Technical note: On seasonal variability of the M_{2} tide
 Geodesy & Geophysics Lab., NASA Goddard Space Flight Center, Greenbelt, Maryland, USA
 Geodesy & Geophysics Lab., NASA Goddard Space Flight Center, Greenbelt, Maryland, USA
Abstract. Seasonal variability of the M_{2} ocean tide can be detected at many ports, perhaps most. Examination of the cluster of tidal constituents residing within the M_{2} tidal group can shed light on the physical mechanisms underlying seasonality. In broadest terms these are astronomical, frictional/advective interactions, and climate processes; some induce annual modulations, some semiannual, in amplitude, phase, or both. This note reviews how this occurs and gives an example from each broad category. Phase conventions and their relationship to causal mechanisms, as well as nomenclature, are also addressed.

Notice on discussion status
The requested preprint has a corresponding peerreviewed final revised paper. You are encouraged to refer to the final revised version.

Preprint
(277 KB)

The requested preprint has a corresponding peerreviewed final revised paper. You are encouraged to refer to the final revised version.
 Preprint
(277 KB)  BibTeX
 EndNote
 Final revised paper
Journal article(s) based on this preprint
Richard Ray
Interactive discussion
Status: closed

CC1: 'Comment on egusphere2022252', Haidong Pan, 01 May 2022
This is a nice paper which clarifies the seasonality of M2 tide. For a long time, I am puzzled about the use of H1 for a semidiurnal wave and it seems that no one (except this paper) can give me the answer. I have few suggestions which may further improve this paper.
First, I strongly suggest the author to also discuss the seasonality of S2, K1 and O1 tides. In most previous studies, they only focus on the seasonality of M2 but ignore S2,K1 and O1. In fact, there are also lots of confusion on the seasonality of S2/K1/O1 tides. Du and Yu(2021) only clarify some confusion on the seasonality of M2. When I discussed with them, I was surprise that they did not know the seasonality of S2, K1 and O1 tides at all. The frequency of the K2 (P1) tide is equal to that of the S2 (K1) plus (minus) the frequency of the semiannual cycle. When we explore the seasonality the K1 and S2 tides, we need to remove P1 and K2 tides first via harmonic analysis (HA). However, HA is a frequencydepend method, it can extract the amplitude and phase of one specific frequency, thus, HA cannot distinguish different origins of a constituent which means that partial semiannual cycles of K1 and S2 tides are also removed. How to solve this problem?
Second, section 3 shows three nice examples of M2 seasonality. Maybe you can add some maps/tables of tide gauges and tidal information which can help readers know more about local environment and tidal dynamics.
Finally, the nonlinear interaction between K1 and O1 tides can generate KO2 tide which has the same frequency as M2. Since K1 and O1 tides show clear seasonality, thus,KO2 should also have clear seasonality which means that the energy of K1/O1 seasonality is transferred to M2 seasonality. Also, the nonlinear interaction between P1 and O1 tides can generate OP2 tide which has the same frequency as MSK2. Thus, the energy of P1/O1 seasonality can be transferred to M2 seasonality. I think above processes may occur in some coastal areas where diurnal tides are very strong and can be added into the paper.

AC1: 'Reply on CC1, regarding tides other than M2', Richard Ray, 09 Jun 2022
As Haidong Pan notes, there are many tidal constituents in addition to (the usually dominant) M2 that could be examined for seasonal variability. For lunar tides like O1 and N2, any analysis of seasonality should begin by checking for the presence of nearby spectral lines, in the manner laid out in my Note for M2. In each case there are small astronomical constituents within the relevant tidal group, as well as compound tides and climatedriven lines at frequencies 1 or 2 cpy away from the central constituent. One simply needs to take care not to overlook an important contributor, such as a compound tide (of which there are many possible).
The solar tides, as Haidong Pan points out, are more problematic. For S2 there is semiannual modulation from K2, but also annual modulation from T2 and R2. Harmonic analysis is probably not recommended in this case, but some rough results can be obtained by a response analysis (e.g., Cartwright, 1968). With a response analysis, the gravitational parts of K2, T2, and R2 can be approximately determined from estimates of the mean S2 tide. Any residual modulations seen in S2 can then presumably be attributed to seasonal climate variability. But this is only a "rough" approach, because it overlooks the radiational forcing of S2, caused by loading by the S2 atmospheric tide, which itself has significant seasonal variability. The radiational forcing of the S2 ocean tide has been studied by Arbic (2005) and others. In the end, even with a response analysis, we may isolate a seasonal S2 signal, but it may not be clear whether it is originating in the ocean (e.g., from seasonal stratification) or in the atmosphere (from air tides).
The diurnal K1 is just as difficult, if not more so. Again a response analysis may be used to determine the gravitational parts of the neighboring constituents P1 and psi1. But the S1 constituent, 1 cpy from K1, is almost wholly radiational, with significant temporal variability. In monthly estimates of K1, modulations from S1 would be difficult to untangle from other seasonal changes.
For these reasons, analysis of lunar tides is more straightforward. Understanding their seasonality can still be difficult and even the spectral approach requires care. For example, O1 seasonality was briefly examined at station Lusi (China) in the course of a review of coastal tides (Ray et al., 2011). Small semiannual variations in O1 (of order 1%) will be induced by the linear constituent tau1, 2 cpy away, but monthly estimates of O1 at Lusi revealed significantly larger variations than that. Unfortunately, in our 2011 discussion, we overlooked the possible presence of the compound tide MP1 (which coincides with tau1). Because both M2 and P1 are large along the China coastline, the compound MP1 is almost certainly responsible for most of the semiannual oscillation seen in monthly estimates of O1 at Lusi. Nonetheless, there is still a significant annual variation in O1, which is probably induced by climatic changes in ocean stratification, in the same manner as seen for M2 in that region (Kang et al., 2002).
Finally, seasonality of the compound KO2 could indeed mimic seasonality in M2. Presumably, in general, the effect is very small, but it is potentially at work in shallowwater regions with large diurnal tides and small M2.
References:Arbic, B. K. (2005), Geophys. Res. Lett., 32, doi: 10.1029/2004GL021668.
Cartwright, D. E. (1968), Phil. Trans. R. Soc., 263, 155.
Kang, S. K. et al. (2002), J. Geophys. Res., 107, doi: 10.1029/2001JC000838.
Ray, R. D., G. D. Egbert, S. Y. Erofeeva (2011), in "Coastal Altimetry" (Eds: S. Vignudelli et al.), Springer, pp. 191216.

AC1: 'Reply on CC1, regarding tides other than M2', Richard Ray, 09 Jun 2022

RC1: 'Comment on egusphere2022252', Qian Yu, 15 May 2022
This manuscript is on the seasonal variability of the M2 tide, which has been widely recognized based on global observations. It provides a comprehensive and clear description of the origin of the seasonal variability of the M2 tide, which has not been found in literatures. In addition, the nomenclature used in the M2 tidal group is clarified, and H1 and H2 should be abandoned.
I believe that it will be a paper of high impact. The concept is fundamentally important to be written in the textbooks on the ocean tides. Thus, I recommand its publication in the current form.

AC2: 'Reply on RC2', Richard Ray, 06 Jul 2022
I thank the two anonymous reviewers, along with Haidong Pan, for their feedback. A number of the points brought up are actually somewhat general questions on tidal analysis, not specifically on the paper itself, so these online discussions are a useful way to address them. To augment the earlier discussion prompted by Dr Pan, I can respond to two topics brought up by Referee 2.
One relates to the length of time series needed for this kind of tidal analysis, and notably what are the consequences of having "less than a year's worth of data." For the examples in the paper, I had purposely chosen tide gauges with many years of highquality data. A long time series ensures that computed spectra (Figure 1) have adequate spectral resolution and that computations of monthly mean amplitudes and phases have relatively small error bars. But there is no hard rule for the minimum amount of data needed before one can proceed.
Of course, it is implausible to study seasonal variability without data spanning most of a full year. Multiple years are required if a computed spectrum is to have sufficient resolution to separate constituents with frequencies differing by 1 cpy. Spectral analysis is not mandatory, but it is certainly helpful to determine whether a spectrum contains isolated lines (as is the case for Figure 1) or instead is simply a wide cusp of energy surrounding M2. One's interpretation of tidal variability would be sharply different in these two cases.
A simple Rayleigh criterion for separating constituents would also call for at least one year of data. Yet that criterion is only a rough ruleofthumb and depends on noise levels. Munk and Cartwright (1966) emphasized this by noting that it is possible to separate two nearby sine waves with only four perfect, noisefree observations, but with realworld noise something like a Rayleigh criterion is probably required. Of course, bottom pressure measurements of tides are usually much less noisy than surface height measurements, so there is flexibility in all this.
A second point of Referee 2 concerns large gaps in the Port Orford time series before May 2002, which I can confirm. These gaps could conceivably impact spectral analyses, but the gaps are followed by at least seventeen years without gaps, which are more than sufficient to support spectral calculations of good frequency resolution. The calculations of monthly mean harmonic constants are unaffected, as each monthly mean is based on between 22 to 24 monthly estimates. Admittedly, the amplitudes in Figure 2 do not perfectly overlay. This likely stems as much from unavoidable estimation error (from inherently noisy measurements) as it does from any gaps. Yet one should not belabor these small amplitude differences. In fact, the amplitudes differ by a span of only 3 mm, and so they are actually quite consistent.
One item left unaddressed in Dr Pan's discussion is his suggestion to include more constituents than M2. While I sympathize with this desire, doing so would turn this "note" into a much longer paper, which I am unprepared to tackle at present. Moreover, any discussion of solar tides would likely be unsatisfactory owing to complications from radiational tides, as noted.
The revised paper now acknowledges the possibility of seasonality in KO2 adding to seasonality in M2, in those rare cases where M2 is very small and KO2 unusually large. Finally, although not prompted by the reviews, I have inserted a sentence at the beginning of Section 3 which gives an additional technical detail on how monthly tides were estimated.

AC2: 'Reply on RC2', Richard Ray, 06 Jul 2022

RC2: 'Comment on egusphere2022252', Anonymous Referee #2, 18 May 2022
This manuscript on the seasonality of the M2 tide is an extremely relevant publication for the Ocean Sciences journal. Although, as the author states, the manuscript really summarises several wellknown points and introduces evidence on these theories that date back to the early 1900s, the manuscript produces a comprehensive description that is of significant value to the tidal community. A general description of the tides, their sources and their relationships is in itself a valuable contribution and something often overlooked in the tidal community. I am very much a fan of Table 1 in providing very simple and important details. This manuscript will, therefore, be a valuable source of knowledge for the greater tidal community. Overall, the manuscript itself is a pleasure to read. Although, in my opinion, the manuscript is publishable as is, I do have a couple of comments which can hopefully clarify some points within the manuscript.
 One point the author highlights is the length of the time series of data needed to separate the tidal constituents within the M2 tidal band. The explanations in Table 1 and Figure 1 demonstrate this nicely. I guess what is not clear, what is the implications when one does not have a long enough time series, on the estimation of the M2 and the overall tidal height prediction? Should one where possible directly estimate these sidelines and if not possible, what are the implications on the accuracy of tidal predictions? I realise the sidelines are usually fractions of the main M2, but the Chittagong application for example demonstrates significantly large modulations. This is of course more critical in tide gauges/bottom pressure sensors that have less than a year's worth of data or less frequent sampling patterns such as altimetry observations.
 When reproducing the tide gauge evaluation in section 3, I found the same results as the author. However, in the period selected by the author, I noted a large temporal gap in the University of Hawaii dataset for Port Orford (shown below). I checked this with the PSMSL data (https://www.psmsl.org/data/obtaining/stations/1640.php) as well as GESLA3 (flagged as 99 below) and UHSLC data. This is not a criticism of the results as these gaps in data are normal, but could this be an explanation for the differences seen in Figure 2? It could also be that the author has appropriate data from this tide gauge.
Figure. SLA over time of the Port Orford tide gauge as obtained from GESLA3.

AC2: 'Reply on RC2', Richard Ray, 06 Jul 2022
I thank the two anonymous reviewers, along with Haidong Pan, for their feedback. A number of the points brought up are actually somewhat general questions on tidal analysis, not specifically on the paper itself, so these online discussions are a useful way to address them. To augment the earlier discussion prompted by Dr Pan, I can respond to two topics brought up by Referee 2.
One relates to the length of time series needed for this kind of tidal analysis, and notably what are the consequences of having "less than a year's worth of data." For the examples in the paper, I had purposely chosen tide gauges with many years of highquality data. A long time series ensures that computed spectra (Figure 1) have adequate spectral resolution and that computations of monthly mean amplitudes and phases have relatively small error bars. But there is no hard rule for the minimum amount of data needed before one can proceed.
Of course, it is implausible to study seasonal variability without data spanning most of a full year. Multiple years are required if a computed spectrum is to have sufficient resolution to separate constituents with frequencies differing by 1 cpy. Spectral analysis is not mandatory, but it is certainly helpful to determine whether a spectrum contains isolated lines (as is the case for Figure 1) or instead is simply a wide cusp of energy surrounding M2. One's interpretation of tidal variability would be sharply different in these two cases.
A simple Rayleigh criterion for separating constituents would also call for at least one year of data. Yet that criterion is only a rough ruleofthumb and depends on noise levels. Munk and Cartwright (1966) emphasized this by noting that it is possible to separate two nearby sine waves with only four perfect, noisefree observations, but with realworld noise something like a Rayleigh criterion is probably required. Of course, bottom pressure measurements of tides are usually much less noisy than surface height measurements, so there is flexibility in all this.
A second point of Referee 2 concerns large gaps in the Port Orford time series before May 2002, which I can confirm. These gaps could conceivably impact spectral analyses, but the gaps are followed by at least seventeen years without gaps, which are more than sufficient to support spectral calculations of good frequency resolution. The calculations of monthly mean harmonic constants are unaffected, as each monthly mean is based on between 22 to 24 monthly estimates. Admittedly, the amplitudes in Figure 2 do not perfectly overlay. This likely stems as much from unavoidable estimation error (from inherently noisy measurements) as it does from any gaps. Yet one should not belabor these small amplitude differences. In fact, the amplitudes differ by a span of only 3 mm, and so they are actually quite consistent.
One item left unaddressed in Dr Pan's discussion is his suggestion to include more constituents than M2. While I sympathize with this desire, doing so would turn this "note" into a much longer paper, which I am unprepared to tackle at present. Moreover, any discussion of solar tides would likely be unsatisfactory owing to complications from radiational tides, as noted.
The revised paper now acknowledges the possibility of seasonality in KO2 adding to seasonality in M2, in those rare cases where M2 is very small and KO2 unusually large. Finally, although not prompted by the reviews, I have inserted a sentence at the beginning of Section 3 which gives an additional technical detail on how monthly tides were estimated.
Interactive discussion
Status: closed

CC1: 'Comment on egusphere2022252', Haidong Pan, 01 May 2022
This is a nice paper which clarifies the seasonality of M2 tide. For a long time, I am puzzled about the use of H1 for a semidiurnal wave and it seems that no one (except this paper) can give me the answer. I have few suggestions which may further improve this paper.
First, I strongly suggest the author to also discuss the seasonality of S2, K1 and O1 tides. In most previous studies, they only focus on the seasonality of M2 but ignore S2,K1 and O1. In fact, there are also lots of confusion on the seasonality of S2/K1/O1 tides. Du and Yu(2021) only clarify some confusion on the seasonality of M2. When I discussed with them, I was surprise that they did not know the seasonality of S2, K1 and O1 tides at all. The frequency of the K2 (P1) tide is equal to that of the S2 (K1) plus (minus) the frequency of the semiannual cycle. When we explore the seasonality the K1 and S2 tides, we need to remove P1 and K2 tides first via harmonic analysis (HA). However, HA is a frequencydepend method, it can extract the amplitude and phase of one specific frequency, thus, HA cannot distinguish different origins of a constituent which means that partial semiannual cycles of K1 and S2 tides are also removed. How to solve this problem?
Second, section 3 shows three nice examples of M2 seasonality. Maybe you can add some maps/tables of tide gauges and tidal information which can help readers know more about local environment and tidal dynamics.
Finally, the nonlinear interaction between K1 and O1 tides can generate KO2 tide which has the same frequency as M2. Since K1 and O1 tides show clear seasonality, thus,KO2 should also have clear seasonality which means that the energy of K1/O1 seasonality is transferred to M2 seasonality. Also, the nonlinear interaction between P1 and O1 tides can generate OP2 tide which has the same frequency as MSK2. Thus, the energy of P1/O1 seasonality can be transferred to M2 seasonality. I think above processes may occur in some coastal areas where diurnal tides are very strong and can be added into the paper.

AC1: 'Reply on CC1, regarding tides other than M2', Richard Ray, 09 Jun 2022
As Haidong Pan notes, there are many tidal constituents in addition to (the usually dominant) M2 that could be examined for seasonal variability. For lunar tides like O1 and N2, any analysis of seasonality should begin by checking for the presence of nearby spectral lines, in the manner laid out in my Note for M2. In each case there are small astronomical constituents within the relevant tidal group, as well as compound tides and climatedriven lines at frequencies 1 or 2 cpy away from the central constituent. One simply needs to take care not to overlook an important contributor, such as a compound tide (of which there are many possible).
The solar tides, as Haidong Pan points out, are more problematic. For S2 there is semiannual modulation from K2, but also annual modulation from T2 and R2. Harmonic analysis is probably not recommended in this case, but some rough results can be obtained by a response analysis (e.g., Cartwright, 1968). With a response analysis, the gravitational parts of K2, T2, and R2 can be approximately determined from estimates of the mean S2 tide. Any residual modulations seen in S2 can then presumably be attributed to seasonal climate variability. But this is only a "rough" approach, because it overlooks the radiational forcing of S2, caused by loading by the S2 atmospheric tide, which itself has significant seasonal variability. The radiational forcing of the S2 ocean tide has been studied by Arbic (2005) and others. In the end, even with a response analysis, we may isolate a seasonal S2 signal, but it may not be clear whether it is originating in the ocean (e.g., from seasonal stratification) or in the atmosphere (from air tides).
The diurnal K1 is just as difficult, if not more so. Again a response analysis may be used to determine the gravitational parts of the neighboring constituents P1 and psi1. But the S1 constituent, 1 cpy from K1, is almost wholly radiational, with significant temporal variability. In monthly estimates of K1, modulations from S1 would be difficult to untangle from other seasonal changes.
For these reasons, analysis of lunar tides is more straightforward. Understanding their seasonality can still be difficult and even the spectral approach requires care. For example, O1 seasonality was briefly examined at station Lusi (China) in the course of a review of coastal tides (Ray et al., 2011). Small semiannual variations in O1 (of order 1%) will be induced by the linear constituent tau1, 2 cpy away, but monthly estimates of O1 at Lusi revealed significantly larger variations than that. Unfortunately, in our 2011 discussion, we overlooked the possible presence of the compound tide MP1 (which coincides with tau1). Because both M2 and P1 are large along the China coastline, the compound MP1 is almost certainly responsible for most of the semiannual oscillation seen in monthly estimates of O1 at Lusi. Nonetheless, there is still a significant annual variation in O1, which is probably induced by climatic changes in ocean stratification, in the same manner as seen for M2 in that region (Kang et al., 2002).
Finally, seasonality of the compound KO2 could indeed mimic seasonality in M2. Presumably, in general, the effect is very small, but it is potentially at work in shallowwater regions with large diurnal tides and small M2.
References:Arbic, B. K. (2005), Geophys. Res. Lett., 32, doi: 10.1029/2004GL021668.
Cartwright, D. E. (1968), Phil. Trans. R. Soc., 263, 155.
Kang, S. K. et al. (2002), J. Geophys. Res., 107, doi: 10.1029/2001JC000838.
Ray, R. D., G. D. Egbert, S. Y. Erofeeva (2011), in "Coastal Altimetry" (Eds: S. Vignudelli et al.), Springer, pp. 191216.

AC1: 'Reply on CC1, regarding tides other than M2', Richard Ray, 09 Jun 2022

RC1: 'Comment on egusphere2022252', Qian Yu, 15 May 2022
This manuscript is on the seasonal variability of the M2 tide, which has been widely recognized based on global observations. It provides a comprehensive and clear description of the origin of the seasonal variability of the M2 tide, which has not been found in literatures. In addition, the nomenclature used in the M2 tidal group is clarified, and H1 and H2 should be abandoned.
I believe that it will be a paper of high impact. The concept is fundamentally important to be written in the textbooks on the ocean tides. Thus, I recommand its publication in the current form.

AC2: 'Reply on RC2', Richard Ray, 06 Jul 2022
I thank the two anonymous reviewers, along with Haidong Pan, for their feedback. A number of the points brought up are actually somewhat general questions on tidal analysis, not specifically on the paper itself, so these online discussions are a useful way to address them. To augment the earlier discussion prompted by Dr Pan, I can respond to two topics brought up by Referee 2.
One relates to the length of time series needed for this kind of tidal analysis, and notably what are the consequences of having "less than a year's worth of data." For the examples in the paper, I had purposely chosen tide gauges with many years of highquality data. A long time series ensures that computed spectra (Figure 1) have adequate spectral resolution and that computations of monthly mean amplitudes and phases have relatively small error bars. But there is no hard rule for the minimum amount of data needed before one can proceed.
Of course, it is implausible to study seasonal variability without data spanning most of a full year. Multiple years are required if a computed spectrum is to have sufficient resolution to separate constituents with frequencies differing by 1 cpy. Spectral analysis is not mandatory, but it is certainly helpful to determine whether a spectrum contains isolated lines (as is the case for Figure 1) or instead is simply a wide cusp of energy surrounding M2. One's interpretation of tidal variability would be sharply different in these two cases.
A simple Rayleigh criterion for separating constituents would also call for at least one year of data. Yet that criterion is only a rough ruleofthumb and depends on noise levels. Munk and Cartwright (1966) emphasized this by noting that it is possible to separate two nearby sine waves with only four perfect, noisefree observations, but with realworld noise something like a Rayleigh criterion is probably required. Of course, bottom pressure measurements of tides are usually much less noisy than surface height measurements, so there is flexibility in all this.
A second point of Referee 2 concerns large gaps in the Port Orford time series before May 2002, which I can confirm. These gaps could conceivably impact spectral analyses, but the gaps are followed by at least seventeen years without gaps, which are more than sufficient to support spectral calculations of good frequency resolution. The calculations of monthly mean harmonic constants are unaffected, as each monthly mean is based on between 22 to 24 monthly estimates. Admittedly, the amplitudes in Figure 2 do not perfectly overlay. This likely stems as much from unavoidable estimation error (from inherently noisy measurements) as it does from any gaps. Yet one should not belabor these small amplitude differences. In fact, the amplitudes differ by a span of only 3 mm, and so they are actually quite consistent.
One item left unaddressed in Dr Pan's discussion is his suggestion to include more constituents than M2. While I sympathize with this desire, doing so would turn this "note" into a much longer paper, which I am unprepared to tackle at present. Moreover, any discussion of solar tides would likely be unsatisfactory owing to complications from radiational tides, as noted.
The revised paper now acknowledges the possibility of seasonality in KO2 adding to seasonality in M2, in those rare cases where M2 is very small and KO2 unusually large. Finally, although not prompted by the reviews, I have inserted a sentence at the beginning of Section 3 which gives an additional technical detail on how monthly tides were estimated.

AC2: 'Reply on RC2', Richard Ray, 06 Jul 2022

RC2: 'Comment on egusphere2022252', Anonymous Referee #2, 18 May 2022
This manuscript on the seasonality of the M2 tide is an extremely relevant publication for the Ocean Sciences journal. Although, as the author states, the manuscript really summarises several wellknown points and introduces evidence on these theories that date back to the early 1900s, the manuscript produces a comprehensive description that is of significant value to the tidal community. A general description of the tides, their sources and their relationships is in itself a valuable contribution and something often overlooked in the tidal community. I am very much a fan of Table 1 in providing very simple and important details. This manuscript will, therefore, be a valuable source of knowledge for the greater tidal community. Overall, the manuscript itself is a pleasure to read. Although, in my opinion, the manuscript is publishable as is, I do have a couple of comments which can hopefully clarify some points within the manuscript.
 One point the author highlights is the length of the time series of data needed to separate the tidal constituents within the M2 tidal band. The explanations in Table 1 and Figure 1 demonstrate this nicely. I guess what is not clear, what is the implications when one does not have a long enough time series, on the estimation of the M2 and the overall tidal height prediction? Should one where possible directly estimate these sidelines and if not possible, what are the implications on the accuracy of tidal predictions? I realise the sidelines are usually fractions of the main M2, but the Chittagong application for example demonstrates significantly large modulations. This is of course more critical in tide gauges/bottom pressure sensors that have less than a year's worth of data or less frequent sampling patterns such as altimetry observations.
 When reproducing the tide gauge evaluation in section 3, I found the same results as the author. However, in the period selected by the author, I noted a large temporal gap in the University of Hawaii dataset for Port Orford (shown below). I checked this with the PSMSL data (https://www.psmsl.org/data/obtaining/stations/1640.php) as well as GESLA3 (flagged as 99 below) and UHSLC data. This is not a criticism of the results as these gaps in data are normal, but could this be an explanation for the differences seen in Figure 2? It could also be that the author has appropriate data from this tide gauge.
Figure. SLA over time of the Port Orford tide gauge as obtained from GESLA3.

AC2: 'Reply on RC2', Richard Ray, 06 Jul 2022
I thank the two anonymous reviewers, along with Haidong Pan, for their feedback. A number of the points brought up are actually somewhat general questions on tidal analysis, not specifically on the paper itself, so these online discussions are a useful way to address them. To augment the earlier discussion prompted by Dr Pan, I can respond to two topics brought up by Referee 2.
One relates to the length of time series needed for this kind of tidal analysis, and notably what are the consequences of having "less than a year's worth of data." For the examples in the paper, I had purposely chosen tide gauges with many years of highquality data. A long time series ensures that computed spectra (Figure 1) have adequate spectral resolution and that computations of monthly mean amplitudes and phases have relatively small error bars. But there is no hard rule for the minimum amount of data needed before one can proceed.
Of course, it is implausible to study seasonal variability without data spanning most of a full year. Multiple years are required if a computed spectrum is to have sufficient resolution to separate constituents with frequencies differing by 1 cpy. Spectral analysis is not mandatory, but it is certainly helpful to determine whether a spectrum contains isolated lines (as is the case for Figure 1) or instead is simply a wide cusp of energy surrounding M2. One's interpretation of tidal variability would be sharply different in these two cases.
A simple Rayleigh criterion for separating constituents would also call for at least one year of data. Yet that criterion is only a rough ruleofthumb and depends on noise levels. Munk and Cartwright (1966) emphasized this by noting that it is possible to separate two nearby sine waves with only four perfect, noisefree observations, but with realworld noise something like a Rayleigh criterion is probably required. Of course, bottom pressure measurements of tides are usually much less noisy than surface height measurements, so there is flexibility in all this.
A second point of Referee 2 concerns large gaps in the Port Orford time series before May 2002, which I can confirm. These gaps could conceivably impact spectral analyses, but the gaps are followed by at least seventeen years without gaps, which are more than sufficient to support spectral calculations of good frequency resolution. The calculations of monthly mean harmonic constants are unaffected, as each monthly mean is based on between 22 to 24 monthly estimates. Admittedly, the amplitudes in Figure 2 do not perfectly overlay. This likely stems as much from unavoidable estimation error (from inherently noisy measurements) as it does from any gaps. Yet one should not belabor these small amplitude differences. In fact, the amplitudes differ by a span of only 3 mm, and so they are actually quite consistent.
One item left unaddressed in Dr Pan's discussion is his suggestion to include more constituents than M2. While I sympathize with this desire, doing so would turn this "note" into a much longer paper, which I am unprepared to tackle at present. Moreover, any discussion of solar tides would likely be unsatisfactory owing to complications from radiational tides, as noted.
The revised paper now acknowledges the possibility of seasonality in KO2 adding to seasonality in M2, in those rare cases where M2 is very small and KO2 unusually large. Finally, although not prompted by the reviews, I have inserted a sentence at the beginning of Section 3 which gives an additional technical detail on how monthly tides were estimated.
Peer review completion
Journal article(s) based on this preprint
Richard Ray
Richard Ray
Viewed
HTML  XML  Total  BibTeX  EndNote  

168  56  7  231  3  3 
 HTML: 168
 PDF: 56
 XML: 7
 Total: 231
 BibTeX: 3
 EndNote: 3
Viewed (geographical distribution)
Country  #  Views  % 

Total:  0 
HTML:  0 
PDF:  0 
XML:  0 
 1
The requested preprint has a corresponding peerreviewed final revised paper. You are encouraged to refer to the final revised version.
 Preprint
(277 KB)  Metadata XML