Shear, Strain, and Richardson Number Variations in the Thermocline

264 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 27

Shear, Strain, and Richardson Number Variations in the Thermocline. Part I: Statistical Description

ROBERT PINKEL

Marine Physical Laboratory, Scripps Institution of Oceanography, La Jolla, California

STEVEN ANDERSON

Woods Hole Oceanographic Institution, Woods Hole, Massachusetts

(Manuscript received 4 February 1994, in ®nal form 18 June 1996)

ABSTRACT z, (2-m, 2.1-ץ/␩ץ) ,Quasi-continuous depth±time observations of shear (5.5-m, 6-min resolution) and strain min resolution) obtained from the R/P FLIP are applied to a study of Richardson number (Ri) statistics. Data were collected off the coast of central California in the 1990 Surface Waves Processes Experiment. Obser- vations are presented in Eulerian and in isopycnal-following frames. In both frames, shear variance is found to scale as N2 in the thermocline, in agreement with previous ®ndings of Gargett et al. The probability density function for squared shear magnitude is very nearly exponential. Strain variance is approximately uniform with depth. The magnitude of the ¯uctuations is suf®cient to in¯uence the Ri ®eld signi®cantly at ®nescale. To model the Richardson number, the detailed interrelationship between shear and strain must be speci®ed. Two contrasting hypotheses are considered: One (H I) holds that ¯uctuations in the cross-isopycnal shear are independent of isopycnal separation. The other (H II) states that the cross-isopycnal velocity difference is the quantity that is independent of separation. Model probability density functions for Ri are developed under both hypotheses. The consideration of strain as well as shear in the Richardson calculation increases the probability of occurrence of both extremely low and high values of Ri. The observations con®rm this general prediction. They also indicate that, while neither hypothesis is strictly correct, H II appears to be a much better approximation over the most commonly observed values of Ri.

1. Introduction Munk (1981, hereafter M81), and Desaubies and Smith (1982, hereafter DS82). Here we resume the discussion The dissipation of mechanical energy and the mixing armed with of scalar quantities is found to occur in isolated patches /␩ץ ,in the ocean thermocline. Microscale turbulent motions 1) an improved statistical model for ®nescale strain -z (where ␩ is the vertical displacement of an isoץ initially derive their energy from larger (®nescale 1± 10 m) ¯ows, which are prone to instability. The Rich- pycnal surface) and the resulting variations in VaÈis- 2 2 2 ardson number, Ri ϭ N /S , where N is the squared aÈlaÈ frequency (Pinkel and Anderson 1992, hereafter 2 2 2 (z) is the PA92ץ/vץ) z) ϩץ/uץ) VaÈisaÈlaÈ frequency and S ϭ squared shear, mediates one class of these instabilities 2) new observations of shear and strain obtained from (Miles 1961; Howard 1961). To predict the space±time the Research Platform FLIP during the Surface variability of the Richardson number, the coevolution 2 Waves Processes Program (SWAPP). Coded pulse of the ®nescale shear and strain (N ) ®elds must be Doppler sonar (5.5-m vertical resolution over 45± understood. 300 m) and pro®ling CTDs (2-m vertical resolution, The statistical modeling of the Richardson number 720 pro®les/day to 420 m) provide a fresh view of has been addressed in pioneering studies by Bretherton the upper-ocean shear and strain ®elds. (1969), Garrett and Munk (1972, hereafter GM72), The modeling effort here differs from previous work in that: Corresponding author address: Dr. Robert Pinkel, Marine Physical Laboratory, Scripps Institution of Oceanography, University of Cal- ● It is based on observations that are quasi-continuous ifornia, San Diego, La Jolla, CA 92093-0213. in depth and time. E-mail: [email protected] ● The focus is shifted from a study of the gradient

᭧1997 American Meteorological Society

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Richardson number to a study of the ®nite difference isopycnal displacement Richardson number Ri(⌬z). ● The contribution of N2 variability to the pdf of Ri is ␩(␳, t) ϭ z(␳, t) Ϫ z(␳) (1a) considered explicitly (as in DS82). While this con- isopycnal separation tribution is not large at the ϳ10 m scales associated ⌬z (t) ϭ z(␳ , t) Ϫ z(␳ , t) (1b) with most existing observations, its importance in- ij i j creases at the smaller scales more directly relevant normalized separation to mixing processes. (t) z(t)/ z (1c) ● The distinction is made between statistical averages ␥ijϭ⌬ ij⌬ ij formed in Eulerian and isopycnal following (hereafter and semi-Lagrangian or s-L) frames. finite difference strain The Eulerian±semi-Lagrangian distinction is moti- ␥Ã(ijt)ϭ␥ ij(t)Ϫ1 (1d) vated by the observation (Fig. 1) that ®nescale shear is both advected and strained by the vertical motion of the are given in terms of the instantaneous depth z(␳,t)of internal wave®eld. This is most easily seen in the depth an isopycnal surface of density ␳. range 80±200 m, where both the low frequency shear As a matter of notation, expected values of ¯uctuating and high frequency internal waves are energetic. Typ- quantities are denoted by angle brackets. The subscript ically, wave±shear interaction is discussed in terms of ϽϾE or ϽϾL are used to specify the reference frame the refraction of high frequency waves by the shear (Eulerian or semi-Lagrangian) in which the averages are associated with lower frequency motions (e.g., Breth- accumulated. Basic-state variables are denoted by the overbar, for example, z ( ), (z), N2(z). They correspond erton 1966). Here we see that high frequency motions ␳ ␳ to averages formed in a wave-free environment. Dis- can signi®cantly distort the lower frequency back- tortion of the basic state by the motion ®eld is repre- ground. This observation is possible because the vertical sented multiplicatively, for example, N2(t) ϭ N2 ␥Ϫ1(t). resolution of the velocity observations is comparable to 2 2 Ϫ1 2 In an Eulerian frame, ͗N ͘E ϭ N ͗␥ ͘E ϭ N , ren- the typical vertical displacement scale of the internal dering the de®nitions consistent. In a semi-Lagrangian 2 waves. Self-deformation of the wave®eld should be ex- 2 2 Ϫ1 frame, ͗N ͘L ϭ N ͗␥ ͘L Ն N (Pinkel et al. 1992). plicitly accounted for in any simple model of the Rich- In this work, with a focus on the variability of both ardson number. shear and strain, it is of value to consider an s-L de- An introduction to the modeling of shear and strain scription of the horizontal velocity ®eld uL(␳,t). The is presented in section 2. Probability density functions motivations for this approach are dual. First, Anderson (pdfs) of Richardson number are obtained as two-line (1992) has observed that the time evolution of the hor- derivations once the necessary groundwork is laid. The izontal velocity ®eld is qualitatively more consistent observations of shear, strain, and Richardson number with linear internal wave theory when viewed in an s-L are presented in section 3. Estimated pdfs of Ri are frame than it is when viewed in the traditional Eulerian 1 found to be in good agreement with model predictions. (uE(z,t)) frame. The critical issue of the statistical independence of Time evolution in an Eulerian frame has been termed shear and strain is discussed in section 4. The various ``contaminated'' by the presence of ®ne structure in in- modeling options lead to signi®cant differences in the stantaneous pro®les of ␳(z,t) and u(z,t) (Phillips 1971; pdfs for Richardson number, as well as differences in Garrett and Munk 1971; McKean 1974). The implica- the values of statistical correlations that can be derived tion is that observations in an s-L frame are not con- from the models. taminated. However, the existence of contamination in an Eulerian frame is no guarantee of apparent linear behavior in the s-L frame. Thus, Anderson's observation 2. A reversible ®nestructure approach to shear is of signi®cance. and strain modeling The second motivation for considering an s-L description of horizontal velocity stems from interest in Desaubies and Gregg (1981) ®rst suggested that much the Richardson number (or alternatively, its inverse), of the ®nescale detail in vertical pro®les of scalar prop- which embraces both shear and strain (VaÈisaÈlaÈ frequency) ®elds. In the s-L description of ®nite differenced erties is a consequence of the intense straining of the quantities, we have thermocline by the internal wave®eld. They termed this deformation-related detail ``reversible ®ne structure.'' More recently, Pinkel and Anderson (1992) have adapt- ed the reversible ®nestructure perspective in examining 1 The two frames are related by uE(zo,t) ϭ uL(␳(zo,t)), where ␳(z,t) depth±time evolution of strain. A kinematic framework is the inverse of z(␳,t). This inverse always exists in the reversible is de®ned whereby ®nestructure approximation as density is monotonic with depth.

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y)2]1/2 as estimated by the 161-kHz Doppler sonar. Colors represent shearץ/␷ץ) x)2 ϩץ/uץ)] FIG. 1. A 12-h sample of shear magnitude S ϭ magnitude. The solid lines give the depths of a selected set of isopycnals, as determined by the pro®ling CTD. Shear estimates in the upper mixed layer are corrupted by a variety of technical problems. Beyond 350 m, weak echoes result in degraded estimates. In the intervening region a clear picture of the shear ®eld is seen. The shear ®eld, in addition to in¯uencing the propagation of high frequency waves, is also vertically advected by them.

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22Ϫ1 Nij(t)ϭN´␥ ij (t) (2) under H II in the sense of PA92, averaging Eq. (6) for isopycnals constrained to bracket a ®xed reference Ϫ1 Sij(t) ϭ (⌬u ij(t)/⌬z ij)␥ ij (t). (3) depth. We term this average `Èulerian II,'' or `È II,'' Early modeling efforts (e.g., GM72, M81) considered and include an expression for the Eulerian II pdf of only the role of shear variability in determining the val- Richardson number below for completeness. The cross- ue of the Richardson number. The VaÈisaÈlaÈ frequency isopycnal study required to verify its accuracy would was taken as a climatological constant. We term this the be an interesting complement to traditional Eulerian ob- ``traditional model'' of the Richardson number servations. In their earlier study DS82 modeled shear and strain 22 2 2 2 2 Ri(t;␳i,␳ j) ϭ N /S ij(t) ϭ⌬zN ij␥ ij(t)/⌬u ij(t). (4) as independent Gaussian quantities. Thus S2 is described If one wishes to extrapolate from large (ϳ10 m) by the ␹2 probability density, which at two degrees of scales, where the traditional model is perhaps adequate, freedom (corresponding to the sum of the squares of the to smaller scales of more immediate relevance to the two shear components) has exponential form: overturning process, VaÈisaÈlaÈ frequency ¯uctuations 1 22 must be properly considered (DS82). P(S2)ϭ eϪS/͗S͘ (7a) ͗S2͘ At this point, the issue of the statistical independence of shear and strain becomes central. DS82 assumed that or shear squared ¯uctuations are in fact independent of P(r) ϭ eϪr, (7b) ¯uctuations in strain. Presumably, cross-isopycnal dif- 2 2 ferences in velocity, ⌬uij, adjust to changing isopycnal where r ϵ S /͗S ͘. separation such that shear is unaffected. We refer to this Recently, Gregg et al. (1993) have presented con- as the hypothesis I (H I) model. An alternative hypoth- vincing evidence that, when statistics are accumulated esis (H II) is that the velocity differences are themselves in an Eulerian frame, 10-m differenced S2 is well de- independent of isopycnal separation. The local shear scribed by the ␹2 distribution. Their data were obtained squared is then dependent on strain. Clearly, if one of from a free-fall vertical pro®ler. Statistical con®dence these plausible models is true, the other must be false. was gained through averaging in depth rather than time However, it is possible that neither of these limiting at ®xed depth/density (as here). hypotheses is true (§ 4). When considering statistics accumulated in an s-L Quite different statistical models result from these frame, either S2 or ⌬u2 might be described by (7). Ap- contrasting hypotheses. If Ri is expressed as a product plying the SWAPP data to this issue, both variables are of statistically independent quantities, one has seen to exhibit near exponential pdfs. We proceed with the modeling, taking (7) for the pdf of both S2 and ⌬u2, 22 RiI ϭ N /(S (t)´␥(t)) H I (5) as is convenient. Ri ϭ N22⌬z ´␥(t)/⌬u 2(t). H II (6) Modeling strain as a Gaussian quantity leads to po- II tential dif®culty. The Gaussian pdf admits the possibility Interestingly, not all mean properties (such as the ex- that isopycnal surfaces can cross. This corresponds to pected value of Ri) are sensitive to this fundamental an inversion in a density pro®le whose monotonicity is issue. However, the nature of many important correla- an inherent aspect of the reversible ®nestructure model. tions (such as ͗N2S2͘ or ͗N2Ri͘) is strongly affected (ap- DS82 truncated their Gaussian pdf to prohibit the ex- pendix A). istence of density inversions. This distinction between statistical models is not rel- More recently, PA92 identi®ed the gamma pdf as ap- evant when considering traditional Eulerian observa- propriate for the description of strain statistics. Using tions (e.g., Eriksen 1978). If velocity differences ⌬u2 ϵ CTD-derived isopycnal data (comparable to those de- 2 (u(z1) Ϫ (u(z2)) are independent of strain, then the scribed below), they found squared shear u2/ z2 is as well. ⌬ ⌬ ␬Ϫ1 Ϫ␬␥ PA92, however, introduce an alternative form of `Èu- PL(␥) ϭ ␬(␬␥) e /⌫(␬), (8a) lerian'' average. As in an s-L study, they partition the P (␥) ϭ (␬␥)␬ eϪ␬␥/⌫(␬). (8b) thermocline using a set of isopycnal surfaces of mean E separation ⌬z. Rather than considering time averages Here ␬ ϭ ␬0⌬z, where ␬0 is a constant of order 1.1 along (or between) ®xed isopycnals, they consider only mϪ1, and the subscripts L and E refer to the semi-La- that pair of isopycnals that bracket a ®xed reference grangian and Eulerian frames. In addition to describing depth zo. From realization to realization, the identity of the data accurately, the gamma pdf is tractable analyt- the isopycnal pair changes. Pinkel and Anderson im- ically, as will be seen below. plied that, for passive scalars, such an Eulerian cross- It is convenient to introduce a scale Richardson num- 2 isopycnal average is relevant to the traditional Eulerian ber Ri* ϭ N2/͗S2͘ (H I), or Ri* ϭ N2 ⌬z /͗⌬u2͘ (H II). ®nite difference over a vertical separation ⌬z. When The pdf of the normalized Richardson number R ϭ Ri/ considering products of strained quantities, such as Ri, Ri* and its inverse RϪ1 ϭ RiϪ1´Ri* can then be calcu- this is not the case. We can de®ne an Eulerian average lated, as a function of mean separation ⌬z.

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TABLE 1. Probability density functions of normalized Richardson number R ϭ Ri/Ri* and inverse Richardson number as a function of

normalized separation ␬ ϭ ␬0⌬z. Hypothesis I Hypothesis II S22is ␹ , independent of ␥ ⌬u22is ␹ , independent of ␥ 2 Ri*(⌬z) ϭ N22/͗s ͘ Ri*(⌬z) ϭ N22⌬z /͗⌬u ͘ ␬ϩ1 2 ␬ (␬ϩ1)/2 1 ␬R s-L P (R ͦ ⌬z) ϭ K (͙␬/R) P (Rͦ⌬z)ϭ IL R⌫(␬)΂΃R ␬Ϫ1 IIL R2 []␬Rϩ1 ␬ϩ2 2 ␬ ␬/2 (␬ϩ1) ␬R Eulerian P (R ͦ ⌬z) ϭ K (͙␬/R) P (R ͦ ⌬z) ϭ IE R2⌫(␬)΂΃R ␬ IIE ␬R2 []␬Rϩ1 ␬ϩ1 2 1 s-L I (RϪ1ͦ ⌬z) ϭ (␬RϪ1() kϩ1)/2 K (͙␬RϪ1) I (RϪ1 ͦ ⌬z) ϭ IL RϪ1⌫(␬) ␬Ϫ1 IIL []1ϩRϪ1 /␬ ␬ϩ2 2 ␬ ϩ 11 Eulerian I (RϪ1ͦ ⌬z) ϭ (␬RϪ1)␬/2 K(2͙␬RϪ1) I(RϪ1ͦ⌬z)ϭ IE ⌫(␬) IIE ␬[]1ϩRϪ1/␬

Applying the scale Richardson number to expressions P (R ⌬z) ϭ r[eϪr][␬(␬Rr)␬Ϫ1eϪ␬Rr/⌫(␬)] dr. (13) (5) and (6), one obtains IIL ͦ ͵ Ϫ1 Rϭ(rI␥) HI (9a) 22 This integral is easily evaluated, yielding rIϵS/͗S͘ ␬ RϭrϪ1␥ HII (9b) ␬␬R II P(Rͦ⌬z)ϭ . (14) rϵ⌬u22/͗⌬u ͘ IIL II R(␬Rϩ1)[]␬R ϩ 1 Corresponding expressions for the Eulerian pdf as The present task is to produce model pdfs of these well as pdfs of inverse Richardson number are presented expressions, given the accepted expressions [Eqs. (7) in Table 1. and (8)] for r and ␥ variability. At this point, it is a Note that the vertical scale enters these expressions

textbook problem. in two ways. First, ␬ ϭ ␬0⌬z. Also, the scale Richardson Given two independent random variables, a and b, number Ri* depends on the ®nite difference of either 2 2 with joint probability density P(a,b) ϭ Pa(a)Pb(b), the ͗S ͘ (H I) or ͗⌬u ͘ (H II). Both are dependent on ⌬z. probability density of the ratio c ϭ a/b is The pdfs derived under H I and H II are plotted in

ϱ Figs. 2a, b. Several aspects of these plots deserve com- P (c) ϭ ͦbͦP (cb)P (b) db (10) ment. First, the differences between observations in the c ͵ ab s-L and Eulerian frames are quite signi®cant at small 0 mean separations. The differences diminish rapidly as (Papoulis 1984). separation increases. For ␬ Ͼ 12 it will be challenging We apply this result to (9a), ®rst introducing g ϭ rϪ1 to detect differences between reference frames experi- with pdf G(g) ϭ gϪ2eϪ1/g. We have for the s-L case mentally. Under H I, (S2 and N2 are independent), the Eulerian ϱ 1(␬␥)␬Ϫ1 P (R ⌬z) ϭ ␥ eϪ1/R␥ ␬ edϪ␬␥ ␥. pdf is more skewed than its s-L counterpart. Differences ILͦ 22 ͵␥R ⌫(␬) diminish with increasing mean separation ␬0⌬z. Under 0[][] H II, the Eulerian pdf for Ri is identical to its H I form. (11) However, the s-L pdf now has a greater skewness than This integral is evaluated in terms of the modi®ed Bessel the Eulerian (2b). While the differences between the s-L and true Eulerian observations are slight, they do not function ␬⌬ (e.g., Prudnikov et al. 1986, integral 2.3.16: 1), yielding diminish at large differencing scales. In contrast, the differences between the Eulerian II (where statistics are 2 ␬ (␬ϩ1)/2 accumulated between isopycnal pairs constrained to PIL(R ͦ ⌬z) ϭ K␬Ϫ1(2͙␬/R). (12) bracket ®xed reference depths) and s-L models are sig- R⌫(␬) R ΂΃ ni®cant at small ␬, decreasing substantially as ␬ in- Expressions for the pdf in an Eulerian frame, as well creases. as for pdfs of inverse Richardson number, are presented It is also apparent in Fig. 2 that extremely low values in Table 1. of R occur less frequently under H I than H II. To Under H II, ⌬u2 is a ␹2 random variable that is pre- investigate statistical behavior at small R, it is attractive sumed independent of strain. Applying (10) to (9b) we to consider pdfs of normalized inverse Richardson num- have, in the s-L case, ber RϪ1 ϵ RiϪ1Ri* (Fig. 3, Table 1). For reference, the

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FIG. 2. Model probability density functions of Richardson number derived under hypotheses I (a) and II (b). The full pdfs are plotted over the range R ϭ Ri/Ri* ϭ 0 Ϫ 5. An enlargement of the low Richardson number R Ͻ 1 portion of the pdf is overplotted, to enable detailed examination of the differences. Under H II, an Eulerian average is formally identical to that under H I. One can de®ne an alternative Eulerian average, embracing cross-isopycnal differences (§ 2). For completeness we present the alternative Eulerian model in (b) termed Eulerian II, as well as the standard Eulerian result. pdf of the ``traditional'' inverse Richardson number is (SWAPP). The experiment site was in water 4 km also plotted. Since only variations in S2 are considered, deep, approximately 600 km west of Pt. Conception, the traditional pdf is exponential (Anderson 1992; Pol- California. The mean VaÈisaÈlaÈ pro®le at the site is giv- zin 1992). en in Fig. 4. As an aspect of this effort, two pro®ling It is seen that, relative to the traditional model, there CTDs and a downlooking Doppler sonar were oper- are more occurrences of extremely large and extremely ated. The CTDs, Seabird Instruments SBE-9, pro®led small RϪ1 when the effects of strain are considered. For at 130-s intervals from 2 to 220 m (upper) and 200 example, at ␬ ϭ 4 (Fig. 3a) an Eulerian ``RϪ1 ϭ 5'' to 420 m (lower). The fall rate of the instruments was occurrence is more than twice as likely when strain is approximately 3.5 m sϪ1. At this speed it was not considered versus neglected. A similar situation exists necessary to pump the conductivity cells to achieve under H II in the s-L frame (Fig. 3c). As one progresses adequate temporal response. To minimize ``salinity to larger mean separations (Figs. 3b, 3d), the signi®- spiking,'' the conductivity data required response cor- cance of strain diminishes and all models approach the rection. A section of test data of nearly uniform sa- traditional exponential form. linity was selected. The transfer function between the temperature gradient and conductivity gradient was estimated for this section. This transfer function was 3. Observations then used to ``match'' the conductivity data to the In February and March 1990, the Research Platform temperature signal throughout the depth±time domain FLIP was trimoored at 35ЊN, 127ЊW, collecting data of the experiment. Both the temperature and corrected in support of the Surface Waves Processes Program conductivity signals were low-pass ®ltered (1.6-m

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FIG. 3. Model probability density functions of RϪ1 ϭ RiϪ1 * Ri*. Models are derived under H I (a, b) and H II (c, d). The traditional exponential model is plotted as a reference. cutoff) prior to the calculation of density. The pro- of the code was 5 kHz. The associated vertical resolution cedure is discussed in more detail in Anderson (1992) of the sonar was 5.5 m. The theoretical velocity pre- and Sherman (1989). cision of the instrument was 0.74 cm sϪ1 after one min- The Doppler sonar used in this study was a 161-kHz ute of averaging, as limited by the performance of the four beam device constructed at the Marine Physical code. In fact, the dominant limit to precision was the Laboratory of Scripps. Originally designed for Arctic surface wave®eld, which moved both the upper ocean research, the system was here mounted on FLIP's hull and FLIP at speeds far greater than typical thermocline at a depth of approximately 15 m. A repeat sequence velocities. Six-minute average velocity pro®les (30 pe- code (Pinkel and Smith 1992) consisting of six repeats riods of a 12 second wave) were subsequently formed of a 7-bit Barker code was transmitted. The bandwidth in an effort to reduce the ``wave noise'' in our signals.

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During the ®rst week of the experiment, however, shear variance was low, approximately 50% of GM75. Strain variance was also reduced relative to G±M by a factor

of .75 (Anderson 1992). The strain scaling constant ␬0 has a value of 2 mϪ1 for the SWAPP observation, in contrast to the suggested ``canonical'' value of 1.2 mϪ1 (PA92). Probability density functions of strain formed in an s-L frame closely resemble the Gamma pdf, scaled as suggested by PA92. Pdfs formed in an Eulerian frame were signi®cantly in¯uenced by the vertical advection of subinertial features. A basic tenet of the reversible ®ne structure model is that, in the absence of straining, underlying scalar property pro®les are smooth. This is not the case here. While ®xed irregularities can be accounted for in the modeling, for simplicity we will focus on the comparison of the s-L model pdfs and the corresponding s-L observations.

a. Vertical variability of shear variance and mean Richardson number As an initial study, pro®les of mean square shear and inverse Richardson number are formed over the depth range 120±296 m (below the N2 maximum). We plot these as a function of N2(z), rather than depth itself, to investigate their scaling behavior. In Fig. 5a, ®nite difference shear variance, uncor- FIG. 4. The SWAPP mean VaÈisaÈlaÈ pro®le. rected for sonar resolution, is plotted as a function of N2. Averages are formed in both Eulerian and semi- Issues of resolution and precision are further discussed Lagrangian frames. Typically the s-L averages exhibit in appendix B. slightly greater variance. In spite of signi®cant vari- SWAPP was sited in a region of both surface and ability, the shear variance in both frames appears to scale subsurface frontal activity. Energetic subinertial features linearly with N2, consistent with previous ®ndings (e.g., were present in the upper 120 m early in the experiment. Gargett et al. 1981). Later, yeardays 67±70 (8±11 March), a major subsurface In Fig. 5b, the s-L shear variance is compared with front was seen at 200±300 m (Anderson 1992). To min- the related quantity, the variance of the s-L velocity 2 imize in¯uence of these thermohaline features, a subset difference, ⌬u (t,␳i,␳j), normalized by the square of the of the overall observations was selected for the Rich- mean isopycnal separation. The variance pro®les are ardson number study. Analysis efforts focus on the nearly congruent at the 18- and 10-m mean separations. depth region 150±250 m over the ®rst 7.5 days of the At 6 m, there is some indication of enhanced variance experiment, days 60±67.5 (1±8 March). of true shear relative to ͗⌬u2͘/⌬z2 at depth. In restricting the data to avoid ``anomalous'' condi- Pro®les of mean inverse Richardson number ͗S2/N2͘ tions the statistical precision of the results was reduced. are presented in Fig. 6. In addition to the s-L and E II Speci®cally, the subset extended over only 8 periods of averages, pro®les of the ``traditional'' buoyancy nor- the dominant near-inertial shear, and 1±2 dominant malized shear variance ͗S2͘/N2 are also presented, av- wavelengths in depth. The bandwidth of the near-inertial eraged in an Eulerian frame. These inverse Richardson ®eld was suf®ciently narrow that relatively few inde- number estimates are far from critical, varying from .2 pendent realizations of the shear (and strain) process to .4 as spatial separations decrease from 18 to 6 m. were captured. Briscoe (1977) suggested an 11-h inter- The estimates appear approximately independent of val is required between independent shear observations, VaÈisaÈlaÈ frequency. The pro®les are suf®ciently irregular based on data obtained at 28ЊN. Thus, the statistical with depth that weak dependencies cannot be detected. stability of the SWAPP probability estimates was limited. However, relative to previous 1-d time series or b. Shear variance and Richardson number as a vertical pro®les, the present 2-d observations represent function of vertical differencing scale an advance. Climatologically, the overall SWAPP shear variance A second issue is the rate at which shear variance was 1.3 times the Garrett±Munk (1975) nominal value. decreases with increasing mean isopycnal separation

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FIG. 5. Plot of ͗S2͘ as a function of VaÈisaÈlaÈ frequency at mean vertical differencing intervals of 6, 10, and 18 m. Lines of scale Richardson number are drawn for reference. In (a) Eulerian and semi-Lagrangian averages are compared. In (b) true s-L shear is compared with the normalized velocity difference ͗⌬u2͘ /⌬z2. Pro®les are computed without correction for instrument resolution and noise, using data from below 120 m.

(s-L) or differencing interval. This is examined in Fig. instruments of improved precision and resolution. Cor- 7a for data obtained at 150- and 250-m mean depths. responding plots of inverse Richardson number are pre- Independent of reference frame or data type, shear vari- sented in Figs. 8a,b. Estimates of RiϪ1 at 250 m are ance decreases with increasing vertical mean separation, uniformly less than their 150-m counterparts (8a). The approaching a ⌬zϪ1 dependence at separations greater difference increases with decreasing mean separation. than 10 m for the shallow data. The deep data appear To the extent that mean values of RiϪ1(⌬z) vary with to undergo this transition more gradually, a fact gen- depth, the variation will be most apparent at small ⌬z. erally consistent with WKB scaling and inconsistent The enhanced depth variability at 6-m mean separation with the empirical dictum that the vertical wavenumber relative to that at 10 and 18 m could be seen more clearly spectrum of shear does not change form with depth in Fig. 6 if the abscissa were plotted on a linear scale. (Gargett et al. 1981). As with the straight shear estimates, sonar noise and A major concern is that both noise and the ®nite resolution signi®cantly affect the inverse Ri estimates resolution of the sonar affect these results. These pro- at small mean separations. In Fig. 8b the variation of cesses are modeled in appendix B. A comparison of the the inverse Richardson number is plotted versus mean 150-m data, uncorrected and ``corrected'' for resolution separation for 150-m data. Also plotted is a second set and noise is presented in Fig. 7b. Here it is assumed of estimates, where shear variance is enhanced to correct that the rms noise in a given sample of sonar data is 1 for ®nite sonar resolution and reduced to account for cm sϪ1 and that the true shear spectrum is as given in the expected noise level (appendix B). The correction Gargett et al. (1981). While the ultimate ⌬zϪ1 depen- is signi®cant for scales below 10 m. Even as corrected, dence of shear variance is unaffected by noise and res- the inverse Richardson number falls short of Munk's olution, changes at small vertical differencing intervals (1981) hypothesized value of unity at 10-m vertical are signi®cant. A 2-m ®rst difference shear variance scales by a factor of 50%. Relative to the stability of estimate obtained from a noise-free sensor of extremely the thermocline, shears in the early part of SWAPP were high resolution would be approximately a factor of 1.8 mild. greater than the SWAPP sonar estimate. When considering differencing intervals of 6 m and above, the effects c. Probability density functions of R and RϪ1 are considerably more modest. Nevertheless, Fig. 7b Probability density functions of R and RϪ1 were clearly suggests that much can be gained by developing formed and averaged (in an s-L frame) over yeardays

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60±67.5, depths 150±200 m. The restricted sampling substantially avoids observed frontal activity. Observed pdfs are compared with modeled forms (section 2) in Fig. 9. The comparison is conducted for mean separation of 6, 10, and 18 m. Values of Ri* are determined from the 50-m average of ͗S2͘ or ͗⌬u2͘. Other than the ob-

served shear variance and the strain parameter ␬0 ϭ ␬/ ⌬z, which is assigned the value 2 mϪ1 for all observations, there are no adjustable parameters in these ®ts. In general, the H II model (⌬u2 independent of strain) does a better job of ®tting the observations for commonly observed values of R. The model is slightly skewed relative to the observations at ⌬z ϭ 6m, (Fig. 9a), overpredicting the occurrence of small R events. The H I model severely underestimates the peak value of the observed pdfs. The observed pdfs of inverse Richardson number RϪ1 ϭ RiϪ1 R* emphasize the regions where instability can be expected. In Figs. 9d±f, the logarithm of the pdf is plotted in an attempt to focus on the incidence of large RϪ1 occurrences. Reference lines for the H I, H II, and traditional models are drawn. Unusually large RϪ1 events occur more frequently than are predicted by the traditional model, particularly at small separation. The H I model is generally more consistent with the observations in this region. The H II model overpredicts rel- FIG. 6. The expected value of inverse Richardson number plotted ative to the observations. as a function of N2. These pro®les are computed without correction We note, that the pdf estimates are signi®cantly ``data for instrument resolution and noise, using data from below 120 m. starved'' at large RϪ1. Accurate estimates of the value

FIG. 7. Finite difference ͗S2͘ as a function of the mean differencing interval ⌬z. In (a) estimates are centered on depths of 150 and 250 m. In (b) the 150-m averages are shown, as calculated from the raw data and following corrections for instrument noise and resolution (appendix B). An rms error of 1.0 cm sϪ1 is assumed for the velocity.

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FIG. 8. The expected value of inverse Richardson number as a function of the mean differencing interval ⌬z used in the shear and strain calculations. Estimates are presented (a) at observed depths of 150 and 250 m and (b) as corrected for sonar noise and resolution (appendix B). An rms velocity error of 1 cm sϪ1 is assumed for the correction. of the pdf governing these extremely rare events require Estimates of the correlation are presented as a function longer data records and/or higher-resolution (larger Ri*) of mean separation in Fig. 11. Signi®cant correlation sensors. precludes statistical independence. Surprisingly, the observed correlations between S2 2 2 4. The statistical independence of shear squared and N are large, in spite of the fact that N variations and N2 are relatively weak over much of the range of ⌬z investigated. At 2-m mean separation, the correlations are The independence of S2 and N2 is an issue central to .2±.4 in the s-L frame, .3±.5 in the Eulerian frame. With both the statistics and the dynamics of the ®nescale increasing mean separation, the correlation grows slight- ®elds. To explore this matter, the joint probability den- ly in both frames. In contrast, the correlation between sity function of S2 and N2 can be formed in both Eulerian ⌬u2 and N2 is weakly negative (Ϫ.2) at the smallest and s-L frames. Additionally, the joint pdf of cross- separations. It grows toward zero as separation increas- isopycnal squared velocity ⌬u2, and N2 can be generated es. (Fig. 10). The joint pdfs are formed over 100 ϫ 100 The strong S2±N2 correlation signi®cantly complicates bins in r and ␥Ϫ1. Figure 10 is further smoothed using efforts to model Richardson number statistics. Several a3ϫ3 bin convolution to aid in visualization. Distinct processes that contribute to this correlation can be iden- from the pronounced central peak, a ridge of the pdf ti®ed: extends to high S2 values along ␥Ϫ1 ϭ .05 Ϫ 1(N2 (i) If, in the s-L frame, it really is the case that ⌬u2 slightly less than average). A second, less visible ridge is independent of N2, then the covariance of S2 and N2 extends to high values of N2 along r ϭ 0. This ``two- is given by armed'' form for the joint pdf cannot be expressed as a product of univariate pdfs for N2 and ⌬u2. This pre- cov(S22,N ) cludes the statistical independence of these quantities. ϭ͗SN22͘Ϫ͗S 2͗͘N 2͘ The apparent ridges in the pdf describe the occurrence (16) of rare events. The typical behavior of the thermocline ⌬u22⌬u can be investigated using the correlation between the ϭ␥Ϫ22Ń␥Ϫ1Ϫ␥Ϫ22Ń͗␥Ϫ1͘ Ό΍Ό΍⌬z22⌬z observed ⌬u2 (or S2) and N2: 22 ϭ͗⌬u2͘N/⌬z [͗␥Ϫ3͘Ϫ͗␥Ϫ2͗͘␥Ϫ1͘] Ͼ 0. ͗S22Ń͘Ϫ͗S 2͗͘N 2͘ C(⌬z) ϭ (15) (͗S4͘Ϫ͗S 221/2͘)(͗N 4͘Ϫ͗N 221/2͘) The quantities necessary to evaluate this covariance

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Ϫ1 Ϫ1 FIG. 9. Model probability density functions of R (a)±(c) and R (d)±(f), compared with SWAPP 150±200-m s-L data. With ␬0 ϭ 2 m established for the entire dataset, the shear variance is the only free model parameter at each separation. These ®ts are accomplished with the model variance set to the observed values. The H II models (dashed lines) describe the observations better than the H I models (dotted lines) at commonly experienced values of R, RϪ1.

2 2 can be easily calculated from model pdfs presented in regions of low ͗N ͘L, ͗S ͘L water up to a ®xed reference 2 2 PA92, (or appendix A, here). depth. Troughs bring regions of high ͗N ͘L, ͗S ͘L water 2 2 2 (ii) Variations in the ``background'' N pro®le refract down from above. A strong positive ͗N S ͘E covariance internal waves as they propagate vertically. Regions of can be expected if ͗͘E corresponds to a time average at high S2 can be expected in regions of high N2,inaWKB ®xed depth. approximation of the refraction process. It is important to note that an Eulerian depth average, Climatologically, both N2 and ͗S2͘ decrease with such as might be obtained from an instrument that makes depth (below 120 m). Crests of internal waves lift isolated vertical pro®les, would not see an apparent co-

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FIG. 10. An estimate of the joint probability density function of ⌬u2 and N2. Estimates are formed with a mean differencing interval of 10 m using data from 150±200 m. variance caused by vertical advection. Nevertheless, sig- The strong correlation between S2 and N2 convinc- ni®cant coherences between N2 and S2 have been ob- ingly refutes hypothesis I, that the quantities are statis- served (Rubenstein et al. 1982; Evans 1982). Vertical tically independent. From a modeling perspective, it advection is not the sole cause of the Eulerian corre- would be extremely attractive for hypothesis II to be lation. Our observed correlation in the S-L frame sup- valid. Otherwise, additional physical information is re- ports this conclusion. quired to fully describe the observations. It is thus dis- turbing to see the negative correlation between ⌬u2 and N2 in Fig. 11. We suggest that this correlation is in part due to the ®nite resolution of the Doppler sonar. When a given pair of isopycnals is close togetherÐrelative to the resolution of the sonarÐthe estimated value of ⌬u2 will be less than the actual value. When the same pair is widely separated, the ⌬u2 values will be unbiased. Thus, a ten- dency for low ⌬u2 to be associated with high N2 can result from ®nite sonar resolution. An apparent negative correlation is created. This apparent correlation is easily modeled (appendix B) and is plotted in Fig. 11 as a reference line. The observed correlation is well described by the model, except perhaps at those lags so small that the ®nite resolution of the CTD becomes signi®cant. The preliminary correlation analysis is consistent with the hypothesis that ⌬u2 and N2 are independent 2 2 FIG. 11. The observed correlation between squared shear and VaÈis- quantities, while S and N are not. However, visual aÈlaÈ frequency for the depth range 150±200 m. The Eulerian corre- inspection of the joint pdf (Fig. 10) suggests a more lation is positive and relatively independent of spatial lag. The vertical complicated picture. As a metric of the ``degree'' of advection of the mean S2 and N2 ®elds, both of which decrease with increasing depth, might account for a signi®cant fraction of this cor- independence of shear and strain we can apply the joint relation. Semi-Lagrangian estimates experience advective effects dif- pdfs to estimate ͗S2͘ or ͗⌬u2͘ at each value of ␥Ϫ1.If ferently. Again, a positive correlation is observed whose magnitude the ®elds are statistically independent, the observed increases with increasing mean separation. In contrast, estimates of ͗S2͘(␥Ϫ1)or͗⌬u2͘(␥Ϫ1) will be independent of ␥Ϫ1,as the correlation of ⌬u2 Ϫ N2 in an s-L frame are weakly negative at small lag, increasing toward zero as the lag increases. A model de- well. These estimates, normalized by the overall 2 2 Ϫ1 scribing the effect of ®nite sonar resolution on the correlation estimate ͗S ͘,͗⌬u ͘ are presented in Figs. 12 a±c. The pdfs of ␥ (appendix B) produces a qualitatively similar result. are overplotted to give some indication of the relative

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( ␥ ⌬ P ∫ / 2 dS ) 1 Ϫ

␥ , , and (c) s-L 2 2 S S ( P 2 S , is plotted as the solid 1 ∫ Ϫ

, ␥ 1 Ϫ , (b) s-L 2 ␥ S are independent, this moment 1 Ϫ

␥ and 2 S estimates at smaller isopycnal separation (ap- 2 u ⌬ as well. For reference, the pdf of 1 at each value of inverse strain Ϫ

␥ ͘ 2 S ͗ for the depth interval 150±200 m. If 2 , normalized by the overall mean squared shear, is here plotted as a function of 2 N . 12. The mean / 2 dS IG ) N 1 F Ϫ should be independent of ϭ are considered. In thethe s-L sonar, which data preferentially anpendix reduces attempt B). is The made upper to dashed line account gives for the the ``corrected'' ®nite values. resolution of ␥ line, with arbitrary scale. Three distinct cases, (a) Eulerian

Unauthenticated | Downloaded 09/26/21 09:54 AM UTC 278 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 27 number of samples contributing to each estimate of ͗S2͘ processes that precede a mixing event and those that or ͗⌬u2͘. If these quantities are truly independent of ␥Ϫ1, follow. the shear variance estimates should be horizontal lines. To model the Richardson number ®eld more precisely Even accounting for ®nite sonar resolution and noise the effects of strain must be included. It is also necessary (appendix B) there is signi®cant variability in these to specify the statistical relationship between shear and curves. strain. Two contrasting hypotheses are considered. The In both the Eulerian and s-L frames, the expected ®rst holds that, in an s-L frame, cross-isopycnal vari- value of S2 increases monotonically with increasing ␥Ϫ1. ability in shear, ⌬u(t)/⌬z(t), is independent of isopycnal The rapid increase in ͗S2͘ in the s-L frame (12b) is not separation, ⌬z(t). The alternative hypothesis maintains the signature of measurement error. A ®xed velocity that cross-isopycnal velocity differences ⌬u(t) are them- precision corresponds to a shear noise variance that selves independent of separation. scales as ␥Ϫ2, rapidly increasing as isopycnal separation Using a Gamma probability density for the strain ®eld decreases. Such a noise can be modeled and subtracted (PA92), model pdfs of Ri are derived under both hy- from the observed ͗S2͘ estimates. Even when a suf®- potheses. The inclusion of N2, as well as S2 ¯uctuations, ciently large noise is hypothesized that the residual sig- is seen to result in the increased occurrence of extremely nal becomes negative at large ␥Ϫ1 (͗S2͘ less than zero low and high values of Ri in the models. The data are for ␥Ϫ1 Ͼ 3.5, Fig. 12b), the basic increasing trend adequate to con®rm this general result. remains at smaller ␥Ϫ1. The interesting physics here relates to the depen- In contrast to the s-L case, the in¯uence of sonar noise dence/independence of shear and strain. Depending on is uniform as a function of strain in the Eulerian ob- the hypothesis selected, regions of low Ri are more like- servations (Fig. 12a). The dramatic increase in ͗S2͘ for ly to occur in layers of low (H I) or high (H II) VaÈisaÈlaÈ Ϫ1 Ϫ1 ␥ Ͼ ␥mode is seen here as well. frequency. The observed pdfs of Ri are generally more The previous correlation study suggests that the s-L consistent with H II, especially at the more commonly ͗⌬u2͘ estimates might be independent of instantaneous observed values of Ri. However, closer examination of strain. However, a more complicated picture appears in the joint pdfs of shear and strain suggest that neither Fig. 12c, consistent with the visual interpretation of Fig. hypothesis is strictly correct. 10. The mean square velocity difference is largest for The dependence of shear2 ¯uctuations on strain is both uncommonly large (␥Ϫ1 Ͼ 1.7) and small (␥Ϫ1 Ͻ such that when N2 is much less than its average value, .4) values of inverse strain. The apparent zero corre- S2 is as well (Fig. 12a,b). Thus, the occurrence of low lation between ⌬u2 and N2 is a consequence of the Ri events is reduced. This is consistent with Polzin's roughly symmetric variation of ͗⌬u2͘ relative to the peak (1992) ®nding based on vertical pro®ling observations. in the pdf of N2. We emphasize that, while low Ri events are less com- The data support the proposition that neither S2 nor mon than if N2 and S2 were independent, they are still ⌬u2 is independent of N2. Investigations at larger vertical more common than predicted by a model that neglects separations support this general conclusion. The hy- strain altogether (Figs. 9d±f). pothesis that ⌬u2 and N2 are independent (H II) is a The statistical dependence of shear and strain at small much better approximation than the corresponding as- scales is not surprising. One can imagine two adjacent sertion on S2 and N2 (H I). Signi®cant departures from ¯uid layers de®ned by three isopycnal sheets. If two of constancy in the ͗⌬u2͘ versus N2 curve occur at rarely the sheets converge locally (a low strain event), ¯uid experienced values of N2. In contrast ͗S2͘ increases rap- must be exhausted laterally from the region. The re- idly with N2 for all but the largest values of N2. sulting shear2 depends on 2ץ ␥ . zץtץDiscussion and summary ΗΗ .5 Observations of ®nescale shear, strain, and Richard- Here S2 and N2 might be uncorrelated, given that only son number have been obtained using Doppler sonar the absolute value of strain variation matters. However, and pro®ling CTD systems mounted on the Research they are not independent. Platform FLIP. A signi®cant qualitative observation is The two-dimensional nature of the SWAPP dataset that the ®nescale shear ®eld appears to be advected by has yet to be fully exploited. Speci®cally, the combined the vertical displacement of both low- and high-fre- depth±time variability of Ri should be explored. The quency internal waves. goal is to establish the appropriate timescale for low Ri Statistical measures of the shear and Ri ®elds are events and to quantify the depth dependence of this formed in both Eulerian and isopycnal following frames. scale. To a ®rst approximation (neglecting strain), the pdfs of S2 and RiϪ1 are exponential. While there is some ques- Acknowledgments. The authors thank Eric Slater, tion as to whether turbulence in the sea is distributed Lloyd Green, Michael Goldin and Chris Neely for the lognormally, the inverse Richardson number certainly design and construction of the CTD and sonar systems is not. This emphasizes the distinction between those used in this effort, as well as their at-sea operation.

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Discussions with Jerome Smith and Mark Merri®eld TABLE A1. Moments of normalized strain. proved extremely helpful. This work was supported by ␬ ϭ ␬0⌬z ␥ ϭ⌬z/⌬z Ϫ1 the Of®ce of Naval Research under Contract N00014- ␬0(m ) Semi-Lagrangian Eulerian 94-1-0046 and the National Science Foundation Grant ͗␥3͘ (␬ ϩ 2)´(␬ ϩ 1)/␬2 (␬ ϩ 3)(␬ ϩ 2)(␬ ϩ 1)/␬3 OCE91-10553. The support and cooperation of Bob ͗␥2͘ (␬ ϩ 1)/␬ (␬ ϩ 2)´(␬ ϩ 1)/␬2 Weller, Al Plueddemann, the other SWAPP investiga- ͗␥͘ 1(␬ϩ1)/␬ tors, and the crew of the Research Platform FLIP is ͗␥Ϫ1͘ ␬/(␬ Ϫ 1) 1 appreciated. ͗␥Ϫ2͘ ␬2/(␬ Ϫ 1)(␬ Ϫ 2) ␬/(␬ Ϫ 1) ͗␥Ϫ3͘ ␬3/(␬ Ϫ 1)(␬ Ϫ 2)(␬ Ϫ 3) ␬2/(␬ Ϫ 1)(␬ Ϫ 2)

APPENDIX A Shear, Strain Covariance Modeling It is more instructive to examine covariances of speci®c interest. For example, if ⌬u2 and ␥ are independent Pinkel and Anderson (1992) have determined empir- in an s-L frame (H II), then the covariance of S2 and ically that isopycnal separation statistics are well ap- N2 is given by Eq. (16). Using Table A1, this expression proximated by Gamma probability density functions. is easily evaluated, yielding They consider the normalized isopycnal separation ␥ ϵ SN22S 2N 2 ⌬z(t)/⌬z and consider the subset of the Gamma family ͗ ͘Ϫ͗IILLL͗͘ ͘ Ϫ1 (Erlang pdfs) constrained such that ͗␥͘sL ϭ 1, ͗␥ ͘E ϭ 1: 222 Ϫ3 Ϫ2Ϫ1 ϭ͗⌬u͘LLN/⌬z [͗␥ ͘Ϫ͗␥ ͗͘␥ ͘] ␬Ϫ1 Ϫ␬␥ PL(␥ ͦ ⌬z) ϭ ␬(␬␥) e /⌫(␬) (A1a) 222 2 ϭ2␬oL͗⌬u ͘ N [␬/(␬ Ϫ 1) (␬ Ϫ 2)(␬ Ϫ 3)]. (A3) ␬␬␥ PE(␥ͦ⌬z)ϭ(␬␥) e /⌫(␬). (A1b) 2 This is a positive covariance which, assuming ͗⌬u ͘L ϳ⌬zϩ1, decreases with increasing mean separation. Here ␬ ϭ ␬0⌬z, where ␬0 is a constant of the process. Ϫ1 In turn, if H I is in fact correct, one can evaluate the In the more familiar Gaussian framework␬0 is the apparent covariance between ⌬u2 and N2 that must re- analog of the correlation scale of the strain ®eld and ␬0 is the bandwidth of the vertical wavenumber spectrum sult: of strain. In this non-Gaussian model,␬Ϫ1 also sets the 22 2 2 0 ͗⌬uN͘LLL Ϫ͗⌬u ͗͘N͘ magnitude of spectrum. All higher moments of the pro- 222 2Ϫ1 cess are speci®ed by ␬0, as well. Table A1 lists the ®rst ϭ͗S͘LLN ⌬z [͗␥͘Ϫ͗␥͗͘␥ ͘] several moments of the process, as these are useful in the statistical modeling of strain, shear, Richardson ͗S22͘N␬ 2 ϭϪ2L . (A4) number, and related quantities. It is important to note ␬␬2 Ϫ1 o [] that this model is not supported by observations at scales Ϫ1 2 smaller than approximately 3 m. In turn, many of the This covariance is negative. If ͗S ͘L ϳ⌬z , the co- statistical quantities derived become singular at small ␬. variance will have but weak dependence on ⌬z. Using the moments of the strain pdfs it is relatively Do large values of RiϪ1 occur in sheets (regions of easy to model a variety of statistical properties of in- higher than average N2) or in layers? Since RiϪ1 ϵ S2/ terest. For example, using Eqs. (7) and (8), the expected N2, the covariance of RiϪ1 and N2 is just values of Richardson number are cov(RiϪ12 ,N ) ϭ͗S 2͘Ϫ͗RiϪ12͗͘N ͘. (A5) 22 ͗S␥͗͘LLS͘ Considering averages formed in an s-L frame, the ͗RiϪ1͘ϭ ϭ ϭRi*Ϫ1 (A2a) IL NN22IL appropriate expression under H I is

2Ϫ12 Ϫ12 2 2 Ϫ1 ͗⌬u ␥ ͗͘⌬u͘␬␬covILLLLL (Ri ,N ) ϭ͗S͘Ϫ͗S͗͘␥͗͘␥ ͘ ͗RiϪ1͘ϭLL ϭ ϭRi*Ϫ1 IIL 22 22 IIL ⌬zN ⌬zN␬Ϫ1 ␬Ϫ1 2 ϭϪ͗S͘L /(␬Ϫ1). (A6) (A2b) The corresponding covariance formed in an Eulerian ͗S22␥͗͘S͗͘␬ϩ1͘ ␬ϩ1 frame is just ͗RiϪ1͘ϭEE ϭ ϭRi*Ϫ1 IE 22 IE Ϫ1 2 2 NN␬␬ covIE(Ri , N ) ϭϪ͗S͘E/␬. (A2c) Under H II the s-L covariance becomes ͗⌬u2␥Ϫ12͗͘⌬u͘ ͗⌬u2͘ ͗RϪ1͘ϭEE ϭ ϭRi*Ϫ1 . (A2d) cov (RiϪ12 ,N ) ϭ L [͗␥Ϫ2͘Ϫ͗␥Ϫ12͘] IIE ⌬zN22⌬zN 22 IIE IILL⌬z2

22 2 In practice, it is dif®cult to make precise estimates of ϭ␬oL͗⌬u ͘ /[(␬Ϫ1) (␬ Ϫ 2)]. (A7) Ri at scales less than 10 m in the thermocline. Thus, ␬ The corresponding Eulerian II covariance is Ͼ 5±10 for typical observations, and the differences Ϫ1 2 2 2 between the various averages are slight. covIIE(Ri N ) ϭ͗⌬␬0 u͘E/[␬(␬Ϫ1)]. (A8)

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Under H I, the s-L covariance is negative. Regions the variance of the ␹2 distribution as a function of ␥Ϫ1. of low dynamics stability (large RiϪ1) are more likely In particular, as isopycnal separation is decreased, the to occur in regions of low static stability (s small N2). variance of ͗⌬u2͘ in the ␹2 distribution can be decreased. Under H II the reverse is true! The conventional Eu- A recipe for implementing this variance decrease can lerian H I covariance is negative, as is its s-L counter- be created, assuming that the true underlying shear spec- part. The nonconventional Eulerian II covariance, based trum is known. If, for example, the ``true shear spec- on observations from pairs of isopycnals that bracket a trum'' is known, then it is easy to apply a model ®lter ®xed reference depth, is also negative. that mimics the resolution of the sonar, to produce a While this result emphasizes the fundamental impor- ``measured shear spectrum.'' From these ``true'' and tance of the shear-strain independence issue, several ``measured'' spectra, ``true'' and ``measured'' velocity cautions are worthwhile. Speci®cally, it is possible that difference variances ͗⌬u2͘(⌬z) can be derived: neither H I nor H II are valid. Also, as vertical scales decrease to the point where hydrodynamic instability is k⌬z ͗⌬u22͘(⌬z) ϭ S(k)sinc dk. (B3) likely, the model strain pdfs are known to be in error. ͵ ΂΃2␲ Here S(k) is the shear spectrum and sinc (x) ϵ sin (␲x)/ APPENDIX B (␲x). The ®nite resolution effect is modeled by choosing Instrument Resolution and Noise the Gargett et al. (1981) model shear spectrum as the It has proven dif®cult to obtain reliable estimates of true spectrum and a version multiplied by Richardson number variability on scales comparable to k´5.5m typical overturn scales in the thermocline (1±2 m). An sinc2 objective of the present modeling effort is to infer the ΂΃2␲ statistical behavior of Ri at small scales based on consistent application of principles determined from larger- as the measured spectrum. The ®nite difference velocity variance of each is then calculated according to (B3). scale observations. 2 This renders the issue of verifying the models prob- A set of ␹ (exponential) pdfs is then generated, lematic. The primary concern is with the velocity es- 1 ␹22(⌬u ,␥Ϫ1) ϭ eϪ⌬u22/␴ (␥Ϫ1) (B4) timates, where both accuracy and resolution are an issue. ␴(␥Ϫ1) The range resolution of a Doppler sonar is given by with variances adjusted according to the modeled vari- ⌬zres ϭ 0.5c͗Tp Ϫ ␶͘ cos ␾ (B1) ance ratio, assuming a mean isopycnal separation ⌬z. (e.g., Pinkel and Smith 1992). The joint pdf (B2) is then generated, using Eq. (12) for 2 p␥. Since the ®rst moment of ⌬u is now a function of Here c is the speed of sound, Tp is the duration of Ϫ1 the transmitted acoustic pulse, ␶ is the processing time ␥ , the pdf is no longer separable. This process can be 2 lag, and ␾ is the angle of the sonar beam relative to repeated at a variety of ⌬z, with the covariance of N and ⌬u2 vertical. For the 161-kHz system used in SWAPP, ⌬zres equals 5.5 m (Anderson 1992). The ®nite resolution of the sonar reduced the variance of the observed velocity cov(N22,⌬u ) ϭ N 2 ␥Ϫ12⌬uP(␥Ϫ12,⌬u ͦ⌬z)d␥⌬Ϫ12u and shear estimates, presumably without affecting the [͵͵ Gaussianity of these signals. In the attempt to develop velocity time series in is- Ϫ␥Ϫ1P(␥Ϫ12,⌬uͦ⌬z)d␥⌬Ϫ12u opycnal following coordinates, interesting problems can ͵͵ arise. As mentioned in section 5, if one asserts that H II is correct, velocity differences between isopycnal sur- ϫ⌬͵͵ uP2 (␥Ϫ12,⌬u ͦ⌬z)d␥⌬Ϫ12u] faces should be independent of the separation of these surfaces. However, as isopycnal pairs converge, ap- (B5) proaching the resolution limit of the sonar, estimates of calculated at each mean separation. The resulting ap- ⌬u2 will be biased low. When the isopycnals diverge, parent correlation due to ®nite sonar resolution is plotted the ⌬u2 estimate will experience decreased bias. A spu- in Fig. 11. The model variance reduction ratio 2 2 2 2 rious negative correlation between N and ⌬u results. ͗⌬u ͘measured/͗⌬u ͘true was used to ``correct'' the observed This effect can be modeled by assuming that the true estimates of S2 and Ri in Figs. 7b and 8b. correlation is zero, with ␥Ϫ1 and ⌬u2 statistically in- Noise in the sonar measurements is also a signi®cant dependent. If the true joint pdf is given by the separable concern. SWAPP represents the ®rst open ocean ex- product periment where coded-pulse technology was used to im- prove sonar precision (Pinkel and Smith 1992). The 161- P(␥Ϫ1,⌬u2ͦ⌬z) ϭ p (␥Ϫ1ͦ⌬z)␹2 (⌬u2ͦ⌬z), (B2) ␥ kHz sonar transmitted six repeats of a 7-bit Barker code. we can create a resolution limited joint pdf by changing The duration of each bit was 0.25 ms, corresponding to

Unauthenticated | Downloaded 09/26/21 09:54 AM UTC FEBRUARY 1997 PINKEL AND ANDERSON 281 a nominal bandwidth of 4 kHz. The expected velocity fornia, San Diego. [Available from University of California, San precision of such a code is 0.74 cm sϪ1 after a one- Diego, 9500 Gilman Dr., La Jolla, CA 92093.] Bretherton, F., 1966: The propagation of groups of internal gravity minute (80 pulse) average. waves in a shear ¯ow. Quart. J. Roy. Meteor. Soc., 92, 466± In fact, the observed precision is considerably worse, 480. more nearly 1 cm sϪ1 after six minutes of averaging. , 1969: Waves and turbulence in stably strati®ed ¯uids. Radio The major problem is not associated with the sonar, but Sci., 4, 1279±1287. Briscoe, M. G., 1977: Gaussianity of internal waves. J. Geophys. rather the velocity ®eld of the upper ocean. Surface Res., 82, 2117±2126. wave velocities are two orders of magnitude more en- Desaubies, Y. J. F., and M. C. Gregg, 1981: Reversible and irreversible ergetic than the low frequency motions of interest here. ®nestructure. J. Phys. Oceanogr., 11, 541±556. While wave motion decays rapidly with depth, the , and W. K. Smith, 1982: Statistics of Richardson number and wave-induced motion of the sonar imparts a relative instability in oceanic internal waves. J. Phys. Oceanogr., 12, 1245±1259. velocity, which is uniform with range. If the rms hor- Eriksen, C. C., 1978: Measurements and models of ®ne structure, izontal FLIP motion is of order 5 cm sϪ1 and associated internal gravity waves, and wave breaking in the deep ocean. J. primarily with the dominant swell at a 12-second period, Geophys. Res., 83, 2989±3009. then a six-minute (30 wave) average results in an rms Evans, D. L., 1982: Observations of small-scale shear and density Ϫ1 structure in the ocean. Deep-Sea Res., 29, 581±595. velocity noise of order 5/͙ 30 ϭ .9 cm s . With more Gargett, A. E., P. J. Hendricks, T. B. Sanford, T. R. Osborn, and A. advanced processing, one can attempt to remove this J. Williams III, 1981: A composite spectrum of vertical shear range-coherent noise on a pulse-by-pulse basis. How- in the upper ocean. J. Phys. Oceanogr., 11, 1258±1271. ever, this was not done in SWAPP. Garrett, C. J. R., and W. H. Munk, 1971: Internal wave spectra in The velocity imprecision has a signi®cant effect on the presence of ®ne structure. J. Phys. Oceanogr., 1, 196±202. ,and , 1972: Ocean mixing by breaking internal waves. estimates of shear: Deep-Sea Res., 19, 823±832. 2 , and , 1975: Space-time scales of internal waves, A progress Ã22 ͗S͘ϭ͗(u1122ϩ⑀ Ϫu ϩ⑀)͘/⌬z report. J. Geophys. Res., 80, 291±297. 22 Gregg, M. C., H. E. Seim, and D. B. Percival, 1993: Statistics of ϭ͗(⌬u)22͘/⌬z ϩ2͗⑀͘/⌬z. (B6) shear and turbulent dissipation pro®les in random internal wave- ®elds. J. Phys. Oceanogr., 23, 1787±1799. As separation ⌬z decreases, the true velocity differ- Howard, L. N., 1961: Note on a paper of John W. Miles. J. Fluid ence tends toward zero, leaving the noise to play a dom- Mech., 10, 509±512. inant role. Conversely the noise contribution decays as McKean, R. S., 1974: Interpretation of internal wave measurements ⌬zϪ2 with increasing separation, whereas the signal in the presence of ®ne structure. J. Phys. Oceanogr., 4, 200± Ϫ1 213. tends to diminish as ⌬z . To model the effect of noise, Miles, J. W., 1961: On the stability of heterogeneous shear ¯ows. J. a factor of 2͗⑀2͘/⌬z2 was subtracted from the shear es- Fluid Mech., 10, 496±508. timates in Figs. 7b and 8b, prior to the correction for Munk, W. H., 1981: Internal waves and small scale processes. Evo- sonar resolution. lution of Physical Oceanography, B. A. Warren and C. Wunsch, Eds., The MIT Press, 264±290. Velocity imprecision can also in¯uence covariance Papoulis, A., 1984: Probability, Random Variables, and Stochastic estimates of physical interest. For example, considering Processes. 2d ed. McGraw-Hill, 576 pp. the covariance of S2 and N2 under H II, the role of noise Phillips, O. M., 1971: On spectra measured in an undulating layered can be easily seen: medium. J. Phys. Oceanogr., 1, 1±6. Pinkel, R., and S. Anderson, 1992: Toward a statistical description cov(S22,N ) of ®nescale strain in the thermocline. J. Phys. Oceanogr., 22, 773±795. ϭ͗SN22͘Ϫ͗S 2͗͘N 2͘ , and J. A. Smith, 1992: Repeat sequence codes for improved performance of Doppler sonar and sodar. J. Atmos. Oceanic N2 Technol., 9, 149±163. ϭ[(͗⌬u22͘ϩ2͗⑀͘)(͗␥Ϫ3͘Ϫ͗␥Ϫ2͗͘␥Ϫ1͘)] 2 , J. T. Sherman, J. A. Smith, and S. Anderson, 1991: Strain: ⌬z Observations of the vertical gradient of isopycnal vertical dis- N2 placement. J. Phys. Oceanogr., 21, 527±540. ϭ͗⌬u2͘[(1 ϩ NSR(⌬z))(͗␥Ϫ3͘Ϫ͗␥Ϫ2͗͘␥Ϫ1͘)]. Polzin, K. L., 1992: Observations of turbulence, internal waves, and ⌬z2 background ¯ows: An inquiry into the relationship between (B7) scales of motion. MIT/WHOI, WHOI-92-39, Woods Hole Oceanographic Institution, 243 pp. Here NSR ϭ 2͗⑀2͘/͗⌬u2͘ is the noise-to-signal ratio. Prudnikov, A. P., Y. A. Brychkov, and O. I. Marichev, 1986: Integrals Thus, while the noise contribution is signi®cant, the and Series, I. (English Transl), Gordon and Breach, 798 pp. Rubenstein, D., F. Newman, and R. Lambert, 1982: Coherence be- role of the noise is easily modeled in these simple tween strati®cation and shear in the upper ocean. Science Ap- estimators. plications International Tech. Rep. SAI-82-614-WA, 20 pp. [Available from Science Applications International, P.O. Box 1303, McLean, VA 22102.] REFERENCES Sherman, J. T., 1989: Observations of ®ne-scale vertical shear and strain in the upper-ocean. Ph.D. thesis, University of California, Anderson, S. A., 1992: Shear, strain, and thermocline vertical ®ne San Diego, 145 pp. [Available from University of California, structure in the upper ocean. Ph.D. thesis, University of Cali- San Diego, 9500 Gilman Dr., La Jolla, CA 92093.]

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