Earthquakes and Plate Tectonics Note 3 T. Seno (Earthquake Res Inst, Univ of Tokyo) (Revised on April 6, 2006; * indicates part given in the lecture)

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1 Earthquakes and Plate Tectonics Note 3 T. Seno (Earthquake Res Inst, Univ of Tokyo) (Revised on April 6, 2006; * indicates part given in the lecture) 3. Plate kinematics 3.1 Motion on the sphere Let Plate A move around on the surface of a sphere (Earth). Plate A moves to another position as A' (Fig. 1). There is a rotation, i.e., a pole P (fp, lp) and a rotation angle q, which shifts A to A', and this is unique (except for antipode). This is the Euler's theorem, which is the most important for plate motion description. Although it may be difficult to prove this theorem, it would be enough for you to imagine that it is correct intuitively. If you accept this correct, any movement of a plate on the Earth's surface can be described by rotation, i.e., a set of threee parameters fp, lp, and q. Reference frame* However, in any description of motions, it is necessary to specify from what you are looking the motion. It is the matter of reference frame. In plate motions, we use two types of reference frames. In the first case, which is called relative plate motion, we take one plate as a reference. For example, we are riding on plate B and look the motion of plate A. In this case, plate B is fixed and used as a reference. In the second case, which is called absolute plate motion, we take deep mantle as a reference. However, this is practically inconvenient because the deep mantle is not easy to reach and invisible. Instead of taking the deep mantle, we use hotspots as proxy, because hotspots are volcanoes produced by plumes rising from the deep mantle and probably fixed to it. Because hotspots are on the surface and visible, and produce a trace along with the motion of a plate over the hotspots (Fig. 2). We can get information on plate motions from hotspot traces. Finite rotation* Let fix one reference frame, either plate B or hotspots, and assume plate A moves to another position A' during a geological period, for example, from 100 Ma to 90 Ma. This kind of motion is called finite rotation. For plate reconstruction for the geological past, for example, a reconstruction at the breakup of Pangea, these finite rotations are used. We only need to specify, the time period, pole, and a rotation angle. A successive series of these three parameters are put in a table called stage poles (Table 1). If you rotate by at once, plate A from t0 to tn, around a pole and rotation angle, it is called total reconstruction. Instantaneous rotation* Although finite rotations are useful for geological problems, it is not so for tectonics occurring at present or, in other words, for earthquakes. We rather want to know present velocities of one plate to another plate at specific locations. For this, there is an analogie to the motion within a 3-dimensional Euclid space (Fig. 3). A point x moves to another location x'. This is a finite motion. We can obtain the

2 velocity v, by taking the time derivative of x', as dx'/dt. A similar procedure in the rotation is the time derivative of the finite rotation, and we obtain w = dq/dt(1), where w is called the angular velocity. The relationship between w and q is exactly the same as that between v and x'. However, of course, w is not a velocity at some specific point. To obtain a velocity v at position x on the sphere, we can use a geometry of a rotation around a pole as shown in Fig. 4. Let the angle between the pole P and position x be D, and the radius of the earth be R, the arm length of rotation, L, is RsinD. Then the absolute value of the velocity is v=lw = RsinDw(2). The direction of the velocity v is perpendicular to the plane which contains the pole P, the earth's center O, and position x. Then, v OP, x (3). There is a more elegant way to desribe these two relations. Let us define a vector which is called Euler vector (or instantaneous rotation vector, or w-vector). This vector is directing from the center of the earth to the pole P, and its magnitude is w. Then v is written as v=w x x(4). This can be proved as follows. Generally, let c be a cross-vector of aand b (i.e., c = a x b), then c satisfies the two following relations, c=absind(5) and c a, b (6). It is seen, from (2) and (3), that v satisfies both of these relations, if a = w, and b = x. Therefore velocity v at any location x can be obtained, if w is given, using (4). To determine plate motions is, therefore, in other words, to obtain w-vectors. A w-vector is specified by the three parameters (fp, lp, w). An example of a set of w-vectors is given in Tables 2~4. Usefulness of using w- vectors is that they are vectors and hence they are additive. Let A, B, C be three independent plates, and AwC, for example, be the w-vector representing the motion of plate A to plate C, a relation AwC = AwB + BwC (7)

3 holds. This is proved as follows. At arbitrary point x, three is a relation AvC = AvB + BvC(8), where v is a velocity vector in an Eucrid space. Replacing v by w x x, we obtain (AwC - AwB - BwC) x x= 0(9). This holds for arbitrary x. Then (7) results. Using this additiveness, we can get AwC between A and C even if no geophysical or geological information exists at the boundary A-C, if there is informaiton at the boundaries A-B and B-C. In fact, the convergence of the Pacific plate beneath the Eurasian plate along the Japan trench was first obtained by addition of EUwNA and NAwPA, without any information on the Japan Trench (Le Pichon, 1968). Relation (9) holds for three arbitrary plates. If you note that the additiveness of the vectors holds for arbitrary number of velocities at x, we obtain the following general relation for n plates, A1... An, A1wAn = A1wA2 + A2wA An-1wAn(10). 3.2 Calculation of plate motions Although relation (4) is a simple formula to give a velocity vector, it is not convenient to calculate the velocity by hand because it is written in a vector form. Using the cosine law of the spherical trigonometry is an easier way to calculate the velocity at x. Let A, B, and C the vertices of a triangle on the sphere (Fig. 5). There is a relation between a, b, c and C, as cosc = cosacosb + sinasinbcosc(11), where a, b, c are the angles corresponding to A, B, and C (Fig. 5a). Let P, x, and Norh Pole (N.P.) (Fig. 5b) correspond to A, B and C. P and x have geographic coordinates (latitude, longitude) as (fp, lp) and (f, l). Then a = 90 - f, b = 90 - fp, c = D, C = l - lp in Fig. 5b. Put these into (11), we obtain cosd = cos(90 - f)cos(90 - fp) + sin(90 - f)sin(90 - fp)cos(l - lp) = sin fsin fp + cos fcos fpcos(l - lp) (12) Given P(fp, lp) and x(f, l), we can calculate D from (12). Because R = 6371 km and w = w is given, we can obtain v from (2). One thing to note is that w is usually given by an unit of /m.y. (Tables 2-4), then we need to convert w to radian/m.y. using 1 = p/180 radian. After v is obtained, its direction in the geographic coordinate can be calculated as follows. The direction of v is perpendicular to Px (Fig. 5b). Then let the angle at vertex x be a, the velocity direction is 90 - a. a can be obtained using equation

4 (11) again, but in this case, x, N.P., P correspond to C, A, and B of Fig. 5a. Now a = D, b = 90 - f, c = 90 - fp, and C = a. Put these into (11), we obtain cos(90 - fp) = cos(90 - f)cosd + sin(90 - f)sindcosa (13), whichi gives, sin fp = sin fcosd + cos fsindcosa (14) From this, we can obtain a and the velocity direction. One thing to note is when P is located to the east of x, the direction of veolcity is 90 + a. Practice Given the Eurasian (EU)-Philippine Sea (PH) plate motion as pole: 45.5 N, E, w = 1.2 /m.y. (Seno, 1977), calculate the velocity of EU to PH at 35 N, 139 E (Izu Peninsula). Practice Select a region near your country, and select an appropriate plate-pair, which may play an important role in tectonics in this region. Calculate relative motions between these plates at several locations. Do the same by selecting another plate-pair, if possible. 3.3 Determination of plate motions The plate tectonics as a theory to explain the deformation on the earth's surface started when McKenzie and Parker (1967) determined the pole of the Pacific-North American plate motion, and Morgan (1968), almost at the same time, determined poles and w among a few major plates (American, African, Pacific, and Antarctic). Since then, a number of plates for which motions are determined has increased, and their w-vectors have been revised. One of such model, NUVEL-1, is listed in Table 2. Recently, space-geodetic data are also used to determine plate motions (e.g., Prawirodirdjo and Bock, 2004). The method to determine plate motions, i.e., to determine w-vectors is the procedure opposite to the procedure to calculate plate motions. To do that, data of plate motions, i.e., some data on v (direction and magnitude) should be given. Let's first try to determine a pole P when two direction data, vx and vy, are given at locations x and y, respectively (Fig. 6). Draw a great circle Gx, which is perpendicular to vx and goes through position x. Draw another great circle Gy, which is perpendicular to vy and goes through position y. Then a cross point between Gx and Gy gives P. This is because vxop and x, then P must be on Gx (Fig. 6), and similarly P must be on Gy. Therefore P must be on both Gx and Gy, and the cross point becomes the pole. If the number of data of the direction is more than two, we can make a number of cross points, as shown in Fig. 7 (Morgan, 1968), and take an average of them. After P is obtained, the procedure to get the value of w is as follows. Let, at location z, a datum of magnitude of v be given as v' in the direction making an angle b to the direction calculated from the pole.

5 Then using (2), we obtain v'/cosb =RsinDw (15), which gives the value of w, because D can be calculated from (12). If we have many data of v', we can take an average in this case too. However, in such a case when a number of data are obtained for both the directions and velocities, we usually use a least-squares method. Let xi be the location where datum doi is obtained (i =1,..., n). First assume an initial model of w (fp, lp, w), and calculate a predicted value dci at location xi from this model, and make a sum of the differences between the observed and predicted data, as c2 = Sdoi - dci2/si2 (16), where si is the observation error. By making c2 the least, we obtain the best-fit solution of w (fp, lp, w). References for figures and text Figures without reference are from Seno, T. Basics of Plate Tectonics, Asakura Pub., 1995, 190. pp (in Japanese) or are the originals by Tetsuzo Seno for this lecture. Chu, D., and R. G. Gordon, Evidence for motin between Nubia and Somalia along the Southwest Indian ridge, Nature, 398, 64-67, DeMets, C. R., R. G. Gordon, D. Argus, and S. Stein, Current plate motions, Geophys. J. Inter., 101, , Engebretson, D. C., A. Cox, and R. G. Gordon, Relative motions between oceanic and continental plates in the Pacific Basin, Geol. Soc. Am. Spec. Pap., 206, 59 pp., Harada, Y., and Y. hamano, Recent progress on the plate motion relative to hotspots, The History and Dynamics of Global Motions, Geophysical Monograph, 121, , Le Pichon, X., Sea-floor spreading and continental drift, J. Geophys. Res., 73, , McKenzie, D. P., and Parker, R. L., The north Pacific: an example of tectonics on a sphere, Nature, 216, , Morgan, W. J., Rises, trenches, great faults, and crustal blocks, J. Geophys. Res., 73, , Prawirodirdjo, L., and Y. Bock, Instantaneous global plate motion model from 12 years of continuous GPS observations, J. Geophys. Res., 109, B08405, doi: /2003jb002944, Seno, T., The instantaneous rotation vector of the Philippine Sea plate relative to the Eurasian plate, Tectonophysics, 42, , Seno. T., S. Stein, and A. E. Gripp, A model for the motion of the Philippine Sea plate consistent with NUVEL-1 and geological data, J. Geophys. Res., 98, , 1993.

6 Table 1 Reconstruction of major continents with respect to hotspots (Engebretson et al., 1985)

7 DeMets et al. (1990)

8 Seno et al. (1993) Table 4 Rotation vectors between Nubia and Somalia SM-NB N 36.2 E /my Chu et a. (1999)