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On A Particular Solution of the Non-Linear Liouville
Equations
G. Spada, R. Sabadini, Y. Ricard
To cite this version:
G. Spada, R. Sabadini, Y. Ricard. On A Particular Solution of the Non-Linear Liouville
Equa-tions. Geophysical Journal International, Oxford University Press (OUP), 1993, 114 (2), pp.399-404.
�10.1111/j.1365-246X.1993.tb03927.x�. �hal-02046767�
Geophys. J . Int. (1993) 114,399-404
R E S E A R C H NOTE
On a particular solution
of
the non-linear Liouville equations
Giorgio Spada,1.2 Roberto Sabadini1,3 and Yanick Ricard2
’
Dipartimento di Fisica-Sertore Geofisica, Universita di Bologna, Viale Berti Pichat 8, 40127 Bologna, Italy’
Dipartement de Geologie, Ecole Normale Supgrieure. 7523 1 Paris. France’
Istituto di Mineralogia, Universrta di Ferrara, 44100 Ferrara, ItalyAccepted 1992 December 14. Received 1992 November 19; in original form 1992 May 12
S U M M A R Y
T h e rotational behaviour of a stratified viscoelastic planet is analysed by
means of
a
quasi-analytical method. Our approach is particularlyappropriate
to study t h e long-termpolar
wander induced by internal loads, a n d in particular t o study t h e effects due t o time-dependent mantle convection. W e focuson
a simple explicit solution of t h e Liouville non-linear equations, in o r d e r t o establish t h e relationships between internal rheological constitution of t h e planet a n d polar motion. B o t h t h e rate a n d t h e direction of polar wander are found t o be extremely sensitive t o t h e mantle stratification a n d in particular t o t h e nature of the 6 7 0 k m d e p t h seismic discontinuity.Key
words: E a r t h rotation, long-term polar motion, mantle dynamics.1 I N T R O D U C T I O N
The linearized version of the Liouville equations, valid for small displacements of the earth’s axis of rotation, has been widely employed in the past to study the dynamical behaviour of the planet subject to surface forcings such as the cryospheric loadings associated with the Pleistocenic ice ages (Nakiboglu & Lambeck 1980; Sabadini & Peltier 1981; Yuen, Sabadini & Boschi 1982; Wu & Peltier 1984; Spada et al. 1992a). Recent investigations based on the same linearized scheme (Ricard, Sabadini & Spada 1992) or on the fully non-linear equations (Lefftz 1991; Spada 1992) have demonstrated the key role of the dynamic compensa- tion of internal sources for the long-term polar wander.
Three classical solutions of the non-linear Liouville equations for a homogeneous viscoelastic rotating body subject to inertia perturbations have been proposed by Airy (1860), Milankovitch (1934) and Goguel (1950). Although these problems are presented and discussed in Munk &
MacDonald (1960), we will briefly describe their main features, in order to enlighten analogies and differences with our model.
The so-called Airy-Gold problem deals with the polar wander induced by a sudden addition of superficial matter than can be produced, for instance, by a gaseous explosion or by the rise of a continential plateau (Gold 1955). The incipient solution of this problem, which is only valid for small excursions of the earth’s rotation axis, is presented in Munk & MacDonald (1960).
The Milankovitch problem consists in the computation of the displacements of the earth’s rotation axis due to the distribution of continents. The solution has been obtained without placing any restriction to the amplitude of polar wander and the final result, expressed by the ‘Milankovitch theorem’ (Munk & MacDonald 1960; Scheidegger 1958), allows the direct computation of the polar wander in a reference frame which is coincident with the principal axes of the ocean-continent system. In principle, the Milan- kovitch problem could also be extended to stratified viscoelastic Earth models.
Finally, the so-called Goguel-Fermi problem deals with the secular effects of winds and ocean currents. In this case, the source term to the Liouville equations consists in the relative angular momentum associated to the motions of the atmosphere with respect to the earth’s surface. As emphasized by Munk & MacDonald (1960), no analytical solution is known for this problem, although the incipient solution can be easily derived.
These three problems are based on a simple uniform model with Maxwell rheology, and do not consider the internal structure of the earth. In the following we will present the explicit form of a solution of the non-linear Liouville equations for a viscoelastic stratified planet, subject to inertia changes due to internal time-dependent mass distributions. Our aim in solving this basic problem is to clarify the fundamental role played by stratification and to display the main differences between surface and internal excitations in driving the polar wander.
399
400 G. Spada, R. Sabadini and
Y.
RicardThe background equations and approximations necessary in order to solve this problem have been already stated in Ricard et al. (1993) (hereafter indicated as RSS93), where a complete geophysical application has been developed, mainly by means of numerical methods. In a further study (Spada, Ricard & Sabadini 1992b) the long-standing question of how long-term true polar wander can be excited has been addressed, by analysing the effects on the earth’s rotation due to time-dependent subduction episodes.
In this research note, we prefer to focus on a simple analytical solution of the problem, in order to directly show the interplay between the different time scales involved.
2 E Q U A T I O N S OF MOTION
The Liouville equations for a non-rigid rotating body are d
dt -[Ji,(t)wj(t)l= -Eijkwj(trJkl(t)w/(t) + N,(c), (1)
where .Iij([) is the inertia tensor, mi([) the angular velocity of the rotating reference frame, N,(t) the external momentum acing o n the body and the Levi-Civita alternating tensor (e.g. Munk & MacDonald 1960; Lambeck 1980).
The total inertia tensor
.Ii,([)
can be conveniently decomposed in the following wayJ i j W = 16,
+
J f < W c ) +Jf;(c),
(2) where 16,,, J c and J : represent the diagonal inertia tensor of a homogeneous, non-rotating earth, the inertia contribution due to the deformation induced by rotation, and the inertia associated to a given geophysical process taking place in the mantle, respectively. Since the explicit forms of these tems are given in RSS93, they will not be repeated here. Nevertheless, for the sake of clarity, we remark that these expressions can be easily written in terms of the time-dependent tidal and loading Love numbers k T ( t ) and k L ( d , t ) of harmonic degree 1 = 2, which represent the impulse response of a given model earth to perturbations in the centrifugal potential and to a mass heterogeneity located at a depth d within the mantle (Munk & MacDonald 1960; Lambeck 1980).Since we are mainly concerned with mass anomalies embedded in the mantle as possible sources of polar wander, we have genealized the ‘classical’ loading Love number, valid for forcings located at the earth’s surface, to the case of internal heterogeneities (Ricard et al. 1992; Spada 1992). Moreover, since the typical time-scales of the processes which could induce long-term polar wander, such as mass redistribution due to mantle convection, are of the order of 1-10 Ma, it is possible to adopt a simplified approach to the problem, disregarding any viscoelastic relaxation taking place on time-scales shorter than 1 Ma. This approximation is known in the literature as ‘quasi-fluid’ approximation (e.g. Lefftz 1991).
Following, therefore, the way of thinking outlined in RSS93, we arrive, by substitution of eq. (2) into eq. ( l ) , to the following set of non-linear differential equations for the three components of w along the rotating reference frame
( 3 )
where we have assumed that the earth is not subject to any external torque [N,(t) = 01 and the 3 X 3 matrices A,, and B,,
are given by
(4) and
B,(w t ) = -E,,&JkL/(t)wI(c). (5
1
In the above equations the parameters are defined as follows; R is the diurnal angular velocity of the earth, k: the long-time asymptote of the tidal Love number, a the earth’s mean radius, C the universal gravity constant, I the axial inertia of a non-rotating earth (I = 0.33 Ma2, where M is the earth’s mass), and TI is a time constant. The physicalmeaning of T, can be deduced by eq. (16) of RSS93. In fact,
it can be seen that the incipient solutions of o u r equations, valid for short times, coincides with the long-term behaviour of the solutions obtained in the framework of a linearized scheme for the Liouville equations. The time T, is therefore
the inverse of what is usually called steady-state rotational residue A , , as emphasized by the subscript ‘1’ (e.g. Sabadini, Yuen & Boschi 1984; Ricard ei al. 1992).
3 A N A N A L Y T I C A L S O L U T I O N FOR P O L A R W A N D E R
As shown in RSS93, the time constant T, is several orders of magnitudes larger than the average length of day 2z/R (see also Table 1). According to this finding, the diagonal elements of the matrix A,, can be completely neglected with respect to the off-diagonal ones, as it can be seen by direct inspection of eq. (4). Of course, this approximation leads to a singular A,, and therefore it is not possible to retrieve
dw,ldt by eq. ( 3 ) , unless some additional condition on w, is prescribed. In this paper we adopt the same point of view as Lefftz (1991), prescribing the conservation of the norm of w
during the polar motion. In other words, this approximation allows us to deal only with excursions of the vector w ,
neglecting any time evolution of its length.
We choose, for the sake of simplicity, a geophysical
Table 1. Properties of the earth models employed.
Model d q z / q ~ 1
+!$
TI T M I k i m( k m ) (kyr) MY^) (kyr-’)
0.00 1.08 0.18 -9.84 x 0.00 23.38 1.07 1.25 x l O W 5 (a) 0 50 ( b ) 200 1 -0.31 x lo-’ 1.08 0.18 3.07 x lo-* (b) 200 10 1.27 x lo-’ 5.86 0.58 4.41 x (b) 200 50 2.13 x lo-’ 23.38 1.07 -1.05 x lo-’ ( c ) 670 1 0.00 1.08 0.18 1.46 x ( c ) 670 10 0.00 5.86 0.58 1.54 x lo-‘ (c) 670 50 0.00 23.38 1.07 -3.21 x lo-‘ (d) 1600 1 -4.42 x lo-* 1.08 0.18 3.17 x (d) 1600 10 -2.81 x lo-’ 5.86 0.58 2.15 x lo-* (d) 1600 50 -3.71 x lo-’ 23.38 1.07 -3.48 x lo-‘ (4 0 1 (a) 0 10 0.00 5.86 0.58 -9.04 x
Non-linear Liouuille equations 401 process described by a n inertia perturbation of the form
C,,(t) = C , y w , (6)
where H ( t ) is the Heaviside step function and
(7)
A n inertia tensor of this form, already employed by Ricard et al. (1992), is adequate to model a mass heterogeneity 6m
located in the x I x , plane of a Cartesian geographic reference frame. W e assume that, before any inertia perturbation starts t o act, the angular velocity of the earth is directed along the x3 axis. In the following, we assume a mass 6m = 2 X 10'" kg at a latitude of 45", a longitude of 0" and at a given depth d inside the mantle. T h e mass anomaly is comparable with the total mass of the Pleistocenic ice sheets a t their maximum extension (e.g. Sabadini, Yuen &
Boschi 1982).
According to RSS93, J k ( t ) can be expressed as
J ; : ( f ) = I1
+
k;-(d) - k M l ( d ) ~ M l exp (-t/7Ml)lCG, (8) where k , , , 7M1 and ki- denote the amplitude of the ' M l ' viscoelastic mode, its characteristic relaxation time and the long-time asymptote of k L ( f ) , repsectively. T h e isostatic A41 mode appears in the relaxation spectrum of incompressible earth models characterized by a fully non-adiabatic density jump in the mantle (Wu 1978; Peltier 1974; Peltier 1985; Ricard e f al. 1992). Since the relaxation timer M l
may be comparable with the time-scales of mantle convection, it cannot be ignored in our investigations. We observe that both k,, and kf" depend o n the depth of the mass anomaly,d. Values for the parameters k , , , ,,z and kj: are given in Table 1 for the various earth models employed in this study.
Substitution of eq. (4), ( 5 ) and (8) in eq. (3) gives
m2m, - m,m2 = m I m2
7-1
m:
+
m: +rn; = 1, (9c) where we have introduced the adimensional quantitiesm, = w,/Q and the dot denotes the derivative with respect to
time. Eq. (9c) states the invariance of the norm of w and the time-dependent function u ( t ) is defined by
Direct substitution of eq. (10) in (9a-b) and some simple algebra gives the formal solutions
r 9 r l 1
( l l a )
where mi, denotes the components of m, a t time t = 0. Let us assume now that the earth has been spinning around the x, axis at constant angular velocity w = (0, 0, Q) up to time t = 0, when some geophysical process, described by eq. (8), perturbs its rotation. In this particular case, the initial conditions to be imposed t o eq. ( l l a - b ) are simply
rn,,=m,,=O and m3"= 1. As expected, eq. ( l l b ) gives immediately m 2 ( t ) = 0. This means that the earth's axis of
rotation, excited by the inertia perturbation C,,(Z), wanders in the x , x 3 plane and tends to align itself t o the new axis of
maximum inertia. Eq. ( I l a ) gives
= tanh
[
l : u ( t ' ) d t ' ] ,4f)
or, making use of eq. (10)
where we have introduced the angle 6 between the former and the new direction of the rotation axis. T h e constant A is defined a s
3GC" C"
A = - - - -
k , a Q C - A '
where C and A denote the maximum and minimum of the inertia terms of the earth, respectively, and where we identify kfTa'Q2/3G with C - A (Munk & MacDonald 1960).
T h e order of magnitude of A can be easily estimated. A mass 6m = 2 X 10'" kg located o n the earth's surface at a latitude of 45" is associated with a n off-diagonal inertia perturbation c" = - i 6 m a 2 . Since C - A = 1.08 X lo-' Ma2 (e.g. Lambeck 1980), we get A
-
1.5 X lo-, from eq. (14).4 D I S C U S S I O N
Equation (13) describes the behaviour of the earth's axis of rotation perturbed by the time-dependent inertia expressed by eq. (8). To clarify the geophysical meaning of the following discussion, we note that the function 1
+
k f . ( d ) is simply proportional t o the steady-state geoid Green's function of harmonic degree 1 = 2. A theoretical analysis of the geoid in connection with the study of the rotation of the earth has been recently performed by Ricard et al. (1992) in the framework of the linearized Liouville equations. Since the sign of the geoid anomaly detected at the surface of the earth is intimately related to the viscosity profile assumed in the mantle and t o the position of the mass heterogeneity relative t o chemical discontinuities, we expect t o obtain qualitatively different pole excursions for distinct mantle stratifications.We now describe the three different possible time evolutions displayed by eq. (13), according t o the values assumed by the depth-dependent function 1
+
k j - ( d ) . The elastic parameters and density profile adopted in the following calculations are presented in Table 2.402 G . Spada, R. Sabadini and Y . Ricard
-
0.5- b,
0,
Table 2. Elastic parameters and density profile of the earth model employed. The parameters a, b and c denote the earth radius, the upper-lower mantle interface and the core-mantle boundary, respectively. The densities ( p ) and elastic shear moduli ( p ) of the upper mantle, lower mantle and of the inviscid core are denoted by the subscripts U M , L M and C , respectively.
Parameter Value (km) Paramrter Value ( k g / m 3 ) Parameter Value ( N / m 2 )
a 6371 P U M 4300 ~ U M 7.0 x 10'O 5701 P L M 4800 2.5 x 10" b c 3480 PC 10925 ILC 0
(4
d = O k m
19
50
4.1 Perfect isostasy: k;(d)+
1 = 0A loading Love number k i - ( d )
+
1 = 0 reflects the perfect isostatic compensation of the mass anomaly imposed at the depth d. In this case, by the same definition of loading Love number, the perturbed gravity potential at the earth's surface is zero in the asymptotic fluid regime (e.g. Munk &MacDonald 1960; Ricard et al. 1992). Such a kind of
compensation takes palce for density anomalies located close to a non-adiabatic density contrast within the mantle. In particular, this happens for mass heterogeneities placed at the earth's surface, at the 670 km depth transition and at the core-mantle boundary. It is noteworthy that in the presence of an elastic lithosphere the condition k F ( d )
+
1 = 0 is not exactly attained at the surface because of the small stress frozen in this layer even in the asymptotic fluid regime (Wu & Peltier 1982; Wolf 1984; Ricard et al. 1992). In thefollowing computations the lithosphere will not be included. Let us denote by 19" the angular displacement of the axis of rotation in the particular case kF(d)
+
1 = 0, From eq. (13) we obtain the asymptotic displacement.bi
9
4
W -5 - -10Since a positive density anomaly (and therefore a negative inertia perturbation CO) is associated with a negative A (see eq. 14), we conclude from (15) that the pole wanders towards the mass anomaly if k M M l ( d ) > 0 and away from it if k,,(d)<0. The values of k M I ( d ) and of the other parameters which enter to play into eq. (16) and (18) are given in Table 1, where each entry refers to one of the curves drawn in Fig. 1.
In this figure we show the displacement 19 of the earth's axis of rotation driven by a positive mass heterogeneity located at the surface (panel a) or at the compositional 670 km depth discontinuity (panel c). The angle I9 is given in degrees. Our earth model includes an inviscid core and a two-layer viscoelastic mantle with linear Maxwell rheology. We compute the pole displacement for different values of the viscosity contrast q2/v across the lower-upper mantle
interface. In these two panels we observe the relatively small amplitude of 19, of the order of 1". The transition to the asymptotic regime given by eq. (15) is governed by the relaxation time of the M1 mode, ,.z, Of course, similar results are to be expected for mass heterogeneities located close to the core-mantle boundary.
o - ' - I t I
50
10
, , , , , , ( , , 1 , , , , 1 , , , , 1 , , ( , - 1 . 0 1d = 2 0 0 k m
I
I
d = 6 7 0 k m
- 577
2 / q 1=1*
d = 1 6 0 0 k m
- 5 0 , , , , , , , , , , , , , , , , , , , , , , , , 0 10 20 40 50t(MajO
Figure 1. Angular displacement d of the earth axis of rotation induced by a density heterogeneity located at the surface of the earth (panel a) and at a depth of 200, 670 and 1600 km (panels b, c and d, respectively). The parameter
v2/vl
denotes the ratio between the upper and lower mantle viscosities. The mass anomaly is turned on at time t = 10 Ma and its amplitude is kept constant thereafter. For masses located in the proximity of a compositional discontinuity (panels a and c) the pole wanders for a relatively short time and the final displacement is of the order of a few degrees. In panels (b) and (d) the asymptotic displacement is 45" and both the rate and direction of polar motion depend on mantle stratification (see Table 1).Non-linear Liouville equations 403 4.2 Negative geoid anomalies: k:(d)
+
1 < 0Large excursions of the earth's axis of rotation are now allowed. By eq. (13) we obtain immediately
K
l y ( m ) = - 4 '
where if-(=) denotes the final angular displacement of the pole for kfL(d)
+
1 <O. In this case the rotation pole of the earth wanders toward the mass heterogeneity. This is shown in panel (b) of Fig. 1 for q J q , = 1 and in the whole set o f curves of panel (d). In the latter case, because of the relatively largz value of 1 - kfL(d) (see Table l), the rate ofpolar motion is sensibly larger than in (b) and the earth is quickly rotating by 45".
4.3 Positive geoid anomalies: $ ( d )
+
1 > 0This occurs in the presence of a stiff lower mantle where
k i - ( d )
+
1 > 0. Eq. (13) gives, independently from the sign ofk M , ( 4
7r
S+(cc) = -- 4 '
where we denote with 8 + ( ~ ) the limit of if for long times. The condition k j : ( d )
+
1 > 0 indicates that the gravity perturbation induced by the mass anomaly embedded at the depth d has the same sign of that produced by the mass itself (Ricard et al. 1992). The rotational behaviour in this case is strongly different from the previous situation, as depicted by the curves relative to v 2 / q , = 10 and q 2 / v , = 50 in panel (b) of Fig. 1. The earth's axis of rotation wanders away from the mass anomaly, approaching a direction at 45" from the initial rotation axis.5 CONCLUSIONS
In this note we have studied by means of simple analytical methods, the effects of time-dependent internal mass distributions o n the long-term rotational behaviour of the earth. W e have emphasized the role played by the dynarnical compensation of the mass heterogeneities and the different possible evolutions of the direction of the earth's rotation axis. O u r analysis has shown that the axis of rotation wanders towards a geoid low and away from a geoid high and that this dynamical process may take place on time-scales of several million years. Despite the extreme simplicity of the model proposed, we have demonstrated that the efficient masses which may drive long-term polar wander cannot be associated with surface features, as suggested by Goldreich & Toomre (1969), but rather t o density instabilities due to chemical or thermal processes taking place within the mantle. Furthermore, differently from the quasi-rigid model proposed by the same authors, the axis of rotation of our model earth aligns itself to the axis of maximum inertia only after a considerable time, which depends o n both the earth's mechanical structure and on the position of the density heterogeneity.
ACK NOW L E D GM E N T S
This work has been supported by the SCIENCE programme
of the European Economical Community N. SC1*0456. We thank Detlef Wolf for careful review and suggestions. We are indebted t o Ondrej b d e k for careful discussions and to Massimo Bacchetti for technical assistance.
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