Anomalous resistivity and heating in current-driven plasma thrusters *

Transcription

1 PHYSICS OF PLASMAS VOLUME 6, NUMBER 5 MAY 1999 Anomalou reitivity and heating in current-driven plama thruter * E. Y. Choueiri Electric Propulion and Plama Dynamic Laboratory, Princeton Univerity, Princeton, New Jerey Received 0 November 1998; accepted 8 January 1999 A theory i preented of anomalou reitivity and particle heating in current-driven plama accelerator uch a the magnetoplamadynamic thruter MPDT. An electromagnetic dielectric tenor i ued for a current-carrying, colliional and finite-beta plama and it i found that an intability akin to the generalized lower hybrid drift intability GLHDI exit for electromagnetic mode i.e., with finite polarization. Weak turbulence theory i then ued to develop a econd-order decription of the heating rate of particle by the wave and the electron-wave momentum exchange rate that control the anomalou reitivity effect. It i found that the electron Hall parameter trongly cale the level of anomalou diipation for the cae of the MPDT plama. Thi caling ha recently been confirmed experimentally Phy. Plama 5, Polynomial expreion of the relevant tranport coefficient cat olely in term of macrocopic parameter are alo obtained for including microturbulence effect in numerical plama fluid model ued for thruter flow imulation American Intitute of Phyic. S X I. INTRODUCTION Current-driven microintabilitie of a magnetized, current-carrying, and colliional plama and their impact on tranport have not been well tudied in the regime where the plama beta i finite. Aide from it fundamental value, thi problem i relevant to the undertanding of diipation in ome current-driven plama device. The magnetoplamadynamic thruter MPDT i an electromagnetic plama accelerator intended for pacecraft propulion. It i eentially a coaxial device in which a highcurrent dicharge ionize a ga and accelerate it to high exhaut velocitie through the action of the Lorentz force produced by the interaction between the current flowing through the plama and a elf-induced or applied magnetic field. The MPDT can attain exhaut velocitie in the range of km/ with thrut efficiencie exceeding 40%. 1 Reference 1 4 give a review of reearch on MPDT tarting from it inception. 5 The MPDT plama ha an electron temperature between 1 and 3 ev, an ion temperature between 1 and 10 ev and an electron denity between and cm 3 depending on the power level. Current denitie and induced magnetic field can reach a high a 10 3 A/cm and1kg in the megawatt-cla device. The electron Hall parameter and the plama beta both range between 1 and 10. A more detailed decription of the MPDT plama can be found in Ref. 6. The acceleration proce in Lorentz force electromagnetic plama accelerator uch a the MPDT i current-driven with the thrut increaing with the quare of the total current. 4 It i alo known that the current can drive microintabilitie in the thruter plama which may, through induced microturbulence, ubtantially increae diipation and adverely impact the efficiency. The preence of current-driven * Paper F3I.3 Bull. Am. Phy. Soc. 43, Invited peaker. microintabilitie in uch accelerator plama have been etablihed experimentally in the plama of the MPDT at both low and high power level. 7,8 We have previouly preented in conference paper 8,9 a theoretical treatment of microintabilitie and their effect on tranport. Although the tranport coefficient, epecially the anomalou reitivity, derived in that work proved helpful in rendering numerical fluid imulation of MPDT flow more realitic, 10,11 a direct experimental proof of the exitence of anomalou reitivity in uch thruter wa not obtained until the recent meaurement of Black et al In that work, the meaured anomalou reitivity wa found to cale in accordance with the previou theoretical prediction in Ref. 9. Thi experimental confirmation prompted the publication of the theoretical work in the preent paper. Previou attempt to model linear intabilitie in the MPDT Ref. 13, 14 and their effect on tranport 14 have been limited to electrotatic mode, i.e., zero plama beta, a condition difficult to jutify for the plama of uch device. Finite-beta effect have been thought to be globally tabilizing, 15 17,14 epecially for drift velocitie exceeding the Alfvén velocity. We how that, for uch plama, a finite beta can actually reult in the excitation of finite-growth mode with mixed polarization. Thee effect not only alter the character of the linear mode, but affect the magnitude of the reulting anomalou tranport. We tart in Sec. II with the linear tability decription of a magnetoactive, current-carrying, colliional and finite-beta plama and how that finite-beta effect which reult in untable mode with finite polarization are important for the MPDT plama. For the MPDT plama the dominant mode have characteritic frequencie near the lower hybrid frequency lh and the intability i the colliional analog of the colliionle cae termed generalized lower hybrid drift intability GLHDI by Hia et al. 18 In Sec. III we outline the baic formalim we adopt for our formulation of anomalou tranport. In Sec. IV we ue X/99/6(5)/90/17/$ American Intitute of Phyic

2 Phy. Plama, Vol. 6, No. 5, May 1999 E. Y. Choueiri 91 the tatitical decription of the previou ection to derive general finite-beta expreion for the anomalou ion and electron heating rate a well a for the electron-wave momentum exchange rate that control the anomalou reitivity effect. Thee expreion are cat a integral in wave vector pace of quantitie that depend on the variou element of the linear diperion tenor derived in Sec. II, on the root of the linear diperion relation and on the aturation energy denity of the fluctuating turbulent field denoted by E t. We then turn our attention in Sec. V to the difficult quetion of the aturation mechanim that dictate the magnitude and dependencie of E t. We conider four model for E t baed on four poible aturation mechanim. In Sec. VI we how variou calculation of the anomalou heating and momentum exchange rate for plama parameter of interet and compare their magnitude to claical value. We finally conclude in Sec. VII by uing thee calculation to arrive at polynomial expreion of the relevant tranport coefficient cat olely in term of macrocopic parameter for incluion in numerical plama fluid model ued for thruter flow imulation. II. THE DISPERSION TENSOR OF A CURRENT- CARRYING AND COLLISIONAL PLASMA We eek a general kinetic decription of the repone of a colliional and magnetoactive plama carrying a cro-field current to mall perturbation without making the electrotatic aumption. A. Stating the problem Our entire problem can be formulated, a hown below, to be contained in the following matrix equation: R xx R xy R 1 xz 1 R yx R yy R yz E y R zx R zy R zzex E 10, z where the upercript 1 denote the firt-order harmonic part of the linearly perturbed quantitie in thi cae, the component of the electric field vector E. In the above equation, R ij repreent the element of the diperion tenor R,k and are generally complex function of the frequency and wave vector k of the ocillation a well a of all the plama parameter of the problem. A uual the diperion relation i obtained from detr ij,k0. In thi ection we outline the derivation of explicit expreion for the element R ij needed for our tudy. B. Defining the problem in term of the conductivity, dielectric, and diperion tenor The conductivity tenor and the dielectric tenor K are defined by j 1 E 1, 3 1 KE 1 E 1 i 0 j1. 4 By eliminating the perturbed current denity vector, j (1) from the above definition we get K ij ij i ij 0, 5 where ij i Kronecker delta repreenting the identity tenor. If we now recall two of Maxwell linearized equation ke 1 B 1, 6 ikb 1 0 j 1 i 0 0 E 1, 7 and eliminate the perturbed magnetic field vector B (1) from the econd equation uing the firt, we obtain after dividing by k and uing Eq. 4, N kke1 k KE The above equation can be written in the form of Eq. 1, RE 1 0, 9 where we identify R a the diperion tenor whoe element can now be written in term of thoe of the dielectric tenor or, more conveniently, through Eq. 5, in term of thoe of the conductivity tenor where R ij N k ik j k ij ij i ij 0, 10 N ck, 11 i the index of refraction. In order to arrive at the ought expreion for R ij we hall invoke plama kinetic theory to find a relation between the current denity and the electric field which we hall put in the form of Eq. 3 thu allowing u, through Eq. 10, to write the diperion tenor explicitly. 19 C. Derivation of the perturbed ditribution function Our tarting point for formulating a kinetic precription relating the perturbed current denity vector to the electric field i the Vlaov equation with the colliion repreented by the Bhatnagar, Gro, and Krook BGK model, 0 f t v xf q m Ex,tvB v f x,v,t f n 0 n f 0, 1 where q, m, f, v are the charge, ma, velocity ditribution function, and colliion frequency of pecie, repectively. We now linearize by auming that all quantitie with patial and temporal dependence, E, B, f, j and the charge denity are perturbed about their teady-tate value upercripted with 0 by harmonic quantitie upercripted

3 9 Phy. Plama, Vol. 6, No. 5, May 1999 E. Y. Choueiri where J n (a) i the Beel function of integer order n and we have choen, without any lo of generality, the wave vector to be in the y z plane a hown in Fig. 1, namely k (0,k,k z ). We now introduce the teady-tate ditribution function f (0). We chooe to carry our derivation in the laboratory frame by allowing cro-field drift in both the ion and electron ditribution function. For a cro-field drift in a homogeneou plama the teady-tate ditribution function i a drifting Maxwellian f 0 v m 3/ T FIG. 1. The vector j, B, k and u de in the local Carteian coordinate frame. Alo hown i the accelerator fixed cylindrical coordinate frame, r z. with 1 o that for a generic quantity a we have aa (0) a (1) and a (1) /a (0) 1. After replacing the temporal and patial differential operator by i and ik, repectively, the linearization of Eq. 1 reult in the following expreion for the perturbed ditribution function: ii k vf 1 q vb 0 1 m v f q 1 n E 1 vb 1 m v f 0 0 n f If we chooe to work with the cylindrical phae pace coordinate, v,, v z, with the magnetic field aligned with the z-axi, the above equation can be recat into a firt order linear inhomogeneou differential equation in f (1) which can be integrated once to yield 1 f 1 c q m E 1 v E1 kk ve 1 1 n v f 0 0 n f 0dC, where the integrating factor can be written a exp i i k z v z c n i n J n k v e in, c exp m T v x v y u d v z, 16 where u d and T are the cro-field drift velocity and temperature of pecie in the laboratory frame. The drift velocity i taken to be aligned along the y-axi a hown in Fig. 1. Upon ubtituting the above expreion and Eq. 15 in Eq. 14, integrating over the azimuthal angle and uing the following recurive relation: J n1 aj n1 a n a J na, 17 J n1 aj n1 aj n a, 18 where the prime denote the derivative with repect to the argument, we obtain f 1 iq T k u d f 0 ein m i m J m k v / c e im i n c k z v z k u d n 1 C n E 1 T n 0 J q n, 19 where the vector of coefficient C n i defined a C n iv J n ê x v d n c k J n ê y v z J n ê z. 0 D. Switching to the potential formalim We now witch from the electric field formalim exemplified in Eq. 1 to a formalim cat in term of the electrotatic and electromagnetic potential, and A defined below. Thi ha three advantage. The firt advantage i a clear eparation of electromagnetic and electrotatic effect in the diperion tenor making it more natural to a dicuion of electromagnetic correction to an electrotatic mode. 1 Thi will be epecially advantageou in the context of the anomalou tranport theory in Sec. III, where it will be inightful to eparate the electrotatic and electromagnetic contribution to the anomalou heating and momentum exchange rate. The econd advantage i the fact that, in the cae where the ion are taken to be unmagnetized, their contribution to the diperion tenor i much tidier mathematically than in the electric field formalim. Finally, the third advantage i that the effect of colliion on the purely electromagnetic mode can be imply precribed in the potential formalim. The electromagnetic potential A i defined by B A. 1 Any arbitrary choice of A of the form A where i ingle-valued i, in general, poible. Thi arbitrarine of

4 Phy. Plama, Vol. 6, No. 5, May 1999 E. Y. Choueiri 93 gauge, a it i cutomarily called in electrodynamic, i mot appropriately removed by impoing the Coulomb gauge A0. Rewriting the E Maxwell equation uing the above definition we get E A t 0, which implie that the quantity (EA/t) can be repreented by a potential gradient, and for linear harmonic perturbation we thu have E 1 ia 1 ik 1. 3 Taking the divergence of the electric field, uing the correponding Maxwell equation and the Coulomb gauge we alo get k. 4 Finally, upon ubtituting Eq. 84 and 1 in Maxwell B equation, the following relation reult: A 1 c k 1 0 j 1. 5 k c The above two equation become pecified once expreion for the perturbed charge and current denitie, (1) and j (1), are found in term of A (1) and (1). Thi will be done uing the following moment: 1 q f 1 d 3 v, 6 j 1 q f 1 vd 3 v, 7 o that Eq. 4 and 5 can be written a a et of three homogeneou equation in (1), A (1) (1) x, and A z and we have eliminated A (1) y with the Coulomb gauge D 11 D 1 D 13 1 D 1 D D 3 A x D 31 D 3 D 331 A 10. z 8 The above matrix equation and the diperion tenor D ij are the analog of Eq. 1 and the tenor R ij, repectively, in the potential formalim. The perturbed ditribution function f (1) needed to take the moment in Eq. 6 and 7 and cloe the ytem can now be rewritten in the potential formalim f 1 iq T k u d f 0 ein m i m J m k v / c e im i n c k z v z k u d k j 1 C n ia 1 ik 1 T q 0 n J n, 9 where we have alo eliminated n (1) in favor of j (1) uing n 1 k j 1, 30 q which can be directly obtained by taking the divergence of Maxwell B equation. E. The reulting diperion tenor The next tep i to carry the velocity pace integration required by the moment in Eq. 6 and 7 uing the above expreion for f (1). The integration i carried in cylindrical velocity coordinate and thu take the form d v 3 v dv 0 dvz d After the integration over the parallel velocity integral are of the form p v z Gvz dv i n c k z v z k u z, 3 d where p0,1,. Thee integral can be expreed a linear function of the well known plama diperion function Z 1 e t dt, 33 t and it derivative with repect to it argument. The integration over the perpendicular velocity tranform the Beel function into modified Beel function, I n ( ) of the firt kind and of integer order n with the argument being the quare of the normalized perpendicular wavelength k r c /. After much tediou but traightforward algebra, where we ue the following relation: dz 1 d Z, 34 ni n 0, n n I n e, 35 n I n e, 36 n aume the ion to be unmagnetized but keep in full their electromagnetic contribution which ha been demontrated in Ref. to be important and neglect the ion colliion ince for the MPDT plama, i lh e Ref. 6, 3, the element of the diperion tenor D ij can finally be written explicitly in term of the wave and plama parameter, D 11 1 i 1 i Z i e D 1 i pe 1 e0 e e n I n Z en 1i e /k z v te e e n I n Z en, k z k e0 e e e n I n I n 1 e0z en, 37 38

5 94 Phy. Plama, Vol. 6, No. 5, May 1999 E. Y. Choueiri D 13 pe k z k e0e e n I n n Z en 1 en Z en n k k z u de v te e0 k k z en, 39 D 1N pi i Z i pe e0 e e e n n I n I n I n Z en, e D 3 i pe e0 e e u e n I n I n 40 FIG.. Normalized temporal growth rate for colliional and colliionle untable mode in argon a a function of the aniotropy parameter and the electron beta. The olution are growth-maximized over wavelength. The tar on the curve denote the dominant doubly-maximized mode. 1 k z k n enz en, D 33 1N pi i Z i pe e0 e e D 1 D 1, n k k e0 k z k u de v te 41 I n Z en k z k n en, 4 D 31 k k D 13, D 3 k k D 3, where we have ued the following definition: en n cek u de i e k z v te, i k u di kv ti, 45 i e, e0 k z v te e0 k k z u de v te, 46 n k z k en e0, Z en Z en, I n I n e, 47 and the thermal velocity, plama frequency and cyclotron frequency of pecie are, repectively, given by v t T /m 1/, p q n 0 0 m 1/ ; c q B 0. m 48 It i alo ueful to note that the refraction index N appearing in the above diperion tenor can be related to the plama parameter through the following relation: N c k pi e m i k, e m e k 49 where we have introduced, the beta of pecie, in the above equation i et to e for electron defined a the ratio of thermal preure to magnetic preure n kt B. 0 / 0 Finally, by writing pi i i where and pe k k e e, k p k p c k 1 v t k, 5 d d i the Debye length for pecie and by defining a nondimenional parameter related to the propagation angle ee Fig. 1 caled by the ma ratio m e /m i k 1/ m e /m i 1/, 53 k z co it can be verified that the following et of even dimenionle parameter pe ce, e lh, T i u de m i,,, T e v e,, 54 ti m e completely pecify the problem uch that, for a given real wave number, kr ce where r ce i the electron cyclotron radiu we eek the root, / lh and / lh of the diperion relation 4 detd ij It i alo worth mentioning that the electrotatic diperion relation obtained in the limit 0 i imply D Dominant mode. We now how the reult of calculation uing the above diperion relation to illutrate ome of the characteritic of the untable mode and their parametric dependencie. The olution hown in Fig. have been growth-maximized over wavelength and thu repreent the dominant mode. The parameter U de repreent the ratio u de /v ti.

6 Phy. Plama, Vol. 6, No. 5, May 1999 E. Y. Choueiri 95 FIG. 3. Normalized frequencie for colliional and colliionle untable mode in argon a a function of the aniotropy parameter and the electron beta. The olution are growth-maximized over wavelength. The tar on the curve denote the dominant doubly-maximized mode. The enhancement of electromagnetic coupling with increaing e reult in a damping of the dominant mode a can be een from that figure. The damping i not dratic ince more than a three order of magnitude increae in e correpond to only a factor of decreae in the growth rate of the dominant mode. The intability not only perit under finite-beta effect but, a can be een from the e 1 and e / lh 1 curve in that figure, encompae a omewhat wider range of propagation angle than it did for the purely electrotatic cae. The finite-beta effect therefore are not globally tabilizing a wa previouly peculated, 15 17,14 even for drift velocitie exceeding the Alfvén velocity, but rather reult in the excitation of finite-growth mode with mixed polarization. Thi corroborate the finding of Ref. 5, 18, 6 and extend the validity of their argument to the colliional cae. The colliionle olution are alo hown in the ame figure for both the e 0 electrotatic and e 1 cae. It i obviou that the effect of finite beta are qualitatively different for the colliionle and colliional dominated cae. In the abence of colliion the electromagnetic effect reult in a hifting of the intability to more oblique propagation. For argon with e 1 and U de 0, for intance, the dominant mode hift by more than 6 toward the magnetic field vector from the orientation of the purely electrotatic dominant mode. The hift i more pronounced for lighter atom approaching 50 for hydrogen. Thi effect wa firt dicovered by Wu et al. 6 who noted that electromagnetic effect actually tabilize nearly perpendicular wave and detabilize more oblique one. Since many of the preceding tudie that addreed the finite-beta effect on the electrotatic modified two-tream intability focued on either perpendicular or nearly perpendicular propagation, electromagnetic effect which become important when u de v A were generally thought to be tabilizing. The frequencie of the untable mode, growthmaximized over wavelength, are of the order of the lower hybrid frequency a hown in Fig. 3. The dotted curve repreent the colliionle cae that correpond to the generalized lower hybrid drift intability GLHDI previouly dicued in Ref , 6. Conitent with the above reult, we find that the angular dependence of the frequencie of the colliional mixed polarization mode reemble more that of the electrotatic colliionle mode than that of the colliionle mixed polarization mode. Thi indicate that untable electrotatic ocillation couple differently with electromagnetic ocillation when colliion are important. We conclude from the above linear tability analyi that plama parameter typical of the colliional and tranvere current-carrying plama of a Lorentz force plama accelerator, 10U de 100, T i /T e O(1), ce and O1, a mixed polarization mode can be eaily driven untable by the cro-field current. The tability boundarie are broad enough and the evolution time cale are fat enough that the energy aociated with the untable ocillating electric and magnetic field pectra may be expected to ignificantly alter baic tranport procee a will be een next. III. WEAK TURBULENCE KINETIC FORMALISM FOR ANOMALOUS TRANSPORT The kinetic theory of weak turbulence wa firt developed by Vedenov, Velikhov, and Sagdeev , a well a Drummond, Pine and Roenbluth A good treatment i given by Galeev and Sagdeev in Ref The ue of weak turbulence theory i generally jutified when E t n T We can relate E t /n 0 T i to the experimentally meaurable denity fluctuation ñ/n 0, where the tilde denote a fluctuating quantity by noting that ñ/n 0 e /T e and e eẽ/k, E t n 0 T i T e T i kr ce 4 ñ n 0 rm. 57 Experimental evidence of turbulent fluctuation caued by cro-field current-driven intabilitie wa recently found in the low-power teady-tate MPDT plama at variou condition and location in the plume. 7 Thee meaured turbulent fluctuation had mot of their power in the lower hybrid mode with ome power appearing ometime in the electron cyclotron harmonic. Meaured value of (ñ/n 0 ) rm when uch turbulence wa oberved were on the order of 0.1 with magnitude ranging between 0.05 and 0.7. For thee value, with 1T i /T e 6, E t 0 Ẽ k / and auming the doubly growth-maximized (kr ce )**0.1, we obtain from Eq. 57 an etimate for E t /n 0 T i ranging between 10 3 and 10 6 implying that the weak turbulence aumption i generally valid. A. Governing equation: The moment-generating equation In thi ubection we preent an outline of the derivation of the general form of fluid-like equation governing the evolution of macrocopic quantitie under the condition of

7 96 Phy. Plama, Vol. 6, No. 5, May 1999 E. Y. Choueiri weak turbulence. Detailed dicuion of uch a derivation have already been preented in numerou article ee Ref. 30, 31, for intance, for a tutorial review. We do thi in preparation to our derivation of anomalou tranport preented in the following ection. We hould mention at the outet that our interet lie not in the evolution equation itelf but rather in it ue a a moment-generating equation. Therefore, for the ake of implicity and in order to keep a connection with the literature, we hall neglect colliion in the kinetic evolution equation. The effect of colliion will be reintroduced later when we ue the explicit form of the diperion tenor element. The underlying idea 3 i to conider the ditribution function of the pecie, f, a the um of a lowly varying enemble-average part and a rapidly varying fluctuating part f x,v,tf v,t f x,v,t, 58 where F (v,t) f (x,v,t) and denote an enembleaverage while the tilde denote a quantity fluctuating due to the effect of untable wave. When imilar partition are effected on the electric and magnetic field vector, the kinetic Vlaov equation for a patially uniform equilibrium yield F t F c q ẼvB m v F f t v f f c q ẼvB m v f, 59 where, like in Sec. II we have choen to work with the cylindrical phae pace coordinate v,, v z. Taking the enemble-average of the above equation, while noting that f 0, reult in F t F c F, 60 t AN where the right-hand ide repreent the anomalou contribution that i the repone of the average ditribution function to the microturbulent fluctuation and can be written explicitly a F t q ẼvB m v f. AN 61 By ubtracting Eq. 61 from Eq. 59 and, in the pirit of weak turbulence theory, neglecting all term that are quadratic in the fluctuation amplitude which i tantamount to the neglect of nonlinear wave particle and wave wave interaction the following governing equation i obtained for a weakly turbulent plama: f t v f f c q ẼvB m v F. 6 The tandard procedure in weak turbulence theory expounded in Ref. 3 for intance i to olve Eq. 6 along with Maxwell equation for Ẽ, B, and f then ubtitute the reult into Eq. 60 to obtain the evolution of F in the preence of microturbulence. We hall not, however, need to do all that for our particular problem of deriving expreion for the momentum and energy exchange rate. Such expreion can be arrived at by taking moment of the governing equation Eq. 60 a outlined below. B. Evolution of average macrocopic propertie under microturbulence To obtain the macrocopic evolution equation we take moment of Eq. 60, i.e., we multiply the equation by the generic quantity of tranport which could repreent ma, momentum or energy and integrate over velocity pace to get t F dvq m vb 0 v F dv q m ẼvB v f dv, 63 where we have ued integration by part in order to move the ditribution function outide the operator. Taking ucceive moment of Eq. 60 i equivalent to ubtituting 1, v, vv for ma, momentum and energy, repectively in Eq. 63 and integrating over v-pace. Thi yield n 0, t t c e z q Ẽñ m B, W t c e z W q Ẽ W m B, 66 where e z i the unit vector along the z-axi, and we have ued the following definition: n f dv, n v d vf dv, W m vvf dv, for the average number denity, the particle flux denity, and the kinetic energy denity tenor, repectively with v d a the drift velocity vector of pecie. IV. MOMENTUM EXCHANGE AND HEATING RATES We now proceed to define and derive explicit relation for the anomalou rate of interet. The right-hand ide of Eq. 65 repreent the rate of momentum exchange (P /t) AN where the momentum denity vector i P m between the particle and the fluctuating field. Since we hall be intereted in the momentum exchange along the drift velocity vector, we write P t v d p AN P v d, AN 70

8 Phy. Plama, Vol. 6, No. 5, May 1999 E. Y. Choueiri 97 where we have defined ( p ) AN a the effective anomalou momentum exchange rate or frequency between pecie and the fluctuating field. Uing the explicit expreion for (P /t) AN from Eq. 65 in the above equation we obtain p AN q n m v d Ẽ v dñ v d v d B v d, 71 where, unlike mot derivation in the literature, we are retaining the full electromagnetic character of the microturbulence. We now pecialize the above expreion for our particular problem according to the MPDT configuration hown in Fig. 1. We thu obtain the effective anomalou momentum exchange rate for electron along the current after etting e, taying in the ion reference frame and aligning the relative drift u de along the y-axi, p e AN e Ẽ n 0 m e u y ñ e n 0 ũ dez B xn 0 ũ dex B z, 7 de where we have ued the relation ñ v d n 0 ṽ d. The frequency ( p e ) AN can be thought of a an effective colliion frequency between the electron and the fluctuating field and can thu be aociated with a reitivity called anomalou reitivity the ame way that the coulomb colliion frequency ei i aociated with the claical Spitzer reitivity. By analogy the anomalou reitivity () AN i proportional to ( p e ) AN and i given by AN m e p e AN n 0 e. 73 The effective colliion frequency ( p e ) AN i therefore a direct meaure of anomalou reitivity. Similarly, for the temperature T m 3n vv d f dv we define a heating rate for pecie by 74 T 1 T T t, 75 and obtain, after combining Eq. 65 and 66 and pecializing for the MPDT configuration, T i AN e n 3n 0 T 0 Ẽ ũ di, i for the ion and 76 T e AN e n 3n 0 T 0 Ẽ ũ de n 0 u de ũ dez B xũ dex B z, e 77 for the electron. Equation 7, 76, and 77 will be the focu of our remaining analyi and calculation. In order to proceed with more ueful form for thee expreion we need to eliminate the fluctuating denity, velocity and magnetic field in favor of the fluctuating electric field. For thi purpoe we invoke, in the pirit of a quailinear decription, relation between the fluctuating quantitie and the fluctuating electric field that follow thoe of their linearly ocillating counterpart. From a generalized Ohm law we can write for pecie, j k i 0 R Ẽ k, 78 where j i the current denity, R () i the diperion tenor of pecie and i the wave frequency which, combined with the continuity relation, n 1 k j 1, 79 q give a ueful expreion for the fluctuating denity of pecie ñ k k j i 0 q q k l l R lm Ẽ mk, m 80 where the ubcript k i a reminder that thee relation are for the pectrally reolved i.e., Fourier tranformed fluctuation. In thi expreion, R () lm are the element of the tenor repreenting the dielectric repone of pecie and can be readily obtained from Eq through tranformation that will be decribed further below. In a imilar fahion we can derive an expreion for ũ d from the following relation: j q n 0 ũ d ñ u d, and Eq. 80, yielding 81 ũ k i 0R Ẽ k 0 i k R Ẽ q n 0 q n k u d. 8 0 We hall not need to worry about the econd term on the right-hand ide of the above equation in the context of the MPDT configuration hown in Fig. 1, becaue thi term vanihe for the ion u di 0, having choen to tay in the ion ret frame and for the electron it i alo zero for the component that figure in Eq. 7 and 77 i.e., the x and z component o that we are left with ũ dlk i 0 q n 0 m R lm Ẽ mk, 83 where lx,z for e; and lx,y,z for i. Having related the fluctuating denity and velocity to the fluctuating electric field we need to do the ame for B.To thi end, the following equation ẼiÃik, give, for our particular configuration, Ẽ xk iã xk, Ẽ yk ik ki k z k à zk, Ẽ zk ik z kiã zk

9 98 Phy. Plama, Vol. 6, No. 5, May 1999 E. Y. Choueiri Furthermore, combining the above equation with the definition of the electromagnetic potential, Eq. 1, and Coulomb gauge, yield the deired relation B xk 1 k Ẽ zk k z Ẽ yk, B yk k z Ẽ x k, B z k k Ẽ x k. 90 We are now in a poition to evaluate the term of Eq. 7, 77, and 76 by carrying the enemble-average uing the random phae approximation which i a tandard technique of tatitical phyic commonly ued in the pectral reolution of fluctuation, ee Ref. 33, pp , for intance. For the firt term in Eq. 7 uing Eq. 80 we have, Ẽ y ñ e i 0 e l k l e R lm Ẽ mk Ẽ yk eik xk x dkk, which yield, under the aumption of random phae, Ẽ y ñ e 0 e J k l l R e lm Ẽ m Ẽ m y dk, 91 9 where J denote the imaginary part of a complex quantity and, for the ake of implicity, we have dropped the ubcript k from the fluctuating quantitie. Similarly, we find for the other two term in Eq. 7, ũ dez B x 0 e J m k Ẽ z k z Ẽ y ũ dex B z 0 e J m R e zm Ẽ m dk, R e xm Ẽ mk Ẽ x dk If we now ubtitute the above three equation in Eq. 7, expand and collect the term in the ummation while taking advantage of the following ymmetry propertie of the diperion tenor: R xy R xy, we arrive at R xz R zx, e p AN 0 u de m e n e J k l R e yz Ẽ y Ẽ z dk. R yz R zy, R e ll Ẽ l We hall find it convenient, for our particular intability, to cat the the above expreion in term of the pectrally reolved fluctuating field energy denity in the perpendicular direction, E k. Thi can be done by uing the following relation obtained from Eq. 95 and Eq. 9 to yield A Ẽ x Ẽ y R yyr xz R xy R yz R xy R xz R xx R yz, 97 B Ẽ z R xyr xz R xx R yz 98 Ẽ y R xx R zz R xz R xz where each element R lm i the um of the correponding contribution from the electron, ion and vacuum to eliminate Ẽ x and Ẽ z and give p e AN u de m e n e E k k JR e xx A R e yy R e zz B R e yz Bdk. 99 We have carried out the derivation above under the electric field formalim where the relevant diperion tenor i R ee Eq. 9. The diperion tenor D that we derived explicitly in Sec. II, however, wa obtained under the potential formalim. A tated in that ection, witching to the potential formalim ha ome advantage. In the context of anomalou tranport, the potential formalim allow a more phyical inight by expreing the momentum exchange and heating rate in term of an electrotatic contribution plu a finitebeta correction. The reult obtained o far can readily be recat in term of the element D lm of Eq , through the following linear tranformation obtained from combining Eq. 9, 85, 86, and 87, R xx D, R xy R yx k k k z k D 3D 1, R xz R zx k k D 3 k z k D 1, R yy k k D 11 k z k 13 D k z k D 33, R yz R zy k k z k k k k D 13 nonumber 104 k k z k D 11 D 33, 105 R zz k k D 33 k z k D 11 k k z k k D We alo need to eparate the contribution of electron, ion and vacuum in the diperion tenor, which can be done following D e lm D lm D 0 lm D i lm, 107 where ubcript l and m cover the indice 1,, and 3 and where the upercript 0 denote the contribution of vacuum. The element D (0) lm and D (i) lm are given by

10 Phy. Plama, Vol. 6, No. 5, May 1999 D 11 1, D D 33 1N, D 1 D 13 D 3 0, 108 and e p AN lh v ti u de T i T e 1/ m i m e E k n 0 T i E. Y. Choueiri Jk r ce R e xx A R e yy R e zz B 99 i pi k 1 v i Z i, 109 tli D 11 D i D i 33 pi i Z i, D i 1 D i 13 D i When the above tranformation Eq are ued (e) in Eq. 99 to eliminate R lm in favor of D (e) lm, we finally obtain after ome traightforward algebra p e AN p e AN L u de m e n e E k k JD 11 D 33 e B k z e k z B k k k z k B k k z k k k B B k k z k z k k k k k k z k 4 k D e A k e D 13 B k k z k 3 dk, 11 where ( e p ) AN L i the well-known electrotatic longitudinal contribution to the anomalou electron momentum exchange rate p e AN L u de m e n e E k k J e dk, 113 (e) and e D 11 i the electrotatic uceptibility of the electron. In the electrotatic limit 0 it can be verified that the integrand in Eq. 11 vanihe o that we are left with ( p e ) AN ( p e ) AN L and J e JD e 11 JD i 11 J i. 114 We hall demontrate through the calculation of Sec. VI that the tranvere electromagnetic or finite-beta correction to ( p e ) AN in Eq. 11 can be ubtantial, epecially for a finite-beta plama like that of the MPDT. Equation 99 and 11 are equivalent but, for the preent analytical dicuion, we prefer from here on to ue the former i.e., the electric field formalim becaue the reulting expreion are more compact. For our numerical calculation we hall apply the tranformation in Eq in order to obtain the R tenor from the D tenor derived in Sec. II E. It i convenient to expre ( e p ) AN in unit of a natural frequency. We chooe, a we did in Sec. II, the lower hybrid frequency, lh ci ce, and normalize Eq. 99 to get R e yz Bdk. 115 We have focued, above, on the anomalou electron momentum exchange rate. Similar derivation, with no conceptual difference, tart from Eq. 76 and 8 and lead to the following expreion for the ion and electron heating rate ( i T ) AN and ( e T ) AN : i T AN lh e T AN lh E k n 0 T i J R i yz Bdk, T i T e E k n 0 T i J R i xx A R i yy R i zz B lh k r lh ce T i T e 1/ u de v ti R e xx A R e yz BR e zz B k z u de k k v r ti ce T i 116 1/ T e R e zx AR e zy R e zz B R e yy R e yz B lh Ak r ce T i T e 1/ u de R edk. v xy 117 ti The above three equation are the ought expreion for our analyi of anomalou tranport. 35 V. SATURATION MECHANISMS For the numerical analyi of anomalou tranport in the MPDT plama, the lat three equation, along with the tenor element in Eq , the linear diperion relation in Eq. and the tranformation in Eq form an almot complete et of equation in term of kr ce, / lh, / lh and the even parameter in Eq. 54. The only lacking equation i one that relate the level of aturated microturbulence E k /n 0 T i to thoe parameter. The rigorou formulation of thi relation i difficult a it concern the nonlinear aturation mechanim through which the fluctuation, initiated by the intability, reach a teadytate. The aturation mechanim dictate the magnitude and dependencie of the correponding fluctuating energy denity. Since the quetion of which aturation mechanim i relevant i uually bet anwered by computer particle imulation and ince thee imulation have yet to be made for the particular intability tudied here, we conider and compare four different aturation mechanim; ion trapping, electron trapping, ion reonance broadening, and thermodynamic bound. For our purpoe here we only quote the reulting expreion for each of thee aturation model. A. Thermodynamic limit: The Fowler bound An upper limit for E t wa firt derived by Fowler from thermodynamic argument,

11 300 Phy. Plama, Vol. 6, No. 5, May 1999 E. Y. Choueiri E t FB n 1 e m e u de, 118 and imply tate that the energy in the turbulent field cannot exceed the kinetic energy of the electron drift that i fueling the intability. For a convenient incorporation in our particular formulation, we recat the inequality above in term of the problem dimenionle parameter and get E t FB m e n 0 T i m i B. Saturation by ion trapping u de v ti. 119 When the excited wave pectrum i narrow due to the dominance of a ingle wave mode, a monochromatic wave aturation model, uch a that behind particle trapping, can prove to be a viable mechanim for aturation. In uch a cae the aturation dynamic can be governed by the trapping of the particle in the potential well of the growing mode thu limiting it growth. At aturation one can imply write e 1 i m r k, 10 where r /k i the phae velocity of the dominant mode and we have aumed that the ion are the particle being trapped. Again, we normalize the aturation model for compatibility with tranport theory o that, uing e eẽ/k and E k 0 Ẽ k /, the above equation can be rewritten a E k IT 1 n 0 T i 4kr ce ce pe T e T i C. Saturation by electron trapping 4 r lh. 11 Electron trapping i probably not a viable aturation mechanim for an intability in which electron are colliional and are in broad-reonance with the untable wave. We hall, however, include a model for electron trapping aturation in our calculation for the ake of reference. In analogy with ion trapping, we can write for the electron a viewed from the ion ret frame e 1 e m r u k z de, 1 and after ome algebraic manipulation, 1 4kr ce E k ET n 0 T i ce pe T e T r u de kr i ce lh D. Saturation by reonance broadening 4 v te. 13 Thi mechanim relie on the broadening of wave particle reonance by the random motion of particle in the turbulent electric field etup by the microintability. If reonance broadening i to be important in our cae, it would mot probably rely on ion dynamic, ince the electron are already broadly reonant with the wave due to colliion and finite-beta effect while the ion are very narrowly-reonant. Following Gary and Sanderon 37 who applied the Dum Dupree reonance broadening formula 38 to the ion and found, after taking the velocity average f i0 dv/ f i0 dv, E k IRB 1 0B 0 r k, 14 we pecialize the ion reonance broadening model for our dimenionle parameter and obtain E k IRB m e /m i T e n 0 T i kr ce T i ce pe r lh. 15 VI. CALCULATIONS OF ANOMALOUS TRANSPORT Armed with the expreion for anomalou tranport in Eq. 115, 116, and 117 along with the tenor element in Eq , the linear diperion relation in Eq., the tranformation in Eq and the aturation model in Eq. 119, 11, 13, 15, we can now conduct a comparative numerical tudy of anomalou diipation. A. Claical benchmark For benchmark we hall ue the following claical expreion for the momentum and energy exchange rate. For the momentum exchange rate we take the claical Coulomb electron ion colliion frequency 19 for momentum relaxation ( e p ) CL p n e e 4 ln e CL 3 3/ 0 m 1/ 3/, 16 e T e 3 where 9(4/3)n 0 de i the plama parameter. Thi colliion frequency determine the claical Spitzer reitivity CL m e e p CL n 0 e. 17 For compatibility, we normalize with the lower hybrid frequency and cat the reult in term of our dimenionle parameter, to get e p CL lh m i m e 1/ pe ce ln. 18 For a heating rate benchmark we define a claical heating rate, ( e T ) CL, for Joule heating T e CL 1 n 0 T e which yield T CL 8 lh u de 3 v ti T i T e t n et e 3 CL, n 0 T e j m e m i 1/ pe ln ce u de v ti T i m e p e CL, 130 T e m i lh where the econd equation how the explicit dependence on the colliion frequency.

12 Phy. Plama, Vol. 6, No. 5, May 1999 E. Y. Choueiri 301 FIG. 4. The anomalou momentum exchange frequency, ( e p ) AN, normalized by it claical counterpart and plotted v the electron beta according to four aturation model. Argon with u de /v ti 0, T i /T e 1, e / lh 1, and pe / ce 100. Finally, we note that in calculating the anomalou rate we approximate the integral, a commonly done in the literature, by the contribution of the dominant mode only i.e., for k**, meaning that all the Fourier-decompoed propertie are etimated at the doubly maximized growth i.e., maximized over wavelength and propagation angle. B. Numerical reult Since c and u de /v ti are the two parameter that vary mot within the plama of the MPDT, they were choen a the varying parameter of the calculation. When e i varied, u de /v ti i kept at 0, and when u de /v ti i varied, e i et at unity. The other parameter are m i /m e 73,300 for argon, T i /T e 1, e / lh 1, and pe / ce 100 for continuity with the calculation in Sec. II. 1. Effect of plama beta The effect of plama beta on the reitivity are hown in Fig. 4, where the ratio of anomalou to claical momentum exchange frequency which i eentially the ratio of the correponding reitivitie Eq. 73 and 17 i plotted v beta for the parameter lited above. We note from that plot that with increaing e, the curve correponding to the trapping model ignificantly deviate from their e 0 aymptote which are practically reached at e Thee deviation are due to the electromagnetic correction to the electrotatic limit, a eparated in Eq. 11. The ion trapping model i of pecial interet a dicued in Sec. V B epecially ince it wa the only one aumed in the purely electrotatic tudy of Ref. 14. We ee that, when e i on the order of unity or greater, a i commonly the cae of the MPDT plama, the anomalou reitivity i an order of magnitude le than that predicted by the purely electrotatic limit. The reaon the anomalou reitivity decreae with increaing beta according to trapping model can be traced to the coupling with the finite polarization mode we dicued in Sec. II E 1. A beta increae, the diturbance to the magnetic field do not have the time to diipate low Alfvén velocity and ignificant electromagnetic coupling arie. The untable wave acquire ome of the characteritic of the more electromagnetic mode and conequently the mot untable mode hift to lower frequencie. Since the aturation level due to trapping cale with frequency to the fourth power ee Eq. 11, the end effect i a ubtantial reduction in the anomalou reitivity. We note from the plot that the Fowler bound on the calculated rate allow, in principle, for a wide latitude for anomalou reitivity to be important. We hould not expect the electron trapping model to dictate the tranport a per argument already made in previou ection. Furthermore, we hould mention that more careful tudie of reonance broadening than thoe made at the time the mechanim wa firt propoed, have hown that it effect are limited to a reditribution of energy in k-pace at low plama beta, and that it doe not reult in enough diipation to aturate intabilitie uch a thoe being conidered here. Therefore, for low e, ion trapping eem to be the mot viable mechanim. At thee condition, the anomalou reitivity can be quite dominant a i oberved in Fig. 4, more than two order of magnitude larger than the claical value, in agreement with the finding of Ref. 14. A e increae, aturation by reonance broadening can become more viable epecially ince the turbulent aturation level are coniderably lower than thoe for ion trapping a i clear from the ame plot. Whether one or the other mechanim control aturation depend, at leat partly, on whether the pectrum i narrow or broad. Even though experimental data on turbulent fluctuation in the MPDT Ref. 7 give evidence of a dominant narrow peaked pectrum of turbulence in the lower hybrid range, the coniderably lower level of aturation energy implied by the ion reonance broadening mechanim warrant it conideration a a contender in the control of turbulent tranport. Of coure, thi quetion i perhap bet anwered by computer particle imulation. If ion reonance broadening doe take over the control of aturation, anomalou tranport can, for the parameter in the above calculation, be brought down below claical level. One hould therefore expect, by virtue of the ubtantial variability of the plama beta within the MPD thruter plama, that there are region where anomalou reitivity dominate over it claical analog a there are region where the convere i true. The ame comment we made above alo apply for the anomalou electron heating rate that were normalized by the Joule heating rate and plotted in Fig. 5. It i clear from thi plot that when ion trapping dominate, the anomalou heating rate i ubtantially larger than the electron Joule heating rate. Ion heating rate are not hown here but were found to be imilar in both magnitude and dependence to their electron counterpart. To compare the two rate we have calculated their ratio and plotted the reult in Fig. 6. There i only one curve in thi figure becaue the variou aturation model cancel out in the diviion. Since thi ratio i independent of the aturation detail, it i more accurate

13 30 Phy. Plama, Vol. 6, No. 5, May 1999 E. Y. Choueiri FIG. 5. The anomalou electron heating rate, ( e T ) AN, normalized by the Joule heating rate and plotted veru the electron beta according to four aturation model. Argon with u de /v ti 0, T i /T e 1, e / lh 1, and pe / ce 100. FIG. 7. The anomalou momentum exchange frequency, ( e p ) AN, normalized by it claical counterpart and plotted v the normalized drift velocity according to four aturation model. Argon with e 1, T i /T e 1, e / lh 1, and pe / ce 100. than the other quantitie we have calculated. We note from thi figure that, in the electrotatic limit, the two anomalou heating rate are baically equal. Thi feature i in contrat to the way electron and ion are heated claically epecially for a heavy atom like argon and i a well-known characteritic of the electrotatic modified two-tream intability a noted in Ref. 15, 16. Since the ion, are heated by the intability-induced turbulence at rate comparable to thoe of the electron, and ince in the MPDT, the electron energy i trongly tied to excitation and ionization through inelatic colliion, anomalou heating may offer an anwer to the long tanding quetion of why the ion temperature in thoe thruter i often higher than the electron temperature. Of coure, for thi argument to be true not only ( e T ) AN mut be comparable to ( i T ) AN, but the aturation level mut be high enough to warrant the dominance of anomalou heating over claical heating. Such i the cae when the intability aturate by trapping ion. The above argument about the relative temperature i tronget in the electrotatic limit and i in agreement with Ref. 14. When electromagnetic effect tart to become important with increaing beta, the ame figure how a degradation of the heating parity toward a progreively preferential heating of electron. Thi finding i in agreement with that of Ref. 34, where only the colliionle limit wa tudied. Thi degradation in heating parity i not trong enough, however, to weaken the ground for the above argument concerning the relative temperature, epecially for a heavy atom like argon. Indeed, we ee from the ame figure that a four-octave increae in beta doe not change the order of magnitude of the relative heating ratio. The increae of preferential electron heating with increaing beta may be partly due to the fact that, at low beta, the intability ha it dominant mode oriented at mall angle to the magnetic field k z /k(m e /m i ) 1/ or 1 and conequently perceive the electron with an effective ma comparable to that of the ion. 16 A beta increae, electromagnetic coupling with oblique more electromagnetic mode caue the dominant mode to propagate more obliquely, a firt noted by Ref. 6 and 39 for the colliionle cae. Thi i alo the cae when colliion are important a een from Fig.. Conequently, the effective electron ma decreae and the electron become much eaier to heat than the ion. FIG. 6. The anomalou electron heating rate, ( e T ) AN, normalized by the anomalou ion heating rate, ( i T ) AN, plotted v the electron beta. Argon with u de /v ti 0, T i /T e 1, e / lh 1, and pe / ce Effect of the drift velocity The effect of the drift velocity are illutrated in the plot of Fig. 7 and 8 for the ame parameter a above but with e et to unity and u de /v ti varying between 10 and 100. In reference to Fig. 7 we note that the general decreaing trend of anomalou reitivity with the drift velocity, once the intability i onet i not intuitive the u de /v ti threhold for the onet of the intability are not marked on thee plot becaue they are on the order of unity. One would expect that an increae in the free energy ource of the intability would enhance the anomalou reitivity effect. In Ref. 14 the ame trend wa found but no explanation wa given. Thi trend can be undertood once we realize that the caling of the linear growth rate of the dominant mode

14 Phy. Plama, Vol. 6, No. 5, May 1999 E. Y. Choueiri 303 FIG. 8. The anomalou electron heating rate, ( e T ) AN, normalized by the anomalou ion heating rate, and plotted for the ame condition a in Fig. 7. which doe increae with the drift velocity doe not necearily reflect in weak turbulence quailinear tranport caling ince the dependencie of the aturation mechanim which i extraneou to linear theory can overwhelm linear trend. Thi become clearer by noting that although an increae in the drift velocity doe enhance the linear growth rate of the dominant mode, it alo hift the mode to more oblique propagation and lower their frequencie. 13 Even though the intability goe to longer wavelength, the dependence of the aturation level for a trapping mechanim ee Eq. 11 cale with the frequency to the fourth power o that the frequency caling of the aturation mechanim overpower the growth caling of the linear mode. Thi trend i further accentuated for aturation by electron trapping becaue, in addition to the above argument, the aturation level cale with 4 and decreae coniderably oblique propagation with increaing drift velocity. At very high drift, the Doppler hift term in the aturation model ee Eq. 13 become more ignificant and revert the trend, which explain the rie of the electron trapping curve in Fig. 7 at high value of u de /v ti. A expected from the above beta-dependence tudy, anomalou electron heating rate exceed thoe of the ion for the preent cae of e 1. The preferential electron heating i further enhanced by increaing drift velocity a can be een in Fig. 8. The reaon for thi behavior i imilar to the one given above in the context of electromagnetic enhancement of electron preferential heating. Thi i o becaue both increaing beta and increaing drift velocity act to hift the intability toward more oblique propagation thu reducing the large m e-effect where m e i the effective electron ma that cale with the quare of and ubjecting the now lighter electron to more heating. C. MPD thruter calculation FIG. 9. The anomalou momentum exchange frequency, ( e p ) AN, normalized by it claical counterpart and plotted v the normalized electron drift velocity according to three aturation model. The upper line of each band correpond to the moderately colliional condition e / lh 5 T e 3 ev, n m 3 and the lower broken line correpond to the trongly colliional condition e / lh 500 T e 1.5 ev, n m 3. Argon with e 1, T i /T e 1, and pe / ce 100. We have, in the above calculation, choen a et of parameter that i generally repreentative of the MPDT plama. There i, however, one exception. It i the value of e / lh which we have et equal to unity a a compromie between having to repreent a colliional plama and providing a link with previou tudie. Moreover, many of the complex interaction between the natural plama mode, the free energy ource and colliion are mot pronounced when the colliion frequency i on the order of the ocillating frequency. We now, upplement our calculation with reult obtained at colliional level more appropriate of the MPDT plama. In order to approximate a typical range for MPDT plama colliionality, we conider the typical range for the variation of temperature and denity. For more detail on how the variou parameter of interet vary within the MPDT dicharge the reader i referred to the parameter review in Ref. 6. Auming that T e varie between 1.5 and 3 ev, while n 0 range between 10 0 and m 3, we can calculate a lower and upper bound for e / lh in argon from Eq. 18 to be 5 and 500, repectively, where we have fixed pe / ce at 100 for compatibility with the above calculation. For the reult hown in Fig. 9 we have choen to fix beta at unity to preerve electromagnetic effect and varied u de /v ti from 100 down to the threhold of the intability, which, although lightly exaggerated in the figure vertical hahed region, wa at u de /v ti 1.5. For each of the three conidered mechanim the plot how a band whoe upper line correpond to the moderately colliional condition e / lh 5 T e 3 ev, n m 3 and whoe lower broken line correpond to the trongly colliional condition e / lh 500 T e 1.5 ev, n m 3. We note from the figure that, although the Fowler bound allow for a large microturbulent contribution to the reitivity, ion reonance broadening might caue the intability to aturate at low level. Even though argument have been advanced recently dicounting the importance of uch a mechanim, it hould not be totally dicounted pending trong evidence from computer particle imulation and/or dedicated experiment. We furthermore ee that, in the cae of ion trapping aturation, once the intability i onet, the importance of anomalou reitivity in the MPDT plama i not a much dictated