Vcstnik , L i o ~ k o v ~ k o g o
Univetsite.ta. Geologiya
U D C 552.43
.Moscow University
G e o l o g y Bulletin
Vol. 48, AVO. 5 , ? p . 17-29. 1993
P-T P*.THS OF NIETAiMORPHISM AND THE RELATED
GEODYNAMIC b1ODELS
L. L. Perchuk and Yu. Yu. Podladchikov
To explain t h e specific features of P-T p a t h s in t h e evolution of metamorphism
of t h e Precambrian shields a n d folded areas of different ages, formed in different
geotectonic conditions, has b e e n carried out numerical modeling of (1) gravitational ordering of multilayep polyrhythmic sections related with the growth
of t h e r m a l activity of t h e mantle, (2) emergence of large diapirs in thick E a r t h
crust, causing t h e circulation of rocks in t h e lithosphere accompanied by t h e
formation of high-pressure complexes.
INTRODUCTION
The evolution of metamorphic complexes, revealed using the met hods of phase correspondence and
local electron analysis of zonality of coexisting minerals, was found to be different in various geodynarnic
conditions [lo, 30, 311. It turned out t h i t rnetamorp hism is closely related with abyssal magmatic processes,
and the rock structures (lineation, schistosity, the growth of porphyroblasts, etc.) correlate very distinctly
with the transformation of mineral parageneses, i.e., depend on the variation of thermodynamic parameters,
temperature (T) and pressure (P), first of all. The recording of P-T paths shows that one and the same
metamorphic complex goes through many mineral facies and subfacies during its development, and the
resulting facies is determined by a specific geodynarnic environment belonging to it.
We can provisionally distinguish three geodynamic types of metamorphic complex, for which stable
specific P-T paths are revealed [30,311: (1) low-temperature, high- and ultrahigh-pressure complexes (glaucophane schists, eclogites, white schists), confined to' plate junction areas; (2) medium temperature, highand some ultrahigh pressure, often zonal, metamorphic complexes of folded areas; (3) high- temperature
cratonized areas with widely developed complexes of granulite facies of metamorphism.
In mineral assemblages and structural indicators of most complexes of the first two geodynarnic conditions we can find traces of progressive as well as of retrograde metamorphism. Only ultrahigh-pressure
complexes [15, 20, 29, 37, 421 not always preserve the structural and paragenetic information about the
progressive stage of metamorphism. Granulitic complexes, as a rule, do not carry this information. They
preserve only reactions, structures and fluid inclusions of the retrograde stage. The examples are the granulite complexes of t he Yenisei ridge, of the Khankai and Ukrainian massifs, Aldan and Canadian shields, Kola
Peninsula, Southwestern Baikal region. The nature of these complexes has been widely discussed. However,
th'e origination and evolution could not be traced without a comprehensive analysis. Only after a special
study of the solid phase diffusion coefficients it became clear that at a temperature above 700°C the traces
of the progressive reactions of metamorphism disappear in many cases because of the rapid diffusion of Fe,
Mg, and Ca. That is why it is impossible to determine the inversion zonality in garnets from granulites that
are so common for the folded area rocks.
The problem of the relationship between granulite complexes and greenstone belts is of a special interest.
Direct relationships are not virtually observed, but there are geochemical [11] and certain geological [l]
evidences of a possible development of granulite complexes on the basis of the Early Archean greenstone
belts. In this connection, of interest are the geodynamic models of cratonization as the most general process
of regeneration of mobile belts into granulite areas, accompanied by conso1i;dation of the Earth crust.
P-T records of the evolution of metamorphism in some complexes of folded areas not only an evidence
of the complete cycle of their subsidence and uplift but also give information about the advection rate.
To make a numerical advection model. we need, besides P-T parameters, data on the rheoiogical rock
properties, reaction rates, etc. It is obvious that the number of advection variants and stages can be as large
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as is wished. That is why, we will discuss only several models reflecting the most typical results of advection,
characteristic sf che development of granulites and high-pressure complexes.
PETROLOGICAL PROBLEM STATEMENT
Greenstone Belts and Granulites
There is no yet strict proof of the relationships between greenstone belts and granulites, despite the
sound geologic research in the adjacent areas of development of greenstone belts and granulites [I]. Incidentally, tonalite domes in these belts are, as a rule, rimmed with gneissic and migmatized rocks, and the degree
of metamorphism in these rocks sharply reduces in the direction away from the contacts with intrusions.
Tonalites are supposed to be the products of remobilization of the underlying Early Archean sialic rocks [3.
28, 341 or of a partial melting of a very thick Early Archean lithosphere of the basaltic composition [26], and
this corresponds to &I.J. Bickle's [18].
The initial heterogeneity of the Archean greenstone belts is their remarkable feature. The composition
of the interlayered rocks is strikingly various-from rhyolites to komatiites. Moralev et al. [I] distinguish
three types of complexes in greenstone belts according to the rock sequences: (1) direct, where basitekornatiite volcanites tend to the bottom of sections (Barbertone, Murchison, Sutherland, some belts of the
Rhodesian Craton, the Pilbara Block in Western Australia) [23]; (2) reversed, where basite-komatiite series
make less than 50% of the volume of volcanites and tend to the upper part of sections (some belts of the
YiIgari block in Western Australia, most belts of the Canadian Shield, Zimbabwe, Brazil, Karelia). The
most concise review of the Karelian greenstone belts is given in [3]. Here, the polyrhythmic structure of
the general Precambrian section is clearly manifest, and its nature is determined by reversed volcanism
within each cycle (rhythm); (3) in the complexes of the third type, basites-ultrabasites are absent or are
strictly subordinate, prevail terrigenous rocks and effusive products of intermediate or acidic composition
(the greenstone belts of Southern India and South America, the Yenisei chain of hills, some belts of the
Yilgari block). The comprehensive review [3] on .the greenstone belts of the East-European Platform gives a
sirnilarclassification.
However, greenstone belts cannot always be mapped with an accuracy required to make a complete
stratigraphic section. Therefore, there are different conclusions about the sequence of volcanic events in one
and the same region in the Precambrian (compare the works by G. Drugova, A. Smelev, e t al. [8], which are
devoted to the Olondinsky Belt). 1n some cases, the polyrhythmic nature of the development of magmat ism
is not evident, and there appear indefinite terms (such as "direct reversed evolution" (111).
Thus, the rhythmicity of volcanism and sedimentation is a very important feature of all greenstone
belts. It always manifests itself during the metamorphism o f the belts, resulting in a number of cases in a n
intense gravitational redistribution of the material under a thermal field applied.
Komatiitic .volcanism is the second and certain priviledge of greenstone belts. It is an evidence of a
considerable temperature growth in the lit hmp6ere, i.e., of the uplifting of mantle diapirs and destruction
of the Earth crust of any type. The diameter of such diapirs should not be less than 100.200 km, otherwise
their temperature would equal the temperature of the upper mantle, and they would not be able to reach the
Earth surface. According to H. Ramberg [32], such diapirs have a drop-like shape, hence, their projection
on the crust surface should be round. Such was the data obtained by &
Glukhovskii
I.
who has interpreted
space photographs and generalized the details of the geologic structure of the ancient shields of-the former
USSR and Scandinavia. He "has revealed basically new structural elements of the lithosphere: ovoid-ring
systems, or nuclears, which are the main elements of the ancient platform basement tectonics" [I].
. Following the regularities in the development of the mantle magmatism and in the interaction of the
deep-seated diapirs with the Earth crust, one should expect mainly reversed sequence of volcanism against
the background of the crust being thinned, i.e., a stable growth of the heat flow related with the mantle
diapirism, first of all must cause the appearance of more acidic magmas due to eutectic melting out, the
subsequent total melting of the rocks of intermediate and basic composition with the formation of dacites and
andesites, and the following appearance of toleiitic basalts. In a number of cases, the latter may be preceded
by slightly alkaline basalts. As the result, reversed sections arise. While for the Precambrian complexes the
problems of a stratigraphic plan arise, because of a substantial reworking of the initial sections, for the young
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~MoscowUniversity
Geology Bulletin
Vol. Ad, rvo. 5
structures this is unequivocally stated. That was the volcanism on the floor of the Sea of Japan and South
China Sea [14], in the Pannonian basin [9], and in other volcanogenic-sedimentary basins, those comprising
complex contrast series (e.g., South Urals, etc.) in their number. All these sections are obviously potentially
gravitationally unstable.
It is a common knowledge that in some regions basaltic volcanism develops on the Earth crust of the
continental or transitional type. These are, for example, the trapps of Western Siberia, South America, the
Deccan Plato in India. This way or other, basaltic volcanism is related with the mantle diapirism, or in
terms of geophysics, with the rise of the asthenolithic boundary. The tectonic result of this rise at the early
stages is the formation of rifts.
These processes are typical of all periods in the geological history of the Earth, and they proceed
up to now. But in all the cases, they result in the formation of geological sections that are potentially
unstable in the gravitational field of the Earth. It means that with the rise of temperature the rock's fluidity
will drastically grow creating the favorable conditions for gravitational redistribution of the material in
metamorphic processes.
The traces of the advective migration of plastic rocks are distinctly rnappedby the methods of structural
petrology [16, 381. They carry a quantitative information about the horizontal and vertical geodynamical
components for some regions. Depending on the initial stratigraphy, P-T parameters, frequency and duration of metamorphic processes, there arise various complexes which reflect the transformation stage of the
geological sections that -were originally unstable in the gravitational field. Particularly, a complete advec tion
cycle can involve the appearance of granulites due to the reworking of greenstone belts [I, 11, 181. Thus,
cratonization process can develop on the Ar chean mobile belt, caused by gravitational redistribution of the
rocks.
Retrograde metamorphis T-P paths for granulite complexes are generalized by the equation [31] :
P = 0.02(f 3 . 7 loe3). T°C - 6.8(f 2.5), kbar,
(1)
which can be used for calculating the depth of granulite metamorphism at a given temperature, it reflects
a iow thermal gradient (at high T) -and is a generalized characteristic of the conditions of surfacing of the
- .
Precambrian lower crust.
We may assume, with much confidence, that Eq. 1. corresponds to the geothermal gradient of the
Precambrian crust, although it is 100-200° lower than the geotherm obtained earlier [lo]. In any event, the
high-temperature regime of metamorphism (with a relatively low gradient 8T/aP e 19) was characteris tic
of this crust, in the conditions of granite-gneissic domes developing in greenstone belts also. The contrary
conclusion has been made by G. F. West and J. Marecshal [28], M. J. Bickle [la] and others, who assumed
a high value of aT/BP for the Archean greenstone belts. But this gradient reflects rather the lateral
temperature change than its connection with the depth.
The low value of aT/BP in no way matches the classic ideas about its role in regional metamorphism.
A rapid pressure drop, with a relatively weak cooling of the granulite complexes, is an evidence of the high
rates of their migration to the surface. Consequently, the surface erosion mechanism is inappropriate for the
explanation of transportation of these complexes to the Earth surface, and Eq. 1 can reflect the tectonic
denudation or the rise of crust diapirs. Geological mapping methods can be rather useful in elucidating one or
other mechanism of the geodynamics of metamorphism. For example, in the first case these can be imbricate
structures found, in the second-ring structures discovered by thorough measurements of orientations of the
elements of metamorphic rock thin structure (original laenation, schistosity, linearity, banding, etc.).
Thus, the basic purpose of the work is to simulate numerically the development of gravitational instability of the polyrhythmic stratified section and to elucidate the nature of the low bT/BP (at a high Tj
gradienb, which is never theless characteristic of different granulite complexes [30,311.
P-Tpaths of fold areas of different age, unlike those of cratons, have the rising and falling vectors
(30, 311. Already in the first version [lo] of these paths, a temperature rise was noted during the transition
to the retrograde stage. This has been explained much later. First, P. S. England and S. W. Richardson [25],
and then G. Draper and R. Bone [24], B. Coffie and B. Velde [22] showed that in the conditions of erosion
or tectonic denudation, when the rocks just start rising to the surface, they began to be warmed up.
For some ultrahigh-pressure complexes, the falling (retragrade) vector is close, by the P-T parameters, to the third group of complexes, defined earlier. So, according to C. Chopin [20], the regressive
Moscow University
G e o l o g y Bullettn
Vol. 48, N o . 5
(Western Alps) is close to that of many
metamorphism path of pyrope-talc-coesite rocks in Dora
galucophane-schis t complexes [lo, 30, 311. The paths of the progressive and regressive st ages of metamorphism in kyanite-biotite-garnet schists of the Kokchetav massif can have a higher temperature (at the same
a T / d P gradient [Is]). Here, the parageneses talc kyanite garnet and kyanite
zoisite
quartz are
also developed. These rocks are noted by the presence of diamond included into Fe-hig garnet. The metamorphic nature of these garnets is of no doubt because along with diamonds there are also inclusions of
hydrogen-containing minerals (phengite, amphibole, zoisite, etc.).
Unlike the Dora Maira complex, where garnet is represented by the practically pure pyrope, in t h e .
Kokchetav massif the schists contain the usual Fe-Mg garnet, the magnesium content in which reduces
from the center towards the edges of the grain (e.g., Xhfg= 0.235 in the center and 0.144 in the marginal
part). The temperature of the amphibole-garnet and biotite-garnet equilibria falls from 650 to 525' in the
metamorphic evolution process. The diamond's nature is enigmatic. By morphological features, it is close
to garnets from kimberlites [15] and plutonic inclusions in them, which seems to support the idea of A. A.
Marakushev [6] about the endogenous nature of such diamonds.
However, in garnets from the related eclogites and peridotites in the same outcrops, diamond is not
found. Nevertheless, the high-pressure nature of the mineral assemblages is undoubtable (sphene contains 11
mass.% A1203, clynopyroxene is rich in potassium (up to 1.44 mass.%), phengite contains up to 3.7 mass.%
of Ti02,rutile is rich in alumina). There are no geological criteria of the overthrust tectonics here [2, 12,
151, but the same age (- 530 mln. yrs) of schists and garnets of the Zerendin massif is of interest (Fig. 1).
This is the only geological feature that can be taken into consideration in creating a geodynamic model of
metamorphism of the Iiolcchetav massif, based on the emergence of diapirs in the Earth crust.
+
+
+
+
-
Fig. 1
Simplified geologic map of the region of the discovery of eclogites and their diamond-bearing
schists in the northern Zerendin pluton, the Kokchetav massif, North Kazakhstan (according to V. G. Kushev and D. P. Vinogradov [5]): the Proterozoic (PRIe2) metamorphic
formations: 1 ) Berlyk; 2) Zholldybai; 3) Daulet; 4) Uyalin; 5) the Upper Proterozoic and
Paleozoic whole formations; 6) the Cenozoic sediments; 7) the Proterozoic basic intrusive
rocks; 8) the Paleozoic granites; 9) Zerendin pluton granitoids; 10) faults; 11-12) axes:
11) of anticlinal folds, 12) of anticlinal and synclinal zones; 13) outcrops of eclogites and
high-pressure assemblages of mica schists.
,Mo~cowUniversity
G e o b g y Bulletin
The second purpose of this work-to simulate numerically the submergence to low depths and metamorphism of sedimentary-volcanogenic complexes using the advection mechanism.
GEODYNAMIC PROBLEM STATEMENT
There are many models of thermal convection in the deep mantle and the Earth lithosphere [ I T , 21,
32, 411. The structural evidences of thermal convection in the Archean crust were presented by C. J.
Talbot [38]. The application of hydrodynamic models to the continental crust rocks entail, however, serious
problems. The mechanism of the material redistribution under the effect of the heat flow is limited by the
rock viscosity and an insignificant thermal expansion. The models of a purely thermal convection cannot
be virtually realized within the Earth ctust, without the chemical heterogeneity taken into account. On
the other hand, the normal sequence of volcanic manifestations in the Archean volcanogenic-sedimentary
sections can hinder to a much extent the development of convective events, if there is no rhythmicity in
stratigraphic sections.
The model of an early stage of gravitational instability of stratified systems subjected to thermal
excitation was discussed by H. Ramberg [32]. A numerical model of instability was developed by H. Schmeling
[35] in the frames of the Rayleigh-Taylor viscous fluid theory for a two-layer system which is also subjected
to thermal convection. A special contribution to the problem of a stratified model development was made
by V. L. Novikov in his numerous works widely known in this country. An attempt to create a convection
mechanism was made by K. F'Veber [39] fir progressive metamorphism, basing on the rock r b l o g i c data
with high P-T parameters.
All these models have a quite narrow range of application to decoding the geodynamic history of the
granulite complex formation. Supplementary superimposed conditions are required to accelerate the process
of gravitational material redistribution, bringing it to actual ages dated by isotopic methods.
The original rhythmicity of volcanogenic-sedimentary rock sections [I] is one of the conditions. In this
case, the problem reduces to numerical modeling of not simple thermal convection but to its accelerated
variant due to the initial gravitational instability, This problem solution allows one also to look into the
mechanism of granulite formation on the basis of the volcanogenic-sedimentary material of the most ancient
marginal seas and their fringing island arcs, and on the crystalline basement of the Archean greenstone belts.
PHYSICAL MODELS, METHODS AND COMPUTER PROGRAMS
H. Rarnberg [32] has experimentally studied the model of gravitational materid redistribution within
a stratified mass. However, he did not succeeded in making the conclusion about a considerable growth of
the redistribution rates. Computer models 127, 35, 401 turned to be more efficient because they allow one to
virtually instantaneously observe the process by introducing and ~ h a n g i ~the
g necessary parameters in the
problem statement, which makes the model close to a red process.
To calculate the evolution of thermal and mechanical processes in rocks, a model of a heat-conducting
heterogeneous viscous medium and the basic laws of continuum mechanics were used: the equilibrium equation
where i = 1 , 2 and j = 1,2; g'is free-fall vector; p ( X , Y , 2) is density, g/cm3; the rheologic relation for the
Newtonian medium
and d-deformation rates in Xiand X j coordinates, respectively; T (X, Y, 2)where uij is stress tensor;
temperature; po exp (-<TI-viscosity as a function of T, p; €-constant; P-pressure; the mass conservation
law (the condition of incompressibility)
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University
Geology Bulletin
iai;emiczi f ie!ct,
m
e
-r
:in~
:
7i?J.
Vol. 48, X o . 5
300
Chemical f i e l d . T i m e r 4dB.UQd
I* a
Chcmicaf
f i e l d . Time : 278,000
Chemical
f i e l d . Time
04
0.6
0.4
0.2
3.0
I. 0
; FTB.ff0O
0.8
-"IU
1
I
I
-.-
.
Fig. . 2
A typical scenario of a 12-layer four-rhythm model
a'dikrete
variation of density and viscosity toward
---- - - wiih
-- .-- - - correlates with t-he.eolbr density. L = 12 krn,
t h e top of every rhythm. The growth of these parameters
im
=-1019
,p, Ap = p, - /rmin = lo2. The computer calcu~ationswere based 'on the
- by
H. Sehmeling adapted to the laser printer, the memory 'of whkh is not large enough to print the results-ie
accordance with the program [19].
~ C ~ O S CwO Unlverstty
Geology Bulletin
where
Vol. 48, ,Vo. 5
? is velocity, crn/s; heat
transfer equations for the density and viscosity constants
where Cp is heat capacity at a constant pressure, J.g-'; g = 9.8 m-sU2;r-time; Q-radioactive heat,
J - ~ r n - ~ s - ' ;*temperature conductivity, cm2/s; aA/Br and BA/ar-partial derivatives with respect to time
and space, correspondingly; A-density or viscosity field parameters. Limiting conditions: V, = 0 and
b,t
= 0.
The modeling was based on the program developed for the operating station SUN,using the finite
element method [IS]. Successive approximation allowed the researcher to find the most effective variant of
the program in P-?-T
variables and the Cartesian coordinates; chosen were a triangular finite element.
continuous square basic functions for the velocity field, and discontinuous linear functions for the pressure
field.
The efficient rock viscosities were used for modeling on the basis of Newtonian rheology [33], with the
following circumstances taken into consideration: (1) low accuracy of the efficient viscosity data for differentcomposition rocks; (2) the influence of phase transitions in the rocks of granulite facies of metamorphism on
their rheological properties cannot be yet quantitatively evaluated; (3) the actual values of the deformation
rates are not known, and they are needed to estimate the efficient viscosity when the exponential rheologic
law is applied; (4) the rock efficient viscosity grows with the melting temperature.
Thus, the choice of the efficient viscosity values of the application of the Newtonian rheology are
problematic. However, for a section with a total thickness H km, the results of numerical modeling (time r
and velocity
for the given p,.,
and Ap can be converted into other p',, and Ap', using the following
rules:
v)
Special calculations revealed a weak (not over one order) dependence of these values on the viscosity
difference in the layers. Hence, their variations do not limit the possibility to solve the problem set.
To calculate-thegravitational ordering on geologically verified ages, the viscosity range has been chosen
equal to 1 0 ' ~ - 1 0p~, ~corresponding to the acidic rock subsolidus, i.e., the range that differs from the solidus
of basic rocks. These viscosity values correlate with geotherm (1). The chemical inceraction between the
rocks of different composition was not taken into account.
NUMERJCAL MODELING RESULTS
Three-Layer Four-Rhythm Model
In accordance with Eq. (I), the temperature difference within a section 1-3 km thick is 15-50'. Consequently, the variations of the rock proper ties as a function of temperature are insignificant, and numerical
modeling of the rock gravitational redistribution can be isothermally approximated.
Figure 2 shows the numerical modeling results for a four-rhythm 12-layer system. Each rhythm consists
of three layers 250 m thick, and density/viscosity (g-cm-3/p) grows in the upward direction from 2.6/1016
(rhyolites) through 3/10L9(basalts) up to 3.3/1020 (komatiites). These rock properties are given in correlation
with the P-T parameters calculated by Eq. (I), and with the efficient viscosity taken into consideration.
In Fig. 2 we see a largescale redistribution of seams during a geologically reasonable time, caused by the
chain reaction effect. A sharp growth of the seam dislocation rate results in the interaction of rhythms, and
within each of them convection arises. In Fig. 2 we see a substantial accumulation of the granitic material
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Geology Bulletin
Fig. 3
The inner structure of a small dome-like fold in the rocks of the Sharyzhalgai complex,
open mine stripped along the Krugo-Baikal railway, the Baikal Lake shore (after A. blelnikov): 1)biotite-garnet gneisses; 2) hornblende-bipyroxene migmatized gneisses (1850 Ma);
3) migmatized biotite-hypersthene gneisses; 4) enderbite gneisses; 5) subsolidus enderbite,
charnockites, and garnets with a weakly expressed gneissoid structure; 6) boudins and xenoliths of melanocratic crystal schists and amphibolites (2620 Ma); 7) fissures; 8) the direction
of active forces.
. -. .
- (isolated light portions) at the different levels of the Earth crust. We also see (Fig. 2) that acidic rocks ( the
material of a lower density) are being involved into the lower part of the section by the sinking seams of the
denser rocks. This effect is linked with the inverse (against gravity) flows within local advective cells.
The period of gravitational redistribution depends inversely on the rhythm thickness. As was noted
' above, a 3-km section, being 'a product of the reversed sequence of volcanic events, can remain stable in
gravitational field during geologic time even in the conditions of the granulite facies of metamorphism. At
best, there can arise dome structures, without the complete gravitational material redistribution, which
is characteristic, for example, of a number of sections in the Sharyzhalgai complex in Southwestern Baikal
region (Fig. 3). At the same time, there naturally occurs the complete gravitational material redistribution
in the P-T conditions of the granulite facies of metamorphism, and it was necessary to determine the effect
that causes and accelerates this process.
L. L. Perchuk has supposed that the acceleration of gravitational ordering of masses is possible when
a thermal disturbance is being superimposed on stratified polyrhythmic sections. This will result in the
interaction of the rhythms in a stratified system, and a chain reaction will boost convection within the
whole section. The original section's rhythmicity is the basis for the creation of large heterogeneities at the
beginning of gravitational redistribution and for the reduction of the efficient viscosity of the host environment
at the advection stage. This statement of the problem is not abstract. The greenstone belts, confined to
intercontinentd structures, can transform into granite-gnessic .complexes during the lengthy evolution (11,
there are the geochemical evidences of it [ll]. A special role here belongs to the greenstone belts with a
polyrhythmic volcanogenic-sedimentary sequence.
We have not yet used the heat transfer Eq. (5). Consider a 30-km polyrhythmic section a s a model of
the heterogenous Earth crust prior to gravitational ordering.
Equation (1) determines the gradient which is lower than the geotherm calculated on the basis of
adiabatic approximation
.
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where dl denotes dimensionless, Z is depth, km. This equation follows from ( 5 ) with the limiting conditions
T = 0 for P = 1 bar and T = 800°C for P = 10 kbar and with the actual value of the produced radioactive
heat (see, for example, [48]). According to (I), the rocks of granulite facies of metamorphism at low depths
(P = 1.2-1.5 kbar) have the temperature T = 400-415'C, whereas according to the conductive model,
T = 100-120' C. We may assume, with a sufficient degree of assurance, that such a substantial difference of
temperatures is the result of viscous friction during gravitational ordering of the heterogeneous Laminated
crust. Numerical modeling of this process for such a highly heterogeneous medium is limited by the capacities
of the operating station SUN. However, the viscous friction thermal contribution in the first approximation
can be evaluated within the frames of a one-dimensional model, with the results of numerical modeling of
gravitational ordering of a four-rhythm stratified section taken into account.
To solve the one-dimensional problem, consider Eq. (5) in a dimensionless form. We use H for the
length scale, H2/x-for the time scale, A T = 800°C is the temperature scale, h p the density scale. Then we
get the expression
and AT are the density and temperature of the rocks differing in
where g is the free-fall acceleration; i p
composition in a stratified section; rtemperature conductivity; exp (yT)-thermal contribution to viscosity
p ; Rt = (ApgX3)/~pmu-the Rayleigh-Taylor number.
The numerical experiment discovered the following properties of Eq. (10): (1) the dimensionless value
of the viscous flow heat A = aij eij is close to the constant
lod3; (2) the second number to the left and
the last one in the right-hand part of the equation are close to zero, hence, the equation can be rewritten in
the form:
-
.
This equation is well known in the combustion theory. It describes the chain reaction mechanism of ignition
[4]. We use Eq. (9) as the initial condition for solving Eq. (11). ~ c c o r d i hto~ the principal results of the
combustion theory, the initial geotherm (9) can become unstable under the two following conditions which
can be applied to the lower rhythms of a stratified section of a thickness, say, H = 10 km:
The first member
Kond= xH-I
2 9.
~rn-~ear'lreflects the conductive heat loss, and the second
determines hydrodynamic velocity (the velocity of a mass flow). The third member in Eq. (13) is the
dissipative number -
Dis = ( A ~ H ~ ~ ) / ~1.67
,..
(15)
determines dissipation under the viscous fluid flow.
Thus, the conditions (12) and (13) are satisfied, and the geotherm (9) is unstable and can deviate into
the high-temperature area, close to the conditions of Eq. (1). This is the result of gravitational redistribution
and the related unsteady-state heat front (viscous friction effect). The development of this front controls a
,lfoscow University
Geology Bulletin
Fig. 4
A typical scenario of the rise of relatively small blocks of rocks with density p = 4 g/cm3
(black color) following a large volume of granitic rocks (p = 2.4 g/cm3), emerging to the
surface of the Earth crust. The arrows show the direction and relative velocity of the
material moving within the advective cell. L = 100 km,pm, = 1019 p, A p = lo2. The
P-T parameters correspond to the retrograde trend obtained by C. Chopin [20].
rapid thermo- and hydrodynamic evolution of granulite complexes, during which the P-T paths are being
recorded by metamorphic reactions.
In Eq. (13), A is the only unknown parameter found empirically during numeric modeling. It mirrors
the interaction between the layers of different density and their configuration. The higher it is, the higher is
Vhyd.
Note that a rough differentiation of the Earth crust into provisionally granitic and basaltic laye-rs can
also be explained with the aid of this mechanism of chain reactions that has been completed in geologically
real time. Figure 2 clearly shows the accumulation of granitic material (light colors) at the different levels of
ikf'osco w University
Geology Builetin
the Earth crust. Such granite quantities are unpossible to obtain by means of partial melting of the Earth
crust.
The Advective Model for Ultrahigh-pressure Complexes
The geologic position of these complexes allows one, in many cases, to explain their origin by collision
processes with the subsequent tectonic denudation [36]. There are known, however, the complexes, for which
the plate tectonic models need not be applied for the lack of any geologic ground. The above mentioned
ultrahigh-pressure complex of the Kokchetav massif in Northern Kazakhstan is one of them. As was noted,
according to the data by 0. M. Rozen [12], V. G. Kushev and D. P. Vinogradov [5], Yu. A. Zaitsev [2],
V. S. Shatskii [15], here there are no criteria of a large-scale horizontal displacement of the rocks. Moreover,
the contacts of the Zerendin granite massif with the hosting mica schists, where diamond grains are found
in garnets, are not clearly defined. The age of metamorphism and granitic magmatism is the same (530 mln.
yrs). There is no metamorphic zonality around the pluton, the diamond-bearing localities outcrop along the
exocontact of the Zerendin massif (see Fig. 1).
Numerical modeling has been carried out within the frames of a complete advective cycle, as well as in
the two-phase regime. The plate tectonic mechanism is sufficient for the sedimentary-volcanogenic rocks to
be subduced to the low depth [36]. The mechanism of consolidated rock surfacing presents a serious problem.
It is this mechanism that is shown in Fig. 4-emerging of high-density rocks in relatively low-density rocks
due to hydrodynamic "withdrawal" (convektive circulation) of a discrete seam with p = 4 g. ~ r n - provided
~,
by the rise of the large volumes of the magmatic material to the Earth surface from a depth lower than 100
km within a thick continental crust. In the frames of a such model we can also solve the problem of the
ultrahigh-pressure assemblages of the Western Alps, Norway, Central China, and Northern Kazakhstan.
CONCLUSIONS
1. Numerical modeling revealed that a polyrhythmic structure of the many-layered primary stratigraphic sections in the conditions of granulite facies of metamorphism speeds up the process of gravitational
material redistribution within the whole section, following the chain reaction mechanism [4]. A similarity in the mathematical descriptions of the acceleration of gravitational ordering and of the combustion
mechanism has been found out.
Numerical experiments showed that during gravitational ordering possible are the formation of large
diapirs of the tonalite composition, the complete reworking of greenstone belts, and the introduction of
granulite diapirs into the rocks of arnphibolite facies of metamorphism. Such relationships can be interpreted
in the Kola Peninsula, where the Laplandish granulites pierce the Korvatundra amphib,oles and superimpose
on them [7].
Gravitational material redistribution in multilayer polyrhythmic sections of the original volcanogenicsedimentary crust can be considered as a craton formation mechanism (at the place of greenstone belts),
through endogenic heat and mass transfer. We mean the introduction of fluids, first of all carbonic, from the
mantle diapirs, giving rise to basic and ultrabasic intrusive magmatism in the lower part of the continental
crust. This model can explain the formation of granulite facies in the central and middle parts of continental
plates, unlike continental margins, and rising of these complexes almost to the surface of the Earth crust,
which is evidenced by geothermobarometrie data. Greenstone belt cratonization process in the middle part
of the continental plate was modeled following the chain reaction mechanism.
2. The universally low c3T/aP (at a high T) gradient for the rocks of granulite facies of metamorphism
(Eq. 1) can be a result of the rapid nonconductive thermal front dist.ribution. This front was created due
to viscous friction in the rocks +duringgravitational ordering ofsthe polyrhythmic multilayer stratigraphic
sections, i.e., due to the transformation of gravitational energy into thermal. This phenomenon is well
described within the frames of the mathematical model of the combustion theory as the chain reaction
mechanism [4].
3. The problem of ultrahigh-pressure metamorphic complexes can also be solved within the frames of the
developed advective model. In this case the rise of magmatic masses of any composition in a thick continental
crust creates a convective cell and provides the "drawing" of the low-density volcanogenic-sedimentary rocks
to great depths and their consolidation; under multi-phase magmatism they can be transported upward.
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~ M o s c o wUniversi&y
Geology Builctin
The complete advection cycle for a 100-km diameter cell, with the rock viscosity reaching 10" p and their
subsequent (while submerging) compaction u p to p x 4 g-cm-3 can be realized for the geologically accepted
time (some million years) at a rate amounting to 30-40 cm/year.
Two-dimensional numerical models (the finite-element method) have been developed. They allow one
to solve the problems related with the dynamics of high-viscosity media applied to geodynamic problems.
In conclusion we express our gratitude to A. M. Polyakov (Uppsala University, Sweden), who actively
participated in the discussions and in the numerical modeling program.
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26 May 1992