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Mixing times in the mantle of the early Earth derived
from 2-D and 3-D numerical simulations of convection.
Nicolas Coltice, J. Schmalzl
To cite this version:
Nicolas Coltice, J. Schmalzl. Mixing times in the mantle of the early Earth derived from 2-D and 3-D numerical simulations of convection.. Geophysical Research Letters, American Geophysical Union, 2006, 33, pp.L23304. �hal-00342402�
Mixing times in the mantle of the early Earth derived from 2-D and
3-D numerical simulations of convection
N. Coltice1 and J. Schmalzl2
Received 28 July 2006; revised 12 October 2006; accepted 1 November 2006; published 6 December 2006.
[1] The Earth’s mantle is chemically heterogeneous at all
scales, as shown by elemental and isotopic analysis of oceanic basalts. This heterogeneity is continuously destroyed by convective stirring and slow diffusion. It has been argued that 3-D time-dependent convection is less efficient than 2-D convection, except in the presence of a toroidal component. In this study we question this conclusion with numerical simulations and show that mixing in 3-D time-dependent convection is as efficient as in 2-D, and only depends on convective vigor. We compute mixing times of thermal and chemical heterogeneities in the early Earth of 10 and 100 Myrs respectively.
Citation: Coltice, N., and J. Schmalzl (2006), Mixing times in the mantle of the early Earth derived from 2-D and 3-D numerical simulations of convection, Geophys. Res. Lett., 33, L23304, doi:10.1029/2006GL027707.
1. Introduction
[2] The Earth’s mantle is chemically heterogeneous at
all scales, as witnessed by basalts from mid-ocean ridges [Agranier et al., 2005] and hotspots [Blichert-Toft et al., 2003; Abouchami et al., 2005]. This heterogeneity is supposed to originate from early Earth fractionation [Boyet and Carlson, 2005] and subduction of tectonic plates [Hofmann and White, 1982]. Convective mixing tends to erase chemical heterogeneities. Therefore it is a pivotal issue to understand the chemical evolution of the planet especially in its early stages [van Keken et al., 2002]. Fluid dynamical approaches showed that chaotic pathlines are necessary for an efficient mantle mixing [Olson et al., 1984; Christensen, 1989] and are favored mostly by time-dependence of the flow [Christensen, 1989; Schmalzl et al., 1995, 1996] and/or a significant toroidal component [Ferrachat and Ricard, 1998; van Keken and Zhong, 1999]. Without a toroidal field, it was suggested that time-dependent convection in 3-D would be unable to mix efficiently the mantle [Davies, 1990; Schmalzl et al., 1996]. Indeed, it was proposed that the plume shape of instabilities of 3-D convection would be less efficient structures for mixing compared to the sheet-like shape of instabilities in 2-D. This paper is a reassessment of this idea and shows that 2-D and 3-D convective flows are identically efficient even without a toroidal field. Among the numerous existing methods to quantify mixing [Farnetani and Samuel, 2003], we use a Lagrangian strain rate (i.e.
Lyapunov exponent) description of mixing that allows the computation of true mixing times in the mantle of the young Earth.
2. Mixing Time
[3] Mixing occurs when chemical equilibrium is attained,
meaning that chemical diffusion has taken place and erased any heterogeneity. For the Earth’s mantle, mixing is difficult to reach since solid-state diffusion of heat and chemical elements is extremely slow: heat diffusivity is only 106m2s1, Helium diffusivity is as low as 1013m2s1 [Trull and Kurz, 1993] and for lithophile elements it is even lower1019m2s1[Hofmann and Hart, 1978]. However, convective motions in the mantle stretch the creeping rocks and heterogeneities can be reduced to a size where diffusion can eventually occur. A final stage of homogeni-zation can also take place with melting and melt extraction in the shallow mantle on lengthscales comparable to the size of magma chambers.
[4] A simple 1-D model can be proposed for the
evolu-tion of a scalar field C with strain and diffusion [Kellogg and Turcotte, 1987]. As in by Ricard and Coltice [2005], we consider a homogeneous stripe of thickness a(t), experienc-ing pure shear, which initial concentration differs from the background. In the direction of pure shear z, perpendicular to the stripe, the conservation of C across the deforming heterogeneity writes: dC dt ¼ Di a 0ð Þ a tð Þ 2 d2C d~z2; ð1Þ
Dibeing the diffusivity, and ~z the Lagrangian variable ~z =
z/a(t). Strain is increasing the apparent diffusivity by a factor (a(0)/a(t))2. If mixing is chaotic, shrinking along the z axis is exponential [Olson et al., 1984]:
a tð Þ ¼ a 0ð Þ exp _etð Þ; ð2Þ
_e being the Lagrangian strain rate, also called Lyapunov exponent. We can then derive a typical diffusion time, i.e. mixing time: tm 1 2 _elog _ea 0ð Þ2 Di : ð3Þ
To estimate the bulk mixing time of a system with equation (3), one could replace _e by the bulk Lagrangian strain rate of the flow and a(0) by the depth of the system h. [5] As noted by Kellogg and Turcotte [1987], the mixing
time does not depend a lot on the diffusivity. For instance, changing the diffusivity by 13 orders of magnitude (the
1
Laboratoire de Sciences de la Terre, UMR 5570, CNRS, Universite´ de Lyon Universite´ de Lyon 1, Villeurbanne, France.
2
Department of Geophysics, University of Mu¨nster, Munster, Germany. Copyright 2006 by the American Geophysical Union.
ratio between heat and lithophile elements diffusivities) only changes the mixing time by a factor of 10 with equation (3). For a strain rate of 5 1016s1, the mixing time of heat is 300 Myrs which compares well with the age of the oldest slab identified by seismic tomography [Richards, 1999]. The corresponding mixing time of Helium is 0.9 Gyrs and 1.3 Gyrs for a lithophile element. The fundamental parameter for the mixing time is then the Lagrangian strain rate. Its dependence upon convection parameters is the purpose of the following section.
3. Numerical Modeling of Convective Stretching
[6] We set up numerical models of mantle convection in
which we compute the Lagrangian strain rate described above. We solve the Navier-Stokes and energy equations for an incompressible and isoviscous fluid in Cartesian geom-etry using the code ConMan in 2-D [King et al., 1990] for 2:1 and 8:1 aspect ratios, and TDCON in 3-D [Houseman, 1990] for a 2:2:1 aspect ratio (see Figure 1). Because there is no lateral viscosity variations in the models, no toroidal field is generated and the flow can be described by its poloidal component only. The vigor of convection depends on the Rayleigh number defined for basal heating as
Rab¼
rgaDTh3
km ; ð4Þ
r, g, a, DT, k and m being the density, gravity acceleration, thermal expansivity, temperature drop between top and bottom, thermal diffusivity and viscosity respectively. For internal heating the Rayleigh number is given by
Rai¼
r2gaHh5
kkm ; ð5Þ
H and k being the heat production per unit of mass and thermal conductivity.
[7] To compute the Lagrangian strain rates, we follow the
distance d(t) between particle twins advected in the flow using a 4th-order Runge-Kutta scheme. The Lagrangian strain rate, i.e. finite-time Lyapunov exponent, is computed
assuming that for each couple of tracers, stretching d(t)/d(0) occurs as
d tð Þ=d 0ð Þ ¼ exp _etð Þ: ð6Þ
Averaging _e over the tracer couples gives the bulk Lagrangian strain rate characterizing the stretching ability of the flow [Coltice, 2005]. If the computed bulk Lagrangian strain rate tends toward 0, mixing is regular, otherwise it is chaotic. To start the experiment, the distribution of the couples of tracers is homogeneous throughout the box. In the 3-D calculations, up to 10,000,000 tracers are used for an accurate description of the Lagrangian strain-rate field in 3-D.
[8] As shown in Figure 2, there are three regimes of
mixing as a function of Rabwith basal heating, which are
identical in 2-D and 3-D. If Rab< 104, mixing is regular and
the Lagrangian strain rate tends toward zero because con-vection is steady. At high Rab(>105), convection is highly
unsteady, mixing is chaotic and the Lagrangian strain rate scales with Rab2/3, which is the same scaling as the rms
velocity. For instance, an increase by a factor of 10 of Rab
implies 4.6 quicker velocities and 4.6 higher strain rates. For Rabbetween 104and 105, there is a transition regime for
oscillatory convection, in which regular mixing islands coexist with chaotic mixing [Christensen, 1989; Coltice, 2005]. Other quantities measuring the statistical steady state of the flow like heat flow or rms velocity do not display such a transition. Hence it is only the time-dependent properties of the flow that are changing. The region of transition is switched to slightly higher Rabwhen the aspect
ratio of the box is 8:1. It shows that the long wavelength structure at low Rabis responsible for regular mixing.
[9] Figure 3 shows similar results for 2-D and 3-D
calculations for internal heating. At high Rai, the
Lagrang-ian strain rates scales with Rai1/2, which is again the same
scaling as the rms velocity. The transition from regular to chaotic mixing is abrupt and depends on the maximum wavelength of the flow. The large aspect ratio simulations Figure 1. Isotherms (0.7 in green and 0.3 in blue) in a 3-D
convection simulation heated from below at Rab= 107. The
horizontal tracer cloud used in the dispersion experiment is represented in orange.
Figure 2. Lagrangian strain rate as a function of the Rayleigh number for basally heated convection. We artificially set the strain rate to 1 instead of 0 when mixing is regular because of the log-scale.
L23304 COLTICE AND SCHMALZL: MIXING TIMES IN THE EARLY EARTH L23304
shows that for a given Rai the time dependence of the
flow increases with the maximum wavelength. In a 3-D box, the longest wavelength corresponds to the horizontal diagonal, therefore it is not surprising to observe time dependence contributing to chaotic mixing at lower Raiin
3-D than in 2-D for a similar aspect ratio.
4. 3-D vs. 2-D Mixing
[10] The major observation is that the same scalings
apply for 2-D and 3-D convection, as well as the transitions from regular to chaotic mixing. It demonstrates that con-vective mixing in both geometries is identical which seems to contradict a previous study [Schmalzl et al., 1996]. The authors used a different technique to quantify mixing efficiency: they monitored in time the dispersion of a tracer cloud by counting the number of boxes forming a regular grid, that contain at least one tracer. In other words, they estimate the time it takes for a patch of dye to color the whole fluid. The authors observed a factor of 10 longer time of dispersion in 3-D than in 2-D and concluded that 3-D mixing is less efficient.
[11] In their simulations, the tracer cloud in 3-D has a
comparable cross section to that of the 2-D simulation. However, the concentration of ‘‘dye’’ in the experiment (volume of the tracer cloud/volume of the box) is 25 times lower in 3-D than in 2-D. Therefore we propose the choice of the size of their initial cloud explains the difference between their conclusions and ours since a small patch of dye takes a longer time to color the whole fluid than a large one. To test this hypothesis, we performed dispersion experiments choosing tracer clouds that represent 1% of the box volume, filled with 5,000 tracers in 2-D (a rectangle) and 5,000,000 in 3-D (an elongated bar as in Figure 1) in order to preserve the tracer density. We divide the box into a 20:10 grid in 2-D and 20:20:10 in 3-D for the box counting. [12] Figure 4 shows that the clouds of tracers are entirely
dispersed into 100% of the boxes, whatever the heating mode or geometry. The calculated dispersion times in 3-D can be a factor of 2 shorter than in 2-D if the bar is horizontal. For a vertical bar, the dispersion time is identical
to that of the 2-D experiment. In 2-D the initial tracer cloud is trapped within a convection cell. To be dispersed, mixing between neighboring convection cells has to take place. It is the same for the vertical bar in 3-D but not for the horizontal bar which initially cuts through several convection cells. Hence, cross-cell mixing dominates in 2-D and 3-D with the vertical bar, and mixing within cell dominates with the horizontal bar. Schmalzl et al. [1996] proposed that intra-cell mixing is more efficient than cross-intra-cell mixing. It explains the shorter dispersion times for the horizontal bar of tracers.
[13] The dispersion time td is inversely proportional to
the rms velocity of the flow [Schmalzl et al., 1996]. Hence the dispersion time is also inversely proportional to the Lagrangian strain rate. The ratio of the mixing time over the dispersion time is:
tm td / log_ea 0ð Þ 2 Di : ð7Þ
Taking Earth-like values shows dispersion in the mantle is always faster than mixing. Therefore, a heterogeneity can be ubiquitous in the mantle before being erased by diffusion.
5. Discussion
[14] Before this study, mixing in 3-D convection was
supposed to be inefficient, depending mostly on the pres-ence of a toroidal field which can only be generated in a convective flow with lateral viscosity variations [Ferrachat and Ricard, 1998; van Keken and Zhong, 1999]. We show here that in isoviscous 3-D convection, mixing is as effective and of the same type as in 2-D. The different geometry of boundary layer instabilities between 2-D and 3-D does not modify the efficiency nor regime of mixing. The apparent contradiction between this study and [Schmalzl et al., 1996] alternatively points out the fact that 2-D calculations over-estimate the size of heterogene-ities since they implicitly assume heterogeneheterogene-ities are infinite in the third dimension. Dispersing a 3-D blob Figure 3. Lagrangian strain rate as a function of the
Rayleigh number for internally heated convection.
Figure 4. Fraction of the boxes containing at least 1 tracer as a function of adimensional time for dispersion experi-ments in 2-D and 3-D. (left) Internal heating at Rai = 107
and (right) basal heating at Rab= 10 7
would take a longer time than a 2-D one having the same cross section.
[15] We can use the scalings derived above to obtain the
mixing time as a function of potential temperature T as long as the mantle is convecting very vigorously (Ra > 107), which is the case in the early Earth. The Lagrangian strain rate is: _e Tð Þ ¼ _e T0 ð Þ T T0 m Tð Þ m Tð Þ0 2=3 ð8Þ
where T0is present-day potential temperature (1600 K), the
viscosity m being a function of the temperature T and activation energy E:
m Tð Þ / exp E RT
: ð9Þ
The value of _e(T) is a lower bound since on Earth the toroidal component is comparable to the poloidal at the surface [Cˇ adek and Ricard, 1992; Lithgow-Berteloni et al., 1993].
[16] We proposed above a present-day Lagrangian strain
rate of5 1016s1and we choose an activation energy of 500 kJ mol1. From equations (8) and (9), the mixing time in a 200 K hotter mantle which probably corresponds to that of the Archean [Grove and Parman, 2004], is about 10 times faster than today. The calculated lifetime of thermal heterogeneities is reduced to 20 Myrs and chemical heterogeneities should not survive more than 90 Myrs. The evidence of strong fractionation in the Hadean mantle which survived 0.5Gyrs [Albare`de et al., 2000; Boyet et al., 2003; Caro et al., 2003] suggests either that convection was not much vigorous than today, or that a compositional heterogeneity associated with density and viscosity con-trasts [Jellinek and Manga, 2002] already existed in the young Earth.
[17] Acknowledgments. The authors thank M. Jellinek and M. Manga for constructive reviews, P. Allemand, F. Dubuffet and Y. Ricard for useful discussions. This project was funded by DyETI-CNRS. References
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J. Schmalzl, Department of Geophysics, University of Mu¨nster, Corrensstrasse 24, D-48149 Mu¨nster, Germany.
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