RUSSIAN JOURNAL OF EARTH SCIENCES VOL. 7, ES3001, doi:10.2205/2005ES000179, 2005
[3] If convection were absent in the mantle, deep heat would be removed solely
through the conductive heat transfer with a flux equal to
q0 kT1/H ( k is heat conduction and
H is the mantle thickness). With
k
6 W (mK)
-1,
T1
4000 K and
H = 3000 km, this yields
q0
8 mW m
-2,
which is about ten times smaller than the observed value. Therefore, the mantle is
involved in vigorous thermal convection enhancing the heat removal process.
[4] A high gradient of density in the D'' layer is due to an abrupt increase in the
concentration of heavy components at the core-mantle boundary. This hinders much
convection inside this layer that could make the temperature distribution
smoother. Accordingly, the temperature change across this layer can amount to
1000 K with mantle heated from below and 2000 K with mantle heated by its inner
radioactive sources. If the middle of the D'' layer is taken as an effective
lower boundary of the origination of plumes and ascending mantle flows, the
temperature difference responsible for the heating of the mantle should be
DT3000 K. Thermal convection arises only if an ascending material element remains
invariably lighter and thereby hotter than the surrounding mantle. However, due
to decompression, hot material ascending from the mantle base to the Earth's
surface cools by about
DT ad = 1200 K. Therefore, mantle convection is due not
to the total temperature difference but only to its potential (superadiabatic)
part
T p=DT-DT ad. This yields the value
T p = 1800 K for
the potential temperature of mantle plumes.
[5] The effectiveness of heat transfer in a convective layer is characterized by the
Nusselt number Nu, equal to the ratio of the convective heat flux
q to the
conductive heat flux
q0 caused by the potential temperature difference:
q0=kT p/H3.6 W (mK)
-1. To determine the convective heat flux,
we find the conductive heat flux due to the temperature difference in the lower part of the
D'' layer and the adiabatic difference (the overall difference
DT = 2200 K):
Dq=kT p/H
4.4 W (mK)
-1. Subtracting this value from the observed heat
flux, we obtain that the convective heat flux from the mantle amounts to 85 W (mK)-1.
As a result, the Nusselt number in the Earth's mantle is Nu =
q/q0
85/3.6
24.
[6] As is known from the theory of convection in a layer heated at its lower boundary
[Schubert et al., 2001],
the convective velocity
V is connected with the Nusselt number through the relation
V8(k/H) Nu2, where
k is the diffusivity defined as the thermal conductivity
divided by density and specific heat; with
k=10-6 m2 s-1, this formula yields a value
of about 5 cm yr-1 for the velocity of circulation and mixing of mantle material.
[7] Apart from the Nusselt number Nu characterizing the effectiveness of convective
heat transfer, another important quantity is the Rayleigh number Ra
characterizing the intensity of convection (see formula (8) in Appendix):
Ra =
agDTH3/(kn). Here,
a is the thermal expansion
coefficient,
g is gravity,
n is the kinematic viscosity (defined as dynamic
viscosity divided by density), and
DT is the superadiabatic temperature
difference (the potential temperature). Setting in the mantle
a=210-5 K-1,
n = 3
1018 m2 s-1 and
DT = 1800 K, we obtain Ra = 2
106.
In the convection theory, the Rayleigh and Nusselt numbers are interrelated through the approximate
formula Nu = 0.2 Ra1/3; hence, setting Ra = 2
106, we obtain Nu = 25 in
agreement with the estimate obtained from the observed heat flux of the Earth.
[8] In energy terms, the Earth can be compared with a heat engine in which the heat is permanently radiated into the space and is partially converted into the energy of convective motion. Presently, it is believed that four main sources contribute to the heat flux of the Earth: radioactive decay in the crust (~15%), solidification heat of the growing inner core (~10%), radioactive decay in the (predominantly lower) mantle (~50%), and heat of the cooling Earth (~25%) that was accumulated during its formation [Schubert et al., 2001]. This balance corresponds to an Earth's cooling rate of 80 K Gyr-1.
[9] Previously, mantle flows were sometimes attributed to chemical, rather than thermal, convection associated with sinking heavy components of mantle material (e.g. remaining iron) and ascending light components (e.g. liberated from the core). Hot light material rises during thermal convection. Cooling at the Earth's surface, this material forms the thickening lithosphere, which sinks when it becomes heavier than the hot mantle material. However, if the lower density of the rising material were due to its chemical composition rather than its higher temperature compared to the surrounding mantle, magma of mid-ocean ridges would have room temperature, which contradicts data of observations. Moreover, a lithosphere formed by this lighter material would not have thickened with time and would not have sunk into the mantle. As mentioned above, thermal convection is due to the nonzero potential temperature of the ascending material. Calculations of the global heat flux distribution also indicate that mantle density inhomogeneities (apart from the continental lithosphere) determined from seismic tomography data are of thermal, rather than chemical, origin because they are consistent with the observed anomalies of the terrestrial heat flux. If a negative density anomaly were of chemical origin, a higher heat flux would not be observed at the corresponding point of the Earth's surface.
[11] The vertical heat transfer is purely conductive in the upper and lower boundary layers (where velocities are nearly horizontal). However, because the heat flux in a stationary state should be depth independent, the Earth's temperature first increases very rapidly with depth, from 0o C to ~1400o C in the upper 100 km, after which its gradient drops to a value smaller than 1o C km -1. As a result, the upper cold layer of the mantle (lithosphere) is highly viscous and is underlain by a low viscosity asthenosphere. The boundary of these two layers is formally associated with the solidus temperature 1300o C. At greater mantle depths, the effect of slow temperature rise is dominated by the pressure increase, and the mantle viscosity increases with depth by one or two orders. A lower conductive boundary layer with a high temperature gradient arises at the mantle base due to heat flux from the core. As a result, viscosity in the D'' layer, as in the asthenosphere, drops and this leads to instability of flows and generation of plumes in this layer. Possibly, the viscosity in the lower 1000 km of the mantle (but above the D'' layer) is higher than the value presently accepted.
[12] Due to strong lateral variations in temperature, both lithosphere and asthenosphere vary in thickness. Hot mantle material reaching mid-ocean ridges solidifies, after which it moves horizontally, further cooling and forming the thickening oceanic lithosphere. As the lithosphere becomes heavier, the ocean floor deepens. On the contrary, the underlying asthenosphere becomes thinner away from the ridges and disappears completely near subduction zones. In continental regions, the asthenosphere can only exists in the form of isolated lenses.
[13] A fundamental distinction exists between the oceanic and continental lithosphere. The oceanic lithosphere is involved in the convective circulation of the upper mantle material and, for this reason, its lifetime at the surface is no more than 200 Myr. The continental lithosphere is frozen up to a continent from below, making with it a coherent structure due to mass transfer and becoming lighter due to phase transformations. Therefore, the continental lithosphere (adjacent to the base of the continental crust) exists for more than 3 Gyr. Since the mantle beneath continents (at depths of about 200 km) is 100-200o C colder than beneath oceans, the continental lithosphere is much thicker than the oceanic lithosphere. Accordingly, asthenospheric lenses beneath continents arise generally in places where the mantle is heated by hot plumes intruding from below.
[15] With such a high viscosity, the oceanic lithosphere (like solidified tar) is fairly brittle, albeit capable of experiencing very slow viscous deformation. When subjected to a variable stress, the lithosphere is broken into a system of horizontally moving plates. When the lithosphere becomes sufficiently heavy, it starts sinking into the mantle in subduction zones. The subducting lithospheric plates (slabs) heat and dissolve in the mantle. Recent tomography data visualize not only the slabs but also their nondissolved remnants in the lower mantle and at its base.
[16] Strictly speaking, the idea of rigid lithospheric plates is incompatible with their bending in subduction zones. When elastically deformed, the lithospheric plate cannot bend even by a few degrees (an elastic tangential stress cannot exceed the one thousandth of the shear modulus) and should finally break up. Oceanic plates do not break in subduction zones because their material under shear stress conditions becomes ductile. Therefore, the classical tectonics of rigid lithospheric plates is only applicable to horizontal plate movements and small bending associated with underthrusting or overthrusting. At mid-ocean ridges, the ascending mantle flow easily turns to assume a horizontal direction because its material did not solidify as yet. In subduction zones, a change in the movement direction of a plate is associated with a loss of rigidity under conditions of higher stresses.
[17] Plate tectonics is understood, in a narrow sense, as kinematics of quasi-horizontal movements of rigid brittle lithospheric plates including overthrusts, underthrusts, and motions along transform faults, as well as processes of folding and brittle fracture at plate contacts. This theory is deficient in that it does not examine driving forces or treats them as external factors.
[18] The plate tectonics understood in a broad sense is the theory of origination and evolution of lithosphere, including its kinematics and dynamics. Oceanic lithosphere in this theory arises self-consistently as a result of the solution of equations of thermal convection in the mantle with temperature dependent viscosity. This theory describes numerically the processes of the solidification of magma at mid-ocean ridges, seafloor spreading, thickening of the lithosphere with age, and its sinking in subduction zones. However, the theory has not been accomplished as yet. For example, we fail so far to get a self-consistent description of breakage of the lithosphere into plates with the formation of transform faults.
[19] The viscosity of mantle material can be represented as an exponential function
of temperature and pressure and as a power function of tangential stress:
h=As1- n2 exp[(E+pv)/R(T+q)], where
E is the
activation energy,
v is the activation volume,
R is the gas constant,
A is a prescribed constant,
q is an additional constant chosen from a given
value of the effective viscosity of the cold oceanic lithosphere
h|T=0<,
and
s2 is the second invariant of the viscous stress tensor,
s2=(sijsij)1/2.
[20] As mentioned above, the full range of the viscosity variation in the mantle encompasses more than 20 orders of magnitude. Without regard for melting processes, this range amounts to 5-8 orders. The exponent n in a dislocation creep model is usually taken equal to 3. The ideas of mantle convection and oceanic lithospheric plates spreading away from mid-ocean ridges and sinking into the mantle in subduction zones were formulated by the 1970s and replaced notions of a solid mantle with separate magma sources. Plate tectonics arose first as a kinematic theory. However, it was found out in the 1980s that constraints on properties of the oceanic lithosphere (except its breakup into plates) can be directly gained from the theory of mantle convection incorporating the temperature dependence of viscosity. Thus, the theories of mantle convection and oceanic lithosphere are presently united into the general theory of mantle convection with variable viscosity.
[21] The above concept of mantle convection with oceanic lithosphere has been developed over the last three decades by geologists, geophysicists and geochemists up to the stage of its numerical description in terms of mathematical modeling [Anderson, 1989; Fowler, 1996; Schubert et al., 2001].
[23] Likewise, the theory of lithospheric plate tectonics fails to account for such phenomena as the assemblage and breakup of supercontinents, the causes for formation and long-term existence of continental lithosphere and roots of continents, the mechanism of oblique underthrusting of oceanic lithosphere beneath continents, and others.
[24] A new geodynamic model including mechanical and thermal coupling of mantle with floating continents has been developed in [Trubitsyn, 2000, 2004; Trubitsyn and Mooney, 2002; Trubitsyn and Rykov, 2000, 2001; Trubitsyn et al., 2003]. The lifetime of a continent (~3 Gyr) is much longer than the lifetime of a lithospheric plate. The Earth's surface temperature (~300 K) being much less than the melting point of mantle material (~1500 K), continents can be compared with huge rafts floating in a varying ice field. "Ice'' temporarily frozen up on the sides of continents corresponds to the oceanic lithosphere at passive continental margins. The "ice'' frozen up from below for a long time corresponds to the continental lithosphere. However, as distinct from the ordinary ice (which is lighter than water), the heavy oceanic lithosphere at active continental margins can sink into the mantle, and its material is involved in the convective mantle circulation. Due to thermal screening, the heat flux from the mantle in continental regions is three times as small as in oceanic regions. Therefore, floating continents change basically the whole evolution of thermal mantle convection. In order to describe the coupling between viscous mantle and rigid continents, the system of the mantle convection equations is complemented with the equations of motion of rigid continents (in a spherical model, these are the Euler equations of dynamics of a solid). Continents of varying thickness and arbitrary sizes and shape can move on the surface, rotate under the action of viscous forces applied to their submerged surfaces, and collide. Unknown are the velocities of mantle flows and continents, as well as the temperature in the mantle and the continents. Solution of these equations also provides the distributions of viscosity, viscous stress tensor, and heat flux, the seafloor topography, the gravitational field, and movements of continents.
[25] The Earth was sometimes compared with a heat engine, with the mantle being similar to a boiler, and oceanic plates, to connecting rods. However, without actuator valves, the heat engine cannot ensure an ordered motion. It was found out that heat-screening continents play the role of floating valves that redistribute the heat flux in the mantle and control the geologic history of the Earth. Within the framework of the mantle convection model with floating continents, global processes in the Earth are not a result of chaotic convection (as was recently believed in tectonics of lithospheric plates) but can be consistently calculated and thereby predicted. The evolution of the mantle-continents system develops between two quasi-equilibrium states: the state in which each continent is close to adjacent downwelling mantle flow and the state in which all continents are assembled into one supercontinent above a system of mantle downwellings. One might say that continents actually control the evolution of mantle convection and the tectonics of lithospheric plates.
[26] The present study, complementing the papers [Trubitsyn, 2000; Trubitsyn et al., 2003] and presenting modeling results for mantle convection with two floating continents, treats in greater detail the formation of a mantle plume system beneath a supercontinent and gives analysis of uplift and subsidence of continents with related fluctuations in the ocean level incorporating data of recent reconstructions of the continental drift.
Citation: 2005), Evolution of mantle plumes and uplift of continents during the Pangea breakup, Russ. J. Earth Sci., 7, ES3001, doi:10.2205/2005ES000179.
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