1.

Introduction

Waves on the surface of the ocean are a common sight, but internal waves are less familiar. Internal waves propagate along density gradients within the ocean much like surface waves propagate along the air/water density interface. In the ocean, solar heating at the surface and/or fresh water input from terrestrial runoff or melting ice create a layer of water with lower density at the surface. This layer is typically mixed with the water below by the action of surface waves and turbulence, but this is only effective to some finite depth that depends on the strength of the wind. This tends to create a layer of less dense water on top of the more dense water below. The boundary, known as the pycnocline, is the region of large density gradients on which internal waves propagate. Of course, the real ocean is more complex, and multiple layers may form from episodic heating and mixing events.

Internal waves are an important process in mixing in the ocean.^1–3 One of the most common processes for the generation in internal waves is the flow of tidal currents over bottom topography,⁴ especially at the edge of the continental shelf. These waves can propagate over long distances, carrying this tidal energy with them. This energy is eventually released and dissipated where the waves break.⁵

The surface signatures from internal waves have been observed in microwave-radar data.^6–9 The primary mechanism is the modulation of surface roughness, which is the source of the radar signal, by the surface currents produced by the waves.^8,10,11 Where surface currents are diverging, the roughness is decreased, and the radar return is reduced. Where currents are converging, the roughness is increased, and the radar return is increased.

For these waves to be apparent in a lidar signal, there has to be some tracer, such as a layer of phytoplankton to scatter the light. Fortunately, such a layer often exists near the pycnocline. In the simplified two-layer model of the ocean, one can imagine that the nutrients in the upper layer would be consumed rapidly since there is plenty of sunlight for photosynthesis. The lower layer represents a large reservoir of nutrients, but there is little sunlight. The result is a narrow zone near the pycnocline that will support phytoplankton growth.¹² This is oversimplified, and other processes that contribute to thin-layer formation include stretching of a plankton patch into a layer by current shear and sinking of plankton cells until they reach the depth where their density matches that of the surrounding water.^13–15 Regardless of the mechanism, thin plankton layers associated with the pycnocline have been observed,^13,14 and the detection of scattering layers, including thin plankton layers, by airborne lidar is well documented.^12,16–19 We have demonstrated that these thin layers can be associated with upwelling events and with fresh water influx.¹⁸

Acoustic measurements have shown that these thin layers move up and down under the influence of internal waves,¹⁹ and internal waves also have been observed in airborne lidar data.^18,20 However, neither of these lidar observations had coincident in-water measurements for comparison. This paper reports on lidar and photographic measurements of several internal waves in West Sound on Orcas Island in Washington state, and the comparison with in situ measurements of water stratification and optical properties. The primary objective was to determine whether or not the energy of an internal-wave packet can be obtained from remote sensing data collected by lidar and aerial photography. A secondary objective was to determine if quantitative information could also be obtained about turbulence within the plankton layers.

In previous studies, it was observed that the lidar signal from a plankton layer has higher contrast (i.e., stands out more clearly against the lidar signal from the surrounding water) when the laser transmitter is linearly polarized and the receiver is filtered to collect scattered light in the orthogonal polarization.^17,18,20 The reason is that plankton are much larger than the 532-nm wavelength that is generally used for oceanographic lidar and have enough irregularly shaped structures to depolarize backscattered light through multiple scattering within each cell or colony. Previous work has suggested that circularly polarized light will provide better contrast for large underwater targets,^21–23 but does not have a corresponding advantage for a collection of small particles.^24–27 A third objective of this work was to compare the contrast for linear and circular polarizations.

2.

Measurements

East Sound on Orcas Island in Washington state has been the site of a number of studies on thin layer formation and evolution.^{12,14,28–30} This history of plankton layer formation led us to return to East Sound in September 2011 with the airborne lidar. No significant evidence of plankton layers was found in the lidar data from flights over East Sound in September of 2011, so we moved to West Sound to make the measurements. Only data from West Sound are considered in this analysis. While both East Sound and West Sound are shallow fjords that are protected by other islands to the south (Fig. 1), West Sound is smaller and more open to internal waves that can be generated by the strong tidal flows through the channel outside the sound.

Fig. 1

Map of Orcas Island and surroundings. (a) Location of Orcas Island (black) in the northwestern United States between Vancouver, British Columbia, and Seattle, Washington; (b) Orcas Island and surrounding islands, showing East Sound, which is about 10 km long, and West Sound, which is about half of that. The large bay on the east side of West Sound is White Beach Bay.

The NOAA Fish Lidar,^31–33 developed in the Earth System Research Laboratory over a number of years, was installed in the back of a four-seat Cessna-177 aircraft with the rear seats removed (Fig. 2). Two flights were made each day of the deployment at a flight altitude of about 300 m and a speed of about $40\mathrm{m}{\mathrm{s}}^{-1}$. The lidar transmitted 100 mJ of green (532 nm) light in a 12-ns pulse at a rate of 10 pulses per second. It was pointed 12 deg off nadir to minimize the specular reflection from the sea surface. The laser beam divergence was set so that the diameter of the laser spot on the surface was 5 m, which is large enough that the power density at the surface is safe for humans and marine mammals.³⁴ The scattered light from the water column was collected by two telescopes whose fields of view were matched to the laser beam divergence. Each telescope was equipped [as shown in Fig. 2(a)] with a polarizing filter, a field-of-view aperture, and an interference filter to reduce background light. The light collected by each telescope was detected by a photomultiplier tube, logarithmically amplified to increase the dynamic range, and digitized at a rate of ${10}^9$ samples per second. Between the morning flight and the afternoon flight on September 16, we changed the polarization characteristics of the lidar. In the morning, the transmitter was linearly polarized, and the two telescopes were filtered to receive light co-polarized with the transmitter and cross-polarized to it. In the afternoon, the transmitter was changed to right-hand circular polarization, and the receiver telescopes were filtered to receive co- and cross-circularly polarized light.

Fig. 2

Lidar system. (a) Schematic diagram of the laser transmitter optics and the receiver optics for one of the two receiver channels; (b) wide-angle photo of the lidar looking toward the rear of the aircraft. On the left is the receiver side of the optics package with the two telescopes at the bottom and the photomultiplier tube modules near the center. The laser and transmitter optics are out of view on the far side of the mounting plate. On the right, behind the pilot’s seat, is the stack of electronics, including the laser cooling system (bottom), laser power supply, time-delay generator (top left), and data-acquisition package (top right).

Identification of internal waves in the lidar data was by visual inspection. The raw cross-polarized lidar data was used for this because of the improved contrast, which facilitated layer detection; the layers were much less visible in the co-polarized return. Data segments with visible wave structure were processed further to correct for the exponential attenuation and to subtract the background scattering to enhance the visibility of the layers. The depth of the peak return was identified for each lidar shot, and the positions of the maximum depths of the internal-wave oscillations were found. Because of the large number seen on September 16, the analysis concentrated on that day.

In situ instrumentation comprised a Seabird conductivity, temperature, depth (CTD) profiler with a Wetlabs AC-9 to measure the optical absorption and beam attenuation coefficients. Conductivity was used to calculate salinity, which was combined with temperature to provide density as a function of depth. The scattering coefficient was estimated as the difference between the attenuation and absorption coefficients.

Photographs of the surface manifestations of the internal waves were taken through the aircraft window with a Nikon D300 digital camera using an 18- to 200-mm focal length zoom lens. The combination of a periodic displacement of a subsurface scattering layer with a corresponding surface slick at the surface provides strong evidence for internal waves; other mechanisms could produce perturbations of the scattering layer or surface slicks, but it would be highly unlikely that these would occur together. The orientation and length of the surface slicks in the photographs also provide information on the direction of propagation and the width of the wave packet that are difficult to infer from the lidar alone.

To infer the width of the internal wave packet from one of the photographs (not presented), we had to correct for the geometric distortion caused by the oblique view angle. The field of view is easily obtained from the focal length of the zoom lens, which was recorded for each image. This leads directly to the angle with respect to the optical axis represented by each pixel in the image. We assumed that the camera was pointed at some angle ${\alpha }_0$ below the horizon with the horizontal axis of the image parallel to the horizon. The position of each pixel on the surface can then be written as

We also looked for evidence of stratified turbulence in plankton layers by calculating the power spectrum of fluctuations in the lidar return along center of the layer. This investigation used data from a previous deployment to East Sound, because those data could be compared with in situ samples of zooplankton. Biological sampling was not done in 2011.

3.

Internal Waves

Figure 3 is an example of the processed lidar data for one of the internal waves. The background layer depth here was about 4 m, and the maximum depth of the internal wave was 7.8 m. This large perturbation, almost equal to the layer depth, suggests nonlinear propagation. Furthermore, two additional traits provide supporting evidence for this hypothesis. The first is the nonsymmetric shape of the wave, which is sharper on the trough and more rounded on the crest. The second is the fact that these waves were visible from the aircraft, because of the modulation of surface roughness by the currents induced by the waves.

Fig. 3

Example of the relative lidar return across an internal wave (white to black represents 0 to 1) as a function of distance along the flight track and depth. Each lidar profile has been normalized by its maximum value, and the depth of that maximum is plotted as a white line.

In all, 10 observations of internal waves were clearly seen in the lidar data of September 16, two in the morning and eight in the afternoon. The positions of the troughs (maximum depths) of the afternoon observations are shown in Fig. 4, along with the afternoon flight tracks. All of the waves identified were near the mouth of White Beach Bay.

Fig. 4

West Sound with flight tracks (red or gray lines), crests of internal waves ($\mathrm{black}\times$), and locations of in situ measurements (black circles) from observations on the afternoon of September 16.

The in situ profile measured at the northern of the two locations (Fig. 5) shows a scattering layer at a depth of 3.4 m, which is just below the depth where the density gradient is greatest. The density profile has several step-like features, which are often related to different episodes of wind mixing followed by re-stratification. It is useful to consider a two-layer approximation as a simple model of the complex structure in the density profile. This was done by simply picking density values that seemed representative of the shallowest and deepest waters. The transition was set at the depth where the actual density was equal to the midpoint between the shallow- and deep-water values.

Fig. 5

Profiles of density, ${\sigma }_t$ (short-dashed line), a rough two-layer approximation to ${\sigma }_t$ (long-dashed line), and optical scattering coefficient $b$ (black line) at a wavelength of 532 nm. The measurement location was the northernmost of the two in situ measurements marked in Fig. 4. The total water depth at this location was 19.1 m.

For a two-layer ocean, the speed of propagation of a small (i.e., linear) internal wave is given by:

Consider the internal wave of Fig. 3. The water depth is about the same (19.1 m), but the thickness of the upper layer is 4 m, and the wave amplitude is 3.8 m. The layer densities are likely to be similar to the measured profile, so the KdV equation would predict a propagation speed of $5\mathrm{cm}{\mathrm{s}}^{-1}$. We made a second pass over the same area along the same direction 31 min later. The wave had clearly evolved in that time, but we could line up the main peaks by shifting the second record south by 87 m. Dividing 87 m by 31 min, we obtain a wave speed of $4.7\mathrm{cm}{\mathrm{s}}^{-1}$, which is in good agreement with the KdV prediction. This is more evidence for the nonlinear nature of the wave, since the linear approximation predicts a speed of $16\mathrm{cm}{\mathrm{s}}^{-1}$.

Numerical simulations,³⁶ field measurements,³⁷ and remote sensing observations³⁸ have all shown that the polarity of a nonlinear internal wave in a two-layer ocean depends on the relative thickness of the two layers. If the upper layer is thinner, as in the case of Fig. 3, the wave will manifest as a downward perturbation of the original layer. This polarity is clear from the figure. If the bottom layer is thinner, the perturbation will be upward. Figure 6 depicts one such example, where the top layer is thicker than the bottom layer; in this case, the scattering layer is deeper than the example in Fig. 3 and is perturbed upward by the internal wave.

Fig. 6

Example of the relative lidar return across an internal wave (white to black represents 0 to 1) as a function of distance along the flight track and depth. Each lidar profile has been normalized by its maximum value, and the depth of that maximum is plotted as a white line.

The energy within an internal-wave field can be inferred from the aerial observations as well. We note that the potential energy of a small volume of water $dV$ is given by

4.

Turbulence

In a strongly stratified body of water, the energy contained within the internal waves is converted to turbulent energy at large scales through nonlinear interactions, and this energy cascades through smaller scales. The resulting spectra of potential and kinetic energy are predicted to be⁴⁰

To the extent that plankton drift with the local fluid motion, we can consider them to be passive additives that would have a horizontal spatial spectrum similar to that of the turbulent energy. That implies that $\beta$, the volume scattering function at the lidar wavelength and scattering angle ($\pi$ rad), would also have a similar power spectrum⁴²

Previous lidar measurements⁴⁵ of the horizontal spatial power spectrum of lidar backscatter produced a $-3/2$ dependence rather than the $-5/3$ dependence predicted by stratified-turbulence theory. Horizontal power spectra in East Sound and West Sound also failed to produce a $-5/3$ power law. In the theory, the turbulent layer is at a constant depth where the background density is assumed to be constant. In the real ocean, the layer is expected to follow the constant-density surface, which is not at a constant depth. If, instead of the return from a fixed depth, we select the return from the center of the layer and calculate the power spectrum of this quantity, we can sometimes see the predicted $-5/3$ dependence over the extent of a layer (Fig. 7).

Fig. 7

Example of measured power spectral density of lidar return along a layer as a function of spatial frequency. For clarity of presentation, spectral values have been averaged for wavelengths less than 100 m and the positive error bar represents one standard deviation of the initial values. The line is a $-5/3$ power law.

Some layers failed to produce a clear power-law spectrum, however. The data presented in Fig. 7 were taken during a previous deployment to East Sound where extensive in situ sampling confirmed that almost no zooplankton were present within the layer. Other layers in the previous measurements contained significant numbers of zooplankton, and no clear power law spectrum was observed. Generally, zooplankton are more mobile than phytoplankton and are less likely to be true passive additives. In addition, some phytoplankton (e.g., dinoflagellates) have limited mobility, which has been shown to affect ocean color patterns⁴⁶ and may be enough to violate the passive-additive assumption. In other instances, the reason was not as clear, but lidar data may be useful in deciding whether or not the dominant mixing process within a layer is stratified turbulence.

5.

Polarization Effects on Contrast

The layers that allowed detection of the internal waves were easier to see in the cross-polarized return than the co-polarized return. This effect can be evaluated quantitatively by considering layer contrast under different polarization conditions. For each shot, we define the contrast of the depolarized signal in terms of $S$, the cross-polarized lidar signal after correction for attenuation, as

To ensure that we were focusing the analysis on the scattering layers, the data were separated into 10 s segments, and the shot with the highest contrast within each segment was selected for further analysis. This time period corresponds with a horizontal distance of about 400 m, and visual inspection of the data suggests that this is sufficient to include a strong layer in most cases. For the selected data, we calculated the contrast of the co-polarized signal at the same depths as $C_x$ and the depolarization as the ratio of $S{x}_{\text{max}}$ to the co-polarized signal at the same depth.

The contrast, plotted in Fig. 8, shows variability from flight to flight, but the depolarized contrast is clearly higher. For the linear polarization, average contrast was $0.13\pm 0.03$ for the co-polarized signal and $0.25\pm 0.04$ for the cross-polarized case. This difference is statistically significant ($p<0.001$ in a two-sided $t$-test). For the circularly polarized cases, the corresponding values are $0.18\pm 0.04$ and $0.26\pm 0.01$, which also represents a statistically significant difference ($p=0.010$). Interestingly, the contrast values for the linear and circular cross-polarized signals were not significantly different ($p=0.77$). For the co-polarized signals, the difference between linear and circular polarization was larger ($p=0.046$), but still only bordering on significant even if a low standard of $p<0.05$ is adopted.

Fig. 8

Optical contrast by flight as a function of year day, 2011. Symbols denote linear cross-polarized (+), linear co-polarized (×), circular cross-polarized (•) and circular co-polarized (○) values. Error bars denote $\pm$ one standard deviation of the values measured during each flight.

The lack of a significant difference in contrast between linear polarization and circular polarization suggests that both linear and circular depolarization carry the same information. In 1957, van de Hulst²⁴ used symmetry arguments to show that the Mueller matrix for the case of interest for lidar (scattering angle of 180 deg) from a collection of particles was diagonal with only three parameters as long as each particle is accompanied by its mirror image. Hu et al.²⁵ showed that two of these parameters were related for backscattering for even a single dielectric particle. Mishchenko and Hovenier²⁶ combined these two results to obtain the following Mueller scattering matrix:

This implies that the depolarization of an initially polarized beam will be

Average depolarization values (Fig. 9) suggest that the circular depolarization is greater than linear, as predicted. Overall, we have $D_L=0.06\pm 0.01$ and $D_C=0.11\pm 0.03$, and the difference is statistically significant ($p=0.028$). For the cases with linear polarization, we have also estimated what the circular polarization would have been if Eq. (10) is valid. The overall average is $0.13\pm 0.02$, which is not statistically different ($p=0.28$) from the actual measurements using circular polarization. This is consistent with recent measurements of atmospheric scattering.^47–50

Fig. 9

Layer depolarization by flight as a function of year day, 2011. (×) is linear depolarization, (○) is circular depolarization, and (•) is circular depolarization predicted from the measured linear depolarization. Error bars denote $\pm$ one standard deviation of the average of each flight.

6.

Surface Manifestations

As noted in the introduction, internal waves can be detected by microwave radar because the induced surface currents modulate the surface roughness. This same process affects the amount of skylight reflected from the surface. The reason is that the Fresnel reflectivity from the surface is not a linear function of incident angle, so the average reflectivity of a roughened surface at some nominal view angle is not the same as the reflectivity of a flat surface at the same angle. The magnitude of the effect depends on the view angle, surface roughness, and polarization of the skylight, but can be a few percent for view angles of 60 deg to 70 deg from nadir and root-mean-square surface angles near 5 deg.

A photograph taken from the aircraft (Fig. 10) shows the surface signature of one of the wave packets observed. While the information is in the original image, it is much more visible after contrast enhancement. The black line added to the original image crosses three slicks associated with the wave. Figure 11 is a plot of relative pixel value in the green channel of the original color image from the bottom of this line to its top. A linear regression of pixel value was used as the background value, since there was a nearly linear change in illumination along the line. The relative value at each pixel is then estimated as the difference between the pixel value and the regression value, normalized by the regression value. The three slicks are clearly visible, with peak values of about 2%, as expected from the Fresnel reflectivity estimates.

Fig. 10

Photo of the surface signature of one of the observed internal waves: (a) original with a black line marking the section plotted in Fig. 11; (b) same image after contrast enhancement.

Fig. 11

Relative pixel value above background level as a function of position along the black line in Fig. 10(a) (bottom to top).

The presence of the surface manifestation is important for interpreting the lidar data, because it provides the direction of the wave fronts. With this information, we were able to calculate internal wave wavelength and phase speed in Sec. 3, knowing that our flight track crossed the waves nearly normal to the wave front. Without this information, the angle between the phase front and the lidar track is uncertain, and the estimate of wavelength is subject to error. We should point out that the wavelength could also be inferred from the image in Fig. 10, but to do this quantitatively would require a correction of the geometric distortion of the image as described in Sec. 2. This was not done for this image, because we obtained the wavelength from the lidar data.

Remotely sensing the surface manifestation alone misses important information. There have been attempts to infer the depth of the pycnocline from the surface manifestations of internal waves, but with limited success.^51,52 These have only been made for a two-layer model of ocean density profile and require multiple observations. The amplitude of the wave is not possible to measure. The primary advantage of radar detection of the surface manifestation is that it is possible from space; there are currently no orbiting lidar systems capable of internal-wave detection. The closest is the NASA Cloud-Aerosol Lidar with Orthogonal Polarisation (CALIOP), but it lacks both the vertical and horizontal resolution for this application.⁵³

7.

Conclusions

Airborne lidar can easily detect nonlinear internal waves in a shallow fjord when conditions are right to produce a plankton layer associated with a density gradient. The propagation of these waves seems to be fairly well modeled by a two-layer ocean model and the KdV equation. While there were almost certainly linear internal waves in West Sound during our measurement period, their amplitudes would have had to have been much less than 1 m for nonlinear effects to be negligible. Detection of these would require more sophisticated processing to discriminate between internal waves and random variability.

The observed $-5/3$ power law spectrum of lidar backscatter along the layer suggests that stratified turbulence played an important role in producing the variability in lidar backscatter. This spectral dependence is a necessary, but not sufficient condition for stratified turbulence, and more work is needed to establish the reliability with which turbulent mixing within a layer can be identified.

The layers that create the lidar signal can most easily be observed using a polarized transmitter and a cross-polarized receiver, but there is no significant difference in contrast between lidars using linear or circular polarization. The reason for this seems to be that the single-parameter Mueller matrix that has been used for atmospheric lidar studies is also valid for scattering from plankton layers in the ocean.

When surface manifestations of internal waves are visible, these can provide additional information about the direction of propagation and width of the wave front. This can be done reliably with visible images, even if the contrast is low. The combination of lidar and surface imagery provides the total energy in an internal-wave packet.