Bounded physical systems support normal modes. This is true in quantum mechanical, astronomical, and terrestrial systems such as harbors and bays, and owing to the central role that harbors play in human endeavors, there is a rich history analyzing resonant modes of bays and harbors (seiches); see, for example, , and .

Monterey Bay, California (Fig. ), is a dynamic and ecologically rich system influenced by Monterey Submarine Canyon, the California Current, seasonal upwelling, and inshore countercurrents (California undercurrent, Davidson current). Monterey Submarine Canyon is the prominent bathymetric feature, where tidally coherent internal waves are nearly an order of magnitude stronger than the open ocean, with the most intense waves characterized as bores , and where hydrodynamic mixing (turbulent kinetic energy dissipation) reaches 3 orders of magnitude greater than the open ocean . Interaction of the regional coastline and bathymetry with the California Current establishes a persistent anticyclonic mesoscale vortex adjacent to the bay that is readily observed in satellite ocean surface temperature images and in high-resolution hydrodynamic models . For example, Fig.  clearly depicts the gyre expressed in sea surface temperatures from satellite thermal imagery. Upwelling driven by local wind forcing interacts with this gyre, resulting in a bifurcated flow of upwelled water with one branch advected northward near Point Año Nuevo just north of the bay, and the other directed equatorward along the outside edge of the bay .

The bay supports commercial fishing, diving and marine recreation industries serviced from harbors in Monterey, Moss Landing and Santa Cruz. Water level oscillations in the bay and harbors, along with their associated hydraulic currents, play a significant role in the safety and operation of these interests, and, from an oceanographic perspective, have posed an open question regarding the continuous forcing of these modes. That is, seiches are normally excited by transient forcings such as seismic or meteorological events, then tend to lose energy and dissipate; however, modal oscillations in Monterey Bay are observed to be continuously present. Furthermore, noted that “it is difficult to conceive that such oscillations occur only in Monterey Bay, and, if it turns out that the excitation is global in nature, then our results may apply to other resonant basins around the world as well”. We therefore have two open research questions before us: what is the origin of these continuous modes, and, are they peculiar to Monterey Bay?

The seminal study of bay modes was contributed by , who applied analytical and numerical models of increasing sophistication to characterize the oscillations. While some of the numerical results were unsatisfying, the breadth and depth of the analysis were pioneering, and many of the fundamental results quantifying bay modes have been corroborated over ensuing decades. Wilson et al. (1965) assumed that “the surge phenomenon in Monterey Harbor is the consequence of surf-beats or of genuine long-period waves”, concluding that the latter was likely the cause, and it is notable that previous studies did indicate the continuous presence of oscillations. For example, analyzed a 6-month wave gauge record and found that 8–22 s period waves were present nearly 100 % of the time, and examined a 3-year tide gauge record, finding that “shorter waves (1.5–2 min) are recorded almost continuously”; however, we believe that were the first to conclusively observe that long-period bay-wide oscillations are effectively stationary and to question their genesis.

contributed a comprehensive review and analysis of Monterey Bay oscillations, and based on measurements over an 18 month period determined primary bay modes at the Monterey tide gauge of 55.9, 36.7, 27.4, 21.8, 18.4 and 16.5 min, broadly consistent with the work of Wilson et al. (1965). There is general agreement that the 55.9 min mode represents the fundamental longitudinal mode (north–south), while the 36.7 min harmonic is attributed to the primary transverse mode (east–west; refer to ). It is also accepted that Monterey Submarine Canyon acts to decouple the bay into two weakly coupled oscillators, one north of the canyon and one south. Regarding Monterey Harbor, estimates of modal periods are more variable, with most sources suggesting periods of 1–2 to 13.3 min, and several making specific mention of 9–10 min.

Previous observational studies (reviewed in ) used water level data sampled at (or averaged to) daily, hourly, 6, 4 or 1 min intervals such that periods below several minutes are not resolved. Here, we examine a 63 day record of 1 Hz water level recorded at the National Oceanic and Atmospheric Administration (NOAA) Monterey tide gauge allowing spectral characterization of water level variance to periods as short as 2 s, which to our knowledge is the highest-resolution analysis of modal oscillations in the bay. This high-resolution data are used to quantify and attribute water level oscillation modes in Monterey Bay and Harbor to physical processes and boundary conditions. We also analyze a 17.8 year record of 6 min water levels to characterize modes associated with bay-wide resonances, which to our knowledge is the longest continuous record of water levels analyzed for modal oscillations in the bay. This novel combination of data allows us to examine potential mode drivers of both harbor and bay-wide oscillations, clarifying the roles of potential mode drivers suggested by and suggesting a new one.

The dispersion relation for surface gravity waves dictates length scales corresponding to water depth and oscillation period (resolved from spectral analysis), and we characterize water level oscillations as belonging to bay, bight or harbor modes according to spatial scales appropriate to each domain as shown in Table . We define harbor oscillations as modes with periods less than 180 s and wavelengths less than 1 km matching spatial scales within the breakwater of Monterey Harbor (Figs.  and ). Modes with periods between 2 and 15 min and length scales between 2 and 10 km are considered bight modes associated with resonances between Point Pinos at the tip of Monterey Peninsula and the eastern shore of the bay. Bay modes have periods longer than 15 min and scales from 10 to 40 km. The lowest-frequency bay modes correspond to the longitudinal and transverse lengths of the bay.

Observations consist of a 63 day record (14 September through 29 November 2013) of 1 Hz water level from a microwave ranging sensor at the NOAA tide station located on Monterey Municipal Wharf no. 2, a 17.8 year record of 6 min water levels (23 August 1996–30 June 2014) from an acoustic ranging tide gauge located 4 m shoreward of the microwave sensor, and offshore wave height estimated every 30 min over the 63 day record of 1 Hz data from the Coastal Data Information Program (CDIP) buoy located approximately 15.2 km WSW of Moss Landing above Monterey Submarine Canyon. Station information and coordinates for the CDIP buoy are provided in , and for the tide gauges in . Gauge locations are shown in Fig.  with the white stars.

Since the bay and harbor oscillations are at much higher frequencies than the tides, we remove the tidal signal from the 1 Hz water level and analyze the water level nontide residual (NTR). The tidal response is obtained from standard NOAA tidal predictions at the Monterey tide gauge derived from 37 harmonic constituents over the tidal datum epoch of 1983 to 2001 .

Continuous availability of the 1 Hz data was not achieved, resulting in five segments of lengths 12.1, 12.3, 14.3, 10.5 and 14.1 days as shown in Fig.  exhibiting a relationship between nontide residual and offshore wave height. The magnitude of nontide residual is observed to be strongly correlated with wave height, and should be related to the canonical definition of significant wave height Hm0=4σ, where Hm0 is the zeroth moment of the water elevation spectrum and σ the standard deviation of water level. However, here, the water level variance and the significant wave height are not collocated, and it is known that non-wave-driven processes such as wind-driven setup and local oscillations also contribute to the variance such that the canonical relationship is not expected to be realized. Nonetheless, it is worth noting that water level variance estimates from tide gauges are robustly related to wave height and do have potential as proxies of wave height estimates .

Figure  presents 1 Hz water level data over 14 days of November 2013, the corresponding spectrogram of a 1 Hz nontide residual computed with 60 min windows and 50 % overlap, and a power spectral density (PSD) estimate of the 14 day, 1 Hz nontide residual. Power spectral densities are estimated by periodogram with a split cosine bell taper and modified Daniell smoother . The PSD indicates that dominant harbor energy is found at periods of 112, 60, 41, 31, 16 and 12 s, and are marked with vertical dashed lines. Bight modes are also identified with dash-dot lines (10.1, 9.0 and 4.2 min), and bay modes with dashed lines (55.9, 36.7, 27.4, 21.8, 18.4 and 16.5 min). Bight and bay modes are discussed in a following section.

The bay modes are clearly identified in 1 Hz spectra (Fig. ) and we estimate their water level amplitudes over the 63 day period by computing mean PSD amplitudes over all 1 Hz data over a sliding window of 24 h advanced in 2 h increments. The modes are ranked in terms of decreasing amplitude in Table  indicating that the 36.7 min mode exhibits the highest average power. Improved spectral resolution is afforded by a longer record and Fig.  presents a spectrogram and PSD of a 17.8 year record of 6 min water level data. Spectrogram PSDs are from records of 2048 points (8.53 days) with 50 % overlap. The two dominant features are the annual occurrence of increased broadband variance from winter storms (vertical bands), and the continuously present energy at the bay periods quantified by .

The PSD reveals that the 36.7 min transverse bay mode (east–west) is not only the most energetic, but is actually also a series of closely spaced modes. The 27.4 min mode exhibits a high quality (Q) factor (ratio of energy stored in the mode resonance to energy supplied driving the resonance) as evidenced by the high signal-to-noise ratio and narrow bandwidth of the spectral peak indicating a resonance highly tuned to the source. According to and the spatial harmonics of this mode correspond to partitioning of the bay into thirds with northern, central (canyon) and southern regions, which would suggest that this mode is efficiently tuned to the decoupling of the southern and northern parts of the bay by the canyon, whereas the fundamental longitudinal mode (55.9 min) is not. It is not clear why this would be the case. The dominance of these modes suggests that they are the modes most directly coupled to the unknown continuous forcing of bay oscillations.

Monterey Bay is an intensely studied oceanic body, with the Naval Postgraduate School, Moss Landing Marine Laboratories, Hopkins Marine Station and Monterey Bay Aquarium Research Institute providing decades of physical oceanographic research. Water level oscillations in the bay have been studied since at least the late 1940s, yet it seems that were the first to notice that bay-wide oscillations are continuously present. Based on a 17.8 year record of water levels at the NOAA tide station, we substantiate their observation and validate the accuracy of oscillation periods of the six primary bay modes determined by . Amplitudes of the bay modes indicate that the fundamental transverse mode (36.7 min) is the mode that is most directly driven by the unknown source. Kinematics of this oscillation coupled with the resonance amplitude lead us to estimate that a power source of roughly 52.4 MW drives this mode. Comparison of this energy rate to prospective forcings from microseisms, internal waves and the associated turbulence indicate that neither of these mechanisms has sufficient power to sustain the mode. We also find that surface waves are not coherently related to the bay modes.

A potential mode driver is the anticyclonic mesoscale gyre situated just offshore the bay. The potential and kinetic energy it contains are sufficient to sustain the fundamental mode; however, we find that turbulent instabilities such as Kelvin–Helmholtz waves generated in the shear interface of the gyre do not have spatial scales consistent with the fundamental mode. It is intriguing to note that the nodal line of the fundamental transverse mode would be located roughly λ/2=26.7 km from the eastern shore, corresponding roughly to the eastern edge of the gyre. It would seem a peculiar circumstance if this spatial arrangement along with the potential and kinetic energy scales of the gyre were merely coincidental to the continuous harmonic driver of bay modes, but presently we cannot conceptualize a supportable mechanism for such a coupling. While the present study is purely observational, high-resolution non-hydrostatic coupled ocean–atmosphere models could clarify the roles of potential mode drivers, and should be pursued.

Regarding oscillations in Monterey Harbor, we present the first high-resolution analysis resolving spectral components to periods as short as 2 s. The spectral nature of these modes indicates that they represent a continuum of harmonic oscillators closely spaced in wave number. For example, the 112 s dominant mode can be attributed to standing waves between the tide gauge and multiple boundaries including the rocky coast to the east, the breakwater to the north and mooring docks to the northeast, while the 41 s mode is associated with the breakwater to the south. Concerning wave forcing, it is demonstrated that the primary harbor mode amplitude grows as the square root of offshore significant wave height, and that there is a mode-specific dependence on wave-arrival direction. It is also observed that tidal phase serves to frequency modulate the harbor modes, with evidence of enhanced mode energy during high tide.

A temporal analysis monitoring the evolution of harbor modes in response to wave forcing supports the idea that amplitude modulation of specific harbor modes arises from modal interference, and it is observed that when such modulations synchronize, they can have a significant impact on water level amplitude. Identification of specific modes with associated physical sources raises the possibility of engineering solutions to mitigate specific oscillations.