THE REDUCTION IN FINESTRUCTURE CONTAMINATION OF INTERNAL WAVE ESTIMATES FROM A TOWED THERMISTOR CHAIN

Transcription

1 DANIEL C. DUBBEL THE REDUCTION IN FINESTRUCTURE CONTAMINATION OF INTERNAL WAVE ESTIMATES FROM A TOWED THERMISTOR CHAIN Estimates of internal wave displacements based on towed thermistor array data have historically sffered badly from finestrctre contamination. It is hypothesized that a sbstantial amont of sch contamination may be removed by sampling the ocean with thermistors spaced in a close vertical pattern and by sing appropriate internal wave displacement estimation techniqes. Sch a techniqe is described, and preliminary reslts based on data from a 5-centimeter vertical resoltion chain are presented that indicate that a sbstantial redction of finestrctre contamination is achievable. INTRODUCTION Calclation of flid motion from oceanic temperatre measrements has always been difficlt; each of the two commonly sed measrement systems, towed thermistor chains and dropped thermistors (thermistors are temperatre-sensitive resistors), has an inherent limit to the accracy of flid motion estimates that can be prodced from their respective data. In the case of dropped thermistor-type instrments, the horizontal and vertical wavenmber range over which flid motion may be estimated is severely limited by the vertical drop rate and the attainable drop repetition rate. Conseqently, sch instrments are of limited vale for estimating flid displacements over a broad range of horizontal and vertical length scales. Towed thermistor chains do not sffer from sch mechanicallimitations, althogh there is a maximm depth to which they can be deployed. Rather, the limits to accrate flid motion estimates are de mainly to the degree to which sch estimates are contaminated by smallscale vertical temperatre strctres. This effect, known as finestrctre contamination, has been discssed extensively in the oceanographic literatre 1-6 and was also described in a previos Ocean Science isse of the Johns Hopkins APL Technical Digest. 7 The impact of this contamination on the accracy of chain-based flid displacement estimates is shown to be decreased by sing proper processing techniqes and adeqate vertical resoltion. FINESTRUCTURE MODEL Any sccessfl processing techniqe for redcing finestrctre contamination mst be based on an accrate model of the nderlying oceanic processes. One sch model is the passive finestrctre model, which states that small-scale vertical temperatre featres are advected passively by the backgrond internal wavefield (very little or no active mixing occrs) so that a horizontally towed thermistor then not only measres 186 the nderlying flid displacements bt also aliases into this measrement all the short-wavelength temperatre variability contained in the vertical profile. (These temperatre featres are formed by a variety of mechanisms, many of which are discssed in Ref. 8.) Modeling of finestrctre contamination in this manner is qite common in the literatre. The temperatre field is given by T (x,t) = To [z + r (x,t)] where ris the internal wave displacement field, explicitly a fnction of x and time, Z is vertical position, To is the ndistrbed vertical temperatre profile, and x is the horizontal position at fixed depth. No horizontal space or time variation is inclded in the model, and all the small-scale vertical variability is contained in the ndistrbed temperatre profile. Actal oceanic finestrctre shold, of corse, have a space and time dependence that is not entirely attribtable to the space and time variations of the internal wavefield; bt, for the prposes of this analysis, it is assmed that the space and time dependence of To(z) can be ignored. 3 It is difficlt to estimate exactly the amont of a towed thermistor temperatre signal that is de to passive finestrctre contamination and the amont that is de to actal, irreversible mixing events. It does seem highly likely, however, that the passive finestrctre hypothesis is valid over some vertical and horizontal scales. The determination of the scales was the objective of the work described in this article. Past simlation and modeling reslts have indicated that vertical fine- and microstrctres can be resolved, and their effects on the measrement of the internal wave displacement field redced, if the temperatre profile is sampled at high enogh vertical resoltion. Accrate comptation of the nderlying internal wavefield is then possible by tracking specific, identifiable profile featres for long time intervals. These profile featres may, for a monotonic profile, be simply spe- Johns Hopkins APL Technical Digesc, Volme 6, Nmber 3

2 cific temperatres, i.e., isotherms. Assming that the passive finestrctre model is correct over the scales appropriate to sch towed measrements (i.e., a few meters to a few hndred meters in the horizontal), isothermal displacements in these wavebands shold exactly track the nderlying internal wave displacements. If the character of the internal wave displacement field at these scales in the pper ocean were known independently, an evalation of the performance of the above scheme for redcing the effects of fine- and microstrctre contamination on towed chain measrements wold be straightforward. However, de to measrement difficlties, no sch independent knowledge is crrently available. Therefore, the approach taken here is to hypothesize the simplest possible character for this field and to test the degree to which the internal wave displacement field derived from the measrements matches the hypothesis. Specifically, it is assmed that the tre pper-ocean internal wavefield may be described as a piecewise stationary, homogeneos, gassian random field for horizontal scales of a few to a few hndred meters and is vertically homogeneos for vertical scales of a few centimeters to a few tens of meters. It shold be noted that at longer horizontal scales (several hndreds of meters), these assmptions have been shown in general to hold. 9 With the above hypothesis, the gassianity of the internal wave estimates is a measre of how accrately the flid motion has been reprodced. This measre will be assessed by examining the high-order moment strctre of the displacement estimates and will be described more flly in the Reslts section. correlation method. The conceptal basis for the method is simple: The ocean is believed to consist, to a large degree, of a series of layers of nearly isothermal water where the vertical-to-horizontal length-scale ratio is of the order 10-2 If a vertical array of closely spaced thermistors is plled horizontally throgh the water while being raised and lowered by approximately the vertical separation distance of the thermistors every few seconds, the differences in temperatres sensed by each thermistor from the top to the bottom of its path shold allow calclation of a temperatre gradient for that layer of water, for each thermistor. Many sch gradient estimates are compted over, say, half an hor of data, and the least-sqares estimate of the average gradient seen by each thermistor is calclated. These gradients are vertically integrated from top to bottom, ths constrcting an estimate of the average temperatre profile nbiased by calibration offset errors for that time segment. The profile is then compared to the temperatre profile prodced by simply averaging the temperatre vales of each thermistor. The thermistor-by-thermistor differences between these two profiles represent the bias corrections for each thermistor. It is easy to see that this method is independent of initial bias errors in both temperatre and depth. Figre 1 demonstrates the effectiveness of the intercalibration algorithm in correcting an average profile from an nreasonable crve to one that is more physically plasible. There is no known way to estimate the tre residal calibration error, bt estimates from the variation of the corrected average profile indicate CALIBRATION The practical aspects of calclating isotherms are more tedios than the theory. One sch aspect is that of thermistor calibration. Accrate estimates of local temperatre gradients are necessary to compte isotherms. Since sch estimates involve sing adjacent thermistor temperatre differences in the method sed in the analysis, accrate interthermistor calibration is reqired. Typical predeployment thermistor offset accracy is on the order of 20 to 50 millidegrees celsis. Since it is necessary to estimate locations of isotherms to within a few centimeters, it is obvios that a relative accracy between thermistors of greater than 20 millidegrees celsis is not adeqate to resolve the typical thermocline temperatre gradients of 50 millidegrees celsis per meter. Sfficient accracy can only be obtained by in sit calibration. The in sit calibration algorithm applied to the thermistor data sed in the analysis estimated only an offset correction for each thermistor. Any gain errors in the data were corrected by forcing the vertical temperatre variance profile of the thermistors to be smooth, a techniqe that is effective only if the original thermistor-gain vales were more or less accrate and if only the occasional thermistor reqires correction. The specific in sit interthermistor calibration algorithm actally sed is known variosly as the average gradient method or the depth-temperatre crossfo hns H opkins APL Technical Digest, Vo l me 6, N mber r----r----r-----'----:::o"""'-----' (f)... (l) E E (l) o L...-_----L_----.:::::2..._.A..-_---J, _ ' J Temperatre (O el Figre 1-Vertical profile of temperatre averaged over 50 meters horizontally. The smooth crve represents a calibrated profile. 187

3 D. C. Dbbel - Redction in Finestretre Contamination of Internal Wave Estimates that the maximm residal error is something on the order of 1 millidegree celsis or less. The freqency with which the data need to be intercalibrated is a fnction primarily of the stability of the thermistor system electronics. In the case described here, the thermistors were intercalibrated every half hor. ISOTHERMS Conceptally, the process of calclating isotherms is simple: A temperatre vale is selected that falls within the range of temperatres measred by the array, and its location in depth is calclated by linearly interpolating between thermistors (the isotherm's temperatre vale will not, in general, be one of those exactly measred by any thermistor). However, there are problems with this simple method. The local temperatre profile is not always monotonic; regions occr where the gradient changes sign. These are called temperatre inversions. Wherever the thermistor array enconters an inversion, there may be several locations where a given temperatre may occr. A simple isotherm algorithm will typically scan the chain to locate a given temperatre and select the first position where that temperatre can be located. The process gives rise to the sort of behavior exhibited in Fig. 2; note the clearly nphysical step-like strctres in the lower isotherms. One way to alleviate the problem is to calclate all locations for each isotherm and then select the one closest in depth to the previos position. This is called the minimm displacement method. However, inversions are not all of the same type since they are prodcts of a variety of mechanisms sch as salt fingering, wave breaking, and doble diffsion; conseqently, this procedre does not eliminate all nphysical behavior. The problem mst be approached from a more global point of view. In general, the problem may be stated as: What shold be done in regions in which the passive finestrctre model is clearly not an accrate description of the physics? From the point of view of internal wave displacement processing, an adeqate answer to that qestion is that some regions of temperatre inversion are generated by nonreversible, active processes, and the passive finestrctre model clearly does not apply in those areas. Since internal wave displacement estimates are meaningless in regions of trblent mixing, sch regions shold be avoided in the estimation process, i.e., isotherms shold not be tracked throgh regions of temperatre inversion. To be effective, the inversion-avoiding algorithm implemented for this analysis relies on the statistical behavior of the small-scale oceanic trblence. In particlar, the rate of incidence and the vertical extent of temperatre inversions are sch that it is extremely nlikely that a sitation will occr in which most of the array is inverted. 10 It is possible, then, to calclate many isotherms over short time intervals and then select a sbset of them based on the freqency of occrrence of inversions encontered by individal isotherms. At each new short time interval, a completely new set of isotherms is selected. Each of these short intervals (or data windows) is selected to overlap the previos one by 50 percent. In the overlap regions, the sbset of selected isotherms is merged from the two windows with a trianglar smoothing filter. The reslt is smooth, inversion-avoiding, composite isotherms that will behave essentially as tre isotherms over all physically significant scales. These series have been descriptively named modified inversion-avoiding composite isothermal displacements (MIACID). It will not always be possible to choose isotherms that enconter no inversions at all. In these cases, the pair of thermistors yielding the negative gradient is , 80 Figre 2-Simple isotherms. Note the step-like strctres in the lower isotherms.....c c. w o L- 130 o Time (seconds) 188 Johns Hopkins A PL Technical Digest, Volme 6, Nmber 3

4 D. C. Dbbel - Redction in Finestretre Contamination of Internal Wave Estimates eliminated from consideration and the next wider pair that show a positive gradient is sed for the interpolation. Figre 3 is a plot of some representative MIACIDs that can be seflly compared with Fig. 2. Note the complete absence of the step-like strctres observed in Fig. 2 and the well-behaved natre of the time series. These particlar MIACIDs were selected to track throgh no inverted regions at all, a condition that will not be possible overall becase an average MIACID ill enconter abot 20 navoidable inverted points In the corse of abot 5 kilometers of data. However, the MIACIDs shold still be better estimates of internal wave displacements than simple isotherms since a simple isotherm will typically track throgh many hndreds of inversions in the same time period. The nui?ber of inversions encontered in approximately 5 kilometers of Sargasso Sea data dring or investigations was 633 sing simple isotherms (abot 8 percent of the approximately 8000 measrements) verss 20 sing MIACID (abot 0.2 percent) r----, r r =-_--'-_--L L ::...L...-_---L L-_...J T ime (seconds) Figre 3-Modified inversion-avoiding composite isothermal displacements (MIACIO). Note the lack of step-like strctres. Motion compensation system and winch DATA CHARACTERISTICS Or reslts are based on data acqired in the seasonal thermocline in November in the Sargasso Sea with an APL low-drag thermistor chain. II The chain sed to gather these data has a vertical thermistor apertre of 10 meters with thermistors spaced nominally at 5 centimeters in the vertical (see Fig. 4). A tow speed of 6 knots and a data rate of 5 hertz (after decimation) yield a Nyqist wavelength of 1.2 meters. Relative change in chain depth was calclated from the average of 11 Entran pressre transdcers. Chain motion cased by wave-indced ship motion was redced by sing the APL-developed passive motion compensation system, 12 yielding a root-mean-sqare vertical motion of abot 5 centimeters over the primary wavelength band of srface waves (i.e., 10 to 30 meters). Figre 5 gives a representative segment of intercalibrated temperatre data; Fig. 6 indicates the average spectral levels for two temperatre channels and a noise resistor. Atospectral density plots sch as Fig. 6 provide information abot the relative contribtions to the overall signal power from the varios spectral components of the signal. Oceanic temperatre spectra typically exhibit slopes of k -2, where k is the wavenmber. The noise resistor spectrm characterizes the instrment noise floor power levels. The temperatre signal shold be (and is) well above the noise floor across the entire freqency band if the sensor is measring the ocean and not its own self-noise. These data have been preprocessed to remove wild points and electronics drift. All reslts presented are based on ensemble averages taken over abot 75 kilometers of horizontal tow. RESULTS Figre 3 showed a representative sample of MIACID; note again the behavior of these series relative to the simple isotherms shown in Fig. 2, namely, the Johns H opkins APL Tech nical Digest, Vo l me 6, N m ber 3 (not to scale)...-depressor Figre 4-A schematic of the APL high-density towed thermistor chain sed to acqire these data. lack of step-like strctres and the generally more reglar appearance. Qalitatively, the estimates of internal wave displacement certainly appear sperior to the simple isotherms and, obviosly, to temperatre displacements (normalized temperatre). Figre 7 is an atospectral plot of both MIACID and the normalized temperatre (the temperatre has been scaled by the local average temperatre gradient so as to yield displacements in nits consistent with the MIACID). The spectral slope for the MIACID is -2, as wold be expected since the typically observed internal wave spectrm (in these bands) has a slope of k -2 and the spectral level is lower than for temperatre, as wold also be expected if an actal redction in noise power variance were achieved. In other words, the spectral slope of the MIACID is consistent with other, ncontaminated estimates of internal wave slope, and the spectral levels show that some of the power in the temperatre series (presmably attribtable to finestrctre contamination) has been removed by the MIACID_ 189

5 D. C. Dbbel - Redction in Finestrctre Contamination of Internal Wave Estimates Z Q) E Q) r Q) E Q) L L T ime (seconds) Figre 5-A representative segment of intercalibrated temperatres, plotted on the bottom half of the figre, and isotherms calclated from those temperatres, plotted on the top. A qantitative measre of finestrctre contamination redction is now reqired. One sch measre is the lagged flatness factor, (3 (L), which directly reflects the gassianity and independence of a time series. It is defined as (3(L) ([T(x+L) - T(X)]4) ([T(x+ L) - T(x)] 2 ) 2 where T may be either temperatre or MIACID and L is the lag in meters. Lag refers to the difference in horizontal position between two measred temperatre vales (or calclated MIACID vales) that are to be differenced. The procedre of calclating lagged flatness is similar to that of calclating atocovariance, only sing different operations; a time series is displaced horizontally across itself, differenced, raised to the forth power, and then averaged. For an independent gassian process, (3(L) = 3.0; for a nongassian process, (3(L) > 3. Since the data sample length in this case is limited, the estimate of (3(L) will be biased low (see Ref. 13) by an amont, b(n), where -6 b(n) = - M+2 and M is the nmber of independent intervals of length N contained in the data sample. The problem of nonindependent data samples occrs becase the finite size of each data record limits the nmber of points in the expectation vale sm for large lags, ths prodcing a bias becase of the arbitrary limits set on the nmber of data points. This bias has been removed from all reslts. The flatness excess, (3(L) - 3, is a measre of the horizontal independence (the independence of the spectral components). If the data are horizontally independent, then r ' (3(L) - 3 [(3(4>:) - 3] ( where Ax is the horizontal distance corresponding to the digital sampling rate. Ths, if the data are horizontally independent, the excess flatness will have a slopr of -1 when plotted against log L on a logarithmic axis. Figre 8 is a plot of the lagged flatness factor for MIACID calclated with thermistors spaced at 5 centimeters and at 50 centimeters; the flatness has been 190 Johns H opkins APL Technical Digest, Volme 6, Nmber 3

6 D. C. Dbbel - Redction in Finestrctre Contamination of Internal Wave Estimates r::: '- "0 C'C :::J '" :::J 'en '" :s C 'en c "0 (f) bits 17 bits 18 bits '" S 10 C'C LL centimeter isotherms Freqency (hertz) 10 1 Figre 6-The atospectral density of two thermistors and the noise resistor from gradiometer 10, Eqivalent bit noise levels are indicated on the right for a 20 operating range, :::J c E > ' C 'en 10-2 c " S 0 Cl Wavenmber (cycles per m inte) Figre 7-Atospectral density plots of the normalized temperatre and isotherms, calclated for a vertical ensemble average of temperatres. Note the sbstantial drop in vale for the MIA Lag (meters) Figre a-lagged flatness compted for three series: isotherms calclated at 5 centimeters, isotherms calclated at 50 centimeters, and the temperatre (75 kilometers of data averaged), CID at 50 centimeters as compared to temperatre, and the frther drop for the MIACID at 5 centimeters. The faster the flatness drops to near the theoretical gassian vale of 3.0, the more gassian is the series. The horizontal scales for which the series appears gassian are indicated by the lags over which {3 is nearly 3.0. It is clear from Fig. 8 that the MIACIDs are more nearly mltivariate gassian and therefore are better estimators of internal wave displacement than is jst pre temperatre. It is also clear that the hypothesized vertical resoltion effect is important becase the 5- centimeter-based MIACIDs are better beh,aved (Le., are more nearly gassian over a broader range of horizontal wavelengths) than are the 50-centimeterbased MIACIDs. Figre 9 is a plot of flatness excess for the same three qantities shown in Fig. 8. It demonstrates the improvement in horizontal independence achieved with MIACID at 50 centimeters as compared to temperatre, and the frther improvement for MIACID at 5 centimeters. The two straight lines represent slopes of -1, implying complete independence, and -0.25, a typical vale for temperatre. 14 It is obvios from the plot that even the MIACIDs at 5 centimeters are not completely independent, bt it is also clear that they are more nearly so than either of the other two series. The reason for the change in slope of the flatness excesses at high lag vale is more likely attribtable to the smaller nmber of samples sed in the estimate, Johns H opkins APL Technica l D igesl, Volme 6, N mber 3 191

7 D. C. Dbbel - Redction in Finestrctre Contamination oj Internal Wave Estimates en en S 10 0 en en (,) X W 10'ro=r ,-- 10' 10 2 Lag (meters) Figre 9-Lagged flatness excess compted for three series: isotherms calclated at 5 centimeters, isotherms calclated at 50 centimeters, and the temperatre (75 kilometers of data averaged). combined with a change in the actal independence at the higher wavenmbers. In any event, the plot is meant to convey only the qalitative assessment made above, and the vales of the fnction at high lag are not of great importance. CONCLUSIONS A sbstantial redction in the levels of finestrctre contamination fond in estimates of internal wave displacements has been achieved. This redction is attribtable to the very-high-resoltion temperatre data from which the displacements were estimated and to the actal displacement and calibration algorithms applied to the data. Redctions were seen both in the ato spectral estimates and in the lagged flatness estimates, which spports the original passive finestrc- tre hypothesis. The scales over which the contamination redction seems to be most effective are the 20-meter and longer horizontal wavelengths. This is not srprising becase it is clear from theoretical considerations that trblent events shold become very prevalent at meter scales. The wavelength region between the two domains is a gray area where waves may exist given specific environmental conditions, bt these short-scale waves are mch more likely to be attribtable to local sorces and will, therefore, be nonindependent and less gassian. Frther improvements in the redction of finestrctre contamination may be achievable becase the present reslts are based on data that reqired sbstantial corrections for wild points and voltage drift. It is considered likely that the data were corrpted to some degree even after these corrections. The reslts may be considered, therefore, a least pper bond for finestrctre contamination redction and they demonstrate that reasonably accrate estimates of flid motion are achievable in an efficient manner from high-resoltion towed-thermistor-chain data. REFERENCES 10. M. Phillips, "On Spectra Measred in an Undlating Layered Medim," J. Phys. Oceanogr. I, 1-6 (1971). 2 R. O. Reid, "A Special Case of Phillips' General Theory of Sampling Statistics for a Layered Medim," J. Phys. Oceanogr. I, (1971). 3 C. Garrett and W. Mnk, " Internal Wave Spectra in the Presence of Finestrctre," J. Phys. Oceanogr. I, (1971). 4 R. S. McKean, "Interpretation of Internal Wave Measrements in the Presence of Finestrctre," J. Phys. Oceanogr. 4, (1974). 5T. M. Joyce and Y. J. F. Desabies, "Discrimination between Internal Waves and Temperatre Finestrctre," J. Phys. Oceanogr. 7,22-32 (1977). 6c. C. Eriksen, "Measrements and Models of Finestrctre, Internal Gravity Waves, and Wave Breaking in the Deep Ocean," J. Geophys. Res. 83, (1978). 7 M. W. Roth, "The Separation of Oceanic Temperatre Finestrctre and Internal Motion," Johns Hopkins APL Tech. Dig. 3, (1982). 8y. Desabies and M. C. Gregg, " Reversible and Irreversible Finestrctre," J. Phys. Oceanogr. 11, (1981). 9 M. G. Briscoe, "Gassianity of Internal Waves," J. Geophys. Res. 82, (1977). lot. M. Dillon, "Vertical Overtrns: A Comparison of Thorpe and Ozmidov Length Scales," J. Geophys. Res. 87, (1982). II F. F. Mobley, A. C. Sadilek, C. J. Gndersdorf, and S. D. Speranza, "A New Thermistor Chain for Underwater Temperatre Measrement," MTS IEEE Oceans '76 Con! Proc., (1976). 12E. H. Kidera, "A Motion-Compensated Lanch/ Recovery Crane," Ocean Eng. 10, (1983). 13 R. A. Fisher, Contribtions to Mathematical Statistics, John Wiley & Sons, New York (1950). 14 C. L. Hindman, Fine and Microstrctre Effects on Towed Temperatre Measrements, TRW Report I-UT-00 (1983). 192 Johns Hopkins APL Technical Digest, Volme 6, Nmber]

8 D. C. Dbbel - Redction in Finestrctre Contamination of Internal Wave Estimates THE AUTHOR DANIEL C. DUBBEL is a physicist in the Hydrodynamics Grop of the Sbmarine Technology Department. Born in St. Pal in 1955, he stdied at the University of Minnesota, where he received his B.S. degree in physics in He received his M.S. degree from the University of Wisconsin in While there, he specialized in high-energy experimental physics, with emphasis on the hadron jet phenomenon. Mr. Dbbel joined APL in After completing the Associate Staff Training Program, he joined the Wave Physics Grop, where he began working in physical oceanography. Since that time, his work has been oriented toward nderstanding small-scale pperocean trblence and internal-wave phenomena. He is crrently stdying the relationship between small-scale trblence and shortwavelength internal waves. fohns Hopkins APL Technica l Digesc, Volme 6, Nmber J 193