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Ridge migration and asymmetric sea-floor spreading
Earth and Planetary Science Letters, 36 (1977) 51-62 O Elsevier Scientific Publishing Company, Amsterdam - Printed in The Netherlands
51
[11
RIDGE MIGRATION AND ASYMMETRIC SEA-FLOOR SPREADING SETH STEIN, H.J. MELOSH and J.B. MINSTER Seismological Laboratory, Division o f Geological and Planetary Sciences, California Institute o f Technology, Pasadena, Calif. 91125 (USA) Received October 15, 1976 Revised version received April 28, 1977 We propose that asymmetric sea-floor spreading occurs as a consequence of the relative motion between ridges and slow-moving mantle material below. A fluid mechanical model of asymmetric spreading predicts that the trailing flank of a ridge migrating with respect to the mantle spreads fastest. Observed asymmetries are compared to those predicted by ridge migration velocities. Although the magnitude of the asymmetry appears to depend as much on local effects as on the migration of the ridge, the direction of asymmetry agrees with our prediction in most locations. In contrast, models in which the ridge attempts to remain above a source fixed in the mantle predict the opposite direction of asymmetry. Other models, which attribute asymmetric spreading to asymmetric cooling, require large deviations from the standard depth-age relationship, while our model does not.
1. Introduction Asymmetric sea-floor spreading was noted as early as 1968 by Dickson et al. [1 ] in the South Atlantic. It has since been reported south o f Australia [2], north o f Iceland [3] and at many other locations. Suggestions as to its cause have been offered by Weissel and Hayes [4], Solomon et al. [5], Hayes [6], and Rona [7]. Our model, in which the migration o f oceanic ridges with respect to the deep mantle causes asymmetric spreading, uses absolute motions to predict a direction o f asymmetry both o f magnetic anomalies and b a t h y m e t r y . These predictions are tested against published data and found to be in good agreement in most places. The magnitude o f the asymmetry, however, seems to vary dramatically over short distances along a ridge and to change with time. These details o f the asymmetric spreading process are too complicated for our simple fluid mechanical model to describe. We compare the predictions of the ridge migration model to those o f asymmetric cooling models [6] and models Contribution No. 2789 of the Division of Geological and Planetary Sciefices, California Institute of Technology, Pasadena, California 91125, U.S.A.
in which the ridge spreads asymmetrically so as to remain above a source fixed in the mantle.
2. Geometry of ridge migration We first derive expressions to describe the relative motion between the lithosphere and mantle at a ridge. Using absolute plate velocity vectors we construct a vector which measures the migration o f the ridge in the chosen absolute reference frame. This frame may be viewed as one fixed with respect to a slowly moving mantle [8]. Using this migration vector, we then model the mechanics o f a ridge to predict the eventual occurrence and direction o f asymmetric spreading. Consider two plates, i and ], with instantaneous rotation vectors ~ i and ~ j in a chosen absolute reference frame. At a point on the ridge r, velocities on either side o f the ridge are:
rIj
(l)
At some time later, the ridge has migrated to a new point. As shown in Fig. 1, the velocity o f the ridge is
52
3. A fluid dynamic model for asymmetric spreading
// P o,ei Vi
j~
t+ZXt
'
Time Fig. 1. Geometry of ridge migration. Pi, Pl are absolute motion vectors, i~I, ~j are spreading vectors a'nd ~a is the asymmetry vector, rr7 is the migration vector, giving the component of the ridge's motion in the spreading dkection with respect to the mantle.
Fig. 2a shows a simple two-dimensional model of an asymmetrically spreading ridge. Consider a frame of reference attached to the ridge, and a region of the crust and upper mantle enclosing it. The velocity flow field in the mantle, modeled as a Newtonian viscous fluid of constant viscosity 7/, is steady only in this frame. The left-hand plate moves away from the ridge with velocity ~zwhile the right hand plate moves with ~ . At a depth h in the mantle (which may coincide with the base of the asthenosphere) there is a general flow of mantle material to the right with velocity P'm. In general we assume that this relative motion is produced by the migration of the ridge, rather than by motion within the stationary or slowly moving mantle. The mantle motion ~ _ is then the negative of the rmgratlon vector m, Vm = - - m . The general framework of this model is that of a passively spreading ridge whose total spreading rate:
v= given by:
a*= (~'/+ ~
-
2i~a)/2
(2)
where ~a is the correction vector due to any asymmetry of spreading. It is defined as: =
(3)
where ~1 and ~ I are vectors in the spreading directions whose magnitudes are equal to the half-spreading rates on their respective plates. We define these vectors using a unit vector f which points away from the ridge toward the direction of spreading on plate i. In terms of the absolute velocity vectors ~ a n d ~/this vector is defined as f = (i~i - ~)/[~i - ~1. The s'preadin~velocities of the plates relative to the ridge are thus VI = Vfi and ffs= - V f i . This convention has been chosen to ensure that Vz and Vs are both positive. The projection of a* on the spreading direction f is:
i
(5)
is controlled by external forces. Note that V is the relative velocity of the two plates: it is independent of the velocity of either the plates or the ridge with respect to the deep mantle. The ridge may spread either symmetrically or asymmetrically, without altering the absolute velocities of either plate, as long as the total spreading rate is fixed. The asymmetry of spreading is described by the parameter A (note that ~I and ~ j are colinear): (~J-
~I) A = ( ~ + ~t)
(6)
Thus, for symmetric spreading ~jr= - ~ l a n d A = 0. If
s'
~ Vot a
m = (~" 0t
I= = ( ~ + ~/21~-2ffa)-~[2"(ffi- ~i) ( ~ i -
~)
(4)
The migration vector, rn, is the component of the ridge's migration in the spreading direction. We use it to identify the trailing flank of the ridge and predict the direction of asymmetric spreading.
(a)
L
-I
(b)
~m
Fig. 2. (a) Geometry of the ridge dynamics model. The ridge is fixed in this frame of reference. Plate ] is predicted to accrete faster. (b) Boundary conditions for the fluid dynamical model. The position of the ridge is fixed in this frame. Accretion of the plates is modeled by a vertical outflow of mantle material. Schematic contours of the stream function are shown.
53 plate ] does not accrete new sea-floor material (]~j = 0), while plate i accretes at the total spreading rate V (V z = If), then A = - 1 . Conversely, if plate i does not accrete (]~I = 0 ) w h i l e ] spreads (Vjr= V),A = +1. A positive asymmetry A indicates that the trailing flank of the ridge spreads fastest, as illustrated in Fig. 2a. The fluid mechanical model described in the Appendix demonstrates that the total viscous dissipation rate = fvolume ei]olid x of the flow beneath the ridge is smaller for positive A than for negative A. We thus predict that asymmetric spreading with the trailing flank spreading fastest is favored (minimum dissipation theorems are often used to predict the forms of slow viscous flow [9]). This result is easy to demonstrate qualitatively. Fig. 2a shows that the total shear beneath plate i is of order (l•xl + IJ~ml)/h, while the shear beneath plate ] is of order (IJ~l - IJ~ml)/h. The total shear beneath plate i is always larger than that beneath plate ]. Each plate accretes new material from the mantle at a rate dependent upon the total spreading rate and the degree of symmetry. As a plate spreads away from the ridge the flow field induced by its accretion interacts with the shear flow beneath it. Since the energy dissipated is the square of the total strain rate, the total dissipation is largest where the shear flow is largest. Thus, plate i should accrete more slowly than plate ], since the shear (hence the energy dissipation rate) beneath it is larger than the shear (energy dissipation rate) beneath plate ]. This situation is shown schematically by Figs. 2b and 3. Alternately, the same result can be seen by assuming that the higher shear stresses on plate i cause the ridge axis to make small shifts in that direction, resulting in plate ] spreading faster. The fluid mechanical model described in the Appendix makes it plausible that the motion of a ridge with respect to the mantle favors asymmetric spreading with the trailing ridge flank spreading faster. The detailed mechanism of this process, and the possible significance of the different stresses in either flank of the ridge merit more investigation than we can give here. As we shall discuss later, this effect appears to be only a tendency that biases asymmetric spreading toward one direction. The actual magnitude o f the asymmetry seems to be controlled by local effects, so that we can only predict asymmetries qualitatively. Finally, we note that the asymmetric accretion of
• I3
RIDGE FIXED FRAME V LEADING
~
Plate
Plate
i~
J
SIDE FAST MAXIMUM DISSIPATION V/2
V/2
4
SYMMETRICAL SPREADING
V TRAILING SIDE FAST PREFERRED ON GROUNDS OF MINIMUM DISSIPATION V : T O T A L SPREAOING RATE
Fig. 3. Schematic illustration of the fluid mechanical model of asymmetric spreading. The large shear stresses which develop beneath the leading flank of the ridge may be readily seen. Two extreme cases of total asymmetric spreading and the case of symmetric spreading are shown. The detailed model described in the Appendix is in accord with this schematic argument pictured here: the trailing flank of a migrating ridge should spread faster than the leading flank.
material which we have described does not involve any thermal perturbations on either flank of the ridge. The temperatures of the two flanks will be affected only by the different spreading rates. Thus we predict no deviation from the general relation between ocean depths and the square root of age shown by Sclater and Francheteau [10].
4. Computation of migration vectors The model described in the preceding section predicts a relation between ridge migration vectors and
54 asymmetric spreading. To compute migration vectors an absolute motion model must be chosen. Such models are determined by finding relative plate motions and then imposing a rigid body rotation of the entire lithosphere. A variety of different approaches have yielded generally similar absolute motion models, so the migration vectors differ only slightly for different models. The model we chose is model AM2 of Jordan [ 11 ] which requires zero absolute motion for Africa. (The idea that Africa is stationary with respect to the mantle has also been proposed on geologic grounds by Burke and Wilson [12].) A wide variety of absolute motion models are all generally consistent with the hot-spot trace data [11 ]. Selection of any one of these produces only minor changes in the direction of ridge migration vectors, except along the South Atlantic ridge and south of Africa. Our asymmetric spreading model yields best results when the South Atlantic ridge is migrating west, which occurs if Africa is fixed or is moving slowly (1 cm/yr). The migration velocities computed from AM2 are shown in Fig. 4. The arrows point in the direction of ridge migration with respect to the deep mantle (assumed fixed in the hot-spot frame) and have lengths proportional to the migration velocity. For convenience, the migration vectors are computed from equation (4)
~2ts'u\ ,I
?i "~" ) t'-113i5
3~1 ,
"
.I
~"
5"!
Fig. 4. Computed migration vectors and observed asymmetries. Asymmetries are quoted as percentages: differences in spreadhag rates divided by the sum. Asymmetries are given over the longest period available at each location, but never beyond anomaly 5. (In areas of recent asymmetries and longer term gross symmetry, e.g. FAMOUS, the recent asymmetry is plotted.) In general the plate trailing the migrating ridge seems to spread faster.
neglecting the small correction factor ~a, which is always much smaller than the total spreading rate. We also plot a small data set of asymmetries taken from published results discussed below. The shaded side of the rectangle shows the fast-spreading side. Unshaded rectangles indicate symmetric spreading. The percentage of asymmetry is also indicated. (In areas of dense measurements the range is shown.)
5. Data
Since the absolute motion model is based on instantaneous motions, we use only relatively recent data. Nothing beyond anomaly 5 (9 Myr) is included in this data set. Possible consequences of this choice are explored below. We searched the literature for published spreading rates, determined independently for both limbs. Only those compiled from profiles, rather than from processed contour maps, are used. In measuring and quoting asymmetries we attempt to exclude major shifts of the spreading axis, as a result of which entire anomalies are either repeated or lacking on one flank. On the other hand, smaller ridge jumps are difficult to separate from continuous asymmetric spreading, especially as one may be the limit of the other [13]. As we are attempting to predict asymmetric accretion with a simple model, rather than explore the precise mechanisms involved, such a distinction is not crucial for our purposes. Spreading rates are available for several sites in the North Atlantic. Johnson et al. [3] report asymmetry on the Kolbeinsy ridge, north of Iceland: the east flank spreads at 8.2 mm/yr and the west at 7.7 mm]yr out to anomaly 5. (This interpretation has been questioned by Palmason [14]). South of Iceland, on the Reykjanes ridge, Talwani et al. [15] show symmetric spreading. The FAMOUS area (36°N, 33°E) has been extensively studied using both surface and deep-tow instruments [16-18]. Both rifts show evidence of asymmetric spreading. The north FAMOUS rift spread at a half-rate of 7.0 mm/yr to the west, and 13.4 mm/ yr to the east, until anomaly 2 (1.7 Myr). Prior to this the direction of asymmetry reversed: 13.3 mm/yr to the.~est and 10.8 mm/yr to the east. These average to a smaller asymmetry over the last 4 Myr, still with the
55 east flank fast. The south FAMOUS rift spreads at 9.8 mm/yr to the west, and 10.6 mm/yr to the east (complete data is only available over Brunhes time). Despite these asymmetries surface magnetic data [19] for the FAMOUS area show gross symmetry over a much longer time period (10 Myr). At 26°N, Lattimore et al. [20] report half-rates of 13 mm/yr to the east, and 11 mm/yr to the west out to anomaly 5. (Prior to this the direction of asymmetry appears to have oscillated.) In the South Atlantic at 6°-8°S, Van Andel and Heath [21] show the east flank spreading faster with different rates out to anomalies 3 and 5. Further south, of seven measurements to anomaly 5 by Dickson et al. [1 ], five show spreading faster to the east, one shows symmetric spreading, and one shows spreading faster to the west. These are over a ridge length of 22 ° (28 ° 50°S). Similar results are given by Loomis and Morgan [22] for Project Magnet flights in the same area. Thus along the mid-Atlantic ridge there seems to be a strong tendency for the east flank to spread faster than the west during the last several million years. This effect appears real, despite the difficulties involved in measuring spreading rates to the necessary accuracy over a slow-spreading ridge. It also appears that over longer periods, tens of millions of years, the asymmetry has averaged out to produce a general symmetry. Asymmetric spreading occurs at many locations in the Pacific. Klitgord et al. [23] describe symmetric spreading on the Cocos-Pacific boundary at 21 °N. Rea [24] shows slight asymmetry at 9°-12°S, on the East Pacific rise: 80 mm/yr to the west and 77 mm/yr to the east. Herron's [25] two profiles further south (19°S, 28°S) show the Nazca plate to be spreading faster than the Pacific plate over the past 5 Myr. At 31°S Rea [26] finds average east and west flank rates of 86 and 77 mm/yr, respectively, since 2.41 Myr ago. The Costa Rican rift at 83°W spreads faster to the south (Nazca) side than to the north [23]. Further west (86°W) the Galapagos ridge shows the reverse direction out to Jaramillo time [23]. Hey [13] also finds the north flank spreading faster near the Galapagos (99°-93°W), and the south flank faster at 83 ° 84°W on the Costa Rican rift. The Chile ridge shows symmetric spreading [27]. The profiles of Molnar et al. [28] in the South Pacific show the Antarctic side (out to anomaly 2 or
2') to be spreading faster in almost all cases over a range from 110°W to 180°W and from 35°S to 65°S. (Measuring only out to 2 or 2' avoids confusion with ridge jumps which appear on the same profiles.) This asymmetry was noted by Herron [29]. Falconer [30] reports symmetric spreading at 61°S, 161°E on the India-Antarctica boundary. South of Australia, Weissel and Hayes [2] show one profile with the Indian side fast, and another with the reverse out to 17 Myr. (Prior to 17 Myr both profiles had the north side fast.) Further west, at 40°S, 80°W Schlich and Patriat [31 ] (profile GA1-6, anomaly 1-2) shows Antarctica spreading faster than India. It appears, then, that asymmetric spreading is a common worldwide phenomenon. Frequently the direction of asymmetry over the last several million years is the same for substantial distances along the ridge (for example, the mid-Atlantic and Pacific-Antarctic ridges).
6. Analysis The majority of these measurements are consistent with the prediction of our model: the flank trailing the migrating ridge spreads faster. Only a small fraction of the data show asymmetries opposite those predicted. Frequently the spreading is symmetric, but our model predicts a preferred direction of asymmetric spreading rather than its certain occurrence. The most convincing evidence for a fundamental process of asymmetric spreading is the data [1,21,22] in the South Atlantic and [28,29] in the South Pacific. The consistency of the direction of asymmetry over thousands of kilometers argues against its being a purely local phenomenon. On the other hand, the magnitude of the asymmetry varies considerably. Measurements (central anomaly to 2 or 2', or any more sophisticated scheme) on the South Pacific profiles yield asymmetries varying between 5 and 25%. Some of this variation may be due to noise in the data, but it still appears that the magnitude of the asymmetry is being controlled by local effects. Many processes at the ridge, unrelated to ridge migration, can influence the pattern of spreading. These include structural control of intrusion patterns, the shape and cooling history of magma chambers, and local irregularities in mantle flow.
56 The full complexity of the process of asymmetric spreading can be seen in detailed studies of the FAMOUS area [17,18]. Two adjacent ridge segments approximately 50 km apart have very different asymmetries. The direction and magnitude of asymmetry changes quite rapidly (in less than 0.15 Myr). The spreading is quite oblique (17°). At least in this area, asymmetric spreading occurs on a very fine scale, with no evidence of discrete ridge jumps of more than several hundred meters. Clearly no simple model of the dynamics of a ridge can adequately describe these local effects. Thus it would be surprising to find a simple relation between migration velocity and asymmetry. It would not be surprising to find occasional sites where local effects counteract or even reverse the effects of ridge migration. Another possible problem is our assumption that the ridge is migrating much faster than any motion in the underlying mantle. If this assumption is violated then the migration vector will not predict the asymmetry. This may be the case near the Galapagos hotspot. Hey [13] suggests that asymmetric spreading on-the Galapagos ridge is related to the hot-spot to the south. If there is substantial outflow of material from the hot-spot then the direction of net motion between the ridge and asthenosphere would be the reverse of that predicted by the migration vector. In such a case the north side, as observed, would be the fast-spreading side. This area generally coincides with the anomalous petrology of the Galapagos melting anomaly [32], often attributed to the hot-spot. It is interesting to note that of two profiles shown by Klitgord et al. [23] the one on the Galapagos ridge (86°W) shows the north side fast, while further away (83°W) the south side is fast. Hey's [13] profiles at 99°-93°W show the north side fast, and at 83 ° - 8 4 ° w also have the south side fast. Thus the Costa Rican rift may be far enough away that it is not affected by any outflow from the hot-spot. An alternative theory of asymmetric spreading has been advanced by Weissel and Heyes [4] and Hayes [6]. This model is based on their observation that bathymetric data south of Australia shows depths on the south (Antarctic) flank of the ridge consistently shallower than to the north. The conventional depth-age relation [10] would then imply that the south flank is spreading faster. This is the direction of asymmetry
predicted by our migration vector. Yet before 17 Myr their analysis of magnetic anomalies shows the reverse the north flank fast (both in "zone A" and "zone -
C"). To explain this discrepancy Hayes invokes a therma model of asymmetric cooling in which the Antarctic plate is "hotter" and spreads more slowly. This model has been criticized in some detail by Sleep and Rosendahl [34]. Using Sleep's [33] thermal model of an asymmetrically spreading ridge, they show that higher temperatures persist at greater distances from the axis on the fast side. This causes preferential intrusions which reposition the ridge axis and thus oppose the asymmetric spreading. They also show that if a thermal perturbation related to a "hot-spot" causes asymmetric cooling, thermal conduction would prevent the process from continuing for any appreciable length of time. A second difficulty with this asymmetric cooling model is the prediction that in areas of asymmetric spreading the usual depth-age relation would be violated. Trehu's [35] study of the South Pacific and Galapagos ridges shows that on both ridges the bathymetric asymmetries are in complete accord with magnetic asymmetries. This contradicts an asymmetric cooling model but agrees with a ridge migration model which predicts no anomalous thermal effects. The final problem with the asymmetric cooling model is that it would require a symmetric and relative. ly uniform distribution of hot-spots on the slower side in order to account for the apparent consistency of the direction of asymmetry over large segments of ridges. Thus it seems unlikely that the phenomena observed south of Australia are representative of the general process of asymmetric spreading. Furthermore, the structural and bathymetric complexities of this area may indeed be due to some local effects, but asymmetric cooling is probably not an adequate explanation. It is also worth noting that the direction of asymmetry predicted by any model in which the ridge attempts to remain above a source fixedin the mantle is the opposite of what a migration model predicts. Essentially, in such a model the leading flank of the migrating ridge would spread faster so as to prevent the ridge from moving away from its previous location. Most of the data seems to show the reverse - the trailing flank spreads fastest. Our model does not allow us to predict asymmetries
57 beyond about 9 Myr. Many authors [21-23] have shown spreading rate changes between the present and that time. Presumably, by the time of anomaly 5, plate rates were different enough to make the calculation of absolute motion models difficult. There is also the related difficulty that prior to that time hot-spots may have had substantial relative motion [36]. It is thus difficult to determine the relation between older asymmetries and ridge migration. Despite these difficulties, on a regional or global scale, the directions of the recent asymmetries seem well correlated with the direction of ridge migration. Further tests of this theory can be made by examining larger and more homogeneous data sets of magnetic anomalies. Additional data would also allow increased resolution of the local details of the process of asymmetric spreading.
7. Conclusion Ridge migration with respect to the mantle and asymmetric spreading appear related. The direction of the observed asymmetry seems to be that predicted by considerations of ridge mechanics and "absolute" motion models. Although our simple model does not
completely describe the dynamics of the spreading process, we expect that more complex and accurate models will also yield asymmetric spreading with a preferred direction resulting from ridge migration. Despite the noise in the data and random local effects, observations show a definite tendency for the plate trailing the migrating ridge to spread faster.
Acknowledgements We thank Hiroo Kanamori, Robert Geller, Emile Okal, David Hadley, Carl Johnson, Dan Kosloff, Kevir Burke, and Tom Jordan for advice and assistance. Peter Molnar provided us with the South Pacific profiles. Dallas Abbott, Norm Sleep, Rob Cockerham, and Dick Hey offered us critical comments on an earlier draft. Norm Sleep also kindly allowed us to use his results prior to publication. We have benefited from reviews by Ken Macdonald and an anonymous reviewer. Seth Stein was supported by a fellowhip from the Fannie and John Hertz Foundation. This research was also supported by National Science Foundation Grant EAR76-14262. Contribution No. 2789, Division of Geological and Planetary Sciences, California Institute of Technology, Pasadena, California.
Appendix. A fluid mechanical model of asymmetric spreading In this appendix we consider the fluid mechanical model in more detail. Although a number of approximations are necessary, the model contains the essential features of the previous argument. One of the approximations is to treat the region of flow beneath the ridge as if it had a horizontal top as in Fig. 2b, rather than the complex upper boundary shape in Fig. 2a. This approximation is justified for small asymmetries (when A ~ 0.3 the plate thicknesses begin to differ appreciably, thus distorting the flow). Moreover, the region of integration must be sufficiently close to the ridge that the thickness of the lithosphere is much smaller than the thickness of the asthenosphere (which is well satisfied within a few hundred kilometers of the ridge). Given these approximations, a fluid mechanical model can be constructed. This model shows that asymmetric spreading with the trailing flank spreading fastest is favored when a ridge moves with respect to the mantle below. The total flow field beneath an asymmetrically spreading ridge can be represented by a superposition of three components (Fig. 2b). The first component is the shear flow due to differential motion of the ridge and mantle. The shear strain rate S due to this flow is S = ~ = - I T , n • t / h , where f is the unit vector in the ~" direction. The flow itself is described by the stream function xI,l(x, z) from which the velocities and strain rates may be derived by the general relations: vx(x, z) = aq,,/az o,(x, z) = - a,J.q/ax
(A-l)
58 an d:
exx(X, z) = %z(X, z) = a2,~/axaz
1 [~2XItI C~2xltI~ exz(X, z) = ~ ~ z2
"
3x 2 ]
(A-2)
Using these equations, the shear flow is described by:
*l(x, z)
=
S(z2/2)
OxO)
= Sz
(A-3b)
oz(1)
=0
(A-3c)
~xx(1)
= -~zz(1) = 0
(A-3d)
~xzO)
= S/2
(A-3e)
(A-3a)
where x is the horizontal distance from the ridge axis measured along plate j and z is depth below the surface. The velocity components corresponding to this flow are not symmetric about the ridge axis. The second component is induced by separation of the two plates. The velocity of the mantle material just below plate / must be ~s, and that below plate i must be ]~z. The stream function, velocities and strain rates corresponding to this component of the flow are [9] :
~2(x, z)= V
tan -1
+
(A-4a)
Vx(2)
= V
tan -x
- x - T ~ + z2 +
(A-4b)
0,(2)
V[- - z 2 ] = 7r[.x2 + z2 j
(A4c)
V[
xx(2)
=
=
2zx 2 "] 2
+ z2) 3
V[x_(z:-x~)7 These equations describe, the flow only in the immediate vicinity of the ridge: the velocity does not vanish at the lower boundary z = h. We have chosen (A'4) because of its simplicity. Since most of the interaction between the asymmetric spreading of the plates and qz2 occurs near the ridge (in fact, qs2 is singular at the ridge itself), this ~2 is a good approximation. The third component of the flow is due to the accretion of mantle material onto the plates. We model this outflow of mantle material by imposing a vertical component of velocity Vout at the top of the mantle (Fig. 2b). This velocity is determined by the rate at which the plate thickens with age. The plate thickness is given by t = e w e , where c is a constant (c is about 8 km/Myr i/2 if the lithosphere is 70 km thick at age 80 Myr). Mass conservation then requires: 21'
Vout(X)
x
£1/i~£ 21' - x
21'
2x
)
x< 0
59 The stream function describing this outflow is the sum of 4 3 j and 431 , where: ix/i-
4aj(x, Z)
+ z : + , 0 3/2
N/~+Z 2 _ cx/'-ff
43/(X, Z)
4
-
(A-6)
(~]~ + Z2 -- X)3/2
~
Using the shorthand notation 43.i = 43+ and 431 = 4 a _ , the flow fields are described by: 43+(x, z)
-
+ z + x)
4
Vx(3+)
-
~
~
(A-7a)
(X2 + Z2)3/2
2(~iT
2X)
(A-7b)
0z(3+)
3cx/~X/~+Xz(z2
~(xax+-)
=_~(3-+)=
~3z+-)(x,z)
= ~(ax+)(-x, z )
~zz
- 1 6 c ' v v V l ±A (x 2 +z2)S/2
-
3x2+- 2xx/x2 +z 2)
(A-7d) (A-7e)
This stream function, 43f(x, z), is derived from the stream function d4out(X, z) for outflow at velocity Vout through a point sink of width dx centered at x = X~o: ~
tan-1
"~ (x - X o ) 2 +z 2)
(A-8)
Vout(x) is obtained from equation (A-5). Integration of (A-8) from x o = 0 to ~ (or - ~ ) yields 431(x, z) and 4~7(,x,z). Like the 42 stream function, the velocity fields of 43o, and 43z do not vanish at z = h. 4 : ~ and 4 3 i thus describe the flow near the ridge, but not at great depths below it. This failure of the simple model is more serious. The interaction between the asymmetrically spreading plates and the shear flow field 41 decreases slowly with increasing distance from the ridge. The total interaction energy depends upon the area of integration, and thus upon the thickness of the asthenosphere h. The fact that 4 ~ and 43z are not good descriptions of the flow in this region means that errors occur when 4 ~ and 43~r are used. On the other hand, the modifications in the flow field are not so profound that the sign of the interaction energy can change. Since our major conclusion about the asymmetry depends only upon this sign, we shall use equation (7), cutting the integrals off at a distance h from the ridge to crudely model the correct velocity boundary condition. Although this may lead to errors in the size of the interaction energy, such errors do not affect our major conclusion. The stream function uniquely specifies the flow field. The stream function of the entire flow field beneath the ridge is the sum of the separate components:
4total(X, Z) = 41 + 42 + 43,/+ 431
(A-9)
The total energy dissipation rate E in a volume near the ridge is: /~=f
• , 3 6i]ojid x
d
(A-10)
volume where o~i = 2r/e/i is the deviatoric stress. Thus:
= 27 f volume
eije]idax= 4~1f volume
(~x + ~ z )
dax
(A-1 I)
60 The strain rates in (A-11) are the sums of the strain rates due to each component of flow. For example:
1 ~ 02 ~xz ='2~22
02 ) a72 (X~tl +xI*2 + xIt3J + xt't3I)
(A-12)
When the strain rates are squared in (A-11) cross terms appear which represent interactions between the various flows. The total energy dissipation rate thus depends upon the asymmetry, ~7 = kT(A), since the coefficients of xI'as and qt3* depend upon A. Substitution of equations (A-3d, e), (A-4d, e), and (A-7d, e) into the expression (A-11) for E yields: L'(A) = r/V2 {(VII + A + x/1 - A ) I l - (x/1 + A - x / 1 - A ) [(c2S)/112]/2 } + (terms which do not depend upon A)
(A-13)
where the dimensionless integrals are:
8? I 1=if0 12=4 o
? x(2zx - z 2 + x 2) _x (X2+z2)2 f~ ,z) dxdg --~
(A-14a)
? f(x, z) dx dz
(A-14b)
__ot~
in which: 3 ~/x/~+z2+x 2 f(x, z) = 1-6 - ~ + z £ ) gig z(z - 3x 2 + 2 x ~ )
(A-IS)
These integrals may be evaluated in polar coordinates, x = - r sin 0, z = r cos 0:
3 c ~ dr ~/2 I1 - 2-7rx ~ / ~ 3 r/:x/1 - sin 0(-1 + 4 sin2 0 + 2 sin 0) .(1 - 2 sin 2 0 + 2 sin 0 cos 0) sin 0 cos 0 dO (A-16a)
/=-4
¢ ./~ o
~l-sin0(-l+4sin20+2sin0)-cos0de
(A-16b)
-7r/2
The (divergent) radius integrals are cut off at l and h respectively, while the angular integrals are finite and can be computed numerically, yielding: 0.3601c ? dr Ii-~}, 2~=0-3601~
12 = 0.404!x/-ffjh c
c
d__Lr= 0.4041 x / r ~ o 2rl/2 c
(A-17a)
(A-17b)
(r is distance from the ridge axis, r = x/~- + z2). The integral [1 is due to the interaction of q~2 and ~3J and ~aI. It diverges at small r, near the ridge axis and is cut off at a small distance l (generally taken to be 5 km)./2 is due to the interaction of xlq and ~aJ and ~3I- It diverges at large distances, and it is cut off at h. We are mainly interested in the term proportional to 12 in (A-13). As Fig. A-1 shows, the energy per unit time dissipated by the spreading is smaller when A is positive than when it is negative (for sufficiently large S > 0). In other words, when the ridge moves with respect to the mantle it is easier (in terms of energy dissipated per unit
61
.75
olaJ
.74
.73
g .72 c) "O
o~ .71
",,,,
.N m
.70
\
Z
.69 .68
t '.3
I
I
I
I
I
I
-.2
-.I
0
.I
.2
.3
Asymmetry, A
Fig. A-1. Plot of dissipation versus asymmetry for a ridge migrating with respect to the deep mantle. Curve a is for the case of no shear in the mantle below the ridge (ridge at rest). Even is this case symmetric spreading is not favored; positive or negative asymmetry is favored with equal probability. Curves b and c are for increasing amounts of shear (0.1 V/x/-~ and 0.2 V/x/~ respectively). Both curves show that positive asymmetry (trailing flank spreads fastest) is favored. The dissipation E is normalized by
n V3[2c/l.
time) for the ridge to spread asymmetrically with the trailing flank spreading faster. The fluid dynamical model described here predicts that minimum dissipation occurs for complete asymmetry, A = I. Complete asymmetry, however, violates the assumptions of the model (one plate would be infinitely thick, thus altering the domain of integration of the flow field) so that we can only conclude that motion of the ridge with respect to the mantle favors asymmetric spreading with A > 0 if S > 0. Another interesting conclusion of the model is that, even if the ridge is stationary with respect to the mantle (S = 0), symmetric spreading is not favored; asymmetric spreading in one or the other direction requires slightly less dissipation per unit time than symmetric spreading. This result follows from the coefficient o f l l in (A-13) and may be seen in Fig. A-1. This suggests that asymmetric spreading (and reversals in its direction) may occur even on a ridge which is stationary with respect to the deep mantle. When the ridge moves over the mantle sufficiently fast (on the order o f 0 . 4 V w h e n l = 5 km, h = 200 km and IAI < 0.3) the 12 term in (A-13) dominates the 11 term and the trailing flank should consistently spread faster. The forces driving this tendency toward asymmetry may be roughly estimated. The total energy dissipated per u n i t time by the passively spreading ridge is of order 2r/I/2/unit length along the ridge. For V= 10 cm/yr, 77 = 1021 poise this corresponds to 0.25 HFU when averaged over 100 km on either side of the ridge. When the migration velocity of the ridge is nearly equal to the total spreading rate I~ml ~ V, the difference in dissipation between A = +0.3 and A = - 0 . 3 is a few percent of the total dissipation. Thus, the direction of asymmetric spreading may be strongly influenced by the motion of the ridge with respect to the mantle below.
References G.L. Dickson, W.C. Pitman, III and J.R. Heirtzler, Magnetic anomalies in the South Atlantic and ocean floor spreading, J. Geophys. Res. 73 (1968) 2087.
2 J.K. Weissel and D.E. Hayes, Asymmetric spreading south of Australia, Nature 213 (1971) 518. 3 G.L. Johnson, J.R. Southall, P.W. Young and P.R. Vogt, Origin and structure of the Iceland Plateau and Kolbeinsey Ridge, J. Geophys. Res. 77 (1972) 5688.
62 4 J.K. Weissel and D.E. Hayes, The Australian-Antarctic discordance: new results and implications, J. Geophys. Res. 79 (1974) 2579. 5 S.C. Solomon, N.H. Sleep, and R.M. Richardson, On the forces driving plate tectonics: inferences from absolute plate velocities and intraplate stress, Geophys. J.R. Astron. Soc. 42 (1975) 769. 6 D.E. Hayes, Nature and implications of asymmetric seafloor spreading, Geol. Soc. Am. Bull. 87 (1976) 994. 7 P.A. Rona, Asymmetric fracture zones and sea-floor spreading, Earth Planet. Sci. Lett. 30 (1976) 109. 8 W.J. Morgan, Deep mantle convection plumes and plate motions, Bull. Am. Assoc. Pet. Geol. 56 (1972) 203. 9 G.K. Batchelor, Introduction to Fluid Dynamics (Cambridge University Press, Cambridge, 1970) 228. 10 J.G. Sclater and J. Francheteau, The implications of terrestrial heat flow observations on ~urrent tectonic and geochemical models of the crust and upper mantle of the earth, Geophys. J.R. Astron. Soc. 20 (1970) 509. 11 T.H. Jordan, The present-day motions of the Caribbean plate, J. Geophys. Res. 80 (1975) 4433. 12 K. Burke and J.T. Wilson, Is the African plate stationary? Nature 239 (1972) 387. 13 R.N. Hey, Tectonic evolution of the Cocos-Nazca Rise, Ph.D. Thesis, Princeton University, Princeton, N.J. 14 G. Palmason, Comments on 'Origin and structure of the Iceland Plateau and Kolbeinsey Ridge' by G.L. Johnson, J.R. Southall, P.W. Young and P.R. Vogt, J. Geophys. Res. 78 (1973) 7019. 15 M. Talwani, C.C. Windisch and M.G. Langseth, Jr., Reykjanes Ridge crest: a detailed geophysical study, J. Geophys. Res. 76 (1971) 473. 16 H.D. Needham and J. Francheteau, Some characteristics of the Rift Valley in the Atlantic Ocean near 36048 ' North, Earth Planet. Sci. Lett. 22 (1974) 29. 17 K.C. Macdonald, Near-bottom magnetic anonaalies, asymmetric spreading, oblique spreading, and tectonics of the accreting plate boundary on the Mid-Atlantic Ridge near 37°N, Bull. Geol. Soc. Am. 88 (1977) 541. 18 K.C. Macdonald and B.P. Luyendyk, Deep-tow studies of the structure of the Mid-Atlantic Ridge crest near 37°N (FAMOUS), Bull. Geol. Soc. Am. 88 (1977) 621. 19 J.D. Phillips and H.S. Fleming, The Mid-Atlantic Ridge west of the Azores, 35 ° -39°N, EOS, Trans. Am. Geophys. Union 56 (1975) 374 (abstract). 20 R.K. Lattimore, P.A. Rona, and O.E. DeWald, Magnetic anomaly sequence in the central North Atlantic, J. Geo-
phys. Res. 79 (1974) 1207. 21 T.H. van Andel and G.R. Heath, Tectonics of the MidAtlantic Ridge, 6 ° - 8 ° south latitude, Mar. Geophys. Res. 1 (1970) 5. 22 T.P. Loomis and W.J. Morgan, Sea-floor spreading rate changes in the South Atlantic, Mar. Geophys. Res. 2 (1973) 3. 23 K.D. Klitgord, S.P. Huestis, J.D. Mudie and R.L. Parker, An analysis of near-bottom magnetic anomalies: sea-floor spreading and the magnetized layer, Geophys. J.R. Astron. Soc. 43 (1975) 387. 24 D.K. Rea, Analysis of a fast spreading rise crest: the east Pacific Rise, 9 ° to 12 ° south, Mar. Geophys. Res. 2 (1976) 291. 25 E.M. Herron, Two small crustal plates in the South Pacific near Easter Island, Nature 240 (1972) 35. 26 D.K. Rea, Local axial migration and spreading rate variations, East Pacific Rise, 31°S, Earth Planet. Sci. Lett. 34 (1977) 78. 27 K.D. Klitgord, J.D. Mudie, P.A. Larson, and J.A. Grow, Fast sea-floor spreading on the Chile Ridge, Earth Planet. Sci. Lett. 20 (1973) 93. 28 P. Molnar, T. Atwater, J. Mammerickx and S.M. Smith, Magnetic anomalies, bathymetry and the tectonic evolution of the South Pacific since the Cretaceous, Geophys. J.R. Astron. Soc. 40 (1975) 383. 29 E.M. Herron, Crustal plates and sea-floor spreading in the Southeastern Pacific, in: Antarctic Oceanology, I, Antarctic Res. Ser. 15 (1971) 229. 30 R.K.H. Falconer, The Indian-Antarctic-Pacific triple junction, Earth Planet. Sci. Lett. 17 (1972) 151. 31 R. Schlich and P. Patriat, Anomalies magnetiques de la branche Est de la dorsale medio-Indienne entre les iles Amsterdam et Kerguelen, C.R. Acad. Sci. Paris, Ser. B 272 (1971) 773. 32 R.N. Anderson, D.A. Clague, K.D. Klitgord, M. Marshall and R.K. Nishimori, Magnetic and petrologic variations along the Galapagos spreading center and their relation to the Galapagos melting anomaly, Geol. Soc. Am. Bull. 86 (1975) 683. 33 N.H. Sleep, Formation of oceanic crust: some thermal constraints, J. Geophys. Res. 80 (1975) 4037. 34 N.H. Sleep and B. Rosendahl, (in preparation). 35 A.M. Trehu, Depth versus (age)l/2: a perspective on midocean rises, Earth Planet. Sci. Lett. 27 (1975) 287. 36 P. Molnar and T. Atwater, Relative motion of hotspots in the mantle, Nature 246 (1973) 288.
51
[11
RIDGE MIGRATION AND ASYMMETRIC SEA-FLOOR SPREADING SETH STEIN, H.J. MELOSH and J.B. MINSTER Seismological Laboratory, Division o f Geological and Planetary Sciences, California Institute o f Technology, Pasadena, Calif. 91125 (USA) Received October 15, 1976 Revised version received April 28, 1977 We propose that asymmetric sea-floor spreading occurs as a consequence of the relative motion between ridges and slow-moving mantle material below. A fluid mechanical model of asymmetric spreading predicts that the trailing flank of a ridge migrating with respect to the mantle spreads fastest. Observed asymmetries are compared to those predicted by ridge migration velocities. Although the magnitude of the asymmetry appears to depend as much on local effects as on the migration of the ridge, the direction of asymmetry agrees with our prediction in most locations. In contrast, models in which the ridge attempts to remain above a source fixed in the mantle predict the opposite direction of asymmetry. Other models, which attribute asymmetric spreading to asymmetric cooling, require large deviations from the standard depth-age relationship, while our model does not.
1. Introduction Asymmetric sea-floor spreading was noted as early as 1968 by Dickson et al. [1 ] in the South Atlantic. It has since been reported south o f Australia [2], north o f Iceland [3] and at many other locations. Suggestions as to its cause have been offered by Weissel and Hayes [4], Solomon et al. [5], Hayes [6], and Rona [7]. Our model, in which the migration o f oceanic ridges with respect to the deep mantle causes asymmetric spreading, uses absolute motions to predict a direction o f asymmetry both o f magnetic anomalies and b a t h y m e t r y . These predictions are tested against published data and found to be in good agreement in most places. The magnitude o f the asymmetry, however, seems to vary dramatically over short distances along a ridge and to change with time. These details o f the asymmetric spreading process are too complicated for our simple fluid mechanical model to describe. We compare the predictions of the ridge migration model to those o f asymmetric cooling models [6] and models Contribution No. 2789 of the Division of Geological and Planetary Sciefices, California Institute of Technology, Pasadena, California 91125, U.S.A.
in which the ridge spreads asymmetrically so as to remain above a source fixed in the mantle.
2. Geometry of ridge migration We first derive expressions to describe the relative motion between the lithosphere and mantle at a ridge. Using absolute plate velocity vectors we construct a vector which measures the migration o f the ridge in the chosen absolute reference frame. This frame may be viewed as one fixed with respect to a slowly moving mantle [8]. Using this migration vector, we then model the mechanics o f a ridge to predict the eventual occurrence and direction o f asymmetric spreading. Consider two plates, i and ], with instantaneous rotation vectors ~ i and ~ j in a chosen absolute reference frame. At a point on the ridge r, velocities on either side o f the ridge are:
rIj
(l)
At some time later, the ridge has migrated to a new point. As shown in Fig. 1, the velocity o f the ridge is
52
3. A fluid dynamic model for asymmetric spreading
// P o,ei Vi
j~
t+ZXt
'
Time Fig. 1. Geometry of ridge migration. Pi, Pl are absolute motion vectors, i~I, ~j are spreading vectors a'nd ~a is the asymmetry vector, rr7 is the migration vector, giving the component of the ridge's motion in the spreading dkection with respect to the mantle.
Fig. 2a shows a simple two-dimensional model of an asymmetrically spreading ridge. Consider a frame of reference attached to the ridge, and a region of the crust and upper mantle enclosing it. The velocity flow field in the mantle, modeled as a Newtonian viscous fluid of constant viscosity 7/, is steady only in this frame. The left-hand plate moves away from the ridge with velocity ~zwhile the right hand plate moves with ~ . At a depth h in the mantle (which may coincide with the base of the asthenosphere) there is a general flow of mantle material to the right with velocity P'm. In general we assume that this relative motion is produced by the migration of the ridge, rather than by motion within the stationary or slowly moving mantle. The mantle motion ~ _ is then the negative of the rmgratlon vector m, Vm = - - m . The general framework of this model is that of a passively spreading ridge whose total spreading rate:
v= given by:
a*= (~'/+ ~
-
2i~a)/2
(2)
where ~a is the correction vector due to any asymmetry of spreading. It is defined as: =
(3)
where ~1 and ~ I are vectors in the spreading directions whose magnitudes are equal to the half-spreading rates on their respective plates. We define these vectors using a unit vector f which points away from the ridge toward the direction of spreading on plate i. In terms of the absolute velocity vectors ~ a n d ~/this vector is defined as f = (i~i - ~)/[~i - ~1. The s'preadin~velocities of the plates relative to the ridge are thus VI = Vfi and ffs= - V f i . This convention has been chosen to ensure that Vz and Vs are both positive. The projection of a* on the spreading direction f is:
i
(5)
is controlled by external forces. Note that V is the relative velocity of the two plates: it is independent of the velocity of either the plates or the ridge with respect to the deep mantle. The ridge may spread either symmetrically or asymmetrically, without altering the absolute velocities of either plate, as long as the total spreading rate is fixed. The asymmetry of spreading is described by the parameter A (note that ~I and ~ j are colinear): (~J-
~I) A = ( ~ + ~t)
(6)
Thus, for symmetric spreading ~jr= - ~ l a n d A = 0. If
s'
~ Vot a
m = (~" 0t
I= = ( ~ + ~/21~-2ffa)-~[2"(ffi- ~i) ( ~ i -
~)
(4)
The migration vector, rn, is the component of the ridge's migration in the spreading direction. We use it to identify the trailing flank of the ridge and predict the direction of asymmetric spreading.
(a)
L
-I
(b)
~m
Fig. 2. (a) Geometry of the ridge dynamics model. The ridge is fixed in this frame of reference. Plate ] is predicted to accrete faster. (b) Boundary conditions for the fluid dynamical model. The position of the ridge is fixed in this frame. Accretion of the plates is modeled by a vertical outflow of mantle material. Schematic contours of the stream function are shown.
53 plate ] does not accrete new sea-floor material (]~j = 0), while plate i accretes at the total spreading rate V (V z = If), then A = - 1 . Conversely, if plate i does not accrete (]~I = 0 ) w h i l e ] spreads (Vjr= V),A = +1. A positive asymmetry A indicates that the trailing flank of the ridge spreads fastest, as illustrated in Fig. 2a. The fluid mechanical model described in the Appendix demonstrates that the total viscous dissipation rate = fvolume ei]olid x of the flow beneath the ridge is smaller for positive A than for negative A. We thus predict that asymmetric spreading with the trailing flank spreading fastest is favored (minimum dissipation theorems are often used to predict the forms of slow viscous flow [9]). This result is easy to demonstrate qualitatively. Fig. 2a shows that the total shear beneath plate i is of order (l•xl + IJ~ml)/h, while the shear beneath plate ] is of order (IJ~l - IJ~ml)/h. The total shear beneath plate i is always larger than that beneath plate ]. Each plate accretes new material from the mantle at a rate dependent upon the total spreading rate and the degree of symmetry. As a plate spreads away from the ridge the flow field induced by its accretion interacts with the shear flow beneath it. Since the energy dissipated is the square of the total strain rate, the total dissipation is largest where the shear flow is largest. Thus, plate i should accrete more slowly than plate ], since the shear (hence the energy dissipation rate) beneath it is larger than the shear (energy dissipation rate) beneath plate ]. This situation is shown schematically by Figs. 2b and 3. Alternately, the same result can be seen by assuming that the higher shear stresses on plate i cause the ridge axis to make small shifts in that direction, resulting in plate ] spreading faster. The fluid mechanical model described in the Appendix makes it plausible that the motion of a ridge with respect to the mantle favors asymmetric spreading with the trailing ridge flank spreading faster. The detailed mechanism of this process, and the possible significance of the different stresses in either flank of the ridge merit more investigation than we can give here. As we shall discuss later, this effect appears to be only a tendency that biases asymmetric spreading toward one direction. The actual magnitude o f the asymmetry seems to be controlled by local effects, so that we can only predict asymmetries qualitatively. Finally, we note that the asymmetric accretion of
• I3
RIDGE FIXED FRAME V LEADING
~
Plate
Plate
i~
J
SIDE FAST MAXIMUM DISSIPATION V/2
V/2
4
SYMMETRICAL SPREADING
V TRAILING SIDE FAST PREFERRED ON GROUNDS OF MINIMUM DISSIPATION V : T O T A L SPREAOING RATE
Fig. 3. Schematic illustration of the fluid mechanical model of asymmetric spreading. The large shear stresses which develop beneath the leading flank of the ridge may be readily seen. Two extreme cases of total asymmetric spreading and the case of symmetric spreading are shown. The detailed model described in the Appendix is in accord with this schematic argument pictured here: the trailing flank of a migrating ridge should spread faster than the leading flank.
material which we have described does not involve any thermal perturbations on either flank of the ridge. The temperatures of the two flanks will be affected only by the different spreading rates. Thus we predict no deviation from the general relation between ocean depths and the square root of age shown by Sclater and Francheteau [10].
4. Computation of migration vectors The model described in the preceding section predicts a relation between ridge migration vectors and
54 asymmetric spreading. To compute migration vectors an absolute motion model must be chosen. Such models are determined by finding relative plate motions and then imposing a rigid body rotation of the entire lithosphere. A variety of different approaches have yielded generally similar absolute motion models, so the migration vectors differ only slightly for different models. The model we chose is model AM2 of Jordan [ 11 ] which requires zero absolute motion for Africa. (The idea that Africa is stationary with respect to the mantle has also been proposed on geologic grounds by Burke and Wilson [12].) A wide variety of absolute motion models are all generally consistent with the hot-spot trace data [11 ]. Selection of any one of these produces only minor changes in the direction of ridge migration vectors, except along the South Atlantic ridge and south of Africa. Our asymmetric spreading model yields best results when the South Atlantic ridge is migrating west, which occurs if Africa is fixed or is moving slowly (1 cm/yr). The migration velocities computed from AM2 are shown in Fig. 4. The arrows point in the direction of ridge migration with respect to the deep mantle (assumed fixed in the hot-spot frame) and have lengths proportional to the migration velocity. For convenience, the migration vectors are computed from equation (4)
~2ts'u\ ,I
?i "~" ) t'-113i5
3~1 ,
"
.I
~"
5"!
Fig. 4. Computed migration vectors and observed asymmetries. Asymmetries are quoted as percentages: differences in spreadhag rates divided by the sum. Asymmetries are given over the longest period available at each location, but never beyond anomaly 5. (In areas of recent asymmetries and longer term gross symmetry, e.g. FAMOUS, the recent asymmetry is plotted.) In general the plate trailing the migrating ridge seems to spread faster.
neglecting the small correction factor ~a, which is always much smaller than the total spreading rate. We also plot a small data set of asymmetries taken from published results discussed below. The shaded side of the rectangle shows the fast-spreading side. Unshaded rectangles indicate symmetric spreading. The percentage of asymmetry is also indicated. (In areas of dense measurements the range is shown.)
5. Data
Since the absolute motion model is based on instantaneous motions, we use only relatively recent data. Nothing beyond anomaly 5 (9 Myr) is included in this data set. Possible consequences of this choice are explored below. We searched the literature for published spreading rates, determined independently for both limbs. Only those compiled from profiles, rather than from processed contour maps, are used. In measuring and quoting asymmetries we attempt to exclude major shifts of the spreading axis, as a result of which entire anomalies are either repeated or lacking on one flank. On the other hand, smaller ridge jumps are difficult to separate from continuous asymmetric spreading, especially as one may be the limit of the other [13]. As we are attempting to predict asymmetric accretion with a simple model, rather than explore the precise mechanisms involved, such a distinction is not crucial for our purposes. Spreading rates are available for several sites in the North Atlantic. Johnson et al. [3] report asymmetry on the Kolbeinsy ridge, north of Iceland: the east flank spreads at 8.2 mm/yr and the west at 7.7 mm]yr out to anomaly 5. (This interpretation has been questioned by Palmason [14]). South of Iceland, on the Reykjanes ridge, Talwani et al. [15] show symmetric spreading. The FAMOUS area (36°N, 33°E) has been extensively studied using both surface and deep-tow instruments [16-18]. Both rifts show evidence of asymmetric spreading. The north FAMOUS rift spread at a half-rate of 7.0 mm/yr to the west, and 13.4 mm/ yr to the east, until anomaly 2 (1.7 Myr). Prior to this the direction of asymmetry reversed: 13.3 mm/yr to the.~est and 10.8 mm/yr to the east. These average to a smaller asymmetry over the last 4 Myr, still with the
55 east flank fast. The south FAMOUS rift spreads at 9.8 mm/yr to the west, and 10.6 mm/yr to the east (complete data is only available over Brunhes time). Despite these asymmetries surface magnetic data [19] for the FAMOUS area show gross symmetry over a much longer time period (10 Myr). At 26°N, Lattimore et al. [20] report half-rates of 13 mm/yr to the east, and 11 mm/yr to the west out to anomaly 5. (Prior to this the direction of asymmetry appears to have oscillated.) In the South Atlantic at 6°-8°S, Van Andel and Heath [21] show the east flank spreading faster with different rates out to anomalies 3 and 5. Further south, of seven measurements to anomaly 5 by Dickson et al. [1 ], five show spreading faster to the east, one shows symmetric spreading, and one shows spreading faster to the west. These are over a ridge length of 22 ° (28 ° 50°S). Similar results are given by Loomis and Morgan [22] for Project Magnet flights in the same area. Thus along the mid-Atlantic ridge there seems to be a strong tendency for the east flank to spread faster than the west during the last several million years. This effect appears real, despite the difficulties involved in measuring spreading rates to the necessary accuracy over a slow-spreading ridge. It also appears that over longer periods, tens of millions of years, the asymmetry has averaged out to produce a general symmetry. Asymmetric spreading occurs at many locations in the Pacific. Klitgord et al. [23] describe symmetric spreading on the Cocos-Pacific boundary at 21 °N. Rea [24] shows slight asymmetry at 9°-12°S, on the East Pacific rise: 80 mm/yr to the west and 77 mm/yr to the east. Herron's [25] two profiles further south (19°S, 28°S) show the Nazca plate to be spreading faster than the Pacific plate over the past 5 Myr. At 31°S Rea [26] finds average east and west flank rates of 86 and 77 mm/yr, respectively, since 2.41 Myr ago. The Costa Rican rift at 83°W spreads faster to the south (Nazca) side than to the north [23]. Further west (86°W) the Galapagos ridge shows the reverse direction out to Jaramillo time [23]. Hey [13] also finds the north flank spreading faster near the Galapagos (99°-93°W), and the south flank faster at 83 ° 84°W on the Costa Rican rift. The Chile ridge shows symmetric spreading [27]. The profiles of Molnar et al. [28] in the South Pacific show the Antarctic side (out to anomaly 2 or
2') to be spreading faster in almost all cases over a range from 110°W to 180°W and from 35°S to 65°S. (Measuring only out to 2 or 2' avoids confusion with ridge jumps which appear on the same profiles.) This asymmetry was noted by Herron [29]. Falconer [30] reports symmetric spreading at 61°S, 161°E on the India-Antarctica boundary. South of Australia, Weissel and Hayes [2] show one profile with the Indian side fast, and another with the reverse out to 17 Myr. (Prior to 17 Myr both profiles had the north side fast.) Further west, at 40°S, 80°W Schlich and Patriat [31 ] (profile GA1-6, anomaly 1-2) shows Antarctica spreading faster than India. It appears, then, that asymmetric spreading is a common worldwide phenomenon. Frequently the direction of asymmetry over the last several million years is the same for substantial distances along the ridge (for example, the mid-Atlantic and Pacific-Antarctic ridges).
6. Analysis The majority of these measurements are consistent with the prediction of our model: the flank trailing the migrating ridge spreads faster. Only a small fraction of the data show asymmetries opposite those predicted. Frequently the spreading is symmetric, but our model predicts a preferred direction of asymmetric spreading rather than its certain occurrence. The most convincing evidence for a fundamental process of asymmetric spreading is the data [1,21,22] in the South Atlantic and [28,29] in the South Pacific. The consistency of the direction of asymmetry over thousands of kilometers argues against its being a purely local phenomenon. On the other hand, the magnitude of the asymmetry varies considerably. Measurements (central anomaly to 2 or 2', or any more sophisticated scheme) on the South Pacific profiles yield asymmetries varying between 5 and 25%. Some of this variation may be due to noise in the data, but it still appears that the magnitude of the asymmetry is being controlled by local effects. Many processes at the ridge, unrelated to ridge migration, can influence the pattern of spreading. These include structural control of intrusion patterns, the shape and cooling history of magma chambers, and local irregularities in mantle flow.
56 The full complexity of the process of asymmetric spreading can be seen in detailed studies of the FAMOUS area [17,18]. Two adjacent ridge segments approximately 50 km apart have very different asymmetries. The direction and magnitude of asymmetry changes quite rapidly (in less than 0.15 Myr). The spreading is quite oblique (17°). At least in this area, asymmetric spreading occurs on a very fine scale, with no evidence of discrete ridge jumps of more than several hundred meters. Clearly no simple model of the dynamics of a ridge can adequately describe these local effects. Thus it would be surprising to find a simple relation between migration velocity and asymmetry. It would not be surprising to find occasional sites where local effects counteract or even reverse the effects of ridge migration. Another possible problem is our assumption that the ridge is migrating much faster than any motion in the underlying mantle. If this assumption is violated then the migration vector will not predict the asymmetry. This may be the case near the Galapagos hotspot. Hey [13] suggests that asymmetric spreading on-the Galapagos ridge is related to the hot-spot to the south. If there is substantial outflow of material from the hot-spot then the direction of net motion between the ridge and asthenosphere would be the reverse of that predicted by the migration vector. In such a case the north side, as observed, would be the fast-spreading side. This area generally coincides with the anomalous petrology of the Galapagos melting anomaly [32], often attributed to the hot-spot. It is interesting to note that of two profiles shown by Klitgord et al. [23] the one on the Galapagos ridge (86°W) shows the north side fast, while further away (83°W) the south side is fast. Hey's [13] profiles at 99°-93°W show the north side fast, and at 83 ° - 8 4 ° w also have the south side fast. Thus the Costa Rican rift may be far enough away that it is not affected by any outflow from the hot-spot. An alternative theory of asymmetric spreading has been advanced by Weissel and Heyes [4] and Hayes [6]. This model is based on their observation that bathymetric data south of Australia shows depths on the south (Antarctic) flank of the ridge consistently shallower than to the north. The conventional depth-age relation [10] would then imply that the south flank is spreading faster. This is the direction of asymmetry
predicted by our migration vector. Yet before 17 Myr their analysis of magnetic anomalies shows the reverse the north flank fast (both in "zone A" and "zone -
C"). To explain this discrepancy Hayes invokes a therma model of asymmetric cooling in which the Antarctic plate is "hotter" and spreads more slowly. This model has been criticized in some detail by Sleep and Rosendahl [34]. Using Sleep's [33] thermal model of an asymmetrically spreading ridge, they show that higher temperatures persist at greater distances from the axis on the fast side. This causes preferential intrusions which reposition the ridge axis and thus oppose the asymmetric spreading. They also show that if a thermal perturbation related to a "hot-spot" causes asymmetric cooling, thermal conduction would prevent the process from continuing for any appreciable length of time. A second difficulty with this asymmetric cooling model is the prediction that in areas of asymmetric spreading the usual depth-age relation would be violated. Trehu's [35] study of the South Pacific and Galapagos ridges shows that on both ridges the bathymetric asymmetries are in complete accord with magnetic asymmetries. This contradicts an asymmetric cooling model but agrees with a ridge migration model which predicts no anomalous thermal effects. The final problem with the asymmetric cooling model is that it would require a symmetric and relative. ly uniform distribution of hot-spots on the slower side in order to account for the apparent consistency of the direction of asymmetry over large segments of ridges. Thus it seems unlikely that the phenomena observed south of Australia are representative of the general process of asymmetric spreading. Furthermore, the structural and bathymetric complexities of this area may indeed be due to some local effects, but asymmetric cooling is probably not an adequate explanation. It is also worth noting that the direction of asymmetry predicted by any model in which the ridge attempts to remain above a source fixedin the mantle is the opposite of what a migration model predicts. Essentially, in such a model the leading flank of the migrating ridge would spread faster so as to prevent the ridge from moving away from its previous location. Most of the data seems to show the reverse - the trailing flank spreads fastest. Our model does not allow us to predict asymmetries
57 beyond about 9 Myr. Many authors [21-23] have shown spreading rate changes between the present and that time. Presumably, by the time of anomaly 5, plate rates were different enough to make the calculation of absolute motion models difficult. There is also the related difficulty that prior to that time hot-spots may have had substantial relative motion [36]. It is thus difficult to determine the relation between older asymmetries and ridge migration. Despite these difficulties, on a regional or global scale, the directions of the recent asymmetries seem well correlated with the direction of ridge migration. Further tests of this theory can be made by examining larger and more homogeneous data sets of magnetic anomalies. Additional data would also allow increased resolution of the local details of the process of asymmetric spreading.
7. Conclusion Ridge migration with respect to the mantle and asymmetric spreading appear related. The direction of the observed asymmetry seems to be that predicted by considerations of ridge mechanics and "absolute" motion models. Although our simple model does not
completely describe the dynamics of the spreading process, we expect that more complex and accurate models will also yield asymmetric spreading with a preferred direction resulting from ridge migration. Despite the noise in the data and random local effects, observations show a definite tendency for the plate trailing the migrating ridge to spread faster.
Acknowledgements We thank Hiroo Kanamori, Robert Geller, Emile Okal, David Hadley, Carl Johnson, Dan Kosloff, Kevir Burke, and Tom Jordan for advice and assistance. Peter Molnar provided us with the South Pacific profiles. Dallas Abbott, Norm Sleep, Rob Cockerham, and Dick Hey offered us critical comments on an earlier draft. Norm Sleep also kindly allowed us to use his results prior to publication. We have benefited from reviews by Ken Macdonald and an anonymous reviewer. Seth Stein was supported by a fellowhip from the Fannie and John Hertz Foundation. This research was also supported by National Science Foundation Grant EAR76-14262. Contribution No. 2789, Division of Geological and Planetary Sciences, California Institute of Technology, Pasadena, California.
Appendix. A fluid mechanical model of asymmetric spreading In this appendix we consider the fluid mechanical model in more detail. Although a number of approximations are necessary, the model contains the essential features of the previous argument. One of the approximations is to treat the region of flow beneath the ridge as if it had a horizontal top as in Fig. 2b, rather than the complex upper boundary shape in Fig. 2a. This approximation is justified for small asymmetries (when A ~ 0.3 the plate thicknesses begin to differ appreciably, thus distorting the flow). Moreover, the region of integration must be sufficiently close to the ridge that the thickness of the lithosphere is much smaller than the thickness of the asthenosphere (which is well satisfied within a few hundred kilometers of the ridge). Given these approximations, a fluid mechanical model can be constructed. This model shows that asymmetric spreading with the trailing flank spreading fastest is favored when a ridge moves with respect to the mantle below. The total flow field beneath an asymmetrically spreading ridge can be represented by a superposition of three components (Fig. 2b). The first component is the shear flow due to differential motion of the ridge and mantle. The shear strain rate S due to this flow is S = ~ = - I T , n • t / h , where f is the unit vector in the ~" direction. The flow itself is described by the stream function xI,l(x, z) from which the velocities and strain rates may be derived by the general relations: vx(x, z) = aq,,/az o,(x, z) = - a,J.q/ax
(A-l)
58 an d:
exx(X, z) = %z(X, z) = a2,~/axaz
1 [~2XItI C~2xltI~ exz(X, z) = ~ ~ z2
"
3x 2 ]
(A-2)
Using these equations, the shear flow is described by:
*l(x, z)
=
S(z2/2)
OxO)
= Sz
(A-3b)
oz(1)
=0
(A-3c)
~xx(1)
= -~zz(1) = 0
(A-3d)
~xzO)
= S/2
(A-3e)
(A-3a)
where x is the horizontal distance from the ridge axis measured along plate j and z is depth below the surface. The velocity components corresponding to this flow are not symmetric about the ridge axis. The second component is induced by separation of the two plates. The velocity of the mantle material just below plate / must be ~s, and that below plate i must be ]~z. The stream function, velocities and strain rates corresponding to this component of the flow are [9] :
~2(x, z)= V
tan -1
+
(A-4a)
Vx(2)
= V
tan -x
- x - T ~ + z2 +
(A-4b)
0,(2)
V[- - z 2 ] = 7r[.x2 + z2 j
(A4c)
V[
xx(2)
=
=
2zx 2 "] 2
+ z2) 3
V[x_(z:-x~)7 These equations describe, the flow only in the immediate vicinity of the ridge: the velocity does not vanish at the lower boundary z = h. We have chosen (A'4) because of its simplicity. Since most of the interaction between the asymmetric spreading of the plates and qz2 occurs near the ridge (in fact, qs2 is singular at the ridge itself), this ~2 is a good approximation. The third component of the flow is due to the accretion of mantle material onto the plates. We model this outflow of mantle material by imposing a vertical component of velocity Vout at the top of the mantle (Fig. 2b). This velocity is determined by the rate at which the plate thickens with age. The plate thickness is given by t = e w e , where c is a constant (c is about 8 km/Myr i/2 if the lithosphere is 70 km thick at age 80 Myr). Mass conservation then requires: 21'
Vout(X)
x
£1/i~£ 21' - x
21'
2x
)
x< 0
59 The stream function describing this outflow is the sum of 4 3 j and 431 , where: ix/i-
4aj(x, Z)
+ z : + , 0 3/2
N/~+Z 2 _ cx/'-ff
43/(X, Z)
4
-
(A-6)
(~]~ + Z2 -- X)3/2
~
Using the shorthand notation 43.i = 43+ and 431 = 4 a _ , the flow fields are described by: 43+(x, z)
-
+ z + x)
4
Vx(3+)
-
~
~
(A-7a)
(X2 + Z2)3/2
2(~iT
2X)
(A-7b)
0z(3+)
3cx/~X/~+Xz(z2
~(xax+-)
=_~(3-+)=
~3z+-)(x,z)
= ~(ax+)(-x, z )
~zz
- 1 6 c ' v v V l ±A (x 2 +z2)S/2
-
3x2+- 2xx/x2 +z 2)
(A-7d) (A-7e)
This stream function, 43f(x, z), is derived from the stream function d4out(X, z) for outflow at velocity Vout through a point sink of width dx centered at x = X~o: ~
tan-1
"~ (x - X o ) 2 +z 2)
(A-8)
Vout(x) is obtained from equation (A-5). Integration of (A-8) from x o = 0 to ~ (or - ~ ) yields 431(x, z) and 4~7(,x,z). Like the 42 stream function, the velocity fields of 43o, and 43z do not vanish at z = h. 4 : ~ and 4 3 i thus describe the flow near the ridge, but not at great depths below it. This failure of the simple model is more serious. The interaction between the asymmetrically spreading plates and the shear flow field 41 decreases slowly with increasing distance from the ridge. The total interaction energy depends upon the area of integration, and thus upon the thickness of the asthenosphere h. The fact that 4 ~ and 43z are not good descriptions of the flow in this region means that errors occur when 4 ~ and 43~r are used. On the other hand, the modifications in the flow field are not so profound that the sign of the interaction energy can change. Since our major conclusion about the asymmetry depends only upon this sign, we shall use equation (7), cutting the integrals off at a distance h from the ridge to crudely model the correct velocity boundary condition. Although this may lead to errors in the size of the interaction energy, such errors do not affect our major conclusion. The stream function uniquely specifies the flow field. The stream function of the entire flow field beneath the ridge is the sum of the separate components:
4total(X, Z) = 41 + 42 + 43,/+ 431
(A-9)
The total energy dissipation rate E in a volume near the ridge is: /~=f
• , 3 6i]ojid x
d
(A-10)
volume where o~i = 2r/e/i is the deviatoric stress. Thus:
= 27 f volume
eije]idax= 4~1f volume
(~x + ~ z )
dax
(A-1 I)
60 The strain rates in (A-11) are the sums of the strain rates due to each component of flow. For example:
1 ~ 02 ~xz ='2~22
02 ) a72 (X~tl +xI*2 + xIt3J + xt't3I)
(A-12)
When the strain rates are squared in (A-11) cross terms appear which represent interactions between the various flows. The total energy dissipation rate thus depends upon the asymmetry, ~7 = kT(A), since the coefficients of xI'as and qt3* depend upon A. Substitution of equations (A-3d, e), (A-4d, e), and (A-7d, e) into the expression (A-11) for E yields: L'(A) = r/V2 {(VII + A + x/1 - A ) I l - (x/1 + A - x / 1 - A ) [(c2S)/112]/2 } + (terms which do not depend upon A)
(A-13)
where the dimensionless integrals are:
8? I 1=if0 12=4 o
? x(2zx - z 2 + x 2) _x (X2+z2)2 f~ ,z) dxdg --~
(A-14a)
? f(x, z) dx dz
(A-14b)
__ot~
in which: 3 ~/x/~+z2+x 2 f(x, z) = 1-6 - ~ + z £ ) gig z(z - 3x 2 + 2 x ~ )
(A-IS)
These integrals may be evaluated in polar coordinates, x = - r sin 0, z = r cos 0:
3 c ~ dr ~/2 I1 - 2-7rx ~ / ~ 3 r/:x/1 - sin 0(-1 + 4 sin2 0 + 2 sin 0) .(1 - 2 sin 2 0 + 2 sin 0 cos 0) sin 0 cos 0 dO (A-16a)
/=-4
¢ ./~ o
~l-sin0(-l+4sin20+2sin0)-cos0de
(A-16b)
-7r/2
The (divergent) radius integrals are cut off at l and h respectively, while the angular integrals are finite and can be computed numerically, yielding: 0.3601c ? dr Ii-~}, 2~=0-3601~
12 = 0.404!x/-ffjh c
c
d__Lr= 0.4041 x / r ~ o 2rl/2 c
(A-17a)
(A-17b)
(r is distance from the ridge axis, r = x/~- + z2). The integral [1 is due to the interaction of q~2 and ~3J and ~aI. It diverges at small r, near the ridge axis and is cut off at a small distance l (generally taken to be 5 km)./2 is due to the interaction of xlq and ~aJ and ~3I- It diverges at large distances, and it is cut off at h. We are mainly interested in the term proportional to 12 in (A-13). As Fig. A-1 shows, the energy per unit time dissipated by the spreading is smaller when A is positive than when it is negative (for sufficiently large S > 0). In other words, when the ridge moves with respect to the mantle it is easier (in terms of energy dissipated per unit
61
.75
olaJ
.74
.73
g .72 c) "O
o~ .71
",,,,
.N m
.70
\
Z
.69 .68
t '.3
I
I
I
I
I
I
-.2
-.I
0
.I
.2
.3
Asymmetry, A
Fig. A-1. Plot of dissipation versus asymmetry for a ridge migrating with respect to the deep mantle. Curve a is for the case of no shear in the mantle below the ridge (ridge at rest). Even is this case symmetric spreading is not favored; positive or negative asymmetry is favored with equal probability. Curves b and c are for increasing amounts of shear (0.1 V/x/-~ and 0.2 V/x/~ respectively). Both curves show that positive asymmetry (trailing flank spreads fastest) is favored. The dissipation E is normalized by
n V3[2c/l.
time) for the ridge to spread asymmetrically with the trailing flank spreading faster. The fluid dynamical model described here predicts that minimum dissipation occurs for complete asymmetry, A = I. Complete asymmetry, however, violates the assumptions of the model (one plate would be infinitely thick, thus altering the domain of integration of the flow field) so that we can only conclude that motion of the ridge with respect to the mantle favors asymmetric spreading with A > 0 if S > 0. Another interesting conclusion of the model is that, even if the ridge is stationary with respect to the mantle (S = 0), symmetric spreading is not favored; asymmetric spreading in one or the other direction requires slightly less dissipation per unit time than symmetric spreading. This result follows from the coefficient o f l l in (A-13) and may be seen in Fig. A-1. This suggests that asymmetric spreading (and reversals in its direction) may occur even on a ridge which is stationary with respect to the deep mantle. When the ridge moves over the mantle sufficiently fast (on the order o f 0 . 4 V w h e n l = 5 km, h = 200 km and IAI < 0.3) the 12 term in (A-13) dominates the 11 term and the trailing flank should consistently spread faster. The forces driving this tendency toward asymmetry may be roughly estimated. The total energy dissipated per u n i t time by the passively spreading ridge is of order 2r/I/2/unit length along the ridge. For V= 10 cm/yr, 77 = 1021 poise this corresponds to 0.25 HFU when averaged over 100 km on either side of the ridge. When the migration velocity of the ridge is nearly equal to the total spreading rate I~ml ~ V, the difference in dissipation between A = +0.3 and A = - 0 . 3 is a few percent of the total dissipation. Thus, the direction of asymmetric spreading may be strongly influenced by the motion of the ridge with respect to the mantle below.
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