The Stability of a Constricted Gas Discharge · STABILITY OF A CONSTRICTED DISCHARGE 161 Figure 1. Equilibrium configuration of the pinched discharge. The discharge is pinched to

P/33 UK

The Stability of a Constricted Gas Discharge

By R. J. Tayler

One of the most obvious methods of attempting toconfine a hot plasma is to use the pinch effect of theself magnetic field of an axial current. This isillustrated in Fig. 1. A cylindrical system is mosteasily studied theoretically, although it is prob-ably necessary to close the system into a torus,or similar shape, to avoid heat and particle losses atelectrodes. Although some consideration has beengiven to toroidal geometry, most complete resultshave been obtained for infinite cylindrical systemsand only these will be discussed here. This paper isto be regarded as a progress report on an unfinishedproblem and an indication is given of aspects of theproblem which, it is hoped, will be studied soon.

The original idea of using the pinch eñect was thatof constricting a gas discharge to an extremely smallradius when the only heat losses from it would beradiation losses, principally bremsstrahlung.1 It washoped that power would eventually be achieved bypassing a high enough current to reach thermonucleartemperatures, and by containing the system for along enough time.2 At an early stage, it becameclear that this simple idea must be modified becauseof the presence of instabilities (see Fig. 2). Thesewere observed 3 and predicted 4'5 at about the sametime. When attempts were made to understand theinstabilities and to predict methods of stabilisation,it was found that a high degree of constriction did notappear to be possible if the discharge were to bestable.6'7 Thus, study is now made of rather fatdischarges in a large tube.8

It should be noted that a linear stability theory,such as was considered in Refs. 6 and 7, does notimmediately prove that a highly constricted dis-charge cannot be used. Although such a discharge isunstable, non-linear stabilising forces may prevent itfrom approaching the walls of its containing vessel.At the same time, the mass motions of the dischargemay lead to additional heating, which would be anadvantage at high temperatures when Joule heatingbecomes relatively inefficient.9 The non-linear pro-blem has not received adequate investigation todate. What evidence there is suggests that the non-linear forces are not very effective unless the plasmais fairly close to stability.10'11

U.K. Atomic Energy Authority, AERE, Harwell.

Observations are, unfortunately, very easy only forcompletely unstable discharges. Thus, the earlywork at Harwell was done in Pyrex tubes and theentire current channel could be photographed.3 Sucha photograph is shown in Fig. 3. Generally the mostdeveloped instability appeared to be a helix of pitch45°: this result is remarkably independent of theparticular apparatus used. A very detailed analysisof instabilities has been made by Allen, usinga racetrack torus.12 When stabilisation is seriouslyconsidered, the tube has a conducting wall and visualobservations can be made only through slits. In thiscase, the presence of the slits may effect stabilitylocally.13 Probe and microwave measurements aredifficult to interpret, unless their time resolution isvery good, and visual measurements may be ambiguousbecause the most frequent occurrence is for the lightto be given out by impurities, which may not occupythe same volume as the main current channel.

The full problem of discharge stability, even for acylindrical system, is extremely complicated. Inparticular, a discharge will not be in simple hydro-static equilibrium and, even in the absence of instabi-lities, the finite electrical conductivity will causegradual changes in the magnetic field configurations.In most of the work described in this paper, it isassumed that steady motions and field penetrationoccur so slowly that, in the stability problem, thedischarge can be supposed ideally conducting and atrest. Two theoretical models are considered here.The first approach is extremely naïve : plasma dyna-mics is completely neglected and only the forcesexerted by the electromagnetic field are considered.The second approach is to use Maxwell's equationscombined with a simple one-fluid hydrodynamics—classical hydromagnetics. A more sophisticated pro-cedure would be to start from Boltzmann's equationfor the several types of particle present and to tryto derive a hydrodynamic system. As an exampleof this procedure there is the work of Watson andco-workers 14~16 and of Chew, Goldberger and Low.17

In the simplest problems that have been considered,essentially similar results have been obtained fromthe Kinetic Theory, Hydromagnetic and even theMagnetostatic approach.

The remainder of the paper is arranged as follows.In the next section the magnetostatic or extensible

160

STABILITY OF A CONSTRICTED DISCHARGE 161

Figure 1. Equilibrium configuration of the pinched discharge.The discharge is pinched to a radius r0 by the inward radialforce Fr = ]z Be; the discharge may contain an axial field bobx and

be surrounded by conducting walls of radius Л г 0

wire model of the discharge is studied. The fol-lowing two sections are based on the hydromagneticequations: in one, the gradual loss of stability of thestabilised pinch,7'18-20 due to field diffusion, is dis-cussed, and, in the other, another type of stabledischarge configuration is introduced. In the finalsection there is a discussion of further problems whichmust be studied. Gaussian units are used throughoutfor the electrical quantities.

Figure 2. Perturbed configuration of the pinched discharge.When the surface of the discharge is distorted as shown, theazimuthal field lines crowd together on the inside of the kink,leading to a force tending to increase the perturbation. Thetension in the axial field lines and ¡mage currents in the walls

tend to resist the perturbation

THE WRIGGLING DISCHARGE

The first model of the discharge considered is onein which all the plasma dynamics is neglected andonly electromagnetic forces are considered. Thus thedischarge is replaced by an extensible wire carryinga current. When this model was introduced, interestwas concentrated on the highly constricted dischargeand the discharge was treated as a very thin wire.Given perturbed configurations of the wire wereconsidered and the forces acting on the configurationwere calculated in a magnetostatic approximation.First calculations were made for the outward, in-stability-producing forces on a free discharge ; 4 latercalculations were made of restoring forces due toconducting walls.10 The results obtained from thismodel are much better than might be expected andfor the linear terms the stability criteria are in goodagreement with those found for a hydromagneticmodel. In addition it is quite easy to calculate non-linear terms in the forces and to obtain an estimate of

the amplitude to which given perturbations cangrow before the net force is inward. This is a some-what academic calculation because as soon as non-linear terms are considered configuration-changingforces are introduced and all the modes of instabilityinteract. However, it does give a first clue to theimportance of non-linear forces and suggests that theycannot be very effective unless the plasma is fairlyclose to stability; that implies that it is not highlyconstricted.

The results for one particularly simple configurationare as follows. Suppose a discharge of radius r0 is ina tube of radius Ar0 and carries a current / . Theperturbed discharge takes a helical form given bythe equations

% = A cos kz, у = A sin kz. (2.1)

The outward (self) force and inward (image) forceare expanded for small values of Ak and 1/Л and takethe form:

= AkX0[- lnZ0 + In 2 - y + ¿]

- AWX0 [ - lnX0 - In 2 - y + f]243 264 S

P

and(2.2)

\K0(AX0) + K2(AX0)l .Л К Л ° [ 1 0 \

- 2i¥In

(АХ0)+12(АХ0)\

Кг(2АХ00 L /x(2AZ0) + /3(2ЛХ(

^ 0 ( Л Х 0 ) + К 2 ( Л Х 0 )

0)1

0) J

I0(AX0) + I2(AX0)1364

243 Iif2(3AX0) + К,(ЗАХ0)]\6 4 [ / 2 ( З Л Х 0 ) + / 4 ( З Л Х 0 ) J / ' [ • '

Í^ /О Л V \ I Z?" /О Л V \П

i j (ZYVAQJ -j- A 3 (Z7VA0/)

Л(2ЛХ0) + /3(2ЛХ0) J

Figure 3. An unstable discharge in a pyrex torus

162 SESSION A-5 P/33 R. J. TAYLER

where F' is the radial force, y is Euler's constant0.5772..., In and Kn are modified Bessel functions ofthe first and second kinds and Xo is a non-dimensionalwave number kr0. The correcting terms to (2.2) and(2.3) are of order [Ak)1 and 1/Л2.

Results have been calculated from (2.1) and (2.2)for several values of Л and they are shown for onevalue of Л (Л = 20) in Fig. 4. It can be seenfrom this figure that the change in wave number,Xo> between a perturbation which is completelystable and one for which the outward force vanisheswhen it is halfway across the tube, is very small.Values of wavelengths and amplitudes, for which thenet force vanishes, are shown in Table 1 whereRo = Ar0 is the tube radius.

Table 1. Critical Values of Amplitudes for WhichOutward Force Vanishes

i/л*. . 1.490.0

1.460.2

1.390.4

1.290.6

1.140.8

Fuller details of this work are given in Ref. 10.It would also be possible to include the effect of an

axial magnetic field inside the discharge channel, byintroducing a tension into the wire ; however, this pro-blem has been studied only on the hydromagneticmodel. It is possible to show that an externalmagnetic field cannot completely stabilise helicalperturbations of the type considered here.

In more recent work, by Curtis and Roberts,11 anattempt has been made to follow the time developmentof the wriggling discharge by using an electroniccomputer. The model of this section is furthersimplified, by replacing the exact static forces by anapproximation better suited to numerical work. Theequation of motion

= I"self _j_ pimage (2.4)

is then solved.The parameters taken were: tube radius, Ro] length

of discharge, 10 RQ; and number of points alongdischarge channel, either 100 or 300. The dischargeradius was not included as an explicit parameter, asthe logarithmic term in the self force was replaced bya constant ; this is roughly equivalent to saying thatthe correct value of the force is used for some meanwavelength of instability. The value of the loga-rithm taken corresponded to a discharge fatter thanthe one considered in the static problem above(A ^ 1 0 ) .

Because of the finite number of points along thedischarge, there is a cut-off in possible perturbationwavelengths. Some smoothing had also to beintroduced into the integration procedure, because ofthe propagation of rounding-off errors, and thisintroduced a further wavelength cut-off. This cut-off might be considered in some sense equivalent tothe introduction of an axial magnetic field whichstabilises short wavelengths. With these conditionsthe discharge moved across about 50% of the tube

Self force

1/(ЛХ0)

1.5 ^ — ^ .

2.5 *

A.1.0 Ro

0.5 *

/

/

2.5/

Image force

= 0.5 \

o¿5—*

\ -

Д

-4.0

/

-2.0 /

/

--2.0

--4.0

Net force

/ 0.5

-

• ^ - ^ - ^ S ^ 4 ^ v 1.0

\ N 1 5 -

\ 2.5

, 33.4

Figure 4. Forces on helical discharge Л = 20. The outward selfforce. The inward image force and the net force, on a helicalcurrent channel, are shown as functions of the amplitude and

wavelength of the helix

radius. Although the fatter discharge should bemore stable than the results given in Fig. 4, it appearsto be even more stable than would be expected. Thefull interpretation of these results has not yet beenmade.

THE STABILISED PINCH

In this section the stability problem is studied,with the hydromagnetic equations as starting point.These are, for an ideally conducting fluid of pressurep, density p, and ratio of specific heats y, carryingelectric and magnetic fields E and В and current andcharge densities j and Q,

dt

dpdt

1 dpp~dt~~

V x B -

V X £ =

7_

P

4iT

с

ТуV

dp

dt

I

1

с

>

евdt'

= 0,

andv x B

-o.

(3.1)

(3.2)

(3.3)

(3.4)

(3.5)

(3.6)

(3.7)

(3.8)

It is supposed that an infinite cylinder of conductingfluid, of radius r0, is surrounded by a vacuum con-taining a magnetic field B0(0, ro/r, be) [in (r,0, z) co-ordinates]. In turn, the vacuum is surrounded by aconducting wall at radius Ar0. The stability of a


system in which none of the equilibrium quantitiesdepend explicitly on в and z is investigated by consi-dering departures from equilibrium in which anyvariable q has the form

= q0(3.9)

By solving the complete set of equations (3.1) to(3.8) for perturbations of the form (3.9), it has beenshown by several authors 7>18Д9 that there existstable configurations of this system, if the magneticfield inside the conducting fluid is constant and purelyaxial, having the value (0, 0, Bobi). In this casedisturbances corresponding to m = 0 and m = 1are the most difficult to stabilize but there are setsof values of (bi, be, Л) for which complete stability isobtained. These results are illustrated in Fig. 5.Chandrasekhar, Kaufman and Watson have obtainedthe same stability criteria, for a plasma in whichcollisions are neglected and Boltzmann's equation istaken as the starting point.20

In fact, such discontinuous magnetic fields will notbe realised in practice. Even if they could beobtained, the finite conductivity of the dischargewould lead to a gradual interpénétration of theaxial and azimuthal fields and to changes in thedischarge radius. As it has also been shown 7 thatthe discharge is unstable if the current density in theplasma is uniform and b\ and be are equal, it appearsthat at some stage in the interpénétration stability islost. The field diffusion proceeds at a rate determinedby a penetration time, тр = 4ттаго

2/с2 where cris the relevant electrical conductivity. If instabi-

Figure 5. Stability diagram for surface current. For several valuesof wall radius Л the regions of stability for m = 0 and m — 1 areshown. Stability occurs in the region above the curve markedwith the given values of m and Л and below the curve 1 + be2

= bi2. Complete stability for given Л occurs when a point liesin the stability region for m = 0 and m = 1

lities occur, a characteristic instability time n is thetime taken by a sound wave crossing the plasma. Forhigh temperature plasmas тр >̂ ть Because of thisinequality, it seems possible to divide the probleminto two parts; a, the penetration problem, in whichstability is assumed and the rate of field diffusion iscalculated; and b, the stability problem, in whichstability is investigated assuming that the plasma isstatic and ideally conducting.

Penetration

The penetration problem has not been studied ingreat detail. Neglecting the complications due toplasma motions and anisotropic electrical con-ductivity, a simple calculation has been made of thediffusion of an axial field out of a plasma and of anazimuthal field into it.21 With the added assumptionthat the axial magnetic flux and the axial currentremain constant, the equations for the time and spacedependence of the plasma magnetic fields become:

(3.10)ocp[J2{ocp)]2

X

Вг/В0 = ôe0 + (6ю - be0)/j

Jifà)J0(№)

ho - бво)

where: Л(ар)=О;

(A2-l)ft

(3.11)

(3.12)

T = t/rv; R = r/r0; b\0 and beo are the initial values ofbi and be] and the Jn are Bessel functions.

Results have been calculated for the special caseof Л = 2 and ¿eo — ^ a nd these are shown in Figs. 6and 7. The radial dependence of both axial andazimuthal fields is shown for several values of thetime. It can be seen that quite considerable fieldpenetration can occur in time 5 X lO~3rp and, infact, it will be seen that times less than this are ofimportance in stability problems. However, for avery hot plasma, even lO~3rp will be greater than TÍ.Although an actual discharge does not have a constantscalar conductivity, the results of Figs. 6 and 7 areprobably qualitatively correct in showing that arather small fraction of the relevant penetration timesuffices for noticeable field diffusion. The results canbe generalised slightly; if G is time- but not space-dependent, exactly the same equations (3.10),(3.11) hold, provided that a new time coordinate isintroduced, defined by

/• T

Jo °V")Г ' = I —£—¿Г» (3.13)

where a0 is the initial value of the conductivity.


Stability

The stability problem has been treated by anenergy principle method due to Bernstein, Frieman,Kruskal, and Kulsrud.22 This has been developedfrom one first used by Lundquist ;23 similar resultshave also been used by Hain, Lust and Schlúter.24

Bernstein et al. show that if a perturbation of aplasma is considered m which the element initially atr(0) moves to r(0) + Ç, the stability of the plasmaagainst this perturbation can be determined byexamining the sign of the change SW in the potentialenergy of the system. If SW is minimised withrespect to all perturbations \ satisfying certainconditions on the plasma-vacuum interface, and theminimum SW is negative, then the system is unstable.Otherwise it is stable. The energy principle will notbe described in detail here ; the results for the pincheddischarge, which will be briefly discussed, are givenfully in Refs. 25 and 26.

Figure 6. The diffusion of azimuthal magnetic field into acylindrical conductor. The field is shown as a function of the di-mensionless radius R, for several values of the dimensionless time T

Figure 7. The diffusion of axial magnetic field out of a cylindricalconductor. The field is shown as a function of the dimensionless

radius R, for several values of the dimensionless time T

Because of the absence of dependence on в and zin the equilibrium quantities, any perturbation can beresolved into the harmonic form (3.9) and each pairof values of m and k can be considered separately.It is easy to minimise SW with respect to £e and ¿.When this is done, it is found that V#Ç vanishes,except at a discrete set of points, unless the pitch ofthe equilibrium magnetic field is constant in someregion. This does not occur in the present problem.When V#Ç = 0 is introduced into the expressionfor SW, all explicit dependence on the equilibriumplasma pressure is removed; the pre-eminence ofincompressible perturbations is the reason why theresults obtained in the last section are valid. Anotherfeature that deserves mention is that for values of mgreater than zero, the perturbation giving the lowestenergy has an infinite shear at one point within theplasma.

With the forms for the magnetic fields (3.14),(3.15) and the £e and | z minimisations performed,SW becomes, apart from a numerical constant:

It is supposed that in the central region of theplasma there is only an axial field ВОЬЪ but thataxial and azimuthal fields are mixed in the outerfraction e of the discharge radius. For mathematicalsimplicity, the field profiles taken are not those givenby the solution of the penetration problem (Figs. 6and 7). Instead it is supposed that both Bz andBe/r are linear with radius in the current layer. Thus

whereГ = Уо(1 — e) + Го ex. (3.15)

For small e, it can be seen from Figs. 6 and 7 thatthis approximation to the interpenetrating fields isprobably quite good; in particular it appears justifiedto take the same € for both Be and Bz. The fieldand pressure profiles for one particular set of valuesof Ъъ be> A and e are shown in Fig. 8.

В = {0, Bor%lro, BQ [6, + (6e - bi)*]} , (3.14)

/ —\ \

• f \1 \

1 ' 33 8

8"Po/Bo / / \ \

\ / / \\ 1

1 [

Figure 8. Sketch of discharge configuration with partiallyinterpenetrated fields.The normalised magnetic fields (BelBotBzIBo)and plasma pressure (8тгро/Во

2) are shown as functions of theradius г


SW=-(m

O b i v"

Km(X0)Im'(AX0) - Im(X0)Km'(AX0

Km'(X0)Im' (AX0) - Im'(X0)Km'(AX{

t x Im[X0(l - e)]

'•4o)J

Mi-*)

2B,

+

+ k2r2 \ r

1 2

In equation (3.16), Km, Im are modified Besselfunctions, £ is the radial component of Ç and f0 and£]_ its values at the inner and outer boundaries of thecurrent sheet, Xo = £f0, and the prime denotesdifferentiation with respect to r in the integral andwith respect to the argument of the Bessel functionselsewhere. The first term in SW is the vacuumcontribution, the second comes from the plasmacentre and the third from the current sheet. Ife tends to zero, it can be shown the third term ap-proaches the value — 1. In this case, the conditionthat SW vanishes reduces to the stability criterionfor the stabilised pinch given in Ref. 7.

The expression for SW is minimised for f's of theform

so that(3.17)

(3.18)

With the form (3.17) for £ and (3.14) for the magneticfield, the integral in (3.16) can be evaluated eitherexactly or as a power series in e.

One particular problem has first been considered.It is assumed that initially

= 0, Л = 2, be = (3.19)

and that field diffusion occurs subject to the followingthree conditions:

(i) Bo remains constant,(ii) Л remains constant,

(iii) The total axial flux remains constant.Perturbations have been considered with m = 0 andm = 1 ; these were the types of perturbations mostdifficult to stabilise with surface currents. At first,trial functions are considered with only one arbitraryparameter; afterwards the number of parameters isincreased.

In the case of ж = 0, it is easy to see that SW ismonotonically increasing with XQ; thus we need onlydetermine its sign at Xo = 0. For m — 1, no such

simple result holds. For m = 0, one-parameter trialfunctions have been considered for several values ofn and €. The resulting minimum values of bW areshown in Fig. 9. It can be seen that for eachvalue of € considered, 8W has its minimum value forn = 3. It is also easy to show algebraically that theminimum SW, for one-parameter trial functions,approaches the asymptotic value shown as n -> oo.It is possible to find the value of e for which theminimum point on the curve, corresponding to thoseshown in Fig. 9, lies at SW = 0. This is the criticalvalue at which stability ceases in the one-parameterapproximation. Corresponding critical values of ehave been found for trial functions with up to fourparameters and these are shown in Table 2. It canbe seen that the value of e changes only very slightlyin going from one to four parameters. These resultssuggest that the critical value of e converges well asmore parameters are added, and the trial functions £also appear to be converging.

Table 2. Values of efor Which Stability Is Lost for One-,Two-, Three- and Four-Parameter Trial Functions; m=0

30.157

3,40.156

2,3,40.150

2,3,4,50.150

In the case of m — 1 it is more difficult to obtaincomplete results because the value of Xo whichgives the minimum SW is unknown. Results musttherefore be obtained for several values of Xo and theminimum value deduced. For one-parameter trialfunctions, it is found that SW has its minimum valuefor n = 1, and for such trial functions the criticalvalue of e at which stability ceases has been found.Corresponding values of с have also been found fortwo- and three-parameter trial functions. Theseresults are shown in Table 3; they are not as accurateas those in Table 2 because of the amount of inter-Table 3. Values of с for Which Stability Is Lost for One-,

Two- and Three-Parameter Trial Functions; m = 1

n€crit

10.25

2,30.19

1,2,30.19

polation involved. However, the critical values forthe two- and three-parameter cases are very close,both being between 0.185 and 0.190. It is clear that,in this case, the one-parameter trial function does notgive an accurate idea of the critical value of e; it isimpossible to judge whether closeness of the two- andthree-parameter values is a sign of convergence,without also doing calculations for n = 4 and above.It seems likely, for this special problem, that the e atwhich stability is lost is close to 0.15 and that theaxisymmetric modes are the least stable.

In addition to this special problem, qualitativestability criteria for general field configurations havebeen obtained, using only one-parameter trial func-tions. For the case of m = 0, calculations have been


made for e = 0.1 and 0.2 and for several values of Л.A one-parameter trial function with n = 3 was takenand, to this approximation, stability curves weredrawn in the bu be plane. These correspond to thecurves for e = 0 in Fig. 5 and they are themselvesshown in Figs. 10 and 11. The upper curve in thesefigures is obtained from the condition that, for givenbe, there is a maximum value of b\ for which a plasmacan be contained. The equation of this curve is

1 + ¿>e2 + f € - W = № (3.20)

These stability curves should probably be a goodapproximation to the actual stability curves; in anycase they always overestimate the region of stability.

Finally some calculations have been made withone-parameter trial functions with m = 1 and Л = 2.The results of Table 3 indicate that stability curvesobtained from these results cannot be more thanqualitatively correct. Nevertheless, we have illus-trated in Fig. 12a how, on the one-parameterapproximation, the region of stability for both m = 0and m — 1 shrinks as с increases.

Although the results obtained above indicate thatstability is lost for quite a small value of e, the temp-tation to consider only the terms in the energy that

0.5

0.25

8 W

0 0

0.25

0.5

1

\ \ х^

V 2

\

1

i

е = 0.00

0.10

0 . 1 5 ^ к ^

i

4n

À

у 0.20 /

33.9

Asymptotic -»—values —*•

-

-

/

-

Figure 9. Minimum values of $W for one-parameter trial func-tions; m = 0. The value of 8W/B0

2 £02 is plotted as a function of

n for four values of s. It can be seen that the one-parametertrial function with n = 3 gives the lowest value of 8W

Figure 10. Stability diagram for one-parameter trial functions;m = 0.1, m = 0. In the one-parameter approximation, the regionof stability is shown for m = 0, s = 0.1 and several values of thewall radius Л . Stability occurs in the region above the curvemarked with the given value of Л and below the curve 1 + be2 4-fs — £e = bi2. The stability curve for m = 1, e = 0 and Л =

1.5 is also shown

are linear in e must be resisted. For the specialproblem, the critical value of e for axisymmetricperturbations, obtained by linearising in e, is 0.333compared with 0.157 obtained above.

All the results reported here have been obtained byhand computation and it is unlikely that the problemwill be studied any further by this method. Miss S. J.Roberts is at present coding the problem for anelectronic computor; it is hoped that when this isdone, more general field configurations and trialfunctions will be able to be considered. It is alsohoped that further evidence will be obtained aboutthe convergence of the variational procedure.

Recent Developments in the Theory of the StabilisedPinch

Since the previous section was written it has beenshown that the convergence problem mentioned in thelast paragraph is a real one. Rosenbluth27 andSuydam 28 have studied the mathematical characterof the third Euler equation and have shown that,because of a singularity in this equation, the stabilisedfinch is unstable against surface instabilities for anydepth of penetration of the current. This singularityoccurs at the radius in the plasma where the magneticfield helix coincides with the perturbation helix. Itcan be asked whether this singularity is physicallyreal; in particular what happens to it when particlemotions and dissipative processes are considered? Afirst attempt at a study of this problem has recentlybeen made by Hubbard 29 at Harwell.

Hubbard assumes that every particle is acted onnot by a point magnetic field but by the field that has


been averaged over some averaging length (8) andhe has replaced the singular term in the differentialequation by its value averaged over this distance.He has solved the resulting differential equation onthe Mercury computer, for many values of theaveraging length and the depth of the current layer,and has found stability diagrams corresponding toFig. 12a. One such stability diagram is shownin Fig. 126. In the limit of zero averaging lengthhe has found results agreeing with those of Rosen-bluth and Suydam. For finite values of the averag-ing length he has calculated the critical depth offield penetration at which stability ceases. He hasalso given arguments for choosing the averaginglength and a discussion as to whether it is determinedby the electrical conductivity, viscosity or particlegyration. He believes that the Larmor radius of theions provides a suitable averaging length in manycases and his results sometimes show greater stabilitythan the conventional stabilised pinch.

This work is at the moment based on physicalintuition rather than on a rigorous mathematicalfoundation. He has merely modified the final secondorder differential equation and has supposed thatmarginal stability is still determined by the vanishingof the disturbance growth rate. In fact, if finiteviscosity and electrical conductivity are included inthe original equations, the order of the mathematicalproblem is considerably increased and complex growthrates occur.

GENERAL STABLE DISCHARGECONFIGURATIONS

The conventional stabilised pinched discharge is notan obviously stable configuration; as has alreadybeen seen, very stringent conditions have to be satisfiedby the parameters Ъ-ъ be and Л for stability to beobtained. In the energy-principle formulation it is seenthat the change in potential energy is positive inthe vacuum and the centre of the plasma, and negativein the current layer and, for stability, an accurate

o.oo

Figure 11. Stability diagram for one-parameter trial functions;s = 0.2, m = 0. In the one-parameter approximation, the regionof stability is shown for m = 0, s = 0.2 and several values of thewall radius, Л. Stability occurs in the region above the curvemarked with the given value of Л and below the curve 1 + be2

4- | £ —1£2 — bi2. The stability curve for m = 1, s = 0,and Л = 1.5 is also shown

balance has to be obtained. In this section the pos-sible existence of obviously stable configurations isinvestigated. These are configurations in which thechange in energy is positive in all parts of the discharge,for all perturbations, or even more restrictively onesin which all terms in the energy are positive every-where. This problem has been studied at Harwell byLaing.30

Suppose an ideally conducting plasma fills a tubeof radius Ar0. If the plasma contains a magneticfield (0, Be, Bz), the change in its potential energy Wdue to a perturbation \ el (mâ+kz)f after minimisationwith respect to £e and \Zi is

bW = kBA - 1Be (Be/r)'- IB

where £ is the radial component of the displacement Çand differentiation is denoted by a prime. As theconducting fluid fills the entire tube, the radialperturbation £ must vanish at the radius Ar0. Usingthis condition, an alternative form can be obtained forbW by performing an integration by parts on the termin № in (4.1).

Thus,

(4.2)

(mBe

(m* + kh*)

(4.1)

where

rA = [mBe krBz)2 + [тВв - krBz)

2/(m2 + k2r2)- 2B2 - 2m2k2r\Be

2 + B2)\(m2 + k2r2)2

- k2r*(Be

2 + B2y\(m2 + k2r2),В = r{mBe + krBz)

2/(m2 + k2r2). (4.3)

If the integrand in expression (4.1) is to be positiveat all points in the discharge, the coefficient of | 2 mustbe positive. Thus,

(mBe + krBz)2 > 2B0[rBey. (4.4)

168 SESSION A-5 P/33 R. J. TAYLEK

Figure 12a. Stability diagram for one-parameter trial functions;Л = 2. Curves are drawn for three values of s and for m = 0and m = 1. It can be seen how the region of total stability, in

this approximation, decreases as s increases

This condition has previously been obtained from theperturbed differential equations in Ref. 7. If Eq.(4.4) is to be satisfied for all perturbations m, k, theright-hand side of the equation must be negative orzero at all values of r. Thus,

о, (4.5)

where Iz is the total axial current within radius r.This condition cannot be satisfied everywhere; inparticular, at the discharge centre the current and itsgradient must be in the same direction.

If the integrand in expression (4.2) is to be positiveat all points in the discharge, A must be greater thanzero. Using the hydrostatic equation yp = j хВ/4тг,this condition becomes

8тггр'а 2bBeBz + cBz

2, (4.6)

where

а = т\ - 1 + 1/(ж2 + X2) + 2X2/(w2 + X2)2],Ъ = тХ[ - 1 + 1/(ж2 + X2)],с = Х2[ - 1 - l/(w2 + X2) + 2ж2/(ж2 + X2)2],d = X2/(w2 + X2),

X = kr. (4.7)

It can be shown that the worst perturbation for theinequality (4.6) is that with k — 0 and m — 1. Ifthe inequality is to be satisfied for this perturbation,

rp' > Be

2/6ir. (4.8)

If the inequality (4.8) is satisfied at all values of theradius, the plasma cannot be contained by the magne-tic field. Equation (4.8) has been obtained by mini-mising the right-hand side of inequality (4.6) for allpossible values of Be, Bz.

There remains the possibility that the pressure canincrease outwards near the centre of the dischargeand that the current and its gradient are in oppositedirections in the outer region. Thus SW is dividedup, so as to use the form (4.2) in the central regionand the form (4.1) outside. There is also an addi-tional term in bW, from the integration by parts onthe interface r = r0. Thus

J 0

+ rdr kBz) -2Be(Belr)

? X+ kB,\r? XI(m« +

1J r_ r, '

(4.9)

It is now insisted that all parts of this expression arepositive. If the surface term is to be positive for allm and k, Be must vanish on r = r0; thus the totalaxial current inside r = rQ vanishes. In addition,there can be no axial current outside r = r0 ; otherwisethe inequality (4.5) would once again fail. Thefinal requirements are

rp' > £//6тг, for 0 < r < r0

Be - 0, p' = - {B/y/Sn for r0 < r < Ar0. (4.10)

A completely stable configuration satisfying theconditions (4.10) has been found by Laing.30 The

O.75 -

as -

O.O

Figure 12b. Stability diagram with averaged fields ô = 0.05Л = 2. The m = 1 stability curves are drawn for several valuesof s and for the averaging distance (ô) 5 per cent of the dischargeradius. The m — 0 curves refer to zero averaging lenghts as

there is no singularity in this case


Figure 13. A stable configuration with reserved current. Curvesare shown of the radial variation of B#, Bz and p for an "ob-

viously " stable discharge configuration

values of magnetic fields and pressure for this con-figuration are shown in Fig. 13. It is not necessaryfor the axial magnetic field to drop to zero where thecurrent has its maximum, as it does in this example;the example was obtained by assuming that Be hasa simple analytical form. It is hoped that this typeof configuration will be studied further and thatneighbouring stable configurations can be found, whichdo not satisfy the very restrictive conditions of thissection.

When the theoretical results of this section wereobtained, it was not realised that Burkhardt andLovberg31 had obtained a configuration similar inprinciple to the ones described above. In their casethe reversed axial current arose unexpectedly and thedischarge appeared to be stable.

FURTHER PROBLEMS

Plasma Motions

The only problem in which plasma motions havebeen considered in detail is that of the wrigglingdischarge. However, there is no reason to supposethat plasma motions are absent, even in an essentiallystabilised discharge. Three such motions are: theinitial contraction ; motion of expansion or contractionduring field diffusion; and steady oscillations, whichmay occur even for a stable discharge. The presenceof these motions may lead to new types of instabilities.Many problems arising from the presence of equili-brium motions are essentially non-linear in thevelocity; certainly, perturbations no longer grow ordecay harmonically with time.

The oscillation problem can be studied in a linearway, as a first approximation. If it is imagined thata force is instantaneously applied to counteract theplasma acceleration, at the time in its motion whenit is momentarily at rest, then effectively the problemis the same as that of a plasma supported againstgravity by a magnetic field. A simple problem ofthis type has been studied by Kruskal and Schwarz-schild.5 In their problem the plasma was very

unstable, because there was no magnetic field in theplasma. In the presence of crossed magnetic fields,which are required to stabilise the initial configuration,many of the Rayleigh-Taylor types of instabilityare also removed.

The next approach to the oscillation problem con-siders the plasma to be oscillating in one mode andconsiders small time dependent perturbations on this.The study of the stability of such a system leads todifferential equations of a generalised Hill's type.Some progress has been made on one simple problemalong these lines. The problem of the stability of aplane plasma, which is executing rigid body oscillationsbetween plane conducting walls, can be reduced, ina certain approximation, to the solution of twosimultaneous Mathieu equations. This is still onlya very poor approximation to the real problem, inwhich there may be interaction between very manymodes on equal terms and the oscillation energy maybe passing back and forth between them.

Finite Dissipation

It has been stated that, on qualitative grounds, it isplausible to treat the electrical conductivity as infinitein stability calculations. Similar arguments can bemade in support of the neglect of viscosity and thermalconduction in some problems. However, even ifnumerically the importance of dissipative processeslooks small, it must be realised that their introductioncompletely alters the mathematical character of thestability problem. One particular problem, in whichviscosity may be important, has already been men-tioned; in the energy principle method the lowestenergy, for non-symmetrical perturbations, is obtainedfor a perturbation \ with an infinite velocity shear.As the energy can be made arbitrarily close to thisminimum, with a perturbation with large but finiteshear velocity, the result is probably virtually un-

VV ^ A s y m p t o t i c v

\ \ Invisc\ \ S solutio

/ x->\

/ Viscous/ solution

кI Asymptotic

/ value

a lue

dn

Figure 14. Influence of viscosity on discharge stability. A quali-tative plot is given of the relationship between growth rate со andwavelength X for the m = 1 instability of a plasma carrying asurface current. This shows the ¡nviscid solution and the effectof the introduction of viscosity. The asymptotic solutions atsmall wavelength intersect at a point which gives a first approx-

imation to the wavelength of maximum instability


altered for small enough viscosity; but this requiresfurther investigation. For axisymmetric perturba-tions the trouble does not arise and, for sufficientlysymmetrical systems and perturbations, a generalisedform of energy principle still appears to be valid. Asolution has been given to one problem involvingviscosity in Ref. 6. For the earliest problem of theinstability of the pinch effect studied by Kruskal andSchwarzschild 5 there is an infinite instability growthrate at zero wavelength. The introduction of anyviscosity, however small, reduces this growth rate tozero. This is illustrated in Fig. 14. An attempt isalso being made to study the stability of a plasma,with finite electrical conductivity carrying a uniformaxial current and a uniform axial magnetic field.With no axial field, the problem is simple; with an

axial field, it is only easy to obtain asymptotic resultsand, at present, no general results on the effect offinite electrical conductivity have been deduced.

Two other factors are likely to become much moreimportant to us in the future: the geometrical factor,which arises both through toroidal shape and throughincomplete symmetry produced by gaps in conductingwalls; and the problem of the range of validity of thehydromagnetic equations. It is true to say that theproblem of gas discharge stability is still in its earlystages and much remains to be done.

ACKNOWLEDGEMENTSI am grateful to Dr. E. W. Laing, Miss S. J. Roberts

and Dr. W. B. Thompson for many helpful discussionson various aspects of plasma stability.

REFERENCES

1. R. S Pease, Equilibrium Characteristics of a Pinched GasDischarge cooled by Bremsstrahlung Radiation, Proc.Phys. Soc, В 70, 11-23 (1957).

2. J. D. Lawson, Some Criteria for a Power ProducingThermonuclear Reactor, Proc. Phys. Soc, В 70, 6—10(1957).

3. R. Carruthers and P. A. Davenport, Observations of theInstability of Constricted Gaseous Discharges, Proc Phys.Soc , В 70, 49-51 (1957)

4. S. J. Roberts, P. A. Sturrock, W. B Thompson andR. T. P. Whipple, The Wriggling Discharge in FreeSpace, AERE T/R, 1792 (1955).

5. M. Kruskal and M. Schwarzschild, Some Instabilities ofa Completely Ionised Plasma, Proc. Roy. Soc , A 223,348-60 (1954).

6. R. J. Tayler, Hydromagnetic Instabilities of an IdeallyConducting Fluid, Proc Phys. Soc, В 70, 31-48 (1957).

7. R. J. Tayler, The Influence of an Axial Magnetic Fieldon the Stability of a Constricted Gas Discharge, Proc.Phys Soc, В 70, 1049-63 (1957).

8. P. С Thonemann, E P. Butt, R. Carruthers, A. N Dellis,D. W. Fry, A Gibson, G. N. Harding, D. J. Lees,R. W. P. McWhirter, R. S. Pease, S A. Ramsden andS Ward, Production of High Temperatures and NuclearReaction ш a Gas Discharge, Nature, 181, 217 (1958).

9. L. Spitzer Jr , Cooperative Phenomena in Hot Plasmas,Nature, 181, 221-2 (1958).

10. S. J. Roberts and R. J. Tayler, The Influence of Con-ducting Walls on the Wriggling Gas Discharge, I. Wallsof Infinite Conductivity, AERE T/R, 2138 (1957).

11. A. R. Curtis and S. J. Roberts, The Time Developmentof the Wriggling Gas Discharge Confined within a Con-ducting Cylinder, unpublished.

12. Т. К Allen, Instabilities in High Current Discharges,Proceedings of the Third Conference on Ionisation Phe-nomena m Gases, Venice, 40-9 (1957).

13. W. B. Thompson, On the Image Field in a Cut Wall,AERE T/R, 2300 (1958).

14. К. М. Watson, Use of the Boltzmann Equation for theStudy of Ionised Gases of Low Density I, Phys Rev., 10212-19 (1956).

15. K. A Brueckner and К M. Watson, Use of the BoltzmannEquation for the Study of Ionised Gases of Low Density II,Phys. Rev, 102, 19-27 (1956).

16. S. Chandrasekhar, A N Kaufman and К. М. Watson,Properties of an Ionised Gas of Low Density in a MagneticField III, Ann. Phys, 2, 435-70 (1957).

17. G F Chew, M. L. Goldberger and F E Low, The Boltz-mann Equation and the One Fluid Hydromagnetic Equa-tions in the Absence of Particle Collisions, Proc. Roy. Soc,A 236, 112-8 (1956)

18 M Rosenbluth, The Stabilized Pinch, Proceedings of theThird Conference on Ionisation Phenomena in Gases,Venice, 903-7 (1957).

19. V D Shafranov, On the Stability of a Cylindrical GaseousConductor in a Magnetic Field, Atomnaya Energ., 1,38-43 (1956).

20. K. M. Watson, A N. Kaufman and S. Chandrasekhar,The Stability of the Pinch, Los Alamos report, unnumbered(1957).

21. J. H. Adlam and R J Tayler, The Diffusion of MagneticFields in a Cylindrical Conductor, AERE T/M, 160(1958).

22 I. В Bernstein, E. A Fneman, M. D. Kruskal andR. M. Kulsrud, An Energy Principle for HydromagneticStability Problems, Proc Roy. Soc, A 244, 17-40 (1958).

23 S. Lundquist, On the Stability of Magneto-hydrostaticFields, Phys Rev, 83, 303-11 (1951).

24. К Ham, R Lust and A. Schluter, On the Stability of aPlasma, Z Naturforsch., 12a, 833-41 (1957).

25 M. H Boon, E W. Lamg, J. Randies, S. J. Roberts andR J. Tayler, Hydromagnetic Instabilities of a CylindricalGas Discharge 6. Energy Principle Calculations forAxisymmetric Perturbations, AERE T/R, 2503 (1958)

26. E W. Lamg and R J. Tayler, Hydromagnetic Instabilitiesof a Cylindrical Gas Discharge. 7. Energy PrincipleCalculations for Wriggling Perturbations, AERE T/R,to be published (1958).

27. M. Rosenbluth, Theory of Pinch Effect, Stability andHeating, YjSAl', this Volumes, these Proceedings.

28. В R. Suydam, Stability of a Linear Pmch, P/354,this Volumes, these Proceedings.

29. J. Hubbard, The Instabilities of a Cylindrical Gas Dis-charge with Field Penetration, AERE T/R, 2668, to bepublished in 1958.

30. E. W. Lamg, Stable Configurations of a Cylindrical GasDischarge, AERE T/M, 161 (1958).

31. L C. Burkhardt and R. H. Lovberg, New ConfinementPhenomena and Neutron Production in a Linear StabilisedPmch, Nature, 181, 228-30 (1958).

Documents

The Stability of a Constricted Gas Discharge · STABILITY OF A CONSTRICTED DISCHARGE 161 Figure 1. Equilibrium configuration of the pinched discharge. The discharge is pinched to