# Properties of Coherent Structures

As we can see, coherent structures are important elements for understanding the processes of turbulence formation (appearance) and further evolution of the turbulence structure. Therefore, here we list briefly the revealed properties of single coherent structures and the properties of the mixtures (sums) of different coherent structures.

## Properties of Single-Coherent Structures

The properties of single coherent structures were discussed thoroughly in References 82, 86—90, 95, 117, 128. We studied characteristics of a single coherent structure experimentally (by small-size acoustic sensor) with the following theoretical analysis of experimental spectra of fluctuations of random temperature and velocity components.

- 1. As a result of action of thermodynamics gradients (temperature or pressure) at boundaries of some selected volume, a spatial vortex structure (cell, energycarrying vortex) arises in a fluid medium. There can be one cell or many cells. The cells are results of transformation of energy perturbations at the volume boundaries into the motion of fluid medium. In our extended definition, one such (usually long-lived) cell together with products of discrete coherent cascade breakdown of this cell is referred to as a coherent structure [82,86-90,95,117,128].
- 2. The breaking-down spatial structure, being the main energy-carrying vortex, can be called the parent cell (structure). The frequency of the breaking-down coherent main vortex (parent cell) is the main attribute of a coherent structure [82,86-90,95,117,128].
- 3. The size of the coherent structure is blurred. Currents, external with respect to the main vortex, can transport breakdown products to significant distances, thus forming a long turbulent wake [117,128].
- 4. The lifetime of the coherent structure is determined by the time of action of thermodynamic gradients (temperature and pressure gradients) [36,37,86-89,95,117,118,128].
- 5. As a limiting case of high stability, the coherent structure can consist of one long-lived parent structure. Then the parent structure is some configuration of laminar flow (undecomposing topological soliton) [95,128].
- 6. In the convective coherent structure arising in a closed room, all the main scenarios of turbulence formation from laminar flows [82,86-90,95,128] (Landau—Hopf, Ruelle-Takens, Pomeau-Menneville, and Feigenbaum stochastization scenarios) are confirmed. The Rayleigh-Benard scenario of breakdown of a convective coherent structure (Rayleigh-Benard convection [1,2]) is confirmed as well. The Feigenbaum scenario can be considered as the main scenario. Based on our results [82,86-90,95,112115,128,130,133,134,137,138], we can state that all these scenarios reflect different aspects of the process of turbulence formation.
- 7. The breakdown of the main energy-carrying vortex of a convective coherent structure follows the Feigenbaum scenario. The main vortex in a cell breaks down into smaller ones as a result of the series of period-doubling bifurcations (in the atmosphere, about 10 bifurcations). The resultant turbulence is coherent and deterministic [82,86—90,95,128].
- 8. The spectrum of passive admixture (temperature) is the breaking-down cell is fractal (locally self-similar) [82,86—90,95,128].
- 9. Turbulence arising in a coherent structure, as was shown in References 82, 86—90, 95, 128, satisfies all the attributes characterizing the appearance of chaos in typical dynamic systems. These attributes usually include formation of irregular long-lived spatial structures, whose type (character) is determined by dissipative factors, local instability and fractality of the phase space of such structures, appearance of the central (at zero frequency) peak in the spectrum. As was found, the central peak arises in this spectrum due to nonstationarity of random processes in the coherent structure [86—89,95,128].
- 10. The known processes of transformation of laminar flows into turbulent ones (Rayleigh—Benard convection, flowing-around of obstacles by fluid, and others) can be considered as processes of formation of either single coherent structures or a sum of different coherent structures [82,86—90,95,113,117,120,128].
- 11. A coherent structure contains both large-scale and small-scale turbulence. The 1D spectrum of turbulence (velocity components and temperature) is characterized by the faster decrease in the inertial range (usually, the 8/3-law decrease, which in the high-frequency part of the inertial range transforms into even faster 12/3-law decrease) in comparison with the Kolmogorov 5/3- law decrease [82,86-90,95,128].
- 12. The outer scale of turbulence in a single coherent structure can be considered as a product of the first breakdown event of the coherently breaking-down main vortex [82,86-90,95,128].