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Keywords:

  • turbulence;
  • intermittency

Abstract

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Structure Functions and Intermittency Models
  5. 3. Measurements and Background Hydrology
  6. 4. Method
  7. 5. Results
  8. 6. Discussion
  9. 7. Conclusions
  10. Acknowledgments
  11. References

[1] The higher-order structure functions of vertical velocity fluctuations (transverse structure functions (TSF)) were employed to study the characteristics of turbulence intermittency in a reversing tidal flow on a 19 m deep shallow shelf of the East China Sea. Measurements from a downward-looking, bottom-mounted Acoustic Doppler Velocimeter, positioned 0.45 m above the seafloor, which spanned two semidiurnal tidal cycles, were analyzed. A classical lognormal single-parameter (μ) model for intermittency and the universal multifractal approach (specifically, the two-parameter (C1 and α) log-Levy model) were employed to analyze the TSF exponent ξ(q) in tidally driven turbulent boundary layer and to estimate μ, C1, and α. During the energetic flooding tidal phases, the parameters of intermittency models approached the mean values of equation image ≈ 0.24, equation image1 ≈ 0.15, and equation image ≈ 1.5, which are accepted as the universal values for fully developed turbulence at high Reynolds numbers. With the decrease of advection velocity, μ and C1 increased up to μ ≈ 0.5–0.6 and C1 ≈ 0.25–0.35, but α decreased to about 1.4. The results explain the reported disparities between the smaller “universal” values of intermittency parameters μ and C1 (mostly measured in laboratory and atmospheric high Reynolds number flows) and those (μ = 0.4–0.5) reported for oceanic stratified turbulence in the pycnocline, which is associated with relatively low local Reynolds numbers Rλw. The scaling exponents ξ(2) of the second-order TSF, relative to the third-order structure function, was also found to be a decreasing function of Rλw, approaching the classical value of 2/3 only at very high Rλw. A larger departure from the universal turbulent regime at lower Reynolds numbers could be attributed to the higher anisotropy and associated intermittency of underdeveloped turbulence.

1. Introduction

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Structure Functions and Intermittency Models
  5. 3. Measurements and Background Hydrology
  6. 4. Method
  7. 5. Results
  8. 6. Discussion
  9. 7. Conclusions
  10. Acknowledgments
  11. References

[2] By its very nature, turbulence in geophysical flows is highly intermittent in space and time. Turbulence characteristics such as the kinetic energy etr, its dissipation ɛ, eddy diffusivities K, scalar dissipation χ, turbulent scales Ltr are subjected to sharp variations with typical spatial scales of tens/hundreds meters vertically/horizontally and temporal scales ranging from minutes to hours. Such mesoscale inhomogeneity of hydrophysical fields is called “external or outer intermittency,” and is associated with variations of mean fields, patchiness of turbulent regions and presence of interfaces that separate turbulent and nonturbulent regions [Sreenivasan, 2004]. Conversely, the small-scale, fine-scale intermittency of turbulence or internal intermittency occurs at spatial scales from meters to millimeters, and is usually confined within turbulent regions (layers, patches, wakes, plumes, etc.). It is attributed to random inhomogeneous spatial distribution of vortex filaments within turbulent regions, where they stretch and dissipate energy in isolation [Kuo and Corrsin, 1971].

[3] Mesoscale intermittency is a phenomenon characteristic of turbulent mixing in oceans, seas, large lakes and reservoirs, which may not be observed in other turbulent flows. The measured vertical and horizontal sizes of turbulent zones in the upper ocean are subjected to specific statistical regularities. Their probability distributions appear to be approximately lognormal while the distances between turbulent regions follow a double exponential distribution [Lozovatsky et al., 1993; Pozdinin, 2002]. Internal intermittency, however, is inherent to any high Reynolds number turbulent flow [Monin and Yaglom, 1975] due to its inhomogeneous microstructure.

[4] Internal intermittency of turbulence in oceans and lakes affects processes at the scales of inertial-convective and diffusive subranges [Tennekes and Lumley, 1972]. Among them are the viscous dissipation of energy, biochemical processes (planktonic mating, predator-prey contacts, chemical reactions [Seuront and Schmitt, 2005]), thermal convection and redistribution of salinity concentration or multi/double diffusive convective fluxes [Sánchez and Roget, 2007]. The influence of small-scale turbulent fluctuations on the propagation of light and sound in the ocean is an important problem for various applications of ocean optics and acoustics [Tyson, 1991; Colosi et al., 1999].

[5] In application to aquatic ecosystems, turbulent oscillations of various scales influence aggregation, incubation and foraging processes of small-scale planktonic organisms [Druet, 2003]. Internal intermittency can affect phyto and zooplankton species less than several millimeters in size [Peters and Marrasé, 2000], specifically, floating microscopic algae that are responsible for photosynthesis in coastal oceans [Margalef, 1985, 1997]. Zooplankton larger than ∼1 cm usually do not react to small-scale intermittency of turbulence [Squires and Yamazaki, 1995, 1996].

[6] Intermittency of biochemical (plankton, nitrites) and scalar (fluorescence concentration, temperature) variables have been studied by Seuront and coauthors in a series of papers [e.g., Seuront et al., 1999, 2001, 2002; Seuront and Schmitt, 2005]. In particular, it was found that phytoplankton patchiness substantially increased the predator-prey encounter rates, but the encounter was much less influenced by turbulence when ɛ was considered as an intermittent variable rather than a mean value [Seuront et al., 2001]. The patchiness of small-scale phytoplankton distribution in a tidal current (the Eastern English Channel) increased with decreasing turbulence intensity [Seuront and Schmitt, 2005] and it varied depending on the phase of tidal cycle. This finding is directly consistent with the present results.

[7] The first results on intermittency of ocean turbulence at scales of inertial-convective subrange, with application to scalar dissipation χ, were presented by Fernando and Lozovatsky [2001] and for the velocity field and ɛ by Seuront and Schmitt [2001] and Yamazaki et al. [2006]. Seuront and Schmitt [2001] concluded that fluorescence is more intermittent than the velocity, but less intermittent than the conductivity fields in the Neko Seto Sea, offshore the Japanese coast. The distributions of ɛ and χ in deep ocean and shallow waters, at the scales from tens of centimeters to several meters that are affected by internal as well as external intermittencies, were found to be approximately lognormal [e.g., Baker and Gibson, 1987; Gibson, 1991; Gregg et al., 1993; Rehmann and Duda, 2000; Lozovatsky and Fernando, 2002; Lozovatsky et al., 2006; Yamazaki and Lueck, 1990; Davis, 1996], but they disputed the applicability of lognormal approximation to the distribution of ɛ in the ocean.

[8] Most theoretical studies on internal intermittency (see reviews of Lesieur [1990], Frisch [1995], and Seuront et al. [2005]) employed a suite of scaling models, either of the fluctuations of ɛr or qth-order statistical moments of velocity increments 〈ΔVrq〉, which are also called the qth-order structure functions (SF). The angle brackets indicate ensemble averaging over a specific volume in the inertial subrange with a characteristic radius r. Laboratory experiments, DNS, and atmospheric measurements have produced voluminous literature on internal intermittency (see reviews of Sreenivasan and Antonia [1997], Anselmet et al. [2001], Tsinober [2001], Vassilicos [2001], Seuront et al. [2005], and Lovejoy and Schertzer [2007]). Specific findings of previous theoretical and laboratory studies will be given in sections 5 and 6 in relation to our results.

[9] In all, despite recent progress, small-scale intermittency within turbulent patches of the pycnocline or in the surface and bottom boundary layers has not been extensively studied and remains a relatively unexplored area in physical oceanography, though its oceanic applications abound. The goal of this paper is to investigate internal intermittency of marine turbulence near the seabed during different phases of a nonstratified reversing tidal flow and determine whether the intermittency parameters depend on the boundary layer and microscale Reynolds numbers. The analysis is based on measurements of vertical velocity w using a bottom mounted Acoustic Doppler Velocimeter (ADV). An overview of the scaling concepts in relation to structure functions analysis is given in section 2. Section 3 contains a brief summary of the measurement site and its hydrography as well as averaged turbulence parameters. The data have already been analyzed for mean flow and tidally induced temporal variations of averaged dissipation rate and friction velocity [Lozovatsky et al., 2008a, 2008b]. The methodology of the SF analysis and calculation of the scaling exponents of the transverse structure functions (TSF) as well as the dissipation rate are presented in section 4, followed by the results in section 5. This includes the evolution of basic turbulence parameters (section 5.1) and transverse structure function exponents (TSFE) during the tidal cycle (section 5.2), a comparison of scaling exponents with log-Levy and lognormal intermittency models (section 5.3) and a discussion of dynamical relevance of model parameters (section 5.4). The dependence of the second-order TSFE on microscale turbulent Reynolds number is presented (section 5.5). The possible influence of Taylor hypothesis on evaluating TSFE is addressed in section 6 as well as other sources of uncertainty that may affect results. Conclusions are given in section 7.

2. Structure Functions and Intermittency Models

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Structure Functions and Intermittency Models
  5. 3. Measurements and Background Hydrology
  6. 4. Method
  7. 5. Results
  8. 6. Discussion
  9. 7. Conclusions
  10. Acknowledgments
  11. References

[10] The wide range of scales of ocean processes affected by turbulent motions naturally calls for a scaling approach of studying intermittency of ocean turbulence. To paraphrase Landau on Kolmogorov's [1941a] turbulent cascade, “… in a turbulent field the presence of curl of the velocity was confined to a limited region …” (cited by Frisch [1995]), which indicates the essence of turbulence intermittency. To account for fluctuations of ɛ (and χ) at the scales r of locally isotropic turbulence in inertial-convective and viscous spectral subranges, Kolmogorov [1962] and Oboukhov [1962] suggested a refined similarity hypothesis (RSH). It argues that the velocity increment ΔVr over a separation distance r is specified not by the mean dissipation rate equation image but the dissipation ɛr averaged over a specific volume of radius r < Lo, which leads to the following scaling relation

  • equation image

where Lo is an external turbulent scale and ΔVr is the increment of any component (u, v or w) of velocity fluctuations in the longitudinal (x) and transverse (y) and (z) directions, respectively. Accordingly, the qth-order velocity increment

  • equation image

The probability distribution of ɛr was considered lognormal, which is based on Kolmogorov's [1941b] postulate that the sizes of particles resulting from a series of successive statistically independent breaking must be asymptotically lognormal. Yaglom [1966] and Gurvich and Yaglom [1967] theoretically derived an explicit cascade intermittency model, wherein transfer of kinetic energy down the cascade occurs with the breakdown of turbulent eddies, which produce lognormally distributed ɛr with the variance

  • equation image

where Aɛ depends on the large-scale motions and μ is an intermittency factor that accounts for stretching of the probability distribution function of log ɛr [e.g., Monin and Yaglom, 1975; Frisch, 1995].

[11] For the scalar dissipation rate χr (e.g., fluctuations of temperature, conductivity, fluorescence concentration) [Gibson, 1981]

  • equation image

where μsc is the intermittency factor of the scalars corresponding to χr and Aχ depend on the characteristics of mean flow. Both intermittency factors μ and μsc are assumed universal, with canonical values μ ≈ 0.25 and μsc ≈ 0.35 for very high Reynolds number turbulence, with μsc having lesser statistical confidence than μ [Sreenivasan and Kailasnath, 1993].

[12] The lognormal model of intermittency led to the modification of Kolmogorov-Oboukhov original scaling (the −5/3 laws for the spectral densities and 2/3 laws for the second-order structure functions) by small additives μ/9 and μsc/9. The modified spectral functions are [Monin and Yaglom, 1975]

  • equation image

where cK and csc are universal constants. Equations (2a) and (2b) have been utilized for empirical estimation of μ and μsc in laboratory experiments [e.g., Gibson et al., 1970] and for ocean turbulence. For mesoscale intermittency, μ and μsc ranged between 0.44 and 0.5 (see review of Gibson [1998]). The same numbers (μsc = 0.44) were obtained by Fernando and Lozovatsky [2001] by analyzing microstructure of oceanic conductivity (temperature) in the thermocline. Although the lognormal model of intermittency is simple and convenient, as discussed by Novikov [1970, 1990], Frisch [1995], and Seuront et al. [2005], it is mathematically ill posed.

[13] The application of multifractal theory [Mandelbrot, 1974] to study turbulence was the next important step for understanding and quantifying the multiscale nature of the intermittency phenomenon. The cascade of energy flux can be thought of as a multiplicative process where larger eddies are randomly modulated by smaller eddies to determine the fraction of energy transfer from larger to smaller scales [Seuront and Schmitt, 2005], which naturally entertains a multifractal approach as a tool. This approach associates each intermittency level pr (a fraction of the volume Lo3 occupied by turbulence of characteristic scale r = γnLo, 0 < γ < 1) with its own fractal dimension D = log N(n)/log n, where N is a number of self-similar structures and n−1 is a fraction (reduction) factor. In the simplest case of a β model [Fournier and Frisch, 1978], pr is assumed to decrease at all levels by the same constant factor 0 < β < 1, so that after n (= log(r/Lo)/log γ) breakdowns, pr = βn = (r/Lo)3−D, where 3 − D = log β/log γ (see Frisch [1995, chapter 8] for details).

[14] The multifractal models are based on the scaling assumption that if ɛrq/3 has a power law variation with r in the inertial subrange then all qth-order statistical moments of velocity increments (i.e., qth-order structure functions) can be written as

  • equation image

where the constant Cq may depend on large-scale flow characteristics. The exponent function ξ(q) signifies the multiscale transfer process specified by statistical moments of order q. The scaling exponent, which is universal (ξ(q) = q/3) for nonintermittent Kolmogorov turbulence under very high local Reynolds numbers rrr)1/3/ν, is thought to be universal for intermittent turbulence whence ξ is a nonlinear function of q [Sreenivasan and Antonia, 1997].

[15] A number of models (see review of Seuront et al. [2005]) have been suggested to specify ξ(q) in (4). Here we choose the log-Levy multifractal model [Schertzer and Lovejoy, 1987], since it has already been used in several oceanographic applications cited above. Note that the log-Levy model is stable under addition of the logarithm of the process variable [Feller, 1971; Mandelbrot, 1983]. For the scaling exponent ξ(q) this model gives

  • equation image

where C1 and α reflect the multifractal nature of intermittency in point. Seuront et al. [2005] suggested that C1 and α have universal values close to 0.15 and 1.5, respectively, for very high Reynolds numbers. The Levy distribution index 0 < α < 2 and the multifractal codimension of mean C1 characterizes the intermittency of the turbulent field in a way that sparseness and variability of the localized events (singularities) increase with increasing C1 and decreasing α. Note that in a d-dimensional space with fractal dimension D, the quantity d-D is called the codimension [Frisch, 1995]. For a quadratic polynomial function (α = 2), equation (5) gives the scaling exponent for the lognormal intermittency model

  • equation image

which has the same intermittency factor μ as in equation (2a).

3. Measurements and Background Hydrology

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Structure Functions and Intermittency Models
  5. 3. Measurements and Background Hydrology
  6. 4. Method
  7. 5. Results
  8. 6. Discussion
  9. 7. Conclusions
  10. Acknowledgments
  11. References

[16] The data taken in a shallow water tidal current, including a bottom-mounted Acoustic Doppler Velocimeter (ADV) measurements and hourly CTD profiles, were used in the study. The measurements were conducted on 14 December 2005, about 1.2 km offshore the northeastern coast of China (36.04°N, 120.32°E) at a water depth of 19 m. The measurement volume of a downward looking Nortek 6 MHz ADV “Vector” was set up at ζs = 0.45 m above the bottom for a 25 hour period, covering two complete semidiurnal tidal cycles. The ADV sampling rate was 16 Hz; the data were recorded continuously.

[17] A nearly unidirectional reversing tidal flow [Lozovatsky et al., 2008a, Figure 6] dominated mesoscale dynamics at the test site (see also Figure 1). The M2 amplitude of the west-directed flood current u(t) of ∼0.35–0.42 m/s was twice that of the eastern ebb current; the amplitude of transversal horizontal component v(t) was much smaller ∼0.05 m/s. The M2 amplitude of surface elevation was 1.1 m. The reversing tidal flow was affected by seiches of ∼2.3 hour period generated in the semienclosed Jiaozhou Bay located 2 km away. The seiching modulation of zonal velocity during the ebb tide was comparable with the tidal magnitude. The shallow water column was well mixed due to winter cooling from the sea surface and tidal mixing in the bottom boundary layer (BBL). The variations of the turbulent kinetic energy, averaged dissipation rate and friction velocity in the flow are given by Lozovatsky et al. [2008b].

image

Figure 1. The surface elevation δ; the magnitude U; and direction ϕo of near-bottom current, its streamwise u and transverse v components. The segment-averaged “−5/3” dissipation rate equation imagenb is represented by bars and the tidal dissipation ɛtd is represented by stars. The near-bottom Reynolds number Renb (circles with crosses) coincides with the U curve.

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4. Method

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Structure Functions and Intermittency Models
  5. 3. Measurements and Background Hydrology
  6. 4. Method
  7. 5. Results
  8. 6. Discussion
  9. 7. Conclusions
  10. Acknowledgments
  11. References

[18] To analyze the intermittency in tidal boundary layer turbulence, 25 hour records of the ADV current components were subdivided into 44 segments. Each segment contained 215 = 32628 individual samples (time interval ∼ 34 min), which is relatively long to provide sufficient multiplicative averaging of higher-order SF calculations (see section 4.2), thus minimizing errors to an acceptable level. On the other hand, turbulent fluctuations in the segments should be relatively stationary to yield reliable spectra and structure functions. Our tests showed that a 34 min segment is a good compromise between the above competing factors.

4.1. Spectra and the Dissipation Rate

[19] It is known [Stapleton and Huntley, 1995; Voulgaris and Trowbridge, 1998] that vertical velocity fluctuations w′ obtained from a bottom-mounted ADV have a lower noise level at higher frequencies compared to those of u′ and v′; this was also observed in our measurements. The spectral densities of all velocity components contained a relatively wide inertial subrange, and at frequencies f > 3–4 Hz the spectra of horizontal components were affected by noise while it was not the case for the spectra of the vertical component [Lozovatsky et al., 2008b, Figure 2a]. Therefore, only the w(t) records were used in further analysis.

[20] The mean dissipation equation imagenb at each segment i (see Figure 1) was estimated from the inertial subranges of the frequency counterpart of the Kolmogorov wave number spectra

  • equation image

with cw = 0.67 [Sreenivasan, 1995] for one dimensional spanwise velocity spectrum and the magnitude of mean current Ui = (〈ui2 + 〈vi2)1/2. Several examples of the compensated k5/3equation image(k) spectra (k = κ/2π) are shown in Figure 2 for high, low, and intermediate Renb, where Renb = (ΔU/ζs)Ltr2/ν is the near-bottom Reynolds number; here ΔU is the ADV velocity at ζs (thus, Uζ=0 = 0), ν the molecular viscosity, Ltr = κζs the characteristic turbulence scale above the seafloor, and κ the von Karman's constant. At each segment, the power spectral densities (PSD) were calculated using 1024 consecutive samples of w′ with a 128-point spectral window and by further averaging PSD of 32 nonoverlapping sections. Although the initial records wi(t) were quasi-stationary, they were detrended before spectral processing. The w measurements of ADV were of high quality [see Lozovatsky et al., 2008a, 2008b], and thus the despiking procedure of Goring and Nikora [2002] was not used, which affects the signal structure.

image

Figure 2. Examples of compensated spectral densities k5/3equation image(k). The spectra are arranged with respect to the near-bottom Reynolds number Renb from the highest (segment 41) to the lowest (segment 8). The inertial subranges are shown by horizontal lines. The 95% confident bounds (valid for every spectrum) are shown for the spectrum 8.

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[21] The inertial subrange of PSDs was typically well identifiable (Figure 2), which allowed investigations of “internal” intermittency using equations (5) and (6). At several segments, however, the −5/3 subrange was ill defined (e.g., segments 8, 24, and 30; an example is given in Figure 2), which were excluded from further analysis.

4.2. Calculation of the Structure Function Scaling Exponents

[22] The streamwise structure function of the vertical velocity (transverse SF or TSF) is defined as

  • equation image

where x is the along-flow distance, r = n(Uif) is the SF increment in the x direction, Ui is the magnitude of mean velocity at every segment (i = 1–44) containing 32628 samples each, Δf = 16 Hz the sampling frequency and n = 1, 2, … 32. Taylor's “frozen turbulence” hypothesis was employed to transform w(t) at each segment to the spatial series w(x). The applicability of Taylor's hypothesis was tested by calculating the ratio rmsi(w′)/Ui; it never exceeded 2.5% at segments close to high and low tides (minimum advecting velocity) and mostly took values below 1%. Possible impacts of Taylor hypothesis on SF exponents is further addressed in section 5.

[23] Periods close to high and low tides (low Renb) showed a narrow inertial range (see Figure 2) and hence evaluation of ξ(q) directly from the inertial-dissipative subrange of siq(r) could be subjected to substantial uncertainty. To overcome this problem, Benzi et al. [1993] suggested to scale the modulus of the qth-order SF not with respect to r, but with the modulus of the third-order structure function ∣s3(r)∣ ≡ S3(r). It is known [Kolmogorov, 1941c] that for rη, where η = (ν3/ɛ)1/4 is the Kolmogorov scale, S3(r) ∼ ɛr is accurate and therefore Si3(r) should not be affected by intermittency. In Figure 3, examples of log Si3(r) against log(r) are shown, and a wide “+1” subrange is evident for high Renb segments. This subrange almost disappears when Renb is low. Based on this rigorous result of the classic self-similarity theory, Benzi et al. [1993] argued that the velocity increment (Δwrq in our case) satisfies the equation

  • equation image

which is “… somehow more fundamental than the self-similar scaling with respect to r…” because it is valid even at moderate and low Reynolds numbers. Here Aq and Bq are two different sets of constants and the modulus accounts for the negative SF of odd orders. Equation (9) is sometimes referred to as the extended self-similarity, ESS [e.g., Benzi et al., 1993, 1995; van de Water and Herweijer, 1999; Hao et al., 2008]. The ESS allows for encompassing a wide scaling range even if the turbulence is not fully developed and the separation between the dissipative and the integral turbulent scales is narrow (e.g., experiments of Camussi et al. [1996] (laboratory) and Briscolini et al. [1994] (numerical); also detailed discussions on this topic are given in these papers). Formally, it was suggested to plot log Siq(r) versus log Si3(r) instead of log(r). Such plots (Figures 4 and 5) show linear sections of SFs with good accuracy for moderate and low Renb; on the other hand, the inertial subrange in spectral densities was hardly detectable for the same Renb.

image

Figure 3. The third-order transverse structure functions for several segments. The “+1” subranges are highlighted by bold lines. The arrows correspond to the turbulent integral scale Lint.

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image

Figure 4. Examples of TSF plots showing Siq(r) versus Si3(r) for relatively (a) high Renb ∼ 4 × 104 (segment 36) and (b) low Renb ∼ 7 × 103 (segment 11). The numbers near the curves (2, …, 14) designate the even order TSFs. The range Lint > r > LK is highlighted by solid circles. The linear sections of the structure functions (in log10-log10 scale) are well defined up to q = 11–12 in Figure 4a but only up to q = 7–8 in Figure 4b.

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image

Figure 5. Examples of TSF plots showing Siq(r) versus Si3(r) (q = 1–7) for segments with high ((a) 17 and (b) 39), moderate ((c) 35 and (d) 43), and low ((e) 23 and (f) 33) Renb. The linear sections (in log10-log10 scale) are well defined and coincide with Lint > r > LK range highlighted by solid circles.

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[24] The scaling Siq(r) versus Si3(r) was applied in the range Lint > r > LK, where

  • equation image

is the integral turbulent scale and LK = cdsη is the scale of maximum of dissipation. An approximate match between Lint and the low-wave number end of the inertial subrange in Ew(κ) spectra was attained with co = 0.6. Although the constant cds that defines LK could be as small as ∼5–7 [Gregg et al., 1996; Camussi et al., 1996] or as large as 30 [van de Water and Herweijer, 1999], an intermediate estimate cds = 15 [Monin and Yaglom, 1975] was used to fit the log10Siq(log10Si3) functions inside the inertial subrange, which did not necessarily cover all scales r. This also justifies the use of Lint and LK in evaluating SF of all orders, although the inertial subrange of higher-order structure functions extends to smaller scales [Frisch, 1995].

[25] Our tests with high-order Siq(r) (qmax = 14; rmax = 32 × Uif) showed that a confident linear fit in the range Lint > r > LK can be applied to log10Siq versus log10Si3 functions at almost all observational segments if q < 7–8. A few segments with the highest Renb (e.g., Figure 4) showed perfect linear fits for SF of much higher q = 11–12. To obtain confident scaling functions ξ(q) for different tidal phases and evaluate intermittency models given by equations (5) and (6), Siq(r) for all segments were calculated using the absolute values of streamwise increments of vertical velocity 〈∣Δwrq∣〉 for q = 1–7 in the r range (1–32) × Uif. Note that final estimates of Siq(r) at the largest separation scale were obtained by averaging more than 1000 individual samples of Δw′(r), ensuring minimal statistical error of Siq(r). Indeed, the error of the seventh-order SF after averaging is equal to the error σw of an individual original sample of w, namely 7σwequation imageσw. Several examples of Siq(r) versus Si3(r) are shown in Figure 5, which exhibit clear linear sections (solid circles) in log10 scale. Scaling exponents ξ(q) for every segment were determined by the least squares fits applied to these sections.

5. Results

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Structure Functions and Intermittency Models
  5. 3. Measurements and Background Hydrology
  6. 4. Method
  7. 5. Results
  8. 6. Discussion
  9. 7. Conclusions
  10. Acknowledgments
  11. References

5.1. Basic Characteristics of Turbulence Near the Seafloor

[26] The averaged kinetic energy dissipation rate near the bottom equation imagenb (spectral estimates, equation (7)) generally followed the mean flow magnitude U, which specified the near-bottom Reynolds number Renb in the range 5 × 103 – 8 × 104 (see time variations of equation imagenb, U, and Renb in Figure 1). The highest equation imagenb = 3 × 10−5 W/kg was associated with tidal flooding and the lowest with ebbing. The periodic modulation of 〈ɛnb〉 during the ebb tide with a period of ∼2 hours was consistent with seiching in the nearby Jiaozhou Bay [Lozovatsky et al., 2008a]. The near-bottom diffusivity Knb = equation imagenb1/3Ltr4/3 was ∼3 × (10−4 – 10−3) m2/s [Lozovatsky et al., 2008b], which is about an order of magnitude larger than that observed on nontidal shelves [Lozovatsky and Fernando, 2002; Roget et al., 2006].

[27] The tidal dissipation rate ɛtd = ctdUbt3/equation image, which is also shown in Figure 1, was calculated using the barotropic tidal velocity Ubt (ADCP data) and the mean water depth equation image = 19 m [MacKenzie and Leggett, 1993]. The constant ctd ≈ 0.003 that needed to match the spectral estimates of equation imagenb in Figure 1 was two times smaller than that (0.006) of Bowers and Simpson [1987]. The correspondence between equation imagenb and ɛtd is especially good during two periods (6.5 < t < 12.5 hours and t > 20.5 hours) of almost unidirectional westerly (ϕ ∼ 270°) flow with relatively high advection velocity (Figure 1).

[28] The turbulent kinetic energy etr and friction velocity u* were calculated by Lozovatsky et al. [2008b] using the near-bottom ADV covariance measurements. It was found that the drag coefficient Cd = u*2/U2 was approximately constant, with median value 1.65 × 10−3. The classical parameterization of wall dissipation equation imagenb = equation image appeared to be valid as well as equation imagenb = cɛequation image, with the constant cs being close to the generally acceptable value of 0.08 [e.g., Mellor and Yamada, 1982; Lozovatsky et al., 2006].

5.2. TSF Exponents in a Tidal Cycle

[29] The five-point running averaged estimates of ξ(q = 1–5) are shown in Figure 6 (ξ(3) = 1 by definition). The exponents ξ(1) and ξ(2) are above the classical values 1/3 and 2/3, respectively, in all segments. The low-frequency variations of the first two scaling exponents (Figure 6 (bottom)) generally follow the tidal cycle, showing a decreasing trend as the tidal velocity increases and reaching maxima at t ∼ 9 and 21 hours, corresponding to the maxima of Renb shown in Figure 6 (middle). The near-bottom Reynolds number Rnb was compared with the microscale turbulent Reynolds number

  • equation image

which is the major governing parameter for turbulence in the inertial-dissipative subrange [Tennekes and Lumley, 1972]. The Rλw is based on rms(w′) and the modified Taylor microscale

  • equation image

Rλw was selected over the conventional Rλ = rms(u′)λ/ν, where λ = equation image is the longitudinal Taylor microscale, because of the measurement accuracy considerations mentioned in section 4.2. The variations of Rλw almost mimic those of Renb (Figure 6 (middle)) suggesting a quasi-homogeneous energy cascade at scales between Ltr and λw.

image

Figure 6. The running averaged estimates of the scaling exponents ζ(q) for (bottom) q = 1, 2 and (top) q = 4, 5, and (middle) the near-bottom (Renb) and local microscale (Rλw) Reynolds numbers; note ζ(3) = 1.

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[30] The higher-order exponents ξ(4) and ξ(5) are always below their classical values of 4/3 and 5/3, respectively, (Figure 6 (top)) but unlike ξ(1) and ξ(2) they show a growing trend with increasing Renb and Rλw. This tendency is well correlated with the curvature of the empirical ξi(q) functions that cross the classic q/3 curve at q = 3 (see Figures 7 and 8). A departure of ξi(q) from the classical values appears to be pronounced at lower Renb (underdeveloped turbulence). In section 5.4, a physical interpretation of this tendency with decreasing Rλw is given.

image

Figure 7. Examples of the empirical scaling exponents ξ(q) for q = 1–7 at several segments (numbered in the insert) and their approximations by (a) the multifractal (equation (5)) and (b) lognormal (equation (6)) models. The symbols show empirical results and the lines show model predictions. The best-fit values of model parameters are given for each segment in the insets (C1 and α in Figure 7a and μ in Figure 7b).

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image

Figure 8. The scaling exponents ξ(q) for q = 1–7 for segments with the highest Renb. The (a) multifractal (equation (5)) and (b) lognormal (equation (6)) model approximations are shown. The segment numbers (17–41) and the corresponding model parameters (C1 and α (Figure 8a) and μ (Figure 8b)) are given in the insets.

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5.3. Intermittency Parameters

[31] The scaling exponents ξi(q) of TSF were approximated by equations (5) and (6) using the standard cftool utility of Matlab. The examples of ξi(q) with respective multifractal (equation (5)) and lognormal (equation (6)) fits are shown in Figure 7 for representative segments. All empirical exponents ξi(q) can be nicely fitted by equation (5) with a coefficient of determination above 0.98. In segments with the highest integral turbulent Reynolds number Rint (Figure 8)

  • equation image

where Lint is given in equation (10), the scaling exponents yield the mean and the rms boundaries of the codimension parameter equation image1 = 0.148 ± 0.016 and the Levy parameter equation image = 1.46 ± 0.13. Both are in good agreement with the “universal” values equation image1 = 0.15 and equation image = 1.5 obtained in previous high Reynolds number laboratory and atmospheric measurements cited in section 2.

[32] At the segments with low Rint (segments 5, 13, 23, 29 in Figure 7a), the scaling exponents ξi(q) substantially depart from the classic function ξi(q) = q/3, demonstrating an amplification of C1 compared to equation image1. This tendency is not monotonic, and at few segments with low Reint, C1 was also low (e.g., segments 28 and 31 not shown in Figure 7a).

[33] The lognormal approximation (equation (6)) of the same experimental ξi(q) functions yielded high-confidence fits (Figures 7b and 8b). The intermittency parameter μ followed the variations of C1, generally out of phase with Renb and Rint (see Figure 9). In section 2, it was mentioned that a universal value of equation image ∼ 0.25 is expected [Sreenivasan and Kailasnath, 1993] in well-developed high Reynolds number turbulent flows. For the highest Reint segments, the scaling exponents follow the lognormal intermittency model with a mean equation image = 0.237 ± 0.033 (Figure 8b), which is close to the suggested equation image = 0.25. This shows that internal intermittency of turbulence during energetic tidal flow phases can be satisfactorily described by either a multifractal intermittency model or a classical lognormal model with universal intermittency parameters. In segments with moderate and low Reint, μ generally exceeded equation image, sometimes increasing to 0.5 or beyond (see Figure 7b), although, as in the case of C1, this tendency is not monotonic.

image

Figure 9. Parameters of the multifractal (equation (5), C1 and α) and lognormal (equation (6), μ) intermittency models. The near-bottom (Renb) and integral turbulent (Rint) Reynolds numbers are mainly in phase with α and out of phase with C1 and μ. The intermittency parameters are shown with 95% confidence bounds. Two periods of flooding current with highest Renb and Rint recorded are highlighted.

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[34] The running average of the intermittency parameters shown in Figure 9 indicate that the temporal variations of μ, C1 and α are all affected by the semidiurnal tidal cycle and seiching, which is evident from temporal variations of Renb and Reint.

5.4. Dependence on the Turbulent Reynolds Number Rλw

[35] In order to illustrate the dependence of μ, C1, and α on internal flow parameters, they are plotted in Figure 10 as a function of the microscale Reynolds number Rλw (equation (11)), which is a variant of Rλ = rms(u′)λ/ν. This modification, however, precludes direct comparison of our results with most laboratory experiments, where Rλ is specified. Despite high scatter, the data show a general increase of μ and C1 with decreasing Rλw (Figures 10 (top) and 10 (middle)). The best least squares power trends are given with 95% confidence bounds and the coefficients of determination r2. The empirical functions so obtained are

  • equation image

implying that μ and C1 have similar dependencies on Rλw, attaining asymptotic values μo = 0.23 and C1o = 0.13 at high Rλw > 500–700. Both estimates are close to the universal values of equation image and equation image1 expected from turbulence intermittency models at high Reynolds numbers. A weak (statistically insignificant) dependence of α on Rλw is shown in Figure 10 (bottom). The mean value of α, equation image = 1.53 ± 0.39, however, matches well with the universal equation image = 1.5–1.55 cited by Seuront et al. [2005] for log-Levy multifractal intermittency model. It should be reminded that the analysis here was based on TSF.

image

Figure 10. The dependencies of intermittency parameters μ, C1, and α on the local turbulent Reynolds number Rλw. The least-squared fits with 95% lower and upper confident bounds (for μ and C1) are shown. The laboratory data of Hao et al. [2008] are shown by stars (see text for details).

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[36] Qualitatively similar dependence of ξi(q) on Rλ has been reported by Hao et al. [2008, Figure 6] for TSF. These authors, however, found that the scaling exponents for the longitudinal structure functions are independent of Rλ. In order to compare laboratory findings with our results, Figure 6 of Hao et al. [2008] was digitized, wherein the scaling exponent ξ(q = 0–8) of TSF is shown for Rλ = 120, 184, 250, and 350. The empirical functions ξ(q) were then approximated by multifractal (equation (5)) and lognormal (equation (6)) intermittency models. The resulting intermittency parameters are shown in Figure 10 by stars. The laboratory and field data agreed well in the range of the variability of μ and α. We conclude that the oceanographic and laboratory measurements of TSF show an increasing tendency of intermittency parameters with decreasing Rλw, which is evident from Figure 10.

5.5. Scaling Exponent of Second-Order SF as a Function of Rλw

[37] As pointed out before, refined similarity hypothesis (RSH) suggests that spectral densities of velocity components and the corresponding second-order SF must deviate in the inertial subrange from the classic “−5/3” and “2/3” laws, respectively, due to internal intermittency. If the intermittency parameters C1 and μ are dependent on Rλw at relatively low Rλw < 500–700 (equation (14)), then the scaling exponent of the second-order SF ξ(2) can be a function of Rλw as well. Based on the concept of incomplete self-similarity, Barenblatt [1996] showed that the limiting behavior of the first moments of the velocity field takes the form of an inverse logarithmic dependence on the Reynolds number [see also Barenblatt et al., 1999; Barenblatt and Chorin, 2004]. Therefore, the higher moments are expected to converge to a limit at the same or a lower rate with increasing Re. For the second-order structure function S(q=2), the following formula has been suggested [Barenblatt and Goldenfeld, 1995; Barenblatt et al., 1999]

  • equation image

where c0, c01, and γ1 are constants, and ln Re ≡ log Re is the natural logarithm of a Reynolds number Re. If Re is taken as Rλw and empirical values of S(q=2) exponent ξi(2) are plotted against log Rλw, then a decreasing trend of ξ(2) is well evident with increase of log Rλw (see Figure 11a). In the spirit of equation (15), this trend could be approximated as

  • equation image

The least squared fit of equation (16) to data gave γ1 = 0.77 and n = 1.63 with r2 = 0.76 for the entire range of log Rλw variability (log Rλw = 4–7.5). The rate of increase of ξ(2) at low Rλw is substantially higher than the inverse logarithmic rate that equation (15) predicts. If only high Rλw are considered and the approximation (16) with n = 1 is enforced for log Rλw > ∼ 5.1 (Rλw > 150), the inverse logarithmic function fits the data reasonably well with γ1 = 0.24.

image

Figure 11. (a) Scaling exponent of the second-order TSF ζ(2) as a function of the natural logarithm of local turbulent Reynolds number log Rλw approximated by equation (15) (dotted line) using the adjustable parameter of equation (16)γ1 = 0.24 (25 samples for log Rλw > 5.1). A modification for this formula (with n = 1.63 and γ1 = 0.77) was applied to the entire Rλw variability (bold line). (b) The ratio between the “Taylor separation distance” dm = η and LK = 15η, where η = (ν3/equation image)1/4 and τη = (ν/equation image)1/2 are the Kolmogorov length and time scales, respectively, and U the mean velocity for each segment.

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6. Discussion

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Structure Functions and Intermittency Models
  5. 3. Measurements and Background Hydrology
  6. 4. Method
  7. 5. Results
  8. 6. Discussion
  9. 7. Conclusions
  10. Acknowledgments
  11. References

6.1. Universality of ξ(2)

[38] The behavior of ξ(2) at low Reynolds numbers suggested by Barenblatt and Goldenfeld [1995] appears to be born out by our data, but the rate of increase is not in par with that predicted by the inverse logarithmic rate (equation (15)). Note that the 2/3 power law for ξ(2) is expected only at very high Rλw. For the range of Reynolds numbers considered in this paper, ξ(2) somewhat exceeded 2/3 and was close to 0.7 for log Rλw > 6, which is consistent with the laboratory results of Benzi et al. [1995]. Benzi et al. [1999] also concluded that the universal (Reynolds number independent) value of ξ(2) is close to 0.7 rather than 2/3. This assertion is not supported by the presented data, which show the possibility of further reduction of ξ(2) toward classical 2/3 at very high Rλw.

6.2. Test of Taylor Hypothesis

[39] Our analysis of turbulence intermittency in a tidal BBL and several laboratory experiments [e.g., Shen and Warhaft, 2000; Pearson and Antonia, 2001; Cleve et al., 2004; Arenas and Chorin, 2006; Hao et al., 2008] have shown that ξ(q) is a function of Re. The Re dependence of ξ(q), however, has been disputed [e.g., Benzi et al., 1999; Castaing, 2006] mainly on the basis that Taylor hypothesis may fail at low Rλ by introducing a bias to SF exponents. Responding to the concern of Castaing [2006], Sreenivasan [2006] questioned whether the relationship between ξ(q) and Rλ can be affected by Taylor hypothesis, calling for direct and accurate spatial turbulent measurements to resolve the concern. To our knowledge, no such measurements exist, but the numerical simulations on SF without using Taylor hypothesis [e.g., Briscolini et al., 1994] have confirmed the applicability of ESS in flows with Rλ as low as 32.

[40] Although the rmsi(w′) in the tidal flow never exceeded several percents of the mean advection velocity Ui, we decided to study the influence of Taylor hypothesis on our data by running an additional test recommended by Castaing [2006]. Restricting frozen turbulence hypothesis to the Kolmogorov time scale τη = (ν/equation image)1/2, Castaing [2006] suggested that the “Taylor separation distance” dm = η must exceed the dissipation scale (LK = 15η) in order to develop a reasonable linear range in SF. Using our measurements, the ratio dm/LK was calculated for each data segment by employing spectrally obtained estimates of the mean dissipation rate equation image. The ratio dm/LK is shown in Figure 11b as a function of log(Rλw), which can be inspected vis-à-vis the plot of ξ(2) versus log(Rλw) in Figure 11a. For log(Rλw) > 5, the results show dm/LK > 6, suggesting that even for not very high turbulent Reynolds numbers Taylor hypothesis should work well (Rλw > 150). As such, the weak inverse logarithmic dependence of ξ(2) on Rλw observed for log(Rλw) > 5 can be accepted with confidence. The samples of ξ(2) at log(Rλw) < 5, which show a faster increase of the scaling exponent with decreasing Rλw, were in the range 3 < dm/LK < 6, which is not sufficient to invalidate Taylor hypothesis. The break of ξ(2) behavior observed at log(Rλw) ≈ 5, nevertheless, raises concerns of whether there is a breakdown of Taylor hypothesis when log(Rλw) < 5.

6.3. Influence of Anisotropy and the Nature of the Flow

[41] The scaling exponents of TSF can be differently affected by the Reynolds number compared to those of the longitudinal SF due to anisotropy, which was found in some laboratory studies [Antonia et al., 2002]. The nature of the flow may also influence the Rλw dependence of ξ(q) and intermittency model parameters. For example, in the oceans, the intermittency in a reversing highly sheared tidal flow over the seabed (current case) may have different scaling compared to wave-generated turbulence, flow near the sea surface (drift currents), currents around and behind obstacles (seamounts), parallel shear and jets flows (equatorial counter currents) and river plumes. After all, large-scale structures of these flows are different, and the assumption was made that the selected averaging period was appropriate to analyze only the universal turbulent structures and energy cascade. The use of isotropic assumption in calculation of the dissipation rate instead of utilizing the full three-dimensional energy dissipation tensor [e.g., Hosokawa et al., 1996; Wang et al., 1996] may affect Rλw and Rλ calculations, and hence the dependence of ξ(q) on corresponding Reynolds numbers. All these factors may influence specific forms of the scaling laws given by equations (14) and (16), but the tendencies of increasing μ, C1, and ξ(2) with decreasing Rλw is expected to hold. The intermittency parameters and the scaling exponent of the second-order SF approach their universal asymptotic values at very high Reynolds numbers.

7. Conclusions

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Structure Functions and Intermittency Models
  5. 3. Measurements and Background Hydrology
  6. 4. Method
  7. 5. Results
  8. 6. Discussion
  9. 7. Conclusions
  10. Acknowledgments
  11. References

[42] The ADV measurements of vertical velocity (w) in a tidal current on a shallow shelf of the East China Sea were used to calculate higher-order transverse structure functions (TSF) in the streamwise direction, based on which the intermittency characteristics of turbulence near the seafloor were studied. It was found that at fairly low Reynolds numbers Re the TSF scaling exponents ξ(q) (with respect to the third-order TSF) progressively deviate from the classical universal turbulent regime ξ(q) = q/3 as Re decreases, possibly due to higher intermittency of underdeveloped turbulence. The second-order TSF scaling exponent ξ(2) shows an inverse logarithm dependence on the microscale turbulent Reynolds number Rλw for Rλw > 150, supporting the theoretical formulation of Barenblatt and Goldenfeld [1995].

[43] The application of the classical single-parameter (μ) lognormal model and two-parameter (C1 and α) log-Levy multifractal model for scaling of SF exponents ξ(q) shows that during high-speed flooding phases of tidal flow the mean intermittency parameters are equation image ≈ 0.24, equation image1 ≈ 0.15, and equation image ≈ 1.5. These are close to the canonical universal values for well-developed turbulence at high Reynolds numbers. When the turbulent Reynolds number Rλw drops below ∼100, μ and C1, showed a tendency to increase up to μ ∼ 0.5–0.6 and C1 ∼ 0.25–0.35.

[44] The relationships between μ, C1 and Rλw were approximated by power law functions (equation (14)) with asymptotic values of μo = 0.23 and C1o = 0.13 at very high Reynolds numbers. The dependence of intermittency parameters on Reynolds number delineated in this study helps to resolve the controversy between small universal values of μ and C1 obtained previously in high Reynolds number laboratory and atmospheric flows and relatively large values of μ = 0.4–0.5 reported for stratified ocean turbulence [Baker and Gibson, 1987; Gibson, 1991; Fernando and Lozovatsky, 2001]. Note that turbulent patches in the pycnocline are usually associated with relatively low Rλw. The dependence of intermittency parameters on microscale Reynolds number obtained in laboratory experiments of Hao et al. [2008] is in general agreement with the results of the present study for tidally induced turbulence in the marine bottom boundary layer. The influence of Reynolds number Rλ on the scaling exponents of transverse and longitudinal structure functions is expected to depend on the nature of the flow and anisotropy of turbulence, which must be addressed in future studies. Small-scale as well as mesoscale intermittencies of turbulence in natural waters should be taken into account in developing new parameterizations of vertical and lateral mixing for advance numerical models of oceanic and atmospheric dynamics.

Acknowledgments

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Structure Functions and Intermittency Models
  5. 3. Measurements and Background Hydrology
  6. 4. Method
  7. 5. Results
  8. 6. Discussion
  9. 7. Conclusions
  10. Acknowledgments
  11. References

[45] This study was supported by the U.S. Office of Naval Research (grant N00014-05-1-0245), the Spanish Ministry of Education and Science (grant FIS2008-03608), and the Major State Program of China for Basic Research (grant 2006CB400602). The first author also received financial support during his temporary affiliation with the Catalan Institute for Water Research (ICRA).

References

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Structure Functions and Intermittency Models
  5. 3. Measurements and Background Hydrology
  6. 4. Method
  7. 5. Results
  8. 6. Discussion
  9. 7. Conclusions
  10. Acknowledgments
  11. References