Download Mechanism of Formation of Active Margins.

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Mechanism of formation
•
of active margllls
AClive margin s
Ho t manlle
Subduc tion
Basal te-eclogitc transi tion
Marges active,
Manteau chaud
Subduction
T ransilion basalte-êclogitc
E. V. Artyushkov
(nstilute of PhySÎcs of the Earlh , Moscow, USS R
ABSTRACT
The upper mantle mate riaJ is strongl y healed benealh active rnargins as compa red 10 stable
continental and occanle areas. The energy associa ted with ils a bno rmal hcating is by IWO
o rders of magnitude hig he r than 3ny e nerge lic effc e ls of subduction. This hot material has
becn Înjected from a large depth. most Iikel y from the core-mantle boundary.
Low-velocity and low-denSÎl y ma ntle mate r ial is segregaled from the hOI upper mantle benealh
ac tive margins. It forms a la yer severa l tens of kilo mete rs thic k bcneath the lithosphere.
Oceanic lithos phe re is denser than anomalous man tle a nd il s inks into Ihe a nomalous manlle
layer. Heavy e clogite is formins in the oceanic c rus t after it contac ts a nomalous mantle. This
permits the s ubduc tio n to occur a lo ng an inc lined boundar y be tween nor mal mantle of high
viscosity and hot lig ht material of lo w viscosity located beneath active marsins.
Deep depressions of marginal seas are formed by the basall-eclogite tra nsition after the
contac t of low.velocity mantle wi th the continental crust in a number of reg io ns. Local centers
of sea floor spreading can arise in Ihese depressions when low-velocity ma ntle te mperat ure is
ve ry high unde r the crus!.
Oceanul. Acta, 1981 . Proceed ings 26'h Internatio nal Geological Congress, Geology of
continenta l margi ns s ymposium, Paris , July 7-17, 1980. 245 ·250.
RÉSUMÉ
Mécanisme de formatio n des marges acti ves.
Le manteau supérieur sous les marges aç ti ves est porté à une tempéra ture s upérieure à ceIJe
régnant sous les zones océaniques ou les marges pa ssive~. L'énergie associée à ce
réchauffcm e nt ano rm;11 est environ deUil fo is plus importanl que I·énergie assoc iée à la
s u bduction. Le maté riel chaud est issu de niveaull très profond s, très probablement de la limite
noyau-manteau.
Sous les marges act ives, il y a ségrégation d·un m;tnteau à basse vi tesse sismique e t faible
de nsité à partir du ma nteau supérieur anormalement récha uffé. Celui-ci fo rme une couche de
quelques dizaines de kilomètres sous la lithosphère.
La lithosphère océanique. plus dense. s'enfo nce dans le mantea u anor malement c haud. Il ya
formmion d'éclogile au contac t c roûte océanique·mante au c haud. Ceci cond uit à une
subduct ion le lo ng d'une limite inclinée entre le manteau normal fortement visque ull. e t le
maté riel léger et c ha ud p lu ~ flu ide situé sous les ma rges a ctives.
Les dépressio ns corresponda nt aux me rs marginales appa ra issent dans les domaines où la
transilion basalte-éclogite a lieu après conta ct entre un manteau à basse vÎlesse s ismique et la
croû te continentale. Lo rsque la tem pérature du ma nteau basse vitesse est s uffisamme nt
é le vée, il y a for mation d' un centre d·accrétion océaniq ue.
Oceanol. A t·ta, 1981. Aç tes 26< Congrès International de Géologie. colloque Géologie des
marges conlinentales, Paris. 7· 17 juil. 1980. 245-250.
245
E. V. ARTYUSHKOV
MA NTLE INHOMOGENEITIES BENEATH ACTIVE
MARG INS
ENERGY RELEASE BY SUBDUCTION
Il is known that the subducting lithosphere cools the mantle
(McKenzie . 1969; Toksoz el al.. 1971). This is hecausc its
upper boundary is at a temperature T- O"C before the
su bduction. As a res ult the amount of energy necessary to
heat the oceanic lithosphere up 10 the temperature of the
surroundi ng mantle appears to he lower than the release in
the potentiaJ energy by subduction. Hence s ubduction
cannot he the cause of formation of the above mentioned
inhomogeneities. Let us neve rtheless compare the amount
of heat energy stored in the m with the release of energy by
subd uction.
Suppose (hat the lithospheric s lab of the thickness d is
subducted down to a deplh 0 ,tI an angle q; 10 a horizonlal
plane. Then the re lease of t he potential energy is
According 10 Ihe recenl ideas ( Mc Kenzie, 1969) the oceanic
lithosphere Îs heavier than the underlying asthenosphere aU
over the oceans except at the mid-ocean ridges regions. This
is considered to be a cause of the lithosphere s ubduction on
the active margins. The position of a heavier layer on a
lighter one is convectivel y unstable . Hence subduction
s hould arise in many regions from the above point of view.
Actually it takes place only in the regions of large and
strongly pro nounced inhomogeneities in t he uppe r manlle.
This difficulty is usuaUy avoided by assuming that the
inhomogeneities themselves a re produced by subduction.
Let us consider if it is possible or not.
Seismic wave velocities are considcrably dec reased and
their al1e nuation is strongly incrcased beneath active mar·
.!lins as compared to s table cont inental a reas and 10 the
adjacent oceanic regions (Baraza ngi . lsacks. 1971). Sud
properties of the mantle are normally observed from the
Benioff zo nes tothe conti nen ts for a d istance - 1000 km in a
hori zontal direct ion and from the Moho boundary d own 10 a
dep lh - 250-300 km in a vertical direct io n. Thus the volume
of the inhomogeneities is very large.
A ty picaJ c rust and mantle structure on an active margin is
shown in Figure 1. As it foUows from t his section P·wave
ve!ocities are strongly reduced henea th the sea of Japan as
compared to the mantle on the oceanic sidc of the Benioff
zone. The minimum V values are - 7.7 km/sec. in a layer
located at a depth - 100-200 km whic h is by - 1 km/sec.
lowcr than in stable areas.
d
C\L=lJ.p . g · - - ·D ,
cos 1{>
(1)
per unit vertical section of the slab. Here!J.p is the density
contrast between the slab and the mantle. g is the gravity.
Let us neglect the heating of the slab itself and assume that
one half of the encrgy is spent on the heating of the mantle
below the sJab and the other half of eoergy prod uces the
heating of the mantle above the slab.
Then the colomn of mantle of the height 0 above the s Iab is
heated by
lJ.T = !J. p.g.d
2C. cosl{>·
(2)
where C~ is the heal cap,lCilY under constant pressure.
Taki ng the ave rage value of !J. p a long the slab as 0.1 g/cm ' .
d = 50 km . Cf = 45 CI' = 1 eni/cm' we find IhM
Q
,
(3)
T his quantity is by 1.5-2 order!; of magnitude lo wer than the
add itional healing .6. T- 500·C of the manlle in the regions
under consideration. Hence this heating cannol reprcsent
the resull of subduction and it s hould he of qu ite another
origin.
T HE CAUSE OF INCLINED SU BDUCTION
If the lithosphe re is subducting ioto a lighler mantle which is
origioally uniforrn. it tends to bend down to subside
vertically. Large deformations of the lithosphere -y;;t Hr - 1
occur beneath the deep·sea tre nchcs where the lithosphere
cha nges its position from horizonlal to an inclined one. In
this case elaSlic stresses would he a - M-y where M is the
shear modu lus. M - 4.1W bar/c m1. For the above value of "y
the stresses are a - l 00 kbar which is far heyond the limit of
elastic ity. Hence the lithos phere should be broken under the
deep·sea trenc hes a nd il s hould s ubside into the mantle as a
set of blocks. Sud a Ii thos pheric slab has a very 10w rigidity
which cannOI be re sponsible for an indined subduction .
Figure 1
Selfemmit: seÎsmic cross·sectÎOl! (llolIg tire profile Jupan Se(l •
H O/flll· Pucific oce(ll! (Alexul'. R)"(lho}". 1976).
Strong decrease in P·wave veloôties lakes place when
mantle tempe rature approaches the melting po int of basic
inclusions ( Magnitsky, 1967). This temperature is T-t300' C
for the above mentioned depth . T he temperature in stable
regions is estimated as T- 700-800°C fo r the same depth.
Hence an additio nal heating by .6. T - 500°C (or by several
hundred degrees at least) in a very big volume is necessary
to form the mantle inhomogeneilies beneath the aClive
margins.
Let us consider now. whether the oceanic lithosphere can
be sustained in an inclioed posilion without considerable
bending by visco us forces in the mantle. The rate of
movement along the slab v. (Fig. 2) is controlled by viscous
stresses at the slab boundarics :
a.,
246
_ (av.
~)
a)' + ax .
-TI
(4)
MECHANISM OF FORMATION OF ACTIVE MAAGINS
ord er of magnitude at least. The upper man tle temperature
is by A T -300-S00· higher abovc Ihe Be nioff zones than a l
the same d e pth in stable a reas. He nce the viscosity should
be lower by 3-5 orders o f magni tude a t le ns t above the
Benioff zones as compared to the above mentioned values.
'"
If the viscosity li is much higher below the slab tha n above
it. the vy-co mponenl s hould be two times greater below t he
s lab as compared to the value de termined from (8). He nce
r
-v
,
_ L\p .R . d.cosl/' . L
4"
(lO)
Fiiure 2
ln o rd e r 10 prevenl a cons iderable bcnding o f the s lab, V y
s ho uld be much le ss Ihan V •• The lalter quantity is no rmall y
- j-l0 c mlyea r. Taking V r '" 1 cm/ year . the ave rage v11lue
o f de nsit y contraSI
p _ 0.1 gicm J , d o. 50 km , L ...
1000 k m we find from (10):
Grlll'Ïly !orCt actillg 011 Iht tltment of Iht Iilhosphuic pl/llt
subduf l ing inlo Ihe mlltll/e II/Id visCOUJ .f/rusu 1111 il S bOllndariu.
T he rate o f bending of the s lab \Ir in the d irect io n norm al 10
the slab is controlled by viscous s tress Cl rr and il can be
eSlima led from the fOllowing rela tion
u,,· 2~
l'la y'
li ~
(5)
5 _ Yr.
CI ,
(Ix
L '
!!t _ ~
L ' CI,
L
-
2'1 (2T+
v" Y...)
-!1p . g , d.cos/(' - 4'1
L
2V
'L
'
J
poise.
(11)
C HEMICA L CO NVECTION IN T H E MANTL E
T he formatio n of the manlle inho mogeneities on the ac tive
margins can he ex plained by c hemical con vection
(A rt yushko v, 1968; 1970). This convection results from a
density diffe rentiat ion a l t he core-mande bou nda r y. The
lowe r man lle conta ins Ihe outer core mate rial in a form of
inclus io ns in a solid Slale . This mate rial (most likely FeO
(Dub rovsky. Pankov. 1972) is in li Jiquid s la le in the oute r
core. The isot herm of solid dielectric - Jiquid high condu ctin g phase transition ro ughly coinc ide s with the core -mantle
boundar y.
(6)
The (J"y-com ponents acti ng on the upper a nd lo wer s lab
boundaries s ho uld compensale on the average the fo rce
F" .. L\ p .g.d.sinlf' acting alo ng the x-ax is on a un il col umn
of Ihe s lab which is perpendicular to Ihe x-axis. Sim ilarly
Ihe (J y-components o n the upper 1lnd lo we r bound a ries of
the s r.'\b sho uld balance the force Fr .. L\ p.g. d.cosq: acti ng
o n t he same column along the y-ax is. Then using (4)-(6) wc
fin d
• p . g . d • Sln/('
"
1er
Il is the same estimate aS lhat fo r the upper manlle viscosi t y
below the as the nosphere in stoble a reas. Thus Ihe ma ntle
s ho ulJ indeeJ ha ve qu ite d irrerent properties below a nd
above Ihe BenioH zones on Ihe ac tive margins.
The componenl o f the mantle velocit y v~ is constant a l the
sla b boundaries whe re v~ _ V. and ". is variable wit hin the
manlle. The componen! 'Ir varies both aJo ng the slab and in
the normal direction. The flow induccd by subduction in Il
homogeneous ma nlle is maioly concentrated in the area
locll ied al a d ista nce - U 2 from the sJab, w here Lis the
lenglh o f the slab. Hence tak ing the charac tcrist ic s cale of
v~ as V y we have :
Clv, _ 2V •.
3.
The pressure of the trans ition s ho uld de c rease a s the
lempera ture Tise s ( Magnits ky. 1967). T he lower ma nt le
tempera lu re increase s wilh lime bec,lUse of radi oact ive
healing. As a res ult Ihe phase bound ary ascends in the lower
mantle . The solid core malcrial bccomes liquid bclow Ihis
boundary prod uc ing a d ensity differentia tion of the lower
mantle ma lter. In the process o f differenliation Ihe heavy
core material sinks a nd joins Ihe core. The residual mixt ure
o f subst!l nces !lccumulates unde r the lower m:tntle. This
mi xture - light male r ial - is lightc r tha n the overlying
ma ntle whie h still has a normal conce ntrat ion o f hea vy core
matcrial.
(7)
(8)
The le fl -hand s ides o f (7). (8) are a pproxim1ltcJy equal w hen
q;- 4.'j °. He nce the right-hand s idcs o f (7). (8) s hould ha ve
close values too. which gives
(9)
As a res ull o f conve c ti ve ins ta bility light ffi;lterial intr udes
t he lower mantle a nd il ascends up to Ihe uppe r ma nt le
Ihrough the lowe r mantle . This movemenl is associa ted wi th
a hlrge releas e in t he potentinl encrgy :
This mcans that the velocity-components of the slab are of
the !lame order of magnit ude both ulong the slab and
no rm all y to il. He nce an o riginn lJ y straigh t slab will mpidl y
bend in ra homogeneous mantle ta kin g li position close to a
ve nical one . Therdore an indined subsidence is possible
o oly w hen the man tle below the slab has a muc h hig he r
viscosit y than above il.
The lIs the nosphe re viseos ity is ,,- 10\'- lcro poise in s tab le
continental lu eas (Artyushko v, 1966 ; 1971 a). The upper
m1lOtie unde r the a sthenos phcre has a viscos it y lI0!!3. 10 u
poise. An increase in t he mantle tempe ratu re by o ne
hundre d degrees results in a d ecrease o f ils viseosity by o ne
U - A p.g. H .
(12)
per unil volume o f lig ht mate ria l. Here A p is the densil Y
contras l between the lower man tle a nd the light mate rial. H
is the lower mantle th ic kness. Taking .6 p = 0 .2 g/cm l ,
li _ 2000 km we find Chut
U - 1000 kalfc m' .
247
(13)
E.V. ARTYUSHKOV
6. T - 1ooo" .
~_ ~ t
Figure 3
Scheme {JI Cru SI
a/Id
.
r~.
~
• •
marr/le SIrI/cl l/rt Îlr I/rt regiO lr /11 SIIbdru·/ iIJII (II
Ille malll/ .. o n the <letÎ I·t margilr.
tire plaie 01 O"ea"ie lilllfMp /Jere ÎIilO
considerably lower than thal of a normal mantle under the
c r ust p - 3.35 g/cm ' . Dceanie lithosphe re is usually composed of a thick mantle peridotite layer a nd li thin oeeanic
c rust near continental margins. The mean !ithosphere den·
s ity is p = 3.27 glcm ' when its thickness is 50 km. this layer
including basallic crust of the thickness 7 km and of the
mean density p '" 2,8 g/cm J . Anomalous mande is lighler
than the oceanic lithosphere. When it cornes to the lithosphere, subduction of the oceanic lithosphere arises tnto the
low-velocity mantle layer (the possibil ity of subduc tion into
li light material wilho ut indication of its oriSin was earlier
s uggested by G. Lo mnit z and C. Lo mni tz).
(14)
A certain amount of heat is spenl on the s pinel-olivine
transition in the upper mantle cooling the light material by a
few hund red degrees. Eventually the light material should
reach the upper layers of the upper mantle heated by
6. T - 500_700" .
, , ..... " , , ,
, , ____
... , ,
, , , _-..rf,·
,
• .,.. __
, , , , , , , ,
,
r
T his e ne rgy is spent on the ovcrcoming of visco us friction
and it is event ually transfor med into heat. Rock viscosity
slrongly decreases with temperature. After the uplift of a
large block of light material through the lower mantle a zone
of decreased viscosity appears. An uplift of new blocks
bccomes easier along this zone s ubsequently decreasing it s
viscosity. Th is is a kind of a Slrong instability concent rating
an ascending movement of light material in a relative!y small
amounl of pipes of very slrong!y reduced viscosity. It can
be s hown that their diameter s hould be - 100 km
(Artyushkov. 1979).
Because of the low temperalure conduclivity of the lower
man tle X - 10- 1 cml/scc. Ihe heat cannot penetra te far
through the walls of the pipe. For example the regions are
heated a l a distance of ~ 50 km from the walls during the
lime of - 50 M . Y. The main portion of the potential energy
is released in a relatively small volume of a pipe and il is
spe nt on Ihe heating of the lower ma nt le substance and of
the light material in the pipe. When the vol ume of light
material passing through the pipe is much greater than that
of the pipe, the main portion of the heat release is consumed
by light material. Taki ng t his quantity as (13) and
Cp - 1 kal/cm l wc fi nd that a n increase in tempera HJ re of
light matcrial after passi ng th rough the lower mantle should
be
The thickness of the low-vclocity mantle layer dues not
usually exceed a few tens of ki lometers. The underlying
mantle shou ld have the de nsity typical of normal mantle.
uCl>l'ite it il> sll"UIlgly hcalcu. Otllcrwise <In isu.\oIa lb; upli!!
many ki lometers high. o r extremely intense negative isoslatic anomal ies would arise on the surface. Hence wilhout an
increase in its density the oceanic lithosphere cannot
subs ide decply below the low-veloc ity mantle layer.
(15)
with respect tO the adjacent stable areas. This resull
coincides weil with the above estimates o f the mantle
tempe rature on the active margins .
Intrusions of large masses of light material from the
core-mantle boundary into the upper m<lntle OCCUf in many
regions of the Earth : bcneath the mid-ocean ridges. high
plateaux on the continents . geosyncline belts and continenlai margins.
The oceanic lithosphere includes basaltic c rustal layer.
Basait is unstable under:1 pressure p ;;?: 10 kbar and it s hould
he transformed inlo edogite when the temperature Îs
T - 700-904rC (11 0, Kennedy. 1970). AI such a lemperature
the rate of phas e transition becomes very high (Sobolev.
1978). As a resull a heavy edogîte of the density
p - 3.s5 glcm1 can be formed afte r a contact with the
low-velocity mantJe during the time t ~ 1 M. Y. Eciogite is
heavier than the normal mantle. Hence the mean dens ity of
the lithosphere becomes higher than in Ihe normal mantle
thus permitting the developmenl of s ubduction into deeper
Jayers .
A hOI low-velucity manlle substance is segregating from
light ma terial in the asthenosphere a nd it ascends to the
lithosphe re and c rustal bottom. Interaction of th is s ubs tance
with the crust a ppenrs to be the main cause of tec tonic
movements both horizontal and vertical (Artyushkov.
1979). The· low-velocity man tle lenses exist under the crust
in mOSI regions of high teClonic activity. The best known
examples are the Basin and Range province (Cook, 1966)
and t he mid-ocean ridge s (Talw<lni el al .. 1965). Decreased
seismic velocities are also observed under active marBins
(Yoshii. Asa no. 1971 : James. 1971).
Phase transitions were earlier suggested a~ the cause of
subduction (R ing wood, 1%9). However thi s phenomenun
cannot itse[f produçe subduçtion. Otherwise subdUc tion
would take place in any region. An impulse is necessary to
subside the lithosphere for a few ten kilometers 10 put the
gabbro-eclogite tra nsition into operation. Thi s impulse is
provided by a contact of the low-velocity mantle with the
lithosphcre .
MECHAN ISM OF SUBDUCTIO N
U~ing
the above sc heme of the structure of the upper mantle
beneath active nUlrgins the mechanism of subduction can be
presented as follow s. Large masses of light and strongly
heated material arc injected into Ihe upper m<lntlc below
sorne continental margins as a consequence of density
differenliation on the corc-mantle boundary ( Fig. 3).
Low-velocity mantle is segregated from this material. The
low.ve!ocity mantle density i~ p - 3.1-3,25 glcm J which is
An inçrease in density and subsidence of the oceanic
lithosphere into the mantle s hould he accompanied by ilS
bending under the ocean. The wid th of a zone of a surficial
subs idence s hould be a few ti mes greater than the lithosphere thickness (WaJcott. 1970). As a result deep-seil
trenches arise in the re gions of subduction on the surface.
248
MECHANISM OF FORMATION OF ACTIVE MARGINS
acting on the oceanic lit hosphere fro m subduction when the
fo rce resuhi ng from the spreading of Ihe ridges is known
(Artyushkov, 1979):
The bounda ry between a strongly heated light material and a
cool nor mal manlle o n t he oceanic s ide of the aClive margins
can be of a rather arbitrar y shape . No rmal mantle viscosity
is at leas l several o rders of mag nilude higher than that of t he
light ma terial. Oceanic lithosphere s ubsiding into Ihe mantle
fro m the s urface meels the above boundary and begins 10
move a long it. Thus il is separating the afeas of hot light
material and cool no rmal ma ntle.
L
= (0.3-0.7).109 bar. c m .
(16)
T he latter quanlity is [j few times s maller th[jn the compressional force acting o n the oceanic bas ins f rom the ridges.
Separation of a portion of oceanic cruSI from the s ubsiding
lithosphere al a depth - 150-200 km may explain the island
a rcs volcanis m o bser ved on the surface.
The real density contras t between Ihe subd uc ling plates a nd
the surro und ing mantle is a very uncertain q ua nli ty.
Deep-focus earthquakes occur in some regions like Pa mirsHindu- Kush zone where s ubduction of oceanic lithosphere
from the s urface should have been terminated a long lime
ago. In this case oceanic lilhosphere would be expected to
s ubside very deeply under a high dens ity contrac!. Hence its
density may appear to be very c lose to that of the
sur rou nding medium .
FORMA TlON O F MARGINAL SEAS
The main po rtion of the te rritory of active margins is
occupied by ma rgi nal seas . They include both deeply
s u bme rged continen tal bloc ks and the regions underla in the
oceanic crus t. Formation of marginal seas can be expJained
as folJo ws (Art y ushkov. 1979).
Ah er the 10w-veloCÎty mamie cornes to the crus t on
continen tal margins, gabb ro-eclogite phase trans ition occurs
in the basaltic layer of the continental crust in ma ny regio ns.
Heavy eclogite tears away of the c rust a nd si nks into the
mam ie . This results in crustal thinning and s ubsidence.
Interac tion of a hot low-velocity mande wilh the upper.• gra nitic" geophysical layer of the crust can strongly
increase seismic velocities in it up to Ihose typical o f the
basaltic laye r in ce rtain cases.
Large lenses of a 10w-veloCÎty ma ntle o f a lemperalu re
T - 1200"C ma y then [jrise under a strongly atte nua ted crust
in ma rginal seas. They produce the same force in the
lithosphere as that crealed by the mid-ocean ridges. In th is
case marginal seas can become local centers of sea-fl oor
spreading. As a res ull the regions underlain by typical
oceanic crust can a rise wi thin t he basins produce d by
ve rtical subsidence of Ihe continental cr ust.
Thus the main processes taking place on the active margins
are associated wit h the injection of light heated male rial
from the core-manlle boundary into t he upper mantle. More
exactly continental margins beco me active aller this injec·
tion occurs beneath them . S ubduction is a phenomenon of a
minor energetic scale as compared 10 the injection of light
material. It is o nly one of t he conseque nces of t he injec tion.
This conclus io n is opposite to a popuJar idea that s ubd uc tio n
is the main cause of d ynamic processes on the active
ma rgins.
DRI VING FORCE FRO M S UBDUCTION
A nu mber of fa ults are formed in the oceanic lithosphere
wh il e ilS be nding under a deep-sea trenc h before subduction. Hence even a large gravit y force (if il exists) ac ti ng o n
a long subduc ting plate in Ihe upper mantle cannot be
tra nsfe rred 10 t he lithosphere [ocated o n the surface. Thus
the tensile force fro m s ubduc tion pulling the oceanic plates
canno t be ve ry large .
The other major fo rce acting in Ihe oceanic Iithos phere
results fro m the spread ing of the lo w-velocity ma nlle
beneath the mid-ocean ridges (Art yus hkov. 1971 b ; 1973).
T his force is compressio nal and it can be calculated, when
the crust and the low-ve locity mantle s truc tu re is known
beneath the ridge. In this case the above force can be
represe nted as a function of the relief of the ridge.
The data o n Ihe focal mechanisms on the ridges s how that
tens ile force is a cting in the lithosphere in their axial zones
(S ykes, Sbar, 1973). It c hanges to compressio nal fo rce
towards the margins of the ridges betwe en the magne tic
a no malies 5 a nd 10. These data permit to find the fo rce L
REFERENCES
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