469 30 An Application of the Lognormal Theory to Moderate Reynolds Number Turbulent Structures Hidekatsu Yamazaki and Kyle D. Squires CONTENTS 30.1 30.2 Lognormal Theory 470 30.3 Simulations 471 30.4 Discussion 474 30.4.1 Surface Turbulent Layer 475 30.4.2 Subsurface StratiÞed Layer 477 Acknowledgments 477 References 478 30.1 Introduction Kolmogorov (1941) proposed one of the most successful theories in the area of turbulence, namely, the existence of an inertial subrange. Successively, Kolmogorov (1962) revised the original theory to take the variability of the dissipation rate in space into account. The process of this reÞnement introduced a lognormal model to describe the distribution of dissipation rates. The inertial subrange theory requires an energy cascade process, whose length scale is much larger than that of the viscous dominating scale. Thus, the types of ßows to which the theory applies occur at high Reynolds numbers. Geophysical ßows provide an example in that they typically occur at high Reynolds numbers because the generation mechanism is usually much larger than the viscous dominating scale. In fact, the Þrst evidence of the existence of an inertial subrange came from observations of a high Reynolds number oceanic turbulent ßow (Grant et al. 1962). Gurvich and Yaglom (1967) further developed the lognormal theory that described the probability distribution of the locally averaged dissipation rates. In their work, the theory was also intended for high Reynolds number ßows to simplify the development (see also Monin and Ozmidov, 1985). Although both the inertial subrange and lognormal theories successfully describe high Reynolds number turbulence, an important question arises: To what degree are these theories appropriate to turbulence occurring over a moderate Reynolds number range, whose power spectrum does not attain an inertial subrange? Clearly, the inertial subrange theory is out of the question; i.e., there is a limited range of scales at moderate Reynolds numbers. However, is it possible that the dissipation rate in moderate Reynolds number turbulence obeys the lognormal theory? Relevant to the present chapter is that turbulence generated at laboratory scales in many facilities does not attain high Reynolds numbers; thus, energy spectra do not typically exhibit an inertial subrange. Microorganisms, such as zooplankton in the ocean, may be transported in the water column by a large- scale ßow that is clearly occurring at high Reynolds numbers, but the immediate ßow Þeld surrounding © 2004 by CRC Press LLC Introduction 469 470 Handbook of Scaling Methods in Aquatic Ecology: Measurement, Analysis, Simulation the individual organism in a seasonal thermocline is another example of moderate Reynolds number turbulence (Yamazaki et al., 2002). The lognormal theory provides a simple statistical representation of the ßow, as well as yielding a tool to predict the local properties of the strain Þeld. If lognormality holds at moderate Reynolds numbers, it would enable one to predict the probability of the strain Þeld in many ßows of practical interest. Turbulence dissipation rates reported in the literature are normally values averaged over a scale of a few meters. On the other hand, a relevant scale for the encounter rate of predator/prey is normally much shorter than 1 m. It is important to note that the volume-averaged dissipation rate associated with this length scale will not be identical to that obtained for the original domain since the dissipation rate for this length scale is an additional random variable that obeys a different probability density function from the mother domain. The lognormal theory assists in understanding the local properties of velocity strains. Direct numerical simulation (DNS) is well suited for investigating the applicability of the lognormal theory at moderate Reynolds numbers. A signiÞcant advantage of DNS relevant to this study is that all components of the strain rate can be directly computed and the dissipation rate can be calculated as a function of position and time. DNS studies, e.g., Jiménez et al. (1993), show that the strain Þeld of turbulence is dominated by Þlament-like structures. These coherent structures are crucial to understanding ßow dynamics. Yamazaki (1993) proposed that planktonic organisms may make use of these structures to Þnd mates and prey/predator. Presented in this chapter is a demonstration that the lognormal theory is consistent with the strain properties associated with the Þlament structures, at least, for moderate Reynolds numbers. 30.2 Lognormal Theory A complete discussion of the lognormal theory can be found in Gurvich and Yaglom (1967). The theory can be developed by considering a domain, Q, with energy-containing eddies of size, L, where Q is proportional to L 3 . The volume-averaged dissipation rate over Q is denoted and is deÞned as (30.1) where e(x) is the local dissipation rate. The original domain, Q, is successively divided into subdomains denoted q i , whose length scale is l i . This successive division process is referred to as a breakage process. The average dissipation in a volume q i is then (30.2) The dissipation rate e i is a random variable representing the average within q i . The breakage coefÞcient, a, is deÞned as a ratio of two successive e i : for (30.3) where N b is the number of breakage processes. In the original lognormal theory, the ratio of length scales l i–1 and l i for two successive breakages is a constant, l b = l i /l i–1 . At the N b breakage, the volume averaged dissipation rate in a single cell, e r , for the averaging scale can be expressed in terms of by (30.4) where r might be considered as an encounter rate length scale, such as perception distance/reaction distance. Gurvich and Yaglom (1967) assumed that the random variable log a i follows a normal distri- bution. One drawback of the Gurvich and Yaglom theory is that, if a is lognormal, the maximum value of a is inÞnity. Yamazaki (1990) argues that the maximum value of a cannot exceed and proposes e ee= () - Ú Qxdx Q 1 ee ii q qxdx i = - Ú 1 () aee iii = - / 1 iN b = 1, , rl N b = e log log logee a ri i N b =+ = Â 1 l b 3 © 2004 by CRC Press LLC An Application of the Lognormal Theory to Moderate Reynolds Number Turbulent Structures 471 the B-model, which assumes a beta probability density function for a. The B-model predicts high-order statistics of velocity well. An important question arises in the above development: Is the assumption of high Reynolds number required in the lognormal theory? There are two constraints: a i is mutually independent and N b is large. However, in practice, the Þrst condition is not so strict, and the second requirement may be as small as 2 or 3 (Mood et al., 1974). In other words, the sum of a few random variables, e.g., log a i , tends to approach a normal distribution as the central limit theorem predicts. Therefore, there is no explicit requirement for the existence of an inertial subrange to satisfy these conditions. Hence, it may be reasonable to expect that the lognormal theory might be applicable to turbulence occurring at modest Reynolds numbers in which there is no inertial subrange. It should be noted that, while Gaussian statistics is an approximation, increasingly less accurate for the higher-order moments as shown by Novikov (1971) and Jiménez (2000), the lognormal theory has provided a reasonable model for some applications (e.g., see Arneodo et al., 1998). The practical advantages offered via assumption of Gaussian statistics outweigh the inaccuracies in many instances, e.g., as applied to positive-value statistics such as temperature and rainfall. In this chapter, we emphasize the practical aspects of application of the lognormal theory for analyzing the dissipation rate for turbulent ßows at moderate Reynolds numbers, bearing in mind the limitations of the theory as shown by other investigators. 30.3 Simulations We have simulated isotropic turbulence using DNS of the incompressible Navier–Stokes equations (Rogallo, 1981). A statistically stationary ßow was achieved by artiÞcially forcing all nonzero wave- numbers within a spherical shell of radius K F (Eswaran and Pope, 1988). For the simulations presented here, , corresponding to 92 forced modes. The small-scale resolution is measured by the parameter k max h, where h is the Kolmogorov length scale and k max is the highest resolved wavenumber. The value of h is obtained from (n 3 /e) 1/4 where n is the kinematic viscosity of the ßuid. In this study, k max h was approximately 2. Several preliminary computations were performed to ensure the adequacy of the numerical parameters and to test the data reduction used to acquire the dissipation rate. Most of the results presented in this chapter are from simulations performed using 64 3 collocation points, corre- sponding to a Taylor-microscale Reynolds number Re l = 29 (Case C64). Although a single simulation (sampled over time) should be sufÞcient for testing the hypothesis that the lognormal theory is applicable to a moderate Reynolds number ßow, simulations performed at higher resolution were desired to give some conÞdence that conclusions from this study were relatively free of resolution effects and not adversely inßuenced by the scheme used to maintain a statistically stationary state. Therefore, calcula- tions were also performed at a higher resolution 96 3 (Case C96) and used to conÞrm the trends observed at the lower resolution, in which there is less separation between the peaks of the energy and dissipation The calculations were run using a Þxed time step, chosen so that the Courant number remained approximately 0.40. The ßow was allowed to evolve to a statistically stationary state; ßow-Þeld statistics time T e = L f /u¢, in which L f is the longitudinal integral time scale and u¢ is the root-mean-square velocity, for subsequent postprocessing of the dissipation rate. For each grid resolution, an ensemble of ten velocity Þelds was processed to determine the minimum averaging scale at which lognormality was satisÞed as well as to calculate breakage coefÞcients. Each velocity Þeld was subdivided into smaller volumes, and the dissipation rate within a given subdomain was calculated by integrating over the grid point values within a given volume. B-spline integration (de Boor, 1978) was used for calculation of the dissipation rate within subvolumes to faithfully follow the deÞnition of local averaging given in Equation 30.2. Note that Wang et al. (1996) averaged grid point dissipation rates arithmetically. K F = 22 © 2004 by CRC Press LLC spectra (Figure 30.1). The Taylor-microscale Reynolds number for the higher-resolution ßow is 42. were then acquired over a total time period T (Figure 30.2). Flow Þelds were saved every eddy turnover 472 Handbook of Scaling Methods in Aquatic Ecology: Measurement, Analysis, Simulation Lognormality for the compiled data is tested by making use of the Kolmogorov– Smirnov test (KS test) at a 5% signiÞcance level. The KS test is a powerful tool to distinguish if the samples are drawn from a hypothesized distribution; however, the target distribution must be free from the estimation of para- meters or without parameters involved in the distribution (Mood et al., 1974). In other words, if the hypothesized distribution contains some parameters, e.g., the mean and the variance, the KS test is not, rigorously speaking, applicable. As usual, in the practical application of statistical theories, since no other simple test is available to determine if the samples come from the hypothesized distribution, the KS test is employed in this work, albeit with the limitations described above. If the theory is applicable to the present simulations, locally averaged e r should be lognormal, but no shows the quantile–quantile plot (qq-plot) of instantaneous dissipation rates, equivalent to grid-level dissipation rates, for Case C64. The distribution is clearly different from a lognormal distribution. Yeung and Pope (1989) and Wang et al (1996) also show a similar distribution for the grid-level dissipation rates, but at higher Reynolds numbers, Re l = 93 in Yeung and Pope and Re l = 151 in Wang et al. There is of course no a priori knowledge of the probability distribution of the instantaneous dissipation rates and, hence, it should not seem surprising that the grid-level values do not distribute as lognormal. The lognormal theory is only applicable to a locally averaged quantity; therefore it is necessary to consider a locally averaged dissipation rate, e r . The grid-level dissipation rate exhibits features remarkably similar to instantaneous dissipation rates observed in geophysical data (Yamazaki and Lueck, 1990). Stewart et al. (1970) measured the velocity in the atmospheric boundary layer over the ocean. They attributed the departure from lognormality to be caused by a limited cascade process with an insufÞcient Reynolds number. They presumed that to satisfy the lognormal theory, it was necessary for the turbulence Reynolds number to be very high. Because there was no local averaging applied to their data, the reported values were essentially the same as the grid-level dissipation rates in the present DNS. They also argued that the departure from log- normality at the low end of the distribution was caused by instrument noise. The DNS results, however, FIGURE 30.1 Three-dimensional energy and dissipation spectra. Case C64: dotted line is energy and chain dot line is dissipation; Case C96: solid line is energy and dashed line is dissipation. FIGURE 30.2 Temporal variation of the volume-averaged dissipation rate. Case C64, solid line; Case C96, dashed line. 10 -1 10 0 0.0 0.2 0.4 0.6 k η 2kE(k)/q 2 , kD(k)/< ε > 010203040 -1.00 -0.75 -0.50 -0.25 0.00 0.25 0.50 0.75 1.00 t/T e (<ε( t )>-<ε>)/<ε> © 2004 by CRC Press LLC information is given in the theory on how the instantaneous dissipation rate, e(x), distributes. Figure 30.3 An Application of the Lognormal Theory to Moderate Reynolds Number Turbulent Structures 473 do not suffer from analogous problems. Small-scale resolution of the velocity Þeld has been carefully maintained. Therefore, the concave nature of the grid-level dissipation rates (the instantaneous values) is possibly a more universal characteristic of the kinetic energy dissipation rate. If one is interested in extremely high values of the local dissipation rate, the lognormal theory provides an upper bound for the estimate. In other words, the actual value should be smaller than the predicted value. On the other hand, if one is interested in extremely low values, the lognormal theory overpredicts the values compared to the actual dissipation rate. To investigate what averaging scale satisÞes the lognormal theory, we have computed the local average of dissipation rates with varying averaging scales for each of the ten Þelds, as well as compiled all data. Lognormality is tested for these compiled data sets. The minimum averaging scale for lognormality to hold in terms of the Kolmogorov scale for the two cases are similar, 9.5 for Case C64 and 10.2 for Case C96. Because statistics may change from one realization (i.e., velocity Þeld) to the next, lognormality of the dissipation rate for each of the ten different Þelds has also been examined. Shown in Table 30.1 are the numbers of individual Þelds passing lognormality for Case C64. The minimum averaging scale for the entire ensemble of ten Þelds is 9.5, but there are several individual Þelds satisfying lognormality at smaller averaging scales. Although one Þeld at r/h = 7.9 failed the KS test, all individual Þelds follow lognormality for an averaging scale as small as 6.3. This is roughly 30% smaller than that obtained using the entire ensemble. FIGURE 30.3 The quantile–quantile plot of grid-level dissipation rate and prediction from lognormal distribution for Case C64. TABLE 30.1 Number of Individual Fields Passing KS Test for C64 Case No. of Cells for Local Averaging No. of Fields Passing KS Test 10 3 9.5 10 11 3 8.6 10 12 3 7.9 9 13 3 7.3 10 15 3 6.3 10 16 3 5.9 8 20 3 4.7 6 25 3 3.8 3 27 3 3.5 3 30 3 3.2 1 32 3 3.0 1 -4 -3 -2 -1 0 -4 -3 -2 -1 0 1 Theoretical values Computed values 1 r / h © 2004 by CRC Press LLC 474 Handbook of Scaling Methods in Aquatic Ecology: Measurement, Analysis, Simulation To consider why individual cases can satisfy lognormality at smaller averaging scales compared to the entire ensemble of ten Þelds, we consider the nature of the KS test. The test statistic is the maximum difference between the observed cumulative distribution function and the hypothesized cumulative distribution function. The critical value for the test statistic is deÞned as, , where d g is the critical value at a certain signiÞcance level g, and n is the number of samples. When the test statistic exceeds d, the hypothesis that the samples come from the proposed probability density function is rejected at the speciÞed signiÞcance level. For a signiÞcance level of 5%, as used in this study, the value of d g is 1.36. As the number of samples increases, the test value decreases. Thus, the test is more difÞcult to pass for larger sample sizes. As we have mentioned earlier, the KS test is developed for a parameter- free distribution. However, we are using an estimated mean and variance for the hypothesized lognormal distribution, so we are violating the assumptions for the KS test. Therefore, the observed minimum averaging scale difference between the ten-Þeld case and single-Þeld cases is, most likely, due to the violation of the KS test assumption. Unfortunately, we do not have any other simple way to test the hypothesized distribution. Practically speaking, the observed dissipation rate is very close to a lognormal distribution even at the smallest averaging scale obtained from the single-Þeld case. It is further interesting to note that one Þeld satisÞes lognormality at an averaging scale r/h = 3.0. This is almost identical to the minimum averaging scale for oceanic data (Yamazaki and Lueck, 1990). Despite the difference in the nature of the data source, the minimum averaging scales obtained from the present moderate Reynolds number ßow calculated using DNS, which are roughly between 5 and 10, are remarkably close to the geophysically observed values. Making use of a laboratory air- tunnel experiment, van Atta and Yeh (1975) report 36h as the length scale that assures statistical independence between successive observations. The sample independence length scale should be larger than the corresponding minimum averaging scale for lognormality. The laboratory experiment also provides a similar minimum averaging scale to the present simulation results. Recently, Benzi et al. (1995, 1996) show velocity scale similarity as small as 4h using both wind-tunnel experiments and direct numerical simulations, and propose a new scaling notion: extended self-similarity (ESS). These observations are consistent with each other, showing that the lognormal theory is fairly robust at moderate Reynolds numbers. How the breakage coefÞcient distributes is an important issue in the lognormal theory. However, no previous investigation has been made to examine the appropriate distribution of this coefÞcient. Yamazaki (1990) proposed the Beta distribution and developed the B-model. The minimum averaging scale at which lognormality holds for each individual Þeld has been used as a child domain length scale, i.e., l c = 6.32. The corresponding mother domain for l = 5, which is the recommended value, is then l m = 31.6. Thus, the entire volume is subdivided into 15 3 cells for the child domain and 3 3 cells of the mother domain. The breakage coefÞcient, a, is tested against both the Beta distribution (the B-model) and the lognormal observed statistics well. The lognormal distribution, on the other hand, exhibits a poor Þt to the data. 30.4 Discussion Although the lognormal theory is not developed from a vigorous ßuid mechanical point of view, the theory seems to work remarkably well even if the ßow occurs at moderate Reynolds numbers, which lack an inertial subrange. Therefore, it offers the possibility of a practical tool for predicting locally averaged dissipation rates at spatial scales larger than 10h and the minimum averaging scales as small as three times h. The theory can be extended to smaller averaging scales bearing in mind that the theory overpredicts high value of dissipation rates. A perception distance of larval Þsh may be taken as the local averaging scale of dissipation rate in order to predict the upper band for encounter rate with prey. Another example is that an ambient ßow Þeld around a single organism can be extrapolated from the average dissipation rate of a turbulent water column. Incze et al. (2001) observed that several copepod species avoided high turbulent water column when the dissipation rate exceeded 10 –6 W kg –1 and they interpreted this observed feature via the dd n= g / © 2004 by CRC Press LLC distribution (the Gurvich and Yaglom model). As shown in Figure 30.4, the Beta distribution predicts the An Application of the Lognormal Theory to Moderate Reynolds Number Turbulent Structures 475 behavioral response of the organisms to the ßow Þeld. The majority moved from the surface to a stratiÞed intermediate water column where the dissipation rate was reduced to 10 –8 W kg –1 or less. Haury et al. (1990) also observed that a shift in the community structure of zooplankton took place when the average dissipation rate of the water column exceeded 10 –6 W kg –1 . The Kolmogorov scale associated with 10 –6 W kg –1 is 10 –3 m, roughly the size of a copepod. Is this the reason the community structure of zooplankton is responding to the turbulence level at 10 –6 W kg –1 ? According to the universal spectrum for oceanic turbulence, the peak in the shear spectrum takes place at no higher than 30 cycles m –1 at this dissipation rate (Gregg, 1987; Oakey, 2001). At h scale, the kinetic energy is virtually exhausted. The dissipation rates reported in the literature are normally based on at least 1-m scale averaging, but the highly intermittent nature of instantaneous dissipation rates is masked (Yamazaki et al., 2002). Clearly, the average dissipation rate does not describe the ambient ßow Þeld for a single organism. To provide an estimate of the representative ambient ßow Þeld around a single plankter, we make use of the lognormal theory. We assume that the plankter is a sphere whose radius is 1 mm. Based on the observed evidence (Haury et al., 1990; Incze et al., 2001), we consider the following scenario: the assumed organism moves from a surface turbulent layer whose dissipation rate is 10 –6 W kg –1 and whose thickness, L 1 , is 10 m to a subsurface stratiÞed layer whose dissipation rate is 10 –8 W kg –1 and whose thickness, L 2 , is 1 m. Then we consider two levels of averaging scales for the lognormal theory: r 1 = 10h and r 2 = 3h. For low-order moments, such as mean and variance, any lognormal models give nearly identical predictions; thus, we make use of the Gurvich and Yaglom model with the intermittency coefÞcient m = 0.25 (Yamazaki et al., 2002). The model provides the following relationship for the local average dissipation rate e r and the domain average dissipation rate <e>: m r = log<e> – 0.125 log(Lr –1 ) (30.5) s r 2 = 0.25 log(Lr –1 ) (30.6) where m r is the mean and s r 2 the variance of log e r . 30.4.1 Surface Turbulent Layer In this layer, we use the following values: <e> = 10 –6 W kg –1 L 1 = 10 m FIGURE 30.4 (A) The qq-plot of the breakage coefÞcient for Case C64 and l = 5; the Beta distribution is assumed. (B) The qq-plot of the breakage coefÞcient for Case C64 and l = 5; the lognormal distribution is assumed. 01 2345678 0 1 2 3 4 5 6 7 8 Theoretical values Computed values 01 2345678 0 1 2 3 4 5 6 7 8 Theoretical values Computed values AB © 2004 by CRC Press LLC 476 Handbook of Scaling Methods in Aquatic Ecology: Measurement, Analysis, Simulation Thus, h = 1.0 ¥ 10 –3 m r 1 = 10h = 1.0 ¥ 10 –2 m r 2 = 3h = 3.0 ¥ 10 –3 m The turbulence rms velocity q may be expressed in terms of L and <e> (Tennekes and Lumley, 1972): q = (<e>L) 1/3 (30.7) These values lead to q = 2.15 ¥ 10 –2 m s –1 . The Reynolds number based on L is Re = (qL)/n (30.8) and is related to the Taylor scale Reynolds number as followed (Levich, 1987). Re l ª (8Re) 1/2 (30.9) For the speciÞc example considered here, Re = 2.15 ¥ 10 5 and Re l = 1311. Since log e r distribute as normal, the following z value distributes as a standard normal distribution: (30.10) For a given L/r, the probability that local e r exceeds the global mean, <e>, can be assessed by taking log<e> = log e r in Equation 30.10 (Figure 30.5). For r 1 and r 2 , the probability is 0.256 and 0.238, respectively. Hence, nearly 75% of spatial volume is occupied by the local average dissipation rate less than <e>. Large values are taking place in less than 25% of the total volume. To estimate an extreme value of the local average dissipation rate for each averaging scale r 1 and r 2 , we suppose that the extreme values take place at a probability that is equivalent to the volume occupancy of the assumed organisms. As a typical number of copepod observed in Þeld, we assume ten individuals per liter. The volume occupied by organisms is 4.19 ¥ 10 –2 m 3 and the corresponding probability, p r , is 4.19 ¥ 10 –5 . This probability is equivalent to an extreme event that takes place for less than 0.15 s in FIGURE 30.5 Probability exceeds the global mean against log 10 (L/r). z m rr r = -loge s 10 0 10 1 10 2 10 3 10 4 10 -1 10 0 log 10 ( L / r ) Probability exceeds the global mean © 2004 by CRC Press LLC An Application of the Lognormal Theory to Moderate Reynolds Number Turbulent Structures 477 1 h. The lognormal theory provides e r = 7.5 ¥ 10 –5 W kg –1 and 1.6 ¥ 10 –4 W kg –1 for r 1 and r 2 , respectively. When we equate this dissipation rate with the isotropic formula (e = 7.5s m 2 ), the mean cross stream turbulence shear, s m , is 3.3 and 4.6 s –1 for each case. These are substantial values, although the volume occupation of such high values is low. Where do these high strain rates take place? Unfortunately, the lognormal theory does not predict the actual ßow structures. Thus, we relate the lognormal theory to the coherent structure studies with DNS. Numerical simulations show the strain Þeld of turbulence is dominated by a Þlament-like structure (Vincent and Meneguzzi, 1991; Jiménez et al., 1993). Jiménez (1998) shows that the mean radius of Þlament R is roughly 5h and a maximum azimuthal velocity u q is roughly q. A maximum vorticity w max is 3(q/R). The volume fraction of Þlament p f is related to the Taylor scale Reynolds number: p f = 4 Re l –2 (30.11) For our case, p f is 2.33 ¥ 10 –6 so that the actual volume occupied by the Þlament in 10 3 m 3 is 2.33 ¥ 10 –3 m 3 . Thus, if we assume the cross section of the Þlament is a circle whose radius is 5h and that the remaining length scale of a “typical” Þlament is the same as the Taylor microscale, then there are roughly 250 Þlaments for this particular volume. According to the development above, the maximum dissipation rate associated with the Þlament is 6.13 ¥ 10 –4 W kg –1 . The lognormal theory predicts that the local dissipation rate based on p f is 1.7 ¥ 10 –4 and 2.5 ¥ 10 –4 W kg –1 for r 1 and r 2 . The maximum dissipation rate for the Þlament should be larger than the local average value; thus two independent assessments for the local shear values are consistent. 30.4.2 Subsurface Stratified Layer We use the following values for this layer: <e> = 10 –8 W kg –1 L 2 = 1 m Thus, h = 3.16 ¥ 10 –3 m r 1 = 10h = 3.16 ¥ 10 –2 m r 2 = 3h = 9.48 ¥ 10 –3 m The probability that local average values exceed the global mean is 0.32 and 0.29 for each averaging scale. Thus, nearly 70% of space is occupied by the local dissipation rate that is below the global mean. Based on the same argument for extreme values, the volume occupancy ratio by the organism, p f = 4.19 ¥ 10 –5 , provides e r = 2.4 ¥ 10 –7 W kg –1 and 3.9 ¥ 10 –7 W kg –1 for r 1 and r 2 , respectively. The mean cross stream turbulence shear, s m , is 0.18 and 0.23 s –1 for each case. The number of Þlaments expected in 1 m 3 in this case is smaller, roughly three, and the maximum dissipation rate occurring within the Þlament is 6.16 ¥ 10 –7 W kg –1 . The lognormal theory predicts that the dissipation rates associated with the Þlament occupancy ratio are 1.5 ¥ 10 –7 and 2.3 ¥ 10 –7 W kg –1 . Zooplankton in the surface mixed layer may be reacting to the intermittent high shear that can be argued quantitatively from the lognormal theory as presented in the chapter. The local quantities should be used to investigate the effects of turbulence on individual microscale organism behaviors. Acknowledgments We are indebted to A. Abib for his patient work running the simulation codes. This work was supported by Grant-in-Aid for ScientiÞc Research C-10640421. © 2004 by CRC Press LLC 478 Handbook of Scaling Methods in Aquatic Ecology: Measurement, Analysis, Simulation References Arneodo, A., Manneville, S., and Muzy, J.F., Toward log-normal statistics in high Reynolds number turbulence, Eur. Phys. J. B, 1, 129, 1998. Benzi, R., Ciliberto, S., Baudet, C., and Chavarria, G.R., On scaling of three-dimensional homogenous and isotropic turbulence, Physica D, 80, 385, 1995. Benzi, R., Struglia, M.V., and Tripiccione, R., Extended self-similarity in numerical simulations of three- dimensional anisotropic turbulence, Phys. Rev. E, 53, R5565, 1996. de Boor, C., A Practical Guide to Spline, Springer-Verlag, New York, 1978. Eswaran, V. and Pope, S.B., An examination of forcing in direct numerical simulations of turbulence, Comput. Fluid, 16, 257, 1988. Grant, H.L., Stewart, R.W., and Moilliet, A., Turbulence spectra from a tidal channel, J. Fluid Mech., 12, 241, 1962. Gregg, M.C., Diapycnal mixing in the thermocline: a review, J. Geophys. Res., 92, 5249, 1987. Gurvich, A.S. and Yaglom, A.M., Breakdown of eddies and probability distributions for small-scale turbulence, boundary layers and turbulence, Phys. Fluids, 10, 59, 1967. Haury, L.R., Yamazaki, H., and Itsweire, E.C., Effects of turbulent shear ßow on zooplankton distribution, Deep-Sea Res., 37, 447, 1990. Incze, L.S., Hebert, D., Wolff, N., Oakey, N., and Dye, D., Changes in copepod distributions associated with increased turbulence from wind stress, Mar. Ecol. Prog. Ser., 213, 229, 2001. Jiménez, J., Small scale intermittency in turbulence, Eur. J. Mech. B Fluids, 17, 405, 1998. Jiménez, J., Intermittency and cascades, J. Fluid Mech., 409, 99, 2000. Jiménez, J., Wray, A.A., Saffman, P.G., and Rogallo, R.S., The structure of intense vorticity in isotropic turbulence, J. Fluid Mech., 255, 65, 1993. Kolmogorov, A.N., The local structure of turbulence in incompressible viscous ßuid for very large Reynolds numbers, Dokl. Akad. Nauk SSSR, 30, 299, 1941. Kolmogorov, A.N., A reÞnement of previous hypotheses concerning the local structure of turbulence in a viscous incompressible ßuid at high Reynolds number, J. Fluid Mech., 13, 82, 1962. Levich, E., Certain problems in the theory of developed hydrodynamical turbulence, Phys. Rep., 151, 129, 1987. Monin, A.S. and Ozmidov, R.V., Turbulence in the Ocean, D. Reidel, Dordrecht, the Netherlands, 1985. Mood, A.M., Graybill, F.A., and Boes, D.C., Introduction to the Theory of Statistics, 3rd ed., McGraw-Hill, New York, 1974. Novikov, E.A., Intermittency and scale similarity in the structure of a turbulent ßow, Prikl. Mat. Mech., 35, 266, 1971. Oakey, N.S., Turbulence sensors, in Encyclopedia of Ocean Sciences, J.H. Steele, S.A. Thorpe, and K.K. Turekian, Eds., Academic Press, San Diego, 2001, 3063. Rogallo, R.S., Numerical experiments in homogeneous turbulence, Tech. Rep. NASA TM 81315, NASA Ames Research Center, 1981. Stewart, R.W., Wilson, J.R., and Burling, R.W., Some statistical properties of small-scale turbulence in an atmospheric boundary layer, J. Fluid Mech., 41, 141, 1970. Tennekes, H. and Lumley, J.L., A First Course in Turbulence, MIT Press, Cambridge, MA, 1972. van Atta, C.W. and Yeh, T.T., Evidence for scale similarity of internal intermittency in turbulent ßows at large Reynolds numbers, J. Fluid Mech., 71, 417, 1975. Vincent, A. and Meneguzzi, M., The spatial structure and statistical properties of homogenous turbulence, J. Fluid Mech., 225, 1, 1991. Wang, L.P., Chen, S., Brasseur, J.G., and Wyngaard, J.C., Examination of hypotheses in the Kolmogorov reÞned turbulence theory through high-resolution simulations. Part 1: Velocity Þeld, J. Fluid Mech., 309, 113, 1996. Yamazaki, H., Breakage models: lognormality and intermittency, J. Fluid Mech., 219, 181, 1990. Yamazaki, H., Lagrangian study of planktonic organisms: perspectives, Bull. Mar. Sci., 53, 265, 1993. Yamazaki, H. and Lueck, R.G., Why oceanic dissipation rates are not lognormal, J. Phys. Oceanogr., 20, 1907, 1990. Yamazaki, H., Mackas, D., and Denman, K., Coupling small scale physical processes to biology: towards a Lagrangian approach, in The Sea, Biological-Physical Interactions in the Ocean, A.R. Robinson, J.J. McCarthy, and B.J. Rothschild, Eds., John Wiley & Sons, New York, 2002, 51–112. Yeung, P.K. and Pope, S.B., Lagrangian statistics from direct numerical simulations of isotropic turbulence, J. Fluid Mech., 207, 531, 1989. © 2004 by CRC Press LLC [...]... the role of the copepods swimming behavior (including the body orientation and swimming direction and speed) and morphology (including the morphology of the main body and the morphology and motion pattern of the cephalic appendages) Both are the determining factors of 479 â 2004 by CRC Press LLC 480 Handbook of Scaling Methods in Aquatic Ecology: Measurement, Analysis, Simulation drag forces Observational... density of s* Turbulent òow computed by DNS (solid line); random òow (dotted line) â 2004 by CRC Press LLC 500 32.3.2 Handbook of Scaling Methods in Aquatic Ecology: Measurement, Analysis, Simulation Statistics A total of 24,000 particles were seeded into the òow ịelds The particles were divided into six groups containing 4000 particles each For each group the orientation of gravity was changed in two... larvae in still water and tethered polychaete larvae created ow elds in which particles â 2004 by CRC Press LLC 488 Handbook of Scaling Methods in Aquatic Ecology: Measurement, Analysis, Simulation followed curved trajectories, whereas particles followed straighter trajectories around free-swimming polychaete larvae and bivalve larvae tethered in owing water The dependence of ow geometry on swimming behaviors... diameter of 1 mm We also assumed that the swimming speeds of the organism are of the same order of magnitude as the terminal sinking velocities Any heavier-than-water organism requires a swimming speed that at least exceeds the sinking speed in order to maintain a preferred depth within the water column The speciịc density of planktonic organisms is, in general, slightly larger than that of the surrounding... capture area of a model copepod (A) hovering (like a helicopter) in the water, (B) sinking freely with the anterior pointing upward, at its terminal velocity (4.1 87 mm ã s1 and along its body axis in the present case), (C) swimming forward (in positive x-direction) at a speed of 1.0 47 mm ã s1, and (D) swimming forward (in positive x-direction) at a speed of 4.1 87 mm ã s1 Note that the frame of reference... direction cosine of the intermediate principal axis of the strain rate tensor As shown by Equation 32.20, the magnitude of the swimming vector is proportional to s* while the orientation is dictated by the intermediate principal axis of the strain rate There are numerous possible choices for the orientation of the swimming vector We use the direction cosine of the intermediate strain principal axis... is probable that in the near future numerical experiments on computers will become a powerful tool for studying zooplankton and their â 2004 by CRC Press LLC 490 Handbook of Scaling Methods in Aquatic Ecology: Measurement, Analysis, Simulation interactions with the environment The key point is that numerical studies should be combined with experimental studies in an interactive way in which one complements... the strain ịeld for DNS is quite different from RFS This reòects the difference in both swimming velocity components The strain-based model spends 36% of the total velocity vector in the direction against gravity For the vertical velocity component, the velocity-based model forces more effort in the vertical component â 2004 by CRC Press LLC 502 Handbook of Scaling Methods in Aquatic Ecology: Measurement,. .. Research) Institute of the University of WisconsinMilwaukee, is compared with the counterpart results obtained from the hydrodynamic model and numerical simulation framework In addition, a future application of the numerical simulation method in the study of the on/off or timedependent feeding current is outlined 31.2 Dynamic Coupling 31.2.1 NavierStokes Equations Governing the Flow Field around a Free-Swimming... CRC Press LLC 494 Handbook of Scaling Methods in Aquatic Ecology: Measurement, Analysis, Simulation underlying structures of turbulent òows a great deal One notable feature of the turbulence is the organized structures exhibited in both the velocity and the strain ịelds (e.g., Vincent and Meneguzzi, 1991) Yamazaki (1993) proposed that those organized structures could provide helpful information for planktonic . work was supported by Grant -in- Aid for ScientiÞc Research C-10640421. © 2004 by CRC Press LLC 478 Handbook of Scaling Methods in Aquatic Ecology: Measurement, Analysis, Simulation References Arneodo,. Handbook of Scaling Methods in Aquatic Ecology: Measurement, Analysis, Simulation they used an idealized morphology, consisting of a spherical body and a single appendage represented by a point. cephalic appendages). Both are the determining factors of © 2004 by CRC Press LLC 480 Handbook of Scaling Methods in Aquatic Ecology: Measurement, Analysis, Simulation drag forces. Observational