## Abstract

Near-diurnal internal waves were observed in velocity and shear measurements from a shipboard survey along a 35-km section of the Kaena Ridge, northwest of Oahu. Individual waves with upward phase propagation could be traced for almost 4 days even though the ship transited approximately 20 km. Depth–time maps of shear were dominated by near-diurnal waves, despite the fact that Kaena Ridge is a site of considerable *M*_{2} barotropic-to-baroclinic conversion. Guided by recent numerical and observational studies, it was found that a frequency of ½*M*_{2} (i.e., 24.84-h period) was consistent with these waves. Nonlinear processes are able to transfer energy within the internal wave spectrum. Bicoherence analysis, which can distinguish between nonlinearly coupled waves and waves that have been independently excited, suggested that the ½*M*_{2} waves were nonlinearly coupled with the dominant *M*_{2} internal tide only between 525- and 595-m depth. This narrow depth range corresponded to an observed *M*_{2} characteristic emanating from the northern edge of the ridge. The observations occurred in close proximity to the internal tide generation region, implying a rapid transfer of energy between frequencies. Strong nonlinear interactions seem a likely mechanism. Nonlinear transfers such as these could complicate attempts to close local single-constituent tidal energy budgets.

## 1. Introduction

Inverse calculations using satellite altimetry suggest that 25%–30% of *M*_{2} barotropic tidal energy dissipates in the open ocean (Egbert and Ray 2000, 2001). Primarily, it is believed that this energy is converted into internal (baroclinic) tides and cascades through the internal wave spectrum before being lost as heat. Simmons et al. (2004) found, using an isopycnal model, that 75% of the *M*_{2} global barotropic-to-baroclinic conversion occurred at 20 sites of abrupt topography, which together accounted for less than 10% of the ocean floor. The lowest-mode internal tides have been observed to propagate hundreds of kilometers from their generation regions (e.g., Dushaw et al. 1995; Ray and Mitchum 1996; Cummins et al. 2001), thereby transporting energy into the open ocean.

A potentially significant fraction of the energy removed from the barotropic tide is dissipated near the internal tide generation region. Estimates of the fraction dissipated locally vary widely, in part because of differing definitions of “local.” For example, Althaus et al. (2003) found that at Mendocino Escarpment 1% was dissipated over the 3.5-km-wide ridge, but 50% was dissipated within 100 km; St. Laurent and Garrett (2002) suggest that <30% of the generation at mid-ocean ridges is in higher modes, and this would be all the energy available for local dissipation; around Hawaii, Nash et al. (2006) suggest that 30%–50% was dissipated by the 3000-m isobath, and Klymak et al. (2006) estimate that ∼20% is dissipated within 60 km of the ridge crest. The fate of the high modes, in particular if they are dissipated locally or if their energy is transferred within the internal wave spectrum, is an important question remaining in our understanding of the distribution of energy in the ocean.

Nonlinear interactions are able to transfer energy within the internal wave spectrum. In the late 1970s a small number of weakly nonlinear resonant interactions were identified as being responsible for maintaining the Garrett–Munk open-ocean internal wave spectrum (e.g., McComas and Bretherton 1977). Resonant triad interactions occur between three waves, which each satisfy the appropriate dispersion relation and, in addition, have their frequency and wavenumbers governed by

Parametric subharmonic instability (PSI), which transfers energy from frequency *ω* to ½*ω* and to higher wavenumbers, was one of the interactions identified. PSI lost favor after researchers calculated that the associated transfer rates were in the order of 100 days for mode 1 (e.g., McComas and Müller 1981a, b; Olbers and Pomphrey 1981). However, recent numerical modeling renewed interest in the role of PSI by suggesting that the transfer rates were in the order of 2 days for mode 1 (Hibiya et al. 2002; MacKinnon and Winters 2003, 2005, unpublished manuscript, hereinafter MKW). MKW conclude that the transfer time scales in the models were consistent with coherent wave–wave interactions rather than the random phase limit used in the earlier calculations. The reason they give is that weak nonlinear interactions of coherent waves are described by the first-order term of a perturbation expansion. However, this first-order term is eliminated by the ensemble averaging required in the random phase limit, resulting in the next-order term governing the interactions.

The Hawaiian Ridge is an important site for *M*_{2} barotropic-to-baroclinic conversion. The 2000-km-long ridge lies almost perpendicular to the dominant *M*_{2} barotropic tidal currents. The inverse calculations of Egbert and Ray (2000, 2001) indicate that 18 ± 6 GW are lost from the *M*_{2} barotropic tide along the Hawaiian Ridge, and an unconstrained baroclinic model (Simmons et al. 2004) gave 30 GW. Local flat-bottom baroclinic modes calculated by Nash et al. (2006) found that by the 3000-m isobath the semidiurnal energy flux was dominated by a mode-1 structure. Closer to the ridge crest significant energy flux was in higher modes.

The Hawaiian Ridge lies equatorward of 28.9°; hence, nonlinear interactions, such as PSI, can potentially transfer energy from *M*_{2} to ½*M*_{2} without violating the internal wave requirement that *f* ≤ *ω* ≤ *N*. Numerical simulations have showed that the time evolution of *M*_{2} energy injected into a background wave field differs significantly depending on whether or not they are conducted at a latitude where transfer to ½*M*_{2} is possible (Hibiya et al. 2002; MacKinnon and Winters 2003; MKW).

Considering velocity and displacement data from two 1-month-long R/P *Floating Instrument Platform* (*FLIP*) deployments, Rainville and Pinkel (2006) found significant energy fluxes in the diurnal frequency band both over the Hawaiian Ridge and 430 km south of the ridge. Although these time series were not long enough to distinguish between *K*_{1} and ½*M*_{2} spectrally, they found evidence that significant diurnal energy was at ½*M*_{2}. The fortnightly beating caused by the interaction of *M*_{2} and *S*_{2} was out of phase with the *K*_{1}−*O*_{1} fortnightly beating. The diurnal energy fluxes were strongest when the semidiurnal forcing was largest and the diurnal forcing was smallest. The diurnal energy fluxes observed when the diurnal forcing was largest were also along ridge rather than across ridge, as observed when the semidiurnal forcing was largest.

In this paper, we present observations from a deep-profiling shipboard Doppler sonar system. These data showed that velocity and shear over an energetic section of the Hawaiian Ridge (Kaena Ridge) were dominated by internal waves of near-diurnal frequency. Given that the majority of barotropic-to-baroclinic conversion in this region was semidiurnal (*M*_{2}), this suggests that energy was locally transferred to lower frequencies. In section 2 we present the observations. Phase and plane-wave solutions suggest the existence of ½*M*_{2} waves. Bicoherence, a measure of nonlinear coupling between waves, is used in section 3 to show energy transfer from *M*_{2} to ½*M*_{2} in a narrow depth range (525–595 m). That this depth range corresponded to an *M*_{2} characteristic and the implications of subharmonic transfer are discussed in section 4. The results are summarized in section 5.

## 2. Observations

Topographic variations along the Hawaiian Ridge result in the off-ridge internal tidal energy flux ranging from 1 to possibly more than 40 kW m^{−1} (Rudnick et al. 2003; Lee et al. 2006). Numerical models (Merrifield et al. 2001) and observations (e.g., Rudnick et al. 2003; Lee et al. 2006; Martin et al. 2006) found that the Kaena Ridge, extending northwest of Oahu, is one of the strongest regions of *M*_{2} internal tide generation along the Hawaiian Ridge. The barotropic tide over the Kaena Ridge is dominated by *M*_{2} (Table 1). Kinetic energy observations (Martin et al. 2006) and energy-flux calculations (*F* = 〈*p*′ *u*′〉; Nash et al. 2006) suggest that a significant fraction of the internal tide energy observed off the ridge came from generation at the opposite flank of the ridge crest.

Between 2 and 24 September 2002 we measured velocity over the Kaena Ridge crest, using shipboard Doppler sonar, as part of the Nearfield phase of the Hawaii Ocean Mixing Experiment (HOME; Pinkel et al. 2000). Twelve survey lines were occupied over a 35-km-long section of the ridge (Fig. 1a). The nature of the survey, along with ridge topography, resulted in a large variation in the depth of water being surveyed (from ∼500 to over 1100 m). In general, each line was occupied for two tidal cycles (Fig. 1b). For 3 days in the middle of the experiment (11–14 September) we sampled 30 km south of the ridge.

Velocity measurements were made using the hydrographic Doppler sonar system mounted on the R/V *Revelle*. This system consists of two four-transducer Doppler sonars: a 50-kHz deep-penetrating sonar and a 140-kHz high-resolution sonar. In this paper we only examine data from the 50-kHz sonar, which has an operating range of ≈80–800 m. Velocity measurements are spaced every 8.6 m in the vertical, based on overlapping 25.8-m trapezoidal averaging windows. The transducer beams are aligned 30° from the vertical, resulting in sidelobe interference in the lower 15% of the water column. The configuration used during this experiment resulted in acoustic interference between the two sonars and unusable data between 620- and 680-m depth.

### a. Depth–time maps

A depth–time map of the northward velocity component between 0600 UTC 18 September and 0000 UTC 22 September (Fig. 2a) clearly shows internal waves with upward phase (downward energy) propagation. Although displayed as a time series, we occupied five stations during this time (SM, SW, EW1, CR9, and EW2; see Fig. 1). Sherman and Pinkel (1991) caution that vertical self-advection of the internal wave field can alter the frequency content observed in an Eulerian reference frame. To ensure that these waves were not an artifact of our vertically Eulerian sampling, the northward velocity was replotted in a semi-Lagrangian frame (not shown). Waves of upward phase propagation were also observed in the semi-Lagrangian frame, indicating that they are not resultant from self-advection of the wave field passed an Eulerian sensor. The density profiles, and hence semi-Lagrangian data, however, have a much coarser and variable time resolution (30–130 min) than the Eulerian sonar observations. Therefore, we use the Eulerian reference frame data throughout the paper.

Examination of the phase, Φ = tan^{−1}*υ*/*u* (black dots in Fig. 2b), showed a strong clockwise rotation with a nearly diurnal frequency. Velocities were low passed with an 11-h boxcar filter prior to calculating the phase, in order to minimize the influence of the *M*_{2} barotropic currents. The near-diurnal waves had a period over 7 h shorter than the local inertial period (32.5 h), which effectively eliminates wind forcing as their generation mechanism. Guided by the findings of Rainville and Pinkel (2006), we found that the frequency of these waves compares favorably to ½*M*_{2} (i.e., a period of 24.84 h). The thick gray line in Fig. 2b is the phase of a ½*M*_{2} wave given by Φ = 2*πω*(*t* − *t*_{0}), where *ω* = 24/24.84 cycle per day (cpd) and *t*_{0} = 2230 UTC 18 September.

The persistence of these waves is worth comment, with some of the waves being visible for almost 4 days. The persistence is more surprising, since over this time the ship traveled more than 20 km along the topographically diverse ridge. Over this distance the topography included the slope down from the island of Oahu, a saddle, and a 400-m-high seamount. As noted above, this section of the Hawaiian Ridge is also a site of strong internal tide generation. Numerical models (Merrifield et al. 2001; Merrifield and Holloway 2002) show variation in internal tide fluxes over distances of several kilometers.

It is worth examining shear, as first-differencing removes the barotropic tide. Also in a variably stratified ocean, internal waves follow curved ray paths and their amplitudes vary with depth. To minimize these complications, the Wentzel–Kramers–Brillouin (WKB) approximation is commonly used to plot oceanic data with an equivalent constant stratification. The WKB-stretched depth coordinate is given by

where we have taken *N*_{0} to be the cruise-average stratification (7.0 × 10^{−3} s^{−1}), *N*(*z*) is the isopycnally averaged stratification profile, and *z*′ is a dummy variable of integration. The horizontal velocities are scaled by

This implies that the ratio of nonlinear terms to linear terms [*u*(∂*u*/∂*x*)/(∂*u*/∂*t*)] scales as *N*(*z*), and, therefore, the nonlinear interactions could be expected to be larger in regions of strong stratification. The WKB shears were calculated by differencing the scaled velocities relative to the stretched coordinate [*Ŝ _{x}* = Δ

*û*/Δ

*ẑ*∝

*N*(

*z*)

^{−3/2}].

Figure 3 shows depth–time maps of 3-hour low-passed WKB-stretched and -scaled shear. The picture here is more complicated than is observed in velocity (e.g., Fig. 2a), with coherent upward- and downward-propagating internal waves observed throughout the water column. The general impression is that upward phase propagation dominates shear between 15 and 21 September. However, there appears to be a checkerboard pattern throughout much of the record, indicating concurrent upward- and downward-propagating waves.

In WKB-stretched coordinates, an internal wave can be adequately described by a “plane wave,”

where Ψ̂(*z*) is the depth-varying amplitude, **k** = (*k _{x}*,

*k*,

_{y}*k*) are the wavenumbers, and

_{z}*ϕ*is the phase. Zero velocity contours from eyeballed ½

*M*

_{2}plane-wave solutions are overplotted on sections of Fig. 3b. Prior to sampling south of the ridge, the wave field appears to be well described by a combination of upward- and downward-propagating waves with vertical wavenumber

*k*= ±2

_{z}*π*/200 m. Between 18 and 22 September, it was more difficult to find plane-wave solutions that work well, although the velocity contained obvious near-diurnal waves (Fig. 2a). There appears to be variation in the vertical wavenumber over our observations. The plane-wave solutions shown between 18 and 21 September (Fig. 3b) have a vertical wavenumber of

*k*= +2

_{z}*π*/166 m. Unfortunately, the complicated nature of the wave field combined with the short period of the experiment (i.e., not meaningful to filter for a single wave component) means it is not practical to calculate fits to plane waves as done by Alford and Gregg (2001).

An estimate of the horizontal scale of these waves follows from the vertical wavenumber. The characteristic slope of an internal wave is given by

where *f* (=5.4 × 10^{−5} s^{−1}) is the Coriolis frequency, *N* is taken to be the cruise average *N*_{0}, *ω* is ½*M*_{2}, and *k _{z}* = 2

*π*/200 m is the vertical wavenumber from the plane waves. This gives a horizontal length scale for the observed waves of ∼31 km, or about the total along-ridge distance sampled. This helps to explain the persistence of the waves, as it seems possible that our sampling occurred within one wave.

### b. Frequency spectra

Frequency spectra were calculated using the periodogram method. This was used even though it is susceptible to leakage, as it has the narrowest central lobe (distance between two frequencies that can be distinguished; Percival and Walden 1993). Our interest was constrained to high-energy low-frequency signals (inertial to semidiurnal); therefore, leakage did not play a significant role. The longest continuous time series we had covering a good depth range (>400 m) was only 4.58 days (0419 UTC 6 September–1814 UTC 10 September). This was due to our sampling off the ridge and along the 500-m isobath, a sampling scheme that, although unfortunate for this analysis, was governed by our primary focus—the collection of microstructure data. Theoretically, 4.58 days is just sufficient to distinguish between inertial ( *f* ) and ½*M*_{2},^{1} but not between any components within the diurnal frequency band.

Figure 4a shows shear spectra averaged over 100–500-m depth range for this 4.58-day period. The strongest signal occurred at near-diurnal (½*M*_{2}) frequencies. As the local *M*_{2} barotropic-to-baroclinic conversion dominates diurnal conversion, it is somewhat surprising that there is no signal at *M*_{2}. The individual spectra in each depth bin show the same pattern (Fig. 4b). The apparent decrease in spectral amplitude with depth is due to the spectra being calculated from the measured velocity rather than the WKB-scaled velocities. The largest amplitude in the velocity spectra (Fig. 4c), however, was from the barotropic *M*_{2} tide. A diurnal signal in velocity was also obvious throughout most of the water column.

A strong *M*_{2} signal in velocity but not in shear suggests that the semidiurnal energy is either barotropic or low mode. We know from numerical models (Merrifield et al. 2001; Merrifield and Holloway 2002) and in situ observations (e.g., Rudnick et al. 2003; Lee et al. 2006; Martin et al. 2006; Nash et al. 2006; Rainville and Pinkel 2006) that an *M*_{2} internal tide is generated locally. The topographic interactions that generate internal tides are known to produce energy at a range of scales, not just at the lowest mode. Therefore, we conclude that significant energy has been removed from the high-mode semidiurnal baroclinic tide, over the ridge crest. Indeed, energy-flux observations of Nash et al. (2006) showed more high-mode signal over the ridge flanks than the crest.

## 3. Evidence for nonlinear interactions

In the previous section we demonstrate that ½*M*_{2} waves existed in the velocity and shear fields above the Kaena Ridge. In this section we consider a potential origin for these near-diurnal waves.

### a. Bicoherence

Many waves making up the oceanic internal wave field are not independent, as they interact with each other. Linear spectral analysis assumes that the energy at each frequency or wavenumber is independent and hence is of limited use for determining interactions. Higher-order spectral analysis such as bicoherence (normalized bispectrum) can be used to distinguish between nonlinearly coupled waves and waves that have been independently excited (Kim and Powers 1979; McComas and Briscoe 1980). This method, which looks for the phase coherence resulting from resonant nonlinear interactions, may not identify waves that have evolved separately but were originally coupled and conversely may erroneously identify oscillations with phase-locked [but not in phase; e.g., see (9)] generation.

Here we will very briefly outline the bicoherence method; we refer the reader to Kim and Powers (1979) and McComas and Briscoe (1980) for further details. Let *X _{k}* be the complex Fourier transform of a zero-mean time series

*x*(

*t*), such that

where *T* is the record length, and *ω _{k}* = 2

*πk*/

*T*. The bispectrum is then defined as

where *X**_{k+1} is the complex conjugate [*X*_{−(}_{k}_{+1)}], and *E*[ ] is the expected value. If a fluctuating quantity is the sum of statistically independent random oscillators then the bispectra will be zero. If, however, some of the oscillations were caused by nonlinear coupling between other oscillations, a nonzero value will result. The bispectrum, therefore, is a measure of the dependence within a wave triad (*ω _{k}*,

*ω*,

_{l}*ω*

_{k}_{+}

*).*

_{l}The bispectrum can be normalized to eliminate the influence of wave amplitude, resulting in the bicoherence, *b*(*k*, *l*), defined as

It can be shown that 0 ≤ *b*(*k*, *l*) ≤ 1. Strictly speaking, a nonzero bicoherence identifies two components of a nonlinear triad interaction, but in practice a value nearer to one is required. Elgar and Guza (1988) determined significance levels for the bicoherence method through Monte Carlo simulations [80% ≡ 3.2/*n*_{dof}; 90% ≡ 4.6/*n*_{dof}; 95% ≡ 6/*n*_{dof}, where the number of degrees of freedom (*n*_{dof}) is 2 times the independent realizations]. The third frequency in the resonant triad is the sum of the two components identified by the bicoherence. As mentioned above, a significant bicoherence value is possible if the three components of a triad are forced independently such that a linear combination of their phases is zero.

Bicoherence has not often been applied to oceanic internal waves. Neshyba and Sobey (1975) considered cross-bispectra of 17 30-h temperature records from the Arctic Ocean. These records were separated by 3 m in the vertical and cover the depth range 200–250 m. They report that “cross-bispectra clearly show the existence of nonlinear coupling among the internal wave traces” but then go on to note that about one-half of the identified interactions were harmonics and were more likely to represent “shared mutual power” than nonlinear interaction.

McComas and Briscoe (1980) calculate bispectra for the Garrett–Munk internal wave model. They concluded that it was futile to apply bispectral analysis to the oceanic internal wave field and, consequently, that the results observed by Neshyba and Sobey (1975) cannot be interpreted as evidence of nonlinear interaction. Two features of the oceanic internal wave field, as described by Garrett–Munk, are responsible for this futility: 1) the magnitude of the interactions required to maintain the Garrett–Munk spectra are not very strong, and 2) many combinations of wave triad interactions sum to form a given wavenumber or frequency, whereas bicoherence is looking for a single triad interaction. Importantly, they explicitly do not denigrate the use of bispectral analysis for situations where strong interactions or nonlinearity between three strong components are suspected. Internal tides provide a narrow band of strong energy not described by the Garrett–Munk model and, therefore, could provide distinct triad interactions that dominate those considered by McComas and Briscoe (1980). The ∼2 day time scale for subharmonic energy transfer (MKW) clearly indicates a much stronger interaction than the 100-day PSI time scales used in McComas and Briscoe (1980). Therefore, we conclude that *M*_{2} to ½*M*_{2} subharmonic transfer is likely amenable to bicoherence analysis.

Contamination caused by isopycnals being vertically advected past the Eulerian temperature sensors was suggested by McComas and Briscoe (1980) as the reason for the significant bispectral values of Neshyba and Sobey (1975). We suggest that the dataset of Neshyba and Sobey (1975) is particularly sensitive to this form of contamination. First, it is from the Arctic Ocean (85°15′N, 97°47′W), where the internal wave field is weak (e.g., Levine et al. 1985; D’Asaro and Moorhead 1991) and cannot support *M*_{2} internal tides. Second, when describing the data, they note that it is from a strongly double diffusive region and has a multilayer structure. The layers are separated by thin sheets of fluid with large temperature and salinity gradients. The spacing of the layers is similar to the spacing of the sensors (∼3 m).

### b. Application to Kaena Ridge

Excluding the data taken south of the ridge and along the 500-m isobath gives 10.95 days of sonar data divided into three time series. In total, the data can be divided into seven 50%-overlapping windows of 2.155 days for the bicoherence calculation (Table 2). Using the 2-day time scale estimated by MKW for PSI suggests that five of the seven windows are independent, and consequently *n*_{dof} = 10. Conservatively, we assume that all the depth bins are within the same wave, and hence do not increase the degrees of freedom.

Figure 5a gives the average of the bicoherence of the *u* and *υ* velocities from the seven time intervals and all depth bins between 100 and 600 m. As we do not expect the two velocity components to be independent, this does not increase the degrees of freedom. The largest bicoherence values occur near [2, 2] cpd, which would indicate an *M*_{2} + *M*_{2} = *M*_{4} superharmonic interaction, if they had been significant. Bicoherence analysis of all depths simultaneously (Fig. 5a) showed that no resonant nonlinear interaction occurred. However, significant bicoherence values (i.e., potential nonlinear interactions) are possible, and in fact exist, in limited depth ranges.

Depth–frequency maps of phase (Fig. 5b) show significant phase variation with depth in the ½*M*_{2} band but nearly constant phase in the *M*_{2} band. Given that we had assumed no increase in *n*_{dof} from including more than one depth bin, the analysis can be repeated on individual depth bins. Figure 6a gives the bicoherence values with depth for the frequency bin [0.98, 0.98 cpd] or *M*_{2} = ½*M*_{2}+½*M*_{2}. This profile is the average of the bicoherence profiles calculated from the *u* and *υ* velocities. The bicoherence is significant at the 90% level between 525- and 595-m depth and at the 95% level for depths between 553 and 575 m. This is consistent with a subharmonic transfer of *M*_{2} energy occurring in this limited depth range.

In terms of spectral phase, this region of significant bicoherence corresponds to

For example, in Fig. 5b, *ϕ*_{1/2M2} ≈ 8/9*π* in the depth range 520–580 m, and *ϕ*_{M2} ≈ −2/9*π* over the same range. This additive phase relation is in agreement with Kim and Powers (1979).

Figure 6b shows the bicoherence values versus frequency for the 560-m depth bin. The largest bicoherence value occurred at (0.98, 0.98 cpd). The high bicoherence, however, was not confined to a single frequency bin, although all the significant bins were clustered near (0.98, 0.98 cpd). This is primarily due to zero-padding prior to calculating the spectra; this increases the number of bins available for plotting but not the frequency resolution (e.g., Emery and Thomson 2001).

## 4. Discussion

### a. Nonlinear interaction occurs on tidal beam

In the previous sections we showed that persistent internal waves of near-diurnal frequency were observed over the Kaena Ridge crest. Bicoherence analysis found that the phase of the ½*M*_{2} waves was consistent with nonlinear coupling to the dominant *M*_{2} internal tide, within the depth range 525–595 m.

The *M*_{2} internal tide was generated on both flanks of the ∼20-km-wide Kaena Ridge (e.g., Merrifield and Holloway 2002; Nash et al. 2006; Rainville and Pinkel 2006). Approximately one-third of the energy observed off ridge originated on the opposite flank, that is, had propagated across the ridge (Nash et al. 2006). Average kinetic energy density from repeated cross-ridge sections (Martin et al. 2006) and semidiurnal energy fluxes (Rainville and Pinkel 2006) showed a beamlike intensification that was in remarkable agreement with an *M*_{2} tidal beam generated on the northern edge of the ridge crest. The depth of this tidal beam was between 500 and 600 m as it crossed our sampling region (near the center of the ridge). Along-ridge Doppler sonar sections averaged over a tidal cycle showed elevated average kinetic energy in the 500–600-m depth range (Fig. 7, highlighted by the cross-hatching). This indicates that the *M*_{2} tidal beam was fairly coherent over the saddle region, where most of our observations were concentrated.

The good agreement between the tidal characteristic and the location of the 90% significant bicoherence strongly suggests that nonlinear interactions occurred within the *M*_{2} internal tidal beam to produce ½*M*_{2} waves. Internal wave beams are regions of intensified displacement and energy; consequently, it seems sensible that nonlinear interactions are more likely to occur along the beam than throughout the remaining water column.

### b. Other indications of subharmonic transfer

In this analysis we have concentrated solely on the frequency signal in implying that a subharmonic energy transfer occurred. A triad interaction, however, also requires that a linear combination of the wavenumbers is zero (1) and that the recipient waves have higher wavenumber than the original wave. The observed ½*M*_{2} waves had a higher wavenumber [|*k _{z}*| = (2

*π*/200)−(2

*π*/166) m

^{−1}; section 2a] than the low-mode

*M*

_{2}waves [|

*k*| ∼ (2

_{z}*π*/1000) m

^{−1}; section 2b]. Depth–time maps of rms displacement from a nearby 30-day R/P

*FLIP*deployment also showed a high-mode structure in the diurnal band and a low-mode structure in the semidiurnal band (Rainville and Pinkel 2006). The largest rms displacements they observed in the diurnal band over the ridge were ∼80 m thick and between 500- and 600-m depth.

Using the characteristic slope of an internal wave (5) and our estimates of vertical wavenumber it is possible to determine the propagation direction for the waves involved in the subharmonic transfer of energy from *M*_{2} to ½*M*_{2}. Assuming the *M*_{2} wavelength is the water depth (1000 m) gives *k _{z}*

_{1}= 6.3 × 10

^{−3}m

^{−1}and

*k*

_{h}_{1}= 1.2 × 10

^{−4}m

^{−1}(

*λ*

_{h}_{1}∼37 km). If we take the first ½

*M*

_{2}wave in the triad to have the + 200 m vertical wavelength (

*k*

_{z}_{2}= 3.1 × 10

^{−2}m

^{−1},

*k*

_{h}_{2}= 2.0 × 10

^{−4}m

^{−1}) we observed in shear between 6 and 11 September (section 2a), then vector summation gives the third wave in the triad. The second ½

*M*

_{2}wave, therefore, has

*k*

_{z}_{3}= −3.7 × 10

^{−2}m

^{−1}and

*k*

_{h}_{3}= −3.2 × 10

^{−4}m

^{−1}. The calculated wavelength of this wave (|

*λ*

_{z}_{3}| = 170 m) is in remarkable agreement with the 166-m wavelengths that were estimated from the plane-wave analysis for data between 18 and 21 September. Not only does this give confidence in our subharmonic energy transfer interpretation, it means that the variation in observed vertical wavelength (Fig. 3) corresponds to the same triad interaction. The reason the most easily observed ½

*M*

_{2}wave varied with time is unknown. Filtering and, hence, a longer time series would be required to identify both these ½

*M*

_{2}wavelengths simultaneously. Although there are significant differences in vertical wavenumber within the triad, all three waves propagate nearly horizontally (Fig. 8). Accurate vertical velocities, not measured here, would, therefore, be required to required to prove subharmonic energy transfer with propagation direction.

### c. Implications

The parametric subharmonic instability mechanism is a candidate for explaining our observations, as energy was transferred from *M*_{2} to ½*M*_{2}. However, it seems unlikely that the subharmonic transfer we observed was governed by the classical PSI mechanism. That mechanism was based on weak nonlinear interactions between freely propagating waves with random phase. As the subharmonic transfers we observed were confined to tidal beams, they probably cannot be classed as weakly nonlinear. Another difficulty is that our observations are so close to the generation site that even the 2-day time scale found by MKW may be too long. A mode-1 *M*_{2} internal tide would travel ∼400 km in 2 days; the higher modes travel slower (e.g., a mode-10 wave would travel in the order of 40 km over 2 days).

McComas and Bretherton (1977) noted that “if the wave amplitudes are too large, the nonlinear interactions proceed so fast that the selection of narrow bands of wavenumbers surrounding resonant triads does not occur, and the entire treatment is inadequate.” This suggests an alternative explanation where strong (i.e., not weak) nonlinear interactions are responsible for transferring energy from *M*_{2} to ½*M*_{2} so quickly. Recent laboratory (Teoh et al. 1997) and numerical (Lamb 2004) studies have found that nonresonant nonlinear interactions can generate superharmonics in small regions of time and space, such as where two internal tidal beams cross. The simulations of Lamb (2004) found beams of subtidal frequency emanating from above the ridge. The frequency of the subtidal beams summed to the tidal frequency, suggesting a subharmonic transfer. Interestingly, the subtidal beams were only apparent close to the ridge in the simulations, whereas the superharmonics were observed throughout the domain. Another potential explanation could also be that with the strong and regular forcing provided by the *M*_{2} tide, sufficient “information” remains over the ridge to effectively precondition the wave field for rapid subharmonic transfer.

An implication of this rapid transfer of energy from *M*_{2} to ½*M*_{2} is the potential complication of local *M*_{2} energy budgets. Often energy budgets are considered to be a balance between energy lost from a single barotropic tidal constituent (e.g., *M*_{2}), internal tide energy radiating away at that frequency, and local dissipation. Hence, energy transferred from *M*_{2} to ½*M*_{2} by nonlinear interactions close to the generation region would not be accounted for if it radiated out of the area of the budget calculation. As subharmonic transfer results in higher wavenumbers and slower group velocities, much of the energy is presumably dissipated “locally”; whether or not it is accounted for in the budget, therefore, depends of the size of the domain considered. Recently, a reasonably well closed local *M*_{2} energy budget for the Hawaiian Ridge was proposed (e.g., Rudnick et al. 2003; Klymak et al. 2006). As mentioned earlier, satellite altimetry data suggest that 18 ± 6 GW of energy is lost from the *M*_{2} barotropic tide along the Hawaiian Ridge (Egbert and Ray 2000, 2001). Numerical simulations (Merrifield and Holloway 2002; Lee et al. 2006) found that 9.7 GW was radiated away from the ridge as an *M*_{2} internal tide. Klymak et al. (2006) proposed a simple model for integrating the limited turbulent dissipation measurements over the entire ridge and got a rough estimate of 3 ± 1.5 GW dissipated. Although these numbers are in good agreement, the error estimates still allow for ≳10 GW of energy to be radiated out of the budget domain by other frequencies via nonlinear interactions.

## 5. Summary

Kaena Ridge, the northwest of extension of Oahu, is one of the most energetic regions of

*M*_{2}barotropic-to-baroclinic conversion along the Hawaiian Ridge. Velocity measurements were taken over the ridge crest using a deep-profiling Doppler sonar system mounted on the R/V*Revelle*. Twelve survey lines were occupied over a 35-km-long section of ridge between 2 and 24 September 2002 as part of the Hawaii Ocean Mixing Experiment. The data, which had a depth range of 80–800 m, were divided into three relatively short (2–4.5 day) sections by sampling south of the ridge and in shallow water.Depth–time maps of velocity showed waves with upward phase propagation. Individual waves were observed to persist for almost 4 days. Clockwise rotation with a nearly diurnal period was obvious when 11-h low-passed phase (Φ = tan

^{−1}*υ*/*u*) was examined. This period was over 7 h shorter than the local inertial period, eliminating a wind-driven generation mechanism.Depth–time maps of first-difference shears were dominated by near-diurnal internal waves. Both upward- and downward-propagating waves were visible throughout most of the water column. Plane-wave solutions with ½

*M*_{2}frequency and vertical wavelengths of 166–200 m were a good visual fit to these waves.Recent numerical modeling (Hibiya et al. 2002; MacKinnon and Winters 2003; MKW) as well as observations (Rainville and Pinkel 2006) have found that equatorward of 28.9° significant energy may be transferred to ½

*M*_{2}(i.e., a period of 24.84 h) from*M*_{2}by nonlinear interactions. Applying bicoherence analysis, which can distinguish between nonlinearly coupled waves and waves that have been independently excited, suggested that the ½*M*_{2}waves were consisted with being nonlinearly coupled to the dominant*M*_{2}internal tide between 525- and 595-m depth. This narrow depth range corresponded to where an*M*_{2}characteristic crossed the part of the ridge crest we sampled. Because internal tidal beams are regions of intensified displacement and energy density, nonlinear interactions are more likely to occur in them.Both the close proximity of our observations to the generation region and the confinement of the subharmonic transfer to the tidal beam suggest strong (i.e., not weak) nonlinear interactions were responsible for transferring energy from

*M*_{2}to ½*M*_{2}. This is in line with recent laboratory and numerical studies that have showed the generation of super- and subharmonics from local regions in time and space.Often local energy budgets are considered to be a balance between energy lost from a single barotropic tidal constituent (e.g.,

*M*_{2}), internal tide energy radiating away at that frequency, and local dissipation. Therefore,*M*_{2}energy transfer to ½*M*_{2}by nonlinear interactions close to the generation region would not be accounted for in an*M*_{2}budget if it radiated out of the domain. Considering the error bars on recent attempts to close the*M*_{2}budget around the Hawaiian Ridge would suggest that ≳10 GW could have been transferred to ½*M*_{2}.

## Acknowledgments

The authors are grateful to Matthew Alford, Paul Aguilar, Steve Bayer, Earl Krause, John Mickett, Jack Miller, Avery Synder, Maya Whitmont, Dave Winkel, and the crew of the R/V *Revelle* for their assistance in data collection. Rob Pinkel, Mike Goldin, Eric Slater, and Mai Bui designed and built the Doppler sonar. Helpful discussions were had with Matthew Alford, Eric Kunze, Eleanor Williams, and Dave Winkel. The authors also thank two anonymous reviewers. The TPXO.5 data were available from Gary Egbert’s Web page (www.coas.oregonstate.edu/research/po/research/tide/). Eleanor Williams provided the bicoherence code. This research was funded by National Science Foundation Grants OCE9818693 and OCE9819535.

## REFERENCES

**,**

**,**

**,**

**,**

**,**

**,**

_{2}tidal energy dissipation from TOPEX/Poseidon altimeter data.

**,**

**,**

**,**

**.**

**,**

**,**

**.**

**,**

**,**

**,**

**,**

**,**

**,**

**,**

**.**

**,**

**,**

**,**

**,**

**,**

**,**

**,**

**,**

**,**

**,**

**,**

**,**

## Footnotes

*Corresponding author address:* Glenn S. Carter, JIMAR, University of Hawaii, 1000 Pope Road, MSB 312, Honolulu, HI 96822. Email: gscarter@hawaii.edu

^{1}

A periodogram can distinguish between two frequencies ( *f*_{1}, *f*_{2}) if ( *f*_{1} − *f*_{2}) > *f _{s}*/

*l*, where

*l*is the record length and

*f*is the sampling frequency.

_{s}