Parametric instabilities in magnetized multicomponent plasmas

0.36 MB·English

by  M. P. Hertzberg

Parametric instabilities in magnetized multicomponent plasmas
M. P. Hertzberg, N. F. Cramer, and S. V. Vladimirov∗
School of Physics, The University of Sydney, New South Wales 2006, Australia
This paper investigates the excitation of various natural modes in a magnetized bi-ion or dusty
plasma. Theexcitationisprovidedbyparametricallypumpingthemagneticfield. Heretwoion-like
species are allowed to be fully mobile. This generalizes our previous work where the second heavy
6
species was taken to be stationary. Their collection of charge from the background neutral plasma
0
modifies the dispersion properties of the pump and excited waves. The introduction of an extra
0
mobile species adds extra modes to both these types of waves. We firstly investigate the pump
2
wave in detail, in the case where the background magnetic field is perpendicular to the direction

n of propagation of the pump wave. Then we derive the dispersion equation relating the pump to
a theexcited wavefor modes propagating parallel to thebackground magnetic field. It is found that
J
thereare a total of twelve resonant interactions allowed, whose various growth rates are calculated

5 and discussed.
2
PACSnumbers: 52.35.Bj,52.35.Mw,52.30.Ex,52.25.Vy

]
h
p
I. INTRODUCTION The parametric excitation of waves in a dusty magne-
-
m tizedplasmahasbeeninvestigatedinRef.[12],butinthe
approximation where the dust is taken to be immobile.
s The basic natural modes of magnetized plasmas such
a In that case the oppositely travelling pair of waves are
as those that occur in molecular clouds, cometary plas-
l modified, with the presence of a cutoff frequency in the
p mas and stellar atmospheres are of great interest. When
fast Alfv´en wave. Furthermore, an interaction between
s. the frequencies are low then a class of linear waves, re-
a pair of fast waves or a pair of slow waves was found
c ferredto collectivelyasAlfv´enwaves,areknownto exist
i toexist. Thecorrespondinggrowthratesofthe slow-fast
s and are of importance to the understanding of many ba-
and fast-fast pairs were maximized as a function of the
y sic plasma phenomena [1, 2]. The linear approximation
h dust concentration.
to these waves breaks down at large amplitudes where
p nonlineareffects become importantin theirpropagation. Many space and laboratory plasmas are multicompo-
[
One such large amplitude wave is a magnetoacoustic nent, i.e., contain multiple ion species. It is therefore

1 wave,whichmodifiesthebackgroundmagneticfieldinan of great interest to investigate the effects of the addi-
v oscillatory fashion, and so can be considered as a pump tional species on the linear and nonlinear properties of
3 wave that drives other waves nonlinearly. Such large- the waves in the plasma. For instance, it is often the
9 case that in the presence of an additional ion compo-
amplitude pump waves may occur in conditions such as
1
seeninsolarandspaceplasmas. Forexample,solarshock nent an extra mode is excited, or forbidden regions of
1
frequency are introduced. One area of study of multi-
0 wavescansetuplargeamplitudestandingmagnetoacous-
6 ticwavesincoronalloopsormagneticfluxtubes[3,4,5]. ion plasmas is the bi-ion plasma. The inclusion of one
0 extra ion species often captures the basic information of
By considering perturbations of this large amplitude
/ several extra species. The bi-ion plasma has particular
s pump, we can investigate the possibility of exciting nat-
c importance in plasma fusion, laboratory plasmas and in
ural modes of the system, such as Alfv´en waves, due to
i astrophysicalenvironments, with the secondary ion usu-
s a resonant interaction between the pump wave and the
y ally positively charged.
natural waves. In the single ion species case this basic
h A dusty plasma adds another level of interest to the
phenomenon was predicted by [6, 7], and subsequently
p
topic of multicomponent plasmas, due to the dust prop-
: pursued by several authors [8, 9, 10]. For a pump mag-
v erties. Dust is an additional impurity of large mass and
netic field parallel to the backgroundmagnetic field in a
i often ofnegative charge. Under the simplest approxima-
X single ion species plasma, the excited waves are Alfv´en
tion, all the dust grains may be considered to have the
wavestravellinginoppositedirectionsalongthemagnetic
r
a field [6, 7]. If ion-cyclotron effects are included, the ex- samemassandequilibriumcharge,andsoareequivalent
toasecondionspecies. Thisisthecaseconsideredinthis
cited waves are the fast and slow (ion–cyclotron) Alfv´en
paper. It isknownthat the inclusionofdustin aplasma
waves travelling in opposite directions [8]. More general
may introduce cutoff frequencies into the basic Alfv´en
behavior is allowed in the case where the excited waves
waves, and introduce a low frequency mode, whose na-
are permitted to travel obliquely to the magnetic field
ture is different from that in a bi-ion plasma due to the
[11].
dust grain’s extremely high mass [13, 14]. Qualitative
differencestothebi-ioncasearisewhenchargeperturba-
tions of the grains are included, leading to an additional
∗Electronic address: [email protected]; damping mechanism [15], or when a spectrum of dust
URL:http://www.physics.usyd.edu.au/~vladimi grainsizesandchargesis allowedfor [16]. Dust is found,
2
invaryingamounts,inmanyastrophysicalandspaceen- electroninertia. Thisisvalidifthefrequenciesofinterest
vironmentssuchasmolecularcloudsandtheringsofSat- are well below the electron cyclotronfrequency. In addi-
urn. Dusty plasmas have also been studied closely ex- tionweuseMaxwell’sequationsandtwomasscontinuity
perimentally, since the heavy grain mass introduces low equations for each of the ion species, ignoring Maxwell’s
frequency effects that may be studied in real time. displacement current.
An immediate consequence of the presence of an ad- Theprimaryspecies(assumedpositivelycharged)shall
ditional ion species is its modification of the background be denoted with a “1” subscript, and the secondary
free-electron number density. This influences the propa- species (either positively or negatively charged) shall be
gation of plasma and electromagnetic waves [15, 17, 18]. denoted with a “2” subscript. In this notation we may
If in the case of a dusty plasma we fix the grain charge, write the current density as
then the grain becomes entirely defined in terms of its
mass and charge, and as such acts just like another ion J=Z1en1v1+Z2en2v2 eneve, (2)
−
in the plasma (albeit of negative charge). The state of
where the two ion species and electrons have velocities
charge neutrality may be written as:
denoted by v , v , and v , respectively. We employ the
1 2 e
ene+eZ1n1+eZ2n2 =0. (1) parametersδ1 =ne/Z1n1 andδ2 =ne/Z2n2,whichmea-
− sure the distribution of charge in the plasma amongst
Here ne,1,2 is the number density of plasma electrons the ions. Employing the total charge neutrality condi-
(with the charge e), and the two ion species (of signed tion, given in Eq.(1), we may write this as
−
charge Z and Z ), respectively. For laboratory dusty
1 2
1 1
plasmas, the grain charge is negative (i.e., Z2 < 0) and + =1, (3)
large (Z2 102 103), so that an appreciable propor- δ1 δ2
| | ∼ −
tion of the negative charge in the plasma may reside on
in terms of these parameters. Note that in the limit of a
the dust particles. For astrophysicaldusty plasmas, Z
mayonlybe ofthe orderofunity,whichis oftenthe c|as2e| single (primary) ion species, we have δ1 → 1 and |δ2|→
.
for a canonical bi-ion plasma. In environments such as ∞
Ignoring collisions, but including the effects of pres-
the interstellar medium, where the dust grain is in an
sure, the starting equations for the velocities, electric
electron-proton plasma with little ultra-violet radiation
andmagneticfieldsandeachnumberdensityarethemo-
present, the dust grains acquire a negative charge. On
mentumequations,thetwoioncontinuityequations,and
theotherhand,exposuretoultravioletlightfromnearby
Ampere’s law neglecting the displacement current:
starscancauseionizationofthegrains,leavingaresidual
positive charge. dv
m n 1 = p +Z en (E+v B), (4)
In this paper, we investigate the propagation of plane 1 1 dt −∇ 1 1 1 1×
hydromagnetic waves (Alfv´en waves and magnetoacous- dv
tic waves modified by the presence of a secondary ion m2n2 2 = p2+Z2en2(E+v2 B), (5)
dt −∇ ×
or dust species in a bi-ion plasma), propagating paral- 0= p en (E+v B), (6)
e e e
lel to the pumped magnetic field of a large amplitude −∇ − ×
∂n
magnetoacousticwave. We generalizeRef.[12]wherethe 1 = (n v ), (7)
1 1
∂t −∇·
second heavy space was assumed immobile, to the case
∂n
where both ions are fully mobile. A further generaliza- 2 = (n v ), (8)
2 2
tion is the inclusion of pressure. The background mag- ∂t −∇·
neticfieldandplasmadensityaretakentobeuniform,at B=µ0e(Z1n1v1+Z2n2v2 neve). (9)
∇× −
frequencieswellbelow the electronplasmaandcyclotron
Here, p are the two heavy species and electron pres-
frequencies. First, we find the dispersion equationof the 1,2,e
sures,andm aretheheavymasses. Themagneticfield
pumpwave,andthenconcentrateonpumpwavesoflarge 1,2
B includes the background magnetic field B .
wavelength. We then obtain a coupled pair of equations 0
Finally, by assuming either an isothermal or an adia-
of motion governing the perturbed plasma. Resonant
batic equation of state, we have
interactions are sought, and growth rates of the waves
parametrically excited by the pump are calculated and p =U2m n (10)
discussed. ∇ α α α∇ α
for each species, where U are the individual sound
α
speeds. Though it is not imperative, we shall assume
II. MULTIFLUID MODEL isothermalchanges,whichpermitsustowritedownU in
α
terms of the plasma temperatures, i.e., U2 = k T /m
α B α α
The most general set of equations used to describe is the square of each thermal speed, where T is each
α
twomobileion-likespecies,pluselectrons,isathree-fluid temperature and k is Boltzmann’s constant.
B
model. Inthispictureweemploythreemomentumequa- At this point we choose to eliminate the electron vari-
tions for the electrons and the two ion species, where we ables from Eqs.(4)–(9), and use Faraday’s law to elimi-
include both the ions’ inertia terms, while ignoring the nate the electric field E. There are then two choices for
3
the way to proceed. We can add and subtract the mo- whereωisthewavefrequencyandv ,v aretheAlfv´en
A1 A2
mentum equations and deal with a total fluid velocity speeds associated with each ion. The charge neutrality
and the current density J. Instead we shall employ the conditionimpliesthatthewavefrequenciesarerestricted
mostdirectmethod, whichis to dealwith the ionveloci- to the regime ω ω , which may be restated as
pe
tiesv andv separately,sinceourequationswillexhibit ≪
1 2
the most symmetry this way. By employing the charge Ω ω 2 v2
1,2 A1,2
neutralityconditiontoallorders,thefollowingsystemof |δ1,2|≫ | Ω | Ω c2 . (17)
nonlinear partial differential equations is found: e (cid:18) 1,2(cid:19)
∂ρ There exists a wide range of physical environments for
1 = (ρ v ), (11)
1 1 which all of the above mentioned conditions are met.
∂t −∇·
∂ρ
2 = (ρ v ), (12)
2 2
∂t −∇·
∂B B dv III. MAGNETOACOUSTIC PUMP WAVE
= (v B) 0 1
1
∂t ∇× × − Ω ∇× dt
1
B dv Supposethere isaconstantbackgroundmagneticfield
=∇×(v2×B)− Ω0∇× dt2, (13) in the z-direction, given by B0zˆ. We now periodically
2 pump the field, with a periodic modulation:
dv
δ ρ 1 = (δ U2+α2) ρ α2 ρ
1 1 dt − 1 1 1 ∇ 1− 2∇ 2 B [1+ε¯bcos(ω t)]ˆz, (18)
0 0
ρ Ω 1
+ 2 2(v v ) B+ ( B) B, (14)
1 2
B0 − × µ0 ∇× × where ω0 is the pump frequency. Here ε¯ is a constant
dv dimensionless quantity which determines the amplitude
δ ρ 2 = (δ U2+α2) ρ α2 ρ
2 2 dt − 2 2 2 ∇ 2− 1∇ 1 of the pump. More precisely, ε¯is an expansion param-
ρ Ω 1 eter, which permits us to keep track of our terms, by
+ 1 1(v2 v1) B+ ( B) B. (15) matching powers of ε¯. The parameter b on the other
B − × µ ∇× ×
0 0
hand is a dimensionless quantity that we include to cap-
In Eqs.(11)–(15) ρ = m n are the densities of
1,2 1,2 1,2 tureanynecessaryfrequencyinformation,i.e.,b=b(ω ),
0
each massive component of the plasma and Ω =
1,2 such that the average of b over ω is O(1). By speci-
0
Z eB /m are the corresponding (signed) cyclotron
1,2 0 1,2 fying a particular choice of normalization condition, we
frequencies,withB = B . Alsoα2 =Z U2m /m and
α2 =Z U2m /m a0rep|seu0|do-squar1edthe1rmealsepeed1sas- may solve for b. We later in fact impose the condition
2 2 e e 2 thattheenergydensityinthepumpsystemisaconstant
sociatedwiththeelectronpressure(ifZ <0weallowα
2 2 over ω , and calculate the resulting b. Since we are in-
0
to be imaginary). The presence of both ρ and ρ in
1 2 terestedinalargeamplitude pumpwave,ε¯willtypically
∇ ∇
both equations of motion, is a consequence of the elimi- be O(10−2 10−1).
nationoftheelectronvariablesfromtheequations. Tobe −
However, since this field has no spatial dependence, it
more explicit, the electron density fluctuations generate
does not satisfy our wave equations. Hence this is an
fluctuations in both the ion densities through the charge
approximation to a wave with a large wavelength. We
neutralitycondition,whichproducesthis couplinginthe
must therefore modify the pump with an envelope of
momentum equations. The second term on the right
some wavelength. Since the pump wave magnetic field
hand side of Eq.(13) is the Hall term; it is important
points in the z-direction, the wavevector should point
when the wave frequency is comparable to either of the perpendicular to this to satisfy B = 0. For a planar
cyclotron frequencies. Also, we see that the two species ∇·
geometryplasma,wechoosethe axessuchthatthe wave
are strongly coupled through the momentum equations,
variesinthex-direction,withwavenumberk . Bydenot-
0
viathethirdtermonthe righthandsideineach. This is
ing the pump magnetic field (and all subsequent pump
anadditionalHall-typetermassociatedwiththe relative
fields) with a “0” superscript, we have
motionofthetwospecies. NoteEqs.(11)–(14)reduce,in
the case of a single species plasma (δ1 → 1, |δ2| → ∞, B(0) =B0[1+ε¯bcos(k0x)cos(ω0t)]ˆz. (19)
assuming local charge neutrality is maintained), to the
equations used in Ref.[19, 20, 21, 22, 23] where non-
In a cylindrical plasma, with r denoting the radial dis-
linear Alfv´en waves were investigated. They are then
tance from the cylindrical axis, we have
known as the (collisionless) Hall-MHD (magnetohydro-
dynamics) equations. B(0) =B [1+ε¯bJ (k r)cos(ω t)]ˆz, (20)
0 0 0 0
The neglect of the displacement current in Maxwell’s
equation is justified when the electron current is much
whereJ isthe0thorderBesselfunctionofthefirstkind.
0
greater than the displacement current, which leads to Note that b=b(k ,ω ) in Eqs.(19) and (20).
0 0
the conditions
In the absence of a wave, we shall suppose that the
ω v2 ω v2 plasma is stationary, with ion densities ρ and charge
δ A1, δ A2, (16) 01,2
1 ≫ Ω1 c2 | 2|≫ Ω2 c2 ratios of δ01,2. The effect of pumping will be to modify
| |
4
the velocity, density and charge imbalance to order ε¯, so
that
0.5
v(0) =ε¯v¯ , v(0) =ε¯v¯ , (21) (a)
1 1 2 2 Ω1
ρ(0) =ρ +ε¯ρ¯ , ρ(0) =ρ +ε¯ρ¯ , (22) ω/00.4
1 01 1 2 02 2 y
δ1(0) =δ01+ε¯δ¯1, δ2(0) =δ02+ε¯δ¯2. (23) uenc 0.3
q
e
Thesequantitiesdefinethe pumpwave,andwenowpro- Fr
ceed to solve the equations (11)–(15) to orderε¯(i.e., the p
m 0.2
linear solution). Pu
ByinspectingtheformofthemagneticfieldinEq.(19) d
e
z
[or(20)],andEqs.(14)and(15),wenotethatthevelocity ali 0.1
cannot have a z-component. Hence, the velocities of the m
or
ion species must be of the form N
0.0
0.0 0.1 0.2 0.3 0.4
v¯1 =(v¯x1,v¯y1,0) , v¯2 =(v¯x2,v¯y2,0) (24) Normalized Pump Wavenumber v k/Ω
A1 0 1
for planar waves. From the resulting pair of equations
we ascertain the form of each velocity component and
density: 0.5
(b)
v¯ =A sin(k x)sin(ω t), (25) Ω1
x1,2 x1,2 0 0 ω/00.4
v¯y1,2 =Ay1,2sin(k0x)cos(ω0t), (26) y
c
n
ρ¯1,2 =R1,2cos(k0x)cos(ω0t), (27) ue 0.3
q
e
where Ax1,2,Ay1,2 and R1,2 are amplitudes independent p Fr
of space and time, to be determined. Then, from the m 0.2
u
continuity equations, we have P
d
e
R =ρ k0A , R =ρ k0A . (28) aliz 0.1
1 01ω x1 2 02ω x2 m
0 0 or
N
For the cylindrical plasma case, we make the substi- 0.0
0.0 0.1 0.2 0.3 0.4
tutions: x r, y φ, cos J and sin J , where
→ → → 0 → 1 Normalized Pump Wavenumber v k/Ω
φ is the azimuthal angle and J is the 1st order Bessel A1 0 1
1
functionofthefirstkind. Thenitisfoundthatallthere-
lationships(24)–(28)stillhold. Itfollowsthatthemodes FIG. 1: The normalized pump frequency ω0/Ω1 versus nor-
in the planar and cylindrical plasmas satisfy the same malized pump wavenumber vA1k0/Ω1 with δ01 = 1.1 and
dispersion equation. Ω2/Ω1 =0.1. (a) The warm plasma, with β1 =0.8, B1 =1.5
Upon substitution of Eqs.(25)–(28) into Eqs.(11)– and U2/U1 = 1/2. (b) The cold plasma, with β1 = B1 =
(15), written to first order in ε¯, the following dispersion U2 =0.
equation of the pump wave is obtained:
ω02(ω02−ωc2)=W k02(ω02−X−Yk02). (29) sBpecrifeylattehde tchorlodu/gwha:rmδ pBlasm=a rδegiBme.s, Nwoitthe Bth1atanads
2 01 1 02 2
Here, we have defined δ 1, δ (i.e., a single species) we are left
01 02
→ | | → ∞
with the nondispersive relation
ω =Ω /δ +Ω /δ , (30)
c 1 02 2 01
W =vA21(1+δ021β1+δ01B1)/δ021 ω0 = c2s+vA21k0, (34)
+vA22(1+δ022β2+δ02B2)/δ022, (31) q
X =ω (v2 Ω (1+β +δ B )/δ where c2s = U12 + α21 is the combined sound speed in
c A1 2 1 01 1 01 thislimit. Thisisthefamiliarfastmagnetoacousticwave
+vA22Ω1β2/δ02)/W, (32) characteristic for k B0. However, in the presence of
Y =vA21vA22(β2(1+δ01B1)/δ021 a secondary ion spe⊥cies the fast magnetoacoustic wave
+β (1+δ B )/δ2 +β β )/W, (33) gains an additional mode and the relationship is disper-
1 02 2 02 1 2
sive, see Fig.1.
where ω is a hybrid cutoff frequency as k 0. It is easy to show that the parameters X and W are
c 0
Also we| ha|ve introduced β := U2/v2 , β := U2→/v2 , always positive, while Y = 0 for the cold plasma and is
1 1 A1 2 2 A2
B := α2/v2 and B := α2/v2 which allow us to positive if any of β ,β ,B ,B are nonzero. Although
1 1 A1 2 2 A2 1 2 1 2
5
Fig.1 is for a plasma in which the secondary ion species
ispositive,thesamebasicqualitativefeaturesarepresent
when the second ion species is negative. For the warm vA14
b
lpaltaisomnaisthteo ecffaeucsteotfhethleowYekr02btrearnmchintothinecrdeiaspseerwsiiotnhoruet- A|/x1,y1
bound as k0 , as indicated in Fig.1(a). Physically, d | 3
it can be tho→ugh∞t of as the fast magnetoacoustic mode Fiel
beingconvertedintoanacousticmodeforlargek dueto c
the inclusion of pressure. However, if the plasma0is cold neti 2
g
(i.e., Y = 0) then the lower branch experiences a reso- Ma
nance as k , as indicated in Fig.1(b). In this limit n
X becomes0t→he∞squareofaresonancefrequency,givenby y o 1
cit
o
el
ω v2 Ω δ V
X ω2 = c A1 2 02 , (35) p
→ r δ v2 /δ +δ v2 /δ m 0
02 A1 01 01 A2 02 u 0.0 0.1 0.2 0.3 0.4
P
Normalized Pump Wavenumber v k/Ω
which gives rise to a forbidden frequency regionbetween A1 0 1
ω and ω , since ω < ω whenever δ =1.
r c r c 01
| | | | | | | | 6
The ratio of the x-velocity amplitudes A to the
x1,2 FIG. 2: The ratio of normalized x-velocity amplitude and
magnetic field parameter b are of particular importance y-velocity amplitude to magnetic field Ax1,y1/bvA1 ver-
to our later discussion, so we briefly discuss them here. sus normalized pump wavenumber vA1k0/Ω1, with δ01 =
In terms of the frequency and wavenumber, we find, 1.1, Ω2/Ω1 =0.1, β1 =0.8, B1 =1.5 and U2/U1 =1/2. The
solid curveis for theupperbranch of thedispersion relation,
A ω A ω thedashed curveis for thelower branchof thedispersion re-
x1 0 x2 0
= , = , (36)
b k A b k A lation. Note that on the vertical axis we plot the absolute
0 1 0 2
valueofthisratio, sinceweareinterestedherein theratioof
where amplitudes. Consequentlyoneofthetwodashedcurvesexpe-
riences a discontinuity in its first derivative when it touches
1 1 Ω U2k2+Ω ω ω2 thehorizontal axis.
A := + 2 1 0 1 c− 0,
1 δ δ Ω U2k2+Ω ω ω2
01 02 1 2 0 2 c− 0
1 1 Ω U2k2+Ω ω ω2 magnetoacoustic waves is (e.g. [24]),
A := + 1 2 0 2 c− 0. (37)
2 δ δ Ω U2k2+Ω ω ω2
02 01 2 1 0 1 c− 0 E = B¯2 + ǫ0E¯2 + 1ρ A2 +A2
Finally, to complete the set of pump quantities, we find D 2µ0 2 2 01 x1 y1
that the velocity in the y-direction is given by 1 (cid:0)1 U2 (cid:1)
+ ρ A2 +A2 + αR2, (39)
2 02 x2 y2 2 ρ α
Ay1 = (U22k02−ω02)Ω1−(U12k02−ω02)Ω2 =F , (38) (cid:0) (cid:1) Xα 0α
Ax1 δ02ω0(U22k02+Ω2ωc−ω02) 1 where B¯ = B0b, and the summation in the final term
is over each plasma species. In our approximations the
with Ay2 given similarly. This is quite different to the electric energy term may be neglected compared to the
single ion species case, where Ay 0. magnetic energy. By switching to dimensionless units
≡
choTihceeobfehbarvainocrhoffr|oAmx,tyh|/ebdisisdprearsmioanticraellalytiaolnt;erseedebFyigt.h2e, gainvdenseutntiinqugelµy0EinDt/eBrm02s≡of1t,hethbeansicalpllaosumravaprairaabmleestearrse.
where for brevity we plot Ax1,y1 only. We see that as By using the expressions for Ax1,y1 and Ax2,y2 we may
k0 , the x-component approaches a finite positive solve for b. The result is
→ ∞
value while the y-component approaches zero for the
upper branch of the dispersion relation. On the other √2
b= (40)
hand the x-component approaches large values and the 1+ ω02(1+F12)ω02(1+F22) + β1 + β2 +B δ
y-componentpassesthroughzeroforthe lowerbranchof k2v2 A2 k2v2 A2 A2 A2 1 01
0 A1 1 0 A2 2 1 2
r
the dispersion relation.
Note that the two branches stemming from the up- [usingF1,2 fromEq.(38)]. Aplotofbversuswavenumber
per branch of the dispersion relation become singular as is given in Fig.3 for each branch ω0(k0). It is important
kco0n→diti0o.n rTehlaitsinmgotthiveatpeusmtphefienlededamfoprlitaudneosrmofaltihzaetivoen- tborannocthe.thItatfobll=owOs (tvhAa1tka0/llΩt1h)eavseklo0ci→ty0amfoprlitthuedeusppaerer
locities Ax1,2,Ay1,2, magnetic field b, and density R1,2. well behaved near k0 ≃0.
Areasonablewaytoproceedistoassumethataswevary The remaining pump quantities to solve for are the
(0)
ω the energy density E in the pump system is fixed. charge ratios δ . These may be obtained by imposing
0 D 1,2
Theappropriaterelationinafluiddescriptionforgeneral the charge neutrality condition on both the background
6
ever, this would clearly be inconsistent with wavenum-
ber matching of a propagating pump with finite perpen-
0.6
dicular wavenumber. We therefore envisage this anal-
ysis to apply to a standing wave (in the transverse di-
0.5
rection) pump in a finite geometry system, such as a
b
e cylindrical or toroidal laboratory plasma, or an astro-
d
u 0.4
plit physical magnetic flux tube; by seeking localized solu-
m tions for distances transverse to the magnetic field such
d A 0.3 that k0x = O(ε¯) in a planar plasma, or k0r = O(ε¯) in a
Fiel cylindrical plasma, we can recover the initially proposed
c 0.2 form for the magnetic field Eq.(18). In other words, we
eti
n are taking x small rather than k small, to ensure that
g 0
Ma 0.1 the fieldswilllookapproximatelyuniforminthe perpen-
dicular direction, i.e. that the wavelength is effectively
0.0 very long on the length scale of interest. In the parallel
0.0 0.1 0.2 0.3 0.4
direction,however,thepumpfieldsareuniform,weplace
Normalized Pump Wavenumber v k/Ω
A1 0 1 nosuchrestrictionondistancesandexcitedwavelengths,
and we can apply wavenumber matching of the excited
wave fields in that direction, as discussed in Section VI.
FIG. 3: The magnetic field amplitude b versus normalized
Then in the region near the z-axis we have
pump wavenumber vA1k0/Ω1 due to the normalization con-
dition (40), with δ01 =1.1, Ω2/Ω1 =0.1, β1 =0.8, B1 =1.5
and U2/U1 = 1/2. The solid curve is for the upper branch B(0) =B0(1+ε¯bcos(ω0t))ˆz, (43)
of the dispersion relation, the dashed curve is for the lower k0x∼ε¯
h i
branch of the dispersion relation. v(0) =0, v(0) =0, (44)
1 k0x∼ε¯ 2 k0x∼ε¯
h i h ik
ρ(0) =ρ (1+ε¯ 0A cos(ω t)), (45)
plasmaandontheperturbationstotheequilibrium. This 1,2 k0x∼ε¯ 01,2 ω0 x1,2 0
is justified since we are interested only with frequencies h i k
(0) 0
that are much less than that of the electron plasma fre- δ1,2 k0x∼ε¯=δ01,2(1+ε¯(b− ω0Ax1,2)cos(ω0t)), (46)
quency. This means the electrons have sufficient time to h i
respond to the perturbations from the equilibrium and correct to O(ε¯). Since the magnetic field has now sim-
neutralize the plasma at each point in space. Thus, on ply a periodic variation in time it resembles a canonical
the perturbed plasma we impose the condition, parametric pump. The problem then resembles a para-
metric amplifier, such as a harmonic oscillator with a
en¯ +Z en¯ +Z en¯ =0, (41)
e 1 1 2 2 time-varyingspringconstant. Foruseintheexcitedwave
−
equationswe must alsocompute the derivativesnear the
where we have ignoredthe effects of dust (or ion) charg-
z-axis, given here:
ing, that is, Z is constant. It is then straightforward
2
to obtain the chargeratios in terms of the densities, and ∂B(0)
subsequently in terms of the velocity amplitudes and b, =0, (47)
∂x
as follows: (cid:20) (cid:21)k0x∼ε¯
∂v(0)
δ1(0) =δ01(cid:18)1+ε¯(b− ωk00Ax1)cos(k0x)cos(ω0t)(cid:19), " ∂x1 #k0x∼ε¯=ε¯k0(Ax1sin(ω0t),Ay1cos(ω0t),0),(48)
δ2(0) =δ02(cid:18)1+ε¯(b− ωk00Ax2)cos(k0x)cos(ω0t)(cid:19). (42) "∂∂vx2(0)# =ε¯k0(Ax2sin(ω0t),Ay2cos(ω0t),0),(49)
k0x∼ε¯
This implies that the amount of charge that resides on ∂ρ(0) ∂δ(0)
each species is temporally coupled to the other. 1,2 = 1,2 =0. (50)
∂x ∂x
Using the derived set of quantities characterizing the " #k0x∼ε¯ " #k0x∼ε¯
pumpwavewemay,inprinciple,investigatethepossibil-
ityofexcitedmodespropagatingatarbitraryangleswith Notethatthevelocityhasafinitederivative,eventhough
respectto the z-axis,asin Ref.[11]. However,by remov- the velocity itself is zero in this regime.
ingthex-dependenceinthepumpwavesourcalculations
become much more tractable, with the requirement that
our excited waves are restricted to parallel propagation IV. PERTURBATION METHOD
along the z-axis, see Ref.[8].
For an infinitely extended medium, the perpendicular Nextwe wishtotest the stabilityofthe selfconsistent
wavenumber of the excited waves is then zero. How- linear solution given in Eqs.(43)–(50), now regarded as
7
afinite-amplitude pumpwave,tothe excitationofwaves are interested only in plane waves that travel parallel to
propagatingalongthe magnetic fielddirection. We must thez-axis. Inthiscase,anylongitudinalcomponentswill
however be careful since this is only a linearized pump decouple completely and merely describe a linear acous-
solution. Toproceedwewilladoptthebasicmethodology tic wave, so we may set the z-component of the primed
of Refs.[7, 8], but shall attempt to refine the argument. velocity to zero. Hence, both primed velocities have the
The basic technique is to perturb each quantity, i.e., following form:
B(0), v(0), ρ(0), δ(0), by an arbitrarily small amount as
measured by an expansion parameter ε′. For example, v1′,2 =(vx′1,2(z,t),vy′1,2(z,t),0). (57)
rtaeskeenatantiaornbiotfratrhyefipeuldm,psafiyeXld:, wXit(0h)a=knXo0w+nε¯liXn¯e.arTrheepn- Tcohnutsin,u∇ity·veq1′u=ati∇ons·vw2′e=hav0e, and from each of the two
Refs.[7, 8] proceed by perturbing this quantity in the
following way: ∂ρ′
1,2 = ε¯ρ′ [ v¯ ] , (58)
X =X(0)+ε′X′. (51) ∂t − 1,2 ∇· 1,2 k0x∼ε¯
where we have imposed the k x = O(ε¯) [or k r = O(ε¯)]
However, since X(0) has neglected terms of O(ε¯2), the 0 0
termofO(ε′)isevenmoresonegligibleinthisexpansion condition. Hence we have ρ′1 =ρ′2 =0, i.e., there are no
(recall that ε′ is arbitrarily small). The problem lies in perturbations in the densities of the excited fields. From
this it also follows that δ′ =δ′ =0.
the factthat we areultimately interestedin the stability 1 2
of the exact pump solution, say Y(0) =X +ε¯Y¯, with BytakingthederivedexpressionsforB,v,ρandδand
0
substitutingintoEqs.(13)&(14),andusingthestability
Y¯ =X¯ +O(ε¯). (52) analysis procedure conveyedin Eq.(56), we obtain
Thus, we shall perform the following expansion instead: ∂B′ B ∂v′
(v′ B )+ 0 1
X =Y(0)+ε′X′. (53) ∂t −∇× 1× 0 Ω1∇× ∂t
B ∂v¯
= ε¯ v′ B¯ +v¯ B′ 0v′ 1 (59)
Upon substitution into our set of nonlinear differential ∇× 1× 1× − Ω x1 ∂x
equations (11)–(15), we obtain a set of equations of the (cid:20) (cid:18) 1 (cid:19)(cid:21)k0x∼ε¯
∂v′ 1
following structure (no approximations): ρ δ 1 ρ Ω (v′ v′) ˆz ( B′) B
01 01 ∂t − 02 2 1− 2 × − µ ∇× × 0
0
ε¯A(Y¯)+ε¯2B(Y¯)+ε′C(X′)+ε′2D(X′) ∂v¯
= ε¯ δ ρ v′ 1 +Ω ρ¯ (v′ v′) ˆz . (60)
=ε¯ε′E(Y¯,X′), (54) − 01 01 x1 ∂x 2 2 1− 2 ×
(cid:20) (cid:21)k0x∼ε¯
where A,B,C,D,E are differential operators acting on Here the terms of O(ε¯), which occur on the right hand
their respective arguments which we may treat as func- side, should be thought of as driving terms from the
tions (for brevity we have suppressed their dependence pump wave. Note that there exist two further equations
on fields other than X). Now ε¯A(Y¯)+ε¯2B(Y¯) = 0 by of motion under the index interchange 1 2. Note also
definition of an exact pump solution. Also, the term of thatthereareno acoustictermsonthe le↔fthandsidesof
O(ε′2) can certainly be neglected. Next, we Taylor ex- these equationsbecause ofthe decouplingofthe longitu-
pand E(Y¯,X′) and write dinalmotions: howeverthere arestill acoustic influences
in the pump fields on the right hand sides of the equa-
E(Y¯,X′)=E(X¯,X′)+O(ε¯). (55)
tions.
WearethenpermittedtoneglectthistermofO(ε¯)when Now in order to treat the x and y components of the
inserted back into Eq.(54). This gives us a set of equa- B and v vectors on an equal footing we form a complex
tions for the perturbation X′, in terms of the X¯, to suf- vectoroutofeachcomponentofthese partialdifferential
ficient accuracy, i.e., equations,utilizingthevariablesv± =vx ivy. Thespa-
±
tialvariationisassumedtobe periodicinthez-direction
C(X′)=ε¯E(X¯,X′). (56) withwavenumberk,asfollows: exp(ikz). Giventhis,the
followingpairoflinearordinarydifferentialequationsare
Note that X¯ has served two purposes. First, it gives us
obtained:
an approximate description of the corresponding exact
isnovluetsitoignatYe(i0t)s=staXbi0lit+y εw¯Y¯it,hoauntdesveecrohnadv,initgatlolofiwnsduY¯s.to ∂∂Bt± −ikB0v±1∓ kΩB0∂∂v±t1
1
B k A
0 0 y1 ′ ′
=iε¯ B0bkv±1+k Ω vx1±k0Ay1Bx cos(ω0t)
V. EQUATIONS OF MOTION AND NATURAL (cid:20)(cid:18) 1 (cid:19)
MODES ′ B0k0
A k B +ik sin(ω t) , (61)
∓ x1 0 y Ω 0
(cid:18) 1 (cid:19) (cid:21)
Our task then is to ascertain the primed variables ∂v±1 kB0
which are the excited fields. As stated previously, we δ01ρ01 ∂t ±iρ02Ω2(v±1−v±2)−i µ B±
0
8
=ε¯ δ ρ k v′ (A sin(ω t) iA cos(ω t)) In an analogous way to the classic driven pendulum
− 01 01 0 x1 x1 0 ± y1 0 problem [25], we have found that the velocities of the
h k
iΩ2ρ02 0Ax2cos(ω0t)(v±1 v±2) . (62) two species satisfy a pair of generalized Mathieu equa-
∓ ω0 − tions. Furthermore, the two species and magnetic field
i
arestronglycoupled. Toproceed,wemovefromthetime
Letus define the sense ofpolarizationin referenceto the
domaintothefrequencydomain,undertheFouriertrans-
screwsenseofthefieldsinthedirectionofpropagationin
form:
the z-direction. Thenv =v′ +iv′ correspondstoaleft
+ x y
hand circularly polarized wave for positive frequencies
∞
afrnedquaenrcigiehst, hwahnildecvi−rcu=lavrlx′y−poivlay′riczoerdrewspaovnedfsortoneagaritgivhet V±1(ω)= √12π Z−∞v±1(t)eiωtdt,
hand circularly polarized wave for positive frequencies 1 ∞
and a left hand circularly polarized wave for negative V±2(ω)= v±2(t)eiωtdt, (63)
√2π −∞
frequencies. Also, B± is defined similarly. With these Z
definitionsv′,v′,B′,B′ maybeeliminatedentirelyfrom
x y x y
these expressions. Note that the presence of a second where ω is the frequency ofthe excitedwaves. Using the
moving ion species introduces coupling throughterms of linearityofourpairofdifferentialequations,wecompute
the form: v±1 v±2, andanadditionaltermonthe right theFouriertransformofboth. UponeliminatingB± and
−
hand side in Eq.(62) due to density variations. V±2(ω) in favor of V±1(ω), we obtain:
1 1
F±(ω)V±1(ω)=ε¯ − 2(bvA21k2+P±(ω))(V±1(ω+)+V±1(ω−))+ 2P±(ω)(V±2(ω+)+V±2(ω−))
+Mh±( Ax1,+Ax1,ω+, 1)V+1(ω+)+N±( Ax2,+Ax1, 1)V+2(ω+)
− − − −
+M±(+Ax1, Ax1,ω−, 1)V+1(ω−)+N±(+Ax2, Ax1, 1)V+2(ω−)
− − − −
+M±( Ax1, Ax1,ω+,+1)V−1(ω+)+N±( Ax2, Ax1,+1)V−2(ω+)
− − − −
+M±(+Ax1,+Ax1,ω−,+1)V−1(ω−)+N±(+Ax2,+Ax1,+1)V−2(ω−) , (64)
i
where,
ω Ω δ 1 ω
F±(ω):=−δ01ω2+vA21k2(cid:18)1∓ Ω1(cid:19)± δ10201ω 1− 1∓∓ ΩΩω12!, (65)
Ω δ k V b
1 01 0 x2
P±(ω):=± δ02 ω ω0 − 1∓ Ωω2!, (66)
k v2 k2 1
M±(A1,D1,ξ,ǫ):=− 40(cid:20)(δ01ω± AΩ1i )(A1±Ay1)±(−δ01ξ+ǫ(1−δ01)Ω1) 1− δ02(1∓ Ωω2)!(D1+Ay1)(cid:21), (67)
k k Ω δ ωA A
0 0 1 01 2 y2
N±(A2,D1,ǫ):=± 4 ǫ(1−δ01)Ω1(D1+Ay1)− 4 δ Ω 1 ± ω , (68)
02 2 ∓ Ω2
and ω± :=ω ω0. Eq.(64) is obviously not an algebraic where χ is defined to satisfy,
±
expression from which we can uniquely obtain the dis-
persionequationbetweenω andk. Insteaditprovidesus F±(χ)=0. (70)
withafunctionalrelationshipbetweenthe Fouriertrans-
These define the naturally occurring modes in the ab-
forms with arguments ω, ω ω and ω+ω .
0 0 sence of pumping, (i.e., if ε¯= 0 then ω = χ). The solu-
−
Nevertheless,we mayproceedby noting thatthe right tionsofthesealgebraicexpressionsgivethedispersionre-
handisO(ε¯),whichissmall. Forthelefthandsidetobe lationscharacterizingplanetransversewavespropagating
small we must ensure that the frequency is near a root paralleltothebackgroundmagneticfieldinatwospecies
of the polynomial on this side with a correction of O(ε¯), plasma. The last term of F± is only present when a sec-
i.e., ondary moving ion-like species resides in the plasma. It
isanadditionalHall-typeterm,associatedwiththe rela-
ω =χ+O(ε¯), (69) tivemotionofthetwochargedspecies. Intheonespecies
9
limit this relation simplifies to
0.6
χ
χ2+v2 k2 1 =0, (71) (a)
− A1 (cid:18) ∓ Ω1(cid:19) χΩ/10.5
y
and we are left with the familiar fast and slow (ion- c
n
cyclotron) transverse waves [8]. ue 0.4
q
The important feature of the two-species result in e Lf
Fr
Eq.(70) is that it is a cubic in χ and hence another d 0.3
e
mode of excitation has been added. Moreover, since we cit
smixaysoclhuotioosnesF;t+hroereFa−reiltefftolhloawnsdtchiractutlahrelryepaorleaarizteodtaalnodf ed Ex 0.2
z R
three are right hand circularly polarized. However, only ali 0.1 Ls
m
three of these are physically different. This is because or
if χ satisfies F+(χ) = 0, then χ satisfies F−( χ) = 0, N 0.0
andviceversa. We maythenco−ncentrateonthe−positive 0.0 0.1 0.2 0.3 0.4 0.5
frequency solutions. Normalized Excited Wavenumber v k/Ω
A1 1
Ifthesecondaryspeciesispositivelychargedthenthere
are two left hand modes of excitation [14]. These we de-
notebyω andω ,wherethe“s”denotesslow,andthe
Ls Lf 0.6
“f” denotes fast. There is also a single right hand mode, (b)
dareenoptleodttebdyiωnRF.igT.h4(eac).orNreostpeonthdaintgthdeisrpiegrhstiohnanredlamtioodnes χΩy /10.5 Rf
c
intersects with the fast left hand mode. Also, the two n
e 0.4
left hand modes both experience a resonance as k . qu
→∞ e
F(worheωrLefwiethisavgeivaesnsubmyeΩd1Ω, 1an>dΩfo2r).ωLs it is given by Ω2 ed Fr 0.3
For a negatively charged secondary species there are cit
x
E 0.2
also three modes of excitation, however the combination d L
e
of polarizations has changed. In this case there are two z
right hand modes ωRf and ωRs, and a single left hand mali 0.1 Rs
mode ω ; see Fig.4(b). Here ω has a resonance at or
L Rs N
Ω ,andω hasaresonanceatΩ . Inthedustyplasma 0.0
− 2 L 1 0.0 0.1 0.2 0.3 0.4 0.5
case,further propertiesof these modes were investigated
Normalized Excited Wavenumber v k/Ω
in Refs.[13, 26, 27]. A1 1
Note that the upper curves (fast modes), regardlessof
the sign of the species, experience a nonzero cutoff fre- FIG. 4: The normalized excited wave frequencies χ/Ω1 ver-
quencywhichcoincideswiththatofthepumpfrequency, sus normalized wavenumber vA1k/Ω1 of the three natural
see Eq.(30). modes. (a) The second species is positively charged, with
δ01 = 1.1 and Ω2/Ω1 = 0.1. (b) The second species is nega-
tively charged, with δ01 =0.9 and Ω2/Ω1 =−0.1.
VI. GROWTH RATES OF PARAMETRIC
INTERACTION
transformwouldbe asumofthreeDirac-δ functions; see
From the above analysis we see that χ may be any
Fig.5(a)forarepresentation. Theeffectofpumpingisto
one of ω , ω , ω for a positively charged secondary
Ls Lf R modify this and provide frequency shifts. The modified
species, or ω , ω , ω for a negatively charged sec-
Rs Rf L solution will have some broadened frequency spectrum,
ondary species. Since the frequency describing the ex-
since it is the case that Dirac-δ functions do not solve
citedwaveisaperturbationofanaturalmodefrequency,
the Fourier transformed equations in the presence of a
we denote the change by φ which is allowed to be com-
pump. These types ofmodifications will occur near each
plex, where
naturalfrequency; see Fig.5(b) fora simplified represen-
tation of this effect. In fact an exact solution would be
ω =χ+ε¯φ. (72)
more complicated with the identification of harmonics
etc, but this figure should illuminate the key feature of
Now by returning to the Fourier transform relationship
the interaction.
in Eq.(64) we can see what effect φ (i.e., pumping) will
have on V±. Without any pumping (i.e., ε¯=0) the spa- This is described as a parametric interaction between
tial solutions would be a linear superposition of three the excited fields due to the pump fields. In order to
pure monochromatic exponentials. Hence the Fourier proceed we consider the case in which the interaction is
10
mode interacting with a right hand mode.
To illustrate, suppose the secondary species is nega-
tively charged and ω ω (i.e., χ = ω ). Then we
(a) L L
≈
can have ω ω ω or ω ω ω (left hand
0 Rs 0 Rf
− ≈ − − ≈ −
ω) modeinteractingwithrighthandmode)orω−ω0 ≈−ωL
V( (left hand mode interacting with left hand mode). In
m other words, the resonance condition is that the pump
or
sf frequency satisfies
n
a
Tr ω =ω +ω , ω =ω +ω , or ω =2ω , (73)
er 0 L Rs 0 L Rf 0 L
uri respectively, which is a statement of conservation of en-
o
F ergy. Thatis, the resonanceconditiondescribes the cou-
pling of a pump wave with two daughter waves.
Moreover,ifω ω thenwecanhaveω ω ω ,
Rs 0 L
≈ − ≈−
-ωL 0 ωRs ωRf ω−ω0 ≈−ωRf orω−ω0 ≈−ωRsandthesameconditions
Excited Frequency ω forresonanceapply. Similarrulesapplyforω ωRf and
≈
for a plasma where the secondary species is positive.
Let us now address the issue of the treatment of the
wavenumber k in our pair of interacting waves. First,
note that in the z-direction the pump wavenumber k is
0
(b) zero. For the natural modes we have that a wave given
by (k, χ) is physically equivalent to a wave given by
−
( k,χ). Thusourtwointeractingwaveswillbe givenby
ω) −
V( (k,χ1) and ( k,χ2), where χ1,χ2 are any combination
m of ω ,ω ,ω−,ω ,ω ,ω . Hence the wavenumbers
or L Ls Lf R Rs Rf
sf associated with each interacting wave are equal and op-
n
a posite, i.e.,
Tr
er k +k =0, (74)
uri 1 2
o
F whichisastatementofconservationofmomentuminthe
z-direction. The fact that the right hand side is zero re-
flectsthespatialuniformityofthepumpwave. Thistells
-ω 0 ω ω
L Rs Rf us that if the approximately spatially uniform standing
Excited Frequency ω
pump wave decays, then it does so into two daughter
waves of equal wavelength travelling in opposite direc-
tions. Since the wavenumber magnitudes are equal we
FIG.5: ArepresentationoftheFouriertransformV(ω)versus
may just consider the wavenumber k without referring
frequencyω. (a)Nopump,givingthreeDirac-deltafunctions.
to the sense of polarization. This is a direct generaliza-
(b) The effect of a pump, broadening the spectrum. Note
tion of the previous investigations in [7, 8, 14].
thatwehavechosentoillustratetheideausingonlytheright
handmodes(i.e.,V−),foraplasmawithanegativelycharged Withtheaboveframeworkwenowproceedtosolvefor
secondary species. φ from the Fourier transform relationship. We discuss
the method involved in obtaining φ in the case where
a left handed wave interacts with a right handed wave,
greatest, as in Refs.[7, 8]. It is clear that this effect will denoted φLR. Here we may assume, without loss of gen-
be greatest when the right hand side of Eq.(64) is large, erality, that χ = χL in Eq.(72), where χL is any one of
which is the resonance condition. This occurs when one ωL,ωLs,ωLf. This will interact with χR (i.e., χ ω0
− −
of the Fourier transforms on the right hand side is reso- may be any one of ωRs, ωRf, ωR).
− − −
nant, in other words when their arguments are near one FollowingthemethodologyofRef.[28]weformanother
of the χ’s, as conveyed in Fig.5. Since the right hand equationcorrespondingto(64)underthetransformation
side of Eq.(64) involves terms of the form: V(ω+) and ω → ω−ω0. In the resulting two equations, we neglect
V(ω−), it follows that there will be a large parametric the obviously nonresonantterms V(ω+) and V(ω 2ω0).
−
interactionwhen ω+ω or ω ω satisfies this. Now, as Additionally, due to our particular choice of interaction,
0 0
mentionedearlierwearerestr−ictingourattentiontopos- i.e., left-right, we neglect V+(ω−) and V−(ω), and retain
itiveω (withoutlossofgenerality). Underthiscondition, only V+(ω) and V+(ω−). We find the following:
it is found that only ω ω can satisfy this. Moreover,
there are three choices−for0this interaction: a left hand F+(χL+ε¯φLR)F−(χR−ε¯φLR)
=ε¯2[Y ( χ )+J (+A ,+A , A , χ , 1)]
mode interacting with a right hand mode, a left hand + R + x1 x2 x1 R
− − − −
mode interacting with a left hand mode, or a right hand [Y (+χ )+J ( A , A ,+A ,+χ , 1)],(75)
+ L + x1 x2 x1 L
× − − −