A formal analogy exists between 2D turbulence and 3D turbulence with stratification and rotation. Although the effect of the rotation, to the scale typical of the turbulence, is negligible in the atmosphere, we have found a relationship between the behavior of the intermittency and that of the atmospheric stratification. In order to do that, the intermittency has been characterized through the flatness of the PDFs of velocity increments, for the smallest possible scale, present in our measurements

Arum, Carl-Jørgen

Babigumira, Ronnie

Blindheim, Augun Jodis

Fosså, Sophie Dorothea

Haug, Erik Skaaheim

Kulle, Bettina

Myklebust, Tor Åge

English

Vindel, J. M.

Yagüe, Carlos

Redondo, J. M.

Scientific Open-access Literature Archive and Repository

DOI 10.1393/ncc/i2008-10327-0IL NUOVO CIMENTO Vol. 31 C, N. 5-6 Settembre-Dicembre 2008Relationship between intermittency and stratificationJ. M. Vindel(1)(∗), C. Yagu¨e(2)(∗∗) and J. M. Redondo(3)(1) Area de Modelizacio´n, Agencia Estatal de Meteorolog´ıa (AEMET) - Madrid, Spain(2) Departamento de Geof´ısica y Meteorolog´ıa, Universidad Complutense de MadridMadrid, Spain(3) Departamento de F´ısica Aplicada, Universidad Polite´cnica de Calatun˜a - Barcelona, Spain(ricevuto il 6 Ottobre 2008; approvato il 10 Dicembre 2008; pubblicato online il 9 Aprile 2009)Summary. — A formal analogy exists between 2D turbulence and 3D turbulencewith stratification and rotation. Although the effect of the rotation, to the scale typ-ical of the turbulence, is negligible in the atmosphere, we have found a relationshipbetween the behavior of the intermittency and that of the atmospheric stratification.In order to do that, the intermittency has been characterized through the flatnessof the PDFs of velocity increments, for the smallest possible scale, present in ourmeasurements.PACS 47.27.-i – Turbulent flows.1. – IntroductionThe deviation that turbulence presents [1, 2] with regard to the theory of Kol-mogorov [3] is known as intermittency. This intermittency is associated with the presenceof rare events; this is translated in the scale-dependent stretching of the tails presentedby the PDFs of the velocity increments, compared to the Gaussian form and its approachby stretched exponentials [4-8].2D turbulence is a special type of turbulence characterized by the conservation ofthe vorticity and the absence of intermittency [9, 10]. In conditions of homogeneity andisotropy, 2D turbulence is explained by the theory of Kraichnan [11, 12] according towhich a direct cascade of enstrophy exists with a slope of −3, and other inverse cascadeof energy, with a slope of −5/3.An index that allows characterizing the form of the PDFs is log (F/3) [13], where F isthe corresponding flatness. The number 3 alludes to the value of the Gaussian flatness,so that the larger this index, the bigger the deviation of the PDFs from the Gaussiandistribution is found and, therefore, major will be the grade of existing intermittency (anexample of the evolution of the flatness with the scale can be seen in [14]).(∗) E-mail: jmvindel@inm.es(∗∗) E-mail: carlos@fis.ucm.esc© Societa` Italiana di Fisica 669670 J. M. VINDEL, C. YAGU¨E and J. M. REDONDOIn environmental flows, 2D turbulence does not properly exist. Nevertheless, 3Dturbulence in conditions of stratification and rotation shows a formal analogy with 2D(the geostrophic approach of Charney [15]; see also [16,17]). With regard to the rotationin the atmosphere, to the turbulence typical scale, the effect of the rotation can beconsidered to be negligible, since it presents a very high Rossby number. Concerningthe stratification, in 3D turbulence, the absence of rotation hinders the analogy with2D turbulence. However, the stability produces, precisely, the inhibition of the verticalmotions. And since 2D turbulence is characterized by the absence of intermittency, itis expected that the stable situations present a minor grade of intermittency, or evena type of intermittency in another sense, than the corresponding ones in situations ofneutrality or of convective instability. Consequently, we could find a relationship betweenstratification and intermittency.2. – DataSABLES98 (Stable Atmospheric Boundary Layer Experiment in Spain 1998) data [18]from a sonic anemometer (20Hz sampling rate) at 32m have been used. This fieldcampaign took place in September 1998 (from 10 to 28) at the Research Centre forthe Lower Atmosphere (CIBA) which is situated on the Northern Spanish Plateau [19].The surrounding terrain is fairly flat and homogeneous. In the experiment, differentdegrees of stable stratification were achieved during the night, from near-neutral to verystable conditions, while during the day, the stability ranged from neutral to convectivesituations.The period of study will be between September 10, 2008 at 17:00 GMT and the 17thSeptember at 23:30 GMT, with information taken at intervals of 30 minutes (in total,350 samples to study). For each of these samples we will use intervals of 5 minutes(6000 data), which in the atmosphere constitute a suitable scale for the study of theturbulence. Anyway, due to the sampling rate frequency (20Hz), we will be studyingmainly the spectral interval between the integral length scale and the Taylor microscale,which can be denominated as macroturbulence [20].3. – Relationship between intermittency and stratificationAmong the numerous parameters existing to characterize the degree of stratifica-tion in the atmosphere we will use the difference of temperatures between two levels(50 and 0.22m), which has showed to reflect enough the daily evolution of the stability.Also, the module of the vertical speed has been used, since for the greatest stabilities,such vertical velocity will be inhibited, although sometimes it can be perturbed by thepresence of internal-gravity waves. Figures 1a and 1b show the temporal evolution ofthese variables for our period of study. In fig. 1b, in order to eliminate the irregularcomponent of the series and to make the cyclical component more visible, we also haveperformed a moving average with a 6 hour window (12 points).Figure 1a reflects the biggest stability that the night usually presents opposite to theday, since the night cooling makes the temperature of the air increase with the height.Also it is possible to observe that the last nights of the study (the 14th, 15th, 16th and17th) had lower winds, presenting a major stability than the previous ones. An extendedwork on this stable period of SABLES98 can be found in [21].With regard to intermittency, there exist several models that try to characterize it [22].Some of these models look for an expression of the scaling exponents that appear in theRELATIONSHIP BETWEEN INTERMITTENCY AND STRATIFICATION 671a)b)c)d)Fig. 1. – Temporal evolution along the studied period of a) temperature difference between50m and 0.22m; b) vertical speed module, and moving average using a mean period of 6 h;c) log (F/3) and moving average with a 6 hour period for vertical velocity; d) log (F/3) andmoving average with a 6 hour period for horizontal velocity.672 J. M. VINDEL, C. YAGU¨E and J. M. REDONDOFig. 2. – log (F/3) vs. ΔT50-0.22 for a) horizontal velocity; b) vertical velocity.potential relation existing between the structure functions and the scales, for example,Kolmogorov [23] or Meneveau and Sreenivasan [24]. A previous study on the horizontalvelocity, including a convergence study for the data, showed that it was adequate up tostructure functions of order 2-3 [25]. Other models concentrate on the evolution withthe scale that the PDFs of the velocity increments present. Hnat et al. [26] reportedtwo important approaches in this sense: the model of diffusion, based on the equationof Fokker-Planck, and the model of Castaing. On the other hand, as we have alreadypointed out, the degree of intermittency can be represented from the log (F/3) of thePDFs, where F is the flatness of the PDF (figs. 1c and 1d). Concretely, we have usedthe PDF corresponding to the velocity increments to the smallest possible scale (1 s/20)since, when scale is diminished, the differences from Gaussian form are getting moreevident (at least in terms of tendency).As in the previous figures, in figs. 1c and 1d we observe a daily evolution that confirmsthe relationship between intermittency and stratification. It is noteworthy that thisrelation appears using both vertical (1c) and horizontal velocity increments (1d).In order to quantify the grade of existing correlation between stratification and flat-ness, in fig. 2 we have represented log (F/3) vs. the difference of temperatures betweenthe levels of 50 and 0.22m (both for the horizontal speed and for the vertical one) forall the data. Although, as was expected when working with atmospheric data far fromcontrolled laboratory experiments, the coefficient of determination that is obtained is nottoo high, even when a process of smoothed across a moving average is done, but never-theless the obtained decreasing of flatness (for the minor possible scale) as the grade ofstability is increased is clear.In order to visualize the difference between the PDFs corresponding to situations withdifferent grade of intermittency, we have represented in fig. 3 the PDFs corresponding tothe scale 1 (the minor scale that we can represent, 1 s/20) for some moments of the periodof study (in the same graph, a nocturnal situation and a diurnal of the next morning,are included) in which peaks (minima and maxima, respectively) are observed (in fig. 1).It is observed how the PDFs corresponding to the diurnal situations present morestretched tails (major presence of rare events) than those corresponding to the situationsat night, with minor intermittency, or even, an intermittency in an inverse sense.It is interesting to show not only the PDFs corresponding to the minor scale for whichwe have resolution, but also the change that is taking place in the tails of the PDFs asthe scale is diminishing. In figs. 4 and 5 this evolution has been represented.RELATIONSHIP BETWEEN INTERMITTENCY AND STRATIFICATION 673Fig. 3. – PDFs of the horizontal velocity (a, c, e) and the vertical velocity (b, d, f) for scale1 s/20 for different study situations.In these figures (4 and 5) it is observed that the most intermittent situations, relatedto diurnal convection (those of the left side) are more peaked, and that the PDF of theminor scale is wider than those of major scales. This change of the PDF with the scaleis less pronounced in the situations of the right-hand side (nocturnal), compared to thatthey present in the situations of the left side (diurnal).674 J. M. VINDEL, C. YAGU¨E and J. M. REDONDOFig. 4. – PDFs of the horizontal velocity increments for different scales (61, 41, 21, 1) anddifferent studied situations.To make more clear the evolution that the PDFs of figs. 4 and 5 experience, in fig. 6we have represented the evolution of the flatness (log (F/3)) with the scale. This figureputs in evidence clearly: a) the very top value of the flatness in the situations in thedaytime opposite to the night situations and b) the increase, in general, of the flatnesswhen the scale decreases for the diurnal situations (especially strong for the smallestscales) and the rough constancy, or even the decrease of the flatness for nocturnal stableRELATIONSHIP BETWEEN INTERMITTENCY AND STRATIFICATION 675Fig. 5. – PDFs of the vertical velocity increments for different scales (61, 41, 21, 1) and differentstudied situations.676 J. M. VINDEL, C. YAGU¨E and J. M. REDONDOFig. 6. – Evolution of log (F/3) with scale for different situations for: a), b), and c) horizontalvelocity; d), e) and f) vertical velocity.RELATIONSHIP BETWEEN INTERMITTENCY AND STRATIFICATION 677situations.4. – ConclusionsAlthough the reasons are not known exactly, the experiments in laboratory show that2D turbulence is characterized by the absence of intermittency. It has been shown in thiswork that, in the atmosphere, also the situations of stability are characterized by a minorintermittency, or an intermittency in another sense. Consequently, the different degreeof stratification along the day (normally major stability during the night than during thediurnal hours) will be reflected in a different behavior of the intermittency during the day(minor intermittency for the night, or even in strong stratification situations, an inter-mittency in an inverse sense). To show these relations data from 7 consecutive days havebeen used. For this period, the characteristic daily evolution of the stratification (andof the module of the vertical speed) also becomes clear for the case of the intermittency.To characterize this intermittency we have used as index the log (F/3) of the PDF of thevelocity increments to the resolution scale (1 s/20). This index gives idea of the degreeof stretching that can reach the tails of the PDFs in its evolution along the cascade. Inthe least intermittent cases, or with inverse intermittency, the PDFs turn out to be moreclosed (minor flatness), while in those presenting major direct intermittency (normallydiurnal), the PDFs becomes more pointed and with more elongated tails. This is clearlyshown in the evolution of the flatness with the scale (fig. 6).It would be interesting to perform this type of analysis in much more stable atmo-spheric boundary layer datasets in extreme conditions, such as those in the Antarcticdescribed by Yagu¨e and Redondo in [27].∗ ∗ ∗This research has been funded by the Spanish Ministry of Education and Science(projects ESP2005-07551, CGL2004-03109 and CGL2006-12474-C03-03). IV PRICITprogram (supported by CM and UCM) has also partially financed this work through theResearch Group “Micrometeorology and Climate Variability” (n◦ 910437). Thanks arealso given to Carmen Vasini of SIF Editorial Office for her helpful collaboration.REFERENCES[1] Batchelor G. K. and Townsed A. A., Proc. R. Soc. London, Ser. A, 199 (1949) 238.[2] Landau L. D. and Lifshitz E. M., Fluids Mechanics (Pergamon Press, Oxford) 1959.[3] Kolmogorov A. N., C. R., Acad. Sci. USSR, 32 (1941) 16.[4] Castaing B., Gagne Y. and Hopfinger E. J., Physica D, 46 (1990) 177.[5] Gagne Y., Marchand M. and Castaing B., J. Phys. II France, 4 (1994) 1.[6] Sorriso-Valvo L., Carbone V., Veltri P., Politano H. and Pouquet A., Europhys.Lett., 51 (2000) 520.[7] Mordant N., Metz P., Michel O. and Pinton J. F., Phys. Rev. Lett., 87 (2001) 21,214501.[8] Barndorff-Nielsen O., Blaesild P. and Schmiegel J., Eur. Phys. J. B., 41 (2004)345.[9] Paret J. and Tabeling P., Phys. Fluids, 10 (1998) 3126.[10] Boffeta G., Celani A. and Vergassola M., Phys. Rev. E, 61 (2000) R29.[11] Kraichnan R. H., Phys. Fluids, 10 (1967) 1417.[12] Kraichnan R. H., J. Fluid Mech., 47 (1971) 525.[13] Chevillard L., Castaing B. and Leveque E., Eur. Phys. J. B, 45 (2005) 561.678 J. M. VINDEL, C. YAGU¨E and J. M. REDONDO[14] Bottcher F., Barth S. and Peinke J., Stoch. Environ. Res. Risk Assess., 21 (2007)299.[15] Charney J. G., J. Atmos. Sci., 28 (1971) 1087.[16] Metais O., Bartello P., Garnier E., Riley J. J. and Lesieur M., Dyn. Atmos.Oceans, 23 (1996) 193.[17] Mahanov A., Nicolaenko B. and Zhou Y., Phys. Rev. E, 57 (1998) 11, 6187.[18] Cuxart J., Yagu¨e C., Morales G., Terradellas E., Orbe J., Calvo J., Ferna´ndezA., Soler M. R., Infante C., Buenestado P., Espinalt A., Joergensen H. E., ReesJ. M., Vila´ J., Redondo J. M., Cantalapiedra I. R. and Conangla L., Boundary-Layer Meteorol., 96 (2000) 337.[19] Yagu¨e C. and Cano J. L., Atmos. Environ., 28 (1994) 1275.[20] Rodriguez A., Sanchez-Arcilla A., Redondo J. M. and Mosso C., Exp. Fluids, 27(1999) 31.[21] Yagu¨e C., Viana S., Maqueda G. and Redondo J. M., Nonlinear Proc. Geophys., 13(2006) 185.[22] Frisch U., Turbulence (Cambridge University Press, England) 1995.[23] Kolmogorov A. N., J. Fluid Mech., 13 (1962) 82.[24] Meneveau C. and Sreenivasan K. R., Phys. Rev. Lett., 59 (1987) 1424.[25] Vindel J. M., Yagu¨e C. and Redondo J. M., Nonlinear Proc. Geophys., 15 (2008) 915.[26] Hnat B., Chapman S. C. and Rowlands G., Phys. Rev. E, 67 (2003) 056404.[27] Yagu¨e C. and Redondo J. M., Antarc. Sci., 7 (1995) 421.

Relationship between intermittency and stratification

Aim: Evaluation of treatment and survival of pT1 stage (T1) bladder cancer (BC) patients diagnosed with transitional cell carcinoma of the urinary bladder in Norway. Material and methods: According to the Cancer Registry of Norway, 1,108 patients were diagnosed with T1 BC between 2008-2012. Information on surgical and medical procedures was provided by the Norwegian Patients Registry. Regression and survival models were applied to characterize patients receiving bacillus Calmette-Guerin (BCG) and radical cystectomy (RC) as early and delayed treatment and to estimate overall and cause specific survival rates (OS; CSS). Adjustments for sex, age, WHO grade and concomitant cis were made. Results: In total, 449 (41%) patients received BCG treatment, 162 (15%) as early treatment. RC represented the early treatment in 96 (9%) patients and the delayed treatment in 84 (8%). Overall, 850 (77%) patients received neither BCG nor RC as early treatment, of whom 287 (26%) were treated with BCG and 66 (6%) with RC during follow-up. Patients <75 years and patients with high grade tumors or concomitant cis were more likely to receive BCG and RC as early treatment. 5-year survival rates for all T1 BC patients were 84% (CSS) and 65% (OS). Delayed RC was associated with the lowest 5-year CSS (70%). After adjustment, gender did not impact treatment choice and CSS. Conclusions: The use of BCG as early treatment indicates low adherence to existing guidelines. Delayed RC was associated with low survival rates. An increased focus on the management of T1 patients is needed in Norway

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T1 bladder cancer in Norway: treatment and survival

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Relationship between intermittency and stratification

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