-1
'I ¡ :I
.
-
.
JULY t0, 1975
J O U R N A L O F G E O P H Y S I C A LR E S E A R C H
voL. 80.No. 20
Effectsof Tidal Dissipationin the Oceanson the Moon's Orbit
and the Earth'sRotation
KuRr L¡¡vlsecr
{Jniuersité de Paris' Paris, France
Institut de physique du Globe and Département des Sciencesde la Terre,
b y c a l c u l a t i n gt h e
T h e d i s s i p a t i o n o f t i d a l f r i c t i o n i n t h e o c e a n s a n d a t m o s p h e r eh a s . b e e ne s t i m a t e d
for the principal
models
tide
available
the
using
tides
by
the
orbit
the
moon's
of
,""ut"r f"tiurUations
cause
semidiuinal and diurnal tides. Only the secònd-degreewavelength components in the ocean,tide
l57o among
significant secular changes in the lunar orbit. These components of \h_e(z tideagreewithin
lunar motion
thi different ocean tide models. other frequencies(o,, /fr, ,sr, P,, and K,) have eftcts on thê
acceleralunar
computed
The
45Vo
Mz.
of
of
about
rotation
the
earth's
and
on
tide
úte
M,
of
of about 25Vo
determination of -35 arc sec/(100 yr), agreesto better than l57o with nearly all recent-astronomical
years. The diË
tions. There has been no signifiänt change in the tidal effect during the last 2000 or 3000
ference between the two esiimates of thJlunar acceleration is of the same order as their uncertainties'
and on dissipaHenceonly upper limits can be set onthel/Q of the sotid earth and moon of about l/50
to result
tion in the coìà of about 3 X lgrs ergs s-'. The total tidal acceleration of thç earth is estimated
ancient eclipse
in a 3.7-ms/100 yr increase in the-length of day. The most recent discussion of the
yr. If the nontidal ¿crecords indicates an observed increase in the length of day of only 2.5 ms/100
í000 years,which is
is
about
time
the
decay
eartir,
the
oìf
postglacial
response
ihe
to
celeration is attributed
equivalent to a mean viscosity for the mantle of about 10" P'
I¡lrnoPucrtoN
Tidal dissipation is the recognizedexplanation for the
ofthe moon and the deceleration
observedsecularacceleration
ofthe earth'srotation,but the energysink hasnot beenunambiguously identified lMunk and MacDonald, 1960;'Jefreys'
19621.It is generallyassumedthat the world's oceansmust
piay a dominantrole in the dissipationof tidal energy,but the
most recentcalculationsof dissipationin the oceansby Miller
11966l,Pekerísand Accad [1969],and HendershottU9721are
not entirelyconclusive.Part of the total dissipationmay also
occur in the solid parts of the earth, and an attempt at
separatingthesetwo possibleenergysinksis of interestinsofar
as it would give an estimateof the soiid earth'sglobal anelastic properties,namely, the specificdissipationfunction
and would provide lurther insight
0/Q), at low frequencies,
into the evolutionof the lunar orbit and the earth'srotation.
To calculatethe tidal dissipationin the world'soceans,three
approachesare possiblelMunk and MacDonald, 1960;Kaula,
tiO8l. f¡e first is to calculatethe meanrate of work per unit
areadoneby the sun and moon on the oceansurface.The second is to calculatethe energyflux per unit time and per unit
area acrossthe entrancesto shallow seas.The thi¡d is to
calculatethe rate ofwork per unit surfacedoneby currentson
thê sea floor. Miller [1966] used essentiallythe secondapproach,which, becauseit dependsoniy on the first power of
the tidal velocity,is usuallyconsideredto be the most reliable'
The third methoddependson the third power of the velocity'
The first method has beenusedby severalinvestigators,including Grouesand Munk [1958]and most recentlyPariyskiy
This methodhasthe advanet al. ll912l andKuznetsor:11912)'
of the oceanfloor disnature
the
of
tageof beingindependent
sipationmechanismand doesnot requirea knowledgeof the
oceancurrents.Neither doesit assumethat friction occursin
shallowseasonly. But it doesrequirea kiowledge of the tide
we use this apover the entire oceansurface.Nevertheiess,
as far as the effectof tidal dissipationon
proachherebecause,
in the ocean
the moon is concerned,only certainwavelengths
Copyright@ 1975by the AmericanGeophysicalUnio¡r;
seemto be relatively
tide are important and thesewavelengths
well known for the major tides' Thus our approachdiffers
from the standardmethod in that the ratesof work done by
rather than on
the moon and sun on theselong wavelengths,
the
oceansurface'
over
integrated
are
tide,
the total ocean
Kaula llgíglalso usedthe first approachimplicitly by developing the tidal effectiveLove numbersand the lag anglesas functions of latitude, but his approachdoes not lend itself to a
ready interpretationin terms of the usual descriptionof the
ocean tides. Also, Kauia does not aliow for the frequency
The development
oftheseglobaltidal parameters.
dependence
the similar
eivenherefollows that of Lambecket al. 119741îor
satelartificial
earth
close
of
evolution
the
orbital
f,roblem'of
lites acted upon by the potential of the earth'stides'
Ttonl PoreNrtnls
Solidearrh üdes. The earth, acted upon by an external
force,will deform,and this deformationwill resultin a further
perturbingpotentialAU outsidethe earth that will contribute
io the total force function acting upon the satellite'This adin termsof the equatorial
ditional potential can be expressed
Keplêrian elementsa*e*i*M*os*dl*of the tide-raisingobject
(sun or moon) and of the satellitepositionaeiMul, at which
AU is evaluated,in the following way lKaula, 1964;Lambeck
et al., 19741:
o r ¿ @
AU :
t
Ð t
l-2
t
q'-o
m-O
+
i o. t' -l ¿Å' ). ,' \(nE* t)\ n' ./ ' c ' 'a' -+
/-¿
iÉ0
e--@
"E0
(l -
ffi
m)t' .^
F¡^¡(í)G
(2 - 6 Ft^o(ì*)G
¿¡"@)
¿oo(e*)
"^)
'coS (u¿-oo* -
t)t^ìo *
(1)
.,,,o0)
with
út^pq* = (t -
Zp)of + (/ - 2p * q)M* *
md|*
and
utmre= (l - 2j)o + (l - Zi t g)M + mA
where¡n*-is the.massof the tide'raisingobject,R''isthe mean
2917
L¡rttl¡gcr: TIo¡l
2918
Dtsstperlo¡r
radiusol the earth,and k¿is the Lovenumberof degree/. The w i t h
response
of the earth
€xpl€ssês
the anelastic
phaseangle€¿¿pq
Gm * k ¡
-.
Al
the
time
deiay
in
responseby
to
the
related
is
and
lLambeck ^w' - - - ---------:--:---: l
G
(
M
lm*)al
et al., 19141
We haveignoredany contributionsthat could arisefrom the
ût*pq* - û¡^oo* At =
êt^pc:
on the moon by the
of energydue to tides'raised
dissipation
- ( t - 2 p + s ) M *N + m â a t earth.Substituting
into (l) the valuesof theplanetparameters
givesthe
lunar parameters
for those of the corresponding
aeiMo:Qwill vary in time owinginteralia
The orbitalelements
potentialof the moon'stidal deformationper unit massof the
he¡ewith theeffect
to the tidal potential(1).We areconcerned
planet.The effecton the lunar orbit reiativeto the planetis
of this potentialon the lunar orbit itself and in particulai
then found by multiplyingthis potentialby the factorM/m*,
of tidal
with the secularevolutionof the orbit. In the absence
whereM is the massof the planet and m* is the massof the
elementsof the earth-moonmotion
friction the conservative
satellitelKaula, 19641.
(l) the scalarangularmomentumof the
19661
are lGoldreich,
in the lunar orbit due to the
The principalperturbations
lunarorbitalmotion,(2) the scalarangularmomentumof the potentialaregivenin Table1. To identifythemwith theusual
earth'sspin,(3) the componentnormal to the eclipticof the nomenclature
of the oceantides,we alsogivetheequivalences
total angularmomentumin the earth-moonsystem,and (4) between the orbital indices lmpq and the Darwinian
of tidal disIn the presence
the sum of the potentialenergies.
The major contribution
nomenclaturelLambecket al., 19141.
wiil vary on the long time scale,and
sipationtheseelements
Toà¡^oocomesfrom the M, tide, as hasbeenunderstoodsince
thesechangesarerelatedto the secularchangesin the lunar or- the work of G. H. Darwin.The K, tide doesnot perturbthe
and semimajoraxis.The time
bit's inclination,eccentricity,
semimajor'axis,
sinceI - 2p + 4 = 0. The diurnalO' tiderepvariationsin theseelementscan be found upon substitutingthe
about 5Vo.The
resentsabout 20Voof the effectof M, and 1y'2,
Next,
potentialÂU into the Lagrangianplanetaryequations.
less
tidescontributes
sum of all otherdiurnaland semidiurnal
to estimate the effect on the lunar orbit, we can equate
than LToof the M, contribution.The fortnightly tide correa*e*i*M*'þr*Q*with aeiMoQ. Finally, as v/e are interestedin
spondsto the sum of the termswith lmpq : 2000andlmpq =
the secularchanges,we requireonly terms of zero frequency: 2020,for which the phaseangiesare -2M*Lt and +21'I*At,
The
í)t^pq*- í)t^rs: 0, orp = 7 and q : g lKaula, 19641.
Its contributionto å is about l7o,and that of the
respectively.
e, and
secularchangesin the lunar inclinationi, eccentricity
monthly tide even less. For the eccentricitythe principal
semimajoraxis ¿ are then given by
secularchangecomesfrom the N, tide with signiflcantcontributions îrom Mr, Lr, Or, and M*. For dildt the contribu- 2p + q) sin e¿^on
à,^on = 2Kt^lFr^p])f'zlG,on@¡l'it
tions of LheMr, K,, and Or tidesare of similarmagnitude,but
the last two are of oppositesign and will largeiycancel.
(1 co2\r/2
Oceantides. The global ocean tides are known approx.,
\¡
,tI ñ p q -- ttt,
rLm
ø
LFr^,G)flch"(r)l'[(1 _
imatelyonly for the principaltides,suchasMz, Sr,Kr, and Ot.
Ae
"r)rÆ
at any point in the world's oceanby
Thesetidesare expressed
.(l - 2p * q) - (l - zp)l sin e¡-,0
an amplitude{o and phaseP as
- lr¿]
- É(d, À)]
[(l - 2p) cos i
Eu(ó,À) = åld, À) cos [2a'f
,
2\1/2
j
a
\
L
e
s
l
n
,
\d|/
¡^,q
where/ is the frequencyof the tidal component¡r being
. l F L * , ( Ð ) ' Í G , o o ( e ) f 's i n e¿ - o o analyzed.On the continents,{ : 0. For the availableocean
(+\ - K,^
TABLE l.
lmpq
Amplitudesof the SecularVariationsin aei of the Lunar Orbit Due to the PrincipalFrequencies
in the Solid Earth Tide
Tidal
Component¡.r
MultiplyingFactor*
{
+ c o s i )' n a
da/dt
de/dt
a(t- sê)
-àe
2200
M2
*(l
2201
N2
\ff1t + cosi¡,!
220(-t)
L2
Ð'f
${t + cos
ào"'
2 rt 0
K,
t(sin, ¡ cos,i) {
0
U
2100
ol
lsin'?i(l + cosr¡tí
(l - 5e'z)a
-le
(l - 5e'z)a
î,e
I
2ooo
I
2o2o
rI
Ml
iae'
(ftsin'i) ¿
-(l
)
*tt -¡sin,i*f sin.Ðf
t y =
1k"Gm/laG(M + m)lt/'l(R/a)6 si¡ et-pq.
- Se'z)a
_t^
8s
_L
di/dt
ll,
9 e ' ? \ c o s t I-
t\'-tlsinl1",/coC¿l\
'\ sini
' "/cosiâ¿'\.'ñi
/
I \
/
1"2\ |
- { /l + + l - - :
\
z /sln¡
I l,
4\'
-
9e,\(2 cos¡ - t)
t/--n
l l,
e ' z \ c o si
¿\
¿tstnt
l l, - e'¿\cos i
-Z\'
1) ¡' ¡
t-
L¡,¡r¡¡¡cr: Tronl DrssrperroN
models the {o and É have been evaluatedat grid intervals of
10" in latitude and longitude and the {o cos B and {o sin B are
expanded in series of spherical harmonics lLambeck and
Cazenaue,1973].The tide can then be expressedby
with
V
l
-
| + k"'p" /R\"
3GM
2'+t7\;l
tG(M+ ^*)al'
. (c,,- ),F
" ," ( i ) G" " , ( e )
whereþ is the meandensityof the earth.
Table 2 summarizesthe principalresultsfor the oceantide
. ( C , , ' ) , s i n l 2 t r l T + r À + ( € " , t ) u l perturbations,
and theseresultscanbe compareddirectlywith
perturbations
the
solid
tide
in TableL Therelativeimportance
where the (C,¿r) and (r",1)uare functions of the sphericalharperturbations
will dependlargelyon the
the
two
types
of
of
potential
exmonic expansionsof the ¡¿ tide component.The
phaseanglesoccurringin the two potentials.
pressed in Keplerian elements outside the earth due to this
Table 3 summarizesthe relevantcoefficients(Cr¿1)uand
layer of density p. is
(e"¿1)u
for some of the availableoceantide models.For M",
resultsof threemodelsare available.Hendershottfl972] gives
+Îul(
iii
i 1 + -¡/ r " '
AII'.' :
t
directly coefficientsof the sphericalharmonicexpansionthat
L / , L / 4 4 ^ ^ l
zs
7
4
g=0
Ëo Eo "=;
can
be relatedto the (C",¿),and (e,,r)u.The other modelsused
-ls-reven
l- _¡_
/p\"
are
those
of Bogdanou
and Magarik [967] and PekerisandAc^r
' (i) o,tc",.)*F",.(i)G"..(erLjär_1"_,
occur betweenthese
ooou,u.* Q) cad |9691. Major regionaldiscrepancies
solutions(seethe discussionby Hendershottll973l), but the
with
three individual estimatesof (Crr* cos err+)"differ from their
meanby only 107o.For K, and O, the interpolatedsolutionsof
:
(s
+
2u)u
*
2u
u)M
1"cuu! $
+ (0 - 0) +.ZrfT * (e"")u Díetrich [944], as discussedby Lambecket al. ll974l, are
used.Sincethesecalculationsa numericalsolutionfor the K,
of degrees and al- tide hasbeenpublishedby Zahel[1973].For therelativelyimThe kr' arethe load deformationcoefficients
low for the deformationof the earthunderthe variableocean portant ff, tide, for which we haveneithernumericalnor emload. For the seculareffectsof AUf'on the lunar orbit we re- pirical models, we assumethat (Crr+)u, ãnd, (C22+)¡¡"are
quire that ^yrruut= 0. The correspondencebetween the proportional to the correspondingtide-raisingpotentials,or
equatorialelementsMo:Q and the eclipticelementsin which
au"ot
7 'Ztrfi is usuallyexpressed
is discussed
by Lambecket al. |9141.
( c r r * ) n " È ( G ,:*)' * . ' 1 í # x ,
e(c22")¡a.
For the M, tide, for example,ir vanishesfor 7* only, and
:
=
:
=
t h e nw h e nt
m , s 2 u t , s 2 u * u t , a n d u 0 ; o r and (err+)¡¿,= (err*)u,
in view of the fact that the frequencies
w h e n . r: 2 , u = 0 ; w h e ns : 4 , u = l ; w h e n s: 6 , a , : 2 ; e t c .
of M, and N, are similar and that the solutions by Bogdanors
Because
of the (R/a)" term in A Uu'the principalcontributions
and Magarik [1967] for the Mz and Sz tides show a similar
will occur for s : 2. For closeearth satellites,perturbations
proportionality. For L, with lmpq = 220(-l) we have 1
due to the coefficientsof degree4 can be important, but for
AUrrot-¡ = 1-e/2) AUr"ro, and we assumethat
the lunar orbit theseperturbationswill be smallerthan those
due to the coefficientsof degree2 by a factor'of (l/60)'. The
(C r,* )r" o (- e/ 2)(C zz+
)u,
secularperturbationsin the lunar orbit are given by
¡ : 5 - \ -
IT
L-/
!-0
t,
Ð "",,,tn
Ll
ú-0
l-
a : zK"n,'g - 2u * r)l
and
''l
t?t
|
"-
LSlD-l,-¿
t e"".
. (errn)", = (er"*)u,
(.",*),
ôdd
For the fortnightly tide M ¡ and the monthly tide M ^ no tidal
information is available on a elobal scale.
Evolur¡o¡¡ oF THE Lu¡¡¡rR On¡¡r
di
di
(s -
.,1-.orl'-'"""" , *r
(t"' J,
'u)lLri.rl,-,
ouu
's K
"
tuol
2u\cosí - ¡ll-cosl'-'u"""u rr
/r\
(€"r 'lc (r/
a slnt 0 -Zfrlrinl"-,,o0
Secularchangesin the semimajoraxis. We introducethe
subscriptsM andS to referto lunar and solarparameters,respectively.Table4 summaiizesthe computeda¡¡ for the principal öceanmodels.The threeestimatesobtainedfor the M"
In the subsequent
calculatiåe differ bv a maximumof 157o.
TABLE 2. Amplitude of the SecularVariationsin aei oî the Lunar Orbit Due to the PrincipalFrequencies
in the OceanTide
l^pq
Tide p
2200
M2
2201
N2
220(-t)
Lz
2 ll 0
Kr
2100
ol
* Z = l(l + k,')/lc(M
MultiplyingFactor*
(err+)"
"ori)'zcos
(er2+),
4J +
"ori)'zcos
(e,,+),
*
!0
"ori)'zcos
cosi sini sinier*),
f;
4l *
Z sini (l * cosi) sin(er¡+)¡r
+ m)altt2l(GM/R)(pol ùß)'z/(a)(Ct^*).
q
|
da/dt
de/dt
5¿2\
-f."
7\t-7)
ffiea
ffit - ze'¡
-*æo
9
¡o-
0
0
e(. 5r\
3\'- zlo
-iæ
di/dt
q,. , ^ a./cosi- l\
ï ñ { l f l e ' )' l\ - ,
slnt I
l\
/cositñel"- -"
I
\
sln, /
9
/cosi-
l\
20-\ sini /
9 (l + 2e2\
l0\ sini /
s.. ^ ./2cosi-l\
óÃl - ¿e"
' l' l \|
sln,
/
LAMBEcK:Trp¡l
2920
T A B L E 3 . A m p l i t u d e s a n d P h a s e so i t h e P r i n c i p a l C o m p o n e n t s i n
the Fluid Tides Leading to SecularVariations in aei of
the Lunar Orbit
(c,^,),,
Tide
lJ
Solution
M2
Pekeris and Accad [1969\
Hendershott |9721
Bogdanou and Magarík
.s2
Bogdanoo and Magarik
M2
M2
Kl
ol
D
L2
.s2
u 9671
lt967l
Dietrich ll944l
Dietrich lI944l
Estimated
Estimated
Estimated
tide+
Atmospheric
(c,^"),,
cm
deg
4.4
5.t
4.3
340
3r6
3r8
1.6
332
3.3
1.8
0.7
0.9
- 0 .I
0.34
318
t42
140
320
330
158
* Converted to an equivalent water layer.
tions the mean of the three models is used.The secularchange
in the mean motion n¡r of the moon is related to à¡q by
(4)
hu:-3/2(nu/au)itu
and is also given in Table 4. We have ignored the tide raised on
the moon by the earth's attraction, and the possibilityremains
that the tidal dissipation in the moon may be important. The
Mztide wiil not lead to dissipationin the moon, since it raises
a permanent tide on the moon. Qissipation in the moon wili
occur due to the O, and N, tides, but the actual amount will be
smail. From equation (a8) of Kaula 119641we obtain
(d)-oo,"rr."t
( ¿ i ) " ' . r o erfe c r
:
sin .,
( rt_\'(n"\'.(k,).
/., tt" .
\^-l
\ R )
with (k)¡a : 0.020 lKaula, 19681.The phase angle e¡ais unknown. Seismic data from the Apollo seismic network indicate
a Q lor shear waves for the upper 500-600 km of the moon
higher than several thousand and below this depth a Q of
about 300 with a central region p of about 100 fToksöz et al.,
Dtssrp¡,rroN
d iththe
t 9 7 4 1W
. e t a k eQ : 1 5 0a n d eu : l o , a s c o m p a f e w
below).The
e of the earthabout7.5o(seediscussion
observed
moon effecton àM for the O, and N, tideswill thenbe of the
order of 0.05 X l0-7 cm s-r or smallerand unimportant.
Secular changesin eccentricityand inclination. Table 5
of thelunar
in theeccentricity
the secularchanges
summarizes
rate
The totalestimated
in the oceans.
orbit dueto dissipation
of changein eccentricitycannotbe very certainin view of the
of Crr+
importanceof the N, tide,for whichno directestimates
and err+are available.An importantcontributionIo ey can
come from the N, tide raisedon the moon, but the consequencesof this on the lunar orbit evolution cannot be
withouta reliablevaluefor the Q of the moonat the
evaluated
tidal frequencylGoldreich,19661.The secularchangesin the
inclinationof thelunarorbit on theequatorialplanedueto the
oceantidesarealsogivenin Table5.The tideson themoondo
not appear to be important, since the effectsof Kt and O'
largelycancel.The total computedpresentvalue of di/dt is
years.
small,lessthan 2o/1O'g
Astronomicalobseruations
for the lunar acceleration.The
astronomically observed quantity is the non-Newtonian
of the moon'smeanlongitudeand is due
secularacceleration
in part to the earth'svariablerateofrotationó and in part to
of the moon,it*. We ignoreany possible
the tidal acceleration
contribution from a secularchangein the gravitationalconstant.The it¡aand c,rare separatedby combiningthe observed
lunar accelerationwith observedsolar or planetaryacceleraof the
tions or by observingthe earth'srotation independently
of the pastevolution
lunarmotion.Most previousdiscussions
or of
of Spencer-Jones
of the lunar orbit usethe accelerations
Fotheringham[Munk andMacDonald,1960],but a numberof
indicatethat the accelerations
analyses
recentand independent
deducedfrom theseresultsarein error by a factor of almost2.
Newton [19701 analyzedancient eclipse observationsand
yr)2near200B.C. and-42.4
foundn¿ =41.6+.4 arcsec/(100
t 6 arc sec/(100yr)z near 1000A.D. The more recentinterpretationby Muller and Stephenson
|9751 of the ancient
gives-37.5 + 5 arc sec/(100yr)'. From
eclipseobservations
TABLE 4. Summaryof ComputedTidal Variationsin the Lunar Orbit and in the Earth's
SecularRotation
t:\
Solution
Lunar
M,
M2
Hendershott
ll972l
PekerisandAccad
da/dt, l0-7
cm s-t
t.4l
r.59
- |.4'l
- 1.65
I.JJ
- 1.28
I re6e]
Mz
Bogdanoo and Magarik
ltg67l
Mz
o,
N2
L2
K,
Solar
,S,
P,
Rz
Atmospheric
,Sz
I OtAl
Mean
Díetrich u9441
Estimated
Estimated
Estimated
dn/dt, 10-23
rads-'
i _¡.41
0 .r 0
0.09
-t.4'7
- 0 . 1I
-0.10
l(l-22
tirlo
1^Å
<-2
@Ttt
-6.04
-0.46
-0.01
-0.4r
0.10
-0.02
-0.56
0.t7
-0.02
-0.40
UT
-6.61
-0.29
-0.34
-0.02
-0.40
Bogdanou and Magarik
- 1.34
lt967l
Estimated
Estimated
- 0.05
-0.07
Kertz et al.*
r.60
- 1.68
-6.91
-0.13
-0.8 l
u.¿o
- 8.86
órro is the contribution from the secular change in a, isyl" that from e, and ri71¡ that from i.
*
Quoted in Chapman and Lindzen ll910l.
z92l
L¡ns¡cr: Trp¡,1 DlssrpnrroN
occultationsof stars by the moon since1955,Van Flandern
yr)'z;from merihu = -52 +.4 arcsec/(100
[970] determined
circle
dian
observations
of the sun, moon, and planetssince
1913,Aesterwinter
and Cohenll972l foundi¡¡ : -38 t 8 arc
sec/(t00 yr)'?,and from lunar occultationdata since 1663,
Morrison[973] estimatedh* = - 42 +.6 arc sec/(100yr)'?.
The resultsfrom the eclipse,
occultation,
and meridianobservationsare in good agreement.
Thereis no evidence
that any
significantchangein the tidal effectshasoccurredduring the
last2000years.The oceantidalestimate
of i" is, from Table4
(seealsothe comparison
yr)'z,
Table6), -35 t 4 arc sec,/(100
with the astronomical
in agreement
estimates.
In contradiction
lo Newton's[970, p. 387]argument,a valueof ¡t¡rof about
-40 arcsec/(100yr)'zis in completeharmonywith the energy
dissipationwithin the earth-moonsystem.In particular,as
the tide modelsuseddependon recenttide observations,
the
agreement
betweenthe tidal and the ast¡onomical
estimates
is
furtherevidencethat therehas beenno significantchangein
the rate of dissipationof tidal energybetweenabout 500
B.C.and the present.We canconcludewith confidence
that if
not all, at leasta very major part of the secularchangein the
moon's mean longitude,is causedby dissipationof tidal
energyin the oceans,
and we do not haveto invokesignificant
energysinksin the earth'smantleo¡ core.
TABLE 5. Computed SecularVariations in the Lunar Eccentricity
and fnclination on the Equator Due to the Principal Ocean Tides
Ocean
Tide
de/dt, s-'
-4.90x
6 . 0 0x
- 1.08
x
- 3 . 1 9x
0
x
5.36
M2
L2
o,
Kl
Total
Hs :
lMm5/(M
t m)las'zns(l -
es2)t/2
:
6.
- 5 . 5 7x r 0 - , '
-2.74 x l0-Le
-
rl
(Ë/¡llcos ia I
Hs cos is)
.
ar¡t :
- C1
Mm*
d ,
2
,i. -
nu\t
M + mMfiLam
eu
)
COS ¡Ml
=
1 Mmu
,l .
ñ*
nuaa-\'s cos iu C U + ^*
I
e, cos iuèu I
(5a)
sin ' " * )
The astronomicallyobservedquantity is the iv. Thus the
first term of (5a),
.,
The componentof .Ë1parallel to the rotation axis is 1l cos i.
whereC
The angularmomentumof theearth'srotationis C<o,
is the momentof inertiaabout the spin axis.When it is aswith the spin of
sumedthat the angularmomentaassociated
the moon and sun arenegligiblysmall,that the inertiatensors
of the threebodiesdo not varv with time.and that G is conTABLE
l0-r,
l0-'0
l0-æ
l0-,,
l0-¡,
"
wherethe -Lu and -l,s represent
the lunar and solar tidal
torquesactingon the earth.With (4'),
(4')
l.a
- 3 . 8 5x
- 1 . 3 9x
- 2 . 5 0x
l . l 9x
-Lrtr,*Zs)
ar:ur,, *ór. :
For a circularorbit the angularmomentumof the orbital
motion of the moon and earthabout their centerof massis
The angularmomentumof theorbitalmotionof the earthand
sun about their centerof massis
l0-,0
l0-"
l0-,0
l0-,,
stant,the changein the earth'sacceleration
r,r.dueto the tidal
potentialis
E¡,nrH'sSeculen AcceL¡n,quou
Hy : lMm¡a/(M + mr)layzny(l - eM2)1/2
di/dt, rads-l
6)T yto¡1
:
1
Mm*
3CMlmu
d ¡¡4
2
COSí¡¿n¡¡
can be evaluateddirectly from the astronomicalobservations
or from those oceantides ¡aisedby the moon that transfer
angula¡ momentum through a changein a¡1or ny. Table 4
summa¡izes the computed ó7, for the principal tidal
Summary of the Observed Lunar Acceleration hu, à*,
the Earth's Acceleration ó, and the Energy Dissipated Ê,
ó , l 0 - 2 2r a d s - 2
Solution
L
2.
3.
4.
Spencer-Jonesff
Newton ll910l!
Newton|9701$
Muller and Stephenson
I l 97s]
B e s tm e a n s o l u t i o n
of Muller and
Stephenson
5. Oesterwinterand Cohen
ue12l
6. Morrison[19751
Meanof solutions
4-6
Tidal estimation(Table 4)
h,
[ 0 2 3r a d s - 2
ï
a, l0-1
cm s-r
TidesT
Observed
- 1.09
-2.04+.0.24
1.04
r.98
- 8 . 3 6* r . l 2
- l.8l+ 0.24
r;76
- 7 . 3 9+ t . 0 7
- 1 . 9+
6 0.12
l.91
-8.03+ 0.54 - t0.44! 0.64 -7.05+ 0.35
- 1.8+
5 0.38
t.77
- 7 . 4 9+ l ; 7 0
-2.02r 0.29
- 1 . 9+ 0 . 2
- 1 . 7+ . 0 . 2
1.96
1.8
t.6
- 8 . 2 6* 1 . 3 0 - t 0 . 6 7+ t . 3 4 - 4 . 0 1r 1 . 0
- 7 . 6 8+ 1 . 0 - 1 0 . 0 9+ 1 . 0
- 8 . 9 0+ 0 . 7
- t 0 . 7 7+ t . 2 4 - 5 . 8 0+ 0 . 7
- 6 . 5 2L 0 . 7
-9.80
Nontidal
l 0 r oe r g ss - t
4.9't+. 1.42
4.25r 1.42
3 . r 0+ 1 . 3 2
6.03+ 0.55
3.39I 0.73
5.84* 0.26
- 9 . 9 0+ t . 7 3
5.59+ 0.50
5.54+ 0.75
6.66r t.6',1
6.08+ 0.59
: 5 . 7t 0 . 5 0
5.0+ 0.30
* Acceleration ô¡, due to lunar tides only and estimated directly from åy.
t Total tidal acceleration (ô"ü + ò"s) consisting of the acceleration described in the first footnote plus the additional tidal contributions
discuised in the text.
f Mean of Newton's [970, p. 272l two estimates for the epochs 200 B.C. and 1000 A.D.
$ Newton's |970] value as reinterpreted by Muller and Stephenson.
f Quoted in Munk and MacDonald 11960l.
2922
Lltr{srcr: Ttp¡l DlsslpnrroN
components.
The major contributioncomesfrom the secular
changeín a¡aor n, due to the lunar tideMrwith smallercontributionsfrom ìy', and L". lmportant contributionsto óa"
comefrom thesecularchangein e¡1(principallydueto /fr) anä
from the secularchangein in¡ (essentiallyK,).
The acceleration
due to the solar torqueis
.
ú)?s :
I
Mm"
M
C
+ *,
ns
,
rsas " ( cos ls .\ã
* es cosisi" * ,i" r"
#)
(só)
and althoughi, will be very small,ó7" will be importantbecauseof the as2term occurringin (5å).The perturbations
ris,
ås,and dis/dt areobtainedfrom (3) by consideringthe earthas
the satellitemoving aroundthe sun as planetand computing
the changesin the earth orbit due to the tide raisedon the
earth (as satetlite)by the sun (as planet).The ratio of the
torquesdue to the M" andS, oceantidesthen becomes,ustng
the resultsin Table 2,
(L)u.
:
(L)""
(,ir)n .
(ó')""
:
m* (ar\t
^t
\"")
lU-åar'=.".¡,,+ *tft -- ..
\
t h e m e a n v a l u e ( - 1 0 . I + 1 . 0 ) X 1 0 - 2 2r a d s - 2 f r o m t h e t h r e e
astronomicallybased solutions discussedabove and in Table
6. The uncertainty of the ó7 has been estimatedfrom the lollowing uncertainty estimatesof the C2r* cos er"': l\Vo for Mr;
20Voîor N2, Rz,Kb P,, and O,; l5VoforSr; and l0Zofor the atmospheric tide. The mean of the above two estimatesof r,.r7
correspondsto an averageinc¡easein the length of day (lod) of
3.7 t 0.5 ms/100 yr over the last 3000 years. This value is
somewhat higher than the estimateby Muller and Stephenson
[1975] and represents an improvement for the following
reasons:
l. The contribution due to S, is basedon an actual ocean
tide model rather than on any assumptionsabout the relation
betweendissipationof tidal energy at the M, and ,S, frequencres.
2. Additional solar tides have been considered.
3. Transfer of angular momentum associatedwith the
secularchangesin inclination and eccentricityof the lunar and
solar orbits has been included.
4. The error introduced by Newton [1968] in computing
the loading effect of the atmosphere has been corrected
(Newton assumedthat the load coefficientkr, was numerically
equal to the Love number kr, whereas,ti : -0.¡O and kz :
0.30).
Thereremainsan uncertaintyabout the atmospheric
tidal
contributionin that in the estimates
of the Cr,+,any response
of theoceansurfaceto the pressurevariationhasbeenignored.
.(Crr" cos erz*)u"
If the oceanresponse
is static,an increasein atmosphericpres(Crr* cos ezz*)s,
surewill resultin a loweringof sealevelin the ratio of about I
increase
of I mbar overand above
By means of the coefficientsgiven in Table 3 for the two cm of waterfor a pressure
semidiurnaltide modelsof Bogdanouand Magarik [1967]and the mean pressureover the entire oceansurface[Munk and
MacDonald,19601.Analysesby Wunsch[1972]indicatethat
the error estimatesdiscussed
below,this ratio becomes(4.9 *
.
1.3).This valueis in agreement
with the valueof 5.1givenby the responseis largelystatic for pressurefluctuationsas short
as40 hours.If at the semidiurnalfrequencythe response
is still
the equilibriumtheory [Munk andMacDonald,19601.
Jefreys,
largely
static, the coefficientsin Table 3 will be significantly
[1962]estimate,modifiedby Newton[1968],is 3.g when rhe
argumentof nonlinearfriction in the oceansis used.The effect reducedin amplitude.
Nontidalchangesin the earth'srotation. The astronomical
of thesolartideson theò" is givenin Table4. For pr(tmpq=
observations
of the total accelerations
of the earth are sum2 t00), for which no globaloceandataareavailable,we assume
that its contributionis proportionalto that of O, accordingto marizedin Table 6. Muller and Stephenson
[1975]conclude
that
Newton's
estimate
is
in
error
owing
to Newton'sas[1970]
an equilibriumresponse.
That is,
sumption that the differenceand derivative(ephemeristime
(ôr)", = (rùo,/5.1
minus universaltime) vanishat epoch1900,whereasMuller
and
Stephenson
arguethat the derivativevanishesnearepoch
and similarly for the .R, tide (lmpq : 2201),
l77Q[Munk andMacDonald,1960,p. 1791.
Their reinterprered
resultof Newton[1970]is alsogivenin Table6, and now the
. (ór)", = (i7)y,/5.1
agreementis most satisfactory,indicatingan increasein the
We must alsoconsiderthe atmospherictide, which, because
lod of (2.5+ 0.3)ms/100yr. Morrison'sanalysisof the lunar
of its thermal origin, leadsthe sun and tends to opposethe
occultationssincethe seventeenth
centurygivesan increasein
oceantide effectslHolmberg, 19521.Table 3 givesthe relevant
the lod of about 1.5ms/100yr, but this estimatemay reflect
coefficients,
expressedin centimetersof water on the earth's
not mereiythe secularaccelerationbecauseof the presenceof
surface,estimatedfrom the model of W. Kertz, B. Haurwitz,
large,long-period,or irregularfluctuationsof nontidal origin
and A. Cowleyas given by Chapmanand Lindzen[1970].The
in the earth'srotation spectrum.Oesterwinter
and Cohenhave
accelerationof the earth will be
also estimatedthe variationin the earth,srotation,but their
( r_
time span of 70 yearsis too short to give a meaningfulestic o s a-r-r '*l)'" , * r ^ o " o 0 " . " _
( r r ) " " n . - o " o n " ' " - t * : :2 2 "
(ôr)""o""""
::
:
---mateof the secularchangein lod.
(Crr- cos err*)s,o""""
Comparingthe astronomicalestimateswith the computed
Sincea major part of the tidal accelerationof the earthcan tidal accelerations
indicatesthe existenceof a major secular
be deducedfrom the lunar acceleration,
the total acceleration variationof nontidalorigin in the lod. Despitethe improved
can be computedin two ways:(l) ór_,o plus the computed analysesof the astronomicaldata
and the improvedestimates
contributionsdue to the tidesnot conüibutingro ri' (^S,and of the tidal contributions
we arein thesamejosition asMunk
P,), the contributionsarisingfrom the iecularchangesin e and and MacDonald in 1960in that we have postulate
to
further
di/dt (both lunar and solar),and the atmospherictide S, and mechanismsto explain the observed
secularchangein the
(2) as the sum of all tidal components.
The total purelytidal earth'srotation. The new magnitudesestimatedfor thesenonestimateis (-8.9 + 0.6) X 10-r, rad s-2 as comparedwith tidal contributionsare larger than those
estimatedby Munk
\ I f
åes' cos is f
åt(l
cos is)/cos isl ,/
L¡r¡vts¡cr:Tlo¡l Dtssrp¡rto¡l
2923
and MacDonald,andanyproposed
mechanism
mustexplaina i n t h e e a r t h m u s t b e
nontidaldecrease
in the lod of about 1.2ms,/100yr from the
ii = Ìi,+ È"= C(uà, - nuur,)
M u l l e ra n dS t e p h e n s osno l u t i o no, r a b o u t1 . 6m s / I 0 0y r f r o m
the reinterpreted
Newtonsolution.Of all the acceleration
es_ From Table 4, ,r, = 0.8ór. Also, nn 0.04t:, and .Ë :
timatesmade,the nontidalacceleration
is the leastsatisfac- 0.96C0:u7 = 5.ó x l0noc.r,ergs
s-t. With the essentiallv
tory, sinceit represents
the difference
betweentwo quantities astronomical estimate of òa
the total rate at which .n"rgy
eachof which is known only to about 102o.
must be dissipatedin the earth is 5.7 X lOreergs s-r. The tidal
'best'
Muller and Stephenson's
estimatefor the observed
ó estimate of ó, (Table 4) results in a total dissipation
in the
givesa nontidaldecrease
in lengthof day of 1.3+ 0.4 ms/100 oceansof 5.0 X
lOreergs s-t. The differenceof 0.7 X lOrsergs
yr. This nontidalchangecould be due to (l) furthertorques
s - ' c o u l d b e i n t e r p r e t e da s a s o u r c eo f t h e r m a l e n e r g yi n t h e
actingon the earth'smantle,(2) changes
in the earth'sinertia mantle, but this value is hardly
significantin view of the uncer_
tensor,or (3) a decreasein the gravitationalconstantG.
tainties associatedwith both the tidal and the astronomical es_
Rochester
[1973]givesa shortreviewof someof the possible timatesof Ë. The agreementbetween
thesetwo estimatesagain
contributions.
Dicke [1969]assumes
rhat part ofthe nontidal stressesthe dominant
role of the global oceansin dissipating
part of lod is due to ö and attributes
the remainingpart to the the tidal energy, and any
dissipation in the mantle or in the
recentdeglaciationand its associated
isostaticadjustments. core must be small in comparison.
The energydissipatedin the
O'Connell[97 l] attributesall of the nontidal lod to the
M2 tid,e is
Pleistocene
glaciationand its consequences
and arrivesat a
relationbetween
È:CQiòu"(r.,-nno)
therelaxationtimeof theisostaticadjustment
of the earthdue to a load harmonicin degree2 and the non_
/ \Ð
**
(a*\-t
tidal changein lod. This relationhas two roots, and he es: t r("-n¡n)(n¡r)¡r.
M \Ri
timatesthat the relaxationtime is either about 2000yearsor
about100,000
yearsusingthe nontidalchanges
in ó that range
'
, l l ¡ t , .
,
betweenthe limits0.8 X l0-æ and 2.1 X l0-r, rad s-r. These
n'v)
\a¡t)¡r.
o
]
limits are basedon the discussionby Munk and MacDonald
[960] and on the analysisby Curott [1966]of ancienrsolar
-(s/10),^* (L) r, I k"')(, eclipses.
The aboveestimateof (3.4 + 1.2)X 10-22rad s-2,
"ò Ip
basedon Muller and Stephenson's
discussion
of the eclipse
.(l * cos ¡*)"(l - E
data, givestwo roots for the relaxationtime, one near 4000
*')(crr* cos ee2+)¡¡"
yearsand the other near 40,000years,althoughconsiderable and for the
meanof the threemodelsfor theMrtide therateof
dispersionexistsbetweenthe differentmodelsby O'Connell. dissipationis 3.7 X 101"
ergss-t. Hendershott
[lgjT] found a
Thesedecay times do not differ sensiblyf¡om O'Connell's comparable
3.04X 1Ore
ergss- I by computingtherateof work
values.Of the two roots, O'Connellprefersthe smaller one, done on the
oceanby both the tide-generating
potentialand
and from his discussionit appearsthat a relaxationtime of the seafloor motion
due to the solid tide. Figure I illustrates
4000yearsis in agreementwith a uniform mantleviscosityof the part of the M,
tide that is represented
by the coefficients
about 102 P or a modelin whichtherewould be only a very C"r+ and e22+.
This distribution does not identify the actual
slight increasein the viscositybelowthe asthenosphere.
How- energfsinks.Calculations
of the dissipationby Mitter 11966l
ever,O'Connell'sresultsmay be vitiatedby hisassumption
of usingthe energyflux methodgivea total dissipationof 1.7X
exponentialdecayof the harmonicdeformationcoefficients,
as l0t" ergss-r and identifysomefivelocationswhere,together,
Pellier|9741 arguesthat the decayis stronglynonexponential. half
this amountis dissipated.
It is not obvioushow thesefive
regions(the BeringSea,the Okhotz Sea,Australianto the lesDlsslpeiror.¡
oF TrDALENeRcy
serSundaIslands,the Patagonia
shelf,and the HudsonStrait)
Rateof energydissipation. The actualrate ofdissipationof relateto the distributionof Figurel, in particular,since
four
tidal energycan be computedin severalways,for example,by of the five areasoccur in high latitudes,wherethe tidal
torque
the time averageof the rate of work by the moon on the earth of the truncated.M, tide is a minimum.Apparently,any
disor inverselyby the rateof work by the earthon the moon. In sipationin a limited areawill resultin a phaselag over
much of
view of the fact that ò and á havealreadybeencomputedfor
the varioustide models,the mostdirectway is to considerthe
total energybalanceof the earth and moon. That is. the
rotational energylost by the earth is
-tc# (?)'. -
Ê, : ¿/¿t (vzCof) : Cui,t,
whereboth lunarand solartidescontributeto ttreenergylost
(seeTable 4 or the astronomicallybasedestimategiven in
Table 6). The orbital energygainedby the moon is
d /.
ã\àaa-ní
*
zr:
:
d(
t \-
^*
^,
,!!tuf_ _ GMmy\
e¡¡ /
+ ¡rt
GM\
:
zo*)
--tm v n¡¡Q¡¡''ttr¡4
: -cnuàr-
Fig. l. Globaldistributionof thecoefficient
C""+cosc""+
intheM"
oceantide.The amplitudes(solidlines)arein centimeters.
The phaseii
The energytransferred
to the earth,orbitis only about lZo of indicatedby the dashedlines.The five principalregionsof friðtion in
seasasestimatedbyMiller [1966]areasfollcws:I, BeringSea;
that transf,erred
to the moon orbit. When it is assumedthat ¡latfo1
II, Okhotz Sea; III, Australia to the lesser Sunda Islands: IV.
any dissipationin the moon ís small,the amountof dissipation Patagoniashelf;and V, Hudson Strait.
LAMBECK:TtnnL DtsslPnrlo¡¡
2924
'7
then givesezz= '5o'
to phaselags are independentof frequency
the equatorialregionsif, indeed,dissipationis limited
âø for the
o, u 2 oilessthan ì0. eft., correótingthe observed
dissipashallowseasonly. Munk [1968]does not rule out
in the
change
unexplained
ocean tide there remains a small
tion other than by friction in the shallowseasand concludes
mean
t
h
e
w
e
a
d
o
p
t
i
f
s
'
c
m
X
l
0
?
0
.
1
2
takesplaceby way l u n a r o r b i t o f a b o u t
fractionof the dissipation
that a significant
In view of
än¡'
for
estimates
astronomical
recent
three
the
of
ol scatteiinginto internalmodesand that friction at the deepesby the uncertaintiesin both the observedand the computed
sea boundiry may be important (seealso the reviews
we
but
very
significant'
not
probably
timates this differenceis
Hendershottand Munk [1970]and Hendershottll973l)'
solid
the
for
possible
the
on
Q
bounds
obtain
of the euolution of the lunar or' can use it to
Some consequences
=
dissipationin the moon with p' õal?,
occursprincipallyin the oceans' earth and moon. For
dissipation
the
bit. Because
ôal
with
earth
the
a"
Q6'
an im- 8 x l0-? sin e¡r,and for dissipationin
of the lunar orbit into the pastbecomes
extrapolation
Then
=
e
s
.
s
i
n
X
l
O
u
1
.
2
of
leordering
very
major
possiùl"task,sincewe know that a
distributionhas occurredduring at leastthe õ a l q , ,I õ o l o " : ( 8 X 1 0 - ? s i n e ¡ ¡* l ' 2 2 X 1 0 - u s i n e ¡ )
àcean-continent
constantdissipation,MacDonald
Assuming
m.y.
last 200
: ( 0 ' 2 0 + 0 . 1 7 )X 1 0 - '
musthavebeenverycloseto the
moon
the
that
showed
[96a]
esastronomical
recent
this
àarth about 1.8 b.y. ago. The more
lf we assumethat the Q for the two bodiesarethe same'
about60' This is
of
=
limit
timateswill aggravatethe time scaleproblemevenmore' One
:
lower
a
with
100,
Q*
vi.io, 0"
w a y o u t o f t h i s d i l e m m a w i t h o u t i n v o k i n g n o n t i d a l ätrortã, unsatisfactory
as the estimateobtainedby Lambeck
oceans
A
is to removethe major part of theshallow
mechanisms
of orbitsof closeearthsatellites'
et at. U974)from analyses
observalor longtimeintervalsduringtheearth'spast'If we canbe sure major-uncertaintyin á" is due to the astronomical
is restrictedto the shallowseas,a loweringof tions.For an averagemantleQ of 200the dissipationin the
that diJsipation
could
shelves
belowthepresentcontinental
depth
ergss-I' The dissealevelio a
mantlewould be of the order of 1'8 X 1018
Cn'+and
that estimated
and
conceivablydecìeasethe dissipation'The coefficients
models
sipationestimatedfrom the tidal
a¡e borhprobablycorrect
e . r + a l s o g i v e r i s e t o s e c u l a r c h a n g e s i n t h e i u n a r o r b i t ' bfåm
u t a ttne astronomical
observations
piesentthey are small becauseof the ('R/a)'zfactor' which ap- to within l0 or 157o,
and this placesan upperlimit on the disto Cnr+ and C"'* '
due
ergs
perturbations
of
thi
ratio
in
þears
sipationof tidal energyin the coreof a few unitsof 1018
coupling
As the moon approachesthe earth in the backwardextra- s-t. This presentsa maximumlimit on dissipative
polation, theseterms could conceivablybecomesufficiently betweenthe core and the mantleand is an orderof magnitude
importantto changethepatternof thelunar orbit evolutionas smallerthan that usedby Stacey ll973l in his study of the
vieied by Goldreich and MacDcinald'Of particular interest precessional
couplingof the earth'score' H' G' Rochester
inclination'
the
and
axis
semimajor
the
in
changes
possiúle
are
communication)independentlyarguesthat Stacey's
rates iprivate
From the oceanpotential(2) we can computethe secular
is at least 2 orders of magnitudetoo large' More
only
canbe established
in thesetwo elementsdue to both Cr2* cose,r* and Cn'* cos "rtit"t.
preciselimitsto the aboveparameters
it
if
are'
perturbations
the
with
of
ratios
The
important'
e.r* of the Mz tide.
*ith irnp.ou.doceantide modelsand,more
throughout'
circular
remains
Astronomtthe
orbit
angles'
that
lag
the
for
assumed
is
values
improvedobservational
of longwill not providethe latterbecause
cal
observations
(di/dt)zzoo
4zzoo
(+\'
of the
analysis
The
:
rotation'
y.in
:
variations
:
..
period,
nontidal
Z:
T i"',
Ãzz/tz
5 1 + ka' \Ri
aorro (difdt)A,n'
more
is
a
satellites
iidal evolutionof the orbitsof closeearth
method'
yet
adequate,
promising,thoughnot
#ffiÈos8(;)'
(7 sin" i cos i
\ 1 + * r t
- t)
-\-t
crr*
ci
coS É22'
"*;;
I am gratefulto y'. y' Kaula for some
Acknowledgmen¡s..
for the
verifiedthe expressions
,i,oug¡,fut comments.Anny Cãzenave
Institut de
tides'
solid
and
ocean
the
io
due
äruitäi oottrUations
NS 131'
du Globecontribution
Physiquè
ReP¡neNcEs
in
For an order of magnitudeestimationwe usethe variation
V'
A. Magarik,Numericalsolutionsfor the
K. T., and
inclinationwith decreasingsemimajoraxis-asgivenby Figure Boqdanov,
"iä.iå;t'r.*i-ãi;t;;l
(Mz ands-,) tides,Dokl' Akad' N¿a&ssstR'
(Cn"*
e,,*)f
(Crr+
cos
8 of Gotdreich[1966i.The presentratio
r72, t3l5-t3r7, \96'7.
-3 lLambecket al" 1974'
cos eor'¡ for th! Mzúde is about
S., and R' S. Lindzen, AtmosphericI¡des' D' Reidel'
Cú";;;;:
AnapTable2]. We use-l for this ratio in evaluatingX22¡n2'
D'ordrecht,Netherlands,1970.
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most
at
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be
would
àicatesthat the inclinationof thelunarorbit
the
R. H., Average accelerationof the earth rotâtion and
Dicke,
only
and
Goldreich
by
1969'
a few degreeslarger than that lound
the deepmantle,J. Geophys'Res,"74'.5895-5902'
of
uis"osity
áer halb un-deintägigenTiden
then whenthe moon wascloserto the earththan about6 R¿' Di"ì;ì;h, ô. oi. Schïingungssystemè
and
30o
near
Uniu' Berl.in'Ser' A' 41'
sign
changes
F.rr(Ð
Veroef-Meereskunde
function
Ozeanel,
inclination
den
in
As the
opto
tend
T-68,1944.
againnear 80', the Czz+cos ezz+aîd C.2+cose.r+
SpacePhys''
cofiø.fr, P., History of the lunar orbit' -Reu'Geophys'
oãseeachother betweenthesetwo valuesof the inclination. 4 , 4 1 1 4 3 9 , 1 9 6 6 .
for maintainingthe moon
Mar' Res"17'
a possiblemechanism
ïhis suggests
Groves,G., and W' Munk' A note on tidal friction' J'
feasible'
be
to
this
for
But
earth'
the
from
at const-Ãtdistànce
t99-2t4, 1958.
global
Hendershott,M. C., The effectsof solid earthdeformationson
C.r* cos e.r* has to be at least l0 timesgreaterthan C"+ cos --;;;;;
29'-389-402'1972'
Soc',
Asrron'
Rov.
J'
Geophvs.
tides,
the
to
close
,r)7 un¿onlyth.n when the moon is dangerously
i', Oceantiåes,Eos Trans'AGU' 54' 76-86' 1973'
Ue.n¿erstrott,'M.
FluidMech"
Roche limit oî 2.9 Rø;this is an impiobable situation'
ä."ã.ttft"rt, M' C., and W. H. Munk, Tides,Annu' Reu'
Earth's specifc dissipationfunction' If the tidal accelera- 2,205-224, t9'10.
(1) the '-;;i;;ìd'"ir;tádon
explanationof the presentvalueof the
Hoímberg,R. R', A suggested
tion is attributed to a solid tide of potential
Mon' Notic' Rov'Astron' Soc" 6'
eartÈ',
o-ithe
directly
compared
be
astronomicailydeducedvalueàu can
1952.
325-330'
suPPl.,
geophys.
with the expiessionsfor äv in Table l' Assumingthat the
2925
LrMsecx: Tlo¡rl- DlsstpnrtoN
J e f f r e y s ,H . , T h e E a r t h , 4 t h e d . , C a m b r i d g e U n i v e r s i t y P r e s s ,L o n d o n ,
t962.
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K a u l a , W . M . , T i d a l f r i c t i o n w i t h l a t i t u d e d e p e n d e n ta m p l i t u d e a n d
p h a s ea n g l e , A s t r o n . J . , 7 4 , l | 0 8 - l I 1 4 , 1 9 6 9 .
K u z n e t s o v .M . V . . C a l c u l a t i o n o f t h e s e c u l a rr e t a r d a t i o no f t h e e a r t h ' s
r o t a t i o n l - r o m u p t o d a t e c o t i d e l c h a r t s ,l z u . A c a d . S c i . U S S R P å y s .
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467-54t, 1964.
Miller, G. R., The flux of tidal energy out of the deep oceans,./
Geophys. Res., 71, 2485-2489, 1966.
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(ReceivedSeptember13; 19'74:
revisedJanuary20, 1975:
acceptedFebruary3, l9'15.