Mixed Modes: a more general case p n Including the compression - - PDF document

Waves in Partially Ionized Solar Atmosphere

• K. A. P. Sing

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Kwasan & Hida Observatories, Graduate School of Science, Kyoto University, Yamashina 607-8471, Japan

MHD Equations

Momentum eq. for electron, ion &neutral

     

V V V V B V E V V V

i e e ei n e e en e e e e e e e

c n e t n m                             

     

V V V V B V E V V V

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

c n e t n m                            

     

V V V V V V V

e n n ne i n n ni n n n n n

t n m                    

The electrons are inertialess (i.e. me =0). For δ << 1, the ion dynamics can be ignored. This gives Ohm’s Law in the electron’s case as    

V V γ V V γ B V E

i e e e ei n e e e en e

n e n e c         

The ion force balance equation now becomes

   

V V V V B V E

e i i ie n i i in i i

c n e                 

These equations ultimately lead to

 

c t

n n n n

B J V V V              

and an induction equation

   

                   B B B J B J B V B   

i in e n

c n e t

     

                             

i i i z in en in B n in en B n en B i ei in Ai A e H en ei pe

• i

Ai n

• i

Ai z z A H

V V k k i e cm cm T k n T k n T k Z n V n e B c c B V V k k i

2 2 15 2 15 5 1 5 1 2 24 2 2 2 5 2 2 2 2

k m m m m m 10 5 10 m 8 m 8 10 89 5 2004) al. et ko (Khodachen by given are s frequencie collision the and 4 4 where

_ _ _ _ . . .

, , , , , ~ , ~ , , , . , , ) ( ,                             

   n k n k kn in en

π π π π

The Ｄｉｓｐｅｒｓｉｏｎ Ｒｅｌａｔｉｏｎ

Mixed Modes: a more general case

Including the compression term and the oblique propagation, we show that the modes are mixed. The pressure-perturbation term is and

 

n n s

p C 

From the continuity equation, we have    ) (

1 1

V k

n n n

  Using the continuity equation, momentum equation for the neutrals and the induction equation for a partially ionized plasma, a determinant D (ω) is given by:

 

C k k k V C k k k C i k C V k i k i k i V D

s z H Ai s z s A s Ai A z A Ai 2 2 2 2 2 2 2 2 2 2 2 4 2 2 2 2 2 3 2 4 2 2 2 2

] ) ( ) ( ) ( ][ ) ( [ ) (                                 pn  

 n

n

Cs p

1 2 1 

the sound speed is

Physical parameters in the solar atmosphere

h T ρi ρn B βi (in 105 cm) (K) (g cm-3) (g cm-3) (G)

0 6520 1.0×10-10 1.90×10-7 1200 9.4×10-4 50 5790 1.2×10-11 1.59×10-7 1125.77 1.2×10-4 125 5270 1.18×10-12 1.00×10-7 980.16 1.3×10-5 175 5060 3.39×10-13 7.04×10-8 880.33 4.6×10-6 250 4880 9.37×10-14 3.89×10-8 737.21 1.7×10-6 400 4560 1.12×10-14 1.09×10-8 503.71 4.2×10-7 490 4410 4.37×10-15 4.84×10-9 394.42 2.6×10-7 560 4430 4.72×10-15 2.47×10-9 322.27 4.2×10-7 650 4750 2.29×10-14 1.00×10-9 246.31 3.7×10-6 755 5280 1.08×10-13 3.79×10-10 183.67 3.5×10-5 855 5650 1.75×10-13 1.66×10-10 143.40 1.0×10-4 980 5900 1.78×10-13 6.57×10-11 108.65 1.8×10-4 1065 6040 1.67×10-13 3.61×10-11 90.88 2.5×10-4

It can be seen that the magnetoacoustic and Alfvén-like modes are mixed.

Typical plasma parameters in a weakly ionized solar atmosphere

h δ γei γen γin η ηH ηA (in 105 cm) (s-1) (s-1) (s-1) (cm2 s-1) (cm2 s-1) (cm2 s-1) 

0 5.08 ×10-4 6.22×109 5.92×109 9.78×108 4.46×107 7.74×107 1.2×106 50 7.96×10-5 9.41×108 4.51×109 7.45×108 7.8×107 2.82×108 1.1×107 125 1.18×10-5 1.0×108 2.71×109 4.48×108 1.02×108 6.26×108 1.4×108 175 4.8×10-6 3.07×107 1.86×109 3.07×108 1.08×108 8.84×108 5.9×108 250 2.4×10-6 8.96×106 1.0×109 1.66 ×108 1.09×108 1.38×109 2.7×109 400 1.02×10-6 1.18×106 2.74×108 4.53×107 1.06×108 3.4×109 3.9×1010 490 9.03×10-7 4.87×105 1.19×108 1.97×107 1.02×108 5.94×109 1.4×1011 560 1.91×10-6 5.22×105 6.11×107 1×107 9.86×107 9.06×109 1.7×1011 650 2.27×10-5 2.28×106 2.58×107 4.26×106 8.83×107 1.36×1010 4.9×1010 755 2.86×10-4 9.22×106 1.02×107 1.68×106 5.67×107 9.42×109 1.5×1010 855 1.05×10-3 1.34×107 4.63×106 7.65×105 4.24×107 9×109 1.2×1010 980 2.7×10-3 1.28×107 1.87×106 3.09×105 3.64×107 4.72×109 1.7×1010 1065 4.61×10-3 1.16×107 1.04×106 1.72×105 3.41×107 4.31×109 2.3×1010

Using Cox et al. (2000), Khodachenko et al. (2004) and Table 1 B0 = 1200 G

Conclusions

The damping of Alfvén- like mode in a weakly ionized part of the solar atmosphere is mainly caused by the electron-neutral collisions and the ion-neutral collisions (through Cowling diffusivity). Cowling diffusivity is dominant beyond the height 175 km above the solar surface for the solar model given by Cox (2000) and chosen magnetic field. The Hall effect introduces a strong dispersion to the Alfvén- like mode in a partially ionized solar atmosphere. It has been shown clearly that the symmetry between co- and counter- propagating wave modes breaks at the length scale approaching the Hall length scale In the presence of Hall effect the Alfvén- like mode is circularly polarized whereas in the absence of it, the Alfvén- like mode is linearly

• polarized. The, Hall effect facilitates

propagation of short- wavelength modes required for the heating of the solar plasma.

Acknowledgements

KAPS gratefully acknowledges Prof. Vinod Krishan for enlightening on the topic. A detailed version of the poster has been accepted for publication in the journal NEW ASTRONOMY.

500 1000 1500 2000

• 7
• 6
• 5
• 4
• 3
• 2
• 1

500 1000 1500 2000

• 7
• 6
• 5
• 4
• 3
• 2
• 1

log  height

The ratio of ion to neutral density is shown (log axis) as a function of height (in units of 105 cm) in the solar atmosphere

• 6
• 4
• 2

2 3 6 9 12 15

• 6
• 4
• 2

2 3 6 9 12 15



RH+



RH-



Rt+



Rt-



RAM+ =  RAM-

(c) Dispersion at h = 600 km

  

Dispersion of wave modes at h = 600 km

• 6
• 4
• 2

2

• 12
• 9
• 6
• 3

3 6 9 12

• 6
• 4
• 2

2

• 12
• 9
• 6
• 3

3 6 9 12



IH-



IH+



IA M-



It+

(c) Damping at h = 600 km

  

Damping of wave modes at h = 600 km

The lines ωRH± are the dispersion curves for co- and counter- propagating Hall Alfvén waves

The lines ωRAM± are the the dispersion curves for Alfvén waves including , ambipolar diffusion The lines ωRt± are the the dispersion curves for Alfvén waves including both the Hall and ambipolar diffusion

Figures