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.

It is long recognized that coastal water levels resonate.
Resonances span the ocean as tides

Tides expressed on coasts are significantly altered by coastline and
bathymetry; for example, continental shelves modulate tidal amplitudes and
dissipate tidal energy

Internal waves are known to influence seiches as demonstrated by

Persistent seiching around the islands of Mauritius and Rodrigues was
observed by

Most recently,

The foregoing suggests that coastal seiche are not to be considered solely
transitory phenomena, yet it seems that continuous seiching is not as widely
known as its transitive cousin. For example,

The focus of this paper is to present evidence in pursuit of the questions
posed by

Decomposition of shelf-resonance time series into intrinsic mode functions
(IMFs) allows us to examine temporal characteristics of shelf resonance in
detail, leading to identification of fortnightly tidal signatures in the
shelf modes. An energy assessment of the available power from the shelf modes
as well as the power consumed to drive and sustain observed oscillations in
Monterey Bay indicates that shelf modes are indeed energetic enough to
continually drive seiches. In terms of the query posed by

Let us first recall some essentials of resonance as a general phenomenon. The
simple harmonic oscillator consisting of a frictionless mass

A hallmark of resonance occurs when a resonant system is driven at
or near the resonant frequency, and the resulting amplitude is maximized. This is
conventionally expressed by an amplification factor of the form

This is consistent with our hypothesis that tidally forced shelf resonances can initiate and sustain small-amplitude bay or harbor seiching even though the tidal forcing is not frequency matched to the seiche. An energy analysis below, based on the observed resonant properties of water levels in Monterey Bay, illustrates this. Since the terms mode and resonance represent the same underlying property of the system, specifically a characteristic eigenmode, we use these two terms interchangeably. We apply modifiers of shelf, bay, or harbor to qualify the spatial scale and domain over which the mode operates.

We examine tide gauge water levels from six bays and harbors shown in
Fig.

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 notched coastline, while Honolulu is an inland harbor of Mamala Bay.

Location and approximate dimensions of bays. Tide gauge locations are marked with a star and denoted by latitude and longitude.

Tidal ranges at the six tide gauges. GT is the great diurnal range (difference between mean higher high water and mean lower low water) and MN the mean range of tide (difference between mean high water and mean low water).

Approximate shelf widths and dimensions of bays and harbors, data
period of record, and sampling interval

Data for Hawke and Poverty bays at the Napier (NAPT) and Gisborne (GIST) tide
gauges, respectively, are recorded at a sample interval of

Continuous seiching throughout a 17.8-year period has been observed, but most
studies are limited to periods less than a year, one or a few coastal
locations, and a single geographic region. In Fig.

A close examination of modes at Kahului with periods between 1 and
5

Spectrograms of water level data at each tide gauge. Horizontal bands indicate continuous oscillations, vertical bands are associated with periods of increased wave energy.

Power spectral density (PSD) estimates of water level (WL) at each tide gauge. Horizontal arrows indicate the frequency span of resonant modes associated with spatial scales. Triangles mark the tidally forced shelf resonance. The red curve at Honolulu plots data from outside the harbor.

Spectrograms provide information regarding time dependence of energy, but are
not well suited for obtaining detailed frequency resolution. To identify
resonances in the water level data, we estimate power spectral densities with
smoothed periodograms

Temporospatial scales according to the dispersion relation

To relate temporal modes with spatial scales, we find solutions to the general
dispersion relation

For example, the 55.9 min mode at Monterey and the 160–170 min modes at
Hawke correspond to longitudinal modes between the ends of the bays and are
therefore delineated as closed-boundary

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

Topological similarities between Monterey and Hawke bays are striking, each a
semi-elliptical open bay with aspect ratios of 2.0 and 1.9 respectively,
although a factor of 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.

Estimates of shelf-resonance periods.

Poverty Bay is the other semi-elliptical open bay and exhibits the same
generic modal structure, although the bay modes here 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

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 the water level
variance at periods less than one hour, but rather than a set of discrete,
high-

Monterey Bay seiching has been studied since at least the 1940s

Hawke Bay is approximately 85 km long and 45 km wide with a rich set of
modes at periods between 20 and 180

At Hilo we are afforded full spectral frequency coverage and find that pier
modes have periods below 20

Oscillations at Kahului follow the same general structure as Hilo with wave
and swell excited pier modes at periods less than 20

An interesting feature of the Kahului power spectra is a low energy notch
between periods of 120 and 160

At Honolulu we have the benefit of both short sample times (

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 “harbor paradox” as noted by

Poverty Bay is a small-scale version of Hawke and Monterey bays with a
similar resonance structure. However, the bay is small enough that the lowest
frequency mode is not a longitudinal mode within the bay, but is the
shelf resonance at a period of 79

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 cross-spectral statistics between the two locations, with
power spectra presented in the upper panel of Fig.

Power spectral density (top) of concurrent water levels at Napier in Hawke Bay, and Gisborne in Poverty Bay. Bottom: coherence of the power spectra shown as the upper and lower 95 % confidence interval values.

Coherence at the shallow water tidal periods (373, 288, 240, 199

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

It is clear from Fig.

To assess the relative contribution of individual shelf-mode IMFs (metamodes)
to the total shelf-mode variance, we list the mean period in days (

Shelf-resonance power spectral density (PSD) amplitudes (black) with
low-frequency IMFs (metamodes) in red. The large amplitude in Poverty Bay is
a result of the 27 February 2010 Chile 8.8

Mean period in days (

We note that the fortnightly astronomical tidal constituents, the lunisolar
synodic fortnightly (

The foregoing indicates that fortnightly metamodes are present at all six
stations, suggesting 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 20 days, with
results shown in Fig.

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

Cross-correlation of fortnightly tide and shelf-resonance metamode
IMFs.

Intrinsic mode functions (IMF) of shelf-mode amplitude variance
(metamodes) with mean Hilbert instantaneous frequencies corresponding to
fortnightly periods (highlighted in Table

Correlation coefficients between tide and shelf-resonance metamode IMFs with fortnightly periods. The dashed red lines indicate the 95 % confidence levels.

Knowledge of a mode's temporospatial characteristics allows estimation of the
total energy sustained by the mode. Figure

Estimates of total energy and power generated by resonances in
Monterey Bay. Modal amplitudes (

We estimate the total potential energy to support a mode by assuming a
raised-cosine profile of amplitude

The ratio of energy stored in the mode resonance to energy supplied driving
the resonance is the

Modal length scales (

Results of these estimates are shown in Table

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 amplitudes by transitory phenomena such as weather and tsunamis, and
since large amplitude seiches are easily observable, seiches are often viewed
as transitory given that they dissipate after cessation of the driving force.
Moving from transient to persistent behavior, seasonal weather patterns are
known to sustain nearly continuous seiching for extended periods

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

Simple geometric and dynamical estimates of tidally forced shelf modes are consistent with modes observed in the power spectra at all stations, and their continual presence in water level spectrograms and mode amplitude time series indicates that tidally forced shelf modes are continuously present at each location. Decomposition of shelf-mode amplitude time series identifies metamodes reflecting dynamic behavior of the shelf modes, and we find that fortnightly metamodes are the dominant mode at periods longer than diurnal. Assuming that these fortnightly modulations are 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. We therefore suspect that tidally forced shelf modes are a continuous energy source in harbors and bays adjacent to continental or island shelves.

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 frequencies. It is evident that internal tides play a
role, and it may be that episodic changes in stratification as noted by

A natural question is: does the proposed source contain sufficient energy to
sustain the observed resonant oscillation? Power estimates of the most
energetic modes at Monterey suggest that the shelf mode is fully capable as a
primary driver of continuous seiche, while the low-frequency infragravity
waves suggested by

Examination of six coastal locations around the Pacific with
diverse shelf conditions finds that tidally forced shelf resonances are
continually present. An energy assessment of the shelf mode and primary
seiche in Monterey Bay indicates that the shelf resonance is fully capable of
supplying the power input required to drive the primary bay oscillations, even
though the grave mode produces more output power than the shelf mode, a
consequence of the resonance structure of the bay. Hawke Bay is dynamically
similar to Monterey and we suspect that a similar relation holds there, while
at the other locations the shelf mode is the dominant energy source. Our
conclusion is that tidally forced shelf modes constitute a global candidate
for continuous seiche excitation, a view consistent with that of

Specific to Monterey Bay, these results offer a simpler explanation for
continuous seiche generation than the mesoscale gyre hypothesis proposed by

In the course of attributing tidal forcing as the driver of the observed shelf resonances, we introduced the idea of metamodes, dynamical modes of shelf-mode amplitude determined by empirical mode decomposition. The metamodes exhibited fortnightly modulation, and it is likely that examination of other metamode components may be useful towards understanding the dynamic behavior of modal structure in coastal environments.

We are indebted to Lawrence Breaker of Moss Landing Marine Laboratory for his identification of continuous seiche in Monterey Bay, his questioning of their origin, and fruitful discussions. We gratefully acknowledge additional citations on continuous seiche provided by an anonymous reviewer. Edited by: J. M. Huthnance