geodynamic model of the evolution of the pacific ocean

93

ISSN 1069-3513, Izvestiya, Physics of the Solid Earth, 2006, Vol. 42, No. 2, pp. 93–113. © Pleiades Publishing, Inc., 2006.Original Russian Text © V.P. Trubitsyn, 2006, published in Fizika Zemli, 2006, No. 2, pp. 3–25.

1. INTRODUCTION

Processes ranging in duration from 100 kyr to100 Myr are described by a model of thermal convec-tion in a viscous mantle with a lithosphere broken intoplates. The boundaries of lithospheric plates extendalong subduction zones, ridges, and transform faults[Khain and Lomize, 2005]. The plates cover the entiresurface of the Earth and are linked by friction forces.Being also linked with mantle flows, the plates mustmove at different velocities. In reality, only the middlepart of a plate moves at a certain mean velocity. Edgesof the plate stick to other plates, and the plate isdeformed. When the driving force exceeds the staticfriction force in a segment of the plate boundary, a jum-plike displacement occurs there. These local interplateslips are accompanied by earthquakes. The oceaniclithosphere is born at mid-ocean ridges during coolingof ascending magma. Subsequent portions of magmaare welded (frozen) onto the previously solidified por-tions. As material moves away from the ridge, the thick-ness of the cooling high-viscosity material increases.As a result, a moving thickening oceanic lithosphere

arises. During the breakup of a supercontinent, magmais injected between the diverging continents and iswelded first onto the continental margin and then ontothe preceding already solidified portion of the oceaniclithosphere. Therefore, each of the diverging continentsis attached at its passive margin to a portion of oceaniclithosphere, forming a lithospheric plate. Oceanicplates that adjoin continents at active margins areinvolved in the convective circulation of mantle mate-rial and sink into the mantle in subduction zones. Litho-spheric plates are drawn downward into the mantlebecause plate portions older than 20 Ma becomeheavier than the mantle and, due to high shear stressesin the bending zone, the plate material becomes plastic.

As long as geophysicists studied seismic wave prop-agation, tides, and other relatively rapid processes withperiods shorter than 30 kyr, an elastic model of theEarth corrected for viscous relaxation was sufficient.The model of mantle convection and plate tectonicswas created in order to describe geological processes oflonger duration. This model has been elaborated bymany geophysicists and geologists throughout the

Geodynamic Model of the Evolution of the Pacific Ocean

V. P. Trubitsyn

Schmidt Institute of Physics of the Earth, Russian Academy of Sciences, Bol’shaya Gruzinskaya ul. 10, Moscow, 123995 Russia

e-mail: [email protected]

Received September 12, 2005

Abstract

—The present Pacific Ocean differs significantly in its structure and evolution from the expandingAtlantic Ocean. The Pacific is asymmetric. Its mid-ocean ridge is located not along its median line but is closerto South America and adjoins North America. The Pacific is surrounded by a ring of subduction zones but hasmarginal seas only at its Eurasian margins. After the breakup of Pangea, the Atlantic began to open and thePacific began to close. This paper examines the evolution of the Pacific Ocean and, in particular, the formationmechanisms of its present structures. Numerical modeling of the long-term drift of a large continent is per-formed, with the initial position of the continent corresponding to the state after the breakup of the superconti-nent. At first the continent, driven by the nearest descending mantle flow, begins to approach a subduction zone.Since the mantle flows beneath a large continent have different directions, its velocity is a few times lower thanthat of the mantle flows near the subduction zone. As a result, a zone of extension arises at the active continentalmargin and a fragment is broken off from the continent; this fragment rapidly moves away and stops above thedescending mantle flow as in a trap. A marginal sea forms at the active continental margin. The continent con-tinues its slow movement toward the subduction zone. The oceanic lithosphere, which earlier sank vertically,begins to descend obliquely. This evolutionary stage corresponds to the present position of Eurasia. The mod-eling shows how the interaction of the continent with the mantle causes the subduction zone to roll back towardthe ocean. Subsequently, the continent nevertheless catches up with the subduction zone, and they movetogether for a while. The marginal sea then closes and high compressive stresses arise at the active continentalmargin. This state corresponds to the present position of South America. During the subsequent drift, the con-tinent together with the subduction zone reaches the mid-ocean ridge and partially overrides it. This state cor-responds to North America, which was the first to break off from Pangea and passed through the stages of bothEurasia and South America. The large and slowly moving Eurasia, which formed only at the time of Pangea, isstill in the first evolutionary stage of the Pacific Ocean closure.

PACS numbers: 91.45.By

DOI:

10.1134/S1069351306020017

94

IZVESTIYA, PHYSICS OF THE SOLID EARTH

Vol. 42

No. 2

2006

TRUBITSYN

world. The Kyoto Prize awarded to J.P. Morgan (theUnited States), X. Le Pichon (France), and D. McKen-zie (Great Britain) recognized them as the outstandingcontributors to the development of plate tectonics.Major contributions to the study of the properties ofmantle convection as the driving force of plate tectonicswere made by D.L. Turcotte, G. Schubert, U. Chris-tensen, and others [Schubert

et al.

, 2001].

At present, the theory of plate tectonics and mantleconvection provides a basis for studying geologicalprocesses. We should note, however, that a completedynamic theory of plate tectonics has not yet been cre-ated. So far this theory remains kinematic because itcannot be used to quantitatively describe the breakup ofthe oceanic lithosphere into plates. The plates move andare born due to mantle convection. However, the oce-anic lithosphere in the existing convection theory arisesas a high-viscosity layer due to low temperatures. Thislayer has boundaries along mid-ocean ridges and sub-duction zones, but without transform faults. For about15 years now, the leading scientific institutions of theUnited States and Europe have been attempting to con-struct a theory of mantle convection taking into accountthe rheological properties of material; within the frame-work of this theory, the oceanic lithosphere shouldautomatically break up into plates along linear zoneswhere stresses exceed the strength of the lithosphere[Tackley, 2000; Zhong

et al.

, 2000].

In studying processes of even longer duration (morethan a few hundred million years), a continent with theadjacent seafloor can no longer be treated as one litho-spheric plate. Continents remain on the mantle surfacefor a few billion years, and the oceanic lithosphere bro-ken up into plates participates in the convective circula-tion of mantle material. During the lifetime of a conti-nent, the oceanic lithosphere is frozen to and brokenaway from a continent up to a few dozen times. There-fore, in studies of global geological processes, conti-nents can be treated as floating on a convecting mantleand the oceanic lithosphere broken into plates can beapproximately considered as a surface layer of ananomalously high viscosity.

Actually, these two models have existed in the Earthsciences for a long time. The theory of plate tectonicswas created about 50 years ago. The concepts of float-ing continents have been developed since Wegener’stime, i.e., for a hundred years now. Yet, in contrast to theelaborate theory of plate tectonics, the ideas of conti-nental drift were based solely on paleogeographicreconstructions because no one could mathematicallydescribe the driving forces of moving continents. Thus,there were two problems: to develop a mathematicaltheory for floating continents and to reconcile the twomutually exclusive ideas of continents floating on themantle independently of plates on one hand and conti-nents frozen to plates on the other hand. The equationsof motion of continents taking into account theirmechanical and thermal interactions with mantle con-

vection were derived in a general form for the first timein [Trubitsyn, 2000]. For a 2-D Cartesian model, theyare given in Appendix A.1.

A natural generalization of the theory of lithosphericplates and floating continents can be the model of con-tinents floating among the oceanic plates that periodi-cally (up to 100 Myr) are frozen to continents at theirpassive margins and then break away and sink into themantle [Trubitsyn, 1998, 2000]. This model is trans-formed into the theory of lithospheric plates if pro-cesses shorter than 100 Myr are considered and into thetheory of floating continents in the case of processes oflonger duration.

Water freezes at

T

= 0°ë

and magma solidifies at

T

≈

1250°ë

. Therefore, the oceanic lithosphere (brokeninto plates) and continents resemble a moving ice fieldwith giant rafts onto which ice floes can temporarilyfreeze. However, in contrast to light ice, which remainson the surface, the oceanic lithosphere during coolingbecomes heavy and therefore sinks, participating in theconvective circulation of mantle material. Only theyoung lithosphere can remain frozen to a continent atits passive margin for a period of up to 100 Myr.

Since the melting point of magma is much higherthan the temperature of the mantle surface, mantlematerial can “freeze” onto the continents not only attheir edges but also from below. This process allows usto understand the nature of the continental lithosphereand roots. Due to mantle flows, the continents con-stantly drift toward subduction zones, where the mantletemperature at a given depth is lowest and the viscosityis highest. As a result, the conditions are created for theattachment of mantle material and the growth of conti-nental roots. The attachment of continents to the conti-nental lithosphere is enhanced by the intrusion of man-tle plumes.

Note, however, that the influence of low tempera-tures under the continents can prevent the continentallithosphere and roots from mixing with the mantle onlyat the initial stage. The decreased (by 200 K) tempera-ture beneath continents increases the viscosity of thecontinental lithosphere only by two orders of magni-tude. However, the viscosity can additionally increaseby two orders due to physicochemical processes mak-ing the lithosphere drier. Moreover, the continentallithosphere becomes lighter by 1–2% as it is depleted iniron and garnet [Jordan, 1988]. As a result, the old con-tinental lithosphere, particularly, continental roots,becomes rigidly attached to the continent and movestogether with it without mixing with the mantle for abillion years.

At present, a general model describing the globalgeodynamic processes over the entire time scale andunifying the theories of plate tectonics and floating con-tinents has not yet been completed. The attachment andbreakaway of the viscous oceanic lithosphere at conti-nental margins is already described self-consistently inthe model of floating continents. However, even the the-


Vol. 42

No. 2

2006

GEODYNAMIC MODEL OF THE EVOLUTION OF THE PACIFIC OCEAN 95

ory of plate tectonics without continents does not yethave equations describing the breakup of the oceaniclithosphere into plates. After obtaining these equationsin the theory of mantle convection without continents,they can also be included into the theory with floatingcontinents. It will then be possible to quantitativelydescribe not only the breakup of the oceanic lithospherebut also the breakup of the continents themselves on thebasis of the theory of plastic fracture when the criticalstress due to mantle flows is reached.

Research on the influence of continents on mantleconvection in a 2-D model was initiated by Gurnis[1988]. Yet an artificial tool was used to describe themotion of a continent. Its velocity was determined fromthe motion of a marker without using an equation ofmotion. A mathematical apparatus for describing themotion of continents was developed in [Trubitsyn,1998, 2000]. Since the strain of a continent is muchsmaller than its displacement, it can be treated, in a firstapproximation over a short time interval, as a rigidfloating body. The forces acting on a continent arefound by analogy with a floating ship as the sum of vis-cous stresses over the entire surface of the continentsunk into the mantle. At each time step, the forces act-ing on the continent are calculated and the movement ofthe continent is determined. Then, from the alreadyknown forces, the strain experienced by the continentover this time can be calculated. At the next time step,the displacement of the deformed continent and newactive forces are determined. The model of rigid conti-nents floating on a viscous mantle made it possible towrite for the first time the full system of equations tak-ing into account both mechanical and thermal interac-tions of the continent with mantle convection. In 3-DCartesian and spherical models, this system of equa-tions includes the description of all possible rotations ofcontinents and their collisions [Trubitsyn, 2000, 2005].Thus, Wegener’s ideas and the above model have led tothe development of a theory that can be used to calcu-late the continental drift and the evolution of globalmantle processes spanning billions of years.

Driving forces of continents are the forces of vis-cous interaction with mantle flows. However, conti-nents do not float on the mantle as passive bodies.Numerical modeling showed that continents influencethe entire structure of mantle flows. The continentsequalize mantle flow velocities at their base. However,their main effect is that they hinder the heat flux fromthe mantle. Therefore, the subcontinental mantle isheated, particularly when they are assembled into asupercontinent, beneath which a new giant ascendinghot flow arises in the mantle. This flow breaks up thesupercontinent and pushes the continents apart. Thus, atthe assemblage stage, continents exert the strongesteffect on the mantle and, at the divergence stage, thecontinents drift mainly passively toward subductionzones. The theory of floating continents describes theinteraction processes of continents with the mantle,when global structures of the Earth develop. The theory

of plate tectonics describes the processes operating inthese already formed structures. Thus, we may statethat the continents govern plate tectonics.

In this paper, on the basis of the theory of tectonicsof floating continents, we examine the period of theEarth’s global evolution marked by the divergence ofcontinents after the breakup of the supercontinent ofPangea. The Atlantic Ocean, whose structure and evo-lution have been fairly well studied, formed at this time.However, structures of the present Pacific Ocean, forwhich no quantitative self-consistent global modelshave so far been developed, also formed in this period.The Atlantic and Pacific oceans differ markedly in theseafloor structure. The Atlantic Ocean has no marginalbasins and its seafloor does not sink. The young Atlan-tic Ocean arose at about 100 Ma, and its structure iseasy to explain [Morgan, 1983]. The Pangea supercon-tinent heated the mantle at 200 Ma, and the resultinghot ascending mantle flow under the supercontinent ini-tiated the breakaway of first North America and thenSouth America from Africa and Eurasia. The hot mantlematerial intruding into the newly formed crack (theMid-Atlantic Ridge) was first welded onto the passivemargins of the diverging continents and then began toaccrete onto the cooling plates. As a result, the AtlanticOcean continues to expand. Its lithosphere, being stillyoung and light, cannot break away from the passivecontinental margins. As a result, the contemporary evo-lution of the Atlantic Ocean is controlled by the move-ment of four plates with attached continents (the NorthAmerican, South American, Eurasian, and African plates).Modeling studies [Trubitsyn, 2000; Trubitsyn

et al.

, 2003]showed how in a few tens of millions of years the AtlanticOcean floor will start to separate from the continents withthe formation of subduction zones; the passive margins ofthese continents will become active.

The more complex structure of the older PacificOcean could not be explained in the last few decades[Uyeda, 1982; Schubert

et al.

, 2001] because a signifi-cant role in its formation was played by the previouslyneglected effects of floating continents. In contrast tothe Atlantic Ocean, the Pacific is not expanding; on thecontrary, it is closing. Island arcs and marginal seasabundant at its Eurasian margin are absent at the mar-gins of South and North America. Moreover, the PacificOcean is asymmetric. In the south, its mid-ocean ridgeis two times closer to South America than to Australia,and in the north, it plunges beneath North America (seeFig. 1). What is important, the ridge and the subductionzone closely approach each other in this place. In terms offree mantle convection without continents, the ascendingand descending mantle flows are always separated and lieon the opposite sides of the convection cell.

In plate tectonics, this complex structure of thewestern United States is interpreted in terms of a cha-otic component of mantle convection. However, thenumerical solution of the equations of motion of conti-nents floating on the convective mantle indicates that a

96


Vol. 42

No. 2

2006

TRUBITSYN

structure of the Pacific type should originate as a con-sequence of the general laws of energy, mass, andmomentum transfer.

The ridge of the Pacific Ocean can still be main-tained by the heat of a giant hot anomaly in the lowermantle that could arise due to heating of the mantlebeneath the supercontinent Rodinia. After the breakupof Rodinia at about 800 Ma, the continents drifted overthe Earth’s surface away from the Pacific and subse-quently reassembled elsewhere above a system of colddescending mantle flows, forming Pangea. However,the giant hot anomaly in the lower mantle beneath thePacific has not yet dispersed because its lifetime is ofthe order of 1 Gyr. After the breakup of Pangea, thenewly formed Atlantic Ocean began to expand and thePacific Ocean began to slowly close. In the older ocean,oceanic plates are heavy and sink into the mantle insubduction zones. As a result, on the periphery of thePacific Ocean, a circular belt of descending mantleflows arose toward which not only the Pacific litho-spheric plates but also all continents surrounding thePacific Ocean had to move.

Studies of the continental drift evolution [Trubitsynand Rykov, 1997, 2001; Trubitsyn, 1998, 2000] showedthat, as the ocean closes, each continent can passthrough three stages. However, numerical calculationswere performed for continents of relatively small sizesin thermal convection models with small Rayleighnumbers. Therefore, for quantitative comparison with

the observational data for the present Pacific Ocean, thenumerical results were extrapolated to larger Rayleighnumbers corresponding to more vigorous convection.In this work, a fully self-consistent model with theparameters of convection and continents correspondingto their currently accepted values is constructed.

2. MODEL

The geodynamic evolution of the Pacific was stud-ied with a model of the long-term drift of a continent ona mantle experiencing vigorous thermal convection.Typical features of such convection in the mantle aresubduction zones, mid-ocean ridges, and hotspotplumes. Mantle convection is the superposition of threecomponents. The main component is whole-mantlethermal convection created by internal heat sources inthe mantle. It is characterized by intense descendingmantle flows originating in subduction zones and trace-able in seismic tomography imagery. However, ascend-ing hot flows induced by internal heating are very wide.They are not revealed by seismic tomography and areinconsistent with the narrow ridges observed on the sur-face.

The second component of mantle convection is ther-mocompositional convection, induced by massexchange with the core and its heat (mantle heatingfrom below). This convection takes the form of plumesarising in the D'' layer of the lower mantle [Dobretsov

10 cm/yr (relative velocity)

60°

30°

30°

0°

60°

80°

180° 150° 120° 90° 60° 30° 0° 30° 60° 90° 120° 150° 180° 150°

Eurasian Plate

ArabianPlate

AfricanPlate

PhilippinePlate

PacificPlate

IndianPlate

Antarctic Plate

AmericanPlate

South

Cocos Plate

NazcaPlate

AmericanPlate

North

Caribbean Plate

Fig. 1.

Lithospheric plates on the Earth’s surface (after [Brown and Mussett, 1981]). The arrows indicate the velocities of platesdiverging during the solidification of magma in ridges and converging toward subduction zones.


Vol. 42

No. 2

2006


et al.

, 2001; Lobkovsky and Kotelkin, 2000]. On thesurface, these plumes are observed as volcanichotspots. Their study is important for understanding thetectonic and geochemical processes in the Earth’s his-tory. However, in terms of energy, the role of theseplumes in global convection is not large becausehotspot plumes remove no more than about 8% of theEarth’s heat [Schubert

et al.

, 2001]. We should empha-size that this relates only to the narrow hotspot plumesfed by the core’s heat. The wide (in particular, giant)hot ascending flows from the lower mantle are gener-ated by the radioactive heat of the lower mantle itself;i.e., they are part of the first whole-mantle componentof thermal convection.

Finally, the third component of mantle convection isthe partially forced convection in the upper mantle thattakes the form of ascending hot mantle flows feedingmid-ocean ridges. These mantle flows do not extendinto the lower mantle. They originate in the upper man-tle and rise to fill the gap resulting from the breakup ofthe oceanic lithosphere. These mantle flows of theridges are generated by the heat supplied to the uppermantle from below.

At present, only the first component of mantle con-vection has been carefully studied. There are numerousmodels of whole-mantle thermal convection induced byheating from below and by internal sources with dueregard for phase boundaries and other effects (but with-out continents) [Schubert

et al.

, 2001; Dobretsov

et al.

,2005]. The properties of mantle plumes are extensivelystudied, but self-consistent models in which plumeswith the observed properties arise automatically as thesolution of the general transfer equations have notbeen constructed as yet. The third component ofmantle convection, related to the formation of mid-ocean ridges, remains virtually unexplored. Thedynamic theory of plate tectonics is still under devel-opment and cannot self-consistently describe howthe lithosphere is broken into plates by transformfaults. Therefore, there are no models providing themechanism of the generation and evolution of mantleflows under mid-ocean ridges.

The goal of this work is to study the interaction of afloating continent with mantle convection in variousregions, in particular, near subduction zones and par-tially near mid-ocean ridges. Since narrow ascendingmantle flows of ridges are absent in the model of whole-mantle convection with internal heat sources, we exam-ined a model with the parameters of upper mantle con-vection (with heating from below). This model auto-matically generates not only the narrow descendingmantle flows of subduction zones but also the narrowascending mantle flows beneath mid-ocean ridges. Itshould be kept in mind, however, that this model canonly approximately describe the real ascending flowsthat feed the ridges. In the model under consideration,these ascending mantle flows are produced by the heatcoming from the base of the upper mantle and the con-

vection effect of the breakup of the lithosphere is nottaken into account. Oceanic lithospheric plates in thismodel (as in the vast majority of mantle convectionstudies) are examined only to a first approximation, asa high viscosity layer without transform faults that isborn in mid-ocean ridges and sinks into the mantle insubduction zones.

Thermal convection was calculated with the sim-plest model of a viscous fluid in a 2-D elongated regionclosed into a ring and heated from below with the Ray-leigh number Ra =

10

6

. We considered a region of thick-ness

D

(the dimension along the

z

axis) and length

L

(along the

x

axis). The dependence of the viscosity onthe depth (1 –

z

) and temperature

T

in dimensionlessvariables was taken in the simple form proposed byU. Christensen in 1984:

η

(

T

,

z

) =

η

0

exp[–4.6

T

+ 0.92(1 –

z

)],

where

η

0

is the viscosity at the mantle surface

z

= l.With this depth dependence, the viscosity decreases bytwo orders of magnitude due to temperature andincreases by about two orders due to pressure.

As is noted in the Introduction, the continental litho-sphere is frozen to the continent from below and movestogether with the continent. Since the continent heatsthe underlying mantle and moves over great distances,it can be not only above the coldest regions of the man-tle but temporarily also above its hotter regions. Atthese times, the lower part of the continental lithospherecan soften and mix with the mantle. Only the ancientdrier depleted part of the continental lithosphereremains unchanged and always moves with the conti-nental crust. In the model under consideration, varia-tions in the thickness of the continental lithosphere arenot considered and the continent along with the stablepart of the underlying lithosphere is modeled by a solidthick plate floating on the surface of the mantle.

3. NUMERICAL RESULTS

The system of equations for calculating the evolu-tion of mantle convection with a floating continent isdescribed in Appendix A.1. These equations make itpossible to calculate the evolution of mantle convec-tion, determining at each time moment the velocities ofmantle flows and a floating continent and the tempera-ture distribution in the mantle and within the continent.The form of the equations is simplified if, instead of thecomplete temperature, only its superadiabatic (poten-tial) part is calculated. Hot material of the Earth’s inte-rior having a temperature

T

expands as it ascends to thesurface and therefore partially cools. Upon reaching thesurface, its temperature is

T

p

=

T

–

T

ad

, where

T

ad

is theadiabatic temperature and

T

p

is the potential (or super-adiabatic) temperature. The adiabatic temperature isrelated only to the inevitable cooling of ascendingmaterial and heating of descending material. Therefore,convection develops only if the temperature of heating

98


Vol. 42

No. 2

2006

TRUBITSYN

of the Earth’s interior exceeds the adiabatic tempera-ture, i.e., due to the potential temperature. In the man-tle, the adiabatic temperature increases with depthapproximately linearly from 0 to

1200°ë

at the bound-ary with the core. Hereafter, by the temperature

T

, wemean precisely the potential temperature (as is usuallythe case in most studies dealing with the calculation ofmantle convection), bearing in mind that, in order toobtain the depth distribution of the complete tempera-ture, it is necessary to add the adiabatic temperature tothe calculated temperature.

The intensity of thermal convection is characterizedby the Rayleigh number Ra, which is the combinationof the mantle parameters and the degree of its heating.Without convection, heat would be carried from themantle interior only conductively. In the presence ofconvection, hot material rises to the surface in the formof ascending jets and material cooled at the surfacedescends to the base of the mantle as cold descendingjets. This results in a marked increase in the heat lossfrom the mantle and its cooling. The effectiveness of con-vective heat loss is characterized by the Nusselt numberNu, which is the ratio of the total heat flux to the conduc-tive heat flux. The Nusselt number is actually a relativedimensionless heat flux. Since the heat flux can be differ-ent in different parts of the mantle, the Nusselt number isa function of the lateral position and depth

Nu(

x

,

z

)

. In par-ticular, the function

Nu(

x

,

z

= 1)

gives the heat flux distri-bution over the surface of the mantle and the laterally aver-aged function

Nu(

z

)

gives the value of the heat flux at var-ious depths. The values of the mantle parameters and theformulas for determining the Rayleigh and Nusselt num-bers are given in

Appendix A.2

.Figure 2 presents the results of calculating the

steady-state convection in the mantle. The relation

Nu(

z

)

deviates insignificantly from linearity, varyingwith time near the average. This implies that thermalconvection becomes quasi-stationary at a Rayleighnumber Ra >

10

5

. Figure 2 also shows the calculateddistributions of the superadiabatic (potential) tempera-ture and the viscosity as a function of depth 1 –

z

. Theupper cold highly viscous layer of the mantle corre-sponds to the oceanic lithosphere. A region correspond-ing to the asthenosphere arises under the lithosphere. Ithas a lower viscosity because the temperature hasalready risen but the pressure has not yet increased theviscosity. With the further development of convection,the temperature rises very slowly with depth and theviscosity increases due to pressure. At the base of themantle in the model with heating from below, the tem-perature again rapidly rises and the viscosity drops. Themaximum velocity of mantle flows calculated in termsof upper mantle convection (see Appendix A.2) is

V

m

≈

3700

, or (in dimensional units)

V

m

≈

15

cm/yr.

Evolutionary Stages of the Structure of Mantle Convection with a Floating Continent

After thermal convection attains its steady-statestage, a floating continent is placed into the mantle.This time moment is taken as zero

t

1

= 0 for further cal-culations. The continent is modeled by a solid plateconsisting of two parts: the main plate of the dimen-sionless length

l

0

= 1.5 and a small fragment of thelength

l

1

= 0.05. The thicknesses of both parts of theplate are taken to be identical and equal to

d

= 0.05.Both parts of the plate closely adjoin each other, butwithout attachment. The initial position of the plateshown in Fig. 3 is not far from a descending mantleflow. The initial temperature distribution within the

0.5

10

z

Nu20 0.5 1.0 T 20.0 20.5 21.0 log

η

0

Fig. 2.

Calculated depth distributions of the dimensionless heat flux N

u(

z

)

, temperature

T

(

z

)

, and viscosity

η

(

z

)

.


Vol. 42

No. 2

2006


plate is taken as it was at the time

t

1

= 0 in the mantlevolume initially occupied by the plate. Then, the evolu-tion of the mantle–continent system is calculated over along time by solving the equations of convection with afloating continent (9)–(12) and (24).

First Stage: Movement of the Continent toward the Subduction Zone, Opening of Marginal

Seas, and Evolution of Island Arcs (the Eurasia State)

Figures 3 and 4 show the calculated first stage of thecontinental drift and the structure of mantle convectionat times from

t

1

= 0 to t4 = 30 Myr. The distribution ofthe superadiabatic (potential) temperature in the mantle

and within the continent is shown in shades of gray withthe scale given at the bottom of the figure. The bound-aries of the submerged part of the continent are shownby a white line. The part of the continent protrudingfrom the mantle is shown on an exaggerated scale. Thevelocities of mantle flows are shown by arrows whosemaximum length corresponds to the velocity Vm ≈15 cm/yr. The velocities of all points of the rigid conti-nent during its progressive motion are equal to eachother and to the velocity of its center of gravity, whichis equal to the velocity of mantle flows that stick to thebase of the continent. Since the mantle under a conti-nent can have several convection cells, the velocity of alarge continent is always lower than the maximum

20

40

1.0

z 0.5

04 5 6 7 8 x

Dimensionless temperature

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1

Nu

zN

u

t2

t1

20

40

1.0

0.5

0

60

Fig. 3. Calculated initial evolution of the moving continent and its fragment at the times t0 = 0 and t2 = 10 Myr, corresponding tothe detachment of the fragment from the continent and the initial opening stage of the marginal sea. Dimensionless temperature isshown in shades of gray, and flow velocities are shown by arrows whose maximum length corresponds to 15 cm/yr. The boundariesof the continent are shown by white lines. The protruding part of the continent is shown on an exaggerated scale. The distributionof the relative heat flux from the mantle is shown at the top of the diagrams.

100

IZVESTIYA, PHYSICS OF THE SOLID EARTH Vol. 42 No. 2 2006

TRUBITSYN

velocity of mantle flows. In the model under consider-ation, after the continent is placed on the mantle, thecalculated velocity of the continent is v ≈ 560 ≈2.2 cm/yr, which is a few times smaller than the veloc-ity of mantle flows. As can be seen from Fig. 3, after theinteraction of the continent with the mantle attains thesteady-state stage at t = t2, the velocity of mantle flowsnear the surface to the right of the continent is higherthan the continent velocity. The velocities of mantleflows increase toward the subduction zone becausemantle material is dragged into the mantle in the sub-duction zone. Note that the upper low-temperaturelayer of the mantle is actually the highly viscous oce-anic lithosphere. As a result, an extension region arisesbetween the continent and the subduction zone (mainlyin the oceanic lithosphere), and a drag force should acton the right-hand end of the continent. In order to deter-

mine the drag effects, the model continent consists of themajor portion and a fragment serving as an indicator. Asseen from Fig. 3, as early as the time t2 = 10 Myr, the frag-ment is detached from the continent and moves toward thesubduction zone at the velocity vf ≈ 760 ≈ 3.0 cm/yr,which is higher than the velocity of the continent.

The calculated distributions of the relative mantleheat flux Nu(x) are shown in the figures above the flowpattern diagrams. At zero time t1 = 0, the distribution isthe same as it was in the absence of the continent. In thecourse of time, the heat flux distribution above the con-tinent substantially changes because, due to the absenceof convection in the continent, the heat is transferredthere more slowly. The continent acts as a heat screenfor the mantle and, as seen from Fig. 3, the heat fluxabove the continent has decreased already by the time t2.The oceanic lithosphere between the departing frag-

20

40

1.0

z 0.5

04 5 6 7 8 x


0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1

Nu

zN

u

t4

t3

1.0

0.5

0

60

50

30

10

Fig. 4. Calculated evolution of the moving continent and its fragment and the structure of mantle flows at the times t3 = 25 Myr andt4 = 30 Myr, corresponding to the formation of a structure of the marginal sea type.



ment and the continent is spreading, as is reflected inparticular by the appearance of a local maximum in theheat flux from the mantle.

Figure 4 shows the subsequent evolution of the man-tle–continent system. At the time t3 = 25 Myr, the con-tinent continues to move slowly to the subduction zone,while the continental fragment is already approachingit. The numerical modeling reveals back effects of themoving continent on mantle convection. As the conti-nent with the fragment approaches the subduction zone,the oceanic lithosphere, which up to this moment sankvertically, now begins to be thrust under the continent.In the model under discussion, this effect is obtainedself-consistently, as a result of interaction between thecontinent and the mantle. Up to the time t4 = 30 Myr, astructure very similar to the active margin of presentEurasia forms in the mantle with a floating continent.

The continental fragment is stuck above the subduc-tion zone. It cannot pass through the zone because ofthe oceanic lithosphere moving toward it from theocean.

This model includes solely geodynamic processes,without taking into account many other effects, in par-ticular, andesitic volcanism. If the volume of magmaerupted by such volcanoes is small, the magma can bedragged back into the mantle in the subduction zone.However, magma can be welded onto the lighter conti-nental fragment, producing an unsinkable growingisland arc consisting of continental and volcanic rocks.This process can account for the origin and evolution ofthe Japanese and other islands of the active margin ofEurasia. As seen from Fig. 4, a structure of the marginalsea type forming between the fragment and the conti-nent is characterized by a stretched thinner oceanic

30

70

1.0

z 0.5

04 5 6 7 8 x


0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1

Nu

zN

u

t6

t5

30

70

1.0

0.5

0

Fig. 5. Evolution of the movement of the continent and its fragment and the structure of mantle flows at the times t5 = 34 Myr andt6 = 40 Myr, corresponding to the initial closure stage of the marginal sea and back-rolling of the subduction zone.

102


TRUBITSYN

lithosphere and a pronounced local maximum of therelative heat flux Nu(x) in agreement with data ofobservations.

Second Stage: Back-Rolling of the Subduction Zone, Closure of the Marginal Sea, Overridingof the Subduction Zone by the Continent, and Their Joint Oceanward Movement

(the State of South America)

Figures 5 and 6 present the results of calculation of con-vection in the mantle with a floating continent at the secondevolutionary stage. During the time period t = t5–t7, the con-tinent continues to move toward the subduction zonedue to the forces of viscous cohesion with mantle flowsat its base and ends. The calculated velocity of the con-tinental drift is v ≈ 550 ≈ 2.5 cm/yr. However, the sub-

duction zone itself together with the overlying fragment(island arc) begins to move away toward the ocean. Atthe time t = t5 = 34 Myr, the calculated velocity of thesubduction zone vs is vs ≈ vf ≈ 330 ≈ 1.4 cm/yr.

Back-rolling of the subduction zone toward theocean has long been studied on the basis of geologicalreconstructions. Yet the mechanism of this process wasnot clear until now. In the numerical model considered,this effect arises self-consistently as a result of nonlin-ear interaction of mantle convection with the floatingcontinent, i.e., as a result of processes described by thegeneral equations of energy, mass, and momentumtransfer. This phenomenon was previously attributed tothe end of the old large oceanic plate being broken offunder its own weight. However, according to all paleo-reconstructions [Muller et al., 1997], subduction zonesat the active margins of both South and North America

30

70

1.0

z 0.5

04 5 6 7 8 x


0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1

Nu

zN

u

t8

t7

30

70

1.0

0.5

0

Fig. 6. Evolution of the moving continent and its fragment and the structure of mantle flows at the times t7 = 44 and t8 = 48 Myr,corresponding to the continuing closure of the marginal sea and the attachment of the island arc to the continent.



were constantly moving toward the Pacific. Moreover,in recent time, the oceanic plates (particularly, in frontof North America) were no longer large; i.e., there wasno reason for their simple breakup.

The numerical modeling of continental drift per-formed in this work suggests a different cause of theback-rolling of the subduction zone. The powerfuldescending mantle flow in the subduction zone not onlydraws the oceanic lithosphere down into the mantle butalso moves the floating continents. When the continentapproaches the subduction zone, the convection cellson both sides of the descending mantle flow becomestrongly asymmetric (cf. Figs. 3 and 5). Convectionitself arises in the process of self-organization in a non-linear system due to the coupling of mechanical andthermal processes. In a nonlinear system, there aremany states with different energies. The flow structureestablished in the system corresponds to a minimumenergy of the latter. Energy dissipation is proportionalto the gradient of flow velocities. Therefore, the energyminimum is reached when the structure of convectiveflows is maximally symmetrical. The continentapproaching the subduction zone disturbs this symme-try of flows, and this new state no longer corresponds tothe energy minimum. Such a system must pass intoanother, more symmetrical state. Specifically, the sub-duction zone moves away from the approaching conti-nent to make the flows on both sides of the descendingflow as similar as possible (see Figs. 4 and 5). This isthe essence of the back-rolling process.

In the presence of a continent, however, completesymmetry is not attained because the continent steadilycontinues to move to the subduction zone due to theaction of mantle flows. Therefore, the subduction zonerolls back toward the ocean for a fairly long time. Yetultimately, as can be seen from Figs. 5 and 6, the conti-nent catches up with the subduction zone. Thus, themarginal sea is contracting at t7 = 44 Myr and com-pletely closes at t8 = 48 Myr, while the island arc(formed by the continental fragment and solidifiedandesitic magma) joins the continent. At this time, theextensional setting at the active continental margin isreplaced by a compressional setting, which must causeorogenic folding. As seen from Fig. 6, the distancebetween the subduction zone and the mid-ocean ridgedecreased by approximately three times relative to theinitial state. The state of the mantle with a floating con-tinent calculated at the time t8 = 48 Myr is similar to thestate of the active margin of South America. As is evi-dent from comparison of Figs. 4 and 5, at the time t5 =34 Myr, the descending oblique mantle flow is broken,which agrees with seismic tomography data on thedetachment of the subducting slab and its sinking to thebase of the mantle.

At depths of 100–200 km, the mantle under the con-tinents is known to be colder than under the oceans. Asnoted above, the continent in this model is meant as thecontinental crust together with the attached (frozen)

ancient part of the continental lithosphere. Comparisonof Figs. 3–6 shows that the temperature within andimmediately under the continent decreases with time.This effect is self-consistently obtained in the numeri-cal modeling. A possible cause of the low temperatureof the older continental lithosphere is the fact that thecontinents spend the greater part of their lifetime nearcold subduction zones, toward which they are con-stantly drawn by descending mantle flows [Trubitsyn,1998, 2000].

Third Stage: Overriding of the Mid-Ocean Ridge by the Continent and Collision of the Subduction Zone

with the Ridge (the State of North America)

Figures 7 and 8 illustrate the calculated evolution ofthe mantle with a floating continent after the closure ofthe marginal sea and the arrival of the continent at thesubduction zone. During the time period t = t9–t10, thecontinent continues to move together with the subduc-tion zone, partially covering it; the distance between theridge and the subduction zone rapidly decreases, andthe descending mantle flow under the zone graduallyattenuates.

Figure 8 presents the structure of mantle convectioncalculated at the end of the third stage of evolution atthe times t11 = 58 Myr and t12 = 62 Myr for the uppermantle convection model (t11 = 170 Myr and t12 =180 Myr for whole-mantle convection). At the time t11,the continent touches the ridge and, at the time t12, theridge is drawn under the continent. The descendingmantle flow spreads, and part of the hot ascending man-tle flow makes its way under the continent. At this time,the right edge of the continent is heated by the ascend-ing mantle flow. This must decrease the viscosity andthickness of the continental lithosphere. The state of themantle at this time can correspond to the present stateof the mantle at the active margin of North America.

4. DISCUSSION

During the last three decades, many attempts havebeen made to explain the origin of the marginal seas atthe active margin of Eurasia [Karig, 1971; Chase, 1978;Toksoz and Hsui, 1978; Hsui and Toksoz, 1981; Taylorand Karner, 1983; Hynes and Mott, 1985; Jarrard,1986; DeMets, 1992; Trubitsyn and Belavina, 1992]. Itwas established that there are two different types ofsubduction zones [Molnar and Atwater, 1978; Uyedaand Kanamori, 1979; Nakamura and Uyeda, 1980]. Ifthe continent moves toward the subduction zone (theChilean type), marginal seas do not arise. However, ifthe mobility of the continent is low (the Mariana type),the weakening of the oceanic lithosphere leads to thedevelopment of marginal seas with the back-rolling ofthe deep-sea trench toward the ocean [Schubert et al.,2001]. However, a self-consistent model describing the

104


TRUBITSYN

formation and evolution of marginal seas has not yetbeen constructed.

The numerical solution of the equations of convec-tion with a floating continent indicates that the afore-mentioned two types of subduction zones are obtainedautomatically and correspond to two consecutive evo-lutionary stages of the continental drift after thebreakup of the supercontinent. When a large continentstarts moving toward the subduction zone but is still atsome distance from it, its velocity is lower than that ofthe oceanic lithosphere drawn by the descending man-tle flow. Therefore, the lithosphere between the conti-nent and the subduction zone is stretched and thinned.A marginal sea with a local maximum of heat fluxforms, and the oceanic lithosphere on the side of theocean begins to sink obliquely. The modeling alsoshowed that fragments of the continent can rapidly

swim away and then stop above the subduction zone.These results are in a good agreement with data of geo-logical reconstructions [Maruyama, 1997; Maruyamaet al., 1997], according to which the Japanese islandsseparated from Eurasia at 15–25 Ma. Thus, in contrastto previous models (e.g., see [Schubert et al., 2001]),the model presented here provides a new and self-con-sistent explanation for the formation of marginal seas.Moreover, this model agrees better with data of obser-vations. In particular, it accounts for even such a detailas the stress distribution within a marginal sea (morespecifically, extension near the end of the continent andminor compression near the subduction zone [Jarrard,1986]). As seen from Fig. 3, at the formation time of themarginal sea (t = t4), the horizontal velocities of theoceanic lithosphere moving from the continent to thesubduction zone first increase (leading to extension)

30

70

1.0

z 0.5

04 5 6 7 8 x


0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1

Nu

zN

u

t10

t9

30

70

1.0

0.5

0

Fig. 7. Calculated structure of mantle convection at the times t9 = 52 Myr and t10 = 54 Myr in the upper mantle model (t9 = 150 Myrand t10 = 156 Myr for whole-mantle convection), when the continent moves together with the subduction zone, partially overriding it.



and then drop (leading to compression), vanishingabove the descending mantle flow.

The model predicts the further evolution of marginalseas. Although the fragment gets stuck in the subduc-tion zone, the continent is constantly moving towardthe ocean and, for a certain time, the marginal sea doesnot close. The numerical modeling showed that, in thisstructure of mantle convection, the subduction zonemust inevitably roll back toward the ocean. Later,although the subduction zone is moving away, the con-tinent catches up with it, causing the marginal sea toclose. After the continent has adjoined the subductionzone, they move together. A region of strong compres-sion appears at this end of the continent because, abovethe descending mantle flow, the horizontal componentsof the velocities of mantle counterflows vanish andmaterial from both sides is driven into this place. This

stress state may have caused the formation of the Andesin South America.

Recently, the extension of the lithosphere with theformation of marginal seas was attributed to the ocean-ward back-rolling of the subduction zone. The back-rolling was in turn ascribed to the breaking of the oldoceanic plate under its own weight. However, aspointed out above, in the western part of North Amer-ica, the ridge and subduction zone closely approachedthe continent (see Fig. 1). Consequently, the back-roll-ing involved a very young and light plate that could notbreak under its own weight. The model presented in thispaper also gives a new, more complete and self-consis-tent explanation of the process of back-rolling and thecauses of the extension of oceanic lithosphere with theformation of marginal seas. These phenomena are dueto the mechanical and thermal interactions of the float-

30

70

1.0

z 0.5

04 5 6 7 8 x


0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1

Nu

zN

u

t12

t11

30

70

1.0

0.5

0

Fig. 8. Calculated structure of mantle convection at the times t11 = 58 Myr and t12 = 62 Myr (t11 = 170 Myr and t12 = 180 Myr forwhole-mantle convection), when the continent reaches the ridge and the right-hand edge of the continent is heated.

106


TRUBITSYN

ing continent with mantle convection, described by thegeneral equations of mass, energy, and momentumtransfer.

The most complex tectonic setting is presentlyobserved in the western part of North America, whichis an object of study in numerous publications. A com-parative study of the structure and tectonics of theactive margins of South and North America was per-formed by Romanyuk et al. [2001]. It was concludedthat North America in its tectonic evolution was aheadof South America and, in particular, that the plate sink-ing under North America could have been detached afew tens of millions of years ago. Morgan [1983]pointed out that North America could have movedtoward the Pacific Ocean due to the expansion of theAtlantic. However, in most of the subsequent works,the tectonics of North America was examined not fromthe standpoint of the movement of the continent but interms of underthrusting of oceanic plates under the con-tinent [Bohannon and Parson, 1995] followed by theunderthrusting of the subduction zone and even of themid-ocean ridge. In these studies, data of observationswere analyzed in detail but no self-consistent modelsexplaining the cause of the processes were constructed.

In the present paper, the possible evolution of NorthAmerica is explained on the basis of numerical model-ing. As the main controlling factor, we propose themotion of the floating continent interacting with themantle and changing the structure of mantle convec-tion. After the breakup of the supercontinent, each con-tinent can travel a distance comparable to the sizes ofoceans because the driving force is the whole-mantleconvection, encompassing the whole Earth. Therefore,the continent moves for a fairly long time and can gothrough several evolutionary stages during which itchanges the structure of mantle flows in the vicinity ofthe active margin. According to the modeling results,the evolution of North America is at the third stage ofthe general evolutionary sequence of continental driftafter the breakup of the supercontinent. This conclusionagrees with data of paleoreconstructions. As seen fromFig. 9, the breakup of Pangea started with the separa-tion and rapid movement of North America, followedby South America, while Eurasia moved very slowly.Thus, Eurasia is still at the first evolutionary stage,South America is already at the second stage, and NorthAmerica may have passed through the first and secondstages. Together with its subduction zone, the NorthAmerican continent has already reached the ridge. It isthe interaction between the subduction zone and theridge that can be the cause of the very complex tectonicsetting of western North America. These three stagescan explain why the distances from the mid-ocean ridgeto Eurasia, South America, and North America are sodifferent, i.e., why the Pacific Ocean is strongly asym-metric. As is evident from paleoreconstructions(Fig. 9), it was not the ridge that moved eastward toNorth America (as is assumed by many adherents ofplate tectonics who do not recognize the important role

of continents), but the continent overrode the ridge (asfollows from the model presented here and based on thesolution of the general equations of energy, mass, andmomentum transfer).

The Pacific Ocean evolved due to interactionsbetween many mechanical, thermal, geological, andgeochemical processes [Milanovskii, 1987]. Therefore,this simple model does not claim to explain all specificfeatures of the evolution of the present Pacific belt. Inparticular, the calculated flow velocities and times ofthe evolutionary stages are accurate to only an order ofmagnitude. These values were calculated in dimension-less variables. The times of the stages are mainly deter-mined from the model of upper mantle convection. Inthe case of whole-mantle convection, when passing todimensional variables according to (29)–(31), the cal-culated times must be increased by about three times.

The first approximation model proposed hereignores the attachment of the fragment to the continent.Therefore, under real conditions, its detachment shouldoccur at a later time, when the tensile strength in themarginal sea will exceed the strength of continentalrocks.

A drawback of this model is the approximatedescription of ascending mantle flows giving rise tomid-ocean ridges. In the real Earth, these ascendingflows form not at the base but within the upper mantledue to the gap produced by the rifting of the oceaniclithosphere. Methods used for constructing such mod-els are still under development.

The closure of the Pacific Ocean is expressed in thesimultaneous motion of four continents (Eurasia, Northand South America, and Australia). This paper exam-ines the evolution of only one floating continent, i.e.,without considering the mutual influence of continentson each other. However, this approximation is justifiedby the fact that, at the evolutionary stages considered,all these continents were located sufficiently far fromeach other. Moreover, this model of evolution of onecontinent can clarify the sequence of different stages ofthe general evolution of continents diverging after thebreakup of the supercontinent.

The relation of the theory of floating continents tothermochemical convection is worth noting. The mod-els of thermal convection consider single-componentmaterial, taking into account phase transitions in thegeneral case. Convective flows in the mantle are gener-ated by lateral density heterogeneities, implying thatlight material rises or heavy material descends in someplaces. Density anomalies are caused by nonuniformheating in thermal convection and by a heterogeneouschemical composition in chemical convection. The lat-eral heterogeneities in turn can be caused by initial con-ditions or can arise due to the instability of a nonuni-form depth distribution of density in the gravitationalfield. Flows of thermal convection are constantly main-tained by the heating of material from below or by dis-tributed heat sources.



Models of chemical convection include multicom-ponent material with variable density and take intoaccount the energy of chemical reactions. The energyof chemical reactions in the mantle is much smallerthan the radioactive heat and primary heating and canbe neglected in the calculation of mantle flows. In theabsence of thermal convection, flows develop due to thedescent of denser material and the ascent of lightermaterial of a different composition. Therefore, this typeof convection is more frequently referred to as compo-sitional rather than chemical convection. Composi-tional convection in the mantle continues as long asthere are lateral chemical density heterogeneities. Themain heavy component of the mantle was iron. How-ever, it had already sunk down into the core during thefirst few tens of millions of years after the formation ofthe Earth, when the Earth was partially molten [Schu-bert et al., 2001]. At present, the separation of compo-nents at the molecular level is possible only in magmachambers and in the D'' layer at the boundary with thecore. Within the mantle, at temperatures below themelting point, the differentiation rate of material is very

low and the differentiation process requires hundreds ofmillions of years.

When convection was discovered in the Earth’smantle, global flows were at first attributed to both ther-mal and compositional convection. Yet it was laterestablished that mantle dynamics is controlled pre-cisely by thermal convection, while the effects of chem-ical (or compositional) convection introduce only addi-tional details, without changing the overall structure ofglobal flows [Schubert et al., 2001]. What is this con-clusion based on? The structures of all observed geo-physical fields (thermal and gravitational fields,dynamic topography, and seismic tomography) corre-spond to thermal, rather than chemical, convection. Forexample, if the ascending material becomes lighter dueto overheating, hot magma will erupt above such a flow.If the ascending material becomes lighter due to achange in its chemical composition, this will beobserved at the surface as a cold density anomaly (suchmaterial cannot reach the surface simply because of itshigh viscosity), but this is inconsistent with all observa-tional data. Therefore, it is necessary to distinguish

163 å‡ 119 å‡

84 å‡ 59 å‡

43 å‡ 0 å‡

Fig. 9. Paleoreconstructions of the drift of continents and the configuration of isochrons (after [Muller et al., 1997]).

108


TRUBITSYN

which effects are due to thermal convection and whichare due to compositional convection.

All primary density heterogeneities (with dimen-sions of the order of 10 km and more) should alreadyhave sunk to the mantle base or mixed with surroundingmaterial because the mantle flows have already experi-enced a few tens of cycles. Yet lateral chemical densityheterogeneities can be generated constantly. The mainreservoir supplying lateral chemical (or compositional)density heterogeneities into the mantle is crustal mate-rial and material of the D'' layer [Schubert et al., 2001;Dobretsov et al., 2001; Lobkovsky and Kotelkin,2000]. This material is entrapped by descending mantleflows in subduction zones or by ascending flows fromthe base of the mantle. Thus, it is primarily thermalconvection flows that generate lateral chemical hetero-geneities required for maintaining chemical convec-tion, which is actually a secondary phenomenon rela-tive to thermal convection. Internal chemical heteroge-neities are much smaller in size than thermalheterogeneities. Therefore, the transportation of chem-ical elements within the mantle is effected mainly pas-sively by thermal convection flows. Although the struc-ture of global flows in the mantle is controlled by ther-mal convection, the effects of chemical (compositional)convection should be taken into account in calculatingthe redistribution of chemical elements in the Earth andexamining the process of plume formation and ascentfrom the base of the mantle [Dobretsov et al., 2001].

The above discussion of chemical (or composi-tional) convection was concerned only with lateralchemical heterogeneities within the mantle. It is theseheterogeneities that are usually considered when speak-ing of chemical convection. However, such an impor-tant factor as continents was disregarded. Continentsare the largest lateral chemical heterogeneities, whichhave existed for about 4 billion years. Their size (thou-sands of kilometers) is much larger than that of chemi-cal heterogeneities within the mantle. Moreover, theinfluence of continents on mantle flows is very greatbecause of their high buoyancy. Small chemical heteroge-neities can be transported together with mantle flows,distorting them only weakly, whereas the movement ofcontinents differs basically from mantle flows because,due to their buoyancy, they never sink into the mantle.The influence of continents on the structure of mantleflows is particularly great when they are assembled as asupercontinent. Yet, as shown in this paper, even whencontinents diverge after the breakup of the superconti-nent, they participate in the formation of the observedmantle structures. Thus, the most important manifes-tation of thermochemical convection in the mantle isthe interaction of thermal convection with floatingcontinents, differing in chemical composition fromthe mantle.

5. CONCLUSIONSAfter the breakup of Pangea, the Atlantic Ocean

arose and began to expand, while the Pacific Oceanbegan to close. A circular belt of subduction zones withpowerful descending mantle flows developed on itsperiphery. Therefore, the North and South American,Australian, and Eurasian continents began to movetoward these subduction zones. However, since thestructures of mantle flows beneath the continents weredifferent, the velocities of the continents were alsolargely different. According to available reconstruc-tions, the breakup of Pangea at 170 Ma was followed bythe expansion of the North Atlantic and, therefore,North America was the first to move away from thesupercontinent. Later, at 110 Ma, the South Atlanticbegan to expand and South America began to movetoward the Pacific Ocean. Eurasia began to movetoward the Pacific only recently. The modeling of con-tinental drift showed that, after the breakup of thesupercontinent, a continent can experience three stagescorresponding to the present states of Eurasia andSouth and North America. Therefore, it is possible thatthese continents, like stars, are following the same evo-lutionary sequence. The asymmetry of the PacificOcean can be attributed to the fact that rapidly movingNorth America was the first to approach the ridge,pushing forward the subduction zone; at present, thelatter and the ridge are being destroyed.

ACKNOWLEDGMENTSThis work was supported by the Russian Foundation

for Basic Research, project no. 05-05-64029.

APPENDIX

A.1. EQUATIONS OF MANTLE CONVECTION WITH A FLOATING CONTINENT

Equations of Thermal Convection

The system of equations of thermal convectionincludes the Stokes equation, heat transfer equation,and continuity equation. In the Boussinesq approxima-tion, when the density is taken to be constant every-where, except the term describing the buoyancy force,the equations of motion have the form

, (1)

, (2)

(3)

Here, ρ is the mantle density; gi is the gravitationalacceleration; T is the temperature measured from theadiabatic distribution; k is diffusivity; δij is the Kro-necker delta, which is equal to 1 at i = j and equal to 0at i ≠ j; and σij is the deviatoric tensor of viscousstresses [Landau and Lifshitz, 1986]

(4)

ρdVi/dt ∂p/∂xi– ∂σij/∂x j ρgδi3+ +=

dT /dt ∂ k∂T /∂xi( )∂xi=

∂ρ/∂t ∂ Viρ( )/∂xi+ 0, i 1 2 3., ,= =

σij η ∂Vi/∂x j ∂V j/∂xi+( ),=



where η is the viscosity. Equations (1)–(3) containthree unknown functions: the velocity vector Vi(xi, t),temperature T(xi, t), and pressure p(xi, t).

Equation (1) means that the acceleration of a liquidin a given unit volume is equal to the gradient of thepressure acting from the surrounding liquid, i.e., theresultant of the viscous friction and gravity forces.Equation (2) implies that the temperature variation ofthe given liquid element is controlled by the influx ofconductive heat. The equation of incompressibility (3)means that the density variation of the liquid in thegiven unit volume is controlled by the mass inflow ofthe surrounding liquid.

The relative value of the inertial terms on the leftside of the equation of momentum transfer in a viscousliquid (1) with respect to the terms on the right sides ofthe equations is of the order of kρ/η ≈ 10–23. Therefore,these inertial terms can be neglected. In the Boussinesqapproximation, we set ρ = ρ0(1 – αT) in the last buoyancyterm in Eq. (1) and ρ = ρ0 in all other terms of Eqs. (1)–(3).The pressure is measured from its hydrostatic distribu-tion p(z), which is defined by the condition —p0 = –ρ0g.We introduce dimensionless variables, using the fol-lowing units of measurement: the mantle thickness Dfor the length, k/D for the velocity, D2/k for the time,∆T = T2 – T1 for the temperature, the viscosity value onthe surface for the viscosity, and η0k/D2 for the pressureand stresses.

In these variables, the equations of convection (1)–(3) take the form

(5)

(6)

(7)

where Ra is the Rayleigh number,

(8)

For the 2-D model of convection, Eqs. (5)–(7) canbe written in an explicit form, directing the z axis verti-cally upward:

(9)

(10)

(11)

(12)

The lower boundary of the mantle is usuallyassumed to be impermeable with the no-slip condition.Its upper boundary is free and deformable. Yet, since

0 ∂p/∂xi– ∂σij/∂x j RaTδi3,–+=

∂T /∂t Vi∂T /∂xi+ ∂ ∂T /∂xi( )∂xi,=

∂Vi/∂xi 0,=

Ra αρ0gT0D3/kη0.=

∂p/∂x– ∂ η ∂V x/∂z ∂Vz/∂x+( )[ ]/∂z+

+ 2∂ η∂V x/∂x( )/∂x 0,=

∂p/∂z– ∂ η ∂Vz/∂x ∂V x/∂z+( )[ ]/∂x+

+ 2∂ η∂Vz/∂z( )/∂z RaT+ 0,=

∂T /∂t V x∂T /∂x Vz∂T /∂z+ +

= ∂/∂x k∂/T /∂x( ) ∂/∂z k∂/T /∂z( ),+

∂V x/∂x ∂Vz/∂z+ 0.=

the deformation is small (much smaller than the man-tle’s thickness), its influence on the distribution of con-vective velocities is also small. Therefore, in a firstapproximation, the mantle’s upper boundary can beconsidered as a nondeformable impermeable no-slipboundary. (The deformation of the mantle’s surface isusually determined in the following approximationfrom the ascent of mantle material at a given point due tothe vertical component of the viscous stress at the surfacelevel z = 1.) The impermeability of the upper and lowerboundaries requires that the normal components of flowvelocities vanish there:

Vz(x, z = 0) = Vz(x, z = 1) = 0. (13)

At the free-slip boundaries, the tangential compo-nents of the stress tensor vanish:

σxz(x, z = 0) = σxz(x, z = 1) = 0. (14)

According to (4), this condition gives ∂Vx/∂z +∂Vz/∂x = 0. Since the viscous stress tensor is symmetri-cal, (14) reduces to the condition ∂Vx/∂z = 0 at z = 0 andz = 1. Given that this condition is satisfied at all x values, itcan be differentiated with respect to x: ∂2Vx/∂z∂x = 0 or∂/∂z(∂Vx/∂x) = 0. Since the divergence of the velocity(7) vanishes, we may replace ∂Vx/∂x by –∂Vz/∂z. There-fore, the condition at free-slip boundaries is finallywritten as

∂2Vz/∂z2 = 0 at z = 0 and z = 1. (15)

We preserve the boundary conditions for tempera-ture in the commonly used form, when its values arefixed at the upper and lower boundaries:

T(x, z = 1) = 0 and T(x, z = 0) = 1. (16)

In a ring-type model, the calculations are performedin an elongated rectangular region with periodicity con-ditions set for the velocities and temperature:

(17)

System (10)–(12) contains four equations for fourunknown functions, which are the velocities Vx and Vz,pressure p, and temperature T. In order to determine theintegration constants, we have eight boundary condi-tions for the velocities (13), (15), and (17) and four forthe temperature (16)–(18), as well as one condition forthe pressure; for example, we may require that the pres-sure vanish at the extreme left point of the upper sur-face: p(x = 0, z = 1) = 0. As the initial condition for ther-mal convection, we used a linear temperature distribu-tion with a random disturbance along the x axis at zeroinitial velocities.

Equations of Motion of a Rigid Floating Continent

After thermal convection develops in the calculatedring-like region, a continent in the form of a solid rectangu-lar plate of a length x2 – x1 = l and a thickness z2 – z1 = d is

Vz x z,( ) Vz x L z,+( ),=

V x x z,( ) V x x L z,+( ), T x z,( ) T x L z,+( ).= =

110


TRUBITSYN

placed into the region. Since the continent is consideredto be rigid, the velocities of all its points u(x, z) are thesame and equal to the velocity of its center of gravity u0;the vertical velocity of the continent floating on the sur-face is equal to zero. As a result, we have ux(x, z) = u0,uz(x, z) = 0.

The continent moves in response to the forces of vis-cous cohesion with mantle flows applied to the part ofthe continent submerged in the mantle, at its base andends. Since the part of the continent protruding fromthe mantle is small and has no effect on its horizontalmotion, we will neglect it in what follows. The forcesin the Newton equation responsible for the horizontalmotion of the floating continent can be written as anintegral of the viscous forces over the surface of contactwith the mantle [Trubitsyn, 2000]. This force is equal ateach point to the product of the viscous tensor and thenormal to the surface fi = (p – σij)nj [Landau and Lif-shitz, 1986]. The entire integral is divided into threeparts: integrals over the two end faces and over thebase. The force acting at the points of the end faces isfx = (p – σxx)nx; the force at the base is fx = –σxznz. As aresult, the equation of motion of the rigid floating con-tinent takes the form

(18)

,

where m is the mass of the continental plate per unit oflength along the y axis; x2(t) = x1(t) + l; and x1(t) is thecoordinate of the left end of the plate, which satisfiesthe kinematic condition

dx1/dt = u0. (19)

Since the continent moves at a velocity that is com-parable to the velocities of mantle flows, the left side inEq. (18) is of the order of 10–23 and can be set equal tozero (as in Eq. (5)). As a result, Eq. (18) reduces to anintegral relation for the velocities of mantle flows at theboundaries with the continent:

(20)

.

m∂u0/∂t p( )x x1= 2η ∂V x/∂x( )x x1=–[ ] zd

1 d–

1

∫=

– p( )x x2= 2η ∂V x/∂x( )x x2=–[ ] zd

1 d–

1

∫

– η ∂V x/∂z( )z 1 d–= ∂Vz/∂x( )z 1 d–=+[ ] xd

x1

x2

∫

p( )x x1 l+= p( )x x1=–[ ]{1 d–

1

∫–

– 2η ∂V x/∂x( )x x1= ∂V x/∂x( )x x2=–[ ] }dz

– η ∂V x/∂z( )z 1 d–= ∂Vz/∂x( )z 1 d–=+[ ] xd

x1

x1 l+

∫ 0=

The temperature distribution Tc(x, z, t) within thecontinent is calculated from the equation of heat trans-fer. In a fixed coordinate system, the total derivativedTc/dt consists of the partial derivative with respect tothe time t and the advective heat transfer with the con-tinent moving at a velocity u0:

(21)

where kc is the diffusivity of the continent.

Boundary Conditions for the System of Equations of Thermal Convection

with a Floating Continent

As compared with convection without a continent,the conditions in the case of a floating continent changeonly in the place where the continent is present. At thebase and end faces of the rigid moving continentalplate, the continuity condition must be satisfied at eachtime for the temperature and heat flux:

T = Tc, k∂T/∂n = kc∂Tc/∂n, (22)

where n is the normal to the corresponding surface ofthe continent. Mantle flows are subject to the impene-trability condition; i.e., the normal component of theirvelocity must be equal to the velocity of the center ofgravity of the continent at all points of the end faces andthe base of the continental plate:

Vx = u0, Vz = 0. (23)

Since Vz = 0 all along the end face of the continent,i.e., at all z values, the differentiation of this equalitywith respect to z yields ∂Vz/∂z = 0. Hence, accordingto the incompressibility condition, we obtain ∂Vx/∂x =–∂Vz/∂z = 0 at all points of the end faces. Therefore, thesecond terms in the integrals over the end faces in (18)vanish. Given that Vz = 0 all along the base of the con-tinent, i.e., at all x values, we have ∂Vz/∂x = 0 at z = 1 – d.Therefore, the second term in the third integral van-ishes. As a result, the equation of motion of the conti-nent (18) (or (20)) reduces to a simple equation of thebalance of pressure at the end faces and the balance ofshear stresses at the base [Trubitsyn and Rykov, 1997;Trubitsyn et al., 1999; Trubitsyn, 2000]:

(24)

In the particular case of a thin continent (d = 0), rela-tion (24) reduces to a simple balance condition for the

∂Tc/∂t u0∂Tc/∂x+

= ∂/∂x( ) kc∂Tc/∂x( ) ∂/∂z( ) kc∂Tc/∂z( ),+

p( )x x1 l+= p( )x x1=–[ ] zd

1 d–

1

∫

+ η ∂V x/∂z( )z 1 d–= xd

x1

x1 l+

∫ 0.=



shear stress forces at the base of the continent that wasused in [Gable et al., 1991]:

(25)

Thus, in order to calculate the evolution of mantleconvection with a floating continent, it is necessary tofind seven unknowns: two mantle flow velocities Vx(x,z, t) and Vz(x, z, t); the pressure p(x, z, t); the tempera-ture in the mantle; the temperature within the rigid con-tinent Tc(x, z, t); the horizontal velocity of its center ofgravity u0(x, t); and the x coordinate of any point of thecontinent, for example, belonging to its left end face x1(t).We have seven equations for these seven unknowns:four equations of thermal convection (9)–(12) coupledwith three equations for the continent (the equation ofits motion (20), the equation of heat transfer within thecontinent (21), and the kinematic relation (19) betweenthe velocity and the coordinate of its position). Thissystem of equations is essentially nonlinear because itcontains large terms quadratic with respect to the func-tions Vx∂T/∂x, Vz∂T/∂z, and u0∂Tc/∂x. This problemwith a floating continent differs from the well-knownproblem with a fixed continent in that the impenetrableno-slip boundary conditions at the upper surface at eachtime moment are now set for a floating continent whosevelocity and position are not known a priori but aredetermined at each time step by solving a system ofinterrelated differential equations.

Numerical Method

In the numerical procedure of solving the system ofequations of thermal convection with floating conti-nents, the equations of heat and mass transfer in themantle outside and within the continent are solved ateach time step separately and the solutions are thenadjusted in order to satisfy the boundary conditions atcontinent surfaces submerged in the mantle.

In the general form the algorithm used for thenumerical solution of the system of equations of ther-mal convection with floating continents reduces to thefollowing. Assume that, at a certain time moment t1, weknow the velocities of convective flows and the fields oftemperatures T(t1) and pressures p(t1) in the mantle, aswell as the continent position x1(t1) and velocity u0(t1). Weare to find the solution to the system of equations (5)–(8),(14), (23), and (24) at the time t2 = t1 + ∆t. The newposition of the continent x1(t2) at the time t2 can be eas-ily determined from (19): x1(t2) = x1(t1) + u0(t1)∆t. If wealso knew the new velocity of the continent in this posi-tion u0(t2), it would be possible to solve the equations ofthermal convection (5)–(8) with the boundary condi-tions for the temperature (18) and velocities (19)–(22)corresponding to the new position of the continent andto find the velocities of mantle flows Vx(t2) and Vy(t2),

η ∂V x/∂z( )z 1 d–= xd

x1

x1 l+

∫ 0.=

the temperature T(t2), and the pressure p(t2) at the time t2.However, the complexity of this problem lies in the factthat the velocity of the continent u0(t2) is unknown. Thisvelocity must be such that the newly obtained velocitiesof mantle flows Vx(t2) and Vy(t2) corresponding to u0(t2)should also satisfy the equation of motion of the conti-nent in form (20). Therefore, it is necessary to find aniterative method for determining this value of the veloc-ity of the continent. In principle, we can simply con-sider all possible values of the continent velocityaccording to a certain scheme, calculating the convec-tive velocity fields and the integrals in (20) until we findthe value of u0(t2) at which the right-hand side ofEq. (20) differs from zero by a value ε corresponding tothe given accuracy. Since the right-hand side ofEq. (20) corresponds in physical meaning to the forceacting on the continent, we have ε > 0 if the chosenvelocity of the continent u0(t2) is underestimated andε < 0 if it is overestimated.

The system of equations of thermal convection withfloating continents was solved by the finite differencemethod. The equation of temperature transfer (11) wasnumerically solved by Zalesak’s flux-corrected trans-port method [Zalesak, 1979]. The equations for veloci-ties and pressure (9)–(11) were reduced to elliptic equa-tions with variable coefficients (generalized Poissonequations). They were solved by the triangle method ina three-layer modification alternating with the choice ofiterative parameters by the method of conjugate gradi-ents [Samarskii and Nikolaev, 1978]. The algorithmiccode was written by Rykov [Trubitsyn and Rykov,1997, 2001].

A.2. RAYLEIGH AND NUSSELT NUMBERS

The Rayleigh number Ra (8) characterizes the inten-sity of thermal convection. Another characteristic is theNusselt number Nu, which quantifies the effectivenessof convective heat transport and is equal to the ratio ofthe heat flux in the presence of convection q to the heatflux with no convection and with the same temperaturedifference over the layer:

Nu = q/(κ∆T/D) =VzT – dT/dz, (26)

where κ is the thermal conductivity, connected with thediffusivity through the relation κ = ρcpk. The Rayleighnumber in form (9) is used if the temperature difference∆T over the layer is a priori known. However, if thetemperature at the bottom of the mantle is not exactlyknown (as in the case of the Earth), it is necessary to useanother modification of the Rayleigh number Raq, inwhich the heating of the mantle is characterized not bythe temperature difference ∆T but by the heat flux qobserved on the surface:

Raq = (αρgqD4)/(kκη). (27)

Comparing formulas (8), (26), and (27), we obtain

Raq = NuRa. (28)

112


TRUBITSYN

The total heat flux of the entire Earth is Qtot = 4.43 ×1013 W. It consists of the radioactive heat of the crust(17%), the radioactive and primary heat coming fromthe mantle, and the heat coming from the core (10%)[Davies, 2000]. The heat coming from the mantle isQman = 3.69 × 1013 W. With the Earth’s surface areabeing equal to 5.1 × 108 km2, the average heat flux ofthe Earth is qtot = 0.087 W/m2 and the flux from themantle is qm = 0.0724 W/m2 [Schubert et al., 2001].

According to Schubert et al. [2001], the upper man-tle values of parameters in the equations of convection(1)–(12) are α = 3 × 10–5 ä–1, g = 10 m/s2, ρ =3600 kg/m3, D = 700 km, k ≈ 4 W/m, and κ ≈ 0.8 ×10−6 m/s2. The value of the viscosity is most indefinite,lying within the range η = 1020–1021 Pa s. With theabove values of parameters and the average viscosityη = 5 × 1020 Pa s, the Rayleigh number is Raq =(αρgqD4)/(kκη) ≈ 107. The analytical boundary-layertheory of thermal convection [Schubert et al., 2001]gives the relation between the Rayleigh and Nusseltnumbers Nu ≈ 0.2 Ra1/3. Hence, taking into account(29), we obtain

Nu ≈ 0.3 . (29)

Therefore, with the above values of the parameters,the Nusselt number for the upper mantle is Nu ≈ 17 andthe Rayleigh number corresponding to the average vis-cosity η = 5 × 1020 Pa s is Ra = 6 × 105.

Convection was calculated in dimensionless vari-ables, using the model of a 2-D ring with an aspect ratioof 1:5 and the Rayleigh number Ra = 106. To obtaindimensional values, it is necessary to multiply the com-puted values by the corresponding units of measure-ment specified as follows (with regard for the abovevalues of the upper mantle parameters):

(30)

According to Schubert et al. [2001], the whole-mantle values of parameters in the equations of convec-tion (1)–(12) are α = 2 × 10–5 ä–1, g = 10 m/s2, ρ =3600 kg/m3, D = 2900 km, κ ≈ 5 W/m, k ≈ 10–6 m/s2,and qm = 0.0724 W/m2. With the viscosity equal to η =2 × 1021 Pa s, the corresponding Rayleigh numberaccording to (28) is Raq = (αρgqD4)/(kκη) ≈ 5 × 108.Then, for whole-mantle convection, relation (30) yieldsthe Nusselt number Nu ≈ 50 and, using relation (29),we find the effective Rayleigh number Ra = 107.

The units of measurement for whole-mantle convec-tion are D = 2900 km, V0 = k0/D ≈ 1.1 × 10–3 cm/yr, andt0 = D2/k0 ≈ 270 Gyr. To convert the model resultsobtained in dimensionless variables for Ra = 106 intovalues corresponding to the whole-mantle Rayleighnumber Ra = 107, we can use the boundary-layer theory

Raq1/4

D 700 km, V0 k0/D 4 10 3– cm/yr,×≈= =

t0 D2/k0 20 Gyr.≈=

of convection, according to which the flow velocity V0

and the time t0 are both proportional to Ra2/3. Thus, forwhole-mantle convection with Ra = 107, the dimension-less units of measurement calculated in the model withRa = 106 should be multiplied by 102/3 = 4.7 for veloc-ities and divided by the same value for the time. As aresult, we obtain

(31)

The Nusselt number is actually the dimensionlessheat flux and can be calculated by formula (26) at anydepth within the mantle, characterizing the averageheat flux through the corresponding layer. In the case ofstationary convection and heating from below, thedependence Nu(z) should be represented as a verticalline.

REFERENCES1. B. Blankenbach, F. Busse, U. Christensen, et al., “A

Benchmark Comparison for Convection Codes,” Geo-phys. J. Int. 98, 23–38 (1989).

2. R. G. Bohannon and T. Parson, “Tectonic Implications ofPost-30 Ma Pacific and North American Relative PlateMotions,” Geol. Soc. Am. Bull. 107 (8), 937–959(1995).

3. G. C. Brown and A. E. Mussett, The Inaccessible Earth(Allen and Unwin, 1981; Mir, Moscow, 1984) [in Rus-sian].

4. C. G. Chase, “Extension behind Island Arcs and MotionRelative to Hot Spots,” J. Geophys. Res. 83, 5385–5387(1978).

5. C. DeMets, “A Test of Present-Day Plate Geometries forNortheast Asia and Japan,” J. Geophys. Res. 97, 17627–17635 (1992).

6. N. L. Dobretsov, A. G. Kirdyashkin, and A. A. Kirdyash-kin, Deep Geodynamics (SO RAN, Novosibirsk, 2001)[in Russian].

7. A.-M. Forte and J.-X. Mitrovica, “Deep-Mantle High-Viscosity Flow and Thermochemical Structure Inferredfrom Seismic and Geodynamic Data,” Nature, 1049–1056 (2001).

8. M. Gurnis, “Large-Scale Mantle Convection and Aggre-gation and Dispersal of Supercontinents,” Nature 332,696–699 (1988).

9. A. T. Hsui and M. N. Toksoz, “Back-Arc Spreading:Trench Migration, Continental Pull or Induced Convec-tion?,” Tectonophysics 74, 89–98 (1981).

10. A. Hynes and J. Mott, “On the Causes of Back-ArcSpreading,” Geology 13, 387–389 (1985).

11. R. D. Jarrard, “Relations among Subduction Parame-ters,” Rev. Geophys. 24, 217–284 (1986).

12. T. H. Jordan, “Structure and Formation of the Continen-tal Lithosphere,” J. Petrol., 11–37 (1988).

13. D. E. Karig, “Origin and Development of MarginalBasins in the West Pacific,” J. Geophys. Res. 76, 2542–2561 (1971).

D 2900 km, V0 k0/D 5.2 10 3– cm/yr,×≈= =

t0 D2/k0 57 Gyr.≈=



14. V. E. Khain and M. G. Lomize, Geotectonics and Fun-damentals of Geodynamics (Knizhnyi Dom, Moscow,2005) [in Russian].

15. L. D. Landau and E. M. Lifshitz, Fluid Mechanics, 2nded. (Nauka, Moscow, 1986; Pergamon Press, Oxford,1987).

16. L. I. Lobkovsky and V. D. Kotelkin, “Geodynamics ofMantle Plumes, Their Interaction with the Asthenos-phere and Lithosphere, and Surface Features of Riftingand Trap Formation,” in General Problems in Tectono-physics: Tectonics of Russia (GEOS, Moscow, 2000) [inRussian].

17. S. Maruyama, “Pacific-Type Orogeny Revised: Miyash-iro-Type Orogeny Proposed,” The Islands Arc 6 ((1)),91–120 (1997).

18. S. Maruyama, Yi. Isozak, G. Kimura, and M. Teraba-yashi, “Paleogeographic Maps of the Japanese Islands:Plate Tectonic Synthesis from 750 Ma to Present,” Isl.Arc 6 (1), 121–142 (1997).

19. E. E. Milanovskii, Rifting in the Earth’s History (Nedra,Moscow, 1987) [in Russian].

20. P. Molnar and T. Atwater, “Inter-Arc Spreading and Cor-dilleran Tectonics As Alternates Related to the Age ofSubducted Oceanic Lithosphere,” Earth Planet. Sci. Lett.41, 330–340 (1978).

21. J. W. Morgan, “Hotspot Tracks and Opening of theAtlantic,” J. South Am. Earth Sci., No. 94, 123–129(1983).

22. R. D. Muller, W. R. Roest, I. Y. Royer, et al., “DigitalIsochrons of the World’s Ocean Floor,” J. Geophys. Res.102, 3211–3214 (1997).

23. K. Nakamura and S. Uyeda, “Stress Gradient in Arc–Back Arc Regions and Plate Subduction,” J. Geophys.Res. 85, 6419–6428 (1980).

24. T. V. Romanyuk, W. D. Mooney, and R. J. Blakely, “Den-sity Model of the Cascadia Subduction Zone,” Fiz.Zemli, No. 8, 3–22 (2001) [Izvestiya, Phys. Solid Earth37, 617–635 (2001)].

25. A. A. Samarskii and E. S. Nikolaev, Methods for SolvingGrid Equations (Nauka, Moscow, 1978) [in Russian].

26. G. Schubert, D. L. Turcotte, and P. Olson, Mantle Con-vection in the Earth and Planets (Univ. Press, Cam-bridge, 2001).

27. P. J. Tackley, “Self-Consistent Generation of TectonicPlates in Time-Dependent, Three-Dimensional MantleConvection Simulations, 1. Pseudoplastic Yielding,”Geochem. Geophys. Geosyst. 1 (2000).

28. B. Taylor and G. D. Karner, “On the Evolution of Mar-ginal Basins,” Rev. Geophys. 21, 1727–1741 (1983).

29. M. N. Toksoz and A. T. Hsui, “Numerical Studies ofBack-Arc Convection and the Formation of MarginalBasins,” Tectonophysics 50, 177–196 (1978).

30. V. P. Trubitsyn, “The Role of Floating Continents in theGlobal Tectonics of the Earth,” Fiz. Zemli, No. 1, 3–12(1998) [Izvestiya, Phys. Solid Earth 34, 1–7 (1998)].

31. V. P. Trubitsyn, “Principles of the Tectonics of FloatingContinents,” Fiz. Zemli, No. 9, 3–40 (2000) [Izvestiya,Phys. Solid Earth 36, 708–741 (2000)].

32. V. P. Trubitsyn, “Why Do Venus and Mars Have NoLithospheric Plates? Why Did Eurasian Marginal Seasand South American Mountains Arise? Why Is thePacific, Unlike the Atlantic Ocean, Asymmetric?” NaukaTekhnol. Rossii, Nos. 6–7, 53–54 (2003).

33. V. P. Trubitsyn, “Tectonics of Floating Continents,”Vestn. Ross. Akad. Nauk, No. 1, 3–40 (2005).

34. V. P. Trubitsyn and Yu. F. Belavina, “Geodynamic Mod-els of Subduction Zones,” Fiz. Zemli, No. 7, 3–34(1992).

35. V. P. Trubitsyn and V. V. Rykov, “Formation Mechanismof Inclined Subduction Zones,” Fiz. Zemli, No. 6, 1–12(1997).

36. V. P. Trubitsyn and V. V. Rykov, “3-D Spherical Modelsof Mantle Convection with Floating Continents,” in U. S.Geological Survey Open File Report 00–218, pp. 2–44(2000).

37. V. P. Trubitsyn and V. V. Rykov, “A Numerical of the For-mation of Marginal Seas during the Eurasia Movementtoward the Subduction Zone, Breakaway of Japan, andIts Possible Future Aggregation with Eurasia,” in Com-putational Seismology (Moscow, 2001), Vol. 32,pp. 248–256 [in Russian].

38. V. P. Trubitsyn, W. D. Mooney, and Abbott, “Cold Cra-tonic Roots and Thermal Blankets: How ContinentsAffect Mantle Convection,” Int. Geol. Rev. 45 (6), 479–496 (2003).

39. S. Uyeda, “Subduction Zones: An Introduction to Com-parative Subductology,” J. South Am. Earth Sci. 81,133–159 (1982).

40. S. Uyeda and H. Kanamori, “Back-Arc Opening and theMode of Subduction,” J. Geophys. Res. 84, 1049–1061(1979).

41. S. T. Zalesak, “Fully Multidimensional Flux-CorrectedTransport Algorithms for Fluids,” J. Comput. Phys. 31,335–361 (1979).

42. S. Zhong, M. T. Zuber, L. Moresi, and M. Gurnis, “Roleof Temperature-Dependent Viscosity and Surface Platesin Spherical Shell Models of Mantle Convection,” J.Geophys. Res. 105, 11 063–11 082 (2000).

geodynamic model of the evolution of the pacific ocean

Documents