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Intermittency and nongaussian statistics of air transmittency fluctuations
Phys. Chem. Earth (B), Vol. 24, No. 8, pp. 953-957.1999 Q 1999 EIaavier science Ltd AlIri&talwerved 1464-1909/99/$-soefrontmatter PII: s1464-1909(99)001os-4
Intermittemy
and Nongaussian
Statistics of Air Tranmittency
M. Serio, L. Bergamasco and M. Onorato Dipartimento
di Fisica Generale dell’Universit8 di Torino, Via P. Giuria, l-10125 Torino, Italy
Received 24 April 1998; accepted 29 July 1998 Abstract. We present some preliminary results on the probability density functions of the increments of air transmittency time series which confirm the presence of small scale intermittency, previously observed with multifractal and wavelet methods. A relation to simultaneous measurements on wind intensity is also established. 8 1999 Elsevier Science Ltd. All rights reserved.
independent, to a spiky structure with exponential tails, corresponding to a correlation between the two measurements. In turbulence, it is costumary to assume for the shape of the PDF’s the mathematical representation PDF(Ax’) = A exp[-j&ld)a]: in the gaussian limit, fi = 0.5 and a = 2, while in the exponential limit’/3 > 0.5 and a < 2 (for 1 Ia I 2, the concavity of the exponential tails is downward, while for 0 < a I 1 it is upward) (Tabeling, 1996; Kailasnath et al., 1992). We give in Table 1 the bestfit values for the three coefficients A, p and a computed for Ax’ 1 2 for the two time series: it may be seen that correlation is present already at large values of z for the good visibility sequence, while it does not become evident for the poor visibility time series until z c 50 At = 4 min. The three panels on the right of the same Figures 1 and 2 show the PDF’s of the surrogate of the two experimental time series, obtained by substituting the Fourier phases of the signal with a set of random numbers distributed uniformly between 0 and 2~ (Theiler, 1991). The fact that the PDFs of the surrogate signals are quasi-gaussian for every values of z signify that the correlation between the data taken at small time intervals gets lost with the randomization of the Fourier phases. The transition from the gaussian shape to a stretched exponential with decreasing time lags is in itself generally considered a good indicator of intermittency: the absence of correlation in the surrogate signals reinforces this conclusion. To understand better the origin and the phenomenology of the observed intermittency it is helpful to analyse simultaneous measurements of variables which play a role in the processes of formation and dissipation of fog, such as temperature, wind, humidity, etc. For a poor visibility sequence of the air transmittency data we have measurements on the atmospheric pressure and on the intensity of wind. While for the former there is no reliable indication for intermittency, for the latter, instead, we find correlation at small time scales. Figure 3 compares the PDF’s of the increments of the wind intensity data (panels a, b) with those of the simultaneous measurements on air transmittency (panels c, d). The best-fit values for the coefficients A, /land a for the wind data are reported in the far section of Table 1. Since the intermittency effects are expected to be more strongly pronounced when higher order moments are considered, the relation between atmospheric pressure, wind
1 Introduction In a previous paper (Serio et al., 1998), we presented evidence for multifractal structure in air transmittency time series measured in tbe PO valley (North Italy) by means of a 25 m baseline transmissometer (Richiardone et al., 1995): 100% corresponds to zero attenuation (perfect visibility); average valueslarger than 75% identify the “good visibility” time series (GV), while values lower than 50% identify the “poor visibility” time series (PV). In this paper we elaborate on the small-scale intennittency associated to the multifractal structure of the data, focusing on its connections with the behavior of the amplitude probability density function (PDF) of the increments of the time series cAx(r,z)> = x(r+r) - x(t), where the time lags is a multiple of the experimental sampling interval At. The study of the probability density function (PDF) is generally considered to be a rich diagnostics of small scale intermittency because of its ability to evidence the correlation between two field values measured at different times (Biskamp et al., 1990; Biferale, 1993; Tabeling et al., 1996; Marsh and Tu, 1997). 2 Results
and
discussion
We analyse the normalized time series of the increments AX’+-E)/
Q in order to allow a direct comparison of
the PDF’s among one another and with the Gaussian distribution. Figures I and 2 show in the panels on the left the results on the PDF’s of the increments of two air trasmittency time series recorded respectively during poor and good visibility conditions. It may be seen that, as the time lag r decreases, the shape of the PDF’s evolves from being approximately gaussian, indicating that the two measurements involved in the increments are statistically Correspondence to: Marina Serio, e-mail: [email protected] 953
Air Transmittency
-4
-2
0
2
4
AX'
Fig. 1. On the left: PDF’s of the increments of an air transmittency time series recorded in poor visibility conditions for different time lagz. On the right: surrogate signals.
r
c>
N
0
k
N
A
Probability
Probability
density function
density function
Probability
Probability
density function
density function
Probability
Probability
density function
density function
M. Serio et al.: Air Transmittency
956
-4
I
I
I
I
-2
0
2
4
Fluctuations
-4
I
I
I
I
-2
0
2
4
(b) i
8 I LT. 28 I 2 3 L
Air Transmittency 6 s 4 3 2 I
IO
100 t=nAt
10’
(At=5d
Fig. 3. Panels a, b: PDF’s of the increments of a wind intensity time series recorded in poor visibility conditions for different time lagz. Panels c, d: the same for the air transmittency time series recorded in coincidence. Panel e: Flatness factor F(Z) for the simultaneous measurements on atmospheric pressure, wind intensity and air transmittency.
M. Serio et al.: Air Transmittency intensity and air transmittency has been investigated also through the behavior of the flatness factor F (4-th moment) of their PDF’s as a function of z (Fig. 3e) We notice that the flatness factor for the pressure and for the wind intensity has its maximum value for z = 1, but while for the pressure F decreases rapidly to the Gaussian saturation value (indicating absence of intermittency), for the wind, F decreases slowly, remaining consistently above 3 for all values of 2. For the air transmittency data, F shows significant deviations from the reference gaussian value, with its maximum around time lags of T= 5At = 25s.
3
957
the subject are under course and will be reported in an extended form in a future paper.
Acknowledgements.
We thank the AtmosphericPhysics Group of our Department for the data on air transmittcncy,atmosphericpressure and wind intensity. We acknowledge the financial support of MURSTItaly and of the University of Torino.
References Biferak. L., Probabilitydistributionfunctionsin turbulent flows and the shell model,Phys. Fluids A5,428-435, 1993. Biskamp,D.. Welter, H., and Walter, M., Statisticalpropertiesof twodimensionalmagnetohydrodynamic turbulence,Phys. Ffu;d.r 82, 30243031,199O. Burlaga, L. F.. Multifractal structureof the interplanetarymagneticfold. Ceophys.Res. L&f. I& 69-72, 1991(a); Intermittentturbuknce.in the solar wind, Journ. Gcophy..r. Res. 96, 5847-5851. I991 (b). Kailasnath,P., Sreenivasan,K.R., and Stolovitzky,G.. Probabilitydensity of velocity incrementsin turbulentflows, Phys. Rev. Left. 68,27662769,1992. Marsch, E. and Tu, C.Y., Intermittency,non-gaussianstatisticsand fractal scalingof MHD fluctuationsin the solar wind, Nonlinear Proc. in Geophys. 4, 101-124, 1997. Richiardone.R., Ale&o, S., Canavero.F., Einaudi, F., and Longhetto,A., Experimentalstudyof atmosphericgravity wnvesand visibility oscillationsin a fog episode,Nuovo Cimrnro IRC, 647-662, 1995. Serio, M.. Bergamasco,L., Onorato, M., Osborne,A.R.. Aleasio, S.. Richiardone,R., and Longhetto,A., Multifrsnality of air transmittency at small time scales,Fractals 6, 159-170, 1998 Theiler, J., Some commentson the correlationdimensionof l/p noise, Phys. Len. A IS& 480-493. 199 I. Tabeling, P., Zocchi, G., Belin, F., Maurer, J., and Willaimc, H., Probabilitydensityfunction,skewnessand flatnessin large Reynolds numberturbulence,Phyr. Rev. E 53. 1612, 1996.
Conclusions
The analysis of the properties of the PDF’s of the increments of air transmittency data recorded in the PO valley (North Italy) fully confirms our initial diagnosis of small-scale intetiittency. The transition of the PDF’s from a gaussian shape (e.g. uncorrelated measurements) to a spiky exponential (e.g. correlated measurements) occurs at smaller time lags for the sequences recorded during poor visibility (Z < 50 ds) than for those recorded during good visibility (z c 500 AC). The parameters which may play a role in the processes of formation and dissipation of fog, and thus influence the air transmittency are essentially temperature and wind; from available measurements on the intensity of wind carried out in coincidence with those on air transmittency we may infer the existence of a relation between the intermittency exhibited by the two sets of data. Further investigations on Table 1.
Fluctuations
Best-tit values for the three coefficients in the mathematicalrepresentationof the PDF (see Section 2).
WhNlllltCs&
Air trausmitteney
Poor visibilitysequence
Poorvisibilitysequence
Good visibility sequence
ml
0.09
I .30
I .03
SO
0.08
0.49
2.00
50
0.25
I .23
1.40
100
0.11
I so
0.88
IO
0.10
1.80
0.75
IO
0.30
1.85
0.95
JO
.I2
2.30
0.58
5
0.10
I .80
0.77
3
0.30
2.03
0.83
Intermittemy
and Nongaussian
Statistics of Air Tranmittency
M. Serio, L. Bergamasco and M. Onorato Dipartimento
di Fisica Generale dell’Universit8 di Torino, Via P. Giuria, l-10125 Torino, Italy
Received 24 April 1998; accepted 29 July 1998 Abstract. We present some preliminary results on the probability density functions of the increments of air transmittency time series which confirm the presence of small scale intermittency, previously observed with multifractal and wavelet methods. A relation to simultaneous measurements on wind intensity is also established. 8 1999 Elsevier Science Ltd. All rights reserved.
independent, to a spiky structure with exponential tails, corresponding to a correlation between the two measurements. In turbulence, it is costumary to assume for the shape of the PDF’s the mathematical representation PDF(Ax’) = A exp[-j&ld)a]: in the gaussian limit, fi = 0.5 and a = 2, while in the exponential limit’/3 > 0.5 and a < 2 (for 1 Ia I 2, the concavity of the exponential tails is downward, while for 0 < a I 1 it is upward) (Tabeling, 1996; Kailasnath et al., 1992). We give in Table 1 the bestfit values for the three coefficients A, p and a computed for Ax’ 1 2 for the two time series: it may be seen that correlation is present already at large values of z for the good visibility sequence, while it does not become evident for the poor visibility time series until z c 50 At = 4 min. The three panels on the right of the same Figures 1 and 2 show the PDF’s of the surrogate of the two experimental time series, obtained by substituting the Fourier phases of the signal with a set of random numbers distributed uniformly between 0 and 2~ (Theiler, 1991). The fact that the PDFs of the surrogate signals are quasi-gaussian for every values of z signify that the correlation between the data taken at small time intervals gets lost with the randomization of the Fourier phases. The transition from the gaussian shape to a stretched exponential with decreasing time lags is in itself generally considered a good indicator of intermittency: the absence of correlation in the surrogate signals reinforces this conclusion. To understand better the origin and the phenomenology of the observed intermittency it is helpful to analyse simultaneous measurements of variables which play a role in the processes of formation and dissipation of fog, such as temperature, wind, humidity, etc. For a poor visibility sequence of the air transmittency data we have measurements on the atmospheric pressure and on the intensity of wind. While for the former there is no reliable indication for intermittency, for the latter, instead, we find correlation at small time scales. Figure 3 compares the PDF’s of the increments of the wind intensity data (panels a, b) with those of the simultaneous measurements on air transmittency (panels c, d). The best-fit values for the coefficients A, /land a for the wind data are reported in the far section of Table 1. Since the intermittency effects are expected to be more strongly pronounced when higher order moments are considered, the relation between atmospheric pressure, wind
1 Introduction In a previous paper (Serio et al., 1998), we presented evidence for multifractal structure in air transmittency time series measured in tbe PO valley (North Italy) by means of a 25 m baseline transmissometer (Richiardone et al., 1995): 100% corresponds to zero attenuation (perfect visibility); average valueslarger than 75% identify the “good visibility” time series (GV), while values lower than 50% identify the “poor visibility” time series (PV). In this paper we elaborate on the small-scale intennittency associated to the multifractal structure of the data, focusing on its connections with the behavior of the amplitude probability density function (PDF) of the increments of the time series cAx(r,z)> = x(r+r) - x(t), where the time lags is a multiple of the experimental sampling interval At. The study of the probability density function (PDF) is generally considered to be a rich diagnostics of small scale intermittency because of its ability to evidence the correlation between two field values measured at different times (Biskamp et al., 1990; Biferale, 1993; Tabeling et al., 1996; Marsh and Tu, 1997). 2 Results
and
discussion
We analyse the normalized time series of the increments AX’+-E)/
Q in order to allow a direct comparison of
the PDF’s among one another and with the Gaussian distribution. Figures I and 2 show in the panels on the left the results on the PDF’s of the increments of two air trasmittency time series recorded respectively during poor and good visibility conditions. It may be seen that, as the time lag r decreases, the shape of the PDF’s evolves from being approximately gaussian, indicating that the two measurements involved in the increments are statistically Correspondence to: Marina Serio, e-mail: [email protected] 953
Air Transmittency
-4
-2
0
2
4
AX'
Fig. 1. On the left: PDF’s of the increments of an air transmittency time series recorded in poor visibility conditions for different time lagz. On the right: surrogate signals.
r
c>
N
0
k
N
A
Probability
Probability
density function
density function
Probability
Probability
density function
density function
Probability
Probability
density function
density function
M. Serio et al.: Air Transmittency
956
-4
I
I
I
I
-2
0
2
4
Fluctuations
-4
I
I
I
I
-2
0
2
4
(b) i
8 I LT. 28 I 2 3 L
Air Transmittency 6 s 4 3 2 I
IO
100 t=nAt
10’
(At=5d
Fig. 3. Panels a, b: PDF’s of the increments of a wind intensity time series recorded in poor visibility conditions for different time lagz. Panels c, d: the same for the air transmittency time series recorded in coincidence. Panel e: Flatness factor F(Z) for the simultaneous measurements on atmospheric pressure, wind intensity and air transmittency.
M. Serio et al.: Air Transmittency intensity and air transmittency has been investigated also through the behavior of the flatness factor F (4-th moment) of their PDF’s as a function of z (Fig. 3e) We notice that the flatness factor for the pressure and for the wind intensity has its maximum value for z = 1, but while for the pressure F decreases rapidly to the Gaussian saturation value (indicating absence of intermittency), for the wind, F decreases slowly, remaining consistently above 3 for all values of 2. For the air transmittency data, F shows significant deviations from the reference gaussian value, with its maximum around time lags of T= 5At = 25s.
3
957
the subject are under course and will be reported in an extended form in a future paper.
Acknowledgements.
We thank the AtmosphericPhysics Group of our Department for the data on air transmittcncy,atmosphericpressure and wind intensity. We acknowledge the financial support of MURSTItaly and of the University of Torino.
References Biferak. L., Probabilitydistributionfunctionsin turbulent flows and the shell model,Phys. Fluids A5,428-435, 1993. Biskamp,D.. Welter, H., and Walter, M., Statisticalpropertiesof twodimensionalmagnetohydrodynamic turbulence,Phys. Ffu;d.r 82, 30243031,199O. Burlaga, L. F.. Multifractal structureof the interplanetarymagneticfold. Ceophys.Res. L&f. I& 69-72, 1991(a); Intermittentturbuknce.in the solar wind, Journ. Gcophy..r. Res. 96, 5847-5851. I991 (b). Kailasnath,P., Sreenivasan,K.R., and Stolovitzky,G.. Probabilitydensity of velocity incrementsin turbulentflows, Phys. Rev. Left. 68,27662769,1992. Marsch, E. and Tu, C.Y., Intermittency,non-gaussianstatisticsand fractal scalingof MHD fluctuationsin the solar wind, Nonlinear Proc. in Geophys. 4, 101-124, 1997. Richiardone.R., Ale&o, S., Canavero.F., Einaudi, F., and Longhetto,A., Experimentalstudyof atmosphericgravity wnvesand visibility oscillationsin a fog episode,Nuovo Cimrnro IRC, 647-662, 1995. Serio, M.. Bergamasco,L., Onorato, M., Osborne,A.R.. Aleasio, S.. Richiardone,R., and Longhetto,A., Multifrsnality of air transmittency at small time scales,Fractals 6, 159-170, 1998 Theiler, J., Some commentson the correlationdimensionof l/p noise, Phys. Len. A IS& 480-493. 199 I. Tabeling, P., Zocchi, G., Belin, F., Maurer, J., and Willaimc, H., Probabilitydensityfunction,skewnessand flatnessin large Reynolds numberturbulence,Phyr. Rev. E 53. 1612, 1996.
Conclusions
The analysis of the properties of the PDF’s of the increments of air transmittency data recorded in the PO valley (North Italy) fully confirms our initial diagnosis of small-scale intetiittency. The transition of the PDF’s from a gaussian shape (e.g. uncorrelated measurements) to a spiky exponential (e.g. correlated measurements) occurs at smaller time lags for the sequences recorded during poor visibility (Z < 50 ds) than for those recorded during good visibility (z c 500 AC). The parameters which may play a role in the processes of formation and dissipation of fog, and thus influence the air transmittency are essentially temperature and wind; from available measurements on the intensity of wind carried out in coincidence with those on air transmittency we may infer the existence of a relation between the intermittency exhibited by the two sets of data. Further investigations on Table 1.
Fluctuations
Best-tit values for the three coefficients in the mathematicalrepresentationof the PDF (see Section 2).
WhNlllltCs&
Air trausmitteney
Poor visibilitysequence
Poorvisibilitysequence
Good visibility sequence
ml
0.09
I .30
I .03
SO
0.08
0.49
2.00
50
0.25
I .23
1.40
100
0.11
I so
0.88
IO
0.10
1.80
0.75
IO
0.30
1.85
0.95
JO
.I2
2.30
0.58
5
0.10
I .80
0.77
3
0.30
2.03
0.83