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Ocean tidal effects on Earth rotation
Journal of Geodynamics 48 (2009) 219–225
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Ocean tidal effects on Earth rotation Richard S. Gross ∗ Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Drive, Pasadena, CA 91109, USA
a r t i c l e
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Keywords: Earth rotation Length-of-day Solid body tides Ocean tides
a b s t r a c t Tidal forces due to the tide-raising potential deform the solid and fluid regions of the Earth, causing the Earth’s inertia tensor to change, and hence causing the Earth’s rate of rotation and length-of-day to change. Because both the tide-raising potential and the solid Earth’s elastic response to the tidal forces caused by this potential are well-known, accurate models for the effects of the elastic solid body tides on the Earth’s rotation are available. However, models for the effect of the ocean tides on the Earth’s rotation are more problematic because of the need to model the dynamic response of the oceans to the tidal forces. Hydrodynamic ocean tide models that have recently become available are evaluated here for their ability to account for long-period ocean tidal signals in length-of-day observations. Of the models tested here, the older altimetric data-constrained model of Kantha et al. (1998) is shown to still do the best job of accounting for ocean tidal effects in length-of-day, particularly at the fortnightly tidal frequency. The model currently recommended by the IERS is shown to do the worst job. © 2009 Elsevier Ltd. All rights reserved.
1. Introduction Tidal forces due to the changing gravitational attraction of the Sun, Moon, and planets deform the solid and fluid regions of the Earth, causing the Earth’s rotation to change by causing the Earth’s inertia tensor to change. In fact, solid body tides, caused by the tidal forces acting on the solid Earth, are the dominant cause of lengthof-day variations on intraseasonal to interannual time scales. And ocean tides, caused by the tidal forces acting on the oceans, are the dominant cause of subdaily length-of-day variations and contribute to length-of-day variations at longer periods. In fact, the fortnightly ocean tide is the dominant cause of intraseasonal length-of-day variations once atmospheric, non-tidal oceanic, and body tidal effects are removed. Purely hydrodynamic ocean tide models, that is, models that are not constrained by any type of measurement, were used first to model the effects of ocean tides on the Earth’s rotation (for a review see, e.g., Gross, 2007a). But the accuracy of ocean tide models greatly improved when Topex/Poseidon sea surface height measurements became available and were incorporated into them. Topex/Poseidon-constrained models are now available for the effects on the Earth’s rate of rotation and length-of-day of ocean tides in the semidiurnal and diurnal tidal bands (Ray et al., 1994; Chao et al., 1996; Chao and Ray, 1997) as well as for their effects in the long-period tidal band (Kantha et al., 1998; Desai and Wahr, 1999).
∗ Tel.: +1 818 354 4010; fax: +1 818 393 4965. E-mail address: [email protected]. 0264-3707/$ – see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.jog.2009.09.016
Recently, Benjamin et al. (2006) and Weis (2006) have again used purely hydrodynamic ocean tide models to compute the effect of long-period ocean tides on the Earth’s rotation. The goal of this study is to evaluate these newly available tide models by comparing them to residual length-of-day observations, that is, to observations from which atmospheric, non-tidal oceanic, and body tidal effects have been removed. It will be shown that while these models do a reasonably good job of accounting for the effect of the fortnightly ocean tide on length-of-day, the Topex/Poseidonconstrained model of Kantha et al. (1998) is still superior. It will also be shown that the model currently recommended by the IERS Conventions (McCarthy and Petit, 2004) is inadequate, leaving a large signal at the fortnightly tidal period when it is removed from length-of-day observations. 2. Residual length-of-day observations The length-of-day observations used in this study are from the COMB2006 combined Earth orientation series (Gross, 2007b). COMB2006 is a combination of Earth orientation measurements taken by the techniques of optical astrometry, lunar and satellite laser ranging, very long baseline interferometry, and the global positioning system. The COMB2006 length-of-day series spans 20 January 1962 to 10 February 2007 at daily intervals. Fig. 1a shows the spectrum of that portion of the COMB2006 length-of-day series spanning 2 January 1993 to 10 September 2006. The prominent peaks evident in this spectrum are primarily caused by solid body and ocean tides. The effects of atmospheric winds and surface pressure variations were removed from the COMB2006 length-of-day series using
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Fig. 1. Power spectral density (psd) estimates in decibels (db) computed by the multitaper method from time series of length-of-day observations spanning 2 January 1993 to 10 September 2006 of the: (a) observed COMB2006 length-of-day series; (b) residual length-of-day series formed by removing atmospheric and non-tidal oceanic effects from the observations; and (c) residual length-of-day series formed by additionally removing the Yoder et al. (1981) model for the effect of elastic body tides. The magnitude of the squared coherence between the sum of the AAM and OAM series and length-of-day observations from which just the elastic body tide model of Yoder et al. (1981) has been removed is shown in (d). Vertical dashed lines in (c) and (d) indicate the frequencies of the monthly (13 cpy), fortnightly (27 cpy), termensual (40 cpy), and 7-day (51 cpy) tidal terms. Horizontal dashed lines in (d) indicate the 95% and 99% confidence levels of the magnitude of the squared coherence.
effective atmospheric angular momentum (AAM) functions that were computed from products of the National Centers for Environmental Prediction/National Center for Atmospheric Research (NCEP/NCAR) reanalysis project (Kalnay et al., 1996). The pressure term used here assumes that the oceans respond as an inverted barometer to the imposed atmospheric surface pressure variations. The NCEP/NCAR reanalysis AAM series, which was obtained from the International Earth Rotation and Reference Systems Service (IERS) Special Bureau for the Atmosphere (Salstein, 2003; Zhou et al., 2006), is given at 6-h intervals since 1 January 1948. Daily averages of the 6-h values were formed by summing five consecutive values using weights of 1/8, 1/4, 1/4, 1/4, 1/8. The effects of oceanic currents and bottom pressure variations were removed from the length-of-day observations using effective oceanic angular momentum (OAM) functions computed from the products of an unconstrained oceanic general circulation model (OGCM) that was run at the Jet Propulsion Laboratory (JPL) as part of their participation in the Estimating the Circulation and Climate of the Oceans (ECCO) consortium (Stammer et al., 2002). This model, designated kf066a2, is the unconstrained version of model kf066b. Both kf066a2 and kf066b have the same model configuration and forcing as kf049f, the older constrained ECCO/JPL model that was replaced by the newer kf066b model. That is, both kf066a2 and kf066b are based on the MIT oceanic general circulation model (Marshall et al., 1997a,b), have realistic boundaries and bottom topography, have 46 levels ranging in thickness from 10 m at the
surface to 400 m at depth using height as the vertical coordinate, and span the globe between 80◦ S and 80◦ N latitude with a latitudinal grid-spacing ranging between 0.3◦ at the equator to 1◦ at the poles and a longitudinal grid-spacing of 1◦ . During the 10-year spinup period both models were forced with climatological fields from the Comprehensive Ocean-Atmosphere Data Set (COADS). Subsequently, the models were forced with twice daily wind stress and daily surface heat flux and evaporation–precipitation fields from the NCEP/NCAR reanalysis project. Atmospheric surface pressure fields were not used to force the models. Model kf066b assimilates altimetric sea surface height measurements whereas model kf066a2 does not assimilate any measurements. Even though one might think that kf066b should be the more realistic model of the general circulation of the oceans because it is constrained by data, it was not used here to remove the effects of non-tidal oceanic current and bottom pressure variations from length-of-day observations because, like kf049f, it is contaminated by tidal signals. A tide model was used to remove tidal signals from the altimetric sea surface height measurements before they were assimilated into kf049f and kf066b. Unfortunately, before July 2005 an equilibrium model was used to remove the long-period tidal signal (Fukumori, personal communication, 2007), thereby allowing the non-equilibrium component of the long-period tides to be assimilated into kf049f and kf066b. This introduced a tidal signal into the bottom pressure fields of these models and hence into the oceanic angular momentum computed from the bottom pres-
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Fig. 2. Power spectral density (psd) estimates in decibels (db) computed by the multitaper method from time series of length-of-day observations spanning 2 January 1993 to 10 September 2006 of the: (a) residual length-of-day observations formed by removing the in-phase part of the inelastic body tide model of Wahr and Bergen (WB, 1986) along with atmospheric, non-tidal oceanic, and elastic body tidal effects; and (c) residual length-of-day series formed by additionally removing the out-of-phase part of the Wahr and Bergen (1986) inelastic body tide model. The magnitude of the squared coherence between the sum of the AAM and OAM series and the residual length-of-day observations formed by removing just the elastic and in-phase part of the inelastic body tide model of Wahr and Bergen (1986) is shown in (b), and in (d) it is shown when the out-of-phase part of the inelastic body tide model of Wahr and Bergen (1986) is also removed. Vertical dashed lines indicate the frequencies of the monthly (13 cpy), fortnightly (27 cpy), termensual (40 cpy), and 7-day (51 cpy) tidal terms. The monthly tidal frequency is also indicated by an arrow in (a) and (c). Horizontal dashed lines in (b) and (d) indicate the 95% and 99% confidence levels of the magnitude of the squared coherence.
sure fields. So, the unconstrained model kf066a2, which does not contain any tidal signals, was used here to remove non-tidal oceanic current and bottom pressure effects from the length-of-day observations. The OAM series computed here from the oceanic current and bottom pressure fields of model kf066a2 spans 2 January 1993 to 10 September 2006 at daily intervals. Model kf066a2 is based on the MIT oceanic general circulation model that, because it is formulated using the Boussinesq approximation (Marshall et al., 1997a,b), conserves volume rather than mass. Artificial mass variations can be introduced into Boussinesq models through the applied surface heat and salt fluxes. For example, the changing applied heat flux will change the density, which, since volume is conserved, will artificially change the mass of the modeled oceans. If left uncorrected, this artificial mass change will cause artificial changes to the ocean-bottom pressure and hence to the angular momentum computed from the bottom pressure variations. The artificial mass variations caused by the use of the Boussinesq approximation have been removed from kf066a2 by computing the effect on the bottom pressure of a spatially uniform layer added to the model’s sea surface that has just the right fluctuating thickness to impose mass conservation (Greatbatch, 1994). Fig. 1b shows the spectrum of the COMB2006 length-of-day series from which atmospheric and non-tidal oceanic effects have been removed using the NCEP/NCAR reanalysis AAM and the kf066a2 OAM series, respectively. Compared with the spectrum shown in Fig. 1a, the background power has been greatly reduced,
allowing peaks that were just above the background in Fig. 1a to be much more prominent in Fig. 1b. 3. Solid body tides 3.1. Elastic Yoder et al. (1981) derived a model for the effect of long-period body tides on the Earth’s rate of rotation by assuming that the crust and mantle of the Earth is elastic, that the fluid core is decoupled from the mantle, and that the ocean tides are in equilibrium. More recently, Defraigne and Smits (1999) derived a model for the effect of long-period elastic body tides on the Earth’s rate of rotation using a more accurate Earth model, PREM (Dziewonski and Anderson, 1981), and a more recent tide generating potential, RATGP95 (Roosbeek, 1996), than those used by Yoder et al. (1981). But unfortunately, Defraigne and Smits (1999, Table 1) tabulate their results for only 11 long-period tidal lines. Since Yoder et al. (1981) give results for 62 tidal lines, their more complete, but possibly less accurate, model has been used here to remove the effects of long-period elastic body tides from the length-of-day observations. Fig. 1c shows the spectrum of length-of-day observations from which long-period elastic body tidal effects have been removed using the model of Yoder et al. (1981), atmospheric effects have been removed using the NCEP/NCAR reanalysis AAM series, and
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Fig. 3. Power spectral density (psd) estimates in decibels (db) computed by the multitaper method from time series of length-of-day observations spanning 2 January 1993 to 10 September 2006 of the: (a) residual length-of-day observations formed by removing the ocean tide model of Weis (2006) in addition to removing atmospheric, non-tidal oceanic, and elastic and inelastic body tidal effects; and (c) residual length-of-day series formed by removing the Benjamin et al. (BWRED, 2006) ocean tide model instead of the Weis (2006) model. The magnitude of the squared coherence between the sum of the AAM and OAM series and the residual length-of-day observations formed by removing just the elastic and inelastic body tide model and the Weis (2006) ocean tide model is shown in (b), and in (d) it is shown when the ocean tide model of Benjamin et al. (2006) is removed instead of the Weis (2006) model. Vertical dashed lines indicate the frequencies of the monthly (13 cpy), fortnightly (27 cpy), termensual (40 cpy), and 7-day (51 cpy) tidal terms. Horizontal dashed lines in (b) and (d) indicate the 95% and 99% confidence levels of the magnitude of the squared coherence.
non-tidal oceanic effects have been removed using the kf066a2 OAM series. The most prominent peak at intraseasonal frequencies in this residual spectrum is at the fortnightly tidal frequency (27 cycles per year, cpy). A much smaller peak is evident at the termensual tidal frequency (40 cpy). Peaks at the level of the background are also possibly evident at the monthly (13 cpy) and 7-day (51 cpy) tidal frequencies. Fig. 1d shows the magnitude of the squared coherence between the sum of the AAM and OAM series and the length-of-day observations from which just the effects of the elastic body tides have been removed. Note the dramatic drop in coherence at the fortnightly tidal frequency. Any acceptable model for the dynamic effect of the fortnightly ocean tide on the length-of-day must both reduce the spectral peak evident in Fig. 1c to the level of the background power and must increase the coherence evident in Fig. 1d. But before evaluating different ocean tide models, the effects of mantle inelasticity on the body tides are first examined. 3.2. Inelastic The effects of mantle inelasticity on tidal length-of-day variations are only a few percent of the effects of mantle elasticity at the monthly and fortnightly tidal frequencies. But dissipation associated with mantle inelasticity also causes the deformational and hence rotational response of the Earth to lag behind the forcing tidal potential. As a result, not only does mantle inelasticity modify
the in-phase rotational response of the Earth to the tidal potential, but out-of-phase terms are introduced as well. While Defraigne and Smits (1999) modeled the effects of mantle inelasticity on just the in-phase part of the Earth’s rotational response, Wahr and Bergen (1986) modeled its effects on both the in-phase and out-of-phase parts. Fig. 2a and b shows the results of additionally removing just the in-phase part of the Wahr and Bergen (1986) inelastic body tide model from length-of-day observations that have also had elastic body tidal, atmospheric, and non-tidal oceanic effects removed. By carefully comparing these figures with Fig. 1c and d it can be seen that the spectral peaks at the monthly and fortnightly tidal frequencies are slightly smaller when the in-phase part of the inelastic body tide is removed. Similar results are obtained when the inphase part of the Defraigne and Smits (1999) inelastic body tide model is removed from the residual length-of-day observations (not shown). Fig. 2c and d shows the results of removing both the in-phase and out-of-phase parts of the Wahr and Bergen (1986) inelastic body tide model from the residual length-of-day observations. By carefully comparing these figures with Fig. 2a and b it can be seen that the spectral peaks at the monthly and fortnightly tidal frequencies are reduced even further when the out-of-phase part is additionally removed from the length-of-day observations. The tidal peaks at the termensual (40 cpy) and 7-day (51 cpy) tidal frequencies are not affected when the effects of the inelastic body tides are removed from the residual length-of-day observations because neither the
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Fig. 4. Power spectral density (psd) estimates in decibels (db) computed by the multitaper method from time series of length-of-day observations spanning 2 January 1993 to 10 September 2006 of the: (a) residual length-of-day observations formed by removing the ocean tide model of Kantha et al. (KSD, 1998) in addition to removing atmospheric, non-tidal oceanic, and elastic and inelastic body tidal effects; and (c) residual length-of-day series formed by removing the model recommended by the IERS Conventions (McCarthy and Petit, 2004, Table 8.1) in addition to removing atmospheric and non-tidal oceanic effects. The magnitude of the squared coherence between the sum of the AAM and OAM series and the residual length-of-day observations formed by removing just the elastic and inelastic body tide model and the Kantha et al. (1998) ocean tide model is shown in (b), and in (d) it is shown when the IERS model is removed from the observed length-of-day variations. Vertical dashed lines indicate the frequencies of the monthly (13 cpy), fortnightly (27 cpy), termensual (40 cpy), and 7-day (51 cpy) tidal terms. Horizontal dashed lines in (b) and (d) indicate the 95% and 99% confidence levels of the magnitude of the squared coherence.
Wahr and Bergen (1986) nor the Defraigne and Smits (1999) models include terms at these tidal frequencies. A large peak at the fortnightly tidal frequency in the length-ofday spectrum is still evident after removing elastic and inelastic body tidal, atmospheric, and non-tidal oceanic effects from the observations (Fig. 2c). The ability of selected ocean tide models to account for this presumably ocean tidal signal in residual lengthof-day observations is now examined. 4. Ocean tides Weis (2006) recently developed a high-resolution global purely hydrodynamic ocean tide model and used this model to predict the effects of 12 long-period ocean tidal constituents on the Earth’s rate of rotation. Fig. 3a and b shows the results of removing the Weis (2006) long-period ocean tide model from length-of-day observations that have also had elastic and inelastic body tidal, atmospheric, and non-tidal oceanic effects removed. By comparing these figures with Fig. 2c and d it can be seen that the Weis (2006) model does a good job of removing the tidal signal at the monthly (13 cpy), termensual (40 cpy), and 7-day (51 cpy) tidal frequencies but is unable to completely account for the observed power at the fortnightly tidal frequency. Since the coherence with the sum of AAM and OAM is greatly improved at the fortnightly tidal frequency when the Weis (2006) model is removed from the residual lengthof-day observations (compare Figs. 2d and 3b), the error in the Weis (2006) model at the fortnightly tidal frequency is probably mostly with its amplitude, not its phase.
Benjamin et al. (2006) used the ocean tide modeling system of Egbert and Erofeeva (2002) to test a number of different ocean tide models, both unconstrained and constrained by altimetric sea surface height measurements, for their efficacy in removing fortnightly and monthly ocean tidal effects from the Earth rotation observations that they used in their study of mantle inelasticity. Since all of the models that they tested gave consistent results for the fortnightly and monthly tidal effects on Earth rotation, they chose to use the solution from an unconstrained ocean tide model. Fig. 3c and d shows the results of removing the fortnightly and monthly ocean tide model preferred by Benjamin et al. (2006) from length-of-day observations that have also had elastic and inelastic body tidal, atmospheric, and non-tidal oceanic effects removed. As can be seen, the Benjamin et al. (2006) model does an excellent job of accounting for the observed power at the fortnightly and monthly tidal frequencies (Fig. 3c). It also does a reasonably good job of increasing the coherence with AAM and OAM (Fig. 3d). Of course, the tidal peaks at the termensual (40 cpy) and 7-day (51 cpy) tidal frequencies are not affected when the Benjamin et al. (2006) ocean tide model is removed from the residual lengthof-day observations because this model does not include terms at these frequencies. Kantha et al. (1998) combined an empirical ocean tide model based on Topex/Poseidon altimetric sea surface height measurements with a hydrodynamical tide model to predict the effect of the fortnightly and monthly ocean tides on the Earth’s rotation. Fig. 4a and b shows the results of removing this constrained ocean tide model of Kantha et al. (1998) from length-of-day observations
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that have also had elastic and inelastic body tidal, atmospheric, and non-tidal oceanic effects removed. As can be seen, the Kantha et al. (1998) model also does an excellent job of accounting for the observed power at the fortnightly and monthly tidal frequencies (Fig. 4a). And it does an even better job than the Benjamin et al. (2006) model of increasing the coherence with AAM and OAM (compare Figs. 3d and 4b). Again, the tidal peaks at the termensual (40 cpy) and 7-day (51 cpy) tidal frequencies are not affected when the Kantha et al. (1998) ocean tide model is removed from the residual length-of-day observations because this model does not include terms at these frequencies. Of the three ocean tide models evaluated here, the dataconstrained model of Kantha et al. (1998) does the best job of accounting for the observed fortnightly tidal signal evident in length-of-day observations from which elastic and inelastic body tidal, atmospheric, and non-tidal oceanic effects have been removed. The reason that this model does the best job is probably because it is the only model studied here that is constrained by data. While the unconstrained ocean tide model of Benjamin et al. (2006) does just as well as the constrained model of Kantha et al. (1998) in reducing the fortnightly spectral peak to the level of the background power (compare Figs. 3c and 4a), the coherence with AAM and OAM is much greater when the Kantha et al. (1998) model is removed from the length-of-day observations than it is when the Benjamin et al. (2006) model is removed (compare Figs. 3d and 4b). While the unconstrained model of Weis (2006) does not do as well as either the Kantha et al. (1998) or the Benjamin et al. (2006) models in accounting for the fortnightly ocean tidal signal in length-ofday observations (compare Fig. 3a with either Fig. 3c or Fig. 4a), it does do a reasonably good job of accounting for the termensual (40 cpy) and 7-day (51 cpy) tidal signals. Given that the sum of the Yoder et al. (1981) elastic body, Wahr and Bergen (1986) inelastic body, and Kantha et al. (1998) ocean tide models does an excellent job of accounting for the effects of the body and ocean tides on length-of-day, especially at the fortnightly tidal frequency, how does it compare to the model recommended by the IERS Conventions (McCarthy and Petit, 2004, Table 8.1), a model based on that of Defraigne and Smits (1999)? 5. IERS model Models recommended by the IERS for the removal of solid body and ocean tidal effects from the Earth’s rate of rotation and lengthof-day are given in Chapter 8 of the IERS Conventions (McCarthy and Petit, 2004). Table 8.1 in that document provides recommended corrections for the long-period tidal effects, that is, for tidal variations having periods between 5 days and 18.6 years. Unfortunately, very little information is provided in Chapter 8 about the derivation of the values contained in Table 8.1, with the latest update to Chapter 8, available at, stating only that they are derived from the Defraigne and Smits (1999) model. As discussed above, Defraigne and Smits (1999) derived a model for the effect of the elastic and in-phase part of the inelastic solid body tides on the Earth’s rate of rotation and length-of-day. When comparing their theoretical model to observed length-of-day variations they accounted for the effects of ocean tides using the models of Dickman (1993) and Seiler and Wünsch (1995). Unfortunately, it is not apparent which of these ocean tide models was used when deriving the values given in Table 8.1 of the IERS Conventions (McCarthy and Petit, 2004). In any case, Fig. 4c and d shows the results of removing the model recommended by the IERS from length-of-day observations that have also had atmospheric and non-tidal oceanic effects removed. As can be seen, when the IERS-recommended model is removed, a large signal at the fortnightly tidal frequency remains in the residual
length-of-day observations (Fig. 4c) and there is a dramatic drop in coherence with AAM and OAM at the fortnightly tidal frequency (Fig. 4d). The sum of the Yoder et al. (1981) elastic body, Wahr and Bergen (1986) inelastic body, and Kantha et al. (1998) ocean tide models does a much better job of accounting for the effects of the body and ocean tides on length-of-day, especially at the fortnightly tidal frequency, than does the IERS model (compare Fig. 4c and d with Fig. 4a and b). 6. Discussion and summary In a companion paper, Gross (2008) examined the effect of longperiod ocean tides on polar motion excitation, evaluating a number of different ocean tide models for their ability to account for the fortnightly ocean tidal signal seen in these observations. He concluded that no existing ocean tide model is able to adequately account for the observed signal. So he derived a new empirical ocean tide model by fitting periodic terms at the monthly, fortnightly, and termensual tidal frequencies to polar motion excitation observations from which atmospheric and non-tidal oceanic effects had been removed. Unlike polar motion excitation, it has been shown here that a model currently exists that does adequately account for ocean tidal effects in length-of-day, namely, the model of Kantha et al. (1998). In fact, the sum of the Yoder et al. (1981) elastic body tide model, the Wahr and Bergen (1986) inelastic body tide model, and the Kantha et al. (1998) ocean tide model does an excellent job of accounting for solid body and ocean tidal effects in length-of-day. In particular, and unlike the model recommended by the IERS, it is able to completely account for the fortnightly tidal signal that is seen in length-of-day observations from which atmospheric and non-tidal oceanic effects have been removed. The older Yoder et al. (1981) model was used here to remove the effects of elastic body tides from length-of-day observations because Defraigne and Smits (1999) provided results for only 11 tidal terms in their paper. It would be worth repeating this study using the complete Defraigne and Smits (1999) elastic body tide model to see how it compares with the Yoder et al. (1981) model when used in conjunction with the Wahr and Bergen (1986) inelastic body tide model and the Kantha et al. (1998) ocean tide model to remove tidal effects from length-of-day observations. Acknowledgements The work described in this paper was performed at the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration. Support for this work was provided by the Earth Surface and Interior Focus Area of NASA’s Science Mission Directorate. The supercomputers used in this investigation were provided by funding from the JPL Office of the Chief Information Officer. References Benjamin, D., Wahr, J., Ray, R.D., Egbert, G.D., Desai, S.D., 2006. Constraints on mantle anelasticity from geodetic observations, and implications for the J2 anomaly. Geophys. J. Int. 165, 3–16. Chao, B.F., Ray, R.D., 1997. Oceanic tidal angular momentum and Earth’s rotation variations. Prog. Oceanogr. 40, 399–421. Chao, B.F., Ray, R.D., Gipson, J.M., Egbert, G.D., Ma, C., 1996. Diurnal/semidiurnal polar motion excited by oceanic tidal angular momentum. J. Geophys. Res. 101 (B9), 20151–20163. Defraigne, P., Smits, I., 1999. Length of day variations due to zonal tides for an inelastic earth in non-hydrostatic equilibrium. Geophys. J. Int. 139, 563–572. Desai, S.D., Wahr, J.M., 1999. Monthly and fortnightly tidal variations of the Earth’s rotation rate predicted by a TOPEX/POSEIDON empirical ocean tide model. Geophys. Res. Lett. 26 (8), 1035–1038. Dickman, S.R., 1993. Dynamic ocean-tide effects on Earth’s rotation. Geophys. J. Int. 112, 448–470.
R.S. Gross / Journal of Geodynamics 48 (2009) 219–225 Dziewonski, A.M., Anderson, D.L., 1981. Preliminary reference earth model. Phys. Earth Planet Inter. 25, 297–356. Egbert, G.D., Erofeeva, S.Y., 2002. Efficient inverse modeling of barotropic ocean tides. J. Atmos. Ocean. Technol. 19, 183–204. Greatbatch, R.J., 1994. A note on the representation of steric sea level in models that conserve volume rather than mass. J. Geophys. Res. 99, 12767–12771. Gross, R.S., 2007a. Earth rotation variations—long period. In: Herring, T.A. (Ed.), Physical Geodesy, Treatise on Geophysics vol. 3. Elsevier, Oxford, pp. 239–294. Gross, R.S., 2007b. Combinations of Earth orientation measurements: SPACE2006, COMB2006, and POLE2006. Jet Propulsion Laboratory Publ 07-5, Pasadena, CA. Gross, R.S., 2008. An improved empirical model for the effect of long-period ocean tides on polar motion. J. Geodesy, doi:10.1007/s00190-008-0277-y. Kalnay, E., Kanamitsu, M., Kistler, R., Collins, W., Deaven, D., Gandin, L., Iredell, M., Saha, S., White, G., Woollen, J., Zhu, Y., Chelliah, M., Ebisuzaki, W., Higgins, W., Janowiak, J., Mo, K.C., Ropelewski, C., Wang, J., Leetmaa, A., Reynolds, R., Jenne, R., Joseph, D., 1996. The NCEP/NCAR 40-year reanalysis project. Bull. Am. Met. Soc. 77, 437–471. Kantha, L.H., Stewart, J.S., Desai, S.D., 1998. Long-period lunar fortnightly and monthly ocean tides. J. Geophys. Res. 103 (C6), 12639–12647. Marshall, J., Hill, C., Perelman, L., Adcroft, A., 1997a. Hydrostatic, quasi-hydrostatic, and nonhydrostatic ocean modeling. J. Geophys. Res. 102, 5733–5752. Marshall, J., Adcroft, A., Hill, C., Perelman, L., Heisey, C., 1997b. A finite-volume, incompressible, Navier Stokes model for studies of the ocean on parallel computers. J. Geophys. Res. 102, 5753–5766. McCarthy, D.D., Petit, G. (Eds.), 2004. IERS Conventions (2003). IERS Tech. Note No. 32. Bundesamts für Kartographie und Geodäsie, Frankfurt, Germany.
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Ray, R.D., Steinberg, D.J., Chao, B.F., Cartwright, D.E., 1994. Diurnal and semidiurnal variations in the Earth’s rotation rate induced by oceanic tides. Science 264, 830–832. Roosbeek, F., 1996. A tide generating potential precise to the nanogal level. Geophys. J. Int. 126, 197–204. Salstein, D.A., 2003. The GGFC Special Bureau for the Atmosphere of the International Earth Rotation and Reference Systems Service. In: Richter, B., Schwegmann, W., Dick, W.R. (Eds.), Proceedings of the IERS Workshop on Combination Research and Global Geophysical Fluids. Bundesamts für Kartographie und Geodäsie, Frankfurt, Germany, IERS Technical Note No. 30, pp. 121–124. Seiler, U., Wünsch, J., 1995. A refined model for the influence of ocean tides on UT1 and polar motion. Astron. Nachr. 316, 419–423. Stammer, D., Wunsch, C., Fukumori, I., Marshall, J., 2002. State estimation improves prospects for ocean research. Eos Trans. AGU 83 (27), 289–295. Wahr, J., Bergen, Z., 1986. The effects of mantle anelasticity on nutations, Earth tides, and tidal variations in rotation rate. Geophys. J. Roy. Astron. Soc. 87, 633– 668. Weis, P., 2006. Ocean tides and the Earth’s rotation—results of a high-resolving ocean model forced by the lunisolar potential. Ph.D. Thesis, Universität Hamburg. Yoder, C.F., Williams, J.G., Parke, M.E., 1981. Tidal variations of Earth rotation. J. Geophys. Res. 86 (B2), 881–891. Zhou, Y.H., Salstein, D.A., Chen, J.L., 2006. Revised atmospheric excitation function series related to Earth variable rotation under consideration of surface topography. J. Geophys. Res. 111, D12108, doi:10.1029/2005JD006608.
Contents lists available at ScienceDirect
Journal of Geodynamics journal homepage: http://www.elsevier.com/locate/jog
Ocean tidal effects on Earth rotation Richard S. Gross ∗ Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Drive, Pasadena, CA 91109, USA
a r t i c l e
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Keywords: Earth rotation Length-of-day Solid body tides Ocean tides
a b s t r a c t Tidal forces due to the tide-raising potential deform the solid and fluid regions of the Earth, causing the Earth’s inertia tensor to change, and hence causing the Earth’s rate of rotation and length-of-day to change. Because both the tide-raising potential and the solid Earth’s elastic response to the tidal forces caused by this potential are well-known, accurate models for the effects of the elastic solid body tides on the Earth’s rotation are available. However, models for the effect of the ocean tides on the Earth’s rotation are more problematic because of the need to model the dynamic response of the oceans to the tidal forces. Hydrodynamic ocean tide models that have recently become available are evaluated here for their ability to account for long-period ocean tidal signals in length-of-day observations. Of the models tested here, the older altimetric data-constrained model of Kantha et al. (1998) is shown to still do the best job of accounting for ocean tidal effects in length-of-day, particularly at the fortnightly tidal frequency. The model currently recommended by the IERS is shown to do the worst job. © 2009 Elsevier Ltd. All rights reserved.
1. Introduction Tidal forces due to the changing gravitational attraction of the Sun, Moon, and planets deform the solid and fluid regions of the Earth, causing the Earth’s rotation to change by causing the Earth’s inertia tensor to change. In fact, solid body tides, caused by the tidal forces acting on the solid Earth, are the dominant cause of lengthof-day variations on intraseasonal to interannual time scales. And ocean tides, caused by the tidal forces acting on the oceans, are the dominant cause of subdaily length-of-day variations and contribute to length-of-day variations at longer periods. In fact, the fortnightly ocean tide is the dominant cause of intraseasonal length-of-day variations once atmospheric, non-tidal oceanic, and body tidal effects are removed. Purely hydrodynamic ocean tide models, that is, models that are not constrained by any type of measurement, were used first to model the effects of ocean tides on the Earth’s rotation (for a review see, e.g., Gross, 2007a). But the accuracy of ocean tide models greatly improved when Topex/Poseidon sea surface height measurements became available and were incorporated into them. Topex/Poseidon-constrained models are now available for the effects on the Earth’s rate of rotation and length-of-day of ocean tides in the semidiurnal and diurnal tidal bands (Ray et al., 1994; Chao et al., 1996; Chao and Ray, 1997) as well as for their effects in the long-period tidal band (Kantha et al., 1998; Desai and Wahr, 1999).
∗ Tel.: +1 818 354 4010; fax: +1 818 393 4965. E-mail address: [email protected]. 0264-3707/$ – see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.jog.2009.09.016
Recently, Benjamin et al. (2006) and Weis (2006) have again used purely hydrodynamic ocean tide models to compute the effect of long-period ocean tides on the Earth’s rotation. The goal of this study is to evaluate these newly available tide models by comparing them to residual length-of-day observations, that is, to observations from which atmospheric, non-tidal oceanic, and body tidal effects have been removed. It will be shown that while these models do a reasonably good job of accounting for the effect of the fortnightly ocean tide on length-of-day, the Topex/Poseidonconstrained model of Kantha et al. (1998) is still superior. It will also be shown that the model currently recommended by the IERS Conventions (McCarthy and Petit, 2004) is inadequate, leaving a large signal at the fortnightly tidal period when it is removed from length-of-day observations. 2. Residual length-of-day observations The length-of-day observations used in this study are from the COMB2006 combined Earth orientation series (Gross, 2007b). COMB2006 is a combination of Earth orientation measurements taken by the techniques of optical astrometry, lunar and satellite laser ranging, very long baseline interferometry, and the global positioning system. The COMB2006 length-of-day series spans 20 January 1962 to 10 February 2007 at daily intervals. Fig. 1a shows the spectrum of that portion of the COMB2006 length-of-day series spanning 2 January 1993 to 10 September 2006. The prominent peaks evident in this spectrum are primarily caused by solid body and ocean tides. The effects of atmospheric winds and surface pressure variations were removed from the COMB2006 length-of-day series using
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Fig. 1. Power spectral density (psd) estimates in decibels (db) computed by the multitaper method from time series of length-of-day observations spanning 2 January 1993 to 10 September 2006 of the: (a) observed COMB2006 length-of-day series; (b) residual length-of-day series formed by removing atmospheric and non-tidal oceanic effects from the observations; and (c) residual length-of-day series formed by additionally removing the Yoder et al. (1981) model for the effect of elastic body tides. The magnitude of the squared coherence between the sum of the AAM and OAM series and length-of-day observations from which just the elastic body tide model of Yoder et al. (1981) has been removed is shown in (d). Vertical dashed lines in (c) and (d) indicate the frequencies of the monthly (13 cpy), fortnightly (27 cpy), termensual (40 cpy), and 7-day (51 cpy) tidal terms. Horizontal dashed lines in (d) indicate the 95% and 99% confidence levels of the magnitude of the squared coherence.
effective atmospheric angular momentum (AAM) functions that were computed from products of the National Centers for Environmental Prediction/National Center for Atmospheric Research (NCEP/NCAR) reanalysis project (Kalnay et al., 1996). The pressure term used here assumes that the oceans respond as an inverted barometer to the imposed atmospheric surface pressure variations. The NCEP/NCAR reanalysis AAM series, which was obtained from the International Earth Rotation and Reference Systems Service (IERS) Special Bureau for the Atmosphere (Salstein, 2003; Zhou et al., 2006), is given at 6-h intervals since 1 January 1948. Daily averages of the 6-h values were formed by summing five consecutive values using weights of 1/8, 1/4, 1/4, 1/4, 1/8. The effects of oceanic currents and bottom pressure variations were removed from the length-of-day observations using effective oceanic angular momentum (OAM) functions computed from the products of an unconstrained oceanic general circulation model (OGCM) that was run at the Jet Propulsion Laboratory (JPL) as part of their participation in the Estimating the Circulation and Climate of the Oceans (ECCO) consortium (Stammer et al., 2002). This model, designated kf066a2, is the unconstrained version of model kf066b. Both kf066a2 and kf066b have the same model configuration and forcing as kf049f, the older constrained ECCO/JPL model that was replaced by the newer kf066b model. That is, both kf066a2 and kf066b are based on the MIT oceanic general circulation model (Marshall et al., 1997a,b), have realistic boundaries and bottom topography, have 46 levels ranging in thickness from 10 m at the
surface to 400 m at depth using height as the vertical coordinate, and span the globe between 80◦ S and 80◦ N latitude with a latitudinal grid-spacing ranging between 0.3◦ at the equator to 1◦ at the poles and a longitudinal grid-spacing of 1◦ . During the 10-year spinup period both models were forced with climatological fields from the Comprehensive Ocean-Atmosphere Data Set (COADS). Subsequently, the models were forced with twice daily wind stress and daily surface heat flux and evaporation–precipitation fields from the NCEP/NCAR reanalysis project. Atmospheric surface pressure fields were not used to force the models. Model kf066b assimilates altimetric sea surface height measurements whereas model kf066a2 does not assimilate any measurements. Even though one might think that kf066b should be the more realistic model of the general circulation of the oceans because it is constrained by data, it was not used here to remove the effects of non-tidal oceanic current and bottom pressure variations from length-of-day observations because, like kf049f, it is contaminated by tidal signals. A tide model was used to remove tidal signals from the altimetric sea surface height measurements before they were assimilated into kf049f and kf066b. Unfortunately, before July 2005 an equilibrium model was used to remove the long-period tidal signal (Fukumori, personal communication, 2007), thereby allowing the non-equilibrium component of the long-period tides to be assimilated into kf049f and kf066b. This introduced a tidal signal into the bottom pressure fields of these models and hence into the oceanic angular momentum computed from the bottom pres-
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Fig. 2. Power spectral density (psd) estimates in decibels (db) computed by the multitaper method from time series of length-of-day observations spanning 2 January 1993 to 10 September 2006 of the: (a) residual length-of-day observations formed by removing the in-phase part of the inelastic body tide model of Wahr and Bergen (WB, 1986) along with atmospheric, non-tidal oceanic, and elastic body tidal effects; and (c) residual length-of-day series formed by additionally removing the out-of-phase part of the Wahr and Bergen (1986) inelastic body tide model. The magnitude of the squared coherence between the sum of the AAM and OAM series and the residual length-of-day observations formed by removing just the elastic and in-phase part of the inelastic body tide model of Wahr and Bergen (1986) is shown in (b), and in (d) it is shown when the out-of-phase part of the inelastic body tide model of Wahr and Bergen (1986) is also removed. Vertical dashed lines indicate the frequencies of the monthly (13 cpy), fortnightly (27 cpy), termensual (40 cpy), and 7-day (51 cpy) tidal terms. The monthly tidal frequency is also indicated by an arrow in (a) and (c). Horizontal dashed lines in (b) and (d) indicate the 95% and 99% confidence levels of the magnitude of the squared coherence.
sure fields. So, the unconstrained model kf066a2, which does not contain any tidal signals, was used here to remove non-tidal oceanic current and bottom pressure effects from the length-of-day observations. The OAM series computed here from the oceanic current and bottom pressure fields of model kf066a2 spans 2 January 1993 to 10 September 2006 at daily intervals. Model kf066a2 is based on the MIT oceanic general circulation model that, because it is formulated using the Boussinesq approximation (Marshall et al., 1997a,b), conserves volume rather than mass. Artificial mass variations can be introduced into Boussinesq models through the applied surface heat and salt fluxes. For example, the changing applied heat flux will change the density, which, since volume is conserved, will artificially change the mass of the modeled oceans. If left uncorrected, this artificial mass change will cause artificial changes to the ocean-bottom pressure and hence to the angular momentum computed from the bottom pressure variations. The artificial mass variations caused by the use of the Boussinesq approximation have been removed from kf066a2 by computing the effect on the bottom pressure of a spatially uniform layer added to the model’s sea surface that has just the right fluctuating thickness to impose mass conservation (Greatbatch, 1994). Fig. 1b shows the spectrum of the COMB2006 length-of-day series from which atmospheric and non-tidal oceanic effects have been removed using the NCEP/NCAR reanalysis AAM and the kf066a2 OAM series, respectively. Compared with the spectrum shown in Fig. 1a, the background power has been greatly reduced,
allowing peaks that were just above the background in Fig. 1a to be much more prominent in Fig. 1b. 3. Solid body tides 3.1. Elastic Yoder et al. (1981) derived a model for the effect of long-period body tides on the Earth’s rate of rotation by assuming that the crust and mantle of the Earth is elastic, that the fluid core is decoupled from the mantle, and that the ocean tides are in equilibrium. More recently, Defraigne and Smits (1999) derived a model for the effect of long-period elastic body tides on the Earth’s rate of rotation using a more accurate Earth model, PREM (Dziewonski and Anderson, 1981), and a more recent tide generating potential, RATGP95 (Roosbeek, 1996), than those used by Yoder et al. (1981). But unfortunately, Defraigne and Smits (1999, Table 1) tabulate their results for only 11 long-period tidal lines. Since Yoder et al. (1981) give results for 62 tidal lines, their more complete, but possibly less accurate, model has been used here to remove the effects of long-period elastic body tides from the length-of-day observations. Fig. 1c shows the spectrum of length-of-day observations from which long-period elastic body tidal effects have been removed using the model of Yoder et al. (1981), atmospheric effects have been removed using the NCEP/NCAR reanalysis AAM series, and
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Fig. 3. Power spectral density (psd) estimates in decibels (db) computed by the multitaper method from time series of length-of-day observations spanning 2 January 1993 to 10 September 2006 of the: (a) residual length-of-day observations formed by removing the ocean tide model of Weis (2006) in addition to removing atmospheric, non-tidal oceanic, and elastic and inelastic body tidal effects; and (c) residual length-of-day series formed by removing the Benjamin et al. (BWRED, 2006) ocean tide model instead of the Weis (2006) model. The magnitude of the squared coherence between the sum of the AAM and OAM series and the residual length-of-day observations formed by removing just the elastic and inelastic body tide model and the Weis (2006) ocean tide model is shown in (b), and in (d) it is shown when the ocean tide model of Benjamin et al. (2006) is removed instead of the Weis (2006) model. Vertical dashed lines indicate the frequencies of the monthly (13 cpy), fortnightly (27 cpy), termensual (40 cpy), and 7-day (51 cpy) tidal terms. Horizontal dashed lines in (b) and (d) indicate the 95% and 99% confidence levels of the magnitude of the squared coherence.
non-tidal oceanic effects have been removed using the kf066a2 OAM series. The most prominent peak at intraseasonal frequencies in this residual spectrum is at the fortnightly tidal frequency (27 cycles per year, cpy). A much smaller peak is evident at the termensual tidal frequency (40 cpy). Peaks at the level of the background are also possibly evident at the monthly (13 cpy) and 7-day (51 cpy) tidal frequencies. Fig. 1d shows the magnitude of the squared coherence between the sum of the AAM and OAM series and the length-of-day observations from which just the effects of the elastic body tides have been removed. Note the dramatic drop in coherence at the fortnightly tidal frequency. Any acceptable model for the dynamic effect of the fortnightly ocean tide on the length-of-day must both reduce the spectral peak evident in Fig. 1c to the level of the background power and must increase the coherence evident in Fig. 1d. But before evaluating different ocean tide models, the effects of mantle inelasticity on the body tides are first examined. 3.2. Inelastic The effects of mantle inelasticity on tidal length-of-day variations are only a few percent of the effects of mantle elasticity at the monthly and fortnightly tidal frequencies. But dissipation associated with mantle inelasticity also causes the deformational and hence rotational response of the Earth to lag behind the forcing tidal potential. As a result, not only does mantle inelasticity modify
the in-phase rotational response of the Earth to the tidal potential, but out-of-phase terms are introduced as well. While Defraigne and Smits (1999) modeled the effects of mantle inelasticity on just the in-phase part of the Earth’s rotational response, Wahr and Bergen (1986) modeled its effects on both the in-phase and out-of-phase parts. Fig. 2a and b shows the results of additionally removing just the in-phase part of the Wahr and Bergen (1986) inelastic body tide model from length-of-day observations that have also had elastic body tidal, atmospheric, and non-tidal oceanic effects removed. By carefully comparing these figures with Fig. 1c and d it can be seen that the spectral peaks at the monthly and fortnightly tidal frequencies are slightly smaller when the in-phase part of the inelastic body tide is removed. Similar results are obtained when the inphase part of the Defraigne and Smits (1999) inelastic body tide model is removed from the residual length-of-day observations (not shown). Fig. 2c and d shows the results of removing both the in-phase and out-of-phase parts of the Wahr and Bergen (1986) inelastic body tide model from the residual length-of-day observations. By carefully comparing these figures with Fig. 2a and b it can be seen that the spectral peaks at the monthly and fortnightly tidal frequencies are reduced even further when the out-of-phase part is additionally removed from the length-of-day observations. The tidal peaks at the termensual (40 cpy) and 7-day (51 cpy) tidal frequencies are not affected when the effects of the inelastic body tides are removed from the residual length-of-day observations because neither the
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Fig. 4. Power spectral density (psd) estimates in decibels (db) computed by the multitaper method from time series of length-of-day observations spanning 2 January 1993 to 10 September 2006 of the: (a) residual length-of-day observations formed by removing the ocean tide model of Kantha et al. (KSD, 1998) in addition to removing atmospheric, non-tidal oceanic, and elastic and inelastic body tidal effects; and (c) residual length-of-day series formed by removing the model recommended by the IERS Conventions (McCarthy and Petit, 2004, Table 8.1) in addition to removing atmospheric and non-tidal oceanic effects. The magnitude of the squared coherence between the sum of the AAM and OAM series and the residual length-of-day observations formed by removing just the elastic and inelastic body tide model and the Kantha et al. (1998) ocean tide model is shown in (b), and in (d) it is shown when the IERS model is removed from the observed length-of-day variations. Vertical dashed lines indicate the frequencies of the monthly (13 cpy), fortnightly (27 cpy), termensual (40 cpy), and 7-day (51 cpy) tidal terms. Horizontal dashed lines in (b) and (d) indicate the 95% and 99% confidence levels of the magnitude of the squared coherence.
Wahr and Bergen (1986) nor the Defraigne and Smits (1999) models include terms at these tidal frequencies. A large peak at the fortnightly tidal frequency in the length-ofday spectrum is still evident after removing elastic and inelastic body tidal, atmospheric, and non-tidal oceanic effects from the observations (Fig. 2c). The ability of selected ocean tide models to account for this presumably ocean tidal signal in residual lengthof-day observations is now examined. 4. Ocean tides Weis (2006) recently developed a high-resolution global purely hydrodynamic ocean tide model and used this model to predict the effects of 12 long-period ocean tidal constituents on the Earth’s rate of rotation. Fig. 3a and b shows the results of removing the Weis (2006) long-period ocean tide model from length-of-day observations that have also had elastic and inelastic body tidal, atmospheric, and non-tidal oceanic effects removed. By comparing these figures with Fig. 2c and d it can be seen that the Weis (2006) model does a good job of removing the tidal signal at the monthly (13 cpy), termensual (40 cpy), and 7-day (51 cpy) tidal frequencies but is unable to completely account for the observed power at the fortnightly tidal frequency. Since the coherence with the sum of AAM and OAM is greatly improved at the fortnightly tidal frequency when the Weis (2006) model is removed from the residual lengthof-day observations (compare Figs. 2d and 3b), the error in the Weis (2006) model at the fortnightly tidal frequency is probably mostly with its amplitude, not its phase.
Benjamin et al. (2006) used the ocean tide modeling system of Egbert and Erofeeva (2002) to test a number of different ocean tide models, both unconstrained and constrained by altimetric sea surface height measurements, for their efficacy in removing fortnightly and monthly ocean tidal effects from the Earth rotation observations that they used in their study of mantle inelasticity. Since all of the models that they tested gave consistent results for the fortnightly and monthly tidal effects on Earth rotation, they chose to use the solution from an unconstrained ocean tide model. Fig. 3c and d shows the results of removing the fortnightly and monthly ocean tide model preferred by Benjamin et al. (2006) from length-of-day observations that have also had elastic and inelastic body tidal, atmospheric, and non-tidal oceanic effects removed. As can be seen, the Benjamin et al. (2006) model does an excellent job of accounting for the observed power at the fortnightly and monthly tidal frequencies (Fig. 3c). It also does a reasonably good job of increasing the coherence with AAM and OAM (Fig. 3d). Of course, the tidal peaks at the termensual (40 cpy) and 7-day (51 cpy) tidal frequencies are not affected when the Benjamin et al. (2006) ocean tide model is removed from the residual lengthof-day observations because this model does not include terms at these frequencies. Kantha et al. (1998) combined an empirical ocean tide model based on Topex/Poseidon altimetric sea surface height measurements with a hydrodynamical tide model to predict the effect of the fortnightly and monthly ocean tides on the Earth’s rotation. Fig. 4a and b shows the results of removing this constrained ocean tide model of Kantha et al. (1998) from length-of-day observations
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that have also had elastic and inelastic body tidal, atmospheric, and non-tidal oceanic effects removed. As can be seen, the Kantha et al. (1998) model also does an excellent job of accounting for the observed power at the fortnightly and monthly tidal frequencies (Fig. 4a). And it does an even better job than the Benjamin et al. (2006) model of increasing the coherence with AAM and OAM (compare Figs. 3d and 4b). Again, the tidal peaks at the termensual (40 cpy) and 7-day (51 cpy) tidal frequencies are not affected when the Kantha et al. (1998) ocean tide model is removed from the residual length-of-day observations because this model does not include terms at these frequencies. Of the three ocean tide models evaluated here, the dataconstrained model of Kantha et al. (1998) does the best job of accounting for the observed fortnightly tidal signal evident in length-of-day observations from which elastic and inelastic body tidal, atmospheric, and non-tidal oceanic effects have been removed. The reason that this model does the best job is probably because it is the only model studied here that is constrained by data. While the unconstrained ocean tide model of Benjamin et al. (2006) does just as well as the constrained model of Kantha et al. (1998) in reducing the fortnightly spectral peak to the level of the background power (compare Figs. 3c and 4a), the coherence with AAM and OAM is much greater when the Kantha et al. (1998) model is removed from the length-of-day observations than it is when the Benjamin et al. (2006) model is removed (compare Figs. 3d and 4b). While the unconstrained model of Weis (2006) does not do as well as either the Kantha et al. (1998) or the Benjamin et al. (2006) models in accounting for the fortnightly ocean tidal signal in length-ofday observations (compare Fig. 3a with either Fig. 3c or Fig. 4a), it does do a reasonably good job of accounting for the termensual (40 cpy) and 7-day (51 cpy) tidal signals. Given that the sum of the Yoder et al. (1981) elastic body, Wahr and Bergen (1986) inelastic body, and Kantha et al. (1998) ocean tide models does an excellent job of accounting for the effects of the body and ocean tides on length-of-day, especially at the fortnightly tidal frequency, how does it compare to the model recommended by the IERS Conventions (McCarthy and Petit, 2004, Table 8.1), a model based on that of Defraigne and Smits (1999)? 5. IERS model Models recommended by the IERS for the removal of solid body and ocean tidal effects from the Earth’s rate of rotation and lengthof-day are given in Chapter 8 of the IERS Conventions (McCarthy and Petit, 2004). Table 8.1 in that document provides recommended corrections for the long-period tidal effects, that is, for tidal variations having periods between 5 days and 18.6 years. Unfortunately, very little information is provided in Chapter 8 about the derivation of the values contained in Table 8.1, with the latest update to Chapter 8, available at
length-of-day observations (Fig. 4c) and there is a dramatic drop in coherence with AAM and OAM at the fortnightly tidal frequency (Fig. 4d). The sum of the Yoder et al. (1981) elastic body, Wahr and Bergen (1986) inelastic body, and Kantha et al. (1998) ocean tide models does a much better job of accounting for the effects of the body and ocean tides on length-of-day, especially at the fortnightly tidal frequency, than does the IERS model (compare Fig. 4c and d with Fig. 4a and b). 6. Discussion and summary In a companion paper, Gross (2008) examined the effect of longperiod ocean tides on polar motion excitation, evaluating a number of different ocean tide models for their ability to account for the fortnightly ocean tidal signal seen in these observations. He concluded that no existing ocean tide model is able to adequately account for the observed signal. So he derived a new empirical ocean tide model by fitting periodic terms at the monthly, fortnightly, and termensual tidal frequencies to polar motion excitation observations from which atmospheric and non-tidal oceanic effects had been removed. Unlike polar motion excitation, it has been shown here that a model currently exists that does adequately account for ocean tidal effects in length-of-day, namely, the model of Kantha et al. (1998). In fact, the sum of the Yoder et al. (1981) elastic body tide model, the Wahr and Bergen (1986) inelastic body tide model, and the Kantha et al. (1998) ocean tide model does an excellent job of accounting for solid body and ocean tidal effects in length-of-day. In particular, and unlike the model recommended by the IERS, it is able to completely account for the fortnightly tidal signal that is seen in length-of-day observations from which atmospheric and non-tidal oceanic effects have been removed. The older Yoder et al. (1981) model was used here to remove the effects of elastic body tides from length-of-day observations because Defraigne and Smits (1999) provided results for only 11 tidal terms in their paper. It would be worth repeating this study using the complete Defraigne and Smits (1999) elastic body tide model to see how it compares with the Yoder et al. (1981) model when used in conjunction with the Wahr and Bergen (1986) inelastic body tide model and the Kantha et al. (1998) ocean tide model to remove tidal effects from length-of-day observations. Acknowledgements The work described in this paper was performed at the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration. Support for this work was provided by the Earth Surface and Interior Focus Area of NASA’s Science Mission Directorate. The supercomputers used in this investigation were provided by funding from the JPL Office of the Chief Information Officer. References Benjamin, D., Wahr, J., Ray, R.D., Egbert, G.D., Desai, S.D., 2006. Constraints on mantle anelasticity from geodetic observations, and implications for the J2 anomaly. Geophys. J. Int. 165, 3–16. Chao, B.F., Ray, R.D., 1997. Oceanic tidal angular momentum and Earth’s rotation variations. Prog. Oceanogr. 40, 399–421. Chao, B.F., Ray, R.D., Gipson, J.M., Egbert, G.D., Ma, C., 1996. Diurnal/semidiurnal polar motion excited by oceanic tidal angular momentum. J. Geophys. Res. 101 (B9), 20151–20163. Defraigne, P., Smits, I., 1999. Length of day variations due to zonal tides for an inelastic earth in non-hydrostatic equilibrium. Geophys. J. Int. 139, 563–572. Desai, S.D., Wahr, J.M., 1999. Monthly and fortnightly tidal variations of the Earth’s rotation rate predicted by a TOPEX/POSEIDON empirical ocean tide model. Geophys. Res. Lett. 26 (8), 1035–1038. Dickman, S.R., 1993. Dynamic ocean-tide effects on Earth’s rotation. Geophys. J. Int. 112, 448–470.
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