ARTICLES Electrostatic drift modes in a closed field line configuration

PHYSICS OF PLASMAS

VOLUME 9, NUMBER 2

FEBRUARY 2002

ARTICLES

Electrostatic drift modes in a closed field line configuration J. Kesner and R. J. Hastie Plasma Science and Fusion Center, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139

共Received 7 August 2001; accepted 7 November 2001兲 The stability of electrostatic drift waves in a closed field line configuration in collisionality regimes ranging from collisional to collisionless is compared. The maximum sustainable pressure gradient is found to be dependent on the ratio of the temperature and density gradients ( ␩ ⬅d ln T/d ln n). The eigenmodes are seen to be flute-like. The stability boundary was found to be similar when both ions and electrons are collisional, when they are collisionless, and for collisional electrons and collisionless ions. The largest stable pressure gradients are obtained for ␩ ⭓2/3. As the collisionality is reduced one observes some reduction of the region of stability. © 2002 American Institute of Physics. 关DOI: 10.1063/1.1431594兴

I. INTRODUCTION

For a collisionless plasma and considering low frequency perturbations that conserve the adiabatic invariants ␮ and J, particles that are magnetically trapped 共all the particles for closed field lines兲 change energy as the field lines compress. This effect has been shown to correspond exactly to the MHD plasma compressibility. For passing particles, however, the passing particles sample the entire flux surface and the path integral of perturbed potential will average to zero.8 The description of drift ballooning modes in tokamaks, has been studied extensively.9–11 The study of Ref. 11 indicates the importance of the parameter ␩ in expanding the region of instability in a tokamak. However, the results in this reference apply to a tokamak equilibrium and in particular they do not apply to the case of zero safety factor and shear 共i.e., q⫽s⫽0). In the collisionless regime, ␻ , ␻ j , ␻ d j Ⰷ ␯ j , drift modes * were shown to be stable when the MHD stability condition is met and ␩ ⬃2/312 and stability was seen to improve as k⬜ ␳ i increases, with k⬜ the wave number perpendicular to the magnetic field direction and ␳ i the ion gyro radius. For ␩ ⬎2/3 a drift frequency mode was observed and it became more unstable when k⬜ ␳ i increased. Electromagnetic effects on these modes have been studied. Wong et al. have shown that finite beta effects are stabilizing at low beta.13 They also consider the electromagnetic effects at very high beta, and they have shown that there can be new regions of instability when ␤ Ⰷ1.13 A levitated dipole experiment known as LDX is presently under construction.2 The most relevant parameter regime for LDX would have collisional but long mean free path electrons ( ␻ be ⬎ ␯ e ⬎ ␻ , ␻ de ) and collisionless ions. We will, therefore, consider the stability of low beta electrostatic modes with this intermediate collisional ordering and compare our results with those obtained with collisional and collisionless orderings.

Closed field line systems, such as a levitated dipole, provide a promising new approach for the magnetic confinement of plasmas for controlled fusion.1,2 The plasma in a closed field line system can be stabilized in so-called ‘‘bad curvature’’ regions by plasma compressibility. In magnetohydrodynamic 共MHD兲 theory stability of interchange modes limits the pressure gradient to a value d⬅d ln p/d ln U⬍␥ with U ⫽养dl/B and ␥ the ratio of specific heats ( ␥ ⫽5/3 in threedimensional systems兲. Recent studies which imposed a long mean free path collisional ordering 共defined by ␻ b j Ⰷ ␯ j Ⰷ ␻ , ␻ j , ␻ d j with * ␻ the wave frequency, ␻ b j the bounce frequency, ␯ j the collision frequency, ␻ j the diamagnetic drift frequency, ␻ d j * the curvature drift frequency兲 as compared with the short mean free path collisional MHD ordering ( ␯ j Ⰷ ␻ b j Ⰷ ␻ , ␻ j , ␻ d j ) to both electrons and ions3–5 have shown that * near marginal stability for MHD modes (d⬃5/3) the MHD mode will couple to a potentially unstable drift frequency mode known as the ‘‘entropy mode.’’ The entropy mode is flute-like and stability depends on d and on the ratio ␩ ⫽d ln T/d ln n, with the most stable value being ␩ ⫽2/3. Furthermore it has been shown that ion collisional relaxation 共described as gyro-relaxation in Refs. 6,7兲 can destabilize the entropy mode and produce a weak instability in some otherwise stable regions of d⫺ ␩ space.5 In a short mean free path collisional plasma 共as characterized by MHD兲 the plasma compressibility derives from the local relationship of pressure to changing flux tube volume. The plasma is constrained to move with the field lines but in the case of closed field lines the field does not form flux surfaces and each closed field line must satisfy the equation of state. This imposes a periodicity on the perturbation that results in an important stabilizing term, the plasma compressibility. For open field lines the plasma is free to flow along field lines. This eliminates plasma compressibility. 1070-664X/2002/9(2)/395/6/$19.00

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© 2002 American Institute of Physics

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II. SOLUTION OF DRIFT KINETIC EQUATION

We will compare the MHD prediction with the predictions of the more general plasma drift kinetic equation in the electrostatic limit. To compare the MHD result with kinetic theory we define

␻ˆ

*

p⬅

bÃk•“p ⍀ im in i

共1兲

and

␻ˆ MHD ⬅ d

2c Rk ␪ T 养dl ␬ /RB 2 , e 1⫹ ␥ 具 ␤ 典 /2 养dl/B

共2兲

with R the cylindrical radial coordinate, ␬ the field line curvature, k ␪ the azimuthal part of the perpendicular wave number (k⬜2 ⫽k ␪2 ⫹k R2 ), and k ␪ R⫽mⰇ1. One can show that d ˆ p /␻ ˆ MHD and therefore the defined above is equal to d⫽ ␻ d * MHD stability requirement, d⭐ ␥ , can be written as ˆ MHD ⭐␥␻ . 共3兲 d *p We will consider the solution of the drift kinetic equation in the high collision frequency limit for electrons and low collision frequency for ions. We therefore apply the ordering for electrons:14

␻ˆ

⍀ ce ⬎ ␻ be ⬎ ␯ e ⬎ ␻

*e

⬃ ␻ de ⬃ ␻ ,

共4兲

and for ions, ⍀ ci ⬎ ␻ bi ⬎ ␻ i ⬃ ␻ di ⬃ ␻ ⬎ ␯ i ,

共5兲 * with ⍀ c j the appropriate cyclotron frequency. To derive the stability criterion for electrostatic modes we consider a fluctuating potential ( ␾ ) and ignore any equilibrium electrostatic potential. From Faraday’s law it is possible for a perturbation to leave the magnetic field undisturbed if E⫽⫺“ ␾ 共which is consistent with ␤ Ⰶ1). The gyro kinetic equation was derived under the assumption that the wave frequency ␻ is less than the cyclotron frequency ⍀ c and the perpendicular wavelength ␭ ⫽2 ␲ /k⬜ is short compared to a parallel wavelength, 2 ␲ /k 储 . The appropriate equation for the gyro averaged distribution function ˜f is then8,15,16 ˜f ⫽q ␾ F 0 ⑀ ⫹J 0 共 Z 兲 h,

共6兲

and the nonadiabatic response h satisfies 共 ␻ ⫺ ␻ d ⫹i v 储 b•“ ⬘ 兲 h⫽⫺ 共 ␻ ⫺ ␻ 兲 q ␾ F 0 ⑀ J 0 共 Z 兲 ⫹iC 共 h 兲 .

*

共7兲

In Eq. 共7兲 C(h) is the collision operator, J 0 (Z) is the Bessel function of the first kind, F 0 ( ⑀ , ␺ ) is the equilibrium distribution function, i.e., F 0⫽

冉 冊 m 2␲T

⑀ (2 共 1⫺␭B 兲 b•“b⫹␭“B… ⫽k•bà , ⍀c B⫽“ ␺ Ó ␪ ,

n 0 e ⫺ ⑀ /T ,

共8兲

⳵F0 F 0⑀⬅ , ⳵⑀

bÃk•ⵜ ⬘ F 0 ␻ ⫽ , * m⍀ c F 0 ⑀

共9兲

b⫽B/ 兩 B兩 .

We have defined ⑀ ⫽m v 2 /2, ␮ ⫽m v⬜2 /2B, and ␭⫽ ⑀ / ␮ . The prime on spatial gradients indicates that ⑀ and ␮ are held fixed in the differentiation. The magnetic flux function is ␺ and ␪ is the azimuthal angle. Assuming k⬜2 ␳ 2i Ⰶ1, J 0 共 Z 兲 ⬇1⫺

2 共 k⬜ ␳ i 兲 2 ␭B 0 ⑀ , 2 B Ti

共10兲

2 , B 0 the with the gyro radius ␳ i , defined as ␳ 2i ⫽T i /m i ⍀ 0i magnetic field at the field minimum at the location R⫽R 0 and ⍀ 0 the associated cyclotron frequency. We consider perturbations whose growth time is long compared to a particle bounce time for all species and expand Eq. 共7兲 in powers of ␦ ⫽ ␻ / ␻ b with ␻ b the bounce frequency, i.e., h⫽h 0 ⫹ ␦ h 1 ⫹ ␦ 2 h 2 ⫹••• . Notice ␦ ⬃ ␻ / ␻ b ⬃k ␪ ␳ i and corrections to the perturbed density will * enter as ␦ 2 ⬀k 2␪ ␳ 2i . To zeroth order in ␦ we obtain from Eq. 共7兲:

v 储 b•ⵜ⬘ h 0 ⬇0,

共11兲

i.e., h 0 ⫽h 0 ( ⑀ , ␮ , ␺ ) a constant along a field line. We determine the constant h 0 by taking the bounce average of the first order form of Eq. 共7兲 to annihilate h 1 to obtain ¯ d 兲 h 0 ⫽⫺ 共 ␻ ⫺ ␻ 兲 q ␾ J 0 F 0 ⑀ ⫹iC ¯ 共 h0兲. 共␻⫺␻

共12兲 * Since we will not pursue the ␻ / ␻ b expansion to higher orders, we will suppress the subscript 0 in the following analysis of Eq. 共12兲. The overbar indicates a bounce time average: ¯⫽ ␾

冑m/2⑀ ␶b

冖 冑␾

共 l 兲 dl

1⫺␭B

共13兲

,

␶b defined as ␶b with the bounce time 冑 冑 ⫽ m/2⑀ 养dl/ 1⫺␭B. The electron collision operator conserves particles and energy. With the chosen ordering and these conservation properties the exact form of the electron collision operator does not enter the results. Consider first collisional electrons following Ref. 3. We analyze Eq. 共12兲 in the high collisionality limit ( ␯ e / ␻ Ⰷ1) and expand h e in a subsidiary ordering, h e ⫽h 0 ⫹h 1 ⫹••• . To lowest order we find that h 0 is proportional to a Maxwellian distribution function, F 0 , multiplied by a perturbed density, ␦ N, and having a perturbed temperature T⫹ ␦ T. 14 Expanding to first order we obtain

3/2

and k⬜ v⬜ , Z⫽ ⍀c

共 v 2储 b•“b⫹ ␮ “B 兲 ␻ d ⫽k•bà ⍀c

h 0e ⫽ ␦ N ⬇

冋

冉

␦N n0

m 2 ␲ 共 T⫹ ␦ T 兲 ⫹

冉 冊册

␦T ⑀ T

冊

3/2

T

⫺

3 2

e ⫺ ⑀ /(T⫹ ␦ T) F0 .

共14兲

To next order the drift kinetic equation becomes

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Electrostatic drift models in a closed field line configuration

397

¯ d 兲 h 0 ⫽⫺ 共 ␻ ⫺ ␻ 兲 q ␾ J 0 F 0 ⑀ ⫹iC ¯ 共 h1兲. 共␻⫺␻

共15兲 * We can obtain the perturbed density and temperature for the ¯ (h 1 ) with the two operaelectron species by annihilating C tors:

冖 冕

冉 冊 1

dl B

d 3v

⫽␲

冉冊 冖 冕

m v 2 /2

3/2

2 m

⬁

dl

0

d⑀

冉 冊冕 ⑀ 1/2 ⑀

3/2

d␭

1/B

0

冑1⫺␭B

.

共16兲

Since the collision operator conserves particles and energy these operators will annihilate it, i.e.,

冖 冕 dl/B

¯ 共 h 兲⫽ d 3v C

冖 冕 冉⑀ 冊 dl/B

d 3v

T

⫺

3 ¯ 共 h 兲 ⫽0. C 2

¯ di normalized to FIG. 1. Bounce averaged curvature drift frequency ␻ ˆ ⑀ ␻ di /T 关Eq. 共22兲兴 in a dipole field versus B 0 ␭(⫽B 0 /B bounce).

共17兲

ˆ by We will define ␻

␦ne

*

␻ˆ j ⫽ *

n

TkÃb•“n 0 , n j m⍀

⫽⫺

共18兲 ⫹

and write

␻ ⫽ ␻ˆ „1⫹ ␩ 共 ⑀ /T⫺3/2兲 …F 0 , * *

共19兲

ˆ p⫽ ␻ ˆ (1⫹ ␩ ). Taking with ␩ ⫽d ln T/d ln n. Notice that ␻ * * the flux tube and velocity space integral of Eq. 共15兲 we obtain the following expression for the nonadiabatic perturbed density ␦ N e for electrons from Eq. 共14兲:

␦ N e⫽

␻ˆ de n 0 共 ␦ T e /T e 兲 ⫹q e 共 ␻ ⫺ ␻ˆ e 兲 具 ␾ 典 /T e * , ␻ ⫺ ␻ˆ de

兰 ␾ dl/B

冖

dl/B

冖

B 2R

共 ␬ ⫹“⬜ B/B 兲 ,

Te

⫽

2 3

ˆ de ⫺q e ␩ e ␻ ˆ 共 ␦ n e /n e 兲 ␻

␻ ⫺ ␻ˆ de 7 3

␾ /T *e具 典 .

共24兲

冋 册

5 1⫹ ␩ . 7 1⫺ 37 ␩

共25兲

In Refs. 3–5 the ion response was assumed to be collisional. Here we assume a collisionless ion response and ignore terms of order k⬜2 ␳ 2i to obtain ni

dl

共22兲

ˆ d becomes with U⫽养dl/B. For low ␤ , ␬ ⬇“⬜ B/B and ␻ equal to the MHD definition given in Eq. 共2兲. To obtain ( ␦ T e /T e ) we take the flux tube average and integrate over velocity space for ( ⑀ /T⫺3/2)⫻ Eq. 共15兲 which again annihilates the collision operator to obtain

␦Te

d⫽

␦ni

ˆ d j , the flux tube averaged drift, defined as and ␻ cT j 共 Rk ␪ 兲 ␻ˆ d j ⫽ q jU

qe ˆ e ,␻ ˆ de 兲 …. „⫺ ␾ ⫹ 具 ␾ 典 ⌳ ce 共 ␻ , ␻ * Te

册

A. Collisional electrons, collisionless ions

共21兲

,

冋

7ˆ 7 ˆ ˆ ˆ q e具 ␾ 典 ␻ 2⫺ ␻ 共 3␻ de ⫹ ␻ e 兲 ⫹ ␻ de ␻ e 共 3 ⫺ ␩ e 兲 * * 2 Te ␻ 2 ⫺ 103 ␻ ␻ˆ de ⫹ 35 ␻ˆ de

If we assume both collisional electrons and ions so that the zero gyroradius ion response becomes similar to Eq. 共24兲 ˆ de , ␻ ˆ e→ ␻ ˆ di , ␻ ˆ i 兲 we can apply quasi-neutrality to 共with ␻ * * obtain the marginal stability condition when

共20兲

with the flux tube average defined as

具␾典⫽

⬅

q e␾ Te

共23兲

Equations 共20兲 and 共23兲 determine ␦ N e /n and using Eq. 共6兲 we construct the total electron density perturbation ␦ n e /n ⫽ ␦ N e /n⫺q e ␾ /T e ,

⫽⫺ ⬅

q i␾ q i ⫹ Ti Ti

冕

d 3v

␻ ⫺ ␻ˆ i „1⫹ ␩ i 共 ⑀ /T i ⫺3/2兲 … * ¯␾ F 0 ¯ di 共 ⑀ ,␭ 兲 ␻⫺␻

qi ˆ i ,␻ ˆ 兲 …. „⫺ ␾ ⫹⌳ i 共 ␻ , ␻ * di Ti

共26兲

The pitch angle dependence of the normalized bounce aver¯ di ( ⑀ ,␭)T i /( ⑀ ␻ ˆ di ) is shown in Fig. 1 for the moage drift ␻ tion of a particle in the field of a point dipole. The pitch angle parameter ␭ varies in the range (0⬍␭⬍1/B 0 ) with B 0 the minimum magnetic field on the outer midplane. We ob¯ di T i / ⑀ ␻ ˆ di is relatively constant as a function of serve that ␻ pitch angle in a dipole field 共it varies from 0.6 to 0.75兲 and to obtain the correct MHD response to order ( ␻ di / ␻ ) when ␻ Ⰷ ␻ , ␻ d we will choose

*

¯ di 共 ⑀ ,␭ 兲 ⬇ ␻

2 ⑀ ␻ˆ . 3 T i di

共27兲

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Phys. Plasmas, Vol. 9, No. 2, February 2002

J. Kesner and R. J. Hastie

This approximation is discussed further in the Appendix. To obtain a dispersion relation we apply quasi-neutrality. Taking T i ⫽T e and applying quasi-neutrality to Eqs. 共24兲 and 共26兲 yields 2 ␾ ⫽⌳ i ⫹ 具 ␾ 典 ⌳ ce .

共28兲

Taking the flux tube average of Eq. 共28兲 yields a dispersion relation that does not depend on the spatial dependence of the eigenfunction ␾ (l), 2⫽F i 共 ␻ 兲 ⫹⌳ ce ,

共29兲

where F i共 ␻ 兲 ⫽

冕

d 3v F 0

␻ ⫺ ␻ˆ i „1⫹ ␩ i 共 ⑀ /T i ⫺3/2兲 … * . 2 ⑀ ˆ ␻⫺ ␻ 3 T di

共30兲

It has been shown for the collisional ordering that the solution is flute5 and we will show below that, in the collisionless case the solution is also flute to order k⬜2 ␳ 2i . In the present intermediate collisionality case, a comparison of Eqs. 共28兲 and 共29兲 shows that a flute eigenmode is also present. Equation 共29兲 can be written in the form ⍀3

冉

冊

冉

3d␩ 3d␩ ⫺1⫺ ␩ I 1 共 ⍀ 兲 ⫹⍀ 2 1⫹ ␩ ⫺ 2 2 ⫹

冉

冊 冊

冉

冉

⫹

冊 冊

冊

⫺5 d 5 d ␩ ⫹ I 1 共 ⍀ 兲 ⫽0, 3 2

ˆ de , d⫽(1⫹ ␩ ) ␻ with ⍀⫽ ␻ / ␻ I 1共 y 兲 ⫽

冕 冑␲ 4

⬁ x 2 e ⫺x

0

2

dx

2

y⫹2x /3

*e

共31兲

共32兲

which can be expressed in terms of error functions. Equation 共31兲 takes the form D共 ⍀ 兲⫽

d 关 F 共 ⍀ 兲 ⫹ ␩ F 2 共 ⍀ 兲兴 ⫺F 3 共 ⍀ 兲 ⫽0. 1⫹ ␩ 1

共33兲

It can be shown analytically that there is no marginally stable mode with ⍀⬍0. For modes with ⍀⬎0 there are no driftresonant ions and, at marginal stability, there must be coincident real roots for ⍀ so that ⳵ D/ ⳵ ⍀⫽0, i.e., d 关 F ⬘ 共 ⍀ 兲 ⫹ ␩ F 2⬘ 共 ⍀ 兲兴 ⫽F 3⬘ 共 ⍀ 兲 . 1⫹ ␩ 1 Substituting Eq. 共34兲 into Eq. 共33兲 yields

冊

3 1⫺ ␩ 共 F ⬘1 F 3 ⫺F 3⬘ F 1 兲 ⫽0. 2

d⫽0.65868

ˆ de , and /␻

,

冉

共36兲

共37兲

Substituting ⍀⫽⍀ c into Eq. 共33兲 yields the equation for the stability boundary:

10 7d 3d␩ ⫺ 共 1⫹ ␩ 兲 ⫺ 3 3 2

冉

共35兲

One can show that (F 2⬘ F 3 ⫺F ⬘3 F 2 )⫽⫺(3/2)(F 1⬘ F 3 ⫺F ⬘3 F 1 ) and substituting into 共35兲 yields the result

⍀ c ⫽0.3218.

5 10 d 5 d ␩ ⫺ ⫹ ⫺ 共 1⫹ ␩ 兲 ⫹ I 1共 ⍀ 兲 3 3 2 ⫹

共 F ⬘1 F 3 ⫺F ⬘3 F 1 兲 ⫹ ␩ 共 F 2⬘ F 3 ⫺F ⬘3 F 2 兲 ⫽0.

At marginal stability Eq. 共36兲 yields a real frequency, independent of ␩ ,

10 10 ␩ 7 d ␩ ⫺d⫹ ⫺ I 1共 ⍀ 兲 3 3 2

13 ⫹⍀ ⫺ 共 1⫹ ␩ 兲 ⫹d⫹5 d ␩ 3

FIG. 2. The stability of semicollisional 共collisional electron-collisionless ion兲 mode in 共d,␩兲. The stable region is shaded.

共34兲

1⫹ ␩ . 1⫺0.51214␩

共38兲

For ⫺1⬍ ␩ ⬍2/3 the stability boundary, Eq. 共38兲, is close to, and below, the collisional boundary given by Eq. 共25兲. At marginal stability the mode has a real frequency given by Eq. 共37兲 and it propagates in the electron diamagnetic direction. We have solved Eq. 共29兲 numerically using a zerofinding routine to obtain the eigen frequency and we have utilized a Nyquist analysis to establish that we have found all possible unstable modes. Figure 2 shows the stability diagram in d⫺ ␩ space in the semi-collisional regime given by Eq. 共38兲. The thick line at d⫽5/3 is the MHD stability boundary. This appears when k⬜ ␳ i ⫽0 and the region d⬎5/3 is unstable. Notice that at the point where the pressure gradient vanishes, ␩ ⫽⫺1 and d ⫽0, and this is a marginally stable point. The topology of the stability region is similar to that shown in Ref. 5 for the entropy mode. We can add finite Larmor radius 共FLR兲 terms in the limit k⬜ ␳ i ⬎k ␪ ␳ i 共this limit does not require higher order terms in the bounce expansion兲. For sufficiently small FLR corrections a MHD-like mode appears when d⬎5/3. For the collisional case it was found that increasing FLR terms will raise the stability boundary in the vicinity of ␩ ⫽2/3 and reduce it elsewhere.3 In the semi-collisional case discussed here we find that FLR corrections lead to a degradation of the stabil-

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ity boundary when ␩ ⬎2/3. For example, for ␩ ⫽2, (k⬜ ␳ i ) 2 ⫽0.001 the stability boundary degrades by 5% 共from d ⫽5/3). For ␩ ⬍2/3 no significant degradation of the stability boundary is observed. B. Collisionless mode

Rosenbluth8 considered a collisionless isothermal plasma 共i.e., ␩ ⫽0兲 in a closed field line system. He showed that when T e ⫽T i an instability is always present, with zero real frequency at marginality, when d exceeds a critical value provided that some particles bounce in bad curvature. We now extend this result to arbitrary values of ␩ , and obtain an analytic expression for the stability boundary in a point dipole 关to be compared with Eqs. 共25兲 and 共38兲兴. To obtain a dispersion relationship for collisionless electrostatic modes we assume a collisionless electron response and apply quasi-neutrality. For k⬜2 ␳ 2i Ⰶ1 we obtain the dispersion relation

冕

2␾⫽

⫹

␻⫺␻ 3

d v

*e

1⫹ ␩ e

⑀ 3 ⫺ Te 2

2 ⑀ ␻⫺ ␻ˆ 3 T de

␻⫹␻

冕

冉 冉

d 3v

*

e

冉 冉 1⫹ ␩ i

冊冊

⑀ 3 ⫺ Ti 2

2 ⑀ ␻⫹ ␻ˆ 3 T de

⫺2 ␾ 0 ⫹⌳ i 共 ␾ 0 兲 ⫹⌳ e 共 ␾ 0 兲 ⫽0, ¯␾ F 0e

冊冊

冕冑

Bd␭ 1⫺␭B

⫺2 ␾ 1 ⫹⌳ i 共 ␾ 1 兲 ⫹⌳ e 共 ␾ 1 兲 ⫹⌳ 1i 共 l, ␾ 0 兲 ⫹ ¯␾ F 0i .

共39兲 ⫹

¯␾ ,

共40兲

where we have separated the energy and the pitch angle integrations. Taking the flux tube average 共setting T e ⫽T i , n e ⫽n i ) yields the dispersion relation ⑀ , 2⫽⌳ e⑀ ⫹⌳ i0

共41兲

which may be substituted into Eq. 共40兲 to obtain

␾⫺

1 2

冕冑

Bd␭ 1⫺␭B

¯␾ ⫽0.

共42兲

Multiplying Eq. 共42兲 by ␾ and taking the flux tube average yields

冖 冕冑 dl B

Bd␭ 1⫺␭B

¯ 2兲⫽ 共 ␾ 2⫺ ␾

冕␶

b d␭ 共 ␾

2

¯ 2 兲 ⫽0. ⫺␾ 共43兲

¯ 2 ⭓0 we can conclude that the mode is fluteSince ␾ ⫺ ␾ like, i.e., ␾ ⫽ ␾ 0 to order k⬜2 ␳ 2i . Introducing finite k⬜2 ␳ 2i Ⰶ1 the general dispersion relation takes the form 2

⫺2 ␾ ⫹⌳ i ⫹⌳ e ⫹k⬜2 ␳ 2i ⌳ 1i 共 l 兲 ⫽0,

共45兲

which we have shown requires the flute solution ␾ 0 ⬅ 具 ␾ 0 典 . In first order,

Thus 1 ⑀ 2 ␾ ⫽ 共 ⌳ e⑀ ⫹⌳ i0 兲 2

FIG. 3. The stability of the collisional entropy mode 共dashed兲, the semicollisional mode 共solid兲, and the collisionless mode 共dotted兲 with k⬜2 ␳ 2i ⫽0. The MHD boundary is also shown.

共44兲

where ⌳ 1i does depend on the arc length along the field line (l). Defining t⬅k⬜2 ␳ 2i Ⰶ1, we can expand ␾ ⫽ ␾ 0 ⫹t ␾ 1 ⫹••• and ␻ ⫽ ␻ 0 ⫹t ␻ 1 ⫹••• . Then

⳵ ⌳ e共 ␾ 0 兲 ¯ 2 ⫽0. ␻ 1␾ 2⫺ ␾ ⳵␻

⳵ ⌳ i共 ␾ 0 兲 ␻1 ⳵␻ 共46兲

Integrating 养dl/B annihilates the ␾ 1 terms and determines ␻ 1 from

␻1

⳵ 关 ⌳ 共 ␻ 兲 ⫹⌳ e 共 ␻ 0 兲兴 ⫹ ⳵␻ i 0

冖

dl ⌳ 共 l, ␻ ⬅ ␻ 0 兲 ⫽0. B 1i 共47兲

This gives the shift in ␻ away from ␻ 0 , caused by k⬜2 ␳ 2i ⫽0. A similar perturbation analysis in the collisional and semi-collisional cases show that the low frequency electrostatic mode is flute-like, ␾ 0 ⫽ 具 ␾ 0 典 , in leading order of a k⬜2 ␳ 2i Ⰶ1 expansion, in all collisionality regimes. Returning to the leading order dispersion relation, Eq. 共41兲 and following Ref. 8, we seek a solution with Re关 ␻ 兴 ⫽Im关 ␻ 兴 ⫽0. Inserting this into Eq. 共41兲 the velocity space integrals can be evaluated analytically to determine the stability boundary in the form d⫽

冋 册

1 1⫹ ␩ . 3 1⫺ ␩

共48兲

For ␩ ⫽0, this recovers the stability result of Ref. 8; i.e., instability for d⬎1/3. Figure 3 displays the stability diagram for the collisional, the collisional electron/collisionless ion and the collisionless modes. When d⬎5/3 a vigorous MHD mode becomes unstable. For ␩ ⬍2/3 an electron drift mode is responsible for a reduction of the stability limit 共i.e., reduced d), whereas for ␩ ⬎2/3 the stability boundary matches the MHD boundary, i.e., d⬇5/3.

Downloaded 13 Feb 2002 to 198.125.177.51. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/pop/popcr.jsp

400

Phys. Plasmas, Vol. 9, No. 2, February 2002

J. Kesner and R. J. Hastie

␦ni

III. CONCLUSIONS

Two modes are seen to be present; a drift frequency mode with ⍀⬃O(1) and a MHD mode with ⍀⬃(k⬜ ␳ i ) ⫺1 Ⰷ1. These modes are driven unstable by a combination of curvature and profile effects, characterized by the parameters d and ␩ . We have analyzed the stability boundaries for these modes in the d⫺ ␩ parameter space for varying values of plasma collisionality. We find a somewhat smaller stable region in this parameter space when collisionality is reduced. In the collisional regime the drift mode has been called the entropy mode.5 In the collisionless regime the mode may be identified as an unfavorable curvature driven mode 共for d ⬎0) 8 or a temperature gradient driven mode 共for d⬍0 and ␩ ⬎ ␩ crit). In the LDX experiment we expect the parameters to be such that electrons are collisional but ions are collisionless (n e ⬃5⫻1012⫺1013 cm⫺3, T e ⬃T i ⬃100⫺200 eV兲. We find that as the ions become collisionless the stability boundary is somewhat degraded from the boundary determined by the entropy mode, without the destabilizing corrections that derive from collisional relaxation. In the collisionless case, which is relevant to the fusion reactor regime, we observe a further degradation of the stability boundary observed for ␩ ⬍2/3. In a dipole configuration, as described in Refs. 1,2, “T ⬍0, “n e ⬍0, and therefore ␩ ⬎0 in the region between the pressure peak and the wall. In this region we expect the pressure gradient to adjust so as to be just below the MHD limit, which for ␩ ⬎2/3 indicates dⱗ5/3 and from Fig. 3 we expect to be in a drift stable regime. At the pressure peak d ⫽0 and ␩ ⫽⫺1, which is a marginally stable point. On the contrary, in the region between the pressure peak and the internal coil d⬍0, and ␩ can be positive or negative depending on the sign of “n e 共since “T⬎0) in this region. In this region close to the ring, weak, temperature gradient instabilities are possible, but Figs. 2 and 3 show that stable operation is possible for ⫺1⬍ ␩ ⬍ ␩ crit , with ␩ crit of the order 1→3. ACKNOWLEDGMENTS

APPENDIX: PITCH ANGLE VARIATION OF ␻ ¯d

Consider the collisionless case. From Fig. 1 we observe that a more accurate representation of the curvature drift term would be 2 ⑀ ␻ˆ „1⫹ ␦ 共 ␭B min⫺0.4兲 …, 3 T i di

and ␦ ⬃0.17. Therefore, in the ␻ ⬅0 limit

⫹

q i␾ Ti 3 d 2 1⫹ ␩

冕

d 3 v关 1⫹ ␩ i 共 ⑀ /T i ⫺3/2兲兴 q ␾ F 0 /T i . 共 ⑀ /T i 兲关 1⫹ ␦ 共 ␭B min⫺0.4兲兴 i 共A2兲

For ␦ Ⰶ1 we obtain

␦ni ni

⬇⫺

q i ␾ 3d 1⫺ ␩ ⫹ Ti 2 1⫹ ␩

冕冑

Bd␭ 1⫺␭B

q i ␾ /T i

⫻„1⫺ ␦ 共 ␭B min⫺0.4兲 …,

共A3兲

with a similar expression with q i →q e for electrons. Consider quasi-neutrality with ␾ ⫽ ␾ 0 ⫹ ␦ ␾ 1 (l)⫹••• and d⫽d 0 ⫹ ␦ d 1 ⫹••• with ␾ 0 the flute solution. To lowest order the Re关 ␻ 兴 ⫽Im关 ␻ 兴 ⫽0 mode defines a line d 0 ⫽d 0 ( ␩ ) 关Eq. 共48兲兴 as before. The first order equation becomes ⫺␾1

1⫹ ␩ 3d 0 ⫹ 1⫺ ␩ 2

⫺

3d 0 ␾ 0 2

冕

冕冑

Bd␭ ␾ 1 共 l 兲 1⫺␭B

⫹

3d 1 ␾ 0 2

Bd␭ 共 ␭B min⫺0.4兲

冑1⫺␭B

⫽0.

冕冑

Bd␭ 1⫺␭B 共A4兲

Multiplying by ␾ 0 and taking the flux tube average to anni¯ 1 we obtain hilate ␾

冉

冊

d1 2 B min养dl/B 2 ⫺0.4 . ⫽ d0 3 养dl/B

共A5兲

The flux tube integrals can be evaluated to give d 1 /d 0 ⬇0.06. Thus 1 1⫹ ␩ d⫽ 1.01 . 3 1⫺ ␩

共A6兲

¯ d with pitch angle drives We conclude that the variation of ␻ ballooning structure, ␾ 1 (l), in the eigen function but has a weak effect on the boundary in (d, ␩ ) space. For the weak ¯ d (␭) in the point dipole, ␦ ⬇0.17, the stability variation in ␻ boundary is modified approximately 1%. A. Hasegawa, Comments Plasma Phys. Controlled Fusion 1, 147 共1987兲. J. Kesner, L. Bromberg, D. T. Garnier, and M. E. Mauel, 17th IAEA Fusion Energy Conference, Yokohama, Japan, 1998, Fusion Energy 1998 共International Atomic Energy Agency, Vienna, 1999兲, Paper IAEA-F1-CN69-ICP/09. 3 J. Kesner, Phys. Plasmas 7, 3837 共2000兲. 4 J. Kesner, A. N. Simakov, D. T. Garnier, P. J. Catto, R. J. Hastie, S. I. Krasheninnikov, M. E. Mauel, T. S. Pedersen, and J. J. Ramos, Nucl. Fusion 41, 301 共2001兲. 5 A. N. Simakov, P. J. Catto, and R. J. Hastie, Phys. Plasmas 8, 4414 共2001兲. 6 L. V. Mikhailovskaya, Sov. Phys. Tech. Phys. 12, 1451 共1968兲. 7 A. B. Mikhailovskii and V. S. Tsypin, Sov. Phys. JETP 32, 287 共1971兲. 8 M. N. Rosenbluth, Phys. Fluids 11, 869 共1968兲. 9 R. J. Hastie, and K. W. Hesketh, Nucl. Fusion 21, 621 共1981兲. 10 R. R. Dominguez and R. W. Moore, Nucl. Fusion 26, 85 共1986兲. 11 B-G. Hong, W. Horton, and D-I. Choi, Phys. Fluids B 1, 1589 共1989兲. 12 J. Kesner, Phys. Plasmas 5, 3675 共1998兲. 13 H. V. Wong, W. Horton, J. W. Van Dam, and C. Crabtree, Phys. Plasmas 8, 2415 共2001兲. 14 B. Lane, Phys. Fluids 29, 1532 共1986兲. 15 T. Antonsen and B. Lane, Phys. Fluids 23, 1205 共1980兲. 16 T. Antonsen, B. Lane, and J. J. Ramos, Phys. Fluids 24, 1465 共1981兲. 1

The author would like to thank Professor M. E. Mauel, Dr. D. T. Garnier 共Columbia University兲, Dr. A. N. Simakov and Dr. P. J. Catto 共MIT PSFC兲 for stimulating conversations. This work was supported by the United States Department of Energy under Contract No. DE-FG02-91ER-54109.

¯ di 共 ⑀ ,␭ 兲 ⬇ ␻

ni

⫽⫺

共A1兲

2

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