Abstract

Long-period tides (LPT) are studied using a stratified, primitive equation model on a global domain and in the presence of a fully developed, atmospherically forced ocean general circulation. The major LPT constituents, from termensual to nodal (18.6 yr) periods, are examined. Ocean circulation variability can overwhelm the longest tide signals and make inferring LPT from data difficult, but model results suggest that bottom pressure offers cleaner signal-to-noise ratios than sea level, particularly at low latitudes where atmospherically driven variability is substantially stronger at the surface than at the bottom. Most tides exhibit a significant large-scale dynamic response, with the tendency for weaker nonequilibrium signals in the Atlantic compared to the Pacific as seen in previous studies. However, across most tidal lines, the largest dynamic signals tend to occur in the Arctic and Nordic Seas and also in Hudson Bay. Bathymetry and coastal geometry contribute to the modeled nonequilibrium behavior. Baroclinic effects tend to increase with the tidal period. Apart from short spatial-scale modulations associated with topographic interactions, the excitation of various propagating baroclinic wave modes is clearly part of the modeled LPT, particularly at tropical latitudes, for fortnightly and longer-period tides.

1. Introduction

Aside from the energetic diurnal and semidiurnal ocean tides that are conspicuously present in all sea level records, the astronomical gravitational potential also gives rise to tides of much lower frequencies. Despite their considerably weaker amplitudes, these so-called long-period tides (LPT)—including Mt, Mf, Mm, Ssa, and Sa, at termensual, fortnightly, monthly, semiannual, and annual periods, respectively, and the nodal tide at 18.6 yr (Table 1)—are nevertheless of significant interest. In contrast to the atmospheric fields that drive the ocean general circulation, the LPT forcing is essentially perfectly known, albeit of a very narrowband character in frequency and wavenumber (Cartwright and Tayler 1971). Thus, observations of LPT can provide important insights into how the ocean responds to forcing and how it dissipates energy (e.g., Wunsch et al. 1997). Broader geophysical inference is possible from the study of LPT, as they can measurably affect parameters such as Earth’s axial rotation and polar motion (e.g., Kantha et al. 1998; Ray and Egbert 2012). A short summary of LPT is provided by Le Provost (2001).

Table 1.

Main long-period tidal constituents.

Main long-period tidal constituents.
Main long-period tidal constituents.

In the isostatic limit, the oceanic response to astronomical tidal forcing is sufficiently rapid and efficient to render any resulting pressure gradients negligible compared to the applied forcing, and the tides are said to be at equilibrium. More interesting solutions involve substantial pressure gradients and a dynamic response, which can be dominated by near resonances, as with the diurnal and semidiurnal tides. With the LPT, the quest to understand how much of a nonequilibrium response is expected has focused on Mf, the strongest LPT, with a couple of processes highlighted so far. The excitation of Rossby waves, including those of topographic origin, has been extensively discussed dating back to Wunsch (1967), later revisited by Carton (1983) and others. The influence of gravity wave mechanisms (Miller et al. 1993) and, most importantly, of incomplete interbasin adjustment involving mass fluxes mediated by geostrophic flows between basins (Egbert and Ray 2003) has also been discussed. Ponte (1997) discusses similar mechanisms in the dynamic response of the ocean to the 5-day Rossby–Haurwitz wave in atmospheric pressure.

The mentioned studies (and most others as well) have treated the LPT in a barotropic ocean with no interactions with the general circulation. Numerical simulations of Mf and Mm in baroclinic settings include the two-layer model study of Arbic et al. (2004), but the emphasis has been invariably on the discussion of short-period tides. In this work, rather than trying to simulate the LPT as realistically as possible, we focus on exploring potential nonequilibrium behaviors across the whole set of frequencies. In addition, our modeled ocean is vertically stratified and driven by the atmosphere, thus containing a realistic general circulation on which the LPT are superposed. This setup allows one to examine potential effects of baroclinicity on the LPT. In addition, one can set the LPT variability in the context of the atmospherically driven “noise,” an issue that is important when trying to separate the LPT from other variability in the records.

2. Modeling and analysis details

The main tool of analysis is the MIT general circulation model (MITgcm) described in Marshall et al. (1997). Configuration includes 1) a global grid of varying horizontal resolution (nominal 1° decreasing to ~⅓° at low latitudes and ~40 km in the Arctic) and 50 vertical levels (Forget et al. 2015), 2) treatment of sea ice (Losch et al. 2010), 3) representation of bottom topography using partial cells (Adcroft et al. 1997), and 4) forcing by bulk flux formulations using ERA-Interim surface atmospheric fields (Dee et al. 2011). Atmospheric pressure is not included in the forcing fields. Apart from subgrid-scale parameterizations of Gaspar et al. (1990), Gent and McWilliams (1990), and Redi (1982), quadratic bottom friction is also used. Forget et al. (2015) provide a full description of the MITgcm configuration in the context of optimization experiments done with a comprehensive set of observational constraints. Here, we focus on model experiments done with no optimization.

Our two basic experiments consist of 1) forcing only with atmospheric fluxes to obtain a simulation of the general circulation (denoted as GCE) and then 2) adding the LPT potential to examine the tides in the presence of a fully variable ocean (denoted as GC+TE). The LPT solutions can be approximated by subtracting GCE from GC+TE.1 Forcing with a tidal potential is equivalent to having atmospheric pressure loading, as discussed in detail by Ponte and Vinogradov (2007). The LPT forcing used here is based on a simple code provided by R. Ray (2006, personal communication), which includes the 15 largest LPT constituents listed in Table 1, as tabulated by Cartwright and Tayler (1971) and Cartwright and Edden (1973). All the major frequencies (Table 1) are included, and subsequent analyses emphasize Mt, Mf, Mm, Ssa, and Sa. In an aquaplanet, the equilibrium LPT has the simple form

 
formula

where θ is latitude, and is the equilibrium amplitude given in Table 1, which includes the reduction factor of 0.693 resulting from body tide effects.2 Effects of self-attraction and loading, which can effectively increase the amplitudes of forcing and response but may not be easy to parameterize accurately (Woodworth 2012), are not considered for simplicity.

Initial conditions are taken from one of the preliminary, unconstrained solutions discussed in Forget et al. (2015), and the model is run for 20 yr from 1 January 1992. All results discussed here are obtained by harmonic analysis carried out on the last 18.6 yr of output, which permits the resolution of all the relevant tidal lines in Table 1, including one full cycle of the nodal tide Ln and also the weaker nodal modulations in termensual and fortnightly bands (e.g., Egbert and Ray 2003) not treated here. For these analyses, sea level, bottom pressure, and other relevant diagnostics were archived as daily averages. All equilibrium solutions and respective dynamic deviations are calculated based on self-consistent tides that account for the conservation of total ocean volume. The absolute values of the phase are not important for our purposes, but for reference all values are relative to the starting date of the analyzed time series (1200 UTC 24 May 1993). Given the model’s Boussinesq formulation, effects of variability in global-mean sea level and bottom pressure are not considered; changes in the spatial mean bottom pressure resulting from the Boussinesq assumption are calculated and removed prior to analysis.

3. LPT and large-scale circulation

The extent to which LPT are in equilibrium or not has been a subject of much debate (Le Provost 2001). The difficulty in addressing the issue with observations lies mostly in the presence of the considerable noise that constitutes the variable ocean general circulation. The problem can be particularly acute at annual and semiannual periods but also affects analyses of Mf and other tides (e.g., Ponchaut et al. 2001). Here, because of our coarse model grid, we only address the effects of the atmospherically forced large-scale circulation on the LPT and do not treat possible impacts of small-scale eddies. Nevertheless, given expected large-scale structure of the LPT signals, in reality eddy noise might be less of a problem in separating out LPT signals if spatial smoothing is possible (e.g., in altimetric analyses).

Amplitudes of full sea level ζ at the Mf frequency, inferred from the experiment with both tidal and nontidal forcing, clearly show the zonally banded structure of the tidal forcing with a node around ±35° latitude and maxima in tropical and high latitudes (Fig. 1). Amplitudes are similar but not a perfect match to those expected under a pure equilibrium tide. Results indicate a relatively strong LPT contribution to variability at the Mf period, clearly discernible amid all other nontidal variability, and with hints of nonequilibrium behavior, such as the higher amplitudes in the tropical Atlantic compared to the tropical Pacific. Results at the Mm frequency (not shown) are quite similar in character. In contrast, at the Ssa period, no clear signature of respective LPT emerges in Fig. 1 and ζ variability seems to be dominated by the nontidal component, with amplitudes much larger than expected from a simple equilibrium Ssa tide in the tropics (maximum Ssa amplitude is ~13.5 mm at the pole). Similar conclusions can be drawn at the annual and nodal periods (not shown).

Fig. 1.

Amplitude (mm) of the full sea level response ζ at the (a) Mf and (c) Ssa frequencies and respective equilibrium tidal amplitudes for (b) Mf and (d) Ssa.

Fig. 1.

Amplitude (mm) of the full sea level response ζ at the (a) Mf and (c) Ssa frequencies and respective equilibrium tidal amplitudes for (b) Mf and (d) Ssa.

Results in Fig. 1 confirm the substantial noise associated with the large-scale circulation that can hamper attempts to infer LPT signals from sea level measurements, particularly for Ssa and Sa corresponding to periods of strong atmospheric forcing. In the model setting pursued here, one can readily separate LPT signals from the background continuum by differencing the GCE and GC+TE sea level fields. A quantitative measure of the circulation noise can then be obtained by examining the ratio of the tidal amplitudes to the amplitudes of the GCE fields at the respective tidal frequencies.

Values of such ratios for Mf, Mm, Ssa, and Sa (Fig. 2) can be treated as rough signal-to-noise ratios if one is trying to separate out the tidal signal from the variable circulation. For Mf and Mm, apart from latitudes near the nodal lines, ratios are typically of order 10 and larger. The strongest signals are expected in the tropics, where atmospherically driven variability is relatively weak compared to higher latitudes. For both Ssa and Sa, ratios are of order one and smaller, indicating the relatively strong circulation, particularly for the annual period. Regions with the largest signal-to-noise ratios include the Arctic (away from the Eurasian shelves) and to some extent parts of the Southern Ocean.

Fig. 2.

Logarithm of the ratio of tidal amplitude to the amplitude of variability of the general circulation calculated at the (a) Mf, (b) Mm, (c) Ssa, and (d) Sa periods. Note the different logarithmic scales.

Fig. 2.

Logarithm of the ratio of tidal amplitude to the amplitude of variability of the general circulation calculated at the (a) Mf, (b) Mm, (c) Ssa, and (d) Sa periods. Note the different logarithmic scales.

As seen in the model simulations, bottom pressure variability can be considerably weaker than that of the sea level, particularly at semiannual and annual periods (Ponte 1999). Given the availability of global bottom pressure fields derived from the Gravity Recovery and Climate Experiment (GRACE) satellite mission (e.g., Chambers and Bonin 2012), it is instructive to compute the signal-to-noise ratio based on bottom pressure (Fig. 3). Values are considerably higher than in Fig. 2. For Mf and Mm, using bottom pressure does not change the results at mid- and high latitudes, as expected from the barotropic dynamics that tend to dominate the circulation at the periods of interest. Higher signal-to-noise ratios are obtained in the tropics, however, where considerably more baroclinic activity can lead to larger sea level variability compared to bottom pressure. Similar results are found for Ssa and Sa periods, but improved signal-to-noise ratios at high latitudes are also observed, particularly for Sa. While these ratios are mostly >1 for Ssa, values <1 are still the norm for Sa.

Fig. 3.

Logarithm of the ratio of tidal amplitude to the amplitude of bottom pressure variability associated with the general circulation calculated at the (a) Mf, (b) Mm, (c) Ssa, and (d) Sa periods. Note the different logarithmic scales.

Fig. 3.

Logarithm of the ratio of tidal amplitude to the amplitude of bottom pressure variability associated with the general circulation calculated at the (a) Mf, (b) Mm, (c) Ssa, and (d) Sa periods. Note the different logarithmic scales.

4. Nonequilibrium behavior of LPT

For the purpose of examining in detail any potential departures from equilibrium behavior, rather than focusing on the full sea level ζ, it is instructive to analyze ζd = ζζeq, which represents dynamic pressure fields associated with any deviations from the equilibrium response ζeq (e.g., Ponte 1997). All fields described here are based on the difference between GC+TE and GCE experiments to isolate the LPT solutions. The focus is on constituents with the largest dynamic response (Mt, Mf, and Mm); in particular, Sa and nodal tides have submillimeter dynamic responses and are not discussed.

The amplitude and phase of ζd for Mt, Mf, and Mm, all display similar nonequilibrium characteristics (Fig. 4). In particular, there is a large-scale pattern of weaker (stronger) departures from equilibrium in the Atlantic (Pacific) Oceans, with ζd being approximately out of phase between the two basins but having almost the same phase within each basin. The amplitude and phase of ζd in the Indian Ocean are somewhat between those in the Pacific and Atlantic. Spatial patterns and amplitude values are in general agreement with previous works (Miller et al. 1993; Wunsch et al. 1997; Egbert and Ray 2003). In particular, typical amplitudes of a few millimeters in the tropical Pacific are consistent with observations (cf. Tables A3 and A4 in Miller et al. 1993). The dynamics responsible for the above-noted features have been extensively analyzed by Miller et al. (1993) and Egbert and Ray (2003) in the context of Mf and Mm and represent, in a heuristic sense, incomplete interbasin adjustment of the mass field in the presence of continental barriers and relatively constricted connections between basins through the Southern Ocean. Thus, we will focus below on a couple of other features apparent in Fig. 4 that have not been discussed as much in previous works.

Fig. 4.

Amplitude (mm) and phase (°) of dynamic response ζd for (top) Mt, (middle) Mf, and (bottom) Mm constituents.

Fig. 4.

Amplitude (mm) and phase (°) of dynamic response ζd for (top) Mt, (middle) Mf, and (bottom) Mm constituents.

a. Arctic/Nordic Seas response

The Arctic and Nordic Seas exhibit, by far, the largest deviations from equilibrium of all the ocean basins. Although the response can scale locally with the magnitude of the forcing, which has a maximum at the poles, other factors are likely at play. Notice that the dynamic response at the high southern latitudes is substantially weaker by comparison. In more detailed plots of amplitude and phase of ζd for the Arctic and adjacent Nordic Seas (Fig. 5), one sees that the enhanced amplitudes are sharply reduced at latitudes around Iceland and the Faroe Islands, with amplitudes within the Arctic and Nordic Seas more homogeneous. Phase values do not vary much across the latter domain. In particular, there is no evidence of phase propagation anywhere that would suggest wave processes being prominent in the dynamics. These characteristics are present across all three tides examined in Fig. 5 and also arise in the dynamic response of Ssa and Sa (not shown).

Fig. 5.

(top) Amplitude (mm) and (bottom) phase (°) of dynamic response ζd for (left to right) Mt, Mf, and Mm constituents for the Arctic and subpolar latitudes. The 50- and 200-m isobaths are plotted as dashed and solid contours, respectively.

Fig. 5.

(top) Amplitude (mm) and (bottom) phase (°) of dynamic response ζd for (left to right) Mt, Mf, and Mm constituents for the Arctic and subpolar latitudes. The 50- and 200-m isobaths are plotted as dashed and solid contours, respectively.

Given the relatively homogeneous nature of the response in Fig. 5, the relation between ζ, ζd, and ζeq for a point near the North Pole (Fig. 6) can be taken as representative of the general conditions across the Arctic and high northern latitudes. Amplitudes of ζ are typically smaller than ζeq, with ζ lagging ζeq by ~1–2 days, across all three tidal frequencies examined in Fig. 6. These lags are in agreement with the modeling results of Stepanov and Hughes (2006), who find that global barotropic adjustment relative to the equator takes only a fraction of a day in most basins but up to 2 days in the Arctic. The response gets closer to equilibrium as the period increases; the ratio of the amplitudes of ζd to those of ζeq gets smaller from Mt to Mf to Mm. The large-scale nonequilibrium behavior depicted in both Figs. 5 and 6 is consistent with an incomplete mass exchange with the rest of the ocean, following the dynamics discussed by Egbert and Ray (2003) and, in another context, by Ponte (1997). Similar homogeneous mass fluctuations in the Arctic/Nordic Seas can result from wind-driven dynamics, as reviewed recently by Fukumori et al. (2015).

Fig. 6.

Tidal cycles for ζ (blue), ζd (red), and ζeq (green) representing (top) Mt, (middle) Mf, and (bottom) Mm at a point in the central Arctic indicated by the asterisk in Fig. 5. Abscissa values are given in days.

Fig. 6.

Tidal cycles for ζ (blue), ζd (red), and ζeq (green) representing (top) Mt, (middle) Mf, and (bottom) Mm at a point in the central Arctic indicated by the asterisk in Fig. 5. Abscissa values are given in days.

Another useful measure of dynamic response can be obtained from examination of the velocity field. Maximum speeds U over the tidal cycle at a depth of 50 m (Fig. 7) are typically less than 0.2 mm s−1 over much of the basin. Other depth levels give very similar results. Enhanced speeds of up to 1 mm s−1 are found only across the entrance to the Nordic Seas. These tend to coincide with regions of sharp gradients in amplitude and variable phase for ζd (Fig. 5) and therefore relatively large surface pressure gradients and geostrophic currents.

Fig. 7.

Maximum speed (mm s−1) for the Mf constituent over the Arctic calculated at a depth of 50 m. Contour lines represent isolines of f/H, with gradients dominated by changes in bathymetry.

Fig. 7.

Maximum speed (mm s−1) for the Mf constituent over the Arctic calculated at a depth of 50 m. Contour lines represent isolines of f/H, with gradients dominated by changes in bathymetry.

Bathymetry plays a substantial role in shaping the dynamic response seen in Figs. 6 and 7. Flat bottom model experiments (not shown), with depths H of 1000 and 4000 m, lead to very different spatial characteristics and substantially weaker amplitudes for ζd. In particular, the contrast across the entrance to the Nordic Seas in Fig. 6 is completely absent for constant H. The Denmark Strait and the straits between Iceland and the Faroe and Shetland Islands are regions of shallow sill depths, which together with the island landmasses provide a partial boundary separating the Nordic Seas from the North Atlantic. Changes in H also lead to larger gradients in f/H (Fig. 7), which are dominated by changes in H over those in the Coriolis parameter f. These features tend to inhibit wave propagation across the straits, for both gravity and vorticity waves, and thus contribute to partially isolate the response in the Arctic and Nordic Seas from that in the North Atlantic and the rest of the World Ocean.

Embedded in the basin-scale characteristics of the dynamic response, there are also regional differences worth noting. In particular, the higher amplitudes in the East Siberia, Laptev, and Kara shelves coincide also with some of the largest phase differences across the basin. Apart from the expected coastal trapping of wave energy, the influence of bathymetry is also plausible in this case, given that the largest ζd amplitudes are mostly confined to the shallowest depths (Fig. 5). The bottom depth considerably affects the propagation speeds of waves that are likely to be involved in the mass field adjustment of the Arctic: for gravity waves, speed goes as , and for long Rossby waves, speed goes as H (Gill 1982). Thus, the adjustment is expected to be slower over the shelves than over the deep basin and can lead to differences in behavior observed in Fig. 5.

b. Hudson Bay response

In terms of shallow coastal and semienclosed regions, Hudson Bay stands out as the place of strongest dynamic signals in Fig. 4, with ζd amplitudes >20 mm for Mf and comparable in general to the Arctic signals. Similar behavior is seen for all tidal lines in Fig. 4. Amplitudes tend to be relatively homogeneous in Hudson Bay proper (but generally weaker in the northern Foxe basin and in the Hudson Strait). Phases are also similar inside the bay and over Hudson Strait, indicating a rise and fall of sea level more or less in unison. In fact, phases inside the bay are roughly the same as outside of the strait.

The relation between ζ, ζd, and ζeq in Hudson Bay (Fig. 8) is very similar to that in the Arctic (cf. Fig. 6). In particular, ζ tends to lag ζeq by ~2–3 days and has also smaller amplitudes, with the differences from equilibrium heights largest for the shortest-period tide (Mt). The lagged sea level response implies dynamic amplitudes ζd as large as ζeq, a sign of the strong nonequilibrium nature of the simulated long-period tides in Hudson Bay. Ponte (2006) finds similar behavior at monthly and longer scales in solutions of a shallow-water model under freshwater surface loading. Thus, the tendency for nonequilibrium behavior of the long-period tides in Hudson Bay is not unique and probably not strictly dependent on the large-scale structure of the forcing but more so on the bathymetry and geometry of the domain, with a shallow bay and a constricted exchange with the open ocean through the shallow and narrow Hudson Strait. Consistent with this conjecture, the experiment with constant H = 4000 m, with a much deeper bay and strait, results in a much weaker dynamic response.

Fig. 8.

As in Fig. 6, but at a point in central Hudson Bay.

Fig. 8.

As in Fig. 6, but at a point in central Hudson Bay.

5. Effects of stratification

With few exceptions, long-period tides have been studied in a barotropic context with no effects of stratification considered. In the presence of topography and coastal geometry, even very large-scale forcing like that of the astronomical tidal potential can lead to a baroclinic response, as is the case with the generation of internal tides at short periods. To examine possible effects of stratification on the long-period tides, we follow Ponte and Vinogradov (2007) and consider the relation between ζd, steric height anomaly ζs, and dynamic bottom pressure ζb, defined as

 
formula

where ρ is density anomaly, ρs is a constant surface density, and the vertical integral is over the full water column. Values of ζb, given in equivalent sea level units, can be directly compared to ζd. Differences in the two fields indicate the presence of density variations over the water column typically associated with baroclinic processes and stratification effects.3

Working again on the difference between GC+TE and GCE experiments to focus on the LPT solutions, the ratio of the amplitudes of ζd to ζb (Fig. 9) yields for the most part values of ~1 over the global ocean. Focusing first on Mf (Fig. 9a), deviations from 1 by more than 10% are confined to very small regions that tend to coincide with some continental shelf breaks (e.g., western Australia and Patagonia) and midocean ridges (southeast Pacific) and related topographic features. Differences as large as 50% or more can occur at a few places. Enhancement of baroclinic activity is also present along the equatorial band, but with a somewhat different character; the large-scale zonal patterns are not connected to topographic features, and there is a clear latitudinal trapping.

Fig. 9.

Ratio of the amplitudes of dynamic sea level and bottom pressure ζd/ζb for (a) Mf, (b) Mm, and (c) Ssa.

Fig. 9.

Ratio of the amplitudes of dynamic sea level and bottom pressure ζd/ζb for (a) Mf, (b) Mm, and (c) Ssa.

Impact of baroclinicity can depend on period, and the ζd/ζb ratios for Mm and Ssa (Figs. 9b,c) tend to support this contention. Patterns for Mm are very similar to Mf, in regards to the connection of baroclinic effects to topography and the equatorial regions, but larger deviations and longer zonal scales can be seen in the equatorial Indian Ocean. For Ssa, there is much more baroclinic activity at relatively large scales away from the equator, particularly in the Indian Ocean but also in the Atlantic Ocean.

The baroclinic features associated with topography, clearly seen for Mf and Mm, suggest the possibility of having very short-scale modulations in the LPT behavior. The details of these stratification effects may be sensitive, however, to how topographic interactions are represented in the relatively coarse-resolution model and are, thus, not further treated here. We focus instead on the large-scale features seen in Fig. 9 for all tides. Ratios higher and lower than 1 tend to alternate zonally, which suggests wave modulation. This behavior is clear at low latitudes for the case of Mf and Mm, with equatorial trapping suggesting the excitation of baroclinic equatorial waves and, in the south Indian Ocean for Ssa, with bending patterns in latitude suggesting Rossby wave beta refraction.

Assessment of physical mechanisms and propagation characteristics can be further explored by examining the amplitude and phase of ζs fields. Hovmöller diagrams for Mf (Fig. 10) show ubiquitous eastward propagation at the equator at phase speeds of c ~ 2.5–3 m s−1, roughly consistent with first baroclinic mode Kelvin waves (Gill 1982). Zonal wavelengths of ~30° match roughly the values expected from the Kelvin wave dispersion relation ω = ck, where ω is frequency and k is zonal wavenumber. Amplitudes tend to be strongest near the western boundary, particularly in the Pacific, and decay eastward, suggesting generation by the barotropic tide impinging on the coast.

Fig. 10.

Hovmöller diagrams for steric height anomalies ζs (mm) at (top to bottom) 3°N, 1.5°N, 0°, 1.5°S, and 3°S over one full cycle of the Mf tide.

Fig. 10.

Hovmöller diagrams for steric height anomalies ζs (mm) at (top to bottom) 3°N, 1.5°N, 0°, 1.5°S, and 3°S over one full cycle of the Mf tide.

Examination of the Hovmöller diagrams at off-equatorial latitudes (Fig. 10) reveals a more complex amplitude and phase structure. While eastward propagation is seen at 1.5°N, more of a standing pattern is seen at 1.5°S, particularly in the Pacific. At 3°N and 3°S, there is a mixture of westward propagation (mostly in the Indian and Atlantic basins) and standing patterns (mostly in the Pacific) and also some hints of eastward propagation in the eastern Indian sector. Westward phase speeds and zonal wavelengths are very similar to those seen at the equator. This behavior is generally consistent with the presence of a first baroclinic mode, westward-propagating, mixed Rossby–gravity (Yanai) wave (Gill 1982), in addition to the Kelvin mode.4 Because of the antisymmetric nature of the mixed Rossby–gravity wave with a node at the equator, one expects a clean Kelvin wave signature at the equator and an interference pattern in latitude that depends on the relative amplitudes and phases of the two waves, with standing wave patterns where amplitudes are similar and eastward or westward propagation where Kelvin or mixed Rossby–gravity waves dominate, respectively.

Similar equatorial wave excitation is involved in the case of Mm. The strongest baroclinic signals in the equatorial Indian Ocean occur at the equator and show clear eastward phase propagation at ~2.5–3 m s−1 and a zonal wavelength close to the full basin width (~60°), consistent with a first baroclinic mode Kelvin wave (Fig. 11). The pattern is similar in the western Pacific, although the hint of a standing wave pattern points also to the possible presence of Rossby waves, consistent with westward propagation seen at 3°N (not shown). In the Atlantic, eastward propagation at about half the speed and zonal wavelengths of ~30° point to the presence of second baroclinic mode Kelvin waves. The lack of a strong first baroclinic mode in the Atlantic has to do with the relatively small length of the basin. Weak excitation of the second baroclinic mode is also seen in the central and eastern Pacific.

Fig. 11.

Hovmöller diagram for steric height anomalies ζs (mm) along the equator for the Mm tide.

Fig. 11.

Hovmöller diagram for steric height anomalies ζs (mm) along the equator for the Mm tide.

At longer periods, baroclinic effects become more ubiquitous and can be found at higher latitudes, as for the case of Ssa noted in Fig. 9c. The strongest ζs signals in the south Indian Ocean are highlighted in the Hovmöller diagram in Fig. 12. Westward propagation at ~15 cm s−1 along 16°S is in agreement with estimates of first baroclinic mode Rossby wave phase speeds for this latitude [cf. Fig. 2b in Piecuch and Ponte (2014)]. Zonal wavelengths are consistent with the presence of semiannual baroclinic Rossby waves in the south Indian Ocean. Amplitude patterns are consistent with generation at the Australian shelf break and decay westward from the source region.

Fig. 12.

As in Fig. 11, but along 16°S in the Indian Ocean for the Ssa tide.

Fig. 12.

As in Fig. 11, but along 16°S in the Indian Ocean for the Ssa tide.

6. Conclusions and final remarks

Solutions of the LPT are derived in a vertically stratified ocean with a well-developed atmospherically forced circulation. The surface signatures of LPT are substantially contaminated by the atmospheric noise, particularly for the longest periods. The best signal-to-noise ratios for Mf and Mm are obtained in the tropics. Examining bottom pressure instead of sea level provides less noisy LPT signatures, given the larger atmospherically driven variability present at the surface compared to the bottom. Considerable noise is still the case for Sa, but Ssa signatures could be substantially cleaner in bottom pressure than in sea level. The potential use of GRACE data for examination of the LPT signals is noted. Nominal monthly GRACE solutions could complement altimeter data for better determination of the LPT signals at Ssa and Sa periods; more frequently sampled (7 day) GRACE fields could be explored for shorter-period tides.

The strongest nonequilibrium signals tend to be large-scale and barotropic in nature. All LPT exhibit enhanced dynamics in the Arctic and Nordic Seas and also in Hudson Bay, related to respective shallow bathymetric features and basin geometries. The largest dynamic amplitudes range from a few millimeters for Mt, Mm, and Ssa to a couple of centimeters for Mf. Although nonzero, departures from equilibrium for Sa and Ln (not shown) are very small and at the submillimeter level. These values give an approximate estimate of the errors incurred in using the equilibrium assumption when estimating the LPT. For practical purposes, consideration of nonequilibrium solutions is most important in places like the Arctic and Hudson Bay, for all LPT except at periods longer than semiannual.

Stratification effects add modulations to the dynamic response, which are particularly apparent in the tropics for Mf and Mm and extend to higher latitudes for longer-period tides. A number of weakly forced resonances are identified in the form of propagating baroclinic waves, possibly excited at the coastal boundaries. The potential for shorter-scale baroclinic effects trapped to regions of enhanced topographic gradients is also suggested. Amplitudes of the stratification effects can amount to >20% of the dynamic response in the model. Some of these modulations might be “observable” if sufficiently long records exist in tropical regions (e.g., Miller et al. 1993). The model results suggest that, for the best predictions of LPT in many regions, stratification effects should be considered. However, at the longest periods for which stratification effects are more ubiquitous, the dynamic response is quite weak, and their practical importance is diminished.

No attempt is made in this work to assess in detail the quality of the LPT solutions in relation to data. In this regard, a number of caveats are worth noting, particularly the coarse grid of the model used and the consequent lack of resolution of coastal and topographic features that can influence the details of the ocean response to the tidal forcing. The presence of localized features associated with shallow or variable topography, semienclosed seas, or other factors has been suggested and could affect the nature of LPT solutions in many regions depending on period considered. Future work with finer horizontal and vertical resolution will benefit from comparisons with large-scale observations (altimetry and gravimetry) as well as with available in situ data (tide gauges and bottom pressure recorders) that can shed light on any potential short-scale features implied in the model LPT solutions. Available estimates of the atmospherically driven noise could be applied to filter out such noise in the observations for improved analyses. The present results will hopefully motivate future work along these lines.

Acknowledgments

This work was partly funded by NASA Grant NNX11AQ12G and by NSF Grant OCE-0961507. We thank Richard Ray for providing the tidal forcing code, Patrick Heimbach and Gael Forget for help with MITgcm codes and setup, and Chris Hughes and an anonymous referee for their helpful reviews of the original manuscript.

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Footnotes

1

These derived LPT solutions could include effects of nonlinear interactions between LPT and the ocean circulation, but given the small LPT amplitudes, such effects are expected to be weak.

2

Accounting for body tide effects means that derived tidal amplitudes should be interpreted in terms of relative sea level, as normally measured by a tide gauge.

3

We take the equality of ζd and ζb to mean essentially a barotropic response, although in very special cases one could have density anomalies that cancel out in the vertical integral, leaving ζb unaffected.

4

From the mixed Rossby–gravity wave dispersion relation k = ω/cβ/ω (β = 2.3 × 10−11 m−1 s−1 is the gradient of the Coriolis parameter) (Gill 1982) for the Mf frequency and c ~ 2.5–3 m s−1, one gets negative (i.e., westward propagating) zonal wavelengths of ~30°, very similar to the Kelvin wave zonal wavelength for the same ω and c. In fact, at the Mf frequency, |k| takes the same value for c ~ 2.5 m s−1.