## 1. Introduction

In the stratified ocean interior, mixing is caused primarily by the breaking of internal gravity waves. In the open ocean, away from sources and sinks, the properties of these waves are generally well described by the model spectrum constructed by Garrett and Munk [GM; the specific form presented in Munk (1981) is referred to as GM79]. Mixing rates increase rapidly as the level of internal-wave shear rises above the GM level (Gregg 1989), confirming the strong role of internal waves in controlling mixing.

On the continental shelves, several additional complications arise (MacKinnon and Gregg 2003). First, the water depth, which does not enter the GM model, becomes an important parameter. Levine (2002) introduced a modified GM spectrum with a vertical scale that depends on water depth in shallow water. In this formulation, the internal-wave energy density is not significantly different from that in the deep open ocean. However, because the vertical scale is smaller in shallow water, the shear increases and one might expect a higher mixing rate.

Second, flow past topographic features, including bumps on the shelf and the shelf break, can efficiently generate internal waves and localized mixing, as reported by Moum and Nash (2000). In particular, barotropic tidal currents acting on these features can generate internal tides, often with sufficiently large amplitude to also form high-frequency internal-wave packets, often with the misnomer “solitary waves” (Henyey and Hoering 1997). These can propagate long distances across the shelf and are often associated with enhanced mixing rates (MacKinnon and Gregg 2003; Stanton and Ostrovsky 1998; Moum et al. 2003).

Here, 241 days of data from six neutrally buoyant floats deployed on the Oregon continental shelf are used to investigate the summer climatology of internal waves in this region. These measurements are described in section 2. The floats are particularly effective at measuring vertical velocity, so the internal waves are described through this parameter. Section 3 examines the mean properties of the waves. The major difference between the GM model internal waves and those measured here is the presence of much more energetic waves near the buoyancy frequency. Section 4 shows that this peak is highly variable. This variability is mapped; more energy is found near rough topography, which points toward a topographic source for these waves. Section 5 investigates the role of high-frequency wave packets. Although the largest vertical velocities are associated with these packets, they play a small role in the total energy of the internal-wave field. Several wave packets are analyzed in detail; only the very largest have the right shape to be properly called solitary waves. These results are summarized in section 6.

## 2. Experiments and measurements

Lagrangian floats (Model “MLFII”) were deployed on the Oregon shelf during the upwelling seasons of 2000 and 2001. D’Asaro et al. (1996) describes the float design concept. D’Asaro (2003) describes the design, operation, and performance of the float model used here. D’Asaro (2004) describes the Oregon shelf deployments in detail and analyzes the data in the context of upwelling and circulation. An image of the float is inset in Fig. 1.

For the purposes of this study, the floats are considered to be isopycnal following. On long time scales, these floats are truly isopycnal. This is accomplished by first matching the float’s compressibility to that of seawater, both through the hull design and through active control of the float’s volume, so that its potential density is unchanged by pressure changes. Active control of the float’s volume is then used to remain close to the target isopycnal. A CTD is used to measure the temperature and salinity of the surrounding water. From this the water’s potential density is computed. If the float is at a shallower potential density than the target isopycnal, its density is increased by moving a piston into the float thus decreasing its volume and increasing its potential density. This causes the float to slowly fall toward the target. If the float is deeper than the target isopycnal, it increases its volume so as to move upward toward the target. On short time scales, these floats are Lagrangian (i.e., they follow the water surrounding them). As described in detail in D’Asaro (2003), the transition between Lagrangian behavior at high frequencies and isopycnal behavior at lower frequencies occurs at about *N*/30. This distinction is not important for the internal-wave issues discussed in this paper.

A significant factor limiting the water-following ability of these floats is their finite size, about 1 m. Floats cannot accurately follow water motions smaller than themselves. This is an important consideration for turbulent flows where the smallest scales of motion are much smaller than the floats (Lien et al. 1998). This effect is not important for the internal-wave issues discussed in this paper.

The floats used in this experiment operated in two distinct modes: Lagrangian drift and profiling. During each Lagrangian drift the floats followed the water, as described above. Every 12 h they profiled vertically to the surface. Upon reaching the surface, a GPS receiver obtained position fixes accurate to roughly 10 m. Backup positioning was provided by an Argos radio beacon. A subset of the collected data was telemetered to shore using the Orbcomm satellite system. Commands were relayed to the float using the same system. After roughly 20 min on the surface, the float profiled down to a specified depth 20–50 m below that of the reference isopycnal and then adjusted itself to begin another Lagrangian drift. Lagrangian drifts typically lasted 10.5 h with the remaining 1.5 h of each cycle spent profiling and communicating.

Three floats, numbers 2, 3, and 4, were deployed in the summer of 2000; three, numbers 8, 9, and 10, were deployed in the summer of 2001. Details of float instrumentation and calibration can be found in D’Asaro (2004). Variations in data quality limited the use of some float data. The first parts of the data from floats 2 and 4 are used little; the data from float 10 are used heavily.

Figure 1 shows an example of the float data. During the first half of the record, the float profiles upward to the surface from its target isopycnal (25.6 kg m^{−3} at the lower CTD), communicates with satellites, profiles to 40-m depth, and then returns to its target isopycnal. Data from CTDs on both the top and the bottom of the float are shown. Deviation of the density measured at the bottom of the float from its target isopycnal (dashed line) during the Lagrangian drift is seen to be less than the typical difference in density between the top and bottom CTDs. Accordingly, the bottom CTD deviates from the target isopycnal by less than the distance between the two CTDs, about 1.4 m. Starting at about 10 000 s, a packet of high-frequency gravity waves travels past the float. This results in vertical excursions of the float of up to 30 m. Although significant changes in stratification occur as the packet passes, the density measured at the float appears uncorrelated with the float’s vertical excursions. This shows that the float is accurately following the vertical motion of isopycnals and can thus make an accurate measurement of the vertical velocity. At about 11 500 s, the density of the upper CTD briefly becomes larger than that measured at the lower CTD. This is suggestive of a turbulent overturn.

## 3. Average internal-wave properties

### a. Theory

*ω*is the radial wave frequency,

*f*is the Coriolis frequency, and

*N*is the buoyancy frequency. The bracketed expression,

*B*(

*ω*), integrates to 1, so the total energy per unit mass at

*N*=

*N*

_{0}= 0.0052 s

^{−1}(3 cph) is ε =

*Eb*

^{2}

*N*

^{2}

_{0}· Following Levine the spectrum is normalized with ε = 2.9 m J kg

^{−1}, rather than separately using

*E*= 6.3 × 10

^{−5}and

*b*= 1300 m, since these have little physical meaning in a coastal environment.

*/Φ*

_{w}*,*

_{E}*f*where (4) is singular.

### b. Stratification

The density stratification is a crucial parameter in the GM models. Figure 2 shows stratification surrounding float 10 during its time on the shelf. Figure 2a shows potential density contours computed from the 12-hourly profiles, the smoothed float depth during the Lagrangian drifts (heavy line), and two standard deviations of the float depth from this average (shading). Although the float’s depth accurately tracks that of an isopycnal, the float’s smoothed depth does not follow the smoothed isopycnal exactly because these two quantities are sampled and averaged differently. Figure 2b shows the stratification at the float computed from the difference between the CTDs on the top and bottom of the float averaged over each Lagrangian drift period. This estimate has little statistical or sampling bias because of its long average and because the typical density difference between the CTDs, 0.1 kg m^{−3}, is much larger than the uncertainty in the calibrations, less than 0.01 kg m^{−3}. However, this estimate cannot be made for most of the floats because they have only one CTD. Attempts to compute *N* from the semidiurnal CTD profiles yielded unacceptably large errors.

### c. Velocity spectra estimation and scaling

The frequency spectra of vertical velocity shown in Fig. 3 were computed as follows. The spectrum of each approximately 11-h-long Lagrangian drift segment is shown as the light gray lines. These spectra were computed using half-overlapping, 7.1-h-long Hanning windows and a tapered four-point frequency average. Their average is shown as the heavy black line. The drift segments were joined together, linearly interpolating over the gaps. The spectrum of this record is shown by the light black line. Some biases due to the interpolation are evident in the difference between the light and heavy lines. A special analysis was done for the *M*_{2} tidal frequency. The interpolated time series of float depth was bandpass filtered by Fourier transforming, zeroing frequencies more than 0.5*M*_{2} from *M*_{2}, and inverse transforming to form an *M*_{2} frequency displacement *Z*_{M2}. A time series of *M*_{2} vertical kinetic energy was defined as *E*_{M2}= 0.25(*W*^{2}_{M2} + *ω*^{2}_{M2}*Z*^{2}_{M2}), where *W*_{M2} is *dZ*_{M2}/*dt* and *ω*_{M2} is the radial *M*_{2} frequency. The two terms in *E*_{M2} are nearly in quadrature, so *E*_{M2}, unlike *W*^{2}_{M2}, has little variation at the *M*_{2} frequency. An *M*_{2} tidal peak was appended to the low-frequency end of the vertical velocity spectrum computed for each drift segment. The peak shape is constant and arbitrary so that spectral density is arbitrary, but the variance contained in the peak is correct. Thus, the gray and thick black spectra in Fig. 3 are the composite of a normal FFT-based spectrum at high frequencies and a time-variable *M*_{2} tidal peak.

Figure 4a shows the individual spectra from floats 3, 8, and 10 as a function of time (i.e., the spectrogram of the data). As expected, all spectra decrease rapidly near *N*. The frequency of the falloff varies with *N*. Accordingly, a value of *N*, denoted by *N _{w}*, can be estimated from these spectra. The white–black line in Fig. 4a shows

*N*computed from the two CTDs on float 10; the magenta line shows

*N*estimated from 90% of the frequency at which the spectrum, smoothed by a 10-point Hanning filter in both time and frequency, first falls below 0.001 m

_{w}^{2}s

^{−1}. The two estimates track well for float 10. The frequency-based estimates of

*N*, unlike the CTD-based estimates, can be used with all floats and are therefore used to scale the spectra.

### d. Mean spectral properties

The average internal-wave spectrum (Fig. 3) can be considered to be the sum of a GM spectrum and large tidal and near-*N* peaks. The mean vertical velocity spectrum is near the GM79 level for about a decade of frequency near *ω*/*N* = 0.03. A tidal peak dominates at lower frequencies. At higher frequencies, the spectrum rises to a broad peak centered near *ω*/*N* = 0.5, with about 10 times the GM energy. Spectral peaks near *N* are commonly seen in deep-ocean spectra (Desaubies 1975), but the peak seen here is much larger and broader. It contains most (98%) of the superinertial vertical kinetic energy (i.e., *ω*/*N* > 0.1). This peak also contains substantial contributions from energetic internal-wave packets, an example of which is shown by the thin dashed line. The total internal-wave energy, computed from (4), is shown by the dashed line. It is much larger than the vertical kinetic energy, containing 97% of the total energy for *ω*/*N* > 0.01 and, unlike the vertical kinetic energy, is dominated by low frequencies.

## 4. Internal-wave variability

### a. Spectral view

The internal-wave field on the Oregon shelf is highly variable. This is seen in the spread of the individual gray spectra (Fig. 3), in the raw (Fig. 4a) spectrogram, and in the WKB-scaled (Fig. 4b) spectrogram. The spectrograms also show that the near-*N* peak tends to rise and fall while maintaining its shape and that there is some tendency for the midfrequencies to be higher when the near-*N* peak is higher.

More formal results can be obtained by correlating the energy in the three major frequency regions: the *M*_{2} tidal peak, the near-*N* peak (0.07 ≤ *ω* /*N* ≤ 1), and the internal-wave continuum between these (0.02 ≤ *ω* /*N* ≤ 0.07). There is no correlation between tidal and near-*N* energy, or tidal and continuum energy. A weak correlation exists between the continuum energy and the near-*N* energy (Fig. 5).

Overall, the analysis shows that variations in the vertical kinetic energy spectrum on the Oregon shelf occur as a coherent and broad peak near *N*, a narrow *M*_{2} internal tidal peak, and within the continuum between these. The variations in all three bands are independent except for a weak tendency of the continuum to rise with, but more slowly than, the *N* peak.

### b. Spatial distribution of energy

The GM model describes only spectra and implicitly assumes that the internal-wave field is well described as random and nearly Gaussian. In the open ocean this is often a good assumption (Briscoe 1977). Figure 6 shows the probability distribution of vertical velocity from float 10 on the Oregon shelf. It is obviously not Gaussian, because it is very long tailed. The kurtosis 〈*w*^{4}〉/〈*w*^{2}〉^{2} is 19.5 as compared with 3 for a Gaussian. The infrequent large values of vertical velocity are contributed by occasional events, such as the one seen in Fig. 1. The probability distribution for one such event is shown by the dashed line.

Figure 7 shows the distribution of vertical kinetic energy on the Oregon shelf from all floats. The kinetic energy is highly intermittent, as shown above, but also shows clear geographic patterns. Figure 8 shows examples of the various waveforms. The location of each example is shown by the associated arrow in Fig. 7.

The largest 1-h-average vertical kinetic energies were found near the location labeled A. Three floats, number 4 in 2002 and numbers 8 and 10 in 2001, passed through this region. The data from float 10 are shown in Fig. 8a. The high energy is associated with packets of internal waves at about −0.2 and +0.25 days (242.6 and 243.15 yeardays of 2001). A third packet, possibly at about 0.65, was not well sampled by the float. Both packets occurred on the sharply downward-going phase of the internal tide (heavy black line). This is the classic pattern of a highly nonlinear internal tide such as is found on the north Australian shelf (Holloway 1994). The tidal nonlinearity causes the tide to develop a triangular waveform with a slow rise followed by a rapid descent. As the waveform becomes very sharp, high-frequency internal waves develop upon it. These packets are phase locked with the internal tide, occurring on its downward-going edge. These waves are therefore not freely propagating but are part of the internal tide, drawing their energy from it and thus forming a potentially important path for tidal dissipation and mixing. Henyey and Hoering (1997) note that these waves have both wavelike and borelike features and coin the term “solibore” to describe them.

Low values of vertical kinetic energy were found offshore and nearshore (Fig. 7). Figure 8b shows two examples.

Over the rough topography of Heceta Bank, a generally higher level of vertical kinetic energy was measured. Two examples are shown in Figs. 8c and 8d. Although the vertical kinetic energy shows variability, the peaks are much broader than in Fig. 8a and not generally phase locked with the tide.

A similar higher level of kinetic energy occurred north of Cape Blanco where the floats crossed the shelf break. A striking feature is apparent at day 248.6 (Figs. 8e and 8f). The float descends from 20 to 60 m and back in a single bump lasting only 500 s and with maximum upward and downward vertical speeds of 0.2 and 0.19 m s^{−1}, respectively. The density fluctuations are qualitatively similar to those shown in Fig. 1; the float deviates from an isopycnal by only slightly more than the separation of its two CTDs.

Notably absent in these data are rank-ordered packets of “solitary” waves, which are a major feature of many descriptions of high-frequency internal waves (e.g., Apel 2003). Although short packets of waves are common, the individual waves are rarely arranged with the largest wave first.

Figure 9 shows the spatial distribution of the largest vertical kinetic energy events. An event was defined as having a 1-h-averaged vertical kinetic energy greater than 5 × 10^{−4} J kg^{−1} (i.e., an rms vertical velocity greater than 0.032 m s^{−1}). A 1-h average captures entire solibore events; the threshold was chosen to sample a reasonable number of large events. Data from float 2 and the first part of float 4 were not included because of instability in the control system (see D’Asaro 2004). The events were classified. Those showing a clear peak in vertical kinetic energy on the downgoing part of an asymmetric tidal waveform were called “tidal solibores” and are colored red in the figure. For example, the event at day 234.1 in Fig. 8c was so classified, while the one at 239.3 (Fig. 8d) was not. Those features showing a single isolated downward excursion, such as that in Fig. 8f, were classified as solitary and are colored black in the figure. All other events are colored blue.

The most energetic events were tidal solibores. They were observed at midshelf, both north and south of Heceta Bank. Of the 84 events, 31% were tidal solibores. Isolated events were infrequent; only 7 (8%) were observed. The strongest such event (Fig. 8f) occurred at the shelf break near 43°N. Events were most energetic on the narrow shelf south of Heceta Bank where almost all events were tidal solibores or solitary events. Some of the strongest events occurred near a small (3 km long, 1 km wide, 20 m high) unnamed bump marked by the yellow triangle in Fig. 9, suggesting that this may be an important source.

Although the strongest vertical velocities occurred in the energetic events discussed above, these events were sufficiently rare that they contain only a small fraction of the vertical kinetic energy. The 84 events in Fig. 9 occupy only 1.6% of the total float record length, assuming 1-h duration for each. The tidal solibores occupy only 1%. The vertical kinetic energy for all events is only 12% of the total; tidal solibores contain only 6%. The vertical kinetic energy itself is only a small fraction (see section 3) of the total internal-wave energy; most of this is contained in horizontal kinetic energy.

## 5. Tidal solibores and solitary waves

Figure 10 shows two of the most energetic tidal solibores. Each is composed of many waves of varying amplitude that, on average, decrease in amplitude over time. Similar waves are often called solitary waves, despite their obvious gregarious behavior here. Formally, a solitary wave is one that propagates at a constant speed with a constant shape. These waves are therefore steady in a coordinate system propagating at the wave speed. For two-dimensional, inviscid, adiabatic internal waves propagating in a nonrotating, shear-free, stratified, horizontally homogeneous Boussinesq fluid, steady solutions of arbitrary amplitude can be found by solving the Dubreil–Jacotin–Long (DJL) equation (often called Long’s equation). Other common equations, such as Korteweg–deVries with ∂* _{t}* replaced by

*υ*∂

*, are approximations to DJL valid only at small amplitude.*

_{x}*υ*

^{2}. The Euler–Lagrange equation for this variation principle is the DJL equation:

*ζ*is the vertical displacement,

*υ*is the solitary wave speed, and

*N*

^{2}

_{r}is the reference stratification (i.e., the stratification before the wave arrives).

We found that the iterative algorithm of Turkington et al. (1991) often does not converge, particularly for waves with amplitudes comparable to those found in the data. To solve this problem, the iterative step was changed to go halfway back to the previous iteration when the function that is trying to be minimized increases. If the function continues to increase at the halved step, it is halved again until it decreases. In addition, we impose the constraint more strongly than does Turkington et al. (1991); this is important for the first few iterations if the initial guess is not very good.

Each solution of the DJL equation yields the structure of a single wave of depression, qualitatively similar to that shown in Fig. 8f. These are evaluated along a streamline (i.e., an isopycnal) to yield a depth–time trajectory directly comparable with the measured float trajectories. Figure 11 plots the full width at half-height of these solutions as a function of their height. Height is measured as the distance from the deepest point of each wave to the shallowest point. For very weak solitary waves, the width decreases rapidly as the height increases. Solitary waves reflect a balance of nonlinearity and dispersion. Since small waves have a small nonlinearity, they must also have a small dispersion. The waves achieve this by being wide and thus approaching the nondispersive, long-wave limit. For very energetic waves, the height is limited by the water depth. The wave energy can thus increase only by making the wave wider, but not higher. The width must therefore increase more rapidly than the height, resulting in an increase in width with height. Between these two extremes lies a minimum in width.

Wave widths and height from days 242 and 243 are shown in Fig. 11. Data from both the tidal solibores and from times outside these are shown. Two estimates of the half-width are available for each deepest point (Fig. 10). The average of these is used. Width is measured in seconds (i.e., the physical width divided by the speed). Note that the width observed by a float is different from the physical width divided by the wave propagation speed because the speed of a Lagrangian particle varies due to the velocities of the wave.

Most of the data clearly fall below the theoretical curve, implying that most of the observed waves are not broad enough to be solitary waves. These waves are much steeper than solitary waves of the same amplitude. A few of the data points outside of the solibores (i.e., “ordinary waves”) approach the solitary solutions. This may be due merely to statistical fluctuation in the wave “width” computed in a randomly fluctuating field. Waves from the tidal solibores (diamonds and stars) fall on a line that is distinct from that of the ordinary waves. These waves are both larger and steeper than the ordinary waves. The line generally falls well below the solitary waves’ solutions; that is, these waves are narrower than solitary waves, except at the largest amplitudes. Accordingly, only the largest of the observed waves could strictly be called solitary waves.

The solutions to the DJL equation assume a two-dimensional steadily propagating wave of constant form with no dissipation and no background shear. Although we have little direct evidence, it seems likely from other observations that these waves are approximately two dimensional. However, they are probably neither inviscid nor steady. Because these nonlinear waves occur at the plunging edge of the internal tide, it seems likely that they draw their energy from it and thus are unlikely to be steady. These deviations from the assumptions of DJL undoubtedly account for the shorter aspect ratio of the observed waves relative to the Long’s equation solutions. Further modeling of the unsteady evolution of such waves from the tide is necessary for a more detailed explanation as is the consideration of the effects of background shear.

## 6. Summary and discussion

Measurements of vertical velocity by isopycnal-following, neutrally buoyant floats deployed on the Oregon shelf during the summers of 2000 and 2001 were used to characterize internal gravity waves on the shelf. The data unevenly and coarsely sampled the entire length and breadth of the shelf from the Columbia River to Cape Blanco. Internal-wave data were collected within the pycnocline on isopycnals whose depths varied from 70 m to near the surface. D’Asaro (2004) describes the floats and their trajectories in more detail.

Significant results include the following:

The average spectrum of vertical kinetic energy has the level predicted by GM79 (Levine 2002) plus a narrow

*M*_{2}tidal peak and a broad high-frequency peak extending from about 0.1*N*to*N*and rising a decade above GM79 (Fig. 3).The amplitude of the high-frequency peak varies coherently across its entire bandwidth.

The tidal peak and the high-frequency peak vary independently.

The energy in the “continuum” region between them is weakly correlated with the level of the high-frequency peak energy and independent of the tidal peak energy (Fig. 5).

The vertical velocity is not Gaussian. It is highly intermittent with a calculated kurtosis of 19 (Fig. 6).

The vertical kinetic energy varies geographically (Fig. 7). Low energy is found offshore and nearshore. The highest energy is found near a small seamount. High energy is found over the rough topography of Heceta Bank and near the shelf break.

The highest energy occurs as packets of high-frequency waves, often occurring on the sharp downgoing phase of the

*M*_{2}internal tide and called “tidal solibores.” A few isolated waves with high energy are also found.Of the 801 hour-long periods with the highest vertical kinetic energy, 31% contain tidal solibores, 8% contain isolated waves, and the remainder of the periods appear unorganized (Fig. 9).

The two most energetic tidal solibores (Fig. 10) were examined in detail. Compared with the steady, propagating, two-dimensional inviscid, internal-wave solutions to the equations of motion with no mean shear (i.e., the DJL equation) the observed waveforms are mostly thin for their height and thus cannot be called “solitary waves” (Fig. 11). A few of the most energetic waves have an aspect ratio approaching that of the solitary solutions.

Despite the large near-*N* peak in vertical kinetic energy, the internal-wave energy on the Oregon shelf, as almost everywhere, is dominated by low-frequency waves. The strongest component is the *M*_{2} internal tide with about 18 m J kg^{−1} (Fig. 3). The total energy at frequencies above *M*_{2} is 3 m J kg^{−1}; with about one-half, 1.8 m J kg^{−1}, in the near-*N* peak (i.e., from 0.1*N* to *N*). Of this, only a small part, roughly 0.11 m J kg^{−1} (section 4), is due to tidal solibores. This analysis says nothing about the near-inertial motions because such motions have little expression in vertical velocity. They probably contain significant energy (Levine 2002). Thus, the intermittent packets of very large, high-frequency internal waves observed in tidal solibores contribute little to the overall internal-wave energy. In comparison, the internal tide is omnipresent and has a similar amplitude but much more kinetic energy. Similarly, the incoherent high-frequency waves found over rough topography are much more common than the infrequent solibores and thus contribute much more energy. Internal waves on the Oregon shelf, as almost everywhere else, have a strong, nearly random component.

## Acknowledgments

This work was supported by ONR Grants N00014-94-1-0024, N00014-04-1-0130, N00014-05-1-0282, and N00014-03-1-0927. This work was made possible by the support of the APL-UW Ocean Engineering Department, especially Michael Ohmart and Michael Kenney. Bradley Bell gave helpful advice on numerical methods for variational principles.

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