P/361 USA

Generation and Thermalization of Plasma Waves

By T. H. Stix

A fully ionized gas immersed in an axial magneticfield may exhibit low-frequency oscillations wherethe electric field and macroscopic velocity per-turbations are perpendicular to the axis. Theordinary and extraordinary hydromagnetic waves ofAlfvén 1 and Âstrôm 2 are examples of these transverseoscillations. Ion cyclotron waves3 are anotherexample. The physical environment appropriate tothese low-frequency transverse waves is present inseveral types of proposed thermonuclear reactors,including the stellarator 4 and the pyrotron.5

For several reasons, the thermonuclear experimen-tist may be interested in generating these waves.First, they are a hydromagnetic phenomenon that canbe produced in the laboratory and experimentaldata can be checked against hydromagnetic theory.Second, generating and observing these waves atlow power may be a useful diagnostic technique fordetermining plasma characteristics which are impor-tant parameters of these waves such as magneticfield strength, plasma density and density distribution,plasma temperature and composition, and the stateof ionization of the plasma. Finally, generatingthese waves at high power is a possible scheme forrapid heating of the plasma. Calculations on ioncyclotron wave generation and thermalization in-dicate an efficiency, in a typical example, of morethan 65% for the transfer of power from an rfpower source into effective random ion motion.

An effect which is termed " cyclotron damping "is found to be extremely important in the theory ofion cyclotron waves. Cyclotron damping causes freeion cyclotron waves to decay exponentially withtime, or propagated ion cyclotron waves to decayexponentially with axial distance. Furthermore,cyclotron damping converts the energy of the wavemotion into effectively random transverse ion motion,which makes ion cyclotron heating especially usefulfor a pyrotron-type device.

Cyclotron damping occurs for transverse oscillationswhich are periodic both in time and in axial distance.Ions moving along lines of force will " feel " theoscillations of the transverse electric field at afrequency which differs from the plasma rest frame

* Project Matterhorn, Princeton University, Princeton,New Jersey.

frequency by the Doppler shift. Some ions will" feel " the oscillations at their own cyclotron fre-quency, and Lenard6 has pointed out that theseions will absorb energy at a constant rate from theelectric fields. Dawson 6 has shown that the presenceof classes of such ions leads to the damping withdistance of propagated oscillations. In the excellentpaper of Gershman7 the effects of cyclotron dampingon ordinary and extraordinary hydromagnetic waveshave been computed.

The kinematic effects of cyclotron damping appearin the off-diagonal components,

щт <( wxwz )>Av and щт ( wywz )>AV,

of the stress tensor. The axial ion velocity wz is aconstant of the motion through first order in thesetransverse oscillations, and is a zero-order quantity.The off-diagonal stress tensor terms are then firstorder terms, and must be included in a theory ofsmall-amplitude oscillations. Although the stresstensor does not appear explicitly, it is the calculationof these momentum transport quantities whichoccupies the first section of this paper. A dispersionrelation is obtained at the end of this section.

Then the physical effects of cyclotron dampingare discussed in terms of phase relations, powerabsorption and damping rates. The reduction of thedispersion relation and the phase relations to theundamped results previously obtained (Ref. 3) isshown for frequencies slightly removed from the ioncyclotron frequency.

After that we calculate the energy input from aninducing coil to a cylindrical plasma. The cir-cumstances include excitation of the plasma in theimmediate neighborhood of the ion cyclotron fre-quency and excitation of ion cyclotron waves andhydromagnetic waves. Estimates are made for thewidth of the power absorption resonances, andthe power transferred to the plasma is compared to theohmic losses in the induction coil. For a plasma ofappreciable density, the power transfer is efficient onlyin the cases of wave generation. A plasma heatingscheme proposed in the last section is to generate ioncyclotron waves at a long wave-length, in order toachieve efficient transfer from the rf power source,and allow the waves to propagate through a regionof magnetic field which decreases with axial distance

125

126 SESSION A-5 P/361 T. H. STIX

from the coil The wavelength of the ion cyclotronwaves becomes shorter and shorter, and finally localcyclotron damping starts to occur In this localregion, the waves damp out and the wave energy isconverted into energy of effectively random transverseion motion

DISPERSION RELATION FOR CYCLOTRONDAMPING

Small amplitude oscillations perpendicular to thelines of force are considered for a fully ionized gasimmersed m an axial magnetic field Particle colli-sions are infrequent and the temperature of the gascorresponding to ion motions perpendicular to thelines of force is assumed to be zero The temperatureof ions corresponding to motions parallel to the linesof force is taken to be finite We consider oscillationswhich are periodic m axial distance and m time, andneglect effects of resistivity gravity viscosity andelectron inertia

For transverse oscillations the velocity of anindividual ion along a line of force is approximatelyconstant We divide up the plasma into a masslesselectron fluid and into constituent streams of ions,where each stream is populated by ions with thesame zero-order axial velocity We solve the equa-tion of motion for each constituent stream separatelyin terms of the transverse electric field along a line offorce Adding the streams together gives a chargeand current distribution due to the ion motions mterms of the electric field Inserting this ion chargeand current distribution into Ohm's Law and Max-well's equations yields homogeneous equations m thetransverse electric fields The solution of theseequations determines the x and y dependence of theelectric fields for a given z and t dependence, andthe elimination of the electric field variables from theequation yields a dispersion relation for the system

The coefficients in the dispersion relation areintegrals over the distribution of zero-order parallelion velocities Following a suggestion made byIra Bernstein 8 we can avoid divergences in theseintegrals at resonance by considering an initial valueproblem m the ion motion

A list of symbols used in this paper is given inTable 1, together with their definitions Dimension-less quantities will be used extensively

The first-order linearized equation for the macro-scopic motion of ions in one of the constituent streams is

ыx (1)

We assume that the first-order electric and magneticfields vary as the real parts of ©(#, y) exp (ikz + icot)and $&(x y) exp (tkz + icot), where the componentsof © and Я5 may be complex A consequence of themassless electron motion is that (£z is zero Using thisfact and Maxwell's equations, Я5 m equation (1) canbe expressed m terms of ©# and ©^

A solution to (1) will then express the transversevelocity m terms of the transverse electric fields We

choose that particular solution u ± which fits theinitial values ux = щ = 0 at T = TQ and sum u ±

over all the constituent streams, to obtain v ± Indimensionless form, we have

- A iDJ\

X E±e™z + г С 1 Т +

where

X

Л-О

(2)

Л

к / 1 ± Л

X sin [(Г - To) (1 ± A]dA, (3)

2\xA-Q Л

к / 1 ± Л

X {1 _ cos [(Г - Го) (1 ± A)]}dA (4)

The integrals in (3) and (4) approach long-timeasymptotic values For values of T — To^> \KSQ\~1,which means a time long compared with the time ittakes ions with the rms velocity to travel an axialdistance equal to A/9, the integrals may be reason-ably approximated by their asymptotic values

For a Maxwelhan distribution of axial ion velocitiescharacterized by a mean axial ion energy of kTl¡2i

A± has the long-time asymptotic form

n9 47Л

± e-P±' (5)

where = (1 ±For large values of T — To we may approximate

D± by dropping the cosine term and taking theprincipal part of the remaining integral For aMaxwelhan distribution, the long-time asymptoticform is conveniently given by £)± —> I± — | Agraph of KS0I± versus p± is shown m Fig. 1 Alsoshown on the graph is \xs0A± I± reaches peakvalues of O54|^so|"1 at \p±

at ions to / ± are I±

n)"1 for U

1 0 Useful approxim--1 for \p±\ > 1 and

Figure 1 Graphs of ̂ sol ± and |«|s0A

GENERATION AND THERMALIZATION OF PLASMA WAVES 127

Table 1

Integers and OperatorsA atomic weightZ\ ionic atomic number„ д д àУ €*^T7+ еу-^ + ex

Re real part of

0 €XQX + eyQy + ezQz

Q± exQx + eyQy

Q Q = Re{Qe™Z + iQT}

€x, ey, €z, er, unit vectors

cgs-Gaussian Unitsfirst-order magnetic fieldzero-order axial magnetic field strengthvelocity of lightfirst-order electric fieldelectronic charge

3*kI/coü

mПе, fli

n(wz)dwz

Гс

Чy, 6, z; x,y, z;Tiu

wЯ = | А |СО

COci

sheet current density in " ideal coil "axial wave-number, -¿ 2я/АBoltzmann's constantover-all length of induction coilion massnumber of electrons, ions per unit volumenumber of ions per unit volume with wz inrange dwz

radius of induction coilradius of plasma boundary

t cylindrical and Cartesian coordinates; timeion temperature in °Kfirst order macroscopic velocity for a singleconstituent streammacroscopic plasma velocityindividual ion velocityaxial wave-lengthangular frequencyangular ion cyclotron frequency, ZieB^mc

Dimensionless Units

В

EF±

h, /i,

J *LM, NPmP±Q.u,R

Eq. (3), (5)

Eq. (4)SSo

-)'ФКг

Eq. (6)Eq. (21)

n (wz)dwz

mRéf. 9Fig. 1

* > *- cod

COci ¿coil/cEq. (14)Eq. (22)

(1 ± Q)(xso)-iF, W Section 3

rcocilc

УVкAvS

a

Eq. (23)nc*]ïcod/t

[*, У> *]

4жщтс2В0~2

Eq. (20)Eq. (9)

Eq. (18)kc/coci

il -f- y.wzc~x

Eq. (8)Eq. (23)

of Eq. (7)

We look now at particle energies rather thanvelocities. Let m(u^ + %2>/2 denote the averageover an oscillatory cycle of the transverse energy ofan ion in a single constituent stream. Summingthis average energy over all the constituent streams,we obtain

( 6 )

where F+ has the long-time asymptotic form for aMaxwellian distribution, F± -> Fo + (T — T0)A±¡2.Fo is the contribution to the sum from ions which do

not " feel " their ion cyclotron frequency, and repre-sents the kinetic energy associated with the non-resonant part of the forced ion oscillations. The secondterm is the contribution to the sum from ions whichdo " feel " their ion cyclotron frequency. The energyassociated with a single-constituent stream at exactresonance increases as the square of the elapsed time.However, because of the spread in axial velocities,fewer and fewer streams remain in resonance as timegoes on, so that the total energy associated with allthe streams increases only linearly with time. Simi-larly, the velocity u ± of a single-constituent streamat exact resonance increases linearly with elapsedtime, but the summed velocity v ± asymptoticallyapproaches a constant magnitude.

The physical situation we wish to approximate ina self-consistent manner is a plasma with steady-state oscillations. A low but finite collision rate inthe plasma creates a quasi-equilibrium situation.(There may be a heating of the plasma due to theoscillations, which we can make arbitrarily slow byreducing the amplitude of oscillation.) To helpunderstand this quasi-equilibrium, we consider, forsimplicity, those collisions which interchange theaxial velocities of the colliding ions but leave thetransverse velocities of each ion unchanged. Eachcolliding ion will be removed from one constituentstream and put into another. The transverse velocityan ion carries with it from the first stream may bedifferent from the transverse velocity at that pointassociated with the second stream. If so, this dif-ference will be random. In particular, an ion movingfrom a resonant stream to a non-resonant streamwill bring with it a large transverse velocity, almostall of which will be random with respect to thenon-resonant stream. A large number of such tran-sitions will increase the temperature of the trans-verse motion of the non-resonant ions. Conversely,an ion moving from a non-resonant stream toa resonant stream will bring with it almost notransverse velocity. It will then be acceleratedby the transverse electric fields from an initial stateof almost zero transverse velocity. A large numberof such transitions will cause energy to be absorbedby the resonant ions from the electric fields in theplasma. These two processes are the thermalizationand energy absorption mechanisms of cyclotrondamping.

With these processes as guides, we can construct asimplified model of a quasi-equilibrium plasma towhich our ion orbit calulations would accuratelyapply. We consider a plasma in which each ionundergoes a catastrophe at a time r after its previouscatastrophe. Each catastrophe causes the ion's trans-verse velocity to be reduced to zero. The macroscopicvelocity <V_L> for this plasma is obtained from (2)by averaging over catastrophe times,

This average has the same asymptotic form forlarge T that v± has for large T — To. <V±> gives us

128 SESSION A-5 P/361 T. H. STIX

the plasma ion current density, which is all we require.Plasma electron current is obtained from Ohm's Law(Eq. 5a, Ref. 3) which states that electrons movefreely along magnetic lines of force and with a velo-city cE X ez across magnetic lines of force. InMaxwell's equations (Eq. 6a-9a, Ref. 3) we canneglect displacement current in both the plasma andvacuum for the wave-lengths and frequencies ofinterest to us. We obtain the differential equationfor the transverse electric field in the plasma

[V X (V X E)]JL = crJÉ + icTaE X e* (7)

where ax == oc(iA+ — iA_ — 1 + / + + 7_), and(T2 == oc(iA+ + iA_ + О + / + — /__), and the long-time asymptotic expressions are to be used for A±.

In cylindrical coordinates, with no dependence on в,and when аг and a2 are independent of position, the rand в components of Eq. (7) may be combined togive Bessel's equation. The solution which is finiteat the origin9 is Er~ Ёв~ J^vR), where

- сг2 (8)

v is the radial wave-number. By going to the limitof large radii, Eq. (8) may be thought of as thedispersion relation for a Cartesian coordinate system,where v and к are wave-numbers in the appropriateCartesian directions.

PHASE RELATIONS AND CYCLOTRONDAMPING RATES

We distinguish four regions, determined by fre-quency, wavelength, density, and temperature,

Four parameters are particularly important incharacterizing these regions, \x\s0, p±, y, and 1 0 X2/OL,where

у =/ осЛ±

mk2c2

-2 .03Х10 1 6 (10)

\K\S0 is the ratio of the velocity of an ion of energyIT i to the phase velocity of the oscillation, and issmall in cases of practical interest. It may be eva-luated numerically by inspection of equation (5).p± measures the proximity of the oscillation fre-quency to the ion cyclotron frequency, у large orsmall will determine whether or not the electricfield in the plasma is appreciably affected by theinduced field due to plasma currents. x2/oc large orsmall was the principal criterion found in Ref. (3)separating ion cyclotron waves from hydromagneticwaves. In detail, the conditions in the four regions are :

Case I : \p+\ < 1 or \p_\ <c 1, \x\s0 <C 1, X2/OL > 1

Case I I : \p+\ > 1, \p_\ > 1, x2/oc > 1, O fía ± 1

Case I I I : \p+\ > 1, \p_ > 1, X2\OL <C 1

Case IV: \p+\ > 1, |/>_ > 1, x2/oc <C 1. (11)

Using (2), the radial component of (7), and ap-proximations based on (11 ), we obtain the approximatedispersion relations and phase relations for each ofthe four cases given in Tables 2 and 3. Some addi-tional comments may be made on the separate cases.References will be made to (11), Table 2 and Table 3without specific mention.

Table 2. Approximate Dispersion Relations, аг and <r2 Case I

Dispersion relation

II. =

III. Q.2 =

IV. П2 =

7^1 ±o*i

_сг 2

a_ ^ 2 _

ci

2(1 ± Q)

1 - Ü2

ai l 3

1 - П2

The ions rotate in circular orbits with the samesense of rotation as that for a free ion in a magneticfield. The ion velocity vector is thus circularlypolarized. An electric field with the same circularpolarization and phase exists in the plasma, so thatenergy is being put into the ions. Taking the timeaverage of v*E, and using the phase relations ofTable 3 to obtain v in terms of E, we obtain a power

Table 3. Phase Relations

where the oscillations described by the dispersionrelation (8) may be rather easily characterized. Thespectrum also contains other regions, where the plas-ma behavior is not so easily characterized. The kine-matics in the other regions can, however, also beobtained from the equations of the preceding section.The four regions are labelled:

Case I : Neighborhood of ion cyclotron frequency

Case II : Ion cyclotron waves

Case III : Torsional hydromagnetic waves

Case IV : Compressional hydromagnetic waves.

I.

II.

Ш . а

Er

iEfí

гу

iv.a a 11 +

w- a

vr ±Er ±

2A

T

a Valid for «И » хй.

GENERATION AND THERMALIZATION OF PLASMA WAVES 129

input for Cases I and II which is the same as givenby (6), where the energy in each constituent streamwas summed over all the streams. The same powerinput for Case I was also obtained by Kulsrud andLenard 6 through quite a different calculation.

For Case I, EQ varies as 1г (constant xR), where theabsolute value of the constant lies between 1 andV2. For | и | Д 0 < 1 , Ee~R for R < Ro, and isequal to the EQ that would exist in the region in theabsence of plasma due to the induction coil, i¿0vacuum.This Ee cannot match the boundary condition fora free oscillation (see Ref. 3), so the oscillations canonly be forced or driven.

For sufficiently low density or short wavelength,y < l and Er is very small compared to Ee. Thetotal vector electric field in the plasma is approxi-mately equal to e0Ee

Yacuum. Ee is in phase with ve,and in this circumstance cyclotron damping is astrong effect. For у ^> 1, corresponding to higherdensity or longer wavelength, Er is approximatelyequal in magnitude to EQ. The physical explanationfor this set of circumstances was offered first byKulsrud.6 The currents due to the radial ionmotion and the associated longitudinal electronmotion induce a fluctuating magnetic field and inturn a radial electric field, Er. It turns out that thisradial electric field is phased so that the resultantEr and Ee combination have principally a circularpolarization which is opposite to that of the ion mo-tion, vr and ve. The wrongly polarized electricfield does not put energy into the ions. There is asmall electric field of the correct polarization whichsupplies enough energy to the ions to overcome thenow much reduced cyclotron damping effect. Itwill turn out that at appreciable densities this in-ductive effect greatly reduces the efficiency of ioncyclotron heating in the Case I region.

Cases II, III and IV

These are possible oscillatory modes of the plasma*The simple criterion, \ф+\ ^> 1 and |̂ >_| ^ 1 is suffi-cient to reduce the formalism in this paper to thesimpler approach of Ref. (3), where the stress tensorwas neglected in the equation of motion.

In Cases III and IV and in the undamped form ofCase II, the electric field in the plasma is in quadraturewith the ion velocity. No energy is transferred toor from the plasma wave in the steady state. How-ever, when these waves are externally generated by aninduction coil, the electric field due to the coil willappear in the plasma in phase with the ion velocity andin quadrature with the coil current. The electricfield due to plasma currents associated with thegenerated wave will be in quadrature with the ionvelocity but in phase with the coil current. Thisfield is the " back emf ". Power flows from thecoil to the plasma wave.

Of particular interest is the case for ion cyclotronwaves which are slightly damped by a small amountof cyclotron damping. This situation occurs forvalues oí ft± such that A± is small but not negligiblysmall compared to | / ± | . (See Fig. 1.) In the

constituent stream derivation, к and Q were assumedreal. The dispersion relation is satisfied by com-plex values of v. Physically, the cyclotron dampingpower is furnished by a divergence in the radial (or^-direction) flow of energy. Writing v = vQ + ivx

where v0 and v1 are real, (v^-1 is the ^-direction¿-folding distance for the damping of velocity andfield amplitudes. We are also interested in twoother cases.

When О and v are real, but к is complex, thecyclotron damping energy is supplied by a divergencein the axial flow of power. Writing к = x0 + ixlt

(%1)~1 is the axial damping distance for the wave

amplitudes. Finally, when v and к are real butО = ÇlQ 4- ¿Qx is complex, the cyclotron dampingenergy is supplied by the wave itself, causing it todamp out with time at a rate Clv

For slightly damped waves, the damping rate anddistances are linearly related to one another withcoefficients determined by the undamped waves.Using such linearized perturbations of the dispersionrelation, we obtain the damping rate and reciprocaldamping distance for slightly damped ion cyclotronwaves,

±aA± (v* 4K4)

4v2x2

(12)

(13)

Equation (13), for the case v = 0, has also beenobtained by Dawson.6

ENERGY INPUT TO A PLASMA FROM ANINDUCTION COIL OF FINITE LENGTH

The relations given in the preceding sections applyto steady-state oscillations in a cylinder of plasmaof infinite length. In practice, one has a plasma offinite length and an induction coil considerablyshorter than the plasma. The true resonant modesof such a plasma are periodic in the axial length.However, we shall assume that wave energy which ispropagated axially away from the coil is thermalizedand absorbed without reflection. Assuming rapidthermalization, the rate of temperature rise of a plasmain a finite size reactor (e.g., stellarator or pyrotron)may be found by dividing the net power input to theplasma by the heat capacity of the plasma. Forpractical cases, it is useful to have a ratio of powerinputs rather than an absolute level. We choose tocompare the power input to the plasma with the rfpower which is wasted, in a practical system, in ohmiclosses in the induction coil. This ohmic loss is theprincipal source of inefficiency in a practical plasmaheating system. And in a diagnostic experiment,power absorption by the plasma must be measuredabove the background of this ohmic loss. For-tunately, in both plasma heating and plasma diag-nostic experiments, this power ratio is convenientto measure.

130 SESSION A-5 P/361 T. H. STIX

We call this power ratio W, and break W up intoa product of three more ratios, such that W = QVU.We discuss these ratios in turn.

W

W is the average rf power input to the plasmadivided by the average rf ohmic power loss in thecoil system. The coil system includes not only theexciting coil, but also the connecting leads andthe resonating elements, such as capacitors.

Q is the figure of merit for the coil system. It isdefined as the peak rf magnetic energy stored inthe coil system multiplied by the angular frequency,со, and divided by the average rf ohmic power lossin the coil system. Typical Q values in the mega-cycle range would run from 200 to 400.

A practical coil system stores magnetic energy inregions other than inside the volume normally oc-cupied by plasma. To allow room for the vacuumchamber wall and for electrical insulation, the radiusof the plasma boundary may be only one half thecoil radius. Magnetic energy will also be storedoutside the coil volume in the stray inductances of theresonating elements and connecting leads. We there-fore introduce a storage ratio, V, defined under no-plasma conditions. V is the peak rf magneticenergy stored in an " ideal coil " divided by thepeak rf magnetic energy stored in the coil system.The " ideal coil " is defined to have the followingcharacteristics: (a) its radius is the same as theradius of the plasma boundary, (b) it carries a sheetcurrent proportional to cos (kz + cot), (c) it haszero stray inductance, and (d) internal to the idealcoil, the electromagnetic field produced by thesheet current of the ideal coil is the same as theelectromagnetic field produced by the actual coilsystem.

If the current distribution in a physical coil systemis best approximated by standing waves, we wouldsuperimpose two running-wave ideal coils. Thenumerator of V should be the sum of the magneticenergies for both ideal coils. For a practicalcase, we consider an induction coil constructed ofindividual sections which are spaced axially everyhalf wavelength and which are electrically connectedin series. The azimuthal direction of current flowalternates from section to section. We denote byJiff that part of the total coil system inductance,J>?cs, which is associated with the fundamental(cos kz cos cot) component of the induction-coilcurrent. If each section contains N turns spreadevenly over a 1/Mth part of a wavelength, we obtain,neglecting end effects, the end-to-end Jiff,

and the storage ratio is

F = (15)

л сх Г

= — X Ю-9 MNkrc sin ( ^л | \М

X Г2,

coil

(14)

where Jiff and j£?cs are in henries, and /Coii is in cm.Evaluating F in (14) for typical cases will revealthe high penalty for short wavelengths and smallr0 to rG ratios. In practice, values of V may runfrom 0.05 to 0.2.

UThe numerator of U is the average power input

to the plasma for a given ideal-coil current. Thedenominator of U is the product of the angular fre-quency, со, and the peak rf magnetic energy storedin the ideal coil under a no-plasma condition with thesame ideal-coil current.

Long-coil Approximation

We consider now an induction coil of finite length.We divide the plasma into three sections: to theleft of, underneath, and to the right of the coil. Weconsider the plasma section underneath the coil asthough it were a plasma section underneath an in-duction coil of infinite length. End effects areneglected.

A coil of finite length will excite a band of wave-lengths. If the band is too broad, it may be brokeninto groups of narrower band-widths. A calculationmay be made for each group, and the results super-imposed. In treating Case I, we shall assume thatthe band of excited wavelengths lies entirely withinthe range allowed by the criteria in (11), and weshall neglect the change of plasma kinematics withinthe band. In Cases II, III and IV, the effect of thewide band-width for a short coil will appear in theshape and amplitude of the power absorption curve.

In Case I, the power absorption calculation will bebased on the cyclotron damping integrals (4). Indiscussing these integrals, it was mentioned thations would reach their asymptotic velocity afterbeing subjected to the electric fields for times longcompared with the time it would take an ion withenergy ÏTj/2 to travel the distance A/9. For theseintegrals to apply to ions drifting through the coilsection, the coil length must be long compared toA/9.

Case I

We consider the case \K\R0 <C 1 so that Ев ^Ев

УЖиит. Underneath the induction coil, the powerinput per unit volume may be obtained from (6).We use the phase relation in Table 3 to obtainEx ± iÊy in terms of Ee

vacuum. To obtain the powerratio TJi, we integrate the power input per unitvolume over the plasma volume, and divide by thereactive power stored magnetically in the ideal

ap-

(16)

coil. For кproximation. We

Tl

1, weobtain,

, - У

mayat О

n

use= -

a-1

solenoidor 1,

2A2

GENERATION AND THERMALIZATION OF PLASMA WAVES 131

£7i is a maximum for y = 1. At higher values ofy (higher density, longer wavelength), the plasma-current inductive effect reduces the efficiency. Theshape and width of the resonance depend on themagnitude of y. We derive an improved version ofEr¡iEe in Table 3, Case I, using (Уг = o2 = oc[iA + -\-I+),and obtain by numerical calculation the followinghalf-power frequencies.

= 1 ±7.0 forfor

where

(17)

(18)

The resonant peak in the first two cases occurs at|£2| = 1. In the у = 1 case, the peak occurs at|Í2| = 1 —8.0 77, and the calculated power absorptionat the peak is 2.6 times the absorption at |Q| = 1(Eq. 16). For larger values of y, this peak moves intothe region of ion cyclotron waves. If one evaluatesp±, which measures when damping can occur (seeFig. 1), at the resonant frequency for undamped ioncvclotron waves, its numerical value turns out to beapproximately equal to y. For у <C 1 then, ion cyclo-tron waves cannot exist because they are overdamped.у = 1 is an intermediate case. For у ^> 1, cyclotrondamping will be small, and there is a weak powerabsorption peak at |fi| = 1 and a strong peak asso-ciated with wave generation. The calculation ofpower absorption at the strong peak is done in thefollowing section.

Cases II, III and IV

We consider now the cases where we have a reso-nance of the total plasma. The power which theexciting coil puts into the plasma in the coil sectionis propagated or radiated, axially out each end of thesection as a hydromagnetic or ion cyclotron resonancewave. (For Case II, a small amount of power willalso go into cyclotron damping inside the coil section.)

We expect running waves to squirt out each endof the coil, and to run axially down the plasma toinfinity in each direction. We divide the plasmainto three axial sections, and join the solutions foreach section to one another by making the waveamplitudes continuous at the joint. Further detailsof end effects at the joints are not considered. For aleft-moving running wave, for instance, we assumethat everywhere to the left of the left end of the ex-citing coil the wave motion is a natural mode ofoscillation of the system, and that the wave amplitudeis constant. To the right of the right end of theexciting coil, the wave amplitude for this left movingwave is assumed to be zero. And underneath theexciting coil, we assume that the plasma behavesas though it were underneath an exciting coil ofinfinite length.

The solution underneath the coil must satisfy theboundary conditions at the plasma-vacuum interfaceR = Ro and at the induction coil radius R = Rc.Without loss of generality, we can replace the in-duction coil by an ideal coil at R = Ro carryinga current j * = eeRe\j* exp (ixZ + i£lT)]. We usethe boundary conditions in Ref. (3), Eqs. (4b)-(9b),and expand J^vR), where it occurs, in an infiniteseries of J^VmR), where vm is the radial wave numberfor the wth radial mode of oscillation of the freecylindrical plasma. The various vm are the varioussolutions of Eq. (12), Ref. (3).

For each mode m we choose particular solutionsEe

m for the left moving wave which are zero at theright end of the coil, Z = Zo. We obtain, under-neath the coil and for R < Ro,

È / W f r R ) _ cos{{x _ Xm){z _ Zo)}V< — Vn

where

and

+ i sin {(и - xm) {Z ~~ Zo)}] (19)

2i£Lj*

R¿\ + (G2 -

G= -

-, (20)

(21)

xm is the axial wave-number for the mth radial modeof free oscillation at the frequency Q. Other am-plitudes in the plasma beside Ee may be obtainedfrom Ee via the phase relations of Table 3.

The power input to the plasma from the idealcoil is found by integrating over the surface of theideal coil the product of the coil current densitywith the electric field due to plasma currents. Theaverage power going into the wth radial mode is

(22)

where

and £ = (и — xm)L/2, and where we use the realterm in 5(f ) to represent real power and the imaginaryterm to represent reactive power. S( |) approaches 1as £ -> 0 so that Pm is real at resonance.

The power ratio UwaYe is the ratio of the real partof Pm to the reactive power in an ideal coil. Eva-luating this ratio at resonance, v = vm and к — xm,we obtain

= \(2I1(xR0)K1(xR0)

(24)

132 SESSION A-5 P/361 T. H. STIX

A useful approximation to (24) is the |^ |^0

limit. Then

xR02

r J •(25)

The quantity dv2/dx2 is obtained by differentiatingthe relevant dispersion relation, (8) or Table 2.In the low xR0 limit, vmR0 are the roots of JoivmRo),and are given approximately by vmRa & n{m — J)for m > l . (Compare Ref. 3.) In this limit,v2 ^> x2. It is frequently useful to drop the x2joc ^> 1criterion on ion cyclotron waves. For the undampedwaves, it is sufficient to require only that v2 ^> x2

and v2¡oc ^> £i2 to obtain the approximate dispersionrelation

а/и2]-1. (26)

Using the V2/OÙ ^> Q2 and v2 ^> x2 approximationfor ion cyclotron waves, we obtain the \x\ RO<^1approximation to (24) for cases II, III, IV :

Un =

8/cou r02

яЛ»(т - 1 ) 4

¿coil И,

(27)

(28)

(29)

The resonance curve of plasma load versus fre-quency is given by (22). Since the quantity inabsolute value signs in Eq. (22) varies only slowly inthe vicinity of a resonance, the shape of the resonancecurve is essentially determined by S (f ), Eq. (23). Onemay substitute for £, | = (L/2) (Q - пт) {díl/dx)-1.The half-power points occur at f = i 1.4. dÛ/dx isthe group velocity of the wave, and may be deter-mined from the dispersion relation. For our variouscases, we obtain the half-power frequencies

±-

¿coil

0.45A

¿coi]coil

±1.8f0

2

¿coil A (m — I)2

í]-1 Case II

Case III

Case IV. (30)

THERMALIZAT1ON OF PLASMA WAVES

We can list some of the processes which will actto thermalize the energy of plasma wave motion andwhich are amenable to calculation. In addition tothese known processes, there may occur in an actualplasma a rapid thermalization of wave energy throughnot yet understood processes similar to those observedby Langmuir u and Gabor.12

In all three types of plasma waves, ion-electronсо lisions will transform the wave energy slowly intoohmic heat. A faster thermalization will take placein hydromagnetic compressional waves through theheating processes of magnetic pumping,6 which willact on both ions and electrons.

An interesting situation occurs for ion cyclotronwaves in a gas containing ions with different chargeto mass ratios. Only the resonant ions move withthe wave. One might consider heating a deuterium-tritium mixture in this manner. Even if the relativeaverage velocities of deuterium and tritium wereinsufficient for nuclear reaction without wave ther-malization, the ion-ion collisions will cause the waveto be thermalized fairly rapidly.

Cyclotron damping oners a possibility for therma-lizing ion cyclotron waves extremely rapidly. It willgenerally be highly inefficient to put power into aplasma of appreciable density at the short wave-lengths required for strong cyclotron damping (Case I).At longer wavelengths, however, one can coupleenergy into ion cyclotron waves, and the powertransfer can be extremely efficient (Case II). For aplasma confinement device, such as a stellarator ora pyrotron, one could cause the steady-state con-fining field to decrease with the axial distance fromeach end of the coil section. If the rate of changewith distance of the confining field is sufficientlysmall, the ion cyclotron waves which squirt out eachend of the coil section will change in an adiabaticfashion. As they move through the decreasing Bo,their wavelength will gradually decrease, and whenit becomes sufficiently short, cyclotron dampingwill occur locally. The ion cyclotron waves willdamp out (Eq. (13)), and the wave energy will betransformed into energy of individual ion motionswhich are still perpendicular to the lines of force, butwith effectively random amplitudes and phases. Onemight say that the actual heating of the plasmaoccurs in this region of cyclotron damping.

The author wishes to acknowledge the many usefulideas which came from discussions with Ira Bernstein,John Dawson, Russell Kulsrud, and Andrew Lenard,some of which have been indicated in this text. Inaddition, the author is profoundly grateful to ProfessorLyman Spitzer for his constant encouragement andproductive interest.

Addendum

The author also wishes to thank Dr. J. Neuf eld,who has recently brought to his attention the paperof Gershman.7 It may be noted that the dampingrate computed from the dispersion relation (8) inthe present paper will be equal to Or1 times thedamping rate computed from the dispersion rela-tion (31) in Gershman's work. This difference hasits origin in the assumption here that the tempe-rature of transverse ion motion was zero, whileGershman assumed an isotropic distribution.

GENERATION AND THERMALIZATION OF PLASMA WAVES 133

REFERENCES

1 H Alfven, Arkiv Mat Astron Fysik 29B No 2 (1942)2 E Âstrom, Arkiv Fysik, 2 443 (1951)3 T H Stix, Phys Rev, 106 1146 (1957)4 L Spitzer, The Stellarator Program, P/2170, Vol 32,

these Proceedings5 R F Post, Summary of UCRL Pyrotron (Mirror Machine)

Program, P/377, Vol 32, Pyrotron High Energy Expenments, P/379, Vol 32, these Proceedings

6. J M. Berger, W A Newcomb, J Dawson, E A Frie-man, R M Kulsrud and A Lenard, Heating of a ConfinedPlasma by Oscillating Electromagnetic Fields, P/357,this Volume, these Proceedings

7 В N Gershman, Zhur Eksptl Teoret Fiz 24, 453 (1953).

8 I B Bernstein, private communication

9 G N Watson, Theory of Bessel Functions, CambridgeUniversity Press, New York (1922)

10 An incorrect numerical evaluation of x2¡o¿ was given mRef 3

11 I Langmuir, Phys Rev, 26, 585 (1925), Z Phys, 46,271 (1928)

12 D Gabor, E A Ash and D Dracott, Nature, 776, 916(1955).