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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; t