Plasma heating in reversed field pinches at the fundamental ion cyclotron frequency

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1 PHYSICS OF PLASMAS VOLUME 9, NUMBER 4 APRIL 2002 Plasma heating in evesed field pinches at the fundamental ion cycloton fequency V. A. Svidzinski and S. C. Page Univesity of Wisconsin-Madison, Madison, Wisconsin Received 14 Novembe 2001; accepted 29 Januay 2002 The possibility of plasma heating in evesed field pinches RFP by adio-fequency f waves at ci is studied. A simple cylindical RFP equilibium which is symmetic in poloidal and axial diections is consideed. RF fields ae excited with given poloidal and axial wave numbes by a model antenna. The plasma dielectic popeties ae descibed by a hot plasma dielectic tenso in the limit of zeo Lamo adius. Collisionless absoption by plasma ions is assumed. The powe deposition distibution in the plasma is examined fo diffeent plasma paametes. Results demonstate that effective heating of the cental pat of the plasma at ci is possible fo some ange of paametes when the m 1 poloidal mode is excited. This is a hot plasma effect; it disappeas in the cold plasma limit Ameican Institute of Physics. DOI: / I. INTRODUCTION The heating of fusion plasmas with waves in the ion cycloton ange of fequencies is successfully employed on tokamaks and stellaatos. Seveal scenaios ae identified fo efficient plasma heating on these machines. They include the second o highe hamonic scenaio, o the scenaio involving ion ion hybid esonance especially with a minoity species that is fundamentally heated ; see eviews by Swanson 1 and by Adam, 2 see also Refs. 3, 4. In this pape we investigate the possibility of plasma heating in evesed field pinches RFP by f waves at the fundamental ion cycloton esonance. Ealy expeiments demonstated that this method of plasma heating is inefficient in tokamaks; see Ref. 5. Goup velocity of the slow Alfvén wave esonant at ci is nealy paallel to the magnetic field fo an abitay diection of the phase velocity see Chap. 2 in Ref. 6. Due to this the wave is not accessible adially and its excitation efficiency by an antenna located outside the plasma is low. The wave enegy supplied by an antenna goes into the fast Alfvén wave, which is not esonant at ci. This method of plasma heating is somewhat efficient in stellaatos. Plasma was heated efficiently at ci on the L-2 Stellaato. 7 Some heating was obseved in ecent expeiments on the lage helical device. 8 This effect is explained by coupling between fast and slow waves in the stellaato magnetic field; see Refs. 9, 10. In an RFP the poloidal component of magnetic field is compaable with the tooidal component. Thee is a stong vaiation of the diection of magnetic field with adius. This may esult in a significant diffeence between the f wave stuctues in RFPs and in tokamaks. Anothe impotant distinctive featue of an RFP is that it opeates at low magnetic field. A simple analysis with a cold plasma model in plane geomety is made to study whethe the apid change of the magnetic field diection leads to a esonant wave-plasma inteaction at the location of the esonance. This esult is biefly discussed in the pape. We demonstate that this popety of magnetic field does not lead to the wave-plasma esonance and by itself cannot lead to plasma heating. The pesent investigation concentates on the question whethe the global popeties of the wave stuctue which is not esonant at ci combined with the hot plasma effects can esult in effective plasma heating nea the location whee ci. We conside a simple RFP equilibium in cylindical geomety with poloidal symmety. Field equations ae Fouie analyzed in and z diections. The model antenna induces f fields with the wave numbes m and k z. Because the highe hamonics effects ae not impotant in ou case, a hot plasma dielectic tenso is deived in the limit L 0 whee L is the Lamo adius. We conside collisionless wave damping by plasma species. The heating efficiency is examined fo diffeent plasma paametes and diffeent mode numbes of the wave. Effective heating of the plasma cente is found fo some ange of paametes in the case of the m 1 mode. RFP esult may also be applicable fo othe low field configuations such as spheical tokamaks. Section II contains a detailed desciption of the model used and of the appoximations made in the analysis. In Sec. III the esults of the analysis ae discussed. We summaize in Sec. IV. II. DESCRIPTION OF THE MODEL AND FIELD EQUATIONS We assume a cylindical geomety with vessel adius a. We conside the Bessel function equilibium the Taylo state given by 11,12 B z B 0 J 0, B B 0 J 1, B 0, 1 whee B 0 is the tooidal field on axis, J 0 and J 1 ae Bessel functions. Fo this state B z eveses diection with adius if a 2.4. The plasma density pofile is of geneal fom n n 0 n(), whee n 0 is the density on axis X/2002/9(4)/1342/6/$ Ameican Institute of Physics

2 Phys. Plasmas, Vol. 9, No. 4, Apil 2002 Plasma heating in evesed field pinches This equilibium is symmetic in and z diections. In RFP expeiments the nonunifomity of magnetic field intensity in the poloidal diection is not stong; it is about 30% at /a 0.6 in Madison Symmetic Tous MST expeiment. 13 Also we concentate on heating nea magnetic axis whee the faction of tapped paticles is minimal. Thus the esults of the f study found in ou 1-D model should be applicable to the eal tooidal expeiments. We pefom Fouie tansfom of and z coodinates of Maxwell equations in cylindical coodinates. We study plasma heating at ci but not at the hamonics of this fequency, so that the limit of zeo ion Lamo adius Li is consideed. In this limit the elation between the Fouie components of f cuent and electic field is local with espect to. Assuming time dependence exp( i t) the Maxwell equations ae E im E E m2 2 k z 2 im ik z E z 2 c 2 TE, E mk z 2 E 2E E 1 E 1 2 k z 2 2 c 2 TE, ik z E mk z E z E E z m2 2 E z 1 E z 2 c 2 TE z, whee E, E, E z ae the Fouie components of f electic fields with the azimuthal and axial wave numbes m and k z, and the dielectic tenso T is calculated in cylindical coodinates. In the above equations the deivatives ae with espect to. The hot plasma dielectic tenso in Eqs. 2 is calculated fom lineaized Vlasov equation by integating along unpetubed paticle tajectoies in a Maxwellian plasma with T e T i in the limit L 0. We conside simple tajectoies along the magnetic field lines without difts in which the components of the ion velocity in cylindical coodinates with espect to the diection of the local magnetic field ae v const, v const, ci t. The angle is measued fom the local adial diection e. This calculation leads to the expected esult T 11 K 1, T 12 cos 0 K 2, T 13 sin 0 K 2, T 21 T 12, T 22 cos 2 0 K 1 sin 2 0 K 3, T 23 cos 0 sin 0 K 3 K 1, T 31 T 13, T 32 T 23, T 33 cos 2 0 K 3 sin 2 0 K 1, whee 0 is the angle between the magnetic field and z axis, K K 2 i 2 K p 2 Z Z, p 2 Z Z, 4 2 p 2 2 Z, whee coesponds to ions and electons,, k v T c k v T, v T 2T m, p, c ae the plasma and cycloton fequency, Z is the plasma dispesion function and k m sin 0 k z cos 0. 6 Equations 4 ae the components of hot plasma dielectic tenso in a unifom magnetic field see, e.g., Ref. 6 in the limit L 0 in which k of Eq. 6 is used in place of k z. Simila desciption involving k of Eq. 6 is often used fo the f analysis in tokamaks when the poloidal component of equilibium magnetic field is included in the model. 14 Equations 3 6 povide an accuate desciption fo a. One can see, howeve, that these equations ae not accuate nea 0. At 0 the magnetic field is paallel to the z axis, 0 0. In this case the dielectic tenso is coectly descibed by Eqs. 3 5 with k k z.if 0 then fo magnetic field of Eq. 1 the atio sin 0 / has a finite value and k given by Eq. 6 is not equal to k z. This discepancy is due to the fact that in the deivation of the dielectic tenso the angle in the velocity space is measued fom the local adial diection e at a point in which paticle is located. This diection changes apidly when the paticle moves along field line at small which leads to an inappopiate desciption of the paticle motion. The ate of absoption of f enegy in a hot plasma is sensitive to the value of k in Eqs. 5. It appeas that the exact calculation of the dielectic tenso even with the simple motion of paticles and in the limit of zeo Lamo adius would equie additional numeical teatment which is beyond the scope of ou analysis. This necessity of an accuate teatment of the paticle s tajectoy in ou simple model indiectly suppots the statement made in Ref. 15 that such exact calculation is necessay fo a pope desciption of the plasma dielectic popeties in the esonance egion. Fo a moe accuate desciption we modify k by intoducing the multiplye sin 2 0 in the fist tem in Eq. 6 : k m sin 0 sin 2 0 k z cos 0. 7 When this k is used in Eqs. 5 it gives the coect esult fo 0 at which 0 0 and fo a at which 0 /2. Fom this point of view the use of k given by Eq. 7 is adequate fo the analysis of absoption ates nea the axis which is the egion of ou inteest in ou simple model. Some inaccuacy in the dielectic tenso fo the intemediate should not change the conclusions of ou study. Simila esults ae obtained fom a collisional cold plasma model. We assume a unifom tempeatue distibution in the plasma and find the powe absobed pe unit volume as P j E, whee j is the wave-induced plasma cuent density calculated using the dielectic tenso T. Because of the high themal conductivity of the plasma along the magnetic field lines, the absobed powe should be aveaged ove a magnetic suface; see Sec. 38 in Ref. 16. In ou simple analysis the antenna dives f 5

3 1344 Phys. Plasmas, Vol. 9, No. 4, Apil 2002 V. A. Svidzinski and S. C. Page FIG. 1. ci / ci (0) vs /a. fields with unifom amplitudes in and z diections. Thus the powe P defined by the above equation is a suface aveaged quantity. Fo simplicity we assume that the antenna induces an electic field of a unit amplitude with time dependence exp( i t) and wave numbes m, k z on the inne suface of the vessel paallel to the z diection so that E a e z e im ik z z. As a paticula equilibium we conside the equilibium without field evesal. We take B z (a) 0 which coesponds to a 2.4 in Eq. 1. Fo this equilibium the antenna electic field is pependicula to the equilibium magnetic field. In eal expeiments thee is some, typically small, evesal of the B z component nea the edge. Ou calculations show that such a evesal does not change the conclusions of ou analysis. Equations 2 in which k is given by Eq. 7 ae solved numeically by a finite diffeence method. The condition TE 0 is included in the diffeence equations fo the consistency of the method. III. RESULTS AND DISCUSSION Ou goal is to examine plasma heating efficiency nea the axis by the excited antenna fields with diffeent wave numbes m and k z. Heating is due to the pesence of ion cycloton esonance nea the axis. The adial pofile of nomalized ci found using Eq. 1 is shown in Fig. 1. Ion heating aises fom a finite amount of the left-handed polaization which coincides with the ion gyomotion nea the esonance. The only modes fo which E, E ae nonzeo at 0 ae with m 1 it is a geneal popety of fields in cylindical coodinates. We conside f fields only with these wave numbes. Fist we examined a simple model of a plane geomety with a cold plasma analytically to see whethe thee is a esonant inteaction of the wave and plasma at ci when the wave is launched away fom the esonance and is popagating in the diection pependicula to the magnetic field. In this model the magnetic field intensity and its diection ae allowed to change along the diection of wave popagation. The esult of this analysis is that neithe a stong gadient of the magnetic field intensity no a apid change of the magnetic field diection field otation of /2 ove one wave length esult in the wave esonance at ci. The wave is ight-handed at the location of the esonance. Absence of the esonance in the magnetic field with a spatial gadient elates to a poo heating efficiency of tokamaks at ci. An initial motivation fo this eseach was to see whethe thee is a esonance in the case of a apid change of the diection of magnetic field. This apid change takes place in an RFP in which the magnetic field is tooidal on axis and is poloidal nea the edge. The absence of a esonance in this simple analysis leads to the simila conclusion about poo heating efficiency of an RFP with a cold plasma at ci. Now we conside the poblem of plasma heating fom a diffeent pespective. In the cold plasma limit the polaization is ight-handed at the location of the esonance. If the global mode stuctue is such that in the vicinity of the esonance thee is some amount of left-handed polaization then in the case of a hot plasma when the esonance is boadened heating is possible nea the location whee ci. We examine the heating efficiency fo a hot plasma using the equations of the pevious section. The plasma density pofile used in ou calculations is n cos /a /1.1. Fist we fix the paametes in the model as a 50 cm, B 0 1 kg, n cm 3 and we conside a hydogen plasma. These paametes ae not fa fom the paametes of the MST expeiment. The electic field on axis is epesented as a sum of left and ight-handed polaizations with espect to the diection of magnetic field E E L E R. Let E max be the maximum absolute value of the amplitude of electic field in the cosssection. Then we find the atio E L (0) /E max as a function of k z fo the modes with m 1 and m 1 fo the fequencies close to ci on axis. The esult shows that this atio is pactically zeo fo the m 1 mode fo the paametes of inteest. This means that the m 1 mode is not appopiate fo heating at the plasma cente. Figue 2 shows the dependence of E L (0) /E max vs k z a fo the m 1 mode fo the tempeatues T 200, 300, 500 ev. On this figue the f fequency is / ci (0) One can see that the amount of left-handed component on axis is substantial fo heating puposes fo a ange of wave numbes k z. It is not stongly dependent on plasma tempeatue and can be consideed as a popety of the paticula mode. A simila amount of left-handed component is obtained with a cold plasma dielectic tenso. Theefoe if thee is a lefthanded component nea the esonance in the cold plasma limit then this popety does not change in the hot plasma when the esonance is boadened. Fo the heating on axis to be stong at this fequency, the esonance should be boad enough:

4 Phys. Plasmas, Vol. 9, No. 4, Apil 2002 Plasma heating in evesed field pinches FIG. 2. Dependence of E L (0)/E max vs k z a fo thee tempeatues. m 1, / ci (0) i ci k z v Ti Fo k z a 1.5 and T 300 ev, i 1.5. This value is appoximately in the ange defined in Eq. 8. Fo these paametes the adial pofile of the powe absobed pe unit volume P in abitay units is shown in Fig. 3 a and the FIG. 4. Radial pofiles of a P; b E, E ; c E L /(E L E R ). m 1, k z a 1.5, T 300 ev, / ci (0) FIG. 3. Radial pofiles of a P; b E, E ; c E L /(E L E R ). m 1, k z a 1.5, T 300 ev, / ci (0) pofile of the components E, E is pesented in Fig. 3 b. The heating on axis is much stonge than it is nea the plasma edge even with the assumption of a hot plasma nea the edge. At the cente the powe is absobed by ions and the electon heating is negligible. Nea the edge the powe is absobed mostly by electons. Fom Fig. 3 b one can see that thee is a good penetation of the wave into the plasma with some finite wave amplitude nea the plasma cente. The wave amplitude on axis is elatively small; howeve, its polaization thee is puely left-handed. Figue 3 c shows the adial pofile of the atio E L /(E L E R )(E L,R ae the absolute values of the coesponding amplitudes. At the location of the cold plasma esonance ( ci,/a 0.4) the wave is ight-handed. One should note that the use of Eq. 6 fo the calculation of k instead of the coected Eq. 7 would not esult in significant change of the wave polaization pofile but it would esult in the suppession of the cental heating in this paticula example. We compae the esults of Figs. 3 a 3 c obtained fo the m 1 mode with the esults fo the m 1 mode. Figues 4 a 4 c show the coesponding pofiles fo the m 1 mode fo the same paametes as on Figs. 3 a 3 c. One can see that despite a good penetation of the wave into the plasma, the wave polaization is ight-handed nea the axis

5 1346 Phys. Plasmas, Vol. 9, No. 4, Apil 2002 V. A. Svidzinski and S. C. Page FIG. 5. P tot vs k z a, m 1, T 300 ev, / ci (0) and thee is no heating of the plasma cente. The electon heating is now compaable with the ion heating. The powe P in Fig. 4 a is in the same scale as in Fig. 3 a. In Fig. 3 a the absobed powe is much highe than that in Fig. 4 a. This is patially because the m 1 hamonic is left-handed in the vicinity of the esonance and is patially because an eigenmode of the cylindical RFP plasma is excited fo the conditions of Figs. 3 the wave amplitudes on Fig. 3 b ae significantly lage than those on Fig. 4 b. Figue 5 shows the dependence of the total powe absobed in the plasma P tot pe unit length of the cylinde, in abitay units vs k z a fo the m 1 mode and fo T 300 ev and / ci (0) The two peaks at k z a 0.3 and k z a 1.5 coespond to the excitation of the eigenmodes of the cylindical RFP plasma. k z a is elatively small at the fist naow peak, the condition of Eq. 8 is not satisfied nea axis so that mostly the edge plasma electon component is heated fo these paametes. The lage absoption is due to a vey lage excited wave amplitude. Nea the second boad maximum the condition of Eq. 8 is appoximately satisfied and the plasma cente is heated. Because the maximum is boad, it is feasible to excite such wave fields by an antenna. Fom Fig. 5 one can expect that fo these plasma paametes the most efficient heating is fo a diectional antenna which excites the m 1 mode with the wave numbes coesponding to this maximum. Analysis of diffeent anges of plasma paametes shows, howeve, that geneally thee is no favoable diection of wave excitation fo effective heating. Excitation of the eigenmode of the RFP plasma is pefeable fom the antenna loading consideations. Howeve, the plasma cente is heated effectively also in cases when no eigenmode is excited. Fo a given distibution of em fields we have estimated the powe dissipated in aluminum walls and in a cold collisional edge plasma. Results indicate that fo the m 1 mode the powe absobed in the hot plasma is usually seveal odes of magnitude highe than the powe absobed because of the othe mechanisms. In the case of the m 1 mode both ates can be compaable. Ou analysis shows that the m 1 mode launched with the wave numbe k z satisfying the condition of Eq. 8 and fo which the wave enegy penetates deeply into the plasma, FIG. 6. P vs /a. n cm 3, k z a 3.5, a 50 cm, B 0 1kG, T 300 ev, / ci 0.9. can be used fo an effective heating of the cental pat of the plasma. This esult is suppoted by calculations using the cold plasma dielectic tenso with the esonance esolved by atificially enhanced collision ates. Fo the paametes chosen in the above analysis the f field pofiles coespond to the excitation of the lowest adial wave numbes. Fo highe plasma densities o lage adius of the vessel wave fields with highe adial wave numbes ae excited. In this case the favoable popety of the m 1 mode emains. It is left-handed on the axis. Figue 6 shows the adial pofile of the powe absobed pe unit volume P in abitay units fo highe density n cm 3 and k z a 3.5. On this figue a 50 cm, B 0 1 kg, T 300 ev, / ci (0) 0.9. Because of the excitation of a highe adial wave numbe the absobed powe has an oscillatoy behavio. The most powe is absobed nea the cente. Fo highe density the value k z a is lage and Eq. 8 is bette satisfied. These values of k z a ae feasible fo eal antennas. The value of magnetic field B 0 which we used to demonstate effective heating of the plasma cente was chosen to be elatively small in ode to satisfy the condition of Eq. 8. Fo a fixed value of / ci (0) the mode dispesion elation and polaization pofile is weakly dependent on the value of B 0 while with the incease of B 0 the absolute value of the diffeence ci (0) in Eq. 8 inceases. Thus fo highe magnetic fields the amount of left-handed field component in the cente is substantial but the condition of Eq. 8 is not satisfied and thee is no cental heating of the plasma. This conclusion is suppoted by ou numeical analysis. In Eq. 8 fo a fixed / ci (0) which appoximately fixes the elative amount of the left-handed field component in the cente the diffeence ci (0) is popotional to B 0, while k z found fom a dispesion elation is appoximately popotional to n 0. Thus the condition of Eq. 8 sepaates plasma paametes into egions whee the heating is effective and ineffective. These two egions ae pesented appoximately in Fig. 7 fo a 50 cm and T 200, 400, 800 ev fo a hydogen plasma. Fo highe B 0 egion above sepaation line the heating is ineffective and fo highe densities

6 Phys. Plasmas, Vol. 9, No. 4, Apil 2002 Plasma heating in evesed field pinches left-hand polaization to the plasma cente is bette, howeve, fo the adial pofile of the magnetic field intensity of an RFP type. In tokamaks the magnetic field is stonge than in RFPs so that it is hade to satisfy the condition of Eq. 8. This, when combined with the smalle accessibility of the lefthanded field component to the plasma cente, makes this method of plasma heating ineffective fo tokamaks. FIG. 7. Appoximate sepaation of paamete anges with effective and ineffective heating fo thee ion tempeatues. m 1, a 50 cm. n 0 egion below sepaation line the heating is effective. This chat is found fom a visual analysis of the powe deposition and polaization pofiles fo diffeent / ci (0) and k z and should be consideed as an appoximate one. Thee is no shap sepaation of the two egions. The sepaation lines on this figue appoximately satisfy B 0 n 0 and B 0 T. This chat should not change significantly with the change of the vessel adius a. To ensue an effective plasma heating one should opeate at some distance below the specified line. The sepaation lines in Fig. 7 ae in the ange of paametes of moden RFP expeiments. One can use this chat to detemine whethe this heating method is effective fo paticula opeating conditions. The possibility of plasma heating in the RFP nea the fundamental ion cycloton esonance is an impotant finding. This effect is due to the paticula polaization of the m 1 mode and also because the RFP opeates at low magnetic field. It aises the question of whethe this mode polaization is a unique popety of the evesed field pinch in which the magnetic field diection vaies stongly with adius o is a geneal popety of the m 1 mode in a magnetized plasma. To addess this issue we pefomed simila calculations fo two additional magnetic configuations. In the fist case we assumed that thee is only a B z component of magnetic field pesent which vaies with the adius as the intensity of the RFP field shown in Fig. 1. Now an antenna excites an E component on the suface of the vessel. In the second case we consideed the field configuation simila to the pevious one but assumed that B z is unifom in coss section. These calculations show that in the fist case the wave popeties and heating efficiency is close to those in the RFP case. In the second case the m 1 mode still has some finite lefthanded polaization in the cente fo ci but the effect is weake than in the case of the RFP. One can conclude that the consideed popeties of the m 1 mode ae the geneal to this mode which ae exhibited in diffeent magnetic field configuations. The accessibility of the wave with the IV. SUMMARY The plasma heating efficiency in the evesed field pinch by excited f waves with the fequency in the vicinity of the fundamental ion cycloton esonance is examined. Fo this, appopiate teatment of the dielectic popeties of the hot magnetized plasma is equied. Results showed that the effective cental heating of ions is possible fo some ange of paametes when the m 1 mode with the appopiate wave numbe k z is excited. The m 1 mode is not suitable fo the heating. This effect is due to a themal boadening of the ion cycloton esonance in a hot plasma; it disappeas in the cold plasma limit in which the esonance is naow. Heating is due to a geneal popety of the m 1 mode which has a substantial amount of lefthanded polaization on axis when the wave fequency is close to the ion cycloton fequency at the cente. The method is effective in a low magnetic field. Analysis indicates that the effect is substantial in an RFP configuation athe than in tokamaks. Plasma heating in spheical tokamaks and in othe low field configuations by this method can be effective also. A sepaate analysis fo these configuations is encouaged by the esults of ou study. 1 D. G. 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