Alfven wave heating of solar wind protons using a

generalized non maxwelliandistribution function

byH.A. Shah

Z.Kiran, M.N.S.Qursehi & G.MurtazaGC University Lahore

•Layout

• Non maxwelian distribution functions• Landau damping of oblique Alfven waves• Observed cooling rate• CGL cooling rate• Power requirements• Comparison between observation and

theoretical model

Non maxwelian distribution functions

• Data from real (space) shows distribution functions deviate from maxwellians

• Presence of high energy tails, shoulders in the profile, peaks or flat tops

• The default assumptions of using maxwellians seems no longer valid

• Generally space plasmas are turbulent thus thermodynamic equilibrium does not exist

• Quasi thermodynamic equilibrium state valid for turbulent systems

• Hasegawa et.al. showed how a non maxwellian distribution can emerge as a natural consequence of super thermal radiation fields in plasmas

• Entropy generalization using non extensive statistics

( ) ( )qr

rqr

vv

qrr

q

qqf

−+

⊥

⊥

⊥

+−

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛Ψ

+⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

Ψ−+

⎟⎟⎠

⎞⎜⎜⎝

⎛+

+Γ⎟⎟⎠

⎞⎜⎜⎝

⎛+

−ΓΨΨ

Γ−=

122

2

)1(23

0),( 111

)1(231

)1(234

13

π

( )( )( ) ( )

( ) ( )⎟⎟⎠⎞

⎜⎜⎝

⎛+

−Γ⎟⎟⎠

⎞⎜⎜⎝

⎛+

Γ

⎟⎟⎠

⎞⎜⎜⎝

⎛+

Γ⎟⎟⎠

⎞⎜⎜⎝

⎛+

−Γ−=Ψ

+−

⊥⊥

rq

r

rrqq

vr

t

125

125

123

12313 1

1

,,

2/5)1( >+rq

• Generalized Lorentzian or κ distribution

• Davydov Druyvestien

• Maxwellian•

0→r

∞→q

( ) ( )

⎟⎠⎞

⎜⎝⎛Γ⎟

⎠⎞

⎜⎝⎛ −ΓΘΘ

+Γ

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛Θ

+⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

Θ+=

⊥

−+−

⊥

⊥

25

214

13112

23

)1(22

0

κπ

κκκ

κ

κvv

f

))(exp( 12

2 +−∝ rv

vDD t

f

∞→→ qr ,0

Solar wind

• Stream of charged particles emanating from the sun

• 95% hydrogen plasma + alpha particles• Dominated by Alfven waves – fast streams• Turbulent• Solar magnetic field is dragged out by the Alfven

waves• In situ obsevations between 0.3 – 1 A.U.• Protons should cool as the sw expands

• Adiabatic eqn of state β−∝ rT 3/4=β

• Observations show that protons do not cool adiabatically for fast and slow streams

• Some local heating mechanism at work for fast speed streams

• Landau damping of oblique Alfven waves

3/43.0, ≤≤∝ − ααrT

Power dissipated due to Landau damping of Alfven waves

• We consider small angles of propagation

general expression for power dissipated per unit vol (Stix)

For Landau damping we need only

zBB ˆ00 =ρ

xz EE <<

EMEP jj j

pjjJ

∗∑ Ω=

2

161 ωεπ

zzM

( ) ( ) ( ) ( ) ( ) ( )∑

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

′ΩΨ

+′ΩΨ

+′′ΩΨ

−

′+ΩΨ

+⎟⎟⎠

⎞⎜⎜⎝

⎛ΩΨ

−

Ψ

Ω−=

⊥⊥⊥⊥⊥⊥

⊥⊥⊥⊥

j

j

j

j

j

j

j

qr

j

jqr

j

jqr

j

jjzz

CAkCAkCBAk

CAZkZkZ

ki

M

13

2

22

22

22

32

22

11,4

2

22

2,

2

22

1,2

222

222

ξξξ

ξξξξξξε

ΧΧ

( ) ( ) ( ) ( )( )

( )∫∞+

∞−

−+

+−

−

⎟⎟⎠

⎞⎜⎜⎝

⎛−

+

⎟⎟⎠

⎞⎜⎜⎝

⎛+

+Γ⎟⎟⎠

⎞⎜⎜⎝

⎛+

−Γ

Γ−= ds

s

sq

rrq

qqZ

qr

rqr

ξξ

12

)1(23

1, )1(

11

)1(231

)1(234

13

( ) ( ) ( ) ( ) ( )( )

( ) ( )( )( )

( )∫∞+

∞−

−−−−

+−

−

−−+

++

−+

++−−+

⎟⎟⎠

⎞⎜⎜⎝

⎛+

+Γ⎟⎟⎠

⎞⎜⎜⎝

⎛+

−Γ

Γ−=

dss

sqr

qr

qqFs

qrqqrq

rrq

qqZ

rqrq

qrqr

ξ

ξ

]1,1

1,1

1,1[12

11)1(

)1(231

)1(234

13

1212

)1(23

2,

jj kv

sΧΧΧ

Χ2,

Ψ=

Ψ=

ωξ

p

e

pIIx

z

mm

kk

EE

2

2

Ω≅ ⊥ ω

Wu & Huba 1975

( )

( ) ( ) ( ) ( ) ( ) ( )

2

32

13

3

2

22

22

22

32

22

1

1,

4

4

2

22

2,

2

22

1,

2

2

2

,

2

22

22

81

⎟⎟⎠

⎞⎜⎜⎝

⎛×

Ω×

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜

⎝

⎛

′ΨΩ

Ψ+

′ΨΩ

Ψ+′′

ΨΩΨ

−′Ψ

+

ΨΩΨ

+⎟⎟⎠

⎞⎜⎜⎝

⎛ΩΨ

−Ψ

Ψ−=

⊥

⊥⊥

⊥⊥⊥⊥

⊥⊥⊥⊥

∑

yp

eA

p

j

j

A

j

j

j

A

j

j

j

A

j

j

j

A

qr

j

A

j

jqr

j

jqr

j

A

j

pjjqr

Bmm

vc

kk

CAvk

CAvkCBA

vkCAv

ZvkZkZv

kiP

ξξξ

ωπ

yzp

e

pIIz B

ckmm

kkE ωω

2

2

Ω= ⊥

Expanding the plasma dispersion functions for large values of the argument we obtain

( )

( )( )

2

32

1

2

2

1

2

2

2

22

1

2

2

2

2

2

22

2

22

,

]1,1

1,1

1,1[12

2

111

21

81

⎟⎟⎠

⎞⎜⎜⎝

⎛×

Ω

⎠

⎞

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜

⎝

⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛Ψ

−−+

++

−+×

⎟⎟⎠

⎞⎜⎜⎝

⎛Ψ

′ΩΨ

−

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛Ψ−

+⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛ΨΩ

Ψ+

⎟⎟⎠

⎞⎜⎜⎝

⎛ΨΨ

=

⊥

−−

−−⊥⊥

−+⊥⊥

∑

yp

eA

p

j

r

j

A

qrq

j

A

j

j

qr

j

A

j

A

j

j

j

A

j

pjjqr

Bmm

vc

kk

vqr

qr

qqF

vBk

vq

vk

vkiP ω

pypy kLBLakB −⟩⟨= )()( 023

02 δρ

• Here an isotropic turbent spectrum is taken for the sw (Shah, Iess and Dobrowolny 1986). L is the outer scale of the turbulence and p is the spectral index of the power law and a is a constant

• The total power dissipated per unit volume is

dkkdSindPdkkdPR j

r

jjalf

p

2

1

00

2

0

2. θθφππ

∫∫∫∫ =Ω=

( )

( ) ( )

( )

( )⎥⎦

⎤⎢⎣

⎡ +=

⎥⎦

⎤⎢⎣

⎡ +==

⎥⎦

⎤⎢⎣

⎡ +Ω=Ω⎥

⎦

⎤⎢⎣

⎡ +⎥⎦

⎤⎢⎣

⎡=

=⎥⎦

⎤⎢⎣

⎡ +=

−−

−−

−−

−−−

−−

−−−

2

,2

,

,2

,2

1

,,2

42

0222

21

42

12

022

21

42

0

21

22

1

0

21

42

0

xxBBB

xxxv

vx

xxxxr

xTTxxBB

yy

AApjpj

ppp

jj

δδ

ωω

α

αΧΧ

( ) ( )

qr

j

Ap

p

p

y

p

p

p

p

e

p

A

j

AA

j A

pjjqr

alf

TA

vxq

Txx

BBLrL

pa

mmvv

cv

vAR

−+

−−⎟

⎠⎞

⎜⎝⎛ −

+−

−−

−

−

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

Ψ′⎟⎟⎠

⎞⎜⎜⎝

⎛ +−

+⎟⎟⎠

⎞⎜⎜⎝

⎛ +

⎟⎟⎠

⎞⎜⎜⎝

⎛

−⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛

Ω⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

Ψ⎟⎠⎞

⎜⎝⎛′= ∑

1

1

2

01

02

29

262

62

02

0

20

3

6

0

0

24

0

0

3

0

02

0

0

02

,

~22

11

11~2

1

68δωπ

0/ rrx =

KTcmn

cmLgm

gmBGB

ppacmc

II

e

py

p

53

100

28

24250

10

105.1,/5

102.3,101094.9

,10672.1,25.0,105

2/7,4

3sec,/103

×==

×=×=

×==×=

=−

=×=

−

−− δ

( ) ( )

qr

qqr

jqralf

TA

xq

AqTxx

AAR

−+

−−

+−+−−−−

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛

′⎟⎟⎠

⎞⎜⎜⎝

⎛ +−

+

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠⎞

⎜⎝⎛

′−+⎟⎟

⎠

⎞⎜⎜⎝

⎛ +⎟⎟⎠

⎞⎜⎜⎝

⎛′

′×=

1

12

12

2

1

12

411

254

1722

11

1

19,

~22

11

11

21

11~2

11108.1

0

)(~

II

II

TxT

T =

Data analysis( ) ( )( )[ ]⎥

⎦

⎤⎢⎣

⎡+

−+−=

+1222, sincos

111loglog r

qr bcvq

qaf θθ

⊥

⊥ ==⎟⎟⎠

⎞⎜⎜⎝

⎛=

TT

T

vAc

vvb

AA

a22

22

2

2/32

1 1,,logΧπ

Day a b c q r

21st Feb.2001 -17.4536 0.9 10-13 11.219 -0.7000

22nd Feb.2001 -15.3374 0.7 10-12 05.651 -0.6600

11th Feb.2002 -15.0607 0.8 10-12 05.967 -0.6665

Power requirements of the sw due to adiabatic cooling

• Sw appears to cool less rapidly than expected• Some additional source of heating• We first estimate the rate at which parallel

thermal energy varies on the basis of obs.• Secondly we obtain heating or cooling rate due

to a known theoretical law or eqn of state• The difference between these two will give the

power requirement of the sw protons• This will then be compared power dissipated

due to Landau damping of Alfven waves

Variation of parallel energy density according to obs

3/43.0,0 <<= − ααxTT obsobs

αα −−⎟⎟⎠

⎞⎜⎜⎝

⎛ +−== 3

000

. 2)( xr

TknvTnkdrdvR obs

Bswobs

Bswobs

Heating rate due to known eqn of state

• CGL theory gives a good approximation to a collisionless plasma (SW)

.2

. constnBT adb =⎟⎠⎞

⎜⎝⎛

( ) ( )⎟⎟⎠

⎞⎜⎜⎝

⎛

+−

+

−−= 2322

0

0.

0

14

14

xxxxrTknv

Robs

Bswadb

Power requirements of the solar wind

... obsadbreq RRR −=

sec,/105.4,33.1 7. cmvsw ×==α

1316. sec100.1 −−−×= ergcmRreq

5.9,7.0 =−= qr1316. sec104.4 −−−×= ergcmRalf

Radial evolution of power required and due to Landau damping of Alfven waves