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P/357 USA Heating of a Confined Plasma by Oscillating Electromagnetic Fields By J, M. Berger,* W. A. Newcomb,f J. M. Dawson, * E. A. Fneman,* R. M. Kulsrud * and A. Lenard We wish to consider the heating of a fully ionized, infinitely long, cylindrical plasma confined by an axial magnetic field. The method of heating originally suggested by Lyman Spitzer, Jr. (1953) involves the application of an external oscillating electromagnetic field over a region of extent L in the axial direction. The oscillating electric field vector in the plasma is perpendicular to the lines of force of the main confining field. Since this oscillating electric field is produced by changing the main magnetic field in time, this method of heating is often denoted by the term " magnetic pumping ". Since we are interested in heating the ions to thermonuclear temperatures in a controlled fusion reactor we shall not consider electron heating in the following but it should be noted that some of the methods considered can also be utilized to heat the electrons directly. It is found that simple analytical results can be obtain^, in four limiting cases. These cases are classified in terms of the sizes of four characteristic times. These times are: (1) rCoii = 1/v, the time for a 90° deflection of an ion due to many small angle encounters with electrons and ions, (2) n — 2n/со, the period of the oscillating applied field, (3) Ttr, the time of transit of a typical ion through the length L and (4) rCi = 2п/а)си the cyclotron period of an ion. The four cases which have been analyzed are: (1) (2) (3) (4) collisional heating, rCi <C TCOII ~ Tf <C Ttr transit-time heating, TCI <C Ttr ~ Tf <C TC0H acoustic heating, rCi <C TCOII <C Tf ~ n r ion Cyclotron Tci ~ Tf <C Ttr <C Tcolb resonance heating, In the following four sections we treat the four cases outlined above. In cases (2) and (4), where the collision time is very long compared with the other characteristic times, the question arises as to the time necessary for actual heating in the sense of achieving randomization of the ordered energy delivered to the system. Finally we discuss a mechanism which can act to thermalize the ordered energy by phase or " fine-scale " mixing. A further question which may be raised is, why is the plasma not cooled by the oscillating field? In fact, in transit time heating, for certain initial velocities, particles lose energy in traversing the oscillating field region even after an average over the initial phases of the field relative to the particles has been performed. Although it is intuitively obvious that heating will occur, a simple thermodynamic argument is given here which shows that after averaging over a Maxwellian velocity distribution the energy transfer cannot be negative. Assume that the initial state г is in kinetic equilibrium; i.e., that the velocity distribution is Maxwellian and the spatial distribution is uniform. Denote by / the state of the gas at some future time after the oscillating field has been applied. The transition from state i to state / is adiabatic: i.e., no heat has been added to the plasma (dQ = 0). Now let collisions act to bring the plasma to the equilibrium state /'. Clearly the internal energy of the plasma is the same for states / and /'. Since the entire transition from state г to state /' is adiabatic, by the second law of thermodynamics the entropy of state /', S(f), is greater than S(i). Since the energy of an equilibrium state is a monotonically increasing function of entropy for fixed confining volume, the energy of state /', E(f), is greater than E(i). Therefore, energy has been added to the plasma by the oscillating field. This argument assumes perfect confinement of the plasma, since it applies only when i, /, and /' contain the same number of particles. COLLISIONAL HEATING i-e It is known that the magnetic moment ¡x = mw±2/2B is an adiabatic invariant for a particle in a magnetic field which is sufficiently slowly varying in space and time. 4 Here m is the particle mass and w± the magnitude of the particle velocity perpendicular to the magnetic field B. Thus, in the absence of collisions, a time-varying field will only cause a propor- * Now at IBM Research Laboratory, Yorktown, New York. t Now at University of California Radiation Laboratory, Li ver more, California. î Project Matterhorn, Princeton University, Princeton, New Jersey. 112 HEATING BY OSCILLATING FIELDS tionate variation in the perpendicular energy E±. If we imagine a square-wave pulse field superimposed on the main field and further assume that a collision occurs during the time the pulsed field is non-zero, energy will be transferred from E± to E „. After the field has returned to zero we see that a net energy transfer to the gas has taken place. Taking into account the two assumptions that the magnetic moment is a constant and the departure from equipartition during a collision time is small, we obtain an approximate set of equations governing this effect : dt В dt dE» = -?(£,-*£,.), dt (1) where v is the collision frequency. Note that because of our assumption that rt r ^> TCOIL changes in v or В with time resulting from motion in the axial direction can be neglected. If we assume that В is given by В — B0(l + ecos со/), (2) and treat e as small compared to unity we can solve Eq. (1) by a perturbation treatment. A simple application of the Floquet theory for differential equations with periodic coefficients leads to the increase in energy AE, in a time 2я;/со, 113 Because of the constancy of t h e magnetic moment ¡u, it is clear t h a t in this mode of heating all t h e energy changes will be due t o t h e parallel velocity changes. T h a t the parallel velocity will change is easily seen if E q . (8) is interpreted as representing a mechanical system in which a particle passes through a timevarying potential well. I t is not so clear, however, what t h e net effect of t h e oscillating field on a distribution of particles will be. Equation (8) can be easily solved b y perturbation theory assuming BJB0 to be small. The energy change of a particle AE = ^m(w\\f2 — w¡\i2), where m¿n a n d w\\f are the initial a n d final parallel velocities respectively, can then be obtained a n d averaged over values of t h e phase oc. I n this calculation i t is necessary t o 2 compute w^fto second order in Вг/В0. The result is d (AE\ = - dco [co^j X dz ], (9) where w {l is the zeroth-order parallel velocity. To enable us to analyze this further we have chosen the specific field shape f(z) = exp — (z/a)2. With this choice (9) becomes coa \ 2 - 2 w 2/ TI Í ~m\ ]. (10) (О* (3) The difference Eq. (3) can be replaced by the differential equation dE = [is2co2v/(9v2/4 + co2)]E = ocE. (4) In investigating the rate of heating from (4), note that the energy dependence of v is v ~ E-l\ (5) Equation (10) clearly shows that particles whose transit time is somewhat greater than the field period are heated, while those for which the converse is true are, in fact, cooled. A quantity of greater interest is the flux of energy of heated particles passing out of the field section. We obtain this by multiplying <(АЕУа by w „ n, where n is the particle density, and then averaging over both the perpendicular and parallel velocities. Thus, we find that the flux F is TRANSIT-TIME HEATING * = In the limiting case of transit-time heating, the assumption is made that the particles pass through the heating section sufficiently fast so that they suffer no collisions during the transit time. It is also assumed that the transit is slow enough for the adiabatic approximation to hold. The equation of motion of an ion moving in the z (axial) direction is then m'z = — ¡xdB/dz. (6) (7) where Bo and B1 are constants, f{z) describes the field shape, and a i s a phase angle specifying the time the ion enters into the field region. Equation (6) then becomes z = - {¡LiB1lfn){df{z)/dz) cos {cot — a ) . (8) l nkT (kT/m)iG(x) (11) C, (x) + Кг> {%)] (12) where and x — aco (tn/kT)? / /1 О\ G(x) is a function with a single maximum at x = 1.55 where G(1.55) — 0.24. The limiting forms are given We assume for this treatment that В is given by В = Bo + BJ(z) cos (cot - a), (я/2)* G(x) = x\ 1 — In — 2 x 2" m (14) e When this calculation was originally made a different average from that given by (11) was computed and therefore the results given here do not agree with those of Ref. 2. 114 SESSION A-5 P/357 Simple inequalities for the validity of the second order theory are easy to obtain from the integrated form of Eq. (6). These are: w. wn < 1 if w, Bn «1 if aco w \\ aco w„ The usual equations of motion for small amplitude sound waves are : ' = - Ъфг И = — др/dz 2 P = C Q, (17) 2 (15) One efíect which has ostensibly been neglected in the above treatment is the reflection of some of the particles before a complete transit of the oscillating field region has been made. Necessary conditions for nonreflection can be written down which are satisfied if Eq. (15) is satisfied. Note, however, that in the course of integrating the particle flux over the Maxwell distribution conditions (15) may be violated. Therefore, Eq. (11) should properly be regarded only as an estimate of the flux. There is one further effect which a more sophisticated treatment of transit-time heating should take into account ; namely, that in the analogous situation of mechanical particles passing through a timevarying potential well there will be bunching some distance from the well. In this case, where the particles are charged, it is possible that positive ion waves will be set up in the plasma regions outside the oscillating field section. Under such a circumstance, however, it seems quite likely that either collisions between positive ions of different e/m or the fine-scale mixing phenomena discussed in section (6) would eventually thermalize the energy in the positive ion waves. ACOUSTIC HEATING2 If the effective mean-free path is small, the oscillating field will produce density variations which will result in the propagation of sound waves in the plasma. The small effective mean-free path may be due to high densities or to cooperative phenomena which are at present little understood. We consider here the production of energy in the form of sound waves but do not treat the problem of their absorption by the plasma. The following assumptions are made in this calculation: (1) the plasma is treated as an ideal gas satisfying an adiabatic equation of state, (2) the particles do not diffuse appreciably during the period of oscillation, and (3) there is no appreciable mass motion out of the ends of the field region. Further, only those waves are treated which propagate in the axial direction. Under these approximations, the effect of the oscillating field is to produce a radial mass velocity. Let us assume that the plasma has uniform density within the radius r, and zero density outside. Since the material follows the lines of force, the total magnetic flux through the cross section uzr2 is constant. Thus vr = dr/dt £Ё (r/2B0) dBJdt, J. M. BERGER et al. (16) where we have again assumed that Вг/В0 = s is small compared to unity. where p, Q, and C = ykT/m are the pressure, density and square of sound velocity, respectively. We further assume that the field is given by В = BQ {1 + s exp - [(z/a)2 + icot]} . (18) Noting that we are only interested in the forced oscillations, we obtain the solution to Eq. (17) in the form P = — i ÍQ0CCOS exp + ioot x { J l oo e x P [— (*'A*)2 — Щ* ~ *')¥*' + J/ 0 exp [ - (z'/a)2 + ik (z - z')] dzj (19) where k = со/С. However, since it is sufficient for our purposes to consider the traveling wave solutions outside of the forcing region, we let z-> + oo in the above expression. The result is p = p0ks{7ta2/4)% sin (kz — cot) exp — (ak/2)2. (20) The flux of energy is pvz and, therefore, we easily obtain t h e average flux F = X {BJB0)2 nkT x2 exp (— %2/2y), (21) where x is given by (13). Upon comparison of this expression for the flux with Eq. (11) for the flux obtained for transit-time heating, it is seen that there is good agreement although these two cases represent extremes as far as the approximations are concerned. ION CYCLOTRON HEATING 5 6 In this case the external field is set up in the same way as in the other cases but its frequency is near the ion cyclotron frequency and thus the magnetic moment of the ions need not be constant but increases rapidly. The transit time of the ions across the heating section is long compared to the ion cyclotron period so that the ions undergo many gyrations in their passage through the heating section when collisions are neglected entirely. The heating of ions by this method is very efficient. The ions gain large energies during a single passage through the heating section since the ion particle velocities are always in phase with the electric field and the ions are continually accelerated. Further, the motions of all the separate ions are in phase, being correlated by the external field, so that large currents are developed in the plasma. Therefore, we must consider thefieldproduced by the plasma currents in calculating the heating of the ions. The ions will, in general, have a radial motion relative to the magnetic lines of force due to their correlated motion and their larger Larmor radii. HEATING BY OSCILLATING FIELDS Since the electrons are tied to these lines, a space charge will be set up producing a radial electric field. It can be seen that this radial electric field when combined with the externally produced azimuthal electric field will produce a circularly polarized electric field whose sense is just opposite to the motion of the ions. This circularly polarized component produces no energy increase of the ions. Thus, in some sense we may say that the plasma has shielded itself from the circularly polarized component of the external field which rotates in the correct sense to increase the energy of the ions. The way around this difficulty was pointed out by S t i x 6 who suggested that the space charge could be cancelled by the electrons if the electric field did not have a uniform phase along the axial direction. Thus, in some regions, the electric field has an opposite sign to the field in adjacent regions. In these regions a positive space charge is built up while simultaneously in adjacent sections a negative space charge is built up. Electrons must then flow along the lines of force to cancel these space charges. If the electrons are able to cancel out the space charge totally, one would have the total externally applied field acting to heat the ions. However, the actual current of the electrons oscillating at the cyclotron frequency produces a back emf on the electrons. Since the total electric field along the lines of force must be zero, the total space charge cannot be completely cancelled and there must be a residual space charge to produce a field along the lines to cancel this back emf. This space charge also produces a radial field which combines with the applied electromagnetic field so that even in this case the circularly polarized field, in the favorable sense, is somewhat shielded. Stix 7 has considered this situation under slightly different circumstances and has established that the favorably polarized component is able to penetrate the plasma appreciably. In analyzing this problem, the electromagnetic fields are not known in advance but must be solved for in a self-consistent manner. With complex notation their behavior was assumed to be given by s = Er + iEe = (е+еш + е-е-ш) f{z), (22) with the shape factor f(z) given by sin kz 0 -— a < z < z\> a (23) with ka — 7iN, N being an integer. e_ and e+ represent the two circularly polarized components of the electric field in the favorable and unfavorable senses, respectively. The assumed form is then justified by the self-consistent calculation and the forms of e+ and £_ determined. It should be emphasized that, due to the form of /, the ions will not see the frequency of the impressed electric field but a Doppler shifted frequency со ± kw „, so that for œ different from coCi some ions will always be heated. The motion of the ions is solved in the field (22) neglecting variations of e± (r) across the orbit, neglect- 115 ing the effect of the oscillating magnetic field, and assuming that the axial velocity is uniform. Upon summing over particles with all possible axial velocities, one finds that to SL good approximation the ion current may be given by jr + ije = {е2п/тк¥т) e_ (r) / 0 (a) sinkz в~ш, (24) where n is the ion number density, VT the mean axial velocity, a = со — coCi/kVT, and / 0 (a) is a nearly constant complex quantity of order unity. The transverse electron current is negligible and the axial current is chosen to make V»j = 0 since the space charge must be small compared to V*jj_. Since E l{ = 0 due to the mobility of electrons along the lines of force, we may solve V x ( V x E ) = {4n/c)d)/dt (25) which results Irom Maxwell's equations, to find (26) where у = {4ne2n/m) co/kVT (27) and £ + * is the complex conjugate to s+. For high densities у is large and (26) says that the field produced by the plasma must be very large compared to the favorably circularly polarized component which is driving the ions. Thus it is clear that the s+ component must predominate over £_. On imposing the boundary conditions, it is found that e + is nearly that expected from the impressed field in the absence of plasma so that e_ is shielded by the plasma. On the other hand, if у is much less than unity, the back reaction of the plasma may be neglected. The total energy flux of the plasma passing out the ends is F = >Nne2 2mkWT 71 1 + iy Io (28) This result is in agreement with that obtained by Stix. 6 F seems to be a maximum for a — 0 but a closer examination of the particles shows that those with small parallel velocities are heated to large energies while those with moderate velocities are not heated at all. All the particles would be heated to a somewhat smaller total energy if oc ^ 1, which corresponds to that frequency at which particles with the mean axial velocity see the cyclotron frequency. FINE SCALE MIXING AS A MEANS FOR RANDOMIZING ORGANIZED MOTION In collisional and acoustic heating the possibility exists that the energy transferred to the plasma by the electromagnetic field goes initially into organized motion. It is important to know the rate at which oscillations produced by such pumping are converted into random motion of the ions. If this rate is too low, then the pumping mechanism is inadequate for practical purposes. Of course, collisional damping 116 SESSION A-5 P/357 will always occur, but for the temperatures of interest this damping mechanism is relatively slow. However, there is another damping mechanism which is very effective. It is similar to the damping found by Landau 8 and Van Kampen 9 for electrostatic plasma oscillations. This mechanism depends on phase mixing of the wave due to thermal motions in the plasma. It is this process which will be discussed in this section. The physical picture for this type of damping is the following. Originally one has an organized wave in the plasma and because of thermal motion, ions which are in one region of the wave move into adjoining regions where the phase of the wave is different. Thus the various parts of the wave become mixed and the organized motion disappears. This process does not lead to a Maxwell distribution in general, but it does lead to random motions of the ions. Once this mixing has occurred, collisions become much more effective and probably produce a Maxwell distribution in a short time. Even if this does not happen, however, the random motion is equivalent to a temperature. To illustrate this mechanism we shall consider the case of ion cyclotron heating. A simple model of the plasma which is used is the following: the plasma is taken to consist of a number of cold beams of particles traveling along the magnetic field. This is an approximation to the thermal motion along the field but does not take into account thermal motions perpendicular to the field. This is done since thermal motion perpendicular to the field has little influence on the effect under consideration. The linearized equations of motion for this case are i/dt + = [«/c](E + ViXB0 + i X В ) В V x E == _ dt ¿•VVÍ) v. E = Vx В v. В = 4тт 1 c2 dt ж 0. V Що ei (29) The quantities appearing in the above equations have the following meanings : mi = mass of the particles in the ith beam d = charge of the particles in the ith beam V¿ = the velocity perturbation of particles in the ith beam "У"г = the zeroth-order velocity of the ith beam, with E and В are the electric and magnetic fields respectively due to the disturbance, щ and щ0 are the perturbed and zeroth-order number densities in the ith beam respectively, and B o is the zeroth-order magnetic field with B o = B o e 2 . The beams may be composed of either ions or electrons. In investigating this problem, the simplest type of waves which show the effect were considered. These J. M. BERGER et ai. were transverse, circularly polarized waves traveling along the magnetic field. It should be mentioned that Gershman10 has treated a closely-related problem. He solved the initial value problem for hydromagnetic waves traveling along the magnetic field. His method of treatment was the same as that used by Landau for treating the electrostatic oscillation problem. The approach presented here is closely akin to van Kampen's treatment of electrostatic oscillations. Our problem is also different in that we are interested in the propagation of waves out of a pumping region rather than in the time behavior of an initial perturbation. Our treatment covers waves with frequencies near the cyclotron frequency as well as hydromagnetic waves. The various quantities entering in the calculation were taken to be of the form VilBo, B1BO, E l B o E = E° В = B° (30) When these expressions are substituted in the equations of motion the following relations are found coci со — В а со — CO — kVi (31) (32) where = I ¿г (33) The first equation is the dispersion relation, the second gives the perturbation velocity in terms of the magnetic field. The plus and minus sign depends on the polarization and the charge on the particles. If the positive sign is used for ions then the negative sign is used for electrons. Observe that for certain values of со and k the denominator vanishes in the expression for ViQ. This occurs for those particles which see their cyclotron frequency when the polarization of the wave is appropriate. It is these particles which are close to resonance which are most strongly perturbed. The dispersion relation is most easily analyzed by making a plot like that shown in Fig. 1. This figure is obtained by fixing со and plotting the left- and righthand sides of the dispersion relation. The left-hand side is the parabola (ck)2 and theright-handside is the more complicated curve. Every place one of the denominators in the dispersion relation goes to zero, the right-hand side goes to infinity. The two curves cross at a number of points. Each crossing gives a possible k for the given со. Thus the plasma can transmit many waves with the frequency со. When the plasma is pumped at the frequency со, each of these waves will be excited to some degree. The amplitude of each wave will depend on the details of the pumping mechanism. In the region where the 117 HEATING BY OSCILLATING FIELDS {kc)2/œVi2(oCiF[(œ + œc\)/k]. 6 Stix has obtained a similar result by another method. If the predominant k falls in a region where many beams are near resonance, the damping is fast and occurs in less than a wave length. If one again goes to the limit of an infinite number of beams, one finds that in this case no coherent wave leaves the pumping section. CONCLUSION Figure 1. Typical dispersion current for cyclotron waves pumping occurs the various waves will be forced to be in phase. As one leaves this region the various waves will get out of phase with each other. If the pumping mechanism is such as to excite a large number of waves almost equally, then this phase mixing will result in the disappearance of all macroscopic quantities. This is not true, however, for the microscopic motion. Each beam is excited mainly by one k; that k which is closest to resonance for it. This can be seen from the expression for V{. Thus, what was originally an organized motion is converted into effectively random motion of the ions. More detailed calculations bear out this result. It is found that as the plasma is pumped so that a hydromagnetic wave is produced, then little or no damping results. This is the case if the predominant k is far from resonance for most of the particles. Going to the limit of an infinite number of beams so as to represent a continuous velocity distribution function F(V), normalized to unity, we find that the damping distance for hydromagnetic waves is In summary, it is of interest to examine the temperature dependence of the rates of heating of the various types of magnetic pumping. The collisional heating is more effective at lower temperatures, the rate of heating dropping off as 1Г/* at higher temperatures2 since the collision cross section becomes small as 1/T . However, the transit time heating and acoustic heating improve at higher temperatures; the rate of heating varying as T¡* if the frequency is optimized at each temperature. In ion cyclotron heating the electromagnetic shielding is given by the factor y which decreases at higher temperatures. Thus ion cyclotron heating increases as Ti for y greater than one at higher temperatures and then decreases as 1/Г* for y less than one. It should be remarked, however, that this type of heating is rather sudden; the particles can be heated from near zero temperature to extremely high temperature on passing through the heating section. Finally, it should be mentioned that the randomization of organized motion by fine scale mixing is more effective at higher temperatures. In conclusion the authors must express their deepest gratitude to Lyman Spitzer Jr. for not only initiating the basic ideas involved in this paper but for constant encouragement and advice in the carrying out of their development. REFERENCES 1. L. Spitzer and L. Witten, On the Ionization and Heating of a Plasma, Atomic Energy Comm. Report No. NYO999, PM-S-6 (1953). 2. J. M. Berger and W. A. Newcomb, Heating of a Plasma by Magnetic Pumping, Atomic Energy Comm. Report No. NYO-6046, PM-S-13 (1954). 3. A. Schluter, Der Gyro-Relaxations-Effekt, Z. Naturforsch. 72a, 822 (1957). The concept of collisional heating was arrived at independently in this paper. 4. M. Kruskal, The Gyration of a Charged Particle, Atomic Energy Comm. Report No. 7903 PM-S-33 (1958). References to earlier work are given in this report. 5. A. Lenard and R. Kulsrud, Resonance Heating of Ions in a Strong Magnetic Field, Atomic Energy Comm. Report No. NYO-7902, PM-S-32 (1958). 6. T. Stix, Generation and Thermalization of Plasma Waves, P/357, this Volume, these Proceedings. A similar result to that obtained from fine-scale mixing is given in this reference. 7. T. Stix, Oscillations of a Cylindrical Plasma, Phys. Rev., 106, 1146 (1957). 8. L. Landau, On the Vibrations of the Electronic Plasma, J. Phys., 10, 25 (1946). 9. N. G. van Kampen, On the Theory of Stationary Waves in Plasmas, Physica, 21, 949 (1955). 10. Gershman, Zhur. Eksptl. i Teoret. Fiz.. 24, 453 (1953).