Geophysical Journal (1988) 93, 191-193
RESEARCH NOTE
Effect of a uniform sea-level change on the Earth’s rotation and
gravitational field
B. Fong Chao and William P. O’Connor
Geodynamics Branch, Goddard Space Flight Center, Greenbelt, Maryland 20771, USA
Accepted 1987 September 2. Received 1987 August 7; in original form 1987 April 8.
SUMMARY
Global water redistribution between the oceans, atmosphere and continents causes changes in
the Earth’s rotation and gravitational field. To conserve water mass, the effect of the small
uniform change in sea-level must be considered. Explicit formulae are provided for these
sea-level corrections to the gravitational Stokes coefficients, polar motion and length of day.
In two recent publications, this sea-level correction term for polar motion was given
incorrectly. These errors which arose from normalization conventions with the ocean function
are corrected.
Key words: Earth rotation, gravitational field, length of day, polar motion, sea-level change
1 INTRODUCTION
The purpose of this research note is to provide explicit
formulae for calculating the effect of a uniform change in
sea-level (due to a net change in the total ocean-water mass,
as opposed to any thermal effect) on the Earth’s
gravitational field, length of day and polar motion. This is
motivated by two considerations. (1) Changes in some
gravitational field coefficients (such as J2) due to global mass
redistribution are now detectable from satellite geodetic
data (e.g. Yoder et af. 1983; Rubincam 1984). (2)
Unfortunately, the ‘sea-level Correction’ term for polarmotion excitation has been given incorrectly in the recent
literature.
The sea-level correction that is necessary to conserve
global water mass has been used in polar-motion excitation
studies for some time. Explicit formulae were presented by
Hassan (1961) to account for the seasonal variation in
atmospheric mass load over the ocean. The necessity of the
sea-level correction is also discussed in Munk & MacDonald
(1960) and Lambeck (1980). In this note we shall neglect the
geographical variation in surface gravity, which only
amounts to a few per cent at most. In addition, we shall
consider only small sea-level changes, say no more than a
metre, for which the self-gravitational effects of the
displaced water mass can be neglected. (The more general
problem of water redistribution including self-gravitation
when a large sea-level change is involved (such as during ice
ages) has been studied by, e.g. Nakiboglu & Lambeck
(1980)). Under these conditions, a change in ocean mass
results in a uniform sea-level change.
Small sea-level changes (of the order of 1 cm) typically
result from seasonal variations of continental water storage
and atmospheric water vapour. Other sea-level changes
could result from melting of mountain glaciers, or changes
in the mass of the polar ice-sheets following an enhanced
greenhouse effect, a nuclear or impact-induced winter (e.g.
Muller & Morris 1986) or even the possible influx of
extraterrestrial water from comets (Frank, Sigwarth &
Craven 1986). These changes, however, happen on a time
scale much longer than that of the Earth’s viscous response,
which is of the order of decades depending on the Earth’s
internal rheology (e.g. Gasperini et al. 1986). Strictly
speaking, what we discuss below only applies to situations
where the Earth can be treated as an elastic body-for
instance, for time scales shorter than a few years or when
the transient (as opposed to steady-state) response is
desired.
For polar-motion excitation, Merriam (1982) has called
attention to an error in Lambeck’s (1980) sea-level
correction term. Furthermore, the recent work on the
polar-motion excitation due to continental water variations
by Hinnov & Wilson (1987) contains several errors. The
confusion results from disparities in the normalization
conventions, which we shall proceed to clarify.
2
DEFINITIONS
Let us first give the definitions of the physical quantities and
mathematical symbols:
1, m:
Harmonic degree and order, respectively
P,,,,:
associated Legendre function (equation 1)
normalized associated Legendre function (equations 2,3)
6:
Kronecker delta function
U(r):
Gravitational field at point r = (r, 8, A )
8, 1:
co-latitude and east longitude, respectively
em:
191
B. Fong Chao and W . P . O’Connor
192
solid angle, an abbreviation for (6,A)
gravitational constant
Earth’s mass and mean radius, respectively
Earth’s equatorial and polar moments of inertia,
respectively
C/m Slm : normalized harmonic (Stokes) coefficients of U(r)
(equation 4)
zonal J coefficient of U(r) (equation 5)
JI:
F(Q):
ocean function assuming values 1 over ocean and 0
over land
a/m, blrn : normalized harmonic coefficients of F ( 8 ) (equation 6)
A:
‘the change in’
surface mass load (mass per unit area)
0(Q):
Ah :
(uniform) sea-level change (positive for increase)
AM:
total sea-water mass change
Q:
G:
M, R:
A , C:
9
I
integration over unit sphere with element
dQ = sin 6d6 dA
load Love number of degree 1
k;:
YX>
yy: x (O’E) and y (90”E) components of polar motion
excitation, respectively
LOD:
length of day
d8:
S
The relevant mathematical relationships are
U(r) = G M / r
consideration the elastic yielding of the solid Earth under
the loading of A a through the use of the load Love numbers
k; (in (8) the load Love number effect is balanced out by the
polar motion ‘transfer function’ (Munk & MacDonald 1960,
p. 41)). Note that equation (8) is equivalent to the
ex ression in Munk & MacDonald (1960), p. 106, because
5c,o = -J, = -(C - A)/(MRZ)
and
~,,(cos 0) =
*sin
6 cos 6. Note also that { Yx,Yy}is directly
proportional to {C,,, &}. This relation is generally true
when the polar-motion excitation is caused primarily by
mass redistribution and the motion contribution can be
neglected.
For the problem under consideration (that is, a uniform
sea-level change Ah), we have
P
A a ( 8 ) = (1 g ~ m - ~ ) AF(Q).
h
(9)
To find the result of integration over the ocean, we
substitute equations (6) and (9) into (7) and (8). Using the
orthonormality (3) and substituting in the known values for
.M, R, and J, (e.g. Stacey 1977), we obtain
A{Clm,},S
= 8.54 x 10-”(1+ k;)Ah{a,,
b,}/(21+
1)
(10)
{ Yx,Yy}
= -4nR4(1 g ~rn-~)Ah{a,,,bz1}/[m(C
-A)]
= -42.0Ah {a,,, bzl} milliarcseconds (mas),
(11)
where Ah is in cm.
Adopting Lambeck’s (1980) ocean function values (e.g.,
= -0.060,
a,, = -0.040,
b,,= -0.051, a30 = 0.045,
ado= -0.024, etc.) and Farrell’s (1972) values for the load
Love numbers (k; = -0.31, k; = -0.20, k i = -0.13, etc.),
we find, for example,
2 2 (R/r)‘p1,,, (cos 6)
/=0 m = O
JI = -
dmClo
-
F(Q) =
1
x ( C , cos rnh + S,, sin mh) (4)
(5)
~,,,<cos6)(almcos rnh + b , sin mh).
1=0 m=O
(6)
Note that the orthonormality relation (3) does not apply to
the trivial case where m = 0 and sin mA = 0 identically.
3 GRAVITATIONAL FIELD A N D POLAR
MOTION
The physical relationships between the change in surface
mass load and the changes in the Earth’s gravitational field
and polar-motion excitation are derived in, e.g., Chao &
Gross (1987) and Chao et al. (1987):
x {cos ma, sin mh} dS2 (7)
x {cos A, sin A} dQ.
(8)
Equation (7) arises from Newton’s gravitational law, and
equation (8) from the principle of conservation of angular
momentum. In equations (7,8) we have taken into
AJ, = 15.7 x 10-”Ah
(12)
AJ3 = -11.6
(13)
X
10-12Ah
A14 = 5.9 x 10-12Ah
{ Y x ,Yy}
= {1.68,2.14}Ah (mas).
In their study of continental water storage, Hinnov &
Wilson (1987) incorrectly derived a sea-level correction term
for polar motion excitation (our equation 11). In the second
equation on p. 449 of Hinnov & Wilson (1987), the right
side should be multiplied by 4 n l m ; in the first equation
on p. 450, the right side should be multiplied by 4n; and
their final sea-level correction, expression (14) on p. 450,
should be divided by *.
Their ocean contribution to
polar-motion excitation is thus too large by the factor
3.87. Fortunately, the error is a relatively minor one
in numerical terms (about 15 per cent) because the ocean
contribution is itself small compared with the total
continental water contribution, simply because of the
widespread geographical distribution of the oceans.
There are misprints in the polar-motion excitation
equations for the sea-level correction in Lambeck (1980).
Although Lambeck uses a different notation (his a,, and b,,
are the unnormalized ocean function coefficients), his
equation (7.1.10) should contain the factor 4n,and equation
(7.2.4) should contain the factor (-4n). Despite this, the
magnitudes of the x-components of polar-motion excitation
in both equations (7.1.10) and (7.2.5) are correct. However,
m=
Geodetic effect of sea-level change
the y-components are incorrect, as has been pointed out by
Merriam (1982, p. 52), since the ratios of the components
must always be a z l / b z l . This also applies to equation
(7.1.11) and the equation on p. 157. Lastly, the final values
of equation (7.1.10) should be negative.
4
LENGTH OF D A Y
The situation with length of day (LOD) is a bit more
complex because, unlike Y, its integration kernel is not a
spherical harmonic and hence the orthonormality relation
(3) no longer directly applies. From the conservation of
angular momentum we have (e.g. Munk & MacDonald
1960):
ALOD = LOD(1 + k9R4 Aa(SZ) sin’ edSZ/C.
(16)
Using equations (9, (7) and ~,,(cos 0) = fi(3 cos2 8 1)/2, equation (16) becomes
ALOD = (2MR2/3C)LOD [AJ2+ (1 + k;)AM/M],
(17)
where we have used AM=R’~,Aa(SZ)dSZ. Thus, ALOD
can be divided into two parts: the first term is proportional
to AJ,, and the second term is independent of the
geographical distribution of mass changes. Since a Ah of
1cm corresponds to a AM of 3.6 x 10l8g, the application of
equation (12) in (17) then gives, in ps,
ALOD = 17.4 x 101o(AJ, + 4.16 x
Ah) = 75.1 Ah. (18)
Note that the contribution from the second term in (18) is 26
times larger than the first term.
Often we are also concerned with the source (or sink) of
AM. Take, for example, the case where AM comes from
melting of present-day mountain glaciers. Under the
conservation of water mass, the glaciers will have a total
mass change of -AM; and the contribution of the second
term in equation (17) from the glaciers will exactly cancel
that from the oceans. The net effect on LOD by the
combined (glacier + ocean) system can hence be calculated
using only the first term of equation (17):
ALOD = 17.4 X 10’oAJz (ps),
(19)
+
ocean)
where AJ, refers also to the combined (glacier
system. Substituting expression (12), we obtain the part of
193
(19) that is attributed to the oceans:
ALOD = 2.75 Ah (ps).
(20)
This, of course, must be added to the part that is attributed
to the other source (the glaciers in our example), obtained
from equation (19), to give the total effect. Equations
(12-15) and (19) have been used by Chao et al. (1987) and
Chao & O’Connor (1987) in calculating the global geodetic
effects of the seasonal variations of continental water load.
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