Geopotential 6

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Geopotential 6 . ovninlmdlbsdo h G20 model EGM2008 the on based model Conventional 6.1 3 2 1 http://earth-info.nga.mil/GandG/wgs84/gravitymod/egm2008/ http://www.csr.utexas.edu/grace/ http://op.gfz-potsdam.de/champ/ where nr ucini eae otecascl(nomlzd n by one (unnormalized) classical the to related is function endre and (with as N degree 6.5). to (Section taken up tide be Earth pole field solid should of ocean geopotential 6.3), effects The (Section and variations varying tides 6.4), secular ocean time (Section 6.2), the tide other (Section pole for tides addition, Earth model In solid part background account: static coefficients. into underlying the its describes the of 6.1 and some Section field in model. conventional presented the the is of as that model model conventional EGM2008 The the recom- time data, gravitational altimetry this resolution satellite at and high mends gravimetry with surface data from GRACE obtained of information derived combination models optimal gravitational new the of from development recent avail- the the recognizing to IERS, thanks The (2003), Conventions EGM96, CHAMP IERS model of the geopotential ability of to model analysis conventional respect orbital past with precision the improvement (2010) significant current a in represent used commonly models Gravitational h cln aaees( parameters scaling The ( coefficients unnormalized ( the coefficients to geopotential normalized the Correspondingly, oeta offiins hyaet ecniee sT-optbevle.The gravitational values. its TT-compatible with as parameters considered be scaling value, to as TCG-compatible are used recommended They be coefficients. should potential m) 6378136.3 and (Pavlis model EGM2008 The coefficients normalized the to variations provide 6.5 to 6.2 ( Sections 6.1. Section fbte hn05m o h niae aelts(is 2010). (Ries, satellites accuracy indicated orbit the 6.1 3-dimensional for Sug- Table a in mm provide 0.5 listed model. levels than are the truncation interest better space of these of of in orbit version that the expected users truncated of is a most function It a by 2159, as covered levels order needs truncation and gested their degree find to will complete geodesy at is available with EGM2008 is working The Although estimates) still error respectively). Time those (including (TDB), by Coordinate Time model Dynamical used EGM2008 Geocentric Barycentric be or with (TT) should working Time 398600.4356 Terrestrial when or term (398600.4415 two-body (TCG) the with used order and 2190 The degree to up 2159. coefficients harmonic spherical additional contains and C V ¯ nm ( ,φ λ φ, r, P ¯ , nm S N P C S ¯ ¯ ¯ n nm nm nm nm 0 r h omlzdascae eedefntos h omlzdLeg- normalized The functions. Legendre associated normalized the are = ) ) where 0), = u otepyia ffcsdsrbdi ahsection. each in described effects physical the to due ) = = = GM N N GM s r nm nm ⊕ ( n P C and n X ¯ < − N nm =0 nm 1 m > V , C S , ¯ a )!(2 a nm n,ms motnl,GRACE importantly, most and, e r ttepit(r, point the at ( e GM n ausrpre ihEM08(96041 km (398600.4415 EGM2008 with reported values n nm n + and 1)(2 + m X tal et , m n = =0 a )! S ¯ e C N nm soitdwt h oe r ecie in described are model the with associated ) ,20)i opeet ereadodr2159, order and degree to complete is 2008) ., C nm ¯ nm − nm r h omlzdgooeta coefficients geopotential normalized the are S , δ S ¯ 0 cos GM nm m nm φ ) , ( . δ , by ) λ mλ ⊕ sepne nshrclharmonics spherical in expanded is ) 96041 km 398600.4418 = + ) 0 S m ¯ nm = sin ( if 0 if 1 < ( C mλ 2 ¯ < > nm 3 ) > aai h 2000s. the in data m m , . S P ¯ ¯ 3 0 6= 0 = nm nm /s 2 r related are ) (sin hudbe should , Technical φ (6.1) ) (6.2b) (6.2a) IERS Note (6.3) 3 /s 2 79 No. 36 80 No. 36 Note Technical IERS 4 C http://www.igg.uni-bonn.de/apmg/fileadmin/itg-grace03.html ¯ 20 offiin au t20. eeec ae/yr / Rate Reference 2000.0 at Value Coefficient zr-ie -0.48416948 (zero-tide) al .:Sgetdtucto eesfrueo G20 tdffrn orbits different at EGM2008 of use for levels truncation Suggested 6.1: Table al .:Lwdge offiinso h ovninlgooeta model geopotential conventional the of coefficients Low-degree 6.2: Table C C ¯ ¯ 40 30 0.5399659 0.9571612 e,dpnigo h vrgn nevlue.Tetd-revleof GRACE-based value affect 6.2.2. tide-free Section The will in aliases used. described interval tide-like as averaging GRACE obtained that the of on expected years depending 4 approximately is from ues, data it obtained 2 SLR value as of EGM2008 of uncertainty the data, years an from 17 significantly has of differs and It analysis 2000 the year from on obtained centered zero-tide is the model that tial Note 6.2. Table where oeta offiinst ov o aelt ri,i sncsayt oaefrom rotate to necessary is it orbit, satellite a for solve to coefficients in defined potential frame for reference choice terrestrial This the with consistent 4. Chapter position pole mean the accounted be should rates secular by: the the given use coefficients time, are to of in values order instantaneous it In The project coefficients. for. also and low-degree field properly of static rates field The secular static values. the uses EGM2008 for model original values the geopotential assumes from conventional different the are coefficients, which low-degree values the of some For static following: the Con- the Therefore IERS by the complemented with 180. and complying order (2003) models ventions and assuming developed degree was to field gravitational matrix gravita- covariance GRACE-only error ITG-GRACE03S complete the ( on based model was tional model EGM2008 The eta oe.The model. tential the for Values pee cas rErhsfli oe(ar 97 90.A rsn,teei no is values conventional atmo- there The present, the At important. in are motions 1990). motions 1987; fluid and for such (Wahr, long-period figure that core to mean evidence fluid due independent the Earth’s be time or of same would oceans, positions the pole sphere, averaged over rotation coin- the averaged closely mean pole between should rotation the differences axis the Any of figure the position period. observed years, the many over with averaged cide When axis. figure ri ais/k xml rnainlevel Truncation Example km / radius Orbit • • • topeeadoend-laig O1 L4(lcte,2007). (Flechtner, RL04 AOD1B de-aliasing: 6.5), ocean Section and see atmosphere (2003, Desai tide: pole ocean (Lyard FES2004 tides: ocean C 21 C t ¯ ( 0 × × × n t 60 P 12 20 GPS Lageos 26600 12270 and ) 0 steeohJ000adtevle of values the and J2000.0 epoch the is 10 10 10 31Salte90 Starlette 7331 ( t = ) − − − 3 6 6 < S C C C 4 ¯ 21 21 > n 21 ( 0 Cheng e loMyrGur 07 hc saalbeaogwt its with along available is which Mayer-Gürr, 2007) also see , t ( and r neddt ieama gr xsta orsod to corresponds that axis figure mean a give to intended are ) and C t 0 21 + ) S and S d 21 21 tal et C G20 4 4 EGM2008 EGM2008 ¯ n sraie sflos is,t s h gravitational the use to First, follows. as realized is offiinsaeicue ntecnetoa geopo- conventional the in included are coefficients S 0 21 /dt ,21 11 2010 ., tal et offiinsdsrb h oiino h Earth’s the of position the describe coefficients × C ( ,2006), ., 20 t − offiin ntecnetoa geopoten- conventional the in coefficient t 0 (6.4) ) C . . . ¯ 6 7 9 n × × × 0 ob sdwe optn orbits computing when used be to C ¯ 10 10 10 n 0 − − − × ( − t 12 12 12 1 0 10 and ) − 11 Nerem Cheng Cheng (Cheng d C ¯ Reference n 0 /dt tal et tal et al et Geopotential 6 tal et r ie in given are C ,1993 ., ,1997 ., 1997 ., 20 C ,2010). ., 20 a be can val- 6.2 Effect of solid Earth tides IERS Technical

Note No. 36

the Earth-ﬁxed frame, where the coeﬃcients are pertinent, to an inertial frame, where the satellite motion is computed. This transformation between frames should include polar motion. We assume the polar motion parameters used are relative to the IERS Reference Pole. Then the values √ C¯21(t) = 3¯xp(t)C¯20 − x¯p(t)C¯22 +y ¯p(t)S¯22, √ (6.5) S¯21(t) = − 3¯yp(t)C¯20 − y¯p(t)C¯22 − x¯p(t)S¯22,

wherex ¯p(t) andy ¯p(t) (in radians) represent the IERS conventional mean pole (see Section 7.1.4), provide a mean ﬁgure axis which coincides with the mean pole consistent with the TRF deﬁned in Chapter 4. Any recent value of C¯20, C¯22 and −14 S¯22 is adequate for 10 accuracy in Equation (6.5), e.g. the values of the present conventional model (−0.48416948×10−3, 2.4393836×10−6 and −1.4002737×10−6 respectively) can be used. The models for the low degree terms are generally consistent with the past long- term trends, but these are not strictly linear in reality. There may be decadal variations that are not captured by the models. In addition, they may not be consistent with more recent surface mass trends due to increased ice sheet melting and other results of global climate change.

6.2 Effect of solid Earth tides 6.2.1 Conventional model for the solid Earth tides The changes induced by the solid Earth tides in the free space potential are most conveniently modeled as variations in the standard geopotential coefficients Cnm and Snm (Eanes et al., 1983). The contributions ∆Cnm and ∆Snm from the tides are expressible in terms of the Love number k. The effects of ellipticity and of the Coriolis force due to Earth rotation on tidal deformations necessitate the use of (0) (±) three k parameters, knm and knm (except for n = 2) to characterize the changes produced in the free space potential by tides of spherical harmonic degree and order (nm) (Wahr, 1981); only two parameters are needed for n = 2 because (−) k2m = 0 due to mass conservation. (0) (±) Anelasticity of the mantle causes knm and knm to acquire small imaginary parts (reflecting a phase lag in the deformational response of the Earth to tidal forces), and also gives rise to a variation with frequency which is particularly pronounced within the long period band. Though modeling of anelasticity at the periods relevant to tidal phenomena (8 hours to 18.6 years) is not yet definitive, it is clear that the magnitudes of the contributions from anelasticity cannot be ignored (see below). Recent evidence relating to the role of anelasticity in the accurate modeling of nutation data (Mathews et al., 2002) lends support to the model employed herein, at least up to diurnal tidal periods; and there is no compelling reason at present to adopt a different model for the long period tides. Solid Earth tides within the diurnal tidal band (for which (nm) = (21)) are not wholly due to the direct action of the tide generating potential (TGP) on the solid Earth; they include the deformations (and associated geopotential changes) arising from other effects of the TGP, namely, ocean tides and wobbles of the mantle and the core regions. Deformation due to wobbles arises from the incremental centrifugal potentials caused by the wobbles; and ocean tides load the crust and thus cause deformations. Anelasticity affects the Earth’s deformational response to all these types of forcing. The wobbles, in turn, are affected by changes in the Earth’s moment of inertia due to deformations from all sources, and in particular, from the deformation due to loading by the (nm) = (21) part of the ocean tide; wobbles are also affected by the anelasticity contributions to all deformations, and by the coupling of the fluid core to the mantle and the inner core through the action of magnetic fields at its boundaries (Mathews et al., 2002). Resonances in the wobbles—principally, the Nearly Diurnal Free Wobble resonance associated with the FCN —and the consequent resonances in the contribution to tidal deformation from the centrifugal perturbations associated with the wobbles, cause the body tide and load Love/Shida

81 82 No. 36 Note Technical IERS ato h G ntenraie eptnilcecet aigtesm ( by same domain the having time coefficients the geopotential in normalized given the 6.2.2. are in Section TGP See the this of twice, When computation. part counted the values be EGM96. of not frequency-independent 3 as should Step With such part of permanent model goal this free” the model, the is tide tide” including “conventional coefficient “zero contribution, a a tidal geopotential for using total the adequate the to compute is contribution constituents to which the used (permanent) of be independent few can time a 2 to and applied 1 also. be Steps bands to two other need the corrections value of Similar nominal constant step. first the the the from of band deviations diurnal the the 10 for of corrects level 2 the Step at are they to procedure; tides 2 degree the o h eiaino eoac omleo h om(.)blwt ersn this represent to below (6.9) form Mathews the dependent. see of frequency dependence, formulae strongly frequency resonance become of to derivation tides the diurnal For the of parameters number where to tides 3 degree (2 the of the each tributions for 1, values nominal domain Step independent ∆ time is frequency In changes the using corresponding coefficients the in geopotential and procedure. evaluated ephemerides, is the solar three-step potential to a tidal by contributions the done changes tidal efficiently other the most only of The computation The negligible.) are in coefficients TGP. are these cutoff this to exceeding potential tidal 4 in changes the dependent 3 in time included produce also tides k are 2 corrections degree through These The numbers. incorporated Love below. are of further numbers tables explained Love as load using (1981), the the place to in first take corrections to resonances the equivalent Corrections in the 7.1.2). computed of Section itself (see account numbers is Love loading load in ocean independent given frequency to parameters formulae. due number deformation resonance Love The next. of the the tables using correc- and the small chapter computed in this add included parameters to are number therefore corrections Love necessary These the frequency Earth becomes the It to a among tions makes to. are referred deformation which just frequency induced parameters parameters all deformability tide are to ocean equations contribution wobble the dependent the However entering parameters independent. Earth the that sume 2 (0) GM × GM m k hc a xed10 exceed can which , 10 nm R Φ λ r ⊕ e j j j j ∆ − 12 C ¯ oyfie atlniue(rmGenih fMo rSun. or Moon of Greenwich) (from longitude east body-fixed = degree for number Love nominal = oyfie ecnrclttd fMo rSun, or Moon of latitude geocentric body-fixed Sun, or = Moon to geocenter from distance ( = Moon the for parameter gravitational = Earth, the for parameter gravitational = Earth, the of radius equatorial = nm in C − 4 i m ∆ S and ¯ nm C S = 4 4 m m − 2 and n k 8 through nm C 1 + nmgiue hyas rdc hne exceeding changes produce also They magnitude. in oytide body 3 m S 4 X j and m =2 3 C k through k GM tal et 3 GM − 2 (+) nm S m m 11 oenmes olwn aradSasao and Wahr following numbers, Love 3 m Tedrc otiuin ftedegree the of contributions direct (The . . k and ⊕ j ,(95.Tersnneepnin as- expansions resonance The (1995). ., Se ) hne nue yte( the by induced changes 1), (Step 21 (0) rdcdb h ere3pr fthe of part 3 degree the by produced , R k r k fsvrlo h osiun ie of tides constituent the of several of S n 2 j e 2 (+) m 3 m m n order and n o h respective the for +1 a ecmue yasimilar a by computed be may through j C k P ¯ 21 )n u ( Sun 2)and = 2 nm m sue o hsbn in band this for assumed snΦ (sin n ∆ and C m k 2 m , 3 (0) m m j and S ) n lotoeof those also and e 2 sn ua and lunar using m − Geopotential 6 imλ k S r computed are j 2 (0) 2 m 3), = m m j h con- The . through , atof part ) (6.6) nm nm C ¯ 20 ) ) , 6.2 Effect of solid Earth tides IERS Technical

Note No. 36

Equation (6.6) yields ∆C¯nm and ∆S¯nm for both n = 2 and n = 3 for all m, apart from the corrections for frequency dependence to be evaluated in Step 2. (The particular case (nm) = (20) needs special consideration, however, as already indicated.) One further computation to be done in Step 1 is that of the changes in the degree 4 coeﬃcients produced by the degree 2 tides. They are given by

(+) 3 k2m P3 GMj Re −imλj ∆C¯4m − i∆S¯4m = P¯2m(sin Φj )e , (m = 0, 1, 2), (6.7) 5 j=2 GM⊕ rj

which has the same form as Equation (6.6) for n = 2 except for the replacement (+) of k2m by k2m . The parameter values for the computations of Step 1 are given in Table 6.3. The choice of these nominal values has been made so as to minimize the number of terms for which corrections will have to be applied in Step 2. The nominal value for m = 0 has to be chosen real because there is no closed expression for the ¯ (0) contribution to C20 from the imaginary part of k20 .

Table 6.3: Nominal values of solid Earth tide external potential Love numbers.

Elastic Earth Anelastic Earth (+) (+) n m knm knm Re knm Im knm knm 2 0 0.29525 −0.00087 0.30190 −0.00000 −0.00089 2 1 0.29470 −0.00079 0.29830 −0.00144 −0.00080 2 2 0.29801 −0.00057 0.30102 −0.00130 −0.00057 3 0 0.093 ··· 3 1 0.093 ··· 3 2 0.093 ··· 3 3 0.094 ···

The frequency dependent corrections to the ∆C¯nm and ∆S¯nm values obtained from Step 1 are computed in Step 2 as the sum of contributions from a number of tidal constituents belonging to the respective bands. The contribution to ∆C¯20 from the long period tidal constituents of various frequencies f is

P iθf P R I (6.8a) Re f(2,0)(A0δkf Hf ) e = f(2,0)[(A0Hf δkf ) cos θf − (A0Hf δkf ) sin θf ],

while the contribution to (∆C¯21 − i∆S¯21) from the diurnal tidal constituents and to ∆C¯22 − i∆S¯22 from the semidiurnals are given by

X iθf ∆C¯2m − i∆S¯2m = ηm (Amδkf Hf ) e , (m = 1, 2), (6.8b) f(2,m)

where

1 −8 −1 A0 = √ = 4.4228 × 10 m , (6.8c) Re 4π

m (−1) m −8 −1 Am = √ = (−1) (3.1274 × 10 ) m , (m 6= 0), (6.8d) Re 8π

η1 = −i, η2 = 1, (6.8e)

83 84 No. 36 Note Technical IERS where o h udmna ruet ( arguments fundamental the For h atr102399bigtenme fsdra asprslrdy The day. solar per days sidereal of number the for the herein used being values 1.002737909 factor the lydi nltcltere fntto. npriua,gvntetdlfrequency tidal the em- given that particular, of In opposite nutation.) the of f is theory, theories (prograde) tidal analytical in in retrograde followed ployed represent convention, sign frequencies (This (negative) waves. positive and nutation), that core convention inner free the as known and (also FCN, nutation retrograde the core the (CW), free wobble prograde Chandler of the the account parame- with deformability take associated Earth’s to The frequencies the are inexact.) of (6.9) (They make few which earlier. a ters, to to contributions referred dependent corrections frequency small the for except Love tide body or load any of values dependent parameter frequency number the tides, diurnal amplitude For the chapter. to this conventions of different end to according defined H amplitudes multipliers tidal their choosing of in here followed tion lx The plex. etda ucino h ia xiainfrequency excitation tidal the of function a as sented N δk and F δk δk f ndgesprhu,oehas one hour, per degrees in ¯ β n H ¯ ¯ ¯ θ θ orsodn oteCrwih-alrcneto,ueTbe68gvna the at given 6.8 Table use convention, Cartwright-Tayler the to corresponding ( f R L f f f f f I π α θ L i-etro odo’ udmna arguments fundamental Doodson’s of six-vector = v-etro multipliers of five-vector = v-etro udmna arguments fundamental of five-vector = multipliers of six-vector = n( in g r h orsodn eoac offiins l h aaeesaecom- are parameters the All coefficients. resonance corresponding the are ( ieec between difference = = of part real = ¯ = of part imaginary = mltd i ees ftetr tfrequency at term the of meters) (in amplitude = σ = ) θ teDlua variables Delaunay (the ( steGenihMa ieelTm xrse nangle in expressed Time ( Sidereal units Mean Greenwich the is frequency of nutation the ftefnaetlarguments, fundamental the of g σ ,s ,p N p, h, s, τ, α + h oia value nominal the m loading, ocean from contribution n endacrigt h ovnino atrgtand Cartwright and of (1971), convention potential, Tayler the generating to tide according the defined of expansion harmonic the L · ( and π 0 θ β ¯ snwt erpae y180 by replaced be to now is ) g + i.e. = + α L X σ =1 P 3 π sc as (such 4h=360 = h 24 ) r xrse ncce e ieeldy(pd,wt the with (cpsd), day sidereal per cycles in expressed are i 6 0 ( =1 p , − σ δk s N σ − n L ¯ ), σ f α i α β and , · σ = r rmMathews from are F i α ¯ or , k ) δk f/ k 21 (0) = k , 2 ◦ ,l l, f f m (15 e hpe 5). Chapter see ; m i.e., , or endas defined ntesense the in , 0 n ,l l, ,D, F, , N σ ( i − θ × α k j 0 g frtetr tfrequency at term the (for ,D, F, , ( , f 21 (+) fteDlua aibe for variables Delaunay the of + 1 δk + . α 002737909) π )o uainter n h conven- the and theory nutation of Ω) f ntepeetcnet a erepre- be may context) present the in dθ 1 = ) = − )o uaintheory, nutation of Ω) g k /dt , ◦ N δk 2 (0) P 2 .) m tal et j , F f k R j 5 , e hpe .Frconversion For 5. Chapter see , ) r h epcieresonance respective the are 3), tfrequency at f =1 j + − 20) dpe otesign the to adapted (2002), . , N iδk k σ j 2 F m f I yarsnneformula resonance a by j , lsa plus , , β f i , from f f and Geopotential 6 ) (6.9) 6.2 Effect of solid Earth tides IERS Technical

Note No. 36

convention used here:

σ1 = − 0.0026010 − 0.0001361 i

σ2 = 1.0023181 + 0.000025 i (6.10)

σ3 = 0.999026 + 0.000780 i. They were estimated from a ﬁt of nutation theory to precession rate and nutation amplitude estimates found from an analyis of very long baseline interferometry (VLBI) data.

Table 6.4 lists the values of L0 and Lα in resonance formulae of the form (6.9) (0) (+) for k21 and k21 . They were obtained by evaluating the relevant expressions from Mathews et al. (1995), using values taken from computations of Buffett and Mathews (unpublished) for the needed deformability parameters together with values obtained for the wobble resonance parameters in the course of computations of the nutation results of Mathews et al. (2002). The deformability parameters for an elliptical, rotating, elastic, and oceanless Earth model based on the 1- second reference period preliminary reference Earth model (PREM) (Dziewonski and Anderson, 1981) with the ocean layer replaced by solid, and corrections to these for the effects of mantle anelasticity, were found by integration of the tidal deformation equations. Anelasticity computations were based on the Widmer et al. (1991) model of mantle Q. As in Wahr and Bergen (1986), a power law was assumed for the frequency dependence of Q, with 200 s as the reference period; the value α = 0.15 was used for the power law index. The anelasticity contribution (out-of-phase and in-phase) to the tidal changes in the geopotential coefficients is at the level of 1 − 2% in-phase, and 0.5 − 1% out-of-phase, i.e., of the order of 10−10. The effects of anelasticity, ocean loading and currents, and electromagnetic (0) (+) couplings on the wobbles result in indirect contributions to k21 and k21 which are almost fully accounted for through the values of the wobble resonance parameters. Also shown in Table 6.4 are the resonance parameters for the load Love numbers 0 0 0 h21, k21, and l21, which are relevant to the solid Earth deformation caused by ocean tidal loading and to the consequential changes in the geopotential. (Only the real parts are shown: the small imaginary parts make no difference to the effect to be now considered which is itself small.)

(0) (+) Table 6.4: Parameters in the resonance formulae for k21 , k21 and the load Love numbers. (0) (+) k21 k21 α Re Lα Im Lα Re Lα Im Lα 0 0.29954 −0.1412 × 10−2 −0.804 × 10−3 0.237 × 10−5 1 −0.77896 × 10−3 −0.3711 × 10−4 0.209 × 10−5 0.103 × 10−6 2 0.90963 × 10−4 −0.2963 × 10−5 −0.182 × 10−6 0.650 × 10−8 3 −0.11416 × 10−5 0.5325 × 10−7 −0.713 × 10−9 −0.330 × 10−9 Load Love numbers (Real parts only) 0 0 0 h21 l21 k21 0 −0.99500 0.02315 −0.30808 1 1.6583 × 10−3 2.3232 × 10−4 8.1874 × 10−4 2 2.8018 × 10−4 −8.4659 × 10−6 1.4116 × 10−4 3 5.5852 × 10−7 1.0724 × 10−8 3.4618 × 10−7

The expressions given in Section 6.3 for the contributions from ocean tidal loading 0 (nom) 0 assume the constant nominal value k2 = −0.3075 for k of the degree 2 tides. 0 Further contributions arise from the frequency dependence of k21. These may be expressed, following Wahr and Sasao (1981), in terms of an eﬀective ocean tide (OT ) (0) contribution δk (σ) to the body tide Love number k21 :

4πGρ R δk(OT )(σ) = [k0 (σ) − k0 (nom)] w A (σ), (6.11) 21 2 5¯g 21

85 86 No. 36 Note Technical IERS tnsfor stands where used.) m kg (1025 where al .asostevle of values the shows 6.5a Table frequency of tide ocean s the m of component (9.820 tesseral surface 2 degree Earth’s σ the the for at admittance gravity the to due acceleration h w mltdssol eietclynraie.Wh n aa (1981) Sasao and Wahr normalized. defining in identically form used factorized be harmonics the spherical should employed the amplitudes that two reminder a the being bar the potential, and tide, ocean ζ h ausue for contribution used part values real The magnitude. the same and the the accuracy about term, of of that tide value is ocean order the term half the in this Roughly from from terms disregarded.) comes 10 individual are part least that the imaginary at terms for smaller is used numerous 3 parts the is of the level level the of this at ( either at amplitude which cutoff the for of (A parts those imaginary are and listed real the with along load ocean the from (Chao obtainable tides admittances principal the four of to momenta and angular 1995) (Mathews Wahr, formulae and empirical following (Desai The second, the factors. and al dynamic resonance, et FCN ocean the other of of effect the that represents factor first the wherein eur ml orcin omthteeatvle.I nt f10 of units In were values. formula for exact resonance numbers the (approximate) match resulting is to the the corrections which If computation, small theory require the chapter. nutation-wobble for this of instead of used framework scope the the of outside use involving necessarily ubr 4,4,(,0 for 0) (1, 145,545, Q numbers for 1) (1, out-of-phase) phase, 3 )frJ for 2) (3, aclto ftecreto u oaytdlcntteti lutae ythe by illustrated is constituent tidal any H to table. for the due example of correction following columns the two nonnegligibly last contribute of the would Calculation in corrections the listed which numbers for the tides to only the are These nsbrcigtenmnlvle(0 value nominal the subtracting on 0 21 . 00262 f ncpsd: in 0 = stecmlxapiueo h egto h ( the of height the of amplitude complex the is ,20)wihpoiego t oteFNfcoso e f1 ira tides diurnal 11 of set a of factors FCN the to fits good provide which 2002) ., (∆ (∆ γ G f . i 36870, eqm + 1 = .Euto 68)te yields: then (6.8b) Equation ). C S ¯ ¯ stecntn fuieslgravitation, universal of constant the is 21 21 1 k n 2 )for 1) (2, and , − 21 (0) ) ) steFNfco o lbleulbimocean. equilibrium global a for factor FCN the is K K 3 ), n iial o h te ybl.Ol h elprsne be need parts real the Only symbols. other the for similarly and , f 1 1 θ k OD H f ¯ − × R ( = 470 = 470 = stehih qiaeto h mltd ftetd generating tide the of amplitude the of equivalent height the is ( h 10 σ steErhsma ais(6 radius mean Earth’s the is (1 = ) and θ − k g f 12 21 (0) K + CN F δk 1 π ( f γ entaetdb h cuuae otiuin from contributions accumulated the by affected not be ie that Given . . . Oo σ f ,and ), eqm 9 9 0 . A 3101 No nevaluating in ) ( + 1 = ≡ × × 21 σ A 1 ( 0 = ) ( 10 10 ( 21 n t opno ihDosnnmes185,565. numbers Doodson with companion its and 1 σ k σ 1 (0 , = ) 1 )frO for 1) (1, , 21 ( − (0) k − − = ) σ 21 (0) 12 12 k = ) 0 ( , . 0 72+0 + 1732 σ . − 1 sin( cos( . 8098 f − (0 = 29830 ) CN F )frP for 1) − − ζ h 21 θ A 0 (3 θ ¯ , k g . g i ( m 25746+0 21 ) ( ρ ρ σ + + σ γ − w − δk ) + ) steErhsma est.(Here density. mean Earth’s the is . ( ) 1 = / / 9687 π π σ 1 0 f H (1 f 5¯ n t opno aigDoodson having companion its and ) ¯ 24 9)for 299) (244, , 30 + ) ) . OD ρ tal. et 00144 A δk − . ( r rma xc computation exact an from are ) 1212 σ 1 γ f 21 ( OT 30 . ) ρ 0 . nm 371 σ eqm . 00118 ( = w σ ) . 96 r sdherein: used are 1996) , . ( 2 , 2 ) i − σ stedniyo e water sea of density the is − 2)cmoeto the of component (21) = ) that ) ( × , × × ) σ 3 0 , ) 10 . . 10 i 10 1274 , 6030 o hstd,oefinds one tide, this for ) − A − 6 12 − 12 1 δk δk ) ¯ m), − ψ δk 13 i × cos( sin( 2 ) 1 f f R f ,and ), σ, 1,1)for 12) (12, , − fe round-off. after 10 H ( = Geopotential 6 and 5 g θ θ hyae(in- are they , f − g g .Tetides The ). stemean the is − 8 + + n that and , δk 0 A π π . 48 + 04084 f I 21 ) ) , . would ( σ is ) φ 1 k , 6.2 Effect of solid Earth tides IERS Technical

Note No. 36

R I Table 6.5a: The in-phase (ip) amplitudes (A1δkf Hf ) and the out-of-phase (op) amplitudes(A1δkf Hf ) (0) of the corrections for frequency dependence of k21 , taking the nominal value k21 for the −12 R I diurnal tides as (0.29830 − i 0.00144). Units: 10 . The entries for δkf and δkf are in units of 10−5. Multipliers of the Doodson arguments identifying the tidal terms are given, as also those of the Delaunay variables characterizing the nutations produced by these terms. 0 0 R I Name deg/hr Doodson τ s h p N ps ` ` FD Ω δkf δkf Amp. Amp. No. /10−5 /10−5 (ip)(op)

2Q1 12.85429 125,755 1 -3 0 2 0 0 2 0 2 0 2 -29 3 -0.1 0.0 σ1 12.92714 127,555 1 -3 2 0 0 0 0 0 2 2 2 -30 3 -0.1 0.0 13.39645 135,645 1 -2 0 1 -1 0 1 0 2 0 1 -45 5 -0.1 0.0 Q1 13.39866 135,655 1 -2 0 1 0 0 1 0 2 0 2 -46 5 -0.7 0.1 ρ1 13.47151 137,455 1 -2 2 -1 0 0 -1 0 2 2 2 -49 5 -0.1 0.0 13.94083 145,545 1 -1 0 0 -1 0 0 0 2 0 1 -82 7 -1.3 0.1 O1 13.94303 145,555 1 -1 0 0 0 0 0 0 2 0 2 -83 7 -6.8 0.6 τ1 14.02517 147,555 1 -1 2 0 0 0 0 0 0 2 0 -91 9 0.1 0.0 Nτ1 14.41456 153,655 1 0 -2 1 0 0 1 0 2 -2 2 -168 14 0.1 0.0 14.48520 155,445 1 0 0 -1 -1 0 -1 0 2 0 1 -193 16 0.1 0.0 Lk1 14.48741 155,455 1 0 0 -1 0 0 -1 0 2 0 2 -194 16 0.4 0.0 No1 14.49669 155,655 1 0 0 1 0 0 1 0 0 0 0 -197 16 1.3 -0.1 14.49890 155,665 1 0 0 1 1 0 1 0 0 0 1 -198 16 0.3 0.0 χ1 14.56955 157,455 1 0 2 -1 0 0 -1 0 0 2 0 -231 18 0.3 0.0 14.57176 157,465 1 0 2 -1 1 0 -1 0 0 2 1 -233 18 0.1 0.0 π1 14.91787 162,556 1 1 -3 0 0 1 0 1 2 -2 2 -834 58 -1.9 0.1 14.95673 163,545 1 1 -2 0 -1 0 0 0 2 -2 1 -1117 76 0.5 0.0 P1 14.95893 163,555 1 1 -2 0 0 0 0 0 2 -2 2 -1138 77 -43.4 2.9 15.00000 164,554 1 1 -1 0 0 -1 0 -1 2 -2 2 -1764 104 0.6 0.0 S1 15.00000 164,556 1 1 -1 0 0 1 0 1 0 0 0 -1764 104 1.6 -0.1 15.02958 165,345 1 1 0 -2 -1 0 -2 0 2 0 1 -3048 92 0.1 0.0 15.03665 165,535 1 1 0 0 -2 0 0 0 0 0 -2 -3630 195 0.1 0.0 15.03886 165,545 1 1 0 0 -1 0 0 0 0 0 -1 -3845 229 -8.8 0.5 K1 15.04107 165,555 1 1 0 0 0 0 0 0 0 0 0 -4084 262 470.9 -30.2 15.04328 165,565 1 1 0 0 1 0 0 0 0 0 1 -4355 297 68.1 -4.6 15.04548 165,575 1 1 0 0 2 0 0 0 0 0 2 -4665 334 -1.6 0.1 15.07749 166,455 1 1 1 -1 0 0 -1 0 0 1 0 85693 21013 0.1 0.0 15.07993 166,544 1 1 1 0 -1 -1 0 -1 0 0 -1 35203 2084 -0.1 0.0 ψ1 15.08214 166,554 1 1 1 0 0 -1 0 -1 0 0 0 22794 358 -20.6 -0.3 15.08214 166,556 1 1 1 0 0 1 0 1 -2 2 -2 22780 358 0.3 0.0 15.08434 166,564 1 1 1 0 1 -1 0 -1 0 0 1 16842 -85 -0.3 0.0 15.11392 167,355 1 1 2 -2 0 0 -2 0 0 2 0 3755 -189 -0.2 0.0 15.11613 167,365 1 1 2 -2 1 0 -2 0 0 2 1 3552 -182 -0.1 0.0 φ1 15.12321 167,555 1 1 2 0 0 0 0 0 -2 2 -2 3025 -160 -5.0 0.3 15.12542 167,565 1 1 2 0 1 0 0 0 -2 2 -1 2892 -154 0.2 0.0 15.16427 168,554 1 1 3 0 0 -1 0 -1 -2 2 -2 1638 -93 -0.2 0.0 θ1 15.51259 173,655 1 2 -2 1 0 0 1 0 0 -2 0 370 -20 -0.5 0.0 15.51480 173,665 1 2 -2 1 1 0 1 0 0 -2 1 369 -20 -0.1 0.0 15.58323 175,445 1 2 0 -1 -1 0 -1 0 0 0 -1 325 -17 0.1 0.0 J1 15.58545 175,455 1 2 0 -1 0 0 -1 0 0 0 0 324 -17 -2.1 0.1 15.58765 175,465 1 2 0 -1 1 0 -1 0 0 0 1 323 -16 -0.4 0.0 So1 16.05697 183,555 1 3 -2 0 0 0 0 0 0 -2 0 194 -8 -0.2 0.0 16.12989 185,355 1 3 0 -2 0 0 -2 0 0 0 0 185 -7 -0.1 0.0 Oo1 16.13911 185,555 1 3 0 0 0 0 0 0 -2 0 -2 184 -7 -0.6 0.0 16.14131 185,565 1 3 0 0 1 0 0 0 -2 0 -1 184 -7 -0.4 0.0 16.14352 185,575 1 3 0 0 2 0 0 0 -2 0 0 184 -7 -0.1 0.0 ν1 16.68348 195,455 1 4 0 -1 0 0 -1 0 -2 0 -2 141 -4 -0.1 0.0 16.68569 195,465 1 4 0 -1 1 0 -1 0 -2 0 -1 141 -4 -0.1 0.0

87 88 No. 36 .. ramn ftepraettide permanent the of Treatment 6.2.2 of dependence frequency for Corrections 6.5b: Table Note Technical IERS aeDosndeg/hr Doodson Name M M M M M M M sqm M S stm qm sm S tm sf sa m f a qain(6.8a). Equation ( n mgnr parts imaginary and 10 Units: ip 565053601010010- 2000701-.03 -0.2 -0.00236 -0.2 -0.00237 0.1 3.7 0.1 0.00077 -0.00237 -0.2 0.7 -2 0.00079 -1.2 -0.00237 0 1 0.00080 -0.00242 -2 0.1 0.2 0 -0.1 0 0 -0.3 0.00080 0 -0.00296 0 1 -0.00313 4.3 0.00101 -1 0 0 -0.3 0.1 0.8 0 0 0 -0.00315 0.1 -1 0 0.00326 -2 0 0 0.00398 -0.00349 -5.5 0 0 -1 -2 -0.00488 -6.7 0 -1 2 0.00403 -1.2 1 0 1 0 2 -0.1 -0.00541 -1 -2 -2 0 0.00547 1 0 -2 -1 0.01124 2 -1 0 16.6 1 0 0 0 0 2 0 -2 0 -1 0.01347 -1 0 0 1 0 0 0.55366 0 0 1 -1 0 -1 0 0 65,655 1 0 0 0.54658 0 0 0 0 1 -1 65,465 1 0 1 0.54438 0 0 0 0 -2 0 0 65,455 1 0 0 3 0.54217 0 -1 0 0 65,445 0 0 2 0.47152 0 0 0 63,655 0 2 2 0.12320 0 0 0 58,554 0 1 1 0.08434 0 0 57,565 0 0 0.08214 0 0 57,555 0 0 0.04107 0 56,554 0 0.00441 0 55,575 0.00221 55,565 5352169040- 20- 2-.00 . 00120.1 0.2 -0.00192 0.5 -0.00193 0.1 1.1 -0.00201 0.1 -0.00106 0.2 -2 0.2 -0.00201 0.2 -0.00102 0 -2 -0.00202 0.4 -0.00213 2.6 -0.00071 -2 -2 -1 0 -2 0.1 0.0 6.3 -0.00071 0 -0.00213 0 -2 -2 0.3 -0.00065 -2 0 -0.00019 0 -0.00213 0.2 0 -2 0 0 0.6 -2 -2 -0.00213 0.6 0 0 -0.00019 -1 0 0 -2 -1 -0.00216 0.0 0 0 0 -0.00019 -2 -1 -2 0 -2 0 0 1 0.0 1 0 -0.00018 -2 0 0 0 4 0 0 0 -2 0 -0.00009 -2 0 -1 0 0 4 0 0 0 2.18679 0 0 0 -1 0 -2 0 95,355 3 2 0 2.11394 0 0 0 1 0 93,555 3 1 1.64462 0 0 0 -2 -2 0 85,465 3 0 0 1.64241 0 0 0 0 85,455 2 0 1.56956 0 0 0 0 83,655 2 0 1.10245 0 0 -2 75,575 2 0 1.10024 0 0 75,565 2 1.09804 -2 0 75,555 2 1.08875 0 75,355 1.01590 73,555 o ( No. mltd ( amplitude ) − 12 h oia value nominal The . ae seCatr1.We h ieidpnetcnrbto sicue in included is contribution independent time the of When value adopted 1). the Chapter (see taken Here earlier. to referred constituent, tidal model zonal anelasticity the the of of basis the on eomto n osqettm needn otiuint h geopotential the to contribution independent time coefficient that consequent value ( average) a independent (time and mean time deformation a This has potential nonzero. generating is tide zonal 2 degree The for in 6.5a listed Table constituents with in tidal variations together the tidal The other from the contributions (6.8a). for the Equation of using computed sum 6.5b the Table 1 coefficient Step geopotential of result in variation total value The nominal the and 6.3. formula Table above the from and computed s, 200 of formula the by described is it anelasticity; h aito of variation The A 0 N p h s τ H k δk 20 (0) f δk f R 0 = C f R ¯ and 20 n u-fpae( out-of-phase and ) m . α nfruaigagooeta oe,toapoce a be may approaches two model, geopotential a formulating In . 29525 r optdsmlry xetta qain(.b st eused be to is (6.8b) Equation that except similarly, computed are 0 = δk k 20 (0) f I . 0 5 The 15. − of C costeznltdlbn,( band, tidal zonal the across ¯ k p 20 5 20 δk s . hntevlei emd“eotd”adwl enoted be will and tide” “zero termed is value the then , 796 f o h oa ie stkna 0 as taken is tides zonal the for r itd ln ihtecrepnigin-phase corresponding the with along listed, are m ` ` × f δk m 10 n al .cfor 6.5c Table and 1 = f k 0 sterfrnefeunyeuvln oaperiod a to equivalent frequency reference the is − 20 (0) nTbe65 r h ieecsbetween differences the are 6.5b Table in 4 D F op nm cot mltd ( amplitude ) fteznltdsdet anelasticity. to due tides zonal the of 2)ptnilpoue permanent a produces potential (20) = ) απ 2 Ω 1 C ¯ − 20 δk nm f R sotie yadn othe to adding by obtained is f f A m 2) sdet mantle to due is (20), = ) m 0 Amp. H α 2. = k f ip 20 δk + ( ) 0 = f . I i 09.Tereal The 30190. f ob sdin used be to ) stefrequency the is f . Geopotential 6 f 09 ie in given 30190 m δk C ¯ 2 f I m α and Amp. (6.12) S op k ¯ 2 20 (0) m ) 6.3 Effect of the ocean tides IERS Technical

Note No. 36

(0) Table 6.5c: Amplitudes (A2δkf Hf ) of the corrections for frequency dependence of k22 , taking the −12 nominal value k22 for the sectorial tides as (0.30102 − i 0.00130). Units: 10 . The corrections are only to the real part. 0 0 R Name Doodson deg/hr τ s h p N ps ` ` FD Ω δkf Amp. No.

N2 245,655 28.43973 2 -1 0 1 0 0 1 0 2 0 2 0.00006 -0.3 M2 255,555 28.98410 2 0 0 0 0 0 0 0 2 0 2 0.00004 -1.2

¯zt here C20. If the time independent contribution is not included in the adopted value of C¯20, then the value is termed “conventional tide free” and will be noted ¯tf here C20 . In the case of a “zero tide” geopotential model, the model of tidal eﬀects to be added should not once again contain a time independent part. One must not then use the expression (6.6) as it stands for modeling ∆C¯20; its permanent part must ¯zt ﬁrst be restored. This is Step 3 of the computation, which provides ∆C20, to be used with a “zero tide” geopotential model.

¯zt ¯ ¯perm ∆C20 = ∆C20 − ∆C20 (6.13)

¯ ¯perm where ∆C20 is given by Equation (6.6) and where ∆C20 is the time-independent part:

¯perm −8 ∆C20 = A0H0k20 = (4.4228 × 10 )(−0.31460)k20. (6.14)

In the case of EGM2008, the diﬀerence between the zero-tide and tide-free value −9 of C20 is −4.1736 × 10 . Assuming the same values for A0, H0 and k20, the −3 tide-free value of C20 corresponding to Table 6.2 would be −0.48416531 × 10 . The use of “zero tide” values and the subsequent removal of the eﬀect of the permanent tide from the tide model is presented for consistency with the 18th IAG General Assembly Resolution 16.

6.3 Eﬀect of the ocean tides

The dynamical eﬀects of ocean tides are most easily incorporated as periodic variations in the normalized Stokes’ coeﬃcients of degree n and order m ∆C¯nm and ∆S¯nm. These variations can be evaluated as

− ¯ ¯ X X ± ± ±iθf (t) [∆Cnm − i∆Snm](t) = (Cf,nm ∓ iSf,nm)e , (6.15) f +

± ± where Cf,nm and Sf,nm are the geopotential harmonic amplitudes (see more information below) for the tide constituent f, and where θf (t) is the argument of the tide constituent f as deﬁned in the explanatory text below Equation (6.8e). Ocean tide models are typically developed and distributed as gridded maps of tide height amplitudes. These models provide in-phase and quadrature amplitudes of tide heights for selected, main tidal frequencies (or main tidal waves), on a variable grid spacing over the oceans. Using standard methods of spherical harmonic decomposition and with the use of an Earth loading model, the maps of ocean tide height amplitudes have been converted to spherical harmonic coeﬃcients of the geopotential, and provided for direct use in Equation (6.15). This computation follows Equation (6.21) and has been carried out for the tide model proposed in Section 6.3.2.

89 90 No. 36 .. akrudo ca iemodels tide ocean on Background 6.3.1 Note Technical IERS h ope fcoefficients of couples The where of functions yields: harmonic spherical it in grids) cotidal (6.8e). (from Equation ( below amplitudes text expanding explanatory When the see argument, Doodson where phase and amplitude frequencies. of discrete terms certain in at expressed waves conventionally of are models tide Ocean uhas: such wave main the m of coefficients harmonic spherical malized for necessary 6.3.2. not Section in is selected provided It on is Information use completeness. their geopotential. given and the for is to models here perturbations tidal coefficients included tidal the of is evaluation of and the determination 6.3.2). 6.3.1, Section the Section (see on in FES2004 information of waves background main Some the for developed 6.7 Table where ple nctdlmp.Te opywt h odo-abr convention wave Doodson-Warburg each 1973). For amplitude Eden, the harmonic and the with Cartwright of to sign comply the according They to according defined is maps. which cotidal in applied The (labeled waves, pivot or as lines, 2) main and nearby 1 tidal an two can subscripts of amplitudes spaced using amplitudes with harmonic closely the waves, geopotential from between the secondary derived wave, admittance be by secondary smaller, tidal completed each the of For be variation main to frequencies. can or linear waves perturbations tides of main largest geopotential assumption the the tidal only from of for interpolation spectrum maps The provides model waves. tide ocean an Typically, a ecmue as computed be can n a eatraeyepesdi em famplitude of terms in expressed alternately be can and , χ f ξ ξ ξ H Z f,nm ± ( ( C S S C C S ausarewt h ocle hrmncneto hc straditionally is which convention Shureman so-called the with agree values ¯ ¯ ,λ t λ, φ, t λ, φ, f f,nm f,nm f,nm f,nm ± ± ± ± f,nm f,nm ± ± steatooi mltd ftecniee ae e neapein example an See wave. considered the of amplitude astronomic the is steapiueo wave of amplitude the is ( ,t λ, = ) = ) f = ) ======h coefficients the , X X f f C ¯ C C θ θ 4 4 θ θ ˙ ˙ ˙ ˙ ¯ ˆ ¯ ˆ f f πGρ πGρ 2 2 f,nm n Z X ± f,nm f,nm ± ± − − N g g − − =1 f e e θ θ θ θ ˙ ˙ ( ˙ ˙ 1 1 1 1 w w ,λ φ, m X . . cos( n sin( cos( =0 H H H H C ¯ f f 2 2 1+ 1+ 2 2 o ( cos ) f,nm n n P ± S C ¯ θ Z ε +1 +1 ε k k 2 nm ± 2 f ± f,nm ± n n f,nm ± 0 0 ,nm f ,nm + n hss( phases and ) C , (sin θ S f,nm ± C C χ ¯ f ¯ ˆ ¯ ˆ f + ) + ) f,nm ± , ( f f,nm f,nm ± ± t ψ φ ± ) θ θ θ θ ˙ ˙ ˙ ˙ 2 ) f 2 2 2 − and − − − − X mλ stepaea rewc and Greenwich at phase the is cos( sin( − + θ θ θ θ ψ ˙ ˙ ˙ ˙ ersn rgaeadrtord nor- retrograde and prograde represent f 1 f 1 f + ) . . ξ S ( H H H ε H f,nm ± ε ,λ φ, f,nm ± f,nm ± f f,nm ± 1 f 1 ψ S C S ¯ f 1 f,nm ± ± )(6.17) )) 1 ± ( ,nm ftedffrn ae ftides of waves different the of ) ,nm ,t λ, + ob sdi qain(6.15) Equation in used be to + χ χ (6.18) ) sin( Z f f ) f ) cos f θ C f ¯ ˆ tdegree at f,nm ± + ( ψ χ f H f and ) n phase and ± f Geopotential 6 seTbe6.6 Table (see mλ n Z (6.19) ) n order and f θ sin f (6.16) (6.20) (6.21) sthe is ε ( f,nm ± ψ f ), 6.3 Effect of the ocean tides IERS Technical

Note No. 36

Table 6.6: Values of the phase bias χf according to the sign of Hf

Hf > 0 Hf < 0

n1=0, long period wave π 0 π π n1=1, diurnal wave 2 − 2 n1=2, semi-diurnal wave 0 π

where G and ge are given in Chapter 1, ρw is the density of seawater (1025 kg −3 0 0 m ) and where kn is the load deformation coeﬃcient of degree n (k2 =−0.3075, 0 0 0 0 k3 =−0.195, k4 =−0.132, k5 =−0.1032, k6 =−0.0892).

6.3.2 Ocean tide models The practical implementation of ocean tide models in this form begins with identification of the underlying ocean tide height model. Once this model is identified, further needed information can include the specification of maximum degree and order of the expansion, the identification of the pivot waves for interpolation, the special handling (if necessary) of the solar (radiational) tides, or the long-period tidal bands. For the case of the FES2004 ocean tide model, these details of implementation are provided next. FES2004 The FES2004 ocean tide model (Lyard et al., 2006) includes long period waves (Sa, Ssa, Mm, Mf , Mtm, Msqm), diurnal waves (Q1, O1, P1, K1), semi-diurnal 5 waves (2N2, N2, M2, T2, S2, K2) and the quarter-diurnal wave (M4 ). For direct ± ± use in Equation (6.15), the coefficients Cf,nm and Sf,nm for the main tidal waves of FES2004 can be found at <6> The tide height coefficients can be found in the file <7> , both in the form of the ¯± ¯± ¯ˆ± coefficients Cf,nm and Sf,nm and in the form of the amplitudes Cf,nm and phases ± εf,nm, as defined in Equations (6.19) and (6.20). They have been computed up to degree and order 100 by quadrature method from quarter-degree cotidal grids. Then ellipsoidal corrections to spherical harmonics were applied (Balmino, 2003) in order to take into account that tidal models are described on the oblate shape of the Earth. Table 6.7 provides a list of admittance waves which can be taken into account to complement the model. It indicates the pivot waves for linear interpolation following Equation (6.16), where indices 1 and 2 refer to the two pivot waves.

It is to be noticed that radiational waves like S1 and S2 require special handling, since the common altimetric models (including FES2004) for these tides include the contributions of atmospheric pressure variations on the ocean height (i.e. the radiational tide). As a result, neither S1 and S2 are used as pivot waves for interpolation in Table 6.7. While an S2 wave is available as a part of the FES2004 8 model, a mean S1 wave is given outside FES2004 and available in file < >. The additionally provided mean S1 wave should only be used in case the gravitational influences of mass transport from an ocean circulation model like MOG2D (Carrèreand Lyard, 2003) are not also modeled. This is because the S1 signal is generally part of such ocean circulation models provided with an interval of 6 hours.

Moreover, very long period waves like Ω1 (18.6 yr) and Ω2 (9.3 yr) which are not yet correctly observed can be modeled as equilibrium waves. Their amplitudes (and phases) are computed from the astronomical amplitude Hf considering the elastic response of the Earth through the Love numbers:

5 The χ value for M4, not given in Table 6.6, is 0 6ftp://tai.bipm.org/iers/conv2010/chapter6/tidemodels/fes2004 Cnm-Snm.dat 7ftp://tai.bipm.org/iers/conv2010/chapter6/tidemodels/fes2004.dat 8ftp://tai.bipm.org/iers/conv2010/chapter6/tidemodels/S1.dat

91 92 No. 36 Note Technical IERS awnssmo odo’ number Doodson’s symbol Darwin’s hnte aet elnal interpolated). waves linearly pivot be their to (with waves have secondary they some when for amplitudes and astronomical bold) (in of FES2004 List 6.7: Table M M M M M M 2 M sqm S stm S Q O Q Ω Ω sm S madcnrah2 m ti siae httemi ae fteFES2004 the of waves 109 inclination main km, the 5900 that 49 2008). (altitude inclination estimated (Biancale, several km, 800 effect is order (altitude the of Starlette It For typically of 80% are cm. represent tides 20 typically ocean reach model of can effects the and integration, cm of 98 day inclination one km, 800 for (altitude Stella like satellite a For models tidal of Influence respectively. deformation, where n m epciey o h ieec ewe E20 n h le CSR3.0 older the 2010). and 9 (Ries, FES2004 of between model along-track) difference tide (mostly the difference ocean for RMS respectively, 3-D mm, a 7 showed and respectively, days, 7 and 6 tm σ σ ρ sf τ sa ta m f a 1 1 1 1 1 1 1 2 1 C ¯ ˆ k f, 2 20 0 = 9.5 -.00204 -.01276 093.555 085.455 -.06663 075.555 3.5 -.05020 135.655 4.5 -.26221 145.555 5.5 -.03100 -.00492 057.555 056.554 6.5 -.03518 065.455 5.7 -.00027 .02793 055.575 055.565 6.4 0210755065.455 065.455 065.455 057.555 057.555 057.555 .00231 -.00673 -.00181 065.445 063.655 058.554 7.5 .08 6.5 075.555 075.555 075.555 075.555 065.455 075.555 065.455 065.455 -.00288 065.455 -.00583 065.455 .00188 -.00375 075.355 .00229 073.555 065.655 065.555 065.465 3.4 .04 3.5 145.555 145.555 145.555 145.555 135.655 093.555 135.655 135.655 -.00947 093.555 135.455 -.00802 085.455 -.00664 085.455 -.00194 135.645 085.455 085.455 -.00169 127.555 085.455 125.755 085.455 075.555 -.00529 117.655 075.555 095.355 075.555 -.00100 075.555 -.00242 085.465 -.00258 -.02762 083.665 083.655 075.575 075.565 3.4 .08 3.5 145.555 135.655 -.00180 137.445 4.4 .44 3.5 145.555 145.555 135.655 135.655 -.04946 -.00954 145.545 137.455 5.5 0711555165.555 165.555 165.555 165.555 145.555 145.555 145.555 .00741 145.555 .00194 .00343 .00170 155.455 153.655 147.555 145.755 = . + 1 92 and 29525 √ k 2 4 π − h 2 | h H 2 f H | 0 = , f . . f, 8 H 08aeteLv ubr fptniland potential of numbers Love the are 6078 ◦ 20 f n cetiiy005,itgaintm of time integration 0.005), eccentricity and io ae1Pvtwv 2 wave Pivot 1 wave Pivot = m o anwvsof waves main for (m) − π 2 . 8 ◦ n cetiiy00)adLageos1 and 0.02) eccentricity and . 7 ◦ continued n cetiiy0.001), eccentricity and Geopotential 6 6.4 Solid Earth pole tide IERS Technical

Note No. 36

Darwin’s symbol Doodson’s number Hf Pivot wave 1 Pivot wave 2 155.555 -.00399 145.555 165.555 M1 155.655 .02062 145.555 165.555 155.665 .00414 145.555 165.555 χ1 157.455 .00394 145.555 165.555 π1 162.556 -.00714 145.555 165.555 P1 163.555 -.12203 S1 164.556 .00289 K1− 165.545 -.00730 145.555 165.555 K1 165.555 .36878 K1+ 165.565 .05001 145.555 165.555 ψ1 166.554 .00293 145.555 165.555 ϕ1 167.555 .00525 145.555 165.555 θ1 173.655 .00395 145.555 165.555 J1 175.455 .02062 145.555 165.555 175.465 .00409 145.555 165.555 So1 183.555 .00342 145.555 165.555 185.355 .00169 145.555 165.555 Oo1 185.555 .01129 145.555 165.555 185.565 .00723 145.555 165.555

ν1 195.455 .00216 145.555 165.555 3N2 225.855 .00180 235.755 245.655 2 227.655 .00467 235.755 245.655 2N2 235.755 .01601 µ2 237.555 .01932 235.755 245.655 245.555 -.00389 237.755 245.655 245.645 -.00451 237.755 245.655

N2 245.655 .12099 ν2 247.455 .02298 245.655 255.555 γ2 253.755 -.00190 245.655 255.555 α2 254.556 -.00218 245.655 255.555 255.545 -.02358 245.655 255.555 M2 255.555 .63192 β2 256.554 .00192 255.555 275.555 λ2 263.655 -.00466 255.555 275.555 L2 265.455 -.01786 255.555 275.555 265.555 .00359 255.555 275.555 265.655 .00447 255.555 275.555 265.665 .00197 255.555 275.555 T2 272.556 .01720 255.555 275.555 S2 273.555 .29400 R2 274.554 -.00246 255.555 275.555 K2 275.555 .07996 K2+ 275.565 .02383 255.555 275.555 K2++ 275.575 .00259 255.555 275.555 η2 285.455 .00447 255.555 275.555 285.465 .00195 255.555 275.555 M4 455.555

6.4 Solid Earth pole tide

The pole tide is generated by the centrifugal eﬀect of polar motion, characterized by the potential

Ω2r2 ∆V (r, θ, λ) = − 2 sin 2θ (m1 cos λ + m2 sin λ) (6.22) Ω2r2 iλ = − 2 sin 2θ Re [(m1 − im2) e ].

93 94 No. 36 Note Technical . ca oetide pole Ocean 6.5 IERS 9 ftp://tai.bipm.org/iers/conv2010/chapter6/desaiscopolecoef.txt   ∆ ∆ S C ¯ ¯ nm nm where Ω, and vari- wobble the of relation the including details, further ( for ables 7.1.4 Section (See k ρ coef- geopotential the yields in tide changes polar to equivalent is which ficients potential, external the in perturbation a produces tide this constitutes ( 7.1.4), and h offiinsfo h efcnitn qiiru model, equilibrium self-consistent and the from coefficients The ( ( variables wobble the between relationship of (Values − ( coeffi- variables geopotential wobble pole normalized the ocean of the the function to model, a this perturbations as Using following cients, floor. the tide. ocean produces the pole tide of ocean oceans, the loading the over conservation and of mass self-gravitation, boundaries, model continental for equilibrium accounts self-consistent model This a equipotential presents forcing (2002) the Desai with equilibrium response, in equilibrium is an have surface surface. to ocean expected At displaced is tide variations. the the pole annual where ocean on and the Polar wobble motion periods, Chandler 6.4. polar long 14-month Section these of the from by (6.22) effect Equation dominated centrifugal in is defined the motion is by effect centrifugal generated This is oceans. tide pole ocean The where γ sn h ausdfie bv ilstefloig( tide: following the pole yields ocean above the defined values the Using ( The m x w n 0 0 = p   1 . a oddfraincecet ( coefficients deformation load = y , est fsawtr=1025 = water sea of density = 1032 m , E γ B = ¯ p 2 R ,m n, , nm R ). m m 2 R GM ∆ ∆ + C n r h obeprmtr nrdas ee oScin714frthe for 7.1.4 Section to Refer radians. in parameters wobble the are ) k , 1 n 1 S C 21 = ¯ = ¯ m , iγ (2 = ) 6    0 and 21 21 k , 2 = I B and 2   Ω ¯ g 2 GM 1+ (1 = nm e oteplrmto aibe ( variables motion polar the to ) R 2 and − m and , B A a = = ¯ ¯ 0 E 4 S nm R nm , 2 R + . )cecet r h oiattrso h ca oetide. pole ocean the of terms dominant the are coefficients 1) 0892), 21 ∆ ∆ 4 r nscnso arc. of seconds in are h i − − πGρ sn for Using . C B 2 S ¯ G ¯ ¯   k g 1 2 21 21 nm e I 2 − prpit o h oetd r sgvni etos6.4 Sections in given as are tide pole the for appropriate . . 7232 1778 r endi hpe 1, Chapter in defined are − w m Ω r rvddt ereadodr30at 360 order and degree to provided are , 2 h 1 2 = = r γ 2 2 2 + 1 × × 0 = ) 2 R n i 2 sin − − + 10 10 1 + k k 1 1 m − − 2 n 0 . . . 333 333 80+ 6870 θ 10 10 kgm 2 h au 0 value the k γ Re ( ( 2 0 2 m m I × × = − 2 1 [ + 10 10 3 − k − − , i 2 0   0 − − m 0 0 ( . . 9 9 0036 . . m 3075 03365 01724 1 ( ( B A m ¯ ¯ m , m m . 1 07+0 + 3077 nm I nm I 1 x − 1 2 m , k , 2 p 0 + − n h oa oinvariable motion polar the and ) im y , m m   3 0 2 0 p ,m n, 1 2 ). = 2 . . . h eomto which deformation The ).) ) ) 0115 0115 ) m , , − e . 0036 2 iλ 0 (2 = ) γ . ] m m 2 R 195 2 1 A − ¯ i ) ) nm k , , , prpit othe to appropriate m , )cecet for coefficients 1) 4 0 1 = γ = Geopotential 6 < 2 I A − ¯ 9    > nm R 0 . . 132 + (6.23b) (6.23a) (6.24) k , i A ¯ 5 0 nm I = 6.5 Ocean pole tide IERS Technical

Note No. 36

where m1 and m2 are in seconds of arc. Approximately 90% of the variance of the ocean pole tide potential is provided by the degree n = 2 spherical harmonic components, with the next largest contributions provided by the degree n = 1 and n = 3 components, respectively (see Figure 6.1). Expansion to spherical harmonic degree n = 10 provides approximately 99% of the variance. However, adequate representation of the continental boundaries will require a spherical harmonic expansion to high degree and order. The degree n = 1 components are shown in Figure 6.1 to illustrate the size of the ocean pole tide contribution to geocenter motion but these terms should not be used in modeling station displacements.

−11 x 10 Ocean Pole Tide Effects on Clm & Slm 6 C21 S21 C10 4 C11 S11 C20 C22 S22 2

−2

−4

−6 2003 2003.2 2003.4 2003.6 2003.8 2004 2004.2 2004.4 2004.6 2004.8 2005

Figure 6.1: Ocean pole tide: ﬁrst spherical harmonic components.

95 96 No. 36 al .:Fcosfrcneso oCrwih-alrapiue rmtoedfie codn to according defined those from amplitudes Cartwright-Tayler to conversion for Factors 6.8: Table different to according defined amplitudes tidal of conventions Conversion 6.6 Note Technical IERS odo’ n atanadWne’ conventions. Wenzel’s and Hartmann and Doodson’s rmDosnFo atan&Wenzel f & Hartmann From f f Doodson From f f f f 30 22 21 20 33 32 31 ======m). (from 9.79828685 = radius conven- (1921) heights tide Doodson equivalent the the to follow them (m convert which To potential (1996), the dimensionless. are of Hartmann Roosbeek tion, units of (1971). Tayler in amplitudes high-accuracy and amplitudes recent Cartwright the tabulate the from (1995) in and Wenzel terms other and tidal each of from amplitudes differ the tables for used definition The h olwn ausaeue o h osat pern ntecneso fac- conversion the in appearing constants constant the Doodson for used tors: are 6.5. values Table from following factors The appropriate the by multiply convention, Cartwright-Tayler − − − − √ √ √ 3 10 8 √ 1440 720 96 √ 2 √ √ √ √ √ 15 √ 3 √ 5 2880 20 4 π 7 √ 5 π 24 π 7 7 √ D π π 5 g D D π 7 g e g 1 D D π e g e g 1 1 D e g e 1 0 = 1 D e g = 1 0 = e = 1 0 = = − . = − 695827 . 0 603947 − . 0 683288 − . 426105 0 . 805263 0 . 695827 . 644210 D 1 GM .35325 m 2.63358352855 = f f f f f f f 32 31 30 22 33 21 20 0 0 0 0 0 0 0 3 ======. − − 986004415 √ √ 2 √ 2 g g g g g √ √ 8 8 8 e e e e e √ √ π π π π π g g 8 8 e e π π 0 = 0 = 0 = 0 = 0 = = = . . . . . − 511646 361788 − 511646 511646 361788 × 0 0 2 . . 10 511646 511646 s − 14 2 ; m g e 3 ≡ s − 2 g , tteequatorial the at R e Geopotential 6 6378136 = 2 s − H 2 f ,while ), fthe of . 55 References IERS Technical

Note No. 36

References

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