1. Dissipation of Alfvén Waves in Coronal Structures Coronal Heating Problem T corona ~10 6 K M.F. De Franceschis, F. Malara, P. Veltri Dipartimento di Fisica Università della Calabria T photosphere ~6x10 3 K

2. 23-28 September 2003 Basic Processes in Turbulent Plasmas In the Solar Corona S>10 9 very low dissipation coefficients How to efficiently are waves dissipated before they leave the corona? l = characteristic velocity and magnetic field variation scale An efficient dissipation is possible if small scales are created In a 3D-structured magnetic field small scales can be efficiently creted by phase-mixing mechanism [Similon & Sudan,1986] Energy Dissipation Rate

3. 23-28 September 2003Basic Processes in Turbulent Plasmas The model ▪ Alfvénic perturbations propagating in a 3D magnetic field equilibrium structure ▪ In the Corona Cold Plasma B must be a force-free field ▪ We assumed (linear force-free field)

4. ▪ Planar geometry in which the curvature is neglected ▪ Statistical homogeneity in horizontal directions We assumed periodicity along x and y directions ▪ xy=base of the Corona z=vertical direction L=periodicity lenght

5. Equilibrium Magnetic Field is a superposition of several Fourier components The choice of these parameters determines a particular solution of the problem

6. determines both the current density ▪ and the maximum lenght In order to respect the statistical homogeneity so we used[Pommois et al.,1998] ▪ randomly chosen in the range [0,2π] ▪ The magnetic field is generated by a turbulent process. Assuming a spectral energy density We get

7. Wave evolution equations in a inhomogeneous plasma Alfvénic perturbations propagate in the above magnetic equilibrium. HYPOTESIS: (1)Cold plasma (2)Small wavelenght with respect to the typical lenght scale WKB approximation Alfvénic perturbations are decomposed as a superposition of localized wave packets

8. 23-28 September 2003Basic Processes in Turbulent Plasmas Red tones indicate the field lines flowing out the coronal base, while blue tones the flowing in Statistic homogeneity respected Magnetic field at the coronal base

9. 23-28 September 2003Basic Processes in Turbulent Plasmas This figure is obtained by planning 70 packet trajectories Each line connects a positive polarity zone with a negative one Some lines follow a brief journey, other ones follow longer and more complicated trajectories Magnetic field structure

10. 23-28 September 2003Basic Processes in Turbulent Plasmas “compact” flux tube The initial circle is mapped in a closed curve onto the coronal base “broken” flux tube The magnetic surface separates into various sheets At break points stretching of Alfvénic packets Magnetic Field Topology Flux tubes obtained by calculating the magnetic lines starting from a small circle at the coronal base

11. 23-28 September 2003Basic Processes in Turbulent Plasmas The wavevector k as a function of time t, for a given packet Almost exponential growth The energy e as a function of time t, for a given packet, at S=10 5 Dissipation within few Alfvén times Packet Time Evolution

12. 23-28 September 2003Basic Processes in Turbulent Plasmas The dissipation time t d as a function of the Reynolds number S, for a given packet The scaling law is asymptotically verified for large S Dissipation Time Scaling Law

13. 23-28 September 2003Basic Processes in Turbulent Plasmas Conclusions Coronal heating due to Alfvén waves dissipation Coronal heating due to Alfvén waves dissipation Linear force-free magnetic field in equilibrium configuration Linear force-free magnetic field in equilibrium configuration (statistic homogeneity hypotesis) (statistic homogeneity hypotesis) Evolution equations for an Alfvén waves packet in a inhomogeneous Evolution equations for an Alfvén waves packet in a inhomogeneous cold plasma: small scale generation cold plasma: small scale generation Magnetic field topology: sites of magnetic lines exponential separation Magnetic field topology: sites of magnetic lines exponential separation Wave vector increase and energy decrease Wave vector increase and energy decrease Scaling law of dissipation time recovered Scaling law of dissipation time recovered