Non-Resonant Quasilinear Theory

Non-Resonant Theory

Issue of Interest

• Intrinsic Turbulence• Stable plasma• Effects on physical processes

Basic Considerations

• Fourier transform

ˆ( , t) ( , , t)exp( )t d t i E r kE k k r,

ˆ( , , , ) ( , , , )exp( )s sf t t d f t t i r v k k v k r

ˆ ( , , , ) ˆ ( , , , )

n ( , )ˆ ( , ) 0k

ss

i ts s

s

f t ti f t t

te F t

t em

k vv k k v

vE k

v

Analysis

ˆ( , , , ) ( , , )exp( )s s kf t t f t i t k v k v

( )( ' )n ( , )ˆ ˆ( , , ) ' ( , ) k

ti t ts s

ss

e F tf t dt t e

m

k v vk v E k

v

Expansion in Slow Time

( )( ' )n ( )ˆ ˆ( , , ) ' 1 ( ' ) ( , )

ˆ ˆn ( , ) ( , ) ( )

( 0 ) ( 0 )

k

ti t ts s

ss

s s

s k k k

e Ff t dt e t t t

m t

e i t i t Fi

m i i t

k v vk v E k

v

E k E k v

k v k v v

Quasilinear Kinetic Equation

3 ( , ) ( , )( ) ( )

2s s s s

s

F e f fd

t m

k v k v

k E k E kv v

2 32

2 3 2

2

2

( )(2 )

1 1

2

s sk k

s

k s

k k

F e dE

t m k

E FP

k t

k k

k vv

kk

k v v

Alfven Waves

• In kinetic theory the dispersion equation is

22 2 2 23

2 2

( )

0psz n s

s n s s z z

k c n J b vd R

b n k v i

v

1s z z z s

z

F k v k v FR

v v

ss

k vb

For Thermal Protons• We have 0b

2

1 1 2

s z z s z z sk v k v

22

2

1( )

4ns

nJ b

b

22 2 2 2

2 2 2 2

pz s

s s p A

c k c

v

The general equation may be useful

• If the plasma beta is high and there is temperature anisotropy.

• In case when a tenuous energetic ions are present.

• In case when minor heavy ions exist in the system so that they can result in excitation of Alfven waves due to gyro resonance.

The Corresponding Quasilinear Theory

• The general equation is

2 2 23

2 2

22

1 ( )

1 1

2

s s n ss

ns s

ks z z k

s z z

F e n J bd R v RF

t m v b

En k v E P

n k v t

k

1s z z z s

z

F k v k v FR

v v

For Thermal Protons

• No cyclotron resonance• The equation reduces to

2 2

32 2

1

4s s k

ss p

F e Ed R v RF

t m v t

k

1s z z z s

z

F k v k v FR

v v

For a low beta plasma

• The kinetic equation is greatly simplified.

• Or we have

2 23

2 2

1

4s s k

ss p

F e Ed v F

t m t v v v

k

231 1

2 8s k

sp p

F Bd v F

t m n t v v v

k

Final Equation• Introducing two new variables

• New equation

• “Initial” condition

2

0 0

1

4 8 4k

p p

B Wdk

n T n T

1F Fu

u u u

1/2

02 / pu v T m

2 2

3/2

1( , , 0) zu u

zF u u e

Solution

• It is found

• Or

22

1 4

3/2

1( , )

(1 4 )

zu

u

F u e

2 2

00 22 /

3/2

0

1( , )

2 ( / )

p p z

p

m v m v

TT W n

p

F u eT W n

Issues and Controversy

1. Does the process really represent heating?

2. What is the physical mechanism responsible for the randomization?

3. Is dissipation necessary for heating?

4. How to distinguish MHD waves from Alfvenic turbulence?

5. Is pseudo-heating observable?

Issues 1• In classical MHD theory it is known t

hat a coherent MHD disturbance can induce a fluid velocity.

• The corresponding kinetic energy density is

4 p pm n

B

v

221

2 8p p

Bm n v

• The temperature increase in the present theory we have

• Comparing the two results one may feel that the energy increase is due to fluid motion. Hence there is no heating

2 2

8 8w k

p

B Bn T dk

Issue 2.

• Is randomization sufficient for heating?

Issue 3.

• Is dissipation really necessary?

The notion that heating is always associated with dissipation comes from neutral gas heating.

Of course, the concept is also supported by statistical mechanics. However, it is still not clear that in plasma where wave-particle interactions prevail heating must require dissipation.

Issue 4.

• Conceptually many scientists outside plasma physics do not have very clear notion that MHD waves and turbulent waves have very different physical nature.

• This confusion can often make scientific discussion and exchange of ideas very difficult.

Issue 5.

• To avoid unnecessary dispute and controversy a term “pseudoheating” is introduced recently.

• Of course, this does not mean that controversy would go away.

• Our opinion is :

1. In general coherent waves cannot lead to stochastic particle motion.

2. But turbulent waves can.

Students may study the following slides

Langmuir Waves

• Landau damping is proportional to the population of resonant electrons.

• Waves with low phase velocities are heavily damped.

• Only waves with high phase velocities can survive. Therefore

kthvk

Langmuir Waves

• For Langmuir waves the real part of the dielectric function is

2

21 p

RR

Langmuir Waves

• In the following let us consider Langmuir waves as an example. In this case

• And

2R

R k

23

2( ) ( )2

pe ek k k

Ft d v

k

k v kv

Langmuir Wave Energy Density

• The energy density of electric field

• The total wave energy density

• It implies that particle kinetic energy is

23

8k

E WE

W d k

23 2

8kR

k Wk

EW d k

p WW

For Langmuir Waves

• For resonant electrons

• For non-resonant electrons

22 3

216 ( ) ( )re k k

e

eD d k t

n k v

31( )nr

e ke e

D d k tn m t

Electron kinetic energy affected by Langmuir waves

• We calculate this based on kinetic equation.• We first discuss the non-resonant electrons

so that

e enr

F F

t

D

v v

2 3,

1

2e

e e e e e e nr kF

n m v F n m dvvD d kt v t

Non-resonant electrons

If we define

The implication is that the non-resonant electrons gain energy in the presence of the wave field.

3 2

2e e

en m

d vv Ft

3 s kFd k du u

u t

2

3 32pek k w

k

d k d kt t t

/u k k v

Resonant electrons

But

Therefore we can write

22 ( )k spe k k

Fdk du ku

k u

3 2

2e e

en m

d vv Ft

2

( )pe ek k k

Fdu ku

k u

3 2 4 22e e

e k k kn m

d vv F dk dkt t

Conclusions

In summary we find for Langmuir waves

3 2 22e e

eresonant

kn m

d vv F dkt t

3 2

2 non

e ee

resok

nant

n md vv F dk

t t

3 2

2e e

e ktotal

n md vv F dk

t t

General Discussion for Arbitrary Wave Mode

Let us consider

3 2

2s s

ss

n md vv F

t

3 k Wd kt t

3 2 3

2 22k s k

pss k

Fd k d v

tk

k

vk v

23 3

2 ( )ps sk k k

s

Fd k d v

k

k v kv