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Lunisolar torque on the atmosphere and Earth's rotation
Planetary and Space Science 50 (2002) 323 – 333
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Lunisolar torque on the atmosphere and Earth’s rotation C. Bizouard ∗ , S. Lambert Observatoire de Paris, 61, avenue de l’Observatoire, 75014 Paris, France Received 11 June 2001; received in revised form 8 October 2001; accepted 19 October 2001
Abstract The atmospheric e0ect on Earth’s rotation is classically computed from atmospheric angular momentum (AAM) series by considering that the whole AAM variations are transmitted to the solid Earth. In fact, such a procedure is an approximation because it ignores that the atmosphere exchanges also angular momentum with the moon and the sun. The computation of the lunisolar gravitational torque on the atmosphere is the core of the present study. It is derived analytically and then estimated from pressure data over the period 1975 –2000. The mean surface pressure is the main contributor, at the level of 1017 N m for the equatorial component, 1016 N m for the axial one, whereas the thermally driven pressure variations provide about 10% of the variance. This torque plays a signi:cant role in the budget of the atmospheric angular momentum, but does not account for the inconsistency between the present series of the atmospheric angular momentum and the interaction torque between the solid Earth and the atmosphere. From the tidal waves of this torque we derive accurately the contribution of the atmosphere to the lunisolar precession–nutation of the Earth (0:31 marcsec=year for the precession in longitude, 46 microarcseconds for the nutation in 18.6 years). ? 2002 Elsevier Science Ltd. All rights reserved. Keywords: Earth’s rotation; Atmospheric angular momentum; Lunisolar precession–nutation; Tidal torque on the atmosphere
1. Introduction The e0ect of the atmosphere on the Earth rotation is usually computed from the angular momentum variation of this >uid layer. This approach implicitly assumes that the variations of the atmospheric angular momentum (AAM) are only caused by the interaction torque between the atmosphere and Earth. In other words the exchanges of angular momentum between the Earth and the atmosphere are not in>uenced by forces external to the Earth and to the atmosphere, and the exchanges obey to the balance of the angular momentum between the solid Earth and the atmosphere. However the gravitational interaction with the Moon and the Sun creates a tidal torque acting on the atmosphere. ˜ grav has to be considered in the angular This tidal torque momentum balance of the solid Earth–atmosphere system according to: ˜A dH int ˜ A→T ˜ grav ; = − + (1) dt int ˜ A→T ˜ A is the AAM and the interaction torque where H that the atmosphere exerts on the solid Earth (by virtue of ∗
Corresponding author. E-mail address: [email protected] (C. Bizouard).
the principle of the action and reaction the atmosphere is int ˜ A→T submitted to the opposite torque − ). ˜T The variation of the angular momentum of the Earth H is thus given by ˜T ˜A dH dH int grav ˜ ˜ : (2) = A→T = − − dt dt If the external torque was signi:cant, it should be removed from the AAM variations. This means that part of the AAM variations is not associated with interaction process between the solid Earth and the atmosphere. In the diurnal band and semidiurnal bands, where gravimetric forces are the most important, the magnitude of the tidal torque could compete with that of the time derivative of the AAM. This hypothesis has to be considered in the prospect of earlier studies (Dehant et al., 1996; de Viron, 1999; de Viron et al., 1999), which have put forward a disagreement between the Earth–atmosphere interaction torque and the time derivative of the AAM in the diurnal and subdiurnal band. This discrepancy could be explained by a non-negligible gravitational tidal torque. The purpose of this paper is to investigate the role of this tidal gravitational torque in the AAM balance and the consequences on the Earth rotation modeling. To our knowledge
0032-0633/02/$ - see front matter ? 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 0 3 2 - 0 6 3 3 ( 0 1 ) 0 0 1 2 0 - 9
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C. Bizouard, S. Lambert / Planetary and Space Science 50 (2002) 323–333
such a torque has never been computed rigourously but estimated partly (see for instance Volland (1996)). First we establish the analytical expression of this torque in function of the spherical harmonic coeHcients of the surface pressure (Section 2), then present the numerical computation we have carried out (Section 3). The second part of the paper is devoted to the in>uence of this torque in the AAM budget (Section 4) and the consequences on the Earth rotation modeling (Section 5). 2. Analytical expression of the tidal gravitational torque on the atmosphere The tidal gravitational torque on the atmosphere is given by ˜ ˜ = ˜r ∧ ∇ (3) A dVA ; VA
where A is the density of the atmospheric volume element dVA , is the tidal gravitational potential (TGP), usually given by its spherical harmonic development: (t) =
l l r l=0 m=0
a
This leads to dPlm ˆ ˜ lm = a2 PS (clm (t) cos(m) + c˜lm (t) sin(m)) d ; m − (−clm (t) sin(m) + c˜lm (t) cos(m)) sin ˆ (7) Plm (cos ) sin d d: Then the spherical harmonics development of the surface pressure :eld is also considered: PS (t) = (prs (t) cos s + p˜ rs (t) sin s)Prs ; (8) rs
where (prs (t); p˜ rs (t)) de:ne the time-dependent harmonic coeHcients of the surface pressure. The coordinates of the torque are then expressed in the terrestrial frame. The equatorial components receive the complex representation
(4)
where (r; ; ) are the spherical coordinates in the terrestrial reference system of any volume element dVA of the atmosphere ( is the colatitude, the longitude), Plm the non-normalized associated Legendre polynomials, (clm (t); c˜lm (t)) time-dependent harmonic coeHcients. We have ˆ ˜ = @ ˆ − 1 @ ; ˜r ∧ ∇ @ sin @
r=rS
1 2 lm (t) = lm (t) + ilm (t) = a2 (prs (t) cos s + p˜ rs (t) sin s)Prs (cos )
g(clm (t) cos(m)
+ c˜lm (t) sin(m))Plm (cos );
surface: ≈30 km
A g dr: PS =
(5)
where ˆ and ˆ are the unit vectors along the respective line coordinates. We get for any component (l; m) of the tidal potential the torque l r dPlm ˆ ˜ lm = g (clm (t) cos(m) + c˜lm (t) sin(m)) a d VA m − (−clm (t) sin(m) + c˜lm (t) cos(m)) sin (6) ×Plm (cos )ˆ A r 2 sin dr d d: The atmosphere can be considered as a thin layer of constant radius r = a with respect to the Earth. In the expression here above the integral in function of the radius r from the Earth surface (r = rS ), that is ≈30 km l r r 2 A g dr a r=rS can be then substituted by a2 PS , where PS is the surface pressure given by the weight of the air column above a unity
; rs
(clm (t) cos(m) + c˜lm (t) sin(m))
dPlm sin iei d
− m(−clm (t) sin(m) + c˜lm (t) cos(m)) ×Plm (cos ) cos ei d d;
(9)
whereas the axial component admits the expression 3 lm = a2 (prs cos s + p˜ rs sin s)Prs (cos ) ; rs
×m(−clm (t) sin(m) + c˜lm (t) cos(m)) ×Plm (cos ) sin d d:
(10)
After some laborious computations which are exposed in Appendix A it has been shown that the equatorial torque, associated to the degree l and the order m of the TGP presents a coupling with the same degree of the pressure :eld but with the order m − 1 or m + 1. The expression we have found is lm (t) = D{(clm (t) + ic˜lm (t))(p˜ l; m−1 (t) + ipl; m−1 (t)) ×(1 + m1 − m0 ) − (c˜lm (t) + iclm (t))(pl; m+1 (t) + ip˜ l; m+1 (t))((l + m + 1)(l − m) + m0 l(l + 1))} (11) with D = a2
(l + m)! : 2l + 1 (l − m)!
The axial component induced by the harmonic (l; m) of the TGP is associated with the same degree and order of the
C. Bizouard, S. Lambert / Planetary and Space Science 50 (2002) 323–333
pressure (see Appendix B): 3 (t) = 2D(plm (t)c˜lm (t)(1 + m0 ) lm
− p˜ lm (t)clm (t)(1 − m0 )):
(12)
The tidal potential we have used is the one computed by Roosbeek (1995), whose the general expression is given by n
5 n r (13) Gnm Rnm ; (t) = D1 a n=1
m=0
where D1 = 2:63 m2 =s2 is the Doodson constant, r the distance from the center of the Earth, and a = 6 378 140 m the equatorial radius. For our purpose, a restricted model has been used: the RATGP95c (Roosbeek Analytical Tide Generating Potential 95 c) (Roosbeek, 1995), which is truncated to 1:6 × 10−3 m2 =s2 for the degree 2 and 3:2 × 10−3 m2 =s2 for the degree 3. This model contains 151 terms of degree 2 and 6 of degree 3. The corresponding components of degree 2 of the torque can be deduced from expressions (11) and (12): 12a2 20 = − ic20 (p21 + ip˜ 21 ); 5
22 =
z 22 =
24a2 (c˜22 p22 − c22 p˜ 22 ) 5
G21 = 2 sin cos ; G22 = sin2 ; G30 = G31 =
R22 =
2
24a (c˜30 + ic30 )(p31 + ip˜ 31 ); 7
24a2 {ip30 (c31 +ic˜31 )−5(p32 +ip˜ 32 )(c˜31 +ic31 )}; 7
120a {(c32 + ic˜32 )(p˜ 31 + ip31 ) 32 = 7 − 6(c˜32 + ic32 )(p33 + ip˜ 33 )}; 720a2 (c33 + ic˜33 )(p˜ 32 + ip32 ); 7
= 0;
z 31 =
1 G20 = (1 − 3 cos2 ); 2
1 30
1 31
1 32
24a2 (c˜31 p31 − c31 p˜ 31 ); 7
j
(14)
(16)
with:
R21 =
and the corresponding components of degree 3 follow the expressions:
2
(15)
The tidal potential is given by r 2 [G20 R20 + G21 R21 + G22 R22 + G30 R30 2 = D1 a + G31 R31 + G32 R32 + G33 R33 ]
R20 =
2
12a (c˜21 p21 − c21 p˜ 21 ); 5
z 30
1440a2 (c˜33 p33 − c33 p˜ 33 ): 7
j
z 21 =
33 =
z = 33
and:
z 20 = 0;
31 =
240a2 (c˜32 p32 − c32 p˜ 32 ); 7
cos (3 − 5 cos2 );
30
= 0:89442719;
sin (1 − 5 cos2 );
31
= 1:37706075;
cos sin2 ;
32
= 0:38490018;
G33 = sin3
24a2 (c22 + ic˜22 )(p˜ 21 + ip21 ); 5
30 = −
z = 32
G32 =
12a2 21 = {ip20 (c21 + ic˜21 ) 5 − 2(p22 + ip˜ 22 )(c˜21 + ic21 )};
325
j
R30 =
j
R31 =
j
R32 =
j
R33 =
j
(17)
C0j; 20 cos fj; 20 ; S0j; 21 sin fj; 21 ; C0j; 22 cos fj; 22 ; S0j; 30 cos fj; 30 ; C0j; 31 cos fj; 31 ; S0j; 32 cos fj; 32 ; C0j; 33 cos fj; 33 ;
(18)
where C0j and S0j are the amplitudes of the tidal waves. The arguments fj are given by fj = aj1 % + aj2 s + aj3 h + aj4 p + aj5 N + aj6 ps ;
(19)
where the variables %, s, p, N and ps (called Doodson variables) describe the angular position of the Moon and the Sun. For each tidal wave j they are combined according to the integers aj . The mean lunar local time,%, is given by % = 15◦ t + GMST12 h − s + , where GMST12 h is the
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C. Bizouard, S. Lambert / Planetary and Space Science 50 (2002) 323–333
Greenwich Mean Sidereal Time at 12 h UT, t is the universal time in hours and the longitude. Let us consider the expression of 20 : r 2 1 j; 20 (1 − 3 cos2 ) 20 = D1 C0 cos fj; 20 (20) a 2 j
The Doodson variables have the following expressions (Roosbeek, 1995):
with:
h = 1 009 679:221 + 129 602 771 :0957T + 1 :0916T 2
fj; 20 = aj;2 20 s + aj;3 20 h + aj;4 20 p + aj;5 20 N + aj;6 20 ps = j; 20 :
(21)
D1 j; 22 C sin(30◦ t + 2GMST12 h + j; 22 ); 3g j 0
c33 =
(26)
where T is the time measured in Julian centuries from J2000.0. The Greenwich Mean Sidereal Time at 12 h UT is computed according to GMST12 h = 1 009 658:226 + 129 602 772:193T + 1 :39656T 2 − 0 :000093T 3 :
j
i[m(15 0;j; nm + e
◦
t+GMST12 h )+j; nm ]
with
2D1 j; 31 C sin(15◦ t + GMST12 h + j; 31 ); 3g 31 j 0
D1 j; 32 S sin(30◦ t + 2GMST12 h + j; 32 ); 15g 32 j 0
D1 j; 32 S cos(30◦ t + 2GMST12 h + j; 32 ); 15g 32 j 0
D1 j; 33 C cos(45◦ t + 3GMST12 h + j; 32 ); 15g j 0
c˜33 = −
− 0:000064T 3 − 0 :00001204T 4 :
+
2D1 j; 31 C cos(15◦ t + GMST12 h + j; 31 ); c31 = − 3g 31 j 0
c˜32 =
− 0:007702T 3 + 0 :00005939T 4 ;
j
2D1 j; 30 c30 = − S sin j; 30 ; g 30 j 0
c32 =
N = 845 839:602 + 6 962 890 :5431T − 7 :4722T 2
(27)
Any component nm can be splitted in two circular components, one prograde, the other one retrograde: j; nm ◦ nm = 0; − e−i[m(15 t+GMST12 h )+j; nm ]
D1 j; 22 C cos(30◦ t + 2GMST12 h + j; 22 ); c22 = 3g j 0
c˜31 =
− 0:044970T 3 + 0 :00018948T 4 ;
(23)
D1 j; 21 S cos(15◦ t + GMST12 h + j; 21 ); 3g j 0
c˜22 = −
p = 300 071:6752 + 14 648 449 :0869T − 37 :1582T 2
ps = 1 018 574:428 + 6 190 :0476T + 1 :6448T 2
The computation of the other coeHcients clm is carried out by the same manner: D1 j; 21 S sin(15◦ t + GMST12 h + j; 21 ); c21 = 3g j 0 c˜21 =
+ 0:006665T 3 − 0 :00005522T 5 ; + 0:000072T 3 − 0 :00002353T 4 ;
By comparing 20 to the equivalent expression 20 deduced from Eq. (4), r 2 1 20 = g (3 cos2 − 1)c20 ; (22) a 2 it turns out: D1 j; 20 C cos j; 20 : c20 = − g j 0
s = 785 939:9243 + 1 732 564 372 :30470T − 5 :2790T 2
D1 j; 33 C sin(45◦ t + 3GMST12 h + j; 32 ); 15g j 0 (24)
where j; nm = (aj;2 nm − m)s + aj;3 nm h + aj;4 nm p + aj;5 nm N + aj;6 nm ps : (25)
0;j; 20 − =i
6 D1 2 j; 20 a C0 (p21 + ip˜ 21 ); 5 g
0;j; 20 + =i
6 D1 2 j; 20 a C0 (p21 + ip˜ 21 ); 5 g
0;j; 21 − =−
24 D1 2 j; 21 a S0 p20 ; 15 g
0;j; 21 + =−
48 D1 2 j; 21 a S0 (p22 + ip˜ 22 ); 15 g
0;j; 22 − =
48 D1 2 j; 22 a C0 (p˜ 21 + ip21 ); 15 g
0;j; 30 − =
−24 D1 j; 30 S (p31 + ip˜ 31 ); 7 g 30 0
0;j; 30 + =
24 D1 j; 30 S (p31 + ip˜ 31 ); 7 g 30 0
0;j; 31 − = −i
16 D1 a2 C0j; 31 p30 ; 7 g 31
0;j; 22 + = 0;
(28)
C. Bizouard, S. Lambert / Planetary and Space Science 50 (2002) 323–333
80 D1 a2 C0j; 31 (p32 + ip˜ 32 ); 7 g 31 8 D1 0;j; 32 a2 S0j; 32 (p˜ 31 + ip31 ); − =i 7 g 32 48 D1 a2 S0j; 32 (p33 + ip˜ 33 ); 0;j; 32 + =− 7 g 32 48 D1 2 j; 33 a C0 (p˜ 32 + ip32 ); 0;j; 33 0;j; 33 − = + = 0: 7 g
0;j; 31 + =i
(29)
The axial components can be expressed as: z 20 = 0; z 21 =
24 D1 2 j; 21 a S0 (p21 cos(15◦ t + GMST12 h + j; 21 ) 15 g j −p˜ 21 sin(15◦ t + GMST12 h + j; 21 ));
z =− 22
24 D1 2 j; 22 C0 (p22 cos(30◦ t a 15 g j +2GMST12 h + j; 22 )
+ p˜ 22 sin(30◦ t + 2GMST12 h + j; 22 )); z = 0; 30 z 31 =
16 D1 a2 C0j; 31 (p31 sin(15◦ t+GMST12 h +j; 31 ) 7 g 31 j + p˜ 31 cos(15◦ t + GMST12 h + j; 31 ));
z 32 =
8 D1 a2 S0j; 32 (p32 cos(30◦ t+2GMST12 h +j; 32 ) 7 g 32 j − p˜ 32 sin(30◦ t + 2GMST12 h + j; 32 ));
z 33 =−
96 D1 2 j; 33 C0 (p33 sin(45◦ t a 7 g j +3GMST12 h + j; 33 ) ◦
+ p˜ 33 cos(45 t + 3GMST12 h + j; 33 )):
(30)
In the case of constant pressure coeHcients, the frequency of the torque is that of the tidal argument m(15◦ t + GMST12 h ) + j; nm . Consequently if m = 0 the torque is long periodic, m = 1 the torque is quasi-diurnal and if m = 2 the torque is quasi-diurnal. 3. Pressure eld data and numerical computation of the tidal gravitational torque As showed in the former section, the computation of the external gravitational torque is based on the knowledge of the pressure :eld p. Q Part of the Global Geophysical Fluids Center of the International Earth Rotation Service (IERS), the Special Bureau for the Atmosphere (SBA) provides spherical harmonic coeHcients of the surface pressure every 6 h from 1975 to 2000 (http:==www.aer.com=groups=diag=sb.html) up to degree 4.
327
These data were produced by the National Center for Environmental Predictions (NCEP) and National Center for Atmospheric Research (NCAR) in the frame of a global reanalysis of worldwide atmospheric data of the last 40 years. They are suHcient to compute the expressions (28) – (30) of the torque, since they include pressure coeHcients up to degree 3. In Fig. 1 is drawn the mean pressure :eld reconstructed from spherical harmonics up to degree 4, in a latitude– longitude grid (2:5 × 2:5) degrees. Some general features are revealed. First the pressure is maximum on the ocean, minimum on the continents, especially on the high lands and mountains. Low pressure :elds are especially noticeable on Antartic and Tibet. The mean pressure :eld is a mirror of the continents and oceans. A second feature, less visible, is the fact that the pressure is globally higher in the equatorial regions than in temperate and polar regions. This pattern is associated with the terms (2,0) of the spherical harmonics development. Mean pressure is much larger than its temporal variation, mostly seasonal, diurnal and semidiurnal. In Table 1 are reported the mean coeHcients of the pressure over the period 1975 –2000, as well as their standard deviations. For the (2,0) term the standard deviation amounts about 7% of the mean value, for the (3,0) term the standard deviation is up 15%. The most important contribution to the tidal gravitational torque on the atmosphere originates from the mean pressure :eld. This e0ect has been computed separately from the in>uence of the residual pressure variations, by using the analytical expressions developed in Section 2. The mean torque is displayed in Figs. 2– 4 (x; y; z components, respectively) together with the residual torque over a period of 30 days from 1 January 1999 at 0 h UT. We report also in Table 2, the signi:cant frequency components (larger than 1016 kg m2 s−2 ). As it could be foretold from residual pressure :eld, the residual torque amounts about 10% of the mean torque. This latter one is mostly composed by quasi-diurnal and quasi-semidiurnal waves, which produce variations up to 3 × 1017 kg m2 s−2 in the equatorial components, but 10 times smaller for the axial one. Indeed the largest coeHcients of the pressure, that are zonal, contribute only to the equatorial torque. 4. Lunisolar in!uence in the atmospheric angular momentum budget As pointed out in the introduction, the AAM budget is not closed for diurnal components (de Viron et al., 1999). One explanation could be that some external torque, which has not been taken into account, has a signi:cant magnitude in diurnal domain. The lunisolar torque on the atmosphere is a candidate. The magnitude of the axial component is not relevant, and we shall focus our interest on the equatorial
328
C. Bizouard, S. Lambert / Planetary and Space Science 50 (2002) 323–333 90 .
46.6 -36
72 54
5
-57
-15
36
-57 -119-98-77
2
25.9
-36
5
5.3 5
26
-15.3
-1
-36.0
-
5 18 -15
-15
-15 -3
-0
-56.6
-18 26
5
-36 -54
5
5
-36 -57 -77 -98 -119 -139
-72
.
-90 0
36
72
108
144
180
216
252
-7
-77.3
-36 5
26
288
324
-97.9
-9
-118.6
-1
-139.2
-
-159.8
360
0
1
Fig. 1. Mean pressure :eld 1975 –2000 up to degree 4 of spherical harmonics development minus the normal pressure (1013 mbar). The longitude-latitude grid is 2:5 × 2:5 degrees. Unit is mbar.
Table 1 Mean spherical harmonics coeHcients of the pressure and standard deviation over the period 1975 –2000. Units are Pa (from data of the NCEP and NCAR)
p00 p20 p21 p˜ 21 p22 p˜ 22 p30 p31 p˜ 31 p32 p˜ 32 p33 p˜ 33
Mean coeHcients
Standard deviation
98 523.3 −3645:4 7.7 −850:1 421.6 −194:9 4935.8 337.5 −666:0 355.6 −193:9 −6:0 −67:3
252.7 112.7 135.4 45.7 69.8 385.2 99.0 105.2 25.5 21.5 5.6 4.6
components. For our purpose we have used the AAM series of the NCEP=NCAR Reanalysis project for computing its time derivative and the interaction torque Tint→A (from the Earth on the atmosphere) computed from NCEP=NCAR Reanalysis data by (de Viron et al., 2001). For both series the most important diurnal tidally coherent waves have been estimated (O1, P1, K1 and S1), and then compared to that one of the lunisolar torque (see Table 3). These components are expressed according to Eq. (28): j; nm j; 21 0; − e−i + 0; + ei ; (31) j
j
4 contribution of the pressure variations contribution of the mean pressure
3
2
1
0
-1
-2 0
5
10
15
20
25
30
Fig. 2. x component of the lunisolar torque on the atmosphere from 1=1=1999 0hUT over 30 days. Unit is 1017 kg m2 s−2 .
where the phases are given by: ,O1 = −2s − 2h + 2N + GMST12 h ; ,P1 = −2h + GMST12 h ; ,K1 = GMST12 h + 15◦ t; ,S1 = −h + ps + GMST12 h :
(32)
For waves K1 (sidereal wave, 23:93 h), S1 (solar elliptic wave, 24:00 h), P1 (solar principal wave, 24:07 h) and O1 (principal lunar wave, 25:82 h), the AAM time derivative
C. Bizouard, S. Lambert / Planetary and Space Science 50 (2002) 323–333
Table 2 Most important waves of the equatorial and axial components of the mean lunisolar torque on the atmosphere. Unit is 1017 kg m2 s−2 . The frequency in space (cycles per day) is written inside parentheses for retrograde quasi-diurnal waves
4 contribution of the pressure variations contribution of the mean pressure
3 2
Frequency Tidal waves in ◦ =h
1 0 -1 -2 -3 0
10
5
15
20
25
329
30
Fig. 3. y component of the lunisolar torque on the atmosphere from 1=1=1999 0hUT over 30 days. Unit is 1017 kg m2 s−2 .
0 13.398 13.940 13.943 14.959 15.041 15.043 28.439 28.984 30.000 30.082
Equatorial component
Axial component
0; −
sin
0; +
0.26 0.26 (1=9:12) 0.14 (1=13:63) 0.14 O1(1=13:66) 0.75 −0:17 P1(1=182:6) 0.35 −0:08 + i0:10 −1:06 0:25 − i0:10 K1(0) −0:14 (1=6798) −0:16 −0:85 −0:39 S2 −0:11 K2
cos
0.18 −0:25 −0:21 −0:1
0.10
Table 3 AAM budget for O1, P1, K1 and S1 tidally coherent waves. Values for AAM time derivative and torques are given in 1017 kg m2 s−2
0.4 0.2
dHatm dt
0
O1 prograde 3:0 + i0:3 O1 retrograde −5:4 + i1:6 P1 prograde 29:2 − i69:1 P1 retrograde 0:8 + i5:1 K1 prograde 100:3 + i34:7 K1 retrograde 0:1 S1 prograde −231:2 + i323:1 S1 retrograde 0:1 + i1:7
-0.2 -0.4 -0.6
int −1:4 + i0:9 −15:6 − i6:4
21:1 + i12:6 8:1 + i4:5 2:8 + i37:3 −43:0 + i18:8 −136:6 − i13:4 30:4 − i16:6
grav −0:2
0:8
−0:1 + i0:1
0:4 0:3 − i0:1 −1:1 0:0 0:0
contribution of the pressure variations contribution of the mean pressure
-0.8 0
5
10
15
20
25
30
Fig. 4. z component of the lunisolar torque on the atmosphere from 1=1=1999 0hUT over 30 days. Unit is 1017 kg m2 s−2 .
presents large di0erences with the interaction torque. The diurnal interaction is dominated by prograde components (≈ 1019 kg m2 s−2 ). These ones are much large in the case of AAM time derivative. The retrograde components are an order less, and are larger in the case of the torque. For waves K1, S1 and P1 the consideration of the lunisolar torque on the atmosphere does not modify the AAM budget. This makes the budget signi:cantly closer only in the case of O1 retrograde wave. It should be noted that the O1 wave, the principal lunar wave (period of 25:82 h), is a purely gravitational one, not associated with thermal process. Its presence in the AAM spectrum and in the interaction torque spectrum con:rms the lunar gravitational in>uence on the atmosphere. We have to point out that the inconsistency can partly re>ect the accuracy level of the torque and AAM estimates. Actually the error on these quantities remains unknown, and their values can only be con:rmed by their e0ects on the Earth’s rotation. Prograde diurnal components for both
AAM and interaction torque perturb the polar motion at the level of 10 arcsec (Petrov, 1999; BrzeziTnski and Bizouard, in press), below the accuracy of polar motion estimates (≈ 100 arcsec). On the other hand retrograde diurnal e0ects of the AAM are much more larger, and contribute to nutation terms at the level of (≈ 100 arcsec), and in a consistent way with respect to the observations (Bizouard et al., 1998), whereas torque approach gives too large results. Therefore the retrograde diurnal torque seems to be either erroneous or compensated by some extra external torque, of which the nature still holds enigmatic. 5. Consequence for Earth rotation We would like to solve here the problem raised by Eq. (2). For computing the rigourous atmospheric e0ects from AAM series, it is necessary to remove from the AAM time derivative the lunisolar torque on the atmosphere according to ˜T ˜A dH dH ˜ grav : − (33) =− dt dt
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C. Bizouard, S. Lambert / Planetary and Space Science 50 (2002) 323–333
It can be noticed that everything happens as if the lunisolar torque was directly transmitted to the solid Earth, but this mathematical appearance has not to be confused with the physical reality. By considering the lunisolar torque, one gets the e0ective AAM variations taking part in the Earth rotation. The results of the usual AAM approach will be slightly modi:ed. In this paper the corrections will be evaluated in the case of a rigid Earth. From Earth’s nutation theory, it is a well known fact that the gravitational torque on the equatorial bulge, induced by the tesseral part (2; 1) of the tidal potential, is proportional to the product of its dynamical ellipticity times its equatorial inertia moment. A similar formula can be expected for the torque induced by the tesseral tidal potential (2; 1) action on the equatorial high pressure “ring”, described through the term p20 . By using the thin layer approximation (constant radius over the atmosphere) we have shown that the equatorial inertia moment of the atmosphere is equal to 8a4 p00 Aatm = (34) 3g and the atmospheric dynamical ellipticity can be expressed by: −6p20 + 5(p22 + p˜ 22 ) eatm = : (35) 20 p00 From Table 1 the equatorial inertia moment of the atmosphere is 1:4 × 1032 that is 1:7 × 10−6 times the equatorial inertia moment of the solid Earth, and the dynamical ellipticity is equal to 0:0122 that is about 3.7 times the dynamical ellipticity of the solid Earth. Therefore it can be expected that under the same tidal gravitational potential, the torque on the atmosphere will be 1:7×10−6 ×3:7=6:29×10−6 times the one on the solid Earth. Hence the expected correction on precession in longitude is 50 ×6:29×10−6 =314 arcsec=year, on 18.6 years nutation in longitude: 17 × 6 × 10−6 = 107 arcsec, on 18.6 years nutation in obliquity 9 × 6:29 × 10−6 = 57 arcsec. For other nutations no signi:cant e0ect can be deduced, since their amplitude does not exceed 2 . These estimates are con:rmed by the accurate computation, based upon the mean tidal gravitational torque in frequency domain (Table 2), as presented below. A rigid ellipsoidal axisymetric Earth admits in the terrestrial system the angular momentum [H ] = [A.m1 ; A.m2 ; C.(1 + m3 )]; where A and C are equatorial and axial inertia moments of the Earth, respectively, . the mean angular velocity of the Earth, and m1 , m2 and m3 perturbations on this mean angular velocity. The linearized form of the dynamical Euler equations takes the form: grav ; (36) m˙ − i/r m = A. grav m˙ 3 = 3 ; (37) C.
where m=m1 +im2 ; grav =1grav +i2grav ; /r =(C−A)=A. ≈ .=300 (Euler frequency). In the frequency domain these equations become: m(/) = −i m3 (/) =
1 grav (/); A.(/ − /r )
1 grav (/): C./ 3
(38) (39)
The Euler kinematical equation provides the temporal variation of the Euler angle of the terrestrial pole in function of m(t): ˙ + i0˙ sin = .mei
(40)
with 0 the Euler precession angle reckoned from the mean vernal point of the epoch J2000.0, the Euler nutation angle, and the Earth angle of rotation: ≈ 15◦ t (hour) +GMST0 h . Starting from the fact that the waves on the tidal torque are equal or smaller than 1017 kg m2 s−2 , it can be easily shown that only retrograde diurnal equatorial torque will bring a signi:cant correction (larger than 10 arcsec) on the Earth orientation parameters. Indeed, in such a case the second term of the kinematical Euler equation presents a frequency much smaller than 1 cycle per day, and hence provides signi:cant e0ect by integration; namely, for retrograde diurnal components, Eq. (40) becomes: 1 j; n1 −ij; n1 e ˙ + i0˙ sin = −i : A. 0; −
(41)
More common is the use of the angle 1 = − (nutation in obliquity) and = −0 (precession–nutation in longitude). In the equation here above sin() can be considered approximatively as constant: sin ≈ −sin 10 = −23◦ 7 where 10 is the mean obliquity of the ecliptic at the epoch J2000.0. Then we have: 1˙ − i ˙ sin 10 = i
1 j; n1 −ij; n1 : e A. 0; −
(42)
The numerical results have shown that the main e0ect is produced by the K1 component of the tidal potential. In this case j; n1 = 0, and we have: 1˙ − i ˙ sin 10 = i
1 K1 : A. 0; −
(43)
We have 0;K1− = −1017 N m from Table 2, A = 8:01 × 1037 kg m2 , . = 2=86164:1 s−1 . This is a contribution to the precession in longitude equal to ˙ sin 10 = 123 arcsec=year. It should be noted that the current precession is determined from VLBI analysis with a formal accuracy around 20 arcsec=year. Therefore, the obtained value should be taken into account in the budget of di0erent e0ects contributing to precession. An interesting point is that the atmospheric e0ect increases the theoretical value of the precession rate. For non zero-frequency in space (nutation), the only e0ect above 10 arcsec is associated with the 15:043◦ =h retrograde component of the torque. This contributes to the retrograde
C. Bizouard, S. Lambert / Planetary and Space Science 50 (2002) 323–333
term of the 18.6 years (6798 days) nutation at the level of 46 arcsec, namely X 18:6years sin 10 = −46 arcsec sin . for the nutation in longitude and X118:6years = 46 arcsec cos . for the nutation in obliquity (. is the mean tropic longitude of the Moon’s node with respect to the ecliptic plane). This contribution is at the level of the formal uncertainty on the 18.6 years nutation estimate. It has to be noticed that these corrections are already taken into account in the standard precession–nutation model. Indeed the mass distribution considered in this model is directly taken from the geopotential coeHcients determined by satellite orbitography, and hence involves also the atmosphere. The lunisolar precession–nutation is then computed for the global Earth–atmosphere system, but the associated model has to describe the case of a rigid Earth. Consequently in classical precession–nutation models for a rigid Earth the precession constant is overestimated by 310 arcsec . Same conclusion can be drawn for the 18.6 years nutation. Actually, this error is compensated by the fact that the above-mentioned corrections are not considered when the atmospheric e0ect is estimated from AAM series. According to Volland (1996) the tidal torque on the atmosphere, associated with the thermal tides, exhibits a constant term amounting about 4 × 1015 N m; and thus causes an acceleration of the length of day by about 0:2 ms per century, that is 10% of the secular decreasing of the length of day. But our analysis of the residual torque (that associated with pressure variations) has provided an e0ect ten times smaller. 6. Conclusion In this study we have raised the problem of the e0ect of the tidal gravitational torque on the atmosphere, which was never taken into account in the atmospheric angular momentum balance. We have concluded that it contributes signi:cantly to the atmospheric angular momentum budget at the level of 1017 N m. But it is not big enough for explaining the discrepancy between the time-derivative of the AAM and the interaction torque between the atmospheric and the solid Earth in the diurnal frequency band. However the error on these quantities remains unknown. Therefore any statement for AAM budget in the diurnal band still remains quite speculative. On the other hand this study threw light in the modeling of the lunisolar precession–nutation. The common approach is to consider the atmosphere as an external system, so that the e0ect of its mass distribution on the geopotential harmonic coeHcients should be removed before to compute the lunisolar torque on the “solid Earth”. But this is not done, so that the lunisolar precession-nutation model is not e0ectively restricted to the solid Earth, but account also for an atmosphere, which has been made rigid. This “non-solid Earth” e0ect has been accurately computed from the lunisolar torque on the atmosphere. We have shown that the atmosphere increases the precession in
331
longitude (0:31 marcsec=year) as well the 18.6 year nutation (0:04 marcsec on its retrograde component). The effect on precession is larger than the observational accuracy level (0:02 marcsec=year for precession) and the nutational e0ect is at the accuracy level of the VLBI determination (0:04 marcsec). This is an important clari:cation brought by this paper.
Acknowledgements Olivier de Viron (Royal Observatory of Belgium) and Nicole Capitaine (Paris Observatory) are acknowledged for providing us comments and suggestions. We are grateful to Peter Nelson (Atmospheric and Environmental Research, Inc.) for informations concerning the spherical harmonic developments of the pressure :eld, that we have used. We thank Fabian Roosbeek (Royal Observatory of Belgium) for informations concerning the tidal potential RATGP95.
Appendix A. Expression of the equatorial tidal gravitational torque on the atmosphere in spherical harmonics of the surface pressure and tidal gravitational potential This appendix is devoted to the computation of the component of degree l and order m (9) of the equatorial tidal torque. It can be expressed as rs lm = lm (A.1) rs
with rs lm (t) =
2
=0
=0
(prs (t) cos s+p˜ rs (t) sin s)Prs (cos )
(clm (t) cos(m)+c˜lm (t) sin(m))
dPlm sin i ei d
− m(−clm (t) sin(m) + c˜lm (t) cos(m)) i × Plm (cos ) cos e d d:
(A.2)
The problem consists in computing the following integrals: 2 I 1rs = cos s cos mei d; (A.3) lm =0
I 2rs lm = I 3rs lm I 4rs lm
=0
=
2
=0
=
2
2
=0
cos s sin mei d;
(A.4)
sin s cos mei d;
(A.5)
sin s sin mei d;
(A.6)
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C. Bizouard, S. Lambert / Planetary and Space Science 50 (2002) 323–333
J 1rs lm = J 2rs lm =
pi
=0 pi
Prs
=0
dPlm sin d; d
(A.7)
Prs Plm cos d:
(A.8)
Hence rs rs rs rs lm (t) = a2 {iJ 1rs ˜ rs clm lm (I 1lm prs clm + I 2lm prs c˜lm + I 3lm p rs + I 4rs ˜ rs c˜lm ) − mJ 2rs lm p lm (−I 2lm prs clm rs + I 1rs ˜ rs clm + I 3rs ˜ rs c˜lm )}: lm prs c˜lm − I 4lm p lm p (A.9)
After easy computation we obtain: I 1rs (A.10) lm = (s; m−1 + s; m+1 + s; −m+1 ); 2 i I 2rs (A.11) (−s; m−1 + s; m+1 − s; −m+1 ); lm = − 2 I 3rs (A.12) lm = − (s; m−1 − s; m+1 − s; −m+1 ); 2 I 4rs (A.13) lm = − (−s; m−1 − s; m+1 + s; −m+1 ); 2 where ij = 0 if i = j; ij = 1 if i = j. The expression (57) is reduced to: rs rs (t) = a2 {(I 1rs lm lm + mI 2lm )(iprs clm − prs c˜lm 2 rs + p˜ rs clm + ip˜ rs c˜lm )s; m−1 + (I 1rs lm − mI 2lm ) ×(iprs clm + prs c˜lm + −p˜ rs clm + ip˜ rs c˜lm ) rs ×s; m+1 + (I 1rs lm + mI 2lm )(iprs clm − prs c˜lm
+ − p˜ rs clm − ip˜ rs c˜lm )s; −m+1 }:
(A.14)
rs The integrals J 1rs lm and J 2lm are expressed in function of the variable t = cos : 1 dPlm rs J 1lm = − Prs (t) 1 − t 2 dt; (A.15) dt −1 1 t J 2rs = − Prs (t)Plm (t) √ dt: (A.16) lm 1 − t2 −1
But Plm (t) = (1 − t 2 )m=2
d m Pl0 : dt m
Hence d m Pl0 dPlm m = (1 − t 2 )(m−2)=2 (−2t) m dt 2 dt m+1 d P l0 + (1 − t 2 )m=2 dt m+1 or dPlm −mt 1 − t2 = √ Plm + Pl; m+1 : dt 1 − t2 It results: J 1rs lm
=
mJ 2rs lm
−
1
−1
Prs (t)Pl; m+1 (t) dt:
(A.17)
(A.18)
As the integrals in are can be di0erent of zero only if s = m + 1 or s = m − 1 or s = −m + 1, we can reduce the rs computation of J 1rs lm and J 2lm to those cases: (a) s = m + 1 We obtain immediately J 1r;lmm+1 − mJ 2r;lmm+1 = −rl
2(l + m + 1)! : (l − m − 1)!(2l + 1)
(A.21)
(b) s = m − 1 The problem consists in computing the integral 1 P (t)Pl; m+1 (t) dt. −1 r; m−1
1
−1
Pr; m−1 (t)Pl; m+1 (t) dt
=
1
Pr; m−1 (t)(1 − t 2 )(m+1)=2
−1
d m+1 Pl0 dt: dt m+1
(A.22)
An integration by part leads then to:
1
−1
Pr; m−1 (t)Pl; m+1 (t) dt
=−
1
−1
d d m Pl0 [Pr; m−1 (t)(1 − t 2 )(m+1)=2 ] m dt dt dt
1
dPr; m−1 d m Pl0 (1 − t 2 )(m+1)=2 dt dt dt m −1 1 t √ Plm Pr; m−1 dt + (m + 1) 1 − t2 −1
1 d d m−1 Pr0 =− (1 − t 2 )(m−1)=2 (1 − t 2 )(m+1)=2 dt m−1 −1 dt =−
d m Pl0 dt + (m + 1)J 2r;lmm−1 dt m 1 m−1 d m−1 Pr0 (1 − t 2 )(m−3)=2 =− (−2t) m−1 2 dt −1 d m Pr0 d m Pl0 (1 − t 2 )(m+1)=2 + (1 − t 2 )(m−1)=2 dt m dt dt m ×
+ (m + 1)J 2r;lmm−1 1 t = (m − 1) √ Pr; m−1 Plm dt 1 − t2 −1 1 (m − 1)Pr; m Plm dt + (m + 1)J 2r;lmm−1 − −1
=2mJ 2r;lmm−1 − rl (A.19)
2(l + m)! : (l − m)!(2l + 1)
(A.23)
We obtain :nally (A.20)
r; m−1 J 1r;lmm−1 + mJ 2lm = rl
2(l + m)! : (l − m)!(2l + 1)
(A.24)
C. Bizouard, S. Lambert / Planetary and Space Science 50 (2002) 323–333
(c) s = −m + 1 r; −m+1 J 1lm = mJ 2r;lm−m+1 −
= mJ 2r;lm−m+1 − m1
1
−1
1
−1
− m0
− m1
1
−1
1
−1
Pr1 (t)Pl1 (t) dt
Pr0 (t)Pl2 (t) dt
= mJ 2r;lm−m+1 − m0 rl
In a :rst step the problem consists in computing the following integrals: 2 rs cos s cos m d; (B.3) K1lm = =0 2 K2rs cos s sin m d; (B.4) lm = =0 2 K3rs = sin s cos m d; (B.5) lm =0 2 K4rs = sin s sin m d: (B.6) lm
Pr; −m+1 (t)Pl; m+1 (t) dt
2(l + 1)! (l − 1)!(2l + 1)
=0
Pr0 (t)Pl2 (t) dt:
(A.25)
1 By noting that the integral −1 Pr0 (t)Pl2 (t) dt is equal to 1 P (t)Pl; m+1 (t) dt with m = 1; expression which has −1 r; m−1 been computed for the case (b) (Eq. (A.23)), we obtain: r; −m+1 J 1lm = mJ 2r;lm−m+1 − m0 rl
− m1 2J 2l1 r; 0 − rl
2(l + 1)! (l − 1)!(2l + 1) 2(l + 1)! (l − 1)!(2l + 1)
333
After some easy computations we obtain: rs rs rs K1rs lm = sm (1 + s0 )K2lm = 0K3lm = 0K4lm = sm (1 − s0 ):
(B.7)
It results that the integral in takes part only for s = m, and hence we have only to consider the integral: Prm (cos )Plm (cos ) sin d =0
=rl
(B.8)
It follows that:
r; −m+1 = −m1 mJ 2lm + rl (m1 − m0 )
2(l + 1)! × (l − 1)!(2l + 1)
(r + m)! : (r − m)!(r + 1=2)
3 lm (t) = a2
(A.26)
(l + m)! (plm (t)c˜lm (t)(1 + m0 ) (l − m)!(l + 1=2)
− p˜ lm (t)clm (t)(1 − m0 ))
(B.9)
hence Eq. (12).
that is J 1r;lm−m+1 + mJ 2r;lm−m+1 =rl (m1 − m0 )
2(l + 1)! : (l − 1)!(2l + 1)
References (A.27)
Substituting the expressions (A.21), (A.24), (A.27) in Eq. (A.14), we obtain after some algebra the Eq. (11). Appendix B. Expression of the axial tidal gravitational torque on the atmosphere in spherical harmonics of the surface pressure and tidal gravitational potential This appendix is devoted to the computation of the axial torque of degree l and order m (9). It can be expressed as: 3; rs 3 lm = (B.1) lm rs
with 3; rs lm (t) = a2
;
(prs cos s + p˜ rs sin s)Prs (cos )
×m(−clm (t) sin(m) + c˜lm (t) cos (m)) ×Plm (cos ) sin d d:
(B.2)
Bizouard, C., BrzeziTnski, A., Petrov, S., 1998. Diurnal atmospheric forcing and temporal variations of the nutation amplitudes. J. Geodesy 72, 561–577. BrzeziTnski, A., Bizouard, C. In>uence of the atmosphere on Earth rotation: what new can be learned from the recent atmospheric angular momentum estimates? Surv. Geophys., in press. Dehant, V., Bizouard, Ch., Hinderer, J., Legros, H., Gre0-Le0tz, M., 1996. On atmospheric pressure perturbations on precession and nutations. Phys. Earth Planet. Inter. 96, 25–39. Petrov, S., 1999. Modeling geophysical excitation of Earth rotation: stochastic and non linear approaches. Ph.D. Thesis, Olsztyn University of Agriculture and Technology, Poland. Roosbeek, F., 1995. RATGP95: Technical Note, Royal Observatory of Belgium, 5pp. de Viron, O., 1999. In>uence de l’atmosphre sur la rotation de la Terre: application au calcul des nutations. Ph.D. Thesis, UniversitTe Catholique de Louvain. de Viron, O., Bizouard, C., Salstein, D., Dehant, V., 1999. Atmospheric torque on the Earth and comparison with the atmospheric angular momentum variations. J. Geophys. Res. 104 (3), 4835. de Viron, O., Marcus, S.L., Dickey, J.O., 2001. Diurnal angular momentum budget of the atmosphere and its consequences for the Earth’s nutation. J. Geophys. Res., 26 (B11) 26–747. Volland, H., 1996. Atmosphere and Earth’s rotation. Surv. Geophys. 17, 101–144.
www.elsevier.com/locate/planspasci
Lunisolar torque on the atmosphere and Earth’s rotation C. Bizouard ∗ , S. Lambert Observatoire de Paris, 61, avenue de l’Observatoire, 75014 Paris, France Received 11 June 2001; received in revised form 8 October 2001; accepted 19 October 2001
Abstract The atmospheric e0ect on Earth’s rotation is classically computed from atmospheric angular momentum (AAM) series by considering that the whole AAM variations are transmitted to the solid Earth. In fact, such a procedure is an approximation because it ignores that the atmosphere exchanges also angular momentum with the moon and the sun. The computation of the lunisolar gravitational torque on the atmosphere is the core of the present study. It is derived analytically and then estimated from pressure data over the period 1975 –2000. The mean surface pressure is the main contributor, at the level of 1017 N m for the equatorial component, 1016 N m for the axial one, whereas the thermally driven pressure variations provide about 10% of the variance. This torque plays a signi:cant role in the budget of the atmospheric angular momentum, but does not account for the inconsistency between the present series of the atmospheric angular momentum and the interaction torque between the solid Earth and the atmosphere. From the tidal waves of this torque we derive accurately the contribution of the atmosphere to the lunisolar precession–nutation of the Earth (0:31 marcsec=year for the precession in longitude, 46 microarcseconds for the nutation in 18.6 years). ? 2002 Elsevier Science Ltd. All rights reserved. Keywords: Earth’s rotation; Atmospheric angular momentum; Lunisolar precession–nutation; Tidal torque on the atmosphere
1. Introduction The e0ect of the atmosphere on the Earth rotation is usually computed from the angular momentum variation of this >uid layer. This approach implicitly assumes that the variations of the atmospheric angular momentum (AAM) are only caused by the interaction torque between the atmosphere and Earth. In other words the exchanges of angular momentum between the Earth and the atmosphere are not in>uenced by forces external to the Earth and to the atmosphere, and the exchanges obey to the balance of the angular momentum between the solid Earth and the atmosphere. However the gravitational interaction with the Moon and the Sun creates a tidal torque acting on the atmosphere. ˜ grav has to be considered in the angular This tidal torque momentum balance of the solid Earth–atmosphere system according to: ˜A dH int ˜ A→T ˜ grav ; = − + (1) dt int ˜ A→T ˜ A is the AAM and the interaction torque where H that the atmosphere exerts on the solid Earth (by virtue of ∗
Corresponding author. E-mail address: [email protected] (C. Bizouard).
the principle of the action and reaction the atmosphere is int ˜ A→T submitted to the opposite torque − ). ˜T The variation of the angular momentum of the Earth H is thus given by ˜T ˜A dH dH int grav ˜ ˜ : (2) = A→T = − − dt dt If the external torque was signi:cant, it should be removed from the AAM variations. This means that part of the AAM variations is not associated with interaction process between the solid Earth and the atmosphere. In the diurnal band and semidiurnal bands, where gravimetric forces are the most important, the magnitude of the tidal torque could compete with that of the time derivative of the AAM. This hypothesis has to be considered in the prospect of earlier studies (Dehant et al., 1996; de Viron, 1999; de Viron et al., 1999), which have put forward a disagreement between the Earth–atmosphere interaction torque and the time derivative of the AAM in the diurnal and subdiurnal band. This discrepancy could be explained by a non-negligible gravitational tidal torque. The purpose of this paper is to investigate the role of this tidal gravitational torque in the AAM balance and the consequences on the Earth rotation modeling. To our knowledge
0032-0633/02/$ - see front matter ? 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 0 3 2 - 0 6 3 3 ( 0 1 ) 0 0 1 2 0 - 9
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C. Bizouard, S. Lambert / Planetary and Space Science 50 (2002) 323–333
such a torque has never been computed rigourously but estimated partly (see for instance Volland (1996)). First we establish the analytical expression of this torque in function of the spherical harmonic coeHcients of the surface pressure (Section 2), then present the numerical computation we have carried out (Section 3). The second part of the paper is devoted to the in>uence of this torque in the AAM budget (Section 4) and the consequences on the Earth rotation modeling (Section 5). 2. Analytical expression of the tidal gravitational torque on the atmosphere The tidal gravitational torque on the atmosphere is given by ˜ ˜ = ˜r ∧ ∇ (3) A dVA ; VA
where A is the density of the atmospheric volume element dVA , is the tidal gravitational potential (TGP), usually given by its spherical harmonic development: (t) =
l l r l=0 m=0
a
This leads to dPlm ˆ ˜ lm = a2 PS (clm (t) cos(m) + c˜lm (t) sin(m)) d ; m − (−clm (t) sin(m) + c˜lm (t) cos(m)) sin ˆ (7) Plm (cos ) sin d d: Then the spherical harmonics development of the surface pressure :eld is also considered: PS (t) = (prs (t) cos s + p˜ rs (t) sin s)Prs ; (8) rs
where (prs (t); p˜ rs (t)) de:ne the time-dependent harmonic coeHcients of the surface pressure. The coordinates of the torque are then expressed in the terrestrial frame. The equatorial components receive the complex representation
(4)
where (r; ; ) are the spherical coordinates in the terrestrial reference system of any volume element dVA of the atmosphere ( is the colatitude, the longitude), Plm the non-normalized associated Legendre polynomials, (clm (t); c˜lm (t)) time-dependent harmonic coeHcients. We have ˆ ˜ = @ ˆ − 1 @ ; ˜r ∧ ∇ @ sin @
r=rS
1 2 lm (t) = lm (t) + ilm (t) = a2 (prs (t) cos s + p˜ rs (t) sin s)Prs (cos )
g(clm (t) cos(m)
+ c˜lm (t) sin(m))Plm (cos );
surface: ≈30 km
A g dr: PS =
(5)
where ˆ and ˆ are the unit vectors along the respective line coordinates. We get for any component (l; m) of the tidal potential the torque l r dPlm ˆ ˜ lm = g (clm (t) cos(m) + c˜lm (t) sin(m)) a d VA m − (−clm (t) sin(m) + c˜lm (t) cos(m)) sin (6) ×Plm (cos )ˆ A r 2 sin dr d d: The atmosphere can be considered as a thin layer of constant radius r = a with respect to the Earth. In the expression here above the integral in function of the radius r from the Earth surface (r = rS ), that is ≈30 km l r r 2 A g dr a r=rS can be then substituted by a2 PS , where PS is the surface pressure given by the weight of the air column above a unity
; rs
(clm (t) cos(m) + c˜lm (t) sin(m))
dPlm sin iei d
− m(−clm (t) sin(m) + c˜lm (t) cos(m)) ×Plm (cos ) cos ei d d;
(9)
whereas the axial component admits the expression 3 lm = a2 (prs cos s + p˜ rs sin s)Prs (cos ) ; rs
×m(−clm (t) sin(m) + c˜lm (t) cos(m)) ×Plm (cos ) sin d d:
(10)
After some laborious computations which are exposed in Appendix A it has been shown that the equatorial torque, associated to the degree l and the order m of the TGP presents a coupling with the same degree of the pressure :eld but with the order m − 1 or m + 1. The expression we have found is lm (t) = D{(clm (t) + ic˜lm (t))(p˜ l; m−1 (t) + ipl; m−1 (t)) ×(1 + m1 − m0 ) − (c˜lm (t) + iclm (t))(pl; m+1 (t) + ip˜ l; m+1 (t))((l + m + 1)(l − m) + m0 l(l + 1))} (11) with D = a2
(l + m)! : 2l + 1 (l − m)!
The axial component induced by the harmonic (l; m) of the TGP is associated with the same degree and order of the
C. Bizouard, S. Lambert / Planetary and Space Science 50 (2002) 323–333
pressure (see Appendix B): 3 (t) = 2D(plm (t)c˜lm (t)(1 + m0 ) lm
− p˜ lm (t)clm (t)(1 − m0 )):
(12)
The tidal potential we have used is the one computed by Roosbeek (1995), whose the general expression is given by n
5 n r (13) Gnm Rnm ; (t) = D1 a n=1
m=0
where D1 = 2:63 m2 =s2 is the Doodson constant, r the distance from the center of the Earth, and a = 6 378 140 m the equatorial radius. For our purpose, a restricted model has been used: the RATGP95c (Roosbeek Analytical Tide Generating Potential 95 c) (Roosbeek, 1995), which is truncated to 1:6 × 10−3 m2 =s2 for the degree 2 and 3:2 × 10−3 m2 =s2 for the degree 3. This model contains 151 terms of degree 2 and 6 of degree 3. The corresponding components of degree 2 of the torque can be deduced from expressions (11) and (12): 12a2 20 = − ic20 (p21 + ip˜ 21 ); 5
22 =
z 22 =
24a2 (c˜22 p22 − c22 p˜ 22 ) 5
G21 = 2 sin cos ; G22 = sin2 ; G30 = G31 =
R22 =
2
24a (c˜30 + ic30 )(p31 + ip˜ 31 ); 7
24a2 {ip30 (c31 +ic˜31 )−5(p32 +ip˜ 32 )(c˜31 +ic31 )}; 7
120a {(c32 + ic˜32 )(p˜ 31 + ip31 ) 32 = 7 − 6(c˜32 + ic32 )(p33 + ip˜ 33 )}; 720a2 (c33 + ic˜33 )(p˜ 32 + ip32 ); 7
= 0;
z 31 =
1 G20 = (1 − 3 cos2 ); 2
1 30
1 31
1 32
24a2 (c˜31 p31 − c31 p˜ 31 ); 7
j
(14)
(16)
with:
R21 =
and the corresponding components of degree 3 follow the expressions:
2
(15)
The tidal potential is given by r 2 [G20 R20 + G21 R21 + G22 R22 + G30 R30 2 = D1 a + G31 R31 + G32 R32 + G33 R33 ]
R20 =
2
12a (c˜21 p21 − c21 p˜ 21 ); 5
z 30
1440a2 (c˜33 p33 − c33 p˜ 33 ): 7
j
z 21 =
33 =
z = 33
and:
z 20 = 0;
31 =
240a2 (c˜32 p32 − c32 p˜ 32 ); 7
cos (3 − 5 cos2 );
30
= 0:89442719;
sin (1 − 5 cos2 );
31
= 1:37706075;
cos sin2 ;
32
= 0:38490018;
G33 = sin3
24a2 (c22 + ic˜22 )(p˜ 21 + ip21 ); 5
30 = −
z = 32
G32 =
12a2 21 = {ip20 (c21 + ic˜21 ) 5 − 2(p22 + ip˜ 22 )(c˜21 + ic21 )};
325
j
R30 =
j
R31 =
j
R32 =
j
R33 =
j
(17)
C0j; 20 cos fj; 20 ; S0j; 21 sin fj; 21 ; C0j; 22 cos fj; 22 ; S0j; 30 cos fj; 30 ; C0j; 31 cos fj; 31 ; S0j; 32 cos fj; 32 ; C0j; 33 cos fj; 33 ;
(18)
where C0j and S0j are the amplitudes of the tidal waves. The arguments fj are given by fj = aj1 % + aj2 s + aj3 h + aj4 p + aj5 N + aj6 ps ;
(19)
where the variables %, s, p, N and ps (called Doodson variables) describe the angular position of the Moon and the Sun. For each tidal wave j they are combined according to the integers aj . The mean lunar local time,%, is given by % = 15◦ t + GMST12 h − s + , where GMST12 h is the
326
C. Bizouard, S. Lambert / Planetary and Space Science 50 (2002) 323–333
Greenwich Mean Sidereal Time at 12 h UT, t is the universal time in hours and the longitude. Let us consider the expression of 20 : r 2 1 j; 20 (1 − 3 cos2 ) 20 = D1 C0 cos fj; 20 (20) a 2 j
The Doodson variables have the following expressions (Roosbeek, 1995):
with:
h = 1 009 679:221 + 129 602 771 :0957T + 1 :0916T 2
fj; 20 = aj;2 20 s + aj;3 20 h + aj;4 20 p + aj;5 20 N + aj;6 20 ps = j; 20 :
(21)
D1 j; 22 C sin(30◦ t + 2GMST12 h + j; 22 ); 3g j 0
c33 =
(26)
where T is the time measured in Julian centuries from J2000.0. The Greenwich Mean Sidereal Time at 12 h UT is computed according to GMST12 h = 1 009 658:226 + 129 602 772:193T + 1 :39656T 2 − 0 :000093T 3 :
j
i[m(15 0;j; nm + e
◦
t+GMST12 h )+j; nm ]
with
2D1 j; 31 C sin(15◦ t + GMST12 h + j; 31 ); 3g 31 j 0
D1 j; 32 S sin(30◦ t + 2GMST12 h + j; 32 ); 15g 32 j 0
D1 j; 32 S cos(30◦ t + 2GMST12 h + j; 32 ); 15g 32 j 0
D1 j; 33 C cos(45◦ t + 3GMST12 h + j; 32 ); 15g j 0
c˜33 = −
− 0:000064T 3 − 0 :00001204T 4 :
+
2D1 j; 31 C cos(15◦ t + GMST12 h + j; 31 ); c31 = − 3g 31 j 0
c˜32 =
− 0:007702T 3 + 0 :00005939T 4 ;
j
2D1 j; 30 c30 = − S sin j; 30 ; g 30 j 0
c32 =
N = 845 839:602 + 6 962 890 :5431T − 7 :4722T 2
(27)
Any component nm can be splitted in two circular components, one prograde, the other one retrograde: j; nm ◦ nm = 0; − e−i[m(15 t+GMST12 h )+j; nm ]
D1 j; 22 C cos(30◦ t + 2GMST12 h + j; 22 ); c22 = 3g j 0
c˜31 =
− 0:044970T 3 + 0 :00018948T 4 ;
(23)
D1 j; 21 S cos(15◦ t + GMST12 h + j; 21 ); 3g j 0
c˜22 = −
p = 300 071:6752 + 14 648 449 :0869T − 37 :1582T 2
ps = 1 018 574:428 + 6 190 :0476T + 1 :6448T 2
The computation of the other coeHcients clm is carried out by the same manner: D1 j; 21 S sin(15◦ t + GMST12 h + j; 21 ); c21 = 3g j 0 c˜21 =
+ 0:006665T 3 − 0 :00005522T 5 ; + 0:000072T 3 − 0 :00002353T 4 ;
By comparing 20 to the equivalent expression 20 deduced from Eq. (4), r 2 1 20 = g (3 cos2 − 1)c20 ; (22) a 2 it turns out: D1 j; 20 C cos j; 20 : c20 = − g j 0
s = 785 939:9243 + 1 732 564 372 :30470T − 5 :2790T 2
D1 j; 33 C sin(45◦ t + 3GMST12 h + j; 32 ); 15g j 0 (24)
where j; nm = (aj;2 nm − m)s + aj;3 nm h + aj;4 nm p + aj;5 nm N + aj;6 nm ps : (25)
0;j; 20 − =i
6 D1 2 j; 20 a C0 (p21 + ip˜ 21 ); 5 g
0;j; 20 + =i
6 D1 2 j; 20 a C0 (p21 + ip˜ 21 ); 5 g
0;j; 21 − =−
24 D1 2 j; 21 a S0 p20 ; 15 g
0;j; 21 + =−
48 D1 2 j; 21 a S0 (p22 + ip˜ 22 ); 15 g
0;j; 22 − =
48 D1 2 j; 22 a C0 (p˜ 21 + ip21 ); 15 g
0;j; 30 − =
−24 D1 j; 30 S (p31 + ip˜ 31 ); 7 g 30 0
0;j; 30 + =
24 D1 j; 30 S (p31 + ip˜ 31 ); 7 g 30 0
0;j; 31 − = −i
16 D1 a2 C0j; 31 p30 ; 7 g 31
0;j; 22 + = 0;
(28)
C. Bizouard, S. Lambert / Planetary and Space Science 50 (2002) 323–333
80 D1 a2 C0j; 31 (p32 + ip˜ 32 ); 7 g 31 8 D1 0;j; 32 a2 S0j; 32 (p˜ 31 + ip31 ); − =i 7 g 32 48 D1 a2 S0j; 32 (p33 + ip˜ 33 ); 0;j; 32 + =− 7 g 32 48 D1 2 j; 33 a C0 (p˜ 32 + ip32 ); 0;j; 33 0;j; 33 − = + = 0: 7 g
0;j; 31 + =i
(29)
The axial components can be expressed as: z 20 = 0; z 21 =
24 D1 2 j; 21 a S0 (p21 cos(15◦ t + GMST12 h + j; 21 ) 15 g j −p˜ 21 sin(15◦ t + GMST12 h + j; 21 ));
z =− 22
24 D1 2 j; 22 C0 (p22 cos(30◦ t a 15 g j +2GMST12 h + j; 22 )
+ p˜ 22 sin(30◦ t + 2GMST12 h + j; 22 )); z = 0; 30 z 31 =
16 D1 a2 C0j; 31 (p31 sin(15◦ t+GMST12 h +j; 31 ) 7 g 31 j + p˜ 31 cos(15◦ t + GMST12 h + j; 31 ));
z 32 =
8 D1 a2 S0j; 32 (p32 cos(30◦ t+2GMST12 h +j; 32 ) 7 g 32 j − p˜ 32 sin(30◦ t + 2GMST12 h + j; 32 ));
z 33 =−
96 D1 2 j; 33 C0 (p33 sin(45◦ t a 7 g j +3GMST12 h + j; 33 ) ◦
+ p˜ 33 cos(45 t + 3GMST12 h + j; 33 )):
(30)
In the case of constant pressure coeHcients, the frequency of the torque is that of the tidal argument m(15◦ t + GMST12 h ) + j; nm . Consequently if m = 0 the torque is long periodic, m = 1 the torque is quasi-diurnal and if m = 2 the torque is quasi-diurnal. 3. Pressure eld data and numerical computation of the tidal gravitational torque As showed in the former section, the computation of the external gravitational torque is based on the knowledge of the pressure :eld p. Q Part of the Global Geophysical Fluids Center of the International Earth Rotation Service (IERS), the Special Bureau for the Atmosphere (SBA) provides spherical harmonic coeHcients of the surface pressure every 6 h from 1975 to 2000 (http:==www.aer.com=groups=diag=sb.html) up to degree 4.
327
These data were produced by the National Center for Environmental Predictions (NCEP) and National Center for Atmospheric Research (NCAR) in the frame of a global reanalysis of worldwide atmospheric data of the last 40 years. They are suHcient to compute the expressions (28) – (30) of the torque, since they include pressure coeHcients up to degree 3. In Fig. 1 is drawn the mean pressure :eld reconstructed from spherical harmonics up to degree 4, in a latitude– longitude grid (2:5 × 2:5) degrees. Some general features are revealed. First the pressure is maximum on the ocean, minimum on the continents, especially on the high lands and mountains. Low pressure :elds are especially noticeable on Antartic and Tibet. The mean pressure :eld is a mirror of the continents and oceans. A second feature, less visible, is the fact that the pressure is globally higher in the equatorial regions than in temperate and polar regions. This pattern is associated with the terms (2,0) of the spherical harmonics development. Mean pressure is much larger than its temporal variation, mostly seasonal, diurnal and semidiurnal. In Table 1 are reported the mean coeHcients of the pressure over the period 1975 –2000, as well as their standard deviations. For the (2,0) term the standard deviation amounts about 7% of the mean value, for the (3,0) term the standard deviation is up 15%. The most important contribution to the tidal gravitational torque on the atmosphere originates from the mean pressure :eld. This e0ect has been computed separately from the in>uence of the residual pressure variations, by using the analytical expressions developed in Section 2. The mean torque is displayed in Figs. 2– 4 (x; y; z components, respectively) together with the residual torque over a period of 30 days from 1 January 1999 at 0 h UT. We report also in Table 2, the signi:cant frequency components (larger than 1016 kg m2 s−2 ). As it could be foretold from residual pressure :eld, the residual torque amounts about 10% of the mean torque. This latter one is mostly composed by quasi-diurnal and quasi-semidiurnal waves, which produce variations up to 3 × 1017 kg m2 s−2 in the equatorial components, but 10 times smaller for the axial one. Indeed the largest coeHcients of the pressure, that are zonal, contribute only to the equatorial torque. 4. Lunisolar in!uence in the atmospheric angular momentum budget As pointed out in the introduction, the AAM budget is not closed for diurnal components (de Viron et al., 1999). One explanation could be that some external torque, which has not been taken into account, has a signi:cant magnitude in diurnal domain. The lunisolar torque on the atmosphere is a candidate. The magnitude of the axial component is not relevant, and we shall focus our interest on the equatorial
328
C. Bizouard, S. Lambert / Planetary and Space Science 50 (2002) 323–333 90 .
46.6 -36
72 54
5
-57
-15
36
-57 -119-98-77
2
25.9
-36
5
5.3 5
26
-15.3
-1
-36.0
-
5 18 -15
-15
-15 -3
-0
-56.6
-18 26
5
-36 -54
5
5
-36 -57 -77 -98 -119 -139
-72
.
-90 0
36
72
108
144
180
216
252
-7
-77.3
-36 5
26
288
324
-97.9
-9
-118.6
-1
-139.2
-
-159.8
360
0
1
Fig. 1. Mean pressure :eld 1975 –2000 up to degree 4 of spherical harmonics development minus the normal pressure (1013 mbar). The longitude-latitude grid is 2:5 × 2:5 degrees. Unit is mbar.
Table 1 Mean spherical harmonics coeHcients of the pressure and standard deviation over the period 1975 –2000. Units are Pa (from data of the NCEP and NCAR)
p00 p20 p21 p˜ 21 p22 p˜ 22 p30 p31 p˜ 31 p32 p˜ 32 p33 p˜ 33
Mean coeHcients
Standard deviation
98 523.3 −3645:4 7.7 −850:1 421.6 −194:9 4935.8 337.5 −666:0 355.6 −193:9 −6:0 −67:3
252.7 112.7 135.4 45.7 69.8 385.2 99.0 105.2 25.5 21.5 5.6 4.6
components. For our purpose we have used the AAM series of the NCEP=NCAR Reanalysis project for computing its time derivative and the interaction torque Tint→A (from the Earth on the atmosphere) computed from NCEP=NCAR Reanalysis data by (de Viron et al., 2001). For both series the most important diurnal tidally coherent waves have been estimated (O1, P1, K1 and S1), and then compared to that one of the lunisolar torque (see Table 3). These components are expressed according to Eq. (28): j; nm j; 21 0; − e−i + 0; + ei ; (31) j
j
4 contribution of the pressure variations contribution of the mean pressure
3
2
1
0
-1
-2 0
5
10
15
20
25
30
Fig. 2. x component of the lunisolar torque on the atmosphere from 1=1=1999 0hUT over 30 days. Unit is 1017 kg m2 s−2 .
where the phases are given by: ,O1 = −2s − 2h + 2N + GMST12 h ; ,P1 = −2h + GMST12 h ; ,K1 = GMST12 h + 15◦ t; ,S1 = −h + ps + GMST12 h :
(32)
For waves K1 (sidereal wave, 23:93 h), S1 (solar elliptic wave, 24:00 h), P1 (solar principal wave, 24:07 h) and O1 (principal lunar wave, 25:82 h), the AAM time derivative
C. Bizouard, S. Lambert / Planetary and Space Science 50 (2002) 323–333
Table 2 Most important waves of the equatorial and axial components of the mean lunisolar torque on the atmosphere. Unit is 1017 kg m2 s−2 . The frequency in space (cycles per day) is written inside parentheses for retrograde quasi-diurnal waves
4 contribution of the pressure variations contribution of the mean pressure
3 2
Frequency Tidal waves in ◦ =h
1 0 -1 -2 -3 0
10
5
15
20
25
329
30
Fig. 3. y component of the lunisolar torque on the atmosphere from 1=1=1999 0hUT over 30 days. Unit is 1017 kg m2 s−2 .
0 13.398 13.940 13.943 14.959 15.041 15.043 28.439 28.984 30.000 30.082
Equatorial component
Axial component
0; −
sin
0; +
0.26 0.26 (1=9:12) 0.14 (1=13:63) 0.14 O1(1=13:66) 0.75 −0:17 P1(1=182:6) 0.35 −0:08 + i0:10 −1:06 0:25 − i0:10 K1(0) −0:14 (1=6798) −0:16 −0:85 −0:39 S2 −0:11 K2
cos
0.18 −0:25 −0:21 −0:1
0.10
Table 3 AAM budget for O1, P1, K1 and S1 tidally coherent waves. Values for AAM time derivative and torques are given in 1017 kg m2 s−2
0.4 0.2
dHatm dt
0
O1 prograde 3:0 + i0:3 O1 retrograde −5:4 + i1:6 P1 prograde 29:2 − i69:1 P1 retrograde 0:8 + i5:1 K1 prograde 100:3 + i34:7 K1 retrograde 0:1 S1 prograde −231:2 + i323:1 S1 retrograde 0:1 + i1:7
-0.2 -0.4 -0.6
int −1:4 + i0:9 −15:6 − i6:4
21:1 + i12:6 8:1 + i4:5 2:8 + i37:3 −43:0 + i18:8 −136:6 − i13:4 30:4 − i16:6
grav −0:2
0:8
−0:1 + i0:1
0:4 0:3 − i0:1 −1:1 0:0 0:0
contribution of the pressure variations contribution of the mean pressure
-0.8 0
5
10
15
20
25
30
Fig. 4. z component of the lunisolar torque on the atmosphere from 1=1=1999 0hUT over 30 days. Unit is 1017 kg m2 s−2 .
presents large di0erences with the interaction torque. The diurnal interaction is dominated by prograde components (≈ 1019 kg m2 s−2 ). These ones are much large in the case of AAM time derivative. The retrograde components are an order less, and are larger in the case of the torque. For waves K1, S1 and P1 the consideration of the lunisolar torque on the atmosphere does not modify the AAM budget. This makes the budget signi:cantly closer only in the case of O1 retrograde wave. It should be noted that the O1 wave, the principal lunar wave (period of 25:82 h), is a purely gravitational one, not associated with thermal process. Its presence in the AAM spectrum and in the interaction torque spectrum con:rms the lunar gravitational in>uence on the atmosphere. We have to point out that the inconsistency can partly re>ect the accuracy level of the torque and AAM estimates. Actually the error on these quantities remains unknown, and their values can only be con:rmed by their e0ects on the Earth’s rotation. Prograde diurnal components for both
AAM and interaction torque perturb the polar motion at the level of 10 arcsec (Petrov, 1999; BrzeziTnski and Bizouard, in press), below the accuracy of polar motion estimates (≈ 100 arcsec). On the other hand retrograde diurnal e0ects of the AAM are much more larger, and contribute to nutation terms at the level of (≈ 100 arcsec), and in a consistent way with respect to the observations (Bizouard et al., 1998), whereas torque approach gives too large results. Therefore the retrograde diurnal torque seems to be either erroneous or compensated by some extra external torque, of which the nature still holds enigmatic. 5. Consequence for Earth rotation We would like to solve here the problem raised by Eq. (2). For computing the rigourous atmospheric e0ects from AAM series, it is necessary to remove from the AAM time derivative the lunisolar torque on the atmosphere according to ˜T ˜A dH dH ˜ grav : − (33) =− dt dt
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C. Bizouard, S. Lambert / Planetary and Space Science 50 (2002) 323–333
It can be noticed that everything happens as if the lunisolar torque was directly transmitted to the solid Earth, but this mathematical appearance has not to be confused with the physical reality. By considering the lunisolar torque, one gets the e0ective AAM variations taking part in the Earth rotation. The results of the usual AAM approach will be slightly modi:ed. In this paper the corrections will be evaluated in the case of a rigid Earth. From Earth’s nutation theory, it is a well known fact that the gravitational torque on the equatorial bulge, induced by the tesseral part (2; 1) of the tidal potential, is proportional to the product of its dynamical ellipticity times its equatorial inertia moment. A similar formula can be expected for the torque induced by the tesseral tidal potential (2; 1) action on the equatorial high pressure “ring”, described through the term p20 . By using the thin layer approximation (constant radius over the atmosphere) we have shown that the equatorial inertia moment of the atmosphere is equal to 8a4 p00 Aatm = (34) 3g and the atmospheric dynamical ellipticity can be expressed by: −6p20 + 5(p22 + p˜ 22 ) eatm = : (35) 20 p00 From Table 1 the equatorial inertia moment of the atmosphere is 1:4 × 1032 that is 1:7 × 10−6 times the equatorial inertia moment of the solid Earth, and the dynamical ellipticity is equal to 0:0122 that is about 3.7 times the dynamical ellipticity of the solid Earth. Therefore it can be expected that under the same tidal gravitational potential, the torque on the atmosphere will be 1:7×10−6 ×3:7=6:29×10−6 times the one on the solid Earth. Hence the expected correction on precession in longitude is 50 ×6:29×10−6 =314 arcsec=year, on 18.6 years nutation in longitude: 17 × 6 × 10−6 = 107 arcsec, on 18.6 years nutation in obliquity 9 × 6:29 × 10−6 = 57 arcsec. For other nutations no signi:cant e0ect can be deduced, since their amplitude does not exceed 2 . These estimates are con:rmed by the accurate computation, based upon the mean tidal gravitational torque in frequency domain (Table 2), as presented below. A rigid ellipsoidal axisymetric Earth admits in the terrestrial system the angular momentum [H ] = [A.m1 ; A.m2 ; C.(1 + m3 )]; where A and C are equatorial and axial inertia moments of the Earth, respectively, . the mean angular velocity of the Earth, and m1 , m2 and m3 perturbations on this mean angular velocity. The linearized form of the dynamical Euler equations takes the form: grav ; (36) m˙ − i/r m = A. grav m˙ 3 = 3 ; (37) C.
where m=m1 +im2 ; grav =1grav +i2grav ; /r =(C−A)=A. ≈ .=300 (Euler frequency). In the frequency domain these equations become: m(/) = −i m3 (/) =
1 grav (/); A.(/ − /r )
1 grav (/): C./ 3
(38) (39)
The Euler kinematical equation provides the temporal variation of the Euler angle of the terrestrial pole in function of m(t): ˙ + i0˙ sin = .mei
(40)
with 0 the Euler precession angle reckoned from the mean vernal point of the epoch J2000.0, the Euler nutation angle, and the Earth angle of rotation: ≈ 15◦ t (hour) +GMST0 h . Starting from the fact that the waves on the tidal torque are equal or smaller than 1017 kg m2 s−2 , it can be easily shown that only retrograde diurnal equatorial torque will bring a signi:cant correction (larger than 10 arcsec) on the Earth orientation parameters. Indeed, in such a case the second term of the kinematical Euler equation presents a frequency much smaller than 1 cycle per day, and hence provides signi:cant e0ect by integration; namely, for retrograde diurnal components, Eq. (40) becomes: 1 j; n1 −ij; n1 e ˙ + i0˙ sin = −i : A. 0; −
(41)
More common is the use of the angle 1 = − (nutation in obliquity) and = −0 (precession–nutation in longitude). In the equation here above sin() can be considered approximatively as constant: sin ≈ −sin 10 = −23◦ 7 where 10 is the mean obliquity of the ecliptic at the epoch J2000.0. Then we have: 1˙ − i ˙ sin 10 = i
1 j; n1 −ij; n1 : e A. 0; −
(42)
The numerical results have shown that the main e0ect is produced by the K1 component of the tidal potential. In this case j; n1 = 0, and we have: 1˙ − i ˙ sin 10 = i
1 K1 : A. 0; −
(43)
We have 0;K1− = −1017 N m from Table 2, A = 8:01 × 1037 kg m2 , . = 2=86164:1 s−1 . This is a contribution to the precession in longitude equal to ˙ sin 10 = 123 arcsec=year. It should be noted that the current precession is determined from VLBI analysis with a formal accuracy around 20 arcsec=year. Therefore, the obtained value should be taken into account in the budget of di0erent e0ects contributing to precession. An interesting point is that the atmospheric e0ect increases the theoretical value of the precession rate. For non zero-frequency in space (nutation), the only e0ect above 10 arcsec is associated with the 15:043◦ =h retrograde component of the torque. This contributes to the retrograde
C. Bizouard, S. Lambert / Planetary and Space Science 50 (2002) 323–333
term of the 18.6 years (6798 days) nutation at the level of 46 arcsec, namely X 18:6years sin 10 = −46 arcsec sin . for the nutation in longitude and X118:6years = 46 arcsec cos . for the nutation in obliquity (. is the mean tropic longitude of the Moon’s node with respect to the ecliptic plane). This contribution is at the level of the formal uncertainty on the 18.6 years nutation estimate. It has to be noticed that these corrections are already taken into account in the standard precession–nutation model. Indeed the mass distribution considered in this model is directly taken from the geopotential coeHcients determined by satellite orbitography, and hence involves also the atmosphere. The lunisolar precession–nutation is then computed for the global Earth–atmosphere system, but the associated model has to describe the case of a rigid Earth. Consequently in classical precession–nutation models for a rigid Earth the precession constant is overestimated by 310 arcsec . Same conclusion can be drawn for the 18.6 years nutation. Actually, this error is compensated by the fact that the above-mentioned corrections are not considered when the atmospheric e0ect is estimated from AAM series. According to Volland (1996) the tidal torque on the atmosphere, associated with the thermal tides, exhibits a constant term amounting about 4 × 1015 N m; and thus causes an acceleration of the length of day by about 0:2 ms per century, that is 10% of the secular decreasing of the length of day. But our analysis of the residual torque (that associated with pressure variations) has provided an e0ect ten times smaller. 6. Conclusion In this study we have raised the problem of the e0ect of the tidal gravitational torque on the atmosphere, which was never taken into account in the atmospheric angular momentum balance. We have concluded that it contributes signi:cantly to the atmospheric angular momentum budget at the level of 1017 N m. But it is not big enough for explaining the discrepancy between the time-derivative of the AAM and the interaction torque between the atmospheric and the solid Earth in the diurnal frequency band. However the error on these quantities remains unknown. Therefore any statement for AAM budget in the diurnal band still remains quite speculative. On the other hand this study threw light in the modeling of the lunisolar precession–nutation. The common approach is to consider the atmosphere as an external system, so that the e0ect of its mass distribution on the geopotential harmonic coeHcients should be removed before to compute the lunisolar torque on the “solid Earth”. But this is not done, so that the lunisolar precession-nutation model is not e0ectively restricted to the solid Earth, but account also for an atmosphere, which has been made rigid. This “non-solid Earth” e0ect has been accurately computed from the lunisolar torque on the atmosphere. We have shown that the atmosphere increases the precession in
331
longitude (0:31 marcsec=year) as well the 18.6 year nutation (0:04 marcsec on its retrograde component). The effect on precession is larger than the observational accuracy level (0:02 marcsec=year for precession) and the nutational e0ect is at the accuracy level of the VLBI determination (0:04 marcsec). This is an important clari:cation brought by this paper.
Acknowledgements Olivier de Viron (Royal Observatory of Belgium) and Nicole Capitaine (Paris Observatory) are acknowledged for providing us comments and suggestions. We are grateful to Peter Nelson (Atmospheric and Environmental Research, Inc.) for informations concerning the spherical harmonic developments of the pressure :eld, that we have used. We thank Fabian Roosbeek (Royal Observatory of Belgium) for informations concerning the tidal potential RATGP95.
Appendix A. Expression of the equatorial tidal gravitational torque on the atmosphere in spherical harmonics of the surface pressure and tidal gravitational potential This appendix is devoted to the computation of the component of degree l and order m (9) of the equatorial tidal torque. It can be expressed as rs lm = lm (A.1) rs
with rs lm (t) =
2
=0
=0
(prs (t) cos s+p˜ rs (t) sin s)Prs (cos )
(clm (t) cos(m)+c˜lm (t) sin(m))
dPlm sin i ei d
− m(−clm (t) sin(m) + c˜lm (t) cos(m)) i × Plm (cos ) cos e d d:
(A.2)
The problem consists in computing the following integrals: 2 I 1rs = cos s cos mei d; (A.3) lm =0
I 2rs lm = I 3rs lm I 4rs lm
=0
=
2
=0
=
2
2
=0
cos s sin mei d;
(A.4)
sin s cos mei d;
(A.5)
sin s sin mei d;
(A.6)
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C. Bizouard, S. Lambert / Planetary and Space Science 50 (2002) 323–333
J 1rs lm = J 2rs lm =
pi
=0 pi
Prs
=0
dPlm sin d; d
(A.7)
Prs Plm cos d:
(A.8)
Hence rs rs rs rs lm (t) = a2 {iJ 1rs ˜ rs clm lm (I 1lm prs clm + I 2lm prs c˜lm + I 3lm p rs + I 4rs ˜ rs c˜lm ) − mJ 2rs lm p lm (−I 2lm prs clm rs + I 1rs ˜ rs clm + I 3rs ˜ rs c˜lm )}: lm prs c˜lm − I 4lm p lm p (A.9)
After easy computation we obtain: I 1rs (A.10) lm = (s; m−1 + s; m+1 + s; −m+1 ); 2 i I 2rs (A.11) (−s; m−1 + s; m+1 − s; −m+1 ); lm = − 2 I 3rs (A.12) lm = − (s; m−1 − s; m+1 − s; −m+1 ); 2 I 4rs (A.13) lm = − (−s; m−1 − s; m+1 + s; −m+1 ); 2 where ij = 0 if i = j; ij = 1 if i = j. The expression (57) is reduced to: rs rs (t) = a2 {(I 1rs lm lm + mI 2lm )(iprs clm − prs c˜lm 2 rs + p˜ rs clm + ip˜ rs c˜lm )s; m−1 + (I 1rs lm − mI 2lm ) ×(iprs clm + prs c˜lm + −p˜ rs clm + ip˜ rs c˜lm ) rs ×s; m+1 + (I 1rs lm + mI 2lm )(iprs clm − prs c˜lm
+ − p˜ rs clm − ip˜ rs c˜lm )s; −m+1 }:
(A.14)
rs The integrals J 1rs lm and J 2lm are expressed in function of the variable t = cos : 1 dPlm rs J 1lm = − Prs (t) 1 − t 2 dt; (A.15) dt −1 1 t J 2rs = − Prs (t)Plm (t) √ dt: (A.16) lm 1 − t2 −1
But Plm (t) = (1 − t 2 )m=2
d m Pl0 : dt m
Hence d m Pl0 dPlm m = (1 − t 2 )(m−2)=2 (−2t) m dt 2 dt m+1 d P l0 + (1 − t 2 )m=2 dt m+1 or dPlm −mt 1 − t2 = √ Plm + Pl; m+1 : dt 1 − t2 It results: J 1rs lm
=
mJ 2rs lm
−
1
−1
Prs (t)Pl; m+1 (t) dt:
(A.17)
(A.18)
As the integrals in are can be di0erent of zero only if s = m + 1 or s = m − 1 or s = −m + 1, we can reduce the rs computation of J 1rs lm and J 2lm to those cases: (a) s = m + 1 We obtain immediately J 1r;lmm+1 − mJ 2r;lmm+1 = −rl
2(l + m + 1)! : (l − m − 1)!(2l + 1)
(A.21)
(b) s = m − 1 The problem consists in computing the integral 1 P (t)Pl; m+1 (t) dt. −1 r; m−1
1
−1
Pr; m−1 (t)Pl; m+1 (t) dt
=
1
Pr; m−1 (t)(1 − t 2 )(m+1)=2
−1
d m+1 Pl0 dt: dt m+1
(A.22)
An integration by part leads then to:
1
−1
Pr; m−1 (t)Pl; m+1 (t) dt
=−
1
−1
d d m Pl0 [Pr; m−1 (t)(1 − t 2 )(m+1)=2 ] m dt dt dt
1
dPr; m−1 d m Pl0 (1 − t 2 )(m+1)=2 dt dt dt m −1 1 t √ Plm Pr; m−1 dt + (m + 1) 1 − t2 −1
1 d d m−1 Pr0 =− (1 − t 2 )(m−1)=2 (1 − t 2 )(m+1)=2 dt m−1 −1 dt =−
d m Pl0 dt + (m + 1)J 2r;lmm−1 dt m 1 m−1 d m−1 Pr0 (1 − t 2 )(m−3)=2 =− (−2t) m−1 2 dt −1 d m Pr0 d m Pl0 (1 − t 2 )(m+1)=2 + (1 − t 2 )(m−1)=2 dt m dt dt m ×
+ (m + 1)J 2r;lmm−1 1 t = (m − 1) √ Pr; m−1 Plm dt 1 − t2 −1 1 (m − 1)Pr; m Plm dt + (m + 1)J 2r;lmm−1 − −1
=2mJ 2r;lmm−1 − rl (A.19)
2(l + m)! : (l − m)!(2l + 1)
(A.23)
We obtain :nally (A.20)
r; m−1 J 1r;lmm−1 + mJ 2lm = rl
2(l + m)! : (l − m)!(2l + 1)
(A.24)
C. Bizouard, S. Lambert / Planetary and Space Science 50 (2002) 323–333
(c) s = −m + 1 r; −m+1 J 1lm = mJ 2r;lm−m+1 −
= mJ 2r;lm−m+1 − m1
1
−1
1
−1
− m0
− m1
1
−1
1
−1
Pr1 (t)Pl1 (t) dt
Pr0 (t)Pl2 (t) dt
= mJ 2r;lm−m+1 − m0 rl
In a :rst step the problem consists in computing the following integrals: 2 rs cos s cos m d; (B.3) K1lm = =0 2 K2rs cos s sin m d; (B.4) lm = =0 2 K3rs = sin s cos m d; (B.5) lm =0 2 K4rs = sin s sin m d: (B.6) lm
Pr; −m+1 (t)Pl; m+1 (t) dt
2(l + 1)! (l − 1)!(2l + 1)
=0
Pr0 (t)Pl2 (t) dt:
(A.25)
1 By noting that the integral −1 Pr0 (t)Pl2 (t) dt is equal to 1 P (t)Pl; m+1 (t) dt with m = 1; expression which has −1 r; m−1 been computed for the case (b) (Eq. (A.23)), we obtain: r; −m+1 J 1lm = mJ 2r;lm−m+1 − m0 rl
− m1 2J 2l1 r; 0 − rl
2(l + 1)! (l − 1)!(2l + 1) 2(l + 1)! (l − 1)!(2l + 1)
333
After some easy computations we obtain: rs rs rs K1rs lm = sm (1 + s0 )K2lm = 0K3lm = 0K4lm = sm (1 − s0 ):
(B.7)
It results that the integral in takes part only for s = m, and hence we have only to consider the integral: Prm (cos )Plm (cos ) sin d =0
=rl
(B.8)
It follows that:
r; −m+1 = −m1 mJ 2lm + rl (m1 − m0 )
2(l + 1)! × (l − 1)!(2l + 1)
(r + m)! : (r − m)!(r + 1=2)
3 lm (t) = a2
(A.26)
(l + m)! (plm (t)c˜lm (t)(1 + m0 ) (l − m)!(l + 1=2)
− p˜ lm (t)clm (t)(1 − m0 ))
(B.9)
hence Eq. (12).
that is J 1r;lm−m+1 + mJ 2r;lm−m+1 =rl (m1 − m0 )
2(l + 1)! : (l − 1)!(2l + 1)
References (A.27)
Substituting the expressions (A.21), (A.24), (A.27) in Eq. (A.14), we obtain after some algebra the Eq. (11). Appendix B. Expression of the axial tidal gravitational torque on the atmosphere in spherical harmonics of the surface pressure and tidal gravitational potential This appendix is devoted to the computation of the axial torque of degree l and order m (9). It can be expressed as: 3; rs 3 lm = (B.1) lm rs
with 3; rs lm (t) = a2
;
(prs cos s + p˜ rs sin s)Prs (cos )
×m(−clm (t) sin(m) + c˜lm (t) cos (m)) ×Plm (cos ) sin d d:
(B.2)
Bizouard, C., BrzeziTnski, A., Petrov, S., 1998. Diurnal atmospheric forcing and temporal variations of the nutation amplitudes. J. Geodesy 72, 561–577. BrzeziTnski, A., Bizouard, C. In>uence of the atmosphere on Earth rotation: what new can be learned from the recent atmospheric angular momentum estimates? Surv. Geophys., in press. Dehant, V., Bizouard, Ch., Hinderer, J., Legros, H., Gre0-Le0tz, M., 1996. On atmospheric pressure perturbations on precession and nutations. Phys. Earth Planet. Inter. 96, 25–39. Petrov, S., 1999. Modeling geophysical excitation of Earth rotation: stochastic and non linear approaches. Ph.D. Thesis, Olsztyn University of Agriculture and Technology, Poland. Roosbeek, F., 1995. RATGP95: Technical Note, Royal Observatory of Belgium, 5pp. de Viron, O., 1999. In>uence de l’atmosphre sur la rotation de la Terre: application au calcul des nutations. Ph.D. Thesis, UniversitTe Catholique de Louvain. de Viron, O., Bizouard, C., Salstein, D., Dehant, V., 1999. Atmospheric torque on the Earth and comparison with the atmospheric angular momentum variations. J. Geophys. Res. 104 (3), 4835. de Viron, O., Marcus, S.L., Dickey, J.O., 2001. Diurnal angular momentum budget of the atmosphere and its consequences for the Earth’s nutation. J. Geophys. Res., 26 (B11) 26–747. Volland, H., 1996. Atmosphere and Earth’s rotation. Surv. Geophys. 17, 101–144.