OSOcean ScienceOSOcean Sci.1812-0792Copernicus GmbHGöttingen, Germany10.5194/os-11-439-2015Water level oscillations in Monterey Bay and HarborParkJ.joseph_park@nps.govhttps://orcid.org/0000-0001-5411-1409SweetW. V.HeitsenretherR.National Park Service, 950 N. Krome Ave, Homestead, FL, USANOAA, 1305 East West Hwy, Silver Spring, MD, USANOAA, 672 Independence Parkway, Chesapeake, VA, USAJ. Park (joseph_park@nps.gov)8June201511343945313October201420November20142May201512May2015This work is licensed under a Creative Commons Attribution 3.0 Unported License. To view a copy of this license, visit http://creativecommons.org/licenses/by/3.0/This article is available from https://os.copernicus.org/articles/11/439/2015/os-11-439-2015.htmlThe full text article is available as a PDF file from https://os.copernicus.org/articles/11/439/2015/os-11-439-2015.pdf
Seiches are normal modes of water bodies responding to geophysical
forcings with potential to significantly impact ecology and maritime
operations. Analysis of high-frequency (1 Hz) water level
data in Monterey, California, identifies harbor modes between 10 and
120 s that are attributed to specific geographic features.
It is found that modal amplitude modulation arises from cross-modal
interaction and that offshore wave energy is a primary driver of these
modes. Synchronous coupling between modes is observed to
significantly impact dynamic water levels. At lower frequencies with
periods between 15 and 60 min, modes are independent of
offshore wave energy, yet are continuously present. This is
unexpected since
seiches normally dissipate after cessation of the driving force,
indicating an unknown forcing. Spectral and kinematic estimates of
these low-frequency oscillations support the idea that a persistent
anticyclonic mesoscale gyre adjacent to the bay is a potential mode
driver, while discounting other sources.
Introduction
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 .
Monterey Bay and Canyon. The location of the wave gauge (CDIP) and
water level gauges are indicated with stars. Station information and
coordinates for the CDIP buoy are provided in , and for the
tide gauges in . We classify bight modes as having periods
between 2 and 15 min with length scales between 2 and 10 km,
and bay modes with periods longer than 15 min and scales from 10 to
40 km.
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?
Sea surface temperature image from 13 October 2008 depicting the
persistent mesoscale anti-cyclonic gyre offshore of Monterey Bay. Image from
.
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.
Length scales
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
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 .
Spatial scales of the bay, bight and harbor modes according to the
dispersion relation ω2=gktanh(kd) where ω is
frequency, k the wave number, d the water depth and λ the
wavelength. The bay and bight modes use depths of 60 m, the harbor
modes 7.5 m.
Bay Bight Harbor Periodλ/2Periodλ/2Periodλ/2(min)(km)(min)(km)(s)(m)55.940.710.17.411248036.726.79.06.56025227.419.94.23.14117221.715.83113318.413.4166716.512.01250
Significant wave height (Hs) at the Monterey Canyon CDIP buoy
(30 min data) and 1 Hz nontide residual water (NTR) levels
from the NOAA tide gauge in Monterey Harbor.
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
.
Oscillations in Monterey Harbor
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.
Harbor modes
Harbor modes are typified by smooth broad peaks in the PSD, suggesting
that for a specific harbor component, there are multiple harmonic
oscillators closely grouped in wave-number space, leading us to expect
that there will be a nearly continuous range of spatial scales
contributing to these modes. In addition to the broad spectral peak
centered on 112 s, there are also distinct spectral lines near
112 s indicating specific resonant scales. The spectrogram
reveals a time-dependent intensity of harbor modes, for example, the
generally low amplitudes around 25 November and high amplitudes
following 27 November. Referring to Fig.
suggests that offshore wave height influences harbor amplitudes.
Another interesting feature is the frequency modulation (FM) of modes
coherent with the tidal signal. We believe that this frequency
modulation arises from the changing water depth and shoreline profile
as mean water level rises and falls such that different length scales
for surface waves are realized. These spectral features are
representative of all 1 Hz data.
Figure shows a chart of the harbor
with colored arrows corresponding to resonant mode length scales
governed by the dispersion relation at a water depth of 7.5 m
from mode periods identified in Fig. .
The highest-frequency modes with periods less than 30 s are
not depicted in Fig. ; they are
associated with reflections from wharf infrastructure.
It should be noted that standing waves form nodes at multiples
of λ/2 from a reflective fixed boundary, and at multiples of
λ/4 from an open boundary so that the lowest-frequency
standing wave (mode) between two reflectors corresponds to a
distance of λ/2, and in the case of a mode between an
open boundary such as the tide gauge on the wharf and a reflector,
a distance of λ/4. Given this, we attribute the
30 and 60 s modes to reflections from the
breakwater protecting the mooring docks, the 41 s
mode to a standing wave within the mooring docks, and
the dominant harbor mode near 112 s to
reflections between the inner and outer breakwaters.
Wave height
To assess the wave dependence of harbor mode amplitudes,
Fig. presents NTR PSD estimates during
three 2 h periods when offshore wave height was increasing.
With a significant wave height of 0.8 m (dominant period
Ts=5.9s), the characteristic harbor modes are
broadly observed with the 60 s period reflection from the
breakwater well resolved. As significant wave height grows from 0.8
to 1.1 m, most of the energy increase is contained in the band
between 80 and 300 s, suggesting that it is a combination of
growing reflections from the rocky shore to the west, the breakwater
to the north and a bight mode contributing to increased NTR variance.
When offshore wave height reaches 2.4 m
(Ts=12.5s), there is a significant increase in
energy at all modes, and a conspicuous broadening of the 80 to
120 s resonances suggesting that a rich set of closely spaced
modes is being driven. We also note that spectral shapes are
essentially invariant as offshore waves transition from periods of 6
to 12 s but the amplitudes increase, indicating that these
modes are generated by local resonances in the harbor forced by
offshore wave energy, but independent of wave period.
(a) Raw 1 Hz water levels from 15 November 2013
through 29 November 2013. (b) Spectrogram of the water levels.
(c) Power spectral density estimates of nontide residual water
levels. Dashed vertical lines mark the bay-wide resonance modes (55.9, 36.7,
27.4, 21.8, 18.5 and 16.5 min), dash-dot lines mark bight periods
(10.1, 9.0 and 4.2 min) and dotted lines the harbor modes (112, 60,
41, 31, 16 and 12 s).
Chart of Monterey Harbor with resonant mode length scales corresponding to
periods observed in the power spectra. The tide gauge location is denoted by
the star. Wavelengths are determined from the general dispersion relation
applied at a depth of 7.5 m. Spatial scales of λ/2 are
sustained between two reflective boundaries, while the fundamental length of
λ/4 corresponds to one open boundary such as the tide gauge on the
wharf and one reflective boundary. Chart is number 18 685 from the NOAA
National Ocean Service Coast Survey.
To quantify the amplitude dependence of offshore wave height on harbor
and bay oscillations, we regress PSD amplitudes of the dominant harbor
and bay modes (112 s and 36.7 min, respectively)
against offshore significant wave height (Hs):
PSDM=αHs+βHs1/2,
where PSDM are 1 Hz NTR PSD amplitudes in
dB of the 36.7 min or 112 s modes over a 24 h
sliding window advanced in 2 h increments over all data.
Figure plots the data and model fits,
indicating that the harbor mode responds to offshore wave amplitude
with nonlinear growth (r2=0.70), while the bay mode has no such
dependence (r2=0.03).
Wave direction
To examine offshore wave direction in the forcing of harbor modes,
Fig. a plots NTR PSD estimates from two
periods when offshore significant wave heights were small
(0.5–0.7 m), and the dominant wave direction was either west
(250∘ N) or northwest (295∘ N). With the
exception of the 15 s resonance, harbor modes are
significantly enhanced when low-amplitude waves are arriving from the
northwest instead of the west. This is consistent with the wave
refraction analysis presented by indicating that
wave energy from the west is less efficiently refracted into the
harbor than waves from the northwest.
Figure b presents NTR PSD estimates
from larger waves and arrival directions of west
(265∘ N) and northwest (297∘ N). Here,
a comparison of the NTR spectra finds that wave direction has a smaller
influence on harbor mode amplitudes then when waves are small,
although some specific modes such as those with 9 and 60 s
periods are enhanced.
Tidal phase
Spectrograms of 1 Hz data indicate that tidal phase serves to
modulate the frequency of harbor modes as shown in Fig. b. To examine this, we selected
a period with minimum offshore wave height covering a semidiurnal
tidal cycle (25 November, 04:30 to 14:00 UTC). This cycle
corresponds to the largest
semidiurnal cycle of the day with a range of roughly 1 m, which
is close to the mean tidal range of 1.08 m. The water depth at
the sensor is nominally 9.1 m at MLLW, so that the water depth
to mean tidal range ratio is roughly 9 to 1.
Power spectral density estimates of 1 Hz nontide residual
over 2 h periods during 20 and 21 September 2013 with different deep
water wave heights. Dotted vertical lines mark the harbor modes at 112, 60,
41, 31, 16 and 12 s.
(a) Water level amplitudes at the dominant harbor period of
112 s, and (b) amplitudes at the dominant bay period
of 36.7 min as a function of significant wave height. Each
amplitude is estimated from a PSD computed over an 18 h moving
window with a time increment of 1 h. Solid lines are a least
squares fit to a nonlinear model (PSDM=αHs+βHs1/2).
Note that the water level amplitudes are in dB.
To compare the spectral response of these two tidal regimes, we
computed NTR PSD
estimates over 2 h periods centered on the low (04:30–06:30)
and high water (12:00–14:00) tidal phases as shown in
Fig. . There is a clear shift from
longer to shorter periods at high water in relation to low water,
supporting the idea that water depth and associated shoreline
boundary condition characteristics influence the modal structure
of the harbor.
We note that this
cycle was during a neap tide, and expect that, during spring tide, the
tidal dependence on harbor oscillation frequencies will be even more
pronounced. For example, in Fig. b,
we noted the frequency
modulation of harbor modes with tidal phase where the tidal range
exceeded 2 m, and one can clearly see the changing frequencies of
the harbor modes.
Dynamic mode response
The dynamic characteristics of harbor modes can be assessed by
decomposing the 1 Hz nontide residuals into components
capturing temporal variations at different timescales with a maximal
overlap discrete wavelet transform (MODWT, ). The
MODWT is defined in Appendix A and we employ an 11-level
(J=11) transform based on the least asymmetric mother
wavelet (LA8). Approximate temporal scales for the wavelet levels are
listed in Table .
Figure plots the MODWT decomposition for two of the
2 h periods shown in Fig. . The
bottom panel of each plot shows the NTR time series with panels above
in ascending order plotting the wavelet coefficients for each level of
increasing timescale. The wavelet coefficients of each level are scaled
by their respective magnitude squared so that the amplitude
represents the partial variance contributed by each level. With
offshore significant wave heights of
0.8 m (0–2000 s in Fig. a), the NTR
energy is fairly evenly distributed between the W3, W4,
W5, W6, W10 and W11 levels corresponding to temporal
scales of 15, 31, 59, 99, 900 and 1800 s. As waves grow from
0.8 to 1.1 m (3000–7000 s in Fig. a),
the W6 and W7 levels exhibit the emergence of
oscillatory modes at timescales of 96 and 126 s, consistent
with the spectral perspective shown in
Fig. . When waves have grown to
2.4 m, we find in Fig. b that the
W5, W6 and W7 timescales (58, 101,
117 s) dominate the NTR energy, again consistent with
the Fourier decomposition in Fig. . The
same general behavior with the emergence, growth and dominance of the
50 to 120 s modes in Monterey Harbor during wave events is
robustly observed.
An interesting feature of these primary harbor modes (W5,
W6 and W7) is their temporal amplitude modulation (AM).
These AM effects are generally not synchronous across levels, and
appear to have modulation periods in the neighborhood of
20 min. These periods are not representative of the bay or
bight modes previously identified, and their non-synchronous nature
suggests that they are not forced by a unitary driver, e.g., long-period waves propagating from offshore. However, we previously noted
the broad spectral nature of the harbor modes indicative of multiple
harmonic oscillators closely spaced in frequency/wave number, and this
leads us to speculate that the AM arises from superposition of closely
spaced modes in frequency space. To test this hypothesis, we select 10
spectral amplitudes from the NTR PSD with 2.4 m wave height
from Fig. at periods of 58, 83, 94, 97,
100, 104, 107, 112 116 and 209 s, and create a reconstruction
from these components with a harmonic superposition:
R=∑iAisin(ωit),
where Ai is the NTR spectral amplitude and
ωi the frequency. This reconstruction is compared
to the W6 level in Fig.
where the envelope has been plotted for each time series.
Qualitatively, the overall AM behavior of the reconstructed
superposition and the actual mode is similar, leading to the
suggestion that it is cross-modal interference of closely spaced
harbor modes resulting in the AM of specific resonances. Although the
AM envelopes are generally not synchronous, there is nothing
precluding them from randomly aligning, and in
Fig. we present evidence of such
an alignment between the W6 and W7 levels
suggesting that when modes synchronize, their impact on the overall NTR
can be significant.
Mean water level spectral amplitudes at the bay-wide oscillation
modes estimated from 1 Hz nontide residuals over 63 days. The
95 % confidence interval on spectral amplitude is 3.0 dB.
PeriodPSD dBΔdBWater level(min)(m2Hz-1)(m2Hz-1)(m)36.7-9.30.00.34127.4-12.8-3.40.23055.9-13.3-4.00.21521.7-15.7-6.40.16418.4-17.8-8.40.12916.5-17.9-8.50.128Oscillations in Monterey Bay
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.
(a) Power spectral density estimates of 1 Hz
nontide residual water level over 2 h periods with offshore
significant wave heights of 0.5 and 0.7 m and arrival directions of
250 and 295∘. (b) Power spectral density estimates of
1 Hz nontide residual water level over 2 h periods with
offshore significant wave heights of 2.0 and 2.1 m and arrival
directions of 265 and 297∘. Dotted lines correspond to harbor
modes (112, 60, 41, 31, 16 and 9 s).
PSD estimates of 1 Hz NTR data from 2 h periods
centered on low and high water of a tidal period. Offshore waves were low
during this period. Dotted vertical lines mark harbor modes at 112, 60, 41,
31, 16 and 12 s.
(a) Discrete wavelet transform decomposition of nontide residual
during a 2 h period when deep water significant wave height increased
from 0.8 to 1.1 m. The red box highlights the W6 and W7 wavelet
levels exhibiting emergence of energy in these two bands as wave height
increases. (b) Discrete wavelet transform decomposition of nontide
residual during a 2 h period when deep water significant wave height
was 2.4 m. The red box highlights the W5, W6 and W7 wavelet
levels exhibiting dominance of energy in these bands and amplitude
modulation. Power spectral density estimates of these 2 h period are
shown in Fig. .
Mode forcing
considered six physical mechanisms as prospective
forcings for continuous oscillations of the bay:
edge waves,
long-period surface waves,
sea breeze,
internal waves,
microseisms, and
small-scale turbulence,
and noted that the first three are not likely to be continuously present,
and so are not consistent with the observation of persistent
oscillations. Tidally synchronous internal waves have been observed
propagating up the submarine canyon and episodically transitioning to
bores ; their tidal coupling
suggests a continuous presence and potential mode forcing.
Microseisms are an appealing candidate due to their omnipresence
; however, it was speculated that their energy was
insufficient at the bay frequencies to drive the oscillations.
Breaker et al. (2010) also discounted small-scale turbulence on the basis of
its intermittent nature. These arguments are based primarily on
temporal persistence; however, we offer an alternative perspective
based on kinematic energy scales.
Under the assumption that the 36.7 min mode is the directly
forced fundamental mode, we can ask questions regarding its observed
amplitude and spectral resonance to grossly estimate the energy
required to sustain it. Table indicates that the rms (root mean square) water
level deviation of this mode at the Monterey tide gauge is
hM=0.341m. Neglecting the effect of
shoaling on wavelength, a raised-cosine profile with λ/2=26.7km (Table ) approximates the cross-shore
elevation of the mode, with a profile area of AM=9104m2. assessed spatial harmonics
of bay modes with a regional ocean modeling system (ROMS)
implementation providing an estimate of the
alongshore profile of this mode. They found a strong harmonic response
over the majority of the eastern bay shoreline with a damped response
in an area north of the submarine canyon. Based on this, we estimate
that approximately 70 % of the shoreline responds to this
mode. In the spirit of our gross estimate, we assume the alongshore
spatial dimension of the mode to be LA≈0.7×40km=28km. Combining this with the
cross-shore elevation area, we arrive at an estimate of the volume of
water displaced by this mode of VM=LAAM≈258Mm3. The
energy to move this mass is equivalent to the work performed to change
the potential energy of the mass in the gravitational field, and we
estimate the energy of the mode as EM=ρVMhMg≈894.75GJ, or
an average power of 406 MW over the 36.7 min modal
period. Obviously, this leading-order value does not incorporate
dissipation and momentum, terms that we ignore in all subsequent
energy estimates.
If the Q factor is
large (the resonance signal-to-noise ratio is high), then Q may be
estimated from the power spectrum: Q=fM/Δf, where fM is the mode resonant
frequency and Δf the -3 dB (half power)
bandwidth of the mode. We observe signal-to-noise ratios routinely
exceeding 5 dB at the bay modes
(Figs. and
) and use PSD estimates based
on 120 h records of 1 Hz NTR (95 % CI
2.6 dB) advanced in 4 h increments to find a mean
Δf=5.86×10-5Hz and an estimate of Q=7.74
for the 36.7 min mode. This implies that within the gross
level of estimation in which we are engaged, the driving energy for the
36.7 min mode is roughly 894.75/7.75≈115.5GJ, or a power consumption of 52.4 MW. We will
compare this forcing to estimates of energy available from prospective
mode drivers in the following sections.
Microseisms
Microseisms are pressure (acoustic) waves primarily generated by
nonlinear wave–wave interactions on the ocean surface. They radiate
into the atmosphere where they are globally detected as microbaroms
, into the water column as acoustic modes, and
couple into the seafloor where they travel as Rayleigh/Stoneley waves
presenting a global seismic signature . One can
estimate deep water microseismic energy by considering the acoustic
intensity of a plane wave incident on the seafloor
IA=P2/Z where P is the pressure
and Z the acoustic impedance. In the linear regime, the
characteristic acoustic impedance of a medium is
Z0=ρc where ρ is the density and
c the sound speed, which in the case of seawater is
approximately Z0≈1.5×106Nsm-3.
Spectral amplitudes of the microseism peak at deep water seafloor
sites were found by to be approximately
5000 Pa2Hz-1, giving an intensity of IA≈5×10-3Wm-2. Assuming a source generation
region of radius 100 km, the total power is 157 MW.
This energy is efficiently converted into seismic Rayleigh waves or
ocean acoustic modes , and these waves can
propagate with small attenuation coefficients over large distances,
suggesting that microseismic energy is of sufficient magnitude to
couple to bay resonances. However, demonstrate
that coastal zone microseismic energy is dominated by local wave
reflections from coasts, not deep water arrivals, as propagation from
deep to shallow water is inhibited by the changing seismic waveguide
as refraction of the Rayleigh modes significantly reduces energy
reaching the coastal zone .
found coastal zone pressures of approximately
70Pa2Hz-1, which under the same assumptions as
the deep water case gives a total power of 2.2 MW,
insufficient to drive the fundamental bay mode. Not only is the
estimated energy insufficient, microseismic energy is distributed
around periods of 5 to 7 s, more than 2 orders of magnitude
shorter than the dominant bay mode, and we conclude that microseisms
are not a likely driving mechanism for the continuous bay
oscillations.
Internal waves
observed that internal waves in Monterey Canyon are
nearly an order of magnitude more energetic than in the open ocean and
are tidally locked to M2. Mean horizontal energy fluxes
are steered by canyon bathymetry and are predominantly up canyon, with
depth integrated fluxes of 5 kWm-1 at the
mouth diminishing to ±1 kWm-1 near the head
(Moss Landing), although the internal wave field is highly anisotropic,
with evidence of both sources and sinks along the canyon.
Particularly energetic fluxes have been characterized as bores with
peak currents of 55 cms-1 and 2 h averages
exceeding 30 cms-1.
quantified turbulent kinetic energy dissipation, finding that the
turbulence is primarily in a stratified turbulent layer (STL) along
the canyon floor and is thickest on the canyon axis. Time averaged
values of STL thickness and dissipation were estimated to be
135 m and ε‾=1.36×10-6Wkg-1.
Both and compared mean internal
wave energy flux with dissipation rates, tentatively concluding that
most of the along-canyon internal energy is dissipated as turbulence,
although noted that “large error estimates suggest
this agreement is fortuitous”. A gross estimate of the internal tide
energy can be made from a mean value of energy flux:
2 kWm-1, which for a 20 km
length of canyon gives PI≈40MW
(20 km corresponds to the length of the canyon from the head over
which there is a primarily a single channel, is approximately equal to
the length of the primary transverse mode, and is the length used by
in their global estimates of canyon dissipation).
An estimate of the dissipation over this 20 km section with
a mean STL height of 135 m and width of 4 km
(VSTL=10.8Gm3) gives
PT=ρVSTLε‾≈15.1MW.
Even if these gross estimates of internal wave energy and dissipation,
which ignore the well-documented fine-scale sink/source and spatial
variability, are only accurate to a factor of 2, the residual energy
rate PI-PT=40-15=25MW is insufficient to sustain our estimate of the fundamental
mode (52.4 MW).
(a) Wavelet level W6 of the nontide residual over
a 2 h period on 21 September 2013 when offshore wave height was
2.4 m. (b) Time series reconstruction from ten spectral
amplitudes of Fig. at periods of 58, 83, 94, 97,
100, 104, 107, 112, 116 and 209 s.
Mesoscale eddy
A potential energy source that, to the authors knowledge, has not been
considered as a mode driver, is the persistent mesoscale anticyclonic
gyre offshore Monterey Bay (Fig. ).
The gyre is nominally 50–70 km in
diameter and models suggest that it extends from the surface to at
least 600 m in depth with instantaneous velocities of
70 cms-1 near the surface and 30 cms-1 at
depth . The gyre supports a persistent
elevated dome of sea surface height (SSH) rising approximately
10–12 cm above the coastal levels along the eastern shore of
the bay .
The potential energy of this dome with respect to the coast can be
estimated by assuming the dome has a cosine-bell profile from the
center with λ/2≈30km. Taking the height of
the dome to be hG=10cm, the dome volume is
VG=2π∫0λ/2[hGx+hGxcos(kx)]dx=168Mm3 where
k=2π/λ. The potential energy of this mass is
EG=ρVGghG2=120.7GJ, which is comparable with the 115.5 GJ
estimated to sustain the 36.7 min fundamental mode. However,
even though the geostrophic balance of the gyre will fluctuate due to
wind stress and dynamics of gyre interaction with the California
Current, it is unlikely that the geostrophic balance will fluctuate by
its full amplitude on timescales of 30 min, allowing this mass
of water to relax and propagate as a wave.
The kinetic energy of the equatorward portion of the gyre can be
estimated from the cross-sectional flow of the gyre, which from the
model of during a weak flow regime (April) can be
represented as a velocity of u=5cms-1 from
a depth of at least 200 to 600 m over a width of
30 km. A 30 km long section of this flow
(corresponding to the length of the equatorward portion of the gyre
just offshore the bay) would have a kinetic energy of
EK=12ρVKu2=465.8GJ, so there appears to be sufficient energy in the jet
offshore the bay to sustain the fundamental mode, but it is not clear
how a portion of this energy would couple into the mode.
Discrete wavelet transform decomposition of nontide residual water
level during a 2 h period with an offshore significant wave height of
2.0 m. Dashed vertical lines mark periods of synchronization between
the W7 (T≈118s) and W6 (T≈111s)
levels.
(a) Spectrogram of 6 min water levels at the
Monterey tide gauge from August 1996 through June 2014. (b) Power
spectral density estimate of 6 min water levels at the Monterey tide
gauge from August 1996 through June 2014. Dashed vertical lines mark the
bay-wide resonance modes of 55.9, 36.7, 27.4, 21.8, 18.5 and
16.5 min.
One possibility is that the shear interface between the jet and deeper
water generates Kelvin–Helmholtz instabilities to drive the fundamental
mode. With the assumption of two stratified water masses of density
ρ1 and ρ2 and mean velocities
U1 and U2, the minimum horizontal
wave number of instabilities can be found from a dispersion relation of
the unsteady Bernoulli equation as
kmin<gρ22-ρ12ρ1ρ2U2-U12.
To apply this, we assume that the upper water mass is the warmer,
fresher water of the California Current gyre (ρ1,U1) and the lower layer the colder, saltier deep water
(ρ2,U2), where values of ρ1=1026.70kgm-3 and ρ2=1026.97kgm-3 are mean values from the surface to a depth of
200 m at offshore and canyon locations computed from
temperature and salinities reported in Fig. 13 of
using the method of . The
resultant maximum wavelengths over the range of velocity differences
suggested by (30–70 cms-1)
indicate that instabilities with length scales of several hundred
meters are possible. However, these length scales are much shorter
than the characteristic scale of the 36.7 min fundamental bay
mode (26.7 km), and it seems unlikely that even coherent
trains of such instabilities could effectively drive the fundamental
mode.
Finally, we examined whether there was a relation between the seasonal
upwelling and mode response. Upwelling in the bay typically peaks in
spring/summer (April/May), introducing a tongue of upwelled water
between the gyre and the outside edge of the bay. It was assumed that,
if the bay modes are driven by the gyre, then upwelling that might
partially decouple the bay from the gyre could have an impact on the
bay mode amplitudes. We compared PSD estimates of the 6 min data
averaged over 17 years for April and May with PSD estimates
from September and October, but found no statistically significant
differences. To the extent that upwelling was expressed in the April
and May data, we find no evidence to support the idea that it changes
the oscillating modes.
Conclusions
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=26.7km 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.
The maximal overlap discrete wavelet transform (MODWT,
) is defined by
X=∑jJDj+SJ,
where X is a time series of length N
and j represents a distinct wavelet level. The
Dj are referred to as wavelet details, and
SJ the smooth, each a vector of length N. The
details capture transient and oscillatory behavior at different
timescales; the smooth are the residual energy not captured by the
details and, in the optimal decomposition, correspond to a moving
average of the signal. Each wavelet level is computed with a matrix
transform Dj=WjTΦj and SJ=VjTΨJ where W
and V are N×N matrices of MODWT
coefficients, and Φ and Ψ
are referred to as the wavelet coefficient and scaling coefficient
vectors, respectively. The wavelet and scaling coefficients are the
result of cascaded high-pass (h) or low-pass (g)
wavelet and scaling filters recursively applied to the input
Φj,t=∑l-0Lj-1hj,lXt-1,modN, or
Ψj,t=∑l-0Lj-1gj,lXt-1,modN, where the
filter width at each level Lj=(2j-1)(L-1)+1 is
determined by the length of the mother wavelet L. In terms
of the MODWT matrix and wavelet scaling coefficients, the
input can then be represented as
X=∑jJWjTΦj+VJTΨJ.
The MODWT provides a convenient encapsulation of the signal energy in
terms of the wavelet and scaling coefficient vectors:
‖X‖2=∑jJ‖Φj‖2+‖ΨJ‖2,
which is related to the sample variance of X. It
is useful to plot the wavelet coefficients of each level scaled by its
respective magnitude squared so that the relative amplitude scales
represent the partial variance contributed by each level.
Acknowledgements
The authors gratefully acknowledge insightful discussions and reviews of the
manuscript by L. Breaker of Moss Landing Marine Laboratory, University of
California, and Y.-H. Tseng, National Center for Atmospheric Research (NCAR)
Earth Systems Laboratory. Edited by:
E. J. M. Delhez
ReferencesAMS: Seiches in Lake Garda, Mon. Weather Rev., 31, 532–533,
doi:10.1175/1520-0493(1903)31[532b:SILG]2.0.CO;2, 1903.AMS: The seiche and its mechanical explanation, Mon. Weather Rev., 34,
226,
doi:10.1175/1520-0493(1906)34<226b:TSAIME>2.0.CO;2, 1906.
Bloomfield, P.: Fourier Analysis of Time Series: An Introduction, Wiley, New York,
1st Edn., 261 pp., 1976.
Breaker, L. C., Broenkow, W. W., Watson, W. E., and Jo, Y.: Tidal and non-tidal
oscillations in Elhorn Slough, California, Estuar. Coast., 31,
239–257, 2008.Breaker, L. C., Tseng, Y., and Wang, X.: On the natural oscillations of
Monterey Bay: observations, modeling, and origins, Prog. Oceanogr., 86, 380–395,
doi:10.1016/j.pocean.2010.06.001, 2010.Bromirski, P. D. and Duennebier, F. K.: The near-coastal microseism spectrum:
spatial and temporal wave climate relationships, J. Geophys. Res.-Sol. Ea., 107, ESE 5-1–ESE 5-20,
doi:10.1029/2001JB000265, 2002.Carter, G. S. and Gregg, M. C.: Intense, variable mixing near the head of
Monterey Submarine Canyon, J. Phys. Oceanogr., 32, 3145–3165,
doi:10.1175/1520-0485(2002)032<3145:IVMNTH>2.0.CO;2, 2002.CDIP: Station 156 Monterey Canyon Outer, available at:
http://cdip.ucsd.edu/?nav=recent&stn=156&sub=observed&xitem=info&stream=p1,
last access: 14 November 2014.Chrystal, G.: On the hydrodynamical theory of seiches, T. Roy. Soc. Edin.-Earth, 41, 599–649,
doi:10.1017/S0080456800035523,
1906.
Darwin, G. H.: The Tides and Kindred Phenomena in the Solar System, Houghton,
Boston, 342 pp., 1899.
Forston, E. P., Brown, F. R., Hudson, R. Y., Wilson, H. B., and Bell, H. A.:
Wave and surge action, Monterey Harbor, Monterey California, Tech.
Rep. 2-301, United States Army Corps of Engineers, Waterways Experiment
Station, Vicksburg, MS, 45 Plates, 1949.
Gill, A. E.: Atmosphere-Ocean Dynamics, Academic Press, New York, 662 pp., 1982.Hasselmann, K.: A statistical analysis of the generation of microseisms,
Rev. Geophys., 1, 177–210,
doi:10.1029/RG001i002p00177, 1963.IOOS: A National Operational Wave Observation Plan. Integrated Ocean Observing
System (IOOS) plan for a surface-wave monitoring network for the United
States, Tech. rep., Integrated Ocean Observing System, available at:
http://www.ioos.noaa.gov/library/wave_plan_final_03122009.pdf (last access: 14 November 2014), 2009.
Kedar, S., Longuet-Higgins, M., Webb, F., Graham, N., Clayton, R., and Jones, C.: The origin of deep ocean microseisms in the North Atlantic Ocean, P. R. Soc. A, 464, 777–793,
doi:10.1098/rspa.2007.0277,
2008.Key, S. A.: Internal tidal bores in the Monterey Canyon, M. S. thesis,
Naval Postgraduate School, available at: http://www.dtic.mil/dtic/tr/fulltext/u2/a370949.pdf (last access; 14 November 2014),
1999.
Kundu, P. K.: Fluid Mechanics, Academic Press, San Diego, 1st Edn., 638 pp., 1990.Kunze, E., Rosenfeld, L. K., Carter, G. S., and Gregg, M. C.: Internal waves in
Monterey Submarine Canyon, J. Phys. Oceanogr., 32, 1890–1913,
doi:10.1175/1520-0485(2002)032<1890:IWIMSC>2.0.CO;2, 2002.NOAA: Monterey, CA, National Water Level Observation Network –
Station ID: 9413450, available at:
http://tidesandcurrents.noaa.gov/stationhome.html?id=9413450, last
access: 14 November 2014a.NOAA: Harmonic Constituents for 9413450, Monterey CA,
available at: http://tidesandcurrents.noaa.gov/harcon.html?id=9413450,
last access: 14 November 2014b.
Park, J., Heitsenrether, R., and Sweet, W.: Microwave and acoustic water level
and significant wave height estimates at NOAA tide stations, J. Atmos. Ocean. Tech., 31, 2294–2308, 2014.
Parke, M. E. and Gill, S. K.: On the sea state dependence of sea level
measurements at platform Harvest, Mar. Geod., 18, 105–111, 1995.
Percival, D. B. and Walden, A. T.: Wavelet Methods for Time Series Analysis,
Cambridge University Press, New York, 594 pp., 2006.Petruncio, E., Rosenfeld, L., and Paduan, P.: Observations of the internal tide
in Monterey Canyon, J. Phys. Oceanogr., 28, 1873–1903,
doi:10.1175/1520-0485(1998)028<1873:OOTITI>2.0.CO;2, 1998.Raines, W. A.: Sub-tidal oscillations in Monterey Harbor, M. S. thesis,
Naval Postgraduate School, available at: https://calhoun.nps.edu/bitstream/handle/10945/13214/subtidaloscillat00bain.pdf?sequence=1 (last access: 4 June 2015),
1967.Rosenfeld, L. K., Schwing, F. B., Garfield, N., and Tracy, D. E.: Bifurcated
flow from an upwelling center: a cold water source for Monterey Bay,
Cont. Shelf. Res., 14, 931–964,
doi:10.1016/0278-4343(94)90058-2, 1994.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, 347–404,
doi:10.1016/j.ocemod.2004.08.002, 2005.Ryan, J. P., Davis, C. O., Tufillaro, N. B., Kudela, R. M. and Gao, B. C.:
Application of the Hyperspectral Imager for the Coastal Ocean to Phytoplankton Ecology Studies in Monterey Bay, CA, USA,
Remote Sens., 6, 1007–1025,
doi:10.3390/rs6021007, 2014.Strub, P. T., Kosro, P. M., and Huyer, A.: The nature of the cold filaments in
the California Current system, J. Geophys. Res.-Oceans,
96, 14743–14768,
doi:10.1029/91JC01024, 1991.Tseng, Y.-H. and Breaker, L. C.: Nonhydrostatic simulations of the regional
circulation in the Monterey Bay area, J. Geophys. Res.-Oceans, 112, C12017,
doi:10.1029/2007JC004093, 2007.Tseng, Y.-H., Dietrich, D. E., and Ferziger, J. H.: Regional circulation of the
Monterey Bay region: hydrostatic versus nonhydrostatic modeling,
J. Geophys. Res.-Oceans, 110, C09015,
doi:10.1029/2003JC002153, 2005.Tseng, Y.-H., Chien, S.-H., Jin, J., and Miller, N. L.: Modeling air-land-sea interactions using the integrated regional model system in
Monterey Bay, California, Mon. Weather Rev., 140, 1285–1306,
doi:10.1175/MWR-D-10-05071.1, 2012.Waxler, R. and Gilbert, K. E.: The radiation of atmospheric microbaroms by
ocean waves, J. Acoust. Soc. Am., 119,
2651–2664,
doi:10.1121/1.2191607, 2006.
Webb, S. C. and Cox, C. S.: Observations and modeling of seafloor microseisms,
J. Geophys. Res.-Sol. Ea., 91, 7343–7358,
doi:10.1029/JB091iB07p07343, 1986.
Wilson, B. W., Hendrickson, J. A., and Kilmer, R. E.: Feasibility study for a
surge-action model of Monterey Harbor, California, Tech. Rep. 2–136,
United States Army Corps of Engineers, Waterways Experiment Station,
Vicksburg, MS, 1965.