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TURBULENCE IN THE ANISOTROPIC BOUNDARY LAYER
V.V. Nosov, V.M. Grigoriev*, P.G. Kovadlo*, V.P. Lukin, E.V. Nosov, A.V. Torgaev
Institute of Atmospheric Optics SB RAS, 1, Akademicheskii ave., Tomsk 634055, Russia *Institute of SolarTerrestrial Physics SB RAS, 126, Lermontova str., Irkutsk 664033, Russia ABSTRACT Processes of Benard cell origination and disintegration in air are studied experimentally. It is shown that temperature gradients cause a Benard cell. Our data confirm the main scenarios of turbulence origination (Landau­Hopf, Ruelle­ Takens, Feigenbaum, and Pomeau­Menneville stochastic scenarios). It is ascertained that a Benard cell disintegrates according to the Feigenbaum scenario. In this case, the main vortex in the Benard cell is decomposed into smaller ones as a result of ten perioddoubling bifurcations. It is shown that the resulting turbulence is coherent and determinate; the fractal character (local selfsimilarity) of its spectrum is found. These results allow the definition of a coherent structure as a compact formation containing a longlived threedimensional hydrodynamic cell (originating from longterm action of thermodynamic gradients) and products of its discrete coherent cascade disintegration. The coherent structure answers all the signs of chaos occurrence (turbulence) in typical thermodynamic systems. On the base of the results, presented in this work, a coherent structure can be considered as a key turbulence element. It is shown that the real atmospheric turbulence is the result of mixing of different coherent structures. Key words: turbulence, Benard cell, stochastization scenario, coherent structure, random analysis, spectrums

INTRODUCTION
Processes of Benard cell origination and disintegration in air are studied experimentally. It is shown that temperature gradients cause a Benard cell. Our data confirm the main scenarios of turbulence origination (Landau­Hopf, Ruelle­ Takens, Feigenbaum, and Pomeau­Menneville stochastic scenarios). It is ascertained that a Benard cell disintegrates according to the Feigenbaum scenario. In this case, the main vortex in the Benard cell is decomposed into smaller ones as a result of ten perioddoubling bifurcations. It is shown that the resulting turbulence is coherent and determinate; the fractal character (local selfsimilarity) of its spectrum is found. These results allow the definition of a coherent structure as a compact formation containing a longlived three dimensional hydrodynamic cell (originating from longterm action of thermodynamic gradients) and products of its discrete coherent cascade disintegration. The coherent structure answers all the signs of chaos occurrence (turbulence) in typical thermodynamic systems. Our results show that known processes of transformation of the laminar flows to turbulent ones (RayleighBenard convection, fluxions in pipes, a fluid flow around hindrances, etc.) are the coherent structures (or sum of such structures). Therefore, it is possible to consider coherent structure a basic element of the turbulence.



Origination of a coherent structure in a closed room follows from the experiments, described in Ref. 1. The most interesting are measurement results of turbulence parameters inside the astronomic spectrograph pavilion of the Large Solar Vacuum Telescope (LSVT)1. These results are of primary importance for finding coherent structures. Therefore, in this work (the 1st part) we describe the experiment and its results with necessary additions, following from an analysis of the results with relation to coherent structure origination. Coherent structures in open air (the 2nd part) are considered using data of other measurements, carried out by the authors in 2000th (see, e.g., Ref. 2). In all these measurements, a mobile ultrasound meteosystem was used; it records more than hundred of atmospheric parameters (with sufficient precision1, 2). It is shown that the real atmospheric turbulence is the result of mixing of different coherent structures. It follows from our data, that extended areas are often observed in open air, where one coherent structure exercises a decisively influence. Due to a difference of atmospheric coherent turbulence from the Kolmogorov noncoherent one, the values of Kolmogorov C and Obukhov constants in the Kolmogorov­ Obukhov law are verified. It is shown that the error of their definition can be 93%. This is one of the reasons of large errors in measurements of turbulent atmospheric parameters. ______________________________ Further author information: nosov@iao.ru


1. COHERENT STRUCTURE IN A CLOSED ROOM
The scheme of measurements in the LSVT spectrograph pavilion is shown in Fig. 1. The pavilion is an isolated big closed rectangular room of about 5 x 16 x 7 m (height, length, width) in size, westeastward elongate. Inside surfaces of the room is smooth, walls are without windows. There is an entrance (doubledup) door from the south side and air hole (0.5 m â 0.5 m) in the southwest upper angle of the pavilion. Measurements in the spectrograph pavilion were carried out at two height levels from the floor h: the lower level 1.10 m corresponded to the optical path height in the pavilion (points 1­7 in Fig. 1); the height of the upper one was mainly 3.10 m (points 9­12 in Fig. 1, 2.55 m ­ for point 8). The entrance door and air hole were closed during the measurements. A high level of temperature and refraction index fluctuations in the pavilion follows from the measurement results. The intensity of these fluctuations is usually characterized by the structural characteristics CT2 and Cn2. For example, Cn2 at the - lower level (points 1­7) reaches the value 1.6 10 14 cm­ 2/3. At average, Cn2 decreases with height. Noticeable gradients of average temperature have been recorded in the pavilion. Thus, the vertical gradient between points 2 and 8 (near the east wall) reaches the value d/dh = ­ 0.41 deg/m (the average temperature decreases to the top). The value of vertical gradient, averaged over all the observation points in the pavilion, equals ­0.145 deg/m. The value of longitudinal horizontal gradient (along the eastwestward directed optical path), averaged over all the observation points at the path height 1.1 m, equals ­0.028 deg/m (average temperature decreases when transferring from the east to west wall). However, this averaged gradient essentially decreases at the upper measurement level (at a height of 3.1 m) and becomes equal to +0.009 deg/m. Considering the spectrographmeasured temperature gradients, one can see that the pavilion floor, east and west walls are the hottest. A built smoothed vector of pavilion temperature gradient, corresponding to measured vertical and side gradients, is to be a straight line directed from the ceiling near the southwest angle (air hole) to the floor near the northeast angle of the pavilion (alignment table). Spatial periodicity of some variables (of quincunx structure type) is observed in a vertical plane passing through the optical path in the pavilion (through points 2 and 4­6). For example, the Monin­Obukhov number alternates in sign, the average temperature periodically deviates from the value, smoothed over the points at one height. The structural characteristics CT2 and Cn2 are periodic at the lower level in this plane. Similar periodicity takes place for the inner and outer turbulence scales. Such behavior of the above parameters connected with origination of stable periodic vortex formations in large closed rooms. 1.1. INCIPIENT CONVECTIVE TURBULENCE. BENARD CELLS The map of recorded averaged air motion (wind map) inside the pavilion is shown in Fig. 1. Comparatively strong counter wind flows are observable at the lower measurement level in the pavilion centre 0 (approximately, pavilion lengthwise, East­West A l i gnment +0. 05 tabl e 1 6, 11 1 direction). A similar pattern is observable at the 5, 10 4, 9 2 8 upper measurement level, while the motion speed 2 ­ 0. 03 3 +0. 04 +0. 01 essentially decreases here and motion directions are +0. 02 +0. 01 +0. 03 3 +0. 1 displaced toward the air hole. The solid alignment +0. 00 ­ 0. 00 7, 12 4 table strongly affects air motion near the east wall, N where airflows are mainly directed along and toward 5 +0. 04 the wall.
N orth­ South di stance, m 6 ­ 0. 03 0. 1 m/ s 2

It also follows from the obtained data that the sign of MoninObukhov number , characterizing the temperature local temperature stratification, alternates in sign at the optical path height when passing from the east pavilion wall to the west one (through points 2 and 4­6). Similar spatial periodicity is observed for the deviation T ( T = ­ av) of the average temperature from the smoothed average temperature av, considered as a function of distance in the path through points 2 and 4­6. Thus, = ­ 5.16, T = ­ 0.04 deg at point 2; = + 0.47,

A i r hol e 14 12 10

E ntrance door 8 6 4 0 E ast­ W est di stance, m

S

7 16

Fig. 1. Spectrograph pavilion. Scheme of measurements in the pavilion (overhead view) and wind map. Numbers inside the rectangles are the numbers of measurement points; numbers near the arrays' ends with the signs «+» or «­» are the values of vertical wind (m/s). Solid arrows and nonitalic numbers correspond to the velocity at the floor of pavilion (1­ 7), dashed arrows and italic numbers ­ at the top of pavilion. The velocity scale is shown in the right bottom angle

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T = + 0.29 grad at point 4; = ­ 4.55, T = ­ 0.07 deg at point 5; = + 0.49, T = + 0.06 deg at point 6. These results allow us to ascertain the character of averaged flows between observation points and to build a more detailed motion pattern. Actually, let T exceeds zero at a certain point. This means a higher temperature of air at this point as compared to neighboring adjoining areas at the same level. Warm air is lighter than cold one. Hence, cold air runs under the warm one and displaces it upward. If > 0 at this point, then there is a warmer air over the observation point, which follows from the sense of the parameter , while the air at the point is colder and heavier. Hence, a blocking buoyancy force appears at > 0. Thus, two vertical opposite directed forces affect the air volume at T > 0 and > 0 (stabilizing action of stable stratification competes with destabilizing effect of an unstable temperature profile3). These forces restrict vertical travels of air volume and allow only horizontal movements (e.g., at point 4 in Fig. 1). At T < 0 and < 0, the situation is similar (for example, at point 5 in Fig. 1). However, between these areas, when, e.g., T > 0 and changes from positive to negative ( T > 0, < 0), both vertical force are directed upward and, hence, the air volume moves upward (gap between points 4 and 5 in Fig. 1). Otherwise ( T < 0, > 0), both forces are directed downward and air moves downward. Such situation is probable in the gap between point 2 and 4 and near point 6 in Fig. 1. A more detailed pattern of air motion in the pavilion in a vertical plane passing through the optical path (through observation points 2 and 4­6) can be built on the grounds of the above data. Such approximate pattern is shown in Fig. 2. Here the solid slant line shows the (above plane) projection of smoothed average temperature gradient vector. As is seen from Fig. 2, there are rotating airflows (whirls) into the pavilion. Averaged air motion in the pavilion is similar to vortex toroidal motion of liquid in a space cell, which is the pavilion. The cell axis (torus axis) is parallel to the temperature gradient vector (slant line in Fig. 2). In the pavilion center, air moves upaxis, parallel to the gradient direction, while near the walls ­ downaxis. Fig. 2 reflects only principal properties of air motion in the pavilion, real motions are more complicated. Artificial obstacles to the flows (mirrors, screen, etc.) distort these vortex flows. It is clear, that circulation of averaged flows is caused by the temperature gradient, existing in the pavilion. These equilibrium vortices, observable in a completely closed room, can be interpreted as Benard convection cells3.

5 4 3 2 6 5 4 2 1 0 0 H ei ght of the pavi l i on fl oor, m

16

14

8 6 4 12 10 E ast­ W est di stance, m W E

2

Fig. 2. Air motion inside the pavilion in a vertical plane through Theoretical models, implying existence of Benard convection cells, were built long ago. It follows from the points 2 and 4­6 (their positions are designated by italic numbers). Ellipses show the trajectories of averaged motions, slant solid line ­ theory that origination of the cells (to which the the projection of smoothed average temperature gradient convective motion is decomposed) requires a temperature gradient between opposite planes. The cells can take a shape of hexagonal prisms (with an axis along the gradient) depending on the gradient value. In the centre of such prisms, liquid moves upaxis (parallel to the gradient direction) and along the edges ­ downaxis (d /dT < 0, is the kinematic viscosity), or vice versa (d /dT > 0). In more complex (than two spaceapart planes) space areas, the cells can take other than hexagonal forms, e.g., longitudinal and transverselongitudinal rolls (often with neck) etc. Model (vessel) experiments with water, oil, liquid helium as a medium, confirm the fact of origination of Benard convection cells3. Such experiments for air inside large closed rooms were not carried out earlier, as they require small detection device sensitive to motion of weak airflows.

As is known3, origination of Benard cells (stationary periodic motion) requires that the Rayleigh number Ra exceed the critical number Racr. According to the definition, Ra = g h3(T0 ­ Th)/( ), where T0 and Th are the air temperatures at the bottom and on the top of a hheight level; g is the gravitational acceleration; is the thermalexpansion coefficient ( = 1/T0); is the kinematic viscosity; is the air thermal diffusivity. Substituting values of the parameters (T0 = 285.1 K; Th = T0 + h/dh, h = 5 m, d/dh = ­ 0.145 deg/m; = 1.3 10­5 m2/s, = 0.7 ), obtain Ra = 1.3 1010. The critical Rayleigh number Racr is (according to Ref. 3) within the range 657­1708. Hence, the recorded Ra essentially exceeds the critical number (Ra >> Racr) and stationary periodic motions exist. Thus, the measurement results (Figs. 1 and 2) confirm the presence of Benard convection cells for averaged motions inside closed rooms.

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Theoretical study of stability of liquid motion between two spaceapart planes (implies existence of Benard cells) allows several scenarios of turbulence origination (initiation) to be formulated. In particular, it was ascertained 3,4 that the turbulence, originating in closed volumes, is undeveloped in conditions of instability of convective motions. A part of liquid energy is spent for regular (laminar) motions (vortices in Benard cells), another ­ for turbulent ones. It is clear from this definition that random air motion inside a closed room is an example of incipient turbulence.



Statistical characteristics of the incipient undeveloped turbulence in air are insufficiently studied 3,5. Therefore, it is interesting to point out some characteristic features of undeveloped turbulence, distinguishing it from the developed turbulence mode. Consider here (as the simplest) structure and correlation functions of temperature fluctuations. Figures 3 and 4 show the comparison results of experimental data for main statistical characteristics of air temperature fluctuations in a closed room and in open air. For a closed room, data correspond to measurements in point 5 inside the spectrograph pavilion ( = ­ 4.5, Cn2 = 1.6 10 ­ 14 cm ­ 2 / 3, h = 1.1 m, = 11.95 deg, V = 0.09 m/s). For open air, typical experimental data were chosen. They were obtained separately in summer measurements over an approximately flat underlying surface in the clear dry weather ( = ­ 3.8, Cn2 = 6.5 10 ­ 16 cm­ 2 / 3, h = 3.1 m, < T > = 24.56 deg, V = 0.86 m/s). As is evident from Fig. 3, which shows twominute realizations of random temperature, the random fluctuation process in the open air is close to a steadyflow one. In closed room conditions, fluctuation process is clearly divided into two intervals with different turbulence modes, one of which changes another one with discrete jump. The jump shows an appearance of a steady flow from the previous one, with new characteristics. This phenomenon is called bifurcation of stability change3. As is follows from Fig. 4, the structure functions of temperature fluctuations DT ( ) at small time intervals are Kolmogorov functions (DT 2/ 3) both in developed and undeveloped turbulence. At large time intervals, they essentially differ. The fluctuation correlation coefficient bT inside a closed room has a number of sufficiently large local maxima, in contrast to open air. Each such maximum meets a local minimum of the structure function DT (their arguments agree). The structural function DT ( ) at very small in a closed room has a quadratic segment (see Fig. 4, bottom left subfigure), longer than in open air, which corresponds to increased values of the inner turbulence scale.

Fig. 3. Twominute realizations of the random temperature T in a closed room and in open air

Fig. 4. Normalized structure DT( ) and correlation bT( ) (right bottom subfigure) functions in a closed room and in open air (the initial segment of DT( ) is shown at the bottom left)

1.2. MODELS OF TEMPERATURE FLUCTUATION SPECTRA IN THE INCIPIENT TURBULENCE Models of temperature fluctuation spectra, first of all, of the spatial 3D spectrum ( ), are required in problems of optical radiation propagation in undeveloped turbulence. As is known5, time frequency spectra of temperature fluctuations WT(f) in open air are adequately described by the Karman model. The Kolmogorov developed turbulence spectra have a long inertial interval, where WT f ­5/3 and the energy is transported from largescale vortices to smaller ones.

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Smoothed time frequency spectra of temperature fluctuations WT are shown in Fig. 5 for a closed room (point 5 in Fig. 1) and open air. As is seen, the closed room spectra rolloff much more rapidly than openair ones within the inertial interval; besides, within the interval, there are only short individual frequency regions (in echelon), inside which the turbulence can be considered as Kolmogorov (WT f ­5/3). These regions are observed between jumps of the spectral function at the frequencies, corresponding to local maxima of fluctuation correlation function (or minima of the structure function). In case of smoothed echelons, experimental spectra of undeveloped turbulence have a number of characteristic regions of rapid power decrease. Thus, if WT const at a long energy interval, then first WT f ­8/3 with frequency rise (within the inertial interval) and then WT f ­12/3. Hence, energy transport from large to small vortices is insignificant in the incipient turbulence, i.e., the Fig. 5. Smoothed time frequency spectra of temperature vortices are weakly diffused. Spectra rolloff slows fluctuations WT in a closed room and in open air. ­2/3 down (WT f ) with further frequency rise, in the viscous interval, where the spectral density is close to the noise level. Similar behavior of spectra is observed at other points of the pavilion. The spectrum echelon is pronounced even stronger at some points. To construct an abstract model ( ) of incipient turbulence spectra, the Karman model with rolloff corresponding to Fig. 5 within the inertial interval can be used. Such a crude model was obtained in Ref. 1:

(Ô) = A0CT2 (6,6 Ô 0)2( ­1/3) (Ô2 + Ô02)­( + 3/2) exp(­ Ô2/Ôm2), (1) A0 = 0.033, Ô0 = 2 / L0, Ôm = 5.92 /l0 .
where L0 and l0 are the outer and inner turbulence scales, respectively; = 1/3 for the developed turbulence, then (Ô) Ô­ 11 / 3 in the inertial interval. According to Fig. 5, = 5/6 in the incipient turbulence, hence, (Ô) Ô­ 14/3 in the most part of inertial interval. Further a more rapid spectrum rolloff is described by the exponential factor in Eq. (1). The parameters L0 and l0 for = 5/6 and = 1/3 are given in Table 1, from which it is evident that the inner scale of incipient turbulence l0 (average l0 = 1.9 cm, = 5/6 and l0 = 2.7 cm, = 1/3) is oneorderofmagnitude larger than the inner scale in open air (0.7­4 mm3,5,2). The spectrum (Ô) at = 5/6, which can be considered as a n incipient turbulence spectrum, meets the experiment better that at = 1/3 (developed turbulence spectrum, = 1/3, but with L0 and l0 for incipient turbulence, Table 1). It is evident from the comparison of curves 4 and 5 in Fig. 5. Areas under these curves differ from experimental one by 15% ( = 5/6) and 28% ( = 1/3), respectively. However, model (1) with = 1/3 is widely used; it allows one to transfer solutions of problems of wave propagation in the developed turbulence to the case of incipient one. Hence, this model is preferable.

The maximum error of spectrum approximation by Eq. (1) falls into the region of very large frequencies (viscous interval). Therefore, problems of wave propagation, where this interval is of great importance, require a more detailed model as compared to Eq. (1). The viscous interval does not significantly contribute into problems of optical beam shift, image jitter, etc., i.e., where wave phase fluctuations play a key part; therefore, equation (1) can be used here. The size distribution of outer turbulence scale L0 in vertical plane inside the pavilion is shown in Fig. 6. Periodic behavior of the inner scale l0 is similar to those shown in Fig. 6 for the outer one (see Table 1). In this case, the smaller outer scale corresponds to the smaller inner one. It is known, that outer and inner scales determine maximum and minimum sizes of inhomogeneities. Hence, a spatial periodicity of sizes of temperature field inhomogeneities (of quincunx structure type) inside the pavilion follows from Fig. 6 and Table 1.

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Table 1. Parameters of incipient turbulence spectra
5

South-si de vi ew

Observation point No.

1 2 3 4 5 6 7 8 9 10 11

= 5/6 L0 cm 33.2 47.4 19.8 27.0 62.6 18.5 46.4 18.0 60.3 18.1 30.4

= 1/3 l 0 L0 l 0 10 3 cm 11 cm 9 cm 8 1.2 83.0 1.8 2 2.3 101.6 3.5 2 1.2 1 5 4 1.2 6 53.9 1.8 67.5 2.8 0 16 2.14 12 10 8 125.1 6 4 2 04.1 3 E ast­ W est di stance, m 1.6 46.4 2.1 2.3 126.5 2.9 1.84 36.0 scale L0 in the vertical 2.8 Fig. 6. Distribution of outer turbulence 2.3 through the pavilion center and WestEast line 139.2 3.9 plane passing (according to the data from Table 9 . Circles of larger 1.8 1) 1.8 38. diameter corresponds to larger L0 values. Figures designate 9 2.3 76.1 2. the numbers and positions of observation points 1225.42.354.33.2
4 H ei ght, m

Regions with decreased outer scales can be called turbulence locks (or focuses), where intensified decomposition of the largescale averaged flow to smaller spatial components is observed. The intensity of random temperature variations, characterized by the periodic parameter C, at average, decreases in the focuses, which is caused by smaller temperature differences (of passive admixtures3,5) in smaller vortices there. 1.3. STOCHASTIC SCENARIOS OF CONVECTIVE FLOWS Compare the measurements in the pavilion with the wellknown data on turbulence incipience from laminar flows (stochastic scenarios). The most known stochastic scenarios are Landau­Hopf, Ruelle­Takens, Feigenbaum, and Pomeau­Menneville.3 It will be shown below, that all these scenarios are confirmed in the incipient turbulence. ) the Pomeau­Menneville scenario As is known3,4, as the distance increases (Reynolds number) in laminar flows in pipes, small turbulent regions with non laminar flow first arise. These regions are usually called turbulent locks (or focuses). The locks become longer with increasing the distance and finally merge in a continuous turbulent flow. Turbulent locks are observed in experiments with other schemes as well4. The locks cause alternation of laminar and turbulent modes. Such turbulence incipience via alternation is called the Pomeau­Menneville scenario3, 6. It follows from our measurements, that turbulent locks and alternation (and, as follows from Fig. 3, the corresponding bifurcations of stability change) exist in periodic flows in the Benard cell as well. The parts of locks are played by regions with decreased spatial components (outer L0 and inner l0 scales). The locks turn out to be trapped in the structure of the Benard cell and alternate with regions of large L0 and l0 scales. Hence, our data confirm the Pomeau­Menneville scenario. b) the Landau­Hopf scenario The incipient turbulence in the Benard cell is a convenient model allowing one to trace the decomposition of energy carrying vortices into smaller ones. In fact, the toroidal vortex of averaged movements can be considered as the only energycarrying vortex in the Benard cell. Its sizes are determined by sizes of the room, where it originates. It is difficult to register the sizes of the main energycarrying vortex in open air, because they depend on climateforming factors. The outer turbulence scale is usually considered as this vortex, which itself is the decomposition product.

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The correlation factor bT and the sample nonsmoothed temperature frequency spectrum WT, calculated from the pavilion measurements data, are shown in Fig. 7. The correlation factor bT has been calculated from different sample estimates7­9, which, however, give no agreement between the results and reveal the local maxima of bT. The correlation function can be calculated with an arbitrary small error at the large sample length N (a variance of bT error is proportional to 1/N). In our case, N = 19139; hence, the 95% confidence bound of bT definition, shown in Fig. 7, makes ± 0.014; this is much less than bT maxima. The sample spectrum WT was calculated without smoothing with a rectangular spectral window. As is known7,8, such a window acts like a slit of about 2/T in width (T = 120 s). This resolution is sufficient to reveal the WT spectrum maxima, uniquely corresponding to bT maxima. Arguments of bT and WT maxima are related as kfk = 1, k = 1, 2,... ( 1 and f1 usually determine characteristic scales of bT and WT decrease). The diameter of the main energycarrying vortex 2R1 is easily retrievable from the relation 2 R1 = 1 = /f1; it is equal to 294.4 cm ( = 9 cm/s, f1 = 0.00973 Hz) at point 5 in the pavilion. Virtually the same result is obtained from data of Fig. 2, if the trajectories of averaged movements are considered as circles.
1.0 bT ( ) 0.8 9 0.6 10 0.4 0.2 0.0 ­ 0. 2 2 ­ 0. 4 0 20 40 60 80 100 1 , s 120 8
101 100

W T , deg2/ H z 1 23 45 6 7 10 89

7 65 4 3

10­ 10­ 10

1 2

­3

0.01

0. 1

f, H z

Fig. Comparison of WT spectra in Figs. 5 and 7 shows disappearance of real spectrum maxima at a standard 7. Correlation factor bT and nonsmoothed frequency spectrum spectrum smoothing by a wide window (dispersion of the WT (top right) in the pavilion. Figures designate the numbers of bT and WT maxima, corresponding to each other. smoothed spectrum in Fig. 5 is equal to 1% of the sample spectrum dispersion in Fig. 7). Therefore, to calculate frequencies of spectrum maxima (harmonics), data of Fig. 7 are required. However, the rectangular spectrum window has large side lobes, causing oscillations especially at high frequencies. It is possible to get rid of these lobes using any of commonly used nonrectangular windows (difference between them is negligible), for example, the Welch window9. This window about twofold decreases the dispersion as compared to a rectangular one and similarly widens the frequency band width. The widening is acceptable, because the bandwidth turns out to be less than the mean width of spectrum maxima. This is evident from data for WT in Fig. 7. To improve the sample estimate, a digital threshold filter can be used additionally to eliminate weak (lower than the averaged in Fig. 5) harmonics in spectrum. These harmonics can be interpreted either as high frequency side lobes, incompletely damped by the Welch window, or a weakly pronounced transient process of energycarrying vortex decomposition to smaller ones. It is shown below (see Fig. 10), that the structure of weak harmonics is fractal, it does not coincide with those of side lobes. Therefore, the side lobes are completely damped. The level of the used threshold filter in the form of smoothed spectrum WT (with 1% dispersion) has been chosen on the base of detailed analysis of sample spectra at many measurement points in the pavilion. The threshold filter is thus singles out main maxima (or firstorder harmonics) from the spectrum.

Based on the above analysis, the frequencies of firstorder harmonics (arguments of maxima) in the fluctuation spectrum WT can be interpreted as frequencies of stable vortices observed in the temperature field (Fig. 8). The vortex frequencies fn turn out to be multiple to the frequency of main energycarrying vortex f1 = 0.00973 Hz. Normalized to f1, they are integer natural numbers (n = 1, 2, ...): fn/f1 = 1, 6, 8, 11, 13, 17, 20, 66, 68, 77, 90, 93, 109, 113, 117, 120, 127, 130, 133, 136, 144, 150, 152, 157, 162, 164, 167, 175, 179, 183, .... Multiple frequencies is an exact result of discrete decomposition of the main energycarrying vortex to smaller ones. The multiplicity means also that the phases of different harmonics (oscillations) are rigidly bound (agreed). In this case, the oscillations are called coherent (inphase). Note that the function fn/(nf1), shown in Fig. 8, is tolerant to variations of threshold filter level; it varies insignificantly at essential variations of the filter level. Vortex diameters Rn (cm) for each harmonics can be calculated by the equation 2 Rn = /fn, n = 1, 2,...: Rn = 147.21, 24.52, 18.39, 13.38, 11.32, 8.66, 7.36, 2.23, 2.16, 1.91, 1.63, 1.58, 1.35, 1.30, 1.26, 1.23,

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1.16, 1.13, 1.11, 1.08, 1.02, 0.98, 0.97, 0.94, 0.91, 0.90, 0.88, 0.84, 0.82, 0.80, ... They are shown in Fig. 9. As shows the comparison of data in Fig. 9 and Table 1 (point 5, = 5/6), the second frequency 6f1 approximately corresponds to the outer turbulence scale L0 (2R1 = 294 cm, 2R2 = 49 cm), while the frequencies 127f1 and 130f1 (terminating the inertial interval) ­ to the inner one l0 (2R17 = 2.3 cm). Vortices, which are decomposition products of larger vortices (their frequencies are multiple to lower ones) are observed in the viscous interval and in a part of inertial one. For example, fn/f2 = 11, 15, 20, 24, 25,...(n = 8, 11, 16, 21, 22,...); fn/f3 = 15, 17, 18, 19, 36,...(n = 16, 20, 21, 23, 52, ...). The processes, observed inside the pavilion, are stable. Therefore, both the origination of vortex in the Benard cell (due to a temperature gradient) and its decomposition into smaller ones happen permanently (probably, with simultaneous selfreplication). This evidently results in a threshold Nperiodic flow with the frequencies fn, n = 1, 2, ..., N. Transformation of a small perturbation into a stable periodic flow follows from solutions of the Landau equation; the origination of timeperiodic flows is called normal Hopf bifurcation. The Landau­Hopf scenario describes the incipience of turbulence as a sequence of normal bifurcations, generating a limiting Nperiodic flow (N >> 1) with, generally speaking, incommensurable frequencies. However, the frequency (phase) incommensurability is usually unrealizable (this is seen from our data as well). Therefore, to save idea, it is considered now that normal bifurcations generate consequent subharmonics3, 11. It is easily seen that our results confirm the Landau­Hopf scenario.
H armoni c f requency f n/ ( nf 1) l0 L
0 2

10

700 V ortex di ameter 2R n, cm 100 L 10
0

R R R

n n n

= R n/ 2/ n­ 2,22, = n­ 1, = 2


1 l
0

1 1

f 1 = 0. 00973 H z = 2. 50, 2 = 6. 26, = 4. 67 10 100 n, number of harmoni c 1000

0. 1 0. 02 1

n­

2, 22

n­

1

10 100 n, harmoni c number

1000

4000

Fig. 8. Frequencies of the firstorder stable vortices (harmonics) fn in the fluctuation spectrum WT

Fig. 9. Diameters of firstorder stable vortices 2Rn in the fluctuation spectrum WT

c) the Ruelle­Takens scenario The Ruelle­Takens scenario can be considered as a refinement of the Landau­Hopf scenario. The difference is in the number of normal bifurcations, after which a flow can be considered as turbulent. According to the scenario, the turbulence occurs (a strange attractor appears) already after three normal bifurcations 3,10, i.e., a threeperiodic flow (N = 3) can be already considered as turbulent. Then the convective flow becomes turbulent after origination of the main vortex in the Benard cell and two events of its decomposition. d) the Feigenbaum scenario The Feigenbaum scenario describes the turbulence incipience (appearance of a strange attractor) as a result of infinite consequence of perioddoubling bifurcations11. These bifurcations appear only after magnitude change of a certain controlling parameter µ , e.g., Reynolds or Rayleigh numbers, etc. As is known, the Feigenbaum scenario follows from the universality of location of periodic points x0, (x, x), (x, x, x, x), ... of 2mmultiple cycles. These points x on the x(µ ) curve correspond to tree branches, which symmetrically bifurcate at critical bifurcation points µ m. Asymptotic similarity relations =

{

, m (µ ) = = , = 2.503, = 4.669.

8


(m >> 1, and are the Feigenbaum constants) are correct for both x and µ m. Here x is usually (due to the universality) considered as the main parameter, characterizing a nonlinear dynamic structure, e.g., coordinates (in meters), or standard (m/s) or normalized speed (Hz = 1/s), as in the case of Navier­Stokes equations. Though the equality m(µ ) = is asymptotic, its suitability already after two­three period doublings (up to several percents) has been shown11. The theory has a high predictability due to the large rate of convergence ( = 4.67). For approximate evaluation, this equality can be used at m 0. It is obvious that the equation m(µ ) = has the solution µ m = c ­m + µ , c = const. Taking m = 0, 1 in the solution, we obtain a set of equations to find the constants c and µ . Hence, c = µ 0 ­ µ and µ = µ 0 + (µ 1 ­ µ 0) /( ­ 1). As is known, one can take µ 0 = 1 and µ 1 = 3 for the logistic equation xm+1 = µ xm(1 ­ xm), considered in Ref. 11, hence, µ = 3.54508, which insignificantly differs from the exact value µ = 3.56994, obtained in Ref. 11. Note, that µ m (at µ 0) can both increase (c < 0) and decrease (c > 0) with increase of m. In our case, the value of the controlling parameter µ is fixed (pavilion sizes, temperature gradient, etc. are preset) and, hence, we can observe only products of vortex decomposition. Due to selfreplicating of harmonics in the presence of doubling bifurcations, the harmonics, resulting from these bifurcations, are recordable. Let, for example, the basic system parameter x be a shift with the length dimension. Then x0 (m = 0) can be identified with the radius of the basic vortex R1. The following variables (x, x) are to be the products of x0 decomposition (nonsymmetrical bifurcation at m = 1). The only next largest radiuses R2 and R3 can play their parts (others are too small). Hence, as it follows from the decomposition diagram 11, the pairs R4, R5 or R5, R6 (the first element in each pair is approximately equal to a half of R2 and the second ­ to a half of R3) can be chosen as the pair x , x (m = 2). Substituting these variables in the Feigenbaum equation, we obtain (R2 ­ R3)/(R4 ­ R5) = 2.97; (R2 ­ R3)/(R5 ­ R6) = 2.30. Another pair x , x (m = 2) should be chosen from evident decomposition products of the vortices with radiuses R2 and R3 (their frequencies are multiple to f2 and f3); R8 and R16 correspond to the first decomposition elements (see above). From the Feigenbaum equation, found (R2 ­ R3)/(R8 ­ R16) = 6.11. Thus, though we are at the top of bifurcation tree (m = 1, 2), the obtained values of the modulus of left part of Feigenbaum equation are close to and 2. Note, that Rn satisfy the equation Rn = Rn/2/ (see Fig. 5); in the inertial and viscous intervals 2. However, in the energy interval and in the inertial one near l0, hence, the equation Rn = Rn/2/ agrees with the Feigenbaum similarity equation for the Fourier harmonics of x [Refs. 3 and 11]. The Feigenbaum constants , 2, and are best pronounced in the harmonic frequency plot. As it is seen from Fig. 8, the normalized frequency yn = fn/(nf1) undergoes two large jumps with the rise in n. The first jump is observed near the outer turbulence scale L0 (as a result, the frequency yn is saturated to the level ) and the second one ­ near the inner scale l0 (yn is saturated to the level bypassing the level 2). These jumps correspond to large steps of the function fn/f1 which can be interpreted as an analog of the "devil's staircase" (see Fig. 10). The presence of , 2, and in Fig. 8 and the satisfaction the similarity equation confirm the Feigenbaum scenario. Since ym and m(µ ) at m >> 1, then bifurcation values of the controlling parameter µ m can be connected with the harmonic frequencies fm and vortex radiuses Rm, which are different parameters of the same turbulence incipience process. Actually, |ym ­ | < and | m* ­ | < are should be true for some positive , 1, and 2 as soon as m* > 1 and m > 2. Then |ym ­ m*| = |(ym ­ ) ­ ( m* ­ )| |ym ­ | + | m* ­ | 2 . Consequently, m* ym. The levels 1 and 2 can be connected with each other, considering the domains of the confident convergence at sufficiently large m* and m. It is known3,11 that m* converges rapidly after several iterations, therefore, m* 4­5 can be taken as the domain of confident convergence. This determines the level 1. As is evident from Fig. 8, the frequency ym rapidly converges to near the inner scale l0 at m 17Â 30 (level 2). Hence, levels 1 and 2 and numbers m* and m are approximately related as 2 2 1 and m 2m*. Taking these relations into account and solving the equation m(µ ) = yn, where n 2m, obtain µ m = c yn ­ m + µ


, yn = f n/(n f1) = /(2 n f1 Rn) , c = const , n 2m. (2)

The constant c is determined here after the choice of µ , for example, in the form of the Reynolds number Re, Rayleigh number Ra, or others. For approximate estimates, equation (2) and m(µ ) = are applicable at m 0. As an example, we show how equation (2) can be used provided µ is the Rayleigh number. Let µ m = Ram /const in Eq. (2). The new constant simply leads to the overdetermination of c.

9


Let us observe the decomposition process of the main vortex in the Benard cell to the level, when the existence of steady periodic flows (vortices) becomes impossible. In this case, the Ra number decreases from some maximum Ra0 to the critical Racr. In the process of decomposing, Ram << Racr at a sufficiently large m and, hence, Ra can be taken equal to zero in the first approximation. From Eq. (2), obtain then Ram = Ram0 yn0m0 yn­ m, where m0 is the m value, at which Ram0 is known (since n = 2m, then n0 = 2m0). If m0 = 0, then, according to the Feigenbaum numeration, Ram0 is the Rayleigh number for the main vortex in the Benard cell. This number can be roughly found, taking the layer thickness h in Ra definition equal to the main vortex diameter. The bifurcation diagram of the main vortex decomposition in the Benard cell is shown in Table 2 (the dependence of Ram numbers on the bifurcation number m), as well as the harmonic numbers n and vortex diameters 2Rn (cm) corresponding to m. As is seen from Table 2, the bifurcation numbers Ram decreases with the rise of m and cross the level Racr (Racr = 657Â 1708) at m = mcr 9­10. The same mcr value is obtained from the number of the recorded harmonics (in Fig. 8, the maximal n equals 1849): mcr E( log n/ log 2) = E(10.85) = 10, where E(x) is the integer part of x. Thus, the origination of the main vortex in the Benard cell and about ten events of its discrete decomposition are observed in the pavilion. Vortex diameters, corresponding to the critical values mcr (see Table 2), are within the 0.6­ 1.2 mm range. They agree with sizes of minimal vortices, existing in air3. On the base of the performed analysis, the known stochastic scenarios can be divided into two groups. The first one includes scenarios of origination of periodic flows from laminar ones. Among these scenarios are, first of all, Pomeau­Menneville and Rayleigh­ Benar convection. The second group includes decomposition scenarios (degeneration) of the appeared vortex periodic flows, the main of them is the Feigenbaum scenario. The Landau­Hopf and Ruelle­Takens (origination of the limiting Nperiodic flow) scenarios contain features of both groups. It follows from the obtained results, that scenarios of both groups are confirmed in the incipient turbulence, a periodic flow originates (the main vortex in the Benard cell) and discretely decomposes. These processes in principle (in the simplest situations) are confirmed by known solutions of nonlinear equations of hydrodynamics. However, such solutions for a general case are presently unknown. It is clear, that vortex solutions should exist, since they are experimentally observed. It is also clear, that the stability of vortex solutions will be determined by nonlinear resonances (both between external forces and dissipative processes and between harmonics with commensurable frequencies). Therefore, the mechanism of hydrodynamic turbulence incipience and existence becomes clearer, in which the role of stochastization essentially decreases. Then the turbulence, usually considered as a merely random event, turns to be significantly determinate.
Table 2. Bifurcation diagram in main vortex decomposition 2Rn, m n Ram cm 1.55 10 0 1 294 9 1 2 3 4 5 6 7 8 9 10 11 2 4 8 16 32 64 128 256 512 1024 2048 5.15 10
8

49.1 26.8 4.5 2.5 1.5 0.87 0.50 0.26 0.12 0.06 0.05

2.05 10
8

2.76 10
6

4.89 10
5

1.94 10
5

6.86 10
4

2.65 10
4

9.46 10
3

The turbulence determinacy is much stronger than it could be expected from the above analysis. Actually, it follows from the consideration of the thresholdfilter damped spectrum harmonics WT in Fig. 3, that their greatest maxima (secondorder harmonics) can be extracted with the same filter. It turns out that the local structure of secondorder harmonics location (between the neighboring firstorder ones) is similar to the structure of firstorder harmonics location throughout the spectrum WT. The local selfsimilarity of the spectrum or, in other words, the fractal structure of the turbulence spectrum is observed. The frequency dependence of the first and secondorder harmonics (normalized frequencies fn / f1 of harmonics) is shown in Fig. 10. This dependence is usually called the "devil's selfsimilar staircase" 12­14; here each inner space between main steps is similar to the whole staircase. Each dash (step) answers its fn / f1 value (all these values are integers in Fig. 10). Long dashes correspond to firstorder harmonics and short dashes ­ to secondorder ones. The distances between arguments of the neighboring harmonics fn ­ fn­1 (n = 2, 3, ...) as functions of frequency (of frequencies fn) are shown in Fig. 11. The bottom plot here shows the distances between the arguments of firstorder harmonics for the whole spectrum WT (see Fig. 7). (As is seen from Fig. 5, it is sufficient to consider only the 0­10 Hz frequency range in the spectrum WT, without weak noise component, which is caused by the transfer of the motion energy to heat (and the hardware errors also) and is observed at frequencies higher than 10 Hz.) The middle plot in Fig. 1 shows the distances between arguments of secondorder harmonics for the longest range between the firstorder harmonics (0.19­0.64 Hz) and the upper one ­ for the 23.34­23.44 Hz range. As is seen from the "devil's staircase,"

1474.1 444.7 67.5

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other ranges contain a less number of secondorder harmonics (because of limited experimental facilities) and, hence, are not so convenient for the analysis. The comparison of the data in Fig. 11 confirms a similarity of locations of the local structures of secondorder and first orderharmonics in the whole spectrum WT. The devil's staircase is truly selfsimilar and the spectrum is fractal. Since secondorder maxima are significantly weaker than firstorder ones, the secondorder harmonics can be called the fractal shadows of the firstorder ones. Thus, the process of turbulent flows originating and decomposing is evidently determinate even in weak spectral particulars.

Fig. 10. "Devil's selfsimilar staircase" (normalized frequencies of the first and secondorder harmonics fn /f1). The fragment shows the devil's staircase in the 23.1­23.7 Hz range

Fig. 11. Distances between neighboring harmonics fn ­ fn­1 (n = 2, 3, ...): between the firstorder harmonics of the whole (bottom plot), secondorder one for the 0.19­0.64 Hz (middle plot) and 23.34­23.44 Hz (upper plot) ranges

The conditions of chaos occurrence in typical dynamic systems, laid down in Refs. 12­14, allows one to point out the characteristic universal features observed in turbulence origination (in incipient turbulence), such as 12: origination of irregular longliving spatial structures, form (character) of which is determined by dissipative factors, local instability and fractal character of phase space of such structures, appearance of central (on zero frequency) spectral peak. All these features are observed in our measurements. A convective Benard cell is a spatial structure; its form depends on medium viscosity and space geometry (form), where it originates (dissipative factors). The cell is locally unstable and cascade decomposes to smaller ones (vortices). The passive admixture spectrum (temperature) in the cell is fractal. The average temperature is not constant due to nonstationarity of a random process in the cell (bifurcation of stability change, see Fig. 3). Without additional actions for elimination of the nonstationarity (average value of random function is obtained by time averaging over the length of characteristic scale of function variation, but not over the whole sample) the centered random temperature includes a noncompensated constant component. Just this component raises the low frequency part of Fourier spectrum and gives the central peak on zero frequency. Such phenomenon is observed for the Fourier transform of both correlation and the random nonstationary function.

It is convenient to name the above features "coherent structure". A.S. Monin and A.M. Jaglom3 define a coherent structure as a random nonlinear superposition of considerably stable largescale turbulence components. According to the above results, we can define a coherent structure as following. A hydrodynamic coherent structure is a compact formation containing a longlived spatial structure, resulting from longterm action of thermodynamic gradients, and products of its discrete coherent cascade disintegration. The disintegrating spatial structure can be called a cell (structure) generating a coherent structure. Disintegration of the generating cell is accompanied by origination of the main energycarrying vortex. The frequency of this main vortex can be considered as the main features of both generating cell and the coherent structure. Sizes of the coherent structure are fuzzy. Flows, external with respect to the generating cell, can transport products of its disintegration to significant distances, forming long turbulent wake. The lifetime of a coherent structure is determined by the action time of thermodynamic gradients. As a limit case of very weak local instability (when a generating structure is locally stable and does not disintegrate), a coherent structure can consist of only one longliving generating structure. In this case, the generating structure is some configuration of laminar flow. Such situation we have when observing, for example, Benard cells in a thin layer of very viscous liquid.

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Thus, the process of turbulence incipience can be connected with origination of coherent structures. The compact formation, observed in our experiments, includes a longliving main energycarrying vortex and products of its simultaneous cascade discrete decomposition (coherent vortices with multiple frequencies). The main vortex frequency is determined by convective Benard cell sizes. According to our definition, such formation is a coherent structure. Origination of coherent structures in liquid, e.g., in water, is of interest. An example of smoothed turbulence spectrum in ocean is given in Ref. 3. This spectrum, similar to those in a coherent structure in air (see Fig. 5), has echelon frequency ranges with 5/3 decrease. The echelon structure of such spectrum allows it to be considered as the spectrum of coherent structure in water. Flows in pipes easily yield to experimental investigation. However, they are studied not enough theoretically. In this case it is clear from qualitative consideration that the pressure gradient P in liquid can play the part of temperature gradient required for convective cell origination. This follows already from the simplest equation of state P = const T. The density of incompressible liquid is usually considered as constant, hence, P T. Therefore, cells similar to convective can appear in liquid in presence of the longliving pressure gradient. Disintegration of these cells should generate a coherent structure. In contrast to the Rayleigh­Benard convection, vertical sizes of flows in pipes (pipe length) essentially exceed horizontal ones (pipe crosssection). Therefore, cells are situated not horizontally, but vertically, i.e. along the pipe length. In addition, there is a middle flow. It distorts the cell form and transport disintegration products of generated cells (smaller vortices) to the orifice, where they accumulate and a segment of continuous turbulence (commonly called developed) appears. This situation is described by the stochastic scenarios via alternation. Similar patter arises in flowing different obstacles, which follows from data of numerous observations. A longliving main vortex (generating structure, sometimes several) originates on the obstacle surface (near the backwall). Clear large vortices appear behind the obstacle (at some distance); they are not larger than the main one. The shape of vortices is distorted by an external flow (front surface is indented). At further moving off, several smaller vortices are observed and again continuous turbulent flow. Absence of tight bounds (e.g., pipe walls) results further in transversal flowing of turbulent wake. The described phenomenon is the result of transport of generating structure disintegration products by the external flow. This is a coherent structure.

2. COHERENT STRUCTURES IN OPEN AIR
As is known from meteorology, sufficiently stable vortex formations (cells) of different scales exist in the atmosphere. The Ferrel and Hadley cells are the largest among them (up to 5000 km in radius). They can be considered as modifications of Benard cells in a thin spherical layer (at the Earth scale). Somewhat smaller cells (cyclones, anticyclones, thundercells, tornados, etc.) exist there as well. The decomposition products of these vortices have clearly pronounced deterministic character (corresponding to a coherent structure or the non Kolmogorov incipient turbulence) and are observable in open air. Openair measurement data (The Sayan Solar Observatory, July 5, 2007) are shown in Fig. 12 along with the modeled Fig. 12. Kolmogorov turbulence is the result of mixing of results of the steadywind transfer of a frozen spatial different coherent structures. WT are the smoothed spectra; DT pattern of flows inside the LSVT (sequential slow transfer is the structure functions of temperature fluctuations: curve 1 of data, recorded at seven abreast neighboring points, corresponds to summer daytime measurements in mountains 1 located along a straight line at one height , through point 5, (at a height of 2032 m). curve 2 corresponds to wind transfer of Fig. 1). This straight line includes not only points 5, 4, and a frozen spatial pattern of flows inside the LSVT through one 2 in the spectrograph pavilion, but also another four points, point located in neighboring closed LSVT rooms, which are isolated from the pavilion. Frequencies of main energycarrying vortices are different in these rooms and the pavilion. Comparison of Figs. 12, 5, and 4 shows that the results for open air, given in Fig. 12, indicate the prevailing activity of one coherent structure (WT f ­8/3 within the inertial interval). On the contrary, transfer of determinate vortices, formed inside closed rooms, through one point results in the turbulence close to Kolmogorov (WT f ­5/3). Hence, the conclusion can be drawn that the real atmospheric Kolmogorov turbulence is a mixture of determinate vortices from different




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coherent structures. Inequality (nonmultiplicity, incommensurability) of frequencies of main vortices of different coherent structures results in outof phase (noncoherent) oscillations which are disintegration products. Hence, is naturally to call the turbulence, originating in mixing of coherent structures with incommensurable frequencies of main vortices, noncoherent. As a result of expeditionary works in mountain and flat conditions in 2000th years, the authors gather an extensive experimental database of surface measurements of main turbulence parameters in various geographical areas and meteorological situations. It follows from the data that extensive areas, where one coherent structure exercises a decisively influence, are observed in open air. Moreover, characteristic features of coherent structures (coherent turbulence) are present in the majority of gathered data. The Kolmogorov noncoherent developed turbulence usually appears only over large areas with flat and uniform underlying surface.

The coherent turbulence differs from noncoherent one, first of all, in more rapid rolloff of the smoothed spectrum WT in the inertial interval ( f ­8/3) and less contribution of highfrequency component (smallscale vortices). Therefore, in air, the spectrum in the inertial interval has two pronounced decrease regions within the areas with decisive influence of one (local) coherent structure, i.e., first a sufficiently rapid decrease is observed (usually f ­8/3, sometimes even more rapid), and than the decrease slows down ( f ­5/3) with frequency rise. The second Kolmogorov region is characterized by a mixture of disintegration products of other large generating structures existing in the atmosphere. Sometimes the inertial interval of the spectrum has two pronounced stepwise regions with the 8/3decrease. In this case, one can tell about presence of two local coherent structures in the measurement site. Due to a difference of atmospheric coherent turbulence from the Kolmogorov noncoherent one, the Kolmogorov­ Obukhov law applicability limits are to be specified. In particular, the values of Kolmogorov C and Obukhov constants are to be verified. In accordance with the Kolmogorov­Obukhov law, the structure function of longitudinal velocity fluctuations Drr(r) in the inertial interval of the scales r is expressed via the structural characteristic CV2 of longitudinal velocity fluctuations: Drr(r) = CV 2r2/3. The structure function of temperature fluctuation DT(r) is expressed via the structural characteristic CT2 of temperature fluctuations: DT(r) = CT2r2/3. The structural characteristics CV2 and CT2 determine the intensity of velocity and temperature fluctuations and are important parameters of medium turbulent motion. In their turn, CV 2 and CT 2 depend on the average dissipation rate of kinetic energy and temperature N and Kolmogorov and Obukhov constants: CV 2 = 2/3, CT 2 = ­1/3N. Hence, velocity and temperature fluctuations are to be functions of and at known and N. The constants and were measured in different media by different methods. The values = 1.9 and = 3.0 are recommended3 as the most probable estimates of the constants. This estimate is average with respect to data of different authors. Deviations from the average value are sufficiently large3, e.g., the values 0.9, 1.6, and 2.8 were observed for C and 1.1, 1.4, 2.5, 2.7, 3.3, 3.5, 5.6, 5.8, 6.5, 9.0 ­ for . As is known3,5, the Kolmogorov constant is related with the asymmetry S of probability distribution of longitudinal velocity difference of by the equation = (0.8/ |S| )2/3, S = Duuu/ |Duu|3/2. The third moment of longitudinal difference of wind velocities Duuu(r) = < [u (r +r,t) ­ u (r ,t)]3 > can be obtained from the time moment D uuu( ) = < [u (r ,t+ ) ­ u (r ,t)]3 > and the frozen condition3 Duuu(v ) = D uuu ( ). Here u (r,t)= u(r,t) ­ , where u(r,t) is the random value of longitudinal velocity (usually the projection of random velocity vector to the vector of average velocity) at the point r at the moment t. The constant is related with the asymmetry S of probability distribution of longitudinal temperature difference by the equation3,5 = ­ (4/3) / (C1/2S ), S = DuTT /(Duu1/2DT). The third spatial moment of difference DuTT(r) = <[u (r +r,t) ­ u (r ,t)] [T (r +r,t) ­ T (r ,t)]2 > can also be found from the time moment D uTT ( ) = < [u (r ,t+ )­u (r ,t)] [T (r ,t+ )­T (r ,t)]2 > and the frozen condition DuTT (v ) = D uTT ( ). Here T (r,t) = T(r,t) ­ , u (r,t) = u(r,t) ­ is the centered random temperature and longitudinal velocity. The process of Kolmogorov and Obukhov constants determination is shown in Figs. 13 and 14 for the case of atmospheric nonKolmogorov turbulence. The frequency spectrum, shown in Fig. 13, have a long inertial interval, where WT f ­8/3.

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Fig.13. Third moments of longitudinal differences of velocities and temperatures Duuu (m3/m3) and DuTT (deg2 m/s). The smoothed spectrum is shown in the fragment. Coherent turbulence, summer daytime measurements in mountains at a height of 680 m, July 2, 07

Fig.14. The Kolmogorov and Obukhov constants. Coherent turbulence, measurements in mountains at a height of 680 m, July 2, 07. Vertical lines show the boundaries of the inertial interval, dashed lines ­ average values of and

Analyzing more than 30 observation points, the following has been ascertained. If turbulence is close to Kolmogorov (flat underlying surface, time spectra in the inertial interval Wu(f) f ­5/3 etc.), then can be taken equal to 1.9 with an accuracy of 1­12 % and ­ to 3.0 with an accuracy of not worse than 30%. If the turbulence deviates from Kolmogorov and close to coherent (Wu(f) f ­ 8/3 etc.), then C is within the 0.9­3.7 range and is within the 1.3­5.1 range. In this case, = 1.9 with an accuracy of 19­93%, while the error of = 3.0 can reach 70%. As follows from experience, atmospheric turbulent parameters measurement errors are usually significant. The main reason of these errors is clear from our measurement data. Variations of the Kolmogorov and Obukhov constants within 100% (depending of observation point) result in similar errors in determination of the characteristics CT2, CV2, and Cn2. Another reason of errors is the structure function of temperature fluctuations DT(r), if C Kolmogorov­Obukhov law DT(r) = CT2r2/3. As is known5, the function DT(r) is representable as

2 T

is found using the

DT(r) = 8

d
0

Ô [1 ­ sin(Ôr)/ (Ôr) ] (Ô) Ô2.

Substitute spectrum (1), written in exponential form:
(Ô) = A0 C 2 (6.6 Ô0e) 2( ­ 1/ 3) Ô ­ 2( + 3/ 2) exp( ­ Ô2/Ôm2) [1 exp( ­ Ô2/ Ô0e 2], (3) Ô0e = 2 /L0e, = 5/6, 1/3.

Spectrum (3) differs from between the Karman outer similar to Karman (1). At integrals, obtain asymptotic

(1) only in its behavior in the energy interval, where Ô2/Ô02 << 1. There is a correlation scale L0 and exponential L0e (usually L0e = 0.54 L0), when spectrum (3) gives the results the same time, it significantly simplifies calculations. After calculating the corresponding representations for the structure function DT(r) in coherent turbulence ( = 5/6): DT(r) = 53.45CT2 L0 ­1 l0­1/3 r2, r << l0, f(r/ L0), f (x) = 49.87 (x ­ 1.34 x 4/3), l0 << r << L0, DT(r) = 3.56CT2 L0 2/3[1 ­ 0.63(r/ L0)­1/3], L0e << r.

DT(r) = CT r

2 2/3

The function DT(r) evidently deviates from the Kolmogorov­Obukhov "2/3" law. DT(r) ~ r2 at small r (r << l0) like in the Kolmogorov turbulence; however, the asymptotic coefficient becomes dependent on the outer scale L0. There is an extended initial region in the inertial range (l0 << r << L0), where DT(r) ~ r5/3. The regions ~r 2 and ~r 5/ 3 intersect at r = 0.8 l0 (remind here that the inner scale l0 of the coherent turbulence is one order of magnitude larger than those of the Kolmogorov one). The function f(x) has a maximum; therefore, a 2/3 Kolmogorov region, where DT(r) CT2fmax r2/3, is approximately observed in the inertial range of coherent turbulence 3.0 l0 r 0.4 L0, where f(x) const = fmax = 2.17. Hence, when measuring CT2 from the 2/3asymptotics of DT(r), the values are more than twotime overestimated.

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