Thickness of the mixed bottom layer in the Northern Atlantic

ARTICLE IN PRESS Deep-Sea Research I 53 (2006) 387–407 www.elsevier.com/locate/dsr Sheared turbulence in a weakly stratified upper ocean I.D. Lozovatskya,c,, E. Rogetb, H.J.S. Fernandoa, M. Figueroab,d, S. Shapovalovc a Arizona State University, Environmental Fluid Dynamics Program, MAE, Tempe, AZ 85287-9809, USA b University of Girona, Campus de Montilivi, E-17071 Girona, Catalonia, Spain c P.P. Shirshov Institute of Oceanology, Russian Academy of Sciences, Moscow 117997, Russia d CICESE, Department of Physical Oceanography, Ensenada, B.C., Mexico Received 23 December 2004; received in revised form 30 September 2005; accepted 5 October 2005 Available online 19 December 2005 Abstract Microstructure, ADCP and CTD profiles taken in the North Atlantic along 531N under moderate and high winds showed that the median of log-normally distributed kinetic energy dissipation rate e within the upper mixing layer is 1.5
107 W/kg and the layer depth, on the average, is 40 m. Assuming that mixing efficiency g is a constant (g ¼ 0:2), the following scaling is proposed for the normalized eddy diffusivity: K^ b ¼ K b =ku z ¼ ð1 þ Ri=Ricr Þp Pr1 tr , where K b ¼ ge=N 2 , N2 is the squared buoyancy frequency, u the surface friction velocity, Ri the local Richardson number, Prtr ¼ 1 þ Ri=Rib the turbulent Prandtl number, p ¼ 1 or 2/3, Ricr ¼ 0.1 and Rib ¼ 0.1 or 0.05. The power-law function with p ¼ 1 relates the asymptotes of Kb(Ri) to the buoyancy scale LN ðe=N 3 Þ1=2 at RibRicr and to the shear scale LSh ðe=Sh3 Þ1=2 at Ri5Ricr. If p ¼ 2=3, the lengthscale LR ¼ ðe=N 2 ShÞ1=2 replaces LN in spectra of ocean microstructure than due to the influence of local shear. This mixing regime corresponds to intermediate Richardson numbers (0.25oRio2). Alternatively, if g is not a constant, but an increasing function of Ri for 0oRio1, then K^ b ðRiÞ shows a very weak dependence on Ri for Rio0.25. Numerical experiments using a one-dimensional q2e model with different Kb parameterizations indicate that the measured mixed-layer depth agrees well with model results when the diffusivity is parameterized with an Ri-dependent g (the GISS model approach). The modeled dissipation profiles, however, resembled microstructure measurements better if g is treated as a constant and the proposed formula for K^ b ðRiÞ is used. r 2005 Elsevier Ltd. All rights reserved. Keywords: Turbulence; Dissipation; Diffusivity; Mixing parameterization; Upper ocean; Modeling 1. Introduction Drift currents and associated wind-induced mixing are important components of upper oceanic Corresponding author. Arizona State University, Environ- mental Fluid Dynamics Program, MAE, Tempe, AZ 85287-9809, USA. Tel.: +1 480 965 5597; fax: +1 480 965 1384. E-mail address: [email protected] (I.D. Lozovatsky). 0967-0637/$ - see front matter r 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.dsr.2005.10.002 dynamics. Depending on the wind stress and variability of buoyancy flux at the sea surface, turbulent mixing in the upper layer can produce vertical transports of kinetic energy, momentum and heat down to the underlying pycnocline or across the sea surface to the atmosphere. The upper ocean in high and mid-latitudes is usually well mixed by sustained moderate and high winds. With the exception of relatively short periods of intense ARTICLE IN PRESS 388 I.D. Lozovatsky et al. / Deep-Sea Research I 53 (2006) 387–407 convective mixing, the residual mean stratification of the upper layer is weakly stable with an averaged 2 squared buoyancy frequency N~ ¼ ðg=r0 Þqr̄=qz of 6 2 or less; here r0 is the reference about 10 s density, qr̄=qz the mean vertical density gradient (z is positive downward), and g the gravity. Small density gradients and large vertical shears Sh ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðqu=qzÞ2 þ ðqv=qzÞ2 in the upper layer (Sh2105–104s2) provide favorable conditions for developing Kelvin–Helmholtz (K–H) instability, and thus local turbulent mixing, when the Richardson number, Ri ¼ N 2 =Sh2 , falls below a critical value Ricr. Traditionally such turbulence is parameterized in terms of exchange coefficients also known as eddy viscosity KM and diffusivity Kb. The exchange coefficients are assumed constant (say KMo and Kbo, respectively) for Ri5Ricr, Ri dependent at intermediate Ri whilst decreasing to the molecular values at RibRicr. For locally generated turbulence, this variation of KM and Kb can be modeled as K M K Mo ð1 þ bRiÞm and K b K bo ð1 þ bb RiÞn , (1) where m, n, b, and bb are constants. Some commonly used variables are, m ¼ 1/2, b ¼ 10, n ¼ 3/2, bb ¼ 10/3 (Munk and Anderson, 1948); n ¼ 1, bb ¼ 5 (Monin and Yaglom, 1975); m ¼ 2, n ¼ 3, b ¼ bb ¼ 5 (Pacanowski and Philander, 1981, hereafter PP-81); m ¼ 1.5, n ¼ 2.5, b ¼ bb ¼ 5, for Ri40.25 (Peters et al., 1988, hereafter PGT-88), n ¼ 3/2, bb ¼ 10, (Pelegri and Csanady, 1994); and m ¼ 1/2, n ¼ 3/2, b ¼ 10, bb ¼ 20 (Paka et al., 1999). Based on the direct measurements of microstructure in the ocean, Soloviev et al. (2001) recently reported KM ¼ KbK0(1Ri/Ricr) for RioRicr ¼ 0.25, which signifies a very weak dependence on Ri at Ri50.1. A similar result has been found for atmospheric stratified flows (Monti et al., 2002). On the other hand, PGT-88 showed an extremely steep variation KbRi9.6 for RioRicr, but obviously, on physical grounds, the diffusivities are expected to asymptote to their non-stratified values as Ri-0. The discrepancy between different results, however, is curious as it points to the unreliability of existing scaling for Ri5Ricr. We believe that the difficulties of N2 calculations in quasi-homogeneous layers, the limited bandwidth of airfoil sensors that are used to measure small-scale shear in highly turbulent zones and insufficient averaging of Kb and Ri are major contributors to the high uncertainty of Kb(Ri) at small Ri (see Appendix B for details). In addition, possible deviation of mixing efficiency g (at subcritical Ri) from the assumed constant value g ¼ 0.2 (Oakey, 1982) used for Kb calculations is also an important contributor to the mystery (e.g., Strang and Fernando, 2001a, b; Smyth et al., 2001; Canuto et al., 2001; Fringer and Street, 2003). In this paper, new microstructure measurements of the dissipation rate e, thermohaline, and velocity structure in the upper layer of North Atlantic are presented to further support the notion of relatively weak growth of Kb at Ri5Ricr  0:1 (Paka et al., 1999). Note that the dependence of eddy viscosity KM on Ri is believed to be weaker than of that of Kb, but the critical Richardson number seems to play the same role in the demarcation of weakly and strongly stratified regimes for both cases. Recent atmospheric studies (e.g., Monti et al., 2002) indicate that KM may even grow at Ri4Ricr possibly because of the increase of momentum transfer by internal waves in stably stratified boundary layers (Lee et al., 2005); a parameterization based on this result has produced better predictions in mesoscale atmospheric models. Direct measurements of vertical momentum flux in the marine environment remain technically challenging (Moum, 1998), and thus understanding of the role of momentum transfer in oceans has made only limited progress. The data to be discussed herein (Section 2) were obtained during the 9th cruise of R./V. Akademik Ioffe (P.P. Shirshov Institute of Oceanology, Russian Academy of Sciences) along 531N in April 2001 (Tereschenkov et al., 2002), and they have already been utilized to study the response of mixed-layer depth to atmospheric forcing (Lozovatsky et al., 2005). The present work focuses on statistics of the dissipation rate e, Ri, Kb, and the buoyancy Reynolds number Rb (Section 3) and on the dependence of local Kb on Ri within the upper boundary layer (Section 4). Here, a new version of Kb(Ri) scaling is proposed and we also compared the processed data with several widely used mixing parameterizations. The effects of an Ri-dependent g on the behavior of Kb(Ri), which greatly impacts relevant numerical modeling, are illustrated and discussed in Sections 5 and 6. The main results are summarized in Section 7. ARTICLE IN PRESS I.D. Lozovatsky et al. / Deep-Sea Research I 53 (2006) 387–407 2. Observations ADCP, CTD, and microstructure profiling measurements were conducted at hydrographic stations and were taken roughly every 30 miles along 531N from the Labrador coast to the shelf of Ireland in late April 2001. Vertical profiles of horizontal velocity components u and v were obtained via a ship-mounted ADCP equipped with GPS navigation while a Neil Brown Mark III (NBIS) profiler measured the temperature–salinity–density in conjunction with a microstructure profiler MSS (Prandke et al., 2000) that measured small-scale shear [and therefore e(z)]. Severe weather prevented MSS casts at a number of stations. Technical details of the MSS sensors, a brief description of the processing of small-scale shear signal and the calculation of e are given in Appendix A. Temperature, salinity, and density profiles were processed according to the WOCE standards (Millard et al., 1990). The conductivity channels of NBIS and MSS were controlled with the water sampled by an Autosal 8400B salinometer. Appendix B contains an error analysis of the main variables (Kb, N2, Sh2, and Ri). In the near-surface layer (the depth range down to 12–20 m), ship-based ADCP measurements are noisy (e.g., Firing, 1988; Chereskin et al., 1989) and thus, taking into account 8 m ADCP vertical resolution, the depth range 0–16 m was excluded from our analysis. Microstructure measurements at depths above 10–15 m are usually considered contaminated by ship rolling (e.g., PGT88; Lombardo and Gregg, 1989; Paka et al., 1999), and therefore we decided to exclude microstructure data above zcw ¼ 16 m from the analysis. This 389 allows matching of ADCP and MSS data and to focus on the inner part of the upper mixed layer that is free of surface-wave influence. This choice of zcw is further supported by the probability distribution function of e given in Section 3. Standard meteorological observations were conducted at all stations. Winter storms and series of atmospheric cyclones that cross the Atlantic often in early spring significantly influence upper-layer dynamics. The mean wind speed during the entire cruise was 10.7 m/s. Fig. 1 shows the friction velocity u averaged over the time of measurements at each station (about 2.5 h). The mean wind speed at the stations was 8.7 m/s. Three periods of strong stormy winds were encountered during the sailing across the ocean. The direction of the sea-surface buoyancy flux Jb was convection favorable at all stations except two (Fig. 1), but wind-induced turbulence, rather than convection, dominated mixing in the upper boundary layer. This is evidenced in Section 3 by comparing characteristic dissipation rates with the estimates of mechanical and buoyancy production of turbulence. The calculations of u and Jb were made with the Matlab Air-Sea toolbox (http://sea-mat.whoi.edu), which employs bulk formulae for air–sea fluxes. 3. Turbulence statistics in the mixing layer 3.1. The mixing layer depth and averaged dissipation As pointed out by Brainerd and Gregg (1995), the depths of mixed (hD) and mixing (he) layers are the same for developed, quasi-stationary turbulence. For decaying, or fossil, turbulence, hD should exceed Fig. 1. Friction velocity u at the sea surface at drift stations of the 9th cruise of R./V. Akademik Ioffe, April 17–May 1, 2001. The stations marked by filled circles are shown in Fig. 2. ARTICLE IN PRESS 390 I.D. Lozovatsky et al. / Deep-Sea Research I 53 (2006) 387–407 he. These ideas apply if externally induced mixing is evolved within an existing upper quasi-homogeneous layer (UQHL) of the ocean. When active turbulence penetrates into a sharp pycnocline, he can exceed hD because of incomplete mixing at the pycnocline. In this paper, we have specified he as the depth (below highly turbulent near-surface layer) where e(z) rapidly decreases from about 107 to 108 W/kg. Note that this definition has a certain amount of subjectivity. It is based on our experience in visual analysis of e(z) profiles of the mixed layer while employing a formal threshold level for e (107 W/kg) to identify the bounds of the mixing layer. The highly intermittent nature of dissipation profiles, even after substantial averaging, makes it difficult to objectively determine he. The mixing layer depth calculated semi-objectively is, however, acceptable here, given that we are not focused on the analysis of he itself, but interested in a heterogeneous record of e samples devoid of near-surface effects (waves) and the influence of pycnocline turbulence. The mixed-layer depth hD, nevertheless, was identi- -9 0 -7 σθ -5 -9 -7 fied objectively by the algorithm of Kara et al. (2000), using the threshold level for variations of specific potential density dsy ¼ 0.02sy without any vertical averaging or interpolation. Several MSS profiles of sorted specific potential density sy(z) and averaged kinetic energy dissipation rate e^ðzÞ together with hD and he are shown in Fig. 2 to illustrate different regimes of mixing in UQHL (1 m spaced samples of e(z) were smoothed by a four-point running averaging). At St. 922 (53.121N, 50.801W), e.g., where turbulence is confined to a shallow mixed layer (hD ¼ 18 m) under relatively low winds, hD  he . The depths of mixed and mixing layers are identical at St. 935 (53.481N, 39.781W), where active turbulence generated during the 2nd storm (see Fig. 1) produced a deep he ¼ hD ¼ 82 m. When the wind ceased after this storm (St. 936; 53.401N, 38.451W), the dissipation profile showed a mixing layer (he ¼ 50 m) that was shallower than the well-mixed layer based on density profiles (hD ¼ 67 m). Here, the turbulence starts to decay in the lower part of the mixed layer, log 10 ε, (W/kg) -5 -9 -7 -5 -9 -7 -5 ε 10 20 30 Depth (m) 40 50 60 70 80 90 100 110 27.2 27.4 27.6 27.3 27.4 27.5 27.2 27.3 27.4 27.1 27.2 27.3 σθ 922 935 936 948 Fig. 2. Examples of the dissipation (dotted lines) and specific potential density (solid lines) profiles showing the depths of mixed (larger arrows) and mixing (smaller arrows) layers at several stations. St. 922 (53.121N, 50.801W): turbulence is confined in a shallow mixed layer under relatively low winds. St. 935 (53.481N, 39.781W): active mixing induced by the 2nd storm all over the UQHL. St. 936 (53.401N, 38.451W): decaying turbulence or turbulence penetration to a limited depth in a well-mixed layer after passage of the 2nd storm. St. 948 (52.511N, 28.701W): the development of a mixed boundary layer at the beginning of the 3rd storm due to high-level penetrating turbulence. Positions of the stations along the transect and the associated values of u are shown in Fig. 1. ARTICLE IN PRESS I.D. Lozovatsky et al. / Deep-Sea Research I 53 (2006) 387–407 and surface-generated turbulence penetrates to a limited depth. Finally, St. 948 (52.511N, 28.701W) gives an example of the mixing layer he ¼ 42 m which is deeper than the mixed layer, hD ¼ 38 m. Energetic turbulence at this station, which was associated with the beginning of the 3rd storm, has entrained fluids from the pycnocline to deepen the homogeneous layer. Note at St. 948, he penetrates deeper into the pycnocline than the mixed-layer depth, and he at this station clearly indicates the depth where e(z) within the stratified layer starts to decrease. Fig. 2 illustrates various turbulence regimes encountered, but at the majority of stations hD was in general agreement with he (see Fig. 3) because during the cruise, turbulence in the upper layer was generated predominantly by active wind forcing. Note that only those stations where MLD exceeded 10 m were used in this analysis. A relatively narrow scatter of samples in Fig. 3 suggests that most of our observations belong to quasi-stationary conditions. Only at three stations, the mixing he and mixed hD layer depths differed substantially. This occurred at the beginning of the 2nd storm (St. 933), whence turbulence quickly penetrated to a depth of about 50 m, but without destroying the stratification completely given that the wind forcing has acted only a short time. Therefore, formally detected MLD is still shallow (hD ¼ 15 m). When the storm developed, both he and hD achieved 80 m at St. 935 (see Fig. 2). Conversely, at Sts. 943 and 946, under calm wind conditions, turbulence was confined to the upper 15–20 m, but the thickness of upper homogeneous mixed layer, which was developed during the previous storm, was still about 40 m. 391 The main contribution of wind mixing to the mean level of dissipation within the mixing layer e~ ml is evident from Fig. 4, where the dependencies of e~ ml on the wind stress production ews, buoyancy flux Jb, and a combination thereof esf are shown. We estimated e~ml at each station as the integrated dissipation in the depth range heozozcw divided by the thickness of this layer. The Ekman scale LE ¼ u =f was used to evaluate the possible integral contribution of wind-stress-induced dissipation ews to the measured e~ ml . Using the law of the wall analogy, we specified ews as u3 =0:1kLE , where k ¼ 0.4 is the von Karman constant. The factor 0.1 before LE is used to link integral turbulence scale with the most energy-containing outer turbulent scale L0 within the mixed layer (see, e.g., Monin and Yaglom, 1975). The scaling for combined boundary-induced forcing (wind and convection) can be given as esf ¼ u3 =0:1kLMO , where LMO ¼ u3 =J b is the Monin-Obukhov scale. According to Fig. 4b, the correlation between e~ml and Jb is weak and the buoyancy production accounts for only 9% of the mean level of dissipation within the mixing layer. The wind stress ews correlates well with e~ ml (Fig. 4c) and accounts, on the average, for more than 90% of e~ml . The combined effect, esf , is responsible for 95% of the mean dissipation e~ ml (Fig. 4a). There is a lower correlation between esf and e~ml than between ews and e~ml due to the destructive role of the lowcorrelated pair J b  e~ ml . 3.2. Probability distribution functions In Fig. 5, a histogram of the logarithm of dissipation rate ðlog10 eÞ is shown for the depth Fig. 3. The depth of the turbulent (mixing) layer he vs. the mixed-layer depth hD obtained from MSS profiles. ARTICLE IN PRESS 392 I.D. Lozovatsky et al. / Deep-Sea Research I 53 (2006) 387–407 Fig. 4. The dependences of the mean dissipation within the mixing layer on wind stress production ews (c), buoyancy flux Jb (b), and a combined effect of the two esf (a). Fig. 5. Histogram of the logarithm of kinetic energy dissipation rate in the depth range zcwozohe (b); he is the depth of the upper turbulent layer (mixing layer), zcw ¼ 16 m is the lower boundary of the near-surface layer. Gaussian approximation is shown by heavy line. range zcwozohe (i.e., below the contaminated zone from surface effects). The probability distribution of e within the mixing layer at depths z4zcw is fairly log-normal (Fig. 5) with m ¼ /log eS ¼ 15.73, RMS(log e) ¼ 1.76 and small values of skewness (0.07) and kurtosis (0.44), which must be zero for normal distribution of natural logarithm of the dissipation log e. The corresponding mean and median values of e are /eS ¼ 3.5
107 and med(e) ¼ 1.5
107 W/kg. The Kolmogorov– Smirnov (K–S) goodness of the fit for normal distribution of log e is 0.03, which exceeds a critical value of K–S statistic at the 90% significance level (0.028) while being almost equal to the K–S goodness at the 95% level (0.031) of significance. If log10 e probability distribution is calculated using all 1 m averaged samples pertinent to the upper turbulent layer, including those from the nearsurface layer (not shown here), then it substantially departs from log-normal approximation with a long tail of almost equal probability in the range 4.5olog10 eo2. This indicates that very large values of e, which were mostly observed at zozcw, belong to a different statistical population than those having log10 ep4, formally justifying the separation of near-surface and inner sections of the mixing layer at zcw ¼ 16 m. The turbulence in the upper layer is generated mainly by wind stress, and therefore the generation mechanism below the wave-breaking zone can be construed as due to shear instability of drift currents. As such, it is useful to analyze this layer with the probability distribution function of the gradient Richardson number Ri ¼ N 2 =Sh2 , where N2 and Sh2 are the squared buoyancy frequency and ADCP vertical shear, respectively. The buoyancy frequency was computed with 4 m vertical resolution along monotonically arranged MSS density profiles obtained by a Thorpe-sorted algorithm. The shear was calculated with 8 m vertical separation ARTICLE IN PRESS I.D. Lozovatsky et al. / Deep-Sea Research I 53 (2006) 387–407 (Appendix B) and interpolated to the depth of N2 profiles. The Sh2 and N2 records were low-pass filtered with running averaging over four points and then interpolated to Dz ¼ 1 m, allowing the calculation of eddy diffusivity K b ¼ ge=N 2 , where g is the mixing efficiency traditionally taken as g ¼ 0.2 (Osborn, 1980; Oakey, 1982, 1985; Oakey and Greenan, 2004). Many authors (see, e.g., recent review of Peltier and Caulfield, 2003) suggest, however, that in the range 0oRio1, g is a growing function of Ri. In Section 5, we discuss how this dependence can change the values of Kb and therefore the Kb(Ri) dependence. The cumulative distribution function of the Richardson number F(Ri) given in Fig. 6 shows that in the depth range zcwozohe the median of Ri is as low as 0.1 and the probability of Rio0.25 is above 60%. This distribution as well as that of e (see Fig. 5) is well approximated by a log-normal probability law. Note that we did not find any noticeable correlation between the local Ri in the depth range zcwozohe and the friction velocity u at the sea surface for u p1:5
102 m=s. Only at four stations, where u exceeded 1.8
102 m/s (winds above 15 m/s), was a weak tendency noted at lower Ri. The state and intensity of mixing in UQHL can be specified by two related variables Kb and the 393 buoyancy Reynolds number Rb ¼ e=30nN 2 (Gibson, 1980; Stillinger et al., 1983). The latter can also be expressed as Rb ¼ K b =6n signifying the ratio between the eddy diffusivity and molecular viscosity n. The distribution of buoyancy Reynolds number F(Rb) in Fig. 6 shows a wide range, 10–106, indicating highly energetic turbulence almost everywhere in the mixing layer. This turbulence can be considered as locally isotropic if the criterion ðe=nN 2 Þ3=4 4200 is satisfied (Gargett et al., 1984). More than 95% of the observed Rb samples comply with this threshold, Rb440 (see Fig. 6). The median of Rb in Fig. 6 is as high as 4
103. The corresponding log-normal approximation (the straight dashed line) fits more than 80% of the upper end of F(Rb) distribution starting from Rb ¼ 6
102 . Perhaps, this signifies the lower cutoff for fully developed isotropic turbulence with cascade of turbulence energy, which is characterized by log-normal distribution of the dissipation (Gurvich and Yaglom, 1967). In the layers with Rb4600, Kb exceeds the molecular viscosity by a factor of more than 3500. About 15% of samples at the lower end of F(Rb) depart from log-normal approximation and possibly belong to the base of UQHL, where internal-wave breaking generates patches of weak intermittent turbulence. Fig. 6. The cumulative distribution functions of the Richardson F(Ri) and buoyancy Reynolds F(Rb) numbers in the depth range zcwozohe. The corresponding log-normal approximations are given by straight and dashed lines. The median of Rb is 4
103; for Ri, the median is 0.1, and the probability of Rio0.25 exceeds 60%. ARTICLE IN PRESS 394 I.D. Lozovatsky et al. / Deep-Sea Research I 53 (2006) 387–407 4. The diffusivity Kb calculated with a constant mixing efficiency c 4.1. Vertical profiles The importance of shear instability in generating turbulent mixing within UQHL along the 531N transect is illustrated in Fig. 7 using the vertical profiles of diffusivity Kb and Ozmidov scale, LN ¼ ðe=N 3 Þ1=2 ¼ ðK b =gNÞ1=2 as well as sorted density sy, shear, and the Ri at two typical Sts. 948 (developing turbulence at the beginning of the 3rd storm) and 936 (active, but decaying turbulence after the 2nd storm). The original profiles of the dissipation rate e(z) are also shown in Fig. 7 to emphasize that the general decrease of Kb and LN with depth is not a simple consequence of a stratification change, and also due to decreasing turbulence. It is interesting that the choice of the mixing layer depth he made semi-objectively from e(z) profiles roughly corresponds to the depth where Ri is close to 0.25 (St. 948) or even below 0.25 (St. 936). The averaged depth of the mixing layer calculated over all available e(z) profiles is /heS ¼ 48 m; the depth, where Rio0.1–0.25, is 38 m on the average. At the base of the mixing layer, Ri  1, the non-linear limit for shear instabilities. The laboratory stratified shear flow mixing data of Strang and Fernando (2001b) suggest that Rio0.1 corresponds to a regime of negligible density stratification effects, 0.1oRio0.25 corresponds to K–H instability dominated regime and 0.25oRio1 is characterized by both internal-wave breaking and K–H instabilities. Interestingly, at Ri  1 the K–H instabilities and internal waves may resonate, thus giving rise to the maximum mixing efficiency (also see Strang and Fernando, 2001a; Pardyjak et al., 2002). For Ri41, the flow is dominated by Hölmböe instabilities, which have weaker mixing behavior. It appears that the mixing layer base is at a state of maximum mixing efficiency, and a rise of Ri therein will lead to a rapid decrease of shear-induced mixing. The results also suggest that he is associated with the depth of a critical Richardson number, which varies in the range Ricr ¼ 0.1–1, depending on the specifics of local generation processes. The diffusivities in the mixing layer are high (102–101 m2/s), but Kb sharply decreases to 105–104 m2/s when entering the pycnocline. Slightly below the base of the mixing layer, at depths where Ri  1, the Ozmidov lengthscale decreases to about 1 m, which is a characteristic lower cutoff of the inertial subrange of the kinetic energy spectra of stratified turbulence in the oceanic thermocline (Gargett et al., 1981). In all, local shear instability appears to favor the generation of turbulence in UQHL. 4.2. Kb(Ri) parameterizations 4.2.1. Critical Ri Two Kb(Ri) dependences obtained at Sts. 933 and 939 are given in Fig. 8. Arguably, these examples represent the typical quasi-stationary balance of turbulent kinetic energy in the mixing layer below the near-surface highly turbulent zone. The stations were taken after the 1st and 2nd storm, exhibiting relatively high dissipation in the depth range 24–48 m where Rio1. In both cases, Kb tends to decrease with the increase of Ri. The trends can be approximated as Kb ¼ K Mo Pr1 , ð1 þ Ri=Ricr Þp t (2) where Prt ¼ KMo/Kbo is the turbulent Prandtl number, which in general is a function of Ri and often written as r Pr1 t ¼ at ¼ ð1 þ Ri=Ri b Þ . (3) Here, KMo and Kbo are the eddy viscosity and diffusivity in non-stratified flow, respectively, Ricr and Rib are certain critical Richardson numbers, which do not necessarily have the same numerical values; Ricr is associated with transition from a nonturbulent to a turbulent regime (or from ‘‘weak’’ to ‘‘well-developed’’ turbulence) whereas Rib indicates a specific state of stratified shear turbulent flow when vertical mixing (buoyancy flux) starts to be affected by stratification more significantly than the momentum flux. For Ricr, the canonical values are 0.25 and 0.20 (e.g., Monin and Yaglom, 1975; PP81; PGT-88), although Ricr ¼ 0.1 has also been suggested (e.g., Munk and Anderson, 1948; Lozovatsky et al., 1993; Pelegri and Csanady, 1994; Lozovatsky et al., 2000). The data of Strang and Fernando (2001a, b) and Pardyjak et al. (2002) as well as numerical simulations of various turbulent flows reviewed recently by Peltier and Caulfield (2003), including DNS (Jacobitz et al., 1997) indicate that for Ri40.1 the turbulence is affected by buoyancy forces. The critical Rib has often been set to Ricr (e.g., PP-81; PGT-88; and others), but ARTICLE IN PRESS I.D. Lozovatsky et al. / Deep-Sea Research I 53 (2006) 387–407 395 Fig. 7. Profiles of the dissipation rate (1), diffusivity (2), and Ozmidov scale (3) (left) and the corresponding profiles of sorted density (4), shear (5), and Ri (6) (right) at St. 936 (upper plot) and St. 948 (lower plot). The horizontal long and short dashed lines and the straight line show the depths where Ri ¼ 0.1, 0.25, and 1, respectively. The crosses mark the mixing layer depths he, which is near the depth where RicrE0.1/0.3. The Ozmidov scale decreases to 1 m at the depth, where RiE1 is observed. ARTICLE IN PRESS 396 I.D. Lozovatsky et al. / Deep-Sea Research I 53 (2006) 387–407 Fig. 8. The eddy diffusivity K b ¼ 0:2e=N 2 vs. the Richardson number Ri at St. 933 (upper panel) and St. 939 (lower panel). The symbols are 1 m averaged samples. The modeling dependencies Kb(Ri) given by Eqs. (2) and (3) are shown by the bold (r ¼ 1, p ¼ 2/3, Ricr ¼ 0.1, Rib ¼ 0.05) and thin (r ¼ 1, p ¼ 1, Ricr ¼ Rib ¼ 0.1) continuous lines. The parameterization of Pacanowski and Philander (1981) and Peters et al. (1988) are in the thin and bold dashed line, respectively. different values for Ricr and Rib have also been used in numerical calculations and data analyses (Ricr ¼ 0.1, Rib ¼ 0.3, Munk and Anderson, 1948; Ricr ¼ 0.1, Rib ¼ 0.05, Paka et al., 1999; Toorman, 2000). More elaborate formulations of Kb and at than those of Eqs. (2, 3) have been suggested, e.g., by Henderson-Sellers (1982), who analyzed the measurements of Linden (1979) and Ueda et al. (1981) and introduced KbKbo(1+(Ri/Ricr)2)1 with Ricr0.165 and at ¼ 1 þ 0:74Ri=1 þ 37Ri2 . This leads to the lowest Rib  0:02, if the formula for at is reduced to Eq. (3). Recent atmospheric turbulent measurements of Monti et al. (2002) also support Eq. (3) but require Rib ¼ 0.05 and r ¼ 1/2. The majority of previous studies, however, prefer r ¼ 1. This in turn yields a constant value for the flux Richardson number Rf ¼ aRi at high Ri, which is consistent with the assumption of a constant mixing efficiency g (Oakey, 1982) when Kb is calculated. Any r41 leads to a reduction of Rf at high Ri, as evident from the atmospheric data of Pardyjak et al. (2002). 4.2.2. Kb asymptotics and links to the turbulent scales As was stated in Section 1, empirical values of p range between 0.5 (Munk and Anderson, 1948) and 2 (PP-81). If the eddy viscosity KMo in a nonstratified sheared ocean flow is specified using the vertical shear Sh, and turbulent kinetic energy q2 or the dissipation rate e as K Mo q2 =Sh, or K Mo e=Sh2 (e.g., PGT-88), then the natural choice for p would be p ¼ 1, leading at RibRicr to K M q2 =N and K M e=N 2 , respectively. These are the generally accepted formulae for eddy viscosity in a stratified ocean. The above asymptotic form for KM derived using Eq. (2) is in consonance with the assumption that, in a stratified layer without a local source of shear energy production, the local turbulence scale ltr should be equal to the Ozmidov scale, ltr ¼ LN, thus yielding K M l tr q. Accordingly, in non-stratified shear flows, l tr ¼ LSh ðe=Sh3 Þ1=2 , where LSh is the Tchen (1953) shear scale. In an intermediate range of the Richardson numbers (0.25oRio2), the scale ltr can be specified as LR ¼ ðe=N 2 ShÞ1=2 (Lozovatsky et al., 1993), leading to K M e=ðN 2 ShÞ2=3 , requiring p ¼ 2/3. The lengthscale LR, which can also be written as ul =N, ul being the local friction velocity, replaces LN in the spectra of turbulent kinetic energy E(k), if a production subrange with EðkÞðe=ShÞk1 emerges between the inertial EðkÞe2=3 k5=3 and buoyancy EðkÞN 2 k3 subranges due to locally generated shear-induced turbulence (Lozovatsky, 1996). Therefore, p ¼ 2=3 is a viable choice for diffusivity parameterization, if turbulence is developed by a combination of non-local (wind stress in our case) and local (inertial oscillations, for example) shear sources. For comparison, the exponent p ¼ 1/2 (Munk and Anderson, 1948) gives K M e=ðNShÞ for RibRicr, which does not correspond to any known turbulent spectral lengthscale. Yet it correctly reflects the dependence of KM on governing parameters of stationary stratified turbulence. Two other putative values are p ¼ 2 (PP-81) and p ¼ 3/2 (e.g., PGT-88; Pelegri and Csanady, 1994), which produce K M eSh2 =N 4 and K M eSh=N 3 , respectively. To satisfy stationary turbulent kinetic energy budget, the flux Richardson number (or g), in these cases should be a decreasing function of Ri, e.g., gRi1 for p ¼ 2 and gRi1/2 for p ¼ 3/2, which is unlikely in weakly stratified UQHL. Note neither PP-81 nor PGT-88 links K Mo with e=Sh2 , rather they simply give a specific numerical value for the ARTICLE IN PRESS I.D. Lozovatsky et al. / Deep-Sea Research I 53 (2006) 387–407 nominator of Eq. (2). Because Eq. (2) with p ¼ 2/3 and/or p ¼ 1 links Kb(Ri) to a specific spectral structure of turbulent fluctuations (via corresponding lengthscales LR and LN), we favor these values of p in Eq. (2) compared to traditional p ¼ 1/2, 3/2 and 2. In all, r ¼ 1 is probably the most rational choice for at(Ri) in Eq. (3). Also, Ricr and Rib were selected from a set of values already discussed (Ricr ¼ 0.1 or 0.2, and Rib ¼ 0.05 or Ricr ¼ Rib ¼ 0.1 or 0.2). Eqs. (2) and (3) fit individual data samples of Kb at St. 933 and St. 939 reasonably well with p ¼ 2/3 or 1 (Fig. 8). The scaling values of eddy viscosities KMo are 5
102 and 4.5
102 m2/s for St. 933 and 4
102 and 3.5
102 m2/s for St. 939, respectively. While p ¼ 2/3 requires Ricr ¼ 0.1 and Rib ¼ 0.05 for a best fit, the exponent p ¼ 1 (which corresponds to the Ozmidov scale LN) fits the data equally well with Ricr ¼ Rib ¼ 0.1. The parameterization of PP-81, with the original set of p ¼ 2, r ¼ 1, Ricr ¼ Rib ¼ 0.2, but with a slightly higher K Mo ¼ ð3  3:5Þ
102 m2 =s (compared to 102 m2/s), gives the best fit, but its standard error is more than that of the first two approximations (see the thin dashed lines in Fig. 8). To be able to fit the data by using Eq. (2) with p ¼ 3/2 (e.g., Pelegri and Csanady, 1994), it is necessary to vary the values of Ricr and Rib for different stations, which is not justifiable. 4.3. The normalized diffusivity In order to compare ities observed under individual Kb samples stations with different and parameterize diffusivvarious wind conditions, obtained at a number of friction velocities u were 397 normalized using the Monin-Obukhov (1954) similarity theory, according to which the vertical diffusivity in the surface boundary layer is given by K sf ðzÞ ¼ ku z, where k is the Karman’s constant and u the friction velocity. Based on Eq. (2), the normalized diffusivity Kbn ¼ Kb/Ksf can be written as K bn ¼ Kb 1 . ¼ ku z ð1 þ Ri=Ricr Þp ð1 þ Ri=Rib Þ (4) Fig. 9a shows the bin-median values of K^ bn and ^ fitted by Eq. (4) for two sets of parameters p, Ri Ricr, and Rib. The bootstrap 90% confidence limit for K^ bn was calculated for 1000 resampled points when the actual number of samples was 15 for each bin. The data were taken from Sts. 933, 936, 939, and 948, where MLD ¼ 38–56 m and the wind speed varied between 4.5 and 14.3 m/s. Both lines in Fig. 9a fit the bin-median samples quite well in the ^ range from 1.3
102 to 5
101, where the Ri coefficients of determination are 0.80 and 0.77 for ^ lines 1 and 2, respectively. At very low Rio0:01, the data significantly depart from Eq. (4). This could be attributed to insufficient accuracy of N2 calculations when density gradients become very small. A ^ in Fig. 9a may, decrease of K^ bn at the lowest Ri ^ however, signify a tendency to be independent of Ri ^ as Ri ! 0 because of vanishing buoyancy effects. Also clear from Fig. 9a is that our data do not support an explosive growth of the diffusivity at ^ Rio0:25 as reported by PGT-88 for patch turbulence in the equatorial thermocline. Recently, Soloviev et al. (2001) parameterized the normalized diffusivity K b =ku z as (1Ri/Ricr) for RioRicr ¼ 0.25 and used the PGT-88 scaling for Kb at Ri4Ricr ¼ 0.25. A slightly modified version of Fig. 9. The bin-median estimates of the normalized diffusivity K^ b ¼ K b =kzu at the probability-equal Ri-intervals (large circles) with 90% bootstrap confidence limits shown for both variables. The data are superimposed by the modeling functions given by Eq. (4) (left panel) and by Eq. (5) (right panel). ARTICLE IN PRESS 398 I.D. Lozovatsky et al. / Deep-Sea Research I 53 (2006) 387–407 the Soloviev et al. (2001) original formula for the normalized Kb is Kb K^ bS ¼ ku z ¼ 1  Ri=Ricr þ K^ bS ¼ 5
102 ku zð1  5RiÞ2:5 5
102 ku zð1  5RiÞ2:5 ðRi4Ricr Þ, ðRioRicr Þ, ð5Þ where Ricr ¼ 0.25. The authors showed that the results of microstructure measurements taken in the upper turbulent layer of the western Pacific warm pool are in general agreement with Eq. (5). Because the foundation of Eq. (5) closely corresponds to that of Eq. (4), it is instructive to compare the two formulae with respect to the bin-median diffusivities obtained in the upper layer of the North Atlantic. Function (5) is presented in Fig. 9b for the same experimental data set. If K^ bS is calculated using Ricr ¼ 0.25, the result substantially departs from ^ but when Ricr our measurements at intermediate Ri, is reduced to 0.1, the agreement with experimental data becomes much better (see Fig. 9b). This later fit (using Ricr ¼ 0.1) is almost equally good as that obtained with formula (4), which is shown in Fig. 8a. The corresponding coefficients of determination are 0.7 and 0.74 for Eq. (4) and 0.6 for Eq. (5), if the two samples at the highest and lowest ^ are disregarded. Note that Eq. (5) of Soloviev et Ri al. (2001) with Ricr ¼ 0.25 lies slightly above their averaged data points (see Fig. 10 of Soloviev et al., 2001) and this effect is manifested in our data also. Fig. 10. Mixing efficiency g as a function of the Richardson number Ri: (1) Strang and Fernando (2001a, b); (2) stable nocturnal ABL; (3) Mellor and Yamada (1982); (4) Canuto et al. (2001). This suggests that the use of Eq. (5) with lower Ricr (Ricr ¼ 0.1) should improve the performance of this parameterization, thus making it comparable with Eq. (4). 5. Evaluation of Kb(Ri) using Ri-dependent mixing efficiency c The analyses in Section 4 as well as the calculation of Kb and Rb (Section 3) rely on the hypothesis of a constant mixing efficiency g ¼ 0.2 based on certain previous work (e.g., Osborn, 1980; Oakey, 1982; PGT-88; Moum et al., 1989; Ruddick et al., 1997; Lozovatsky et al., 1999; Finnigan et al., 2002; Lozovatsky and Fernando, 2002; Rippeth and Inall, 2002). Very recently, Oakey and Greenan (2004) confirmed that, on average, g can be treated as a constant 0.2, based on a comprehensive set of microstructure measurements in the coastal waters of New England. Other studies (e.g., Ivey and Imberger, 1991; Moum, 1996; Fringer and Street, 2003; Wuest and Lorke, 2003) suggest different g— ranging between 0.1 and 0.4 (Moum, 1990). Laboratory experiments of Rohr and Van Atta (1987) were among the first to show a growing trend of g with Ri for 0oRio0.25. In addition, according to the DNS results of Smyth et al. (2001), g can be time dependent, e.g., growing from 0 to 0.9 as K–H billows overturn and then decreasing to g0.2 at later stages. Similar observations have been noted by Peltier and Caulfield (2003) (also see Balmforth et al., 1998, for the applications of these results). Recent laboratory results (Strang and Fernando, 2001a) and the GISS mixed model (Canuto et al., 2001) suggest a continuous growth of g with Ri for 0oRio1 (see Fig. 10), and upon reaching a maximum g starts to decrease with Ri. The GISS model employs specific damping functions Sm and Sh (also called structure functions) for momentum and temperature (buoyancy) diffusivities K m;s ¼ Sm;s q4 =e; respectively (Fig. 11), which result in a growing g(Ri) as shown in Fig. 10. These recent theoretical and laboratory results can be compared with our microstructure measurements by calculating the diffusivities Kb using an Ri-dependent g. Fig. 10 shows several examples of g(Ri) obtained in the laboratory (1), in a stably stratified nocturnal atmospheric boundary layer (2), and those utilized in the Mellor–Yamada parameterization Scheme (3) and in the GISS model (4). The GISS related samples of g(Ri) and their damping functions (Fig. 11a) were obtained by digitizing Figs. 5 and 2, ARTICLE IN PRESS I.D. Lozovatsky et al. / Deep-Sea Research I 53 (2006) 387–407 respectively, of Canuto et al. (2001). Continuous growth of g with Ri for Rio1 is a characteristic feature of all data presented in Fig. 10. Strang and Fernando (2001a) measurements exhibit the fastest 399 growth while the Mellor–Yamada function approaches a level of 0.325, corresponding to the flux Richardson number limit of Rf ¼ 0:725ðRi þ 0:186  ðRi2  0:316Ri þ 0:0346Þ1=2 Þ, ð6Þ while being equal to 0.245 when Ri-1. The atmospheric data follow a power fit g ¼ 0:66Ri2=3 for Rio1 (7) and the Canuto et al. (2001) dependence can be approximated by a polynomial function gC ¼ 0:01 þ Ri  0:53Ri2 (8) that allows a decrease of g beyond Ri ¼ 1. Using these approximations for g(Ri) and a 5th-order polynomial fit to Strang and Fernando (2001a) data, we have recalculated diffusivities (Fig. 9) by multiplying the measured values of e=N 2 by g(Ri) rather than g ¼ 0.2. Fig. 12 shows almost constant level of K^ b corresponding to Ri-dependent g calculations for Rio1. By using a polynomial approximation to the Sh damping function of the eddy diffusivity: Fig. 11. (a) The momentum Sm (5) and heat (buoyancy) Sh (6) damping functions for the diffusivities of Canuto et al. (2001); (b) the Sh/Sm ratio approximated by Eq. (10a) (7) and Eq. (10b) (8). S h ðRiÞ ¼ 2ð0:05  0:117Ri þ 0:093Ri2  0:025Ri3 Þ, (9) we obtained a good fit to the K^ b ¼ gðRiÞe=N 2 samples shown by a dashed line in Fig. 12. Note Fig. 12. The normalized buoyancy diffusivities calculated for a constant (0.2) and Ri-dependent mixing efficiency g (details are in the text). The scalings given by Eq. (4) (a bold line) and Canuto et al. (2001) (a dashed line, Eq. (9)) fit well the two versions of diffusivities. The confidence limits of the original data shown in Fig. 9 are removed to make the plot more transparent. ARTICLE IN PRESS 400 I.D. Lozovatsky et al. / Deep-Sea Research I 53 (2006) 387–407 that the samples at highest and lowest Ri deviate from the main trend, which could be attributed to the low statistics at the edge points and insufficient accuracy of e=N 2 calculations at very low Ri. The result implies that our measurements of e=N 2 support parameterizations (4) and (5), which are based on a traditional diffusivity calculation (g is a constant) as well as parameterization (9), which accounts for an Ri-dependent g (Eq. 8). The question, therefore, becomes, which of these approaches leads to a diffusivity that best represents vertical mixing in numerical boundary layer models? Evaluation of the results of a series of numerical experiments that employ each of these parameterizations against a set of high quality field data is needed to make a definitive judgment. Such a comprehensive investigation is beyond the framework of the present study, but below we offer a simple test that sheds light on this issue. using a standard set of constants (Rodi, 1993), viz., C1e ¼ 1.44, C2e ¼ 1.92, C3e ¼ 0, Cm ¼ 0.09, se ¼ 1.3, and sq ¼ 1 (experiments with C3e ¼ 1.4 and 2/3 suggested by Baumert and Peters, 2000, 2004, did not reveal any significant influence of C3e). The following background vertical profiles were used: for turbulent kinetic energy q20 ðzÞ ¼ 106 m2 =s2 , eddy viscosity K 0 ðzÞ ¼ n ¼ 106 m2 =s, and the dissipation rate e0 ðzÞ ¼ C m q40 =K 0 . To incorporate the GISS model approach, we approximated function Sm (Fig. 11a) as Sm ¼ 2ð0:05  0:069Ri þ 0:0021Ri2 Þ; Rio0:95 and Sm ¼ expð2:9RiÞ; Ri40:95 ð10aÞ and used the inverse turbulent Prandtl number Pr1 tr ¼ S h =S m in the form Sh =S m ¼ 1  Ri; Rio0:45 and 6. Mixed-layer modeling Sh =S m ¼ expð1:3RiÞ; Ri40:45. The data obtained at St. 946, where MLD was only 5–6 m, have been used as the initial conditions for a series of numerical experiments that simulate the development of an 45 m mixed layer under the influence of winds of 10–12 m/s. The results of the calculations are compared with the data from St. 948, where the mixed layer was 48 m deep. St. 946 data were taken under calm winds (3 m/s). The wind rapidly increased from 3 to 11 m/s over an 12 h prior to the arrival of the ship at St. 948 (i.e., after the end of the measurements at St. 947, 30 miles south of the transect). During the 12 h of sailing toward St. 948 the wind was almost constant with u ¼ 1:4
102 m=s, which is close to the transectaveraged u ¼ 1:25
102 m=s. Assuming, that the horizontal variability of hydrophysical fields over a distance of 30–60 miles in the middle of the ocean is insignificant (the 1storder approximation), it is possible to use a onedimensional (1D) boundary layer model to evaluate the different mixing parameterization schemes discussed in Sections 4 and 5. Considering the dominance of wind mixing, we may neglect surface buoyancy forcing. An in-house version of a q2  e 1D model consisting of prognostic momentum equations for stratified rotational flows as well as for buoyancy, turbulent kinetic energy q2 and its dissipation e was used (see, e.g., Rodi, 1993). The calculations were made with 4 cm vertical resolution and a 4 s time step. The model was run Approximations (10b) are shown in Fig. 11b vis-à-vis the digitized samples of the original Canuto et al. (2001) ratio S h =Sm . Fig. 13 shows the initial profile (St. 946) of density excess over the sea-surface density and the modeled profiles after 12 h of wind work for the parameters suggested for Eqs. (2) and (3), and those based on PP-81 and the GISS models (Fig. 11). The density excess at St. 948 is also included in the figure for comparison. The results indicate that the closest to the observed density profile was achieved by the GISS mixing model. During 12 h of constant wind with u ¼ 1:25
102 m=s, an MLD of about 40 m was developed in agreement with the basic features of the density profile at St. 948. Other parameterizations used also develop a mixed layer, but of lesser depth. Almost the same 40 m of the upper layer was influenced by the diffusivities given by Eqs. (2) and (3) and PP-81, but below z  20  25 m incomplete mixing was observed. Approximately 22–24 h of wind work was needed for Eqs. (2) and (3) to create a density structure similar to that produced by the GISS model over 12 h. In the sensitivity runs, the influence of the vertical step DZ used for the estimates of velocity and density gradients was verified by changing DZ in the range 4 cm to 8 m (ADCP sampling rate) but keeping all variables at 4 cm vertical grid and centering the gradients and the Richardson numbers appropriately depending on DZ. The modeling density profiles did not exhibit noticeable differences for DZ ¼ 2, 4, and ð10bÞ ARTICLE IN PRESS I.D. Lozovatsky et al. / Deep-Sea Research I 53 (2006) 387–407 Fig. 13. Results of numerical calculations using a q2e 1D model for various values of p and r in Eqs. (2) and (3), Pacanowski and Philander (1981) formulae (PP-81), and for the GISS model approximations (10a) and (10b). A constant wind of 11 m/s worked 12 h to modify the initial density profile measured at St. 946 (before the calculations it was interpolated at Dz ¼ 4 cm). The modeling profiles of density excess dr over the surface density are compared with the measured profile of dr at St. 948 (the dotted line; Dz ¼ 2 m). 8 m at the time period between 12 and 24 h after the onset of calculations. This indicates that the vertical lengthscale of vertical inhomogeneities of mean Sh, N2, and Ri exceeds at least 8 m in the weakly stratified upper layer. The normalized diffusivity K^ b given by Eqs. (4) and (5) can be directly utilized for prognostic calculations of density and velocity while omitting the equations for q2 and e, the results for which [Soloviev et al.’s (2001) parameterization with Ricr ¼ 0.1] are shown in Fig. 14 for 6, 12 and 15 h of wind work for 11 m/s constant wind. It yields a deep quasi-homogeneous layer with MLD of about 40 m. The GISS model has slightly more mixing affinity; its density profile for t ¼ 12 h almost coincides with that produced by Eq. (5) for t ¼ 15 h. The Richardson numbers associated with Eq. (5) continuously grow with depth. The MLD can be identified by RiE1, but over the next 10 m of the pycnocline, Ri(z) increases slowly and then jumps to a high background value. The GISS-based Ri profile clearly distinguishes the MLD, where Ri drops to 0.1, but then sharply increases at the upper boundary of pycnocline. The parameterization based on Eq. (4)—not shown—leads to a shallower mixed layer due to lower values of Ricr and Rib. 401 Fig. 14. Modeling density and Richardson number profiles after 6, 12, and 15 h of the wind work using Eq. (5) for Kb parameterization without q2 and e balance equations. Profiles of the density excess dr and Ri computed by a complete system of q2e model equations using GISS parameterization (Eqs. (9) and (10)) are also shown for comparison. Finally, we compared the modeled profiles of the dissipation rate based on the GISS model with that of Eq. (4), and the results are shown in Fig. 15 (after 12 h of the wind work). The initial dissipation rate was very small across the entire upper layer (z4zw), between 2
109 and 8
109 W=kg with the mean of 4:6
109 W=kg. Turbulence generated by the constant 11 m/s wind substantially changed the vertical structure of e(z), producing a dissipation rate at z ¼ 16 m of more than 106 W/kg that gradually decreases with depth to 107 W/kg at zE45 m (disregarding the fine-scale intermittency) and then sharply drops at z ¼ 50 m to the background level of 5
109 W=kg observed at St. 946 before the wind started. The GISS modeling profile (e2 in Fig. 15) shows an almost constant level of e within the mixing layer down to z ¼ 40 m, coinciding with the MLD of the density profile dr2, but then the turbulence completely vanishes in the pycnocline. The profile e1 obtained with Eq. (4) follows the measured dissipation structure e (St. 948) closely down to z ¼ 50 m, exhibiting a gradual decrease of dissipation in the mixing layer and penetrating into the pycnocline to sustain developed ARTICLE IN PRESS 402 I.D. Lozovatsky et al. / Deep-Sea Research I 53 (2006) 387–407 of the dissipation profile within the mixing layer and allows the penetration of wind-induced turbulence into the pycnocline. Note that we have assumed that the initial density structure at St. 948 is the same as that at St. 946, and then it was modified by a constant wind of 11 m/s that blew for a 12 h period. This is a reasonable, yet not verifiable, assumption. For a more rigorous inter-comparison between the performances of different mixing parameterizations, a dedicated set of time-dependent CTD, microstructure and current measurements are needed. 7. Summary Fig. 15. The modeling profiles of the dissipation rate (e1 and e2) and density excess (dr1 and dr2) after 12 h of wind work (11 m/s) for parameterizations given by Eqs. (2) and (3) and Eqs. (9) and (10), respectively. The star lines are the measured e(z) profile at Sts. 946 and 948. The initial density profile at St. 946 is shown in Fig. 13. The modeling values of e1 and e2 were increased by one order of magnitude to closely overlay e(948). active turbulence. Neither parameterization matches the measured dissipation rate in absolute values, which could be attributed to the oversimplified nature of the numerical experiments. In the plot, all modeling profiles were shifted by one order of magnitude to overlay the measured dissipation data at St. 948. At first glance, the modeling test favors the gdependent diffusivities of Strang and Fernando (2001a) as well as the GISS mixing scheme, if the sole interest is predicting the mixed-layer depth. The modified formula (5) of Soloviev et al. (2001), however, produces a comparable MLD based on g ¼ 0.2. Nevertheless, considering the availability of limited data, it is too early to speculate which of the models are more appropriate, i.e., those relying on a constant mixing efficiency or traditional Kb(Ri) scaling [e.g., PP-81 and those given by Eqs. (2) and (3)]. Specifically, (2)–(3) yields the correct shape In this paper, the statistics of the turbulent kinetic energy dissipation rate e and eddy diffusivity Kb in the upper mixing layer of the North Atlantic were analyzed and the diffusivity was parameterized using the wind stress and local gradient Richardson number Ri. The wind-stress-induced dissipation ews ¼ u3n = 0:1kLE , where LE ¼ u =f is the Ekman scale, accounts for more than 90% of the measured mean dissipation e~ml within the mixing layer. The depth of the upper weakly stratified layer on the average is 45 m. The probability distribution of e below the near-surface layer (z416 m) appeared to be lognormal with mean hei ¼ 3:5
107 W=kg and the median medðeÞ ¼ 1:5
107 W=kg. The Ri distribution function with med(Ri) ¼ 0.1 indicates the presence of shear instability, and the probability of Rio0.25 is above 60%. The depth where Ri is close to 0.25 roughly corresponds to the mixing layer depth; at the bottom of the mixed-layer Ri is close to unity. The eddy diffusivity Kb was quantified as a function of Ri (Eqs. (2) and (3)). We demonstrated that the power-law damping function 1/(1+Ri/ Ricr)p for Kb with p ¼ 1 has clear physical meaning. In stratified flows, it gives K b e=N 2 for RibRicr, whence the Ozmidov scale LN ðe=N 3 Þ1=2 serves as the main turbulent lengthscale determining Kb(Ri). For non-stratified shear flows p ¼ 1 specifies the shear scale LSh ðe=Sh3 Þ1=2 (Tchen, 1953) as the governing turbulent lengthscale. In addition, p ¼ 2/3 educes the lengthscale LR ¼ ðe=N 2 ShÞ1=2 , which separates the buoyancy and production subranges of the spectra of ocean microstructure (Lozovatsky, 1996). Because turbulence in the upper layer is mainly driven by the wind stress, the Monin-Obukhov ARTICLE IN PRESS I.D. Lozovatsky et al. / Deep-Sea Research I 53 (2006) 387–407 theory was used to estimate the non-stratified turbulence in the upper layer by employing the base diffusivities K sf ðzÞ ¼ kun z for Kb normalization. A simple parameterization of Kb/Ksf(Ri) by Eq. (4) successfully collapsed the bin-median estimates of normalized diffusivity in the range 1:3
102 o ^ Rio5
101 . Statistical analysis of diffusivities K b ¼ ge=N 2 and the parameterizations of Kb as a function of Ri relied on a widely used concept of a constant mixing efficiency, g ¼ 0.2. Conversely, we estimated Kb using an Ri-dependent mixing efficiency based on recent laboratory experiments (Strang and Fernando, 2001a) and the GISS mixing model (Canuto et al., 2001) that suggest g as growing functions of Ri for 0oRio1. The diffusivities so obtained are almost independent on Ri for Rio1, in contrast to Kb calculated with a constant g that decreases with increasing Ri. Interestingly, Kb(Ri) calculated with g ¼ 0.2 as well as the diffusivities of the GISS model reckoned with variable g agree well with the same set of data. To further investigate the differences between the two approaches, a series of numerical experiments with a standard 1D time-dependent q2  e model (Rodi, 1993) was conducted to compare the efficacy of various K(Ri) functions. The development of an 40 m mixed layer during 12 h of sustained constant wind stress (11 m/s wind) and zero buoyancy flux was simulated by the GISS model (with Ri-dependent g), which was in close agreement with observations. A modified Soloviev et al. (2001) formula (5) (with constant g ¼ 0.2), however, yielded almost the same density profile as the GISS model. The scaling (2)–(3) suggested in this paper produced the correct general trend of vertical structure for the dissipation rate within the mixing layer and allowed the penetration of wind-induced turbulence into the pycnocline. The results indicate that from an MLD point of view our modeling calculations favor the GISS model, which is also consistent with the laboratory findings of Strang and Fernando (2001a, b). On the other hand, a constant g and the suggested parameterization (2)–(3) are more favorable if the focus is on vertical structure of microstructure variables. Further microstructure data and model evaluations are needed for a more definitive judgment regarding the role of g(Ri) in parameterization of mixing in sheared, weakly stratified, ocean boundary layers. 403 Acknowledgements We wish to thank F. Gomez, L. Montenegro, and K. Kreyman, who participated in the microstructure measurements, and the crew of R./V. Akademik Ioffe. The cruise was organized with financial support of the Russian Ministry of Science and Technology. The authors received partial support from the US Office of Naval Research, grant N00014-97-1-0140 (I.L. and H.J.S.F.), and NATO 2002 Research Fellowship (I.L.); Spanish Government grant REN2001-2239 and Agencia Catalana de l’Aigua (E.R.); ANUIES and CICESE, Mexico (M.F.), and by the Russian Foundation for Basic Research, grant 02-05-64408 (S.S.). Appendix A. The MSS profiler The MSS profiler (Prandke et al., 2000) encapsulates electronics and a set of sensors at the lower end of a 1 m cylinder. The package is protected by a guard. The vertical resolution of the airfoil shear probe (PNS 98) is limited to 2 cm by the sensor geometry. The lowest in situ noise level of shear measurements is equivalent to 3
109 W=kg. The sensitivity of the fast thermistor is 103 C, and its response time is 7 ms. The precision temperature sensor (Pt-10) has a sensitivity of 103 C, accuracy 102 C, and a time response of 160 ms. The corresponding characteristics of the seven-pole conductivity cell are 103 mS/cm, 102 mS/cm, and 100 ms, respectively. The data are transferred to an on-board computer via a neutrally buoyant elastic cable. The profiler operates at a falling speed of about 0.7 m/s. Microstructure data are usually contaminated by ship-induced movements and profiler transients. Since near-surface data segments cannot be recovered by a denoising procedure, the data at zo16 m was removed from the analysis. This near-surface interval can be probed only with uprising (e.g., Soloviev et al., 1999) microstructure profilers. Airfoil data in the oceanic upper 15–20 m layer taken during free fall are usually discarded (e.g., Oakey, 1982; Peters et al., 1988; Hebert et al., 1991; Paka et al., 1999). Below this heavily contaminated depth interval, small-scale shear signal was denoised by removing isolated spikes, bad values, or gaps in the records identified by an interactive graphic interface. Bad samples were replaced using a cubic spline interpolation, if the number of bad or missing points were less than fifty (about 5 cm of the ARTICLE IN PRESS 404 I.D. Lozovatsky et al. / Deep-Sea Research I 53 (2006) 387–407 record). A localized, narrow-frequency noise, which sometimes appear in the signal because of mechanical resonance of the profiler, was eliminated by a very sharp Lanczos filter (Hamming, 1987) tuned to a specific frequency band (usually 40 Hz). After appropriate editing to the small-scale shear signal was applied, the dissipation rate e was calculated by fitting 1D wavenumber shear spectra to the Panchev and Kesich (1969) theoretical spectrum, following the recommendation of Gregg et al. (1996). This spectrum contains a little more power at lower wavenumbers and rolls off slightly faster at high wavenumbers compared to the wellknown Nasmyth (1970) spectrum. The difference between the two spectra affects the measurements at high levels of dissipation, whereupon small-scale shear at low wavenumbers is not well resolved; therefore, a correction is needed when the shear variance is calculated. The experimental spectra showed a good agreement with the universal spectra at intermediate wavenumbers, covering the most important range for our purpose, which includes the maximum of the dissipation spectrum. The dissipation estimates were obtained at each 1 m vertical segment, which were then used for Kb calculations after appropriate averaging. Appendix B. The Kb(Ri) error estimates B.1. e, N2, and Kb The errors of the diffusivity Kb ¼ ge/N2 and Richardson number estimates depend on the accuracy of field measurements and computing algorithms. Assuming that the mixing efficiency g is a constant (not measured in our case) and e, N, and vertical shear Sh are statistically independent variables, it is possible to estimate the relative (normalized) RMS errors of Kb as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (B.1) dK b ¼ d2e þ d2N 2 , where dy ¼ sy =y and s2y is the variance of a random variable y (Korn and Korn, 1961). The laboratory calibration test for the shear probe gives a noise level about 1011 W/kg (Prandke et al., 2000), but the total computational RMS accuracy of the dissipation estimates obtained by MSS does not exceed se ¼ (35)
109 W/kg (Roget et al., 2006). Given the dissipation rate in the mixing layer was above 107 W/kg, the normalized RMS error of the dissipation estimates de ¼ se =e becomes less than 3–5%. The computational error of N2 is governed by the RMS error of density measurements sr and the vertical step Dz of density gradient calculations pffiffi 2gsr sN 2 , (B.2) ¼ 2 rDzN 2 N where sr depends on the RMS noise levels of temperature sT, conductivity sC, and pressure sp channels and T, S, and p partial derivatives of density. According to Prandke et al. (2000), for MSS sensors, sTE103 K over a bandwidth 0–25 Hz, sCE104 S/m over a bandwidth 0–100 Hz and spE1 GPa. In the weakly stratified upper QHL (N2 is less than 105 s2), the partial derivatives are small, salinity spikes due to mismatch of response time between temperature and conductivity sensors are almost unrecognizable and therefore the upper limit for sr can be set as 5
104 kg/m3. Thus, the normalized RMS errors of N2 calculated over 4 m vertical segments using (B.2) are: dN 2 ¼ sN 2 =N 2 ¼ 0:2  2 in the range N2 ¼ 105–106/s2. These numbers are relatively high, showing the possibility of significant uncertainties of individual N2 samples, specifically for very low N2. To improve the quality of the estimates, averaging over larger domains or several profiles should be conducted. To estimate the error dK b of diffusivity from (B.1), we used de ¼ 0:05, which is much less than dN 2 , and therefore dK b is about the same as dN 2 , ranging between 0.2–0.25 and 2. B.2. Shear and Ri The uncertainty of shear measurements is determined by the nominal RMS error of a single bin, which is 30 cm/s for 8 m cell of 75 kHz ADCP, and the number of bins used to obtain an averaging estimate of velocity at a particular depth. Because our measurements were taken only at drift stations (typical drift 1–1.5 knots), the errors related to ship movement were small (Firing, 1988). Modern GPS navigation allowed quite accurate estimates for the absolute velocity profile to be obtained (Lien et al., 1994). For the estimates of vertical shear, the accuracy of absolute currents, however, is not important. Individual profiles of horizontal current components uðzÞ and vðzÞ were obtained for 18 min averaging intervals, which reduced the nominal ARTICLE IN PRESS I.D. Lozovatsky et al. / Deep-Sea Research I 53 (2006) 387–407 velocity error by a factor of 40. Then, the individual profiles were averaged over the measurement time at each station (4–7 profiles during about 1.5–2 h) and ~ such averaged profiles uðzÞ and v~ðzÞ were used to calculate the shear magnitude Sh ¼ jdV =dzj ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ~ ðdu=dzÞ þ ðd~v=dzÞ2 . The RMS error of jV j ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi u~ 2 þ v~2 is about 0.5 cm/s, producing an RMS error of Sho0.001/s. This is consistent, e.g., with the corresponding estimates reported by Peters et al. (1988). Typical magnitude of the shear in QHL was in the range 5
103  2
102 s1 (see Fig. 7, for example). Therefore, the RMS normalized error of the shear estimates is in the range 5–20% and for the squared shear it goes up to 10–30%. Note that a composite spectrum of vertical shear (Gargett et al., 1981) contains a turbulence subrange bounded at low wavenumbers by the Ozmidov scale LN, which is about 1 m in the ocean thermocline. Therein the characteristic scales of local background shear and stratification are on the order of LN. In QHL, however, the Ozmidov scale may grow to several 10 m (see, e.g., Fig. 7), whence other lengthscales become dominant. For example, the largest scales of 3D turbulent eddies in QHL are determined by the external turbulent scale L0, which is of the order of ð0:1  0:2Þhe (Monin and Yaglom, 1975); most recently Canuto et al. (2002) identified L0 as 0:17hD . Because in our measurements the mean thickness of QHL hhD i  hhe i ¼ 45 m, the characteristic L08 m. As such, the 8 m resolution ADCP shear is appropriate to evaluate the Richardson number pertinent to the QHL small-scale turbulence. Some support for this conclusion is provided by our modeling efforts (Section 6), in which we tested the influence of the vertical separation DZ on the model outcome by calculating velocity and density gradients and thus Richardson numbers with DZ ¼ 2, 4, and 8 m. The modeling output was almost insensitive to DZ, indicating that the shear and density structures in the upper weakly stratified layer did not have substantial variations on scales o8 m. Therefore, the calculation of Ri with an 8 m vertical resolution seems appropriate. The relative RMS error for the squared shear dSh2 is considerably lower in most cases than dN 2 . Therefore, the relative error of Ri is mainly determined by dN 2 rather than dSh2 , although a combination of relatively high N2 and Sh2 can be encountered near the base of the upper mixing layer. 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