The ocean tide at each location is usually represented as a combination of harmonic constituents with frequencies corresponding to those of lines in the tidal potential (Cartwright and Tayler, 1971; Cartwright and Edden, 1973). The major lunar constituents are always accompanied by sidebands separated in frequency by ±1/18.61 cycles per year, 18.61 years being the nodal (or draconic) period of regression in the mean longitude of the lunar ascending node (Doodson and Warburg, 1941). The most efficient way of accounting for the sidebands in a harmonic expansion is via the use of “nodal factors” f and u, whereby the simple representation of a single constituent, Hcos⁡(ωt+A-G), in which ω is the angular frequency of the constituent, H and G are its amplitude and phase lag, and A is its astronomical argument at time t=0, is modified to fHcos⁡(ωt+A+u-G), where f and u are time-dependent functions of the longitude of the ascending node (N). For example, in the tidal potential (or equilibrium tide), the main lunar semi-diurnal tide (M2) has nodal factors: f=1.0-0.037cos⁡(N),u=-2.1∘sin⁡(N), retaining only terms in cos⁡(N) and sin⁡(N), and neglecting smaller terms depending on cos⁡(2N), and so on (Doodson, 1928; Doodson and Warburg, 1941; Pugh and Woodworth, 2014).

Because the frequencies of the sidebands are similar to the constituent's central frequency, it is usually assumed that the response of the ocean at the sidebands and at the central frequency will be in proportion to that given in the tidal potential, i.e. that the same admittance will apply. However, nodal factors different from expectations from the tidal potential (or equilibrium tide) have been found at many locations, at least for semi-diurnal tides.

For example, smaller values of f for M2 were found around the UK by Amin (1983, 1985) and were explained as being a consequence of non-linear frictional damping. Similar findings were obtained from measurements of mean tidal range around the UK by Woodworth et al. (1991). Differences from the expected nodal factors were found in data from the west coast of Australia (Amin, 1993) and the Bay of Fundy and Gulf of Maine (Ku et al., 1985; Ray, 2006; Müller, 2011). Feng et al. (2015) found differences for both semi-diurnal and diurnal tides at locations along the coast of China. In a survey of long-term changes in the amplitudes and phase lags of the four main tidal constituents around the world (M2, S2, O1 and K1), Woodworth (2010) pointed to many locations where differences in f from those expected from the equilibrium tide were evident.

Turning to the long-period tides, all of the long-period constituents of the equilibrium tide have amplitudes proportional to 13-sin⁡2(latitude) with no zonal dependence. The amplitudes are twice as large at the poles as at the Equator, they are 180∘ out of phase between high and low latitudes, and they have zero amplitude at 35∘ N/S. Proudman (1960) suggested that, at least for the longest of the long-period tides (the 18.61-year nodal tide), the tide in the real ocean should be a close approximation of its equilibrium form, and that still seems to be a good theory (Woodworth, 2012). However, tidal modelling and observations by tide gauges and satellite altimetry have demonstrated that the long-period tides with shorter periods in the real ocean, such as Mf and Mm with periods of approximately a fortnight and a month, respectively, have significant spatial variations from their equilibrium form (Wunsch et al., 1997; Mathers and Woodworth, 2001; Egbert and Ray, 2003; Lyard et al., 2006; Ray and Egbert, 2012; Ray and Erofeeva, 2014).

Although the spatial variations of the long-period tides are now much better understood, it is also of interest to consider whether their temporal (nodal) variability conforms to expectations. The Mf constituent (period of 13.66 days) is particularly interesting in this regard. In the equilibrium tide, Mf is the largest of the long-period tides and has very large nodal variations: f=1.043+0.414cos⁡(N),u=-23.7∘sin⁡(N) (Doodson and Warburg, 1941). Why the first term in f is not identically 1.0 (for Mf and for many other constituents) arises from the way that Doodson (1928) combined sideband constituents in order to provide simple functions in terms of N only. Doodson's nodal parameterisations, especially those for Mf, are discussed in the Appendix.

As far as we know, the magnitude of this temporal variability for the long-period tides in the real ocean has never been verified properly. In principle, one would have expected that the relatively large amplitude and short period of Mf would have enabled the temporal variation of its amplitude and phase lag to be estimated reliably from two decades of tide gauge data. However, there is always non-tidal background variability at fortnightly timescales to contend with. Most research on long-period tides in tide gauge records has been focused on regions such as the low-latitude Pacific, where the non-tidal background variability is much lower than at higher latitudes (e.g. Miller et al., 1993). However, the long-period tides are also small in these regions (i.e. centimetric; see Fig. 5 of Ray and Egbert, 2012). These studies of Pacific data were primarily concerned with establishing how the non-equilibrium aspects of Mf and Mm varied spatially, rather than temporally (Wunsch, 1967). Even though some long tide gauge records exist at high latitudes (e.g. northern Norway or Canada), where long-period amplitudes are larger, the relatively high background of non-tidal sea level variability means that it is difficult to make an accurate determination of the long-period tides without also modelling the non-tidal background (e.g. Crawford, 1982).

In this paper, we report on the temporal variations of the amplitudes and phase lags of Mf, Mm and Mt (period of one-third of a month) at the Drake Passage to see if they are consistent with equilibrium expectations. These are the three long-period tides, in order of decreasing amplitude in the equilibrium tide, that one is likely to be able to extract from records of about 1 year. The Drake Passage is at a sufficiently high latitude that any long-period tides should be larger than in most parts of the ocean. In addition, our investigation is based on the use of measurements of bottom pressure (BP) obtained over almost three decades, instead of on conventional coastal tide gauge data. It will be seen that bottom pressure recorders (BPRs) are inherently more suitable for providing long-period tidal information than coastal tide gauges. However, as a comparison of different measurement techniques, we also make use of 31 years of hourly sea level data from the nearby Vernadsky Research Base, which has the longest tide gauge record in Antarctica.

Cartwright (1999, chap. 13) provides a history of the development of BPRs, primarily by groups in Germany, France, USA and UK. Cartwright himself and colleagues from the National Oceanography Centre (NOC, as it is now called) made extensive use of BPRs in sets of “pelagic” (pertaining to the open sea) tidal measurements, first in waters around the UK, and then throughout the Atlantic Ocean (Cartwright et al., 1988; Spencer and Vassie, 1997). The same equipment was also used in international studies of non-tidal ocean processes (e.g. Cartwright et al., 1987), culminating in the late 1980s in the deployment of BPRs at the Drake Passage in order to monitor fluctuations in the transport of the Antarctic Circumpolar Current (ACC) as part of the World Ocean Circulation Experiment (Woodworth et al., 2002).

Most of the BPR deployments were made on the north and south sides of the Drake Passage in order to measure changes in the pressure gradient between them. Bottom landers based on the “Mk.IV” or similar designs were used in most cases (Fig. 1a, Spencer and Vassie, 1997). Over half of the deployments took the form of recoveries and redeployments on an annual basis at depths around 1000 m, providing records of 15 min average bottom pressure typically 1 year long. The other half of the deployments were made at greater depths, between 2000 and 4000 m. Three deployments were made using the longer-duration Multi-Year Return Time Level Equipment (MYRTLE) instrument that provided BP records approximately 4 years long (Fig. 1b). The measurement programme was terminated in 2016, resulting in a BP data set spanning almost three decades.

The measurements have been used in studies of ACC variability, as reviewed by Meredith et al. (2011). BP measured at the south side of the Drake Passage has been shown to be particularly useful as a monitor of fluctuations in ACC transport (Hughes et al., 2003; Hibbert et al., 2010). The data have even proved to be useful in studies of tsunami travel times (Rabinovich et al., 2011). In regard to tides, the data have been employed in validation studies of models of the semi-diurnal and diurnal tides observed by satellite altimetry (Ray, 2013).

BP has advantages in tidal studies over sea level recorded by conventional tide gauges at the coast. An obvious factor is that BPRs can be deployed in deep water offshore (pelagic), at some distance from where storm surges and other shallow-water processes are largest. Another factor is that much of the sea level variability due to air pressure changes is compensated automatically by air pressure itself in the bottom pressure measurement (the inverse barometer effect). As a result of these two factors, BP records tend to have a smaller percentage of non-tidal variability (or “noise”) than do tide gauge records.

The main disadvantages of a BP record are instrumental drift (also known as “creep”) and the absence of a geodetic datum. Fortunately for tidal studies, creep is a slow, monotonic process that tends not to impact upon the determination of high-frequency components of the record, such as the semi-diurnal and diurnal tides (Watts and Kontoyiannis, 1990; Spencer and Vassie, 1997; Polster et al., 2009). Instrumental drift does tend to preclude the reliable observation of annual and semi-annual tides in BPR data. However, these long-period tides are not of lunar origin and so are not the concern of the present investigation. The absence of a datum is an important factor when it is required to combine individual yearly records into longer, continuous records. Unless overlapping records are available, from which the datum of one deployment can be related to that of another, then techniques such as “end-point matching” have to be employed (e.g. Meredith et al., 2004). Although we make use of one such combined record below, in order to demonstrate clearly the existence of long-period tides in the data, combinations of records are not required for most of the present study in which we analyse the records from each deployment separately.

Almost all the Drake Passage BPR data obtained by NOC since 1992 have been reanalysed recently as part of a Natural Environment Research Council (NERC) project called “Weighing the Ocean”. (Several NOC deployments in the centre of the Drake Passage and in the Scotia Sea were not included.) Data were subjected to a new set of quality control that identified any suspect measurements and corrected as far as possible for timing uncertainties and instrumental drift. The processed data, consisting of records from 35 individual deployments, are available on the website of the Permanent Service for Mean Sea Level (PSMSL) (http://www.psmsl.org, last access: 1 June 2018). A total of 10 other records were added from deployments before 1992 at the Drake Passage and from the Falkland–Signy (F–S) line (Woodworth et al., 1996). These earlier records can be obtained from http://www.ntslf.org/files/acclaimdata/bprs/ (last access: 1 June 2018).

Figure 2 shows the locations of the 45 deployments, many of which were at essentially the same positions and so overlap on the map. The 35 locations with reanalysed data from the PSMSL website include those on the north and south sides of the Drake Passage south of the Falkland Islands, and that of the first of the three MYRTLE deployments close to Signy Island. The two other MYRTLE deployments were also made on the south side but in more central positions. The 10 earlier deployments include the westernmost north–south pair and those on the F–S line to the east.

We have treated the data from each deployment as separate records, with record lengths from 296 to 1470 days. Each record of BP was first subjected to a tidal analysis consisting of typically 57 semi-diurnal, diurnal and higher-frequency constituents, the exact number of constituents depending on the record length. However, importantly, long-period tides were not included in the tidal analysis. Residuals of the analysis were interpolated to hourly values, and simple arithmetic averaging of the 24-hourly residuals each day provided the time series of daily mean values of BP that are discussed below. For full details of the data processing, see http://www.psmsl.org/data/bottom_pressure/processing_procedures.php (last access: 1 June 2018).

The evidence for long-period tides in the BP data is demonstrated clearly in Fig. 3a. In this case, the individual time series of daily mean BP were de-meaned and de-trended and, when daily values were available from more than one location on the same day (i.e. from both north and south sides of the Drake Passage), they were averaged, so providing a continuous, composite time series spanning over 26 years. Figure 3a shows the resulting power spectrum which indicates clearly the presence of Mm (period of 27.55 days), MSf (14.77 days), Mf (13.66 days) and Mt (9.13 days). Such tidal signals are obviously less well resolved when analysing records individually (Fig. 3b). In this case, we are dealing with records of different lengths, at times when the relative proportions of each long-period component (primarily Mf) will be different and when there will be different proportions of tidal and non-tidal variability. Consequently, spectra were produced for each individual record, normalised to have unit energy in the long-period tidal band (0.02–0.15 cpd) and then averaged into bins of 0.005 cpd, thus providing a spectrum “typical” of an individual record. It can be seen that Mf, and to a lesser extent Mm, are still present, while Mt is less well resolved, and MSf cannot be seen above the background.

MSf is an interesting constituent that occurs for two reasons. It is partly a long-period tide in its own right, with an amplitude in the equilibrium tide 8.7 % that of Mf, and with variations in amplitude through the nodal cycle of ±14 %. It is also partly an interaction constituent (see below), with a nodal variation of ±3.7 % as for M2 in Eq. (3). However, its generally low amplitude suggests that verification of its nodal variation in real data will be much harder than for Mf, Mm and Mt, and we have not considered MSf in detail further.

In order to study the time dependence of the long-period tides, their amplitudes and phase lags were determined for each deployment record independently by means of a regression of the daily means of BP in terms of three harmonics with periods of Mf, Mm and Mt plus a linear trend. The three periods are so different that the amplitudes and phase lags determined for each harmonic are almost the same whether the regression includes all three constituents or each one individually. This procedure assumes that the amplitudes and phase lags of each harmonic do not change during the record, i.e. h(t)=Hicos⁡ωt-φi, where h(t) is BP for a particular harmonic that is a function of time t measured from the start of 1988, ω=2πperiod radians per day, and Hi and φi are the amplitude and phase lag from the regression for deployment i. Therefore, one can investigate how Hi varies as a function of the central date of each record (Ti), and similarly from Eqs. (2) and (5) one can relate φi+A=G-ui, where the variation of φi as a function of Ti can be described by an oscillation (ui) around the average phase lag (G). A is the astronomical argument for the harmonic constituent concerned at the start of 1988. If the start of that year is defined by GMT (UT), then G will be the constituent's Greenwich phase lag.

The regression is made using the G02CGF function of the Numerical Algorithms Group (NAG) library (https://www.nag.co.uk, last access: 1 June 2018). This results in the determined amplitudes for the three harmonics having the same standard errors, while standard errors on each phase lag are defined by the standard error on the amplitude divided by the amplitude itself (times 360∘/2π). The same standard error for each amplitude arises from an assumption of white noise in the residuals of the regression. Consequently, they may be potentially estimated too low (see below). However, the magnitude of scatter of the points relative to the nodal fits in Figs. 5–8, compared to their individual formal errors, suggests that the standard errors will have been estimated fairly reliably.

Some of the findings of the previous section are consistent with expectations from the equilibrium tide, while those that are not require explanation.

As mentioned above, the long-period tides in the equilibrium tide have simple spatial distributions in amplitude and phase, with north–south variations only. However, their spatial distributions in the real ocean are now known to depart considerably from equilibrium expectations, with larger departures at shorter periods (e.g. see Fig. 2 of Ray and Erofeeva, 2014). These differences are most evident when contrasting the Pacific, Atlantic and Indian Ocean low-latitude and midlatitude basins.

If one considers Mf in particular, atlases of this constituent have been available for many years, notably since the data assimilation numerical modelling of Schwiderski (1982). More recent co-tidal distributions for Mf have been obtained from altimeter measurements and models by Kantha et al. (1998, Fig. 7), Mathers and Woodworth (2001, Plate 4) and Egbert and Ray (2003, Fig. 1). These are consistent with Mf phase lag increasing when travelling south down the Pacific coast of South America, with the 180∘ contour around the Drake Passage, and with a complicated amphidromic pattern in the South Atlantic to the NE of the Falklands. More recent studies have included the development of the FES2004 ocean tide model, which also showed these features (Lyard et al., 2006, Fig. 2), with roughly the same Mf amplitude on both sides of the Drake Passage and larger phase lag on the south side than on the north side.

FES2014 (Finite Element Solution 2014) is the latest in the series of state-of-the-art global ocean tide models provided by French groups. It provides elevations and currents (amplitude and phase) and tidal loading information for 34 tidal constituents on a global 1/16∘×1/16∘ grid. FES2014 (2018) provides more detailed information.

Figure S2a and b show the Mf amplitude and phase lag for Mf at the Drake Passage from the FES2014 model. Some points of consistency with our findings are as follows. First, the model has much the same amplitude over the whole area (∼2 cm), and phase lags are essentially zonal, largely justifying our decision to combine amplitudes and phase lags from all deployments in Fig. 5, and the subsequent discussion in terms of north- and south-side values.

Second, we found the amplitudes for Mf to be similar on the north and south sides of the Drake Passage (Fig. 5a), but phase lags were shown to be 22±2∘ larger for the southern deployments (Fig. 5b). The latter is qualitatively consistent with Fig. S2b. Third, the 192∘ average phase lag for Mf from all the BPRs taken together (Sect. 3.1, Fig. 5b) is consistent with the ∼190∘ contour in mid-passage in Fig. S2b. In addition, the Mf harmonic constants estimated above for Vernadsky using DAC-corrected data (2.49 cm amplitude and 208∘ phase lag) are similar to those in FES2014 (2.41 cm and 202∘, respectively). FES2014 amplitudes and phase lags for Mm and Mt (1.31 cm and 190∘ and 0.42 cm and 211∘, respectively) are all consistent with DAC-corrected Vernadsky findings to within ∼1 or ∼2 SD (standard deviations) for amplitudes and phase lags, respectively.

If a tide model such as FES2014 was perfect, then any differences in observed amplitudes and phase lags due to the different deployment locations could be removed by relating each set of findings to those which would have been obtained at a reference location, using an admittance relationship: Hiref=HiHMrefHMi, where Hi is the measured amplitude for deployment i, HMi and HMref are the model amplitudes at the deployment and reference point locations, respectively, and Hiref is the inferred amplitude at the reference point. Similarly, φiref=φi+φMref-φMi, where φi is the measured phase lag for deployment i, φMi and φMref are the model phase lags at the deployment and reference point locations, respectively, and φiref is the inferred phase lag at the reference point. If the model represented the spatial dependence of the tide correctly, then Hiref and φiref should have only a temporal dependence.

Figure 5c shows the resulting model-adjusted values of Mf amplitude, using a reference point location of 57∘ W, 58∘ S, demonstrating satisfactory consistency between values north and south. That was already the case in Fig. 5a, and the similarity of Fig. 5a and c reflects the uniformity of amplitude in the model in this area. The nodal fit in red shows a cosine with an amplitude of 43±3 % of the mean, which is identical to that in Fig. 5a. For phase lag, Fig. 5d demonstrates a considerable improvement compared to Fig. 5b, with values north and south in agreement (weighted south–north difference of 0±2∘). In addition, the nodal fit in red has an amplitude of 23.4±1.4∘, which is closer to Eq. (4) than that for Fig. 5b.

Consequently, the temporal variation of Mf can be seen from Fig. 5 to conform closely to expectations from its equilibrium form shown by Eq. (4). Mf has the largest amplitude of the long-period tides we have investigated, which together with its relatively short period compared to the typically 1-year long records, means that it is the best resolved. Our finding of consistency with equilibrium expectations parallels an observation regarding fortnightly variations in the solid Earth in a study of polar motion data by Ray and Egbert (2012), who concluded that a similar admittance applied to Mf and its nodal sideband (see also earlier work by Gross, 2009). As explained above, the same admittance for a central frequency and its sidebands indicates that the nodal factors of the equilibrium tide apply equally as well to the tide in the real ocean (or solid Earth).

Turning to Mm, its spatial variation in FES2014 is shown in Fig. S2c and d. Once again, amplitudes are much the same over the whole area, and phase lag contours are roughly zonal. However, in this case, Fig. S2d indicates a north–south gradient of phase lag about half that for Mf in Fig. S2b. Our observation of a small south–north difference of 2±3∘ is qualitatively consistent with the smaller gradient in the model (a south–north difference of ∼10∘). The observed average phase lag of 177∘ for Mm from all deployments combined (Sect. 3.1) is a little lower than the ∼185∘ contour in mid-passage in Fig. S2d.

Figure 6a shows that Mm amplitudes for the first decade are lower in the south, but they become more equal to the northern ones thereafter. One may note that five of the six deployments with particularly low amplitudes before 1994 are from the F–S line. However, some kind of general amplitude bias in these early deployments is unlikely, given that their corresponding amplitudes for Mf are consistent with later ones (Fig. 5a). Overall, Fig. 6a does not provide evidence for a temporal dependence of Mm amplitude similar to that of Eq. (8). However, identifying a nodal signal of only ∼0.15 mbar is clearly a challenge given the uncertainties. At least, the absence of any evidence for nodal variation in Mm phase lag (Fig. 6b) is consistent with Eq. (8).

A nodal variation in amplitude of ∼0.15 mbar might be technically within the resolution of the BPR measurements if Mm was accompanied by only a limited amount of non-tidal variability on similar (monthly) timescales. Monthly timescales are more comparable to processes associated with ACC variability. Sheen et al. (2014) showed that eddy kinetic energy is more intense in the north of the Drake Passage, where the main fronts and their meanders occur. However, eddy activity also occurs in the south. In addition, variability in BP in this region has a contribution on 30- to 70-day timescales from the Madden–Julian Oscillation (Matthews and Meredith, 2004), which could potentially impact our determination of Mm.

An attempt was made to reduce the amount of non-tidal variability in the records with the use of 5-day values of BP from the Nucleus for European Modelling of the Ocean (NEMO) 1/12∘ ocean circulation model for 1988–2012 (Hughes et al., 2018), with the aim of better resolving any nodal tidal signals, particularly that for Mm. The model BP was found to have a high correlation with measured non-tidal BP for most of the southern deployments, while correlations were weaker in the north, as Sheen et al. (2014) would suggest. However, subtraction of the model values from the measurements resulted in little change in the determined Mm amplitudes and phase lags.

FES2014 model adjustments for Mm from Eqs. (9) and (10) result in Fig. 6c and d. Figure 6c confirms similar amplitudes north and south, and the nodal fit gives an amplitude of 0.8±4.3 % of the mean value, a little larger than that from Fig. 6a, but still 3σ away from that expected in Eq. (8). As for phase lag (Fig. 6d), the weighted south–north difference is now -11±3∘, as can be readily observed by eye. This indicates that the model overcorrects for spatial variation in phase lag. This suggests that the difference in Mm phase lag across the real Drake Passage is less than in the model.

One might have expected the detection of Mt to be easier than that of Mm, thanks to its shorter period, even though it has a much smaller amplitude. Figure 7a shows an average amplitude of 0.43 mbar, with little evidence for differences between values north and south, while Fig. 7b indicates an average phase lag of ∼197∘, and some evidence for phase lags about 22∘ larger in the south than in the north. The temporal variations in Mt amplitude and phase lag in Fig. 7a and b are consistent with equilibrium expectations within the large uncertainties for this small constituent.

Figure S2e and f give the corresponding information for Mt from the FES2014 model. (This constituent is called Mtm in the model.) Figure S2e shows an amplitude of ∼0.4 cm over most of the area, while Fig. S2f shows a meridional gradient for phase lag similar to that obtained from the BPRs. The observed mean phase lag of 197∘ (Sect. 3.1) is consistent with the mid-passage contour in Fig. S2f.

If one applies the FES2014 model adjustments from Eqs. (9) and (10) to the observed amplitudes and phase lags for Mt, then one obtains Fig. 7c and d. This procedure results in apparent improvements as for Mf. Figure 7c is much the same as Fig. 7a, with similar amplitudes north and south. The nodal fit in Fig. 7c has an amplitude of 31±14 % which is similar to that obtained in Sect. 3.1 for Fig. 7a. For phase lag, Fig. 7d demonstrates an improvement compared to Fig. 7b with a weighted south–north difference of -5±9∘, consistent with zero difference. The nodal fit shows an amplitude of 23.0±6.9∘, closer to Eq. (4) than the value for Fig. 7b obtained in Sect. 3.1.

As an aside, one can mention that Mt is to some extent a “forgotten constituent”. It is represented in harmonic expansions of the tidal potential (Doodson, 1921; Cartwright and Tayler, 1971) as a line with Doodson number 0, 3, 0, -1, 0, 0 (or 085.455 in Doodson's notation) with one major nodal sideband (0, 3, 0, -1, 1, 0). However, Doodson did not usually refer to it explicitly in his own papers (e.g. Doodson, 1928), and it is not included in the standard sets of harmonic constituents used in tidal analysis packages (e.g. Bell et al., 1996), even though Fig. 3 shows that it is resolvable at higher latitudes, at least in BPR data. One supposes that the reason for lack of interest in this constituent by previous tidal analysts has been due to its smaller amplitude at low latitudes and midlatitudes and to the generally higher level of noise in tide gauge records.

There are several complications we are aware of in the above analyses. One is that when measurements are combined from different locations, the observed tidal amplitudes should be adjusted for spatial variations in water density, latitude-dependent variations in acceleration due to gravity and depth-dependent compressibility of seawater. However, these will be at the ∼1 % level (Ray, 2013) and so are much less than other uncertainties.

A second complication concerns whether imperfections in our tidal analyses and subsequent averaging of the BP residuals into daily means of BP could have aliased residual components of the main diurnal and semi-diurnal tides into frequencies similar to those of the three long-period tides. We do not believe this is an important issue. All of the tidal analyses were subjected to quality control to check that tidal and non-tidal components of the records were separated efficiently. However, any residual tidal signals would then have been considerably reduced by the daily averaging. For example, the amplitude of any residual M2 would have been reduced to approximately 3.5 % of its original value and aliased into the period of MSf (14.77 days). Consequently, while it is possible that aliasing could have contributed to some of the MSf in Fig. 3a, we believe most of that to be real. Furthermore, it is hard to see how the observed Mf, Mm and Mt could have been affected to any significant extent by aliasing. In principle, residuals of the tiny constituents OP2, Lambda2 and SNK2 could be aliased into Mf, Mm and Mt, respectively, although reduced to negligibility by the daily averaging. Lambda2 is included explicitly in the tidal analysis. The other two are interaction constituents (see below) and do not appear as significant lines in the tidal potential (Cartwright and Tayler, 1971).

A further complication is that there will be other constituents present in the data (i.e. genuine and not-aliased ones) with a similar period to Mf, Mm or Mt. We have ignored this complication for present purposes as the other constituents are likely to be small. In the case of Mf, the other main constituent will be MSf. MSf has an amplitude 9 % of that of Mf in the equilibrium tide, which is similar to that found in the composite BPR record (Fig. 3a). Similarly, MSm (period of 31.81 days) has an amplitude of 19 % of Mm in the equilibrium tide, and MSt (period of 9.56 days) is 19 % of Mt. However, there is little evidence for significant amounts of either in Fig. 3a. In principle, these other constituents should be separable from Mf, Mm and Mt given a year of data. One might imagine a more sophisticated harmonic expansion in future work in which information on these and other constituents is inferred from ocean tide models.

Another complication is that observations of the three long-period tides considered here can contain contributions from non-linear interactions between shorter-period tides. For example, the difference between K1 and O1 frequencies is identical to that of Mf, and so their interaction can contribute to the observed Mf. K2 and M2 interactions can also contribute. Similarly, M2 and S2 can provide an interaction with the same frequency as MSf, which is similar to that of Mf. N2 and M2 interaction can contribute to Mm. An interaction will have an f and u determined by the product of the individual f and u values of the two short-period tides involved (see Table 4.4 of Pugh and Woodworth, 2014). Therefore, interaction nodal factors will be different from those of the long-period tide. This complication is primarily an issue for shallow waters, rather than the deeper ocean areas of the Drake Passage where our BPR measurements are located. Nevertheless, it should be possible to estimate the contributions from such interactions using tide modelling.

A final complication relates to all BP spectra having a continuous non-tidal background in addition to a tidal line spectrum (e.g. Fig. 3). The background will tend to increase the amplitudes calculated for each tidal constituent (see Appendix B of Munk and Cartwright, 1966 and discussion in Wunsch, 1967). We simply note that this aspect would impact primarily our determination of Mm. Another issue to do with the background is that it is not white noise. As mentioned above, this could lead to the errors in the harmonic analysis regressions being underestimated (e.g. Williams, 2003). In fact, the background spectra for all of the 45 BPR deployments are similar and can be parameterised reasonably well by a (frequency k) dependence where k∼-1.5 (Fig. S3). This suggests similar biases in estimated errors for each constituent for each deployment. Such biases, as long as they are similar in each case, should not significantly affect the fits to determined parameters from all deployments in Figs. 5–7.

If one has several decades (or at least 19 years) of good tide gauge (or, in theory, BPR) data available for a tidal analysis then, if background noise levels allow, it should be possible to avoid having to consider the nodal sidebands as perturbations of the main harmonic via the use of “nodal factors” f and u. Instead, one can treat them as independent constituents and make an explicit determination of their amplitudes and phase lags. Examples of such analyses of long records include those of Amin (1983) and Foreman and Neufeld (1991).

However, in practice, most tidal analyses are made using one or several years of data, for which assumptions are required for f and u. The drawbacks of this approach have been recognised for many years but primarily for the semi-diurnal and diurnal constituents. As far as we know, the question of whether the variation of the long-period tides through the nodal cycle differs from equilibrium expectations has never been investigated properly.

In this paper, we have used data from 45 separate BPR deployments in the Drake Passage, and 31 years of hourly tide gauge data from the Vernadsky Research Base in Antarctica, to estimate how well the nodal variation of the amplitudes and phase lags of Mf, Mm and Mt compares to expectations from the equilibrium tide. Our analysis uses simple harmonic expansions of daily values of BP or sea level at each location.

The combined data set provides information on how the amplitudes and phase lags of each constituent vary between the north and south sides of the Drake Passage. The measurements indicate that amplitudes are similar throughout the region, which is consistent with a state-of-the-art ocean tide model (FES2014, 2018). Phase lags for Mf and Mt are ∼20∘ larger in the south than in the north, which is also consistent with the model. However, the observed south–north difference in Mm phase lag is consistent with zero, compared to ∼10∘ in the model. In fact, the Mm difference is probably consistent with the model given the uncertainties, and at least the BPR data and FES2014 are in agreement on indicating a smaller meridional gradient for Mm phase lag than for the other two constituents. Any detailed differences for all the long-period tides may be understood better by future modelling.

However, our main interest is in the temporal variability of the long-period tides. The variation of the amplitudes and phase lags of Mf and Mt in the BPR data has been found to be consistent with that suggested by the equilibrium tide within their uncertainties. To a great extent this is an expected finding given that, as explained in the introduction, the long-period tides are closer to equilibrium than the diurnal and semi-diurnal tides, and the frequencies of the nodal sidebands are close to that of the central line. Nevertheless, this is a reassuring finding for tidal analysts who might now (in this region at least) be able to employ f and u for the long-period tides as anticipated. The variation in phase lag of Mm (or rather its non-variation) is also consistent with equilibrium expectations. The absence of an expected 13 % variation in the amplitude of Mm (Eq. 8) at 3σ level (or possibly less if, as explained above, our uncertainties were slightly underestimated) is probably due to the background of non-tidal variability in the ocean circulation in this energetic area and/or in our inability to account adequately for spatial variations in Mm amplitude with the use of FES2014.

Our study has shown clearly that BPR data have advantages over conventional tide gauge measurements in long-period tidal studies such as this. Section 3.2 showed that, when Vernadsky coastal tide gauge data were corrected for non-tidal variability, a major improvement in identification of the long-period tides results (e.g. reduction in the uncertainties for Mf in Table 1 by a factor of 2). However, the Drake Passage BPRs, which were located in deeper water where the inverse barometer-related sea level variations are compensated automatically by BP itself, have still provided more accurate estimates of nodal variation, in spite of different locations for deployments. Table 1 demonstrates that the uncertainties for Vernadsky Mf, even when DAC-corrected, are still double those of the FES2014-corrected BPRs. (A similar conclusion can be obtained from inspection of the uncertainties displayed in Figs. 5c, d and 8c, d.) Nevertheless, it is the case that there is a lot more tide gauge data available for study worldwide than BPR data (Woodworth et al., 2017). Therefore, an obvious recommendation following from the present work is that tide gauge data be investigated more completely in order to investigate whether the temporal variation of long-period tides conforms to equilibrium expectations, perhaps by employing “stacks” of records, as has been used previously to investigate other long-period components of tide gauge records (e.g. Trupin and Wahr, 1990), with DAC-type corrections applied to each record.