Current profiles are examined for evidence of nonlinear energy transfers from the M2 internal tide to diurnal waves. The 6 month records, unlike shorter records, produce well-resolved velocity and shear spectra that consistently exhibit maxima at the diurnal tides O1 and K1, with a minimum at the intermediate M2 subharmonic, M2/2. The ratio of velocity spectral energy at M2/2 and M2 is quantified, providing a needed modeling benchmark. Bispectra and bicoherences imply a negligible [−M2/2, −M2/2, −M2] triad interaction, but possibly a significant interaction for the [−O1, −K1, −M2] triad. Numerical simulations, however, indicate that O1 and K1 signals are from internal tides. Tests with synthetic data, linear tides plus random noise, reveal that bispectrum and bicoherence estimators can yield significant values, thus misleading results. Therefore, resolving the diurnal tides from M2/2 is essential to meaningfully assess nonlinear transfer of energy from M2 to diurnal waves.
The flow of energy through the internal wave (IW) field has taken on renewed importance due to the evidence suggesting that IWs drive significant diapycnal mixing in the deep ocean, helping to maintain the abyssal density stratification against the influx of deep and bottom waters formed at high latitudes. The internal tides are considered to be among the most important IWs in this role. Tide models that assimilate satellite altimeter data indicate that up to 1 TW of barotropic tide power is dissipated in the deep ocean over rough topography [e.g., Egbert and Ray, 2000], possibly accounting for 50–100% of the power needed to maintain the abyssal stratification [e.g., Munk and Wunsch, 1998; Decloedt and Luther, 2010]. Some of the barotropic tide dissipation directly results in localized mixing, but most of it drives internal waves (internal tides) that propagate away. The principal locations and pathways (in physical and frequency-wave number spaces) of the subsequent internal tide dissipation are still unknown [e.g., Garrett and Kunze, 2007; MacKinnon et al., 2013].
The internal tide energy cascade in wave number space can potentially span several orders of magnitude as energy is transferred from large-scale waves directly forced by the barotropic tides to smaller scale internal waves which are more easily dissipated [e.g., Polzin, 2004]. Nonlinear processes, of which the Parametric Subharmonic Instability (PSI) resonant triad wave-wave interaction is an example, have been conjectured to be the main mechanisms facilitating this energy cascade [e.g., McComas and Bretherton, 1977]. Numerical and observational studies have suggested that direct and rapid energy transfer from the large-scale M2 tide to small-scale oscillations at the M2 subharmonic (M2/2, or M1) frequency can occur for a range of latitudes equatorward of ~29˚ [e.g., Hibiya and Nagasawa, 2004; Gerkema et al., 2006]. Specifically, studies around the Kauai Channel [e.g., Carter and Gregg, 2006; Rainville and Pinkel, 2006; Sun and Pinkel, 2013] and the Luzon Strait [e.g., Xie et al., 2011; Liao et al., 2012] have inferred from observations that PSI is an identifiable mechanism of nonlinear energy transfer, even at tropical latitudes (~21˚N). The geographical extent of PSI acting on energetic internal tides is an important aspect of its relevance to diapycnal mixing in the global ocean.
Although there is substantial literature on the subject of PSI, numerical experiments have been highly idealized, and observational efforts usually have not explicitly distinguished nonlinearly produced M1 energy from diurnal tides, due to insufficient data lengths. The principal purpose of the present study is to quantify the energy at M1 relative to M2 internal tide energy, with a data set from the Hawaii Ocean Mixing Experiment (HOME) that is long enough to clearly resolve in frequency the energy in M1 horizontal currents as distinct from the principal diurnal O1 and K1 tidal currents.
A secondary purpose of this work is to provide a direct assessment of the occurrence of nonlinear interactions between M2 and diurnal waves in this unique HOME data set, using bispectral analysis. The HOME data have a vertical extent (~1000 m) that encompasses distinct, depth-limited beams of semidiurnal tide energy. Whether such vertically confined beams, with their higher vertical wave number content, result in more rapid transfers of energy from the principal tidal constituents to nonlinear harmonics than just low-vertical-mode constructs has not yet been determined observationally; in idealized numerical models [e.g., Gerkema et al., 2006; Simmons, 2008], both yield resonant nonlinear interactions.
2 Observations and Methods
The data set used in this study comes from two RDI Long Ranger acoustic Doppler current profilers (ADCPs) deployed from 16 November 2002 to 11 June 2003 at 746 m and 1314 m on the HOME A2 mooring in the Kauai Channel (Figure 1), where the water depth was ~1333 m. The data have been cleaned following Boyd et al.  and Guiles . To facilitate examination of internal wave motions, barotropic tidal currents are subtracted from measurements at all depths using the most up to date Oregon State University tidal inversion software (OTIS) [Egbert and Erofeeva, 2002] Regional Tidal Solution (Hawaii, 2010) for the Kauai Channel.
To optimize the accuracy of the discrete Fourier transform (DFT) for the diurnal band, data lengths that result in the least misalignment of DFT harmonics with O1, M1, and Kl are calculated numerically [Chou, 2013], before estimating power spectral density (PSD). The chosen data length is 163.5 days (2 December 2002 to 15 May 2003), and a tapered (10% cosine) window is applied to the truncated time series prior to calculating DFTs for auto-spectra. In order to increase statistical reliability, the PSD estimates are averaged in frequency space (three points) as well as depth (80 m). Additional degrees of freedom (DOF) from depth averaging are calculated by estimating vertical coherence lengths for the horizontal velocity (~96 m) and vertical shear (~40 m) fields [Chou, 2013].
Many observational studies investigating nonlinear interactions between the M2 tide and M1 at tropical latitudes have utilized bispectral techniques in addition to auto-spectral analysis [e.g., Carter and Gregg, 2006; Liao et al., 2012]. Expectation values necessary for bispectral analysis [e.g., Elgar and Guza, 1988] are most often calculated through ensemble averaging in the time domain, and this has been done for short (~10 days) [Carter and Gregg, 2006] as well as long (> 8 months) [Liao et al., 2012] data sets, at the cost of reduced frequency resolution. In order to maintain sufficient resolution to distinguish between M1 and diurnal tidal constituents O1 and K1, expectation values for this study are calculated with only 1 time segment, using a two-dimensional (5 × 5) filter in frequency space and depth averaging over 80 m [Chou, 2013].
3.1 Spectral Energy of Velocity and Shear
The DFT of a 163.5 day time series yields transform harmonics separated by ~0.006 cycles/day (cpd), permitting clear discrimination of O1 (0.930 cpd), M1 (0.966 cpd), and K1 (1.003 cpd) wave motions in frequency space (e.g., Figure 2). The velocity auto-spectra show distinct, significant peaks for the four main semidiurnal and diurnal tidal constituents, and a noticeable valley at the M1 frequency. Near-inertial waves, probably directly wind generated, are appropriately prominent (the local inertial frequency is indicated as f, as usual). Such waves, being highly intermittent in time, tend to have a broad frequency bandwidth, which extends well above the local f as the waves propagate south from their origins. This might account for much of the apparent “background” level of energy in the diurnal band. At 660 m on the HOME A2 mooring, the broad near-inertial peak of the 8 m vertical shear field (Figure 2b) has increased in prominence compared to the velocity spectrum (Figure 2a) since the shear spectrum is weighted toward larger vertical wave numbers, which tend to dominate the near-inertial waves. Depending on depth, the spectral peaks near M2 − f are attributed to either vertical advection of near-inertial motions by the M2 internal tide or nonlinear wave-wave interactions between near-inertial waves and the M2 internal tide [Guiles, 2009].
Deeper in the water column, spectral energy is decreased overall (e.g., at 1124 m; Figures 2c and 2d), but peaks at the tidal frequencies are still clearly distinguishable in both velocity and shear spectra. Spectral levels at M1 remain lower than at O1 and K1. The shear spectrum is no longer dominated by inertial-frequency motions below 700 m (Figure 3), suggesting a lack of effective downward transport of small vertical scale near-inertial motions past the pycnocline. The velocity PSD profile in Figure 3 shows that M2 internal tide kinetic energy has a relative maximum near 700 m (due to an upward-propagating tide beam; e.g., Carter et al. ), while in both velocity and shear, spectral energy at M1 is always lower than the energies at O1 and K1. Within a 250 m depth range of strongest M2 tidal energy (550–800 m), the velocity PSD at M1 relative to M2 is approximately 0.013, with the 95% confidence interval (0.0063, 0.027) roughly a multiplicative factor of 2 around the estimate.
3.2 Bispectra and Bicoherence of Velocity
At 660 m, where the M2 internal tide is energetic and beam-like, elevated bispectral values (exceeding background levels by at least an order of magnitude) are found for potential resonant triads such as [−O1, −K1, −M2] and [−K1, −K1, −K2] (Figure 4a), where K2 is the lunisolar semidiurnal tidal constituent with twice the frequency of K1, and negative frequencies denote anticyclonic rotation. In general, bispectra and bicoherence need to be considered together, with neither one sufficient by itself for indicating nonlinear interactions [e.g., Yao et al., 1975; MacKinnon et al., 2013]. A potential resonant triad may yield suggestive bispectral peaks without significant bicoherence (such as the [−K1, −K1, −K2] triad), or significant bicoherence but negligible bispectral energy (such as the location of the maximum in Figure 4d, or the [+K1, +O1, +M2] triad from Figure 3.11-3.12 of Chou ). At this depth, only the frequency triplet [−O1, −K1, −M2] has both elevated bispectral energy and bicoherence above the 90% significance level (~0.76, calculated following Elgar and Guza  with DOF ~8).
Depth profiles of bispectral energy and bicoherence show that the frequency triplet [−O1, −K1, −M2] consistently has both high bispectral and bicoherence values from 660 to 884 m (Figure 5). In contrast, the [−M1, −M1, −M2] triad has bispectral values at least an order of magnitude less than those of [−O1, −K1, −M2] throughout most of the water column, and its bicoherence values are generally below the 80% significance level (~0.63).
4 Discussion and Conclusions
Equatorward of the “critical latitude” where the inertial frequency equals M1, there is no general consensus on the issue of which frequencies are preferred by resonant wave-wave triad interactions that transfer energy from a large-scale M2 internal tide (the primary wave) to lower frequency, small-scale secondary waves. In the case of PSI, analytical [e.g., Staquet and Sommeria, 2002] and numerical [e.g., Hazewinkel and Winters, 2011] studies suggest that the growth rate of secondary waves approaches a maximum as secondary wave frequencies approach equality. However, there are observational [e.g., Xie et al., 2011] and numerical [e.g., Nikurashin and Legg, 2011] studies that find M2-forced secondary waves at frequencies f and M2 − f, even at latitudes ~21˚ (where Δω, the frequency separation between these two waves, is M2 − 2f > 0.5 cpd). Given that diurnal band spectral energy appears to be concentrated at O1 and K1 (Δω ~ 0.073 cpd) for the A2 mooring data, we consider whether the bispectral and bicoherence evidence presented earlier supports a hypothesis of resonant nonlinear energy transfer from the M2 tide to internal wave motions at O1 and K1 (the M2 frequency equals the O1 plus K1 frequencies).
First, the magnitudes and vertical structures of observed diurnal currents are well predicted by an internal tide simulation model identical to that in Carter et al.  except forced with only the K1 or O1 constituent. The similarity between K1 (and O1, not shown) harmonic fits from the model and from A2 observations suggests that local generation of an internal tide is a sufficient explanation for the observed diurnal currents (Figure 6). Second, the maxima in the diurnal-band velocity PSD do not coincide with the semidiurnal maxima (Figure 7). Because both diurnal and semidiurnal internal tides emanate from the edges of the Kauai Channel Ridge crest, and the group velocity is more nearly horizontal at the lower frequency, the diurnal-band maxima tend to be deeper than the semidiurnal, though there are exceptions. As a whole, these observations suggest that sustained nonlinear interactions between the diurnal and semidiurnal bands at this location are unlikely.
The bispectral analyses presented earlier (Figures 4a and 4b) showed that the [−O1, –K1, –M2] triad has both elevated bispectral energy and significant bicoherence throughout a depth range > 200 m where the M2 tide is energetic and beam-like, evidence that has been interpreted as indicating nonlinear energy transfer in recent studies of PSI [e.g., Xie et al., 2011; Liao et al., 2012]. However, an experiment using the same bispectral techniques applied to a synthetic data set containing only deterministic tidal inputs and white Gaussian noise also resulted in elevated bispectral values and high bicoherence for the [−O1, −K1, −M2] triad (Figures 4c and 4d), showing that these results do not always carry physical meaning [Yao, 1974]. A basic assumption of bispectral analysis is that the processes under study are stochastic and stationary, and it is unclear how the results should be interpreted for nearly deterministic signals such as internal tides. Tests of the bispectral technique can be performed with synthetic data for a given location (e.g., Figures 4c and 4d), or by using actual observations (a composite data set from different locations, for example) that do not have any real nonlinear interaction between semidiurnal and diurnal waves.
Although some observational studies attribute energy in the entire diurnal band to nonlinear transfers of energy from the M2 tide to waves of approximately M1 frequency, the analysis presented here strongly suggests that resolution of the diurnal internal tides from M1 is essential before attempting to assess nonlinear transfer of energy from M2 to diurnal waves. Within a 250 m depth range of strongest M2 tidal energy, the velocity PSD at M1 is approximately 10−2 of the velocity PSD at M2. This quantification of spectral energy at M1 provides a benchmark for comparison with numerical simulations of nonlinear energy transfers from the M2 internal tide to subharmonic waves [e.g., Gerkema et al., 2006; Simmons, 2008]. If PSI or other nonlinear mechanisms are active in transferring energy at the A2 mooring from the semidiurnal internal tides to the diurnal band, they are so weak as to be of little consequence in either velocity or shear. The role of PSI and other nonlinear energy transfers in shaping the ocean's internal wave spectrum remains to be adequately quantified. Progress will require more realistic numerical simulations and comparable observations of adequate duration and spatial coverage.
We thank Captains Dan Arnsdorf and Tom Desjardins of R/V Wecoma and R/V Revelle, respectively, and their outstanding crews, as well as the many OSU and UH techs and students who assisted with the deployments and recoveries of the HOME moorings and with data processing. High-quality density profiles for Station Kaena were obtained from the Hawaii Ocean Time-series program, funded by the National Science Foundation (NSF). The HOME mooring program at UH was supported by NSF Grant OCE9819533, and the modeling work presented in this paper was supported by NSF Grant OCE0425347. We would also like to thank Mark Merrifield, Janet Becker, Eric Firing, and two anonymous reviewers for their helpful comments and suggestions. This work is dedicated to the memories of Murray Levine and Tim Boyd, who contributed greatly to the success of the HOME mooring deployments and interpretations, and with whom it was a distinct pleasure to work.
The Editor thanks two anonymous reviewers for assistance in evaluating this manuscript.