transient creep of the lithosphere and its role in geodynamics

ISSN 1069�3513, Izvestiya, Physics of the Solid Earth, 2012, Vol. 48, No. 6, pp. 496–503. © Pleiades Publishing, Ltd., 2012.Original Russian Text © B.I. Birger, 2012, published in Fizika Zemli, 2012, No. 6, pp. 34–42.

496

1 INTRODUCTION

Until recently, a Newtonian fluid was used to modelslow flows in the mantle. In this model, which describesdiffusion creep, the deviatoric stress tensor is related tothe deviatoric strain rate tensor by a linear law

(1)

The diffusion viscosity η of polycrystalline materialdepends on temperature, pressure and grain size.

Nowadays, a non�Newtonian power�law fluid isusually assumed to model the mantle slow flows. Thismodel adequately describes a steady�state dislocationcreep which is observed in laboratory studies carriedout at constant stresses and at temperatures and pres�sures approximating mantle conditions. The effectiveviscosity of a power�law model depends not only onthe temperature and pressure, but also on the devia�toric stress. Rheology is determined by micromecha�nism which gives the minimal effective viscosity. Forexample, the estimates (Karato and Wu, 1993) showthat the power�law dislocation creep dominates theupper mantle and the lower crust whereas the lowermantle is rather dominated by the diffusion creep.

To investigate geodynamical processes we musthave a rheological model for the mantle which is validwhen stresses change with time. The power�law fluid isnot such a model. Besides, the power�law model doesnot take into account a transient creep observed in lab�oratory studies at small strains.

1 The article was translated by the author.

2 .ij ijσ = ηε�

TRANSIENT CREEP OF ROCK

A typical experimental creep curve, which gives thedependence of creep strain on time at a constant stressapplied at the initial moment, can be divided intothree stages. At the first stage (transient creep), strainrate decreases (strain hardening). At the second stage(steady�state creep) strain rate is constant. At the thirdstage of the strain rate increases (strain softening) thatassociates with the formation and growth of microc�racks leading to the destruction of the test samplerock. Laboratory studies show that the transition to thesteady�state creep occurs at a certain strain and doesnot depend on the stress at which the experiment iscarried out. The lower the constant applied stress, thelonger the transient creep stage. The experiments alsofound that the transient creep strain is linearly depen�dent on the applied stress

(2)

where f(t) is a creep function, and t is a time. For man�tle rocks at high temperatures the creep function iswell described by the Andrade law

(3)

where A is an Andrade rheological parameter, and thetypical value of exponent is m = 1/3. The Andrade lawfor transient creep has repeatedly confirmed in testscarried out at typical mantle pressures and tempera�tures (Berckhemer et al., 1979; Murrell, 1976).

Because Andrade law exponent less than unity, thislaw gives infinite strain rate and zero effective viscosityat the moment of stress application, which of course isnot observed in tests. That’s why Jeffreys (1958) sug�

2 ( ),ij ij f tε = σ

( ) ,mf t t A=

Transient Creep of the Lithosphere and Its Role in Geodynamics1

B. I. BirgerSchmidt Institute of Physics of the Earth, Russian Academy of Sciences,

ul. Bol’shaya Gruzinskaya 10, Moscow, 123995 RussiaReceived June 2, 2011

Abstract—Laboratory experiments with samples of rocks show that at small strains there is transient creep,at which the strain grows with time, and the strain rate decreases. Plate tectonics allows only small strains inthe lithospheric plates, so that the lithosphere creep is transient. In geodynamics, the lithosphere is regardedas a cold boundary layer formed by mantle convection. If we assume that the lithosphere has a steady�statecreep, which is described by power�law non�Newtonian rheological model, the low effective viscosity of thelower layers of the lithosphere, obtained by data on small�scale postglacial flows, is possible only at high strainrates in these layers. However, the high strain rates in the lithosphere induce large strains that contradict platetectonics. Transient creep of the lithosphere leads to its mobility at small strains, removing the discrepancybetween thermal convection in the mantle and plate tectonics, which holds in the case of power�law rheolog�ical model of the lithosphere.

DOI: 10.1134/S1069351312060018

IZVESTIYA, PHYSICS OF THE SOLID EARTH Vol. 48 No. 6 2012

TRANSIENT CREEP OF THE LITHOSPHERE AND ITS ROLE IN GEODYNAMICS 497

gested to use the modified Lomnits law instead ofAndrade law. However, even at very short times (frac�tions of a second) since the stress application, the lawsof Jeffreys and Andrade are indistinguishable (Berck�hemer et al., 1979).

At a constant steady�state creep strain rate dependsnonlinearly on the applied stress

(4)

where σ and are the second invariants of deviatorictensors of stress and strain rate, B is a rheologicalparameter which characterizes the steady�state creep,a typical value of exponent is n = 3.

To generalize the results of creep tests at constantstress for stresses changing with time the Boltzmannlinear theory can be used. This theory, valid for thecase of sufficiently small strains, leads to the integralrelation

(5)

where K(t) is an integral kernel of creep, which isdetermined by the creep function

(6)

As follows from (3) and (6), the kernel of creep, corre�sponding to the Andrade law, has the form

(7)

Rheological model, which is described by equa�tions (5) and (7), is called a model of Andrade. Thismodel generalizes the Andrade law for the case of vari�able stresses. Equations (5) and (7), describing themodel of Andrade, can be rewritten as

(8)

where R(s) is the relaxation kernel

(9)

and Γ(m) is the gamma function.

NONLINEAR INTEGRAL MODEL OF CREEP

Much more difficult to generalize the rheologicalrelation (4) describing steady�state creep for the caseof time�dependent stresses because the rheology ofrocks under variable stresses and large strains has notbeen investigated experimentally. The simplest gener�alization is the assumption that equation (4) holds fortime�dependent stresses. The model described byequation (4) at any depending of stress on time iscalled the power�law non�Newtonian fluid. This

1 ,nij ijB −

ε = σ σ�

1/2( 2) ,kl klε = ε ε� � �

1/2( / 2) ,kl klσ = σ σ

ε�

0

2 ( ) ( ) ,ij ijK s t s ds

∞

ε = σ −∫

.df

Kdt

=

1( ) .mK t mt A−

=

0

2 ( ) ( ) ,ij ijR s t s ds

∞

σ = ε −∫

1( ) / ( ) (1 ),mR s As m m− −

= Γ Γ −

model is commonly used in the study of thermal con�vection in the mantle, but in the case of non�stationaryconvection, its applicability is questionable. Thepower�law model is in good agreement with experi�ments at constant stress but this does not imply thatthis model is applicable under variable stress. In thepower�law fluids, as in the Newtonian model (1), thecurrent stress is determined by the current strain rate,i.e. by strains that exist in an infinitely short period oftime before the current moment. In the real material,the current stress depends on the entire history ofstrains. In other words, the real material has a memory,in contrast to the power�law fluid.

Rheological models with memory are widely usedin mechanics. The theory of simple fluids with fadingmemory is a fundamental rheological theory in mod�ern continuum mechanics (Astarita and Marrucci,1978). Decaying memory means that the current stressdepends on the recent strains much stronger than onthe strains that existed in the distant past.

Birger (1998) proposed a new nonlinear integral(having a memory) model for high�temperature dislo�cation creep of rocks. This model is consistent with thetheory of simple fluids with fading memory and isdetermined by the equation

(10)

where

(11)

ε = (εklεkl/2)1/2 is the second invariant of the strain ten�sor, and R(s) is defined by (9). Thus, equation (10)generalizes equation (8), corresponding to theAndrade model, for the case of large strains measuredfrom the state at the time of observation. At thismoment (s = 0), the strain εij(t) is zero by definition.As follows from (10) and (11), the memory is cut offwhen the second invariant of the strain tensor exceedsthe transition value εtr, which can be determined inexperiments carried out at constant stress. Nonlinearintegral model (10) reduces to the linear Andrademodel (8) for flows associated with small strains. Atstationary flows, causing large strains, the model isreduced, as shown in (Birger, 1998), to the model ofpower�law fluid with rheological parameters B and n,whose values are determined by the rheologicalparameters A, m, and εtr in (10), (11). If m = 1/3 in theAndrade law, the relations are

(12)

The transition value of strain is estimated as εtr ≈ 10–3–10–2. Note that the experimental measurement of therheological parameters have large uncertainties(Korenaga and Karato, 2008), and therefore the used

0

2 ( ) ( ) ( ) ,ij ijR s g t s ds

∞

σ = ε ε −∫

tr

tr

( ) 1, ,

( ) 0, ,

g

g

ε ≡ ε ≤ ε

ε ≡ ε > ε

2tr

313, 1 .3

n B A= ≈ ε

498


BIRGER

numerical values of these parameters are only roughestimates of orders of magnitude.

The most important characteristic is the depth ofmemory (the ratio of the transition strain εtr, fixed inthe rheological model, to the strain rate caused by theconsidered flow): strains that existed before the timedetermined by the depth of memory, do not affect thestress at the time of observation. The nonlinear inte�gral rheological model (10) reduces to a power�lawmodel for flows in which the characteristic period T ofvelocity variation is much greater than the depth ofmemory

(13)

Such flow is called a quasi�stationary. Emphasize thatcondition (13) permits to replace the nonlinear inte�gral model (10) by the power�law fluid model whenconsidering non�stationary flow, velocity of which var�ies with time, since such a flow behaves as stationary atthe depth of memory. If the strain rate is high, thememory depth is small, and even flows with rapidlychanging velocities can be regarded as quasi�station�ary. For flows, causing minor strains and superim�posed on a basic stationary or quasi�steady flow, non�linear rheological model (10) reduces to the linearintegral Andrade model, if the condition

(14)

is valid, where T characterizes the superimposed flowand tM is determined by the basic flow.

The universal rheological model constructed in(Birger, 2007) is a series connection of four rheologicalelements that describe high�temperature dislocationcreep, low�temperature dislocation creep, diffusioncreep and elasticity. As mentioned above, the diffusioncreep can be ignored when considering the processesin the upper mantle and lithosphere, where the contri�bution of this rheological mechanism to deformationis negligible. The contribution of low�temperature dis�location creep, as shown in (Birger, 2007), should beconsidered only in the study of fast processes, such asattenuation of seismic waves.

LITHOSPHERE AND MANTLE RHEOLOGY

The strain rate at stationary convection in the man�tle is estimated to be 10–15 s–1 (Turcotte and Schubert,1985). This estimate corresponds to the memory depthof about 5 × 104 years. Numerical experiments con�ducted in the framework of the power�law fluid modelindicate that mantle convection forming a cold upperboundary layer (lithosphere) is non�stationary. In themantle underlying the lithosphere, a typical strain rateassociated with non�stationary convection an order ofmagnitude higher than in the case of stationary con�vection. The duration of velocity fluctuations deter�mining the time dependence of mantle convection is107–108 years that greatly exceed the depth of mem�ory, which is estimated as 5 × 103 years at the strain rate

tr, .M MT t t = ε ε��

,MT t�

of the order of 10–14 s–1. Thus, within the framework ofnonlinear integral rheological model, non�stationaryconvective flow in the mantle beneath the lithosphereis quasi�stationary and, indeed, can be adequatelydescribed by the power�law fluid model. At a stronglynon�stationary (turbulent) convection in the mantle, inwhich the characteristic strain rate—about 10–14 s–1,the large�scale postglacial flow having a duration ofthe order of 104 years satisfy the condition (13) and,therefore, can be described by the power�law model aswell as the convective flow.

As the lithospheric plate is a boundary layer formedby mantle convection, its deformations are small(strains are large in some regions of the lithosphere,but these regions can be regarded as the boundarybetween undeformed lithospheric plates). Therefore,the transient creep, which is described by the linearAndrade model, takes place in the lithosphere. Thus,the rheology of the lithosphere is fundamentally dif�ferent from the underlying mantle rheology, which isdue to the difference in the levels of strain. It should benoted that in the present paper we consider only thelithospheric plates, where strains are small, rather thanthe boundaries between plates. These boundaries rep�resent a shear zones, characterized by large strains.The Andrade rheological model of the lithosphereapplied in the present paper cannot properly describethe flows occurring at the boundaries between theplates (subduction zones and transform faults). Atthese boundaries large creep strains lead to a strongstrain softening caused by localization of deformation,microfractures, and other factors.

The vertical temperature gradient in the lithos�phere is sufficiently high and the effective viscosity,corresponding to the transient creep in the lithos�phere, is small enough to cause a convective instabilityin those regions of the lithosphere, where its thicknessis large. The regions of the most thickened continentallithosphere are under the cratons, where its thicknessis about 200 km or more. Since the effective viscositycorresponding to the transient creep depends on thefrequency and tends to infinity when the frequencytends to zero, the lithosphere under the cratons (theroots of cratons) has an oscillatory instability. Low�amplitude thermoconvective oscillations of the cra�tonic lithosphere cause oscillatory movements of theEarth’s surface. Taking into account the significantdifference between the rate of sedimentation and ero�sion, the lithosphere thermoconvective oscillationscan be considered as a mechanism for the formationand evolution of sedimentary basins on continentalcratons (Birger, 1998; 2004).

POSTGLACIAL FLOWS AND EFFECTIVE VISCOSITY OF THE LITHOSPHERE

The effective viscosity of the upper crust is so largethat creep can be neglected (the creep strain is smallcompared with the elastic strain). The duration of



postglacial flow, characterized by the recovery time τ,which is associated with the average Newtonian vis�cosity η by the relation

(15)

where ρ is the density, g is the acceleration of gravity,the wave number k = π/L is determined by the size ofglacier L. In equation (15), the flexural rigidity N ofthe elastic crust, which is modeled as a thin elasticplate, is defined as

(16)

where E is Young’s modulus, ν is Poisson’s ratio, μ isthe shear modulus, K the bulk modulus, δ is the thick�ness of the elastic crust. The flow of material caused bythe removal of the ice load occurs in the upper layer ofthe Earth, the depth of which does not exceed the hor�izontal dimensions of the load. Studies of small�scalepostglacial flows lead to the value of the flexural rigid�ity N ≈ 1023 N m (Turcotte and Schubert, 1985). Withsuch a flexural rigidity, it follows from (16) that thepostglacial flows are associated with δ ≈ 25 km sincethe shear modulus is estimated as μ ≈ 7 × 1010 Pa.

Using the estimates τ ≈ 103 years ≈ 3 × 1010 s, L ≈2 × 105 m which are valid for small�scale glaciations,we find from (15) the average viscosity of the lithos�phere beneath the elastic crust: η ≈ 4 × 1019 Pa s. Sincethe model of a viscous Newtonian fluid does notdescribe the rheology of the lithosphere, this viscosityshould be considered as effective. As it follows fromthe equations (1), (2) and (3), the effective viscosity ofthe medium with Andrade rheology is defined as

(17)

where τ is the characteristic duration of the process(the duration of postglacial flow is determined by therecovery time). Putting ηA = η ≈ 4 × 1019 Pa s and τ ≈3 × 1010 s in equation (17), we obtain the Andraderheological parameter for the asthenosphere underly�ing elastic crust: A ≈ 1012 Pa s1/3.

As follows from (1), (4), the effective viscosity of apower�law fluid is given by

(18)

The relation (18) can be rewritten as

(19)

The equation (18) is useful when the stress σ is uni�formly, i.e. constant with depth, but in the case of auniform strain rate is more convenient to write theeffective viscosity in the form (19). The dependence of

42 /( ),k g Nkτ = η ρ +

( )( )

( )[ ]

333

2

1 3,

3 1 3 412 1

3 1 6,

KENK

K

+ μ μδδ= = μδ ≈

+ μ− ν

μ ≈

2 33 ,A Aη = τ

21 .

2pow

Bη =

σ

1 3 2 31 .

2pow

Bη =

ε�

the rheological parameter that characterizes the steady�state creep, of the temperature can be written as

(20)

where H*(p), E*, and V* are the enthalpy, the energy,and the volume of activation, R is the gas constant. Inthe lithosphere, the pressure p is small, and we canassume that H*(p) = E*. Comparison of (18) and (19)shows that in the case of a homogeneous strain�ratethe dependence of the effective viscosity on tempera�ture and, hence, on the depth is much weaker than inthe case of a homogeneous stress. From (12) and (20)follows that the Andrade rheological parameterdepends on temperature as

(21)

where the parameter A∞

is associated with the param�eter B

∞ by (12). Thus, the Andrade parameter depends

on the temperature (and depth) much weaker than therheological parameter B. The effective viscosity of theAndrade medium (17) depends on the temperature(and depth) in the same way as the effective viscosity ofpower�law fluid at a uniform strain rate. Equation (17)is valid only for constant stress. In contrast to thepower�law fluid which has no memory, the effective vis�cosity of the Andrade medium depends on the history ofstrains. For example, it follows from equation (8) that ata constant strain rate the Andrade medium is character�ized by the effective viscosity

However, in contrast to the non�linear rheologicalmodel of power�law fluid, the distribution of the effec�tive viscosity on depth in the linear Andrade medium isdetermined only by the dependence (21) of the rheolog�ical parameter A on temperature and does not dependon the depth distributions of stress and strain rate.

DEPTH HETEROGENEITY OF TRANSIENT CREEP AND BRITTLENESS

IN THE LITHOSPHERE

Series connection of rheological elements, describ�ing the high�temperature dislocation creep and theelastic element, leads to a rheological model of Max�well type. This model differs from a classic Maxwellmodel since the creep element is described by the non�Newtonian rheological law (10), which reduces to theAndrade law in the lithosphere, where the strains aresmall. Maxwell viscoelastic medium behaves like apurely viscous (elastic deformations are negligible), ifthe time scale of the process is much longer than theMaxwell time, which is defined as the ratio of viscosityto the elastic shear modulus

(22)

( )*

* * *,( )1 1 exp ,

H pH p E pV

B B RT∞

= = +

*( )exp ,

3H p

A ART

∞=

2 3.A Aη = τ

.Aτ η μ�

500


BIRGER

Substituting into (22) the effective viscosity of theAndrade medium (17), we arrive at the conditionwhich permits to neglect the elasticity in a layer of thelithosphere, located at a depth z,

(23)

In the case, where

(24)

a layer, located at a depth z, behaves as a purely elastic.

For a more complete description of the lithosphererheology, elastic, creeping (ductile) and brittle(pseudo�plastic) elements must be connected inseries. Such a connection means that the strain rate ofmedium is the sum of the strain rates of rheologicalelements, and the stresses in all elements are the same.When the elements are connected in series the effec�tive viscosity of medium is associated with the effectiveviscosity of each of the elements by the relation

(25)

As follows from (25), if one of the effective viscositiesof the rheological elements is much less than the effec�tive viscosity of the other elements, this element withlow viscosity determines the rheology of the medium.In the case of transient creep, the effective viscosityof creeping element is defined by (17), and the crite�rion (22) follows from the condition: ηcr � ηel, whereηcr = ηA.

The effective viscosities of elastic and pseudo�plas�tic elements are

(26)

(27)

where cf is the coefficient of friction, which is esti�mated as 0.8 for dry continental lithosphere, and p =ρgz is the pressure at depth z. In the continental litho�sphere, the friction coefficient is much higher than inthe wet oceanic lithosphere. This, apparently, is one ofthe main reasons hindering the subduction of conti�nental lithosphere (Lenardic and Moresi, 1999). Theequation (27) is valid when the stress in the lithosphereis significantly lower than the pressure p. Since thestrain rates are small in the lithosphere, the stresses aresmall too, and this condition is satisfied throughoutthe depth of the lithosphere, except for the top layer ofthe crust where the pressure is low. The brittle element,included in the series of rheological elements, does notallow stress to exceed the limit 2cfρgz. If the stress isbelow this limit, the medium behaves like a viscoelas�tic one. Therefore, the brittleness (pseudo�plasticity)of the medium is excluded from consideration if thestress 2ηcrε/τ, where ε/τ is a characteristic strain rate,in the creeping element below this limit. In themedium with transient creep, the effective viscosity of

1 3( ) 3.A z μτ�

1 3( ) 3,A z μτ�

el cr br

1 1 1 1 .= + +η η η η

el ,η = μτ

br f br ,c pη = ε�

which, is defined by (17), pseudo�plasticity does notoccur on the condition

. (28)

The typical strain in the lithosphere is estimated as ε ≈10–3. Substituting into (28) values of ε, ρ, g and cf,rewrite this criterion as

(29)

The crust is different in mineralogical compositionfrom the mantle lithosphere, therefore, these layers arecharacterized by different values of parameters E*, A∞

and B∞. Burov and Diament (1995), adopting thepower�law rheological model of the lithosphere, useddata on the distribution of temperature in the conti�nental lithosphere and laboratory evaluation of therheological parameter B for the minerals of the crustand mantle lithosphere. They found the depth distri�bution of parameter B in the lithosphere and estimatethe thickness of the elastic layer of continental crust atrates of strain lying in the interval: 10–17 to 10–13 s–1.These strain rates correspond to isostatic and, in par�ticular, postglacial flows. However, since the charac�teristic times for these flows are small, the strains asso�ciated with these flows are small as well. Therefore, theisostatic flows are associated with transient creepwhich cannot be described by the power�law rheolog�ical model. As follows from the criterion (22), thelithosphere, which has the effective viscosity τ whichvaries with depth and the shear modulus τ, can behavelike creeping layer in the slow process (characteristictime τ is large) and like elastic layer in the fast pro�cesses (τ is small). Thus, the thickness of the upperelastic layer of the lithosphere (elastic crust) variesdepending on the duration of considered process. Thiseffect occurs at the Andrade rheology but the quanti�tative description of this effect is very different fromthat which gives the power�law rheological model.Application of the Andrade rheological model leads tothe effective viscosity (17) depending on the durationof process. Since the effective viscosity depends on thetime is not too strong (as τ2/3), the thickness of theelastic crust decreases in slow processes in qualitativeagreement with the effect produced in the frameworkof power�law rheological model. However, the effec�tive viscosity of power�law rheological model does notdepend on time, and the elastic thickness of the crustis reduced in slow process much stronger than in themodel of Andrade.

Using the depth distribution of the parameter B inthe lithosphere, obtained in (Burov and Diament,1995) and the equation (12) which gives the relation ofrheological parameters of steady�state and transientcreep, we obtain the depth distribution of Andraderheological parameter. In the continental crust, whichhas the characteristic thickness of 35 km, the Andradeparameter decreases with depth from 1016 to 1011 Pa s1/3.In the mantle lithosphere (thickness of about 200 km),the Andrade parameter decreases from 1014 Pa s1/3 on

f1 3( ) 3A z c gz< ρ τ ε

1Pа m1 3 6( ) 8 10 .A z z −

τ < ×



the crust�mantle boundary to 1011 Pa s1/3 on the lowerboundary of the lithosphere. The criteria (23) and(29) show that at times τ ≈ 103 years ≈ 3 × 1010 s themedium with transient creep does not exhibit elasticityif A(z) 7 × 1013 Pa s1/3 and does not exhibit pseudo�plasticity if A(z)/z < 4 × 1010 Pa s1/3 m–1. At times aboutτ ≈ 3 × 108 years ≈ 1016 s, elasticity and pseudo�plas�ticity can be neglected if A(z) 5 × 1015 Pa s1/3 andA(z)/z < 2 × 1012 Pa s1/3 m–1. These estimates of theAndrade parameter show that at the small�scale isos�tatic flows the lithosphere behaves as a medium withtransient creep along the entire depth, except for theelastic upper crust and the brittle uppermost layers ofthe crust In these layers, the pressure is small, and theirbrittleness is described by the Byerly law (Byerlee,1968) rather than by the equation (27). For convectiveflow in the lithosphere, the thickness of brittle�elasticlayer of the upper crust is significantly reduced.

Substituting (12) into (17), we can express theeffective viscosity of Andrade medium through therheological parameter B characterizing the steady�state creep

(30)

The effective viscosity of Andrade medium dependson the duration τ of the process, but does not dependon the strain rate. The effective viscosity of mediumwith power�law rheology, in contrast, does not dependon the duration of the process, but depends on thestrain rate. Comparison of (30) and (19) shows that theeffective viscosity of Andrade medium is equal to theeffective viscosity of power�law medium when

,

and, consequently, the effective viscosity of Andrademedium at times the order of 103–104 years, whichcorrespond to the isostatic flows, is equal to the effec�tive viscosity of power�law medium with strain rates ofthe order of 10–15–10–14 s–1. At times of the order of108 years, these viscosities are equal at the strain ratesof the order of 10–19 s–1. The estimates of the thicknessof the elastic layers of the lithosphere, one of which islocated in the upper crust, and the second—at theboundary of the crust and mantle lithosphere,obtained in (Burov and Diament, 1995) for the isos�tatic movements, do not change when the power�lawrheology is replaced by the Andrade rheology. How�ever, at the convective flow (τ ≈ 3 × 108 years), theupper elastic layer of the crust is much thinner, and thelayer under the crust�mantle boundary does notexhibit elasticity.

The distribution of Andrade parameter in the con�tinental crust can be written as

(31)

�

�

tr

2 31 3 33 .A B −

⎛ ⎞τη ≈ ⎜ ⎟ε⎝ ⎠

tr24 10−

ετ ≈ × ε�

( )0 exp ,A A z h= −

where A0 ≈ 1016 Pa s1/3, h ≈ 6 km. Substituting (31) into(24), we estimate the thickness of the elastic crust

(32)

At times of the order of 1000 years, the equation (32)gives the thickness of the elastic crust about 25 km,which corresponds to the estimate obtained above bythe postglacial data. For more slow processes charac�terized by high values of τ, the equation (32) gives athinner elastic crust. At the convective flow in thelithosphere (τ ≈ 3 × 108 years), the thickness of theelastic crust is not more than 7 km. Thus, the lithos�phere defined as a cold boundary layer formed by ther�mal convection in the mantle is much thicker than itsupper elastic layer (elastic crust). The convective litho�sphere, defined in such a way, includes not only theelastic crust but the asthenosphere determined by thedata on isostatic flows. Inconsistencies arising fromdifferent definitions of the lithosphere are consideredin detail by Anderson (1995).

TRANSIENT CREEP OF THE LITHOSPHERE AND MANTLE CONVECTION

One of the central problems of modern geophysicsis the problem of consistency of plate tectonics withmantle convection (Bercovici, 2003). Numericalexperiments, in which the mantle convection is mod�eled within the rheological model of a Newtonianfluid, show that the strong dependence of viscosity ontemperature leads to convection in the mantle beneaththe lithosphere. The cold upper boundary layer (litho�sphere) with high viscosity behaves like a immobile lidnot included in the convective motion, i.e. there is nosubduction required by plate tectonics. Convectionwith a mobile boundary layer is possible only in thecase when viscosity varies with depth not more than3000 times (Solomatov, 1995). In order to lower theeffective viscosity of the lithosphere and make itmobile, Moresi and Solomatov (1998) assumed thatthe entire lithosphere has a pseudo�plastic rheologywith a low friction coefficient which characterizes thewet oceanic lithosphere. However, the effective viscos�ity corresponding to pseudo�plasticity increases rap�idly with increasing pressure, so when considering adry continental lithosphere, where the friction coeffi�cient is much higher, pseudo�plasticity should be con�sidered only in the upper continental crust (Birger,1997).

In the framework of nonlinear integral rheologicalmodel used in this paper, the rheology of the lithos�phere where strains are small is reduced to theAndrade rheology and the effective viscosity of thelithosphere at a constant stress is given by (17). Defor�mations occurring in the lithosphere during postgla�cial flows are small due to short duration of theseflows, and not due to low strain rates. In contrast, atlarge�scale convective flow, the duration of which is

1 3

0

ln .3

hA

⎛ ⎞μτδ = − ⎜ ⎟⎝ ⎠

502


BIRGER

very large, deformations occurring in the lithosphereare small due to very low strain rates. Substituting τ ≈3 × 108 years (this time is determines by on the durationof Wilson cycles (Turcotte and Schubert, 1985)) andthe average value A ≈ 1012 Pa s1/3 of the rheologicalparameter of the lithosphere in the relation (17), weestimate the maximum effective viscosity of the lithos�phere as ηlith ≈ 1023 Pa s (recall that ηlith ≈ 4 × 1019 Pa sfor postglacial flows). In the mantle beneath the litho�sphere, deformations are great, and convective flow isdescribed by the power�law rheological model. Sinceat developed convection the temperature under theboundary layer (lithosphere) almost does not changewith depth, a slight increase of the effective viscositywith depth is determined by increasing pressure. Con�vective flow in the mantle beneath the lithosphere isassociated with the effective viscosity ηmant ≈ 1021 Pa sthat agrees well with the value of mantle viscosityfound from the study of large�scale postglacial flows(Cathles, 1975). Thus, the main change in viscositywith depth occurs in the lithosphere, rather than in theunderlying mantle. In the lithosphere beneath the thinupper crust, the effective viscosity varies by almost3 orders of magnitude, due to the dependence (21) ofthe Andrade parameter on the temperature. If theupper crust, which is characterized by very large valuesof the Andrade parameter, would be purely elastic, thislayer, having a very high effective viscosity, would pro�hibit subduction. Accounting for the pseudo�plasticdeformation in the upper crust leads to a strong reduc�tion of its effective viscosity.

The same situation arises in the numerical experi�ments in which the mantle convection is simulatedwithin the power�law rheology and forms an immobileupper boundary layer. If we take into account that thethin brittle upper crust has a low effective viscosity, theboundary layer modeling the lithosphere, becomesmobile. The temperature dependence of rheologicalparameter B characterizing the power�law rheology isdetermined by the activation enthalpy H*. Thermalconvection in the mantle with power�law rheologyproceeds in almost the same way as in the mantle withNewtonian rheology and 3 times decreased enthalpyof activation (Christensen, 1984). Thus, the effectiveactivation enthalpy for the power�law rheology isreduced to the value that corresponds to the Andraderheology. Such a lowering of the effective activationenthalpy for the power�law rheology and the corre�sponding decrease in the effective viscosity drop isassociated with the occurrence of sufficiently highstrain rates in the lithosphere.

Within the power�law rheological model, the effec�tive viscosity, corresponding to the convective flow,only 3 times higher than the effective viscosity, whichdemonstrated by postglacial flow (Turcotte and Schu�bert, 1985). (For the rheological model of the lithos�phere, describing its transient creep, the effective con�vective viscosity is three orders of magnitude higherthan the postglacial.) Data on postglacial uplifts lead

to the estimation of 4 × 1019 Pa s for the effective vis�cosity of the lithosphere under the upper elastic�brittlecrust. However, the low effective viscosity of the lowerlayers of continental lithosphere with power�law rhe�ology is possible only at large strains and high strainrates, contrary to plate tectonics, not allowing largestrains in the lithosphere (Bercovici, 2003). In otherwords, the use of power�law rheological model leads tothe convective boundary layer whose thickness ismuch smaller than the thickness of continental lithos�phere considered in plate tectonics. Transient creep ofthe lithosphere decreases its effective viscosity at lowstrain rates (power�law rheological model of the litho�sphere leads to a very high viscosity at low strain rates)and makes the lithosphere to be mobile at smallstrains.

CONCLUSION

If we assume that the creep of rocks, that form thecrust and mantle of the Earth, is described by Newto�nian rheological model, the viscosity found in thestudy of postglacial flows can be used in the simulationof thermal convection. If the creep of rocks is charac�terized by non�Newtonian power�law rheologicalmodel, similar situation arises. With this rheology, theeffective viscosity corresponding to postglacial flows isonly 3 times lower than the effective viscosity corre�sponding to the convective flow. If the creep is tran�sient and, hence, time�dependent, the effective vis�cosity, found in the study of postglacial flows or otherflows that restore isostasy, can be several orders ofmagnitude different from the effective viscosity whichcharacterizes the convective flow since characteristictimes of these processes are very different. Earlier, theauthor proposed a nonlinear integral (having a mem�ory) rheological model, which for stationary or quasi�stationary flows associated with large strains, reducesto a power�law fluid, and at small strains—to a linearintegral Andrade model describing transient creep.According to plate tectonics, strains in the lithosphereare small. Therefore, in the lithosphere, in contrast tothe underlying mantle, creep is transient. The nonlin�ear integral rheological model establishes a simplerelationship between the rheological parameters char�acterizing steady�state and transient creep and permitsto find the lithosphere effective viscosity correspond�ing to its convective flow if we know the distributionover depth of the lithosphere effective viscosity corre�sponding to isostatic flows. The lithosphere, where thetemperature is lower than in the underlying mantleand the effective viscosity is larger, is mobile only inthe case when its effective viscosity is not too high. Ifwe assume that the lithosphere creep is steady�stateand is described by power�law model, sufficiently loweffective viscosity of the lower layers of the lithosphere,necessary for its mobility, is possible only at high strainrates in these layers. However, the high strain rates leadto large strains, contrary to plate tectonics, not allow�



ing large strains in the lithosphere. Transient creep ofthe lithosphere decreases its effective viscosity at lowstrain rates (power�law rheological model of the litho�sphere leads to a very high effective viscosity at lowstrain rates) and allows the mobility of the lithosphereat small strains, removing the discrepancy, whicharises in the case of power�law rheological model ofthe lithosphere, between thermal convection in themantle and plate tectonics.

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transient creep of the lithosphere and its role in geodynamics

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