# Coherent Structures in Open Air

## Coherent and Incoherent Turbulence

As is known from meteorology, there are rather stable vortex formations (cells) of different scales in the atmosphere. Ferrel and Hadley cells are the largest, with a radius up to 5000 km. They can be considered as a sort of Benard cells in a thin spherical layer (in Earth’s scales). There are also smaller cells (cyclones, anticyclones, storm cells, tornado, and others). The products of breakdown of these vortices having the pronounced deterministic character (corresponding to a coherent structure or non-Kolmogorov incipient turbulence) can be observed in open air.

As above, we compare the result of measurements carried out in the open air conditions and in the rooms’ areas. Thus, Figure 3.31 presents the data of measurements in open air and the result of simulation of the wind transport of the frozen spatial pattern of flows in different LSVT rooms of the data recorded at seven neighboring points located along a straight line at the same height [80,81]. This straight line, in

Figure 3.31 Kolmogorov turbulence as a result of mixing of different coherent structures. *W*_{T}—smoothed spectra, *D*_{T}—structure functions of temperature fluctuations. (1) summer daytime measurements in mountains (at a height of 2032 m), (2) wind transport of the frozen flow pattern in different LSVT rooms through the same point.

addition to points 5, 4, and 2 in the spectrograph pavilion, includes also four other points located in neighboring closed LSVT rooms. These rooms are isolated from the pavilion, and the frequencies of the main energy-carrying vortices in them and in the pavilion are different. Comparison of data presented in Figures 3.31, 3.24, and 3.23 shows that the results depicted in Figure 3.31 for open air are indicative of the predominant influence of one coherent structure *(W _{T} — f*

*in the inertial range). To the contrary, the transport of deterministic vortices formed in closed rooms through one point gives ultimately the turbulence close to Kolmogorov*

^{-8/3}*(W _{T} —* /~

^{5/3}).

Thus, we can conclude that the actual Kolmogorov atmospheric turbulence is a result of mixing of deterministic vortices from different coherent structures. Inequality (nonmultiplicity, incommensurability) of the frequencies of main vortices of different coherent structures leads to out-of-phase (incoherent) oscillation of vortex families, being products of vortex breakdown. That is why the t urbulence arising at the mixing of coherent structures with incommensurate frequencies of the main vortices is naturally referred to as incoherent.

As a result of research missions in the 2000s under mountain and valley conditions, we have accumulated a large experimental database of near-surface measurements of the main turbulence parameters in different geographic regions and different meteorological situations. It follows from these data that extended areas, in which one coherent structure has the decisive influence, are often observed in open air. Moreover, characteristic attributes of coherent structure are present, in some or other degree, in the most of accumulated data. The incoherent Kolmogorov developed turbulence is usually observed only at wide areas of smooth and homogeneous underlying surface.

Coherent turbulence (characterizing an area with the decisive influence of one coherent structure) differs from incoherent one, first of all, by the faster decrease of the smoothed *W _{T}* spectrum in the inertial range (—f

^{-8/3}) and by the smaller contribution of high-frequency components (small-scale vortices). Therefore, in the atmosphere in areas with the decisive influence of one (local) coherent structure, the spectrum in the inertial range has two pronounced parts of decrease: first, the rather fast decrease is observed (usually, —f

^{-8/3}, sometimes even faster) and then the decrease becomes slower (—

*f*

^{-5/3}) as the frequency increases. The second Kolmogorov section characterizes the mixture of breakdown products of other largest parent structures present in the atmosphere. In some cases, the entire inertial range of the spectrum has two pronounced parts with the 8/3-law decrease located by steps. Then, we can speak about the presence of two local coherent structures in the measurement area.

Thus, the data of our measurements show that the actual atmospheric turbulence can be considered as a result of mixture of different coherent structures with incommensurate frequencies of the main vortices. The coherent structure in this case is understood in its extended meaning, including the Feigenbaum breakdown scenario. Every coherent structure contains a long-lived parent structure arising under effect of local thermodynamic gradients and products of its coherent cascade breakdown.

The parent structures can take different forms (from a solitary ordered structure like a Benard cell to systems of periodically spatially distributed hydrodynamic perturbations like systems of various shafts). The sizes of parent structures (cells) in the atmosphere can differ 10^{8}—10^{9} times: from few centimeters (near-wall turbulence) to several thousands of kilometers (Ferrel and Hadley cells).

In the cases of high stability, parent structures not always break down. However, in the atmosphere due to the low viscosity of the medium (and, correspondingly, high values of the Rayleigh number), the lifetime of nondecomposing cells is not long. These cells are usually observed at the relatively short stage of their generation (at the stage of origination and formation). Sometimes large nondecomposing cells with the sizes larger than the outer scale of turbulence can be considered and studied as laminar flows. However, in any case, regardless of the size, nondecomposing cells should be considered as a part of the single process of turbulence formation and evolution.

In the area with the decisive influence of one large coherent structure, large- scale products of its breakdown are coherent (coherent turbulence). Small-scale products are mixed with products of breakdown of other small coherent structures present in this area, thus forming the incoherent mixture (incoherent Kolmogorov turbulence). If several mixing coherent structures have comparable sizes of parent structures (which is usually observed over regions with smooth underlying surface), then their mixture is incoherent (Kolmogorov) in the ranges of both small and large scales (up to scales comparable with the size of the parent structure).

The data of our measurements [80—82,86,87,91—93,95,98,102—106,128] and known data of other authors on the problem of turbulence formation (both experimental works on flow visualization [43,44,50—52,62,64,71,172,173,181] and theoretical works presenting analytical results [42,45,46,48,49,52—55,61,66,72— 75,78,79,161—163,168—171] and numerical solutions of the Navier—Stokes equations [47,48,56—58,63,65—70,182—184]) show that the coherent structure in its extended meaning can be considered as a structure element in hydrodynamic turbulence. This structure element is a vortex formation arising as a result of transition of the energy of thermodynamic perturbation into the energy of motion of a continuous medium. A vortex can have various shapes and sizes, and under conditions of instability, it breaks down coherently and in a cascading manner into smaller vortices. From the mathematical point of view, the structure element of turbulence is a soliton-like solution of flow-dynamics equations. It can be either a solitary one- soliton solution or one soliton in a many-soliton solution.

This view of coherent structures and, in general, the problem of turbulence formation became possible due to the indoor study of the processes of formation and breakdown of a solitary ordered structure (solitary Benard cell arising in the air medium of a closed room) with the use of small-size sensors and methods of spectral analysis.

For the further investigation of the local structure of actual turbulence, it is necessary to distinguish successfully the products of breakdown of different coherent structures. It can be possible, because the products of breakdown of one particular coherent structure have their own, inherent of only this structure, harmonic composition.