Uncertainties and Limitations in Measuring and T

Transcription

1 NOVEMBER 1999 GREGG 1483 Uncertainties and Limitations in Measuring and T M. C. GREGG School of Oceanography and Applied Physics Laboratory, University of Washington, Seattle, Washington (Manuscript received 10 April 1997, in final form 14 May 1998) ABSTRACT Extending the range of microstructure measurements from the upper thermocline of the open ocean to shallow waters near shore and to the abyss has greatly increased in the range of turbulent intensities being observed. As a result, it is necessary to reexamine the adequacy of microstructure probes to resolve dissipation-scale gradients of velocity and temperature. These variances are needed to estimate directly the viscous dissipation rate,, with airfoil probes and the diffusive dissipation rate, T, with thermistors. 1. Introduction Most measurements of temperature and velocity microstructure are made with probes and techniques developed 0 30 years ago to resolve turbulent gradients in the upper thermocline of the open ocean. Cox et al. (1969) took advantage of the high sensitivity of thermistors by mounting them on a rotating winged profiler that fell much like a helicopter in autorotation. At speeds of w m s 1, their glass-rod thermistors spatially resolved nearly all of the temperature gradients encountered, permitting direct measurement of one component of T, the rate of diffusive smoothing of temperature fluctuations. By adapting airfoils to operate in the ocean, Osborn (1974) enabled direct measurement of one component of, the rate of viscous dissipation of velocity fluctuations. In doing so, he increased w to 0.5 m s 1 because the sensitivity of airfoils increases linearly with speed. Together, thermistors and airfoils provide the data for quantifying turbulent dissipation in the ocean. In most of the open ocean, internal waves are close to the background state modeled by the Garrett and Munk (1975) spectrum. Interactions between background internal waves produce weak turbulence, with decreasing from 10 9 Wkg 1 in the seasonal thermocline to Wkg 1 at 9 MPa (Gregg and Sanford 1988). In this pressure range the largest dissipation samples were about 100 times the average dissipation rate,. While falling freely at w 0.35 Corresponding author address: Dr. Michael C. Gregg, Applied Physics Laboratory, University of Washington, Box , Seattle, WA gregg@apl.washington.edu ms 1, the multiscale profiler (MSP) resolved the largest of these gradients. Dissipation rates exceeding 0.1 W kg 1 have been found using loosely tethered profilers in rapid tidal currents near coasts and in estuaries and straits. Reaching 3 mpa before the tether runs out often requires w 1ms 1, which increases the difficulty in resolving the variance. Resolving T with thermistors is hopeless at these drop speeds, even for moderate dissipation rates. Abyssal measurements with the high-resolution profiler (HRP) have detected average dissipation rates as low as Wkg 1 (Toole et al. 1994), extending the low end of the reported range. Values this low are well resolved spatially but raise questions about the behavior of airfoil probes at low wavenumbers. Measuring dissipation rates over 11 decades of magnitude with one type of velocity probe warrants a review of the uncertainties and limitations of the measurements. In addition, we need to consider the prospects for extending T measurements to higher wavenumbers than can be done at present. This is essential if we are to address seriously the efficiency of smallscale mixing.. Universal spectra as references Observations from a variety of sensors have demonstrated that spectra are close to universal shapes when the turbulence is well developed. Oakey (198) published the dimensionless shear spectrum compiled by Nasmyth (1970) to describe observations in strongly turbulent tidal channels near British Columbia. This spectrum is widely used to check data quality and to estimate variance at scales smaller than can be resolved by the airfoils. Figure 1 contains two examples from the Strait of Gibraltar. Nasmyth s spectrum is similar to one derived theo American Meteorological Society

2 1484 JOURNAL OF ATMOSPHERIC AND OCEANIC TECHNOLOGY VOLUME 16 FIG.. Spectra of one-dimensional transverse shear in fully developed turbulence for Wkg 1 to1wkg 1. The spectra are derived from the universal three-dimensional form derived by Panchev and Kesich (1969). As increases, the variance occurs at higher wavenumbers and with larger spectral amplitude. FIG. 1. Spectra of transverse velocity, each averaged over successive 0.5-m intervals in layers 0 m thick. The vertical dashed lines show where integration was cut off by our processing algorithm. The spikes at higher wavenumbers were produced by vibration. Shading contains the 95% confidence limits, and the thin lines are Nasmyth spectra having the same variance when integrated to k c (from Wesson and Gregg 1994). retically by Panchev and Kesich (1969), which we use owing to its theoretical basis. The Panchev and Kesich expression must be evaluated numerically and has one adjustable parameter, a, which is set to 1.6 to produce the one-dimensional spectra of transverse shear in Fig.. These spectra include the reported range and span 1 decades in amplitude and 5 decades in wavenumber. When wavenumber is normalized by the viscous, or Kolmogorov, wavenumber 1/4 1 k [cpm], (1) 3 the universal spectrum peaks at 0.15k (Fig. 3). Integrating to k, 0.5k, and 0.513k returns 0.1, 0.5, and 0.9 of the variance. Consequently, k is the upper bound required to resolve turbulent shear fluctuations. Temperature gradient spectra from the thermocline are sometimes less regular than those of shear, particularly at wavenumbers smaller than the spectral peak (Gregg 1977; Dillon and Caldwell 1980). However, the one-dimensional form of the three-dimensional spectrum derived by Batchelor (1959) is a good approximation, particularly in the diffusive rolloff at wavenumbers larger than the spectral peak (Dillon and Caldwell 1980; Oakey 198). Gargett (1985) argues otherwise, but the basis of her argument is a misinterpretation of Batchelor s result (Seim et al. 1995). Figure 4 shows the universal temperature gradient spectra for the range of dissipation rates reported in the literature and for T K s 1, a typical value. In practice, spectral amplitudes depend on the mean temperature gradient as well as on. Their tendency, however, is for the spectral amplitude to decrease as increases, in contrast to the increase in amplitude exhibited by shear spectra. FIG. 3. Normalized universal turbulent shear spectrum and its cumulative integral. Wavenumber k and the viscous cutoff k are in cpm.

3 NOVEMBER 1999 GREGG 1485 FIG. 4. Temperature gradient spectra for from Wkg 1 to1wkg 1 and T K s 1.As increases, the temperature gradient variance occurs at larger wavenumbers and the spectral amplitude decreases. Normalizing by the diffusive, or Batchelor, wavenumber 1/4 1 kd [cpm], () D reveals that the spectrum peaks at 0.k D (Fig. 5). Integrating to 0.105, 0.304, and 0.613k D yields 0.1, 0.5, and 0.9 of T ; again the reference wavenumber is the upper bound required to resolve the variance. 3. Measuring with airfoils Figure 6 is a schematic of an airfoil constructed in the manner Oakey adapted from the original design by Osborn (1974), which Osborn and Crawford (1980) describe in detail. The sensing element, known as a piezoelectric bimorph beam, is also used in phonograph cartridges. When deflected slightly along its sensitive axis, which is vertical in the figure, the beam produces a voltage. The beam is sealed in heat-shrink tubing to insulate it electrically from seawater and then encased in a tip of silicone rubber. The parabolic shape provides constant lift along the exposed length of the beam. The voltage produced when a cross-velocity bends the beam by generating lift is up u (t) E 0(t) S [V], (3) g where S is the probe sensitivity, g is the gravitational acceleration, u p is the probe s velocity, and u is a fluctuation in velocity along the sensitive axis of the probe. Typical sensitivities are S V m 1. For S 5 and u p 0.5 m s 1, u 1mms 1 produces E mv. When mounted on a profiler falling through the ocean u p FIG. 5. One-dimensional temperature gradient spectrum, in arbitrary units, and its normalized cumulative integral. The spectrum is obtained by integrating the three-dimensional form derived by Batchelor (1959). The wavenumber is in cpm, as is the Batchelor wavenumber, k D. at speed w, the probe is exposed to the oceanic velocity spectrum u (k 3 ), where k 3 f/w is the vertical wavenumber. The signal is conditioned by its dynamic transfer function, a gain and differentiating circuit, and an antialias filter, details of which are described by Moum et al. (1995). As a result the frequency spectrum of the recorded signal is u(k 3) Sw R( f ) H Oakey( f, w) H gain( f, C S) w g ] Hz [ V H antialias( f, f c ), (4) where the raw signal R is inversely proportional to w. Two dynamic response functions have been published for airfoils. Oakey (1977) guesses that airfoils respond as single-pole low-pass filters and estimates the effective wavelength as c 0.0 m. Consequently, he estimates the amplitude-squared response as 1 H Oakey( f, w). (5) 1 ( f /w) The second dynamic response was obtained by extensive laboratory observations of the airfoil response to grid turbulence, which was measured independently with a laser Doppler velocimeter. By comparing the two responses, Ninnis (1984) obtains n 3 f /w H Ninnis( f, w) A. (6) n 0 k0 The parameters are k cpm, the wavenumber of the first amplitude minimum, A 0 1, A , A 4.763, and A The response is meant to be used only to k 0 and resembles a sinc function. Figure 7 compares Oakey s and Ninnis s response c n n

4 1486 JOURNAL OF ATMOSPHERIC AND OCEANIC TECHNOLOGY VOLUME 16 FIG. 6. Schematic of an airfoil probe. functions with the resolution required to obtain 0.1, 0.5, and 0.9 of. The two upper curves using Oakey s response are to demonstrate the increased resolution with smaller probes. For wavenumbers up to 80 cpm, Oakey s curve for c 0 mm does not differ significantly from Ninnis s function. At higher wavenumbers, however, Ninnis s response rolls off much more steeply before rising again after the first null at K 0. Ninnis s response, however, does not appear to be correct at high wavenumbers because spectra from intensely dissipative regions do not exhibit the first null in his transfer function. An example is shown in Fig. 8. By comparison, spectra of conductivity measured with Neil Brown and SeaBird cells clearly show notches where expected (Gregg et al. 198; Gregg and Hess 1985). Owing to this discrepancy, Oakey s transfer function is preferable and produces corrected spectra that agree well with the universal spectrum. Because Oakey s response is not justified by anything other than agreement with universal spectral shapes, the situation is not satisfactory and is a critical limitation in extending observations toward the very high dissipation rates sometimes found in shallow water. At 150 cpm H Oakey shows attenuation by a factor of 10. In view of a lack of direct measurements, this is a reasonable upper bound for its use and returns 0.9 of the shear variance for 10 5 Wkg 1. Assuming that Oakey s estimate is approximately correct, it can be used to estimate improvements in resolution with smaller airfoils. Reducing c by factors of and 4 increases the wavenumber for 0.9 resolution by approximately the same factors. This in turn increases resolution by 4 16 and , or approximately to 10 4 Wkg 1 and Wkg 1. Knowing H well enough to correct spectra attenuated by a factor of 100 would further increase resolution to 0.05 W kg 1 and 0.4 W kg 1. Therefore, reducing the length and diameter of airfoils by a factor of 4 and obtaining accurate calibrations to amplitude-squared attenuations of 100 would resolve 0.9 of the variance of the largest dissipation rates reported so far. Because spectral amplitudes increase with, smaller probe sensitivities can be accepted for smaller probes. FIG. 7. The diagonal lines show the wavenumbers at which 0.1, 0.5, and 0.9 of are resolved as a function of along the vertical axis. The thick curved lines are the amplitude-squared response to Oakey (198) evaluated with c 0.0, 0.01, and m. The thin curve is the response found by Ninnis (1984) for Oakey s probes and uses the same abcissa. FIG. 8. Spectrum of the x channel, x, of raw airfoil data in a region of intense dissipation compared with the dynamic transfer functions of Ninnis and Oakey plotted with the same scale. The fall rate was 0.74 m s 1. Because x does not dip where Ninnis s function has a notch, the function cannot be not correct at these frequencies.

5 NOVEMBER 1999 GREGG 1487 FIG. 9. Shear spectra observed with the MSP by Gregg and Sanford (1988). Dotted curves are matching universal turbulent spectra for Wkg 1 (upper) and W kg 1 (lower). A fourfold reduction in size and extension of the calibrations to H 0.01 is probably the best we can hope for from airfoils. Different types of sensors may be needed for 1Wkg 1, which should be expected in intense tidal flows. At the other end of the dissipation range, we have arbitrarily set Wkg 1 as the noise level for the MSP, our slowest and quietest profiler. We did this after observing spectra for Wkg 1 depart from the universal shape, as shown in Fig. 9. Determining the inherent noise of airfoils is of considerable importance for understanding abyssal measurements, as signals smaller than Wkg 1 constitute 80% of the samples at 10 MPa when internal waves are at the Garrett Munk background state (Fig. 10). When Wkg 1, 0.9 of the shear variance occurs at k 7 cpm (Fig. 7). Spatial resolution is not a problem; considerably larger airfoils could be used. Sensitivity and noise, however, are issues for these weak signals because the low-frequency noise rises with decreasing frequency. This has not been examined carefully, but it resembles f 1 noise. 4. Measuring T with thermistors and cold films Thermistors enclosed in glass and cold films have been used to resolve dissipation-scale temperature gradients in the ocean. Figure 11 compares the fastest commercial thermistor, the FP07 by Thermometrics, with a cold film. To minimize the thickness of the boundary layer as well as the amount of glass, the thermistor is encased in the small tip at the upper end of the glass rod. More detailed views reveal considerable variation in the thickness of the tip, the location of the thermistor bead, and the size and number of bubbles within the glass. Cold films can be made more uniform than glassencased thermistors, but we have found frequent leaks in the quartz passivation layer insulating the platinum element from seawater, even though typical resistances are R T 10. Thermistors can be mounted without significant leaks, even with resistances of several megohms. The trade-off between thermistors and cold films is one of sensitivity versus spatial resolution: cold films respond faster but are much less sensitive than thermistors. If the probes are operated at approximately constant current, i, the voltage across the sensor is de 0 i TR T [V], (7) dt where T (1/R T )(dr T /dt). For thermistors, T 0.05, about 10 times larger than for platinum. Moreover, thermistors can have resistances megohms. Owing to their higher sensitivity and resistance, thermistors can achieve noise levels of 3 K (Gregg et al. 1978) compared to a 0. mk for cold films (Oakey 1977). Fabula (1968) modeled the response of a cold film to a thin heated plume as where H exp[ ( f / T ) 1/ cold film ], (8)

6 1488 JOURNAL OF ATMOSPHERIC AND OCEANIC TECHNOLOGY VOLUME 16 FIG. 10. Activity classes of samples observed during the Patches Experiment (PATCHEX) (Gregg and Sanford 1988). Internal waves were at the Garrett Munk background state throughout the upper 10 MPa. Consequently, decreased as the stratification weakened to N 0.00 s 1 at 10 MPa. The leftmost line, labeled 10 10, is a conservative estimate of the MSP noise. The middle line, labeled 16 N, is for dissipation rates at the threshold intensity for producing a net bouyancy flux (Stillinger et al. 1983). The rightmost line, labeled 00 N, appears as a few dark bars and is the threshold for dissipation-scale turbulence to be isotropic (Gargett et al. 1984). ( T ) 1/ (u p /1.7) 0.3 (9) is the boundary layer thickness. For a probe speed of u p 0.5 m s 1, H 0.1 at 00 cpm and 0.01 at 700 cpm (Fig. 1), resolving 0.9 of T for W kg 1 and 10 4 Wkg 1, respectively. These resolutions are usually not achieved in the open ocean because of the low sensitivity and resistance of cold films. Several response measurements have been made for thermistors, and none are fully consistent. At least part of the variability is due to differences in glass coatings between individual thermistors of the same type. Each thermistor should be calibrated, but the calibrations are FIG. 11. An FP07 FastTip thermistor and a cold film with a millimeter scale at the top. FIG. 1. Diagonal lines show the wavenumbers at which 0.1, 0.5, and 0.9 of T are resolved as functions of on the vertical axis. For comparison, the thick curved lines are the amplitude-squared response for a cold film probe moving at 0., 0.5, and 1.0 m s 1. The thin curved lines are equivalent responses for FP07 thermistors.

7 NOVEMBER 1999 GREGG 1489 FIG. 13. Ratio of the amplitude-squared response of an FP07 thermistor having 1 3 ms divided by the response of another FP07 having 1 5 ms. too costly for this to be done on many probes. Instead, we use a nominal response based on tests of a few probes passing through sharp interfaces (Gregg et al. 1978). The function is a double-pole, 1 HFP07 (10) [1 ( f )] with a time constant of u p [s]. (11) Note that (9) and (11) both vary with speed to the same power, indicating similar behavior of their boundary layers. At 0.5 m s 1, H 0.1 at 70 cpm and 0.01 at 140 cpm, yielding 0.9 of T for 10 8 Wkg 1 and 10 7 Wkg 1. In view of the uncertainty about H, T is usually not reported except for slower speeds and/or lower dissipation rates. As an example, Fig. 13 shows ratios of H for an FP07 thermistor having 3mstoH for a similar thermistor having 5 ms. These ratios would be the errors if fast probes were overcorrected using slow responses and demonstrate that spectral shapes would be severely distorted in addition to the error in T. The limitations on resolving T also affect estimates of based on the magnitude and shape of the diffusive rolloff of the temperature gradient spectrum. Dillon and Caldwell (1980) estimated k D over short sections of data by fitting the universal shape to observed spectra at wavenumbers beyond the spectral maximum. They then 4 obtained the dissipation rate from (). Because k D, these estimates are very sensitive to errors in k D. Although their profiler fell at 0.1 m s 1, Dillon and Caldwell had to skip sections where the spectrum was not adequately resolved before it fell to the noise level. The fit was typically made between the spectral maximum and the wavenumber at which the spectrum was 0.1 of the maximum. This range corresponds to 0.5 and 0.9 of the total variance and hence requires fully resolved spectra. These limitations on measuring T are severe and preclude estimates in most situations of interest. One approach to overcoming the limitations is to resume using cold films where signals have large amplitudes. As an alternative, we are trying to develop a thinistor, made by depositing thermistor material on a glass rod as is done for cold films. The thinistor was attempted in the early 1980s at the Applied Physics Laboratory of The Johns Hopkins University but was not completed. The challenge is to coat, or passivate, the thermistor film to insulate it adequately from seawater. Owing to the large resistances of thermistors, this is a much more difficult task than for cold films, and we have not had good luck with commercial cold films. Acknowledgments. The Office of Naval Research funded the writing of this paper and most of the research described in it. REFERENCES Batchelor, G. K., 1959: Small-scale variation of convected quantitites like temperature in turbulent fluid. J. Fluid Mech., 5, Cox, C., Y. Nagata, and T. Osborn, 1969: Oceanic fine structure and internal waves. Bull. Japan. Soc. Fish. Oceanogr., (Special Issue: Prof. Uda s Commemorative Papers), Dillon, T. M., and D. R. Caldwell, 1980: The Batchelor spectrum and dissipation in the upper ocean. J. Geophys. Res., 85, Fabula, A. G., 1968: The dynamic response of towed thermometer. J. Fluid Mech., 34, Gargett, A. E., 1985: Evolution of scalar spectra with the decay of turbulence in a stratified fluid, J. Fluid Mech., 159, , T. R. Osborn, and P. W. Nasmyth, 1984: Local isotropy and the decay of turbulence in a stratified fluid. J. Fluid Mech., 144, Garrett, C. J. R., and W. H. Munk, 1975: Space time scales of internal waves: A progress report. J. Geophys. Res., 80, Gregg, M. C., 1977: Variations in the intensity of small-scale mixing in the main thermocline. J. Phys. Oceanogr., 7, , and W. Hess, 1985: Dynamic response calibration of Sea-Bird temperature and conductivity probes. J. Atmos. Oceanic Technol.,, , and T. B. Sanford, 1988: The dependence of turbulent dissipation on stratification in a diffusively stable thermocline. J. Geophys. Res., 93, , T. Meagher, A. Pederson, and E. Aagaard, 1978: Low noise temperature microstructure measurements with thermistors. Deep-Sea Res., 5, , J. Schedvin, W. Hess, and T. Meagher, 198: Dynamic response calibration of the Neil Brown conductivity cell. J. Phys. Oceanogr., 1, Moum, J. N., M. C. Gregg, R. C. Lien, and M. E. Carr, 1995: Comparison of turbulent kinetic energy dissipation rate estimates from two ocean microstructure profilers. J. Atmos. Oceanic Technol., 1, Nasmyth, P. W., 1970: Oceanic turbulence. Ph.D. thesis, University of British Columbia, 69 pp. [Available from the Department of Physics, University of British Columbia, 075 Wesbrook Place, Vancouver, BC V6T 1W5, Canada.] Ninnis, R., 1984: The effects of spatial averaging on air-foil probe measurements of oceanic velocity microstructure. Ph.D. thesis,

8 1490 JOURNAL OF ATMOSPHERIC AND OCEANIC TECHNOLOGY VOLUME pp. [Available from Department of Oceanography, University of British Columbia, 075 Wesbrook Place, Vancouver, BC V6T 1W5, Canada.] Oakey, N. S., 1977: An instrument to measure oceanic turbulence and microstructure. Report Series BI-R-77-3, Bedford Institute of Oceanography, Dartmouth, NS, Canada, 5 pp. [Available from the Bedford Institute of Oceanography, Dartmouth, NS BY 4A, Canada.], 198: Determination of the rate of dissipation of turbulent energy from simultaneous temperature and velocity shear microstructure measurements. J. Phys. Oceanogr., 1, Osborn, T. R., 1974: Vertical profiling of velocity microstructure. J. Phys. Oceanogr., 4, , and W. R. Crawford, 1980: An airfoil probe for measuring turbulent velocity fluctuations in water. Air Sea Interactions: Instruments and Methods, F. Dobson, L. Hasse, and R. Davis, Eds., Plenum, Panchev, S., and D. Kesich, 1969: Energy spectrum of isotropic turbulence at large wavenumbers. Comp. Ren. l Acad. Bulg. Sci.,, Seim, H. E., M. C. Gregg, and R. Miyamoto, 1995: Acoustic backscatter from turbulent microstructure. J. Atmos. Oceanic Technol., 1, Stillinger, D. C., K. N. Helland, and C. W. V. Atta, 1983: Experiments on the transition of homogeneous turbulence to internal waves in a stratified fluid. J. Fluid Mech., 131, Toole, J. M., K. L. Polzin, and R. W. Schmitt, 1994: Estimates of diapycnal mixing in the abyssal ocean. Science, 64, Wesson, J. C., and M. C. Gregg, 1994: Mixing at Camarinal Sill in the Strait of Gibraltar. J. Geophys. Res., 99,