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On the Theory of an Inverse Energy Cascade in Helical Turbulence of a Nonmagnetic Astrophysical Disk

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Abstract

In this article, the possibility of influence of the hydrodynamic helicity appearing in a rotating disk on synergetic structuring of cosmic substance and on the emergence of the negative turbulent viscosity effect in it is investigated. It is shown that prolonged turbulence damping in a disk can be partly associated with the absence of reflection symmetry of the anisotropic field of turbulent velocities relative to its equatorial plane. It is shown that negative viscosity in the rotating disk system is apparently a manifestation of cascade processes in helical turbulence, when the inverse energy transfer from small to larger vortices occurs.

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Notes

  1. As the averaging operation, we will henceforth use the statistical–analytical averaging over the ensemble of possible realizations of random thermo- and hydrodynamic fields [9].

  2. It should be recalled that the true 2D turbulence is not realized in actual fluid flows because the mechanism of the vertex field intensification due to extension of vortex tubes, which underlies the energy transfer to small scales (with simultaneous increase of vorticity), is basically three-dimensional by nature.

  3. It should be noted that a huge number of publications devoted to simulation of the evolution of gyrotropic MHD turbulence in astrophysical disks has appeared in recent decade.

  4. It should be noted that equation (1.3) for heat influx is written here for developed turbulence, when a quasi-stationary state is stabilized in the pulsation field structure, for which turbulent energy b is approximately conserved in time as well as in space [49].

  5. The spatial averaging of hydrodynamic equations for an instantaneous flow precluding the Reynolds postulate concerning the commutativity of averaging and differentiation operations gives an averaged motion equation with asymmetric Reynolds tensor Rij \( \equiv {{Q}_{{ij}}}({\mathbf{x}},0,t,0)\), where Qij(x, ξ, t, τ) = \(\left\langle {u_{i}^{{\text{'}}}({\mathbf{x}},t)u_{j}^{{\text{'}}}({\mathbf{x}} + \xi ,t + \tau )} \right\rangle \) is a second-order asymmetric correlation tensor [55]. It should also be noted that even in the original publication by Reynolds [56], where velocity fields were averaged over the volume and different average values were ascribed to the center of mass of this volume, components Rij and Rji of turbulent stresses were assumed to be different.

  6. Asymmetric hydromechanics of the Kosser brothers was widely recognized long ago, for example, in the theory of liquid crystals and in the liquid helium theory.

  7. This opinion contradicts the Moffat conception [20] that preserves the symmetry of the turbulent stress tensor.

  8. It is important to note that the dissipative helicity scale lh generally does not coincide with Kolmogorov scale lν, but ratio lν/lh \( \approx \) ν9/28 of these two scales tends to zero for large Reynolds numbers (low viscosity). This means that helicity does not reach the small-scale part of the spectrum [73].

  9. It should be recalled that when individual vortex filament Cj is wound around itself before forming a loop, it acquires a knot. Vortex helicity precisely determines the number of knotted and entangled vortex tubes in the volume occupied by the fluid: h = \(\sum\nolimits_{ij} {2{{\alpha }_{{ij}}}{{\Gamma }_{i}}{{\Gamma }_{j}}} \); here, αij are the looping coefficients of vortex filaments, which are positive and negative numbers associated with the number of turns of filament Ci around another filament Cj; Γj is the circulation of an individual vortex filament [18, 19]. Therefore, large-scale loops of vortex filaments appear in the turbulent flow under investigation during the generation of vortex helicity.

  10. It should be noted, however, that there exist other numerical calculations of a turbulent flow of an isotropic inhomogeneous fluid, in which such a correlation between local helicity and dissipation of small-scale kinetic energy is not observed [83, 84].

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    A. V. Kolesnichenko

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Kolesnichenko, A.V. On the Theory of an Inverse Energy Cascade in Helical Turbulence of a Nonmagnetic Astrophysical Disk. Sol Syst Res 57, 767–782 (2023). https://doi.org/10.1134/S0038094623070080

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