Continuous seiche in bays and harbors

Abstract. Seiches are often considered a transitory phenomenon wherein large amplitude water level oscillations are excited by a geophysical event, eventually dissipating some time after the event. However, continuous small-amplitude seiches have been recognized which raises a question regarding the origin of continuous forcing. We examine six bays around the Pacific where continuous seiches are evident and, based on spectral, modal, and kinematic analysis, suggest that tidally forced shelf resonances are a primary driver of continuous seiches.


Introduction
It is long recognized that coastal water levels resonate. Resonances span the ocean 10 as tides (Darwin, 1899) and bays as seiches (Chrystal, 1906). Tides expressed on coasts are significantly altered by coastline and bathymetry, for example, continental shelves modulate tidal amplitudes and dissipate tidal energy (Taylor, 1919) such that tidally-driven standing waves are a persistent feature on continental shelves (Webb, 1976;Clarke and Battisti, 1981). While tides are perpetual, seiches are associated 15 with transitory forcings and are considered equally transitory. A thorough review of seiches are provided by Rabinovich (2009) wherein forcing mechanisms are known to include tsunamis, seismic ground waves, weather, non-linear interactions of wind waves or swell, jet-like currents, and internal waves. Excepting strong currents and internal waves, these forcings are episodic and consistent with the perception that the Philippines finding that periods of enhanced seiche activity are produced by internal bores generated by arrival of internal wave soliton packets from the Sulu Sea. However, as one would expect from soliton excitation, their analysis suggests that seiches are not continually present. Breaker et al. (2008) noticed that seiches in Monterey Bay are continously present, 5 leading Breaker et al. (2010) to consider several possible forcing mechanisms (edgewaves, long period surface waves, sea breeze, internal waves, microseisms, and smallscale turbulence) and to question whether or not "the excitation is global in nature" such that continuous oscillations would be observed in other bays. Subsequent analysis by Park and Sweet (2015) confirmed continuous oscillations in Monterey Bay over 10 a 17.8 year record, and presented kinematic analysis discounting potential forcings of internal waves and microseisms while suggesting that a persistent mesoscale gyre situated outside the Bay would be consistent with a jet-like forcing. However, jet-like currents are not a common feature along coastlines and could not be considered a global excitation of continuous oscillations. 15 Wijeratne et al. (2010) observed that seiches with periods from 17 to 120 min were persistent throughout the year at Trincomalee and Colombo, Sri Lanka, finding a strong fortnightly periodicity of seiche amplitude at Trincomalee, but no discernible seasonal variability at Columbo. The fortnightly modulations of seiche energy were attributed to forcing by astronomical tides, while the overall seiche generating mechanisms were 20 thought to include diurnal weather, tides and currents.
Most recently, MacMahan (2015) analyzed 2 years of data (2011)(2012) in Monterey Bay and Oil Platform Harvest, 270 km south of Monterey, concluding that low-frequency "oceanic white noise" within the seiche periods of 20 to 60 min is directly and continuously forcing the bay modes. The oceanic noise was hypothesized to consist of 25 low-frequency, free, infragravity waves forced by short waves, and that this noise was of O(mm) in amplitude. So while the term "noise" applies in context of a low amplitude background signal, and the qualifier "white" expresses a uniform spatial and wide range of temporal distributions, the underlying processes are coherent low-frequency 2363 Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | infragravity waves. Based on a linear system transfer function between Platform Harvest and Monterey Bay water levels, he concluded that the bay amplifies this noise by factors of 16-40 resulting in coherent seiche. It was also suggested that the highest amplification, a factor of 40, is associated with the 27.4 min mode, however, as discussed below and in agreement with Lynch (1970), we find that this is not a bay-mode, 5 but a tidally-forced shelf-mode, and find an amplification factor (Q) of 12.9. Further, as discussed below, we find low-frequency infragravity waves may not have sufficient energy to drive the observed oscillations, but are likely a contributor to observed seiche amplitude variability.
It seems remarkable that, while tides and seiches have been studied for over a century, seiches have only recently been recognized as continually present, yet to the authors knowledge with exception of the work by MacMahan (2015) and Park and Sweet (2015), the question posed by Breaker et al. (2010) has not been previously addressed. For example, Bellotti et al. (2012) recognized the importance of shelf and bay-modes to tsunami amplification, yet considered them to be independent processes, and the com- 15 prehensive review by Rabinovich (2009) falls short of continuous seiche recognition by noting that "in harbours and bays with high Q-factors, seiches are observed almost continuously." Perhaps the lack of clear recognition of continuous seiche is partly due to the requirement for long term, digital records of coastal water levels capable of precisely resolving small amplitude, long period oscillations, while the episodically forced, large 20 amplitude oscillations are readily apparent, and that such records have only become available in the last few decades. Bays and harbors offer refuge from the open ocean by effectively decoupling wind waves from the bay or harbor, although offshore waves are effective in driving resonant modes of bays and harbors in the infragravity regime at periods of 30 s to 5 min (Okihiro 25 and Guza, 1996;Thotagamuwage and Pattiaratchi, 2014). Even though bays and harbors can appear quiescent in relation to the sea, they can act as efficient amplifiers of long period waves with periods between 5 min and 2 h (Miles and Munk, 1961), and if tidally-forced long period waves from shelf resonances are continuously present, there is potential to continuously excite bay and harbor resonances.
The focus of this paper is to present evidence in pursuit of the questions posed by Breaker et al. (2010), namely, is there a continuous global excitation of seiche, and what is the source? We find that perpetual bay oscillations are indeed present at six bays, 5 suggesting that there is a global excitation, thereby negating the specific mesoscale gyre hypothesis of Park and Sweet (2015) at Monterey Bay. Spectral analysis of water levels allows us to identify resonances from the shelf down to harbor and pier scales, and to identify the shelf-modes as tidally-forced standing waves in agreement with Webb (1976) and Clarke and Battisti (1981). Given the lack of plausible forcing mech-10 anisms based on the analysis of Breaker et al. (2010) and Park and Sweet (2015), we suggest that long period shelf resonances driven by tides are a primary excitation of continuous bay and harbor modes, while internal waves and free infragravity waves are secondary contributors serving to modulate seiche amplitudes. 15 We examine tide gauge water levels from six bay/harbors shown in Fig. 1 with the tide gauge location denoted with a star. Three of the bays (Monterey, Hawke and Poverty) can be characterized as semi-elliptical open bays with length-to-width ratios of 1.9, 2.0 and 1.4 respectively. We therefore anticipate a degree of similarity between their resonance structures. Bays at Hilo and Kahului are also similar with a triangular or 20 notched coastline, while Honolulu is an inland harbor of Mamala Bay.

Locations and data
Data for Hawke and Poverty bays at the Napier (NAPT) and Gisborne (GIST) tide gauges respectively are recorded at a sample interval of T s = 1 min, and are publicly available from Land Information of New Zealand (LINZ) at http://apps.linz.govt.nz/ftp/ sea_level_data/. Data for Monterey and Hilo at a sample interval of 6 min are available 25 from the National Oceanic and Atmospheric Administration (NOAA) tide gauges at http://tidesandcurrents.noaa.gov/stations.html?type=Water+Levels. In addition to this Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | publicly available data, we also analyze water level data from independent wave studies at Honolulu, Hilo and Monterey sampled at 1 s intervals. At Honolulu data was collected by Seabird 26+ wave and water level recorders using Paroscientific Digi-quartz pressure sensors at two locations, one collocated with the NOAA tide gauge inside the harbour, and the other at 157.865 • W 21.288 • N outside Honolulu harbor. At Monterey 5 and Hilo data was recorded at 1 s intervals by WaterLog H-3611 microwave ranging sensors co-located with the NOAA tide gauges. Table 1 lists the approximate bay and harbor dimensions along with the periods of record and sampling intervals. Breaker et al. (2008) noticed that seiches in Monterey Bay were continuously present 10 over 14 months in 2002and 2003, motivating Breaker et al. (2010 to contribute a comprehensive review of Monterey Bay oscillations and to question whether such continuous oscillations were peculiar to the bay. In Fig. 2 we present water level spectrograms at five other bays where vertical bands are associated with seasonal or episodic wave energy, and horizontal bands indicate the presence of persistent oscillations. These os- 15 cillations appear to have essentially invariant amplitudes suggesting that time varying processes such as weather or waves are not likely forcings. For example, inspection of the Honolulu or Kahului data at periods near 0.2 min (12 s) reveals time varying amplitudes from wind waves and swell, whereas the longer-period oscillations are essentially constant. We therefore have reason to suspect that there is a continuous global forcing 20 of bay and harbor oscillations. A close examination of modes at Kahului with periods between 1 and 5 min does reveal a time-dependent frequency modulation. This behavior is also observed at Monterey (Park and Sweet, 2015), and at Hilo, where high-resolution (1 Hz) data was available. These modulations are coherent with the tides, and are a manifestation of chang-25 ing boundary conditions (water depth, exposed coastline and spatial resonance boundaries) as water levels change with the tide.

Mode identification
Spectrograms provide information regarding time dependence of energy, but are not well suited when detailed frequency resolution is desired. To identify resonances in the water level data we estimate power spectral densities with smoothed periodograms (Bloomfield, 1976) as shown in Fig. 3. The Monterey and Hilo estimates are compos-5 ites of 6 min and 1 s data with periods longer than 12 min represented by spectra of the 6 min data. Horizontal arrows indicate the range of modes associated with their respective spatial domains as discussed below. Triangles mark the tidally-forced shelfresonances, also discussed below.
To relate temporal modes with spatial scales we find solutions to the general disper-10 sion relation ω 2 = gktanh(kd ) where ω is the mode frequency obtained from power spectra, k the wavenumber, and d the water depth which are representative values over the bay or harbor from nautical charts. This provides estimates of the modal wavelength λ = 2π/k, which we list as λ/2 or λ/4 in Table 2 for all prominent modes. λ/2 corresponds to spatial modes between two fixed boundaries, for example between two bor maps. We cannot assure that all entries in Table 2

Shelf resonance
The period of a shallow water wave resonance supported by a fixed-free boundary condition is expressed in Merian's formula for an ideal open basin as T = 4L/ gd where L is the shelf width corresponding to λ/4, and d the basin depth (Proudman, 1953). In addition to a shelf-mode standing wave based solely on geometric wave 5 reinforcement, a shelf-resonance is dynamically supported when the shelf width is approximately equal to gα/(ω 2 − f 2 ) where g is the gravitational acceleration, α the shelf slope, ω the frequency of oscillation and f the Coriolis parameter (Clarke and Battisti, 1981). Table 3 lists solutions for shelf-mode period (inverse of frequency) for each of the bays where the shelf slope is approximated as the depth of the shelf break divided by the shelf width, and where the basin depth is taken as one half the shelf break depth. Also listed are modal periods deemed to represent the shelf-resonances obtained from the power spectra. The agreement is reasonable given the simplistic formulations and crude spatial representations, and when viewed from the perspective of the apparently time invariant modal energy evident in the spectrograms and with recognition of tidal 15 energy as a driver of shelf-resonances, suggests that tidally-forced shelf-resonances are continually present.

Dynamic similarities
Topological similarities between Monterey and Hawke bays are striking, each a semielliptical open bay with aspect ratios of 2.0 and 1.9 respectively, although a factor of 20 2 different in horizontal scale. One might expect that these similarities would lead to affine dynamical behavior in terms of modal structure, although not the specific modal resonance periods, and that indeed appears to be the case as seen in Fig. 3 respond to longitudinal bay oscillations. The semi-elliptical topology of these bays is such that boundaries of the longitudinal modes are not parallel as in an ideal rectangular basin, but are crudely represented as semi-circular boundaries. The range of spatial scales between these boundaries is reflected in the longitudinal spectral peaks with broad frequency spans at the base and evidence of a series of closely spaced 5 modes corresponding to a range of wavelengths. This is contrasted to the shelf-modes where the resonances are remarkably narrow indicating the narrow-range of spatial scales reflected in the relatively uniform widths of the shelves at Hawke and Monterey Bays. Poverty Bay is the other semi-elliptical open bay and it exhibits the same generic 10 modal structure. Although here, the bay modes are shorter in period due to the significantly smaller size, and the shelf-mode is the longest period mode. It is also evident here that the shelf-mode is mixed with other modes as it does not have a high Q-factor as found at Monterey and Hawke, although part of this difference could result from poorer trapping or more radiation or other energy loss associated with this mode. 15 Hilo and Kahului bays also share structural similarity, but lack the high degree of topological symmetry found in the semi-elliptical bays that support both longitudinal and transverse modes. As is the case for the semi-elliptical bays, the power spectra of these two bays are conspicuously similar with the substantial difference being the precise frequencies of their associated modes. Here, shelf-modes appear to dominate 20 the water level variance, but rather than a set of discrete, high-Q shelf-resonances as found at Monterey and Hawke, they are energetic over a broad range of frequencies and spatial scales. This suggests that the shelves here are not well represented by a uniform width, but encompass a range of scales to the shelf break as evidenced in bathymetric data. In the following sections we examine specific resonance features at Introduction

Monterey
Monterey Bay seiche has been studied since at least the 1940s (Forston et al., 1949) with a comprehensive review provided by Breaker et al. (2010). The primary bay modes at the Monterey tide gauge have periods of 55.9, 36.7, 27.4, 21.8, 18.4 and 16.5 min, where the 55.9 min mode represents the fundamental longitudinal mode, 5 while the 36.7 min harmonic is attributed to the primary transverse mode. We identify the 27.4 min mode as a shelf-resonance, also recognized by Lynch (1970), and consider it to be a potential continuous forcing of long period water level oscillations throughout the bay. The harbor modes (Fig. 3) have been associated with resonances between breakwaters, and are amplified by wave energy, whereas the bay modes are 10 weakly-dependent on wave forcing (Park and Sweet, 2015).

Hawke
Hawke Bay is approximately 85 km long and 45 km wide with a rich set of modes at periods between 20 and 180 min. Modes at periods of 170.6, 167.1 and 160.1 min correspond to longitudinal oscillations, while the 105.8 min oscillation is identified as 15 a shelf-resonance (Table 2).

Hilo
At Hilo we are afforded full spectral frequency coverage and find that pier modes have periods below 20 s corresponding to spatial scales less than 100 m. These modes are excited by waves and swell just as the harbor modes at Monterey. Harbor modes at 20 periods of 3, 4 and 5.9 min correspond to standing waves within the breakwater and spatial scales of 1, 1.3, and 1.9 km respectively. The shelf offshore Hilo is not a uniform width, but transitions from less than 2 km just south of the bay to roughly 18 km along the northern edge with the spectra revealing a corresponding plateau at periods between 10 and 30 min with a rather broad shelf-resonance centered on a period Introduction

Kahului
Oscillations at Kahului follow the same general structure as Hilo with wave and swell 5 excited pier modes at periods less than 20 s, and within-harbor pier-breakwater modes at periods of 51 and 63 s. The primary harbor mode has peak energy at 188 s (3.1 min) corresponding to a λ/2 spatial scale of 1.1 km which is the dominant lateral dimension of the harbor. An interesting feature of the Kahului power spectra is a low energy notch between 10 periods of 120 and 160 s. This lack of energy corresponds to a lack of standing wave reflective boundaries at scales of λ/2 from 650 to 1000 m. Such low energy features are present in all spectra indicating spatial scales where standing waves are not supported.
The dominant shelf-mode at Kahului has a period of 35.5 min, similar to that of Hilo.

15
At Honolulu we have the benefit of both short sample times (T s = 1 s) and two gauge locations, one inside the harbor and one on the reef outside the harbor. The offshore power spectrum is shown in red in Fig. 3 exemplifying an open ocean or coastal location dominated by wind waves and swell. The rejection of wind wave energy inside the harbor is impressive, revealing a set of pier-modes in the 8 to 20 s band supported by 20 rectangular basins around the gauge. Modes with periods of 82 and 88 s correspond to waves with λ/2 of approximately 500 m, which is the fundamental dimension of the basin. While the harbor is quite efficient in rejection of wind waves and swell, amplification of the shelf-mode and other long period resonances is a striking manifestation of the not a longitudinal mode within the bay, but is the shelf-resonance at a period of 79 min. The 57.3 min mode is not explicitly a Poverty Bay mode, but is a longitudinal mode of the open bay between Table Cape to the south and Gable End to the north, inside which Poverty Bay is inset. We also note that the 42.1 min mode is a shelf edge-wave evident in both Hawke and Poverty bays as discussed below. The reader is referred to Bellotti et al. (2012) for a detailed numerical evaluation of Poverty Bay shelf and bay modes.

Hawke and Poverty
Hawke and Poverty bays are located approximately 35 km apart along the southeast coast of northern New Zealand. Concurrent 7 month records allow examination of 15 cross-spectral statistics between the two locations, with power spectra presented in the upper panel of Fig. 4 and coherence in the lower panel plotted as the upper and lower 95 % confidence intervals. Power spectra reveal that the two bays share shallow water tidal forcings at periods longer than 180 min, but are essentially independent in terms of major oscillation frequencies between 20 and 180 min. There are coincident 20 spectral peaks near periods of 42 and 58 min, however the coherence of the 58 min energy is low indicating that is likely independent between the two locations.
Coherence at the shallow water tidal periods (373, 288, 240, 199 min) is quite high and as expected has near zero phase shift (not shown). Shelf-modes with periods from 100 to 160 min also share coherence in the 0.5 range, which is sensible since they have 25 quarter wavelengths that are as long, or longer than, the 35 km separation distance. Introduction The only other energy with coherence reliably above the 0.5 range is the 42 min mode. This mode has a phase shift of −160 • from Napier (Hawke Bay) to Gisborne (Poverty Bay) indicating a traveling wave moving from south to north along the coast, empirically validating the shelf edge-wave explanation inferred numerically by Bellotti et al. (2012).

Shelf-metamodes 5
Since tidally-forced shelf-modes are a plausible driver of seiches, we expect that tidal amplitude variance should be reflected in seiche amplitudes, a view consistent with the strong fortnightly modulation of seiche amplitude reported by Giese et al. (1990) and Wijeratne et al. (2010). To examine such a dependence Fig. 5 plots time series of the shelf-mode amplitudes from the spectrograms shown in Fig. 2, along with temporal low-pass representations from superposition of the lowest frequency intrinsic mode functions IMFs) of the shelf-mode amplitude time series computed by empirical mode decomposition (EMD, Huang and Wu, 2008). We term these IMFs of shelf-mode amplitudes as metamodes. It is clear from Fig. 5 that shelf-modes are continually present at all stations, albeit with significant temporal variability. The lowest frequency meta-15 modes shown with the thick lines reveal annual modulations in the long period records of Monterey and Hilo, and fortnightly cycles at Kahului and Honolulu.
To assess the relative contribution of individual shelf-mode IMFs (metamodes) to the total shelf-mode variance, we list the mean period in days (T ) of each metamode Hilbert instantaneous frequency vector, and percent variance each metamode IMF 20 contributes to the total variance in Table 4. We note that the fortnightly astronomical tidal constituents, the lunisolar synodic fortnightly (M sf ) and lunisolar fortnightly (M f ), have periods of 14.76 and 13.66 days respectively with IMFs closest to these periods highlighted in Table 4 and shown in Fig. 6 the fortnightly mode is the strongest of the modes at diurnal and longer scales, a relation that holds at all stations except Poverty Bay. We also note in Fig. 6 evidence of a seasonal dependence in metamode amplitude, and will correlate these IMFs with their corresponding fortnightly tidal IMFs below. The foregoing indicates that fortnightly metamodes are present at all six stations sug-5 gesting that tidal forcing of shelf-modes is a likely driver. To assess an assumed linear dependence between fortnightly tidal forcing and metamodes, we compute IMFs on the tidal water level data and cross-correlate the resulting fortnightly tidal IMFs with the fortnightly metamodes. Correlations are computed over a sliding window of length 20 days with results shown in Fig. 7 where dashed red lines indicate the 95 % significance threshold. An interesting feature is that while all stations exhibit near perfect correlation at times, they also episodically transition to near zero or statistically insignificant correlation. This suggests that the fortnightly modulation of tidally-driven shelf-resonances is also influenced by other factors, of which internal tide variability has previously been noted by Giese et al. (1990) and Wijeratne et al. (2010). 15 While there is significant temporal variability in the fortnightly tidal metamode correlations, it appears that the majority of the time correlations are quite high and significant above the 95 % level. The IMF mode numbers and mean correlations statistics are listed in Table 5 where T R % is the percentage of time that the 95 % significance threshold is exceeded, R is the mean correlation of values above the 95 % significance 20 threshold, and Lag the mean lag value of 95 % significant correlations. Overall, these data suggest that correlations significant above the 95 % level are present 76-87 % of the time, and from a linear model perspective that fortnightly tidal oscillations account for 35-50 % of the metamode variance.

25
Knowledge of a mode's temporospatial characteristics allows estimation of the total energy sustained by the mode. Figure 3 indicates  Poverty bays are dominated by energy of the shelf-mode, while at Hawke and Monterey bays the shelf-mode is the second largest amplitude. We are therefore motivated to investigate modal energy in Monterey Bay to test our hypothesis that the shelf-mode is a potential driver of bay oscillations from a kinematic perspective. We estimate the total potential energy to support a mode by assuming a raised-5 cosine profile of amplitude h either orthogonal to the shore for the transverse and shelfmodes, or parallel for the longitudinal modes. Multiplying this profile area (A M ) by the alongshore extent of the mode (L A ) gives the volume of water displaced: V M = L A A M where we have neglected the influence of shoaling on the transverse modes as the wavelength is much longer than the shelf width. (This assumption is supported by the 10 agreement between the shelf-mode spatial scales based on the observed shallowwater frequencies in Table 3.) The energy to move this volume is equivalent to the work performed to change the potential energy of the mass in the gravitational field E M = ρV M h M g, at an average power output of P out = E M /T where T is the modal period. This leading-order value does not incorporate dissipation and momentum, terms that 15 we ignore in subsequent energy estimates. The ratio of energy stored in the mode resonance to energy supplied driving the resonance is the Q factor. If Q is large (the resonance signal-to-noise ratio is high, as is the case for the shelf-mode at Monterey), it may be estimated from the power spectrum: Q = f M /∆f , where f M is the mode resonant frequency and ∆f the −3 dB (half power) 20 bandwidth of the mode. This allows one to estimate the power required to drive the mode P in = E M /(QT ) = P out /Q.
Modal length scales (λ) are taken from Table 2, amplitudes (h) are from bandpass filtering the 17.8 year water level record at the NOAA tide gauge, and Q from 1 Hz water level power spectra (95 % CI 2.6 dB) modal means over 120 h windows over 63 days 25 (Park and Sweet, 2015). The alongshore extent of the modes, L A , are estimated from a regional ocean modeling system (ROMS) implementation in Monterey Bay (Shchepetkin and McWilliams, 2005) as reported in Breaker et al. (2010).
Results of these estimates are shown in Table 6 where we find seiche amplitudes averaged over the 17.8 year period of 0.9, 1.4 and 1.6 cm for the 55.9, 27.4 and 36.7 min modes respectively, although amplitudes of 4 cm in the 27.4 min mode are common during seasonal maximums. The 27.4 min shelf-mode is estimated to produce a total power of 998 kW, which is more than sufficient to supply the required input power of 5 both the primary longitudinal (55.9 min, P in = 23 kW) and transverse (36.7 min, P in = 169 kW) bay modes. This suggests from a kinematic perspective that a tidally-forced shelf-resonance is energetic enough to drive observed seiches in Monterey Bay. Regarding the O(mm) low-frequency infragravity waves suggested by MacMahan (2015), we note that a 27.4 min mode with an amplitude of 3 mm would produce an estimated 10 P out of 41.6 kW (not shown in Table 6), which would be insufficient to drive the observed 27.4 min mode as it requires a power of P in = 77 kW.

Conclusions
Resonant modes are a fundamental physical characteristic of bounded physical systems expressed in bodies of water as seiches. As such, they can be excited to large 15 amplitudes by transitory phenomena such as weather and tsunamis, and since large amplitude seiche are easily observable seiche are often viewed as transitory given that they dissipate after cessation of the driving force. On the other hand, observations of small amplitude continuous seiche are not well documented, to our knowledge those reported in Monterey Bay by Breaker et al. (2008); Park and Sweet (2015) and  han (2015), and in Sri Lanka by Wijeratne et al. (2010) are the only clear expressions of perpetual long period resonances. We have found that in addition to Monterey Bay, each of the five other bays examined exhibit persistent seiches, and to our knowledge, this work represents the first clear recognition of continuous seiche across multiple bays which effectively answers the question posed by Breaker et al. (2008) that indeed of continuous seiche, we are not aware that shelf-modes have been considered as the primary driver of continuous seiche.
In the process of analyzing the resonant structure of these bays and harbors, we have quantified resonant periods and estimated spatial scales corresponding to each mode (Table 2). In some cases, we have identified the physical attributes of a bay or 10 harbor associated with specific temporospatial resonances. In a more general sense, we have also illustrated broad dynamical similarities between bays with affine topologies, such as the clearly defined modes of the semi-elliptical bays when compared to the less structured, shelf dominated bays such as Hilo and Kahului. This analysis also provides empirical verification of the numerically inferred edge-wave by Bellotti et al. 15 (2012) near a period of 42 min along Hawke and Poverty bays.
Although spectrograms of tidal records indicate a continuous presence of shelfmodes, closer examination of shelf-mode amplitude time series identifies metamodes reflecting dynamic behavior, and we find that fortnightly metamodes are the dominant mode at periods longer than diurnal. Assuming that these fortnightly modulations are 20 of tidal origin, cross-correlation of fortnightly IMFs of tidal data with the fortnightly metamodes leads to the conclusion that within the bounds of a linear system model from one-third to one-half of the fortnightly metamode variance is coherent with tidal forcing. From an energy perspective, the suggestion of the shelf-mode as a primary driver of continuous seiche is supported, while the low-frequency infragravity waves suggested 25 by MacMahan (2015) may not have sufficient energy.
Taken together, evidence of continually present tidally-forced shelf-modes, their fortnightly amplitude relation to tidal modes, and assessment of modal energy suggests that long period shelf resonances driven by tides are an important excitation of con-Introduction tinuous bay and harbor modes. However, it is also clear that we do not understand the cyclic nature of fortnightly tidal and metamode correlation. One possibility is that there is a time varying phase-lag between the two such that destructive superposition episodically creates nulls. A linear spectral analysis might use a coherency statistic to identify this, but such an option is not available for IMFs with variable instantaneous fre-5 quencies. It is evident that internal tides play a role, and it may be that episodic changes in stratification as noted by Giese et al. (1990) lead to modulation of the metamodes and contribute to the observed decorrelation, and it is deemed likely that the free, longfrequency infragravity waves suggested by MacMahan (2015) also contribute.
Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Shchepetkin, A. F. and McWilliams, J. C.: The regional oceanic modeling system (ROMS): a split-explicit, free-surface, topography-following-coordinate oceanic model, Ocean Model.,9, Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Table 2. Temporospatial scales according to the dispersion relation ω 2 = g k tanh(kd ) where ω is frequency, k the wavenumber, d the water depth, λ the wavelength and period is 2π/ω. Periods are in min and lengths in km unless otherwise noted. Periods are obtained from the peak modal energy represented in the power spectra shown in Fig. 3. Depths for each bay are taken as representative values from nautical charts, depths for shelf-resonances are assumed to be 150 m, one half a nominal shelf-break depth of 300 m. Spatial scales are listed as λ/2 for modes assumed to be fixed-fixed boundary standing waves, and λ/4 for fixed-open boundaries.    Triangles mark the tidally-forced shelf-resonance. The red curve at Honolulu plots data from outside the harbor.