trapped-particle-mediated damping and transport.pdf

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Trapped-Particle-Mediated Damping and

TransportC. Fred Driscoll , Andrey A. Kabantsev , Terance J. Hilsabeck and

Thomas M. ONeil

Dept. of Physics, University of California at San Diego, La Jolla CA USA 92093-0319

Abstract. Weak axial variations in B( z) or ( z) in Penning-Malmberg traps cause some particles tobe trapped locally. This causes a velocity-space separatrix between trapped and passing populations,and collisional separatrix diffusion then causes mode damping and asymmetry-induced transport.This separatrix dissipation scales with collisionality as 1/ 2 , so it dominates in low collisionallityplasmas. The connement lifetime in the CamV apparatus was dominated by a weak magneticripple with B/ B10

3 , and it appears likely that the ubiquitous ( L/ B)2 lifetime scalings andother applied asymmetry scalings represent similar TPM effects. TPM transport will limit thecontainment of large numbers of positrons or ps, since TPM loss rates generally scale as totalcharge Q2 , independent of length.

INTRODUCTION

Two years ago, trapped particle asymmetry modes were reported to occur whenan applied squeeze voltage causes some particles to be trapped axially; and a simpletheory explained the observed mode frequencies [1]. Now, it appears likely that trapped-particle-mediated (TPM) effects are dominant in plasma lifetime scalings, in transportfrom applied asymmetries, and in diocotron mode damping. This talk will give anoverview of what is known [2, 3, 4, 5], where more experiments are needed, and wherethe theory is lacking.

Electric or magnetic trapping probably occurs in all long apparatuses: unintendedwall potential variations of 0.1 Volts are common, and it is sobering to note that B/ B =103 will trap 3% of the particles. Initial experiments (and all theory to date) consideredelectric trapping; but magnetic trapping is probably more common and important.

Early experiments focused on the new modes (now called "trapped particle diocotron"modes); but the important effect is particles scattering across the trapping separatrix.This breaks the v z adiabatic invariant, allowing 2D potential energy to ow to 3Dkinetics, and enabling external asymmetries to generate strong transport. The effect isdominant in low-collisionality plasmas because this separatrix dissipation scales withcollisionality as 1/ 2 , whereas most other effects scale as 1 . Here, the collisionality canbe electron-electron, electron-neutral, or externally stimulated. The effect can be also bedescribed as dissipation of asymmetry-induced equilibrium currents, as in the analysisof bootstrap current in Tokamaks.


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-V sq-V c -V c

r s R p

Sectors

+

+

--

P h o s p

h o r

S o u r

c e

k z = 1m

= 1,2,...

V a V a

B y x

B

z

E

E

B

FIGURE 1. Schematic of electron plasma with a trapped particle mode in the cylindrical containmentsystem.

Thus, it now appears likely that most of the ( L/ B)2 lifetime scalings from back-ground asymmetries [6] can be given interpretation in terms of the (partially) knownscalings for TPM transport. The measurements of transport from applied electric andmagnetic asymmetries [7, 8, 9, 10] also should be compared to TPM predictions.Anomalous damping of diocotron modes [7, 11, 12] is almost certainly related to TPMeffects, since TPM damping scales as B3. TPM transport also has important implica-tions for containment of large numbers of positrons or pBars, since the TPM loss ratefor magnetic asymmetries scales as total charge Q2 , independent of length .

Theory provides a reasonable picture of trapped-particle-mode damping with electrictrapping [5], but modes in the magnetic trapping case remain enigmatic. Theory cannot yet explain the observed particle transport scalings for either case, but this appearsimminent for electric trapping. Diocotron mode damping has not been worked outtheoretically.

ELECTRIC TRAPPING: NEW MODE

The experiments are performed on magnetized pure electron plasmas conned in thecylindrical CamV apparatus, as shown in Fig. 1. The electron plasmas have density

n

107

cm3

, length L p

40 cm, radius R p

1.5 cm, and temperature T 1 eV.Controlled electric trapping from an applied central squeeze voltage V sq causeselectrons with axial velocity less than the separatrix velocity to be trapped in oneend or the other; here, v s is dened by v 2s (r ) 2em V sq(r ). For small V sq , a fraction N (tr)

L / N L 1.2 (V sq / p) of the electrons are trapped, predominantly at r R p; here, N L 2 r dr n .This trapping enables novel trapped particle diocotron modes with various m =1, 2,... ; but we focus here on m = 1. The mode frequency f a ranges from the edgerotation frequency f E ( R p)(100 kHz ) B[kG]

1 at low V sq , down to the k z = 0 diocotron


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10 2

10 3

0.1 1 10

a

[ s e c

- 1 ]

T [eV]

t h e o r y ( 1 0 k G )

10 kG

4 kG

1 kG

L p = 49 cm

V sq / P = 2/3

N L = 6.4 107 cm -1

FIGURE 2. Measured damping rate a versus temperature for 3 magnetic elds; with theory predictionfor 1 eld.

mode frequency f d at V sq >

p. The modes are anti-symmetric in z, with trapped particleson either end executing E B drift oscillations that are 180 out of phase, while passingparticles move along eld lines in response to the potential from the trapped particles.

At large V sq , the plasma is essentially cut in half, and on either side of the barrier theplasma supports k z = 0 (diocotron) drift orbits which are 180 out of phase. Even forsmall V sq , the trapped particle mode is essentially uniform with z on either side of thebarrier, changing sign at the barrier. A simple kinetic theory model with a zero-lengthtrapping barrier [1] predicts mode frequencies agreeing with measurements to withinabout 10%.

This trapped particle mode is readily excited by m = 1 z-antisymmetric voltages,as shown in Fig. 1. The excited mode then rings down exponentially, and the dampingrate a is unambiguously obtained. Figure 2 shows the measured a for three magneticelds as the plasma temperature is varied; the dash lines represent the generic functionalform a[1 exp(b/ T )]. The modes are strongly damped at low temperatures, but thedamping decreases precipitously as T increases.

Theory analysis of damping from collisional scattering across the trapping separatrixgives factor-of-two agreement with experiments; but signicant discrepancies remain.Since particles on either side of the separatrix are involved in completely different typesof motion, there is a discontinuity in the perturbed particle distribution function. As aresult, electron-electron collisions produce a large ux of particles across the separatrix.The continual trapping and detrapping of particles results in radial transport of particlesand in mode damping, and is readily observed in computer simulations [13].

These collisions at rate have been treated by a Fokker-Planck collision operator[5], in an analysis similar to that used for the dissipative trapped-ion instability byRosenbluth, Ross, and Kostomarov [14]. Velocity space diffusion acting for one modeperiod smoothes out the separatrix discontinuity over a width vt v / f E , and the


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damping includes this dependence. The predicted damping rate can be expressed as

a =

2 Bm

Rw0 rdr | |v f E

2 e f M T c Br

f M r

m f E v= vs

Rw0 dr |

|2

f 2

nt r

, (1)

where (r ) is the mode potential, f E (r ) m f E (r ) f a , nt is the trapped density, and f M is the Maxwellian distribution.Figure 2 shows that this predicts a somewhat less abrupt temperature dependence than

is actually observed. This may be related to another signicant discrepancy: experimentsshow non-zero (20) phase shifts in the mode eigenfunction (r ), whereas no shiftsare predicted. The square root provides a signicant enhancement, since / f E is small.The damping rate is expected to have a strong and complicated temperature dependencethrough the density of particles at the separatrix velocity v s(r ), through the collisionalfrequency , and through the Debye shielding length D.

ELECTRIC TRAPPING AND TILT: TRANSPORT

When -asymmetries exist in the electric or magnetic connement elds, they createtorques which change the canonical angular momentum P of the plasma, causingthe plasma radius to vary. These asymmetry-induced torques are stronger when thesymmetric squeeze trapping is present. If the asymmetry is not static, the sign of thetorque can be positive or negative. The rotating wall connement technique utilizeswall voltages rotating faster than f E to obtain plasma compression [15, 16]. For thepresent experiments, the -asymmetries are static in the lab frame and exert a negative

torque on the electrons, resulting in bulk radial expansion.Here, we focus on the m = 1, k z = 1 asymmetry induced by a magnetic tilt, with B = B( z+ Bx x+ By y); or by the electric tilt induced by static m = 1 voltages V aapplied antisymmetrically in z (Fig. 1). The asymmetry-induced transport rate is denedby the rate of plasma expansion

p 1r 2

d r 2

dt 1

P

dP dt

. (2)

The expansion rate is found to be proportional to the tilt angle 2 B, as shown in Fig. 3.Here, p( Bx) is quadratic about Bx = 0, but the minimum of p( By) is offset by the

separate electric tilt Ey from an applied V ay . Indeed, electric and magnetic tilts addvectorially when the proper z-averaged electrostatic offsets [17] are calculated, as

E (0.51)4 Rw L p

V aeN L

La L p

. (3)

Here, V a is applied to sectors of length La , and the factor 0.51 represents the m = 1Fourier coefcient for the (four) 25 sectors used. The deviation from this quadraticscaling at larger p (dashed line) is due to an increase of the plasma temperature causedby fast radial expansion.


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0

0.01

0.02

0.03

-4 -2 0 2

Bx , By [mrad]

P

[ s e c

- 1 ]

V ay = 4 V

P( Bx)

P( By)

FIGURE 3. Measured transport rate p from a magnetic tilt with simultaneously applied electricsymmetry.

FIGURE 4. Measured normalized transport rate p/ a vs scalings for all plasma parameters.

The expansion rate p shows rather complicated dependencies on plasma parametersn, T , R p, L p and B, unless the ratio p/ a is considered. Essentially, we nd that allthe complicated dynamics is in the separatrix dissipation which causes a . Measuring acoincident with p (by exciting the trapped particle mode and watching it decay) allowsus to accurately obtain the ratio p/ a .

Figure 4 shows that the ratio p/ a has only simple power-law dependencies onplasma parameters, as L2 p, B1, and N 2 L, where N L R

2 pn. We note that all temperature

dependence is in a , that V sq is normalized to the plasma potential p at r = 0, and that p N L.

With the applied electric trapping and the applied tilt, a single trapped-particle-mediated damping and transport process is dominant; and this process exhibits stun-ningly simple and accurate scalings over 3 decades in p/ a , representing 4 decades in


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-0.001

0

0.001

-40 -20 0 20 40

B/B

z [cm]

P

original placement new placement

FIGURE 5. Modied electrode placement relative to magnetic ripples exhibits 5 less backgroundtransport.

p. However, the theory of this transport scaling is still incomplete.

MAGNETIC TRAPPING AND TILT: TRANSPORT BUT NO MODE

Applying a central magnetic squeeze B( z) instead of the electric squeeze alsocauses particle trapping in either end, and causes enhanced transport from electric ormagnetic tilts. However, experiments have not yet identied a corresponding trapped

particle mode. Presumably, this is because the magnetic trapped particle mode has f ( M )a = 0, or ( M )a / f

( M )a 1. This eliminates the conceptual advantage of relating pto a ; but experiments demonstrate conclusively that scatterings across the magnetic

separatrix produce transport, as in the electric trapping case.These magnetic trapping effects have been studied using an axially centered coil

which generates a magnetic mirror of strength ( B/ B)


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0

2

4

6

0 0.05 0.1 0.15 0.2

z /

/

q > 0

q < 0

= =

B

B.

tan .

001

031

V sq

FIGURE 6. The magnetic separatrix with potentials.

by ; and an applied electric squeeze voltage V sq would add in analogously. Thus, themagnetic/electric separatrix is given by

v2 z = B B

v2

+ 2em

+ V sq(r ) . (4)

These hyperbolic separatrices are shown in Fig. 6 for = 103 , showing the 1/ 2

reduction in relevant v z velocities. The lack of radial separation between trapped anduntrapped particles makes f ( M )a = 0 plausible, since the charge separation and E Bdrifts characterizing the electric mode may not occur. These trapped particles can bedirectly detected by selective dumping techniques, but the relevant parallel velocitiesare substantially less than v, so measurements to date are only qualitative.

More incisively, the separatrix can be mapped out by enhancing v z separatrix scat-terings with a resonant RF eld. Figure 7 shows the transport enhancement when a RFwiggle is applied near a magnetic minimum. Here, we have utilized the sheath trans-port resonance to interact with particles with small v z: electrons receive a nonadiabatickick if they have v z L f RF , where L 2 Rw/ j01 is the axial extent of the wiggleelectric elds. The transport response peak in Fig. 7 is as expected for a Maxwellian dis-tribution of particles along the naive (

= 0) magnetic separatrix; surprisingly, recentcalculations show that does not affect this resonance curve [18].

Alternately, adding an electric squeeze at the z-position of a magnetic mirror movesthe separatrix so as to exclude particles with small v z. This causes a reduction in p forsmall V sq , as v z

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