Polar wandering in mantle convection models

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Polar wandering in mantle convection models

M.A. Richards, H.-P. Bunge, Y. Ricard, J.R. Baumgardner

To cite this version:

M.A. Richards, H.-P. Bunge, Y. Ricard, J.R. Baumgardner. Polar wandering in mantle convection

models. Geophysical Research Letters, American Geophysical Union, 1999, 26 (12), pp.1777-1780.

�10.1029/1999GL900331�. �hal-02046747�

GEOPHYSICAL RESEARCH LETTERS, VOL. 26, NO. 12, PAGES 1777-1780, JUNE 15, 1999

Polar wandering in mantle convection models

M.A. Richards

•, H.-P. Bunge

2, Y. Ricard

3, and J.R. Baumgardner

4

Abstract.

We calculate polar motion in models of 3-D spherical

mantle convection at Rayleigh numbers up to 10 s which

include internal heating, radial viscosity variations, and an endothermic phase change. Isoviscous models yield

rapid polar motion of order 3ø/Myr, but a factor of 30

increase in viscosity with depth reduces the rate of po-

lar motion to about 0.5ø/Myr due to stabilization of

the large-scale pattern of convection. Avalanching due to an endothermic phase change causes pulsating iner- tial interchange polar excursions of order 80-110øand of duration 20-70 Myr. A layered viscosity model with an endothermic phase change yields only one inertial inter- change event in 600 million years. These models show that the slow observed rate of post-Paleozoic true polar wander is not incompatible with higher rates inferred

for earlier times.

Introduction

Motion of the Earth's rotation axis determined from

paleomagnetic measurements is called polar wander, and apparent polar wander refers to the motion of the rotation axis with respect to continental plates. "True

polar wander" (TPW) is defined with respect to some

global reference frame, e.g., hotspots, and may repre-

sent motion of the rotation axis with respect to the

deep mantle [Jurdy, 1981]. On timescales long com-

pared to the adjustment time for the Earth's rotational

bulge (of order 1-10 Myr) the position of the rotation axis will coincide with the maximum inertia axis of the

non-rotating Earth [Gold, 1955; Goldreich and Toorare,

1969], so the position of the rotation axis will evolve on

timescales characteristic of the evolution of large-scale

(harmonic degree 2) mass heterogeneities. If the maxi-

mum and intermediate inertia axes become equal the ro- tation axis may become at least momentarily unstable,

leading to rapid polar wander [Fisher, 1974], with the

TPW rate controlled by equatorial bulge adjustment. This process is called "inertial interchange" TPW, re- flecting a possible 90øshift in the rotation axis.

Post-Paleozoic rates of TPW have been small (typi-

cally less than lø/Myr) [Gordon, 1987], but two Pale-

ozoic episodes of rapid apparent polar wander of con- tinental blocks have been described as possible TPW

•Geology &: Geophysics, Univ. of California, Berkeley

2Geological Sciences, Princeton University

a Lab. de Sciences de la Terre, ENS-Lyon, France 4Theoretical Division, Los Alamos National Laboratory

Copyright 1999 by the American Geophysical Union.

Paper number 1999GL900331. 0094-8276/99/1999GL900331 $05.00

events. Meert et al. [1993] identified Cambrian (580-550

Ma) and Devonian (415-379 Ma) intervals in which Lau-

rentia and parts of Gondawana moved as rapidly as 16

cm/yr in a paleomagnetic reference frame. Van der Voo

[1994] suggest that rapid Late Ordovician- Late Devo-

nian movement of Laurentia, Baltica, and Gondwana may represent about 75øof TPW in less than 75 Myr.

Torsvik et al. [1996] suggest a more rapid phase of TPW

during late Silurian-Early Devonian time. Kirschvink

et al. [1997] postulated an Early Cambrian inertial in-

terchange TPW event (90øshift), with apparent plate

motions as high as as 30 cm/yr. Evans [1998] suggests

that the Cambrian through Devonian TPW events rep- resent coaxial shifts of the rotation axis in response to a convection pattern associated with the Rodinia super-

continent. Kirschvink et al. [1997] speculate that ex-

traordinary rates of speciation during the Early Cam- brian resulted from environmental pressure generated by a TPW-induced global shift of the continents with respect to the poles. These inferences are controversial,

and have been challenged by Torsvik et al. [1998].

TPW has been modeled using subduction history

[Richards et al., 1996] or advection of mass anoma-

lies inferred from seismic tomography [Steinberger and

O'Connell, 1997] to infer the history of mantle het-

erogeneity during Cenozoic and Mesozoic time. Since global plate motions cannot be determined reliably for earlier times, other approaches are necessary to study longer-term rotational dynamics relevant to the Paleo-

zoic observations.

In rotational dynamics the timescale(s) for evolu-

tion of thermal mass heterogeneities in convection inter- act with the timescale for rotational bulge adjustment, which itself depends not only on the viscosity struc- ture of the mantle, but also on the strength of the mass

heterogeneities [Ricard et al., 1993]. Here we combine

3-D spherical mantle convection models with solutions for the equations for rotational dynamics in a viscous planet. The convection models are described in detail

elsewhere [Bunge et al., 1996; 1997]. The theory for

rotational dynamics is given by Ricard et al. [1993].

Convection and Polar Motion Models

Four mantle convection models [Bunge et al., 1997]

are used to study polar motion. These models have in- ternal heating, isothermal conditions at the free surface, and insulating conditions at the core-mantle boundary.

We vary the lower/upper mantle viscosity ratio (1 or

30) and the phase buoyancy

parameter (0 or -0.112)

characterizing an endothermic phase change at 670 km

depth [Christensen and Yuen, 1985]. The other physical

parameters of the models (see Table I of Bunge et al.,

1997) yield a "surface"

Rayleigh

number

of 1.1x10

s for

all the models, based on a viscosity of 2.0x1022 Pa-sec

for the upper mantle. The volume-averaged Rayleigh number is lower for models with a high viscosity lower mantle. Temperature fields and. spherical harmonic het-

1778 RICHARDS ET AL.- POLAR WANDERING AND MANTLE CONVECTION

erogeneity spectra for the four models are shown in Fig. 1. All the models are time-dependent and have been run to secular equilibrium.

The rotation axis position is computed with respect to the model reference frame, which is entirely arbi- trary. We solve the Euler equations for conservation of angular momentum subject to viscous adjustment of

the rotational bulge and dynamic compensation of in- ternal mass anomalies by deformation of the Earth's

surface and core-mantle boundary [e.g., Richards and

Hager, 1984]. The bulge adjustment time is inversely

proportional to the strength of forcing, i.e., the differ- ence between maximum and intermediate inertias, so

the character of thermal heterogeneity (i.e., the "style"

R erenee, Ra=lOexp8

e)

Power Spectrum

+phas, chang, (670km

+ layered viscosity

30x

+ ph, se chan:, & layered vise,

sity

z•os oo 0 8 •6 2• 32. 0 0 8 16 24 32 o 8 •6 24 32. 0 8 16 2'4 32

Spherical Harmonic Degree

Figure 1. (a-d) Temperature

fields

for the convection

models

(red=hot, blue=cold). (a) Isoviscous

model

at 10

s

Rayleigh

number. (b) Isoviscous

phase

change

model. (c) Layered

viscosity

model. (d) Layered

viscosity,

phase

change

model (e-h) Harmonic

spectra

of the adjacent

convection

models

(see

Bunge

et al., 1997).

RICHARDS ET AL.' POLAR WANDERING AND MANTLE CONVECTION 1779

of convection) affects the rotational dynamics, as well

as the radial viscosity structure.

We adjust model time to Earth time by scaling the

rms surface

velocity

to that of the Earth's plates (4.7

cm/yr). The isoviscous

model discussed

below yields a

surface rms velocity of 0.22 cm/yr, so we contract the

model run time of 4800 Myr by a factor of 0.22/4.7 to

yield an Earth time of 250 Myr. Likewise, model ro- tational adjustment is appropriate for an upper mantle

viscosity

of 2.0x1022(0.22/4.7)

= 0.9x1021

Pa-sec,

con-

sistent with estimates from post-glacial rebound. Results

The uniform viscosity mode] of Fig. la is character-

ized by downwe]ling plumes and a "blue" heterogene-

ity spectrum dominated by structure at high harmonic

degrees (Fig. le). Fig. 2a shows the position of the

rotation axis in angular distance relative to the final pole position. Polar motion is rapid, typically of order

3ø/Myr. Fig. 2a also gives the total angular distances

of the largest TPW excursions, which appear random

and certainly not clustered about 90 ø. Thus inertial interchange TPW does not occur in this mode]. The

reason for this, and for the rapid TPW rate overall, is

that the low amplitude

degree

2 heterogeneity

pattern

is so unstable that TPW is limited by the rotational

bulge adjustment

time, which is of order 5-10 Myr.

The isoviscous model shown in Fig. lb introduces an

endothermic

phase change

at 670 km depth (Clapey-

ron slope ff =-4 Mpa/øK, phase buoyancy parameter

-0.112) strong

enough

to induce

periodic

"avalanching"

of cold downwelling material across the phase transition

[Tackley

et al., 1994], and also a somewhat

"redder"

heterogeneity spectrum (Fig. If). The rate of TPW re-

mains very high (Fig. 2b), with large, pulsating TPW

excursions whose total angular displacements (96 ø, 85 ø,

111

ø, 94 ø, 94 ø, and 114

ø) indicate inertial interchange

events of duration 20-70 Myr and recurrence interval of

50-100 Myr. However, there is no tendency of the inter-

vening pole positions to be repeated. The TPW pulses

correspond to pulses in the surface heat flux and rms

surface velocity due to phase change avalanches.

A more "Earthlike"

convection

pattern

dominated

by

long-wavelength structure and long-linear downwellings

results

from increasing

the lower mantle viscosity

by a

factor of 30 [Bunge

et al., 1996],

(Figs. lc,g). The vis-

180 , , • Isoviscous I (:]3 150 •- •' 120•- X

<

_Pha

_ -"

96

ø

'-

"%

85

-'

111

ø 94

ø'•"

94

ø'-

-•

14•-

I

Time (millions or years)

80 i i i i i i

Layered Viscosity + Phase Change

5O 7 o 120 90 60 30 (d) o -700 -600 -500 -400 -300 -200 -1 O0 0 Time (millions of years)

Figure 2. Polar wandering

with time zero the end of the model

run. Vertical axis gives

the latitudinal

pole

position relative to time zero calculated in the model reference frame. Total angular distances for individual TPW

excursions

are given

for the bracketed

time intervals.

(a)-(d) correspond

to the models

of Fig. 1 (a)-(d). Model

times scaled

to Earth time by factors

of (a) 0.045, (b) 0.068, (c) 0.064, and (d) 0.070 (see

text).

1780 RICHARDS ET AL.: POLAR WANDERING AND MANTLE CONVECTION

cosity increase yields much slower rates of TPW (Fig.

2c), typically of order 0.5ø/Myr, with no inertial in-

terchange events, although the time series duration of

550 Myr is admittedly short (limited by computing re-

sources). The bulge adjustment time for this model is of

order 10-20 Myr, or only about a factor of two increase

over the isoviscous models. The lower TPW rate re-

sults from a more stable degree 2 heterogeneity pattern induced by the viscosity contrast.

The final model (Figs. ld,h) includes both layered

viscosity and a phase change, and is similar in charac- ter to the previous layered viscosity model. Avalanch- ing is largely suppressed by the viscosity increase with

depth. TPW rates (Fig. 2d) are similar to those ob-

tained for the previous layered viscosity model, with one rapid TPW excursion of 79øapproximating inertial interchange.

Discussion and Conclusions

Isoviscous models yield rapid TPW rates due to in- stability of the degree 2 heterogeneity pattern. TPW is limited by an effective bulge adjustment time of or- der 5-10 Myr. The isoviscous model with phase-change avalanching results in inertial interchange events, demon. strating that such events can occur in coupled convec-

tion/rotational dynamics models.

Layered viscosity models yield smaller TPW rates of

order 0.5ø/Myr due to greater stability of the long-

wavelength structure. One inertial interchange event occured in the layered viscosity phase-change model, suggesting a frequency of one or two events per billion years, but longer model runs are needed to better de- fine this frequency. Layered viscosity models are more "Earthlike" than the isoviscous models, but greater re- alism would require models with plates, which further

stabilize the low-degree structure [Bunge and Richards,

1996].

Rapid TPW events may have durations of order 20- 70 Myr, but pole excursions may not cluster around 90 ø , depending on the nature of convection. Approxi- mate inertial interchange TPW represents a plausible, albeit infrequent, mechanism for rapid continental mo- tions relative to the poles, and for rapid global envi- ronmental change. The low rate of TPW observed for the past 200 Myr is not incompatible with rapid TPW

events inferred for Paleozoic or Precambrian time.

Acknowledgments. We thank D. Evans, J. Kirschvink,

and T. Torsvik for helpful comments. This research was sup-

ported by NSF and IGPP-Los Alamos.

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H.-P. Bunge, Dept. of Geology and Geophysical Sciences,

Princeton Univ., Princeton, NJ 08544

Y. Ricard, Laboratoire de Sciences de la Terre, Ecole Normale Superieure, 69364 Lyon, France

J. Baumgardner, Theoretical Divsion, Los Alamos National Laboratory, Los Alamos, NM 87545

(received November 12, 1998; revised January 25, 1999; accepted February 9, 1999.)