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

23-28 September 200323-28 September 2003 Basic Processes in Turbulent PlasmBasic Processes in Turbulent Plasmasas

Dissipation of Alfvén Waves Dissipation of Alfvén Waves in Coronal Structures in Coronal Structures

Coronal Heating ProblemCoronal Heating Problem

Tcorona~106 K

M.F. De Franceschis, F. Malara, P. Veltri

Dipartimento di Fisica

Università della Calabria

Tphotosphere~6x103K


In the Solar Corona S>109 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]

Std log

Energy Dissipation Rate2

1

ldt

dQ


The modelThe model

1magn

gas

P

P

▪Alfvénic perturbations propagating in a 3D magnetic field

equilibrium structure

▪In the Corona

Cold Plasma

B must be a force-free field

BB

▪We assumed constant (linear force-free field)


▪Planar geometry in which

the curvature is neglected

▪Statistical homogeneity in

horizontal directions

We assumed periodicity along

x and y directions

▪ zB when 0

xy=base of the

Corona

z=vertical

direction

L=periodicity

lenght


Equilibrium Magnetic FieldEquilibrium Magnetic Field

kkk

ykxkizkyxx

kkk

ykxkizkyx

xyy

kkk

ykxkizkyx

yxx

yx

yx

yx

yx

yx

yx

eekkbk

kizyxB

eekkbk

k

kk

kzyxB

eekkbk

k

kk

kzyxB

,

)(

22

,

)(

22

,

)(

22

22

22

22

),(),,(

),(),,(

),(),,(

B

is a superposition of several Fourier components

phases harmonicFourier ),(

amplitudes harmonicFourier ),(

parameter

yx

yx

kk

kka

The choice of these

parameters

determines a particular solution

of the problem


determines both the current density

▪ Bc

Bc

j

11

2

max l and the maximum lenght

In order to respect the statistical homogeneity

Ll max

so we used

L

25.3 [Pommois et

al.,1998]

▪ ),( yx kk randomly chosen in the range [0,2π]

▪ The magnetic field is generated by a turbulent process.

Assuming a spectral energy density3

5)(

kk

34

),(

kkka yx We get


Wave evolution equations in a Wave evolution equations in a inhomogeneous plasmainhomogeneous 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

Equation Evolution Energy

Equation Evolution Wavevector

Equation Evolution Trajectory

2)0(

)0(

)0(

ekdt

de

kx

C

dt

dk

Cdt

dx

j

Ai

Ai

j

i


• 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


• 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


• “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


• 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

Packet Time Evolution


• 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

Std log

Dissipation Time Scaling Law


ConclusionsConclusions

Coronal heating due to Alfvén waves dissipationCoronal heating due to Alfvén waves dissipation

Linear force-free magnetic field in equilibrium configurationLinear force-free magnetic field in equilibrium configuration (statistic homogeneity hypotesis)(statistic homogeneity hypotesis)

Evolution equations for an Alfvén waves packet in a Evolution equations for an Alfvén waves packet in a inhomogeneousinhomogeneous

cold plasma: small scale generationcold plasma: small scale generation

Magnetic field topology: sites of magnetic lines exponential Magnetic field topology: sites of magnetic lines exponential separationseparation

Wave vector increase and energy decreaseWave vector increase and energy decrease

Scaling law of dissipation time recoveredScaling law of dissipation time recoveredStd log

Documents

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