### Abstract

- Top of page
- Abstract
- 1. Introduction
- 2. Structure Functions and Intermittency Models
- 3. Measurements and Background Hydrology
- 4. Method
- 5. Results
- 6. Discussion
- 7. Conclusions
- Acknowledgments
- 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 (*C*_{1} and *α*) log-Levy model) were employed to analyze the TSF exponent *ξ*(*q*) in tidally driven turbulent boundary layer and to estimate *μ*, *C*_{1}, and *α*. During the energetic flooding tidal phases, the parameters of intermittency models approached the mean values of ≈ 0.24, _{1} ≈ 0.15, and ≈ 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 *C*_{1} increased up to *μ* ≈ 0.5–0.6 and *C*_{1} ≈ 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 *C*_{1} (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 *R*_{λw}. 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 *R*_{λw}, approaching the classical value of 2/3 only at very high *R*_{λw}. 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

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

[2] By its very nature, turbulence in geophysical flows is highly intermittent in space and time. Turbulence characteristics such as the kinetic energy *e*_{tr}, its dissipation ɛ, eddy diffusivities *K*, scalar dissipation *χ*, turbulent scales *L*_{tr} 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 ɛ_{r} or *q*th-order statistical moments of velocity increments 〈Δ*V*_{r}^{q}〉, which are also called the *q*th-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

- Top of page
- Abstract
- 1. Introduction
- 2. Structure Functions and Intermittency Models
- 3. Measurements and Background Hydrology
- 4. Method
- 5. Results
- 6. Discussion
- 7. Conclusions
- Acknowledgments
- 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 Δ*V*_{r} over a separation distance *r* is specified not by the mean dissipation rate but the dissipation ɛ_{r} averaged over a specific volume of radius *r* < *L*_{o}, which leads to the following scaling relation

where *L*_{o} is an external turbulent scale and Δ*V*_{r} 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 *q*th-order velocity increment

The probability distribution of ɛ_{r} 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 ɛ_{r} with the variance

where *A*_{ɛ} depends on the large-scale motions and *μ* is an intermittency factor that accounts for stretching of the probability distribution function of log ɛ_{r} [e.g., *Monin and Yaglom*, 1975; *Frisch*, 1995].

[11] For the scalar dissipation rate *χ*_{r} (e.g., fluctuations of temperature, conductivity, fluorescence concentration) [*Gibson*, 1981]

where *μ*^{sc} is the intermittency factor of the scalars corresponding to *χ*_{r} and *A*_{χ} depend on the characteristics of mean flow. Both intermittency factors *μ* and *μ*^{sc} are assumed universal, with canonical values *μ* ≈ 0.25 and *μ*^{sc} ≈ 0.35 for very high Reynolds number turbulence, with *μ*^{sc} 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 *μ*^{sc}/9. The modified spectral functions are [*Monin and Yaglom*, 1975]

where *c*_{K} and *c*_{sc} are universal constants. Equations (2a) and (2b) have been utilized for empirical estimation of *μ* and *μ*^{sc} in laboratory experiments [e.g., *Gibson et al.*, 1970] and for ocean turbulence. For mesoscale intermittency, *μ* and *μ*^{sc} ranged between 0.44 and 0.5 (see review of *Gibson* [1998]). The same numbers (*μ*^{sc} = 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 *p*_{r} (a fraction of the volume *L*_{o}^{3} occupied by turbulence of characteristic scale *r* = *γ*^{n}L_{o}, 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], *p*_{r} is assumed to decrease at all levels by the same constant factor 0 < *β* < 1, so that after *n* (= log(*r*/*L*_{o})/log *γ*) breakdowns, *p*_{r} = *β*^{n} = (*r*/*L*_{o})^{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 ɛ_{r}^{q}^{/3} has a power law variation with *r* in the inertial subrange then all *q*th-order statistical moments of velocity increments (i.e., *q*th-order structure functions) can be written as

where the constant *C*_{q} 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 *r*(ɛ_{r}r)^{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

where *C*_{1} and *α* reflect the multifractal nature of intermittency in point. *Seuront et al.* [2005] suggested that *C*_{1} 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 *C*_{1} characterizes the intermittency of the turbulent field in a way that sparseness and variability of the localized events (singularities) increase with increasing *C*_{1} 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

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

### 3. Measurements and Background Hydrology

- Top of page
- Abstract
- 1. Introduction
- 2. Structure Functions and Intermittency Models
- 3. Measurements and Background Hydrology
- 4. Method
- 5. Results
- 6. Discussion
- 7. Conclusions
- Acknowledgments
- 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 *ζ*_{s} = 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 M_{2} 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 M_{2} 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].

### 7. Conclusions

- Top of page
- Abstract
- 1. Introduction
- 2. Structure Functions and Intermittency Models
- 3. Measurements and Background Hydrology
- 4. Method
- 5. Results
- 6. Discussion
- 7. Conclusions
- Acknowledgments
- 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 *R*_{λw} for *R*_{λw} > 150, supporting the theoretical formulation of *Barenblatt and Goldenfeld* [1995].

[44] The relationships between *μ*, *C*_{1} and *R*_{λw} were approximated by power law functions (equation (14)) with asymptotic values of *μ*^{o} = 0.23 and *C*_{1}^{o} = 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 *C*_{1} 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 *R*_{λw}. 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 *R*_{λ} 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.