solar plasmas T.V. Zaqarashvili and M.L. Khodachenko Space Research - - PowerPoint PPT Presentation

Two-fluid MHD approach for partially ionized solar plasmas

T.V. Zaqarashvili and M.L. Khodachenko Space Research Institute of Austrian Academy of Sciences, Graz, Austria

Solar atmospheric model

FAL93-3 model (Fontenla et al. 1993)

Solar atmospheric model

FAL93-3 model (Fontenla et al. 1993)

ne – electron number density nH – neutral hydrogen number density nHe – neutral helium number density

Plasma is only weakly ionized in the photosphere, but becomes almost fully ionized in the transition region and corona.

Effects of neutral atoms in the solar atmosphere

Neutral atoms may change the plasma dynamics through collisions with charged particles: Damping of MHD waves (Braginkii 1965, De Pontieu et al. 2001, Khodachenko et al. 2004, Leak et al. 2006, Forteza et al. 2007, Soler et al. 2009, 2010, Carbonell et al. 2010, Singh and Krishan 2010, Zaqarashvili et al. 2011a,b) ; Formation of spicules (Haerendel 1992, De Pontieu & Haerendel 1998, James & Erdélyi 2002, James et al. 2004); Influence on energy flux of Alfvén waves in the photosphere (Vranjes et al. 2008); Generation of electric currents and plasma heating (Sen and White 1972, Khodachenko & Zaitsev 2002, Fontenla et al. 2008, Gogoberidze et al. 2009, Krasnoselskikh et al. 2010, Khomenko and Collados, 2012a,b); Emerging magnetic flux tubes (Leake & Arber 2005, Arber et al. 2007); Influence on resonant absorption (Soler et al. 2009). Influence on Kelvin-Helmholtz instability (Soler et al. 2012).

Three ree-Fluid Fluid equations ations

, , , , : ) ( 2 3 , : ) ( 2 3 , : ) ( 2 3 , ) ( , 1 ) ( , 1 ) ( , ) ( , ) ( , ) (

n n n i i i e e e n n n n n n n n n n i i i i i i i i i i e e e e e e e e e e n n n n n n n n i i i i i i i i i i e e e e e e e e e e n n n i i i e e e

kT n p kT n p kT n p Q q V V p T V t T k n Q q V V p T V t T k n Q q V V p T V t T k n R p V V t V n m R B V c E en p V V t V n m R B V c E en p V V t V n m V n t n V n t n V n t n                                                                                                                                                                                                   

(Braginski 1965)

a

R 

is the change of impulse,

a

q 

is the heat flux density,

a

Q

is the heat production. We consider partially ionized plasma, which consists of electrons (e), protons (i) and neutral hydrogen (n).

Maxwell xwell eq eq uations tions

, 1 t B c E        

where The description of the system is completed by Maxwell equations is the current density and

, 4 j c B       

 

i e e

V V en j       .    B 

Plasma is supposed to be quasi neutral

.

i e

n n 

Imp mpulse lse change nge and heat producti duction

• n

                        ,

, , , , ,

n i n ni n e n ne n i n i in i e i ie i e n e en e i e ei e i n ni e n ne i n i in e i ie i n e en i e ei e

V V V V V V Q V V V V V V Q V V V V V V Q V V V V R V V V V R V V V V R                                                                        

ba ab

  

are coefficients of friction between different sort of particles. Impulse change and heat production can be expressed as (Braginskii 1965) For time scales longer than ion-electron collision time, the ion-electron gas can be considered as a single fluid. Then, any additional sort of neutral atoms can be treated as a separate fluid.

Friction ction coefficients icients

 

, 3 2 4

2 / 3 4 e ie ie e i ie

kT m m n n e    

The coefficient of friction between ions and electrons can be expressed as (Braginskii 1965) For elastic collision, the ion-hydrogen collision cross-section is (Braginskii 1965)



where is the Coulomb logarithm . The coefficient of friction between ions and neutral hydrogen atoms is (Braginskii 1965)

, 8

in in in n i in

m kT m n n    

where is ion-hydrogen collision cross-section and is the reduced mass.

in



in

m

  ,

2

2 n i in

r r    

which approximately equals atomic cross section.

Colli llision ion frequ quencies ncies

 

. 3 2 4

2 / 3 4 e e i e e ie ie

kT m n e n m      

Ion-electron collision frequency is expressed by According to this formulation, ion-neutral collision frequency is different than neutral-ion collision

• frequency. But, the mean collision frequency between ions and neutral atoms should be a single

value due to physical basis. Ion-neutral collision frequency is often expressed as

,

i i in in

n m   

while neutral-ion collision frequency is often expressed as

.

n n in ni

n m   

From simple equations of motion of ions and neutrals one can derive the equation for relative velocity between ions and neutrals

   

. 1 1

n i n n i i in n i

V V n m n m t V V                    

This equation gives a single value for the ion-neutral collision frequency (Zaqarashvili et al. 2011)

. ) ( 2 1 1

i in n i n n i i in ni in

m kT n n n m n m                  

Collis ision ion frequen enci cies es in the solar atmosphe sphere re

Let us use FAL93-3 model (Fontenla et al. 1993) for temperature and number densities: z=0: z=900: z=1900: T=6520 K T=6000 K T=8900 K ne =7.67 10

13 cm

• 3

ne =2.60 10

11 cm

• 3

ne =1.27 10

11 cm

• 3

ni =5.99 10

13 cm

• 3

ni =2.43 10

11 cm

• 3

ni =1.20 10

11 cm

• 3

nn =1.18 10

17cm

• 3

nn =8.95 10

13cm

• 3

nn =1.79 10

11cm

• 3

=8 10

8 Hz 10 6 Hz

3.7 10

6 Hz

=8.6 10

6 Hz 6.2 10 3 Hz

24

Hz

ie



in



Ion-electron collision frequency seems to be always much higher than ion-neutral collision frequency in the whole atmosphere. However, this statement may be overestimated in the lower photosphere, where significant number of heavy ions exists.

Two-Fluid luid MHD HD equations uations

 



 



   

 

 

   

     

 

 

 

 

, 1 1 1 ) ( , 1 1 1 1 ) ( , ) ( , 1 ) ( , ) ( , ) (

2 2 2 n n n e en n n i in n n n n n e i e e e e e n e en i n i in e ei i ie ie i ie n i en in e en n n n n n n n i en in e en ie i i i i i n n n i i i

q V V V V V V V p p V t p q q en j p en p j V V V V V V j n e V p p V t p V V j en p V V t V n m V V j en B j c p V V t V n m V n t n V n t n                                                                                                                                                                 

(Zaqarashvili et al. 2011)

e i ie

p p p  

is the pressure of electron-ion gas.

Ind nduction uction equation ation in two-fluid fluid appr proach

• ach

 

. 1 1 1

2 2

B j cen V V n e p en B V c E

e n i e en ei e e i

                 

Ohm’s law is obtained from the electron equation after neglecting the electron inertia Faraday’s law and Ohm’s law lead to the induction equation where

 

2 2 2 2

4 4

e en ei

n e c c        

is the coefficient of magnetic diffusion.

      ,

                                              

e n i en e e e i

en V V c en B j B en p c B V t B          

Singl ngle-Fluid Fluid equations uations

          

 

   

     ,

, 1 2 1 1 1 ) ( ) ( ) ( , 1 ) ( ) ( ) ( ) ( , 1 ) ( , ) (

2 2 2 2

B w en w c en B j B en p c B V t B q q q en j p en p j w j en w j n e w p p w w p p w V p p V t p w j en B j c p p w w w w w V V w t w w w B j c p V V t V V t

n e en e e e n e i e e e e e en en in e en ei ie ie i i n i en in n i e en i n n i ie i n n i

                                                                                                                                                                                                                                   

    ,

j en B j c p p G w

en in e en en in n n n i ie en in

                                  

For time scales longer than ion-neutral collision time inertial terms can be neglected.

X X X X X

where

n i n n i i

V V V          

is the total velocity,

n i

V V w     

is the relative velocity.

Ind nduction uction equation ation in single ngle-flui fluid d appr proach

• ach

     

  

  

, 4 2 1 4

2

                                                                     B B B B G B B n e c B n G p c e c B V t B

en in n en in n e n T e e

                      

and the induction equation becomes

, 4

2 2 2 2

           

en in en en ei e T

n e c       

Then we obtain

    ,

j en B j c p p G w

en in e en en in n n n i ie en in

                                  

where

n i ie n

p p G        .

en in en

     

Next we consider MHD waves in the two-fluid approach.

Line inear ar Alfvén

n waves in two-fl fluid uid approac ach

   

. , , 4 z u B t b u u t u u u z b B t u

iy z y ny iy n in ny ny iy i in y i z iy

                    

We consider linear Alfvén waves in homogeneous medium with uniform magnetic field

.

2 2 2 2 2 3

   

n i in A z i A z n i in

V ik V k i             

Fourie analyses with gives the dispersion relation

 

t i z ikz   exp

The dispersion relation has three solutions: two Alfvén waves damped by ion-neutral collision and purely imaginary solution, which is the vortex mode connected with the vorticity of neutral fluid. The vortex mode has zero frequency in ideal fluid, but gains imaginary part when collisions between ions and neutrals are considered.

Alf lfvén n waves in in two-flu luid id approach

Zaqarashvili et al. 2011

Frequency and damping rate are normalized by

A zV

k

in A zV

k a  0 

is normalized Alfvén frequency. and

Line near ar magneto gneto-ac acousti tic c waves ves in two-fl fluid uid approa

• ach

ch

       

. , ~ , ~ , ~ , 4 4 ~ , ~ , ~

2 2 2 2

z u B t b V V z c t V V V x c t V V V z c t V V V z b B x b B x c t V z V x V t z V x V t

ix z x nz iz n in n n sn nz nx ix n in n n sn nx nz iz i in i i si iz nx ix i in x i z z i z i i si ix nz nx n n iz ix i i

                                                                                              

After Fourier analyses we obtain the dispersion relation for fast and slow MHD waves. Fast waves are similar to Alfvén waves, while slow waves have different properties.

Sl Slow waves s in in sin ingl gle-fl fluid uid approach ch

Forteza et al. 2007

Damping rate of slow waves is different when it is derived from the energy equation (Braginskii 1965) and through a normal mode analysis (Forteza et al. 2007). Slow magneto-acoustic waves show damping for purely parallel propagation in the case of Braginskii, while the damping is absent in Forteza et al. (2007).

Sl Slow waves s in in two-flu luid id approac ach

Zaqarashvili et al. 2011 Red asterisks corresponds to two-fluid solution, dashed line corresponds to Braginskii (1965). The damping rate is zero in Forteza et al. (2007).

• Two-fluid MHD approximation, where electron+ion gas and neutral atoms are the

two components, is important when time scales approach to ion-neutral collision time;

• Single-fluid approach is a good approximation for longer times scales;
• Damping rates of Alfvén and fast magneto-acoustic waves reach their maximum

near ion-neutral collision frequency, but decrease for higher frequency harmonics

• f the wave spectrum;
• No cut-off wave number of MHD waves appears in the two-fluid approach;
• The damping of slow magneto-acoustic waves is not zero in the parallel

propagation in two-fluid approach in coincidence with Braginskii (1965);

• The two-fluid approach could be important for instability processes, when unstable

harmonics have short spatial scales.

Conclusions

Thank you for attention!