Geoscience Reference
In-Depth Information
⎛
⎛
⎞
⎛
⎞
⎞
a
2
cos
λ
cos
θ
sin
λ
sin
θ
d
p
s
d
d
p
s
d
L
p
⎝
⎝
⎠
+
⎝
⎠
⎠
d
=−
(
a
+
h
(θ, λ))
sin
λ
cos
θ
−
cos
λ
sin
θ
θ
d
λ.
λ
θ
−
sin
θ
0
(60)
Equivalent expressions for the equatorial and axial mountain torques can be derived
from Feldstein (
2008
) and Iskenderian and Salstein (
1998
), respectively, after elim-
inating the pressure gradient in Eq. (
60
) using integration by parts. The latter study
also demonstrates the significant contribution of the topographic pressure torque to
the axial component of
L
(
s
)
→
(
a
)
, which may amount to 65% at periods shorter than
15d.
Gravitational Torque
By deploying the thin layer approximation in the set-up of Eq. (
59
), de Viron et al.
(
1999
) showed that the analytical expression for the gravitational torque
L
g
is very
similar to that of the
L
p
. In fact, as it is the case for the pressure torque, the equatorial
gravitational torque can also be divided into two constituents: a global portion arising
from the flattening of the geopotential surface (expressed by Earth's form factor,
J
2
)
and another portion due to the local anomalies of the geoid (comprising all degrees
and orders except that of
J
2
). The contribution of this local gravitational torque to
L
(
s
)
→
(
a
)
has been shown to be negligible at all frequencies (de Viron et al.
2001b
)
and will not be treated any further.
As noted in de Viron et al. (
2005
), the pressure torque on the Earth's bulge as
well as the corresponding gravitational torque on the
J
2
term are both caused by the
pushing of atmospheric surface pressure distributed globally as spherical harmonic
of degree 2 and order 1 (Fig.
8
). A simple expression for this
ellipsoidal torque
L
E
has been deduced by de Viron et al. (
1999
) after applying in Eq. (
59
) spherical
harmonic expansions on the topography, the geopotential and the surface pressure
field. Restricting the topography and geopotential to degree 2 and order 0,
L
E
can
be expressed in terms of a torque acting on the atmosphere
L
20
+
L
20
L
E
=
a
3
⎛
⎞
a
3
⎛
⎞
u
20
˜
p
21
−
p
21
J
2
p
21
0
J
2
˜
12
5
12
5
⎝
⎠
−
⎝
⎠
,
=
−
u
20
p
21
0
(61)
where
u
20
=−
3
f
and
f
represents the dynamical flattening of the Earth (de Viron
andDehant
1999
). The real and imaginary parts of the degree 2 and order 1 coefficient
in the spherical harmonic development of the surface pressure are labeled
p
21
and
2
/
p
21
.
By substituting numerical values for
f
and
J
2
, it can be shown that the gravitational
torque compensates about one half of the ellipsoidal pressure torque. Nonetheless,
the sum of both effects is by far the largest signal in the equatorial part of the total
torque.
˜
