A fiberoptic sensor has been constructed to measure oceanic density fluctuations via their refractive index signature. The resolution (Δz = 1 mm, Δt = 0.2 ms) and precision (Δn < 10−8, Δρ = 3.4 × 10−5 kg m−3) of the device are far better than other methods and are sufficient to resolve the entire turbulent spectrum. Spectra show the salinity Batchelor rolloff at levels undetectable via conductivity measurements. However, the low-wavenumber portion of the spectrum occupied by the turbulent inertial subrange (≈1 m–1 cm scales) is marred by noise resulting from fiber motion in response to turbulent velocity fluctuations. The technique is described, and the first ocean measurements are reported.
Ocean mixing affects many phenomena, including nutrient distributions, biological productivity, pollutant transport, and even the global meridional transport of heat and salt. Though these phenomena occupy a wide range of scales, diffusive smoothing of a scalar, C, is inefficient except at the smallest scales, where strain can increase gradients sufficiently. The strain is that of the turbulent velocity signatures, which are smoothed out by viscosity at scales smaller than the Kolmogorov scale,
where ν is the kinematic viscosity, and ɛ is the kinetic energy dissipation rate. The Batchelor scale, where diffusive and straining effects balance, is
where KC is the molecular diffusivity for scalar C (≈10−7 s−1 m2 for temperature; ≈10−9 m2 s−1 for salinity), and λTB is O(1 cm) for temperature. Owing to salinity’s 100-times-smaller diffusivity, its Batchelor scale, λSB, is 10 times smaller (1 mm). These small scales, together with the requirement of simultaneously measuring temperature and electrical conductivity, have precluded measurement of the full salinity spectrum, except in the strongest turbulent patches (Nash and Moum 1999, 2002).
As a result, many aspects of small-scale mixing remain poorly understood and/or completely unmeasured. Two examples are salt fingering and differential diffusion. Since heat and salt have such greatly different diffusivities, these processes’ potential to transport heat and salt at different rates could have significant consequences for both local processes as well as the global circulation (Gargett and Holloway 1992).
Heat and salt transports can be computed via the Osborn–Cox method, which begins with the evolution equation for scalar variance. In steady, homogeneous, isotropic turbulence with no lateral gradients, this equation reduces to a balance between production of scalar variance by turbulence and destruction by diffusive smoothing:
is the rate of diffusive dissipation of scalar variance. The diffusivity can then be defined via
and estimated by substitution into (3) as
Measurement of χC requires resolution of the spectrum past the Batchelor wavenumber, kCB. Even for temperature, θ, the slow response of thermistor probes has limited the method’s use. Gregg and Meagher (1980) demonstrated that the time response of FP07 glass-bead thermistors is well modeled by a double-pole filter with cutoff near 29 Hz. Nash and Moum (1999) verified this response but found considerable variability between individual thermistors. To resolve all variance at, say, 100 cpm requires quite slow fall speed (10 cm s−1). For faster speeds, high-wavenumber variance is attenuated. From (2), the wavenumber content of the turbulence varies with dissipation rate, leading to a complicated situation in which the fraction of the variance resolved is variable.
For salinity, the situation is much worse given the smaller scales and the additional difficulty of matching the response of two sensors. Consequently, χS is usually estimated via
which is valid provided that turbulence produces the temperature and salinity fluctuations (Gregg 1987). The presence of double-diffusive effects or differential diffusion can invalidate this expression (Nash and Moum 2002). For progress, a sensor with low enough noise and fine enough temporal and spatial resolution is required. Nash and Moum succeeded in resolving the salinity spectrum for very active patches, but noise still precludes measurement of χS for the vast majority of ocean turbulence. No obvious means of reducing the noise inherent in conductivity-based measurements exists.
Another approach is to measure density via the refractive index. This, in principle, has several advantages over electrical conductivity measurements. 1) Refractive index is related directly to fluid density from first principles via the Lorentz–Lorentz equation. By contrast, its relation to conductivity is empirically determined assuming a fixed ratio of salts. 2) As a consequence, variations in salt ratio can lead to errors when using conductivity (Seaver 1987). 3) Conductivity’s contours in θ–S space (Fig. 1, light gray) are almost perpendicular to those of density (dark gray), requiring very precise measurements. The refractive index (black) is much more closely related to density, with a smaller temperature dependence.
Many attempts to measure seawater density via refractive index have been made, which are reviewed by Seaver (1997). Given the extremely small range of oceanic refractive index (Δn = 0.01), the precision of previous efforts (Δn ≈ 10−6) has limited their usefulness. The present sensor (Fig. 2), by contrast, is over 100 times quieter (Δn < 10−8), enabling resolution of density changes of Δρ = 9 × 10−6 kg m−3, or 70 times finer than an SBE-911 conductivity–temperature–depth (CTD) profiler (Sea-Bird Electronics, Inc.).
The precision requirements for both conductivity and index measurements are indicated in Fig. 3. Batchelor turbulent wavenumber spectra of refractive index gradient (which, as seen from Fig. 1, would be nearly identical if expressed as density rather than refractive index) are plotted for several turbulence intensity levels ranging from strong (top) to weak, typical of the ocean thermocline (bottom). Each displays a temperature peak near several centimeters’ scale and a salinity peak 10 [=(KT/KS)] times further out. An interesting aspect is that since kB ≈ ε1/4 [Eq. (2)], stronger signals occur at smaller scales and thus can be harder to observe. Previous refractive index measurements (upper dashed line) are far too noisy for all turbulent levels. The best conductivity-based measurements are capable of resolving the upper two curves. The present device is one–two orders of magnitude quieter and could, in principle, be improved still further (lower line), enabling resolution of many more patches.
However, as will be discussed, the sensor has three major problems in its present form. 1) Poor stability and drift characteristics preclude determination of dc refractive index. 2) As with thermistors, the sensitivity is variable from probe to probe and in time, requiring its determination for each deployment from other measurements. 3) The sensor responds to transverse turbulent velocity fluctuations in a 1–100-cpm wavenumber band (making it an excellent optical shear probe). The first two problems preclude the present sensor’s use as an optical CTD. However, they are easily overcome for turbulence measurements with onboard CTD data. The third issue is more problematic. The velocity response decays for wavenumbers > kK, enabling the high-wavenumber salinity Batchelor spectrum to be resolved. However, the velocity response precludes resolution of the Batchelor peak for temperature, which lies in the same wavenumber band. While there may be uses for an optical shear probe, the velocity response is extremely undesirable for a refractive index sensor. We have been able to reduce the response but have not eliminated it.
We present the techniques and design of the ocean refractometer in the next section. We then show initial results from drops in Puget Sound, Washington, and conclude with a discussion.
2. The evanescent-wave method
The evanescent-wave refractometer (Gerdt and Herr 1996), originally developed for use in immunoassay, is the heart of the measurement technique. The method avoids the need to count interferometric fringes, which limits the resolution of other types of refractometers.
When two single-mode optical fibers are drawn together and fused (Fig. 4), light-emitting diode (LED) or laser light launched into the first couples into the second. If this “coupler” is then fixed in a stable housing, then the ratio of power coupling into the second fiber,
depends only on the wavelength of the light, and the refractive index, n, averaged over a one-optical-wavelength-thick cylinder (roughly λsource = 1.5 μm) surrounding the pair of fibers. Measurement of the ratio removes the effects of fluctuating source intensity, and the simultaneous use of a potted reference coupler (Adkins et al. 1998) allows the effects of thermally induced wavelength fluctuations to be removed as well.
A single-mode optical fiber consists of a quartz core of diameter about 9 μm and refractive index near 1.46, concentrically contained within a thicker and lower-index material called the cladding. The diameter of the core is such that light propagation in each individual fiber is restricted to the lowest-order spatial mode. The cladding’s lower index ensures that light is guided and nearly all remains inside the core. Its thickness is sufficient that variations in the surrounding environment do not affect the propagating mode in the core. Typically, a plastic buffer beyond the cladding protects the fiber from external damage.
If the plastic buffers are removed and two fibers are placed next to each other in a furnace, they fuse. If they are under gentle and constant tension, they elongate and the separation between their cores will reduce. As the process continues, light initially in one fiber will couple into the other and will eventually begin to transfer back again. The essential element for our sensor is that the core and cladding of the two fibers are now one, and the surrounding medium is the new “cladding,” whose refractive index will strongly modulate the coupling ratio. These couplers are described as lightly fused and are very similar to standard communications couplers.
Their behavior can be understood in terms of the solutions to Maxwell’s equations for this geometry, which has been considered by Lacroix et al. (1994) by modeling the fusion region as two offset cores (Fig. 4). The solutions to Maxwell’s equations can be expressed as the sum of one symmetric and one antisymmetric mode that extend into the surrounding medium in the fusion region. One extends slightly farther than the other, and thus they propagate with slightly different speeds. This leads to a phase difference, Θ, which is simply related to the coupling ratio via
For the geometry in Fig. 4 (Lacroix et al. 1994), the coupling ratio, α, is a function of 1) the refractive indices of the core ncore and cladding ncladding; 2) the fused length L; 3) the weighted average of the refractive index of the surrounding medium (seawater), nsw; and 4) the light wavelength λsource. Parameters 1 and 2 do not change; parameter 3 is the desired signal.
Light wavelength (parameter 4) does change slowly in response to thermal fluctuations in the laser cavity. This contribution can be removed by use of a potted reference coupler illuminated with the same light. Since the reference is isolated from refractive index fluctuations, its ratio is only due to the changing wavelength, allowing a subtractive correction (Adkins et al. 1998). Since the effect exhibits a 1/f frequency dependence, it is insignificant at the high wavenumbers in which we are interested; consequently, we will not discuss this technique further here.
b. Design of the ocean module
1) Parameter dependences
Figure 5a shows the model dependence of the coupling ratio on elongation. Initially, all light is in fiber 1, yielding α = 0, but as the pull progresses, coupling increases and decreases as modal interference transfers light back and forth between the output fibers. Each point along this curve has a corresponding curve relating α to nsw. Two of these, at the points indicated at left by circles, are shown at right. The greater the elongation, the higher the sensitivity (∂α/∂n), evident as the slope at right. It is also apparent that the greatest sensitivity results when α ≈ 0.5. Since elongation increases the length of the sensor undesirably, we pulled our couplers to the third equipoint (right circle) for an acceptable balance between sensitivity and spatial resolution.
As can be seen from Fig. 5b, n(α) is multivalued. However, the small oceanic range of refractive index (gray) ensures restriction to one branch. While the Lacroix et al. (1994) model is useful for design and parameter estimates, we found that prediction of the elongation in air required to create a coupler with α ≈ 0.5 in seawater was not possible. Thus, a substantial amount of trial and error was involved.
Due in part to this, the dc value of α and the sensitivity (∂α/∂n) varied from sensor to sensor. Owing to slow accumulation of impurities in the sensing volume, each drifted somewhat with time as well. Given these issues, and our primary interest in turbulent fluctuations rather than mean quantities, our focus for this sensor was refractive index changes, Δnsw = ((∂α/∂n))Δα. For each drop, we computed the large-scale refractive index from collocated CTD measurements and the refractive index equation of state n(S, T, P, λ) (Quan and Fry 1995). The sensitivity (∂α/∂n) was then determined for each drop from a fit.
2) Sensor geometry
The geometry of the sensitive region in Fig. 4 is awkward for ocean deployment, and its size [O(1 cm)] is too large. Considerable effort was devoted to designing and building sensor mounts that overcame both of these shortcomings. Several tips were tested (Fig. 6). All consisted essentially of the coupler in Fig. 4 bent over a “sting” of about 6-mm diameter. Both metal and quartz were used to investigate effects of differential thermal expansion. Each sting was first coated with a layer of RTV-11 silicone rubber, which has sufficiently low refractive index to prevent light loss. The fibers were then mounted on the sting by pulling and fusing a fiber in normal fashion and then gluing the fiber to one side of the sting with epoxy. Then, a jig was used to carefully turn the sting 180°, laying the coupler over the sting tip. It was then affixed to the other side with epoxy.
The sensing region is now a U-shaped region of dimensions still too large for sensing millimeter-scale fluctuations. The solution to this was simple: RTV rubber was painted on the sides of the sensor, leaving only a 2-mm region at the end of the sting in contact with seawater. This procedure affects both the dc value of the coupling ratio and the sensitivity, since part of the sensing volume is now occupied by RTV. The resulting spatial and temporal resolutions are estimated at 1 mm and 0.2 ms, respectively (appendixes A and B).
Three sensor tips were employed: a quartz hemispherical tip, a two-pronged metal mount with the bare fiber extending between them, and a third that is identical except for a thin metal bridge against which the fiber lies (Fig. 6).
One assumption of the Lacroix et al. (1994) model is that the geometry remains fixed. While vibrations, etc., that introduce lateral translation of the entire sensor do not affect the measurements, any distortions in shape also affect the interference pattern of the modes and thus the coupling ratio. With hindsight, therefore, geometry 2 is an exceptionally bad choice, since the fiber is free to deflect in the region between the supports. It will be shown later that transverse velocity fluctuations due to the turbulence itself deflect the fiber spanning the gap, introducing noise. Design 3 was introduced with this in mind, which reduced but did not eliminate the fiber’s ability to move.
Three modules (Fig. 2, bottom) were constructed, shown schematically in Fig. 7. Optical power was provided by a LaserMate distributed-feedback laser in two modules and by an OptoSpeed Superluminescent Diode (SLED) in the other. These were run uncooled and driven at 50 and 180 mA, respectively, which dominated the overall system power consumption. Each delivered about 1.6 mW of power and was pigtailed to a 90/10 tap coupler. The 10% output directed light to a reference coupler to correct for low-frequency thermal wander of the source wavelength, as discussed above. The 90% output was connected to the input of the signal coupler. The outputs of the signal and reference couplers were connected to photodiodes. Standard communications-grade Corning SMF-28 fiber was used for all of the couplers.
Optical intensity at each output of the signal and reference couplers is measured with gallium phosphide (GaP) photodiodes. These produce a current proportional to the intensity, i1 = S I1, where S = 0.9 A/W is the photodiode sensitivity. These currents are then converted to voltages via transimpedance amplifiers:
where the photodiode shot noise and transimpedance noise at each channel are lumped together into nTA, and I(t) is the total. These two voltages for each coupler (signal and reference) are then passed through antialias filters and digitized at 50 Hz.
At 16-bit resolution, digitization noise precludes computing α in software from I1 and I2 with the required precision. Consequently, we use a high-accuracy, low-noise analog divider to compute the transformed ratio
The gains, G1 and G2, are placed on the original outputs to bring the numerator of (11) close to zero, allowing a large gain, Gdiff, to be used that maximizes the signal-to-noise ratio of this stage. The divider output is then passed through preemphasis and antialias filters and digitized at 800 Hz.
The noise in the various stages is indicated spectrally in Fig. 8, which shows the measured spectrum of coupling ratio from the device on the bench using a test coupler that was potted to isolate it from refractive index fluctuations. The measured noise (black) is consistent with that expected for the divider component (dashed), which shows an f −1 dependence at low wavenumber and is white for f > 30 Hz. The response of the antialias filter is seen at the highest frequency and can be corrected for (gray).
The precision gained by taking the ratio to divide out laser intensity fluctuations is evident by considering the spectrum of the ratio, αbad([I2(t)]/〈I1 + I2〉) (gray). The f −1 laser noise dominates, with a level far above the measured noise using the ratio.
The divider circuit’s noise limits the present system’s precision. Future designs (e.g., by digitizing at 24 bits and/or autoranging to allow higher gain, Gdiff) should be able to obtain noise floors as low as the photodiode shot noise (Fig. 8, lower dashed line).
Integration of the noise across the frequency range yields the precision of the system. Multiplying by the appropriate factors, precision is expressed in terms of index, temperature, salinity, and density in Table 1. It is clear that higher precision is possible using this method than with other techniques. However, drift, the motion-response problem, and other factors severely limit the accuracy of the system: the dc values of the index and, consequently, the salinity or density are unmeasured with the present device.
a. Experiment location
We focus on two drops (Table 2) to demonstrate the different aspects of the sensor’s performance. The first measurements were taken with a type-2 (open tip; Fig. 6) probe during summer. The second, with tip type 3, were taken during the following winter.
Measurements were conducted in Puget Sound, a wide, deep estuary to the west of Seattle, Washington. Salinity is lower than in the open ocean (20–25 psu), but stratification (Figs. 10a and 11a) and shear (Alford et al. 2005) are similar. Its proximity to Seattle and its wide range of kinetic energy and scalar dissipation rates made it an ideal test bed.
During drop 12162, the water column was double-diffusively stable. Consequently, (7) is expected to hold unless differential-mixing effects are strong. However, temperature gradients are weak below 40 m, making (7) error prone. In drop 13561, the upper 70 m are weakly susceptible to double-diffusive convection [the density ratio Rρ ≡ ((α∂θ/∂z)/(β∂S/∂z)) ≈ 0.06, where α = −((1/ρ))((∂ρ/∂θ)) and β = ((1/ρ))((∂ρ/∂S))], possibly invalidating (7).
b. Platform and turbulence measurements
For ocean deployment, the refractometer was mounted on the Modular Microstructure Profiler (MMP, Fig. 9), a loosely tethered instrument developed by Professor Michael Gregg at the Applied Physics Laboratory. It was ballasted to fall freely at ≈ 0.5 m s−1, transmitting data up a twisted-pair, Kevlar-coated cable. Onboard sensors include a Sea-Bird Electronics CTD, accelerometers, an FP07 thermistor, and two airfoil probes. As seen in Fig. 2, all sensors are within several centimeters of each other.
Temperature dissipation rate, χθ (Figs. 10d and 11d), is measured with the FP07 thermistor and (4). As discussed above, the response of the FP07 is limited by the diffusion time through the glass bead.
by integrating the shear spectra. It is commonly seen here and elsewhere that Φuz closely resembles the theoretical Panchev form (Gregg et al. 1993), which rolls off near kK.
c. Large-scale profiles and calibration
The coupling ratio, α, for each drop is shown in Figs. 10b and 11b. As indicated above, the refractive index profile is computed from the refractive index equation of state, nCTD = n(S, T, P) (Quan and Fry 1995) and MMP’s CTD (Fig. 10c, gray). A linear fit, nCTD = no + [(∂α−1/∂n)] α (thin black), yields the sensitivity for each drop (Table 2). The two sensors’ different dc coupling-ratio values and sensitivities of opposite sign indicate their different locations on the elongation characteristic (Fig. 5). For various sensors, |(∂α/∂n)| varied from 10 to over 100, bracketing the values expected from theory.
The fit is good at large scale, but fluctuations are evident in the refractive index quantity that are due to fiber motion, as discussed below. For this reason, as well as their drift and variable sensitivity, the present sensors are not an optical substitute for a CTD. However, the high-wavenumber portion of the spectrum is well resolved, as examined next.
Lacking other means of verifying the high-wavenumber response of the sensor, we will use the Batchelor salinity-gradient spectrum as a benchmark. In this section, we first briefly review this form; then we describe our calculation of the refractive-index-gradient spectra; then we describe the low- and high-wavenumber portions of the measured spectrum in turn. To demonstrate, we have selected three sample spectra from weak, medium, and strong turbulent regions.
1) Theoretical spectral forms
The one-dimensional form of the Batchelor spectrum for scalar gradient Cz may be written as
and q is the strain parameter. Estimates of q vary but lie in the range 2 to 5 (as used by Batchelor). Following Oakey (1982), we use 3.7 here. Temperature has been shown to obey (13) in the laboratory (Gibson and Schwarz 1963) and in the ocean (Dillon and Caldwell 1980). Nash and Moum (1999) found better agreement with the slightly different Kraichnan (1968) form.
Spectra of refractive index gradient have a temperature and a salinity contribution, whose relative strengths are set by χθ and χS (Fig. 3):
Using in situ values, (∂n/∂θ) ≈ − 1 × 10−4 °C−1 and (∂n/∂S) ≈ 1.8 × 10−4 psu−1 are determined from the equation of state (Quan and Fry 1995). [In the temperature range, contributions from the θ–S cross-spectrum can be significant (Washburn et al. 1996; Nash and Moum 1999). They are neglected here, since our interest is the high-wavenumber portion k > kθB.]
2) Observed refractive-index-gradient spectra
Wavenumber spectra of refractive index gradient are computed in half-overlapping, 1-m windows that match those used in computing ɛ. For each block, the raw time series of rraw are preconditioned with a Hanning window. (The preemphasis filter ensures zero mean and trend.) The frequency spectrum of refractive index is then
where Φraw(f) is the raw spectrum, and hAA and hpre are the transfer functions of the antialias and preemphasis filters. The wavenumber spectrum,
is then computed using Taylor’s hypothesis, ω = wk, using the mean fall speed, w, over that block. The spectrum of the gradient is then
Finally, the refractive-index-gradient wavenumber spectrum is
Typical spectra of refractive index gradient are shown for medium, weak, and strong turbulent patches in Figs. 12–14. The general features are similar; we will first focus on the medium patch (Fig. 12). At the highest wavenumbers, the spectrum is k+2, overlying the predicted analog divider noise (dashed line). Elsewhere, a two-peaked structure similar to that expected from Fig. 3 is seen. The high-wavenumber peak, as will be shown, is the Batchelor salinity-gradient spectrum. However, the low-wavenumber peak is much too high for the temperature Batchelor spectrum (lower gray), measured from an FP07 thermistor. Instead, it overlies the spectrum of shear measured from the airfoil probes, when appropriately scaled, indicating that the probe responds to the turbulent velocity signals in this range rather than their refractive index signal. We discuss these two wavenumber regimes in turn.
3) Low-wavenumber peak: Motion response
The low-wavenumber peak is in the correct wavenumber range to be the Batchelor peak for temperature. However, the FP07 temperature-gradient spectrum, expressed as refractive index gradient via ΦFP07ng(m) = (∂n/∂θ)2ΦFP07T(m), is much weaker, owing to the strong salinity stratification. Instead, the signal overlies that of the shear spectra, when they are multiplied by the factor (∂n/∂υ)2. Both the airfoil and refractometer output in this range lie atop the theoretical Panchev spectra. Since there is no expected relation between refractive index and shear, this correspondence indicates a motion problem. Specifically, the coupling ratio depends not only on refractive index but also on the turbulent velocity:
where Δυ is the turbulent velocity in the direction transverse to the fiber.
To investigate further, the spectral level from 2 to 20 cpm for each depth bin is plotted in Figs. 10e and 11e and compared to ɛ. Near-perfect agreement over the whole profile and several orders of magnitude variation is seen. Scatterplots (Fig. 15a) indicate a linear relationship. It is clear that (∂α/∂υ) is lower for drop 13561 (tip type 3, gray pluses), as discussed below.
The similarity of the spectra and the tight agreement of the scaled profiles indicate a linear response of the coupling ratio to velocity signals. The raw signals are coherent with the shear signals but are incoherent with MMP’s onboard accelerometers (not shown). This indicates that it is the turbulent velocities themselves, and not a vibration or vehicle-induced motion, that generates the signal. It appears that the transverse turbulent velocity flexes the fiber between its mounts, modifying the geometry of the sensor and leading to the spurious signals.
Several attempts were made to reduce this effect. First, a modified mount was constructed that included a metal strut beneath the fiber (Fig. 6, tip 3), which reduced the response by a factor of 2–5, as evident by the lower spectral level in drop 13561 for the same ɛ (Fig. 15a, gray pluses). Falling more slowly also reduced the response. Finally, we attempted to pull the fiber before the layer of RTV had cured completely. In each of these iterations, the final sensor was examined under a microscope. It was clear that over the 1-mm sensitive length, irregularities in the metal surface of the sting led to regions where the fiber was not touching it. We believe that the fiber was still free to move in these regions, modifying the coupler geometry and inducing a spurious signal.
We conclude that fiber motion in this wavenumber range appears to produce a coupling-ratio signal that is proportional to the transverse turbulent velocity. This response swamps the refractive index response in this range. Thus, we have, without wanting to, created a fiber optic shear probe. We hope in the future to communicate with appropriate people in materials science or nanotechnology who are better equipped to construct a mount that will immobilize the fiber.
4) High-wavenumber peak: Turbulent salinity signals
Above kK, the turbulent velocity signals are viscously damped. Without the motion response, the signals once again reflect refractive index, which for k > kTB is due to salinity alone. The Batchelor spectrum for salinity (Fig. 12, thin black) was computed from (13) using in situ values of ν(S, T, P) and ɛ = 1.5 × 10−8 W kg−1 measured from the shear probes. The measured spectrum closely overlies the Batchelor spectrum. [Agreement appears marginally worse with the Kraichnan (1968) spectrum, also shown.] The slope, shape, and position of the rolloff are all in good agreement, until at 500 cpm the divider noise becomes dominant. The noise from the best conductivity-based measurements (Nash and Moum 1999; Fig. 12, dashed line) is over 10 times higher and would have precluded measurement of this particular turbulent patch. As discussed, a redesign of the electronics would reduce the system noise to that of the “shot noise” from the photodiodes (Fig. 12, lower dashed line), gaining an additional factor of over 10 and allowing resolution of still weaker patches.
As indicated above, the easiest patches to observe are those with large χS and weak ɛ, focusing variance at relatively high levels and low wavenumbers. Unfortunately, χS and ɛ generally covary, making such patches rare. To illustrate the effect of turbulent strength on our ability to observe it, we also present sample spectra of weaker (ɛ = 1.9 × 10−9 W kg−1, Fig. 13) and stronger (ɛ = 1.5 × 10−8 W kg−1, Fig. 14) patches. Both of these are from drop 13561. As in the first example, the velocity response is seen in these patches at low wavenumber, and the Batchelor salinity spectrum at high. In the weaker patch, the peak occurs at lower wavenumber, enabling slightly better resolution than in the first example. In this single realization, the spectrum only loosely appears to follow the Batchelor form but exhibits a cutoff in the predicted location, well above the noise. In the stronger case (Fig. 14), the high-wavenumber spectrum lies atop the Batchelor form. However, since the peak is shifted to a higher wavenumber, the noise floor is reached first, and the cutoff is not resolved. In both cases, agreement appears somewhat better with the Batchelor than with the Kraichnan form, but uncertainty in these single-realization spectra precludes saying more.
e. Profiles of χS
We now wish to compute profiles of χS from these measurements. In principle, χS could be determined by integration of the salinity-gradient spectrum. For the patch in Figs. 12 and 13, 90% and 100% of the variance was resolved before the noise floor was reached, but in Fig. 14 only 52% was resolved. In general the relative signal and noise level is variable, and it is more prudent to use an alternate procedure involving fits to the theoretical spectral form, in the resolved range. As in Nash and Moum (1999), we fit the measured spectral level in a wavenumber range kmin < k < kmax to the salinity Batchelor spectrum, ΦBS(k). Using measured ν(S, T, P) and ɛ, the only remaining parameter determining ΦBS(k) is χS. The range of wavenumbers used (thick line) is that sandwiched between the velocity response at low wavenumber (determined by kmin > 2kK) and the divider noise at high wavenumber (determined by Φobs > 4Φdiv).
In this manner, χS is determined for each overlapping 1-m bin and overplotted with χ′S determined from (7) in Figs. 10f and 11f. The latter is computed by estimating the temperature and salinity gradients over each spectral window. As can be seen from Figs. 10a and 11a, the range of temperature differences can be quite small, leading to spuriously small denominators (∇T ≈ 0; e.g., 60, 78, 87, and 112 m in drop 12162) and errors in χ′S. We have partly countered this by removing regions where |∇T| < 10−4 °C m−1. Still, the resulting profiles of χ′S (gray) are noisier than those of χS (black). Computing gradients over larger windows avoids this difficulty but runs the risk of not being representative of the gradients against which the turbulence is working.
Even with these caveats regarding the use of (7), there is good general agreement over the whole profile in drop 12162. A large difference occurs at 70 m, in a well-mixed region where ∇S ≈ 0. In drop 13561, agreement is good deeper than 80 m but poor shallower. As noted above, decreasing temperature and salinity above 70 m enable double-diffusive convection, effecting differential heat and salt fluxes and potentially invalidating (7). Thus, poor agreement there is not surprising, even though the fluxes are probably small owing to the low density ratio (Rρ ≈ 0.06).
Scatterplots of χS versus χ′S (Fig. 15) are not particularly tight but are sufficient to indicate correspondence between the two quantities. Since the profiles of optically determined χS appear to be less noisy than those determined from (7), we interpret most of the scatter as difficulty in determining the latter in this weak-temperature-gradient region.
We emphasize that aside from (∂α/∂n), which is determined for the entire profile from the large-scale fit, there are no adjustable parameters. Thus, the agreement in both level and general features between χS and χ′S, together with the agreement of observed and theoretical spectra, prove that the sensor responds to turbulent salinity fluctuations for high wavenumbers.
We have developed a fiber optic refractive index sensor for ocean turbulence measurements. Its small size and relatively low power allow it to be deployed from a variety of platforms. Its spatial resolution (1 mm), temporal resolution (0.2 ms), and precision (Δn < 10−8, Δρ = 3.4 × 10−5 kg m−3) are superior to previous measurement techniques. We have demonstrated that it is capable of resolving χS and the Batchelor cutoff of salinity for turbulent signals over 10 times weaker than possible with other methods.
However, because the fiber is not sufficiently immobilized, the probe unfortunately senses turbulent velocity signals in addition to refractive index. While our unintended fiber optic shear probe may have uses, it mars our refractive index spectra in a 1–100-cpm wavenumber range. This effect, together with the need to infer calibration and sensitivity from collocated CTD measurements, limits its usefulness to a high-wavenumber salinity-spectrum sensor.
It may be, therefore, that conductivity sensors remain our most useful tool for small-scale salinity measurements. However, we suspect that while the immobilization of a 30-μm fiber has defeated our efforts, the right nano- or micromachining specialists would consider the problem trivial. If the problem can be overcome, the 100-times-finer precision over conductivity-based measurements will represent substantial progress. The ability to measure thermal microstructure at faster drop speeds, and density microstructure in nonconstant salt-ratio environments, presents additional possibilities.
This work was supported by NSF Grant OCE0002292. Many thanks are expressed to Mike Gregg for allowing his MMP to be modified to accommodate these sensors. We also wish to thank Jim Taylor for his dexterity and ingenuity in learning to pull couplers over the mounts, and to engineers Jack Miller, Earl Krause, and Vern Miller for aid in the design of the electronics, optics, and pressure case for the system.
The 30-μm diameter of the sensing volume is much smaller than the smallest salinity fluctuations. However, its finite length (Ls = 2 mm) will cause attenuation. To compute the transfer function of the anisotropic volume when it is dragged through isotropic turbulence in the viscous-convective subrange, we integrate the three-dimensional spectrum as a function of vector wavenumber, Φ(k), over the two horizontal directions. We follow the notation of Tennekes and Lumley (1972). Vector wavenumber Φ(k) is related to the three-dimensional spectrum, E(k), via the isotropy relations (Hinze 1975)
where k = (k1, k2, k3), and k = k21 + k22 + k23. All wavenumbers are in radians for the calculation. For the viscous-convective subrange,
A perfect sensor would measure the one-dimensional spectrum
Our sensor does not measure variance in the k2 direction greater than ks = 2πL−1s:
The resulting transfer function is then
(The real spectrum, which is attenuated approaching the Batchelor wavenumber, is redder. We therefore compute the response for the viscous-convective range as a pessimistic bound on the true response.)
As it falls the sensor develops laminar boundary layers of thickness δ ≈ (νR/w), where R is an appropriate radius, and w is the fall rate. For the glass-rod tip (R = 3 mm), δ = 77 μm; the radii for the metal tips (0.015 and 0.03 mm) are much finer, yielding δ = 8 and 5.4 μm, respectively. For all of these, the thickness of the optical sensing region (δo = 1.5 μm) is thinner. If diffusion alone transmits temperature and salinity signals across these boundary layers to the sensing volume, unacceptably slow response times τ = δ2 K−1 result for salinity (Table B1). However, the boundary layer flow replenishes the water in the boundary layer much more quickly, substantially speeding the response (Lueck et al. 1977).
We believe that the response of the glass-tip probe is rather slow, owing to the fiber’s placement in a stagnation region of flow past a large cylinder. However, the much smaller metal-tip probes are much faster. To properly model their response, flow past each of the tip geometries should be modeled using computational fluid dynamics techniques. To demonstrate that the order of magnitude of the response is acceptably fast, we adopted a simpler technique, wherein we assumed a known form for the boundary layer velocity field. We used the Blasius solution for flow past a plate (Fig. B1), imagining that the 30-μm fiber (tip 2) and the 30-μm fiber plus the 60-μm metal bridge plate (tip 3) are roughly modeled as such. Then, using this prescribed field, we solved the advection–diffusion equation for each (passive) scalar separately. We assumed that at t = 0 a step signal of Δθ = 1 or ΔS = 1 swept past the sensor.
As time progresses, the shear in the boundary layer deforms the front, as seen in the snapshot (Fig. B1, grayscale). Molecular diffusion acts to spread it out. The combined effects of diffusion and advection determine the sensor’s response time. To determine the impulse response (Fig. B2), we computed the time series at the outside of the optical sensing volume (dashed line, black circle). By 0.2 ms, the salinity within the optical sensing volume has risen to 1. Falling at 0.5 m s−1, the sensor will have swept through 0.1 mm. Though a proper study would be more satisfying, the response time obtained in this manner is sufficiently fast that we believe we would obtain similar results using the full solution with the correct geometry.
Corresponding author address: M. Alford, Applied Physics Laboratory, 1013 E. 40th St., Seattle, WA 98105. Email: email@example.com