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Core-mantle coupling and viscoelastic deformations
•
r,
Ptt'~ S I~-'S ()J:TtI[ [ A R T H , \ N I ) PLA NETAR'~ INI[RIORS
ELSEVIER
Physics of the Earth and Planetary Interiors 90 (1995) 115-!35
Core-mantle coupling and viscoelastic deformations Marianne Greff-Lefftz *, Hilaire Legros Institut de Physique du Globe, 5, rue Ren~ Descartes, 67084 Strasbourg Cedex, France
Received 27 September 1993; accepted 22 April 1994
Abstract
Writing the angular momentum theorem for the Earth and for its fluid core, we show that there are couplings between the core and the mantle induced by viscomagnetic torque, by external active torque, by topographic torque acting at the core-mantle boundary (CMB) but also by viscoelastic deformations of the CMB which may perturb the axial rotations of the Earth and of the core. We compute these deformations at the CMB induced by the Pleistocenic deglaciation. The time-dependence of inertia tensor perturbations, i.e. the rheology of the mantle, is very important in the calculation of the coupling. Taking into account the passive viscomagnetic torque of tangential traction acting at the CMB, we investigate, for different values and various temporal evolutions of the topographic torque, the perturbations in the rotations of the Earth and of the core induced by the deglaciation, by the constant torque of tidal friction and by the 18.6 year tidal potential. We show that, for these excitation sources, the existence of a constant topographic torque involves the core oscillating with respect to the mantle and thus forbids any large drift of the core with respect to the mantle. However, it seems theoretically possible to have an excitation source with enough energy which involves a shift of the core with respect to the mantle. If the pressure within the fluid core varies with time, the motion of the core with respect to the mantle could be drastically different.
I. Introduction
The rotation of a viscous planet has often been considered, in order to investigate secular variations induced by the last deglaciation (e.g. Munk and McDonald, 1960; Peltier, 1974; Spada et al., 1992). On the other hand, the influence of the c o r e - m a n t l e coupling (viscomagnetic and topographic torque) has been studied by geomagneticists for a rigid mantle (for a review see Jault, 1990). In this paper, we simultaneously take into account the viscoelasticity of the mantle and the
Corresponding author.
different couplings which appear at the c o r e mantle boundary (CMB) because of the magnetic motions within the fluid core, in order to investigate the perturbations of the axial rotations of the Earth and of its core induced by various external sources. The axial rotations of the Earth and its fluid core depend on the exciting sources, i.e. on the resulting variations of torques and inertia tensors, and of couplings acting at the c o r e - m a n t l e interface. We analyse the relative influence of these couplings on the rotations of the mantle and of the core and especially the fundamental role played by the topographic torque on these rotations when the variation of the length of the day
0031-9201/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSDI 0031-9201(95)05078-7
116
M. Greff-Lefftz, H. Legros / Physics of the Earth and Planetary Interiors 90 (1995) 115-135
results from a surface load (owing to Pleistocenic deglaciation), from the torque of tidal friction or from the 18.6 year tide.
pC the density of the homogeneous fluid core, H~ may be written HC-3 - e'3 fcorepcTA ( ~ ' + d A Y )
2. Theoretical approach
2.1. Axial angular momentum of the Earth and its fluid core The equations governing the variations of the rotation velocities for a planet having a homogeneous fluid core with differential motions with respect to a viscoelastic mantle are the equations of conservation of angular momentum. In our earth model, we must write on one hand the conservation of the angular momentum of the global Earth (core + mantle), and, on the other hand, the one for the core, in the Tisserand frame of the mantle. We denotes 12 the mean axial rotation of the Earth: ~ = 12~3, where e'3 is the axial axis. The axial rotation of the global Earth is f~ + o93 where o93/~ is small with respect to unity. Den o t i n g C33 the perturbations of inertia moment of the Earth (small with respect to the mean Earth inertia moment C) owing to the deformations and to the change in mass distributions, the axial component of the angular momentum of the Earth can be written n 3 = C(~
+ 0)3) -'t- ~'~C33
h( O, q~) = ~
where V is the radius vector and Pff the associated Legendre functions, contribute to the axial component of the angular momentum of the core H 3" ¢ In the Tisserand frame of the mantle, noting
( h'2 cos m ~ + g 2 sin mqQ
n=0 m--0
× P ~ ( c o s 0)
(3)
where 0 and q~ are, respectively, the colatitude and the longitude. In a linear approximation (hm/b and g ~ / b << 1), we obtain corePC~A U' d v
7 3]
~ n(n + 1) + 4~0Cb4 ~ to an + 1 F/odd
(n - 1)(n - 2 ) ( n + 2) X (2n - 1 ) ( 2 n - 3 ) ( 2 n - 5)
The fluid core, with respect to the mantle, does not have any quasi-rigid rotation but a combination of various motions: these last ones may be inferred at the CMB from the secular variations of the magnetic field of the Earth, using the geostrophic approximation (Jault, 1990; Gire and Le Mou~l, 1990; Hulot, 1992). Only the zonal toroidal motions of the core with respect to the mantle, noted
n=l
(2)
It obviously depends on the velocity field ~" and on the shape of the interface characterized by the vector V(topography of the CMB). This topography at the CMB (Morelli and Dziewonski, 1987), noted h(O, ~), small with respect to the core radius b, is introduced and expanded in spherical harmonics
(1)
~= - r ~_, t°(F'A VP°)
dt,
+
+
(n - 1)(n + 2 ) ( n + 3)
ho n-3
h°
--n 3 + 6n z + 5n -- 15 ( 2 n - 3 ) ( 2 n - 1)(2n + 3) n 3 + 9n 2 + 10n - 13 (2n-
1)(2n + 3 ) ( 2 n + 5)
h°, h° n+a
] (4)
which may be written, for simplicity 8rr
-igp-Cb5o9c, = cc09 ) where 0)~ is an equivalent rigid rotation. We add also some contributions resulting from the perturbations of the core inertia moment C~3
M. Greff-Lefftz, H. Legros / Physics of the Earth and Planetary Interiors" 90 (199.5) 115-135
(related to the degree 2 deformations at the CMB) small with respect to the mean moment of the core C c. Consequently, the Euler linear equations governing the axial rotation variations may be classically written in the Tisserand frame of the mantle - - for the global Earth r~CtbC = L 3
(5)
for the core CC(tb3 + o5~) + ~qC~3 = Lc3 + F~,
(6)
where L 3 is an external torque acting at the Earth's surface (e.g. the tidal torque), F~ is a passive viscomagnetic torque of tangential traction acting at the CMB and L~ is a topographic torque. Notice in the Euler equations that on one • hand, C~3, L c3 and F 3c play a similar role in term of excitation of w 3. On the other hand, in terms of perturbation of to~, C~3 acts as an excitation source whereas L~ and F~ are passive torques (i.e. related themself to to~). We consider these equations on time-scales from a decade to the secular variations of the geomagnetic field. On one side, on a long timescale, only the time derivative of the perturbations of the inertia tensors of the Earth and of the core owing to viscoelastic deformations (such as the ones resulting from the Pleistocenic deglaciation) or the secular tidal torque can contribute to the angular momentum budget of the core. On the other side, on the time-scale of a decade we have to take into account the elastic effects of the 18.6 year zonal tidal potential. To analyse the different terms that appear in these equations, we have to develop the various couplings that exist at the CMB and to model the viscoelastic deformations. In this paper, we point out the influence of the viscomagnetic and the topographic torque at the CMB on the variations of the axial rotation of the Earth and of the axial differential rotation of the core when the Earth is submitted to external geophysical excitations on a secular time-scale such as the Pleistocenic deglaciation and the tidal torque, or a shorter time-scale such as the 18.6 year tidal potential.
117
2. 2. External geophysical excitation sources
The excitation sources may appear in the angular momentum theorem through perturbations of the inertia tensors of the Earth C33 and of the core C~3 (i.e. deformations) or through an external torque L 3. In this section, we model first the effects of the Pleistocenie deglaciation, next those of the tidal torque on the axial rotations and finally those of the 18.6 year tide. 2.2.1. Pleistocenic deglaciation
Choosing an earth model with geometrical and physical parameters must be done before the computations of deformation resulting from a surface load. Our model consists of four incompressible homogeneous layers: an elastic lithosphere, two linear viscoelastic layers describing the mantle, and an inviscid fluid core (Table la). We use a Maxwell model of rheology for the mantle and from the elastogravitational equations in the Fourier domain, we write the viscoelastic deformations as functions of generalized Love numbers• Various Love numbers, especially k'(A) and h'C(A) (relative, respectively, to the surface mass redistribution potential and to the radial displacement at the CMB induced by a surface load) may be written in the frequency domain (Peltier, 1974)
M k'(A) = k '~ + ~
k', ih + 1/z~
and
i=1
M
h,ic
h'C(a) =h'ce+ E iX + 1/ri
(7)
i~l
The first term (with the superscript e) is the elastic Love number; k'i, h',c and r~ are the residues and the relaxation times (given in Table lb, from Lefftz et al., 1994) of the M modes of the earths' model. We compute the perturbations in the inertia moments of the Earth and of the core when the Earth is submitted to a zonal degree 2 loading potential V~o( t ) 633(t )
6(t) +k'(t)
-2a'
- f~Za2
C~3(t)
c----v -
=
~
*l/~0(t )
kf
qo h,C ( t ~
,
, , , v;,,(t)
and
(8)
118
M. Greff-Lefftz, H. Legros / Physics of the Earth and Planetary Interiors 90 (1995) 115-135
Table la (a) Geometrical and physical parameters for a four-layer model consisting of an elastic lithosphere, a viscoelastic upper mantle, a viscoelastic lower mantle and an inviscid fluid core (both rigidity and density are obtained from the PREM model (Dziewonski and Anderson, 1981)) Radius (km)
Density (kg m -3)
6221 < r < 6371 pl = 3232 5701 < r < 6221 192 = 3666 3480 < r < 5701 P3 = 4904 0 < r < 3480 P4 = 10987
Rigidity (10 II Pa)
Viscosity (1022 Pa s)
0.1=0.6114 /z 2 = 0.9169 0.3 = 2.225 0.4 = 0
b'l~ v2 = 1 v3= 5 v4 = 0
of C33/C and C~3/C c are represented, respectively, in Figs• 2(a) and (b). Today, almost 7000 years after the end of the deglaciation, it remains a perturbation of C33 (i.e. of the J2 coefficient of the geoid) which is in good agreement with the observations and of C~3 (i.e. there is a topography at the CMB of about 10 m which is decreasing in order that the deformations tend to be completely relaxed) (for more details see Lefftz et al., 1994).
2.2.2. Tidal torque where 6(t) is the Dirac distribution, a is the mean radius of the Earth, a is the flattening of the Earth, q0 is the geodetical constant and kf denotes the fluid tidal Love number. In C 3 3 ( t ) there is a combination of a direct instantaneous effect owing to the mass anomaly itself (term in 60)) and of a deformational effect function of the viscoelastic Love number k'(t), whereas in C~3(t) there is only a deformation effect. We use a simplified model for the time-dependent loading potential V~o(t) describing the last glaciation-deglaciation (Fig. 1). The glaciation is characterized by a slow build up over a period of about 90 000 years and the deglaciation vanishes in 10 000 years (Peltier et al., 1986). Taking into account the different ice sheets at the surface of the Earth (Laurentides, Fennoscandia and Antarctica) leads to a zonal degree 2 loading potential at the end of the glaciation, V~o = 235 m 2 s -2 (Spada et al., 1992). The time derivatives
We introduce an external constant torque L 3 acting at the Earth's surface which is responsible for the observed secular deceleration of the Earth owing to the lunar tidal potential• Its value is of about L 3 / C = - 6 x 10-22S -2 (for more details see Hinderer and Legros, 1990).
2.2.3. 18.6 year tidal potential We investigate the effects of the 18.6 year tide on o03 and o0~. At this frequency, the tidal potential is purely zonal (because of the Earth's rotation the other terms in the spherical harmonic expansion of the potential are quasi-diurnal, semi-diurnal...). We have shown that a zonal potential acting on the topography of the CMB cannot create any axial torque (cf. Eq. 14 hereafter). Consequently, the 18.6 year tide can con-
1C
'
I
' t ' f ' ( , End of glaciation.-~-~"~
I
~
f
0.~ 04 O?
Table lb (b) Six relaxation modes associated with the viscoelastic Love numbers for the four-layer model described in Table l(a) Mode name
Relaxation modes ( s - l)
Relaxation times (103 years)
T~
0.8997 x 10-10 0.8429 x 10-10 0.1839 x 10-10 0.4849 x 10- II 0.1682 X 10- It 0.2521 X 10 -13
0.35 0.38 1.7 6.5
T2 M0 C L M~
18,8 1257
Ok: 4
6
g
I0
2
101,4 lime in year Fig. 1. Temporal evolution of the normalized surface loading potential V~o(t)/235 describing the Pleistocenic glaciationdeglaciation, where 235 m 2 s -2 is the maximum value of V~0 at the end of the glaciation. In this simplified model, the glaciation is characterized by a slow build up over a period of about 90000 years and the deglaciation by a collapse lasting 10000 years. Present time is almost 7000 years after the end of deglaciation.
M. Greff-Lefftz, 1t. Legros / Physics of the Earth and Planetary Interiors 90 (1995) l 15-135 '
"~
'
~
I
'
'
'
End ot deglaciation
I
'
'
119
The excitation function in the Euler equations is thus sinusoidal.
'
2.3. Torques acting at the CMB
0
In this section we describe the torques acting at the CMB resulting from magnetic motions within the core (for more details see Jault, 1990) i.e. E~ and L c3"
¢
10
11/ /
10+4
12
/
2.3.1. Viscomagnetic torque
10
11
12
10+4 Time in year Fig. 2. Viscoelastic deformations induced by the glaciation model described in Fig. 1, for the four-layer Earth of Table 1. (a) Time derivative of the inertia tensor of the Earth C33/C. The present value is in good agreement with the observed one deduced from the J2 coefficient of the geoid (J2 = - 8.87 + 2.5 x10-19s-1 (Gegout and Cazenave, 1991)). (b)Time derivative of the inertia tensor of the core C~3/C c. At the present, the deformations are not completely relaxed and its remains a zonal degree 2 term of the radial displacement at the CMB with an amplitude of about 10 m.
tribute to the angular momentum budget only by the perturbations of the inertia moments of the Earth and of the core which are related to degree 2 deformations and thus we only consider the degree 2 zonal potential V20(t) which is assumed to be sinusoidal V20(t ) = V2o sin(~rt + ~P0)
(9)
where ~r is the frequency of 18.6 year tide, ~0 the initial phase and with V2o = 0.17 m 2 s -2. Neglecting the viscoelastic effects, we computed the elastic deformations induced by V2o(t) C33(t ) C 6~3 ( t )
Cc
-2a
q0
F 3¢ = - K" f~CC~oc3
2.3.2. Topographic torque Models of the flow at the top of the Earth's core have been calculated from the secular variation of the geomagnetic field (Gire and LeMou61, 1990; Jault, 1990), and the core-mantle topography h(O, ~o) have been computed by seismologists (Morelli and Dziewonski, 1987). The tangentially geostrophic hypothesis allows computation of the pressure p(O, ~o) at the top of the core from the surface motions: - V p = 2pI~AE. This pressure is expanded in spherical harmonics
(P2c°sm~+qm
sin m p )
n=0 m=0
V20( t ) and
•2a2hCeV2o(t)
(11)
The frictional constant K" is strongly dependent on both the viscosity model within the core and the conductivity model in the mantle. Its value is between 10 -s and 10 -~. In this paper, we take K " = 10 -7 as a referred value.
p(O,~o)= ~
ke
O2a 2 kf
We introduce a viscomagnetic torque of tangential traction, F~, acting at the CMB in order to take into account the viscous friction existing at the CMB (Loper, 1975) and the electromagnetic coupling between the fluid core (assumed to be a perfect conductor) and the slightly conducting mantle (Stix and Roberts, 1984; Jault, 1990). This torque is a function of the existing velocity field in the fluid medium and can be expressed by
x pm(cos 0) (10)
(12)
An axial pressure torque acting on the core, denoted L~, results from the action of the pres-
M. Greff-Lefftz, H. Legros ~Physics of the Earth and Planetary Interiors 90 (1995) 115-135
120
sure field p(O, q~) on the core topography h(O, ~) (Jault and Le Mou~l, 1990)
G = - f fcMBp( O,
°h( °'
dS
(13)
This torque may be theoretically computed (Hinderer et al., 1990) L ~ = _2~.b2 S. m 3 = m=o 2 n + l ( ×
-p,,
. g,,. . . +qnhn)
(n+m)!
mantle. This mechanism described by Jault and Le Mou61 (1990) is summarized in scheme 1. Hereafter we assume, for simplicity, that the core has a rigid differential rotation with respect to the mantle and we note ~bc the phase shift between the flow pattern and the CMB topography, i.e. the longitude variation such as ~ ~¢..O c3. The topographic torque may thus be written ~ m - .g,. + q .mh . m ) L 3c = _ 2 7 r b 2 ~ ....= m = 0 2 n + l [ ( pm_m Xcos m~b¢(t) + ( h ,mp , m + qm_,~x , g, )
(14)
(n - m ) !
Notice that the zonal toroidal motions or the zonal pressures (i.e. m = 0) in the core do not induce any topographic torque. This torque changes the angular momentum (see Euler equations), i.e. perturbs the zonal toroidal motions which drive the pressure field. As a consequence, it involves the changes in the torque itself: there is a new topographic torque between the perturbed flow pattern of the core and the unchanged topography relative to the
L
xsin m6¢(t)]
(n+rn)! (n - m ) !
Jault (1990) has computed the amplitude of L~(~b¢) for the epoch 1980 and the CMB model of Morelli and Dziewonski (1987) as a function of the phase shift ¢¢ and has shown that it looks like a sine curve (Fig. 3) L~(~b¢) = - L o sin(3~b~) with L o = 2 X 1019N m (16)
ZONAL MOTION WITHIN THE CORE RIGID ROTATION OF THE MANTLE
ROTATION OF THE MAGNETIC PRESSURE WITH RESPECT TO THE TOPOGRAPHY FIXED IN THE TISSERAND FRAME OF THE MANTLE
n=o rn~o
NEW TORQUE OF THE NON ZONAL PRESSURE ON THE TOPOGRAPHY
~
i_2rrb2Z
(15)
(qnh -png.)cosrndz(t)+(hnp.+qng.) . . . . ~ ( )
n~0 m=0
Scheme 1.
(-~_~-~
M. Greff-Lefftz, 14. Legros/ Physics of the Earth and Planetary Interiors 90 (1995) 115-135
and of the core when the Earth is submitted to different excitation sources.
2 ~OE*I9
LU
200E+19 150E • 19
0 1~
3. Results
500£+18
t~-
0
121
0.00
-500£*10 - I OOE * I9 - I 50E~19 - 2 0 0 E ' ~ 19 - 2.,50£ * 19
tU
-300E÷I9 0 OC - 3 5 0 E + 1 9
,
,
.
,
.
I00
2 O0
PHASE
SHIFT
,
,
,
300
Fig. 3. Changes in the topographic torque acting on the core with an imaginary rotation of the pattern of the flow with respect to the model of topographyof Morelli and Dziewonski (1987), from Jault and Le Mou~l (1990).
2.4. Equations The conservation of the angular momentum of the core (Eq. 6) may be written as a differential equation of the time-dependent phase shift &c ~c + c K " I ) ~ c + - -
sin(3~b c)
Cm
Cm
C [
L3(t)--
C~
-F
~')C33__ ~~C~3c c
C
C
(17)
It governs an oscillating system with a period dependent on the amplitude of the topographic torque (i.e. L0). From &c(t) and the conservation of the angular momentum of the global Earth (Eq. 5), we can compute the variation of the angular rotation of the Earth to3(t) C
L0
033 + C m K"~t°3 - ~-~- sin(3&c) C [L3(t ) + K , , E I f t L 3 ( t ' ) dt'
= c';t
c
+~
C
.,, C
c
(18)
These equations (18) and (17) govern the temporal evolution of the axial rotation of the Earth
Eqs. (17) and (18) have been integrated using a R u n g e - K u t t a method, assuming that for t = 0 both the phase and the angular velocity are equal to zero. We present the results when the Earth is submitted to a surface load (Pleistocenic glaciation) in Figs. 4-8, and to an external torque (tidal torque) in Figs. 9-13, for various values of the amplitude of the topographic torque ( L 0 = 0 in parts (a) of the figures, L 0 = 2 x 10 Is, 2 x 1016, 2 x 1017, 2 x 1018 and 2 x 1019 N m, respectively in (b), (c), (d), (e) and (f)). For each excitation source, we present the resulting perturbations of to3(t) (in Figs. 4 and 9, respectively, for the deglaciation and the tidal torque), to~(t)(in Figs. 5 and 10) as a function of time but also as a function of respective phase & (in Figs. 6 and 11) and &c (in Figs. 7 and 12). The time derivative of the axial rotation of the Earth 033 is also represented (in Figs. 8 and 13) in order to investigate the influence of the topographic torque on the observed secular deceleration of the Earth (induced by the tidal torque) and on the acceleration of the Earth induced by the postglacial rebound. We first comment on the results when there is no topographic torque (L 0 = 0 N m) and then investigate the influence of the amplitude of L~3 c on ~o3 and w c3"
3.1. Without topographic torque In the case where there is no topographic torque, the viscomagnetic coupling (F~) and the viscoelastic deformations of the CMB ( f l ( ~ 3) will control the temporal evolution of w 3 and to~. Because of the Pleistocenic glaciation, we have at the present (almost 7000 year after the end of the deglaciation) a westward drift of the core with respect to the mantle (to~ < 0) (Fig. 5(a)) which may be correlated with the observed westward drift of the geomagnetic field of the Earth. Both the viscomagnetic coupling ( K " = 10 -7) and the
122
M. Greff-Lefftz, H. Legros / Physics of the Earth and Planetary Interiors 90 (1995) 115-135 c : Lo= 2.E+15 N.m
b : Lo= 2.E+15 N.m
Lo=O N.m
o :
l-
,.q
15
-1
1
~. X _ T . "l
I
~ L t I t I
2 4 10+4
6
12
8
× -10
×-
,I~ILI~I, 0 2 4 X 10+4
6
8
-1 4
6
10
8
e : Lo= 2.E+18 Nm "[ , i , i , i i I ,A ,
d : Lo= 2.E+17 N.m
~
0 2 X 10+4
10 12
t I ,
0 2 x 10+4
I ,
I
t I
4
6
8
12
i
i
i [ , L,
0 2 4 X 10+4
11i
10
~
10 12
i I i I i I , IVl-]]
2
12
4
B
8
I0 12
10+4 Time in year
Time in year
Time in ye0r
8
: Lo= 2.E-.l-19 N.m
iLI
,
I ,
6
Fig. 4. T e m p o r a l evolution of the axial rotation of the E a r t h induced by the Pleistocenic deglaciation for various values of the topographic torque: L 0 = 0 N m (a), L 0 = 2 × 1015 N m (b), 2 × 1016 N m (c), 2 x 1017 N m (d), 2 x 10 TM N m (e) and 2 x 1019 N m (f).
20 ' I
I
I
c : Lo=2.E+16 N.m
b : L0=2,[+15 N.m
a : Lo=O N.m
~
I
'
10~20 1~r-'5 I] ' ~I -' I ' I
g
1
o
o'
~-5
x
8
0 2 4 x 10+4
10
12
I ' I f I ' I ~
~
-
~
if
I
'
I
'
I
'-44
---t -i
,
-~I
I L
0 2 x 10+4
i 1 i I , 6 8 10
1~ 1 ~
12
0
2 x 10+4
e : Lo=2.E+18 N.m
d : Lo=2.E+17 N.m I'
I
, l t I ~ I t ~ 6
8
10 12
f : Lo=2,E+19 N.m
_
t~4
X
,
L ,
2 X 10+4
, I t I
6
8
Time in year
x
10 12
...I_L I i 2 10+4
t I i I 6
8
lime in year
10
12
I i 0 2 X 10+4
, I ~ I 6 8
10 12
lime in year
Fig. 5. T e m p o r a l evolution of the axial differential rotation of the core with respect to the mantle induced by the Pleistocenic deglaciation for various values of the topographic torque: L 0 = 0 N m (a), L 0 = 2 × 1015 N m (b), 2 x 1016 N m (c), 2 × 1017 N m (d), 2 x 1018 N m (e) and 2 × 1019 N m (f).
M. Greff-Lefftz, H. Legros / Physics of the Earth and Planetary Interiors 90 (1995) 115-135 0 : Lo= ON.m I
b : Lo = 2.E+15 N,m
I
I
'
c : Lo = 2,E+16 N.m
I
I
i
50
100
]
t
o
q
x
× 150
100
50
x
5O
d : Lo = 2E+17 N.m
-
150
0
e : Lo = 2.E+18 Nm '
o
10O
~
I
'
I
123
~50
f : Lo = 2.E-f19 Nm
'
'
I
'
1
o
, x
50
1O0
150
50 I O0 Phase shift of mantle
Phase shift of mantle
150
0
50 100 Phase shift of mantle
150
Fig. 6. Axial rotation of the Earth in the phase space induced by the Pleistocenic deglaciation for various values of the topographic torque: L 0 = 0 N m (a), L 0 = 2 x 1015 N m (b), 2 x 1016 N m (c), 2 x 1017 N rn (d), 2 x 1018 N m (e) and 2 x 10 I° N m (f).
l!I
o:Lo=ON.m ¢,
15
p,
b : Lo = 2.E+15 N,m
I
2O-
10
x
-20
0
20
40
60
d : Lo = 2.E+17 N.m -,
i
~
i
,
[
,
j
,
]
'
'
I
e : Lo = 2.E+16 N,m
'
2 ~ - - ~
'
I
'
i
1 1
I
-20
O
20
40
e : Lo = 2.E+18 Nm
60
-20
0
20
40
60
f : Lo = 2.E+19 Nm
,
5 0 ~
~ x
-5
-1[
-0.04-0.02&0O0.020.040.06 Phase shift between core and mantle
:15 -¢ -2 [~ Z 4 6 8 -6 -4 -2 0 2 4 6 8 X 10-,] X 10-4 Phase shift between care and mantle #hose shift between core end mun[le
Fig. 7. Axial rotation of the Earth in the phase space (o~(~c)) induced by the Pleistocenic deglaciation for various values of the topographic torque: L 0 = 0 N m (a), L 0 = 2 x 1015 N m (b), 2 x 1016 N m (c), 2 x 1017 N m (d), 2 x 10 Is N m (e) and 2 x 10 m N m (f).
M. Greff-Lefftz, H. Legros / Physics of the Earth and Planetary Interiors 90 (1995) 115-135
124
[
10
11
c : Lo: 2£+16 N.m
b : Lo= 2.E~15 N.m
a : Lo=O N.m I
2C
J
--
i
,
-
._c:
~-2 12
10+4
9 X 10+4
10
11
12
g x lO+4
e : La= 2.E+18 N.m
d : Lo= 2E+17 N.m
20i
i
10
11
12
N.m ,
f : Lo= 2£+/£ [ i i
-2 11
10
12
x 10+4
9 X 10+4
10
Time in yeer
11
12
Time in year
I ~ r , I 9 10 11 X 10+4 Time in year
,
12
Fig. 8. Temporal evolution of time derivative of the axial rotation of the Earth induced by the Pleistocenic deglaciation for various values of the topographic torque: L 0 = 0 N m (a), L 0 = 2 × 1015 N m (b), 2 × 1016 N m (c), 2 × 1017 N m (d), 2 × 1018 N m (e) and 2×1019 N m ( f ) .
e : Lo : 0 N.m
b : Lo : 2£+15 N.m
c : Lo = 2.E+16 N.m i
i
r
tJ
"5
-lC
-1C
-1~
15
e,::~
-20 '
2
4
6
8
10
1
× lO+4
× 10+4 d : Lo = 2£+17 N.m
'
'
'
'
'
e : LD ~
.
_
~
=
' 2 ,~ 10+4
2.E+18 Nm I
'
]
'
4
10
8
6
f : La = 2£+19 N.m J
J
i
T
i
-10 -15 c>
"- -2C o
4 X 10+4
x 10+4 Time in year
× 10+4 Time in year
Time in year
Fig. 9. Temporal evolution of the axial rotation of the Earth induced by the tidal torque for various values of the topographic torque: L 0 = 0 N m (a), L o = 2 × 10 is N m (b), 2 × 1016 N m (c), 2 × 1017 N m (d), 2 × 1018 N m (e) and 2 × 1019 N m (f).
M. Greff-Lefftz, H. Legros / Physics of the Earth and Planetary Interiors 90 (1995) 115-135
y ]
a :Lo= I
'
I
ON.m '
I
b : Lo = 2.E+15 N.m
125
c : Lo = 2£+16 N.m
'
==
%
1
~×o
,
I
,
2 10+4
1 4
t 6
~
I 8
J
x
,
10
I ~
I
'
J
I
I
10
4
d : Lo = 2£+17 N.m '
,
2 10+4
0 2 × 10+4
e : Lo = 2£+18 N.m
4
6
8
I0
f : L:~ = 2.E+19 N.m
Z_ 2 10+4
><
4
6
8
;0
2
0
6
10
8
0 2 x 10+4
X 10+4
Time in year
Time in year
4
6
8
10
*time in year
Fig. 10. Temporal evolution of the axial differential rotation of the core with respect to the mantle induced by the tidal torque for various values of the topographic torque: L 0 = 0 N m (a), L o = 2 X l0 Is N m (b), 2 x 10 t~ N m (c), 2 x 1017 N m (d), 2 x l0 Is N m (e) and 2 x 1019 N m (f).
c : Lo = 2£+16 Nm
b : L0 = 2£+15 N.m
a:Lo=ON.m I
I
'
I
'
t
'
-10
-1 o
-1 x
-5 ~2 X 10+5
~
-1
d : Lo - 2.E+17 N.m i ~--r , i , [ ,
15
-10
5
x 10+4 e : Lo : 2E+18 Nm
0
-25 -20 -15 -10 X 10+2
-5
0
f : Lo = 2E+19 Nm
-1 o
q--1
o
/
x
-25 -20 -15 -10 -5 x 10+2 Phase shift af mantJe
-25 -20 -15 -10 -5 × 10+2 Phase shift of mantle
-25 -20 -15 - ! 0 -5 x 10+2 Phase sh?[t of mantle
0
Fig. 11. Axial rotation of the Earth in the phase space induced by the tidal torque for various values of the topographic torque: L0=0Nm(a), L0=2X1015 Nm(b),2x1016 Nm(c),2x1017Nm(d),2x1018 N m ( e ) a n d 2 X l 0 I~ N m ( f ) .
126
M. Greff-Lefftz, 11. Legros / Physics of the Earth and Planetary Interiors 90 (1995) 115-135
. '
I
'
b : Lo = 2£+1§ N.m
a : Lo = 0 N,m I ' I ' I 'I]]
t
I
I
I
I
I
I
c : L0 = 2£+16 N.m I_
g B
&_ 10
5
15
20
25
0 5 X 10+2
10+2
I
I
d : Lo = 2£+17 N.m I '
.!I'
10
0.00
15
0.05
e : Lo = 2£+18 N.m
0,10
0.15
f : Lo = 2.E+19 N.m
Z-
x
0.000
0.005
0.010
0.015
0.020
Phase shift between core and mantle
0 5 X I0-4 Phase
10
15
20
-0
5
10
15
20
X 10-5
shift between
core and mantle
Phase shift between
core and mantle
Fig. 12. Axial rotation of the Earth in the phase space (to~(~bC)) induced by the tidal torque for various values of the topographic torque: L 0 = 0 N m (a), L 0 = 2 x 1015 N m (b), 2 x 1016 N m (c), 2 x 1017 N m (d), 2 x 1018 N m (e) and 2 X 1019 N m (f).
a : La = 0 N.m
b
Lo = 2.[+15 N.m I'
c : Lo = LE+16 N.m ' I ' I ' _
oq
-z
( -1[
5 X I0+3
10
15
20
5 x 10+5
d : Lo = 2£+17 N,m , , ,--_
, -,
5
10
x 10+3
15
10
20
2 ~ . 5 X 10+3
e : Lo = 2E+18 Nm
20
IU X 10+3
Time in year
15
Time in year
13
I
10
I
h
15
I
20
I : Lo = 2.E+19 Nm
Z~
b X 10+3
IU
13
Zg
Time in year
Fig. 13. Temporal evolution of time derivative of the axial rotation of the Earth induced by the tidal torque for various values of the topographic torque: L o = 0 N m (a), Lo = 2 x 1015 N m (b), 2 X 1016 N m (c), 2 X 1017 N m (d), 2 x 1018 N m (e) and 2 × 1019 N m
(f).
)IlL Greff-Lefftz, 1t. Legros / Physics of the Earth and Planetary Interiors 90 (1995) 115-13 5
viscoelastic deformations of the CMB must exist simultaneously to obtain this negative value of oL~. As a matter of fact, if the mantle is elastic or if there is no viscomagnetic torque (K" = 0), the core will rotate faster than the mantle and consequently an eastward drift of the core with respect to the mantle will be observed (Lefftz and Legros, 1992). We note also at the present time an acceleration of the angular velocity of the Earth (o53 > 0) which is in good agreement with the observations (Fig. 8(a)). The value of o53 is not significantly affected by the viscomagnetic coupling. These results concerning the temporal variations of w~ and o53 induced by the deglaciation have been reinforced by analyses based on stratified mantle models with steady state or transient rheologies (Lefftz et al., 1994). In general, a viscosity increment in the lower mantle acts to decrease the westward drift of the core whereas a viscosity decrement in the D" layer is responsible for faster relaxation of the CMB topography which increases the westward drift of the core.
a:Lo
127
On the other hand, the tidal torque induces a secular deceleration of the Earth ( o 5 3 = - 6 x 10 -22 rad s -2) (Fig. 13(a)). This torque gives also a perturbation of toc3 (Fig. 10(a)). If there is no viscomagnetic torque ( K " = 0) between core and mantle, the core will not be affected by the deceleration of the mantle and will move in a westward direction. On the contrary, if there is a viscomagnetic torque ( K " = 10 7), w~ tends to a constant positive limit which cancels a part of the negative differential rotation of the core resulting from the deglaciation.
3.2. With topographic torque If there is a topographic torque at the CMB with a constant amplitude in time, the relative magnitudes of the terms F~, L~ and l l ( ~ will govern the solutions of to3 and to~. For the axial rotation of the Earth, the presence of a topography at the CMB does not change significantly the solutions in terms of secular vari-
= ON.m
b : Lo = 2.E+18 N.m
'
'
'
I
'
'
'
l
~
~
~
I
~
J
,
~ 4
'
"S
.5
0 ,_,-, -1
7=
. ~ - 2
x
0 X 10+2
2
'
'
I
~
2
0
2
4
X 10+2
'
1
c : LD = 2.E+19 N.m ' ' ' 1 '
d : Lo
=
2.E+20 N.m
0
><
J
l
J 2
l
l
l
10+2
l 4
0
2
4
X 10+2 Time in year
T]me in year
Fig. 14. Temporal evolution of the axial rotation of the Eart h induced by the 18.6 year tidal potential for various values of the topographic torque: L 0 = 0 N m (a), L 0 = 2 x 10 ~8 N m (b), 2 x 1019 N m (c) and 2 × 102~ N m (d).
128
M. Greff-Lefftz, H. Legros/ Physics of the Earth and Planetary Interiors 90 (1995) 115-135
ations (Figs. 8 and 13), There are some added oscillations (Figs. 4 and 9) with amplitude increasing and period T decreasing, respectively, when the torque L o increases (T---2500 years, 800 years, 250 years and 80 years for, respectively, L 0 = 2 X 1016 N m, 10 ]7 N m, 10 t8 N m and 1019 N m). For small values of L o (until 2 × 1016 N m), the perturbations on to 3 owing to the topographic torque are negligible (in comparison with the curve L 0 = 0) (Figs. 4(b) and (c) and 9(b) and (c)). For higher values of L 0, we obtain nonnegligible perturbations of oJ3 which may be correlated (for the case L 0 = 2 × 10 t9 N m, in Figs. 4(f) and 9(f)) with the observed decade variations in the length of the day (Jault, 1990). Consequently, the existence of a topographic torque at the CMB involves periodic variations in to 3 damped by the viscomagnetic torque (i.e. following the value of K") and finally to3 tends to the same secular value for the deceleration induced by the tidal torque (Fig. 13) and the acceleration induced by the deglaciation (Fig. 8).
On the other hand, the influence of the topographic torque is very important on the differential rotation of the core with respect to the mantle, since two kinds of solutions are possible. If L 0 < 2 X 1016 N m (Figs. 5(b) and (c), 7(b) and (c), 10(b) and (c) and 12(b) and (c)), the energy brought by the external excitation sources is more important than that owing to the topographic torque and consequently the core has enough energy to move with respect to the mantle. On the contrary, for the largest values of L0, the energy brought by the excitations is not sufficient in comparison with L 0 and the core cannot drive but oscillates around a stable phase ~b~ with the period related to L 0 (as given above). The core 'is blocked between two bumps' of the CMB topography. For the case L 0 = 2 x 1016 N m, under the action of the surface load, it is of interest to notice in Fig. 7(c) representing to~(~bc) in the phase space the transition between the state of drift and the state of oscillation: until the end of
a:Lo=ON.m e~
'
' ' 1 ' ' ' 1
"s
'
b : Lo = 2.E+18 I , ~ T
N.m i
' 1
2
o
q kSx
1
i
l
k
l
t
L
×
t
2
4
10+2
~ ×
c : Lo = 2.E+19
,
q 2
,
,
~
d : Lo = 2.E+20
N.m
'
'
I
'
'
'
I
'
i i 0 X 10+2
J
I 2
I
I
i
I 4
I
Time in year
IL 0 X 10+2
4
'
0 X 10+2
'
'
l
'
'
2
'
I 4
L J
N,m J
4
Time in yeer
Fig. 15. Temporal evolution of the axial differential rotation of the core with respect to the mantle induced by the 18.6 year tidal potential for various values of the topographic torque: L 0 = 0 N m (a), L 0 = 2 × 10 TM N m (b), 2 × 1019 N m (c) and 2 × 10 2° N m
(d).
M. Greff-Lefftz, H. Legros / Physics of the Earth and Planetary Interiors 90 (1995) 115-135 a : Lo = 0 N.m
120
b : Lo = 2.E+18 N.m
x
5
10
15
20
0 X 10-5
10-6
1
2
'
I
'
l
'
I
4
d : Lo = 2£+20 N.rn
c : Lo = 2£+19 N.m I
~
'
I
r~
o' ><
x
O-X 10-5
1
2
3
0 X 10-5
4
?
4
Phase shift of man~le
Phase shift d rnontle
Fig. 16. Axial rotation of the Earth in the phase space induced by the 18.6 year tidal potential for various values of the topographic torque: L 0 = 0 N m (a), L{~ ~ 2 × l0 TM N m (b), 2 × 1019 N m (c) and 2 x 10 2~ N m (d).
a:Lo=
b : Lo = 2.E+18 N.m
ON.m
4
!
2
--~
0
2 to-2
='T ~_~-4
×
0
5
10
15
2()
-1
-2
X 10-6
0
1
X 10 5 d : Lo = 2£+20 N.m
c : Lc. = 2£+19 N.m
_7x
-2 -1 0 1 2_ X 10-5 Phase shift between core and m•ntle
2 tQ -4-
-2
0
2
4
X 10-5 Phase shift between core end mon'~/e
Fig. 17. Axial rotation of the Earth in the phase space (w~(~c)) induced by the 18.6 year tidal potential for various values of the topographic torque: L o = 0 N m (a), L o = 2 X 10 TM N m (b), 2 × 10 ~9 N rn (c) and 2 × 102° N m (d).
130
M. Greff-Lefftz, H. Legros/ Physics of the Earth and Planetary Interiors 90 (1995) 115-135
the deglaciation, the core drifts with respect to the mantle and suddenly, the inertia moments pertubations having decreased, it has not enough energy to continue its drift and is blocked with oscillations. Consequently, for the value of L o = 2 × 1019 N m proposed by Jault and Le Mou~l (1990), there is no possible secular drift of the core with respect to the mantle induced by deglaciation nor by tidal torque.
If we linearize the Eqs. (17) and (18), assuming that ~bc does not vary a great deal on the decade (assumption confirmed by the results), we obtain a damped oscillator
3.3. Forced rotations induced by the 18.6 year tidal potential
Neglecting the damping effect (the characteristic frequency of damping c / C m K " f ~ is small in comparison with too), the solutions are
Assuming that we have q~0 = 0, we have integrated Eqs. (17) and (18) for various values of the topographic torque: L o = 0 N m L 0 = 2 × 1018 N m, L 0 = 2 × 1019 N m and L o = 2 × 1020 N m. We present the results for the axial rotation of the Earth in Figs. 14 and 16, for the differential rotation of the core in Figs. 15 and 17 and for the time derivative of the axial rotation of the Earth in Fig. 18.
C ~c+ ~K
IV
"
(19)
with a frequency C Lo 0)20= 3 C---~ C---S
Fo
(sin0.t_
~rsin0)0 t) 0)0
Fo'~
0)c
3
(0)2__0.2)
(COS ~ r t - cos 0)ot)
It is interesting to note that for L 0 around 10 2o N m, 0)o is very close to the frequency 0. of the excitation function and thus we may have a reso-
a:Lo=ON.m
b : Lo = 2.E+18 N.m 20F
2O
I
I
'
'
I
'
'
'
'
~
':t//////////;/////////////f
c
~
l)~b C +0)2~bC=Fo sin(o't+q~o)
-IC -
4
-2C 0 X 10+2
2
4
0
c : Lo = 2.E+19 N.m
2CI
,
,
2
4
X 10+2
I
,
,
,
: Lo = 2.E+20 N.m i
I
i
i
I
lO
c
-213 0
i l l l l ~ l l l 2 X 10+2 Time in year
4
o
2
4
X 10+2 Time in year
Fig. 18. Temporal evolution of time derivative of the axial rotation of the Earth induced by the 18.6 year tidal potential for various values of the topographic torque: L 0 = 0 N m (a), L 0 = 2 × 10 TM N m (b), 2 × 1019 N m (c) and 2 × 10 2o N m (d).
M. Greff-Lefftz, H. Legros/ Physics of the Earth and Planetary Interiors 90 (1995) 115-135 h
a
Temporal evoMion of the torque : To=500[30 yr 1019[~q -'-' t ' I ' I ' ± '
Temporal evolution of the torque : To=IODO00 yr ~019~--~-~--q--' I ~.~L.... ~ I017~ - ~ ~
ld5t- \
,S-
/
0 2 X 10+4
\ 4
/
6
X-
8
10
~
0 2 x 10+4
12
4
6
8
/ 4
2 6
8
10
12
Differential rotation of the core
Oifferential rotation of the core
0 2 X 10+4
131
10
0 2 × 1044
12
4
6
6
10
Time in year
Time m year
Differential rotation of the core
Differential rotation of the core I i
12
It,
_' _2k-U----_ x E-, II ~ I -6 -4 -2 Phase between core end mantle
-2.0
-1,0 -0.5 -1.5 Phase between core and mantle
0.0
Fig. 19. Axial d i f f e r e n t i a l r o t a t i o n of the core with respect to the m a n t l e as a function of time a n d in the p h a s e space i n d u c e d by the P l e i s t o c e n i c d e g l a c i a t i o n for a t i m e v a r i a b l e t o p o g r a p h i c t o r q u e with a respective p e r i o d T{~ of 5(1 × 10 :~ years in (a) and 100 x 103 years in (b).
a Temporal evolution of the torque : To=D0000 yr 1019~q~r~--F -~ ] ' I ' ~L..~
/
0 2 X 10+4
4
6
8
10
12
Temporal evaMion of the torque
,at7L " ~
0 2 X 10+4
Tc 1130000 u
--~i~-~
~0t'~ -~---~-~
/
6
4
8
--...
t'.',
Axial rotation of the Earth
Axial rotation of the Earth
'oF''
, ,--T~--,--~
Tq
T & 2 X 10+4
0
x-10~
4
8
10
12
0 2 X 10+4
4
6
8
Time in year
Time in year
Axial rotation of the Earth
Axial rotation of the Earth
, , , I 0
6
, , , I , ,'v,,,',_~ld
50 100 Phase of the mantle
150
x-10N 0
, ,
,_A_,
10
!£
, , , L=__~
50 IO0 Phase of the mantle
150
Fig. 20. A x i a l r o t a t i o n of the E a r t h as a function of t i m e and in the p h a s e space i n d u c e d by the P l e i s t o c e n i c d e g l a c i a t i o n for a t i m e v a r i a b l e t o p o g r a p h i c t o r q u e with a r e s p e c t i v e p e r i o d T O of 50 × 103 years in (a) and 100 × 103 years in (b).
132
M. Greff-Lefftz, H. Legros/ Physics of the Earth and Planetary Interiors 90 (1995) 115-135
nance effect. This explains the increase of the amplitude of to3 and toc3 between Figs. 14(a) and (d) and 15(a) and (d). 3.4. Time-dependent pressure within the core
In the previous calculation we have assumed that the magnetic pressure within the core is constant in time. This hypothesis seems to be reasonable for excitation sources with decade time-scales such as the 18.6 year tidal potential, but this is not clear for phenomena with timescales of some thousands of years. As a matter of fact, the temporal evolution of the pressure field resulting from the variations of the motions is related to geodynamo processes which cannot be neglected at the time-scale of some thousands of years. Consequently, the topographic torque has to vary with time and its amplitude itself changes. Under these conditions the final differential equations (17 and 18) and their solutions will be completely different.
In order to investigate these effects, we have assumed the pressure to be time-dependent as a sine curve around an averaged value of 2 x 10 [6 N m with a period To, and thus the topographic torque acting at the CMB may be written 27"rt
LC3 = - 10[16+3 c°s(To-0)] sin 3~b c N m
When the excitation source is the Pleistocenic deglaciation introduced in Section 2.2.1, we have represented the axial differential rotation of the core (Fig. 19) as a function of time and in the phase space and the axial rotation of the Earth (Fig. 20) as a function of time and in the phase space, for various values of the period To (the period To is, respectively, 50 kyear and 100 kyear for curves (a) and (b)). In Fig. 19, we see that there is a combination of two effects: the first one is an oscillation on a short time-scale of the core around a stable position similar to the one introduced in the previous
a
b
2 n Temporal evolution of the torque : Ta=5OO00 ),r
0
5
10
(20)
15
20
x 10+.3
201emporal evolution of the toraue : fo=lO0000 yr 1019[~ ' ] ' I " I '
0 x 10+3
5
10
15
20
Differential rotation of the core
Differential rotation of the core
_4
"2 o X 10+3
0 × 10-5
5
lO
15
2O
0 X 10+3
5
10
15
20
lime in year
Time in year
Oifferentiel rotation of the core
Differential rotation of lhe core
2
4
0
2
4
x IO-3
Phase betweencoreand mantle
Phase between core and mantle
Fig. 21. Axial differential rotation of the core with respect to the mantle as a function of time and in the phase space induced by the tidal torque for a time variable topographic torque with a respective period TOof 50 x 103 years in (a) and 100 × 10 3 years in (b).
M. Greff-Lefftz, tl. Legros / Physics of the Earth and Planetary Interiors 90 (1995) 115-135
section; the other one is relative to the time variable amplitude of the topographic torque. At the beginning of the glaciation, L~ is maximum and consequently the core is 'blocked' and oscillates with a weak amplitude between two bumps of the mantle. But progressively, the amplitude of L~ decreases until 2 x 1013 N m allowing a shift of the core with respect to the mantle (between 15 000 and 35 000 years). After that, L~ increases and reaches a critical value, there is an oscillation of the core with respect to the mantle with a large magnitude around a stable position (&~ = - 4 rad). The amplitude of this oscillation decreases with time because of the frictional viscomagnetic torque acting at the CMB. This phenomenon (a shift of the core followed by a large oscillation around a stable position) is reproduced for t greater than 85 000 years. For T O= 100 kyear, the comments are similar but during and after the deglaciation, L~ is maximum and almost constant and thus the curve obtained is very close to that in Fig. 5(f) obtained
a i0!~i ~lt~ITeBpOra' e~aJutlon, at the [orqlu8 : TIO=500,O0]r
for a constant topographic torque with an amplitude of 2 x 1019 N m. The stable position of the oscillation of the core is evidently affected by the amplitude of the surface load. Notice that whatever the value of TO, the core could drift with respect to the mantle: in the horizontal axis the phase variation is about 7r and 27r. On the contrary, if the topographic torque is constant (Fig. 5(f)), the phase varies between - 5 × 10 - 4 and 7 x 10 - 4 tad. The results of Fig. 20 for the axial rotation of the Earth are very close to that obtained for a constant topographic torque with an amplitude of 2 X 1019 N m (Fig. 4(f)), but an oscillation on a short time-scale has to be added because of the time variation of L~. In Figs. 21 and 22, we have represented the axial differential rotation of the core and the axial rotation of the Earth as a function of time and in the phase space, when the excitation source is the tidal torque introduced in Section 2.2.2, for two values of the period T0 (50 kyear and 100
3 20Temporal evolution at the torque : To=I000C'O yr
10 r
,
101198~
~ - ~ ,
/
O X 10+,3
§
10
15
20
,
0 x 10+3
~
I
0 x 10+3
5
'
10
I
15
I~--T
I
,
~~
§
I ~_L
'
,
10
J
15
--~ :,0
Axial rolation of the Earth
Axial rotation of the Earth
0
133
'
q
20
0 X 10+3
5
10
15
Time in year
Time in year
Axial rotation of the Earth
Axial rotation of the Earth
?0
°
-80
-60 -40 -20 Phase of the mantle
0
-80
-60 -40 -20 Phase of [he mantle
0
Fig. 22. Axial rotation of the Earth as a function of time and in the phase space induced by the tidal torque for a time variable topographic torque with a respective period T o of 50 x 103 years in (a) and 100 x 103 years in (b).
134
M. Greff-Lefftz, H. Legros/ Physics of the Earth and Planetary Interiors 90 (1995) 115-135
kyear, respectively, for curves (a) and (b)). The horizontal scale is of about 20000 years and during this time, the topographic torque only decreases. For the differential rotation of the core (Fig. 21), the results for curves (a) and (b) are very similar: during the first 12 kyear for T 0 = 50 kyear (or the first 20 kyear for T O= 100 kyear) the core oscillated around a stable position ~b~ of about 0.2 rad, and after that, the amplitude of L~ being small, we observe a shift of the core with respect to the mantle in an eastward direction. In Fig. 22, we see that there is a secular deceleration of the Earth induced by the tidal torque which is weakly perturbed by the time variation of L c3 (comparison between Figs. 22 and 9). Notice, in Figs. 20 and 22, that the variation of the axial rotation of the Earth is sensitive to the time variation of the pressure within the core and could, in principle, give some information concerning the temporal evolution of the core motion. To conclude, if the topographic torque varies with time, it would be difficult to predict the evolution of the core motions and the rotation of the Earth.
to take into account the different terms of degree n and order m appearing in the computation of the torque. First, we have assumed the magnetic pressure to be constant in time (i.e. L 0 is a constant) which is not true on a long time-scale. The temporal evolution of the pressure field resulting from the variations of the motions is related to geodynamo processes which cannot be neglected on a time-scale of some thousands of years. Consequently, the topographic torque has to vary with time and its amplitude itself can change. We have tested a simple temporal variation taking the pressure as a sine curve with various periods. U n d e r these conditions, it would be difficult to predict the evolution of the motions of the core with respect to the mantle: they would be chaotic.
Acknowledgements We thank Dominique Jault for important suggestions. This research has been supported by C N R S - I N S U - D B T grant (contribution number 16).
References 4. Concluding remarks The axial rotations of the Earth and its fluid core are perturbed by external sources (deglaciation, tidal torque or tidal potential) and by internal sources (magnetic pressures which deform the Earth and the core and which act on the CMB topography). We have investigated the influence of the topographic torque on o) 3 and oJ~ excited by the Pleistocenic glaciation, by the tidal torque and by the 18.6 year tidal potential. T h e constant topographic torque proposed by Jault and Le MouEl is large but even a smaller topography (with a factor 100) would maintain its influence. It introduces the problem of a time variable topography, such as the topography induced by the deglaciation itself (of about 10 m) which cannot be neglected, and also the necessity
Dziewonski, A.M. and Anderson, D.L. 1981. Preliminary reference model PREM. Phys. Earth Planet. Inter. 25: 297356. Gegout, P. and Cazenave, A., 1991. Geodynamic parameters derived from 7 years of laser data of Lageos. Geophys. Res. Lett., 18: 1739-1742. Gire, C. and Le MouEl, J., 1990. Tangentially geostrophic flow at the core-mantle boundary compatible with the observed geomagnetic secular variation: the large-scale component of the flow. Phys. Earth Planet. Inter., 59: 259-287. Hinderer, J. and Legros, H., 1990. Global Consequences of the tidal secular deceleration for the solid earth and its fluid core. In: P. Brosche and J. Sundermann (Editors), Earth's Rotation from Eons to Days. Springer, Berlin, Heidelberg, New York. Hinderer, J., Legros, H., Jault, D. and Le MouEl, J., 1990. Core-mantle topography torque: a spherical harmonic approach and implications for the excitation of the Earth's rotation by core motions. Phys. Earth Planet. Inter., 59: 329-341. Hulot, G., 1992. Observations g~omagn~tiqueset g6odynamo. Thesis, Universit6 Paris VII.
M. Greff-Lefftz, H. Legros/ Physics of the Earth and Planetary Interiors 90 (1995) 115-135 Jault, D., 1990. Variation s6culaire du champ g6omagn6tique et fluctuations de la longueur du jour. Thesis, Universit6 Paris VII. Jault, D. and Le Mou~l, J., 1990. Core-mantle boundary shape: constraints inferred from the pressure torque acting between the core and the mantle. Geophys. J. Int., 101: 233-241. Lefftz, M. and Legros, H., 1992. Influence of viscoelastic coupling on the axial rotation of the Earth and its fluid core. Geophys. J. Int., 108: 725-739. Lefftz, M., Sabadini, R. and Legros, H., 1994. Mantle theology, viscomagnetic coupling at the CMB and differential rotation of the core induced by Pleistocenic deglaciation. Geophys. J. Int., 117: 1-18. Loper, D.E., 1975. Torque balance and energy budget for the precessional driven dynamo. Phys. Earth. Planet. Inter., 11: 43-60.
135
Morelli, A. and Dziewonski, A.M., 1987. Topography of the core-mantle boundary and lateral homogeneity of the liquid core. Nature, 325: 678-683. Munk, W.H. and MacDonald, G.J.F., 1960. The Rotation of the Earth. Cambridge University, New York. Peltier, W.R., 1974. The impulse response of a Maxwell earth. Rev. Geophys., 12: 649-669. Peltier, W.R., Drummond, R.A. & Tushingham, A.M., 1986. Post-glacial rebound and transient lower mantle rheology. Geophys. J. R. Astro. Soc., 87: 79-116. Spada, G., Sabadini, R., Yuen, D.A. & Ricard, R., 1992. Effects on post-glacial rebound from the hard rheology in the transition zone. Geophys. J. Int., 109: 683-700. Stix, M. and Roberts, P.H., 1984. Time-dependent electromagnetic core-mantle coupling. Phys. Earth Planet. Inter., 21: 181-201.
r,
Ptt'~ S I~-'S ()J:TtI[ [ A R T H , \ N I ) PLA NETAR'~ INI[RIORS
ELSEVIER
Physics of the Earth and Planetary Interiors 90 (1995) 115-!35
Core-mantle coupling and viscoelastic deformations Marianne Greff-Lefftz *, Hilaire Legros Institut de Physique du Globe, 5, rue Ren~ Descartes, 67084 Strasbourg Cedex, France
Received 27 September 1993; accepted 22 April 1994
Abstract
Writing the angular momentum theorem for the Earth and for its fluid core, we show that there are couplings between the core and the mantle induced by viscomagnetic torque, by external active torque, by topographic torque acting at the core-mantle boundary (CMB) but also by viscoelastic deformations of the CMB which may perturb the axial rotations of the Earth and of the core. We compute these deformations at the CMB induced by the Pleistocenic deglaciation. The time-dependence of inertia tensor perturbations, i.e. the rheology of the mantle, is very important in the calculation of the coupling. Taking into account the passive viscomagnetic torque of tangential traction acting at the CMB, we investigate, for different values and various temporal evolutions of the topographic torque, the perturbations in the rotations of the Earth and of the core induced by the deglaciation, by the constant torque of tidal friction and by the 18.6 year tidal potential. We show that, for these excitation sources, the existence of a constant topographic torque involves the core oscillating with respect to the mantle and thus forbids any large drift of the core with respect to the mantle. However, it seems theoretically possible to have an excitation source with enough energy which involves a shift of the core with respect to the mantle. If the pressure within the fluid core varies with time, the motion of the core with respect to the mantle could be drastically different.
I. Introduction
The rotation of a viscous planet has often been considered, in order to investigate secular variations induced by the last deglaciation (e.g. Munk and McDonald, 1960; Peltier, 1974; Spada et al., 1992). On the other hand, the influence of the c o r e - m a n t l e coupling (viscomagnetic and topographic torque) has been studied by geomagneticists for a rigid mantle (for a review see Jault, 1990). In this paper, we simultaneously take into account the viscoelasticity of the mantle and the
Corresponding author.
different couplings which appear at the c o r e mantle boundary (CMB) because of the magnetic motions within the fluid core, in order to investigate the perturbations of the axial rotations of the Earth and of its core induced by various external sources. The axial rotations of the Earth and its fluid core depend on the exciting sources, i.e. on the resulting variations of torques and inertia tensors, and of couplings acting at the c o r e - m a n t l e interface. We analyse the relative influence of these couplings on the rotations of the mantle and of the core and especially the fundamental role played by the topographic torque on these rotations when the variation of the length of the day
0031-9201/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSDI 0031-9201(95)05078-7
116
M. Greff-Lefftz, H. Legros / Physics of the Earth and Planetary Interiors 90 (1995) 115-135
results from a surface load (owing to Pleistocenic deglaciation), from the torque of tidal friction or from the 18.6 year tide.
pC the density of the homogeneous fluid core, H~ may be written HC-3 - e'3 fcorepcTA ( ~ ' + d A Y )
2. Theoretical approach
2.1. Axial angular momentum of the Earth and its fluid core The equations governing the variations of the rotation velocities for a planet having a homogeneous fluid core with differential motions with respect to a viscoelastic mantle are the equations of conservation of angular momentum. In our earth model, we must write on one hand the conservation of the angular momentum of the global Earth (core + mantle), and, on the other hand, the one for the core, in the Tisserand frame of the mantle. We denotes 12 the mean axial rotation of the Earth: ~ = 12~3, where e'3 is the axial axis. The axial rotation of the global Earth is f~ + o93 where o93/~ is small with respect to unity. Den o t i n g C33 the perturbations of inertia moment of the Earth (small with respect to the mean Earth inertia moment C) owing to the deformations and to the change in mass distributions, the axial component of the angular momentum of the Earth can be written n 3 = C(~
+ 0)3) -'t- ~'~C33
h( O, q~) = ~
where V is the radius vector and Pff the associated Legendre functions, contribute to the axial component of the angular momentum of the core H 3" ¢ In the Tisserand frame of the mantle, noting
( h'2 cos m ~ + g 2 sin mqQ
n=0 m--0
× P ~ ( c o s 0)
(3)
where 0 and q~ are, respectively, the colatitude and the longitude. In a linear approximation (hm/b and g ~ / b << 1), we obtain corePC~A U' d v
7 3]
~ n(n + 1) + 4~0Cb4 ~ to an + 1 F/odd
(n - 1)(n - 2 ) ( n + 2) X (2n - 1 ) ( 2 n - 3 ) ( 2 n - 5)
The fluid core, with respect to the mantle, does not have any quasi-rigid rotation but a combination of various motions: these last ones may be inferred at the CMB from the secular variations of the magnetic field of the Earth, using the geostrophic approximation (Jault, 1990; Gire and Le Mou~l, 1990; Hulot, 1992). Only the zonal toroidal motions of the core with respect to the mantle, noted
n=l
(2)
It obviously depends on the velocity field ~" and on the shape of the interface characterized by the vector V(topography of the CMB). This topography at the CMB (Morelli and Dziewonski, 1987), noted h(O, ~), small with respect to the core radius b, is introduced and expanded in spherical harmonics
(1)
~= - r ~_, t°(F'A VP°)
dt,
+
+
(n - 1)(n + 2 ) ( n + 3)
ho n-3
h°
--n 3 + 6n z + 5n -- 15 ( 2 n - 3 ) ( 2 n - 1)(2n + 3) n 3 + 9n 2 + 10n - 13 (2n-
1)(2n + 3 ) ( 2 n + 5)
h°, h° n+a
] (4)
which may be written, for simplicity 8rr
-igp-Cb5o9c, = cc09 ) where 0)~ is an equivalent rigid rotation. We add also some contributions resulting from the perturbations of the core inertia moment C~3
M. Greff-Lefftz, H. Legros / Physics of the Earth and Planetary Interiors" 90 (199.5) 115-135
(related to the degree 2 deformations at the CMB) small with respect to the mean moment of the core C c. Consequently, the Euler linear equations governing the axial rotation variations may be classically written in the Tisserand frame of the mantle - - for the global Earth r~CtbC = L 3
(5)
for the core CC(tb3 + o5~) + ~qC~3 = Lc3 + F~,
(6)
where L 3 is an external torque acting at the Earth's surface (e.g. the tidal torque), F~ is a passive viscomagnetic torque of tangential traction acting at the CMB and L~ is a topographic torque. Notice in the Euler equations that on one • hand, C~3, L c3 and F 3c play a similar role in term of excitation of w 3. On the other hand, in terms of perturbation of to~, C~3 acts as an excitation source whereas L~ and F~ are passive torques (i.e. related themself to to~). We consider these equations on time-scales from a decade to the secular variations of the geomagnetic field. On one side, on a long timescale, only the time derivative of the perturbations of the inertia tensors of the Earth and of the core owing to viscoelastic deformations (such as the ones resulting from the Pleistocenic deglaciation) or the secular tidal torque can contribute to the angular momentum budget of the core. On the other side, on the time-scale of a decade we have to take into account the elastic effects of the 18.6 year zonal tidal potential. To analyse the different terms that appear in these equations, we have to develop the various couplings that exist at the CMB and to model the viscoelastic deformations. In this paper, we point out the influence of the viscomagnetic and the topographic torque at the CMB on the variations of the axial rotation of the Earth and of the axial differential rotation of the core when the Earth is submitted to external geophysical excitations on a secular time-scale such as the Pleistocenic deglaciation and the tidal torque, or a shorter time-scale such as the 18.6 year tidal potential.
117
2. 2. External geophysical excitation sources
The excitation sources may appear in the angular momentum theorem through perturbations of the inertia tensors of the Earth C33 and of the core C~3 (i.e. deformations) or through an external torque L 3. In this section, we model first the effects of the Pleistocenie deglaciation, next those of the tidal torque on the axial rotations and finally those of the 18.6 year tide. 2.2.1. Pleistocenic deglaciation
Choosing an earth model with geometrical and physical parameters must be done before the computations of deformation resulting from a surface load. Our model consists of four incompressible homogeneous layers: an elastic lithosphere, two linear viscoelastic layers describing the mantle, and an inviscid fluid core (Table la). We use a Maxwell model of rheology for the mantle and from the elastogravitational equations in the Fourier domain, we write the viscoelastic deformations as functions of generalized Love numbers• Various Love numbers, especially k'(A) and h'C(A) (relative, respectively, to the surface mass redistribution potential and to the radial displacement at the CMB induced by a surface load) may be written in the frequency domain (Peltier, 1974)
M k'(A) = k '~ + ~
k', ih + 1/z~
and
i=1
M
h,ic
h'C(a) =h'ce+ E iX + 1/ri
(7)
i~l
The first term (with the superscript e) is the elastic Love number; k'i, h',c and r~ are the residues and the relaxation times (given in Table lb, from Lefftz et al., 1994) of the M modes of the earths' model. We compute the perturbations in the inertia moments of the Earth and of the core when the Earth is submitted to a zonal degree 2 loading potential V~o( t ) 633(t )
6(t) +k'(t)
-2a'
- f~Za2
C~3(t)
c----v -
=
~
*l/~0(t )
kf
qo h,C ( t ~
,
, , , v;,,(t)
and
(8)
118
M. Greff-Lefftz, H. Legros / Physics of the Earth and Planetary Interiors 90 (1995) 115-135
Table la (a) Geometrical and physical parameters for a four-layer model consisting of an elastic lithosphere, a viscoelastic upper mantle, a viscoelastic lower mantle and an inviscid fluid core (both rigidity and density are obtained from the PREM model (Dziewonski and Anderson, 1981)) Radius (km)
Density (kg m -3)
6221 < r < 6371 pl = 3232 5701 < r < 6221 192 = 3666 3480 < r < 5701 P3 = 4904 0 < r < 3480 P4 = 10987
Rigidity (10 II Pa)
Viscosity (1022 Pa s)
0.1=0.6114 /z 2 = 0.9169 0.3 = 2.225 0.4 = 0
b'l~ v2 = 1 v3= 5 v4 = 0
of C33/C and C~3/C c are represented, respectively, in Figs• 2(a) and (b). Today, almost 7000 years after the end of the deglaciation, it remains a perturbation of C33 (i.e. of the J2 coefficient of the geoid) which is in good agreement with the observations and of C~3 (i.e. there is a topography at the CMB of about 10 m which is decreasing in order that the deformations tend to be completely relaxed) (for more details see Lefftz et al., 1994).
2.2.2. Tidal torque where 6(t) is the Dirac distribution, a is the mean radius of the Earth, a is the flattening of the Earth, q0 is the geodetical constant and kf denotes the fluid tidal Love number. In C 3 3 ( t ) there is a combination of a direct instantaneous effect owing to the mass anomaly itself (term in 60)) and of a deformational effect function of the viscoelastic Love number k'(t), whereas in C~3(t) there is only a deformation effect. We use a simplified model for the time-dependent loading potential V~o(t) describing the last glaciation-deglaciation (Fig. 1). The glaciation is characterized by a slow build up over a period of about 90 000 years and the deglaciation vanishes in 10 000 years (Peltier et al., 1986). Taking into account the different ice sheets at the surface of the Earth (Laurentides, Fennoscandia and Antarctica) leads to a zonal degree 2 loading potential at the end of the glaciation, V~o = 235 m 2 s -2 (Spada et al., 1992). The time derivatives
We introduce an external constant torque L 3 acting at the Earth's surface which is responsible for the observed secular deceleration of the Earth owing to the lunar tidal potential• Its value is of about L 3 / C = - 6 x 10-22S -2 (for more details see Hinderer and Legros, 1990).
2.2.3. 18.6 year tidal potential We investigate the effects of the 18.6 year tide on o03 and o0~. At this frequency, the tidal potential is purely zonal (because of the Earth's rotation the other terms in the spherical harmonic expansion of the potential are quasi-diurnal, semi-diurnal...). We have shown that a zonal potential acting on the topography of the CMB cannot create any axial torque (cf. Eq. 14 hereafter). Consequently, the 18.6 year tide can con-
1C
'
I
' t ' f ' ( , End of glaciation.-~-~"~
I
~
f
0.~ 04 O?
Table lb (b) Six relaxation modes associated with the viscoelastic Love numbers for the four-layer model described in Table l(a) Mode name
Relaxation modes ( s - l)
Relaxation times (103 years)
T~
0.8997 x 10-10 0.8429 x 10-10 0.1839 x 10-10 0.4849 x 10- II 0.1682 X 10- It 0.2521 X 10 -13
0.35 0.38 1.7 6.5
T2 M0 C L M~
18,8 1257
Ok: 4
6
g
I0
2
101,4 lime in year Fig. 1. Temporal evolution of the normalized surface loading potential V~o(t)/235 describing the Pleistocenic glaciationdeglaciation, where 235 m 2 s -2 is the maximum value of V~0 at the end of the glaciation. In this simplified model, the glaciation is characterized by a slow build up over a period of about 90000 years and the deglaciation by a collapse lasting 10000 years. Present time is almost 7000 years after the end of deglaciation.
M. Greff-Lefftz, 1t. Legros / Physics of the Earth and Planetary Interiors 90 (1995) l 15-135 '
"~
'
~
I
'
'
'
End ot deglaciation
I
'
'
119
The excitation function in the Euler equations is thus sinusoidal.
'
2.3. Torques acting at the CMB
0
In this section we describe the torques acting at the CMB resulting from magnetic motions within the core (for more details see Jault, 1990) i.e. E~ and L c3"
¢
10
11/ /
10+4
12
/
2.3.1. Viscomagnetic torque
10
11
12
10+4 Time in year Fig. 2. Viscoelastic deformations induced by the glaciation model described in Fig. 1, for the four-layer Earth of Table 1. (a) Time derivative of the inertia tensor of the Earth C33/C. The present value is in good agreement with the observed one deduced from the J2 coefficient of the geoid (J2 = - 8.87 + 2.5 x10-19s-1 (Gegout and Cazenave, 1991)). (b)Time derivative of the inertia tensor of the core C~3/C c. At the present, the deformations are not completely relaxed and its remains a zonal degree 2 term of the radial displacement at the CMB with an amplitude of about 10 m.
tribute to the angular momentum budget only by the perturbations of the inertia moments of the Earth and of the core which are related to degree 2 deformations and thus we only consider the degree 2 zonal potential V20(t) which is assumed to be sinusoidal V20(t ) = V2o sin(~rt + ~P0)
(9)
where ~r is the frequency of 18.6 year tide, ~0 the initial phase and with V2o = 0.17 m 2 s -2. Neglecting the viscoelastic effects, we computed the elastic deformations induced by V2o(t) C33(t ) C 6~3 ( t )
Cc
-2a
q0
F 3¢ = - K" f~CC~oc3
2.3.2. Topographic torque Models of the flow at the top of the Earth's core have been calculated from the secular variation of the geomagnetic field (Gire and LeMou61, 1990; Jault, 1990), and the core-mantle topography h(O, ~o) have been computed by seismologists (Morelli and Dziewonski, 1987). The tangentially geostrophic hypothesis allows computation of the pressure p(O, ~o) at the top of the core from the surface motions: - V p = 2pI~AE. This pressure is expanded in spherical harmonics
(P2c°sm~+qm
sin m p )
n=0 m=0
V20( t ) and
•2a2hCeV2o(t)
(11)
The frictional constant K" is strongly dependent on both the viscosity model within the core and the conductivity model in the mantle. Its value is between 10 -s and 10 -~. In this paper, we take K " = 10 -7 as a referred value.
p(O,~o)= ~
ke
O2a 2 kf
We introduce a viscomagnetic torque of tangential traction, F~, acting at the CMB in order to take into account the viscous friction existing at the CMB (Loper, 1975) and the electromagnetic coupling between the fluid core (assumed to be a perfect conductor) and the slightly conducting mantle (Stix and Roberts, 1984; Jault, 1990). This torque is a function of the existing velocity field in the fluid medium and can be expressed by
x pm(cos 0) (10)
(12)
An axial pressure torque acting on the core, denoted L~, results from the action of the pres-
M. Greff-Lefftz, H. Legros ~Physics of the Earth and Planetary Interiors 90 (1995) 115-135
120
sure field p(O, q~) on the core topography h(O, ~) (Jault and Le Mou~l, 1990)
G = - f fcMBp( O,
°h( °'
dS
(13)
This torque may be theoretically computed (Hinderer et al., 1990) L ~ = _2~.b2 S. m 3 = m=o 2 n + l ( ×
-p,,
. g,,. . . +qnhn)
(n+m)!
mantle. This mechanism described by Jault and Le Mou61 (1990) is summarized in scheme 1. Hereafter we assume, for simplicity, that the core has a rigid differential rotation with respect to the mantle and we note ~bc the phase shift between the flow pattern and the CMB topography, i.e. the longitude variation such as ~ ~¢..O c3. The topographic torque may thus be written ~ m - .g,. + q .mh . m ) L 3c = _ 2 7 r b 2 ~ ....= m = 0 2 n + l [ ( pm_m Xcos m~b¢(t) + ( h ,mp , m + qm_,~x , g, )
(14)
(n - m ) !
Notice that the zonal toroidal motions or the zonal pressures (i.e. m = 0) in the core do not induce any topographic torque. This torque changes the angular momentum (see Euler equations), i.e. perturbs the zonal toroidal motions which drive the pressure field. As a consequence, it involves the changes in the torque itself: there is a new topographic torque between the perturbed flow pattern of the core and the unchanged topography relative to the
L
xsin m6¢(t)]
(n+rn)! (n - m ) !
Jault (1990) has computed the amplitude of L~(~b¢) for the epoch 1980 and the CMB model of Morelli and Dziewonski (1987) as a function of the phase shift ¢¢ and has shown that it looks like a sine curve (Fig. 3) L~(~b¢) = - L o sin(3~b~) with L o = 2 X 1019N m (16)
ZONAL MOTION WITHIN THE CORE RIGID ROTATION OF THE MANTLE
ROTATION OF THE MAGNETIC PRESSURE WITH RESPECT TO THE TOPOGRAPHY FIXED IN THE TISSERAND FRAME OF THE MANTLE
n=o rn~o
NEW TORQUE OF THE NON ZONAL PRESSURE ON THE TOPOGRAPHY
~
i_2rrb2Z
(15)
(qnh -png.)cosrndz(t)+(hnp.+qng.) . . . . ~ ( )
n~0 m=0
Scheme 1.
(-~_~-~
M. Greff-Lefftz, 14. Legros/ Physics of the Earth and Planetary Interiors 90 (1995) 115-135
and of the core when the Earth is submitted to different excitation sources.
2 ~OE*I9
LU
200E+19 150E • 19
0 1~
3. Results
500£+18
t~-
0
121
0.00
-500£*10 - I OOE * I9 - I 50E~19 - 2 0 0 E ' ~ 19 - 2.,50£ * 19
tU
-300E÷I9 0 OC - 3 5 0 E + 1 9
,
,
.
,
.
I00
2 O0
PHASE
SHIFT
,
,
,
300
Fig. 3. Changes in the topographic torque acting on the core with an imaginary rotation of the pattern of the flow with respect to the model of topographyof Morelli and Dziewonski (1987), from Jault and Le Mou~l (1990).
2.4. Equations The conservation of the angular momentum of the core (Eq. 6) may be written as a differential equation of the time-dependent phase shift &c ~c + c K " I ) ~ c + - -
sin(3~b c)
Cm
Cm
C [
L3(t)--
C~
-F
~')C33__ ~~C~3c c
C
C
(17)
It governs an oscillating system with a period dependent on the amplitude of the topographic torque (i.e. L0). From &c(t) and the conservation of the angular momentum of the global Earth (Eq. 5), we can compute the variation of the angular rotation of the Earth to3(t) C
L0
033 + C m K"~t°3 - ~-~- sin(3&c) C [L3(t ) + K , , E I f t L 3 ( t ' ) dt'
= c';t
c
+~
C
.,, C
c
(18)
These equations (18) and (17) govern the temporal evolution of the axial rotation of the Earth
Eqs. (17) and (18) have been integrated using a R u n g e - K u t t a method, assuming that for t = 0 both the phase and the angular velocity are equal to zero. We present the results when the Earth is submitted to a surface load (Pleistocenic glaciation) in Figs. 4-8, and to an external torque (tidal torque) in Figs. 9-13, for various values of the amplitude of the topographic torque ( L 0 = 0 in parts (a) of the figures, L 0 = 2 x 10 Is, 2 x 1016, 2 x 1017, 2 x 1018 and 2 x 1019 N m, respectively in (b), (c), (d), (e) and (f)). For each excitation source, we present the resulting perturbations of to3(t) (in Figs. 4 and 9, respectively, for the deglaciation and the tidal torque), to~(t)(in Figs. 5 and 10) as a function of time but also as a function of respective phase & (in Figs. 6 and 11) and &c (in Figs. 7 and 12). The time derivative of the axial rotation of the Earth 033 is also represented (in Figs. 8 and 13) in order to investigate the influence of the topographic torque on the observed secular deceleration of the Earth (induced by the tidal torque) and on the acceleration of the Earth induced by the postglacial rebound. We first comment on the results when there is no topographic torque (L 0 = 0 N m) and then investigate the influence of the amplitude of L~3 c on ~o3 and w c3"
3.1. Without topographic torque In the case where there is no topographic torque, the viscomagnetic coupling (F~) and the viscoelastic deformations of the CMB ( f l ( ~ 3) will control the temporal evolution of w 3 and to~. Because of the Pleistocenic glaciation, we have at the present (almost 7000 year after the end of the deglaciation) a westward drift of the core with respect to the mantle (to~ < 0) (Fig. 5(a)) which may be correlated with the observed westward drift of the geomagnetic field of the Earth. Both the viscomagnetic coupling ( K " = 10 -7) and the
122
M. Greff-Lefftz, H. Legros / Physics of the Earth and Planetary Interiors 90 (1995) 115-135 c : Lo= 2.E+15 N.m
b : Lo= 2.E+15 N.m
Lo=O N.m
o :
l-
,.q
15
-1
1
~. X _ T . "l
I
~ L t I t I
2 4 10+4
6
12
8
× -10
×-
,I~ILI~I, 0 2 4 X 10+4
6
8
-1 4
6
10
8
e : Lo= 2.E+18 Nm "[ , i , i , i i I ,A ,
d : Lo= 2.E+17 N.m
~
0 2 X 10+4
10 12
t I ,
0 2 x 10+4
I ,
I
t I
4
6
8
12
i
i
i [ , L,
0 2 4 X 10+4
11i
10
~
10 12
i I i I i I , IVl-]]
2
12
4
B
8
I0 12
10+4 Time in year
Time in year
Time in ye0r
8
: Lo= 2.E-.l-19 N.m
iLI
,
I ,
6
Fig. 4. T e m p o r a l evolution of the axial rotation of the E a r t h induced by the Pleistocenic deglaciation for various values of the topographic torque: L 0 = 0 N m (a), L 0 = 2 × 1015 N m (b), 2 × 1016 N m (c), 2 x 1017 N m (d), 2 x 10 TM N m (e) and 2 x 1019 N m (f).
20 ' I
I
I
c : Lo=2.E+16 N.m
b : L0=2,[+15 N.m
a : Lo=O N.m
~
I
'
10~20 1~r-'5 I] ' ~I -' I ' I
g
1
o
o'
~-5
x
8
0 2 4 x 10+4
10
12
I ' I f I ' I ~
~
-
~
if
I
'
I
'
I
'-44
---t -i
,
-~I
I L
0 2 x 10+4
i 1 i I , 6 8 10
1~ 1 ~
12
0
2 x 10+4
e : Lo=2.E+18 N.m
d : Lo=2.E+17 N.m I'
I
, l t I ~ I t ~ 6
8
10 12
f : Lo=2,E+19 N.m
_
t~4
X
,
L ,
2 X 10+4
, I t I
6
8
Time in year
x
10 12
...I_L I i 2 10+4
t I i I 6
8
lime in year
10
12
I i 0 2 X 10+4
, I ~ I 6 8
10 12
lime in year
Fig. 5. T e m p o r a l evolution of the axial differential rotation of the core with respect to the mantle induced by the Pleistocenic deglaciation for various values of the topographic torque: L 0 = 0 N m (a), L 0 = 2 × 1015 N m (b), 2 x 1016 N m (c), 2 × 1017 N m (d), 2 x 1018 N m (e) and 2 × 1019 N m (f).
M. Greff-Lefftz, H. Legros / Physics of the Earth and Planetary Interiors 90 (1995) 115-135 0 : Lo= ON.m I
b : Lo = 2.E+15 N,m
I
I
'
c : Lo = 2,E+16 N.m
I
I
i
50
100
]
t
o
q
x
× 150
100
50
x
5O
d : Lo = 2E+17 N.m
-
150
0
e : Lo = 2.E+18 Nm '
o
10O
~
I
'
I
123
~50
f : Lo = 2.E-f19 Nm
'
'
I
'
1
o
, x
50
1O0
150
50 I O0 Phase shift of mantle
Phase shift of mantle
150
0
50 100 Phase shift of mantle
150
Fig. 6. Axial rotation of the Earth in the phase space induced by the Pleistocenic deglaciation for various values of the topographic torque: L 0 = 0 N m (a), L 0 = 2 x 1015 N m (b), 2 x 1016 N m (c), 2 x 1017 N rn (d), 2 x 1018 N m (e) and 2 x 10 I° N m (f).
l!I
o:Lo=ON.m ¢,
15
p,
b : Lo = 2.E+15 N,m
I
2O-
10
x
-20
0
20
40
60
d : Lo = 2.E+17 N.m -,
i
~
i
,
[
,
j
,
]
'
'
I
e : Lo = 2.E+16 N,m
'
2 ~ - - ~
'
I
'
i
1 1
I
-20
O
20
40
e : Lo = 2.E+18 Nm
60
-20
0
20
40
60
f : Lo = 2.E+19 Nm
,
5 0 ~
~ x
-5
-1[
-0.04-0.02&0O0.020.040.06 Phase shift between core and mantle
:15 -¢ -2 [~ Z 4 6 8 -6 -4 -2 0 2 4 6 8 X 10-,] X 10-4 Phase shift between care and mantle #hose shift between core end mun[le
Fig. 7. Axial rotation of the Earth in the phase space (o~(~c)) induced by the Pleistocenic deglaciation for various values of the topographic torque: L 0 = 0 N m (a), L 0 = 2 x 1015 N m (b), 2 x 1016 N m (c), 2 x 1017 N m (d), 2 x 10 Is N m (e) and 2 x 10 m N m (f).
M. Greff-Lefftz, H. Legros / Physics of the Earth and Planetary Interiors 90 (1995) 115-135
124
[
10
11
c : Lo: 2£+16 N.m
b : Lo= 2.E~15 N.m
a : Lo=O N.m I
2C
J
--
i
,
-
._c:
~-2 12
10+4
9 X 10+4
10
11
12
g x lO+4
e : La= 2.E+18 N.m
d : Lo= 2E+17 N.m
20i
i
10
11
12
N.m ,
f : Lo= 2£+/£ [ i i
-2 11
10
12
x 10+4
9 X 10+4
10
Time in yeer
11
12
Time in year
I ~ r , I 9 10 11 X 10+4 Time in year
,
12
Fig. 8. Temporal evolution of time derivative of the axial rotation of the Earth induced by the Pleistocenic deglaciation for various values of the topographic torque: L 0 = 0 N m (a), L 0 = 2 × 1015 N m (b), 2 × 1016 N m (c), 2 × 1017 N m (d), 2 × 1018 N m (e) and 2×1019 N m ( f ) .
e : Lo : 0 N.m
b : Lo : 2£+15 N.m
c : Lo = 2.E+16 N.m i
i
r
tJ
"5
-lC
-1C
-1~
15
e,::~
-20 '
2
4
6
8
10
1
× lO+4
× 10+4 d : Lo = 2£+17 N.m
'
'
'
'
'
e : LD ~
.
_
~
=
' 2 ,~ 10+4
2.E+18 Nm I
'
]
'
4
10
8
6
f : La = 2£+19 N.m J
J
i
T
i
-10 -15 c>
"- -2C o
4 X 10+4
x 10+4 Time in year
× 10+4 Time in year
Time in year
Fig. 9. Temporal evolution of the axial rotation of the Earth induced by the tidal torque for various values of the topographic torque: L 0 = 0 N m (a), L o = 2 × 10 is N m (b), 2 × 1016 N m (c), 2 × 1017 N m (d), 2 × 1018 N m (e) and 2 × 1019 N m (f).
M. Greff-Lefftz, H. Legros / Physics of the Earth and Planetary Interiors 90 (1995) 115-135
y ]
a :Lo= I
'
I
ON.m '
I
b : Lo = 2.E+15 N.m
125
c : Lo = 2£+16 N.m
'
==
%
1
~×o
,
I
,
2 10+4
1 4
t 6
~
I 8
J
x
,
10
I ~
I
'
J
I
I
10
4
d : Lo = 2£+17 N.m '
,
2 10+4
0 2 × 10+4
e : Lo = 2£+18 N.m
4
6
8
I0
f : L:~ = 2.E+19 N.m
Z_ 2 10+4
><
4
6
8
;0
2
0
6
10
8
0 2 x 10+4
X 10+4
Time in year
Time in year
4
6
8
10
*time in year
Fig. 10. Temporal evolution of the axial differential rotation of the core with respect to the mantle induced by the tidal torque for various values of the topographic torque: L 0 = 0 N m (a), L o = 2 X l0 Is N m (b), 2 x 10 t~ N m (c), 2 x 1017 N m (d), 2 x l0 Is N m (e) and 2 x 1019 N m (f).
c : Lo = 2£+16 Nm
b : L0 = 2£+15 N.m
a:Lo=ON.m I
I
'
I
'
t
'
-10
-1 o
-1 x
-5 ~2 X 10+5
~
-1
d : Lo - 2.E+17 N.m i ~--r , i , [ ,
15
-10
5
x 10+4 e : Lo : 2E+18 Nm
0
-25 -20 -15 -10 X 10+2
-5
0
f : Lo = 2E+19 Nm
-1 o
q--1
o
/
x
-25 -20 -15 -10 -5 x 10+2 Phase shift af mantJe
-25 -20 -15 -10 -5 × 10+2 Phase shift of mantle
-25 -20 -15 - ! 0 -5 x 10+2 Phase sh?[t of mantle
0
Fig. 11. Axial rotation of the Earth in the phase space induced by the tidal torque for various values of the topographic torque: L0=0Nm(a), L0=2X1015 Nm(b),2x1016 Nm(c),2x1017Nm(d),2x1018 N m ( e ) a n d 2 X l 0 I~ N m ( f ) .
126
M. Greff-Lefftz, 11. Legros / Physics of the Earth and Planetary Interiors 90 (1995) 115-135
. '
I
'
b : Lo = 2£+1§ N.m
a : Lo = 0 N,m I ' I ' I 'I]]
t
I
I
I
I
I
I
c : L0 = 2£+16 N.m I_
g B
&_ 10
5
15
20
25
0 5 X 10+2
10+2
I
I
d : Lo = 2£+17 N.m I '
.!I'
10
0.00
15
0.05
e : Lo = 2£+18 N.m
0,10
0.15
f : Lo = 2.E+19 N.m
Z-
x
0.000
0.005
0.010
0.015
0.020
Phase shift between core and mantle
0 5 X I0-4 Phase
10
15
20
-0
5
10
15
20
X 10-5
shift between
core and mantle
Phase shift between
core and mantle
Fig. 12. Axial rotation of the Earth in the phase space (to~(~bC)) induced by the tidal torque for various values of the topographic torque: L 0 = 0 N m (a), L 0 = 2 x 1015 N m (b), 2 x 1016 N m (c), 2 x 1017 N m (d), 2 x 1018 N m (e) and 2 X 1019 N m (f).
a : La = 0 N.m
b
Lo = 2.[+15 N.m I'
c : Lo = LE+16 N.m ' I ' I ' _
oq
-z
( -1[
5 X I0+3
10
15
20
5 x 10+5
d : Lo = 2£+17 N,m , , ,--_
, -,
5
10
x 10+3
15
10
20
2 ~ . 5 X 10+3
e : Lo = 2E+18 Nm
20
IU X 10+3
Time in year
15
Time in year
13
I
10
I
h
15
I
20
I : Lo = 2.E+19 Nm
Z~
b X 10+3
IU
13
Zg
Time in year
Fig. 13. Temporal evolution of time derivative of the axial rotation of the Earth induced by the tidal torque for various values of the topographic torque: L o = 0 N m (a), Lo = 2 x 1015 N m (b), 2 X 1016 N m (c), 2 X 1017 N m (d), 2 x 1018 N m (e) and 2 × 1019 N m
(f).
)IlL Greff-Lefftz, 1t. Legros / Physics of the Earth and Planetary Interiors 90 (1995) 115-13 5
viscoelastic deformations of the CMB must exist simultaneously to obtain this negative value of oL~. As a matter of fact, if the mantle is elastic or if there is no viscomagnetic torque (K" = 0), the core will rotate faster than the mantle and consequently an eastward drift of the core with respect to the mantle will be observed (Lefftz and Legros, 1992). We note also at the present time an acceleration of the angular velocity of the Earth (o53 > 0) which is in good agreement with the observations (Fig. 8(a)). The value of o53 is not significantly affected by the viscomagnetic coupling. These results concerning the temporal variations of w~ and o53 induced by the deglaciation have been reinforced by analyses based on stratified mantle models with steady state or transient rheologies (Lefftz et al., 1994). In general, a viscosity increment in the lower mantle acts to decrease the westward drift of the core whereas a viscosity decrement in the D" layer is responsible for faster relaxation of the CMB topography which increases the westward drift of the core.
a:Lo
127
On the other hand, the tidal torque induces a secular deceleration of the Earth ( o 5 3 = - 6 x 10 -22 rad s -2) (Fig. 13(a)). This torque gives also a perturbation of toc3 (Fig. 10(a)). If there is no viscomagnetic torque ( K " = 0) between core and mantle, the core will not be affected by the deceleration of the mantle and will move in a westward direction. On the contrary, if there is a viscomagnetic torque ( K " = 10 7), w~ tends to a constant positive limit which cancels a part of the negative differential rotation of the core resulting from the deglaciation.
3.2. With topographic torque If there is a topographic torque at the CMB with a constant amplitude in time, the relative magnitudes of the terms F~, L~ and l l ( ~ will govern the solutions of to3 and to~. For the axial rotation of the Earth, the presence of a topography at the CMB does not change significantly the solutions in terms of secular vari-
= ON.m
b : Lo = 2.E+18 N.m
'
'
'
I
'
'
'
l
~
~
~
I
~
J
,
~ 4
'
"S
.5
0 ,_,-, -1
7=
. ~ - 2
x
0 X 10+2
2
'
'
I
~
2
0
2
4
X 10+2
'
1
c : LD = 2.E+19 N.m ' ' ' 1 '
d : Lo
=
2.E+20 N.m
0
><
J
l
J 2
l
l
l
10+2
l 4
0
2
4
X 10+2 Time in year
T]me in year
Fig. 14. Temporal evolution of the axial rotation of the Eart h induced by the 18.6 year tidal potential for various values of the topographic torque: L 0 = 0 N m (a), L 0 = 2 x 10 ~8 N m (b), 2 x 1019 N m (c) and 2 × 102~ N m (d).
128
M. Greff-Lefftz, H. Legros/ Physics of the Earth and Planetary Interiors 90 (1995) 115-135
ations (Figs. 8 and 13), There are some added oscillations (Figs. 4 and 9) with amplitude increasing and period T decreasing, respectively, when the torque L o increases (T---2500 years, 800 years, 250 years and 80 years for, respectively, L 0 = 2 X 1016 N m, 10 ]7 N m, 10 t8 N m and 1019 N m). For small values of L o (until 2 × 1016 N m), the perturbations on to 3 owing to the topographic torque are negligible (in comparison with the curve L 0 = 0) (Figs. 4(b) and (c) and 9(b) and (c)). For higher values of L 0, we obtain nonnegligible perturbations of oJ3 which may be correlated (for the case L 0 = 2 × 10 t9 N m, in Figs. 4(f) and 9(f)) with the observed decade variations in the length of the day (Jault, 1990). Consequently, the existence of a topographic torque at the CMB involves periodic variations in to 3 damped by the viscomagnetic torque (i.e. following the value of K") and finally to3 tends to the same secular value for the deceleration induced by the tidal torque (Fig. 13) and the acceleration induced by the deglaciation (Fig. 8).
On the other hand, the influence of the topographic torque is very important on the differential rotation of the core with respect to the mantle, since two kinds of solutions are possible. If L 0 < 2 X 1016 N m (Figs. 5(b) and (c), 7(b) and (c), 10(b) and (c) and 12(b) and (c)), the energy brought by the external excitation sources is more important than that owing to the topographic torque and consequently the core has enough energy to move with respect to the mantle. On the contrary, for the largest values of L0, the energy brought by the excitations is not sufficient in comparison with L 0 and the core cannot drive but oscillates around a stable phase ~b~ with the period related to L 0 (as given above). The core 'is blocked between two bumps' of the CMB topography. For the case L 0 = 2 x 1016 N m, under the action of the surface load, it is of interest to notice in Fig. 7(c) representing to~(~bc) in the phase space the transition between the state of drift and the state of oscillation: until the end of
a:Lo=ON.m e~
'
' ' 1 ' ' ' 1
"s
'
b : Lo = 2.E+18 I , ~ T
N.m i
' 1
2
o
q kSx
1
i
l
k
l
t
L
×
t
2
4
10+2
~ ×
c : Lo = 2.E+19
,
q 2
,
,
~
d : Lo = 2.E+20
N.m
'
'
I
'
'
'
I
'
i i 0 X 10+2
J
I 2
I
I
i
I 4
I
Time in year
IL 0 X 10+2
4
'
0 X 10+2
'
'
l
'
'
2
'
I 4
L J
N,m J
4
Time in yeer
Fig. 15. Temporal evolution of the axial differential rotation of the core with respect to the mantle induced by the 18.6 year tidal potential for various values of the topographic torque: L 0 = 0 N m (a), L 0 = 2 × 10 TM N m (b), 2 × 1019 N m (c) and 2 × 10 2° N m
(d).
M. Greff-Lefftz, H. Legros / Physics of the Earth and Planetary Interiors 90 (1995) 115-135 a : Lo = 0 N.m
120
b : Lo = 2.E+18 N.m
x
5
10
15
20
0 X 10-5
10-6
1
2
'
I
'
l
'
I
4
d : Lo = 2£+20 N.rn
c : Lo = 2£+19 N.m I
~
'
I
r~
o' ><
x
O-X 10-5
1
2
3
0 X 10-5
4
?
4
Phase shift of man~le
Phase shift d rnontle
Fig. 16. Axial rotation of the Earth in the phase space induced by the 18.6 year tidal potential for various values of the topographic torque: L 0 = 0 N m (a), L{~ ~ 2 × l0 TM N m (b), 2 × 1019 N m (c) and 2 x 10 2~ N m (d).
a:Lo=
b : Lo = 2.E+18 N.m
ON.m
4
!
2
--~
0
2 to-2
='T ~_~-4
×
0
5
10
15
2()
-1
-2
X 10-6
0
1
X 10 5 d : Lo = 2£+20 N.m
c : Lc. = 2£+19 N.m
_7x
-2 -1 0 1 2_ X 10-5 Phase shift between core and m•ntle
2 tQ -4-
-2
0
2
4
X 10-5 Phase shift between core end mon'~/e
Fig. 17. Axial rotation of the Earth in the phase space (w~(~c)) induced by the 18.6 year tidal potential for various values of the topographic torque: L o = 0 N m (a), L o = 2 X 10 TM N m (b), 2 × 10 ~9 N rn (c) and 2 × 102° N m (d).
130
M. Greff-Lefftz, H. Legros/ Physics of the Earth and Planetary Interiors 90 (1995) 115-135
the deglaciation, the core drifts with respect to the mantle and suddenly, the inertia moments pertubations having decreased, it has not enough energy to continue its drift and is blocked with oscillations. Consequently, for the value of L o = 2 × 1019 N m proposed by Jault and Le Mou~l (1990), there is no possible secular drift of the core with respect to the mantle induced by deglaciation nor by tidal torque.
If we linearize the Eqs. (17) and (18), assuming that ~bc does not vary a great deal on the decade (assumption confirmed by the results), we obtain a damped oscillator
3.3. Forced rotations induced by the 18.6 year tidal potential
Neglecting the damping effect (the characteristic frequency of damping c / C m K " f ~ is small in comparison with too), the solutions are
Assuming that we have q~0 = 0, we have integrated Eqs. (17) and (18) for various values of the topographic torque: L o = 0 N m L 0 = 2 × 1018 N m, L 0 = 2 × 1019 N m and L o = 2 × 1020 N m. We present the results for the axial rotation of the Earth in Figs. 14 and 16, for the differential rotation of the core in Figs. 15 and 17 and for the time derivative of the axial rotation of the Earth in Fig. 18.
C ~c+ ~K
IV
"
(19)
with a frequency C Lo 0)20= 3 C---~ C---S
Fo
(sin0.t_
~rsin0)0 t) 0)0
Fo'~
0)c
3
(0)2__0.2)
(COS ~ r t - cos 0)ot)
It is interesting to note that for L 0 around 10 2o N m, 0)o is very close to the frequency 0. of the excitation function and thus we may have a reso-
a:Lo=ON.m
b : Lo = 2.E+18 N.m 20F
2O
I
I
'
'
I
'
'
'
'
~
':t//////////;/////////////f
c
~
l)~b C +0)2~bC=Fo sin(o't+q~o)
-IC -
4
-2C 0 X 10+2
2
4
0
c : Lo = 2.E+19 N.m
2CI
,
,
2
4
X 10+2
I
,
,
,
: Lo = 2.E+20 N.m i
I
i
i
I
lO
c
-213 0
i l l l l ~ l l l 2 X 10+2 Time in year
4
o
2
4
X 10+2 Time in year
Fig. 18. Temporal evolution of time derivative of the axial rotation of the Earth induced by the 18.6 year tidal potential for various values of the topographic torque: L 0 = 0 N m (a), L 0 = 2 × 10 TM N m (b), 2 × 1019 N m (c) and 2 × 10 2o N m (d).
M. Greff-Lefftz, H. Legros/ Physics of the Earth and Planetary Interiors 90 (1995) 115-135 h
a
Temporal evoMion of the torque : To=500[30 yr 1019[~q -'-' t ' I ' I ' ± '
Temporal evolution of the torque : To=IODO00 yr ~019~--~-~--q--' I ~.~L.... ~ I017~ - ~ ~
ld5t- \
,S-
/
0 2 X 10+4
\ 4
/
6
X-
8
10
~
0 2 x 10+4
12
4
6
8
/ 4
2 6
8
10
12
Differential rotation of the core
Oifferential rotation of the core
0 2 X 10+4
131
10
0 2 × 1044
12
4
6
6
10
Time in year
Time m year
Differential rotation of the core
Differential rotation of the core I i
12
It,
_' _2k-U----_ x E-, II ~ I -6 -4 -2 Phase between core end mantle
-2.0
-1,0 -0.5 -1.5 Phase between core and mantle
0.0
Fig. 19. Axial d i f f e r e n t i a l r o t a t i o n of the core with respect to the m a n t l e as a function of time a n d in the p h a s e space i n d u c e d by the P l e i s t o c e n i c d e g l a c i a t i o n for a t i m e v a r i a b l e t o p o g r a p h i c t o r q u e with a respective p e r i o d T{~ of 5(1 × 10 :~ years in (a) and 100 x 103 years in (b).
a Temporal evolution of the torque : To=D0000 yr 1019~q~r~--F -~ ] ' I ' ~L..~
/
0 2 X 10+4
4
6
8
10
12
Temporal evaMion of the torque
,at7L " ~
0 2 X 10+4
Tc 1130000 u
--~i~-~
~0t'~ -~---~-~
/
6
4
8
--...
t'.',
Axial rotation of the Earth
Axial rotation of the Earth
'oF''
, ,--T~--,--~
Tq
T & 2 X 10+4
0
x-10~
4
8
10
12
0 2 X 10+4
4
6
8
Time in year
Time in year
Axial rotation of the Earth
Axial rotation of the Earth
, , , I 0
6
, , , I , ,'v,,,',_~ld
50 100 Phase of the mantle
150
x-10N 0
, ,
,_A_,
10
!£
, , , L=__~
50 IO0 Phase of the mantle
150
Fig. 20. A x i a l r o t a t i o n of the E a r t h as a function of t i m e and in the p h a s e space i n d u c e d by the P l e i s t o c e n i c d e g l a c i a t i o n for a t i m e v a r i a b l e t o p o g r a p h i c t o r q u e with a r e s p e c t i v e p e r i o d T O of 50 × 103 years in (a) and 100 × 103 years in (b).
132
M. Greff-Lefftz, H. Legros/ Physics of the Earth and Planetary Interiors 90 (1995) 115-135
nance effect. This explains the increase of the amplitude of to3 and toc3 between Figs. 14(a) and (d) and 15(a) and (d). 3.4. Time-dependent pressure within the core
In the previous calculation we have assumed that the magnetic pressure within the core is constant in time. This hypothesis seems to be reasonable for excitation sources with decade time-scales such as the 18.6 year tidal potential, but this is not clear for phenomena with timescales of some thousands of years. As a matter of fact, the temporal evolution of the pressure field resulting from the variations of the motions is related to geodynamo processes which cannot be neglected at the time-scale of some thousands of years. Consequently, the topographic torque has to vary with time and its amplitude itself changes. Under these conditions the final differential equations (17 and 18) and their solutions will be completely different.
In order to investigate these effects, we have assumed the pressure to be time-dependent as a sine curve around an averaged value of 2 x 10 [6 N m with a period To, and thus the topographic torque acting at the CMB may be written 27"rt
LC3 = - 10[16+3 c°s(To-0)] sin 3~b c N m
When the excitation source is the Pleistocenic deglaciation introduced in Section 2.2.1, we have represented the axial differential rotation of the core (Fig. 19) as a function of time and in the phase space and the axial rotation of the Earth (Fig. 20) as a function of time and in the phase space, for various values of the period To (the period To is, respectively, 50 kyear and 100 kyear for curves (a) and (b)). In Fig. 19, we see that there is a combination of two effects: the first one is an oscillation on a short time-scale of the core around a stable position similar to the one introduced in the previous
a
b
2 n Temporal evolution of the torque : Ta=5OO00 ),r
0
5
10
(20)
15
20
x 10+.3
201emporal evolution of the toraue : fo=lO0000 yr 1019[~ ' ] ' I " I '
0 x 10+3
5
10
15
20
Differential rotation of the core
Differential rotation of the core
_4
"2 o X 10+3
0 × 10-5
5
lO
15
2O
0 X 10+3
5
10
15
20
lime in year
Time in year
Oifferentiel rotation of the core
Differential rotation of lhe core
2
4
0
2
4
x IO-3
Phase betweencoreand mantle
Phase between core and mantle
Fig. 21. Axial differential rotation of the core with respect to the mantle as a function of time and in the phase space induced by the tidal torque for a time variable topographic torque with a respective period TOof 50 x 103 years in (a) and 100 × 10 3 years in (b).
M. Greff-Lefftz, tl. Legros / Physics of the Earth and Planetary Interiors 90 (1995) 115-135
section; the other one is relative to the time variable amplitude of the topographic torque. At the beginning of the glaciation, L~ is maximum and consequently the core is 'blocked' and oscillates with a weak amplitude between two bumps of the mantle. But progressively, the amplitude of L~ decreases until 2 x 1013 N m allowing a shift of the core with respect to the mantle (between 15 000 and 35 000 years). After that, L~ increases and reaches a critical value, there is an oscillation of the core with respect to the mantle with a large magnitude around a stable position (&~ = - 4 rad). The amplitude of this oscillation decreases with time because of the frictional viscomagnetic torque acting at the CMB. This phenomenon (a shift of the core followed by a large oscillation around a stable position) is reproduced for t greater than 85 000 years. For T O= 100 kyear, the comments are similar but during and after the deglaciation, L~ is maximum and almost constant and thus the curve obtained is very close to that in Fig. 5(f) obtained
a i0!~i ~lt~ITeBpOra' e~aJutlon, at the [orqlu8 : TIO=500,O0]r
for a constant topographic torque with an amplitude of 2 x 1019 N m. The stable position of the oscillation of the core is evidently affected by the amplitude of the surface load. Notice that whatever the value of TO, the core could drift with respect to the mantle: in the horizontal axis the phase variation is about 7r and 27r. On the contrary, if the topographic torque is constant (Fig. 5(f)), the phase varies between - 5 × 10 - 4 and 7 x 10 - 4 tad. The results of Fig. 20 for the axial rotation of the Earth are very close to that obtained for a constant topographic torque with an amplitude of 2 X 1019 N m (Fig. 4(f)), but an oscillation on a short time-scale has to be added because of the time variation of L~. In Figs. 21 and 22, we have represented the axial differential rotation of the core and the axial rotation of the Earth as a function of time and in the phase space, when the excitation source is the tidal torque introduced in Section 2.2.2, for two values of the period T0 (50 kyear and 100
3 20Temporal evolution at the torque : To=I000C'O yr
10 r
,
101198~
~ - ~ ,
/
O X 10+,3
§
10
15
20
,
0 x 10+3
~
I
0 x 10+3
5
'
10
I
15
I~--T
I
,
~~
§
I ~_L
'
,
10
J
15
--~ :,0
Axial rolation of the Earth
Axial rotation of the Earth
0
133
'
q
20
0 X 10+3
5
10
15
Time in year
Time in year
Axial rotation of the Earth
Axial rotation of the Earth
?0
°
-80
-60 -40 -20 Phase of the mantle
0
-80
-60 -40 -20 Phase of [he mantle
0
Fig. 22. Axial rotation of the Earth as a function of time and in the phase space induced by the tidal torque for a time variable topographic torque with a respective period T o of 50 x 103 years in (a) and 100 x 103 years in (b).
134
M. Greff-Lefftz, H. Legros/ Physics of the Earth and Planetary Interiors 90 (1995) 115-135
kyear, respectively, for curves (a) and (b)). The horizontal scale is of about 20000 years and during this time, the topographic torque only decreases. For the differential rotation of the core (Fig. 21), the results for curves (a) and (b) are very similar: during the first 12 kyear for T 0 = 50 kyear (or the first 20 kyear for T O= 100 kyear) the core oscillated around a stable position ~b~ of about 0.2 rad, and after that, the amplitude of L~ being small, we observe a shift of the core with respect to the mantle in an eastward direction. In Fig. 22, we see that there is a secular deceleration of the Earth induced by the tidal torque which is weakly perturbed by the time variation of L c3 (comparison between Figs. 22 and 9). Notice, in Figs. 20 and 22, that the variation of the axial rotation of the Earth is sensitive to the time variation of the pressure within the core and could, in principle, give some information concerning the temporal evolution of the core motion. To conclude, if the topographic torque varies with time, it would be difficult to predict the evolution of the core motions and the rotation of the Earth.
to take into account the different terms of degree n and order m appearing in the computation of the torque. First, we have assumed the magnetic pressure to be constant in time (i.e. L 0 is a constant) which is not true on a long time-scale. The temporal evolution of the pressure field resulting from the variations of the motions is related to geodynamo processes which cannot be neglected on a time-scale of some thousands of years. Consequently, the topographic torque has to vary with time and its amplitude itself can change. We have tested a simple temporal variation taking the pressure as a sine curve with various periods. U n d e r these conditions, it would be difficult to predict the evolution of the motions of the core with respect to the mantle: they would be chaotic.
Acknowledgements We thank Dominique Jault for important suggestions. This research has been supported by C N R S - I N S U - D B T grant (contribution number 16).
References 4. Concluding remarks The axial rotations of the Earth and its fluid core are perturbed by external sources (deglaciation, tidal torque or tidal potential) and by internal sources (magnetic pressures which deform the Earth and the core and which act on the CMB topography). We have investigated the influence of the topographic torque on o) 3 and oJ~ excited by the Pleistocenic glaciation, by the tidal torque and by the 18.6 year tidal potential. T h e constant topographic torque proposed by Jault and Le MouEl is large but even a smaller topography (with a factor 100) would maintain its influence. It introduces the problem of a time variable topography, such as the topography induced by the deglaciation itself (of about 10 m) which cannot be neglected, and also the necessity
Dziewonski, A.M. and Anderson, D.L. 1981. Preliminary reference model PREM. Phys. Earth Planet. Inter. 25: 297356. Gegout, P. and Cazenave, A., 1991. Geodynamic parameters derived from 7 years of laser data of Lageos. Geophys. Res. Lett., 18: 1739-1742. Gire, C. and Le MouEl, J., 1990. Tangentially geostrophic flow at the core-mantle boundary compatible with the observed geomagnetic secular variation: the large-scale component of the flow. Phys. Earth Planet. Inter., 59: 259-287. Hinderer, J. and Legros, H., 1990. Global Consequences of the tidal secular deceleration for the solid earth and its fluid core. In: P. Brosche and J. Sundermann (Editors), Earth's Rotation from Eons to Days. Springer, Berlin, Heidelberg, New York. Hinderer, J., Legros, H., Jault, D. and Le MouEl, J., 1990. Core-mantle topography torque: a spherical harmonic approach and implications for the excitation of the Earth's rotation by core motions. Phys. Earth Planet. Inter., 59: 329-341. Hulot, G., 1992. Observations g~omagn~tiqueset g6odynamo. Thesis, Universit6 Paris VII.
M. Greff-Lefftz, H. Legros/ Physics of the Earth and Planetary Interiors 90 (1995) 115-135 Jault, D., 1990. Variation s6culaire du champ g6omagn6tique et fluctuations de la longueur du jour. Thesis, Universit6 Paris VII. Jault, D. and Le Mou~l, J., 1990. Core-mantle boundary shape: constraints inferred from the pressure torque acting between the core and the mantle. Geophys. J. Int., 101: 233-241. Lefftz, M. and Legros, H., 1992. Influence of viscoelastic coupling on the axial rotation of the Earth and its fluid core. Geophys. J. Int., 108: 725-739. Lefftz, M., Sabadini, R. and Legros, H., 1994. Mantle theology, viscomagnetic coupling at the CMB and differential rotation of the core induced by Pleistocenic deglaciation. Geophys. J. Int., 117: 1-18. Loper, D.E., 1975. Torque balance and energy budget for the precessional driven dynamo. Phys. Earth. Planet. Inter., 11: 43-60.
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Morelli, A. and Dziewonski, A.M., 1987. Topography of the core-mantle boundary and lateral homogeneity of the liquid core. Nature, 325: 678-683. Munk, W.H. and MacDonald, G.J.F., 1960. The Rotation of the Earth. Cambridge University, New York. Peltier, W.R., 1974. The impulse response of a Maxwell earth. Rev. Geophys., 12: 649-669. Peltier, W.R., Drummond, R.A. & Tushingham, A.M., 1986. Post-glacial rebound and transient lower mantle rheology. Geophys. J. R. Astro. Soc., 87: 79-116. Spada, G., Sabadini, R., Yuen, D.A. & Ricard, R., 1992. Effects on post-glacial rebound from the hard rheology in the transition zone. Geophys. J. Int., 109: 683-700. Stix, M. and Roberts, P.H., 1984. Time-dependent electromagnetic core-mantle coupling. Phys. Earth Planet. Inter., 21: 181-201.