Comment [on “Polar motions excited by a convecting viscous mantle”]

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Comment [on “Polar motions excited by a convecting

viscous mantle”]

Y. Ricard, R. Sabadini, G. Spada

To cite this version:

Y. Ricard, R. Sabadini, G. Spada. Comment [on “Polar motions excited by a convecting viscous

mantle”]. Geophysical Research Letters, American Geophysical Union, 1993, 20 (22), pp.2495-2496.

�10.1029/93GL02687�. �hal-02046768�

GEOPHYSICAL

RESEARCH

LETTERS,

VOL. 20, NO. 22, PAGES

2495-2496,

NOVEMBER

19, 1993

COMMENTS ON

"POLAR MOTIONS EXCITED BY A CONVECTING VISCOUS MANTLE" BY J. MOSER, D. A. YUEN AND C. MATYSKA

Yanick Ricard

Departement TAO, Ecole NormMe Sup6rieure, Paris, France

Roberto Sabadini and Giorgio Spada

Dipartimento

di Fisica,

Settore

Geofisica,

Universit•

di Bologna,

Italy

In the last few years there has been a renewed inter-

est in the problem of true polar wailder (TPW) induced by slowly varying mass redistribution [Sabadini and Yuen,

-1989• Ricard et al., 1992, 199:3; Spada et al, 1992; Moser et al., 1992]. One of these papers, hereafter called MYM

[Moser

et al., 1992]

disagrees

with our own

results

(RSS)

on three different points.

- The amplitude of the non-diagonal inertia perturbations

that should be considered

- The importance of a possible relative angular momentum

carried by the mantle

- The rotational equations for the Earth

In this comment, we want to briefly restate the basic equa- tions and to discuss the three points of disagreements.

The behavior of a deformable rotating body is controlled

by the Liouville equation that reads in the absence of ex-

ternal torques:

d '

Z(j

+

A (J + n) =0.

In ttlis equation, J is tile inertia tensor of the rotating

body and h the relative angular momentum due to mass displacement in an Earth-fixed system. With respect to a reference frame fixed in space, the Earth-fixed frame has a angular velocity w. Moreover, the inertia tensor J is itself

a function of the rotation vector and can be written:

kT(t)a•

, (wi(t)wj(t)

-

where I is the main inertia of the Earth, k7r(t) and

the tidal and isostatic Love numbers, a tile Earth's radius,

(; the gravitational constant, * the time convolution mid Iq the inertia changes due to a given geophysical process without taking into account m•y dynamic deformation.

The amplitude of inertia changes

The inertia tensor Jq is simply related to the gravity field of degree 2. Calling z the axis parallel to the axis of rotation, MYM states that the non-diagonal terms and/..• that drive the polar wander can be deduced from the S, and C• terms that appear in some geoid models

and are of order 10 -xø - 10-•iMa • where M is the Earth's

mass. MYM clearly confused Z1 and hi. The Ji1 tensor

mea•sured by geodesists, includes the rotational deforma- tion in addition to hi. Terms such as/c•'(t) and/c•(t) in Copyright 1993 by the American Geophysical Union.

Paper number 93GL02687 0094-8534/93/93 GL-02687503.00

depend on the rheologicM stratification of tile Earth and

are not measurable quantities. No direct estimation of the

I..• components can be obtained on tile basis of the ob-

served geoid. Tile only "proof" of the inertia perturbation

I:• consists of tile existing polar wander.

On a time scale larger than the time scale of viscous re-

laxation (a few 1000 yrs), the vectors J.• and w are parallel. In other words, the non diagonal terms Z..•, J=• and the co-

efficients ½:a•, .92• are equal to zero. Thus, wily are tilere

non-zero $a• and Ca• in some geoid models? Geodesists compute the geoid through modeling of satellite trajecto-

ries and they face the following alternative. They can use

a time-variable frame that follows the instantaneous rota-

tion axis. In this case, geodesists ascribe ,%.• and ½:.• to zero. Geodesists can, on tile other hand, chose an Earth- fixed reference frame. In this case, tile wandering rotation axis does not always coincide with the geographical pole. To explain the apparent motion of tile satellite trajectories

which is in fact due to the real motion of tile observatories

in the direction opposite to the polar wander, geodesists introduce time-dependent $a• and C.•. Oil the time scale of modern satellite observations, these terms are simply

proportional to time. If after 10 years of data processing they are of order 10-•øMa 2, they will be of order I0-SMa 2

in the 2090's and so on if tile same reference frame of tile

!990's is still in use.

As just observed the components of I•i cannot be di-

rectly mea•sured but can be easily modeled. As an exam- pie, the inertia perturbation due to tile sudden melting

of Laurentide corresponded to I..• = 1.3 10-SMa • (2.10 ts kg at 250 of the North pole). Oil the time-scale of mantle

convection, the inertia perturbation associated with a new

sinking slab can be of order I..• = 2.2 10-½Ma • (for a trench located at 450 of the North pole, a length of 5000 km and a slab pull of 5.10•aN/m). The non-diagonal components of Iq in any Earth-fixed reference frame can be as large as the il•ertia associated with the equatorial bulge of tile geoid, (Izi --- 5. 10-•Ma2). Our estimates of the inertia pertur-

bations due to mantle convection can be 104 times larger

than what has been considered in MYM.

Relative angular momentum

MYM claim they can define a reference frame only at-

tached to the Earth in which the mantle has an important

angular momentum. We think this frame t•as no physicM meaning. We state that in the only Earth-fixed reference

frames we can practically use, the angular mantle momen- tum plays no role.

!n a convective planet the characterization of an Earth-

fixed frame is not an easy task. In fact, the most rea-

2496 Ricard et M.' Comment

sonable choice is the reference frame in which the angular

momentum of the convective mantle is equal to zero, the

so-called Tisserand frame. As we define the TPW as the

motion of the rotation axis with respect to the Tisserand reference frame, there is no need to consider a relative an- gular momentum of tile mantle.

An alternative view can be that the lower mantle gives a

fixed reference frame (the hotspot frame) on top of which the upper mantle may have an angular momentum •,. In this case the TPW represents the motion of tile rotation axis with respect to the lower mantle only. However the

angular momentum h carried by the upper mantle is neg-

ligible. For example, the present-day global plate motion requires a global rotation with respect to the hotspots of

about v = 2 cm/yr [Minster and jordan, 1978]. Assuming

that all the upper mantle with a mass m,, is rotating, this only corresponds to h/a < m•,va/•q = 3.3 10-•aMa 2.

Long term TPW

In MYM, the polar wander velocity with components

m• in the linear approximation is (neglecting h)

d -i(lza: + iI.,.y

•("• + •"•) =

c

'

(3)

!(,,• + •.•) =

(4)

where C and A are the polar and equatorial inertia of tile

Earth, and w!•ere r characterizes the time delay of the equatorial bulge in rotational readjustment. MYM state

that (3) is valid for a young Earth with high Rayleigh num-

ber and (4) for an old Earth with low Rayleigh number.

MYM does not really address the problem of what a young

or an old Earth is.

Equation (3) is based on the assumption that the equa-

torial bulge of tile Earth does not play any role. This would be the case if tile mantle viscosity were so low that the bulge would not offer any resistance to its readjustment

during TPW, as if the Earth were a sphere. Equation (4) expresses the fact that the motion of the equatorial bulge controls the polar wandering [Sabadini and Yuen, 1989; 1%icard et al., 1992, 1993; Spada et al., 1992].

For the Earth, the time r in equation (4) is of order 104 yrs [Sabadini and Yuen, 1989] so that •'• is 74000 (c-zF times

slower

than •

This simply

means

that it is easier

by

(3) would

only

hold

for a planet

with an average

viscosity

74000 times smMler than the present. The Earth had such

a low viscosity

only

shortly

after

its formation

[Schubert

et

conclusion

that

a mantle

angular

momentum

h may

play

a role

in TPW excitation

is based

on equation

(3) which

does not apply for the Eartl•.

Conclusions

-The amplitude

of the inertia

terms

that drive

the

TPW

cannot be deduced from the geoid.

-Tile importance

of a possible

angular

n•omentum

of

the

mantle

is negligible

in tile Earth-fixed

reference

frames

in

-Tile TPW is controlled

by tile ability

of the equatorial

bulge

to move

(equation

(4)). Tile inertia

perturbations

The first two points

of our conclusion

are totally

in-

dependant of the mantle theology. We established the

last point using

visco-elastic

models

[Ricard

et al., 1992.

1993;

Spada

et M., 1992].

Tile nature

of the 670

km

depth

boundary, permeable or not to tile mantle flow, does not affect this conclusion. Tile suggestion made by MYM that

non-linear

rheologies

can

fasten

by some

4-5

orders

of mag-

References

Minster, J. B., and T. H. Jordan, Present-day plate mo-

tion, J. Geophys. Res., 83, 5331-5354, 1978.

Moser, J., D. Yuen, and C. Matyska, Polar motions excited

by a convecting viscous mantle, Geophys. Res. Left., 19,

2251-2254, 1992.

Ricard, Y., R. Sabadini, and G. Spada, Isostatic defor- mations and polar wander induced by redistribution of mass within tile Earth, J. Geophys. Res., 97, 14223-!4236, 1992.

Ricard, Y., G. Spada, and R. Sabadini, Polar wandering of a dynamic Earth, Geophys. J. Int., 113, 284-298, 1993.

Sabadini R. and D. A. Yuen, Mantle stratification and

long-term polar wander, Nature, 339, 373-375, !989. Schubert, G., D. Stevenson and P. J. Cassen, Whole planet

cooling and the radiogenic heat source contents of the

Earth and moon, J. Geophys. Res., 85, 2531-2538, 1980. Spada, G., Y. Ricard, and R. Sabadini, Excitation of true polar wander by subduction, Nature, 360, 452-454, 1992. Y. Ricard, D•,partement TAO, Ecole Normale Sup•.rieure,

24, rue Lhomond, 75231 Paris Cedex 05, France.

R. Sabadini and G. Spada, Dipartimento di Fisica, Settore di Geofisica, Universirk di Bologna, Viale Berti Pichat 8, 1-40127 Bologna, Italy.

(Received June 14, 1993; Accepted July 26, 1993)