Rayleigh Instabilty

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J. Plasma Physics (2005), vol. 71, part6, pp. 747751. c 2005 Cambridge University Press

doi:10.1017/S0022377805003648 Printed in the United Kingdom

747

Rayleigh instability in non-uniformmulti-ion species magnetoplasmas

K H . H . E L - S H O R B A G Y and P. K. S H U K L A

Plasma Physics Department, Nuclear Research Center, Atomic Energy Authority,Cairo, Egypt

Ruhr-Universitat Bochum, D-44780 Bochum, Germany([email protected])

(Received 9 August 2004)

Abstract. We present the Rayleigh instability of low-frequency (in comparison

with the ion-gyrofrequency) electrostatic waves in multi-ion species plasmas con-

taining equilibrium density gradient and transverse ion-sheared flows caused bythe dc electric field. For this purpose, multi-fluid equations are used to obtain

the governing equations for two-dimensional convective cells and pseudo-three-

dimensional drift waves in a non-uniform magnetoplasma. Standard procedures

are then employed to study the Raleigh instability. Thresholds and growth rates

are obtained. Applications to Hall thrusters are discussed.

1. Introduction

The RayleighTaylor (RT) instability [1,2] occurs when a heavy fluid is supported

by a lighter fluid. Any perturbation of the interface grows and leads to spikes of the

heavier fluid penetrating into the lighter one. In a magnetized plasma, the Rayleigh

instability can occur because the magnetic field acts as a light fluid supporting a

heavy fluid (the plasma). In curved magnetic fields, the centrifugal force on the

plasma due to the charged particle motion along the curved field lines acts as an

equivalent gravity force. When forces associated with the density gradient and

gravity oppose each other, the RT instability sets in.

However, multi-species plasmas in a Hall thruster [3] are confined in crossed

electric and magnetic fields that are non-uniform. The ions are accelerated in the

imposed electric field to velocities on the order of tens of km s1. Hall thrusters arebeing developed at low power for use on micro-spacecraft, and at high power for

spacecraft orbit raising. Recently, Litvak et al. [4,5] studied the Rayleigh instabil-

ity of high-frequency (in comparison with the lower-hybrid resonance frequency)

electrostatic waves in a non-uniform electronion plasma. They carried out a non-

local analysis and presented stability criteria that are relevant for Hall thrusters.

In this paper, we investigate the Rayleigh instability of low-frequency (in com-

parison with the ion-gyrofrequency) electrostatic waves in a non-uniform multi-ion

plasma that is held in crossed electric and magnetic fields. We focus on the two-

dimensional convective cells and pseudo-three-dimensional drift waves. The multi-

fluid equations are used to derive the governing equations, which are analyzed toobtain thresholds and growth rates of the Rayleigh instability. Possible applications

to Hall thrusters are discussed.


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748 Kh. H. El-Shorbagy and P. K. Shukla

2. Formulation

We consider a non-uniform magnetoplasma whose constituents are electrons and

two ion species of different charge to mass distributions. The plasma contains

non-uniform electric (E0=x0/x, where 0 is the dc electric potential) anda uniform magnetic field zB0. In such a plasma we have a non-uniform particledrift velocityv0= yu0(x), whereu0(x) =(c/B0)0/x and c is the speed of lightin vacuum.

We first study instability of our equilibrium against two-dimensional electrostatic

convective cells whose electric field is E = , where is the wave potential.Assuming that the convective cell frequency is much smaller than the gyrofrequency

c= eB0/m c, namely, |/t|c, where e is the magnitude of the electroncharge, m is the mass, and stands for the particle species ( equals e for theelectrons and1, 2for the two different ion species), we obtain for the perpendicularcomponent of the electron and ion fluid velocities which are, respectively,

ve =

c

B0 z , (1)

and

v = c

B0z

c

B0c

t+

c

B0z

, (2)

where = 0+ .Substituting (1) into the electron continuity equation, letting ne= ne0(x) +

ne1exp(it+iky), we obtain for the electron density perturbation ne1ne0

ne1= ck

B0( kv0)

ne0x

, (3)

where and k are the frequency and the y component of the wavevector. Fur-thermore, substituting (2) into the ion continuity equation, letting n =n0(x) +n1exp(it+iky), we obtain for the ion density perturbation n1n0

n1= ck

B0( ku0)

n0x

+ cn0B0c

2

x2 k2

+

ckn0

B0c ( ku0)

2u0x2

. (4)

Inserting (3) and (4) into Poissons equation

2

x2 k2= 4e

ne1 =1,2

Z n1

, (5)

we obtain the equation for the convective cells

Z n0c

2

x2 k2

+

n0c

k

( ku0)u0

(x) = 0, (6)

where we have assumed that

=1,22

p /2c1. Herep = (4Z

2 e

2n0/m )1/2

is the ion plasma frequency, and

=1,2Z n0=ne0.Equation (6) is similar to the Rayleigh equation, well known in fluid dynamics [6].

In fluid dynamics, instability of flows with a step-like transverse profile of velocity is

well known [7], and is customarily called the KelvinHelmholtz instability. A sim-

plified case of such instability without the density and magnetic field gradientshas been considered earlier [8]. When the kinks in the drift velocity profile and

the inhomogeneity factor are smeared out, the profiles become similar to those


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Rayleigh instability in non-uniform multi-ion species magnetoplasmas 749

existing in Hall thrusters, and the KelvinHelmholtz-type instability described

below becomes a Rayleigh instability. By using a method similar to [6], we can

derive the following dispersion relation for azimuthally propagating unstable mode

[( kV1)2 + ( kV2)

2] cosh(ka) = 0, (7)

where V1 and V2 are the flow speeds in the regions x 0, respectively.From (7) we find that

=k

2[(V1+V2) ik(V2 V1)], (8)

which admits two modes with the frequency =k[(V1+ V2)/2], and one of themis unstable with the growth rate

= 12

k(V2 V1). (9)

The obtained solution represents a wave which is in phase with the electron flow at

the center point, thus satisfying the necessary condition for the Rayleigh instability.Second, we study instability of our equilibrium against low-frequency (in com-

parison with c ) electrostatic drift perturbations whose parallel phase speed ismuch smaller than the electron thermal speed. Here, inertialess electrons rapidly

thermalize along the magnetic field direction and establish a Boltzmann distribu-

tion. The corresponding density perturbation is

ne1=ne0e

Te, (10)

whereTe is the electron temperature.

Inserting (4) and (10) into (5) we obtain the drift wave equation 2

x2 k2

+

c k

n0( ku0)

n0x

+ k

( ku0)u0

(x) = 0, (11)

wherek2 =k2+1/2 and1/2 =

(e B0cne0/c n0Te). Here we have assumedthat is much larger than the electron Debye radiusrD= (Te/4ne0e2)1

/2.

The necessary condition for the Rayleigh-type instability can be readily obtained

by multiplying (11) with and assuming the complex frequency =r+ii, wherethe subscripts r and i stand for the real and imaginary parts, respectively, and the

asterisk denotes the complex conjungate. We have

u0

(x) +c

x(ln n0) = 0. (12)

Thus, an absolute instability is possible if at any pointxcin the transverse direction(perpendicular to the direction of the flow), the condition (12) is satisfied.

Further linear stability shall be discussed in detail for some specific profiles of

the shear plasma flow and multi-ion species density. We suppose that the plasma

flow profile is described by the following well-behaved analytic function

u0(x) =u+a tanh(x), (13)

where1

defines the characteristic width (slope) of the shear flow. We now proceedcalculating the appropriate derivatives ofu0(x)by introducing a new variable =tanh(x), and by assuming a physically interesting case when u = /k. We then


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750 Kh. H. El-Shorbagy and P. K. Shukla

rewrite (11) in the form

(12)d2

d22

d

d+

2

k2 + 1

21

1 2+

1

21

1 2

cn0

1

2

dn0d

= 0. (14)

Obviously, forn0 satisfying

n0 = (1 2)1/2 c (1 tanh2(x))1/2 c (15)

the solution of (14) can be written in the form of the Legendre functions [911] of

the degree1 and of the order= (k2 + 1)1/2/, i.e.

() =C1P1

() +C2Q1

(), (16)

where

P1

() =

(2 )

+ 1

1

/2,

Q1

() =

2(2 )

exp(i)

sin()

(+)

1

+ 1

/2 (+ 2)

+ 1

1

/2.

The only localized solutions for 1, i.e. for x , are possible if

2 k2 + 1

2 = 1. (17)

The wave becomes convectively unstable (growing in space) because of the resonant

interaction with the ion flow, provided that

k >

2 12 1

1/2. (18)

We obtain from (11) the following local dispersion relation for the multi-ion

species that is modified due to sheared ion flows

=c k

k2

x(ln n0) +ku0+

k

k2u0

. (19)

Assuming a small deviation k of the stable value of the wavenumber k, letting = r +ii in (14), and following the analysis in [9, 10], we readily obtain thegrowth rate

i =4ak(2 1/2 1)

(2a2 1)

du0dx

x=0

. (20)

Now, for the profiles ofu0(x)and n0, given by (13) and (15), using the Rayleighcondition (12), we obtain the following relation between the flow width1 and thecritical point for the instability [11]

tanh( xc) =

1

1

2a2 (21)

Thus, an absolute instability at the critical point xc for modes with wavenumbers

satisfying the condition (18) can be expected. Furthermore, it follows from (21)that there appears a limiting value ofL 0.71 above which the critical point xcbecomes complex, i.e. the Rayleigh-type instability disappears.


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Rayleigh instability in non-uniform multi-ion species magnetoplasmas 751

3. Conclusions

In this paper, we have investigated the Rayleigh instability of two-dimensional

convective cells and pseudo-three-dimensional drift perturbations in a non-uniform

multi-ion species plasma containing equilibrium density and transverse velocity

gradients. The latter appears due to a non-uniform dc electric field, which produces

the E0B0 drift of the plasma particles. Such a situation is quite common for

Hall thrusters comprising multi-ion species. Our results show that the presence of

sheared ion flows can cause instability of flute-like convective cells and pseudo-

three-dimensional drift wave perturbations. However, we find that instability cri-

teria is sensitive to the choice of equilibrium density and sheared ion flow profiles.

Physically, instability is attributed to resonance interactions between electrostatic

modes and sheared ion flows. In conclusion, we stress that in order to have a

successful operation of Hall thrusters, low-frequency macroscopic instabilities, as

discussed here, should be avoided.

AcknowledgementThis research was partially supported by the International Atomic Energy Author-

ity, Vienna.

References

[1] Kadomtsev, B. B. 1965Plasma Turbulence. New York: Academic.

[2] Goldston, R. J. and Rutherford, P. H. 1995 Introduction to Plasma Physics. Bristol:Institute of Physics.

[3] Raitses, Y., Keidar, M., Staack, D. and Fisch, N. J. 2002 J. Appl. Phys. 92, 4906.

[4] Litvak, A. A., Raitses, Y. and Fisch, N. J. 1999 Proc. 38th Joint Propulsion Conf.,Indianapolis, IN, 2002. Paper AIAA 2002-3825. Washington, DC: American Instituteof Aeronautics and Astronautics.

[5] Litvak, A. A. and Fisch, N. J. 2004Phys. Plasmas11, 1379.

[6] Lin, C. C. 1955 Theory of Hydrodynamic Stability. Cambridge: Cambridge UniversityPress.

[7] Helmholtz, H. 1868 Ueber discontinuirliche FluessigkeitsBewegungen. Akad. Wiss.Berlin: Monatsber.

[8] Kapulkin, A. M. and Prisnyakov, V. F. 1995 Proc. 24th Int. Electric Propulsion Conf.,Moscow, Russia, 1115 September 1995, IEPC Paper 95-37.

[9] Vranjes, J. and Jovanovic, D. 1996Phys. Plasmas 3, 2275.

[10] Vranjes, J. and Tanaka, M. Y. 2002Phys. Plasmas 9, 4379.[11] Vranjes, J., Petrovic, D. and Shukla, P. K. 2001Phys. Lett. A 278, 231.

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