Enhanced ion-damping in an ion-acoustic instability

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Enhanced ion-damping in an ion-acoustic instability

Tu Khiet

To cite this version:

Tu Khiet. Enhanced ion-damping in an ion-acoustic instability. Journal de Physique Lettres, Edp

sciences, 1979, 40 (18), pp.469-471. �10.1051/jphyslet:019790040018046900�. �jpa-00231667�

L-469

Enhanced

_ion-damping

in

an

ion-acoustic

_instability

Tu Khiet

Laboratoire de _Physique des gaz et des plasmas (*), Université Paris-Sud, 91405 Orsay, France

(Reçu le 19 avril 1979, revise le 20 juillet 1979, accepté le 24 juillet 1979)

Résumé. 2014 On établit _une

équation

quasi linéaire renormalisée pour la fonction de distribution des ions et on

calcule le taux d’amortissement dû aux ions

_énergiques.

Le _spectre

_d’énergie

se _comporte comme k-2 sans _coupure

artificielle.

Abstract. 2014 We derive

a renormalized

_{quasilinear equation}

for the ion distribution function and we calculate

the modified

_damping

rate from

_energetic

_{ions. The energy spectrum} behaves as k-2 without artificial cut-off.

LE JOURNAL DE PHYSIQUE - LETTRES _TOME

40, 15 SEPTEMBRE _1979,

Classification

Physics Abstgacts 52.35

Ion acoustic turbulence is of _great interest to _many

problems

in

_laboratory

and _space

_plasma

_physics.

In

turbulent

_{heating experiments}

it acts to slow down

the

_heating

current, in the solar wind and

laser-produced plasma,

it

_{struggles against}

the

_instability

by

reducing

the heat flux. As a _consequence the

plasma

resistivity

is

_anomalously

enhanced in the first case

and the

_conductivity

diminished in the second one.

In both cases the

_transport

coefficients are shown to

be

_directly

connected to the turbulent _{energy spectrum} which remains ill-known _up to date. In this short

communication,

we examine

_explicitly

the saturation

mechanism of an ion-acoustic

_instability

driven

_by

an

electron

_{drift-current;}

the results can be

_easily

extended to an

_{instability-dominated}

laser

_plasma.

This

_study

is aimed to

_{provide insight}

on the role of

the ion in the saturation. In this

_optics,

induced

scattering

[1]

and modification of ion orbit

_[2]

have been

_previously

studied but

_they

both lead to a too

high

saturated _energy in a _system

_having

a

_large

electron-to-ion

_temperature

ratio. We shall examine the

_production

mechanism of

_energetic

ions which

are

_speculated

to

_quench

the

_instability.

For that purpose we shall derive an

equation

for the

ion-background

function

_using

a Fourier transformation in

_velocity

_space in

_conjunction

with a

_technique

of

expansion

in cumulants in the same manner as for

the calculation of a renormalized _propagator in

strong turbulence

_theory.

Our

_equation

is shown to

be a renormalized

quasilinear

one which _presents an

advantage

that it does not

_imply

a further resolution

(*) Laboratoire associe au C.N.R.S.

of a diffusion

_equation.

_Finally

we shall calculate

the modified ion

_damping

which

_{subsequently gives}

the saturated state

_by

_requiring

an exact balance with electron

_growth

rate.

We consider a

_{homogeneous plasma}

with

_{7~ ~> 7~}

where

_Te

_(7"J

is the electron

_(ion)

_temperature.

The electron distribution function is chosen to be a shifted

Maxwellian with a drift

velocity

_vd

largely exceeding

in

_magnitude

the ion-acoustic

_speed

_Cs =

(7~/M)~,

M

_being

the ion mass. In these

conditions,

ion-acoustic waves are

_generated

with

_frequency

Àe being

the electron

_{Debye length,}

and

_growth

rate

y =

ye +

y~ where

where we have

_only

assumed that

_Fi

is

_isotropic

in

velocity

space. It is also

interesting

to note that the ion

_damping

rate is

_essentially

determined

_by

ion

population

in the _range of

_velocity

between

_Cg/~/2

and cs. As time progresses, the waves can drive the

particles

to

_{higher velocity}

and

_{subsequently modify}

the ion distribution in the domain of interest. This effect can be taken into account

_by

_using

the

conser-vation on the number of

_{particles along}

a

_particle

orbit,

i.e.,

L-470 JOURNAL DE _{PHYSIQUE -} LETTRES

where

_{:F(r, v)}

=

f (r,

_{v, t} =

0), r(t)

and

v(t)

_are the

position

and

_velocity

of a

particle

at t =

_0,

knowing

that

_they

take the values r and _v,

_{respectively,}

at

time t. From the fact that ,~ is a

superposition

of a

homogeneous

part,

Fo(v),

and an

_{inhomogeneous}

part,

bf,

with

_{bf Fo,}

the time

_dependent

_average

distribution

_{function, F(v, t),}

which results from the

ensemble _average of

₍₂₎

can be

_put

in the form

where we have used a Fourier

analysis,

H( p) being

the Fourier transform

_{of Fo(v).}

Calculation of

₍₃₎

can

now be carried out

_by

_using

an

_expansion

into

cumu-lants. The relevant

_quantities

in our

_problem

are

z Av >

and

Av

_{ev ~}

where Av =

v(t) -

v. This can

be done

_{by assuming}

a W.K.B. variation of the wave

field

during

the relaxation process in the form

giving

in the lowest

_{approximation}

It is worth

_noticing

that the usual

_quasilinear

equa-tion can be recovered

_by

_linearising

the

_exponential

term in

₍₃₎

and

_identifying

_p with - i

_olav.

We further

assume an

_isotropic

_spectrum,

_i.e.,

_{I 9k}

12

=

I(k)

and

integrate

(4)

over

_angles,

this

_gives

where

_subscripts

refer to the direction of v and the various

_{quantities appearing}

in the RHS of

₍₅₎

take different values

_accordingly

to the value of v. The two

limiting

cases are :

1. v ~ Cs’ Jn this domain calculations are made

_by

neglecting

v in

_(4),

this

_gives

where x = e2

e

is _the characteristic

energy ratio. 2. v >

Cs/~/2~

In this

domain,

calculations are

greatly simplified by replacing

the kernel of

₍₄₎

_by

a delta function we then obtain

where the notations

have been used and the lower limit of

_integration

which

_originates

from the resonance condition has to

be taken as K = 0 for v >

Cs and

The

_background

function obtained

_by

_integrating

(3)

over _p now takes the form

where

Refering

to the

_expressions

_{of all}

_{and al}

_given

previously

we conclude that the ion distribution is heated to an effective

_temperature

in the

_body

_part

and it contains now a finite

population

with an effective

_temperature

_Th

=

Te(rJ.Jr¡)1/2 in

the

range of

velocity

I v ~

Cg. The

population

of ions

at v =

c~~/~

which determines the

damping

_rate

according

to

₍₁₎

is

_{obviously given}

_by

₍₈₎

and

₍₇₎

in which- K = k.

Balancing

the so-obtained

damping

rate with the linear electron

_growth

rate, we obtain

at saturation the

_equation

where #

=

r¡/Log (C,,/i7Vd)’

It is

_easily

seen from the above formula that the wave

_spectrum,

_{i.e., W(k),}

behaves as

k-2

in the limit

of _very small

_{k ;}

for

_larger

values of

k,

this function

strongly damps

to zero. In addition the reduced

tur-bulent _energy deduced from the above formula is

As a conclusion we have

developed

a

_simple

method which enables us to

_explain

the formation of

experimen-L-471 ENHANCED ION-DAMPING IN AN ION-ACOUSTIC INSTABILITY

tally [3].

Under the

_hypothesis

that these

_particles

are

responsible

of the mechanism of

saturation,

the wave

spectrum

does not necessitate to be cut-off at some

value of k and the total turbulent energy

depends

little on the temperature ratio.

It is also

_tempting

to _compare our method with

those

_developed

_previously

in the field of renormalized

theory.

1. In the

_Dupree

_{[4]-Weinstock [5]}

formalism the

action of the

_operator

U _upon the

_velocity

_space was

considered as

weak,

the

fluctuating

distributed

func-tion

_given

_by

this

theory

in the lowest

_{approximation}

as

was calculated

_by

_expanding

the electric field in terms

of its Fourier

_components

and

_considering

the

back-ground

function as a constant. This

_imperfection

has

been remedied on the one hand

_by

_Misguich

and Balescu

_[6]

_using

the _{operator formalism} and

_by

Horton and Choi

_[7]

on the other

_hand,

_using

the

formalism of Green’s function. As a

_result,

the

_particle

propagator

in the

k-space,

i.e.,

Gk(v,

v’)

defined as

has been found to be non local in the

_velocity

_space.

This feature _may be

_easily

recovered

_by

our method

if we use a double Fourier

_expansion

_(in

real and

velocity spaces)

of the

_integrand

of

₍₁₂₎

before

ope-rating

U and

_using

the cumulant

_expansion.

2. As for the

_{dynamical description}

of the

back-ground

function which leads to _{the appearance of} a

tail

_population,

our calculations differ with those of

Horton and Choi

_mainly

in the

_following

_{points :}

(i)

We have considered the

_production

of

_energetic

particles

as

occuring during

the relaxation

phase

itself

but not in a

_steady

state.

_(ii)

The

_background

func-tion

_appeared

in our work

quite naturally,

unlike in

that of Horton and Choi who have been led to

_require

that the

_particle

_propagator

be a function of the

difference of its _arguments in order to solve the

quasi-linear

_equation.

In our

opinion

this

_subsidiary

assump-tion is rather difficult to

_justify.

The author wishes to _{express many} thanks to

Dr. J. H.

_Misguich

for very useful

suggestions.

References

[1] KADOMTSEV, B. B., Plasma turbulence (Academic Press,

New York) 1965, p. 68.

[2] SLEEPER, A. M., WEINSTOCK, J. and _{BEZZERIDES, B., Phys.} Fluids 16 (1973) 1508.

[3] PRONO, D. S. and WHARTON, C. _B., Plasma _Phys. 15 (1973)

253.

[4] DUPREE, T. _{H., Phys.} Fluids 9 (1966) 1773.

[5] WEINSTOCK, J., Phys. Fluids 12 (1969) 1045.

[6] MISGUICH, J. H. and _{BALESCU, R.,} J. Plasma Phys. 13 ₍₁₉₇₅₎ 385.

[7] HORTON, W. Jr. and CHOI, D. I., Phys. Reports 49 (North