are working parallel to the direction of the relative motion of the plates. We shall consider here two different types of colliding resistances (FCR1 or FCR2), corresponding approximately to those of FORSYTH and UYEDA (1975) and CHAPPLE and TULLIS (1977), respectively. The first colliding resistance per unit length of plate boundaries (fcrl) is assumed to specify the force direction, expressed as where Vr is the relative plate velocity. The second colliding resistance per unit length (fcr2) is assumed to be proportional to the magnitude of the relative plate velocity, expressed as We shall determine if each of the plate boundaries is convergent or not, from the sign of the relative plate velocity at the boundary. 2.5 Slab surface resistance and slab edge resistance The upper mantle provides viscous resistive forces on subducting slabs. There are discontinuities in the seismic velocity structure at depths of about 400 km and 700 km in the mantle. These suggest that the viscosity of the mantle also varies at these depths. ISACKS and MOLNAR (1971) proposed a qualitative model, in which mantle viscosity increases with depth, in order to explain the focal mechanisms of mantle earthquakes. RICHTER (1977) also proposed an idealized model for subducting plates that could best explain the observed velocity of plate motions. He suggested that the distribution of stresses within the downgoing slabs requires that the viscosity in the upper mantle should increase with depth. SMITH and TOKSOZ (1972) showed that the resistive force working on the leading edge of the slab should be three to five times as large as viscous traction applied to the surfaces of the descending plate. For the above reasons, we assume that there are two different types of slab resistance; the first slab resistance (FSSR), which is the resistance exerted on the surfaces of the slab. The resistance per unit length of the trench (fssr) is stated as The second slab resistance (FSER) is considered to work on the leading edge of the slab, below a certain depth. We consider two different slab edge resistances (FSER1 or FSER2). The first slab edge resistance, per unit length of the trench (fser1), may be proportional to the absolute plate velocity, expressed as

8

9

10 378 S. SEKIGUCHI Following the formulation of the slab resistance by FORSYTH and UYEDA (1975), the second slab edge resistance per unit length (fser2)is assumed to be expressed as where Va Û, is the component of the absolute plate velocity perpendicular to the strike of the trench, assuming that no slab resistance is exerted in the direction of the strike of the trench. We shall now consider the following five cases, including; 1) FSSR, 2) FsER1, 3) FssR, and FSER1, 4) FSER2, and 5) FSSR and FSER2. Viscosity distribution in the mantle is not very well known at present. On the other hand, the depth of the leading edge of the downgoing slab may be reasonably well estimated. In the above four cases, 2), 3), 4), and 5), calculations are made for models with three different depths (D=400 km, 500 km, and 600 km), below which the resistive force (FSER1 or FSER2) is exerted on the leading edge of the slab. We shall try to determine the best combination of the slab resistances and the depth, D, judging from the residuals in Eq. (1), for each model. The absolute plate velocity will not be exactly the same as the subducting slab velocity, because the upper plate overlying the subducting plate moves with some velocity. However, since the upper plate moves slowly, we may adopt the absolute plate velocity as the subducting slab velocity. 2.6 Suction The subduction of the slab will induce the flow of the mantle, which yields the force attracting the upper plate toward the trench. Here, we consider both of the following two different situations. The first one is that where the slab is subducting without changing its dip angle and induces the mantle flow. The second one is that where the slab sinks towards the center of the earth and induces the mantle flow. We shall represent the above two situations by the following formulations. In the first case, the suction force, per unit length of the trench (fsc1), may be expressed as In the second case, the suction force (FSC2) has the magnitude, per unit length (fsc2), expressed as which works in a direction perpendicular to the strike of the trench. In both cases, the suction force is assumed to work in an opposite direction to the absolute velocity of the subducting plate. 3. Results The total torque magnitudes of driving forces obtained by solving the equation

11 The Magnitude of Driving Forces of Plate Motion 379 Fig. 3. The slab pull forces and the ridge push forces are drawn to be tangent to the great circles, perpendicular to the torque vectors of these forces. (The ridge push force working on the Antarctic plate is shown by a small dot on the southern portion of the Pacific plate, close to the Antarctic plate, because the total sum of the force has been almost cancelled.) of torque balance are summarized in Table 1. For known quantities i.e., the slab pull and ridge push forces, the total torque magnitudes defined as and ムiƒ j(c3i+j)2 are given in units of 1027 Nm, where b3i+j and ムiƒ j(b3i+j)2 e3i+j are the vector coefficients of Eq. (1). For unknown quantities, the solutions xk of Eq. (1), together with their standard deviations, are also given in the above unit, in such a way that the square root of the square sum of the column vectors for the coefficient matrix becomes unity. Forces with a negative sign imply resistive forces to the plates motions. Table 1 is divided into (a)-(f), which correspond to 6 different cases given in table captions. Figure 4 shows the results for the combination of the colliding resistance FCR1, the suction FSC1, the slab surface resistance FSSR, and the slab edge resistance FSER2 (D=500 km), for which the largest variance improvement is obtained from among all the combinations shown in Table 1(a)-(d). Figure 4(a) shows the x, y, and z components of the torques, of the forces exerted on the plates, arranged from left to right for each plate, and Fig. 4(b) gives the absolute magnitudes of the torques. The obtained values of the torques are summarized in Table Ridge push and slab pull First, we shall describe the results calculated for the slab pull and ridge push forces, which were obtained before solving the equation of torque balance, and hence are not affected from combinations of the other forces (Fig. 3 and Table 2). It is found in Table 2 that the total magnitude of the torques calculated

12 380S. SEKIGUCHITable 2. Calculated values of driving force to each plate for one model.

13 The Magnitude of Driving Forces of Plate Motion381 Table 2. (continued) The directions of the Cartesian coordinate (X, Y, Z) are taken 0 N, 0 E; O N, 90 E; 90 N, respectively. A indicates a root of square sum. Unit: 1027 N Em. for the ridge push force is ~1027 Nm, while that of the slab pull force is 2.53 ~ 1027 Nm. The former is found to be about five times larger than the latter. The magnitude of the torque of the slab pull working on the Pacific plate reaches 46 % of the sum of all the torques of the slab pull forces (ƒ i ムj(c3i+j)2, which is the largest one among other slab pull forces, as may be understood from Table 2. The corresponding magnitudes for the Indian plate, the Nazca plate and the Philippine Sea plate occupy 26 %, 12 %, and 7%, respectively. The largest torque of the ridge push is the one working on the Pacific plate, the magnitude of which is 25 % of the sum of all the torques of the ridge push forces. The magnitude is 17 % for the South American plate, 16 % for the Indian plate, 13 % for the North American plate and 13 % for the African plate, respectively. The Antarctic plate has oceanic ridges with extended lengths, but the sum of the ridge push is quite small (2 %). The reason for this may be that since the Antarctic plate is surrounded by the oceanic ridges, the total ridge push force coming from the surroundings have been almost cancelled, as suggested by FORSYTH and UYEDA (1975). 3.2 The combination of the forces with the largest variance improvement Now, we try to determine the best combination of the slab resistances and the depth D, with the largest variance improvement. Variance improvements, defined as 1-ƒ iƒ j(ƒ ka3i+j,k xk+b3i+j+c3i+j)2/(ƒ iƒ j(b3i+j-c3i+j)2), are given in Table 1, where xk obtained here is the solution of Eq. (1). It is found that the calculated variance improvements appear to be very large, suggesting that the formulations described in the previous section are appropriate.

14 382 S. SEKIGUCHI Fig. 4. (a) The x, y, and z components of the torques of different forces exerted on each plate, which are obtained for the case where the colliding resistance FCR1, the slab edge resistance FSER2 (D=500 km), and the suction FSC1 are taken into account. (b) The magnitudes of the torques of different forces acting on each plate, and their residuals. Solid bars indicate the ridge push and slab pull forces. 1: FRP; 2: FDF; 3: FCD; 4: FCR1; 5: FSP; 6: FSSR; 7: FSER2 (500); 8: FSC1; 9: residual. Table 1(a) gives the calculated results for the models with the colliding resistance FCR1, the suction FSCI, and various combinations of the slab resistances for different depths of D. The variance improvements calculated for different combinations of 1) FSER1, 2) FSSR, 3) FSSR and FSER1, 4) FSER2, and 5) FSSR and FSER2 become successively larger, in this order. The results show that the models with the combination of FSSR and FSER2 are the best ones. Comparison between the slab edge resistances FSER1 and FSER2, shows that the residuals obtained for the case of FSER2 tends to be smaller than that in FSERI (Table 1(a)). This might

15 The Magnitude of Driving Forces of Plate Motion 383 suggest that the models with the slab edge resistance FSER2 are more preferable. Differences in the variance improvements between the models with different depths of D are quite small (Table 1(a)), indicating that it seems difficult to determine the best combination of the depth of D. In the case of D=500 km, however, the largest variance improvement is obtained, as shown in Table 1(a) (for details, see Fig. 4 and Table 2). Comparing Tables 1(a) and (b), the variance improvements for the former are larger than those for the latter, suggesting that the models with the colliding resistance FCR1 are more preferable Tables 1(b) and (c) show that the variance improvement for the various slab resistances given in Table 1(a) are not affected, even if we replace the colliding resistance FCR1, and the suction FSC1 by FCR2 and FSC2, respectively. 3.3 Ridge push driving South and North American plates, colliding resistance, continental drag and suction forces From Tables 1(a), (b), and (c), we can immediately see that the slab pull and ridge push forces drive the plates, and that the other forces, except the suction, resist the plate motions. The colliding resistance and the continental drag force are mostly smaller than the other resistive forces. The suction is also small, and in all of the cases discussed here, is almost in the same order as the standard deviations. One of the remarkable findings from Tables 1(a), (b), (c), and Fig. 4(b), is that the North American plate and the South American plate, which have short and shallow slabs but long oceanic ridges, appear to be driven by the ridge push force. 3.4 High correlation between slab resistances and mantle drag forces Tables 1(a), (b), and (c) suggest that most of the above slab pull and ridge push forces appear to be cancelled by the slab surface and slab edge resistances, and the mantle drag force. As may also be understood from Tables 1(a), (b), and (c), the estimates for the slab surface, slab edge resistances, and the mantle drag force are subject to considerable changes. Table 3 shows the calculated correlations among the various torques exerted on the plates, defined as ƒ iƒ jasi+j,k a3i+j,l/. For the ridge push or slab pull, we ( ムiƒ j(a3i+j,k)2( ムiƒ j(a3i+j,l)2 used b3i+j or c3i+j, instead of asi+j,k. From Table 3(a), we see that the correlations between the mantle drag force, and the sum of the ridge push and slab pull forces, are very high. This makes it difficult to estimate the magnitude of the net driving force (slab pull-slab resistances) exerted on the lithospheric plates, because small changes in the vector coefficients a3i+j,k, in Eq. (1), would affect the estimate for the magnitudes of these forces. We also see high correlations between the slab resistances and the sum of the above two forces. Therefore, each of the three forces could equally be a resistive force, but it may be difficult to discriminate which one is the main resistance.

16 384 S. SEKIGUCHI Table 3. Calculated correlations of driving forces. * A recalculated parameter in the way described in text. Besides the above, there could be some variations in these forces, due to possible changes in the parameters assumed. For this reason, we examined these variations, which refer to different motions of the Pacific plate, of which the rotational pole passes through the latitude 2 and the longitude co, with the angular velocity of co (deg/m.y.). These are expressed by Ď= , ė= , Ě= , where the first terms in each quantity are the original values and the second terms are their standard deviations (MINSTER and JORDAN, 1978). The solutions are shown in Table 1(d). As expected, they are somewhat different from the previous ones, in that the mantle drag force becomes smaller (compare Table 1(d) with 1(a)), although correlations between the resistive forces previously obtained and those obtained in this case are still very high (Table 3(b)). From the above results, we see that it is difficult to determine exactly how much the mantle drag force FDF, the slab surface resistance FSSR, and the slab edge resistances FSER contribute to cancelling the slab pull FSP, and ridge push FRP. However, from Tables 1(a)-(d), we can at least say that the sum of the magnitudes of FSSR and FSER is larger than that of FDF in almost all cases, and that more than half of FSP seems to be cancelled by FSSR and FSER. The conclusions obtained in 3.1 and 3.3 are not affected by this calculation.

17 The Magnitude of Driving Forces of Plate Motion Stress estimate Now, we evaluate the possible range of the stress value for the forces given in Tables 1(a)-(d). To evaluate the magnitude of the stress, we proceeded with the following assumptions. For the following six forces with velocity dependence (FDF, FCD, FCR2, FSSR, FSER1, and FSER2), we assumed a plate velocity of 10 cm/year. For the colliding resistances (FCR1 and FCR2), the width of contact area between the two plates was assumed to be 100 km, in the calculations. For the slab edge resistances (FSER1 and FSER2),we assumed that the thickness of all subducting slabs would be 100 km, and that the resistance works on the upper and lower surfaces of the downgoing slab. The evaluated magnitudes under these assumptions range from -1 to -2 bars for the mantle drag, -2 to -33 bars for the continental drag force, and from -100 to -220 bars and -160 to -230 bars for the two types of colliding resistances, F0R1 and FCR2, respectively. The slab surface resistance has a magnitude of about -30 to -140 bars, comparable to that of the above forces, while two different slab edge resistances, FSERI and FSER2, have larger magnitudes, ranging from -0.7 to -3.6 kbars and -2.6 to 6.1 kbars, respectively. From the above assumptions, and from possible variations in the many parameters included, it should be understood that the above - estimated values indicate only the order of magnitude. Nevertheless, we are able to make the following comparisons. Since KANAMORI and ANDERSON (1975) show that the stress drop for interplate earthquakes is about 30 bars, the above magnitudes, estimating for the colliding resistances, are not inconsistent with this obervation, if the interplate earthquakes releases only a part of the tectonic stress applied to the plate boundaries. The resistive stress of the mantle drag is weaker than that of the slab surface resistance. This suggests that the viscosity under the plate may be smaller than that of the deeper mantle. The stress values of the slab edge resistances indicate that high stress may be concentrated around the leading edge of the downgoing slab. 4. Discussion The relative magnitude obtained here for the torque of the ridge push, compared to that of the slab pull, is appreciably larger than that evaluated by FORSYTH and UYEDA (1975) and CHAPPLE and TULLIS (1977). The reason for this may be that the magnitude of the ridge push was treated as an unknown parameter in the above two studies. It has been confirmed for the model shown in Figs. 4(a) and (b) that, if we take the magnitude of the ridge push as an unknown parameter, the obtained magnitude becomes very small as in the case of the above two studies (Table 1(e)). For this reason, we have taken the ridge push as a known parameter, in order to obtain more reliable estimates. HAGER and O'CONNELL (1981) showed that the magnitude of the ridge push is of comparable order with that of the slab pull, by modelling the mantle as a Newtonian fluid with radially symmetrical viscosity distribution. They attempted to quantify the driving forces, taking into

18 386 S. SEKIGUCHI account the density contrast arising from the thickening and the subduction of plates. Although these forces correspond approximately to the ridge push and the net slab pull now in consideration, there are some problems in their treatment, because they assumed the same viscosity in the subducting slab as in the surrounding mantle. For this reason, their results cannot be directly compared with those from this study. We have shown in 3.4 that the net driving force appears to be smaller than about half of the slab pull force. On the other hand, CHAPPLE and TULLIS (1977) calculated the slab pull force, taking into account the focal mechanism of earthquakes that occurred within the slab. In their calculation, however, most part of the slab pull has been cancelled so that the net driving force becomes very small. To examine whether this result still holds when the slab resistance is included into their forces, we made further calculations. We incorporate the slab resistance working on the surfaces of slabs in the same way as introduced in this study. Because the focal mechanism of earthquakes at depths below 350 km indicates down-dip compression, we assume here that the portion of slabs deeper than 350 km would not pull down the slab. It has been suggested that the focal mechanisms in the Tonga and IZU-Mariana regions show down-dip compression throughout the downgoing slabs. Since the slab pull forces from the above two slabs will have considerable importance on our calculations, we shall consider the following two different cases ; in case 1 the slabs in Tonga region are not included, and in case 2 the slabs in the Izu-Mariana and Tonga regions are not included. The suction force is assumed, in these cases, to be proportional to the product of the trench length and the downgoing plate velocity. The results given in Table 1(f) indicate that a large slab pull force works on the lithospheric plates. This implies that the slab pull force might not be completely cancelled, and that a large part of the slab pull might be transmitted to the horizontal portion of the plate. We did not consider mantle convection, driven by plate motion. Evidence to justify the above treatment comes from the work done by CHASE (1979) and HAGER and O'CONNELL (1979). They calculated the mantle convection in kinematic models, and showed that a better agreement in the magnitude and the direction between plate motions and mantle flows, just under plates, is found for large and fast-moving lates, rather than for small or slow-moving plates. It has thus been confirmed that our formulations for the mantle drag force can be appropriately applied to large and fast-moving plates. We can also see, from the formulation of driving forces in this study, that dominant plates with long subducted slabs, high plate velocity and large area, such as the Pacific and Indian plates, will have controlling effects on the results. Therefore, even if we take the mantle convection into the present calculation, it would not have serious effects on the main results obtained in this study. There leaves a possibility that the formulations for the colliding forces might not be appropriate, or might be too simple to represent the true physical processes, where these forces are exerted on the plates. It is possible that the collid-

19 5 The Magnitude of Driving Forces of Plate Motion 387 ing resistances at the continent-continent boundary, continent-ocean boundary, and ocean-ocean boundary might be working in different ways. Moreover, there could be some difference in the colliding resistance, even within a single plate boundary, because the convergence or subduction rate, and the age of the lithosphere could influence seismic coupling in the subduction zone, as has been suggested by RUFF and KANAMORI (1980), and PETERSON and SENO (1984).. Conclusions The main point of the present study, which is significantly different from the previous studies, was to evaluate the magnitude of the ridge push and slab pull forces from known quantities, before solving the equations of torque balance for all forces. We also took into account the slab surface and slab edge resistances. The conclusions we have obtained from the present study are summarized as follows. 1. The magnitude of the slab pull forces is estimated to be about five times larger than that of the ridge push forces. 2. The magnitude of the slab pull force exerted on the Pacific plate exceeds 40 % of the total slab pull forces. The magnitude of the ridge push force working on the Pacific plate is the largest among the forces exerted on the other plates, but their difference is not so large as in the case of the slab pull forces. 3. There is a high correlation between the mantle drag force and the sum of the slab pull and ridge push forces, which makes it difficult to estimate the net driving force definitely. However, the slab resistances appear to contribute more to cancelling the driving forces than the mantle drag force. 4. The North American and South American plate, which have short and shallow slabs but long oceanic ridges, appear to be driven by the ridge push force. 5. Higher stresses appear, concentrating around the leading edge of the downgoing slabs. The following conclusions are almost the same as in those obtained by earlier workers, but have been confirmed in this study. 6. The slab pull and the ridge push act as driving forces of the plate motions, and the mantle drag, the slab surface resistance and the slab edge resistance act as resistive forces to the plate motions. The continental drag and the colliding resistances are quite small but also work as resistive forces. The suction force is so small, that it is comparable to the standard deviation. I am grateful to Prof. T. Mikumo for his encouragement, and many useful discussions. I thank Dr. K. Shiono, who sent me a list of papers useful for determination of slab shapes, and Dr. K. Hirahara, who allowed me to use his computer program for the maximum likelihood method. I also wish to thank anonymous reviewers for their constructive comments. Computations were made at the Information Processing Center of Disaster Prevention Research Institute, Kyoto University.

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