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The Chandler annual resonance
166
Physics of the Earth and Planetary Interiors, 11(1975) 166 168 © Elsevier Scientific Publishing Company, Amsterdam — Printed in The Netherlands
DISCUSSION THE CHANDLER ANNUAL RESONANCE KURT LAMBECK Institut de Physique du Globe, Université Paris VI and Département des Sciences de Ia Terre, Université Paris VII, Paris (France) (Received July 25, 1975; revised and accepted September 16, 1975)
In a recent paper Cannon (1974) examines the interesting question of the consequences of a possible resonance between the seasonal forcing function acting on the Earth and the free nutation or Chandler wobble. Cannon estimates that such a resonance could have there occurred 185 million ago andbesuggests that may about have been a causalyears connection tween this resonance and the onset of continental drift. However, Cannon’s estimate of the energy dissipated at the time of resonance is in error by at least two orders of magnitude and the actual energy available is quite insignificant compared with other known energy dissipation processes. In addition most of the energy can be expected to be dissipated in the ocean rather than in the mantle. The equations describing the polar motion for a linearly damped system are given by Cannon (p. 84) as: dm —--iwm—--iwf dt —
— —
(1)
where the complex motion is:
corresponds to an equivalent but undamped Earth. The excitation function is assumed to be of frequency w
1. The two componentsm1 and m2 can be decoupled to give an expression of the form: 2~ d~ 2 —4-2cs-~-—+w 2e(~1t+a) (3) d dt2 0m=fjwo
Cannon’s equivalent equation on p. 84 contains some minor errors that are of no further consequence. The solution of eq. 1 is of the form: pe~(~t~’)
where the amplitude factor p is given by P = w 2/[(w 2
0
1
wo2)2
+
(4)
4a2wi2]b’2
The frequency of this motion w~is that of the driving force. Cannon does not appear to distinguish between this frequency and the w of the expression (2). This is also of no further consequence as w does not enter into the subsequent discussion. If the forcing function resonates with the free
m=m
nutation of the damped system, w1 = Wo amplitude factor Pr becomes:
1 + iJfl2
the excitation is:
=
W~the
2/4a2
T=f~ i-if2 and the complex frequency of the free nutation is:
PrWo
—
(2) where r is the relaxation’time. 2 + a2)U2. The Themagnitude frequency of w the frequency is w0 = (w
and the ratio of the amplification at the driving frequency w 1 and the resonance frequency Wr is: O(Wi) O(Wr)
~(
Pr
— —
(c~i2— w~ 2)2 + 4a2w 0 12)
1/2
(5)
THE CHANDLER ANNUAL RESONANCE (DISCUSSION)
Cannon’s equivalent expression is dimensionally incorrect. For the linear system the specific dissipation
167
Ew(tr)
—
~(t)
2 (Wi QQrU~
\/~2(tr)_\7
1)l~,~2(t 0))
function is
1 —
=
Q
defined as:
2aw1 2 w
The present rate of dissipation is of the order (Munk and MacDonald, 1960):
____
E~(t
0)-’ 1.2~l0’~Q
and is frequency dependent. If the oscillator is driven at resonance: 120~r Qr
erg sec~
and at resonance: Ew(tr)
3.5
-
1014
Qr
erg sec~
This is smaller than Cannon’s estimate by a factor of
Wr
where 1/ar is the relaxation time at the Chandler annual resonance. In Cannon’s equivalent expressions a factor 2 has been dropped. The amplitude ratio 5 becomes: 2 12 1 1/2 0(w1) [Wi 0(w 2) = Qr lj + [~
and this differs from his expression (p. 84) by a factor 2wiQIWoQr. Also it does not necessarily follow that Q = Qr. In particular, we cannot assume that both Q = Qr and a = ar. Cannon’s last expression on p. 85 for this ratio should then read: 0(wi)/0(w2)
=
0.40
Qr
instead of his 0.882 Q. His expression for the time at which reaonance occurs remains unchanged but the length of time r for which the resonance condition will persist becomes (p. 87): 2sI = 2~2(tr)/3Q~I~
about 10. The observed time of relaxation for the Chandler motion has been estimated by Jeffreys (1968) to be between 14 and 73 years. A more recent analysis by Wilson suggests wobble Q may lie (1975) between 85 andthat 120.the TheChandler nature of the dissipation mechanism remains uncertain but probably occurs in the ocean (Miller and Wunsch, 1973). Wunsch (1974) estimates a world wide value of 4 l0i2~4. l0’~ erg sec~for the frictional losses in the shallow seas giving an ocean Q of between 25 and 250. Dissipation in deep oceans may also be important if the results of Lambeck (1975) for the dissipation of diurnal and semi-diurnal tides are also indicative for the pole tide dissipation. If the dissipation does occur primarily in the oceans any extrapolation of the Q into the past becomes quite uncertain as this parameter will depend on past ocean—continent distributions. There is no reason to consider that the nature of the dissipation mechanism has changed over the last few hundreds of millions of years. If we assume that r = Tr then: .
.
.
2Q 1/wo) or Qr> Q. Taking Qr 100 gives Ew(tr) 3.5 1016 erg sec1 whereas Cannon’s estimate is about two orders of magnitude larger. This dissipation is quite small compared to the dissipation of the rotational energy of about 5 erg seci associated with the present secular tidal deceleration of the Earth (Lambeck, 1975). Even Cannon’s inflated estimate remains smaller than this more important energy sink. In consequence any thermal impact of the Chandler annual resonance is small and unlikely to be a trigger for the onset of continental drift as Cannon Qr
The total energy dissipated during the passage through resonance is Ew(tr)r’ (p. 88) and is of the order 5 ~ erg sec_i (see below) as compared with Cannon’s estimate of about l0~~ erg sec_i if we assume that the present rate for the secular deceleration &2S of lO2i rad sec2 has persisted throughout the past although a synthesis of growth rates of the skeletal parts of invertebrates by Pannella (1972) suggests that the average value in the past may have been about one half of this value, The ratio of energy dissipation of the wobble energy E~at the time of resonance tr to that at the present time t -
0 should be:
(w
.
.
suggests.
168
K. LAMBECK
References
Miller, S.P. and Wunsch, C., 1973. The pole tide. Nature, 246:
Cannon, W.H., 1974. The Chandler annual resonance and its possible geophysical significance. Phys. Earth Planet. Inter., 9: 83—90. Jeffreys, J., 1968. The variation of latitude. Mon. Not. R. Astron. Soc., 141: 255—268. Lambeck, K., 1975. Effects of tidal dissipation in the oceans on the Moon’s orbit and the Earth’s rotation. J. Geophys. Res., 80: 2917—2925.
Munk, W.M. and MacDonald, G.J.F., 1960. The Rotation of the Earth. Cambridge University Press,. London, 323 pp. Pannella, G., 1972. Paleontological evidence on the Earth’s rotational history since early Precambrian. Astrophys. Space Sci., 16: 212—237. Wunsch, C., 1974. Dynamics of the pole tide and the damping of the Chandler wobble. Geophys. J. R. Astron. Soc., 39: 539—550.
98— 102.
Physics of the Earth and Planetary Interiors, 11(1975) 166 168 © Elsevier Scientific Publishing Company, Amsterdam — Printed in The Netherlands
DISCUSSION THE CHANDLER ANNUAL RESONANCE KURT LAMBECK Institut de Physique du Globe, Université Paris VI and Département des Sciences de Ia Terre, Université Paris VII, Paris (France) (Received July 25, 1975; revised and accepted September 16, 1975)
In a recent paper Cannon (1974) examines the interesting question of the consequences of a possible resonance between the seasonal forcing function acting on the Earth and the free nutation or Chandler wobble. Cannon estimates that such a resonance could have there occurred 185 million ago andbesuggests that may about have been a causalyears connection tween this resonance and the onset of continental drift. However, Cannon’s estimate of the energy dissipated at the time of resonance is in error by at least two orders of magnitude and the actual energy available is quite insignificant compared with other known energy dissipation processes. In addition most of the energy can be expected to be dissipated in the ocean rather than in the mantle. The equations describing the polar motion for a linearly damped system are given by Cannon (p. 84) as: dm —--iwm—--iwf dt —
— —
(1)
where the complex motion is:
corresponds to an equivalent but undamped Earth. The excitation function is assumed to be of frequency w
1. The two componentsm1 and m2 can be decoupled to give an expression of the form: 2~ d~ 2 —4-2cs-~-—+w 2e(~1t+a) (3) d dt2 0m=fjwo
Cannon’s equivalent equation on p. 84 contains some minor errors that are of no further consequence. The solution of eq. 1 is of the form: pe~(~t~’)
where the amplitude factor p is given by P = w 2/[(w 2
0
1
wo2)2
+
(4)
4a2wi2]b’2
The frequency of this motion w~is that of the driving force. Cannon does not appear to distinguish between this frequency and the w of the expression (2). This is also of no further consequence as w does not enter into the subsequent discussion. If the forcing function resonates with the free
m=m
nutation of the damped system, w1 = Wo amplitude factor Pr becomes:
1 + iJfl2
the excitation is:
=
W~the
2/4a2
T=f~ i-if2 and the complex frequency of the free nutation is:
PrWo
—
(2) where r is the relaxation’time. 2 + a2)U2. The Themagnitude frequency of w the frequency is w0 = (w
and the ratio of the amplification at the driving frequency w 1 and the resonance frequency Wr is: O(Wi) O(Wr)
~(
Pr
— —
(c~i2— w~ 2)2 + 4a2w 0 12)
1/2
(5)
THE CHANDLER ANNUAL RESONANCE (DISCUSSION)
Cannon’s equivalent expression is dimensionally incorrect. For the linear system the specific dissipation
167
Ew(tr)
—
~(t)
2 (Wi QQrU~
\/~2(tr)_\7
1)l~,~2(t 0))
function is
1 —
=
Q
defined as:
2aw1 2 w
The present rate of dissipation is of the order (Munk and MacDonald, 1960):
____
E~(t
0)-’ 1.2~l0’~Q
and is frequency dependent. If the oscillator is driven at resonance: 120~r Qr
erg sec~
and at resonance: Ew(tr)
3.5
-
1014
Qr
erg sec~
This is smaller than Cannon’s estimate by a factor of
Wr
where 1/ar is the relaxation time at the Chandler annual resonance. In Cannon’s equivalent expressions a factor 2 has been dropped. The amplitude ratio 5 becomes: 2 12 1 1/2 0(w1) [Wi 0(w 2) = Qr lj + [~
and this differs from his expression (p. 84) by a factor 2wiQIWoQr. Also it does not necessarily follow that Q = Qr. In particular, we cannot assume that both Q = Qr and a = ar. Cannon’s last expression on p. 85 for this ratio should then read: 0(wi)/0(w2)
=
0.40
Qr
instead of his 0.882 Q. His expression for the time at which reaonance occurs remains unchanged but the length of time r for which the resonance condition will persist becomes (p. 87): 2sI = 2~2(tr)/3Q~I~
about 10. The observed time of relaxation for the Chandler motion has been estimated by Jeffreys (1968) to be between 14 and 73 years. A more recent analysis by Wilson suggests wobble Q may lie (1975) between 85 andthat 120.the TheChandler nature of the dissipation mechanism remains uncertain but probably occurs in the ocean (Miller and Wunsch, 1973). Wunsch (1974) estimates a world wide value of 4 l0i2~4. l0’~ erg sec~for the frictional losses in the shallow seas giving an ocean Q of between 25 and 250. Dissipation in deep oceans may also be important if the results of Lambeck (1975) for the dissipation of diurnal and semi-diurnal tides are also indicative for the pole tide dissipation. If the dissipation does occur primarily in the oceans any extrapolation of the Q into the past becomes quite uncertain as this parameter will depend on past ocean—continent distributions. There is no reason to consider that the nature of the dissipation mechanism has changed over the last few hundreds of millions of years. If we assume that r = Tr then: .
.
.
2Q 1/wo) or Qr> Q. Taking Qr 100 gives Ew(tr) 3.5 1016 erg sec1 whereas Cannon’s estimate is about two orders of magnitude larger. This dissipation is quite small compared to the dissipation of the rotational energy of about 5 erg seci associated with the present secular tidal deceleration of the Earth (Lambeck, 1975). Even Cannon’s inflated estimate remains smaller than this more important energy sink. In consequence any thermal impact of the Chandler annual resonance is small and unlikely to be a trigger for the onset of continental drift as Cannon Qr
The total energy dissipated during the passage through resonance is Ew(tr)r’ (p. 88) and is of the order 5 ~ erg sec_i (see below) as compared with Cannon’s estimate of about l0~~ erg sec_i if we assume that the present rate for the secular deceleration &2S of lO2i rad sec2 has persisted throughout the past although a synthesis of growth rates of the skeletal parts of invertebrates by Pannella (1972) suggests that the average value in the past may have been about one half of this value, The ratio of energy dissipation of the wobble energy E~at the time of resonance tr to that at the present time t -
0 should be:
(w
.
.
suggests.
168
K. LAMBECK
References
Miller, S.P. and Wunsch, C., 1973. The pole tide. Nature, 246:
Cannon, W.H., 1974. The Chandler annual resonance and its possible geophysical significance. Phys. Earth Planet. Inter., 9: 83—90. Jeffreys, J., 1968. The variation of latitude. Mon. Not. R. Astron. Soc., 141: 255—268. Lambeck, K., 1975. Effects of tidal dissipation in the oceans on the Moon’s orbit and the Earth’s rotation. J. Geophys. Res., 80: 2917—2925.
Munk, W.M. and MacDonald, G.J.F., 1960. The Rotation of the Earth. Cambridge University Press,. London, 323 pp. Pannella, G., 1972. Paleontological evidence on the Earth’s rotational history since early Precambrian. Astrophys. Space Sci., 16: 212—237. Wunsch, C., 1974. Dynamics of the pole tide and the damping of the Chandler wobble. Geophys. J. R. Astron. Soc., 39: 539—550.
98— 102.