Geophys. J. Int. (2003) 152, 124–138
Perturbations of the Earth’s rotation and their implications
for the present-day mass balance of both polar ice caps
Masao Nakada1 and Jun’ichi Okuno2
1 Department
2 Earthquake
of Earth and Planetary Sciences, Faculty of Science, Kyushu University, Fukuoka, 812-8581, Japan. E-mail: [email protected]
Research Institute, University of Tokyo, Tokyo, 113-0032, Japan. E-mail: [email protected]
Accepted 2002 July 29. Received 2002 April 9; in original form 2001 August 4
SUMMARY
The present-day perturbations of the Earth’s rotation are sensitive to the glacial isostatic adjustment (GIA) arising from the Late Pleistocene glacial cycles and also to the recent mass
balance of polar ice caps. In this study, we evaluate the polar wander and the change of degreetwo harmonic of the Earth’s geopotential ( J˙2 ), proportional to the rotation rate, for four Late
Pleistocene ice models. We examine these perturbations as a function of lower- and uppermantle viscosities and lithospheric thickness and rheology (elastic or viscoelastic), in which a
compressible Earth model with elasticity and density given by the seismological model PREM
is used. By considering the observations and predictions including the GIA process arising
from the Late Pleistocene ice and recent mass balance of polar ice caps, we discuss the recent
mass balance of the Antarctic and Greenland ice sheets. We also examine the effects of internal
processes and the melting of mountain glaciers, although this work is only preliminary. The
results shown below seem to be supported even if these effects are included. Two solutions are
obtained for source areas of the recent Antarctic melting. We denote an equivalent sea level
(ESL) rise (mm yr−1 ) from the Greenland and Antarctic ice sheets as ζ˙ GL and ζ˙ AA , respectively,
being positive for melting and negative for growth. One, solution s1, is a solution satisfying
the relationship ζ˙ GL ∼ ζ˙AA (ζ˙GL > 0, ζ˙ AA > 0), and the other, solution s2, generally satisfies
the relationship ζ˙ GL < 0 and ζ˙AA > 0. In most cases, the magnitude of ζ˙ AA for solution
s2 is larger than that for solution s1. The melting area of the Antarctic ice sheet for solution
s1 roughly corresponds to the Weddell Sea region, approximately located on the symmetric
part of Greenland to the Equator. The area for solution s2 is located on the symmetric part of
Greenland to the centre of the Earth. In both solutions, therefore, the polar wander direction
caused by the mass imbalance of each ice sheet is in an opposite direction. The reason for this
is that the observed polar wander direction is nearly identical to the prediction from the GIA
process for the Late Pleistocene ice models. However, it is difficult to independently examine
which solution is better. If we consider a recent ESL rise of ∼0.6 mm yr−1 from the Greenland
ice sheet, then a similar ESL rise of 0.5–1.0 mm yr−1 is also suggested for the Antarctic ice
sheet around the Weddell Sea region. This solution also suggests the lower-mantle viscosity to
be ∼1022 Pa s.
Key words: Antarctica, Earth’s rotation, Glacial isostatic adjustment, Greenland, ice sheet,
sea level.
1 I N T RO D U C T I O N
Perturbations of the moments of inertia associated with the mass
redistribution on and/or within the Earth cause a time-dependent
change of the Earth’s rotation, i.e. a secular wander of the rotation
pole relative to the surface geography (true polar wander, TPW)
and a non-tidal acceleration of the rate of rotation (e.g. Munk &
MacDonald 1960; Lambeck 1980). The glacial isostatic adjustment
(GIA) associated with the Late Pleistocene glacial cycles has been
124
considered to be the main cause of the observed perturbations (e.g.
O’Connell 1971; Nakiboglu & Lambeck 1980; Sabadini & Peltier
1981; Yuen et al. 1982; Yoder et al. 1983; Wu & Peltier 1984;
Rubincam 1984; Peltier 1985; Vermeersen et al. 1996, 1997;
Mitrovica & Milne 1998; Johnston & Lambeck 1999). Perturbations of the Earth’s rotation to surface load redistribution are longwavelength responses of the Earth, and therefore are very sensitive to
the viscosity of the lower mantle. In fact, the above-cited studies indicate that the rotation rate, proportional to the change of degree-two
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2003 RAS
Perturbations of the Earth’s rotation
zonal harmonic of the Earth’s geopotential ( J˙2 ), is predominantly
sensitive to the lower-mantle viscosity.
On the other hand, the polar wander rates are sensitive to both the
lower-mantle viscosity, the density jump at the 670 km density discontinuity, the thickness of the elastic lithosphere and lithospheric
rheology (elastic or viscoelastic) (e.g. Peltier & Wu 1983; Wu &
Peltier 1984; Vermeersen et al. 1996, 1997; Mitrovica & Milne
1998; Nakada 2000, 2002). The response of the 670 km density
discontinuity is called the M1 mode (Peltier 1976). The M1 mode
arises from the deflection of the density discontinuity, and is caused
by the buoyancy between the upper and lower mantle. That is, excitation of the M1 mode is dependent on the magnitude of the density
jump at 670 km depth (ρ670 ), and ρ670 = 388.57 kg m−3 for
the Preliminary Reference Earth Model (PREM) (Dziewonski &
Anderson 1981). Mitrovica & Milne (1998) examined the sensitivity of the polar wander rates to the M1 mode in detail (see also
Nakada 1999). Thus they evaluated the contribution of the M1 mode
to the polar wander rates, and indicated that its contribution is significant for an Earth model with a lower-mantle viscosity (ηlm ) of
less than 5 × 1021 Pa s.
The contribution of the M1 mode depends on the magnitude of
tidal Love number in the fluid limit (fluid Love number; kfT ) mainly
determined by the lithospheric thickness (H) and viscosity (ηlith )
(Nakada 2002). The magnitude of kfT for Earth models with a viscoelastic lithosphere is larger than that for an elastic lithosphere,
and it is smaller for a thicker elastic lithosphere than for a thinner
one. The relative strength of the M1 mode to polar wander rates is
given by k2T (M1)/kfT , where k2T (M1) is the magnitude of the
tidal Love number (k2T ) for the M1 mode. For Earth models with
a viscoelastic lithosphere, the polar wander rates are insensitive to
the lithospheric thickness and the viscosity, because kfT is constant
regardless of its thickness and viscosity. Then, the polar wander
rates are mainly sensitive to the lower-mantle viscosity, particularly
for ηlm ≥ 2 × 1021 Pa s (Nakada 2002). For Earth models with an
elastic lithosphere, the value of k2T (M1)/kfT is larger for a thicker
lithosphere than for a thinner one. As a result of this relation, the
magnitude of polar wander rates increases with increasing elastic
lithosphere thickness for Earth models with ηlm ≤ 1022 Pa s.
The recent mass imbalance of polar ice caps and mountain
glaciers also influences perturbations of the Earth’s rotation. The
present-day mass imbalance of the Antarctic and Greenland ice
sheets contributes significantly to the Earth’s rotation (e.g. Gasperini
et al. 1986; Yuen et al. 1987; Peltier 1988; Sabadini et al. 1988;
Mitrovica & Peltier 1993; Trupin 1993; James & Ivins 1997). Peltier
(1988) also suggested that the present-day melting of mountain
glaciers tabulated by Meier (1984) is also important in predicting J˙2 and polar wander. The state of the recent mass balance of
the Antarctic and Greenland ice sheets is, however, uncertain (e.g.
Douglas et al. 1990; Jacobs et al. 1992; Trupin 1993; James & Ivins
1995, 1997; McConnell et al. 2000). We define the equivalent sea
level (ESL) as a change in ice mass divided by the area of the ocean.
Then, the present-day ESL rise inferred from the tide gauge data
seems to be 1–3 mm yr−1 (Barnett 1984; Peltier & Tushingham
1989; Nakiboglu & Lambeck 1991; Lambeck et al. 1998; Douglas
2001), although the recent sea level rise may be partly contributed
by thermal expansion of the oceans (e.g. Wigley & Raper 1987).
More recently, Mitrovica et al. (2001) suggested that the melting of
the Greenland ice complex over the previous century was equivalent
to the 0.6 mm yr−1 sea level rise. They applied the GIA correction
to the tide gauge data, and evaluated the residuals between the observations and GIA predictions. By comparing the spatial pattern
between the sea level predictions owing to the recent mass balance
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2003 RAS, GJI, 152, 124–138
125
and the residuals, they suggested a melting of the Greenland ice
sheet of ∼0.6 mm yr−1 .
Observed polar wander is also sensitive to the mass redistribution associated with mantle convection (Spada et al. 1992;
Steinberger & O’Connell 1997) and mountain building (Vermeersen
et al. 1994). We initially assume that the GIA process arising from
the Late Pleistocene glacial cycles and the recent mass balance of
polar ice caps cause the present-day perturbations of the Earth’s rotation. By considering the predictions of the polar wander and J˙2
as a result of four Late Pleistocene ice models proposed by several
studies, we investigate the recent mass balance of both polar ice caps.
We also examine the effect of TPW induced by internal processes
(Steinberger & O’Connell 1997) on estimates of the recent mass
balance. We cannot estimate the ESL rise of each polar ice cap independently, but it may be possible to estimate its value by considering
the results obtained for the Greenland ice complex (Mitrovica et al.
2001).
2 M AT H E M AT I C A L F O R M U L AT I O N
Perturbations of the Earth’s rotation associated with the Late Pleistocene glacial cycles are predicted based on a Maxwell viscoelastic
Earth model in this study. Nakada (2002) has discussed the details
of the method adopted here, and we give a brief review here. In the
unperturbed state, the Earth rotates with an almost constant angular
velocity about the mean rotation pole. In the perturbed state, the
angular velocity is written in terms of dimensionless quantities
m 1 , m 2 and m 3 as (e.g. Munk & MacDonald 1960; Lambeck 1980)
= (m 1 , m 2 , 1 + m 3 ).
(1)
The quantities m 1 and m 2 describe the displacement of the rotation
axis in the directions 0◦ and 90◦ E, respectively. The rotational perturbations associated with the surface load distribution are given by
(e.g. Lambeck 1980; Wu & Peltier 1984)
i
k T (t)
1 i
δ(t) + k2L (t) ∗ I − İ + 2 T ∗ m
ṁ + m =
σr
C−A
kf
(2)
and
ṁ 3 = −
1 d R I33 ∗ δ(t) + k2L (t) ,
C dt
(3)
R
R
+iI23
and σr = (C − A)/A. A
where m = m 1 +im 2 , I = I13
and C are the equatorial and polar moments of inertia, respectively.
R
R
R
I13
, I23
and I33
are components of inertia tensor associated
with the surface load distribution on a rigid Earth. σr is the Chandler
wobble frequency for a rigid Earth. k2T , kfT and k2L are the degree-two
tidal Love number, the fluid limit of k2T and the degree-two load
Love number, respectively. The asterisk denotes a time convolution.
The İ term on the right-hand side of eq. (2) can be neglected for
periods much longer than a day (e.g. Wu & Peltier 1984). The first
term on the left-hand side, corresponding to the Chandler wobble, is
safely neglected in predicting the secular term of the rotation vector
(Vermeersen & Sabadini 1996; Mitrovica & Milne 1998). Then,
equations describing the secular terms of the Earth’s rotation are
expressed as
t
R
R
m 1 (t) =
(t) +
k2L (t − τ )I13
(τ ) dτ
I13
Aσr
0
t
1
T
k (t − τ )m 1 (τ ) dτ
+ T
(4)
kf 0 2
126
m 2 (t) =
M. Nakada and J. Okuno
t
R
R
(t) +
k2L (t − τ )I23
(τ ) dτ
I23
Aσr
0
t
1
k T (t − τ )m 2 (τ ) dτ
+ T
kf 0 2
and
ṁ 3 (t) = −
t
1 d
R
R
(t) +
k2L (t − τ )I33
(τ ) dτ .
I33
C dt
0
3 R E S U LT S
(5)
(6)
The first term on the right-hand side of eqs (4)–(6) represents the
perturbations of the inertia tensor caused by the surface load on a
rigid Earth. The second term represents the perturbations related to
the deformation of the Earth caused by the surface load. The third
term of eqs (4) and (5) is associated with the deformation induced by
the shift of the rotation axis (Munk & MacDonald 1960; Lambeck
1980; Wu & Peltier 1984). Eq. (6) can be related to the rate of change
of the degree-two zonal harmonic of the Earth’s geopotential, J˙2 ,
and is given by (Wu & Peltier 1984)
−3C
J˙2 (t) =
ṁ 3 (t),
(7)
2Me a 2
where Me and a are the mass and mean radius of the Earth,
respectively.
In this study, we evaluate Love numbers caused by a Heaviside
load H (t), k2L,H (t) and k2T,H (t), using an initial-value approach developed by Hanyk et al. (1996, 1998), and solve eqs (4)–(6) using
a finite-difference approach (Nakada 2002). We model the tempoR
ral terms of m j (t) and I j3
(t) ( j = 1, 2) as a series of Heaviside
increments:
∞
δm ij H (t − it)
(8)
m j (t) =
i=0
and
R
(t) =
I j3
∞
R,i
δI j3
H (t − it).
(9)
i=0
Then, the convolutions in eqs (4)–(6) are expressed as
∞
δm ij k2T,H (t − it)
k2T (t) ∗ m j (t) =
(10)
i=0
and
R
(t) =
k2L (t) ∗ I j3
∞
R,i L,H
δI j3
k2 (t − it),
(11)
i=0
where k2L,H (t) = k2L (t) ∗ H (t) and k2T,H (t) = k2T (t) ∗ H (t) are used.
By applying eqs (8)–(11) to eqs (4) and (5), increments δm nj ( j =
1, 2) at time t = nt are given by
−1
k2T,H (0)
n
δm j = 1 −
kfT
n
R,i L,H
R
×
δI j3 k2 (nt − it)
I j3 (nt) +
Aσr
i=0
n−1
k2T,H (nt − it)
i
(12)
+
− 1 δm j .
kfT
i=0
m n3 at time t = nt is given by
m n3 = m 3 (nt)
n
1
R,i L,H
R
=−
δ I33 k2 (nt − it) .
I33 (nt) +
C
i=0
(13)
Eqs (12) and (13) with initial values m j = 0 ( j = 1, 2, 3) are solved
using a finite-difference approach with t = 1 yr.
3.1 Earth and load models
A compressible Earth model with elasticity and density structure
given by the seismological model PREM (Dziewonski & Anderson
1981) is used in predicting the perturbations of the Earth’s rotation
caused by the Late Pleistocene glacial cycles and mass imbalance of
the Antarctic and Greenland ice sheets. The density jump at 670 km
depth (ρ670 ) is 388.57 kg m−3 for PREM. The core is taken to be
inviscid. The parameters defining the rheological structure of the
Earth’s mantle are: (1) the lithospheric thickness (H); (2) the lithospheric rheology; (3) the upper-mantle viscosity between the bottom
of the lithosphere to 670 km depth (ηum ); and (4) the lower-mantle
viscosity from 670 km depth to the core–mantle boundary (ηlm ). The
polar wander rates are sensitive to the lithospheric rheology (elastic
or viscoelastic) as indicated by Nakada (2000, 2002). We therefore
adopt Earth models with an elastic lithosphere (ηlith = ∞) and with
a viscoelastic lithosphere of ηlith = 1024 Pa s. The following results
are insensitive to the lithospheric viscosity for Earth models with
a viscoelastic lithosphere, even if we adopt a depth-dependent viscosity for the lithosphere (Nakada 2002). The lithospheric thickness
adopted in this study is 50, 100 and 200 km.
We adopt several ice models describing the melting histories
during the last deglaciation in order to examine the perturbations
of the Earth’s rotation. One is the ICE3G ice model proposed by
Tushingham & Peltier (1991). The second model is ARC3 + ANT4b
by Nakada & Lambeck (1987, 1988, 1989). The Arctic ice model
ARC3 includes the Laurentide, Fennoscandia and Barents–Kara ice
sheets, in which the ice model for the Laurentide and Fennoscandia
ice sheets corresponds to ICE1 of Peltier & Andrews (1976). The
ESL, defined by the (meltwater volume)/(area of ocean surface), is
89 m for ARC3. The ESL contribution from the Barents–Kara ice
sheet is 12 m, corresponding to the maximum model of Denton &
Hughes (1981). ANT4b was constructed from the reconstruction at
the last glacial maximum (LGM) by Denton & Hughes (1981), the
present ice sheet thickness (Drewry 1982) and sea level observations in the far field (Nakada & Lambeck 1988, 1989). The ESL
contribution is 24 m, and there is a minor Mid- to Late-Holocene
melting of approximately 3 m. The ESL from 2 kyr BP to the present
is assumed to be zero to clearly show the effect of the recent melting of polar ice caps. The predictions of the Earth’s rotation based
on the ANT4 ice model (Nakada & Lambeck 1989), no Mid- to
Late-Holocene melting, are identical to those based on the ANT4b,
although we do not show that here.
The melting histories for the Barents and Kara sea regions have
been examined extensively based on the sea level observations for
the past 15 kyr (Lambeck 1993, 1995; Kaufmann 1997). Denton
& Hughes (1981) proposed two models for the Barents and Kara
ice sheet. One is a maximum model where the Barents and Kara ice
sheet was continuous to the Fennoscandian ice sheet, corresponding
to ARC3. The other is a minimum model where the ice sheet existed
in the Barents sea region only. Lambeck (1995) indicated that the sea
level observations are consistent with the predictions based on the
minimum model. The northern hemisphere ice model constructed by
Lambeck (1995) is referred to as ARC4 (see also Okuno & Nakada
1999). The difference of ESL between ARC3 and ARC4 is 9 m. We
also use ARC4 + ANT4b to predict the perturbations of the Earth’s
rotation.
More recently, Nakada et al. (2000) have examined the sea level
observations at eight sites along the coast of Antarctica to investigate
the melting histories of Antarctic ice complexes. They suggested that
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2003 RAS, GJI, 152, 124–138
Perturbations of the Earth’s rotation
the ice thicknesses removed from the LGM around the Weddell Sea
are significantly thicker than those around the Ross Embayment. The
ice model constructed from the sea level observations is referred to
as ANT5. The ESL contribution is 17 m, and the melted ice is mainly
from the Weddell Sea region. The ice model of ARC4 + ANT5 is
also used in this study. We therefore examine the perturbations of the
Earth’s rotation based on four ice models; ICE3G, ARC3 + ANT4b,
ARC4 + ANT4b and ARC4 + ANT5.
It is necessary to construct the melting history before the LGM.
The method of extending the ice sheet history back in time prior
to the LGM is identical to the approach discussed by Nakada &
Lambeck (1987) and Lambeck & Nakada (1992). For the earlier
cycles, the ice covered area waxes and wanes according to the amplitude of ESL by amounts that are proportional to the height of
the ice column at the LGM for an adopted ice model. The surface
load history with N saw-tooth load cycles is adopted for the melting
history before the LGM. Each cycle is characterized by a 90 kyr
glaciation (loading) phase and by a 10 kyr deglaciation (unloading)
phase. The melting history from the LGM is an ice model proposed
by several studies, for example, ICE3G. We adopt N = 10 in this
study.
The perturbations of the moments of inertia are derived from the
surface load redistribution arising from the Late Pleistocene glacial
cycles, i.e. melting of the ice sheets and its related sea level variations. In this study, we calculate the sea level variations including
the effects of palaeotopography, the distribution of the ice sheet
and also the meltwater influx in ice-covered and subgeoidal geographic regions (Milne et al. 1999; Okuno & Nakada 2001). The
pseudospectral approach formulated by Mitrovica & Peltier (1991)
is adopted to evaluate the sea level variations associated with the
GIA.
3.2 Perturbations of the Earth’s rotation arising
from the Late Pleistocene glacial cycles
A precise evaluation of ice and water loads is required in order to
predict the perturbations of the Earth’s rotation arising from the Late
Pleistocene glacial cycles (e.g. Mitrovica & Peltier 1993; Johnston
& Lambeck 1999). Fig. 1 depicts the predictions of polar wander
speed, polar wander direction and J˙2 as a function of the lowermantle viscosity. The ice model adopted here is ICE3G. The uppermantle viscosity and lithospheric thickness are 5 × 1020 Pa s and
100 km, respectively. The lithospheric viscosity for the predictions
of (a), (c) and (e) is infinite (elastic lithosphere) and that for (b), (d)
and (f) is 1024 Pa s. The model results with ‘el’ and ‘ve’ correspond
to an Earth model with an elastic lithosphere and a viscoelastic lithosphere (ηlith = 1024 Pa s), respectively. The results denoted by the
notation of ‘full’ (solid line) are based on the surface load distribution including ice and water loads. We also show the contributions
of ice (dashed line) and water loads (dotted line) separately to understand the change in the polar wander direction. The shaded regions
show the ranges of the observations for the polar wander (McCarthy
& Luzum 1996) and J˙2 (Nerem & Klosko 1996). In this study, we
use these values as observational constraints.
These figures indicate that the contribution of water load is significant for the polar wander and less significant for J˙2 . The contribution of water load to polar wander speed amounts to 40 per
cent, and it is significant for Earth models with a viscoelastic lithosphere. The polar wander direction changes significantly for Earth
models with ηlm ≥ 5 × 1022 Pa s. The ice load term is constant
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2003 RAS, GJI, 152, 124–138
127
regardless of the lower-mantle viscosity. The distribution of water load is, however, sensitive to the lower-mantle viscosity as inferred from the sensitivity of sea level variations to the mantle viscosity and lithospheric thickness (e.g. Nakada & Lambeck 1989;
Tushingham & Peltier 1991). Thus the direction owing to water
load depends on the lower-mantle viscosity (Figs 1c and d). Moreover, the contribution of water load to polar wander speed increases
relative to the increasing lower-mantle viscosity (Figs 1a and b).
As a result, the predicted polar wander direction changes significantly for ηlm ≥ 5 × 1022 Pa s. It is therefore important to evaluate the water loads precisely to examine polar wander speed and
direction.
Fig. 2 shows the sensitivity of these predictions to the lithospheric
thickness and viscosity. The lithospheric thickness adopted here is
50, 100 and 200 km. The upper-mantle viscosity is 5×1020 Pa s, and
the ice model is ICE3G. The model result denoted by M(el, 100), for
example, corresponds to an Earth model with an elastic lithosphere
of H = 100 km. Predictions of J˙2 are less sensitive to the thickness
and viscosity of the lithosphere. Those for the polar wander direction are insensitive to the viscosity of the lithosphere, but are sensitive to the lithospheric thickness for Earth models with ηlm ≥ 2 ×
1022 Pa s. The polar wander speed is sensitive to the thickness of
an elastic lithosphere, particularly for Earth models with ηlm ≤
1022 Pa s (Yuen et al. 1982; Peltier & Wu 1983; Peltier 1984; Wu
& Peltier 1984; Mitrovica & Milne 1998). For Earth models with a
viscoelastic lithosphere, however, the predicted polar wander speed
is less sensitive to the thickness as discussed in Section 1. This is
true for Earth models with a depth-dependent lithospheric viscosity
(Nakada 2002). As a result, the predictions with a viscoelastic lithosphere (ηlith = 1024 Pa s) are insensitive to the lithospheric thickness
for a range of ηlm ≥ 2 × 1021 Pa s (Fig. 2a). For Earth models with a
viscoelastic lithosphere, we also note that the predicted polar wander speed is smaller than the observation, particularly for ηlm ≥ 2 ×
1021 Pa s.
Fig. 3 illustrates the sensitivity of the predictions to the uppermantle viscosity. The upper-mantle viscosities adopted here are
2 × 1020 , 5 × 1020 and 1021 Pa s. The model result denoted by
M(el, 2e20), for example, corresponds to an Earth model with
ηum = 2 × 1020 Pa s and ηlith = ∞. The lithospheric thickness
is 100 km, and the ice model is ICE3G. These figures indicate
that the predictions are relatively insensitive to the upper-mantle
viscosity.
Predicted perturbations of the Earth’s rotation will depend on
an adopted ice model describing the melting histories of the Late
Pleistocene ice sheets. An evaluation of the sensitivity to the ice
model is therefore required in examining the mass balance of polar
ice caps based on these observables. Fig. 4 illustrates the predictions based on four ice models proposed by several studies. The
lithospheric thickness is 100 km, and the upper-mantle viscosity is
5 × 1020 Pa s. The ice models denoted by AC3AT4b, AC4AT4b
and AC4AT5 correspond to ARC3 + ANT4b, ARC4 + ANT4b
and ARC4 + ANT5, respectively. These predictions show a similar
trend, but there are significant differences among these predictions.
The differences of model results between ARC3 + ANT4b and
ARC4 + ANT4b are attributed to ice sheet distribution of the Barents and Kara ice sheet. That is, the ARC3 and ARC4 correspond
to the maximum and minimum ice models by Denton & Hughes
(1981), respectively. Those for the results between ARC4 + ANT4b
and ARC4 + ANT5 are insignificant. We discuss the mass balance of
the Antarctic and Greenland ice sheets using these results for four ice
models.
128
M. Nakada and J. Okuno
elastic lithosphere
viscoelastic lithosphere
2.0
2.0
(a)
M(el,full)
M(el,ice)
M(el,water)
PW Rate (°/Ma)
1.5
ICE3G
ηum=5x1020 Pa s
H=100 km
ηlith=∞
1.0
0.5
0.5
1022
1023
150
PW Direction (°E)
H=100 km
1022
1023
1022
1023
1022
Lower mantle viscosity (Pa s)
1023
150
(d)
100
100
50
50
0
0
-50
-50
-100
1021
1022
1023
10
-100
1021
10
(f)
(e)
-dJ2/dt (x1011 yrs−1)
ηum=5x1020 Pa s
ηlith=1024 Pa s
0.0
1021
(c)
8
8
6
6
4
4
2
2
0
1021
M(ve,full)
M(ve,ice)
M(ve,water)
1.5
1.0
0.0
1021
ICE3G
(b)
1022
Lower mantle viscosity (Pa s)
1023
0
1021
Figure 1. Predictions of (a) polar wander speed, (c) polar wander direction and (e) J˙2 as a function of lower-mantle viscosity, in which an elastic lithosphere (infinite viscosity) is assumed (results denoted by ‘el’). Those for (b), (d) and (f) correspond to Earth models with a viscoelastic lithosphere
of 1024 Pa s (results denoted by ‘ve’). The upper-mantle viscosity and lithospheric thickness are 5 × 1020 Pa s and 100 km, respectively. The ice
model is ICE3G from Tushingham & Peltier (1991). The results denoted by ‘full’ (solid line) are based on the surface load distribution including ice
and water loads. The contributions owing to ice (dashed line) and water loads (dotted line) are also depicted to understand the change of the polar
wander direction. The shaded regions show the range of the observations for the polar wander (McCarthy & Luzum 1996) and J˙2 (Nerem & Klosko
1996).
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2003 RAS, GJI, 152, 124–138
Perturbations of the Earth’s rotation
2.0
2.0
ICE3G
M(el,50)
M(ve,50)
M(el,100)
M(ve,100)
M(el,200)
M(ve,200)
1.5
PW Rate (°/Ma)
(a)
1.0
0.5
ηum=5x1020 Pa s
H=100 km
1022
0.0
1021
1023
-70
-70
-75
-75
PW Direction (°E)
PW Direction (°E)
0.0
1021
(b)
-85
-90
-80
1023
1022
1023
1022
Lower mantle viscosity (Pa s)
1023
(b)
-90
-95
-100
1021
1022
1023
-100
1021
10
10
(c)
(c)
8
-dJ2/dt (x1011 yrs−1)
-dJ2/dt (x1011 yrs−1)
1022
-85
-95
6
4
2
8
6
4
2
1022
Lower matle viscosity (Pa s)
1023
Figure 2. Predictions of (a) polar wander speed, (b) polar wander direction
and (c) J˙2 as a function of the lower-mantle viscosity and lithospheric thickness. The lithospheric thickness adopted here is 50, 100 and 200 km. The
upper-mantle viscosity is 5 × 1020 Pa s and the ice model is ICE3G. The
results denoted by ‘el’ and ‘ve’ correspond to Earth models with an elastic
lithosphere and viscoelastic lithosphere of 1024 Pa s, respectively. The observations for polar wander (McCarthy & Luzum 1996) and J˙2 (Nerem &
Klosko 1996) are represented by shaded regions.
C
M(el,2e20)
M(ve,2e20)
M(el,5e20)
M(ve,5e20)
M(el,1e21)
M(ve,1e21)
1.0
0.5
0
1021
ICE3G
1.5
PW Rate (°/Ma)
(a)
-80
129
2003 RAS, GJI, 152, 124–138
0
1021
Figure 3. Predictions of (a) polar wander speed, (b) polar wander direction
and (c) J˙2 as a function of the lower-mantle viscosity and the upper-mantle
viscosity. The upper-mantle viscosity adopted here is 2 × 1020 , 5 × 1020 and
1021 Pa s. The lithospheric thickness is 100 km and the ice model is ICE3G.
The results denoted by ‘el’ and ‘ve’ correspond to Earth models with an
elastic lithosphere and viscoelastic lithosphere of 1024 Pa s, respectively. The
observations for polar wander (McCarthy & Luzum 1996) and J˙2 (Nerem
& Klosko 1996) are represented by shaded regions.
130
M. Nakada and J. Okuno
ηum=5x1020 Pa s, H=100 km, ηlith=1024 Pa s
ηum=5x1020 Pa s, H=100 km, ηlith=∞
2.0
2.0
(a)
1.5
PW Rate (°/Ma)
(b)
M(el,ICE3G)
M(el,AC3AT4b)
M(el,AC4AT4b)
M(el,AC4AT5)
M(el,J92)
1.5
1.0
1.0
0.5
0.5
0.0
1021
1022
1023
-50
0.0
1021
PW Direction (°E)
1023
1022
1023
1022
Lower mantle viscosity (Pa s)
1023
(d)
-60
-60
-70
-70
-80
-80
-90
-90
-100
1021
1022
1023
10
-100
1021
10
(e)
-dJ2/dt (x1011 yrs−1)
1022
-50
(c)
(f)
8
8
6
6
4
4
2
2
0
1021
M(ve,ICE3G)
M(ve,AC3AT4b)
M(ve,AC4AT4b)
M(ve,AC4AT5)
M(ve,J92)
1022
Lower mantle viscosity (Pa s)
1023
0
1021
Figure 4. Predictions of (a) polar wander speed, (c) polar wander direction and (e) J˙2 as a function of the lower-mantle viscosity and the Late Pleistocene ice
models, in which elastic lithosphere is assumed (results denoted by ‘el’). Those for (b), (d) and (f) correspond to Earth models with a viscoelastic lithosphere of
1024 Pa s (denoted by ‘ve’). The upper-mantle viscosity is 5 × 1020 Pa s, and the lithospheric thickness is 100 km. The ice models adopted here are ICE3G by
Tushingham & Peltier (1991), ARC3 + ANT4b (results denoted by AC3AT4b) by Nakada & Lambeck (1988), ARC4 + ANT4b (results denoted by AC4AT4b)
by Lambeck (1995) and Nakada & Lambeck (1988) and ARC4 + ANT5 (results denoted by AC4AT5) by Lambeck (1995) and Nakada et al. (2000). The
results denoted by ‘J92’ are based on the ice model J92 scenario describing the recent mass imbalance of the Antarctic ice sheets (James & Ivins 1995). The
equivalent sea level rise is 0.45 mm yr−1 . The observations for polar wander (McCarthy & Luzum 1996) and J˙2 (Nerem & Klosko 1996) are represented by
shaded regions.
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2003 RAS, GJI, 152, 124–138
Perturbations of the Earth’s rotation
3.3 Inference of mass balance of the Antarctic
and Greenland ice sheets
We first assume that the observed polar wander and J˙2 are completely caused by the GIA process arising from the Late Pleistocene
glacial cycles and the recent mass imbalance of the Antarctic and
Greenland ice sheets. Then, the residuals between the observations
and the predictions arising from the Late Pleistocene ice have to be
explained by the predictions owing to the mass imbalance of polar
ice caps. In Fig. 4, we show the predictions of the polar wander
speed based on the ice model J92 scenario describing the recent
mass imbalance of the Antarctic ice sheets (James & Ivins 1995).
The equivalent sea level rise is 0.45 mm yr−1 . The predictions for
the polar wander direction and J˙2 are 166.7◦ E and 1.76–1.90 ×
10−11 yr−1 , respectively. The rates of change of the inertia tensor, İ R13 , İ R23 and İ R33 , are assumed to be constant during the past
100 yr. These predictions depend slightly on the lower-mantle viscosity, but the sensitivity is very low compared with that for the Late
Pleistocene ice models. In the following calculations, we therefore
assume that the predictions of polar wander and J˙2 caused by the
recent mass imbalance are represented by elastic responses with
sufficient accuracy.
We assume a circular ice cap, with mass M and radius α, centred
R
R
on colatitude and E-longitude λ. The moments of inertia, I13
, I23
R
and I33 , associated with this surface load on a rigid Earth are given
by Wu & Peltier (1984)
a21
1
R
I13
cos α(1 + cos α) cos θ sin θ cos λ +
= −Ma 2
(14)
2
5a00
R
I23
= −Ma
2
and
R
= −Ma 2
I33
b21
1
cos α(1 + cos α) cos θ sin θ sin λ +
2
5a00
2a20
1
cos α(1 + cos α) (3 cos2 θ − 1) −
,
6
15a00
(15)
(16)
where a00 , a20 , a21 and b21 are unnormalized coefficients of the
ocean function (e.g. Lambeck 1980). In eqs (14)–(16), a spatially
uniform water load is assumed. Then, the elastic responses of
ṁ j ( j = 1, 2), ṁ Ej , are obtained using eqs (4), (5), (14) and (15),
and are given by
(/Aσr ) 1 + k2L,E R
ṁ Ej =
(17)
İ j3 ,
1 − k2T,E kfT
where k2L,E and k2T,E are the elastic part of the Love numbers. Eqs (6)
and (16) also give the elastic responses, ṁ E3 and J˙E2 , which are expressed as
1 1 + k2L,E İ R33
C
(18)
3 1 + k2L,E İ R33 .
2
2Me a
(19)
ṁ E3 = −
and
J˙E2 =
In this study, we infer the recent mass imbalance of the Antarctic
and Greenland ice sheets by using the χ 2 misfit:
obs
2
3
ṁ j − ṁ PGR
− ṁ Ej
j
χ2 =
.
(20)
σ j2
j=1
PGR
In eq. (20), ṁ obs
is
j is the observed value with an error of σ j , ṁ j
the prediction for an ice model describing the Late Pleistocene ice
and ṁ Ej is caused by the recent mass imbalance. In the following
C
2003 RAS, GJI, 152, 124–138
131
calculations, we assume α = 0 because the influence of α on the
moments of inertia is very small for α < 20◦ . Thus, we search for
the effective areas of melted ice for the polar ice caps.
By considering the previous results for a recent ESL rise (e.g.
Douglas 2001), we adopt a search range for the ESL rise associated
with the recent mass balance: (1) −1 ≤ ζ˙ total ≤ 2 (mm yr−1 ), (2)
−2 ≤ ζ˙ GL ≤ 2 (mm yr−1 ) and (3) −2 ≤ ζ˙ AA ≤ 2 (mm yr−1 ). ζ˙ total
is the sum of ζ˙GL and ζ˙AA . ζ˙ GL and ζ˙ AA are the values of ESL for the
Greenland and Antarctic ice sheets, respectively, being positive for
melting and negative for growth. We compute the χ 2 residual for a
given ζ˙total with a step of 0.1 mm yr−1 . The values of ζ˙ GL and ζ˙AA
satisfying a given ζ˙ total are also changed with a step of 0.1 mm yr−1 .
The recent mass imbalance of the Greenland ice sheet is assumed to
be located on the land region of 10◦ ≤ θ ≤ 30◦ (N) for the colatitude
and 300◦ ≤ λ ≤ 340◦ (E) for the E-longitude. The area examined
for the Antarctic ice sheet is the whole land region. The χ 2 residual
is evaluated with a step of θ = 1◦ and λ = 1◦ . For ice models of
ICE3G, ARC3 + ANT4b, ARC4 + ANT4b and ARC4 + ANT5,
we search for the effective source areas of the melted ice and the
values of ζ˙ GL and ζ˙ AA . The rheological parameters adopted here are
H = 100 km, ηlith = ∞ and 1024 Pa s, ηum = 5 × 1020 Pa s and
1021 ≤ ηlm ≤ 1023 Pa s. Earth models with H = 200 km (ηlith = ∞
and 1024 Pa s) are also examined for the ICE3G.
Figs 5 and 6 illustrate the solutions for ICE3G that give the minimum χ 2 residuals and also satisfy the observations of both polar
wander and J˙2 . The lithospheric thickness is 100 km for Fig. 5
and 200 km for Fig. 6. The values of ζ˙ AA for an adopted ζ˙ total are
depicted in Figs 5(a), (b) and 6(a), (b). Those for the E-longitude
and the lower-mantle viscosity are also plotted. In this study, we
divide the Antarctic ice sheet into three regions, A1, A2 and A3;
(A1), 50◦ ≤ λ ≤ 180◦ (E); (A2), 180◦ ≤ λ ≤ 290◦ (E); and (A3),
290◦ ≤ λ ≤ 360◦ (E) and 0◦ ≤ λ ≤ 50◦ (E). The main regions represented by A2 and A3 roughly correspond to the Ross Embayment
and Weddell Sea, respectively. Two general trends are recognized
in recent sea level rises for both polar ice caps. One is a solution
satisfying the relationship ζ˙GL ∼ ζ˙ AA (ζ˙GL > 0 and ζ˙ AA > 0), and
the other solution generally satisfies ζ˙GL < 0 and ζ˙AA > 0. These
solutions are referred to as s1 and s2, respectively. The magnitude
of ζ˙ AA for s1 is generally smaller than that for s2, and it is slightly
larger for Earth models with H = 200 than for 100 km. The reasons for the existence of these two solutions are discussed in the
next section. The areas for solutions s1 and s2 correspond approximately to the Weddell Sea region and the area of 120◦ ≤ λ ≤ 150◦
(E), respectively. The source areas for the melted ice are effective
only, but similar solutions are obtained for Earth models with a
lithospheric thickness of 100 and 200 km. The relationship for the
solution between the lower-mantle viscosity (ηlm ) and ζ˙ total suggests
that the permissible solutions for s1 are ηlm ≥ 2 × 1021 Pa s and
the lower-mantle viscosity for s1 and s2 increases with increasing
ζ˙total for a range of ηlm ≤ 3 × 1022 Pa s. The above-mentioned two
trends are generally recognized for Earth models with an elastic
and viscoelastic lithosphere, although there exist differences for the
solutions with ηlm ≤ 3 × 1021 Pa s (see also Figs 1 and 2).
Fig. 7 shows the solutions for ice models of ARC3 + ANT4b,
ARC4 + ANT4b and ARC4 + ANT5, in which ηum = 5 ×
1020 Pa s, H = 100 km and ηlith = 1024 Pa s are assumed. The
trends recognized in the results for ICE3G are also detected in these
results. We also evaluate the solutions considering the recent melting
of mountain glaciers tabulated by Meier (1984). This ice model with
an ESL rise of 0.38 mm yr−1 is referred to as MTGLA here. Fig. 7(d)
shows the solutions with an ice model of ICE3G + MTGLA,
in which ηum = 5 × 1020 Pa s, H = 100 km and ηlith = 1024 Pa s
132
M. Nakada and J. Okuno
η um=5x1020 Pa s, H=100 km, η lith=∞
ESL rise for Antarctica (mm yr−1)
2.0
2.0
(a)
1.5
1.0
1.0
0.5
0.5
0.0
0.0
-0.5
-0.5
ICE3G
-1.0
-1.0
A1
A2
A3
-2.0
-1
-0.5
0
0.5
1
1.5
-1.5
2
360
-2.0
-1
-0.5
0
0.5
1
1.5
2
0
0.5
1
1.5
2
0
0.5
1
1.5
2
360
(c)
300
300
240
240
180
180
120
120
60
60
0
-1
-0.5
0
0.5
1
1.5
2
23.0
0
-1
(d)
-0.5
23.0
(f)
(e)
LOG [ηlm (Pa s)]
(b)
1.5
-1.5
E-longitude (°E)
η um=5x1020 Pa s, H=100 km, η lith=1024 Pa s
22.5
22.5
22.0
22.0
21.5
21.5
21.0
-1
-0.5
0
0.5
1
1.5
2
ESL rise for polar ice caps (mm yr−1)
21.0
-1
-0.5
ESL rise for polar ice caps (mm yr−1)
Figure 5. Solutions that give the minimum χ 2 residuals in eq. (20) (see the text) and also satisfy the observations of both polar wander (McCarthy & Luzum
1996) and J˙2 (Nerem & Klosko 1996). The upper-mantle viscosity is 5 × 1020 Pa s, the lithospheric thickness is 100 km and the ice model is ICE3G. (a), (b)
Relationship between the present-day total ESL rise (ζ˙ total ) and the ESL rise of the Antarctic (ζ˙ AA ); (c), (d) relationship between the ζ˙total and the location
(E-longitude) of the Antarctic melted ice; and (e), (f) relationship between ζ˙ total and the lower-mantle viscosity. Results for (a), (c) and (e) correspond to Earth
models with an elastic lithosphere, and those for (b), (d) and (f) are for Earth models with a viscoelastic lithosphere of 1024 Pa s. The results with ‘A1’ indicate
the source area of 50◦ ≤ λ ≤ 180◦ (E) for the recent melting of the Antarctic ice sheet. The areas denoted by ‘A2’ correspond to 180◦ ≤ λ ≤ 290◦ (E), and
those denoted by ‘A3’ are for 290◦ ≤ λ ≤ 360◦ (E) and 0◦ ≤ λ ≤ 50◦ (E).
C
2003 RAS, GJI, 152, 124–138
Perturbations of the Earth’s rotation
η um=5x1020 Pa s, H=200 km, η lith=∞
η um=5x1020 Pa s, H=200 km, η lith=1024 Pa s
ESL rise for Antarctica (mm yr−1)
2.0
2.0
(a)
1.5
1.0
1.0
0.5
0.5
0.0
0.0
ICE3G
-0.5
-0.5
A1
A2
A3
-1.0
-2.0
-1
-0.5
0
0.5
1
1.5
-1.0
-1.5
2
360
-2.0
-1
-0.5
0
0.5
1
1.5
2
0
0.5
1
1.5
2
0
0.5
1
1.5
2
360
(c)
300
E-longitude (°E)
(b)
1.5
-1.5
240
180
180
120
120
60
60
-0.5
(d)
300
240
0
-1
0
0.5
1
1.5
2
23.0
0
-1
-0.5
23.0
(f)
(e)
LOG [ηlm (Pa s)]
133
22.5
22.5
22.0
22.0
21.5
21.5
21.0
-1
-0.5
0
0.5
1
1.5
2
ESL rise for polar ice caps (mm yr−1)
21.0
-1
-0.5
ESL rise for polar ice caps (mm yr−1)
Figure 6. As in Fig. 5, except for the case of a lithospheric thickness of 200 km.
are assumed. The trends for the solutions are less clear compared
with those with no contribution of mountain glaciers, which may
be caused by incomplete estimates of the melting from mountain
glaciers (Trupin et al. 1992; Mitrovica et al. 2001).
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2003 RAS, GJI, 152, 124–138
Preliminarily, we examine the effect of TPW induced by internal processes on estimates of the mass imbalance of both polar
ice caps. This component is referred to as ṁ MC
here. Steinberger
j
& O’Connell (1997) estimated the TPW owing to the mantle flow
134
M. Nakada and J. Okuno
ESL rise for Antarctica (mm yr−1)
2.0
2.0
(a)
1.5
1.5
1.0
1.0
0.5
0.5
0.0
0.0
ARC3+ANT4b
-0.5
A1
A2
A3
-1.0
-1.5
-2.0
-1
-0.5
0
0.5
1
1.5
ESL rise for Antarctica (mm yr−1)
-0.5
-1.0
-1.5
2
2.0
1.5
-2.0
-1
ARC4+ANT4b
-0.5
0
0.5
1
1.5
2
2.0
(d)
(c)
1.5
1.0
1.0
0.5
0.5
0.0
0.0
-0.5
-0.5
-1.0
-1.0
-1.5
(b)
ARC4+ANT5
-2.0
-1 -0.5 0 0.5 1 1.5 2
ESL rise for polar ice caps (mm yr−1)
-1.5
ICE3G+MTGLA
-2.0
-1 -0.5 0 0.5 1 1.5 2
ESL rise for polar ice caps (mm yr−1)
Figure 7. Relationship between the present-day total ESL rise (ζ˙ total ) and the ESL rise of the Antarctic (ζ˙AA ) for ice models of ARC3 + ANT4b, ARC4 +
ANT4b, ARC4 + ANT5 and ICE3G + MTGLA (see the text). The ice model MTGLA corresponds to the recent melting of mountain glaciers tabulated by
Meier (1984). The upper-mantle viscosity is 5 × 1020 Pa s and the thickness of a viscoelastic lithosphere (1024 Pa s) is 100 km.
induced by mantle density heterogeneities inferred from seismic tomography data. The polar wander direction and speed estimated by
them are 24◦ W and 0.37 deg Myr−1 , respectively, for a viscosity
structure obtained through geoid modelling by Forte et al. (1994).
We evaluate the mass imbalance of both polar ice caps by setting
MC
ṁ obs
instead of ṁ obs
j − ṁ j
j in eq. (20). The contribution of internal
processes to J˙2 is assumed to be zero. We adopt the values estimated by Steinberger & O’Connell (1997) for ṁ MC
j ( j = 1, 2). The
χ 2 residuals are also computed for a model with 60◦ W and 0.37 deg
Myr−1 because the estimates by them are dependent on the viscosity structure of the mantle. Results for these models are depicted in
Fig. 8. Two trends obtained for ṁ MC
= 0 are also detected in the
j
solutions with contributions of internal processes (ṁ MC
= 0). In
j
=
0
is
generally
larger
solution s1, the magnitude of ζ˙ AA for ṁ MC
j
= 0. Moreover, the trends of the solutions includthan that for ṁ MC
j
ing contributions of mountain glaciers become clear compared with
those for ṁ MC
= 0 (see Figs 5b, 7d and 8b).
j
4 D I S C U S S I O N A N D C O N C LU D I N G
REMARKS
Perturbations of the Earth’s rotation owing to the GIA process were
examined for several melting history models describing the Late
Pleistocene ice. In these calculations, we adopt a compressible Earth
model with elasticity and density given by the seismological model
PREM. The predictions for ice models examined here show a similar trend, but there are significant differences among these predictions. We also separately examined the effects of ice and water loads
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2003 RAS, GJI, 152, 124–138
Perturbations of the Earth’s rotation
ESL rise for Antarctica (mm yr−1)
2.0
1.5
2.0
(a)
1.0
0.5
0.5
0.0
0.0
-0.5
24°W
-1.5 0.37°/Ma
-2.0
-1
-0.5
0
0.5
1
ICE3G
-0.5
A1
A2
A3
-1.0
1.5
24°W
-1.5 0.37°/Ma
2
ESL rise for Antarctica (mm yr−1)
2.0
1.5
-2.0
-1
-0.5
ICE3G+MTGLA
0
0.5
1
1.5
2
2.0
(c)
1.5
1.0
1.0
0.5
0.5
0.0
0.0
-0.5
-0.5
-1.0
(b)
1.5
1.0
-1.0
135
-1.0
60°W
-1.5 0.37°/Ma
(d)
ICE3G
-2.0
-1 -0.5 0 0.5 1 1.5 2
ESL rise for polar ice caps (mm yr−1)
-1.5
60°W
0.37°/Ma
ICE3G+MTGLA
-2.0
-1 -0.5 0 0.5 1 1.5 2
ESL rise for polar ice caps (mm yr−1)
Figure 8. Relationship between the present-day total ESL rise (ζ˙total ) and the ESL rise of the Antarctic (ζ˙AA ) for models with the effect of internal processes.
The polar wander direction and speed for (a) and (b) are 24◦ W and 0.37 deg Myr−1 , respectively, corresponding to the estimates by Steinberger & O’Connell
(1997). Those for (c) and (d) are 60◦ W and 0.37 deg Myr−1 , respectively. The adopted ice model is shown in each figure. The upper-mantle viscosity is 5 ×
1020 Pa s and the thickness of the viscoelastic lithosphere (1024 Pa s) is 100 km.
associated with the GIA process. These calculations indicate that
the predictions of polar wander are very sensitive to the water load
redistribution associated with the sea level variations in the GIA
process.
The sensitivity of the polar wander and J˙2 to the parameters
defining the Earth’s rheology was also investigated. The prediction
of J˙2 is mainly sensitive to the lower-mantle viscosity, and less
sensitive to the upper-mantle viscosity, lithospheric thickness and
viscosity as indicated by previous studies (e.g. Wu & Peltier 1984;
Mitrovica & Milne 1998; Johnston & Lambeck 1999). On the other
hand, the polar wander speed is also sensitive to the thickness of
an elastic lithosphere (e.g. Wu & Peltier 1984). For Earth models
with a viscoelastic lithosphere, however, the predicted polar wander
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2003 RAS, GJI, 152, 124–138
speed is less sensitive to the lithospheric thickness for a range of
ηlm ≥ 2 × 1021 Pa s as indicated by Nakada (2002). In this case,
the predictions of the polar wander and J˙2 associated with the GIA
are mainly sensitive to the lower-mantle viscosity. The recent mass
balance of the Antarctic and Greenland ice sheets was examined
considering these sensitivity tests.
The recent mass balance of the Antarctic and Greenland ice sheets
and the mountain glaciers also influence the present-day Earth’s rotation (e.g. Peltier 1988; Sabadini et al. 1988; James & Ivins 1997).
In this study, we have inferred the source areas of the melted ice
by minimizing the χ 2 residuals between the observations and predictions including the Late Pleistocene ice and the recent mass
imbalance (see eq. 20). We adopt the J˙2 observation by Nerem
136
M. Nakada and J. Okuno
Figure 9. Schematic figures showing the polar wander for solutions of (a) s1 and (b) s2 (see the text). The polar wander vectors arising from the recent mass
imbalance of both polar ice caps are in opposite directions.
& Klosko (1996) and polar wander observations by McCarthy &
Luzum (1996). The contribution associated with the recent imbalance is assumed to be an elastic response. Then, we searched for
the possible source areas of the mass imbalance from the Antarctic and Greenland ice sheets. Our study suggests two solutions for
the ESL rises from Antarctica (ζ˙ AA ) and Greenland (ζ˙GL ) regardless of the melting history models. One, solution s1, is a solution
satisfying a relationship of ζ˙ GL ∼ ζ˙ AA (ζ˙ GL > 0, ζ˙ AA > 0), and the
other, solution s2, satisfies a relationship of ζ˙ GL < 0 and ζ˙AA > 0.
The magnitude of ζ˙AA for s2 is generally larger than that for s1. The
melting area of the Antarctic ice sheet approximately corresponds
to the Weddell Sea region for s1 and to the area of 120◦ ≤ λ ≤ 150◦
(E) for s2. These two regions are roughly symmetrical to the South
Pole. These two solutions are also supported even if we take account
of the effect of TPW induced by internal processes (Steinberger &
O’Connell 1997). In solution s1, the magnitude of ζ˙AA with internal processes (ṁ MC
= 0) is slightly larger than that for ṁ MC
= 0.
j
j
For models with the melting of mountain glaciers, the trends of
the solutions for ṁ MC
= 0 become clear compared with those for
j
ṁ MC
=
0.
j
We interpret these solutions by denoting the representative location of the melted ice for Greenland as (θg , λg ). For the solution s1
with the melted ice mainly around the Weddell Sea region, the representative location (θa , λa ) is given by θa ∼ 180◦ − θg and λa ∼ λg .
This is easily obtained by considering a geometrical relationship
between the Greenland and the Weddell Sea that these two regions
are approximately located on symmetric positions relative to the
Equator. The polar wander vectors arising from these meltings with
ζ˙ GL ∼ ζ˙ AA are in an opposite direction, and have similar magnitudes. On the other hand, the solution s2 satisfies a condition of
θa ∼ 180◦ − θg and λa ∼ λg − 180◦ . That is, the position of each
melted ice region is located on a symmetric position with respect
to the centre of the Earth. Then, the polar wander direction arising
from the growth of the Greenland ice sheet is in an opposite direction
to that for the melting of the Antarctic ice sheet. Schematic figures
representing polar wander for these two solutions are illustrated in
Fig. 9.
The reason for the existence of two possible solutions is that the
predicted polar wander directions based on the Late Pleistocene ice
models are nearly identical to the observed value by McCarthy &
Luzum (1996). This probably works as a boundary condition in adjusting the polar wander speed and J˙2 in eq. (20). It is, however,
difficult to examine independently which solution is better. More
recently, Bouin & Vigny (2001) investigated the GPS data, and suggested that the plate tectonics framework can reasonably explain
the horizontal movement except for the site of the Weddell Sea
(O’Higgins site). The large difference between the observation and
prediction associated with the plate motion is not reconciled even if
we consider the plate deformation associated with the GIA arising
from the melting of the Late Pleistocene Antarctic ice sheet (James
& Ivins 1998; Nakada et al. 2000). Then, Bouin & Vigny (2001)
suggested the recent melting around the Weddell Sea region. The
solution s1, with a significant melting around the Weddell Sea region
supported by Jacobs et al. (1992), may be a plausible solution.
We tentatively adopt the solution s1 as an acceptable solution.
Then, the ESL rise is caused by the recent melting of the Greenland
ice sheet and the Antarctic ice sheet around the Weddell Sea region.
More recently, Mitrovica et al. (2001) investigated the tide gauge
data and suggested a recent melting of the Greenland ice sheet with
ζ˙ GL ∼ 0.6 mm yr−1 . Then, the solution of s1 gives a similar ESL rise
for the melting around the Weddell Sea region, resulting in an ESL
rise of 1–1.5 mm yr−1 from Greenland and Antarctica. Nakada &
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2003 RAS, GJI, 152, 124–138
Perturbations of the Earth’s rotation
Lambeck (1988) suggested a minor Mid- to Late-Holocene melting
of ESL ∼3 m from the Antarctic ice sheet. This minor melting,
with an average value of ζ˙ AA ∼ 0.5 mm yr−1 , may therefore be
contributing to the ESL rise inferred from this study. Moreover,
this solution suggests that the lower-mantle viscosity is ∼1022 Pa s
(Figs 5 and 6).
ACKNOWLEDGMENTS
We are grateful to the anonymous reviewers for their constructive
comments. This work was partly supported by the Japanese Ministry
of Education, Culture, Sports, Science and Technology (Grand-inAid for Scientific Research nos 11640417 and 14540396) to MN.
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