Scaling laws in geophysical flows

Phys.

Pergamon

Chem.Earth (B), Vol. 26, No. 4, pp. 281-285,200l 0 2001 Elsevier Science Ltd. All rights reserved 1464-1909/01/$ - see front matter

PII: s1464-1909(01)00007-7

Scaling Laws in Geophysical Flows 0. B. Mahjoub’, T. Granata’ and J. M. Redondo’ ‘Dept. Fisica Aplicada, Univ. Polit&nica de Catalunya, Barcelona, Spain ‘Centre de Estudis de Blanes, C. Santa Barbara s/n 17300, Blanes, Spain Received 23 April 1999; accepted 25 April 2000

Abstract. Statistical analysis of velocity structure functions are presented for turbulent flows at lm above the bottom in a shallow (2m) bay in Denmark. High frequency (25 Hz) time series were collected mid day of 9 and 12, September 1997. The turbulent flow on day 12 was more energetic than on day 9 as a result of strong wind waves. The absolute scaling exponent &, was shown to have scale-dependent behavior. In contrast, the relative scaling exponents cP,,calculated using the extended self similarity (ESS) method and calculating the third order structure function with the modulus, was found to have a scale-independent behavior and deviated from the Kolmogorov K41 law. Moreover, values of cp on day 12 were more intermittent than on day 9. This result indicates the influence of the wind and waves on the scaling laws of the velocity structure functions. 0 2001 Elsevier Science Ltd. All rights reserved

1

a detailed a shallow,

9 and a height 1 m mean E and

a small a narrow

Introduction

a “down-looking”

a metal

The 3 transducer h ave

2

Energy

and PDFs

The behavior of the energy spectrum E(f) was analyzed in Fourier space. The energy spectrum was computed using a Fast Fourier Transform. The spectral process has been made with 128 blocks of 256 points and averaging the obtained spectral densities show the prototype spectrum E(f). Figure 1 shows the two energy spectra

0. B. Mahjoub et al.: Scaling Laws in Geophysical

282

10.’

Flows

1’

-4.0

I -2.0

0.0

2.0

4.0

Au

IO”

,

A

Fig.

1. The energy

spectrum

E(f)

for days

9 and

12.

for days 9 and 12 of September 1997. Frequencies of the energy peak associated to wave periods were 0.3 Hz for day 9 and 0.5 Hz for day 12. On the other hand, the amplitude energy of day 12 was 1.5 times higher than on day 9 as a result of higher velocity fluctuations associated with wind wave activity. It should be noted that there is a very small inertial range and no important deviation from the -5/3 Kolmogorov power slope for either day. Quantitative information can be obtained by analyzing the shapes and characteristic parameters of probability distribution functions (PDF) for the random-field. For the probability distribution function of any randomfield u, it is necessary to first calculate the average, U, and standard deviation, CT(U), of the random-field. Dividing (u-U) by cr(u) , a new random-field of Au = $$ is obtained which can easily analyzed and compare statistically. Figures 2a and 2b show the probability distribution functions of the normalized density fluctuations for days 9 and 12. For comparison, the dotted lines show the Gaussian distribution: P(Au)

= &e&Au

- (a~))~)

(1)

where (...) represents an ensemble average. The two curves deviate from Gaussian distribution and this deviation is more pronounced on day 12 than on day 9. This anomalous behavior, especially on day 12, is related to the strong wind waves activity, which are more energetic than on day 9 (figure 1). It also confirms the non-homogeneous character of turbulence generated by the irregular wind waves.

3

Structure

ap”ss(en

functions

The velocity structure function is a basic tool to study the intermittent character of turbulence. The pth order

10”

L

-4.0

I -2.0

0.0

2.0

4.0

AU

Fig. 2. Probability on days 9 and 12.

velocity S,(r)

structure

= ((Su(r)P)

distribution

function

functions

is defined

of velocity

fluctuations

as

= ((u(t + r) - u(t))P)

(2)

where the brackets indicate averages over time. Velocity structure functions require the measurement of velocity at two different times. Using the Taylor’s hypothesis (Taylor (1938)), the correspondence between spatial increments I and temporal increments r may be given as 1= UT

(3)

where U is the local mean velocity of the flow at the measured location. In fact, the use of this relation is limited to the low turbulence intensity. More information about the structure functions is given in Frisch (1995) and others (Vincent and Meneguzzi (1991), Maneveau and Sreenivasan (1991)). For data analysis, p may range from 1 to 10 but this higher order requires long time series in order to examine the statistical behavior of the structure function accurately (Gagne et al. (1994)). Following the Kolmogorov’s theory (Kolmogorov (1941)), the self-similarity of the velocity structure function is attained in the inertial range, which is physically defined as a range of scales where both the forcing and the dissipation process are irrelevant. A general expression of the pth order structure function in time is written as: (6?_&(7)P)N TEP

(4)

0. B. Mahjoub et al.: Scaling Lawsin Geophysical Flows

-g s

B

10' lo* 10' 102 10'

Fig.

IO”

4. comparison

283

between

1(&A(T)~) 1and (I au(~) 13).

lute scaling exponents as

10-l 1

o-2

1

o-=

1

o-4

1o-2

(’ = dlo9(l du(r) P dlogr IO"

16

I”) .

(5)

10'

z

4 Fig. 3. absolute velocity structure functions of ) (au(~ 1 (a) and velocity structure functions of absolute (I &J(T) IP) (b) on day 9.

where &, is the scaling exponent. For the Kolmogorov theory K41 (Kolmogorov 1941), &, = p/3. Yet, the nonlinearity of the scaling exponent with the order p of the statistical moment has been observed in many theoretical, experimental and numerical investigations (Antonia et al. (1982), She and Leveque (1994), She and Waymire (1995) and Sreenivasan and Antonia (1997)). In fact this violation, is referred to as the intermittency, indicating that the average value of the energy dissipation E will be different at different points in space (Frisch (1995)). Figures 3a and 3b show the velocity structure function, for p from 1 to 6, without using the absolute value 1 (6u(~)P) 1 and using the absolute value (I Su(r) 1”) for day 9. Even moments of (6u(~)P) show good scaling to r. In contrast, odd moments of I (Su(.r)P) I seem to be less correlated with 7. By taking the absolute value of the odd moments, we can show that (I &(r) 1”) has a monotonous increasing behavior on p. These results are consistent with non-homogeneous and non-isotropic flow measurements in the numerical investigation (Babiano et al. (1995), Babiano et al. (1997)) and in laboratory measurements (Mahjoub et al. (1998)). Clearly the choice of the absolute values has some serious consequences in non-homogeneous and non-isotropic flows. Therefore, we define the velocity structure functions with an absolute value (I 6u(r) 1”) and their abso-

The absolute scaling exponent cp

The determination of the inertial range boundaries using 513 power law may not be accurate when intermittency corrections apply to the scaling exponents (Vincent and Meneguzzi (1991), Chen et al. (1993) and Gagne et al. (1994)) A better way may be to use the third order structure function, where the third order scaling exponent &, is assumed to be 1, so that it is not affected strongly by the intermittency, note that both in K41 and in K62 I& = 1, as done by (Anselmet et al. (1984)). Using this methodology in geophysical flows, which most of time are non-homogeneous and non-isotropic in nature, would imply that an inertial range does not exist, so the scaling exponent can not be evaluated (Babiano et al. (1995)). To further investigate this scaling dependence, we performed a comparison between the absolute third order structure function I (&(T)~) ( and the third order absolute structure function (I &(r) 13) for day 12. It can be seen that the scaling properties of the two quantities increasingly differ (figure 4). This contradicts the results of Benzi et al. (1993) who assume that the scaling properties of the two quantities nearly coincide. Our results are also in contrast to those of Boratav and Pelz (1997), who found that (Sr~(r)~) is proportional to (I &(r) 13), and to the results of Pearson and Antonia (1997) who noted that (SZL(~)~)N (I &(r) 13)” with the constant of proportionality less than 1 and (Y of order 0.9. The behavior of the absolute scaling exponent 6 is dependent on T for day 9 (figure 5). The same is true for day 12 (not shown). This agrees with a two dimensional numerical simulation of non-homogeneous turbulence by Babiano et al. (1995) and Babiano et al.

284

0. B. Mahjoub et al.: Scaling Laws in Geophysical Flows

6.0

Fig. 5. absolute scaling exponents

on day 9. 10’

(1997) and with experimental results of three dimensional non-homogeneous turbulence by Mahjoub et al. (1998). These results also suggest that the procedure to obtain an inertial range and the scaling laws of the velocity structure functions in non-homogeneous flows can not be directly applied according to the phenomenological theory suggested by (Kolmogorov (1941), Kolmogorov (1962))) even at very high Reynolds numbers.

10’ IO0

lo2

10’



I

10s

Fig. 6. (I &(T) IP) as a function of (I &J(T) 13) on days 9 (a) and 12 (b).

So in this work the relative scaling exponent is calculated as 5

The relative scaling exponents tP

To overcome the difficulties in determining the scaling exponent, we have used the Extended Self Similarity (ESS) property, suggested by Benzi et al. (1993). This method was effective in determining accurate scaling exponents. Moreover, the existence of ESS could be used as a way to define an inertial range, even in situations where the phenomenological theory suggested by (Kolmogorov (1941), Kolmogorov (1962)) does not hold. This would apply to situations where there is a strong deviation from local homogeneity and isotropy, such as in geophysical flows Babiano et al. (1995), Babiano et al. (1997) and Mahjoub et al. (1998). ESS also enables more accuracy in the determination of the scaling laws even in magneto-hydrodynamics (MHD) turbulence (Carbone et al. (1994), Carbone et al. (1996), Marsch and Tu (1993), politano and Pouquet (1998)). The ESS idea considers the scaling of velocity structure functions under the form

(I du(7) I”)

+a (I 6U(T)

18)CJCs

cp =$=~&dlW~) I”)

dZog(l &A(T) 13)

(8)

With the aim of applying ESS to complex geophysical flows, we have plotted the pth order velocity structure functions (I &U(T) 1’) against the third order structure function (I &u(r) 13). ESS exists for all values of p, for both days 9 and 12 (figure 6). The departure from the linear law Q = p/3 predicted by (Kolmogorov (1941)) is observed in the two days. Comparisons of the measured relative scaling exponents for days 9 and 12, obtained by a linear best fitting of the log-log distribution, with the Kolmogorov theory (K41) are presented in Table 1. Values of the relative scaling exponents on the two days deviate from the Kolmogorov K41 law. On the other hand the values of I, on day 12 deviate further from the K41 values p/3 indicating more intermittency than on day 9, this confirms that the scaling laws of the velocity structure functions are affected by the strong non-homogeneity of the turbulent Aow induced by high waves motions on day 12.

(6)

where Cp/CBis the general relative scaling exponent and can be expressed as

For comparison with other results, we have taken s = 3.

6

Conclusions

The scaling properties of velocity structures functions have been investigated in turbulent, geophysical (nonhomogeneous) conditions. For the two times series studied, there was no clear inertial range when the absolute scaling exponent, cp, had a scale-independent behavior.

0. B. Mahjoub et al.: Scaling Laws in Geophysical Flows

Table 1. Comparison of the measured relative scaling exponents of Knebel Vig experiment on days 9 and 12 of September with respect to the Kolmogorov theory (K41).

Consequently we used the ESS concept to measure the relative scaling exponents. We show that for ESS it is necessary to take the absolute value of the velocity structure function. Using this method we found a generalized type of inertial range where the relative scaling exponent was scale-independent, over a significant range. ADV measurements of waves and turbulence captured the peaks in wind and current motions in the bay. On both days 9 and 12, an energetic wave field was driven by local winds, where orbital wave velocities dominated over the mean current (Granata et al. (1999)). The intermittency correction, defined as the departure from the Kolmogorov K41 law, is more pronounced for wind waves on day 12. In fact, the non-universality of the scaling exponent probably depends on the nonhomogeneity conditions of the turbulent flow, as a result of the wave motions. Acknowledgements. This work was financed by the European Science Foundation (TAO grant) and by the European Union projects (UE95-0016), MAS3- CT950016 and MAS3-CT960049. We also thank the French Embassy in Spain (CTT-10269). We would like to thank J. Wojciechowski and A. Babiano for helpful discussions.

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