J. Plasma Fusion Res. SERIES, Vol.4 (2001) 564-569

Analytical Description of T. Sato's Mechanism ofTransformation of lon-Acoustic Double Layer into
Strong Bunemann's One in Gosmic and Laboratory
Nonequilibrium Plasmas
LAPSHIN Vladimir, MASLOV Vasyl* and STOMIN Vadim
NSC Kharkov Institute of Physics & Technology, Kharkov 61108, Ukraine(Received: 5 December 2000 / Accepted:27 September 2001)

Abstract
The model of a tranSformation of ion-acoustic double layer with small amplitude into strong double
layer is presented in [1]. In this paper this model and all stages of evolution are described analytically.
Earlier by numerical simulation in [2] and analytically in [3] it has been shown that an ion-acoustic
instability development leads to a formation of a nonmonotoneous ion-acoustic double layer with small
amplitude if an electron drift velocity is a little smaller than an electron thermal velocity. But this double
layer accelerates ions. Hence on first front of ion-acoustic double layer an ion density becomes smaller.
But into this region the electron flow penetrate with electron velocity only a little smaller than electron
thermal velocity. Hence on the first front of the potential hollow the electron drift velocity becomes morethan electron thermal velocity due to flow continuity law []. Due to Bunemann mechanism interactionof electron flow with this region an electric potential hump of large amplitude is excited with growthrate, proportionalto a*(m.lm)rt3.Here a!" is the electron plasma frequency; tn", midta electron and ion
masses. This strong electric potential hump can be unstable relative to radiation of electron reversal jump.

After that former hump becomes strong double layer.

Keywords:
wave structure, solitary wave, double layer, nonequilibrium plasma

1. Excitation of lon-Acoustic Solitary
Perturbation of Finite Amplitude in
Current-Carrying Plasmas
In present paper a plasma with electron current

modulation. This modulation instability results in turbulence splitting into widely spaced short perturbations.
The width of the latter is of order of the wavelength. On
nonlinear stage of evolution these short perturbations
become the solitary types. Therefore properties ofrelative to nonpropagating positive and negative ions is
considered with electron current velocity a little smaller
than electron thermal velocity, V6 < %r,". This plasma is
nonequilibrium. Ion-acoustic perturbations are excited.

electrostatic potential hollow of solitary kind areinvestigated in this paper (see Fig. la). The hollow
reflects electrons with energy smaller than the hollow

It is shown in [4] that homogeneous ion-acoustic

depth. This leads to hollow depth (amplitude of electric
potential, gto) growth.
The equation describing shape and time evolution

turbulence in one-dimensional nonequilibrium currentcarrying plasma is saturated on low level. Futher thision-acoustic turbulence is unstable relative to the@2001 by The Japan Society of Plasma
Science and Nuclear Fusion Research564

Lapshin Y ' et al" Analytical Description of r. Sato's

Mechanism of rransformation of Ion-Acoustic Doubleof the electric field structure of the hollow is derived inthis paper. It is obtained that the ion_acoustic hollowofelectrostatic potential is excited due to current_carrying

instability. The case of large amplitude of excited
perturbation is considered, when there are no traditional
small parameters, permitting to describe properties andexcitation of perturbation. It is shown that the hollow
propagates with velocity which is close to ion_acoustic

velocity of positive ions (T"lmi)t/2. Here Z" is thetemperature of plasma electrons; rus, is the mass of

positive ion.

We use hydrodynamic equations for densities n11
and velocities V;* of positive and negative ions, Vlasov

equation for electron distribution functionf, and poisson
equation for the electrostatic potential g.

Electrons propagate relative to ions with somecurrent velocity V6. Due to reflection of resonant
electrons, with nonsymmetric relative to hollow velocity

V6 distribution function, from potential hollow thequasineutrality brakes near the hollow: before theLayerrto*, N" = ns"/n6*, Qt = exle, Vs= (TJml*)tt2. We
normalize z on Debye radius of electrons 16", V6 ori

electron thermal velocity Vx,", time t on plasma
frequency of positive ions cro*-I, velocity of solitary
perturbation V" on ion-acoustic velocity (7"/mi*)u2

ofpositive ions. ne_, nsa nta unperturbed densities ofnegative and positive ions; 4*, nti+ €lta the charges andmasses of positive and negative ions.

Integrating the expression (1) on velocity, one canderive the electron density in first approximation on V6n" = ns"exp(Q)ll - (2LQ/G)13 ax exp(-xz)- 2Vo(2h)1/2l| a.e - Q)tt2exp(-f)
F=[e(q,+ q)/7"]1t2.

e)

Far from the hollow the plasma is quasineutral. It
means in approximation nu= 0 that far from the hollow
we have I/"(z)1.-- = N"(z)|.-* = l. From here one canhollow the electron density decreases and after the hollow the electron density increases. The quasineutrality isrealized due to formation of electrostatic potential jump
A,E near the hollow (see Fig. la).

At increasing of hollow amplitude up to criticalvalue, when inverse time of resonant electron (withvelocities lV - Vol 3 Vr,(rps)((2eEo/m")u2) interaction
with the hollow becomes larger than growth rate y(r6) =dlnqldt of hollow amplitude Vr,(ei > y(q,j,6z(cpi aslow evolution of the hollow starts in comparison withelectron dynamics. Here & is the width of the hollow, eis the electron charge. The resonant electron distributionfunction changes. The front with this changed
distribution function propagates from the hollow withrelative velocity equal Vo.
We use the approximation of a hollow slow
evolution for its description, using small parametern:

T@i6z(qilVr,(qi. In zero approximation on thisparameter, taking into account that the resonant

electrons are reflected from the hollow, one can derivefrom Vlasov equation the expression for electron
distribution function on velocity, V,

f. = fw[-(V2 - 2e (E + A,E)lm")u2 t Vil ,y'.A(E)sign(z),A(E) = I2e(W + g)lm"ltn.

Fig. 1 Transformation of ion-acoustic double layer into

strong Bunemann,s one in current_carrying(l)plasma: a) small-amplitude ion_acoustic solitary

perturbation; b) small_amplitude ion_acoustic

Here/s" is the initial unperturbed distribution function ofelectrons; z = 0 corresponds to g =-g10.
We use the normalized values: Q= etp/T", N_= no_lsolitary perturbation with nonhomogeneous tails;c) electron double layer separation from strong
hump; d) strong double layer.
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Transformationof lon-AcousticDoubleLayer
Lapshin Y.etaL,AnalyticalDescriptionof T.Sato'sMechanismof

derive, using (2), the expression for potential

jump near@ Q I a a2 D = (vthe hollow"tv,-)zf

N -((l + Q-zQv

2"-lv "2)tt2- l)+ (r - Q.2QV2*1v"211t2 - 11
LQ = Vo(2ln)1/2(1+ N" {exp(@) - |- exP(-@6))/

I - Q:G)!P dx exP(-*)l(3)

te*p(-00){00(Qo+ Q)3t2 - Qo3t\213+ exP(-@6)(l + Qo + Qo22B\

From hydrodynamic equations for ions one canobtain for perturbations of densities of positive and- exp(@)(1 - Q + f2l3) + (1 - exp(-@s))I - Q.G)!Y dx exlel)l-l

negative ions
fl.i+ = ll11ly *- 2sign(z)Vs(21 n)1t2

flXr ,

texp(-00)((00 + Q)3t2- Qo3t\213/4NL = notlll - (!q)2Elma,V"'f''' 'O nl/ 0 z = + 2 (Oq I Ot ) (nslq 1l m EV"3

+ fiexP(-@6)0 + Qo2l3)\-.Qexn(0) 0 - Q2t3)(4)
1t - (+q)Elmitvc2llll - (+qt)2ElmoV"'13''-!E ay.*pey\ltGll

Substituting (2), (4) in Poisson equation one canSubstituting (7) in approximation of small

derive nonlinear evolution equation
03 Q | 0 z3

+ {Q*2v2"*(r

amplitude into (6) one can derive- 2Q Q*v2 "*lv "27-ztz

Tnr(L - OQ*V2"*1V.2)

t2(q * | e)(n * | n

12

")t

2

Q

threshold one. The threshold decreases atdecreasingn"l+ QIN-V2"-(1 + 2QQ-v2"-lv.')-t''ln"lni* < q*l3e' The maximumQ.
ni-=threshold is realized at
So, we have shown that due to one-dimensional
rx11 orld equal zero atsign(z)Vs(2ln)tt2

t(fu(00 + @))1/2exP(-@s)

ion-acoustic instability development the electric- Iffi+ (1 - exp(-00)tl - Q.G)JY dx exp(*))a2Y2)exP(-Yz11ri2 + Q)1t2

potential hollow with the potential jump near the hollow
is excited.

texp(-0o)/(00 + Q)tt2 +2(Qo+ 4)exp(-00)

2. Description of Solitary Perturbation
Excitation by Bunemann Instability+ +JPo dy v0'+ Q)tt2exP(-Y\lt6tl

N"dQl}z = 0

DeveloPment
Now we describe the further evolution of theexcited ion-acoustic hollow with jump' The potential
jump on the first front of the ion-acoustic hollow is(s)From nonlinear equation (5), using condition (d@/dz)lo=-oo= 0, one can show that the hollow propagateswith thJ slow velocity V" = (T"lmp)tt2(n*ln")tt2(q*le)'From (5) one can get also the growth rate 7nl of thenonmonotonous due to excitation of hollow in time(seeFig. 1b). The potential jump is maximum one nearhollow and it decreases slowly up to zero far from thehollow. So, the shape of the electric potential on the firsthollow small amplitude. For that we select in (5) 0 = -fr
in approximation of small amplitude.
d Qotdt = (LQ tz\[ Q,irimo +-06 l a Q t a rl KQo + Q)1(8)

One can see that the hollow of large amplitude isformed at electron current velocity Vs larger than

Q

avO

o

- (n"ln*)(elq*)f(etpslr")(nl2)tt2(v,h"l2vi)

0 + 0 Q -v 2,-/v,2 ) | @ I a t )21V,3+ ((o / d z) | v ) { Q *' v'- 2 Q *V "* I V "', )t''"*(L"'- {exP(@) -= g*(V I V n)3{ 1 + t1/3

+ QIN-V2"-(l + 2QQ-V2*lv"zr-t''

QQ)

front of the hollow is a wide hump' This humpt2 (6)

accelerates ions from its region' Hence on the first frontof the ion-acoustic double layer an ion density becomes
smaller. For the neutral condition support the electrons'

For determination of the expression limE--qrldQl&V@o + Q)tt2 we use the equation (5) in quasistationary

penetrating in this region, should be accelerated' If into
this region the electron flow penetrate with electron

approximation

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Lapshin Y ' et al', Analytical Description of r. Sato's

Mechanism of rransformation of Ion-Acoustic Double

velocity only a little smaller than electron thermalvelocity, hence on the first front of the hollow theaccelerated electron drift velocity becomes more thanelectron thermal velocity due to flow continuity law[l].
Therefore, on the first front of the hollow the condition
for Bunemann instability is realized. Due to Bunemannmechanism interaction of electron flow with this region
an electric potential hump is excited. Let us describeasolitary perturbation in type of electric potential hump.

We will show that it represents a nonlinear perturbation
on a slow electron-sound mode. Such nonlinear
perturbation has been observed in laboratory
experiments and in numerical simulation[5]. As it isslow, resonant electrons can be trapped by suchperturbation. Thus, it is possible to expect that kinetics
of resonant electrons is of a large role in determinationof this perturbation properties. Hump of electric
potential provides hollow of electron density inaneighbourhood of maximum potential.

(11)

6z =

qld I q=_qon= r6"f2erps/7.(,,[, - D]il2 (I2)

From (12) one can conclude that the soliton width, dz,
grows with qo. Therefore, it is necessary to take into

account electrons, trapped by the soliton field.

Assuming distribution of their density as ntr\z)=n2expfeg(a)lTr,l, we derive similarly to (12't, that widthand velocity of the soliton grow with amplitude growth(in difference from case of small amplitudes of thesolitary perturbation).

Such properties of the soliton and theirdependencies on amplitude have been observed inexperiments and in numerical simulation[5].

the next order to a spatial derivative from electronfR(y)density+ Adlr - y2 + (3/2 _ /r)(R(y) _ 1)ln"1' = ()E/at)ly + (tR(y) = t + OtG)li dx exp(-p)/(t _ y) ,velocity and electric potential of soliton;l[r+QlQ-y)rlu2_t|
From (l 1) one can approximately derive

amplitude, as a result of Bunemann instability de_velopment. In the following order of the theory of
disturbances from (9) one can derive the correction ofsimilarly to [6]- 2yr)(t _ R(y))/yl (13)This expression, as follows from a spatial derivativefrom Poisson equation, should be equal to a spatial
derivative from ion density perturbation ni*,. ni*, ispossible to find in linear approximation from ion(e)Here prime means a spatial derivation; Vs,(f )2 = -O + QIG)I,,L atg - y)2exp(-p)

taking into account of ion mobility it is necessary to
expect occurrence of slow growth of the perturbation'sIntegrating the latter expression on velocities in case ofsmall amplitudes of the solitary perturbation g0 one canderive the expression for perturbation of electron densitv

y = Vs/Vr6"',1,

soliton shapeThus, this solitary perturbation is stationary andelectron one, if ion mobility is neglected. However at

From Vlasov equation the expression for per_turbation of electron distribution function follows.

$n' = (dg/dt)ty + (t - 2yr)(t _ R(yD/yl +

Layer

e are

Q = erp/T"; Vr6.

hydrodynamic equations

is the electron thermal velocity. Substituting (9) in

d2ni*lOf = (m./mi*)il'

Poisson equation, one can derive, in approximation no_=0, an equation, describing spatial distribution ofelectric
potential:(14)

Equating the second time derivative from (13) andfirst spatial derivative from (14), we obtain(f ), = fno) _ [1 + (2y2 _ 3)R(y)]f t6 (10)a3Qlaf = (6m"tmi*)f,,

From (10) and Q'16=6s = 0 one can obtain the(15)

The solution of (15) we search as

expression Vs = 1.32V$. [6).

Let us determine approximately the soliton width

Qk, t) = Qo?)qtz -from (10): 6z = cpslq'lr=prz = 16"(4g T"leE )u2. The

I- dt16vo(fue,;y1 ,(16)4(z) is quasistationary shape of the perturbation,

soliton width decreases with amplitude growth.

assuming, that dQoe)l}t = The).In (16) the change of

In case of large amplitudes, eeolTe > 1, from
Vlasov equation we have the expression for electron

soliton velocity, dv6, with change of its amplitude istaken into account.

distribution funcrion..if =.;fu[(u2 - 2etplm")r," +Vo sign(z)]Substituting aOlA ttuovgh le _ 6voQ , we obtainfrom (15)for faf = lV - Vol > (ZeE/m")tt2. Here /s" is Maxwell

distribution function. Thus we obtain an equation for the567

Lapshin Y. et al., Analytical Description of T. sato's Mechanism

y= 1m.lmi*)tt3Qott2(r7)

Thus electrons and ions interaction of Bunemanntype has resulted in excitation of the solitary hump ofelectrostatic Potential.

3. Potential Jump Separation from the
Hump and HumP Transformation into
Strong Electric Double LaYer
Let us consider, in approximation ns- = 0' thepossibility of the electric potential jump formation andits separation from the electric potential hump (see Fig'lc) and transformation due to this separation of theelectric potential hump into strong electric double layer(see Fig. 1d). We will show that it is possible if the
distribution function of electrons, trapped by the field of
the electric potential hump, has three-hump-type' So, letof Transformation of Ion-Acoustic Double Layer

quasineutrality condition in front of the jump requires V;< %r," (is the thermal velocity of plasma electrons) andthroughthe density of the high-energy electrons passing
plasma
the
is
n"'n"
11
nro
small
jump
be
shouldtheenergyequalsapproximately
15
electron density. Henceof high-energY electrons.
Further we use normalized values' Namely' wenormalize particle densities on unperturbed plasma
electron density, nge' electron velocities on electron
thermal velocity, electric potential on T"le, electron
temperatures on plasma electron temperature' fe 'In the electric field of jump the ion impulse grows'

Its flow equals n;*qs. All electrons transfer impulse to
jump. Flows of impulse, transfered to jump by plasma
electrons, by high- and low-energy electrons equal

I' is thenuo)' Here
2Vigott2,2n6gV,o2' n"7", 4Vo2(nois the unperturbed
n56electrons;
slow
of
the
temperature-density of the high-energy electrons' Plasma electrons

us assume that the distribution function of theseelectrons have three-hump-type. It includes low-energy
and high-energy electrons' Let us show that densities ofand ions obtain energy from jump' Energy flows'
obtained by electrons and ions, equal ZtpsVi,2EsVini''jump'

Electrons lose energy at interaction with potential
Energy flows, transferred to jump by low- and highenergy electrons, equal Vin"T", 4ViV,o2(na - nw), %Vrnw'
Using the momentum- and energy-balance equations
and also the quasineutrality conditions, we obtainlow-energy and high-energy electrons should be closed
for transformation of potential hump into strong double
layer.

We assume that the potential jump separates fromthe hump and moves with a velocity Vi' Thus, thepotential jump may be formed from the value 15 to zero

V/Vo= n6sl(2 + ni+) << I 'in a small space interval.

We investigate the properties of this jump in lDn"= (ni*l2)fl + T"1216- 2(2Tolrpllt2l ,approximation.
The jump formation needs a charge separation in
space. For selfconsistent charge separation in space twogroups of electrons are necessary. Let us consider highenergy electrons with velocity, V6, and thermal velocity,
V.6, satisfying Vt)) Va., Vn'6, and low-energy electrons
with velocity, V,1, and thermal velocity, V,6,, satisfying( V,r,. << V6, and with densities 116 aod n.' V6"= (Tsln,o= (ni*!4)ft -7"/2tps+2(2Tolcp)tt2l '(18)

From (18) one can see, that n" * 0, i'e' the
distribution function of electrons, trapped by potential
hump, has three-hump-type. From (18) it follows thatfor jump separation from the potential hump n5 should
be large. It is should be noted that during humpformation time the plasma electrons have no time for the%rresponse to the field initiation, and are trapped by them.)ttz is a thermal velocity of the plasma electrons' The
velocity of low-energy electrons V,1 can be chosen equal

hump field. In this way the slow electron group isformed, which is necessary for the jump formation
Similar processes for the potential distribution andparticle behaviour have been observed in a numericalto %.

The density of the low-energy electrons

exponentially falls off, resulting in positive charge in thelrurge 9a < q < q) of the jump electric potential, I' while
the densities of both the high-energy and plasma

simulation t7l. The beani injection into the plasma leads
in numerical simulations [7] in certain conditions to thehump formation, to jump separation from the hump andmoving it inside the plasma at a thermal velocity ofplasma electrons' After that former hump becomes

electrons grow in accordance with the power law' This
leads to negative charge at0 < <p < 9"' Here rpo is theamplitude of the jump electric potential' At E" the beam
starts to be reflected. As a results, the quasineutrality is
restored outside the jump. Since the nonresonant highenergy electrons propagate into the plasma, where theystrong double laYer (see Fig. 1d)'tll

are decelerated, their density grows' Therefore, the568

References
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Takamaru
H.
Sato,
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