Long-Period Tides in an Atmospherically Driven, Stratified

JULY 2015 P O N T E E T A L . 1917

Long-Period Tides in an Atmospherically Driven, Stratiﬁed Ocean

RUI M. PONTE AND AYAN H. CHAUDHURI Atmospheric and Environmental Research, Inc., Lexington, Massachusetts

SERGEY V. VINOGRADOV Earth Resources Technology, Inc./National Oceanic and Atmospheric Administration, Silver Spring, Maryland

(Manuscript received 7 January 2015, in ﬁnal form 4 May 2015)

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 over- whelm 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 geophysical inference is possible from the study of LPT, as they can measurably affect parameters such as Earth’s Aside from the energetic diurnal and semidiurnal axial rotation and polar motion (e.g., Kantha et al. 1998; ocean tides that are conspicuously present in all sea level Ray and Egbert 2012). A short summary of LPT is records, the astronomical gravitational potential also provided by Le Provost (2001). gives rise to tides of much lower frequencies. Despite In the isostatic limit, the oceanic response to astro- their considerably weaker amplitudes, these so-called nomical tidal forcing is sufficiently rapid and efficient to long-period tides (LPT)—including Mt, Mf, Mm, Ssa, render any resulting pressure gradients negligible com- and Sa, at termensual, fortnightly, monthly, semiannual, pared to the applied forcing, and the tides are said to be and annual periods, respectively, and the nodal tide at at equilibrium. More interesting solutions involve sub- 18.6 yr (Table 1)—are nevertheless of significant in- stantial pressure gradients and a dynamic response, terest. In contrast to the atmospheric fields that drive the which can be dominated by near resonances, as with the ocean general circulation, the LPT forcing is essentially diurnal and semidiurnal tides. With the LPT, the quest perfectly known, albeit of a very narrowband character to understand how much of a nonequilibrium response is in frequency and wavenumber (Cartwright and Tayler expected has focused on Mf, the strongest LPT, with a 1971). Thus, observations of LPT can provide important couple of processes highlighted so far. The excitation of insights into how the ocean responds to forcing and how Rossby waves, including those of topographic origin, has it dissipates energy (e.g., Wunsch et al. 1997). Broader been extensively discussed dating back to Wunsch (1967), later revisited by Carton (1983) and others. The influence of gravity wave mechanisms (Miller et al. Corresponding author address: Rui M. Ponte, Atmospheric and Environmental Research, Inc., 131 Hartwell Avenue, Lexington, 1993) and, most importantly, of incomplete interbasin MA 02421. adjustment involving mass fluxes mediated by geo- E-mail: [email protected] strophic flows between basins (Egbert and Ray 2003)

DOI: 10.1175/JPO-D-15-0006.1

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TABLE 1. Main long-period tidal constituents. subgrid-scale parameterizations of Gaspar et al. (1990), Gent and McWilliams (1990),andRedi (1982),quadratic Doodson No. Name Period (days) Amplitude A (mm) bottom friction is also used. Forget et al. (2015) provide a 085.465 9.12 2.31 full description of the MITgcm configuration in the con- 085.455 Mt 9.13 5.58 083.655 Mst 9.56 1.06 text of optimization experiments done with a compre- 075.575 13.61 1.13 hensive set of observational constraints. Here, we focus 075.565 13.63 1.21 on model experiments done with no optimization. 075.555 Mf 13.66 29.11 Our two basic experiments consist of 1) forcing only 075.355 13.78 1.26 with atmospheric fluxes to obtain a simulation of the 073.555 Msf 14.77 2.54 065.465 27.44 1.00 general circulation (denoted as GCE) and then 2) add- 065.455 Mm 27.55 15.37 ing the LPT potential to examine the tides in the pres- 065.445 27.67 1.01 ence of a fully variable ocean (denoted as GC1TE). The 063.655 Msm 31.81 2.94 LPT solutions can be approximated by subtracting GCE 057.555 Ssa 182.62 13.52 from GC1TE.1 Forcing with a tidal potential is equiv- 056.554 Sa 365.26 2.15 055.565 Ln 6798.4 12.20 alent 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 has also been discussed. Ponte (1997) discusses similar includes the 15 largest LPT constituents listed in Table 1,as mechanisms in the dynamic response of the ocean to the tabulated by Cartwright and Tayler (1971) and Cartwright 5-day Rossby–Haurwitz wave in atmospheric pressure. and Edden (1973). All the major frequencies (Table 1)are The mentioned studies (and most others as well) have included, and subsequent analyses emphasize Mt, Mf, Mm, treated the LPT in a barotropic ocean with no in- Ssa, and Sa. In an aquaplanet, the equilibrium LPT has the teractions with the general circulation. Numerical sim- simple form ulations of Mf and Mm in baroclinic settings include the 1 3 two-layer model study of Arbic et al. (2004), but the z 5 A 2 sin2u , (1) emphasis has been invariably on the discussion of short- eq 2 2 period tides. In this work, rather than trying to simulate u A the LPT as realistically as possible, we focus on ex- where is latitude, and is the equilibrium amplitude given in Table 1, which includes the reduction factor of ploring potential nonequilibrium behaviors across the 2 whole set of frequencies. In addition, our modeled ocean 0.693 resulting from body tide effects. Effects of self- is vertically stratified and driven by the atmosphere, thus attraction and loading, which can effectively increase containing a realistic general circulation on which the the amplitudes of forcing and response but may not be LPT are superposed. This setup allows one to examine easy to parameterize accurately (Woodworth 2012), are potential effects of baroclinicity on the LPT. In addition, not considered for simplicity. one can set the LPT variability in the context of the at- Initial conditions are taken from one of the pre- mospherically driven ‘‘noise,’’ an issue that is important liminary, unconstrained solutions discussed in Forget when trying to separate the LPT from other variability et al. (2015), and the model is run for 20 yr from 1 January in the records. 1992. All results discussed here are obtained by har- monic analysis carried out on the last 18.6 yr of output, which permits the resolution of all the relevant tidal 2. Modeling and analysis details lines in Table 1, including one full cycle of the nodal tide The main tool of analysis is the MIT general circula- Ln and also the weaker nodal modulations in termen- tion model (MITgcm) described in Marshall et al. sual and fortnightly bands (e.g., Egbert and Ray 2003) (1997). Configuration includes 1) a global grid of varying not treated here. For these analyses, sea level, bottom pressure, and other relevant diagnostics were archived horizontal resolution (nominal 18 decreasing to ;1/38 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 1 These derived LPT solutions could include effects of nonlinear using partial cells (Adcroft et al. 1997), and 4) forcing by interactions between LPT and the ocean circulation, but given the small LPT amplitudes, such effects are expected to be weak. bulk flux formulations using ERA-Interim surface at- 2 Accounting for body tide effects means that derived tidal am- mospheric fields (Dee et al. 2011). Atmospheric pres- plitudes should be interpreted in terms of relative sea level, as sure is not included in the forcing fields. Apart from normally measured by a tide gauge.

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FIG. 1. Amplitude (mm) of the full sea level response z at the (a) Mf and (c) Ssa frequencies and respective equilibrium tidal amplitudes for (b) Mf and (d) Ssa. as daily averages. All equilibrium solutions and re- eddies. Nevertheless, given expected large-scale struc- spective dynamic deviations are calculated based on ture of the LPT signals, in reality eddy noise might be self-consistent tides that account for the conservation of less of a problem in separating out LPT signals if spatial total ocean volume. The absolute values of the phase are smoothing is possible (e.g., in altimetric analyses). not important for our purposes, but for reference all Amplitudes of full sea level z at the Mf frequency, values are relative to the starting date of the analyzed inferred from the experiment with both tidal and non- time series (1200 UTC 24 May 1993). Given the model’s tidal forcing, clearly show the zonally banded structure Boussinesq formulation, effects of variability in global- of the tidal forcing with a node around 6358 latitude and mean sea level and bottom pressure are not considered; maxima in tropical and high latitudes (Fig. 1). Ampli- changes in the spatial mean bottom pressure resulting tudes are similar but not a perfect match to those ex- from the Boussinesq assumption are calculated and re- pected under a pure equilibrium tide. Results indicate a moved prior to analysis. relatively strong LPT contribution to variability at the Mf period, clearly discernible amid all other nontidal variability, and with hints of nonequilibrium behavior, 3. LPT and large-scale circulation such as the higher amplitudes in the tropical Atlantic The extent to which LPT are in equilibrium or not has compared to the tropical Pacific. Results at the Mm been a subject of much debate (Le Provost 2001). The frequency (not shown) are quite similar in character. In difficulty in addressing the issue with observations lies contrast, at the Ssa period, no clear signature of re- mostly in the presence of the considerable noise that spective LPT emerges in Fig. 1 and z variability seems to constitutes the variable ocean general circulation. The be dominated by the nontidal component, with ampli- problem can be particularly acute at annual and semi- tudes much larger than expected from a simple equi- annual periods but also affects analyses of Mf and other librium Ssa tide in the tropics (maximum Ssa amplitude tides (e.g., Ponchaut et al. 2001). Here, because of our is ;13.5 mm at the pole). Similar conclusions can be coarse model grid, we only address the effects of the drawn at the annual and nodal periods (not shown). atmospherically forced large-scale circulation on the Results in Fig. 1 confirm the substantial noise associ- LPT and do not treat possible impacts of small-scale ated with the large-scale circulation that can hamper

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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. attempts to infer LPT signals from sea level measure- sea level, particularly at semiannual and annual periods ments, particularly for Ssa and Sa corresponding to pe- (Ponte 1999). Given the availability of global bottom riods of strong atmospheric forcing. In the model setting pressure fields derived from the Gravity Recovery and pursued here, one can readily separate LPT signals from Climate Experiment (GRACE) satellite mission (e.g., the background continuum by differencing the GCE and Chambers and Bonin 2012), it is instructive to compute GC1TE sea level fields. A quantitative measure of the the signal-to-noise ratio based on bottom pressure circulation noise can then be obtained by examining (Fig. 3). Values are considerably higher than in Fig. 2. the ratio of the tidal amplitudes to the amplitudes of the For Mf and Mm, using bottom pressure does not change GCE fields at the respective tidal frequencies. the results at mid- and high latitudes, as expected from Values of such ratios for Mf, Mm, Ssa, and Sa (Fig. 2) the barotropic dynamics that tend to dominate the cir- can be treated as rough signal-to-noise ratios if one is culation at the periods of interest. Higher signal-to-noise trying to separate out the tidal signal from the variable ratios are obtained in the tropics, however, where con- circulation. For Mf and Mm, apart from latitudes near siderably more baroclinic activity can lead to larger sea the nodal lines, ratios are typically of order 10 and level variability compared to bottom pressure. Similar larger. The strongest signals are expected in the tropics, results are found for Ssa and Sa periods, but improved where atmospherically driven variability is relatively signal-to-noise ratios at high latitudes are also observed, weak compared to higher latitudes. For both Ssa and Sa, particularly for Sa. While these ratios are mostly .1 for ratios are of order one and smaller, indicating the rela- Ssa, values ,1 are still the norm for Sa. tively strong circulation, particularly for the annual period. Regions with the largest signal-to-noise ratios 4. Nonequilibrium behavior of LPT include the Arctic (away from the Eurasian shelves) and to some extent parts of the Southern Ocean. For the purpose of examining in detail any potential As seen in the model simulations, bottom pressure departures from equilibrium behavior, rather than fo- variability can be considerably weaker than that of the cusing on the full sea level z, it is instructive to analyze

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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.

zd 5 z 2 zeq, which represents dynamic pressure fields Egbert and Ray (2003) in the context of Mf and Mm and associated with any deviations from the equilibrium represent, in a heuristic sense, incomplete interbasin ad- response zeq (e.g., Ponte 1997). All fields described here justment of the mass field in the presence of continental are based on the difference between GC1TE and GCE barriers and relatively constricted connections between experiments to isolate the LPT solutions. The focus is on basins through the Southern Ocean. Thus, we will focus constituents with the largest dynamic response (Mt, Mf, below on a couple of other features apparent in Fig. 4 that and Mm); in particular, Sa and nodal tides have sub- have not been discussed as much in previous works. millimeter dynamic responses and are not discussed. a. Arctic/Nordic Seas response The amplitude and phase of zd for Mt, Mf, and Mm, all display similar nonequilibrium characteristics (Fig. 4). In The Arctic and Nordic Seas exhibit, by far, the largest particular, there is a large-scale pattern of weaker deviations from equilibrium of all the ocean basins. (stronger) departures from equilibrium in the Atlantic Although the response can scale locally with the mag-

(Paciﬁc) Oceans, with zd being approximately out of nitude of the forcing, which has a maximum at the poles, phase between the two basins but having almost the same other factors are likely at play. Notice that the dynamic phase within each basin. The amplitude and phase of zd in response at the high southern latitudes is substantially the Indian Ocean are somewhat between those in the weaker by comparison. In more detailed plots of am-

Paciﬁc and Atlantic. Spatial patterns and amplitude plitude and phase of zd for the Arctic and adjacent values are in general agreement with previous works Nordic Seas (Fig. 5), one sees that the enhanced am- (Miller et al. 1993; Wunsch et al. 1997; Egbert and Ray plitudes are sharply reduced at latitudes around Iceland 2003). In particular, typical amplitudes of a few milli- and the Faroe Islands, with amplitudes within the Arctic meters in the tropical Paciﬁc are consistent with obser- and Nordic Seas more homogeneous. Phase values do vations (cf. Tables A3 and A4 in Miller et al. 1993). The not vary much across the latter domain. In particular, dynamics responsible for the above-noted features have there is no evidence of phase propagation anywhere that been extensively analyzed by Miller et al. (1993) and would suggest wave processes being prominent in the

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FIG. 4. Amplitude (mm) and phase (8) of dynamic response zd for (top) Mt, (middle) Mf, and (bottom) Mm constituents. dynamics. These characteristics are present across all Another useful measure of dynamic response can be three tides examined in Fig. 5 and also arise in the dy- obtained from examination of the velocity field. Maxi- namic response of Ssa and Sa (not shown). mum speeds U over the tidal cycle at a depth of 50 m 2 Given the relatively homogeneous nature of the re- (Fig. 7) are typically less than 0.2 mm s 1 over much of sponse in Fig. 5, the relation between z, zd,andzeq for a the basin. Other depth levels give very similar results. 2 point near the North Pole (Fig. 6) can be taken as rep- Enhanced speeds of up to 1 mm s 1 are found only across resentative of the general conditions across the Arctic the entrance to the Nordic Seas. These tend to coincide and high northern latitudes. Amplitudes of z are typically with regions of sharp gradients in amplitude and variable smaller than zeq,withz lagging zeq by ;1–2 days, across phase for zd (Fig. 5) and therefore relatively large surface all three tidal frequencies examined in Fig. 6. These lags pressure gradients and geostrophic currents. are in agreement with the modeling results of Stepanov Bathymetry plays a substantial role in shaping the and Hughes (2006), who find that global barotropic ad- dynamic response seen in Figs. 6 and 7. Flat bottom justment relative to the equator takes only a fraction of a model experiments (not shown), with depths H of 1000 day in most basins but up to 2 days in the Arctic. The and 4000 m, lead to very different spatial characteristics response gets closer to equilibrium as the period in- and substantially weaker amplitudes for zd. In particular, creases; the ratio of the amplitudes of zd to those of zeq the contrast across the entrance to the Nordic Seas in gets smaller from Mt to Mf to Mm. The large-scale Fig. 6 is completely absent for constant H. The Denmark nonequilibrium behavior depicted in both Figs. 5 and 6 Strait and the straits between Iceland and the Faroe and is consistent with an incomplete mass exchange with the Shetland Islands are regions of shallow sill depths, which rest of the ocean, following the dynamics discussed by together with the island landmasses provide a partial Egbert and Ray (2003) and, in another context, by Ponte boundary separating the Nordic Seas from the North (1997). Similar homogeneous mass fluctuations in the Atlantic. Changes in H also lead to larger gradients in f/H Arctic/Nordic Seas can result from wind-driven dynam- (Fig. 7), which are dominated by changes in H over those ics, as reviewed recently by Fukumori et al. (2015). in the Coriolis parameter f. These features tend to inhibit

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FIG. 5. (top) Amplitude (mm) and (bottom) phase (8) of dynamic response zd 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. wave propagation across the straits, for both gravity and behavior is seen for all tidal lines in Fig. 4. Amplitudes vorticity waves, and thus contribute to partially isolate tend to be relatively homogeneous in Hudson Bay the response in the Arctic and Nordic Seas from that in proper (but generally weaker in the northern Foxe basin the North Atlantic and the rest of the World Ocean. and in the Hudson Strait). Phases are also similar inside Embedded in the basin-scale characteristics of the the bay and over Hudson Strait, indicating a rise and fall dynamic response, there are also regional differences of sea level more or less in unison. In fact, phases inside worth noting. In particular, the higher amplitudes in the the bay are roughly the same as outside of the strait.

East Siberia, Laptev, and Kara shelves coincide also The relation between z, zd, and zeq in Hudson Bay with some of the largest phase differences across the (Fig. 8) is very similar to that in the Arctic (cf. Fig. 6). In basin. Apart from the expected coastal trapping of wave particular, z tends to lag zeq by ;2–3 days and has also energy, the influence of bathymetry is also plausible in smaller amplitudes, with the differences from equilib- this case, given that the largest zd amplitudes are mostly rium heights largest for the shortest-period tide (Mt). confined to the shallowest depths (Fig. 5). The bottom The lagged sea level response implies dynamic ampli- depth considerably affects the propagation speeds of tudes zd as large as zeq, a sign of the strong non- waves that are likely to be involved in the mass field equilibrium nature of the simulated long-period tides in adjustmentpffiffiffiffiffi of the Arctic: for gravity waves, speed goes Hudson Bay. Ponte (2006) finds similar behavior at as H, and for long Rossby waves, speed goes as H (Gill monthly and longer scales in solutions of a shallow- 1982). Thus, the adjustment is expected to be slower water model under freshwater surface loading. Thus, the over the shelves than over the deep basin and can lead to tendency for nonequilibrium behavior of the long- differences in behavior observed in Fig. 5. period tides in Hudson Bay is not unique and probably not strictly dependent on the large-scale structure of the b. Hudson Bay response forcing but more so on the bathymetry and geometry of In terms of shallow coastal and semienclosed regions, the domain, with a shallow bay and a constricted ex- Hudson Bay stands out as the place of strongest dynamic change with the open ocean through the shallow and signals in Fig. 4, with zd amplitudes .20 mm for Mf and narrow Hudson Strait. Consistent with this conjecture, comparable in general to the Arctic signals. Similar the experiment with constant H 5 4000 m, with a much

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21 FIG. 7. Maximum speed (mm s ) for the Mf constituent over the Arctic calculated at a depth of 50 m. Contour lines represent iso- lines of f/H, with gradients dominated by changes in bathymetry.

where r is density anomaly, rs is a constant surface density, and the vertical integral is over the full water

column. Values of zb, given in equivalent sea level units, can be directly compared to zd. 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 GC1TE and z z z GCE experiments to focus on the LPT solutions, the ratio FIG. 6. Tidal cycles for (blue), d (red), and eq (green) repre- z z senting (top) Mt, (middle) Mf, and (bottom) Mm at a point in the of the amplitudes of d to b (Fig. 9) yields for the most central Arctic indicated by the asterisk in Fig. 5. Abscissa values are part values of ;1 over the global ocean. Focusing first on given in days. Mf (Fig. 9a), deviations from 1 by more than 10% are confined to very small regions that tend to coincide with deeper bay and strait, results in a much weaker dynamic some continental shelf breaks (e.g., western Australia and response. 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 baro- 5. Effects of stratification clinic activity is also present along the equatorial band, With few exceptions, long-period tides have been but with a somewhat different character; the large-scale studied in a barotropic context with no effects of strat- zonal patterns are not connected to topographic features, ification considered. In the presence of topography and and there is a clear latitudinal trapping. coastal geometry, even very large-scale forcing like that Impact of baroclinicity can depend on period, and the of the astronomical tidal potential can lead to a baro- zd/zb ratios for Mm and Ssa (Figs. 9b,c) tend to support clinic response, as is the case with the generation of in- this contention. Patterns for Mm are very similar to Mf, ternal tides at short periods. To examine possible effects in regards to the connection of baroclinic effects to to- of stratification on the long-period tides, we follow pography and the equatorial regions, but larger Ponte and Vinogradov (2007) and consider the relation between zd, steric height anomaly zs, and dynamic bottom pressure z , defined as b 3 z z ð We take the equality of d and b to mean essentially a baro- 1 tropic response, although in very special cases one could have z 5 z 2 z 5 z 1 r dz density anomalies that cancel out in the vertical integral, leaving b d s d r , (2) s zb unaffected.

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FIG. 9. Ratio of the amplitudes of dynamic sea level and bottom pressure zd/zb for (a) Mf, (b) Mm, and (c) Ssa.

FIG.8.AsinFig. 6, but at a point in central Hudson Bay.

amplitude and phase of zs fields. Hovmöller diagrams for deviations and longer zonal scales can be seen in the Mf (Fig. 10) show ubiquitous eastward propagation at 2 equatorial Indian Ocean. For Ssa, there is much more the equator at phase speeds of c ; 2.5–3 m s 1, roughly baroclinic activity at relatively large scales away from consistent with first baroclinic mode Kelvin waves (Gill the equator, particularly in the Indian Ocean but also in 1982). Zonal wavelengths of ;308 match roughly the the Atlantic Ocean. values expected from the Kelvin wave dispersion re- The baroclinic features associated with topography, lation v 5 ck, where v is frequency and k is zonal clearly seen for Mf and Mm, suggest the possibility of wavenumber. Amplitudes tend to be strongest near the having very short-scale modulations in the LPT behavior. western boundary, particularly in the Pacific, and decay The details of these stratification effects may be sensitive, eastward, suggesting generation by the barotropic tide however, to how topographic interactions are represented impinging on the coast. in the relatively coarse-resolution model and are, thus, not Examination of the Hovmöller diagrams at off- further treated here. We focus instead on the large-scale equatorial latitudes (Fig. 10) reveals a more complex features seen in Fig. 9 for all tides. Ratios higher and lower amplitude and phase structure. While eastward propa- than 1 tend to alternate zonally, which suggests wave gation is seen at 1.58N, more of a standing pattern is seen modulation. This behavior is clear at low latitudes for the at 1.58S, particularly in the Pacific. At 38N and 38S, there case of Mf and Mm, with equatorial trapping suggesting is a mixture of westward propagation (mostly in the the excitation of baroclinic equatorial waves and, in the Indian and Atlantic basins) and standing patterns south Indian Ocean for Ssa, with bending patterns in (mostly in the Pacific) and also some hints of eastward latitude suggesting Rossby wave beta refraction. propagation in the eastern Indian sector. Westward Assessment of physical mechanisms and propagation phase speeds and zonal wavelengths are very similar to characteristics can be further explored by examining the those seen at the equator. This behavior is generally

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FIG. 10. Hovmöller diagrams for steric height anomalies zs (mm) at (top to bottom) 38N, 1.58N, 08, 1.58S, and 38S over one full cycle of the Mf tide. consistent with the presence of a first baroclinic mode, the Atlantic, eastward propagation at about half the westward-propagating, mixed Rossby–gravity (Yanai) speed and zonal wavelengths of ;308 point to the pre- wave (Gill 1982), in addition to the Kelvin mode.4 Be- sence of second baroclinic mode Kelvin waves. The lack cause of the antisymmetric nature of the mixed Rossby– of a strong first baroclinic mode in the Atlantic has to do gravity wave with a node at the equator, one expects a with the relatively small length of the basin. Weak ex- clean Kelvin wave signature at the equator and an in- citation of the second baroclinic mode is also seen in the terference pattern in latitude that depends on the rela- central and eastern Pacific. tive amplitudes and phases of the two waves, with At longer periods, baroclinic effects become more standing wave patterns where amplitudes are similar ubiquitous and can be found at higher latitudes, as for the and eastward or westward propagation where Kelvin or case of Ssa noted in Fig. 9c. The strongest zs signals in the mixed Rossby–gravity waves dominate, respectively. south Indian Ocean are highlighted in the Hovmöller di- 2 Similar equatorial wave excitation is involved in the agram in Fig. 12.Westwardpropagationat;15 cm s 1 case of Mm. The strongest baroclinic signals in the along 168S is in agreement with estimates of first baro- equatorial Indian Ocean occur at the equator and show clinic mode Rossby wave phase speeds for this latitude [cf. 2 clear eastward phase propagation at ;2.5–3 m s 1 and a Fig. 2b in Piecuch and Ponte (2014)]. Zonal wavelengths zonal wavelength close to the full basin width (;608), are consistent with the presence of semiannual baroclinic consistent with a first baroclinic mode Kelvin wave Rossby waves in the south Indian Ocean. Amplitude (Fig. 11). The pattern is similar in the western Pacific, patterns are consistent with generation at the Australian although the hint of a standing wave pattern points also shelf break and decay westward from the source region. to the possible presence of Rossby waves, consistent with westward propagation seen at 38N (not shown). In

4 From the mixed Rossby–gravity wave dispersion relation k 5 2 2 2 v/c 2 b/v (b 5 2.3 3 10 11 m 1 s 1 is the gradient of the Coriolis 2 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 ;308, very similar to the Kelvin wave zonal wavelength for the same v and c. In fact, at the Mf frequency, jkj takes the same value FIG. 11. Hovmöller diagram for steric height anomalies zs (mm) 2 for c ; 2.5 m s 1. along the equator for the Mm tide.

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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 8 FIG. 12. As in Fig. 11, but along 16 S in the Indian Ocean for the ubiquitous, the dynamic response is quite weak, and Ssa tide. 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 6. Conclusions and final remarks regard, a number of caveats are worth noting, particu- Solutions of the LPT are derived in a vertically stratified larly the coarse grid of the model used and the conse- ocean with a well-developed atmospherically forced cir- quent lack of resolution of coastal and topographic culation. The surface signatures of LPT are substantially features that can influence the details of the ocean re- contaminated by the atmospheric noise, particularly for sponse to the tidal forcing. The presence of localized the longest periods. The best signal-to-noise ratios for Mf features associated with shallow or variable topography, and Mm are obtained in the tropics. Examining bottom semienclosed seas, or other factors has been suggested pressure instead of sea level provides less noisy LPT sig- and could affect the nature of LPT solutions in many natures, given the larger atmospherically driven variabil- regions depending on period considered. Future work ity present at the surface compared to the bottom. with finer horizontal and vertical resolution will benefit Considerable noise is still the case for Sa, but Ssa signa- from comparisons with large-scale observations (altim- tures could be substantially cleaner in bottom pressure etry and gravimetry) as well as with available in situ data than in sea level. The potential use of GRACE data for (tide gauges and bottom pressure recorders) that can examination of the LPT signals is noted. Nominal shed light on any potential short-scale features implied monthly GRACE solutions could complement altimeter in the model LPT solutions. Available estimates of the data for better determination of the LPT signals at Ssa atmospherically driven noise could be applied to filter and Sa periods; more frequently sampled (7 day) out such noise in the observations for improved analy- GRACE fields could be explored for shorter-period tides. ses. The present results will hopefully motivate future The strongest nonequilibrium signals tend to be large- work along these lines. scale and barotropic in nature. All LPT exhibit enhanced dynamics in the Arctic and Nordic Seas and also Acknowledgments. This work was partly funded by in Hudson Bay, related to respective shallow bathy- NASA Grant NNX11AQ12G and by NSF Grant OCE- metric features and basin geometries. The largest dy- 0961507. We thank Richard Ray for providing the tidal namic amplitudes range from a few millimeters for Mt, forcing code, Patrick Heimbach and Gael Forget for Mm, and Ssa to a couple of centimeters for Mf. Al- help with MITgcm codes and setup, and Chris Hughes though nonzero, departures from equilibrium for Sa and and an anonymous referee for their helpful reviews of Ln (not shown) are very small and at the submillimeter the original manuscript. level. 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