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1. Kinetic equation for the spectral density of Alfvén waves in a shear flow: effective heating mechanism of accreting turbulent plasmaT. Mishonov, Y. Maneva, T. Hristov

2. Task: • To analyse a possible mechanism of the heating of a magnetized turbulent plasma. Probable applications in the solar corona and accretion discs Method: • To use the self-consistent Burgers approach to the turbulence, which allows an analytical solution in special cases and requires Monte Carlo simulations in general. In short: statistical methods applied to a turbulent MHD system.

4. Aims: • Derivation of kinetic equation for the spectral density of the Alfvén (slow magnetosound) waves. • Calculation of the dissipation, the energy density and the energy flux following from the solution of the master equation.

5. Navier-Stocks’s equation : • Ohm’s law: • Maxwell’s equations: • Evolution of the magnetic field:

6. Geometry used: • In a shear flow the radius vectors of tagged particles are time-dependent: • This leads to the dependence on time of the wave vectors in the momentum space:

7. After reduction the set of magneto-hydrodynamic (MHD) equations for the special case of turns into: where • The solution of the upper system provides the analysis of the evolution of the Alfvén waves in a shear flow

8. Langevin-Burgers approach – suggests a white noise correlator for the external random force in the Navier-Stocks’s equation: whereis the Burgers parameter. • In the spirit of quantum mechanics we introduce “alfvons”, whose number is:

9. Kinetics of “alfvons” – application of the Boltzmann equation with “attenuation’’ coefficient: Maximal effective energy Attenuation Amplification of the waves Birth of “alfvons’’ Heating of the fluid

10. Kinetic (master) equation for the spectral density of Alfvén waves Spontaneous generation of waves by the turbulence “lasing”in a shear flow

11. Derived formulas: • After statistical averaging and summation over the wave vectors the solution of the Boltzmann equation gives an assessment of the wave power, driven by the turbulence: • Energy density of the Alfvén waves: • Transmitted energy by the waves:

12. Results: • The MHD equations for a shear flow in the presence of an external random force, modeling the influence of the turbulence are considered; • A proper change of variables reduces the linearized MHD system of PDEs to a set of ODEs; • A WKB (short wave-length) approximation is used for the analysis of the slow magnetosound mode and the theory of perturbations accounts for a evanescent initial (“bare”) viscosity; • The Boltzmann equation for the spectral density of the Alfvén waves has been derived and solved

13. We have derived an explicit expression for the heating wave power in an isothropic approximation via summation over all wave vectors; • The energy flux of Alfvén waves driven by the turbulence is derived under the Burgers’s approach. This may be considered as a model for the heating of the solar corona, where the Alfvén waves are generated from the granulation in the chromosphere.

14. Conclusion and Perspectives: • The WKB approximation does not account for the significant amplification of the energy of the waves in a shear flow which is observed in numerical calculations. The incorporation of this amplification of the waves’ amplitudes in the statistical analysis can give an explanation of the giant increase of the dissipation and the effective shear tension.

15. A. Rogava et al.: Amplification of Alfvén waves in swirling astrophysical flows (2003) T. Hristov et al.: PhD thesis (1993)

18. This is a mathematical model of the Shakura-Sunyaev phenomenology in the framework of the Burgers approach to the turbulence applied to the MHD of shear flow fluids. The rest is silence

19. THANK YOU FOR THE ATTENTION!