Generalized Impurity Pinch in Partially Magnetized Multi-Ion Plasma
aa r X i v : . [ phy s i c s . p l a s m - ph ] F e b Generalized Impurity Pinch in Partially Magnetized Multi-Ion Plasma
M. E. Mlodik, a) E. J. Kolmes, I. E. Ochs, and N. J. Fisch Department of Astrophysical Sciences, Princeton University, Princeton, New Jersey, USA,08543 andPrinceton Plasma Physics Laboratory, Princeton, New Jersey, USA, 08540 (Dated: 8 February 2021)
In a two-ion-species plasma with disparate ion masses, heavy ions tend to concentrate in the low-temperature regionof collisionally magnetized plasma and in the high-temperature region of collisionally unmagnetized plasma, respec-tively. Moreover, collisional magnetization can be determined as the ratio of the light ion gyrofrequency to the collisionfrequency of light and heavy ion species, and the behavior of this effect in the intermediate regime of partially mag-netized plasma is predominantly dependent on this Hall parameter. Multi-ion cross-field transport has been describedbefore in the collisionally magnetized plasma regime, and generalized pinch relations, which describe densities of ionspecies in equilibrium in that plasma, are found in the literature. In this paper, the role of collisional magnetization andLarmor magnetization in multi-ion collisional transport is clarified and generalized pinch relations are extended to thepartially magnetized regime, in which the ion Hall parameter may be small, as long as electrons remain collisionallymagnetized. Equilibrium ion density profiles have the same dependence on external forces and on each other regardlessof collisional magnetization of ions. The expansion of the range of validity of multi-ion collisional transport modelsmakes them applicable to a wider range of laboratory plasma conditions. In particular, ion density profiles evolve suffi-ciently fast for radial impurity transport to be observable around stagnation on MagLIF, leading to expulsion of heavyion impurities from the hotspot as long as plasma becomes sufficiently collisionally magnetized during the implosion.
I. INTRODUCTION.
In fusion devices, it is typically advantageous to concen-trate fuel ions in the hot core of the plasma and to flush outimpurities and fusion products . This is true for a broadrange of devices. For instance, a particular device where im-purity transport is of interest is MagLIF , a magnetizedZ-pinch device in Sandia. MagLIF features a cylinder of deu-terium plasma, which is premagnetized by applying an ax-ial B field, heated in the center by a laser and compressedby applying a large current to the beryllium liner which en-velops the fuel in order to reach fusion conditions. As such,there has been significant experimental effort to identify impu-rity mix properties , as well as technological developmentssuch as preheat protocols to decrease the amount of impurityintroduced to the fuel hotspot . Nevertheless, multi-iontransport effects, i.e. impurity mix dynamics, have been over-looked in the currently available modeling of MagLIF implo-sions. Therefore, a relatively simple model of impurity dy-namics is topical.There is ample literature on similar plasmas, although theyare different in at least one qualifier. Equlibrium in multi-ionmagnetized plasmas in the absence of temperature gradientsbut in the presence of external forces was found by Kolmeset al . Multi-ion plasma transport in the presence of tem-perature gradients, but in a very strong magnetic field (suchthat Hall parameters are much greater than 1) was studied byOchs and Fisch . More recently, the transport code MITNShas been developed to study evolution of multi-ion transportin collisionally magnetized plasmas. Transport in single-ion-species plasma with arbitrary Hall parameter of electrons wasstudied by Velikovich, Giuliani, and Zalesak . Multi-ion a) Electronic mail: [email protected] species transport in unmagnetized plasmas was studied by Ka-gan and Tang , as well as by Zhdanov and Alievskii .Some progress was also made in a study of multi-ion-speciestransport in a magnetized plasma in the presence of temper-ature gradients was made by Vekshtein et al , although theauthors did not have external forces and they had a specificconfiguration in mind, which is different from this paper. Theexpansion of the range of applicability of muti-ion collisionaltransport models is also useful for plasma mass filters, whichare designed to separate the components of a plasma accord-ing to mass for a variety of applications .This paper provides a simplified model to describe multi-ion transport on devices like MagLIF. In particular, it includesconditions on ion densities in equilibrium in partially magne-tized (large Hall parameter electrons, arbitrarily Hall parame-ter ions) multi-ion species plasma subjected to external forcesor a temperature gradient, and clarifies what dimensionlessparameters have the largest impact on multi-ion cross-fieldtransport. Also, it suggests that collisional cross-field multi-ion transport is sufficiently fast to expel impurities from thehotspot during late stages of implosion. The effect is moreprofound if fuel plasma has collisionally magnetized ions, i.e.if the light ion gyrofrequency is much larger than the collisionfrequency between light and heavy ions.The question of interest in this paper is how differentspecies adjust to a change, such as change of temperature onthe boundary (laser preheat after fuel magnetization) or thechange of external potential (corresponding to Z-pinch com-pression and/or rotation). The problem is to find the forceequilibrium state after that change happened, as well as to es-timate how quickly ions adjust to that change relative to eachother.The paper is organized as follows. In Sec. II, differentregimes of multi-ion cross-field transport with respect to thestrength of the magnetic field, and, in particular, the Hallparameter, are clarified. In Sec. III, generalized pinch rela-tions, which relate ion densities in the force equilibrium, arederived in presence of both temperature gradients and exter-nal forces, such as the centrifugal force, in a two-ion-speciesplasma which has arbitrarily magnetized ions with large massdisparity m a ≪ m b as long as the Hall parameter of elec-trons is large, i.e. Ω e / ν ei ≫
1. Generalized pinch condi-tions turn out to depend predominantly on the Hall parame-ter Ω a / ν ab . In Sec. IV, the multi-ion collisional cross-fieldtransport timescale τ ab , eq is derived in the low- β limit. Themulti-ion cross-field transport timescale turns out to be suffi-ciently fast for impurity expulsion to occur at stagnation andat the later stages of implosion for MagLIF-relevant plasmaparameters, even though it is much slower at the earlier stagesof implosion. In Sec. V there is a summary of these results. InSec. VI there is a discussion of some other potential applica-tions of these results. The procedure to find relevant transportcoefficients is outlined in Appendix A. II. PLASMA MAGNETIZATION.
Consider a plasma slab in a homogeneous magnetic field B = B ˆ z with species-dependent external potential Φ s ( y , t ) with F s = − ∇Φ s (and all other gradients) in the ˆ y direction. Thisplasma can be described by a multiple-fluid model . Thefluid momentum equation for species s is m s d u s dt = q s E + q s u s × B − ∇ p s n s − ∇ · π s n s + ∑ s ′ R ss ′ n s + F s . (1)Here u s is the flow velocity, π s is the traceless part of thepressure tensor of species s , and R ss ′ = R uss ′ + R Tss ′ is the fric-tion force, comprised of flow friction and Nernst (“thermal")friction, between species s and s ′ . In the limit where m s ≪ m s ′ and the Hall parameter of the light species is large, R Tss ′ = n s ν ss ′ ˆ b × ∇ T s / Ω s . The magnetic field enters themomentum equation in two ways: explicitly in the Lorentzforce and implicitly in transport coefficients. The magneti-zation can be understood in two ways: in the ρ s / L sense, asthe smallness of the gyroradius compared to the characteris-tic length scale of perpendicular dynamics, and in the Ω s / ν s sense, as a ratio of gyrofrequency to collision frequency. Plas-mas which exhibit these two types of magnetization can becalled Larmor magnetized ( ρ s / L ≪
1) and collisionally mag-netized ( Ω s / ν s ≫ Ω s / ν s ∼ partially magnetized . In principle,there can be a few different choices of a particular collisionfrequency or a gyrofrequency, but later in this paper it is foundthat in the case of light ion species a and heavy ion species b the relevant Hall parameter is Ω a / ν ab . In order to see howmagnetization impacts Eq. (1), compare the inertia term andthe pressure term: | m s ( u s · ∇ ) u s || q s u s × B | ∼ u s / Lq s u s B / m s ∼ u s Ω s L ∼ u s u th , s ρ s L . (2)The ratio above shows the velocity response of plasma speciesto an external force: in ρ s / L ≪ ( a ) BF d u / dt ( b ) BFu ( c ) BFu
FIG. 1. Classification of the response of species s to an externalforce F that is applied to a multi-ion-species plasma in a magneticfield. From left to right, in order of increase of the magnetic field:(a) Larmor and collisionally unmagnetized plasma ρ s / L ≫
1; (b)Larmor magnetized, but collisionally unmagnetized plasma ρ s / L ≪ ,/ Ω s / ν ss ′ ≪
1; (c) Larmor and collisionally magnetized plasma ρ s / L ≪ , Ω s / ν ss ′ ≫
1. Partially magnetized plasma is an interme-diate regime between (b) and (c). portional to the external force, while in ρ s / L ≫ F × B driftin the direction perpendicular to the external force F , and thedrift velocity is such that the Lorentz force and the force F arebalanced. In collisionally unmagnetized plasma (Fig. 1b) theion-ion friction force is much larger than the Lorentz force.Therefore, the external force F is balanced by the ion-ion fric-tion force. Since the direction of the ion-ion friction force andthe difference in the ion flow velocities are close to each other,the difference in the ion flow velocities is pointing in the sim-ilar direction to the external force F . The Lorentz force andthe ion-ion friction force are comparable in partially magne-tized plasma. In many systems of interest, such as Z-pinches,plasma is Larmor magnetized but collisionally unmagnetized.Similarly, timescales are such that 1 / Ω s · ∂ / ∂ t ≪ u s × ˆ b = − E B + Ω s d u s dt + ∇ p s m s n s Ω s + ∇ · π s m s n s Ω s − ∑ s ′ R ss ′ m s n s Ω s − F s q s B . (3)In order to have the closure of Eq. (3), expressions for π s and R ss ′ are needed. Note, however, that the viscosity π s is smallcompared to other terms so it affects only long-term behavior.As such, in many cases it can be ignored. As far as the frictionforce R ss ′ goes, it can be found from the expression R ss ′ = Z d u m s u C ss ′ ( f s , f s ′ ) . (4)Here C ss ′ ( f s , f s ′ ) is a collision operator which describes colli-sions between species s and s ′ . Note that the friction forcedepends on the distribution functions f s . In the case of ε = ρ L , i / L being small, a perturbative expansion of distribu-tion functions in powers of ε can be performed around non-perturbed Maxwellian with zero mean velocity, f s = f s + f s + ... , as long as the plasma is sufficiently collisional toenforce f s ≪ f s . Then f s satisfies the following equation : ∑ s ′ (cid:2) C ss ′ ( f s , f s ′ ) + C ss ′ ( f s , f s ′ ) (cid:3) + Ω s ∂ f s ∂ξ = u · "(cid:18) ∇ p s p s − q s E T s − F s T s (cid:19) + u u th , s − ! ∇ T s T s f s . (5)Here ξ is the gyrophase. The friction force, inertia term, andviscosity do not enter the right-hand side of Eq. (5) as they areordered down as O ( ε ) , O ( ε ) , O ( ε ) , respectively.Note that f s depends on collisions with all other speciesand on same-species collisions too, as long as collisions aresufficiently frequent compared to Larmor gyration. Therefore,if there are multiple unmagnetized or partially magnetized ionspecies, the friction force R ss ′ depends not only on the behav-ior of species s and s ′ , but on the behavior of all species. In themagnetized case, Eq. (5) is solved by Hinton . The unmag-netized case is solved by Zhdanov and Alievskii , and ithas been successfully applied by Kagan and Tang . In thepartially magnetized case, however, it is a challenging task tofind even a closed form expression for friction force. Never-theless, simplifications can be made if: (1) there are two ionspecies with disparate masses or (2) all ion species except forone are in trace quantities. The aim of this paper is to elucidatecase 1.In the case of two ion species with disparate masses m a ≪ m b , the expression for the friction force can be derived in thefollowing way, as long as electrons are collisionally magne-tized, i.e. Ω e / ν ei ≫
1. The distortion of the light ion dis-tribution function f a is much larger than the distortion of f b .As such, magnetization is determined entirely by the light ionspecies. The most general form of the components of the fric-tion force between light and heavy species which are perpen-dicular to magnetic field is R ba = − m b n b ν ba (cid:2) α ⊥ , ba ( u b − u a ) + α ∧ , ba ( u b − u a ) × ˆ b (cid:3) + β ⊥ , ba n a ∇ ⊥ T a − β ∧ , ba n a ˆ b × ∇ T a . (6)Curiously, in the case of large mass disparity between ions,transport coefficients for ion-ion friction (Eq. (6)) in two-ion-species plasma are the same as transport coefficients forelectron-ion friction in single-ion-species plasma up to thefollowing substitution: Ω e / ν ei → Ω a / ν ab , e Z → n b Z b / ( n a Z a ) (see derivation in Appendix A). Coefficients for the electron-ion friction force in the case of arbitrary magnetization havebeen found by Epperlein and Haines . In the limit of colli-sionally magnetized plasma (suppressing indices ba attached B c de u b FIG. 2. Single-particle picture of the origin of non-collinearity of theion-ion friction force and their flow velocity difference in partiallymagnetized plasma. Consider two particles of species a moving withthe opposite velocities and starting at the same point c . The par-ticle which has the velocity in the same direction as flow velocityof species b (blue trajectory) has smaller relative velocity differencethan the particle which has the velocity in the opposite direction (redtrajectory). As collision frequency in plasma decreases dramaticallywith the increase of relative velocity difference, the particle on theblue trajectory is going to collide much faster (at point d ) than theparticle on the red trajectory (at point e ). Therefore, the arc ce ismuch larger than the arc cd . to the transport coefficients), Ω a / ν ab → ∞ : β ⊥ / β ∧ → β ∧ → / · ν ab / Ω a , α ⊥ →
1. In the limit of collisionally un-magnetized plasma, Ω a / ν ab → β ∧ / β ⊥ → β ⊥ → const , α ⊥ → const . α ∧ / α ⊥ goes to 0 in both limits, and never ex-ceeds 0 . α ⊥ , ba ( u b − u a ) × ˆ b in Eq. (6)). Therefore, it is instructive toprovide an intuitive explanation of why this component ex-ists in the first place. In the single-particle picture, its origincan be attributed to the following observation. In plasma withion Hall parameter ∼
1, ions’ trajectories are arcs of a Larmororbit, interrupted by collisions. The length of these arcs is in-versely proportional to the collision frequency. Therefore, ifthere are more collisions on the one side of the orbit than an-other, there is going to be net motion in the perpendicular di-rection. In comparison, in collisionally magnetized plasma aparticle makes many gyrations between successive collisions,so the effect of collisions on the arc length is averaged out. Incollisionally unmagnetized plasma, particles’ trajectories areclose to straight lines, so there is almost no preference in themotion perpendicular to the flow velocity difference. There-fore, this component of the friction force is significant only inpartially magnetized plasma regime.Another way to get an intuition about the role of collisionalmagnetization is to look at the direction of the distortion ofdistribution function f s using Eq. (5) in a plasma where thereis an imbalance of ˆ x -direction components of flow velocitiesof two species. To see that, consider a case of light species a and heavy species b . Then the distribution function of theheavy species f b is relatively unaffected by the species a .Moreover, the spread of f b is much smaller than the spread ofthe distribution function of light species f a . Therefore, lightspecies essentially see the delta-function distribution of heavyspecies. Collisions with heavy species essentially provide aforce that pushes light species distribution toward the meanvelocity of heavy species. Moreover, it also means that thebehavior of f b can be ignored and all the essential physicsare concentrated in the distortion of the distribution functionof the light species f a .In collisionally magnetized plasma, the reaction of f a isgoing to be in ˆ y direction, perpendicular both to ˆ x , the direc-tion of the flow velocity imbalance and to ˆ z , the direction ofthe magnetic field. Then this ˆ y -directed distortion providesˆ x -directed Lorentz force, balancing the unlike-species frictionforce.In collisionally unmagnetized plasma, like-species colli-sions push the distribution function of the light species back toMaxwellian, and the Lorentz force is comparatively too weak.Therefore, f a is in the ˆ x -direction.In partially magnetized plasma, both mechanisms of push-ing the distribution function of the light species back toMaxwellian are equally important. Therefore, f a is stretchedout in both ˆ x and ˆ y directions. The stretching of f a in both di-rections creates an asymmetry in average relative velocity dif-ference between particles moving in the positive and the neg-ative ˆ y -direction, which in turn results in the asymmetry of thecollisional drag. Summed over all the light-species particles,this is seen as the component of the friction force perpendicu-lar to the direction of flow imbalance. Note that both the pres-ence of magnetic field (creating a component of distortion in ˆ y direction) and the collision frequency dependence on relativevelocity is essential for this component of the friction forceto appear. Also, in the collisionally magnetized plasma casethe distortion of f a in ˆ y -direction is much smaller than in thepartially magnetized plasma case, so the α ⊥ , ba ( u b − u a ) × ˆ b component of the friction force is much smaller, too. III. EQUILIBRIUM AND GENERALIZED PINCHRELATIONS.
Consider a magnetized multiple-ion species slab of plasmawhich is subjected to a slow ( ∂ / ∂ t ≪ ν ss ′ , Ω s ) change in theapplied potential or temperature in the direction perpendicularto the magnetic field. Then cross-field dynamics are governedby collisional transport.On the fast transport timescales (compared to the vis-cous transport timescale) momentum equation for species s (Eq. (1)) has the following form in the equilibrium:0 = q s E + q s u s × B − ∇ p s n s + ∑ s ′ R ss ′ n s + F s . (7)Suppose all the gradients are in the x -direction and the mag-netic field is in the z -direction. Then in equilibrium, u sx = s and Eq. (7) becomes0 = q s E y + ∑ s ′ R ss ′ , y n s . (8)Summing Eq. (8) over all species, the condition that E y = E field in the equilibrium). Therefore,there are s equilibrium conditions: ∑ s R ss ′ , y =
0. These equi-librium conditions can be shown to provide relations between densities of plasma constituents, which can be called gener-alized pinch relations . At much longer timescales, viscosityplays a role, and plasma evolves to the state of thermodynamicequilibrium.
A. Transitivity of Generalized Pinch Relations.
In general, the force equilibrium conditions are ∑ s ′ R ss ′ , y =
0. However, in two important cases: (a) no more than threespecies or (b) uniform temperature ∇ T =
0, they can be re-placed with pairwise friction force cancellation R ss ′ , y =
0. Ifthere are three or fewer species, the ∑ s ′ R ss ′ , y = R ss ′ , y = − R s ′ s , y provides a sufficient numberof constraints to uniquely determine R ss ′ , y , and R ss ′ , y = R ss ′ , y = R uss ′ , y ∝ u s ′ y − u sy , so in equilibrium all flow velocities u s arethe same, and therefore R ss ′ , y = N species in force equilibrium, the zero-friction-force conditions will provide N − β ; seeSec. IV for more details.In plasma of arbitrary magnetization without temperaturegradients, the maximum-entropy approach gives the follow-ing result: (cid:18) n a e Φ a / T (cid:19) / Z a ∝ (cid:18) n b e Φ b / T (cid:19) / Z b . (9)In collisionally magnetized plasma this condition is equiva-lent to setting the friction force between each pair of speciesto zero, see Ref. . Also, in a collisionally magne-tized plasma there are results about pinch relations with tem-perature gradients . However, this is the first time thatEq. (9) has been derived for plasma that is not in the collision-ally magnetized regime. In fact, the analysis in this sectionprovides the first new, nontrivial test of the maximum-entropyderivation of Eq. (9) since that approach was presented inRef. . A partially magnetized plasma fulfills all of the re-quirements given in Ref. for a system to satisfy Eq. (9), andwe have shown here (independently of any entropy consider-ations) that indeed Eq. (9) is satisfied by the equilibrium forthis system.Transitivity is a convenient property of generalized pinchrelations, as it makes the equilibrium state of any pair ofspecies independent of the properties of the other specieswhich comprise the plasma. B. Generalized Pinch Relations in Two-Ion-Species Plasma.
In the presence of temperature gradients, generalized pinchrelations become less tractable as the expression for the fric-tion force becomes complicated and the maximum-entropyprinciple is no longer directly applicable. However, in thetwo-ion-species plasma case, equilibrium requires pairwisefriction force cancellation: R ab , y = R ae , y = R be , y = , there is sep-aration of timescales in collisional transport. In particular,in collisionally magnetized plasma ions equilibrate betweeneach other faster than with electrons. Such fast transporttimescale equilibrium (in which generalized pinch relationsare satisfied for a subset of species, but not for all of them)will have R ab , y =
0, but not necessarily R ae , y = R be , y =
0. Thequestion of transport timescales is addressed in more detail inSec. IV and V.In the case of two ion species with disparate masses m a ≪ m b and collisionally magnetized electrons (i.e. Ω e / ν ei ≫ R ab , y = u ax = u bx = − m b n b ν ba α ⊥ , ba (cid:0) u by − u ay (cid:1) − β ∧ , ba n a T ′ a = . (10)Here and later in the paper, f ′ = ∂ f / ∂ x for any physical quan-tity f . Define β ⋆, ba = β ⊥ , ba + β ∧ , ba α ∧ , ba α ⊥ , ba . (11)Then Eq. (10) can be used to find that in equilibrium R ba , x = β ⋆, ba n a T ′ a . (12)Note that in collisionally magnetized plasma the friction forcedecays quickly ( ∝ ( Ω a / ν ab ) − / , see Ref. and Appendix A).Since electrons are substantially more magnetized than ions,electron-ion friction force is much smaller than ion-ion fric-tion force in equilibrium as long as electrons are collisionallymagnetized.The momentum equation in x -direction is0 = q s E x + q s u sy B − p ′ s n s + ∑ s ′ R ss ′ , x n s + F sx . (13)Adopting Ω s = q s B / m s , F ⋆ s = F s − p ′ s / n s , u sy = − E x B − F ⋆ s m s Ω s − ∑ s ′ R ss ′ , x m s n s Ω s . (14)Combined with Eq. (10): − F ⋆ b m b Ω b − R ba , x + R be , x m b n b Ω b + F ⋆ a m a Ω a + R ab , x + R ae , x m a n a Ω a + β ∧ , ba n a T ′ a m b n b ν ba α ⊥ , ba = . (15)Together with m a n a ν ab = m b n b ν ba and the fact that electron-ion friction force can be ignored as long as Ω e / ν ei ≫ F ⋆ a m a Ω a + β ∧ , ba Ω a α ⊥ , ba ν ab T ′ a m a Ω a − R ba , x m a n a Ω a (cid:18) + Z a n a Z b n b (cid:19) = F ⋆ b m b Ω b . (16) ( a ) B ∇ T R Tab u a u b a b ( b ) B ∇ T R Tab u a u b a b ( c ) B ∇ T R Tab u a u b a b FIG. 3. Thermal force and impurity pinch. In collisionally unmag-netized plasma (a), where ion Hall parameter is small Ω a / ν ab ≪ Ω a / ν ab ≫
1, thermal force is perpendicularto the direction of temperature gradient. However, ion flow velocityis perpendicular to the thermal force in this case. As a result, heavyions tend to concentrate in a colder region of plasma. In partiallymagnetized plasma (b) , where Ω a / ν ab ∼
1, both thermal force andflow velocity are not collinear to the temperature gradient or to eachother. In all cases, collisional cross-field transport due to temperaturegradient conserves charge locally, i.e. Z a u a + Z b u b = Eq. (16) can be rewritten as1 Z b (cid:18) p ′ b n b − F bx (cid:19) = Z a (cid:18) p ′ a n a − F ax + λ T ′ a (cid:19) , (17)where λ = β ⋆, ba (cid:18) + Z a n a Z b n b (cid:19) − β ∧ , ba Ω a α ⊥ , ba ν ab (18)is the temperature screening coefficient. λ describes the ef-fect of the thermal force on the equilibrium density profiles. Incollisionally unmagnetized plasma, λ is positive, while in col-lisionally magnetized plasma λ is negative. The way in whichthermal force affects multi-ion transport in partially magne-tized plasma is shown in Fig. 3.Eq. (17) can be called the generalized pinch relation in par-tially magnetized plasma . It provides a constraint on densitiesof ion species a and b in force equilibrium, as long as m a ≪ m b and Ω e / ν ei ≫
1. The generalized pinch relation makes it pos-sible to analyze characteristics of impurity transport in plasmawith arbitrary collisional magnetization of ions.In limiting cases, Eq. (17) reduces to the forms known inthe literature. In collisionally magnetized plasma, λ = − / )1 Z b (cid:18) p ′ b n b − F bx (cid:19) = Z a (cid:18) p ′ a n a − F ax − T ′ a (cid:19) . (19)In isothermal plasma it is reduced to Eq. (9), as could beexpected from the maximum-entropy principle .The collisionally unmagnetized plasma limit of Eq. (17)can also be found ( β ⊥ , ba > Z b (cid:18) p ′ b n b − F bx (cid:19) = Z a (cid:18) p ′ a n a − F ax + β ⊥ , ba (cid:18) + Z a n a Z b n b (cid:19) T ′ a (cid:19) . (20)In order to find out the dependence of the heavy species’density profile on temperature and the meaning of the temper-ature screening coefficient λ , consider the case of no externalforces acting on the plasma: F a = F b =
0. Then Eq. (17) be-comes 1 Z b p ′ b n b = Z a (cid:18) p ′ a n a + λ T ′ a (cid:19) , (21)It can be reorganized to see the dependence on density andtemperature gradients separately (using equation of state p s = n s T s ): n ′ b n b = Z b Z a T a T b n ′ a n a + ( λ + ) Z b T a Z a T b T ′ a T a − T ′ b T b . (22)If species a and b have the same temperature T a = T b = T ,Eq. (22) can be simplified further. n ′ b n b = Z b Z a n ′ a n a + (cid:20) ( λ + ) Z b Z a − (cid:21) T ′ T . (23)It is clear from Eq. (23) that the density gradient of light ionspecies a and the temperature gradient act provide two inde-pendent contributions to the density gradient of heavy species b . As follows from Eq. (23), heavy species are expelled fromhigh- T regions when plasma is collisionally magnetized and λ → − /
2. However, they are concentrated in high- T regionsinstead when plasma is collisionally unmagnetized. Eq. (18)determines the boundary between these qualitatively differenttypes of plasma behavior. Note, however, that in general thesign of the effect changes at the point λ = Z a / Z b − λ =
0, as the temperature gradient is also presentin p ′ s terms, while λ describes only the contribution of thethermal force. It turns out that the temperature screening co-efficient λ depends predominantly on the ion Hall parameter Ω a / ν ab , since transport coefficients in Eq. (18) vary signifi-cantly with the change of Ω a / ν ab and comparatively weaklywith the Z e f f = n b Z b / ( n a Z a ) (see Ref. and Appendix A fordetails). Therefore, Ω a / ν ab is the criterion to determine colli-sional magnetization of plasma.Note that the dependence of heavy species’ density on lightspecies’ density and on external forces, such as the centrifugalforce, is the same regardless of whether plasma is collisionallymagnetized or not. Aside from that, this section provides anextensive overview of the impact of collisional magnetizationon temperature gradient-driven multi-ion transport. IV. TIMESCALES IN PARTIALLY MAGNETIZEDPLASMA.
Sec. III describes the equilibrium state of multi-ion-speciesplasma. However, it is of significant interest whether theknowledge of relationship between ion density profiles canalso be extended to laboratory plasmas which are dynamicallyevolving. In this section, there is an estimate of the timescaleof relative relaxation of ion density profiles toward the equilib-rium, as well as an example of laboratory plasma where thistimescale is sufficiently fast to make the multi-ion transporteffects observable.
A. Timescale of Ion-Ion Force Equilibration.
In general, as plasma evolves due to cross-field transport,all densities and temperatures are subject to change. There-fore, it is not easy to describe the resulting evolution ofplasma. However, in the special case of low- β isothermalplasma ( T s = T = const across the plasma for all species s ; β ≪ F s L ≪ T for all species s , where L is a lengthscale of plasma), it is possible to track the plasma behavior asa function of time. In this particular case, Eq. (17) prescribesthat densities are close to being uniform, i.e. | ∇ n s | / n s ≪ / L .Then, in order to linearize the momentum and continuityequations, assume n s = n s + e n s , u s = u s + e u s , where n s and u s are the equilibrium density and flow velocity of species s ,respectively. Moreover, ∇ n s / n s ≈ ∇ e n s / n s . The friction forcecan also be linearized in the same way: R ss ′ = R ss ′ + e R ss ′ .Then in partially magnetized plasma Eq. (3) becomes e u s × ˆ b = T ∇ e n s m s n s Ω s − ∑ s ′ e R ss ′ m s n s Ω s . (24)Continuity equation: ∂ e n s ∂ t + ∇ · ( n s e u s ) = . (25)The expression for the perturbation of the friction force be-tween light species a and heavy species b can be inferred fromEq. (6): e R ba = − m b n b ν ba (cid:2) α ⊥ , ba ( e u b − e u a ) + α ∧ , ba ( e u b − e u a ) × ˆ b (cid:3) . (26)Define transport timescale τ ab , eq as a characteristic timescaleof plasma solely due to friction between species a and b , ig-noring interactions with other species. Then Eqs. (24) and(25) can be combined to get the fact that frictional transport isambipolar: Z a e n a + Z b e n b = . (27)Moreover, Eqs. (24) and (26) can be combined (omitting index ab in α ⊥ and α ∧ ) as Tm a n a Ω a ∇ e n a − Tm b n b Ω b ∇ e n b = − ( e u b − e u a ) × ˆ b + (cid:18) ν ab Ω a + ν ba Ω b (cid:19) (cid:0) α ⊥ ( e u b − e u a ) + α ∧ ( e u b − e u a ) × ˆ b (cid:1) . (28)If all the gradients are in one direction, these equations can becombined to get a diffusion equation ∂ e n a ∂ t + ∇ · ( D ab ∇ e n a ) = , (29)where the diffusion coefficient is D ab = α ⊥ (cid:0) ν ab ρ a + ν ba ρ b (cid:1)h − α ∧ (cid:16) ν ab Ω a + ν ba Ω b (cid:17)i + α ⊥ (cid:16) ν ab Ω a + ν ba Ω b (cid:17) . (30)Here ρ s = T / ( m s Ω s ) is the gyroradius of species s . Inthe limit of collisionally magnetized plasma ( Ω a / ν ab ≫ , Ω b / ν ba ≫ D ab = ν ab ρ a + ν ba ρ b . (31)In the limit of collisionally unmagnetized plasma, α ∧ ≪ α ⊥ ,so the diffusion coefficient is similar to what could be ex-pected ( D ∼ νλ m f p ): D ab = ν ab ρ a + ν ba ρ b α ⊥ (cid:16) ν ab Ω a + ν ba Ω b (cid:17) . (32)In the limit of trace heavy impurities, n b → Ω a / ν ab → ∞ ,so the diffusion coefficient attains limit value D ab = ν ba ν ba + Ω b Tm b . (33)Eq. (29) can be solved by spectral decomposition. In acylinder the timescale of equilibration of the lowest mode dueto radial ion-ion transport is τ ab , eq = j , r D ab . (34)Here 1 / j , = . ... is a geometric factor (the squareof the first zero of the first-order Bessel function); this geo-metric factor should be 1 / π in slab geometry. In the limit ofcollisionally magnetized plasma ( Ω a / ν ab ≫ , Ω b / ν ba ≫ up to a geometric factor.Eq. (34) has been found in the limit of low- β isothermalplasma. In high- β plasma, by analogy to neutral gas ,there can be other ways in which plasma evolves to the equi-librium, such as magnetosonic waves. Another way to intro-duce complications to the dynamics would be to include evo-lution of B and T . Nevertheless, Eq. (34) provides an ideaabout how long it takes for densities of ion species to adjustrelative to each other once some change has been applied tothe system. Another feature to note in Eqs. (30) and (34) isthat τ ab , eq approaches a constant value in the limit of traceimpurity species b .Another way to interpret transport timescale τ ss ′ , eq is thatit is the timescale of relaxation of the friction force R ss ′ to-ward zero, after which generalized pinch condition Eq. (17)for species s , s ′ can be used. If one of the species s , s ′ is elec-trons, temperature screening coefficient is λ = − / B. Timescales of Z-pinch Implosion Around Stagnation.
One important area of applicability of generalized pinchrelations is impurity transport in magnetized Z-pinch experi-ments, such as MagLIF, where plasma is Larmor magnetized,but not necessarily collisionally magnetized. MagLIF implo-sions start with magnetization and preheat of fuel, forming ahotter and less dense region of plasma in the center, called the log ( r / r ) log ( τ ab , eq ) ν ab / Ω a + ν ba / Ω b ∼ τ ab , eq = τ imp ∼ ( r / r ) − / ∼ ( r / r ) / FIG. 4. Evolution of radial multi-ion transport timescale in a typicalmagnetized Z-pinch implosion. Initially, plasma is collisionally un-magnetized and radial particle transport is too slow compared to theduration of the implosion. Then collisional transport becomes muchfaster, until sum of ion Hall parameters becomes ∼
1. Afterward,transport timescale starts to increase. Increase of Hall parameter dur-ing the implosion also means that impurities tend to be expelled fromthe hotspot more and more as Z-pinch approaches stagnation phase. hotspot. Then the implosion itself happens, compressing thefuel until stagnation.Suppose for simplicity that the implosion is a radial metriccompression with convergence ratio C = r / r of a cylindricalplasma column which has adiabatic index γ . Then plasma pa-rameters in the fuel have the following scaling: n ∝ C , T ∝ n γ − ∝ C ( γ − ) , B ∝ C . Gyrofrequencies and collisional fre-quencies have Ω s ∝ C and ν ss ′ ∝ n / T / ∝ C − γ scaling, re-spectively. Dimensionless parameters that characterize mag-netization in various ways have the following scalings: β ∝ nT / B ∝ C ( γ − ) , Ω a / ν ab ∝ C ( γ − ) , ρ a / r ∝ T / B − r − ∝ C γ − . As such, the plasma becomes more collisionally mag-netized during the implosion. In the collisionally unmagne-tized plasma limit τ ab , eq ∝ r ν ab / v th ∝ C − ( γ − ) ; in the colli-sionally magnetized plasma limit τ ab , eq ∝ r / (cid:0) ν ab ρ b (cid:1) ∝ C γ − .Collisional radial transport of heavy impurities in a Z-pinchimplosion can be understood as the following (see also Fig. 4).Initially, even after the axial magnetic field is applied to thefuel, plasma is still collisionally unmagnetized ( Ω a / ν ab ≪ τ ab , eq is too slow compared to the im-plosion time for collisional multi-ion transport to have muchimpact. Then, as implosion progresses, τ ab , eq drops dramati-cally ( ∝ C − / if γ = / τ ab , eq starts toincrease.This analysis could be applied to a general class of mag-netized Z-pinch configurations. For example, assume plasmaparameters similar to the recent high-performance shot z3040on Z machine, as described in Ref. . In that shot, initially ρ R = . mg / cm , r = . mm , B = T . If the impu-rity concentration is n Be = . n D and the ion temperature is T i = eV , then ν D , Be / Ω D = . τ BeD , eq ≈ µ s , i.e.radial multi-ion collisional transport is too slow. It wouldbe even slower if impurity concentration is smaller or fuelis colder. Therefore, collisional radial transport cannot beexpected to be prominent in the beginning of the implosion.Also, Ω e / ν ei ≈ . BR = . MG · cm , r = µ m (which corresponds to B = . kT ),fuel density is ρ R = . g / cm , T i = . keV are considered,both ion Hall parameter and transport timescale are muchmore favorable. In particular, if there is n Be = . n D , then Ω D / ν D , Be = .
78 and τ BeD , eq = . ns . If n Be = . n D , then Ω D / ν D , Be = .
75 and τ BeD , eq = . ns . The multi-ion radialtransport timescale is comparable to the burn time ( ∼ ns )and is much smaller than the implosion timescale ( ∼ ns ).As such, significant impurity transport can be expected to oc-cur. Hall parameter for electrons is Ω e / ν ei ≈
28 in this case,well in the range of applicability of the transport model usedin this paper.This paper describes what happens to impurities when Ω e / ν ei ≫ m a ≪ m b . These conditions are satisfied inlater stages of implosion and around stagnation in the cur-rent experiments. However, this is precisely when impuritytransport matters the most. Moreover, the preliminary resultspresented here are promising: both density and temperaturegradients tend to expel impurities from the hotspot as long as Ω a / ν ab ≫
1, and transport timescale τ ab , eq is sufficiently shortfor this to happen. V. SUMMARY.
Multi-ion cross-field transport has been studied in the lit-erature in the limit of collisionally magnetized ions. There isalso literature on the role of collisional magnetization of elec-trons in the single-ion species plasma, as well as on multi-iontransport in unmagnetized plasma. However, the intermediateregime of partially magnetized plasma has largely not beenaddressed until now.This paper provides a rigorous classification of differentmagnetization regimes and their role in multi-ion transport.Moreover, it gives the conditions on density profiles in equi-librium in the case of partially magnetized plasma, thereby ex-tending multi-ion collisional transport models to a new regimeof applicability. Temperature gradient dependence of gen- eralized pinch relations allows to recover both the collision-ally magnetized limit, where heavy ions are expelled from thehigh- T region of plasma, and the collisionally unmagnetizedlimit, where heavy ions are drawn in the high- T region ofplasma. Moreover, in the case of light ion species a and heavyion species b the sign of this effect depends almost exclusivelyon the Hall parameter Ω a / ν ab , while like-species Hall param-eter Ω a / ν aa changes only the magnitude of the effect. Tosome extent, the results in this paper can be generalized to thecase of multiple heavy impurity species, as long as they are intrace quantities, such that their collision frequencies with oneanother are negligible compared to their collision frequencieswith bulk ions.The extension of the parameter space where multi-ion colli-sional transport is understood expands the range of applicabil-ity of transport models to more laboratory plasmas. In partic-ular, the results in this paper are promising in terms of under-standing impurity transport in Z-pinches, such as the MagLIFexperiment. More generally, multi-ion collisional transportleads to a significant radial expulsion of heavy ion impuritiesif two conditions are satisfied: (i) Hall parameter Ω a / ν ab & τ ab , eq beingcomparable to or faster than the length of the stagnation andlate implosion phases. VI. DISCUSSION.
In low- β collisionally magnetized plasma, there is separa-tion of timescales between ion-ion transport timescale τ ab , eq and electron-ion transport timescale τ ei , eq : τ ab , eq ≪ τ ei , eq .Therefore, plasma exhibits curious effects on τ ab , eq , such ascharge incompressibility and the heat pump effect . Moregenerally, in partially magnetized plasma, when electrons arestill collisionally magnetized, but ions are only partially mag-netized, the ratio of cross-field transport timescales is the fol-lowing: τ ei , eq τ ab , eq = D ab D ei . (35)Eqs. (30) and (35) can be combined to identify the parameterspace where there is separation of transport timescales. Inparticular, in the case of trace heavy ion species b and bulklight ion species a , timescales separation exists if (cid:18) ν ab Ω a + ν ba Ω b (cid:19) . ν ab ν ae . (36)Timescales separation between multi-ion transport andelectron-ion transport potentially makes it possible to observecharge incompressibility and the heat pump effect even in par-tially magnetized plasma.Even though magnetized Z-pinches are providing an excel-lent testbed for the application of the theory of multi-ion col-lisional transport in partially magnetized plasma, they do notexhaust the potential upside of the knowledge of heavy iondensity profiles. In particular, the tendency of heavy ions toconcentrate in the region of Ω a / ν ab ∼ . In Ref. , H ions were used on Omega ex-periment (unmagnetized) to increase the concentration of DTfuel in the hotspot and reduce demixing, as heavier ions tendto concentrate in high- T region of plasma in that case. How-ever, the prescription would be the opposite in the partiallymagnetized or collisionally magnetized plasma. Therefore, inthese plasmas the use of H ions would lead to the oppositeresult of fuel concentration in low- T region of plasma, andheavy ions should be used instead. The model presented inthis paper makes it possible to identify the threshold whenthis effect changes sign.A potential extension of this work is to include finite- β effects on the time-dependent evolution of multi-ion trans-port, which would allow more accurate quantitative predic-tions of the evolution of the density profiles, complementingthe knowledge about the multi-ion equilibrium in this paper.Another interesting extension would be to include the role ofparticle fluxes on the generalized pinch relations to the par-tially magnetized plasma regime. ACKNOWLEDGMENTS
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Appendix A: Expression for Ion-Ion Friction Force throughElectron-Ion Transport Coefficients.
One of the ways to simplify the calculation of density pro-files is to use the same expression for the ion-ion friction forceas for the electron-ion friction in a single-ion-species plasma.Consider plasma that has two ion species a and b , such that m a ≪ m b . R ab = Z d v m a u C ab ( f a , f b ) . (A1)The friction force R ab comes from distortion of distributionfunctions f a and f b . But in the m a ≪ m b case, species a does not see a distortion of f b because the thermal spreadof f b is much smaller than the spread of f a . Therefore,only distortion of f a contributes to the friction force. Sup-pose that the distortion of f a from Maxwellian distribution issmall, f a ≪ f a . Then f a satisfies the following relation (seeEq. (173) in Hinton , or Eq. (5)): C aa ( f a , f a ) + C ab ( f a , f b ) + C ae ( f a , f e ) + Ω a ∂ f a ∂ξ = u · "(cid:18) ∇ p a p a − q a E T a (cid:19) + u u th , a − ! ∇ T a T a f a . (A2)Here u th , a = p T a / m a . Since C ae ∝ ν ae , it is a small correc-tion relative to the other terms on the LHS. As long as C ae term can be ignored, C aa ( f a , f a ) + C ab ( f a , f b ) + Ω a ∂ f a ∂ξ = u · "(cid:18) ∇ p a p a − q a E T a (cid:19) + u u th , a − ! ∇ T a T a f a . (A3)Here C aa is a same-species collision operator, and C ab is acollision operator between light and heavy species (e.g. pitch-angle scattering operator). In comparison, in a single-ion-species plasma equation for f e is C ee ( f e , f e ) + C ei ( f e , f i ) + Ω e ∂ f e ∂ξ = u · "(cid:18) ∇ p e p e − q e E T e (cid:19) + u u th , e − ! ∇ T e T e f e . (A4)The RHS as a function of u / u th , s is the same up to indexchange e → a . Therefore, if the coefficients are such that allthe terms on the LHS of Eqs. A3 and A4 are proportional,distortion of distribution function satisfies the same equation(since operators C aa and C ee are similar, the same is true for C ab and C ei ). Then the solution for f a is also the same as thesolution for f e , or at least these solutions are proportional toeach other. Therefore, the corresponding friction forces R ab and R ei are also proportional to each other.Then C ab can be split into terms C ab and C ab that de-pend on f a and u b , respectively (using Eq. (121) in Ref. as the expression for C ab ), and collision operators and f a canbe rewritten in dimensionless form (adorned by tilde), moving C ab to the other side: ν aa e C aa ( e f a , e f a ) + ν ab e C ab ( e f a ) + Ω a ∂ e f a ∂ξ = u · "(cid:18) ∇ p a p a − q a E T a (cid:19) + u u th , a − ! ∇ T a T a − u th , a u ν ab u b u th , a f a . (A5)Therefore, if the ratios Ω a / ν ab and ν aa / ν ab are the same inthe two-ion-species case and in the electron-ion case, the so-lution e f ( u / u th , a ) is also the same, and the expression for thefriction force R ab is the same. As long as m a ≪ m b , ν ab / ν aa = √ n b Z b / ( n a Z a ) . If species a are electrons, and species b are ions, then Ω a / ν ab = ωτ , ν ab / ν aa = √ n b Z b / n a = √ Z b .Usually (e.g. in Braginskii or in Epperlein and Haines )transport coefficients are found in terms of ωτ and Z .Therefore, the expression for the ion-ion friction force be-tween light species a and heavy species b is the same asthe expression for the electron-ion friction force in single-ion-species plasma up to substitutions Ω a / ν ab → ωτ and e Z = ν ab / ( √ ν aa ) = n b Z b / ( n a Z a ))