Dissipation of Alfvn Waves in Coronal Structures Coronal

Dissipation of Alfvén Waves in Coronal Structures Coronal Heating Problem Tcorona~106 K Tphotosphere~6 x 103 K M. F. De Franceschis, F. Malara, P. Veltri Dipartimento di Fisica Università della Calabria

In the Solar Corona S>109 very low dissipation coefficients How to efficiently are waves dissipated before they leave the corona? Energy Dissipation Rate l= characteristic velocity and magnetic field variation scale An efficient dissipation is possible if small scales are created In a 3 D-structured magnetic field small scales can be efficiently creted by phase-mixing mechanism [Similon & Sudan, 1986] 23 -28 September 2003 Basic Processes in Turbulent Plasmas

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

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

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

▪ 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

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 (3) WKB approximation (4)Alfvénic perturbations are decomposed as a superposition of localized (5)wave packets

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

Magnetic field structure • 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 23 -28 September 2003 Basic Processes in Turbulent Plasmas

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

Packet Time Evolution • 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=105 • Dissipation within few Alfvén times 23 -28 September 2003 Basic Processes in Turbulent Plasmas

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

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