Natural Hot Ion Modes in a Rotating Plasma
NNatural Hot Ion Modes in a Rotating Plasma
E. J. Kolmes, ∗ I. E. Ochs, M. E. Mlodik, and N. J. Fisch
Department of Astrophysical Sciences, Princeton University, Princeton, New Jersey 08540, USA (Dated: January 5, 2021)In steady state, the fuel cycle of a fusion plasma requires inward particle fluxes of fuel ions. Theseparticle flows are also accompanied by heating. In the case of classical transport in a rotatingcylindrical plasma, this heating can proceed through several distinct channels depending on thephysical mechanisms involved. Some channels directly heat the fuel ions themselves, whereas othersheat electrons. Which channel dominates depends, in general, on the details of the temperature,density, and rotation profiles of the plasma constituents. However, remarkably, under relatively fewassumptions concerning these profiles, if the alpha particles, the byproducts of the fusion reaction,can be removed directly by other means, a hot-ion mode tends to emerge naturally.
I. INTRODUCTION
Nuclear fusion devices aim to achieve ignition by heat-ing a plasma to a very high temperature, typically on theorder of tens of keV. The heat losses at these tempera-tures are a significant source of inefficiency in a fusion de-vice. However, the fusion cross-section depends only onthe temperature of the fuel ions. At the same time, hotelectrons incur large power losses, either through radia-tion or heat transport, but do not produce fusion power.Moreover, the capacity of a magnetic confinement deviceto trap plasma is typically limited by total plasma pres-sure; thus, the higher-temperature electrons take up alarge share of that pressure limit without producing anyadditional fusion power. As such, the performance of afusion device can be improved – often dramatically – byachieving a “hot-ion mode,” in which the ions are main-tained at a higher temperature than the electrons [1, 2].However, attaining a hot-ion mode is a significant tech-nical challenge. High-energy ions produced by fusionpreferentially lose their energy collisionally to electronsrather than to fuel ions. If no additional strategy is em-ployed to heat the ion population, the electrons will tendto be at least as hot as the fuel ions, if not hotter. A hot-ion mode can be produced if significant external heatingsources are directed at the ion population. These sourcescould be neutral beams or RF waves. Attaining a hot-ion mode in a reactor, however, where the main heatingis necessarily through the fusion reaction, requires someform of α -channeling, in which the energy from fusionbyproducts is channeled into a wave (avoiding collisionalheating of the electrons), and that wave deposits its en-ergy into the fuel ions [3–11]. In all of these cases, thehot-ion mode requires significant intervention to changethe power balance such that energy is directed to fuelions. In any of these cases, the differential in tempera-tures could be increased if the electron energy confine-ment were reduced, though this strategy is less desirableinsofar as it involves increasing energy losses.This paper will suggest an alternative possibility: a ∗ Electronic mail: [email protected] “natural” hot-ion mode. The notion of “natural” requiresdefinition. By natural , we imagine processes in whichthe ion heating comes from transport processes that arealready happening in the plasma. Note that any steady-state fusion device must have inward flows of fuel ionsand outward flows of alpha particles in order to balancethe fusion reactions. In the case of classical transport,each particle flux is accompanied by dissipation. If thedissipation is directed into the ions, and if it is sufficientlylarge, then it could be possible to reach a hot ion mode naturally , without having to heat the ions externally.A plasma with large electric fields is a logical place tolook for such an effect, since the electric fields provide areservoir of potential energy through which moving par-ticles can exchange energy. If there are transfers of en-ergy between the particles and the electrostatic potential,one might imagine that certain populations of particlescould be preferentially heated or cooled. Exploiting thispossibility, this paper will consider cylindrically symmet-ric configurations with radial electric and axial magneticfields. We will focus on a particularly simple case, inwhich the transport is purely classical and where inho-mogeneities in the direction of the field can be neglected.In other words, consider the cylinder long enough for endlosses to be less important than transport across the field.Perpendicular E and B fields in this geometry causeplasmas to rotate, so any discussion of crossed-field plas-mas is inevitably a discussion of rotating plasmas. Thereare a number of proposals for fusion devices involving sig-nificant rotation. Supersonically rotating magnetic mir-ror devices have promising theoretical properties [12, 13]and have been realized experimentally [14–16]. Thewave-driven rotating torus (or WDRT) is a proposal fora toroidal device which relies on poloidal rotation forconfinement [17, 18]. Additionally, significant rotationvelocities are sometimes observed even in devices (liketokamaks) for which rotation is not an essential part ofthe confinement scheme [19–25].The dissipation due to classical cross-field transport ina rotating plasma can be split into two categories. Vis-cous heating occurs when the rotation profile is sheared.It heats ions. Frictional heating occurs due to interac-tions between particles of different species with differentvelocities. Frictional interactions between ions and elec- a r X i v : . [ phy s i c s . p l a s m - ph ] J a n trons heat electrons. However, in the case of a plasmacontaining more than one ion species, there is also ion-ion frictional dissipation, in which the heat is divided be-tween the ion species, preferentially heating the lighterof any pair.The problem of controlling the relative importance ofthese channels has largely been overlooked, despite thefact that expressions for the heating follow readily fromestablished theories of classical transport. The viscousheating scales with the size of the deviation of the rota-tion profile from solid-body rotation; the frictional heat-ing scales with the deviation of the different species’ pres-sures from a class of dissipationless profiles.We show here that a hot ion mode can arise naturallyby arranging for the dominance of the ion dissipationchannel. For a plasma with a single ion species, classicaldissipation directly heats the ions only if the viscous dis-sipation is large. If there are multiple ion species, thatconstraint is relaxed, since ion-ion friction can then com-pete with ion-electron friction. In general, in order forthe heat dissipated in the rotating plasma to naturallyflow to the ions, the temperature, density, and rotationprofiles of the plasma constituents must all be arrangedcarefully.However, we further show here a remarkable propertyof rotating and fusing plasma, so long as the steady statedensity of ions is maintained through ion fueling balanc-ing the prompt removal by α -channeling of the spent fu-sion byproducts. Any other mechanism that removes thefusion byproducts on a collisionless timescale would pro-duce the same result. Surprisingly, such a steady-stateplasma tends to assume naturally the very favorable hot-ion mode without the necessity of arranging in detailthese profiles.This paper is organized as follows: Section II re-views classical cross-field particle transport in a rotatingplasma. Section III describes the classical heating chan-nels associated with the different mechanisms of cross-field particle transport. Section IV discusses the condi-tions allowing different channels to be dominant. Sec-tion V describes how, specifying only global particle bal-ance and boundary conditions, the ion channel can domi-nate. Section VI summarizes and discusses these results. II. CLASSICAL PARTICLE TRANSPORT IN AROTATING LINEAR DEVICE
Consider a fully ionized cylindrical plasma device withan axial field B = B ˆ z . Suppose the system is homoge-neous in the ˆ θ and ˆ z directions, and that all flows areradial or azimuthal. In a region away from any particlesources or sinks, the momentum equation for species s is m s n s (cid:18) ∂ v s ∂t + v s · ∇ v s (cid:19) = q s n s ( E + v s × B ) − ∇ p s − ∇ · π s + R s , (1) where m s is the mass of species s ; n s is the density; v s isthe velocity; q s is the charge; p s is the pressure; ∇ · π s isthe viscous force density; R s is the friction force density;and E is the electric field. In steady state, the radialflux Γ s . = n s v sr can be obtained [26] by rearranging theˆ θ component of Eq. (1):Γ s = R sθ − ( ∇ · π s ) θ m s Ω s [1 + ( rv sθ ) (cid:48) /r Ω s ] . (2)Here the θ subscript denotes the ˆ θ component of a vector,and Ω s . = q s B/m s is the gyrofrequency. The prime in thedenominator denotes the derivative ∂/∂r . The frictionforce density can be written as R sθ = (cid:88) s (cid:48) R ss (cid:48) θ , (3)where the ˆ θ frictional force on species s due to interac-tions with species s (cid:48) can be modeled by [27] R ss (cid:48) θ = n s m s ν ss (cid:48) (cid:20) ( v s (cid:48) θ − v sθ )+ 32 B m s T s (cid:48) + m s (cid:48) T s (cid:18) m s (cid:48) T s T (cid:48) s q s − m s T s (cid:48) T (cid:48) s (cid:48) q s (cid:48) (cid:19)(cid:21) . (4) T s denotes the temperature of species s . ν ss (cid:48) is thecollision frequency between species s and s (cid:48) . In or-der to ensure momentum conservation, it must satisfy n s m s ν ss (cid:48) = n s (cid:48) m s (cid:48) ν s (cid:48) s , so that R ss (cid:48) θ + R s (cid:48) sθ = 0. Eq. (4)includes both the friction due to differences in flow veloc-ity and the thermal friction, which is driven by tempera-ture gradients. The azimuthal friction force that appearsin Eq. (2) can be written as( ∇ · π ) θ = − r ∂∂r (cid:20) η s r ∂∂r (cid:18) v sθ r (cid:19)(cid:21) , (5)where η s is the corresponding Braginskii viscosity coef-ficient [28], or its analog in a multiple-ion-species plasma[29]. This form of the viscous force is derived in Ap-pendix A.Let δ . = v sθ /r Ω s , and suppose δ (cid:28)
1. ThenΓ s = R sθ − ( ∇ · π s ) θ m s Ω s ∞ (cid:88) k =0 (cid:20) − ( rv sθ ) (cid:48) r Ω s (cid:21) k (6)= R sθ − ( ∇ · π s ) θ m s Ω s (cid:20) O ( δ ) (cid:21) . (7)Define the viscous flux Γ πs byΓ πs . = − ( ∇ · π s ) θ m s Ω s . (8)Define the frictional flux Γ Rs byΓ Rs . = (cid:88) s (cid:48) Γ Rss (cid:48) (9)Γ
Rss (cid:48) . = R ss (cid:48) θ m s Ω s . (10) E B ∂∂r (cid:0) v θ r (cid:1) = 0 v aθ = v bθ FIG. 1. The cartoon on the left shows the kind of flow thatgives rise to frictional cross-field transport: the local velocitiesof different species are not the same. The cartoon on the rightshows the kind of flow that gives rise to viscous cross-fieldtransport: it is radially sheared.
Then Eq. (7) can be rewritten asΓ s = (Γ Rs + Γ πs ) (cid:2) O ( δ ) (cid:3) . (11)The kinds of flows that give rise to these different fluxesare shown in Figure 1. In practice, the viscous flux Γ πs isoften small compared to the frictional flux Γ Rs [26]. How-ever, for the purposes of understanding heat and chargetransport, the two fluxes are comparably important. Thisis partly because the flux described by Γ Rss (cid:48) is ambipolar,in the sense that q s Γ Rss (cid:48) + q s (cid:48) Γ Rs (cid:48) s = 0 . (12)The viscous flux Γ πs satisfies no such condition. In thediscussions that follow, the ability of the flux to carry netcharge will be important, since it determines how effec-tively the particles can exchange energy with an electro-static potential. III. CLASSICAL HEATING
The particle fluxes described by Eq. (2) are accompa-nied by dissipation. The temperature evolution equationfor species s can be written as3 n s (cid:18) ∂T s ∂t + v s · ∇ T s (cid:19) + ∇ · q s + p s ∇ · v s = (cid:88) s (cid:48) m s n s ν ss (cid:48) m s + m (cid:48) s (cid:0) T s (cid:48) − T s (cid:1) + Q πs + Q Rs , (13)were q s is the heat flux, Q πs is the viscous heating, and Q Rs is the frictional heating. Q πs and Q Rs are not the onlyterms in Eq. (13) that could possibly produce a differ-ence between different species’ temperatures. The heatflows can have a significant impact on the evolution of T s ,as can compressional heating. The compressional heat-ing is discussed in more detail in Appendix C. However,the goal of this paper is not to describe full solutions toEq. (13). Rather, our focus will be the ways in which Q πs and Q Rs may preferentially heat different species. Theseare the two terms that are driven directly by classicalcollisional transport; each has a connection to one of theparticle fluxes described in Section II. A. Viscous Heating
The heating Q πs due to the viscous flux Γ πsr is the sim-pler of the two to understand. In this geometry, theleading-order viscous heating for species s can be writ-ten as Q πs = − π s : ∇ v s = η s (cid:20) r ∂∂r (cid:18) v sθ r (cid:19)(cid:21) . (14)Eq. (14) is derived in Appendix A. Q πs vanishes for aspecies undergoing solid-body rotation, when v sθ ∝ r .Integrated over the cross-sectional area of the system toan outer radius at r = R , Eq. (14) becomes2 π (cid:90) R Q πs r d r = 2 πη s r v sθ ∂∂r (cid:18) v sθ r (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) r = R − π (cid:90) R q s Bv sθ Γ πs r d r. (15)The boundary term in Eq. (15) results from any uncom-pensated viscous stress at the edges of the system. Itvanishes if the rotation profile flattens at the edge, or ifa no-slip boundary imposes v sθ ( R ) = 0. If the dominantrotation is an E × B drift, then v sθ ≈ − E/B , and thusthe last term is the Ohmic j · E heating due to the cur-rent carried by the viscous flux of species s . The Ohmicheating term can be rewritten as follows: − π (cid:90) R q s Bv sθ Γ πs r d r = 2 π (cid:90) R v sθ ( ∇ · π s ) θ r d r. (16)The right-hand side integrand is the dot product of theviscous force with v s . In other words, the viscous Ohmicdissipation term is exactly compensated by the energytransferred out of the rotational motion by the viscousforce.In some ways, the two terms on the right-hand side ofEq. (15) – the boundary term and the Ohmic dissipation– are independent quantities which can be manipulatedseparately. For instance, as was noted above, the edgeheating can always be eliminated with an appropriatechoice of boundary conditions. However, these terms arenot fully independent. Note that the second term on theright-hand side of Eq. (15) can be positive or negative,but that Eq. (14) requires that Q πs ≥
0. This implies thatthere are cases in which the boundary term in Eq. (15)must be important. Any time the viscous heating car-ries particles from regions of lower electrostatic potentialenergy to regions of higher potential energy, the bound-ary term must be larger than the Ohmic heating term.For similar reasons, any choice of boundary conditionsthat eliminates the boundary heating term will implic-itly set the sign of q s E Γ πs . This point will be significantin Section V. B. Frictional Heating
The dissipation due to the frictional particle flux Γ
Rss (cid:48) behaves quite differently. The heating of species s due tofrictional interactions with species s (cid:48) can be written as Q Rss (cid:48) = m s (cid:48) m s + m s (cid:48) ( v s (cid:48) − v s ) · R ss (cid:48) (17)= q s B (cid:18) m s (cid:48) m s + m s (cid:48) (cid:19) ( v s (cid:48) θ − v sθ )Γ Rss (cid:48) . (18)This expression satisfies m s Q Rss (cid:48) = m s (cid:48) Q Rs (cid:48) s . The totalfrictional heating Q Rs for species s is then Q Rs = (cid:88) s (cid:48) Q Rss (cid:48) . (19)The frictional heating differs from the viscous heatingin two key ways. First, though both scale with the sizeof the corresponding particle flux, the frictional heatingdepends on the difference in velocities ( v s (cid:48) θ − v sθ ) ratherthan v sθ on its own. In an E × B -rotating device, therotational velocities for all species will be close to the E × B velocity, so ( v s (cid:48) θ − v sθ ) (cid:28) v sθ .Intuitively, why can a viscous particle flux be respon-sible for so much more dramatic heating than a frictionalparticle flux of the same size? It follows from the am-bipolarity of the frictional flux. The frictional flux Γ Rss (cid:48) of species s interacting with species s (cid:48) is always pairedwith a flux Γ Rs (cid:48) s of species s (cid:48) interacting with s . To lead-ing order, this pair of particle fluxes carries no net charge,and therefore exchanges no net energy with the electricalpotential. Exchange of energy with the electrical poten-tial has to come from the higher-order corrections to theflux in Eq. (7). These corrections are discussed in muchgreater detail elsewhere [26, 30]; they can result in flowsof net charge.The second key difference between viscous and fric-tional heating is the way in which heat is divided betweenthe different species. For a viscous particle flux Γ πs , theassociated heating goes entirely into species s , althoughthe viscosity coefficient η s can depend on other species.From a macroscopic standpoint, if viscosity drives a cur-rent up or down an electrostatic potential, the particlescarrying the current pick up (or lose) that potential en-ergy. On the other hand, when a frictional interactionbetween species s and s (cid:48) drives particle fluxes Γ Rss (cid:48) andΓ Rs (cid:48) s , the associated heating is divided between species s and s (cid:48) such that each species receives heat inversely pro-portional to its mass. This happens regardless of whichspecies is actually carrying current. The energy sourcefor heating might be the motion of particles down a po-tential gradient, but that motion is mediated by collisions between particles, and in those collisions energy is trans-ferred in such a way as to heat the lighter species in anygiven pair.It can be useful to compare the frictional dissipationfrom cross-field drifts with the Ohmic heating from cur-rents parallel to B . Parallel Ohmic heating is driven byfrictional dissipation, so it is reasonably well-describedby Eq. (17) (although a more complete kinetic treatmentdoes lead to corrections [31, 32]). In steady state andassuming homogeneity in the parallel direction, the forcebalance can be written in terms of a parallel field E || as q s E || = m s (cid:88) s (cid:48) ν ss (cid:48) ( v s || − v s (cid:48) || ) . (20)The velocity difference can be expressed as a functionof the species’ charge densities, collision frequencies,masses, and E || . This is quite unlike the perpendicularcase, where the E × B flows are the same for all speciesand the velocity differences are instead driven by diamag-netic and inertial effects, and where it is possible to havean electric field without any particle flux or dissipation.Combining Eq. (17) with Eq. (20), it follows that solong as the smallest dimensionless parameter in the prob-lem is the electron-to-ion mass ratio, most of the Ohmicheating must go into the electrons. This can be under-stood by considering a system containing two positive ionspecies, labeled a and b . So long as ν ab (cid:29) ν ae , the dif-ference in velocity between species a and b will be muchsmaller than | v a || − v e || | . The heating due to interac-tions between two species scales linearly with the colli-sion frequency but quadratically with the velocity differ-ence, so a pair of species with a high collision frequencywill have similar velocities and therefore little heating.This is a key difference between parallel and perpendicu-lar frictional dissipation; recall that in the perpendicularcase, there are configurations with multiple ion speciesfor which the heating can be directed into the ions. C. Frictional Cooling
There is an odd possibility worth pointing out inEq. (18): there are cases in which the frictional heat-ing Q Rss (cid:48) can be negative. If the frictional force R ss (cid:48) actsto reduce the velocity difference between species s and s (cid:48) , it is clear from Eq. (17) that this cannot happen,and Q Rss (cid:48) ≥
0. But the frictional force can be split intotwo pieces: a flow friction, which always does act to re-duce velocity differences, and a thermal friction, whichdepends on the temperature gradients and has no par-ticular obligation to align itself with the relative flows ofthe different species. These two parts of the frictionalforce can be seen explicitly in Eq. (4).At first glance, frictional cooling may be a worrisomething to find in a transport theory. It would be reasonableto wonder if this effect is unphysical. On its own, Eq. (17)contains no obvious fix; Q Rss (cid:48) and Q Rs (cid:48) s always have thesame sign, so this is not simply a transfer of energy be-tween species s and s (cid:48) . In fact, in order to demonstratethat this frictional cooling is consistent with the secondlaw of thermodynamics, it is necessary to take into ac-count the entropy production from the heat flow q s . Theentropy production rate for species s can be written [33]as Θ s = Q Rs T s + Q πs T s − q s T s · ∇ T s T s + (cid:88) s (cid:48) m s n s ν ss (cid:48) m s + m s (cid:48) (cid:0) T s (cid:48) − T s (cid:1) . (21)The last term is the entropy produced by local temper-ature equilibration between different species. For sim-plicity, it is helpful to consider the Braginskii single-ion-species limit, in which the frictional heating (and its pos-sible destruction of entropy) affects only the electrons.In this case, Q Re T e = m e n e ν ei T e (cid:0) v iθ − v eθ (cid:1) − m e n e ν ei eBT e ∂T e ∂r ( v iθ − v eθ ) (22)and − q e T e · ∇ T e T e = − m e n e ν ei eBT e ∂T e ∂r ( v iθ − v eθ )+ 4 . m e n e ν ei e B T e (cid:18) ∂T e ∂r (cid:19) . (23)The part of the heat flux that depends on the flow veloc-ities is sometimes called the Ettingshausen effect; it is re-sponsible for entropy production or destruction equal tothat of the thermal friction. Q Re /T e and − ( q e · ∇ T e ) /T e can each be positive or negative. However, they can becombined as follows: Q Re T e − q e T e · ∇ T e T e = 2 . m e n e ν ei e B T e (cid:18) ∂T e ∂r (cid:19) + m e n e ν ei T e (cid:18) v iθ − v eθ − eB ∂T e ∂r (cid:19) . (24)There are scenarios in which Q Re <
0; these scenariosinvolve thermal forces driven by temperature gradients.However, for any case in which Q Re <
0, the entropydestroyed by this cooling effect is balanced by entropyproduced as heat flows down the temperature gradients.Note that positive entropy production from these mech-anisms is not the same as an increase in temperature;the entropy production from the heat flow depends on q s · ∇ T s /T s , whereas the temperature evolution dependson ∇ · q s . Incidentally, Eq. (24) provides an alternateway of understanding the numerical instability describedin Ref. [34]. That instability occurred in simulations inwhich the heat flux was artificially reduced below a cer-tain threshold. Eq. (24) shows that if the heat flux is ar-tificially suppressed, the entropy production can becomenegative. For the purposes of understanding classical mech-anisms for driving temperature differences betweenspecies, the possibility of frictional cooling leads to acaveat: | Q Rs | (cid:29) | Q Rs (cid:48) | would not necessarily mean thatfriction is preferentially heating species s . In principlethere are temperature and velocity profiles for which fric-tion cools rather than heats. In order for this to happen,the temperature gradients must be large enough for thethermal friction to be larger than the flow friction.The frictional cooling is also worth understanding be-cause it explains an otherwise surprising fact that willplay a role in Section V: namely, that (cid:80) s Q Rs = 0 doesnot necessarily imply that Q Rs = 0 for each s . IV. CONTROLLING THE HEATINGCHANNELS
If the different heating terms associated with cross-field particle transport can be directed into either theions or the electrons, then under what conditions will ionor electron heating dominate? Cross-field viscous dissi-pation heats ions. For a plasma containing a single ionspecies, the frictional dissipation heats the electrons, so– apart from the possibility of frictional cooling – thetendency of classical heating to heat ions or electrons de-pends entirely on the relative sizes of Q πi and Q Re . For aplasma containing a mix of ion species, ion-ion collisionsprovide an additional channel for ion heating, and Q Re must instead compete with Q πi + Q Ri .There is more than one way to understand the condi-tions under which different heating channels will domi-nate. One approach is to directly compare the expres-sions for heating given in Eqs. (14) and (17), and to in-fer the spatial density, velocity, and temperature profilesthat maximize or minimize each.Consider the conditions under which each of the heat-ing effects vanishes. The viscous dissipation is suppressedwhen v sθ ∝ r — that is, in the limit of solid-body rota-tion. Any time Q πs vanishes, so does Γ πs . However, thereverse is not true; there are profiles with viscous heatingbut no viscous particle transport. In particular, there isheating without particle transport when η s r ∂∂r (cid:18) v sθ r (cid:19) = nonzero constant . (25)If η s is spatially constant, this condition corresponds to v sθ ∝ /r .The relationship is the other way around for the fric-tional heating and transport: there is never nonzero Q Rss (cid:48) without nonzero Γ
Rss (cid:48) , but there are profiles with particletransport and no heating. These profiles are discussed inmore detail in Appendix B. The particle transport van-ishes when R ss (cid:48) θ = 0. This can be written as a conditionon the pressure profiles, because the friction depends on v sθ and v s (cid:48) θ , which depend on the species’ diamagneticdrifts. If T s (cid:48) = τ T s for some constant τ , R ss (cid:48) θ will vanishwhen (cid:26) p s ( r ) p s (0) exp (cid:20) − (cid:90) r d r (cid:18) m s v sθ rT s + γ ss (cid:48) T (cid:48) s (cid:19)(cid:21)(cid:27) /Z s = (cid:26) p s (cid:48) ( r ) p s (cid:48) (0) exp (cid:20) − (cid:90) r d r (cid:18) m s (cid:48) v s (cid:48) θ rT s (cid:48) + γ s (cid:48) s T (cid:48) s (cid:48) (cid:19)(cid:21)(cid:27) τ/Z s (cid:48) . (26)In the model used for Eq. (4), γ ss (cid:48) = 32 m s (cid:48) m s T s (cid:48) + m s (cid:48) T s . (27)Expressions that are closely related to Eq. (26) havebeen studied in the particle-transport literature [35–40].Eq. (26) is more general than the expressions that havebeen used in the past, since it includes both the effectsof the centrifugal potential and the thermal friction, butthe generalization is straightforward; see Appendix B.The existence of a special class of profiles for which R ss (cid:48) θ vanishes is known in the particle transport liter-ature. However, it has not been recognized that thereis a second special class of dissipationless profiles forwhich there is particle transport but no frictional heat-ing. Note from Eq. (17) that Q Rss (cid:48) vanishes whenever ei-ther R ss (cid:48) θ = 0 or v sθ = v s (cid:48) θ . If T s (cid:48) = τ T s , then v sθ = v s (cid:48) θ when (cid:26) p s ( r ) p s (0) exp (cid:20) − (cid:90) r d r m s v sθ rT s (cid:21)(cid:27) /Z s = (cid:26) p s (cid:48) ( r ) p s (cid:48) (0) exp (cid:20) − (cid:90) r d r m s (cid:48) v s (cid:48) θ rT s (cid:48) (cid:21)(cid:27) τ/Z s (cid:48) . (28)This condition is derived in Appendix B. The profilesdescribed by Eq. (26) and those described by Eq. (28)become the same when the temperature gradients vanish,or more generally when m s (cid:48) T s T (cid:48) s q s = m s T s (cid:48) T (cid:48) s (cid:48) q s (cid:48) . (29)The frictional heating Q Rss (cid:48) can be understood as afunction of how close the different pressure profiles areto the classes of dissipationless profiles described byEqs. (26) and (28). A pair of species that satisfiesEq. (28) will have a cross-field frictional particle flux butno corresponding heating. A pair that satisfies Eq. (26)will have no cross-field frictional particle flux (and nocorresponding heating).Qualitatively, the relative importance of viscous andfrictional heating depends on how far the velocity pro-file is from solid-body rotation compared with how farthe pressure profiles are from satisfying Eqs. (26) or (28).Similarly, the relative importance of frictional ion-ion andion-electron heating can be understood in terms of thedeviations of the different species’ profiles from Eqs. (26)and (28), weighted by the appropriate collision frequen-cies. For species with comparable densities, ion-ion colli-sion frequencies are larger than ion-electron collision fre-quencies by a factor of the square root of the ion-electron mass ratio. As such, in order for electron heating to dom-inate, not only must the velocity profile not be too farfrom solid-body rotation, but the electrons must be sub-stantially further away from satisfying Eqs. (26) or (28)with respect to the ions than the ions are with respect toother ion species.
V. GLOBAL CONDITIONS FOR ION HEATING
Remarkably, there are circumstances under which itis possible to determine which heating terms dominatewithout knowing anything other than the global 0-D par-ticle balance and the boundary conditions. This is to becontrasted with the approach taken in the previous sec-tion, which relied on knowledge of the full spatial profilesof density, velocity, and temperature.Consider a simple model of a cylindrical fusion devicewith total radius R . Pick boundary conditions at the ra-dial boundary r = R , such that either v sθ or ∂ ( v sθ /r ) /∂r vanishes at the boundary (so that the boundary term inEq. (15) can be ignored). From Eq. (15) and the factthat Q πs ≥
0, either boundary condition implies that q s Γ πs E r ≥ a and b are supplied at the edge andsuppose there is a fusion reaction consuming one ion ofspecies a and one ion of species b occurring at some rate S per unit length at r = 0. Finally, suppose the fusionproducts are removed promptly, by a process other thanclassical transport, which we can imagine to be wave-driven, as in α -channeling [3]. Suppose the fusion prod-ucts are removed before they interact with the rest of theplasma, collisionally or otherwise [41]. Then, in order toachieve steady state, there must be injection of fuel ionsinto the system. This is shown schematically in Figure 2. fuel ionsfusion productsfusionFIG. 2. The global particle balance for a cylindrical systemwith fusion reactions taking place at the core. If an effect like α -channeling removes the fusion products, classical transportmust provide a balancing influx of fuel ions. First assume that the wave-driven removal of the fu-sion ash is non-ambipolar, in the sense that only the ashions (and not an accompanying population of electrons)are removed. This is the usual circumstance envisionedin the case of α -channeling, where resonant wave-particleinteractions promptly eject the α particles on a collision-less timescale. Then the total fuel ion fluxes must satisfyΓ a = Γ b = − S πr (30)for all r ∈ (0 , R ), and the electron flux isΓ e = 0 . (31)When new fuel ions are supplied at the edge, they mustbe at rest in order to ensure that the total angular mo-mentum in the system remains constant; the j × B torquesdue to the inward motion of the fuel ions and the outwardmotion of the ash will balance one another.The fluxes given by Eqs. (30) and (31) will in generalbe a mix of frictional and viscous transport. However,almost all of the net flow of charge must be carried bythe viscous fluxes Γ πa and Γ πb . This is not at all obvious.Note that we have not made any assumptions about therelative sizes of Γ πs and Γ Rss (cid:48) for the ions, so one mightimagine a case in which Γ πs is at least O ( δ ) smaller than (cid:80) s (cid:48) Γ Rss (cid:48) and the current is instead carried by the O ( δ )terms in Eq. (7). Indeed, if the O ( δ ) corrections wereentirely arbitrary, this might be possible. But they arenot arbitrary; recall that Eq. (2) can be expanded asΓ s = (cid:18) Γ πs + (cid:88) s (cid:48) Γ Rss (cid:48) (cid:19) ∞ (cid:88) k =0 (cid:20) − ( rv sθ ) (cid:48) r Ω s (cid:21) k . (32)Intuitively, the ( k + 1)th term in the sum in Eq. (32) canbe understood as the F × B flow from the Coriolis andEuler forces that result from the radial motion describedby the k th term [26].The k (cid:54) = 0 corrections in Eq. (32) can producenon-ambipolar corrections to otherwise ambipolar flows.However, a non-ambipolar correction will only appear ifthere is a non-vanishing total flow at the next-lowest or-der. In other words, the k (cid:54) = 0 terms can contribute asignificant part of the current-carrying flow only if thereis a non-vanishing ambipolar flow that is much larger.In this scenario, the leading-order flow is already non-ambipolar, so the flow of charge cannot be explained bythese corrections.As such, to leading order in δ , it follows that the cur-rent is carried by the viscous fluxes: Z a Γ πa + Z b Γ πb = − ( Z a + Z b ) S πr . (33)Then, up to O ( δ ) corrections, the heating associated withthe flow of charge down the electrical potential goes intothe ion species.Even so, there could be large frictional fluxes in thesystem; if there are temperature gradients, then it is pos-sible to have Γ Rs = 0 but Γ Rss (cid:48) (cid:54) = 0. This can happen whenthe thermal friction is at least comparable to the friction from flows. In that case, it is possible to have Q Rss (cid:48) (cid:54) = 0.If Γ Rs = 0 for all species, then (cid:88) s,s (cid:48) Q Rss (cid:48) = (cid:88) s,s (cid:48) q s B (cid:18) m s (cid:48) m s + m s (cid:48) (cid:19) ( v s (cid:48) θ − v sθ )Γ Rss (cid:48) (34)= − B (cid:26) (cid:88) s q s v sθ (cid:88) s (cid:48) Γ Rss (cid:48) + (cid:88) s (cid:48) q s (cid:48) v s (cid:48) θ (cid:88) s Γ Rs (cid:48) s (cid:27) (35)= 0 . (36)The frictional heating (cid:80) s (cid:48) Q Rss (cid:48) for any given species s does not necessarily vanish on its own, so it must repre-sent a transfer of heat between species (recall that Q Rss (cid:48) can be frictional heating or frictional cooling). However,if the thermal friction is on the same order as the totalfriction, it is possible to show that (cid:80) s (cid:48) Q Rss (cid:48) is very smallcompared to the temperature equilibration term betweenany two species, so the energy transfer associated withthe frictional flows will not have a significant effect onthe relative temperatures of the different species.Eq. (33) is an interesting result because it implies that,for every fusion event, the overall population of fuel ionswould be heated by ( Z a + Z b ) e ∆Φ, where ∆Φ is the po-tential difference between the edge and the center. Thisis possible because, for these boundary conditions, all ofthe potential energy liberated by a viscosity-driven cur-rent (i.e., the full j · E ) goes into the ions. For a systemwith very large fields, this would represent a very largeion heating effect.For instance, some proposals, like the WDRT [17], callfor voltage drops that could be O (1) MV across the sys-tem. In such a case, the viscous j · E dissipation coulddeliver O ( Z a + Z b ) MeV to the fuel ions for every fusionevent. The total power going into the ions through clas-sical viscous dissipation could be on the same order asthe fusion power output. The overall power balance ofthe system would involve some significant fraction of thefusion power being diverted through α -channeling intomaintaining the electrostatic potential, and fuel ions tak-ing large amounts of energy from that potential as theymove from edge to core.Note that the scenario put forth in this section is a spe-cial case that relies on the transport being non-ambipolarto leading order in the ratio of the rotation frequencyto the ion gyrofrequency. This ordering can be thoughtof as a limit to the electric field (since δ ∝ E ), whichin turn limits only how much energy can be gained byfalling through the electric potential. Note also that in acase in which the net classical transport were ambipolar(for instance, if the mechanism that removed the fusionash also removed a balancing population of electrons), itwould not be possible to determine the dominant heatingchannel through global particle balance constraints alone.Instead, it would be necessary to consider the full spatialprofiles, along the lines of the discussion in Section IV. VI. DISCUSSION
In a rotating plasma, the classical dissipation can pro-ceed through several very different mechanisms. Bycontrolling which mechanism dominates, the power flowthrough the plasma can be altered significantly. If theright channels dominate – and if the dissipation is large– it is possible to create what we have called a “natural”hot ion mode, produced without any need for auxiliaryheating of the ions.Achieving this kind of control is a complex problem. Inthe most general case, the heating going through each ofthe classical dissipation channels is a nonlinear functionof the different species’ densities, rotation velocities, andtemperatures. Qualitatively, the viscous heating is largewhen the system’s velocity profile is far from solid-bodyrotation. The size of the frictional heating can be under-stood in terms of the deviation of the species’ pressureprofiles from a set of pairwise relations (to which the sys-tem would in principle relax, if it were not driven). In or-der to preferentially heat the ions, it is necessary to directmost of the heating through viscous dissipation and ion-ion frictional dissipation rather than through ion-electronfrictional dissipation. This is to be contrasted with par-allel Ohmic dissipation, in which one dissipation channel(the ion-electron friction) dominates and the electronsare always heated.In the most general case, the heating going througheach of the classical dissipation channels depends in acomplicated way on the density, rotation velocity, andtemperature of each species. However, the viscous heat-ing is large when the velocity profiles are far from solid-body rotation. The frictional heating is large when thepressure profiles deviate from the set of pairwise relationsto which the system would in principle relax, were it notdriven. Thus, ions are preferentially heated when vis-cous dissipation and ion-ion frictional dissipation dom-inate ion-electron frictional dissipation. In contrast, inthe case of parallel Ohmic dissipation, the ion-electronfriction always dominates, so that it is always the elec-trons that are heated.It is quite remarkable that, in a simple 0-D model ofa fusion reactor, it is possible to determine the domi-nant heating channel through global constraints alone.In particular, with appropriate constraints on the bound-ary conditions and the behavior of the fusion products,global constraints, that force the particle fluxes to benon-ambipolar to leading order in rotation frequency overion gyrofrequency, also ensure that heat flows to the ionsthrough viscous dissipation. The ions then receive en-ergy in the form of heat comparable to the electrostaticpotential energy drop from the edge to the core. Thision heating may then produce a hot-ion mode significantenough to facilitate economical nuclear fusion.However, the condition that ion heating dominates wasarrived at through a number of simplifying assumptions.First, note that the transport model used here assumesthat cross-field transport is classical and axial effects can be neglected. Although there are indications that classi-cal transport may be more easily attainable in rotatingplasmas [42, 43], classical transport is by no means as-sured. Second, note that, in the case of the 0-D fusionreactor model, this paper assumes that there is a mech-anism like α -channeling that can remove fusion productswithout incurring collisional effects, and that this mech-anism does not also expel electrons or draw in fuel ions.There are scenarios in which the waves used for α -channeling could damp on the fuel ions directly (for in-stance, if the amplified waves encounter the tritium res-onance [4]). This would increase T i − T e significantly,and could circumvent the need for classical dissipationto produce a hot-ion mode. However, if α -channelingdoes move net charge across field lines, and if E r < α -channeling could be simply totransfer the fusion energy to the potential. This wouldtend to make the classical dissipation mechanisms dis-cussed here more important, since it reduces the amountof available energy in the wave that could heat the fuelions directly.Regardless of precisely what happens to the wave en-ergy, what we have shown here is that in a rotating hotplasma, in which fusion ash is expelled promptly whilefuel ions are drawn in through collisional transport, thereis a tendency for the natural dissipation in maintainingthe plasma configuration to favor heat going into the ionsrather than the electrons. This produces the possibilityof a naturally occurring hot-ion mode, without the needfor external mechanisms that directly heat fuel ions. Thisremarkable circumstance could be of great interest in eco-nomical controlled nuclear fusion. ACKNOWLEDGMENTS
The authors would like to thank S. Jin and T. Ru-bin for helpful conversations. This work was supportedby Cornell NNSA 83228-10966 [Prime No. DOE (NNSA)DENA0003764] and by NSF-PHY-1805316.
Appendix A: Viscous Forces and Heating
Consider a system with cylindrical symmetry, no flowin the ˆ z direction, and a magnetic field in the ˆ z direction.The velocity of species s can be written as v s = v sr ( r )ˆ r + rω s ( r )ˆ θ. (A1)Suppose the plasma is weakly coupled and strongly mag-netized; the viscosity can change significantly if either ofthese conditions is not met [47].For the following calculation, the species index s willbe suppressed. The viscosity tensor π ij can be calculatedin an arbitrary coordinate system using the procedureoutlined in Ref. [26]. It is convenient to pick cylindricalcoordinates, for which the metric tensor g ij is g ij = r
00 0 1 (A2)and the Christoffel symbols are given byΓ rij = − r
00 0 0 (A3)Γ θij = r − r − (A4)Γ zij = 0 . (A5)Define a covariant velocity vector u in this coordinatesystem, satisfying u r = v r , u θ = r ω , and u z = 0.By definition, the corresponding contravariant vector hascomponents u i = g ij u j , so u r = v r , u θ = ω , and u z = 0.In terms of the covariant derivative ∇ i , Braginskii’straceless rate-of-strain tensor W ij is defined by W ij . = ∇ i u j + ∇ j u i − g ij ∇ k u k . (A6)For the velocity given by Eq. (A1), its nonzero compo-nents are W rr = 23 (cid:0) ∂ r u r − r − u r (cid:1) (A7) W rθ = W θr = ∂ r u θ − r − u θ (A8) W θθ = − (cid:0) r ∂ r u r − ru r (cid:1) (A9) W zz = − (cid:0) ∂ r u r + r − u r (cid:1) . (A10)Braginskii’s viscosity tensor can be written as π ij = − η W ij − η W ij − η W ij + η W ij + η W ij , (A11)where the k W tensors can be written in covariant form[26] as W ij = 32 (cid:0) b i b j − g ij (cid:1)(cid:0) b m b n − g mn (cid:1) W mn (A12) W ij = (cid:0) δ ⊥ im δ ⊥ nj + 12 δ ⊥ ij b m b n (cid:1) W mn (A13) W ij = (cid:0) δ ⊥ im b j b n + δ ⊥ nj b i b m (cid:1) W mn (A14) W ij = 12 (cid:0) δ ⊥ im (cid:15) nbk b k − δ ⊥ nj (cid:15) imk b k (cid:1) W mn (A15) W ij = (cid:0) b i b m (cid:15) njk b k + b j b n (cid:15) imk b k (cid:1) W mn . (A16) Here b i = (cid:0) (cid:1) is the unit vector in the direction ofthe magnetic field and δ ⊥ ij . = g ij − b i b j . In this coordinatesystem, the Levi-Civita tensor is defined as (cid:15) ijk . = r ˜ (cid:15) ijk , (A17)where ˜ (cid:15) ijk is the Levi-Civita symbol.These expressions are derived from Braginskii’s model,which assumes a single ion species. For a plasma contain-ing several ion species, the viscosity can still be modeledusing Eq. (A11), but the expressions for the η coefficientsfor each species are modified [29].With these definitions, it is possible to directly calcu-late π ij in cylindrical coordinates for the chosen velocityprofile: π ij = − η r ∂ ( ru r ) ∂r r
00 0 − − (cid:20) η r ∂∂r (cid:18) u r r (cid:19) + η r ∂∂r (cid:18) u θ r (cid:19)(cid:21) − r
00 0 0 − (cid:20) η r ∂∂r (cid:18) u θ r (cid:19) − η r ∂∂r (cid:18) u r r (cid:19)(cid:21) . (A18)Then the viscous force density can be written as ∇ i π ij = ∂ i π ij + Γ iiλ π λj + Γ jiλ π iλ . (A19)The first and second of the three matrices in Eq. (A18)contribute to the ˆ r -directed force. After converting fromthe resulting contravariant force vector back to the orig-inal coordinate normalization,( ∇ · π ) r = − ∂∂r (cid:20) η r ∂ ( rv r ) ∂r (cid:21) − r ∂∂r (cid:20) η r ∂∂r (cid:18) v r r (cid:19) + η r ∂ω∂r (cid:21) . (A20)Only the third matrix in Eq. (A18) contributes to theˆ θ -directed force. Again, in the original normalization,( ∇ · π ) θ = − r ∂∂r (cid:20) η r ∂ω∂r − η r ∂∂r (cid:18) v r r (cid:19)(cid:21) . (A21)For the chosen velocity profile, there is no viscous forcein the ˆ z direction.The relative importance of these different terms de-pends, in part, on the η coefficients. From this pointforward, the species indices will no longer be suppressed.For a plasma containing a single ion species, Braginskiigives ion coefficients [28] η i = 0 . √ n i T i ν ii (A22) η i = 310 √ n i T i ν ii Ω i (A23) η i = 12 n i T i Ω i (A24)0and electron coefficients η e = 0 . n e T e ν ei (A25) η e = 0 . n e T e ν ei Ω e (A26) η e = 12 n e T e | Ω e | , (A27)where Ω s . = q s B/m s . In the multiple-ion-species case,the coefficients scale similarly [29], but in general thereare additional contributions due to collisions with otherspecies.If the radial flow is driven by classical transport, then v sr will be much smaller than v sθ . In the case of a plasmawith a single ion species [26], v ir v iθ ∼ v er v eθ ∼ ν ie Ω i ErB Ω i . (A28)This is because the frictional F × B flow due to the dif-ference between ion and electron azimuthal velocities issmaller than the azimuthal velocity difference by a factorof ν ie / Ω i , and the azimuthal velocity difference is typi-cally small compared to the total azimuthal velocity bya factor of E/rB Ω i . In the multiple-ion-species case, theradial transport could instead be driven by unlike-ion col-lisions, in which case the radial flow would be larger bya factor of O ( ν ii (cid:48) /ν ie ) than what is given in Eq. (A28).Meanwhile, η i /η i ∼ ν ii / Ω i . For this reason, for theflows studied in this paper, Eq. (A21) can be approxi-mated as( ∇ · π i ) θ ≈ − r ∂∂r (cid:20) η i r ∂∂r (cid:18) v iθ r (cid:19)(cid:21) . (A29)The relative sizes of the η coefficients also imply that( ∇ · π e ) θ is small compared to ( ∇ · π i ) θ .The radial viscous forces may be relatively large. Thelargest of the η coefficients to appear in Eq. (A20) is η s .However, note that the part of ( ∇ · π s ) r that depends on η s will vanish for any divergenceless radial flow. More-over, if the flow is not divergenceless, the part of theradial force that depends on η s scales like − ∂∂r (cid:20) η i r ∂ ( rv ir ) ∂r (cid:21) ∼ v ir ν ii L p i L (A30) − ∂∂r (cid:20) η e r ∂ ( rv er ) ∂r (cid:21) ∼ v er ν ie L p e L , (A31)where L is the characteristic gradient scale length. Radialflows driven by classically transport will typically moveparticles much less than the gradient scale length overthe course of a collision time. Therefore, this part of theradial viscous force is negligible compared to the pressureforce. Similar arguments apply for the other terms inEq. (A20); after all, η s and η s are small compared to η s .Note that this scaling argument should not be usedto neglect the ˆ θ component of the viscous force, since it is in a different direction. In an axially magnetizedplasma, azimuthal forces are much more efficient than ra-dial forces at driving radial transport, because azimuthalforces produce F × B drifts in the radial direction. Thisis discussed in greater detail in Ref. [26].The other important effect of viscosity is the viscousheat dissipation. The viscous heating for species s is Q πs = − π ijs ∇ i u j , (A32)which can be evaluated as Q πs = η s r (cid:20) ∂ ( rv sr ) ∂r (cid:21) + η s r (cid:20) ∂∂r (cid:18) v sr r (cid:19)(cid:21) + η s r (cid:20) ∂∂r (cid:18) v sθ r (cid:19)(cid:21) . (A33)The ion η coefficients are large compared to the corre-sponding electron coefficients, so Q πi (cid:29) Q πe . For a di-vergenceless flow, the part of Q πi that depends on η i vanishes. If the flow is not divergenceless, then η i r (cid:20) ∂ ( rv ir ) ∂r (cid:21) (cid:30) η i r (cid:20) ∂∂r (cid:18) v iθ r (cid:19)(cid:21) ∼ (cid:18) Ω i ν ii v ir v iθ (cid:19) . (A34)According to the scaling in Eq. (A28), this ratio is small,and the first of the three terms in Eq. (A33) can be ne-glected. The second of the three terms can be neglectedwhenever the radial velocity is small compared to the az-imuthal velocity, as will be the case if that radial flowis driven by classical transport. Therefore, the viscousheating for ion species i can reasonably be approximatedby Q πi ≈ η i r (cid:20) ∂∂r (cid:18) v iθ r (cid:19)(cid:21) . (A35)This is the expression used elsewhere in the paper. Be-cause of the symmetric geometry and velocity profile usedhere, this treatment includes only the perpendicular vis-cosity; however, note that there are contexts (particu-larly involving MHD fluctuations) in which the parallelviscosity may drive significant ion heating [48–50]. Appendix B: Derivation of Dissipationless PressureProfiles
The frictional heating of species s due to frictional in-teractions with species s (cid:48) can be written as Q Rss (cid:48) = m s (cid:48) m s + m s (cid:48) (cid:0) v s (cid:48) − v s (cid:1) · R ss (cid:48) . (B1)In the typical case where v sr (cid:28) v sθ , this can be writtenas Q Rss (cid:48) = m s (cid:48) m s + m s (cid:48) (cid:0) v s (cid:48) θ − v sθ (cid:1) R ss (cid:48) θ , (B2)1where the friction force density R ss (cid:48) θ is R ss (cid:48) θ = n s m s ν ss (cid:48) ( v s (cid:48) θ − v sθ )+ n s m s ν ss (cid:48) B (cid:18) γ ss (cid:48) T s T s (cid:48) q s − γ s (cid:48) s T s (cid:48) T (cid:48) s (cid:48) q s (cid:48) (cid:19) , (B3)where γ ss (cid:48) . = 32 m s (cid:48) m s T s (cid:48) + m s (cid:48) T s . (B4)The momentum equation for species s is given by Eq. (1).In steady state, its ˆ r component is m s n s v sr ∂v sr ∂r − m s n s v sθ r = q s n s E + q s n s v sθ B − ∂p s ∂r + R sr . (B5) v sr is small compared to v sθ , so m s n s v sr ∂v sr /∂r (cid:28) m s n s v sθ /r . The collision frequencies ν ss (cid:48) are small com-pared to the gyrofrequency Ω s , so R sr (cid:28) q s n s v sθ B .Dropping these two small terms and rearranging, v sθ = − EB + 1 q s Bn s ∂p s ∂r − s v sθ r . (B6)Then v s (cid:48) θ − v sθ =1 q s B (cid:20) q s q s (cid:48) p (cid:48) s (cid:48) n s (cid:48) − p (cid:48) s n s − q s q s (cid:48) m s (cid:48) v s (cid:48) θ r + m s v sθ r (cid:21) (B7)and R ss (cid:48) θ = n s m s ν ss (cid:48) q s B (cid:20) q s q s (cid:48) p (cid:48) s (cid:48) n s (cid:48) − p (cid:48) s n s − q s q s (cid:48) m s (cid:48) v s (cid:48) θ r + m s v sθ r + (cid:18) γ ss (cid:48) T s T (cid:48) s − q s q s (cid:48) γ s (cid:48) s T s (cid:48) T (cid:48) s (cid:48) (cid:19)(cid:21) . (B8)The condition for R ss (cid:48) θ = 0 can be rewritten as1 q s (cid:48) (cid:18) p (cid:48) s (cid:48) n s (cid:48) − m s (cid:48) v s (cid:48) θ r − γ s (cid:48) s T s (cid:48) T (cid:48) s (cid:48) (cid:19) = 1 q s (cid:18) p (cid:48) s n s − m s v sθ r − γ ss (cid:48) T s T (cid:48) s (cid:19) . (B9)In the simple limit where T s (cid:48) ( r ) = τ T s ( r ) for some con-stant τ , (cid:26) p s ( r ) p s (0) exp (cid:20) − (cid:90) r d r (cid:18) m s v sθ rT s + γ ss (cid:48) T (cid:48) s (cid:19)(cid:21)(cid:27) /Z s = (cid:26) p s (cid:48) ( r ) p s (cid:48) (0) exp (cid:20) − (cid:90) r d r (cid:18) m s (cid:48) v s (cid:48) θ rT s (cid:48) + γ s (cid:48) s T (cid:48) s (cid:48) (cid:19)(cid:21)(cid:27) τ/Z s (cid:48) . (B10) where Z s . = q s /e , so electrons would have Z e = −
1. Thecondition for v s (cid:48) θ − v sθ = 0 can similarly be treated sim-ilarly. Under the same assumptions, it reduces to (cid:26) p s ( r ) p s (0) exp (cid:20) − (cid:90) r d r m s v sθ rT s (cid:21)(cid:27) /Z s = (cid:26) p s (cid:48) ( r ) p s (cid:48) (0) exp (cid:20) − (cid:90) r d r m s (cid:48) v s (cid:48) θ rT s (cid:48) (cid:21)(cid:27) τ/Z s (cid:48) . (B11)The condition for R ss (cid:48) θ = 0 and the condition for v sθ = v s (cid:48) θ are identical in the limit where T (cid:48) s = T (cid:48) s (cid:48) = 0.Define P Rss (cid:48) as a profile for species s that satisfiesEq. (B10) with respect to species s (cid:48) : P Rss (cid:48) ( r ) = P Rss (cid:48) (0) exp (cid:20) (cid:90) r d r (cid:18) m s v sθ rT s + γ ss (cid:48) T (cid:48) s (cid:19)(cid:21) × (cid:26) p s (cid:48) ( r ) p s (cid:48) (0) exp (cid:20) − (cid:90) r d r (cid:18) m s (cid:48) v s (cid:48) θ rT s (cid:48) + γ s (cid:48) s T (cid:48) s (cid:48) (cid:19)(cid:21)(cid:27) τZ s /Z s (cid:48) . (B12)Define P vss (cid:48) as a profile for species s that satisfiesEq. (B11) with respect to species s (cid:48) : P vss (cid:48) ( r ) = P vss (cid:48) (0) exp (cid:20) (cid:90) r d r m s v sθ rT s (cid:21) × (cid:26) p s (cid:48) ( r ) p s (cid:48) (0) exp (cid:20) − (cid:90) r d r m s (cid:48) v s (cid:48) θ rT s (cid:48) (cid:21)(cid:27) τZ s /Z s (cid:48) . (B13)The flux Γ Rss (cid:48) tends to make p s relax to P Rss (cid:48) . Note thatthe electric field appears only in the centrifugal v sθ term(unlike in the unmagnetized case, where it can drive dif-ferential transport more directly [51, 52]).The heating Q Rss (cid:48) can be rewritten in a way that moreexplicitly shows how it depends on the deviation of p s from P Rss (cid:48) and P vss (cid:48) : Q Rss (cid:48) = m s (cid:48) m s + m s (cid:48) T s n s m s ν ss (cid:48) q s B × (cid:20) ∂∂r log (cid:18) P Rss (cid:48) p s (cid:19)(cid:21)(cid:20) ∂∂r log (cid:18) P vss (cid:48) p s (cid:19)(cid:21) . (B14)This expression vanishes as p s approaches either P Rss (cid:48) or P vss (cid:48) . Appendix C: Compressional Heating
The temperature evolution equation, Eq. (13), includesa compressional heating term that is largely not discussedin this paper:3 n s dT s dt (cid:12)(cid:12)(cid:12)(cid:12) compressional = − p s ∇ · v s . (C1)In part, this is a question of scope: the focus of the paperis on the classical heating terms Q πs and Q Rs , and on their2connection with the classical particle fluxes Γ πs and Γ Rs .Moreover, the compressional heating behaves identicallyin a rotating plasma as in any other plasma (unlike Q πs and Q Rs , both of which depend on the rotation profile inone way or another).However, it may be helpful to say something aboutthe expected size of the compressional heating. If s s isthe volumetric particle source rate, then the continuityequation is ∂n∂t + ∇ · ( n s v s ) = s s . (C2)Then for a system in steady state, − p s ∇ · v s = − T s ∇ · ( n s v s ) + T s v s · ∇ n s (C3)= − T s s s + T s v s · ∇ n s . (C4)Note that for inflowing particles, v s ·∇ n s is positive if the density is peaked toward the core of the system. Awayfrom sources, if the density has a gradient scale length L , − p s ∇ · v s ∼ T s Γ s L . (C5)This can be contrasted with Eq. (15), which implies thatwithout boundary stresses, Q πs ∼ q s E Γ πs . (C6)For many rapidly rotating systems, q s E is large comparedto T s /L . In the most dramatic cases discussed in thispaper, where the j · E heating is very large and can be di-rected into the ions, the scaling in Eq. (C5) suggests thatthe compressional heating will not be important. How-ever, that does not mean there are no scenarios in whichit matters; a full solution of the temperature evolutionequation would have to take it into account. [1] J. F. Clarke, Nucl. Fusion , 563 (1980).[2] N. J. Fisch and M. C. Herrmann, Nucl. Fusion , 1541(1994).[3] N. J. Fisch and J.-M. Rax, Phys. Rev. Lett. , 612(1992).[4] E. J. Valeo and N. J. Fisch, Phys. Rev. Lett. , 3536(1994).[5] N. J. Fisch, Phys. Plasmas , 2375 (1995).[6] N. J. Fisch and M. C. Herrmann, Nucl. Fusion , 1753(1995).[7] I. E. Ochs, N. Bertelli, and N. J. Fisch, Phys. Plasmas , 112103 (2015).[8] F. Cianfrani and F. Romanelli, Nucl. Fusion , 076013(2018).[9] F. Cianfrani and F. Romanelli, Nucl. Fusion , 106005(2019).[10] C. Castaldo, A. Cardinali, and F. Napoli, Plasma Phys.Control. Fusion , 084007 (2019).[11] F. Romanelli and A. Cardinali, Nucl. Fusion , 036025(2020).[12] B. Lehnert, Nucl. Fusion , 485 (1971).[13] A. A. Bekhtenev, V. I. Volosov, V. E. Pal’chikov, M. S.Pekker, and Yu. N. Yudin, Nucl. Fusion , 579 (1980).[14] R. F. Ellis, A. B. Hassam, S. Messer, and B. R. Osborn,Phys. Plasmas , 2057 (2001).[15] R. F. Ellis, A. Case, R. Elton, J. Ghosh, H. Griem,A. Hassam, R. Lunsford, S. Messer, and C. Teodorescu,Phys. Plasmas , 055704 (2005).[16] C. Teodorescu, W. C. Young, G. W. S. Swan, R. F. Ellis,A. B. Hassam, and C. A. Romero-Talamas, Phys. Rev.Lett. , 085003 (2010).[17] J.-M. Rax, R. Gueroult, and N. J. Fisch, Phys. Plasmas , 032504 (2017).[18] I. E. Ochs and N. J. Fisch, Phys. Plasmas , 092513(2017).[19] S. Suckewer, H. P. Eubank, R. J. Goldston, E. Hinnov,and N. R. Sauthoff, Phys. Rev. Lett. , 207 (1979).[20] Y. B. Kim, P. H. Diamond, and R. J. Groebner, Phys.Fluids B , 2050 (1991). [21] G. D. Conway, J. Schirmer, S. Klenge, W. Suttrop,E. Holzhauer, and the ASDEX Upgrade Team, PlasmaPhys. Control. Fusion , 951 (2004).[22] P. Helander and D. J. Sigmar, Collisional Transport inMagnetized Plasmas (Cambridge University Press, Cam-bridge, UK, 2002).[23] J. E. Rice, A. Ince-Cushman, J. S. deGrassie, L.-G. Eriks-son, Y. Sakamoto, A. Scarabosio, A. Bortolon, K. H.Burrell, B. P. Duval, C. Fenzi-Bonizec, M. J. Greenwald,R. J. Groebner, G. T. Hoang, Y. Koide, E. S. Marmar,A. Pochelon, and Y. Podpaly, Nucl. Fusion , 1618(2007).[24] J. S. deGrassie, Plasma Phys. Control. Fusion , 124047(2009).[25] T. Stoltzfus-Dueck, Phys. Rev. Lett. , 065002 (2012).[26] E. J. Kolmes, I. E. Ochs, M. E. Mlodik, J.-M. Rax,R. Gueroult, and N. J. Fisch, Phys. Plasmas , 082309(2019).[27] S. P. Hirshman and D. J. Sigmar, Nucl. Fusion , 1079(1981).[28] S. I. Braginskii, “Transport processes in a plasma,” in Reviews of Plasma Physics , Vol. 1, edited by M. A. Leon-tovich (Consultants Bureau, New York, 1965) p. 205.[29] V. M. Zhdanov,
Transport Processes in MulticomponentPlasma (Taylor & Francis, London, UK, 2002).[30] J.-M. Rax, E. J. Kolmes, I. E. Ochs, N. J. Fisch, andR. Gueroult, Phys. Plasmas , 012303 (2019).[31] L. Spitzer and R. H¨arm, Phys. Rev. , 977 (1953).[32] L. Spitzer, Physics of Fully Ionized Gases , 2nd ed. (JohnWiley & Sons, 1962).[33] R. D. Hazeltine and F. L. Waelbroeck,
The Frameworkof Plasma Physics (Westview Press, Boulder, Colorado,2004).[34] E. J. Kolmes, I. E. Ochs, and N. J. Fisch, Comput. Phys.Commun. , 107511 (2020).[35] L. Spitzer, Astrophys. J. , 299 (1952).[36] J. B. Taylor, Phys. Fluids , 1142 (1961).[37] M. Krishnan, Phys. Fluids , 2676 (1983).[38] D. A. Dolgolenko and Yu. A. Muromkin, Phys.-Usp. ,
994 (2017).[39] E. J. Kolmes, I. E. Ochs, and N. J. Fisch, Phys. Plasmas , 032508 (2018).[40] E. J. Kolmes, I. E. Ochs, M. E. Mlodik, and N. J. Fisch,Phys. Lett. A , 126262 (2020).[41] L. Chen and F. Zonca, Rev. Mod. Phys. , 015008(2016).[42] R. J. Taylor, M. L. Brown, B. D. Fried, H. Grote, J. R.Liberati, G. J. Morales, P. Pribyl, D. Darrow, andM. Ono, Phys. Rev. Lett. , 2365 (1989).[43] J. E. Maggs, T. A. Carter, and R. J. Taylor, Phys. Plas-mas , 052507 (2007).[44] A. J. Fetterman and N. J. Fisch, Phys. Rev. Lett. ,205003 (2008).[45] A. J. Fetterman and N. J. Fisch, Phys. Plasmas ,042112 (2010). [46] A. J. Fetterman and N. J. Fisch, Phys. Plasmas ,055704 (2011).[47] B. Scheiner and S. D. Baalrud, Phys. Rev. E , 063202(2020).[48] C. G. Gimblett, Europhys. Lett. , 541 (1990).[49] K. Sasaki, P. H¨orling, T. Fall, J. H. Brzozowski, P. Brun-sell, S. Hokin, E. Tennfors, J. Sallander, J. R. Drake,N. Inoue, J. Morikawa, Y. Ogawa, and Z. Yoshida,Plasma Phys. Control. Fusion , 333 (1997).[50] M. G. Haines, P. D. LePell, C. A. Coverdale, B. Jones,C. Deeney, and J. P. Apruzese, Phys. Rev. Lett. ,075003 (2006).[51] G. Kagan and X.-Z. Tang, Phys. Plasmas , 082709(2012).[52] G. Kagan and X.-Z. Tang, Phys. Plasmas21