Over the past decade, a novel free-fall imaging profiler has been under development at the Scripps Institution of Oceanography to observe and quantify biological and physical structure in the upper 100 m of the ocean. The profiler provided the first detailed view of microscale phytoplankton distributions using in situ planar laser-induced fluorescence. The present study examines a recent incarnation of the profiler that features microscale turbulent flow measurement capabilities using stereoscopic particle image velocimetry (PIV). As the profiler descends through the water column, a vertical sheet of laser light illuminates natural particles below the profiler. Two sensitive charge-coupled device (CCD) cameras image a 25 cm × 25 cm × 0.6 cm region at a nominal frame rate of 8 Hz. The stereoscopic camera configuration allows all three components of velocity to be measured in the vertical plane with an average spatial resolution of approximately 3 mm. The performance of the PIV system is evaluated for deployments offshore of the southern California coast. The in situ image characteristics, including natural particle seeding density and imaged particle size, are found to be suitable for PIV. Ensemble-averaged velocity and dissipation of turbulent kinetic energy estimates from the stereoscopic PIV system are consistent with observations from an acoustic Doppler velocimeter and acoustic Doppler current profiler, though it is revealed that the present instrument configuration influences the observed flow field. The salient challenges in adapting stereoscopic PIV for in situ, open-ocean turbulence measurements are identified, including cross-plane particle motion, instrument intrusiveness, and measurement uncertainty limitations. These challenges are discussed and recommendations are provided for future development: improved alignment with the dominant flow direction, mitigation of instrument intrusiveness, and improvements in illumination and imaging resolution.
Over the past few decades, in situ studies of the interactions of turbulent oceanic flows and planktonic particles at the fine- to microscales (from 10s of centimeters to millimeters) have faced significant observational challenges, including 1) the simultaneous sampling of biological and physical variables, and 2) robust sampling of the turbulence field at these scales. A number of studies have made significant progress in addressing these challenges.
Advancements in the resolution capabilities of bio-optical and bioacoustic sensors and the development of multidisciplinary sampling strategies has produced links between fine- and microscale physical structure (e.g., stratification and shear) and concentrations of planktonic organisms at these scales (Cowles et al. 1993; Dekshenieks et al. 2001; Franks 1995; Franks and Jaffe 2001; Alldredge et al. 2002; Holliday et al. 2003; McManus et al. 2005; Stacey et al. 2007). Still, correlations between biological distributions and flow structures are limited in temporal and spatial resolution. There remains a need for observations that clarify the role of physical processes in redistributing planktonic organisms at small scales.
Our ability to reliably measure fine- to microscale turbulence has continued to develop, using, for example, thermal and shear microstructure profilers (Oakey 1982; Carter and Imberger 1986; Kocsis et al. 1999), acoustic Doppler velocimeters (ADVs; Voulgaris and Trowbridge 1998; Geyer et al. 2008) and acoustic Doppler current profilers (ADCPs; Lohrmann et al. 1990; Stacey et al. 1999). However, descriptions of the turbulence field are limited in coverage and resolution. Furthermore, estimates of turbulence quantities such as the dissipation rate of turbulent kinetic energy (ɛ) are subject to assumptions such as isotropy and restrictive conditions on the form of velocity and scalar spectra.
A novel particle imaging profiler has been under development at Scripps Institution of Oceanography to address these two observational challenges (Jaffe et al. 1998; Franks and Jaffe 2001, 2008; Zawada 2002, 2003). The profiler, originally named the Free-Fall Imaging Device for Observing Phytoplankton (FIDO-Φ), provided the first detailed view of microscale phytoplankton distributions using in situ planar laser-induced fluorescence (PLIF; Franks and Jaffe 2001, 2008). In this paper, we describe a recent incarnation of the profiler that features stereoscopic particle image velocimetry (PIV) capabilities.
PIV is a well-known quantitative flow measurement and visualization technique in the laboratory, but has only recently been adapted for the study of in situ flows (e.g., Bertuccioli et al. 1999; Doron et al. 2001; Nimmo Smith et al. 2002; Tritico et al. 2007; Katija and Dabiri 2008; Liao et al. 2009). In PIV, the net displacement of imaged particles over a time interval separating consecutive images is used to estimate the fluid velocity (after calibration) in the object plane (Raffel et al. 1998). Two-dimensional (2D) distributions of velocity may be measured with fine spatial and temporal resolution. When the resolution of the velocity maps is on the order of the Kolmogorov scale, it is possible to directly estimate ɛ with relatively few assumptions.
When the image plane is parallel to the light sheet, as is the case in many single-camera configurations, only the in-plane displacements are recorded. Alternatively, if the image plane is oriented obliquely with respect to the light sheet, the particle motions recorded include both in-plane and out-of-plane displacements (Raffel et al. 1998). Two cameras configured in this way can yield a stereoscopic view of particle motion, from which the three-component velocity may be reconstructed within the 2D object plane (Prasad and Adrian 1993; Soloff et al. 1997; Willert 1997; Raffel et al. 1998; Prasad 2000).
To our knowledge, this study represents the first field adaptation of recently developed stereoscopic PIV techniques and the first open-ocean application of PIV. In section 2 we describe the basic components of the profiler and imaging system, PIV data processing, calibration, and deployment. In section 3 we present sample results, including in situ PIV data quality, ensemble-averaged velocity structure, turbulence measurements, and an analysis of dissipation uncertainty. Finally, in section 4 we summarize the preliminary results, discuss the primary challenges in the open-ocean application of PIV, and provide recommendations for future development.
a. Overview of the imaging profiler
The imaging system is housed in a quasi-conical aluminum frame, approximately 2 m in diameter and 3 m tall (Fig. 1). The profiler has a total mass of approximately 103 kg and is nearly neutrally buoyant in seawater. Positively buoyant elements are concentrated at the top of the profiler and negatively buoyant elements are at the bottom so that the profiler maintains a stable, vertical orientation. The profiler features an active ballast system that employs two Duff–Norton electromechanical actuators that control the position of two custom pistons capable of displacing 2.5 kg of water (Fig. 2).
The profiler is equipped for combined stereoscopic PIV and PLIF (Fig. 2). The laser beam of a 3-W, 532-nm diode-pumped solid-state laser (Melles Griot 58-GSS-309) is passed through a 30° Powell lens (Powell 1987) to generate a vertical sheet of light that illuminates natural particles and stimulates phytoplankton fluorescence below the profiler. Two sensitive charge-coupled device (CCD) cameras (Cooke Corporation, Sensicam QE and Sony ICX285AL, 12 bit, 1040 pixels × 1376 pixels) housed in pressure canisters are positioned on opposite sides of the light sheet and are inclined at 45°. The imaged region is located approximately 43–68 cm below the base of the profiler.
Each camera has a dedicated computer (Shuttle SB52G2, P4 2.4 GHz, 512-MB RAM, 120-GB disk, and running LabVIEW) to control image acquisition and storage. An onboard control computer (PXI, Krontron, running LabVIEW) regulates the active ballasting system responsible for descent and ascent, synchronizes the imaging system, and monitors internal sensors that measure instrument dynamics (pitch, yaw, and heading) and pressure housing vitals (temperature, humidity, and battery voltage). Vertical migration and image acquisition plans as well as real-time internal sensor data are transferred between the ship and profiler via a wireless 802.11g access point (Orinoco AP-2000). Two DeepSea Power and Light batteries (each 24 V, 42 Ah) power the entire system for up to approximately 5 h of continuous operation. The profiler has a working depth range of 100 m.
b. Optical design
To ensure sharp focus throughout the field of view of the obliquely oriented CCD cameras, the lenses (Nikor 20 mm 1.8/f) must be tilted to satisfy the Scheimpflug condition, which requires that the image plane, lens plane, and object plane all intersect along the same line (Raffel et al. 1998; Wheeler 2001). For laboratory experiments, this may be accomplished by gradually tilting the lenses until the entire image is in focus. For our field application, however, the lenses must be focused prior to immersion in water and locked to avoid any change in focus during deployment. Therefore, the lens tilt is determined a priori from the Scheimpflug condition, the “hinge rule,” and trigonometric identities (Merklinger 1993; Raffel et al. 1998). The resulting lens tilt relative to the CCD plane is 1.68° for our system. Once the lenses are tilted appropriately, the focus of each camera is adjusted in air using a target that closely mimics the orientation of the laser sheet in water.
Both of the cameras are fitted with filter wheels (Lambda 4, Sutter Instruments, with filters by Omega Optical) to isolate side-scattered light (520–540 nm) and three fluorescent wave bands (560–570, 580–590, and 670–690 nm). The rotation of the filter wheel allows image sequences to contain scattered light from all particles (used for PIV) as well as fluorescent light to distinguish planktonic particle type. Optical flats are placed in front of all filters (except 670–690 nm) to correct for chromatic aberration.
The geometry of the imaging system reflects trade-offs between image resolution, field of view size, and attenuation of light resulting from scattering and absorption by seawater. As seen in Fig. 3, the resulting joint field of view spans approximately 25 cm × 25 cm × 0.6 cm and the oblique perspective results in variable magnification with pixel resolution varying from approximately 0.15 mm at the top to 0.22 mm at the bottom. Locations in the 2D image planes are defined as Xj(k), where j specifies the image coordinate direction (j = 1, 2) and k specifies the camera (k = 1, 2). Object plane coordinates are defined as xi, where i indicates the coordinate direction. The coordinate system is right handed, with in-plane horizontal, cross plane, and in-plane vertical coordinates specified by i = 1, 2, 3, respectively.
c. Field deployment
The profiler was deployed from the Research Vessel (R/V) New Horizon 16–26 September 2004 primarily at sites 2–15 km southwest of Santa Catalina Island, Santa Rosa Island, and Santa Cruz Island, California (typical local depths of 200–500 m). The profiler was fitted with a number of auxiliary instruments, including a downward-looking 600-kHz RD Instruments ADCP (6 Hz and 1-m vertical bins, with the first bin 0.5 m beside and 1.5 m below the field of view) and a Nortek ADV (16 Hz and a 1-cm vertical bin, with a measurement volume 0.5 m beside and 0.16 m above the field of view). During deployment, the profiler was disconnected from the R/V New Horizon and tethered loosely to a small-bottle buoy to aid in recovery. The system parameters (descent rate, exposure and interframe time, start and stop depth, and spectral channels) were configured via the wireless access point while the profiler was at the sea surface. The profile was then initiated and the profiler descended from the surface (typically at a rate of 5–8 cm s−1). Between the start and stop depths (typically 10 and 60 m, respectively), the profiler acquired images. At the stop depth, the imaging system powered down, the pistons were extended to their maximum stroke, and the profiler returned to the surface. Battery and disk space limited each deployment of the profiler to typically three to four profiles.
The profiler exhibited consistently stable dynamics during all of the deployments. The heading, pitch, and yaw of the system were affected by surface waves from the surface down to roughly 20 m; however, below that region the pitch and yaw were typically less than 0.5°, and the heading varied slowly as the platform descended. This is likely a consequence of the large size of the platform, making it highly stable. Large vertical fins were placed on the side of the profiler to improve alignment of the imaging system with the dominant flow direction.
The CCD cameras were configured to acquire four image pairs within each second (nominally 8 Hz), which was limited by filter wheel rotation. After each deployment, sample images were processed to evaluate PIV data quality and tune image acquisition parameters. The interframe time was varied (25–60 ms) to balance the following two competing objectives for PIV: 1) to minimize out-of-plane loss of particle pairs, occurring for long interframe times, and 2) to maximize turbulent particle displacements between image pairs by extending the interframe time. Given that the optimal interframe time depends on the turbulence and out-of-plane velocity characteristics of the flow field, an a priori determination of the appropriate interframe time is not possible. Instead, we typically alternated between two interframe times during a single profile, with the longer of the two resolving small turbulent displacements and the shorter of the two resolving the velocity field when the cross-plane velocity was large. The exposure times were adjusted (10–20 ms) to optimize particle intensity, while maintaining minimal streaking. Daytime images revealed solar light contamination to the full depth of the profiles (typically 60 m) and the intensity signal-to-noise ratio of imaged particles was typically insufficient to yield high-quality PIV data. Thus, the profiler was deployed primarily after dusk and before dawn to minimize image noise.
d. Velocity reconstruction and shipboard calibration
The three-component object plane velocities Ui were calculated using the imaged particle displacements from each camera ΔXj(k) and mapping functions that relate the image planes Xj(k) and object space xi (Fig. 3). The imaged displacements ΔXj(k) were measured via traditional cross-correlation PIV. The images were first corrected for variations in particle size and scattering properties using intensity capping (Shavit et al. 2007). The optimal level of capping n (number of standard deviations above the median intensity) for these images was typically in the range of 0.5–2. The PIV processing was then carried out with MatPIV, which features iterative subwindow shifting, local consistency filtering, and subpixel refinement using a three-point Gaussian fit to the correlation peak (Sveen 2004). The final velocity vector maps had a resolution of 32 pixels × 32 pixels with 50% overlap (i.e., 16 pixels or approximately 3 mm on average in the object plane). Based on extensive visual examination, only those vectors within two standard deviations of the median of their 24 (52 − 1) neighbors were marked as valid (invalid vectors were removed) and only those vector fields with a percentage of valid vectors P exceeding 87% were selected for detailed analysis. We note also that the mean improvement in the number of valid vectors resulting from intensity capping exceeded 5%, indicating a dramatic improvement in the quality of the results.
The mapping functions that relate the image planes and object space were determined via calibration (Soloff et al. 1997; Prasad 2000). This approach is advantageous compared to geometric-based mapping in that it not only accounts for perspective distortion but also image aberrations and any changes in the optical system at sea. The calibration procedure involves acquiring images of a target with known dimensions. For stereoscopic PIV, the target must occupy both in-plane (x1, x3) and cross-plane (x2) dimensions. In a laboratory setting, this may be accomplished by acquiring images of a 2D target placed at various locations throughout the depth of the illumination sheet, for example, using a precision traverse (Soloff et al. 1997). In a field setting, this method is impractical, being both costly in machinery and time and difficult to set up. As an alternative, an inexpensive 3D calibration target was designed and developed that allowed for rapid (<½ h) shipboard calibration. The custom 3D calibration target consists of a vertical rack of polished acrylic bars with laser-etched + symbol calibration marks that occupy the joint field of view and span all three coordinate dimensions (Fig. 4). During calibration, the target was mounted, the profiler was deployed from a ship and submerged at the sea surface, and both cameras acquired images. The laser light passed through the transparent polished acrylic bars, while scattering from the laser-etched + marks. The scattered light from the calibration marks passed only through seawater (not the acrylic) on its path to the CCD cameras. Thus, the index of refraction represented in the calibration images matched that present in the in situ particle images. Using the image plane coordinates Xj(k) and known object plane coordinates xi of the calibration marks, a set of 3D polynomial mapping functions Fj(k) describing the transformation between the object space and image planes were determined via least squares fitting (Soloff et al. 1997). The resulting uncertainty in the mapping was found to be approximately 0.1 pixel. The implied object plane displacement uncertainty resulting from mapping is two orders of magnitude smaller than the uncertainty in PIV (i.e., locating the correlation peak). Thus, the mapping formulation of Soloff et al. (1997) was very satisfactory. Further, calibrations performed before and after the cruise were essentially identical, indicating that the laser and optics were incredibly stable during deployment.
The three-component object plane particle displacements Δxi are determined by combining the imaged particle displacements ΔXj(k) with the mapping functions Fj(k) (see Soloff et al. 1997 for details). The object plane particle displacements were computed in a vertical grid positioned at the center of the laser light sheet and the in-plane velocity resolution reflected the local resolution of the raw 2D PIV fields, approximately 3 mm on average (i.e., the resolution was neither enhanced nor degraded in the transformation from 2D to 3D). The final velocity maps were 60 vectors × 75 vectors. The object plane fluid velocities were estimated simply as Ui = Δxi/Δt.
3. Evaluation of the PIV system
a. In situ PIV data quality
1) In situ particle image characteristics
In contrast to laboratory PIV in which particle characteristics may be controlled, our open-ocean application of PIV relied on the existence of natural particles, including phytoplankton, zooplankton, and other suspended matter (Fig. 5). The suitability of these particles for PIV was evaluated with regard to seeding density and particle size and shape (as imaged).
Particles were identified as connected pixels (either face to face or at corners) for which the pixel intensity was at least two standard deviations above the median of the background intensity distribution (i.e., the Gaussian portion of the image intensity distribution that characterizes the image background). Figure 6 shows the particle seeding density for a sample profile of scattered light images (520–540 nm) for which the integration time was alternated between 10 and 20 ms. The average seeding density is typically between 5 and 10 particles per 32 pixel × 32 pixel interrogation window for a 10-ms exposure. An integration time of 20 ms yields approximately 10–15 particles per interrogation window. Keane and Adrian (1992) found that five particles per subwindow are sufficient to yield a valid vector detection probability greater than 95%. Thus, for both exposure times, the average seeding density is sufficient to yield high-quality PIV data. Ambient particle concentrations are also depth variable, as seen in Fig. 6. The seeding density detected in the fluorescent images was insufficient for PIV, and thus PIV processing was performed exclusively on scattered light images.
In estimating the imaged size of particles, the shoulders of the particles (low-intensity perimeter) were also included to a level of one standard deviation above the background intensity median. Figures 7a,b show histograms of equivalent particle diameter (Deq = , where Ap is the particle area) for the profiles in Fig. 6. The percentage of particles with Deq between one and four pixels is 70% and 74% for 10 and 20 ms exposure times, respectively. Results from Raffel et al. (1998) and Cowen and Monismith (1997) suggest that the optimal particle diameter for cross-correlation PIV is in the range of 2 to 3 pixels. Thus, a large percentage of the imaged particles are near (<1 pixel) or within the optimal size class. The histograms also show that the abundance of particles declines with increasing particle size. A comparison of Figs. 7a,b shows that the longer integration time allows for improved detection of the smallest particles (1 pixel ≤ Deq < 2 pixels) and thereby alters the overall particle size distribution.
Because of the combination of flows relative to the profiler (typically in the range of 5–8 cm s−1) and finite image exposure time (10–20 ms for the present study), a finite level of particle streaking is inevitable. For the magnification of the system, these conditions result in particle streaking distances ranging between approximately 2 and 11 pixels. The effect of particle streaking on the PIV results is evaluated in the following section.
2) Simulated PIV for evaluation of in situ particle image characteristics
To evaluate the effect of particle streaking as well as in situ particle size and shape on PIV, realistic simulated PIV data were generated using the synthetic image generator described in Shavit et al. (2007). Given that the particle displacement error resulting from mapping from 2D to 3D is small compared to the error associated with cross-correlation peak detection (see section 2d), we focus on the latter in this analysis. Simulated particles were randomly distributed within 12-bit images (intensities of 0–4095) of 1040 pixels × 1040 pixels and with an average seeding density of 15 particles per 32 pixel × 32 pixel subwindow. The intensity distribution of the simulated images was tuned to match that of the in situ images: the peak particle intensity was q = 150, the mean background noise was β = 78, and the random (Gaussian) background noise had a standard deviation of 4.4. Intensity capping was also applied to the simulated images (Shavit et al. 2007). Between consecutive images, each particle was displaced using a unidirectional and variable component, (ΔX1, ΔX2) = (ΔX1,uni, ΔX2,uni) + (ΔX1,var, ΔX2,var). In all cases, the unidirectional displacements were purely horizontal and of a relatively small magnitude (ΔX1,uni = 0 pixels and ΔX2,uni = 5 pixels) to prevent any potential errors from the subwindow shifting procedure. Two parameterizations for the variable component were considered, referred to here as “shear flow” and “turbulent flow.” For shear flow, ΔX1,var = 0 and ΔX2,var = Sf sin(2πX1/32), where Sf is the strength of shear and X1 is the vertical position of the particle. This yields vertical shear in the horizontal displacement that is periodic over each subwindow. Note that the shear flow trials are particularly relevant to stereoscopic PIV because variable magnification over the field of view results in apparent shear. For two-dimensional turbulent flow, ΔX1,var and ΔX2,var were randomized using a normal distribution with 0 mean and a width defined by a turbulence parameter Tu = rms(ΔX1,var)/ΔX2,uni = rms(ΔX2,var)/ΔX2,uni.
Figure 8 shows the PIV displacement errors for various particle characteristics (variable particle streak, diameter, and shape) and flow conditions (variable Sf and Tu). The magnitude of the variable component of particle displacements within a subwindow ranges from 0 pixels at Sf = 0 and Tu = 0 to as much as 1 pixel for Sf = 1 and 0.75 pixels (both in X1 and X2), on average, for Tu = 0.15. For our imaging system and flow conditions, the variable component of particle displacements should typically be on the order of, or less than, 10−1 pixel, and thus the results at the low end of the Sf and Tu range are most relevant.
At the low end of the range of Sf and Tu, the distributions of errors are nearly indistinguishable for variable particle streaking, diameter, and shape. Like the median displacement errors, the 90th percentile errors show little variation with particle streak, size, and shape, indicating that even the worst errors are not substantially altered by the in situ particle image characteristics.
At the high end of Sf and Tu, more substantial changes in the error distributions are observed (Fig. 8). For the turbulent flow trials, particle streaking results in an increase in the error compared to the no-streak case. Interestingly, for shear flow, the streaking improves the particle pattern overlap for the variably displaced particles and thereby increases the magnitude of the true correlation peak. This improves the detection of the appropriate correlation peak and tends to reduce the displacement error even though the correlation peak is broader. The in situ particle sizes (i.e., diameters) result in errors that are comparable to uniform particle diameters of 3 pixels and slightly smaller than uniform diameters of 2 pixels for both shear and turbulent flow. Particle shape has little effect on the PIV results, except for a slight reduction in the errors associated with elongated particles, particularly when the elongation is oriented with the flow (as for streaking). The reduction in PIV error associated with an increase in imaged particle size (either physical or resulting from streaking) is consistent with the finding that optimal particle size may increase in the presence of velocity gradients due to improved particle pattern correlation (Cowen and Monismith 1997). Overall, the simulation results suggest that for realistic flows the in situ image characteristics (including finite streaking and variably sized and shaped particles) do not degrade the PIV results compared to laboratory particle image characteristics.
3) Out-of-plane motion
The out-of-plane loss of particle pairs is known to be a source of PIV data quality degradation (e.g., Raffel et al. 1998). When the cross-plane velocity is sufficiently large, some of the particles appearing in one image of a pair may be absent from the other, resulting in errors in the cross-correlation displacement calculation. In laboratory applications of PIV, the laser light sheet may be aligned with the flow to minimize cross-plane motion. In the ocean application of PIV, the flow field is unknown a priori and alignment with the dominant flow direction (if one exists) can be challenging.
Figure 9a shows the variation in the data quality metric P (percentage of valid velocity vectors) for five profiles with different interframe times Δt versus rms cross-plane particle velocity U2,rms computed for each individual vector field. For reference, high data quality is associated with values of P exceeding around 87% and poor quality is observed for P < 87%. For each profile, there is a decline in data quality for large cross-plane velocities, suggesting that the out-of-plane loss of particle pairs is primarily responsible for the degradation in PIV data quality. For a given cross-plane velocity, a reduction in Δt will result in a smaller cross-plane displacement and hence fewer particle pairs lost and higher data quality. Though, the reduction of Δt comes at the cost of smaller turbulent particle displacements.
When the percentage of valid vectors P is plotted versus the rms out-of-plane displacement Δx2 rms from each vector field, the curves collapse to a single curve (Fig. 9b). Thus, the cross-plane displacement is a fundamental variable regulating the quality of the in situ PIV data. The decline in data quality to the unacceptable level of P = 87% occurs for Δx2 rms slightly above 1 mm (approximately ⅙ of the thickness of the laser light sheet).
The correlation (r 2) values of P and Δx2 rms indicate that the out-of-plane loss of particle pairs explains the great majority of the variability in PIV data quality (about 79%–90% of the observed variability in P). The remaining variability in P and the offset from 100% at low cross-plane velocities is likely due to occurrences of suboptimal particle seeding density, particle size, and particle intensity. The relatively small effect of these parameters on the overall data quality is expected, given the suitability of the particle field properties (Figs. 6 –8).
To mitigate the out-of-plane loss of particle pairs, vertical vane-like fins (approximately 0.5 m × 0.5 m) were placed on the profiler to help align the light sheet with the dominant flow direction and thereby minimize the cross-plane component of velocity. The degree to which the profiler aligned with the dominant flow direction was then assessed by comparing the relative strength of the in-plane and cross-plane rms horizontal velocities U1,rms and U2,rms calculated from each individual vector field.
During typical down–up (DU) deployments, there was no consistent dominance in the two horizontal components of velocity, suggesting that the profiler had no preferential orientation. Thus, the fins appear to have been fairly ineffective in aligning the optics system with the dominant flow direction. This is not unexpected, given depth variations in the velocity field and the slow rotational response of the profiler. Furthermore, the near-surface (≲20 m) was often characterized by large wave-induced cross-plane velocities, resulting in low data quality (P < 87%). Surface waves pose a challenge, as wave-induced velocities may be oriented in a different direction than the mean flow.
The fins were effective at depth when the profiler had ample time to adjust to the local flow field. For a deployment in which the profiler maintained a constant vertical position (25-m depth) for a period of roughly 45 min, the fins effectively aligned the light sheet with the dominant flow direction. For this case, the in-plane horizontal velocity U1,rms was typically more than 3 times larger than the cross-plane horizontal velocity U2,rms and the data quality was consistently high (P > 87%).
b. Ensemble-averaged velocity
We now evaluate the mean velocity structure observed for the following two sample periods: 1) a typical down–up profile in which the profiler descended at approximately 8 cm s−1 to a depth of 65 m; and 2) the above-mentioned constant-depth (CD) deployment in which the profiler performed a slow descent (descent rate ≈ 2 cm s−1) and subsequent hover (descent rate ≈ 0 cm s−1) at approximately 25 m. These sample periods will hereinafter be referred to as drop DU and drop CD. PIV data were collected during the downcasts, beginning at 10-m depth.
Ensemble-averaged velocity fields were constructed to observe the spatial structure of the velocity field relative to the profiler
where Uir×s are the instantaneous velocities in a r vector × s vector block centered about (x1, x3) and N is the number of velocity fields. The in-plane horizontal, cross-plane horizontal, and vertical velocities are given by U1, U2, and U3, respectively (Fig. 3). Given considerations of nonstationarity, a 5-min ensemble period was selected; this averages over high-frequency turbulent fluctuations, but naturally includes lower-frequency motions (e.g., resulting from internal waves). Because only the scattered light image pairs (acquired at 1 Hz) were used in the analysis, the 5-min mean velocities represent 300 block-averaged velocity vectors. For drop DU, the uncertainty in the 5-min mean velocities typically ranged from 0.02 to 0.1 cm s−1 (25th–75th percentile of all standard error estimates) for high-quality data acquired below the surface wave–affected zone (below ∼20-m depth). For drop CD, the uncertainty in the 5-min mean velocities typically ranged from 0.07 to 0.28 cm s−1 for high-quality wave-affected estimates above ∼20-m depth and 0.03 to 0.21 cm s−1 below ∼20-m depth.
Figure 10 shows 5-min ensemble-averaged velocity fields for drop DU with r and s set to 3. The vertical velocity relative to the profiler is oriented upward (〈U3〉 > 0), because the profiler is translating vertically downward. Similarly, the in-plane horizontal velocity is positive (〈U1〉 > 0) resulting from the translation of the profiler relative to the ambient horizontal flow field. The vertical velocity is convergent, as 〈U3〉 decreases with x3. The in-plane horizontal velocity is divergent, as 〈U1〉 increases with x1. The vertical convergence and horizontal divergence is indicative of a flow blockage. Thus, in its current configuration, the profiler influences the measured flow field. Presumably the 3D structure of the blockage would result in a flow divergence in the cross plane as well, though we cannot evaluate this given that 〈U2〉 is only known at x2 = 0.
Figure 11 shows 5-min ensemble-averaged velocity fields for drop CD. In this case, the approach velocity is primarily horizontal, because the profiler is not translating vertically. A jet-like structure is present in 〈U1〉, the dominant horizontal velocity. The jet is inclined, suggesting that it emanates from above the field of view. The cross-plane velocity 〈U2〉 is weaker than 〈U1〉 and gradually decays toward the profiler. The vertical velocity 〈U3〉 is oriented downward and is strongest at the top left-hand side of the field of view. A portion of the horizontal approach flow appears to be diverted vertically downward, around the base of the profiler.
The spatial structure of the mean velocity fields reveals that the profiler influences the observed flow and thus unintrusive oceanic velocity measurements are not yet possible with the profiler as configured. A reconfiguration of the instrument components is necessary to avoid this contamination. However, these observations do illustrate the capability of the PIV system to resolve spatial patterns in the velocity field that are consistent.
The accuracy of the PIV velocity observations are further evaluated by direct comparison of 1-min-averaged velocity from PIV (first averaged over each vector map containing 60 vectors × 75 vectors), the ADV (〈Ui〉ADV), and the first bin of the ADCP (〈Ui〉ADCP) for drop DU and drop CD (Fig. 12). For a sample rate of 1 Hz, the 1-min PIV velocities represent 60 field-averaged velocity vectors. For drop DU, the uncertainty in the 1-min mean velocities was typically in the range of 0.01–0.14 cm s−1 (25th–75th percentile of all standard error estimates) for high-quality data below the wave-affected domain (below ∼20-m depth). For drop CD, the uncertainty in the mean velocities typically ranged from 0.01 to 0.17 cm s−1 below the wave-affected domain (below ∼20-m depth) and from 0.04 to 0.34 cm s−1, where data quality was sufficiently high above 20-m depth. We note that the 1-min velocity estimates average over the high-frequency turbulent motions and permit a comparison of the velocities in the adjacent sampling volumes of the PIV system, ADV, and ADCP. The 1-min mean velocities do not average over lower-frequency motions, for example, resulting from internal waves. Also, recall that the ADV measurement volume and the first bin of the ADCP were centered 0.16 m above and 1.5 m below the PIV field of view, respectively. These small offsets are important in the comparison of the mean velocities from PIV, the ADV, and the ADCP, given the spatial variations in the velocity field illustrated in Figs. 10 and 11.
For drop DU, the vertical velocity from PIV 〈U3〉 is comparable to 〈U3〉ADV, both of which are reduced from 〈U3〉ADCP (Fig. 12). This is consistent with the vertical velocity convergence shown in Fig. 10b. The cross-plane PIV velocity 〈U2〉 tracks 〈U2〉ADV and 〈U2〉ADCP when the magnitude is small. For large cross-plane velocities, the PIV system is unable to accurately measure the cross-plane velocities due to out-of-plane particle pair loss [see section 3a(3) above], and thus there are discrepancies with both the ADV and ADCP. The in-plane horizontal velocity 〈U1〉 appears to follow the same trends as 〈U1〉ADV and 〈U1〉ADCP, though with some deviation during periods of significant out-of-plane motion and also resulting from the flow divergence illustrated in Fig. 10a.
For drop CD, the vertical velocity 〈U3〉 tracks well with 〈U3〉ADV and 〈U3〉ADCP during the descent (Fig. 12). Note that the vertical convergence is less pronounced than in drop DU, because the fall speed is substantially reduced. This is followed by a period of data loss (Fig. 12c) resulting from out-of-plane motion (Fig. 9). At the end of the record, the magnitude of 〈U3〉 is greater than both 〈U3〉ADV and 〈U3〉ADCP, which is consistent with the vertical velocity structure shown in Fig. 11c. Similar to drop DU, the cross-plane velocity 〈U2〉 tracks well with 〈U2〉ADV and 〈U2〉ADCP at small cross-plane velocities, but differs for large cross-plane velocities. The in-plane horizontal component 〈U1〉 matches both 〈U1〉ADV and 〈U1〉ADCP during the descent. At constant depth, 〈U1〉 exceeds the estimates from the ADV and ADCP as a result of the jet-like structure that developed (Fig. 11a).
c. Turbulence measurements
Velocity fluctuations were computed by subtracting the ensemble-averaged velocity from the instantaneous velocity
where the 〈·〉 represent an ensemble average of the spatially varying flow around the profiler. Note that the measured instantaneous velocities are relative to the profiler (i.e., we do not measure the absolute fluid velocity). However, the instantaneous, small-scale gradients in the velocity field are maintained, as long as rotations, pitch, and yaw are small.
Figure 13 shows a sequence of fluctuating velocity fields relative to a 5-min ensemble average from drop CD. While these structures include contamination effects from the instrument frame, they show the capability of the PIV system to resolve the finescale velocity structure. Counterclockwise turbulent eddies are advected from left to right across the field of view.
The length scales of the turbulent flow structures resolved by the system are on the order of the dimensions of the field of view, 25 cm, to the resolution of the velocity measurements, 3 mm. Given that this resolution is sufficient to resolve velocity gradients on scales comparable to the Kolmorogov scale ηk = (ν3/ɛ)1/4 (see section 3d for details), it is possible to directly estimate the dissipation rate of turbulent kinetic energy
where ν is the viscosity and i and m are indices for the coordinate directions.
Using the three-component velocity fields from stereoscopic PIV, it is possible to measure 7 of the 12 independent gradient terms in Eq. (3). Using a central difference scheme, we can directly compute
A square stencil is constructed about each of the vertices of the PIV grid to estimate the gradient terms and avoid resolution biases in the various components. One additional term may be computed with continuity,
The remaining four terms can be estimated by assuming that the out-of-plane cross-stream gradients are similar in magnitude to the in-plane cross-stream gradients (Doron et al. 2001),
As noted by Doron et al. (2001), these relations become identities for isotropic turbulence.
Upon substitution of these estimates into Eq. (3), we obtain a direct estimate for ɛ that utilizes all of the measured velocity gradients,
Equation (7) yields a superior estimate to those based on more severe assumptions about the turbulent flow, including isotropy.
To evaluate the accuracy of the dissipation estimates from PIV, we computed independent estimates from the ADV. The dissipation rate was estimated using a fit to the inertial subrange of the power spectral density of ADV-measured vertical velocity fluctuations,
where C is (1) for spectra normal to (aligned with) the advecting flow, α = 1.5, k is the radial wavenumber, and E33 is the power spectral density of the velocity fluctuations (Pope 2000). Figure 14 shows E33 for a sample period from drop CD. A clear inertial subrange is present (which is well described by the k−5/3 scaling) that spans wavenumbers of k1 = 101 m−1 to k1 = 102 m−1, corresponding to turbulent structures from decimeter to centimeter scale.
Spatial spectra may be transformed to temporal spectra using the convective velocity (Tennekes and Lumley 1972). The dissipation rate may then be expressed as a function of measured temporal spectra
where the temporal frequencies and wavenumbers are related by ω = kU and U is the convective velocity. Note that ɛADV is estimated using the vertical velocity fluctuations and therefore inherently involves an assumption of isotropy. Fits to the spectra were accomplished by finding the region of the spectra that best agreed with a slope in a least squares sense.
Figure 15 shows a comparison of ɛ3DPIV and ɛADV for drop CD. Each PIV estimate is based on a horizontal “strip” of PIV data (3 vectors × 75 vectors). While surface wave effects are negligible during this period of sampling (at ∼25-m depth), the 1-min mean dissipation estimates include internal wave contributions. Dissipation varies with distance from the profiler due to frame-induced contamination. Nonetheless, the spatial and temporal structure of dissipation estimated from PIV is consistent with the observations from the ADV. The magnitude of dissipation at the top of the field of view is comparable to the ADV estimates. Small differences in these records are likely in part due to the spatial variability in the turbulence field around the profiler. Dissipation estimates are not shown for drop DU, because the turbulent dissipation signal was below the uncertainty of the PIV system [O(10−7 m2 s3)] and was insufficient to yield an inertial subrange in the ADV velocity spectra.
Figure 16 shows sample plots of energy spectral density of velocity fluctuations E(k) and dissipation spectra D(k) = 2νk2E(k) for drop CD. These were computed for a single horizontal line of velocity vectors along the x1 direction. For a strongly turbulent section of data (Figs. 16a,c), the spectra show a well-defined inertial subrange and follow k−5/3 and k1/3 canonical scalings for E(k) and D(k), respectively. The magnitude of E11, 3/4E22, and 3/4E33 are comparable, indicating near-isotropic conditions. As expected, the wavenumber extent of the inertial subrange is comparable to that observed by the ADV in Fig. 14 (101 m−1 < k1 < 102 m−1). For a weakly turbulent section, there is no apparent inertial subrange and the dissipation range is well resolved, as is evident by the decay in D(k) at high wavenumbers (Figs. 16b,d). The difference in the magnitude of the power spectra suggests a degree of anisotropy.
Figure 16 illustrates that the wavenumber domain of the dissipation range is variable. For strongly turbulent conditions, the inertial subrange extends through the highest wavenumbers resolved by the PIV system (Figs. 16a,c). In this case, the dissipation range is not resolved. For weakly turbulent conditions, the system resolves the entire dissipation range (Figs. 16b,d). To evaluate the capacity of the system to resolve the smallest scales of motion and accurately measure ɛ, we further examine the uncertainty in the dissipation estimates and resolution relative to the Kolmogorov scale.
d. Dissipation uncertainty
Given the importance of ɛ as a metric for turbulence intensity, we develop an estimate for its noise floor using propagation of error analysis. Assuming isotropy, the instantaneous dissipation rate may be expressed as
where Δx1t and Δx1b are the top and bottom particle (fluid) displacements that are vertically adjacent and separated by a distance Δx3 that is assumed to be O(ηk). Using the propagation of error and assuming that Δt and Δx3 are known precisely, the uncertainty in an instantaneous dissipation measurement is given by
where δ(ΔX) is the uncertainty in the particle displacement (in pixels) and R is the image resolution [in pixels per meter; δ(Δx) = δ(ΔX)/R]. Assuming that the errors in the estimates of log10(ɛ) are normally distributed, we may write the uncertainty for the mean dissipation rate as
where N is the number of samples contributing to the mean and the uncertainty is expressed in terms of the base 10 logarithm.
Figure 17a shows 1-min mean dissipation rate normalized by its uncertainty [Eq. (12)]. Typical values for our imaging system have been assumed for Eq. (11). We note that the displacement uncertainty is taken as 0.1 pixels. However, as illustrated in Fig. 8, the displacement uncertainty will naturally vary with flow conditions and particle image characteristics. Note also that the 1-min averaging period is chosen for consistency with Fig. 15 and represents a period over which the turbulence is assumed to be stationary. The results indicate that for the current system for which the resolution is 5 pixels mm−1, a lower bound on the 1-min mean dissipation that can be measured above the noise level is approximately 10−7 m2 s−3.
Given that pelagic dissipation rates are typically several orders of magnitude smaller than the estimated noise level, it is desirable to reduce the uncertainty. The uncertainty in dissipation is a strong function of the displacement uncertainty in PIV. The displacement uncertainty could be reduced by increasing the camera magnification or increasing the pixel density. An increase in magnification sacrifices field of view and also may result in suboptimal particle seeding density and size. Improvements in pixel density may offer improved uncertainty characteristics with fewer trade-offs. Considering an increase in pixel density such that the imaging resolution is 50 pixels mm−1 (about 25 MP for the current magnification), 1-min mean dissipation rates on the order of 10−9 m2 s−3 could be distinguished from the noise (Fig. 17a).
Figure 17b shows that for a range of dissipation rates that exceed the noise floor (from ∼10−7 to ∼10−5 m2 s−3), the velocity gradients can be resolved at spatial scales on the order of the Kolmogorov scale ηk, which is required for estimating dissipation rates (e.g., Cowen and Monismith 1997). Considering an improvement in the camera resolution to 50 pixels mm−1, the velocity gradients could be resolved below the Kolmogorov scale for dissipation rates below ∼10−4 m2 s−3.
Tanaka and Eaton (2007) conducted an in-depth PIV study of the effects of spatial resolution on dissipation error. Their results indicate that dissipation is underestimated when the PIV spacing is larger than ηk because the small-scale turbulent fluctuations are filtered. In contrast, dissipation is overestimated when the PIV spacing is smaller than ηk because of measurement noise. For an optimal PIV spacing between ηk/10 and ηk/2, the measurement error may be successfully corrected. These resolution constraints pose a fundamental challenge for PIV studies (not only in situ applications) given the variability of ηk in many turbulent flows.
An autonomous open-ocean particle imaging profiler is under development at the Scripps Institution of Oceanography to perform in situ stereoscopic PIV for visualization and quantification of small-scale turbulent ocean flow. To our knowledge, this is the first field adaptation of stereoscopic PIV and the first open-ocean application of PIV. While performing controlled descents through the upper 100 m of the water column, the system illuminates and images planktonic particles. Sequences of particle images are processed to yield the three-component velocity structure within a 2D region below the profiler.
a. Performance of the PIV system
The PIV system was successful in acquiring high-quality PIV images off the southern California coast. A 3-W diode-pumped solid-state laser was effective in illuminating a dense matrix of natural particles, sufficient to yield high-quality PIV data. The signal-to-noise ratio of the particle image intensity was suitable for images acquired at night, when ambient light contamination was minimal. The particle size distribution was found to be variable, with a large fraction of particles near (<1 pixel) or within the ideal particle size class for PIV (about a 2–3-pixel diameter). Analysis of realistic simulated PIV images suggests that the in situ image characteristics—including particle streaking, variable particle size, and variable shape—have a minimal effect on the uncertainty of the particle displacement estimates.
Ensemble-averaged velocity fields showed the capacity of the system to resolve the spatially variable flow field. These observations were also consistent with independent velocity observations from an ADV and ADCP mounted on the profiler frame, except for large, irresolvable cross-plane velocities. Direct estimates of dissipation utilizing seven measured gradient terms were consistent with ADV estimates derived from fits to the inertial subrange. Velocity power spectral density and dissipation spectra from PIV showed canonical structure in the inertial subrange for strongly turbulent conditions. For weakly turbulent conditions, the entire dissipation range was resolved and anisotropies were evident in the magnitude of the spectra.
b. Proposed improvements
The current experiments highlight several salient challenges in adapting PIV for the study of open-ocean turbulence: out-of-plane particle motion, instrument intrusiveness, and uncertainty limitations in turbulence measurements. We now review these challenges in the context of improvements that may allow for the successful measurement of oceanic turbulence.
1) Improved alignment with the dominant flow direction
The variability of the in situ PIV data quality was primarily due to out-of-plane particle motion. High data quality was obtained for cross-plane displacements less than about ⅙ of the laser light sheet thickness. Vane-like fins on the profiler were effective in aligning the light sheet with the dominant flow direction only when the profiler sampled at constant depth, as there was sufficient time for the profiler to respond to the local flow field. Thus, stepwise profiles may be an effective method of maintaining orientation with the dominant flow direction. Stepwise profiles may also allow for improved measurement of mean velocity gradients. The alignment response time could also be improved by reducing the size of the profiler.
Reduction of the interframe and image integration times would limit the out-of-plane loss of particle pairs. However, these adjustments would reduce particle displacements and particle intensity, respectively. Thus parallel developments in camera pixel density and onboard laser power would be beneficial. The out-of-plane loss of particle pairs may be further mitigated by increasing the light sheet thickness, though this must be balanced with the desire for intense illumination and high spatial resolution in the cross-plane direction.
2) Unintrusive instrument configurations
The current instrument configuration modified the flow field within the field of view and thus unintrusive measurements of oceanic flows are not yet possible. The future development of unintrusive imaging configurations will benefit from the miniaturization of instrument components. A porous instrument housing that contains no components directly upstream or downstream of the field of view may be most effective. The design must integrate the alignment capability of the imaging system, because the orientation of the approach flow will dictate the appropriate positioning of instrument components.
3) Higher pixel density
It was shown that the current PIV system is limited to estimating 1-min mean dissipation rates no smaller than O(10−7 m2 s−3). However, this may be reduced with the utilization of higher-resolution cameras. Higher pixel density will reduce the displacement uncertainty in PIV. As a result, velocity gradients and derived quantities such as the dissipation rate may be accurately measured at lower turbulence intensities.
The experiments described herein demonstrate for the first time that PIV is adaptable to the open ocean. We propose a number of developments to the current techniques that will enable us to resolve the instantaneous spatial structure of open-ocean, three-component turbulent velocity fields. Based on such measurements, the anisotropy of the turbulence field can be evaluated to test the validity of the isotropy assumption common in oceanic turbulence profiling. Finally, the novel instrumentation described in this work has the capacity to resolve biotic fields simultaneously and at the same scale as the velocity fields. This combination promises to deliver exciting new insights on the interactions of small-scale turbulence and planktonic organisms.
This project was generously supported by a NSF grant (OCE 0220213). We are grateful to the crew of R/V New Horizon. Three anonymous reviewers provided valuable comments that greatly improved the quality of the paper. JVS acknowledges support from a NDSEG fellowship and EPA STAR fellowship.
# Current affiliation: Civil and Environmental Engineering, Purdue University, West Lafayette, Indiana
@ Current affiliation: Civil and Environmental Engineering, University of Washington, Seattle, Washington
Corresponding author address: Jonah Steinbuck, 473 Via Ortega, Yang and Yamazaki Environment and Energy Building, Civil and Environmental Engineering, Stanford University, Stanford, CA 94305-4020. Email: firstname.lastname@example.org