nonlinear dynamo mode dynamics in reversed field pinchesshell. furthermore, the model only deals...
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Nonlinear dynamo mode dynamics in reversed field pinchesRichard Fitzpatrick and Edmund P. Yu Citation: Physics of Plasmas (1994-present) 7, 3610 (2000); doi: 10.1063/1.1286990 View online: http://dx.doi.org/10.1063/1.1286990 View Table of Contents: http://scitation.aip.org/content/aip/journal/pop/7/9?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Mode- and plasma rotation in a resistive shell reversed-field pinch Phys. Plasmas 11, 647 (2004); 10.1063/1.1639016 Evolution process of the mode wall-locking and phase-locking in a reversed-field pinch plasma Phys. Plasmas 8, 1625 (2001); 10.1063/1.1364674 Effect of a resistive vacuum vessel on dynamo mode rotation in reversed field pinches Phys. Plasmas 6, 3878 (1999); 10.1063/1.873650 Stability of resistive wall modes in reversed field pinches with longitudinal flow and dissipative effects Phys. Plasmas 6, 3868 (1999); 10.1063/1.873191 Feedback stabilization of resistive shell modes in a reversed field pinch Phys. Plasmas 6, 3536 (1999); 10.1063/1.873614
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Nonlinear dynamo mode dynamics in reversed field pinches *Richard Fitzpatrick and Edmund P. YuInstitute for Fusion Studies, Department of Physics, University of Texas at Austin, Austin, Texas 78712
~Received 31 January 2000; accepted 19 May 2000!
The nonlinear dynamics of a typical dynamo mode in a reversed field pinch, under the action of thebraking torque due to eddy currents excited in a resistive vacuum vessel and the locking torque dueto a resonant error-field, is investigated. A simple set of phase evolution equations for the mode isderived: these equations represent an important extension of the well-known equations of Zohmet al. @Europhys. Lett.11, 745 ~1990!# which incorporate a self-consistent calculation of the radialextent of the region of the plasma which corotates with the mode; the width of this region beingdetermined by plasma viscosity. Using these newly developed equations, a comprehensive theory ofthe influence of a resistive vacuum vessel on error-field locking and unlocking thresholds isdeveloped. Under certain circumstances, a resistive vacuum vessel is found to stronglycatalyzelocked mode formation. Hopefully, the results obtained in this paper will allow experimentalists toachieve a full understanding of why the so-called ‘‘slinky mode’’ locks in some reversed field pinchdevices, but not in others. The locking of the slinky mode is currently an issue of outstandingimportance in reversed field pinch research. ©2000 American Institute of Physics.@S1070-664X~00!01309-4#
I. INTRODUCTION
A reversed field pinch~or RFP! is a magnetic fusiondevice in which a thermonuclear plasma is confined via acombination of a toroidal magnetic field,Bf , and a poloidalmagnetic field, Bu , in an axisymmetric toroidalconfiguration.1 The RFP concept derives its name from thefact that the toroidal magnetic field spontaneously reversesdirection in the outer regions of the plasma. This reversal isa consequence of relaxation to a minimum energy statedriven by intense magnetohydrodynamical~MHD! mode ac-tivity during the plasma start-up phase.2 Intermittent, rela-tively low-level, mode activity maintains the reversal, by dy-namo action, throughout the duration of the plasmadischarge.
A conventional RFP plasma is surrounded by a thickconducting shell whose resistive penetration time is muchlonger than the duration of the discharge. Such a shell isnecessary in order to stabilize external kink modes whichwould otherwise rapidly destroy the plasma.3 In the presenceof the shell, the dominant MHD modes arem51 tearingmodes resonant in the plasma core. These modes possess arange of toroidal mode numbers, characterized byn;2 R0 /a. Here,m,n are poloidal and toroidal mode num-bers, respectively, whereasa andR0 are the minor and majorradii of the plasma, respectively. The core tearing modes areresponsible for the dynamo action which maintains the fieldreversal, and are, therefore, generally known asdynamomodes.4
The majority of RFP [email protected]., the ReversedField Experiment~RFX! ~Ref. 5!, and the TPE-RX~Ref. 6!experiment# also feature a thin resistive vacuum vessel situ-
ated between the edge of the plasma and the stabilizing shell.The key to the success of the RFP concept lies in the
control of MHD instabilities; there is clear experimental evi-dence that a relatively modest reduction in ambient modeamplitudes leads to a dramatic improvement in plasmaconfinement.7 However, in order to reliably control dynamomodes in RFPs, it is first necessary to fullyunderstandtheirdynamics.
The complex nonlinear dynamics of dynamo modes inRFPs is a fascinating subject in its own right, as well as atopic of great practical importance. Dynamo modes in RFPsare generally observed tophase lockto one another, at rela-tively low amplitudes, so as to form atoroidally localizedstructure in the perturbed magnetic field known as a ‘‘slinkymode.’’8,9 This effect can be understood as a natural conse-quence of the mutual nonlinear couplings of the variousmodes in the plasma.10 The slinky mode per se doesnotappear to significantly degrade the global plasmaconfinement.11 Instead, it gives rise to a toroidally localized‘‘hot spot’’ on the plasma facing surface, presumably be-cause the radial transport due to the diffusion of chaoticmagnetic field-lines peaks at the toroidal angle where theamplitude of the slinky mode attains its maximum value. Thehot spot is not a problem as long as the slinky moderotatesin the laboratory frame. Conversely, if, for some reason, theslinky mode ceases to rotate then the hot spot is forced tohover over the same point on the plasma facing surface, lead-ing almost inevitably to overheating, the influx of impuritiesinto the plasma, and the premature termination of the dis-charge. The enhanced plasma–wall interaction associatedwith a locked slinky mode is of major concern to fusionresearchers, since it is a limiting factor in virtually all currentRFP experiments.12,13 Recent experiments on RFX havedemonstrated that this problem can be alleviated, to someextent, byforcing the locked slinky mode to rotate via the
*This paper was presented as an invited talk at the 41st Annual Meeting ofthe American Physical Society, Division of Plasma Physics, Seattle 1999.
PHYSICS OF PLASMAS VOLUME 7, NUMBER 9 SEPTEMBER 2000
36101070-664X/2000/7(9)/3610/15/$17.00 © 2000 American Institute of Physics
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imposition of a propagating ripple pattern onto the equilib-rium toroidal magnetic field.14 These experiments seem toconfirm the idea that the slinky mode is only a major prob-lem when it fails to rotate.
The physical mechanisms which are most likely to beresponsible for converting slinky modes, and their constitu-ent dynamo modes, from rotating modes~their natural state!to locked modes in RFPs are ‘‘error-fields’’ and eddy cur-rents excited in the resistive vacuum vessel. Error-fields aresmall nonaxisymmetric perturbations in the equilibriummagnetic field generated by field–coil misalignments, cur-rent feeds, and the presence of insulating gaps in the con-ducting shell. The ability of those helical components oferror-fields whichresonatewith the plasma core to arrestdynamo mode rotation in RFPs is well-known15–17 and wellunderstood.18 However, it was recently pointed out that theeddy currents excited in the resistive vacuum vessels ofpresent-day RFPs by rotating dynamo modes exert slowing-down torques on these modes which are generally suffi-ciently strong to reduce their rotation toextremely lowlevels.19 Of course, eddy current torques cannot, by them-selves, completely halt the rotation of a dynamo mode, sincethere is zero torque when the mode is stationary. It is con-cluded, therefore, that the locking of dynamo modes~and,hence, of slinky modes! in RFPs is most probably acom-bined effect of resonant error-fields and eddy currents ex-cited in the resistive vacuum vessel.
The aim of this paper is to develop a theoretical frame-work within which thephase evolutionof rotating dynamomodes in RFPs, in the presence of resonant error-fields and athin resistive shell surrounding the plasma, can be analyzed.In particular, it is hoped to develop a set of phase evolutionequations which are sufficiently simple that experimentalistscan routinely employ them whilst interpreting data or design-ing new RFP experiments.
The model adopted in this paper is somewhat simplistic.Instead of considering a range of unstablem51 modes, itfocuses on the dynamics of asingle representative dynamomode in the presence of a resonant error-field and a thinresistive vacuum vessel surrounded by a thick conductingshell. Furthermore, the model only deals with zero-b, largeaspect-ratio plasmas. Nevertheless, our model is probablysufficiently realistic to allow some progress to be made inour current understanding of locked mode formation inRFPs.
II. PRELIMINARY ANALYSIS
A. The plasma equilibrium
Consider a large aspect-ratio,20 zero-b,21 RFP plasmaequilibrium whose unperturbed magnetic flux-surfaces mapout ~almost! concentric circles in the poloidal plane. Such anequilibrium is well approximated as a periodic cylinder. Sup-pose that the minor radius of the plasma isa. Standard cy-lindrical polar coordinates (r ,u,z) are adopted. The systemis assumed to be periodic in thez-direction, with periodicitylength 2pR0 , whereR0 is the simulated major radius of theplasma. It is convenient to define a simulated toroidal anglef5z/R0 .
The equilibrium magnetic field is written B5@0,Bu(r ),Bf(r )#, where¹∧B5s~r !B.
B. Outline of the problem
Suppose that the plasma~minor radiusa! is surroundedby a concentric, thin, resistive shell of minor radiusb. Theresistive shell is, in turn, surrounded by a concentric, per-fectly conducting shell of minor radiusc. It follows that a,b,c. The conducting shell is assumed to contain thinvacuum gaps though which a static, externally generated,nonaxisymmetric error-field can leak. This paper investigatesthe effect of such an error-field, and any helical eddy cur-rents excited in the resistive shell, on the rotation of a typicalcore tearing mode; them,n mode, say. All other modes inthe plasma are ignored, for the sake of simplicity.
C. The perturbed magnetic field
The magnetic perturbation associated with them,n tear-ing mode can be written
b~r ,t !5bm,n~r ,t !ei ~mu2nf!. ~1!
In this paper, it is assumed thatm.0 andnÞ0. The linear-ized magnetic flux functioncm,n(r ,t)[2 irb r
m,n satisfiesNewcomb’s equation,22
d
dr F f m,ndcm,n
dr G2gm,ncm,n50, ~2!
where
f m,n~r !5r
m21n2e2 , ~3!
gm,n~r !51
r1
r ~neBu1mBf!
~m21n2e2!~mBu2neBf!
ds
dr
12mnes
~m21n2e2!22rs2
m21n2e2 , ~4!
and e5r /R0 . As is well-known, Eq.~2! is singular at them/n rational surface, minor radiusr s , which satisfiesFm,n(r s)50, where
Fm,n~r 
words, csm,n(r ,d) is a real solution to Newcomb’s equation
~2! which is well behaved asr→0, and satisfies
csm,n~r s ,d!51, ~9!
csm,n~d,d!50. ~10!
It is easily demonstrated thatcsm,n(r ,d) is zero in the region
r .d. In general,csm,n(r ,d) possesses gradient discontinui-
ties atr 5r s and r 5d. The real quantity
Em,n~d!5F rdcs
m,n~r ,d!
drG
r s2
r s1
~11!
can be identified as the standardm,n tearing stabilityindex,23 calculated assuming the presence of a single, per-fectly conducting shell at a minor radiusd.
E. Asymptotic matching
The quantity
Csm,n~ t 
G. Plasma rotation
Let V(r ) represent the toroidal angular velocity profileof the plasma. Likewise,V (0)(r ) represents the unperturbedvelocity profile ~i.e., the profile in the absence of resistiveshell eddy currents and the resonant error-field!. Finally, let
V~r ,t !5V~0!~r !1DV~r ,t !, ~25!
whereDV represents themodificationto the plasma velocityprofile induced by the eddy currents and the error-field.
The perturbed angular equation of motion of the plasmacan be written
rr]DV
]t2
]
]r S rm]DV
]r D5dTfEM
m,n
4p2R03 d~r 2r s!, ~26!
wherer(r ) is the plasma mass density profile, andm(r ) isthe plasma ~anomalous! perpendicular viscosity profile.Here, the radial extent of the nonideal region, centered on therational surface, in which the electromagnetic torque devel-ops is assumed to be negligible, for the sake of simplicity.
The boundary conditions which must be satisfied by theperturbed velocity profile are
]DV~0,t !
]r50, ~27!
DV~a,t !50. ~28!
The second boundary condition implies that the edge plasmavelocity is essentiallyunaffectedby the electromagnetictorque which develops in the vicinity of the rational surface.The assumptions underlying the analysis in this section aredescribed in more detail in Ref. 24.
The no slip condition ~17! can be rewritten
1
n
dws
dt5Vs~ t !5Vs
~0!1DV~r s ,t !, ~29!
whereVs(0)5V (0)(r s).
H. Normalization
Equations~24!, ~26!, ~27!, ~28!, and ~29! form a com-plete set which describe the phase evolution of a typical dy-namo mode in a RFP under the influence of a resonant error-field and eddy currents excited in the resistive shell. It is nowconvenient to rewrite this set of equations in normalizedform.
Let Cs5Cs /(r s2Fs8), Cc5Cc /(r s
2Fs8), r 5r /r s , a5a/r s , r5r/r(r s), m5m/m(r s), t5nVs
(0)t, V5V/Vs(0) ,
Vs5Vs /Vs(0) , DV5DV/Vs
(0) , where Fs85(dFm,n/dr) r s.
The typical hydromagnetic and viscous diffusion time scalescan be written
tH5Am0r~r s!
Fs8, ~30!
tV5r s
2r~r s!
m~r s!, ~31!
respectively.Let
as5Cs
L0, ~32!
ac5Esc
m,n
4@Em,n~c!2Em,n~b!#
Av l
vs
Cc
L0, ~33!
vs5Esb
m,nEbsm,n
nVs~0!tb@Em,n~c!2Em,n~b!#
, ~34!
v l51
4nVs~0!tV~*1
adr/ r m !2 , ~35!
L05F ~nVs~0!tH!2tb
tV
1
2
m21n2es2
n2es2
1
Esbm,nEbs
m,nY E1
adr
r mG1/2
.
~36!
Here,as is the normalized dynamo mode amplitude,ac is thenormalized amplitude of the error-field,vs is the critical nor-malized mode rotation velocity above which the resistiveshell starts to act like a perfect conductor,v l is the criticalnormalized mode rotation velocity above which the oscillat-ing component of the plasma velocity profile, driven by theerror-field, becomes localized around the rational surface,and L0 is a convenient scale amplitude. Under normal cir-cumstances, the parametersas andac areO(1), whereasvs ,v l , andL0 are expected to be much less than unity.
The normalized phase evolution equations take the form
r r
4J2v l
]DV
] t2
]
] r S r m]DV
] r D 52T
Jd~ r 21!, ~37!
T5as
2
4
Vs
vs21Vs
2 1asac
Av l
sin~ws2wc!, ~38!
]DV~0,t !
] r50, ~39!
DV~ a, t !50, ~40!
dws
dt[Vs511DV~1,t !, ~41!
whereJ5*1adr/ r m. Equation~37! is the normalized equa-
tion of motion of the perturbed plasma rotation profile drivenby the electromagnetic torque exerted at the rational surface.Equation~38! specifies the nature of this torque; it is the sumof contributions from eddy currents excited in the resistiveshell and the resonant error-field, respectively. Equations~39! and ~40! represent the boundary conditions which mustbe satisfied by the solution of Eq.~37!. Finally, Eq. ~41! isthe no slip constraint, which relates the phase evolution ofthe dynamo mode to the plasma rotation velocity at its ratio-nal surface.
I. Discussion
This is an appropriate point at which to highlight themajor assumptions made during the derivation of Eqs.~37!–~41!, and also to discuss the relationship between this paperand previously published theoretical research on mode lock-ing effects in magnetic fusion experiments.
3613Phys. Plasmas, Vol. 7, No. 9, September 2000 Nonlinear dynamo mode dynamics in reversed field pinches
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The key assumptions made during the derivation of Eqs.~37!–~41! are as follows:
~1! The m,n dynamo mode is assumed to beconvectedbythe plasma at its own rational surface. Furthermore, theelectromagnetic torques exerted on the plasma by eddycurrents excited in the resistive shell and the resonanterror-field are assumed to belocalizedin the vicinity ofthe rational surface. Both of these results follow imme-diately from standard MHD theory, provided that theresponse of the plasma can be modeled according to lin-earized, inertia-free, ideal-MHD everywhere apart froma narrow region centered on the rational surface. For thetypical mode amplitudes~between dynamo events!,plasma densities and temperatures, and plasma rotationvelocities observed in present-day RFPs, this approach isperfectly reasonable.
~2! The plasma in the vicinity of them,n rational surface,which co-rotates with the mode, is assumed to bestrongly coupledto the rest of the plasma via~anoma-lous! perpendicular viscosity. This assumption is cer-tainly justified, given the typical momentum confinementtime-scales observed in present-day RFPs.25 The oppo-site approach, in which the plasma in the vicinity of therational surface is assumed to be free to slip with respectto the bulk plasma,26,27 would appear to be somewhatunphysical. The only possible exception to this rule oc-curs when the torque acting on the mode varies veryrapidly in time ~e.g., when the mode amplitude growsexplosively28!.
~3! It is assumed that the edge plasma rotation profile isdetermined by the balance of torques which are far largerin magnitude than those which typically develop in theplasma due to error-field or resistive shell eddy currenteffects. It follows that the edge plasma rotation is un-likely to be substantially affected by either error-field oreddy current torques; i.e., the plasma rotation profile actslike it is ‘‘ clamped’’ at the edge. This assumption isjustified theoretically in Ref. 24. It also seems to be inaccordance with experimental observations@see, for in-stance, Fig. 4 in Ref. 29 and Fig. 4~a! in Ref. 30#. Theopposite approach, in which the rotation profile is as-sumed to be subject to ‘‘free slip’’ boundary conditionsat the plasma edge,31 is difficult to justify on physicalgrounds.
~4! The amplitude evolution of them,n mode is assumed totake place on a much slower time-scale than its phaseevolution, and is, therefore, neglected in this paper. Inother words, the amplitude evolution of dynamo modesis assumednot to play adirect role in their locking orunlocking to resonant error-fields or the resistive shell.This assumption is reasonable, given the typical moderotation velocities and resistive growth-rates~betweendynamo events! observed in present-day RFPs. In Ref.24 it was found that the periodic modulation in the widthof a magnetic island chain, as it rotates past a staticerror-field in a tokamak, can give rise to a significant,nonoscillating locking torque acting on that chain. It iseasily demonstrated that this effect is negligible in
present-day RFPs~because of the far larger saturatedtearing mode amplitudes prevalent in RFPs compared totokamaks!.
~5! Any poloidal rotation of the plasma is neglected. Thisassumption is made mostly for convenience~it may ac-tually be justified if RFP plasmas are subject to strongpoloidal flow damping32!, and could fairly easily be re-laxed.
Many previously published papers have studied the dy-namics of tearing modes in magnetic fusion experiments un-der the influence of resonant error-fields and/or eddy currentsexcited in the vacuum vessel. The interested reader is re-ferred to Refs. 33–44, in addition to those papers which havealready been mentioned. The present study differs from pre-vious studies in three respects. First, the analysis is carriedout using RFP, rather than tokamak, orderings. Note that thetokamak ordering is simply a subset of the RFP ordering,obtained by taking the limitm2@n2e2. Hence, in this re-spect, the present study ismore generalthan most previousstudies. Secondly, this paper does not assume the existenceof a fixed-width region of the plasma, centered on the ratio-nal surface, i.e., amagnetic island, which corotates with themode. Instead, a far more physical model is adopted inwhich the width of the plasma region which corotates withthe mode is determined by plasmaviscosity, and, therefore,variesas the mode angular acceleration varies. Finally, thispaper deals with error-field and vacuum vessel eddy currenteffectssimultaneously, and presents, for the first time, acom-prehensive theoryof the influence of a resistive vacuum ves-sel on the error-field locking and unlocking thresholds.
III. DERIVATION OF PHASE EVOLUTION EQUATIONS
A. Introduction
The standard approach to tearing mode dynamics inmagnetic fusion experiments, as exemplified by the classicpaper of Zohmet al.,40 leads to the following set of phaseevolution equations:
df
dt5v, ~42!
I f
dvdt
1~v21!1as
2
4
v
vs21v2 1
asac
Av l
sinf50. ~43!
Here, f[ws2wc is the helical phase of the tearing modemeasured with respect to that of the error-field,v[Vs is thenormalized toroidal angular velocity of the mode, andI f
5I fnVs(0)tV /@4p2r s
2R03r(r s)# is the normalized toroidal
moment of inertia of the fixed region of the plasma which isassumed to corotate with the mode~I f is the actual momentof inertia of this region!. The first term in Eq.~43! representsthe inertia of the corotating region, the second term is therestoring torque due to viscous coupling with the remainderof the plasma, the third term represents the braking torquedue to eddy currents excited in the resistive shell, and thefourth term is the locking torque due to the resonant error-field. Although the above set of equations does not, in gen-
3614 Phys. Plasmas, Vol. 7, No. 9, September 2000 R. Fitzpatrick and E. P. Yu
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eral, possess an analytic solution, its numerical solution isvery straightforward. Hence, this set of equations~or someclosely related variant! is used extensively by experimental-ists to interpret mode locking data obtained from magneticfusion experiments.45–47
Instead of assuming the existence of a fixed-width regionof the plasma which corotates with the mode, Eqs.~37!–~41!take into account the fact that the width of the corotatingregion is actually determined by the viscous coupling of theplasma in the immediate vicinity of the rational surface tothe bulk plasma, and is, therefore,variable. Unfortunately,Eqs. ~37!–~41! are far more complicated than Eqs.~42!–~43!, since the former set of equations contains apartialdifferential equation, whereas the latter set merely consistsof two coupled, first-order,ordinary differential equations.Certainly, in their present form, the improved phase evolu-tion equations, Eqs.~37!–~41!, are too complicated for con-venient use by experimentalists. The aim of the remainder ofthis section is to reduce this set of equations to a more man-ageable form via the judicious use of approximations.
B. Derivation
According to Eq.~38!, the electromagnetic torque actingin the vicinity of the rational surface consists of two parts:thesteadytorque due to eddy currents excited in the resistiveshell, and theoscillating torque due to the resonant error-field. It is, therefore, natural to suppose that the~normalized!perturbed rotation profile of the plasma,DV( r , t), consists ofa steady component,V0( r ), driven by the eddy currenttorque, and an oscillating component,V1( r , t), driven by theerror-field torque. In other words,
DV~ r , t !5V0~ r !1V1~ r , t !. ~44!
Suppose that the steady component ofDV( r , t) is simplytaken to be the perturbed rotation profile calculated in theabsenceof the error-field. It follows that19,24
V0~ r !5~v021!
3H 1 for r<1
Er
ad r/ r mY E
1
adr/ r m for 1, r<a
,
~45!
where
12v05as
2
4
v0
vs21v0
2 . ~46!
Here, v0 is the steady~normalized! angular velocity of themode in the presence of the resistive shell, but the absence ofthe error-field. It is generally convenient to search for thephysical roots of the above expression by converting it into adynamical equation,
d2v0
dt21l
dv0
dt512v02
as2
4
v0
vs21v0
2 . ~47!
Here,l is a positive constant chosen so as to optimize theconvergence of the solution of the above equation towards afinal fixed-point@which is equivalent to the physical root ofEq. ~46!#.
Equations~44!, ~45!, and~46! can be combined with theoriginal phase evolution equations~37!–~41! to give a newset of phase evolution equations,
r r
4J2v l
]V1
] t2
]
] r S r m]V1
] r D52T1
Jd~ r 21!, ~48!
T15as
2
4 H v
vs21v22
v0
vs21v0
2J 1asac
Av l
sinf, ~49!
]V1~0,t !
] r50, ~50!
V1~ a, t !50, ~51!
v2v05V1~1,t !. ~52!
Here,v[df/dt is the ~normalized! angular velocity of themode in the presence of both the resistive shell and the error-field.
According to Eq.~49!, the electromagnetic torque driv-ing the oscillating component of the perturbed plasma rota-tion profile consists of two parts; the time-varying compo-nent of the eddy current braking torque induced by error-field driven oscillations of the mode angular velocityv aboutits steady valuev0 , and the oscillating torque due to theerror-field. The latter torque modulates like sinf, wheref isthe helical phase of the mode measured with respect to thatof the error-field. It is the fundamental assumption of thisderivation that the former torque also modulates like sinf.This assumption can be justifieda posteriori; it is found tohold exactly for locked modes, to be a fairly good approxi-mation for most rotating modes, and to only seriously breakdown in a narrow region of parameter space close to theerror-field locking threshold. Hence, this assumption—whichgreatly simplifies the analysis—can be used, with a fair de-gree of confidence, to map out the nature of the various timeasymptotic solutions of the above set of phase evolutionequations in parameter space.
The fundamental assumption is equivalent to the adop-tion of the following trial solution to Eqs.~48!–~52!:
V1~ r , t !52 iV1~ r !exp~ if!. ~53!
Obviously, the actual solution is thereal part of the aboveexpression. Equations~48!, ~50!, ~51!, and ~53! possesssimple analytic solutions in two limits. The first limit corre-sponds tov!v l , in which case
V1~ r !.VsH 1 for r<1
Er
ad r/ r mY E
1
adr/ r m for 1, r<a
.
~54!
The second limit corresponds tov@v l , in which case
3615Phys. Plasmas, Vol. 7, No. 9, September 2000 Nonlinear dynamo mode dynamics in reversed field pinches
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V1~ r !.VsH exp@1Aiv/v l~ r 21!/~2J!# for r<1
exp@2Aiv/v l~ r 21!/~2J!# for 1, r<a.
~55!
The above expressions indicate that when the mode rotationvelocity v lies well below the critical valuev l then the os-cillating component of the perturbed angular velocity profile,driven by the error-field, extends over the whole plasma. Onthe other hand, when the rotation velocity greatly exceedsv l
then the oscillating component of the perturbed velocity pro-file becomeslocalizedin the vicinity of the rational surface.In this paper, it is assumed that if a magnetic island structureexists at the rational surface then its radial width is alwaysmuch less than the radial width of the oscillating velocityprofile. Hence, it is alwaysviscositywhich determines theextent of the region of the plasma which corotates with themode, rather than the magnetic island width. For the typicalisland widths, mode rotation velocities, and momentum con-finement time scales encountered in present-day magneticfusion experiments, this assumption is perfectly reasonable.
Now, according to Eqs.~54! and ~55!,
T15JF r mdV1
dr G12
11
52VsHAiv/v l for v@v l
1 for v!v l. ~56!
It is a reasonably good approximation to write
T152VsR, ~57!
where
R.A11 iv/v l . ~58!
In the Appendix, the above approximation is discussed andjustified.
Let Vs5uVsuexp(ius) and R5uRuexp(iuR). According toEqs.~52!–~57!,
v2v05V1~1,t !5uVsusin~f1us!, ~59!
T1~ t !52uVsuuRusin~f1us1uR!. ~60!
Now, it follows from the fundamental assumption~53! that
dV1~1,t !
dt5 ivV1~1,t !, ~61!
which yields
dvdt
5dV1~1,t !
dt5vuVsucos~f1us!. ~62!
It is clear that the fundamental assumption boils down to theassertion that the time variation of the mode angular velocityv is dominated by the modulation of the helical phasef,rather than any variation in the amplitudeuVsu or the phaseangleus .
Equations~59!, ~60!, and~62! yield
T1~ t !52uVsuuRucos~f1us!sinuR2uVsuuRusin~f
1us!cosuR ,
52uRusinuR
1
vdvdt
2uRucosuR~v2v0!,
.2$@11~v/v l !
2#1/221%1/2
A2uvu
dvdt
2$@11~v/v l !
2#1/211%1/2
A2~v2v0!, ~63!
where use has been made of Eq.~58!. Finally, the aboveequation can be combined with Eqs.~46! and ~49! to give
$@11~v/v l !2#1/221%1/2
A2uvu
dvdt
1$@11~v/v l !
2#1/211%1/2
A2~v2v0!1~v021!
1as
2
4
v
vs21v2 1
asac
Av l
sinf50. ~64!
C. Discussion
According to the previous analysis, the improved phaseevolution equations~37!–~41! can be reduced to the follow-ing set of four, coupled, first-order, ordinary differentialequations:
df
dt5v, ~65!
$@11~v/v l !2#1/221%1/2
A2uvu
dvdt
1$@11~v/v l !
2#1/211%1/2
A2~v2v0!1~v021!
1as
2
4
v
vs21v2 1
asac
Av l
sinf50, ~66!
dv0
dt5a0 , ~67!
da0
dt1la01~v021!1
as2
4
v0
vs21v0
2 50. ~68!
The physical dynamical variables aref—the helical phase ofthe mode measured with respect to that of the error-field—andv—the normalized angular velocity of the mode. Recallthat Eqs.~67! and ~68! are merely a numerical device forfinding the appropriate roots of Eq.~46!, and have no directphysical interpretation. However, the dynamical variablev0
is interpreted as the steady component ofv obtained by bal-ancing plasma viscosity against the eddy current brakingtorque, whilst neglecting the oscillating error-field torque.The parameteras @defined in Eq.~32!# is the normalizedmode amplitude,ac @defined in Eq.~33!# is the normalizedamplitude of the error-field,vs @defined in Eq.~34!# is thecritical value ofv above which the resistive shell starts to actlike a perfect conductor, andv l @defined in Eq.~35!# is thecritical value ofv above which the oscillating component ofthe perturbed plasma rotation profile, driven by the error-field, becomes localized in the vicinity of the rational sur-
3616 Phys. Plasmas, Vol. 7, No. 9, September 2000 R. Fitzpatrick and E. P. Yu
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face. In addition,l is an optional,O(1), positive parameter,with no direct physical significance, which is used to facili-tate the numerical solution of the above equations.
Virtually all of the physics content of the above set ofphase evolution equations resides in Eq.~66!, which is inter-preted as the angular equation of motion of the mode. Thefirst term represents the inertia of the region of the plasmawhich corotates with the mode, the second term is the restor-ing torque due to viscous coupling between the plasma in theimmediate vicinity of the rational surface and the oscillatingcomponent of the perturbed plasma rotation profile, the thirdterm represents the restoring torque due to viscous couplingbetween the plasma in the immediate vicinity of the rationalsurface and the steady component of the perturbed plasmarotation profile, the fourth term is the braking torque due toeddy currents excited in the resistive shell, and the fifth termrepresents the locking torque due to the error-field.
In the limit v l@1, Eqs.~65! and ~66! reduce to
df
dt5v, ~69!
1
2v l
dvdt
1~v21!1as
2
4
v
vs21v2 1
asac
Av l
sinf50. ~70!
Note that since the variablev0 does not appear in these equa-tions, they form a complete set, and Eqs.~67! and ~68! areredundant. Of course, the above pair of phase evolutionequations has the same form as the Zohm equations,~42! and~43!. However, for the typical momentum confinement timescales and mode rotation velocities observed in present-dayRFPs, the parameterv l is usually much less than unity.@Theapproximate formula for this parameter isv l;x' /Vs
(0)a2,wherex' is the anomolous momentum diffusivity,Vs
(0) isthe typical dynamo mode rotation frequency, anda is theplasma minor radius. The anomolous momentum diffusion inthe core of present-day RFPs is predominately due to mag-netic fluctuations: the typical diffusivity isx';50 m/s2.25
Likewise, the typical mode rotation frequency isVs(0);5
3103 rad/s, and the typical minor radius isa;0.5 m.16
Hence,v l;0.04 in present-day RFPs. This estimate is suffi-ciently small compared to unity that we can be certain thatall present-day RFPs lie in the parameter regimev l!1, asassumed in this paper. Another important assumption, madein this paper, is that the radial width of the oscillating com-ponent of the perturbed velocity profile is much wider thatthe radial magnetic island width,W, at the rational surface.This assumption is valid providedW/r s,Av l;0.2, which isreasonably well satisfied in present-day RFPs.# In this pa-rameter regime, the Zohm equations are replaced by Eqs.~65!–~68!. Fortunately, the numerical solution of this new setof phase evolution equations is almost as straightforward asthat of the Zohm equations. Equations~65!–~68! are morecomplicated than the Zohm equations because they containadditional physics; namely, a self-consistent calculation ofthe radial extent of the region of the plasma which corotateswith the mode; the width of this region being determined byplasma viscosity.
IV. ANALYSIS OF PHASE EVOLUTION EQUATIONS
A. Resistive shell braking and releasethresholds
Suppose that the amplitude of the resonant error-fieldapproaches zero~i.e., ac→0!. In this limit, the phase evolu-tion equations~65!–~68! simplify greatly to givev5v0 ,dv/dt5a050, and
12v5as
2
4
v
vs21v2 . ~71!
Note that in the absence of the error-field the mode rotateswith a steadyvelocity. The solution of the above equation inthe physically relevant regimevs!1 is fairly straightfor-ward. There are two branches of solutions. First, there is arapidly rotating branch,
v. 121 1
2A12as2, ~72!
on which the mode rotation velocity is sufficiently high thatthe resistive shell acts like a perfect conductor. Secondly,there is aslowly rotatingbranch,
v.as
2
8$12A12~A8vs/as!
4%, ~73!
on which the mode rotation velocity is sufficiently low thatthe mode eigenfunction can penetrate through the resistiveshell. Moreover, there is aforbidden bandof ~normalized!mode rotation velocities,
vs,v,1/2, ~74!
separating these two branches. When the normalized modeamplitudeas exceeds theresistive shell braking threshold,
~as!b51, ~75!
then the rapidly rotating branch of solutions ceases to exist,and the mode makes a transition from the rapidly to theslowly rotating branch~assuming that it starts off on theformer branch!. When the normalized mode amplitude fallsbelow theresistive shell release threshold,
~as!r5A8vs, ~76!
then the slowly rotating branch of solutions ceases to exist,and the mode makes a transition from the slowly to the rap-idly rotating branch~assuming that it starts off on the formerbranch!. There is considerablehysteresisin the resistive shellbraking/release cycle, since the release threshold is generallysignificantly smaller than the braking threshold. Note that theinevitable penetration of the mode eigenfunction through theresistive shell on the slowly rotating branch of solutions hasa destabilizing effect on the mode, leading to a somewhatlarger saturated amplitude~i.e., largeras! on the slowly ro-tating branch, and, hence, to some deepening of this hyster-esis.
Figure 1 shows an example hysteresis diagram for theresistive shell braking/release cycle, calculated forvs
50.05.
3617Phys. Plasmas, Vol. 7, No. 9, September 2000 Nonlinear dynamo mode dynamics in reversed field pinches
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Note that the above theoretical results were first derivedin Ref. 24, and were later confirmed experimentally inRef. 45.
B. Error-field locking and unlocking thresholds
Suppose that the parametervs goes to infinity. In thislimit, the resistive shell acts like a perfect conductor, andthere is zero eddy current braking torque. Furthermore, thephase evolution equations~65!–~68! simplify greatly to givev051, a050, and
df
dt5v, ~77!
$@11~v/v l !2#1/221%1/2
A2uvu
dvdt
1$@11~v/v l !
2#1/211%1/2
A2~v21!1
asac
Av l
sinf50.
~78!
The numerical solution of the above pair of equations in thephysically relevant regimev l!1 is fairly straightforward.There are two branches of solutions. First, there is arotatingbranchon which the mode angular velocityv is modulatedperiodically, as the mode rotates, by the oscillating error-field torque. Secondly, there is alocked branchon which themode is held stationary~i.e., v50! via the balance of theerror-field and viscous restoring torques. For the lockedbranch, the above phase evolution equations simplify furtherto give
asac
Av l
sinf51. ~79!
Figure 2 illustrates the nature of the rotating branch ofsolutions. It can be seen that as the normalized mode ampli-tudeas increases the amplitude of the error-field driven os-cillations in the normalized mode rotation velocityv also
increases. When the product of the normalized mode ampli-tudeas and the normalized error-field amplitudeac exceedsthe error-field locking threshold,
~asac! l.0.51, ~80!
then the rotating branch of solutions ceases to exist, and themode makes a transition from the rotating to the lockedbranch ~assuming that it starts off on the former branch!.Note that the error-field locking threshold~which is deter-mined numerically! is a very weak function ofv l . Note,further, that the disappearance of the rotating branch of so-lutions corresponds to the point at which the minimum ve-locity of the mode is reduced to zero by the error-field.
Figure 3 illustrates the nature of the locked branch ofsolutions. It can be seen that as the normalized mode ampli-tude as decreases the locking anglef increases. Accordingto Eq. ~79!, when the product of the normalized mode am-plitude as and the normalized error-field amplitudeac fallsbelow theerror-field unlocking threshold,
~asac!u5Av l , ~81!
FIG. 1. The hysteresis diagram for the resistive shell braking/release cycle,calculated forvs50.05. The normalized mode rotation velocityv is plottedas a function of the normalized mode amplitudeas .
FIG. 2. The rotating branch of solutions, calculated forac51 and v l
50.05. The maximum and minimum values of the normalized mode rota-tion velocity v are plotted as functions of the normalized mode amplitudeas .
FIG. 3. The locked branch of solutions, calculated forac51 and v l
50.05. The locking anglef is plotted as a function of the normalized modeamplitudeas .
3618 Phys. Plasmas, Vol. 7, No. 9, September 2000 R. Fitzpatrick and E. P. Yu
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then the locked branch of solutions ceases to exist, and themode makes a transition from the locked to the rotatingbranch ~assuming that it starts off on the former branch!.Note that the disappearance of the locked branch of solutionscorresponds to the point at which the locking anglefreachesp/2.
There is considerablehysteresis in the error-fieldlocking/unlocking cycle, since the unlocking threshold isgenerally significantly smaller than the locking threshold.The origin of this hysteresis is the strong variation in theradial width of the plasma region which corotates with themode as the normalized mode velocityv changes. Of course,the more efficient interaction between the mode and theerror-field on the locked branch of solutions has a destabiliz-ing effect on the mode, leading to a somewhat larger satu-rated amplitude~i.e., largeras! on the locked branch, and,hence, to some deepening of this hysteresis.
C. Resistive shell modified error-field locking andunlocking thresholds
The simultaneous presence of a resistive shell and astatic error-field necessitates the use of the full set of phaseevolution equations~65!–~68!. Fortunately, the numericalsolution of these equations is a relatively straightforwardtask. A comprehensive parameter scan, in the physically rel-evant regimev l , vs!1, leads to the diagram shown in Fig.4. This diagram indicates theapproximatelocations of all thevarious threshold curves in normalized mode amplitudeas vsnormalized error-field amplitudeac space. Note that the lo-cations of these curves are determined numerically. Thecurve labeled 1 represents the resistive shell release thresh-old: the approximate formula for this curve isas5A8vs for
ac<Av l /8vs. The curve labeled 2 represents the resistiveshell braking threshold: the approximate formula for thiscurve isas51 for ac<Av l . The curve labeled 3 representsthe combined error-field locking and unlocking threshold;the approximate formula for this curve isasac5Av l for as
>1. The curve labeled 4 represents the error-field unlockingthreshold: the approximate formula for this curve isasac
5Av l for as,1. The curve labeled 5 represents the error-field locking threshold; the approximate formula for thiscurve isasac5@Av las
210.51(12as2)# for as,1. Finally, the
curve labeled 6 represents the error-field locking threshold inthe absence of the resistive shell; the approximate formulafor this curve isasac50.51.
Note that the location of the error-field unlocking thresh-old curve ~curves 3 and 4 in Fig. 4! is unaffectedby thepresence of the resistive shell. This is as expected, since aresistive shell exerts zero torque on a locked mode, and,therefore, cannot affect its unlocking threshold. On the otherhand, the location of the error-field locking threshold curvein the presence of the resistive shell~curves 3 and 5 in Fig. 4!is different from that in the absence of the shell~curve 6 inFig. 4!. In fact, it can be seen that the critical value of theproductasac above which a transition to a locked mode istriggered is alwaysreducedin the presence of the resistiveshell. In other words, the resistive shell acts as acatalystforthe formation of an error-field locked mode. Incidentally, theuse of the term ‘‘catalyst’’ here is appropriate, since the re-sistive shell plays no role in the dynamics of the mode onceit has locked to the error-field.
Figure 4 suggests the existence ofthreebroad regimes ofbehavior, depending on the value of the normalized error-field amplitudeac . Theweak error-field regimecorrespondsto
ac,Av l . ~82!
The intermediate error-field regimecorresponds to
Av l<ac<Av l /8vs. ~83!
Finally, thestrong error-field regimecorresponds to
ac.Av l /8vs. ~84!
These three regimes are described in the following.
D. The weak error-field regime
Figure 5 illustrates the typical behavior encountered inthe weak error-field regime. As the normalized mode ampli-tudeas is gradually increased from a small value, the time-asymptotic solution starts off on the rapidly rotating branchof solutions, and then makes a transition to the slowly rotat-ing branch when the resistive shell braking threshold~curve2 in Fig. 4! is reached. The solution makes a second transi-tion to the locked branch when the error-field locking thresh-old ~curve 3 in Fig. 4! is reached. Asas is gradually de-creased from a large value, the solution starts off on thelocked branch of solutions, and then makes a transition to theslowly rotating branch when the error-field unlocking thresh-old ~curve 3 in Fig. 4! is reached. The solution makes a
FIG. 4. A schematic diagram showing the various threshold curves associ-ated with the phase evolution of a dynamo mode in a RFP in the presence ofboth a resistive shell and a resonant error-field. The curves are plotted innormalized mode amplitudeas vs normalized error-field amplitudeac space.Curve 1 is the resistive shell release threshold; curve 2 is the resistive shellbraking threshold; curve 3 is the combined error-field locking and unlockingthreshold; curve 4 is the error-field unlocking threshold; curve 5 is theerror-field locking threshold; and curve 6 is the error-field locking thresholdin the absence of the resistive shell. The extents of the strong~S!, interme-diate ~I!, and weak~W! error-field regimes are also indicated.
3619Phys. Plasmas, Vol. 7, No. 9, September 2000 Nonlinear dynamo mode dynamics in reversed field pinches
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second transition to the rapidly rotating branch when theresistive shell release threshold~curve 1 in Fig. 4! is reached.
Note that in the weak error-field regime there is hyster-esis in the resistive shell braking/release cycle but there isnohysteresisin the error-field locking/unlocking cycle.
In order to help explain the somewhat surprising absenceof hysteresis in the error-field locking/unlocking cycle, Fig. 6shows a typical slowly rotating solution which lies close tothe error-field locking threshold in parameter space. It can beseen that the mode rotatesintermittently. In other words, themode remains locked for a certain period of time, then sud-denly executes a single rotation, remains locked for the sameperiod of time, then suddenly executes another rotation, andso on. As the error-field locking threshold is approached inparameter space, the time interval between these intermittentrotation events increases rapidly, and effectively becomesinfinite as the threshold is attained. Hence, there is a com-pletely reversible~i.e., hysteresis free! transition between aslowly rotating mode and a locked mode as the error-fieldlocking/unlocking threshold is crossed. This reversible tran-sition should be contrasted with the irreversible transitionillustrated in Fig. 8 below. The latter transition is typical ofthe behavior of the error-field locking/unlocking cycle in theabsence of a resistive shell.
It is important to note that the resistive shell is unable toremove those components of the hysteresis in the error-fieldlocking/unlocking cycle which are associated with the desta-bilization of the mode when its eigenfunction is able to pen-
etrate through the shell, or with the destabilization of themode due to the more efficient interaction with the error-fieldwhen it is non-rotating.
It is clear that in the weak error-field regime the catalysisof locked mode formation by the resistive shell operates soeffectively that a locked mode forms as soon as a lockedmode solution becomes a possibility~i.e., as soon as theerror-field unlocking threshold is exceeded!.
E. The intermediate error-field regime
Figure 7 illustrates the typical behavior encountered inthe intermediate error-field regime. As the normalized modeamplitudeas is gradually increased from a small value, thetime-asymptotic solution starts off on the rapidly rotatingbranch of solutions, and then makes a direct transition to thelocked branch when the error-field locking threshold~curve5 in Fig. 4! is reached. Asas is gradually decreased from alarge value, the solution starts off on the locked branch ofsolutions, and then makes a transition to the slowly rotatingbranch when the error-field unlocking threshold~curve 4 inFig. 4! is reached. The solution makes a second transition tothe rapidly rotating branch when the resistive shell releasethreshold~curve 1 in Fig. 4! is reached.
Note that in the intermediate error-field regime theslowly rotating branch of solutions is only accessible whenthe normalized mode amplitudeas is decreasing. Note, also,
FIG. 5. The weak error-field regime.The maximum and minimum values ofthe normalized rotation velocityv areplotted as functions of the normalizedmode amplitudeas for ac50.12, vs
50.1, andv l50.05. The cases wherethe normalized mode amplitudeas isincreasing and decreasing are shownseparately.
FIG. 6. A slowly rotating solutionwhich lies close to the error-field lock-ing threshold. The angular positionfand the normalized rotation velocityvof the mode are plotted as functions oft/t, wheret is the unperturbed moderotation period, for as51.75, ac
50.12,vs50.1, andv l50.05.
3620 Phys. Plasmas, Vol. 7, No. 9, September 2000 R. Fitzpatrick and E. P. Yu
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that there is now a small amount of hysteresis in the error-field locking/unlocking cycle.
F. The strong error-field regime
Figure 8 illustrates the typical behavior encountered inthe strong error-field regime. As the normalized mode am-plitude as is gradually increased from a small value, thetime-asymptotic solution starts off on the rapidly rotatingbranch of solutions, and then makes a direct transition to thelocked branch when the error-field locking threshold~curve5 in Fig. 4! is reached. Asas is gradually decreased from alarge value, the solution starts off on the locked branch ofsolutions, and then makes a direct transition to the rapidlyrotating branch when the error-field unlocking threshold~curve 4 in Fig. 4! is reached.
Note that in the strong error-field regime the slowly ro-tating branch of solutions no longer exists. Note, also, thatthere is strong hysteresis in the error-field locking/unlockingcycle in this regime. In fact, as the normalized error-fieldamplitudeac increases beyond the critical valueAv l /8vs, theresistive shell has progressively less effect on the location ofthe error-field locking threshold, which quickly reverts tothat calculated in the absence of a shell.
V. APPLICATION TO EXPERIMENTS
Consider the typical RFX plasma equilibrium investi-gated in Ref. 19. The parameters for this equilibrium areI f
5600 kA, ea50.23, F[Bf(a)/^Bf&520.2, and Q[Bu(a)/^Bf&51.56. The characteristic dynamo mode forthis equilibrium is them51, n59 mode. Making the sameassumptions as those made in Sec. V B of Ref. 19, exceptthat the plasma viscosity profile is now assumed to be radi-ally uniform ~for the sake of simplicity!, the following esti-mated values for the parameters controlling the phase evolu-tion of the 1,9 mode in RFX are obtained:
vs.0.12, ~85!
v l.0.013, ~86!
as.3.7, ~87!
ac.380S br1,9
uBu Dc
. ~88!
Here, it is assumed that the nominal island width of the 1,9mode is 20% of the plasma minor radius. Furthermore,(br
1,9/uBu)c represents the ratio of the radial component of the
FIG. 7. The intermediate error-fieldregime. The maximum and minimumvalues of the normalized rotation ve-locity v are plotted as functions of thenormalized mode amplitudeas for ac
50.23, vs50.1, and v l50.05. Thecases where the normalized mode am-plitudeas is increasing and decreasingare shown separately.
FIG. 8. The strong error-field regime.The maximum and minimum values ofthe normalized rotation velocityv areplotted as functions of the normalizedmode amplitudeas for ac50.5, vs
50.1, andv l50.05. The cases wherethe normalized mode amplitudeas isincreasing and decreasing are shownseparately.
3621Phys. Plasmas, Vol. 7, No. 9, September 2000 Nonlinear dynamo mode dynamics in reversed field pinches
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1,9 harmonic of the error-field to the magnitude of the equi-librium magnetic field, evaluated at the radius of the con-ducting shell.
According to the analysis of Sec. IV, the locking andunlocking of the 1, 9 dynamo mode in RFX occurs in theweak error-field regime. Hence, the combined locking/unlocking threshold corresponds to the curveasac5Av l inparameter space. In the absence of the resistive shell, theunlocking threshold still corresponds toasac5Av l , but thelocking threshold becomesasac50.51. Hence, the criticalerror-field amplitude required to lock a typical dynamo modein RFX is estimated to be
S br1,9
uBu Dc
;831025. ~89!
In the absence of the resistive vacuum vessel, the criticalamplitude becomes
S br1,9
uBu Dc
;431024. ~90!
In other words, the resistive vacuum vessel in RFX reducesthe critical error-field amplitude required to generate alocked mode by a factor of 5. This result suggests that theprimary cause of the locked mode problems observed inRFX ~and other large RFPs possessing thin resistive vacuumvessels, e.g., TPE-RX! is the high resistance vacuum vessel,which renders otherwise innocuous error-fields problematic.
VI. SUMMARY
This paper investigates the nonlinear dynamics of a typi-cal dynamo mode in a RFP plasma in the presence of aresistive vacuum vessel and a resonant error-field. Theachievements of this paper are as follows:
~1! The derivation of a fundamental set of phase evolutionequations~37!–~41! ~see Sec. II!. These equations de-scribe the phase evolution of a typical dynamo mode in aRFP under the influence of the braking torque due toeddy currents excited in a resistive vacuum vessel andthe locking torque due to a resonant error-field. They arederived using the five physics-based assumptions listedand justified in Sec. II I. Unfortunately, Eqs.~37!–~41!
are too complicated for convenient use by experimental-ists whilst interpreting data or designing new RFP ex-periments.
~2! The reduction of the fundamental phase evolution equa-tions to a more manageable set of four, coupled, first-order, ordinary differential equations~65!–~68! ~see Sec.III !. These equations represent an important extension ofthe well-known Zohm equations,40 ~42!–~43!, whichhave been used extensively by experimentalists to inter-pret mode locking data obtained from magnetic fusiondevices. The nature of the extension is the incorporationof a self-consistent calculation of the radial extent of theplasma region which corotates with the mode; the widthof this region being determined by plasma viscosity.
~3! The numerical solution of Eqs.~65!–~68!, so as to obtaina comprehensive theory of the influence of a resistivevacuum vessel on error-field locking and unlockingthresholds~see Sec. IV!. This theory is encapsulated inFig. 4, which shows the locations of all the variousthreshold curves for resistive shell braking, error-fieldlocking, etc., in mode amplitude versus error-field am-plitude space. Not surprisingly, a resistive vacuum vesselis found to have no effect on error-field unlockingthresholds. On the other hand, such a vessel is found tofacilitate error-field locking; i.e., it reduces the error-fieldlocking threshold. Under certain circumstances~i.e., inthe weak error-field regime indicated in Fig. 4!, this ef-fect is so strong that a locked mode forms as soon as alocked solution becomes a possibility, i.e., as soon as theerror-field unlocking threshold is exceeded. Note that, inthe absence of a resistive vacuum vessel, the error-fieldunlocking threshold usually needs to be exceeded by asubstantial factor before a locked mode forms.
~4! The application of the results obtained in Sec. IV to theproblem of the locking of typical dynamo modes in RFX~see Sec. V!. The thin resistive vacuum vessel in RFX isfound to reduce the critical error-field amplitude requiredto generate a locked mode by a factor of 5. This resultsuggests that the high electrical resistance of the vacuumvessel is the primary cause of the locked mode problemsin RFX, rather than the presence of excessively largeerror-fields.
FIG. 9. The functionsf R andgR , de-fined in Eq. ~A3!, plotted against thenormalized mode phase velocityv/v l .The solid lines show the exact formsfor these functions, derived from Eq.~A2!. The dotted lines show the ap-proximate forms for the functions, de-rived from the approximation~A6!.The calculations are performed fora51/0.7.
3622 Phys. Plasmas, Vol. 7, No. 9, September 2000 R. Fitzpatrick and E. P. Yu
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In conclusion, this paper has developed a basic theoret-ical framework within which the phase evolution of rotatingdynamo modes in RFPs can be analyzed. Hopefully, thephysical insights and model equations obtained in this pro-cess will allow experimentalists to achieve a full understand-ing of why the so-called ‘‘slinky mode’’ locks in some re-versed field pinch devices, but not in others. As explained inthe Introduction, the locking of the slinky mode is currentlyan issue of outstanding importance in reversed field pinchresearch.
ACKNOWLEDGMENTS
This research was funded by the U.S. Department ofEnergy, under Contracts Nos. DE-FG05-96ER-54346 andDE-FG03-98ER-54504.
APPENDIX: AN EXACT FORM FOR THE PHASEEVOLUTION EQUATION
In the limit of a uniform density profile~i.e., r51! anda uniform viscosity profile~i.e., m51!, Eqs.~48!, ~50!, ~51!,and ~53! can be solved exactly to give
V1~ r !.Vs5I 0~Ail r !
I 0~Ail!for r<1
K0~Aila!I 0~Ail r !2K0~Ail r !I 0~Aila!
K0~Aila!I 0~Ail!2K0~Ail!I 0~Aila!for 1, r<a
, ~A1!
where J5 ln a and l5(v/v l)/(4J2). Here, I 0 and K0 arestandard modified Bessel functions. In this limit, the functionR, defined in Eqs.~56! and ~57!, takes the exact form
R5JAilH I 1~Ail!
I 0~Ail!
2K0~Aila!I 1~Ail!1K1~Ail!I 0~Aila!
K0~Aila!I 0~Ail!2K0~Ail!I 0~Aila!J . ~A2!
Repeating the analysis of Sec. III B, the above expression forR yields the following phase evolution equation for themode:
f R
v l
dvdt
1gR~v2v0!1~v021!1as
2
4
v
vs21v2 1
asac
Av lsinf50,
~A3!
where
f R5Im~R!
~v/v l !, ~A4!
gR5Re~R!. ~A5!
The approximate expression
R.A11 iv/v l , ~A6!
employed in Sec. III B, leads to the following approximateforms for the functionsf R andgR :
f R approx5$@11~v/v l !
2#1/221%1/2
A2uv/v l u, ~A7!
gR approx5$@11~v/v l !
2#1/211%1/2
A2. ~A8!
Figure 9 compares the above approximate forms forf R andgR with the exact forms derived from Eq.~A2!. It can beseen that the agreement between the two sets of functions is
excellent. This justifies the use of the approximate formula~A6! ~which is far simpler than the exact formula! through-out the bulk of this paper.
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