Centrifugally driven electrostatic instability in extragalactic jets

0.24 MB·English

by  Z. Osmanov

APS/123-QED
Centrifugally driven electrostatic instability in extragalactic jets
Osmanov Z.
Georgian National Astrophysical Observatory
Kazbegi ave. 2a, Tbilisi, 0160, Georgia
(Dated: February 1, 2008)
Thestabilityproblemoftherotationinducedelectrostaticwaveinextragalacticjetsispresented.
SolvingasetofequationsdescribingdynamicsofarelativisticplasmaflowofAGNjets,anexpression
oftheinstabilityratehasbeenderivedandanalyzedfortypicalvaluesofAGNs. Thegrowthratewas
studied versus the wave length and the inclination angle and it has been found that the instability
process is much efficient with respect to the accretion disk evolution, indicating high efficiency of
theinstability.
8 PACSnumbers: 98.62.Nx,98.54.Cm,94.20.wf,95.30.Qd,95.30.Sf
0
0
2 I. INTRODUCTION ing the problem of creation of the outflows, pointed out

n thatinthis processthecentrifugalaccelerationmayplay
a an important role depending on the inclination angle of
Consideringtheproblemoftheescapingradiationfrom
J
magneticfieldlines. Aseriesofworksdedicatedtoeffects
active galactic nuclei (AGN), one has to note that we
9 of rotation, examine its role in the nonthermal emission.
haveastrongobservationalevidenceofacomplexpicture
2
Oneoftheimportantexamplesfromthispointofviewis
ofemissionspectra,whichstartsfromtheradio,theopti-

the workofBlandford&Znajek(seeRef. 8),wherethey
] caluptoX-rayandγ-ray,withthebolometricluminosity
h power of AGNs lying in the range: [1040 1047]erg/s show that the rotational energy of the black hole can be
p ∼ − extracted electromagnetically as a Poynting flux. In the
(see Ref. 1). Originofemissionis supposedto be energy
- context of AGN, Gangadhara & Lesch (see Ref. 9) con-
o ofanaccretedmatterandaspinningblackhole,however,
r the details of the conversion process of this energy into sidered the problem for understanding how efficient the
t rotation is for producing X - ray and γ - ray photons.
s electromagnetic radiation is still unknown. The discov-
a It was shownthat nearby the light cylinder surface, cen-
eryofblazars(seeRef. 2)assourcesofultra-highenergy
[ trifugally accelerated particles, due to scattering against
radiation, has revealed that the formation and accelera-
5 tionofrelativisticjetsarekeyprocessesinunderstanding low energetic photons, may provide the mentioned ra-
v the energy conversion mechanism. diation. Reconsidering the same problem for explaining
2 the TeV energy emission from a certain class of blazars,
AninnermostregionofAGNsischaracterizedbyrota-
9 in Ref. 10 it has been argued that the CA might be so
tional motion, and it is obvious that the rotation affects
3
strong, that could provide ultra-relativistic electrons.
0 acceleration of plasmas in jets, which consequently may
. influence a process of radiation. The problem of a role All mentioned cases exhibit the possibility of energy
6
of centrifugal acceleration (CA) as the rotational effect, pumpingfromrotationandincasesofthecentrifugalac-
0
7 on a theoretical level has been studied in Ref. 3 for the celeration they show its high efficiency in the energetics
0 Schwarzschild black hole and it has been found that un- of relativistic electrons in rotating plasmas accelerating
: dercertainconditionsthecentrifugalforcemaychangeits uptoenergiescorrespondingtotheLorentzfactorsofthe
v directionfromcentrifugaltocentripetal. Thesameeffect orderof105−8 (seeRef. 10). Thus,theamountofenergy
i
X was shown in Ref. 4 where authors studied dynamics of contained within a centrifugally accelerated plasma flow
r particles, sliding along rotating straight channels. It has is very big and if there are mechanisms for the conver-
a beenarguedthatduetothecentrifugalforce,theprocess sionofatleastasmallfractionofthisenergyintothethe
ofaccelerationmightbeveryefficientforrotationenergy variety of instabilities - one might find a number of in-
extraction. Ontheotherhanditisobviousthatnophys- teresting consequences related to the problem of plasma
ical system can preserve rigid rotation nearby the light energy conversioninto radiation.
cylindersurface(LCS)(anareawherethe linearvelocity Generally speaking, for the energy pumping process,
of rotation is equal to the speed of light) where mag- one has three fundamental stages: (a) energy conversion
netic field lines must deviate from the straight configu- from rotation into kinetic energy of the plasma back-
ration. GeneralizationofRef. 4for curvedfieldlines has ground flow, (b) kinetic energy conversion into energy
been presented in Ref. 5 and it was found that for the of electrostatic waves and (c) conversion of the energy
Archimedes spiral a centrifugal outflow may reach infin- of waves into radiation by means of the process of the
ity,avoidingthelightcylinderproblem. Accordingtothe inverse Compton scattering (ICS) of electrostatic waves
spin paradigm (stating that, to first order, it is the nor- to photons l + e = l′ + t, which is thought to be re-
malized black hole angular momentum, that determines sponsible for creation of nonthermal radiation (see Ref.
whether or not a strong radio jet is produced) (see Ref. 11). By Tautz & Lerche (see Ref. 12) the electro-
6) rotation is supposed to be fundamental in formation static/electromagnetic mixed mode has been considered
of outflows. Blandford & Payne (see Ref. 7) consider- forrelativisticjets inthe contextofGamma-RayBursts,
2
to determine low-frequency instabilities. But our aim is tion of motion of a particle is given by:
to consider only electrostatic waves and investigate the
dp e
second stage of the process [see (b)] and show how effi- =γg+ (E+v B). (2)
cient it is. Therefore the electrostatic wave, playing an dτ m ×
important role in the process of radiation might be very
whereγ =(1 v2)−1/2istheLorentz-factor,v=dr/dτ is
effective if one makes it unstable and as we will see the −
thevelocityoftheparticledeterminedinthe1+1formal-
centrifugal force may be very sufficient in this context. ism,p p/m-thedimensionlessmomentum,g ∇ξ,
The centrifugally driven parametric instability has → ≡− ξ
γg - the effective centrifugal force (see Ref. 13) and
been introduced in Ref. 13 where it was shown that
the centrifugalforce acting on particles inside the pulsar τ ξt (ξ 1 Ω2 r2) - the universal time. Tak-
≡ ≡ − ef
magnetospheres,causes separationof charges,leading to ingintoaccounqtanidentityd/dτ ∂/(ξ∂t)+(v )and
generationofanelectrostaticfield,whichexcitesthecor- afactthatLorentzfactorsinthec≡o-rotatingfram·e∇ofref-
responding instability. The increment of the linear stage erenceandintheinertial-laboratoryframe(LF)relateto
hasbeenestimatedandanalyzedfortheCrabpulsarand each other: γ = ξγ′ (prime denotes a physical quantity
it turned out that the linear regime was extremely effi- intheLF),onecanrewritetheequationofmotioninthe
cientandshortintime,indicatinganeedofsaturationof LF by following:
a growth rate. This mechanism is called parametric, be-
causean”external”force-thecentrifugalforce,playinga
role of the parameter, acts on plasma particles, changes ∂pi e
+(vi )pi = γξ ξ+ (E+vi B), (3)
in time and gives rise to the instability. By applying ∂t ·∇ − ∇ m ×
the same approach, in Ref. 14 (to be published in Mon.
complemented by the continuity equation:
Not. R. Astron. Soc.) the problemofthe reconstruction
of the pulsar magnetospheres on large scales nearby the
LmCetSrihcams ebceheanncisomnsiadnedreidt.wWase shhaovwensttuhdaietdfoarnceuwrvpataurrae- ∂∂nti +∇·(nivi)=0, (4)
driftwaves,correspondinginstabilitymightbeextremely
and the Poisson equation:
efficient.
In this paper a hydrodynamic approach will be used
to study the parametric mechanism of rotation induced E=4π e n . (5)
i i
electrostaticwavegenerationinextragalacticjets,apply- ∇·
i
ing the approach developed in Ref. 13 and Ref. 14 (to X
be published in Mon. Not. R. Astron. Soc.).
i=e,p,b
The paper is arranged as follows. In II we derive the
§
dispersion relation, in III the corresponding results are
§ here e,p,bdenote electrons,positronsand the bulk com-
present and in IV we summarize our results.
§ ponents respectively. Eq. (3) in the zeroth approxima-
tion (leading state), by applying the frozen-in condition
E0+v0i B0 =0, can be reduced to (see Ref. 4):
II. THEORY ×
Throughout the paper it is supposed that magnetic d2r = Ω2efr 1 Ω2 r2 2 dr 2 . (6)
field lines are straight and inclined by the angle α with dt2 1−Ω2efr2 " − ef − (cid:18)dt(cid:19) #
respectto the rotationaxis. The plasmais inthe frozen-
incondition,co-rotatingtogetherwiththefieldlineswith In Ref. 4 it has been shown that for ultra-relativistic
the angular velocity Ω. particles (γ 1) Eq. (6) has following solutions:
≫
Itiseasytostartourconsiderationbythelocalnonin-
ertial co-rotating frame of reference. The corresponding
V
0
interval in the rigidly rotating frame will have the form r(t)= sinΩ t, (7)
ef
Ω
(see Ref. 10): ef
v (t)=V cosΩ t, (8)
0 0 ef
ds2 = 1 Ω2 r2 t2 dr2, (1)
− − ef −
for initial conditions: r =0 andV 1. As we see from
0 0
(cid:0) (cid:1) ∼
where Ω Ωsinα is the effective angular velocity of Eq. (3), the effective centrifugal force γξ ξ changes
ef
≡ − ∇
rotation. Hereweuseunitsc=1. Inthispaperweapply in time due to Eqs. (7,8) and therefore the parametric
a method developed in Ref. 13 where relativistic plasma mechanism switches on, giving rise to the electrostatic
motion was described by the single particle kinematics. instability (as we will see).
Accordingtothisapproach,byusing1+1formalism(see When studying the problem for the leading terms,
Ref. 15), in the co-rotating frame of reference the equa- plasma oscillations have not been considered and only
3
theeffectsofthecentrifugalaccelerationhavebeentaken
into account (see Ref. 4). Generally speaking, differ-
ω2 ω2
ent species of plasmas [electrons, positrons and protons ω2 e N(ω)= e J (a)N (ω+sΩ ),
s b ef
(bulk)] experience the centrifugal force, as a result they (cid:18) − γ0e(cid:19) −2γ0e s
X
separate,whichwillleadtothegenerationoftheelectro- (18)
static field, creating the Langmuir waves. This process where M and m are masses of protons and elec-
can be considered as a next step of the approximation. trons/positrons respectively, a ck/Ω and ω =
ef e
≡
The aim is to investigate the linear perturbation theory 8πn0e2/m - electron/positron plasma frequency. For
e
oftheelectrostaticinstabilityandestimateitsroleinrel- deriving Eqs. (17,18), the identity
ativistic AGN jets. p
We start the analysis by introducing small perturba-
tions around the leading state: e±ixsinΩeft = Js(x)e±isΩeft, (19)
s
X
Ψ Ψ0+Ψ1, (9) has been used. Direct substitution of Eq. (17) into Eq.
≈ (18), yields:
where Ψ=(n,v,p,E,B).
Perturbing all physical quantities by following:
ω2 ω2 ω+(s l)Ω 2
ω2 e N(ω)= b J (a)J (a) − ef
s l
− γ 2γ ω+sΩ ×
Ψ1(t,r) Ψ1(t)exp[i(kr)] , (10) (cid:18) 0e(cid:19) 0b s,l (cid:18) ef (cid:19)
X
∝
Eqs. (3,4) will get the form:
N(ω+(s l)Ω ). (20)
ef
× −
∂p1 e where ω = 8πn0e2/M is the plasma frequency corre-
i +ikv p1 =v Ω2 rp1+ E1, (11) b b
∂t 0 i 0 ef i m sponding to the bulk component.
i p
Generally speaking in order to solve Eq. (20) one
∂n1 has to consider similar equations, rewriting Eq. (20) for
∂ti +ikv0n1i +ikni0vi1 =0, (12) N(ω±Ωef),N(ω±2Ωef),etc.,thusforsolvingtheprob-
lemexactly,oneneeds tosolveasystemwiththeinfinite
number of equations, which makes the problem impossi-
ikE1 =4π ein1i, (13) ble to handle. Therefore the only way to overcome this
i difficulty and to gain extended view of the general be-
X
havior of the instability, is to consider physics close to
Here E1 is the electric field in the LF, induced by sepa-
the resonancecondition,which providesthe cutoff ofthe
ration of charges.
infinite row in Eq.(20) and makes the problem solvable
Inordertoreducetheabovesystemintoasingleequa-
(see Ref. 16).
tion let us use an ansatz:
If oneconsidersin Eq. (20)only resonanceterms, cor-
1/2
responding to the following frequency ω ω /γ ,
n1i ≡Nie−iΩVeikf sin(Ωeft), (14) thenpreservingleadingtermsandtakinginretsoa≈ccouent0tehe
resonance conditions: ω s Ω , s = l ωe
then from Eqs. (11,12,13) one gets: ≈ − 0 ef 0 0 ≡ (cid:20)γ01e/2Ωef(cid:21)
(here [A] means the integer part of A), the dispersion
∂2N e n0 relationcanbereducedintoafollowingsingletermspec-
i = i i i eiRikE1, (15) ifying the growth rate (∆ ω ω ) of the instability:
∂t2 − miγi0 ≡ − res
ω2ω
ikE1 =4π e N e−iRi, (16) ∆3 b e J2 (a). (21)
i i ≈ 4γ γ1/2 s0
i 0b 0e
X
where R = V0ik sin(Ω t). Since we are interested in imaginary parts of ∆, it is
i Ωef ef easy to see that the following solution (see Ref. 17):
Restoring the speed of light and introducing a new
variable N N N , after making the Fourier trans-
p e
≡ −
formations, one can easily reduce Eqs. (15,16): 1
1 ω2ω 3
∆ 1 i√3 b e J2 (a) (22)
ω2N (ω)= mn0bγ0e (ω sΩ)2J (a)N(ω sΩ ), ≈−2(cid:16) − (cid:17)"4γ0bγ01e/2 s0 #
b −2Mn0eγ0b s − s − ef is responsible for the instability, the increment of which
X
(17) is given by:
4
−4.5 −4.2
λ = 106cm
−4.6 0
−4.3 λ = 106cm
0
−4.7
−4.8 −4.4
−4.9
−4.5
δlog() −5 Γlog()
−4.6
−5.1
−5.2 −4.7
−5.3
−4.8
−5.4
−5.5 −4.9
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0 10 20 30 40 50 60 70 80
∆λ cm αo
FIG. 1: Dependence of logarithm of the instability rate on FIG. 2: Dependence of logarithm of the average instability
the wave length. The set of parameters is: α = 1◦, MBH = rateonthewavelength. Thesetofparametersisthesameas
108M⊙,Ω=3×10−5s−1,γ0b=20,γ0e =105 andn0b =n0e = inFig. 1,exceptawiderrangeofαandfixedλ≡λ0=106cm.
0.001cm−3. As it is seen in the figure, the increment is very
sensitive on the wave length: slightly changing λ, one may
kill theinstability completely. their ambient, obviously having density of the order of
n 1cm−3.
am
∼
An interesting feature of the result shown in Fig. 1
is sensitiveness of the growth rate on the wave length.
1 As it is clear from the plot, a small change of λ, drasti-
δ √3 ωb2ωe J2 (a) 3 . (23) callychangesthesituation: forcertainvaluesofthewave
≈ 2 "4γ γ1/2 s0 # length the increment reaches its maximum level and for
0b 0e
slightlydifferentvalues-itbecomesequalto0. Therefore
it is better to examine an averagevalue for each interval
witha singlepeak,taking intoaccountbothparameters:
III. DISCUSSION λ andα, calculate the growthrate with respectto them:
We consider a central black hole with mass MBH = 1 λ2 α2
108M⊙ and an angularrate of rotationΩ=3 10−5s−1 Γ dλdαδ(λ,α), (24)
× ≡ (λ λ )(α α )
of an AGN wind, making the light cylinder located at a 2− 1 2− 1 Zλ1 Zα1
distance rL 1015cm fromthe center of rotation. These andbasedondiscretedatainterpolateitforawiderrange
≈
values are typicalfor active galactic nuclei (see Ref. 18). ofparameters. Hereλ andα areminimumandmax-
1,2 1,2
AGN winds and extragalactic jets are supposed to be imumvaluesofthewavelengthandtheanglerespectively
composedofabulkcomponenthavingtheLorentzfactor for each interval.
of the order of 20 and relativistic electron-positron Inspite ofthatjets usuallyhavesmallopening angles,
pairs with Loren∼tz factors: 105 (see Ref. 10). it is better to study a dependence of the growth rate on
∼
Asitwasalreadyexplained,the centrifugalforceleads inclinationforawiderrangeofangles,inordertounder-
to the generation of the Langmuir waves with the wave stand a general behaviour of the increment. In Fig. 2
vector −→k( −→B), the growth rate of which we are going we show the dependence of logarithmof the growth rate
||
to estimate versus the wave length λ and the inclination versusα. The setofparametersis the sameas forFig. 1
angle α in order to investigate the corresponding insta- except a wider range of α and a fixed value of the wave
bility. length λ = 106cm. As we see, the increment is a con-
Let us suppose a jet with the opening angle β =2α= tinuously increasing function. This is a natural result,
2◦. In Fig. 1 we show the dependence of logarithm of becausewhenoneincreasestheangle,theeffectiveangu-
the instability increment on the wave length λ λ + lar velocity increases as well, that makes the centrifugal
0
∆λ, where λ = 106cm and ∆λ = [0,0.1]cm. T≡he set forcehigher,leadingto the moreefficientinstability pro-
of parameters is: α = 1◦, MBH = 108M⊙, Ω = 3 cess.
10−5s−1, γ =20, γ =105 and n0 =n0 =0.001cm−×3. As we see from the present figure, the instability is
0b 0e b e
Thevaluesofn0 andn0 aretypicalforextragalacticjets, less efficient for smaller angles. Let us study now a de-
b e
which are supposed to be under dense with respect to pendence of the growth rate on the wave length for the
5
−4.78 15
−4.8 α = 1o
14.5
−4.82
14
Γlog()−4.84 log(t)evol
13.5
−4.86
13
−4.88
−4.9 12.5
1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 0 5 10 15
λ /106cm L erg/s x 1045
FIG. 3: Dependence of logarithm of the average instability FIG. 4: Dependence of logarithm of the evolution time scale
rate on the wave length. The set of parameters is the same on theluminosity power. The set of parameters is: η=0.03,
as in Fig. 1, except a wider range of thewave length. ǫ = 0.1 and MBH = 108M⊙. luminosity power lies in the
range L∈1.4×[1044;1046]erg/s.
inclination angle 1o when the instability is minimal (as
it it is seen in Fig. 2). to the accretion mass rate M˙ and the luminosity L by
In Fig. 3 the behaviour of logarithm of Γ versus λ is the following expression:
shown. The set of parameters is the same as in Fig. 1
exceptawiderrangeofthewavelength. Thefigureshows
L
that by increasing λ, the average growth rate increases ǫ , (27)
≡ M˙ c2
as well. Let us consider a less efficient case, thus the
smallest increment (for λ = 106cm) shown in the figure:
L is the Eddington luminosity
Γ 1.3 10−5s−1(λ=106cm),andseehowpowerfulthe E
∼ ×
instability is. One can easily estimate the corresponding
time scale by inversing the value of Γ: L =1.4 1046 MBH erg/s. (28)
E × 108M⊙
1
t = 105s. (25)
inst
Γ ∼ Thenestimatingthe disk’sevolutiontimescalet
evol
M /M˙ , by taking Eq. (26) into account, one can easil≡y
On the other hand jets are formed due to an accret- sg
get:
ing matter, therefore energetics of jets is defined by the
efficiencyoftheaccretionprocess,andinordertounder-
stand how efficient the considered instability is, one has η −2/27 ǫ 22/27 L −22/27
to estimate the evolution time scale of an accreting disk t =3.53 1013
evol
× 0.03 0.1 0.1L
and compare it with the instability time scale. (cid:16) (cid:17) (cid:16) (cid:17) (cid:18) E(cid:19)
In Ref. 19 the problem of fueling of AGNs is consid-
eredanditisshownthatforagivenmassandluminosity
of the AGN, mass of the self gravitating disk can be ex- MBH −4/27
s. (29)
pressed by: × 108M⊙
(cid:18) (cid:19)
For taking the sense out of the considered parametric
η −2/27 ǫ −5/27 L 5/27 instability, let us examine typical values of the accretion
M =2.76 105
sg × (cid:16)0.03(cid:17) (cid:16)0.1(cid:17) (cid:18)0.1LE(cid:19) derisinkgηt=he0d.0e3p,enǫd=en0c.e1oafndthMe eBvHolu=ti1o0n8Mtim⊙e, tshceanlecoonnstihde-
luminosity,onegetstheplotshowninFig. 4. Forproduc-
M 23/27 ing the plot we have examined the following luminosity
BH
×(cid:18)108M⊙(cid:19) M⊙, (26) lruatnigoenLti/mLeEs∈ca[l0e.0i1s;a1].coAnstiwnueosueselyindtehcerefiagsiunrge,futhnecteiovno-,
where η is the standard Shakura, Sunyaev viscosity pa- andfor the minimum value of the consideredluminosity,
rameter(seeRef. 20),ǫistheaccretionparameterrelated the time scale is of the order of 1015s, whereas for the
6
Eddingtonlimitthetimescalereachesitsminimumlevel 5. Taking into account efficiency of an accretion pro-
1013s. cess in AGNs, and comparing the corresponding
∼
Calculating t /t , by using Eq. (25) one can see time scale to the time scale of the electrostatic in-
evol inst
that the present ratio lies in the range [108;1010]s, stabilityithasbeenfoundthatevenwhenthelatter
∼
which means that the process of the instability is much is less effective (smallanglesandwavelengths)the
faster than typical rates of accretion, therefore the pro- instability process is extremely efficient.
cess of conversionof rotationalenergy into energy of the
Langmuirwavesisveryrapid. Thismeansthatthelinear The electrostatic instability may strongly affect
stage has to be short in time, and the growth rate must processesofradiationinAGN jets by means ofthe
saturate soon due to non linear effects, consideration of ICS,thereforeforextendingthis work,itisnatural
which seems to be essential. to consider a next important step and study the
In the introduction we have noticed that the Lang- last stage: energy conversioninto radiation.
muir waves may strongly influence processes of radia-
An important restriction in the present model is
tionthroughtheICSofthesewavesagainstsoftphotons,
when due to the channel l+e = l′+t the electrostatic thatwestudiedtheproblemforquasi-straightmag-
netic field lines, whereas in real astrophysical situ-
wavesloseenergy,whereasthephotonsgainenergy. This
ationsthefieldlinesarecurvedanditisinteresting
problemwasnotanobjectiveofthepresentpaper,which
to seewhatchangesinthe dynamicsofthe electro-
seemstobethefirststepforunderstandingtheprocesses
static waveinstability, whenthe curvature is taken
of rotation energy pumping into instabilities. As a next
into account.
steponehastostudythisparticularproblemaswelland
see how efficient the radiation (produced via the ICS)
On the other hand the curvature of magnetic field
could be for the extremely unstable electrostatic waves.
lines induces the curvature drift waves, which also
This will be an objective of a work which we are going
may affect the process of energy pumping. In the
to make soon or later.
context of pulsars this problem has been consid-
ered in Ref. 14 (to be published in Mon. Not.
R. Astron. Soc.). It is reasonable to examine the
IV. SUMMARY
same problem for relativistic AGN jets and study
the role of the curvature drift modes in the energy
1. We studied a relativistic plasma of extragalactic
conversion process.
jets composed of the bulk flow and relativistic
electron-positroncomponents. The role ofthe cen-
Whenstudyingtheproblemonthelinearstage,the
trifugalaccelerationingenerationoftheparametric
increment shows high efficiency of the electrostatic
instability of the Langmuir waves has been consid-
instability, thereforethe nonlinear regime also has
ered.
to be examined in order to study the problem for
longer time scales when the linear approximation
2. Perturbing the Euler, continuity and induction
does not work any more. For this purpose we plan
equations, preserving only first order terms, we
toimplementanumericalmagnetohydrodynamical
have derived the dispersion relation governing the
code to study this particular case as well.
instability.
3. Examining the resonance condition of the system,
anexpressionofthegrowthratehasbeenobtained
for a nearby zone of the light cylinder surface. Acknowledgments
4. Studying dependence of increment on the wave
length for typical values of extragalactic jets, we I thank professor G. Machabeli for valuable discus-
foundthattheinstabilitybecomesefficientonlyfor sions. The research was supported by the Georgian Na-
certain and narrow ranges of the wave length. tional Science Foundation grant GNSF/ST06/4-096.
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