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SUMMARY Observations of the ψ1 earth tide yield valuable insights into the earth’s free core nutation, especially if the effects of the ψ1 ocean tide can be removed. The ocean tide is extremely small, with amplitudes rarely more than a few millimetres, and developing an accurate model is challenging. Direct observations are inadequate to support a global model. The alternative—numerical simulation—must account for a multitude of possible effects. The ocean tide is forced by the gravitational tidal potential, by pressure loading from atmospheric tides, by seasonal modulation of the nearby K1 constituent, and possibly by non-linear interactions among several other constituents. Here we construct a model of the ψ1 ocean tide which accounts for (or attempts to bound) each of these effects. The radiational component (from atmospheric pressure loading), although relatively small, is complicated by the presence of non-tidal atmospheric variability in the diurnal band. The ocean’s response is dynamic, but there is also high-wavenumber pressure forcing with a near-isostatic response. A general circulation model, forced by both winds and the tidal potential, suggests that annual variability in K1 leads to pronounced ψ1 amplitudes in some marginal seas, especially in the western Pacific.

New determinations of ocean tides are extracted from high-rate Global Positioning System (GPS) solutions at nine stations sitting on the Ross Ice Shelf. Five are multi-year time series. Three older time series are only 2–3 weeks long. These are not ideal, but they are still useful because they provide the only in situ tide observations in that sector of the ice shelf. The long tide-gauge observations from Scott Base and Cape Roberts are also reanalysed. They allow determination of some previously neglected tidal phenomena in this region, such as third-degree tides, and they provide context for analysis of the shorter datasets. The semidiurnal tides are small at all sites, yet M2 undergoes a clear seasonal cycle, which was first noted by Sir George Darwin while studying measurements from the Discovery expedition. Darwin saw a much larger modulation than we observe, and we consider possible explanations - instrumental or climatic - for this difference.

The Samoan islands are an archipelago hosting a quarter million people mostly residing inthree major islands, Savai'i and Upolu (Samoa), and Tutuila (American Samoa). The islands haveexperienced sea level rise by 2-3 mm/year during the last half century. The rate, however, has dramaticallyincreased following the Mw 8.1 SamoaTonga earthquake doublet (megathrust + normal faulting) inSeptember 2009. Since the earthquake, we found largescale gravity increase (0.5 Gal/year) around theislands and ongoing subsidence (8-16 mm/year) of the islands from our analysis of Gravity Recovery AndClimate Experiment gravity and GPS displacement data. The postseismic horizontal displacement is faster inSamoa, while the postseismic subsidence rate is considerably larger in American Samoa. The analysis oflocal tide gauge records and satellite altimeter data also identified that the relative sea level rise becomesfaster by 7-9 mm/year in American Samoa than Samoa. A simple viscoelastic model with a Maxwellviscosity of 2–3×10(exp 18) Pa s for the asthenosphere explained postseismic deformation at nearby GPS sites aswell as Gravity Recovery And Climate Experiment gravity change. It is found that the constructiveinterference of viscoelastic relaxation from both megathrust and normal faulting has intensified thepostseismic subsidence at American Samoa, causing ~5 times faster sea level rise than the global average.Our model indicates that this trend is likely to continue for decades and result in sea level rise of 30-40 cm,which is independent of and in addition to anticipated climaterelated sea level rise. It will worsen coastalflooding on the islands leading to regular nuisance flooding.

SUMMARY Seven years of GRACE intersatellite range-rate measurements are used to test the new ocean tide model FES2014 and to compare against similar results obtained with earlier models. These qualitative assessments show that FES2014 represents a marked improvement in accuracy over its earlier incarnation, FES2012, with especially notable improvements in the Arctic Ocean for constituents K1 and S2. Degradation appears to have occurred in two anomalous regions: the Ross Sea for the O1 constituent and the Weddell Sea for M2.

Ocean tides produce significant gravitational perturbations that affect near-Earth orbiting spacecraft. The gravitational potential induced by tidal mass redistribution is routinely modelled for global gravity analysis and orbit determination, although generally by assuming a spherical Earth and a uniform seawater density. The inadequacy of these simplifications is here addressed. We have developed an accurate yet efficient algorithm to compute the ocean tidal geopotential, allowing for Earth’s elliptical shape and variable seawater density. Using this new computation, we find that (1) the effect of ellipticity is several percent of the tide signal over mid to high latitude regions, which is comparable to elevation error in the state-of-the-art ocean tide models; (2) the effect of seawater density variations on the potential is as large as 2-3 cm in water-height equivalent, primarily in deep water where density increases 2-3% from compressibility. Our analysis of new GRACE Follow-On (GRACE-FO) laser ranging interferometer measurements reveals evident errors when ellipticity and density variations are ignored. When accounted for, the GRACE-FO residual tidal gravity perturbations are reduced by half, depending on the adopted tide model; only the remaining half likely represents actual model elevation error. The use of a spherical surface and a uniform seawater density is no longer tenable given the precision of gravity measurements from GRACE and GRACE-FO satellites.

Sea level anomaly (SLA) maps are routinely produced by objective analysis of data from the constellation of satellite altimeter missions in operation since 1992. Beginning in 2014, changes in the Data Unification and Altimeter Combination System (DUACS) used to create the SLA maps resulted in improved spatial resolution of mesoscale variability, but it also increased the levels of aliased tidal variability compared to the methodology employed prior to 2014. The present work investigates the magnitude and spatial distribution of these tidal signals, which are typically smaller than 1 cm in the open ocean but can reach tens of centimeters in the coastal ocean. In the open ocean, the signals are caused by a combination of phase-locked and phase-variable baroclinic tides. In the coastal ocean, the signals are a combination of aliased high-frequency nontidal variability and aliased variability caused by erroneous tidal corrections applied to the along-track altimetry prior to objective analysis. Several low-pass and bandpass filters are implemented to reduce the tidal signals in the mapped SLA, and independent tide gauge data are used to provide an objective assessment of the performance of the filters. The filter that attenuates both the small-scale (less than 200 km) and the high-frequency (period shorter than 108 days) components of SLA removes aliased baroclinic tidal variability and improves the accuracy of tidal analysis in the open ocean while also performing acceptably in the coastal ocean.

Since the beginning of the altimeter mission TOPEX/Poseidon (T/P), followed by Jason-1 and Jason-2 on similar orbits, and many other missions on different orbits (ERS, EnviSat, etc.), mean sea level (MSL) products became essential for the comprehension of global ocean circulation. Since early in the T/P mission, a suspicious signal, having a period of near 59 days and amplitude of roughly 5 mm, was apparent in the Global MSL record. Compared with the 4–5-mm amplitude of the annual signal, the 59-day signal has understandably attracted attention. Moreover, the same signal has been subsequently detected in Jason-1 and later in Jason-2 MSLs. In 2010, the Ocean Surface Topography Science Team (OSTST) concluded this signal as the aliasing of a higher frequency error inherited from the tide model correction: the semi-diurnal wave S2. The source of this error was mainly attributed to T/P measurements, which were assimilated in ocean tide models. When these models are used in the computation of T/P MSL, most of the error cancels. However, this error is communicated to Jason-1 and Jason-2 MSLs. In order to gather and publish the OSTST analyses on this matter, this paper first attempts to list the myriad possibilities for the puzzling 59-day error in MSL. Then, this paper goes deeper into the description of the main contributor to this list: the tide models error. Indeed, since 2010, considerable efforts have been undertaken within the ocean tide community in order to correct ocean tide S2-waves from this error, particularly in the Goddard Ocean Tide (GOT) and finite element solution (FES) latest versions. Comparing several GOT and FES versions and a pure hydrodynamic tide model, this paper assesses, quantifies, and describes a reduction of the MSL 59-day error thanks to the latest releases. These analyses also confirm that a large part of this error has its origins in the T/P mission and has contaminated ocean tide solutions and Jason-1 and Jason-2 MSLs. They also suggest that ocean tide is not the only possible vector. Jason-1 and Jason-2 MSLs contain additional 59-day error—though to a lesser extent—that may either come from the measurements themselves or from another vector.

Comparison of satellite altimetry against a high-quality network of tide gauges suggests that sea-surface heights from the TOPEX altimeter may be biased by ±5 mm, in an approximate piecewise linear, or U-shaped, drift. This has been previously reported in at least two other studies. The bias is probably caused by use of an internal calibration-mode range correction, included in the TOPEX “net instrument” correction, which is suspect owing to changes in the altimeter's point target response. Removal of this correction appears to mitigate most of the drift problem. In addition, a new time series based on retracking the TOPEX waveforms, again without the calibration-mode correction, also reduces the drift aside for a clear problem during the first 2 years. With revision, the TOPEX measurements, combined with successor Jason altimeter measurements, show global mean sea level rising fairly steadily throughout most of 24 year time period, with rates around 3 mm/yr, although higher over the last few years.

Internal waves at tidal frequencies, i.e., the internal tides, are a prominent source of variability in the ocean associated with significant vertical isopycnal displacements and currents. Because the isopycnal displacements are caused by ageostrophic dynamics, they contribute uncertainty to geostrophic transport inferred from vertical profiles in the ocean. Here it is demonstrated that a newly developed model of the main semidiurnal (M2) internal tide derived from satellite altimetry may be used to partially remove the tide from vertical profile data, as measured by the reduction of steric height variance inferred from the profiles. It is further demonstrated that the internal tide model can account for a component of the near-surface velocity as measured by drogued drifters. These comparisons represent a validation of the internal tide model using independent data and highlight its potential use in removing internal tide signals from in situ observations.

The presence of significant tidal mixing in the Indonesian seas is well established from both observations and numerical modeling. One indicator is a clear spring-neap cycle in satellite sea-surface temperature (SST) measurements, as first shown by Ffield and Gordon. Their early results are here updated with SST data of considerably higher spatial and temporal resolution. The largest fortnightly signals are found to be localized to relatively small straits, channels, and sills, while the deep basin of the Banda Sea displays little significant signal. A broader region of somewhat enhanced signal surrounds the Seram Sea. The high resolution of the modern SST data is especially critical for mapping the complex fortnightly signals that arise in, and especially south of, the major straits of the Lesser Sunda Island chain.

Near-global and continuous measurements from satellite altimetry have provided accurate estimates of global mean sea level in the past two decades. Extending these estimates further into the past is a challenge using the historical tide gauge records. Not only is sampling nonuniform in both space and time, but tide gauges are also affected by vertical land motion (VLM) that creates a relative sea level change not representative of ocean variability. To allow for comparisons to the satellite altimetry estimated global mean sea level (GMSL), typically the tide gauges are corrected using glacial isostatic adjustment (GIA) models. This approach, however, does not correct other sources of VLM that remain in the tide gauge record. Here we compare Global Positioning System (GPS) VLM estimates at the tide gauge locations to VLM estimates from GIA models, and assess the influence of non-GIA-related VLM on GMSL estimates. We find that the tide gauges, on average, are experiencing positive VLM (i.e., uplift) after removing the known effect of GIA, resulting in an increase of 0.2460.08 mm yr21 in GMSL trend estimates from 1900 to present when using GPS-based corrections. While this result is likely dependent on the subset of tide gauges used and the actual corrections used, it does suggest that non-GIA VLM plays a significant role in twentieth century estimates of GMSL. Given the relatively short GPS records used to obtain these VLM estimates, we also estimate the uncertainty in the GMSL trend that results from limited knowledge of non-GIA-related VLM.

Since the late 1990s the semi-diurnal tide at Churchill, on the western shore of Hudson Bay, has been decreasing in amplitude, with M(sub 2) amplitudes falling from approximately 154 cm in 1998 to 146 cm in 2012 and 142 cm in 2014. There has been a corresponding small increase in phase lag. Mean low water, decreasing throughout most of the twentieth century, has levelled off. Although the tidal changes could reflect merely a malfunctioning tide gauge, the fact that there are no other measurements in the region and the possibility that the tide is revealing important environmental changes calls for serious investigation. Satellite altimeter measurements of the tide in Hudson Bay are complicated by the seasonal ice cover; at most locations less than 40% of satellite passes return valid ocean heights and even those can be impacted by errors from sea ice. Because the combined TOPEX/Poseidon, Jason-1, and Jason-2 time series is more than 23 years long, it is now possible to obtain sufficient data at crossover locations near Churchill to search for tidal changes. The satellites sense no changes in M(sub 2) that are comparable to the changes seen at the Churchill gauge. The changes appear to be localized to the harbour, or to the Churchill River, or to the gauge itself.

Daily harmonics of the ionosphere’s total electron content (TEC) can be mapped using Global Navigation Satellite Systems (GNSS), as recently accomplished by Lean and colleagues. Here those harmonics are determined (over oceans only) by tidal analysis of ionospheric path delay from dual-frequency radar altimetry. Data from TOPEX, Jason-1, and Jason-2 altimeters are used, with focus on periods of medium to high solar activity. Diurnal-band signals are aliased by the altimeter sampling, but TOPEX-Jason orbits were designed to achieve acceptable tidal aliasing. Altimeter maps reveal where corresponding GNSS-based maps are in error, generally near the geomagnetic equator where high-wavenumber features are overly smoothed. When ocean tides are deduced from single-frequency altimetry, a task that relies on accurate GNSS-based ionospheric path corrections, any TEC errors leak directly into the tide solutions. TEC errors near and west of the Weddell Sea are likely problematic for the CryoSat-2 mission.

We review the characteristics of sea level variability at the coast focussing on how it differs from the variability in the nearby deep ocean. Sea level variability occurs on all timescales, with processes at higher frequencies tending to have a larger magnitude at the coast due to resonance and other dynamics. In the case of some processes, such as the tides, the presence of the coast and the shallow waters of the shelves results in the processes being considerably more complex than offshore. However, ‘coastal variability’ should not always be considered as ‘short spatial scale variability’ but can be the result of signals transmitted along the coast from 1000s km away. Fortunately, thanks to tide gauges being necessarily located at the coast, many aspects of coastal sea level variability can be claimed to be better understood than those in the deep ocean. Nevertheless, certain aspects of coastal variability remain under-researched, including how changes in some processes (e.g., wave setup, river runoff) may have contributed to the historical mean sea level records obtained from tide gauges which are now used routinely in large-scale climate research.

Mean sea level (MSL) is a prominent indicator of climatic change (Ablain et al., 2015; Cazenave et al., 2014; Leuliette and Willis, 2011), and is therefore of great scientific and societal interest. Since the beginning of the altimeter mission TOPEX/Poseidon, followed by Jason-1 and Jason-2 on similar orbits, and many other missions on different orbits (ERS, EnviSat, etc.), MSL products became essential to the comprehension of Global ocean circulation. Since early in the TOPEX/Poseidon mission (Nerem, 1995) a suspicious signal, having period near 59 days and amplitude of roughly 5 mm, was apparent in the GMSL record. Compared with the 4–5 mm amplitude of the annual signal (Minster et al., 1999), the suspicious 59-day signal has understandably attracted attention. Moreover, the same signal has been subsequently detected in Jason-1 and later Jason-2 MSLs. In 2010, it was the subject of a dedicated session at the Ocean Surface Topography Science Team (OSTST) meeting in Lisbon. The conclusions were this signal is the aliasing of a higher frequency error inherited from the tide model correction: the semi-diurnal wave S2. The source of this error was mainly attributed to TOPEX measurements which are assimilated in ocean tide models. When these models are used in the computation of TOPEX/Poseidon MSL, most of the error cancels. However, this error is communicated to Jason-1 and Jason-2 MSLs. Since 2010, considerable efforts have been undertaken within the ocean tide community in order to correct ocean tide S2-waves from this error, particularly in the Goddard Ocean Tide (GOT) and Finite Element Solution (FES) latest versions. The present paper aims at assessing, quantifying and characterizing the reduction of the 58.77-day error thanks to the latest releases.

Like most fields of science over the past century or two, the subject of oceanic tides appears to have progressed through a few major phases that, in hindsight at least, can be readily identified. When observing tides with a satellite altimeter, the problem of aliasing can scarcely be avoided. There is an underlying pattern to the tidal aliasing for any satellite, and to understand this it is helpful to return to basic aliasing concepts. Some tidal aliasing problems, especially those involving mapping the broad-scale barotropic tides, can be mitigated somewhat by combining the additional phase observations from neighboring tracks or from ascending and descending tracks. One area where great progress has been made since the Le Provost review concerns the energetics and dissipation of the barotropic tide. Internal or baroclinic tides are internal gravity waves with tidal periodicity. They are generated by the interaction of barotropic tides with bottom topography.

The astronomer Manuel Johnson, a future President of the Royal Astronomical Society, recorded the ocean tides with his own instrument at St. Helena in 1826–1827, while waiting for an observatory to be built. It is an important record in the history of tidal science, as the only previous measurements at St. Helena had been those made by Nevil Maskelyne in 1761, and there were to be no other systematic measurements until the late 20th century. Johnson's tide gauge, of a curious but unique design, recorded efficiently the height of every tidal high and low water for at least 13 months, in spite of requiring frequent re-setting. These heights compare very reasonably with a modern tidal synthesis based on present-day tide gauge measurements from the same site. Johnson's method of timing is unknown, but his calculations of lunar phases suggest that his tidal measurements were recorded in Local Apparent Time. Unfortunately, the recorded times are found to be seriously and variably lagged by many minutes. Johnson's data have never been fully published, but his manuscripts have been safely archived and are available for inspection at Cambridge University. His data have been converted to computer files as part of this study for the benefit of future researchers.

Tidal geocenter motion has been previously ignored when developing ocean tide models from satellite altimetry. Accounting for tidal geocenter motion is necessary because the best orbit determinations for altimetric satellites position sea-surface heights relative to the center of mass of the total Earth system, including the ocean tides. But the ocean tides are presumed relative to the Earth's crust and thus are effectively relative to the center of figure. By accounting for this effect, we find improved agreement between an altimeter-based ocean tide model and bottom pressure recorder observations. The variance of differences between these two observations is reduced by 31% and 43% for the two tidal constituents with the largest contributions to geocenter variations, O1 and K1, respectively. With this accommodation the predicted contribution from altimeter-based ocean tide models to geocenter variations is amplified by 15–22%, providing improved agreement with observations, especially for the K1 component.

A new model of long-period tidal variations in length of day is developed. The model comprises 80 spectral lines with periods between 18.6 years and 4.7 days, and it consistently includes effects of mantle anelasticity and dynamic ocean tides for all lines. The anelastic properties follow Wahr and Bergen; experimental confirmation for their results now exists at the fortnightly period, but there remains uncertainty when extrapolating to the longest periods. The ocean modeling builds on recent work with the fortnightly constituent, which suggests that oceanic tidal angular momentum can be reliably predicted at these periods without data assimilation. This is a critical property when modeling most long-period tides, for which little observational data exist. Dynamic ocean effects are quite pronounced at shortest periods as out-of-phase rotation components become nearly as large as in-phase components. The model is tested against a 20 year time series of space geodetic measurements of length of day. The current international standard model is shown to leave significant residual tidal energy, and the new model is found to mostly eliminate that energy, with especially large variance reduction for constituents Sa, Ssa, Mf, and Mt.

A new set of pelagic tide determinations is constructed from seafloor pressure measurements obtained at 151 sites in the deep ocean. To maximize precision of estimated tides, only stations with long time series are used; median time series length is 567 days. Geographical coverage is considerably improved by use of the international tsunami network, but coverage in the Indian Ocean and South Pacific is still weak. As a tool for assessing global ocean tide models, the data set is considerably more reliable than older data sets : the root-mean-square difference with a recent altimetric tide model is approximately 5 mm for the M2 constituent. Precision is sufficiently high to allow secondary effects in altimetric and bottom-pressure tide differences to be studied. The atmospheric tide in bottom pressure is clearly detected at the S1, S2, and T2 frequencies. The altimetric tide model is improved if satellite altimetry is corrected for crustal loading by the atmospheric tide. Models of the solid body tide can also be constrained. The free corenutation effect in the K1 Love number is easily detected, but the overall estimates are not as accurate as a recent determination with very long baseline interferometry.

For the last decade, it has been known that reflected GPS signals observed with specialized instruments could be used to measure sea level. In this letter, data from an existing geodetic-quality GPS site near Kachemak Bay, Alaska, are analyzed for a one-year time period. Daily sea-level variations are more than 7 m. Tidal coefficients have been estimated and compared with coefficients estimated from records from a traditional tide gauge at Seldovia Harbor, ~ 30 km away. The GPS and Seldovia estimates of M2 and S2 coefficients agree to better than 2%; much of this residual can be attributed to true differences in the tide over 30 km as it propagates up Kachemak Bay. For daily mean sea levels the agreement is 2.3 cm. Because a standard geodetic GPS receiver/antenna is used, this GPS instrument can measure long-term sea-level changes in a stable terrestrial reference frame.

Satellite altimeter measurements of sea surface height include a small contribution from vertical motion of the seafloor caused by crustal loading. Loading by ocean tides is routinely allowed for in altimeter data processing. Here, loading by nontidal fluids of the atmosphere, ocean, and terrestrial hydrosphere is examined. The crustal deformation can be computed from either geophysical models or from Gravity Recovery and Climate Experiment (GRACE) gravity inversions of mass variability. The loading corrections are found to be very small, rarely exceeding a few millimeters. Nonetheless, they form a significant correction to altimetric determinations of global mean sea level. The correction is most important at the annual cycle and should be accounted for when attempting to balance the global sea level budget.

Dickman argues that our model of the Mf ocean tide is unrealistic and that this invalidates conclusions regarding mantle anelasticity. His evidence is based on comparison with his own ocean modelling experiments and on his physical intuition regarding near-equilibrium tides. That evidence is unconvincing. Simple physical arguments alone are enough to rebut his main points. Furthermore, we test here his suggestion of increasing bottom-friction dissipation by a factor of 100 over our preferred solution, and we show that this results in unacceptable polar motion and poor agreement with independent tide-gauge data. In contrast, tests against independent data lend support to the realism of our Mf tide model and give confidence that our estimates of mantle anelasticity are reliable within stated error bounds.

SUMMARY This study of the fortnightly Mf tide comprises three main topics: (1) a new determination of the fortnightly component of polar motion and length of day (LOD) from a multidecade time-series of observed space-geodetic data; (2) the use of the polar motion determination as one constraint in the development of a hydrodynamic ocean model of the Mf tide and (3) the use of these results to place new constraints on mantle anelasticity at the Mf tidal period. Our model of the Mf ocean tide assimilates more than 14 years of altimeter data from the Topex/Poseidon and Jason-1 satellites. Because the Mf altimetric signal-to-noise ratio is very small, it is critical that altimeter data not be overweighted. The polar motion data, plus tidegauge data and independent altimeter data, give useful additional information, with only the polar motion putting constraints on tidal current velocities. The resulting ocean-tide model, plus the dominant elastic body tide, leaves a small residual in observed LOD caused by mantle anelasticity. The inferred effective tidal Q of the anelastic body tide is 90 and is in line with a ω α frequency dependence with α in the range 0.2–0.3.

One approach to reconstructing historical sea level from the relatively sparse tide-gauge network is to employ Empirical Orthogonal Functions (EOFs) as interpolatory spatial basis functions. The EOFs are determined from independent global data, generally sea-surface heights from either satellite altimetry or a numerical ocean model. The problem is revisited here for sea level since 1900. A new approach to handling the tide-gauge datum problem by direct solution offers possible advantages over the method of integrating sea-level differences, with the potential of eventually adjusting datums into the global terrestrial reference frame. The resulting time series of global mean sea levels appears fairly insensitive to the adopted set of EOFs. In contrast, charts of regional sea level anomalies and trends are very sensitive to the adopted set of EOFs, especially for the sparser network of gauges in the early 20th century. The reconstructions appear especially suspect before 1950 in the tropical Pacific. While this limits some applications of the sea-level reconstructions, the sensitivity does appear adequately captured by formal uncertainties. All our solutions show regional trends over the past five decades to be fairly uniform throughout the global ocean, in contrast to trends observed over the shorter altimeter era. Consistent with several previous estimates, the global sea-level rise since 1900 is 1.70 +/- 0.26 mm/yr. The global trend since 1995 exceeds 3 mm/yr which is consistent with altimeter measurements, but this large trend was possibly also reached between 1935 and 1950.

A method of analyzing GRACE satellite-to-satellite ranging data is presented which accentuates signals from diurnal ocean tides and dampens signals from long-period non-tidal phenomena. We form a time series of differences between two independent monthly mean gravity solutions, one set computed from range-rate data along strictly ascending arcs and the other set computed from data along descending arcs. The solar and lunisolar diurnal tides having alias periods longer than a few months, such as K 1, P 1, and S 1, present noticeable variations in the monthly ascending and descending ‘difference’ solutions, while the climate-related signals are largely cancelled. By computing tidal arguments evaluated along the actual GRACE orbits, we decompose and estimate residual tidal signals with respect to our adopted prior model GOT4.7. The adjustment in the tidal height is small yet significant, yielding maximum amplitudes of 4 cm mostly under the Antarctic ice shelves and ~1 cm in general at spatial scales of several hundred kilometer. Moreover, the results suggest there are possible 1-cm errors in the tide model even over oceans well-covered by decades of radar altimetry missions. Independent validation of such small adjustments covering wide areas, however, is difficult, particularly with limited point measurements such as tide gauge.

Seafloor pressure records, collected at 11 stations aligned along a single ground track of the Topex/Poseidon and Jason satellites, are analyzed for their tidal content. With very low background noise levels and approximately 27 months of high-quality records, tidal constituents can be estimated with unusually high precision. This includes many high-frequency lines up through the seventh-diurnal band. The station deployment provides a unique opportunity to compare with tides estimated from satellite altimetry, point by point along the satellite track, in a region of moderately high mesoscale variability. That variability can significantly corrupt altimeter-based tide estimates, even with 17 years of data. A method to improve the along-track altimeter estimates by correcting the data for non-tidal variability is found to yield much better agreement with the bottom-pressure data. The technique should prove useful in certain demanding applications, such as altimetric studies of internal tides.

A somewhat unorthodox method for determining vertical crustal motion at a tide-gauge location is to difference the sea level time series with an equivalent time series determined from satellite altimetry, To the extent that both instruments measure an identical ocean signal, the difference will be dominated by vertical land motion at the gauge. We revisit this technique by analyzing sea level signals at 28 tide gauges that are colocated with DORIS geodetic stations. Comparisons of altimeter-gauge vertical rates with DORIS rates yield a median difference of 1.8 mm/yr and a weighted root-mean-square difference of2.7 mm/yr. The latter suggests that our uncertainty estimates, which are primarily based on an assumed AR(l) noise process in all time series, underestimates the true errors. Several sources of additional error are discussed, including possible scale errors in the terrestrial reference frame to which altimeter-gauge rates are mostly insensitive, One of our stations, Male, Maldives, which has been the subject of some uninformed arguments about sea-level rise, is found to have almost no vertical motion, and thus is vulnerable to rising sea levels. Published by Elsevier Ltd. on behalf of COSPAR.

Non-linear tidal constituents, such as the overtide M 4 or the compound tide MS 4 , are generated by interaction in shallow seas of the much larger astronomically forced “primary” tidal constituents (e.g., M 2 , S 2 ). As such, errors in modeling these “secondary” shallow-water tides might be expected to be caused first of all by errors in modeling the primary constituents. Thus, in the context of data assimilation, observations of primary-constituent harmonic constants can indirectly constrain shallow-water constituents. Here we consider variational data assimilation for primary and secondary tidal constituents as a coupled problem, using a simple linearized perturbation theory for weak interactions of the dominant primary constituents. Variation of the resulting penalty functional leads to weakly non-linear Euler–Lagrange equations, which we show can be solved approximately with a simple two-stage scheme. In the first stage, data for the primary constituents are assimilated into the linear shallow water equations (SWE), and the resulting inverse solutions are used to compute the quadratic interactions in the non-linear SWE that constitute the forcing for the secondary constituents. In the second stage, data for the compound or overtide constituent are assimilated into the linear SWE, using a prior forced by the results of the first stage. We apply this scheme to assimilation of TOPEX/Poseidon and Jason altimetry data on the Northwest European Shelf, comparing results to a large set of shelf and coastal tide gauges. Prior solutions for M 4 , MS 4 and MN 4 computed using inverse solutions for M 2 , S 2 , and N 2 dramatically improve fits to validation tide gauges relative to unconstrained forward solutions. Further assimilation of along-track harmonic constants for these shallow-water constituents reduces RMS differences to below 1 cm on the shelf, approaching the accuracy of the validation tide gauge harmonic constants.

A possible connection between oceanic tides and climate variability arises from modulations in tidally induced vertical mixing. The idea is reexamined here with emphasis on near-decadal time scales. Occasional extreme tides caused by unusually favorable alignments of the moon and sun are unlikely to influence decadal climate, since these tides are of short duration and, in fact, are barely larger than the typical spring tide near lunar perigee. The argument by Keeling and Whorf in favor of extreme tides is further handicapped by an insufficiently precise catalog of extreme tides. A more plausible connection between tides and near-decadal climate is through “harmonic beating” of nearby tidal spectral lines. The 18.6-yr modulation of diurnal tides is the most likely to be detectable. Possible evidence for this is reviewed. Some of the most promising candidates rely on temperature data in the vicinity of the North Pacific Ocean where diurnal tides are large, but definitive detection is hindered by the shortness of the time series. Paleoclimate temperature data deduced from tree rings are suggestive, but one of the best examples shows a phase reversal, which is evidence against a tidal connection.

SUMMARY The manner in which errors in tide models affect monthly gravity solutions from GRACE satellite data is re-examined. We perform a complete simulation of GRACE gravity estimation, including allowance for accelerometer calibrations and satellite state-vector adjustment. Errors in monthly geoid estimates are for most regions of a order of 0.5 mm, or roughly 1 cm of equivalent water height when smoothed over radii of 600 km. In polar regions where ocean tide models are suspect, errors can grow significantly larger: over 2 mm in geoid height, although over very localized regions. Errors from diurnal tides tend to cancel, but they are not completely eliminated, inducing geoid errors of the order of 0.3 mm. The currently accepted GRACE total error budget is roughly an order of magnitude larger than the component arising from tide modelling errors.

The ERS-1, ERS-2 and Envisat series of satellite altimeters provide the only extensive datasets that could conceivably be usedto constrain ocean tide models in high latitudes. Their sun-synchronous sampling, however, severely limits theobservations of solar tides, especially the principal semidiurnal S2 constituent. The Munk–Cartwright response method is anatural choice when attempting to analyze sun-synchronous data. The present study examines various ways a response analysis might be implemented to extract tides from ERS data. Admittances expressed as simple linear or constant functions of frequency cansometimes improve estimates over standard parameterizations, especially if done in conjunction with a reasonably accurate priorsolution. Some form of regularization, such as ridge regression, is also shown to improve the estimates. The approach provesbeneficial in a test for the southern Indian Ocean tides. It offers some promise for regions otherwise void of usefulobservations.

Overtides and compound tides are generated by nonlinear mechanisms operative primarily in shallow waters. Their presence complicates tidal analysis owing to the multitude of new constituents and their possible frequency overlap with astronomical tides. The science of nonlinear tides was greatly advanced by the pioneering researches of Christian Le Provost who employed analytical theory, physical modeling, and numerical modeling in many extensive studies, especially of the tides of the English Channel. Le Provost’s complementary work with satellite altimetry motivates our attempts to merge these two interests. After a brief review, we describe initial steps toward the assimilation of altimetry into models of nonlinear tides via generalized inverse methods. A series of barotropic inverse solutions is computed for the M\(_4\) tide over the northwest European Shelf. Future applications of altimetry to regions with fewer in situ measurements will require improved understanding of error covariance models because these control the tradeoffs between fitting hydrodynamics and data, a delicate issue in coastal regions. While M\(_4\) can now be robustly determined along the Topex/Poseidon satellite ground tracks, many other compound tides face serious aliasing problems.

SUMMARY We use geodetic observations of the Earth to constrain anelasticity in the Earth’s mantle at periods between 12 hr and 18.6 yr. The observations include satellite laser ranging (SLR) measurements of 12 hr and 18.6 yr tides in the J 2 component of the gravity field; spacebased observations of tidal variations in the Earth’s rotation rate; and optical and space-based measurements of the Chandler Wobble period and damping. These geophysical signals are mostly sensitive to the lower mantle. The results suggest the dissipative process could consist of a single absorption band that extends across seismic periods out at least as far as ∼20 yr. The results also require values of the anelastic parameter Q that are smaller than those required by seismic observations. We interpret this as evidence that Q in the lower mantle is frequency dependent. The frequency dependence suggested by the geodetic observations is reasonably consistent with laboratory measurements, though those measurements have only been done on rocks at upper mantle conditions. After fitting and removing the 18.6 yr tide from the SLR J 2 results, we find that the 1998–2002 anomaly present in the original J 2 observations is no longer a singular anomaly in the J 2 residuals, but becomes one of a series of maxima in a quasi-decadal oscillation.

Analyses of long time series of hourly tide-gauge data at four stations in the Gulf of Maine reveal that the amplitude of the M2 tide underwent a nearly linear secular increase throughout most of the twentieth century. In the early 1980s, however, the amplitude of M2 abruptly dropped. Sea level changes alone appear inadequate to explain either the long-term trend or the recent trend discontinuity. Tidal models that account for Holocene sea level rise do predict an amplification of M2, but much smaller than the currently observed trends. Nor do recent annual mean sea levels correlate with the recent trend discontinuity. Some unknown fraction of the open Atlantic may be similarly affected, since the M2 discontinuity, but not the long-term secular increase in the tide, is evident also at Halifax.

The gravity loading effects induced by the diurnal (S 1 ) air tide are investigated. We compare worldwide superconducting gravimeter records belonging to the GGP (Global Geodynamics Project) network with loading predictions computed from recent air and oceanic tidal models. Although discrepancies between the observations and models are large, the global tidal models allow a significant reduction of gravity residuals compared with the classical and empirical local pressure correction as well as with the loading computed with surface pressure provided by meteorological centers under the assumption of an inverted barometer response of the ocean to pressure forcing.

The small terdiurnal pressure oscillation S3(p) is determined over the conterminous United States by analyzing long time series of hourly barometer data from 180 stations. Spectral analysis of these time series reveals that the terdiurnal band is dominated by three or four spectral peaks, separated in frequency by 1 cpy. The central peak at 3 cpd is invariably smaller than the two immediate side peaks, indicative of extraordinarily strong seasonal variations in the tide. The largest terdiurnal tide occurs over the south-central United States in winter where amplitudes exceed 300 μbar. Summertime amplitudes are roughly one-half as large. Summer and winter tides are almost completely out of phase, with rapid 180° shifts occurring in the equinox seasons when amplitudes are very small.

International audience; We have analyzed data from nine superconducting gravimeters in Western Europe and detected the contribution of non-linear ocean tides in the quarter-diurnal and sixth-diurnal frequency bands. Despite their small amplitudes, a few nanogals (10-11 m s-2), we have compared the tidal loading observations with the loading effects inferred from several non-linear ocean tide models over the North-West European continental shelf and found that they are coherent with the recent models. We also confirmed our results with tide gauge data and Topex/Poseidon altimetric data. We have also shown that the differences between the ocean tidal models are significantly larger than the accuracy of recent superconducting gravimeters. Therefore, we suggest that gravity measurements can be used as an independent validation tool for non-linear ocean tidal models, complementary to tide gauge and bottom pressure records.

The small S1 ocean tide is caused primarily by diurnal atmospheric pressure loading. Its excitation is therefore unlike any other diurnal tide; in particular, pressure loading is maximum near the equator where the diurnal gravitational potential is zero. The global character of the S 1 tide is here determined by numerical modeling and by analysis of Ocean Topography Experiment (TOPEX)/Poseidon satellite altimeter data. The two approaches yield reasonably consistent results. Amplitudes exceeding 1 cm in several regions are further confirmed by comparison with coastal tide gauges. Notwithstanding their excitation differences, S 1 and other diurnal tides are found to share several common features, such as relatively large amplitudes in the Arabian Sea, the Labrador Sea, the Sea of Okhotsk, and the Gulf of Alaska. The most noticeable difference is the lack of an S 1 Antarctic Kelvin wave. These similarities and differences can be explained in terms of the coherences between neardiurnal oceanic normal modes and the underlying tidal forcings. Whereas gravitational diurnal tidal forces excite primarily a 28-h Antarctic‐Pacific mode, the S 1 air tide excites several other near-diurnal modes, none of which has large amplitudes near Antarctica.

The solar diurnal and semidiurnal tidal oscillations in surface pressure are extracted from the operational analysis product of the European Centre for Medium Range Weather Forecasting (ECMWF). For the semidiurnal tide this involves a special temporal interpolation, following Van den Dool et al. (1997). The resulting tides are compared with a "ground truth" tide data set, a compilation of well-determined tide estimates deduced from many long time series of station barometer measurements. These comparisons show that the ECMWF (analysis) tides are significantly more accurate than the tides deduced from two other widely available reanalysis products. Spectral analysis of ECMWF pressure series shows that the tides consist of sharp central peaks with modulating sidelines at integer multiples of 1 cycle/year, superimposed on a broad cusp of stochastic energy. The integrated energy in the cusp dominates that of the side-lines. This complicates the development of a simple empirical model that can characterize the full temporal variability of the tides.Key words. Meteorology and atmospheric dynamics (waves and tides)

New empirical estimates of the long-period fortnightly (Mf ) tide obtained from TOPEX/Poseidon (T/P) altimeter data confirm significant basin-scale deviations from equilibrium. Elevations in the low-latitude Pacific have reduced amplitude and lag those in the Atlantic by 308 or more. These interbasin amplitude and phase variations are robust features that are reproduced by numerical solutions of the shallow-water equations, even for a constant-depth ocean with schematic interconnected rectangular basins. A simplified analytical model for cooscillating connected basins also reproduces the principal features observed in the empirical solutions. This simple model is largely kinematic. Zonally averaged elevations within a simple closed basin would be nearly in equilibrium with the gravitational potential, except for a constant offset required to conserve mass. With connected basins these offsets are mostly eliminated by interbasin mass flux. Because of rotation, this flux occurs mostly in a narrow boundary layer across the mouth and at the western edge of each basin, and geostrophic balance in this zone supports small residual offsets (and phase shifts) between basins. The simple model predicts that this effect should decrease roughly linearly with frequency, a result that is confirmed by numerical modeling and empirical T/P estimates of the monthly (Mm) tidal constituent. This model also explains some aspects of the anomalous nonisostatic response of the ocean to atmospheric pressure forcing at periods of around 5 days.

Third-degree diurnal tides are estimated from long time series of sea level measurements at three North Atlantic tide gauges. Although their amplitudes are only a few mm or less, their admittances are far larger than those of second-degree diurnal tides, just as Cartwright discovered for the M(sub 1) constituent. The tides are evidently resonantly enhanced owing to high spatial correlation between the third-degree spherical harmonic of the tidal potential and a near-diurnal oceanic normal mode that is most pronounced in the North Atlantic. By estimating the ocean tidal response across the diurnal band (5 tidal constituents plus nodal modulations), the period and Q of this mode and one nearby mode are estimated.

Most of the tidal energy dissipation in the ocean occurs in shallow seas, as has long been recognized. However, recent work has suggested that a significant fraction of the dissipation, perhaps 1 TW or more, occurs in the deep ocean. This paper builds further evidence for that conclusion. More than 6 years of data from the TOPEX/Poseidon satellite altimeter are used to map the tidal dissipation rate throughout the world ocean. The dissipation rate is estimated as a balance between the rate of working by tidal forces and the energy flux divergence, computed using currents derived by least squares fitting of the altimeter data and the shallow water equations. Such calculations require dynamical assumptions, in particular about the nature of dissipation. To assess sensitivity of dissipation estimates to input assumptions, a large suite of tidal inversions based on a wide range of drag parameterizations and employing both real and synthetic altimeter data are compared. These experiments and Monte Carlo error fields from a generalized inverse model are used to establish error uncertainties for the dissipation estimates. Owing to the tight constraints on tidal elevation fields provided by the altimeter, area integrals of the energy balance are remarkably insensitive to required dynamical assumptions. Tidal energy dissipation is estimated for all major shallow seas (excluding individual polar seas) and compared with previous model and data-based estimates. Dissipation in the open ocean is significantly enhanced around major bathymetric features, in a manner consistent with simple theories for the generation of baroclinic tides.

Energy fluxes for first-mode M(sub 2) internal tides are deduced throughout the central North Pacific Ocean from Topex/Poseidon satellite altimeter data. Temporally coherent internal tide signals in the altimetry, combined with climatological hydrographic data, determine the tidal displacements, pressures, and currents at depth, which yield power transmission rates. For a variety of reasons the deduced rates should be considered lower bounds. Internal tides were found to emanate from several large bathymetric structures, especially the Hawaiian Ridge, where the integrated flux amounts to about six gigawatts. Internal tides are generated at the Aleutian Trench near 172 deg west and propagate southwards nearly 2000 km.

The problem of deducing tidal current velocities from measured tidal elevations is considered. Emphasis is given to global scales for which accurate cotidal charts based on satellite altimetry are readily available. Barotropic tidal currents are deduced from a recent altimetric tide solution by solving the two-dimensional momentum equations and the continuity equation in a least squares fashion. This approach represents a middle ground between commonly used data assimilation schemes and simple local solutions of the momentum equations. The estimated velocities are compared with those of other available global models, with acoustic velocity measurements, and with current meter measurements. Agreement with acoustic measurements is especially good; RMS discrepancies are consistently less than 2 mm s−1. The method appears to be a useful tool for investigating various aspects of tidal dynamics.

The present rate of tidal dissipation in the Earth-Moon system is known to be anomalously high, in the sense that the implied age of the lunar orbit is only 1.5×109 years, though other evidence suggests an age closer to 4×109 years. To assess how long the anomalous dissipation has persisted, we use published estimates of lunar orbital configurations derived from (a) fine grained sediments containing tidal laminations and (b) numerical ocean models averaged over varying ocean geometries. The implied histories of the lunar semimajor axis are surprisingly consistent over the past 109 years. The ocean models imply, on average, reduced dissipation in the past because of a spatial mismatch between tidal forcing and oceanic normal modes of higher frequencies. Webb's ocean model suggests that the “anomalous” oceanic dissipation began about 109 years ago and has been increasing since then.

Observing high-wavenumber ocean phenomena with a satellite altimeter generally calls for “along-track” analyses of the data: measurements along a repeating satellite ground track are analyzed in a point-by-point fashion, as opposed to spatially averaging data over multiple tracks. The sea-level aliasing problems encountered in such analyses can be especially challenging. For TOPEX/POSEIDON, all signals with frequency greater than 18 cycles per year (cpy), including both tidal and subdiurnal signals, are folded into the 0–18 cpy band. Because the tidal bands are wider than 18 cpy, residual tidal cusp energy, plus any subdiurnal energy, is capable of corrupting any low-frequency signal of interest. The practical consequences of this are explored here by using real sea-level measurements from conventional tide gauges, for which the true oceanographic spectrum is known and to which a simulated “satellite-measured” spectrum, based on coarsely subsampled data, may be compared. At many locations the spectrum is sufficently red that interannual frequencies remain unaffected. Intra-annual frequencies, however, must be interpreted with greater caution, and even interannual frequencies can be corrupted if the spectrum is flat. The results also suggest that whenever tides must be estimated directly from the altimetry, response methods of analysis are preferable to harmonic methods, even in nonlinear regimes; this will remain so for the foreseeable future. We concentrate on three example tide gauges: two coastal stations on the Malay Peninsula where the closely aliased K1 and Ssa tides are strong and at Canton Island where trapped equatorial waves are aliased.