Impurity outward particle flux from externally applied torque

In this work a term in the impurity particle flux expression, which arises from the externally applied angular momentum torque, is rederived and analyzed in detail. This contribution to the species particle flux is found to be directed outward for co-current injected torque in conditions pertinent to present devices, which could explain the increasing hollowness of light impurity density profiles observed experimentally as neutral beam injected power is increased. This result is obtained by revisiting the fluid framework to compute the particle flux of a generic ion species.


Introduction
Impurities are present in a fusion-grade plasma due to both interaction with the first wall and plasma-facing-components (intrinsic impurities) and by voluntary injection to protect the same components (seeded impurities).At the same time, the impurity content in the core plasma dilutes the main reactant fuel (deuterium+tritium for example), leading to a degraded fusion power output.
As such, predicting the core impurity density profiles is of importance to correctly evaluate the output fusion power.Moreover, the accumulation of impurities inside the plasma core could lead to a negative feedback loop in which the temperature decrease and the impurity peaks even more, thus causing radiative collapse of the plasma core [1].
Various works, as recently reviewed in [2], have reported a disagreement between the gyrokinetic predictions and the observed light impurity density profiles in conditions in which * Author to whom any correspondence should be addressed.
Original Content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence.Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. the impurity density profiles are observed to be hollow [3][4][5][6][7][8].Analyses of large databases of experimental impurity density profiles have identified a correlation between the observed hollowness of the impurity density profiles and high ion temperature logarithmic gradients and large toroidal rotation gradients [5,7,8], also correlating with large NBI heating.Although impact of roto-diffusion [9] was consistently included in the gyrokinetic modeling of the experimental data [10], this was found to be insufficient to reproduce the most hollow experimental cases [3,4].The observed disagreement has motivated research exploring the impact of the neoclassical equilibrium flows on the turbulent transport [11] as well as the impact of the presence of beam ions in the neoclassical and turbulent impurity transport [12].While in some conditions quantitative agreement can be obtained within uncertainties [12], most recent analyses from high accurate experimental measurements of impurity density profiles and inferred transport coefficients [8] have confirmed that the outward convection measured in neutral beam injection (NBI) dominated plasmas is not well captured by the gyrokinetic simulations, despite the inclusion of fast ions.In some instances, discrepancy is also observed in absence of NBI injection [13].
In this work we tackle in particular the observed discrepancy between predicted and observed impurity density profiles in regimes in which NBI power is applied.At ASDEX Upgrade (AUG), it is observed that the impurity profile displays an increasing hollowness (that is, decreasing density towards the magnetic axis) as the NBI power is increased [7,8,12].Interestingly, in non-axisymmetric geometry, a similar phenomenon is observed, ascribed to the modification of the background electric field [14].
Previous works have considered modifications to the species particle flux due to a direct effect of external sources of momentum [15][16][17][18] and a simplified model for the anomalous part of the stress tensor.The result found was that externally applied momentum injection could lead to a net outward impurity flux, under certain conditions.Application to the Princeton Large Torus (PLT) device in [16] shows that indeed the radial impurity flux can become outward at large injected neutral beam power.In this work, we employ a similar theoretical framework to compute the particle flux of an ion species, including the presence of an externally applied torque, e.g.angular momentum flux, and turbulence-driven terms.We analyze under which conditions an outward radial impurity particle flux arises for co-current injection.This additional flux contribution will be in general proportional to the applied torque (that is, the injected NBI power).
The structure of the present work is as follows: in section 2, the particle flux term for impurity species related to externally driven torque is re-derived.We also discuss the general framework to compute the ion species particle flux.In section 3, application to existing experimental observations at AUG is shown.Finally, in section 4 conclusions are drawn.

Torque-driven impurity particle flux
We start by considering the momentum conservation equation for a generic species j, at stationarity (∂ t = 0): where Π is the stress tensor, Z the charge, n the density, E, B the electric and magnetic fields, F the friction term, and T the externally injected force density.Note that this latter term appears only in the fast ion population balance, since the direct momentum injection arises from the fast neutrals ionization or charge exchange interactions.Thermal deuterium and impurities only indirectly receive momentum via collisional (frictional) exchange with the fast ions, whereas the left-hand-side stress tensor will be influenced by the total absorbed torque once summed over species.From now on we drop the index j, intending that we are solving for a generic ion species.Moreover, we intend to solve for non-majority species, that is the impurities.In case an expression is shown for the main ions (e.g.deuterium), it will be explicitly stated.
Since we want to project equation (1) in 3 different directions, we introduce a coordinate system tangent to the flux surfaces, employing the system of versors e y , e || , which are related to the poloidal and toroidal e θ , e ϕ in the following way: Note that at this level the fields are assumed to be all positive with respect to the coordinate system (r, θ, ϕ) or alternatively (r, y, ||).The explicit sign of the fields will be considered at the appropriate location.A consequence of these definition ( 2) is also the relation between y, ϕ and parallel direction as: b ϕ e || = b θ e y + e ϕ .This latter projection formula will be used often.
The radial particle flux is defined as: which is consistent with the particle continuity equation.
Note that the latter is not the usual parallel projection, which is taken using B • (. ..).Summing equation ( 5) over all species (including the fast ion population), all the charge-related and symmetric terms go away (in the limit of small Debye length/vacuum electrostatic energy), leaving: where the torque here is absorbed by all (thermal) species.At this point, it is now important to address the issue of torque distribution into the various species, and its balance, which will become important later.As such, we now express equation (5) for the fast ion population, ignoring both the lefthand-side stress tensor contribution and the first right-handside electric field term (justified by considering that Larmorscale turbulence effects weakly influence fast ions [19,20]): This relation is the basic expression of the absorbed torque being composed of the JxB contribution (radial fast ions current) [21], and the collisional terms (onto deuterium D and impurity Z).Since we have ignored transients, the time-dependent part of the prompt torque [22] does not appear in this formalism.We also define T D = −⟨RF ϕ ⟩ fast,D , We assume that the stress tensor of the impurities Z is the same as that of deuterium (D), but simply rescaled by the mass density.In practice: Including these assumptions into equation ( 5), for the impurity we get (Z subscript is assumed, with ρ m = Mn + M D n D the total plasma mass density.This expression shows that the impurity feels a piece of particle flux, related directly to the externally applied torque, given as: This flux is defined positive for co-current injected torque, which leads to a more hollow density profile of the impurity. In the case of counter-current injected torque, the effect would be opposite, that is causing a peaking of the impurity density profile.Moreover, from equation (11) it can be seen that the particle flux arising from the externally injected torque scales with the ratio M/Z of the impurity.This ratio is ≈2 for fully ionized elements, typically light impurities.It can be larger for high-Z impurities for which M > 2Z as they are only partially ionized even in the plasma core.Despite this, it is expected that for high-Z impurities, this term would become sub-dominant to standard neoclassical transport, since the dominant convective contribution is proportional to the impurity charge and the neoclassical transport is also increased by rotational centrifugal effects, which are typically large for heavy impurities in the presence of externally applied NBI torque [23][24][25][26][27][28].To estimate the variation in impurity density normalized gradient due to this term, we use the estimate: Γ ≈ S lat Dn/L n , where S lat is the plasma lateral surface, D the particle diffusion coefficient, and L n the impurity density length scale.As such: Coming back to equation (10), we also consider the fast ions-impurity friction term T Z , which instead causes a radial inward convection of impurities.As noted in [15,16], the net convection caused by the combination of the term explicited in equation (11), and the frictional term T Z , depends on how much of the total torque is actually carried by the frictional term.A useful criterion is: which defines when the net effect of the torque contributions gives outward radial impurity flux.This criterion is valid in the limit of strong D-Z coupling, as also expressed in [15,16].For mid-to-low collisionality regimes, a more complicated criterion would arise due to the indirect modifications of the flows themselves due to the different torque distribution between main ions and impurities.However, different from those two previous works, here we have included the perpendicular fast ion particle flux term (also called JxB torque) G, and the consideration of the turbulence-driven toroidal-radial stress component in full form.Note that the thermalization torque term is usually negligible.Finally, in here we neglect particle density poloidal asymmetries and inertial effects, that can lead to an enhancement of the neoclassical flux, as already mentioned.

Microscopic origin of the new flux term
To understand better how the flux contribution (11) arises from microscopic dynamics, we go back to equation ( 4) and focus on the left-hand-side.Microscopically, the perpendicular component of the stress tensor divergence arises from two mechanisms: neoclassical poloidal flow damping and Larmor-scale polarization effects.We thus separate the tensor into these two components that we call respectively Π N and Π p .As such, neglecting perpendicular friction, electric field convection, and the direct torque terms, one gets: We focus on the neoclassical term, and neglect the polarization term, to get: Moving to the parallel direction, the parallel momentum balance, neglecting electric field, friction, and torque (for the impurity), gives: where Π T arises from Larmor-scale turbulent microinstabilities like the ion-temperature-gradient mode.Since in axisymmetric geometry, ⟨Re ϕ • ∇ • Π N ⟩ = 0, we can rewrite it as: where we have neglected the perpendicular turbulent stress linked to polarization effects [29].Finally, substituting this into the particle flux expression (15), we re-obtain the same expression as (11).
The physical interpretation is thus the following: (1) Applied external torque excites turbulence-driven momentum flux, particularly the diffusive component (outward flux), by increasing the toroidal rotation of the plasma, which is carried by both main ions and impurities; (2) We assume that the two fluxes only differ by the ratio of the corresponding mass densities.As such, the impurity momentum flux thus is finite and increasing with increasing applied torque; (3) The poloidal neoclassical viscosity balances the turbulent momentum flux, thus leading to a modification of the neoclassical flow which is proportional to the injected torque; (4) The modified poloidal flow leads to a modified radial particle flux, which is outward and proportional to the injected torque.
As such, this new particle transport mechanism does not come from the perpendicular polarization drift, which is instead studied, with respect to the experimental observations, for example in [30], but it comes from the modification of neoclassical flows by the turbulent momentum flux, leading to a deformed particle flux, which acquires an outward component directly proportional to the applied torque.

Near-axis behavior
A point to clarify is the following: upon looking at (11), it is rather striking that the particle flux does not go to zero at the magnetic axis, as it should due to symmetry arguments.The reason is because that formula is obtained by neglecting parallel friction force and keeping the poloidal damping term.However, close to the magnetic axis, poloidal damping vanishes, while parallel friction dominates.Infact, one has to remember that parallel momentum balance solves for the parallel flow.The poloidal neoclassical stress tensor produces a damping of the poloidal component of the parallel flow, but only that one.
To calculate the asymptotic expression of the particle flux for near-axis behavior, we explicit the linear dependence ∝ r, of all quantities which rely on the existence of the poloidal magnetic field and poloidal variations of the toroidal field (which is the cause of poloidal damping).
where r is the local radial coordinate, P N the parallel neoclassical stress tensor, A T the turbulence parallel acceleration, F the parallel friction force.P T is the turbulent polarization stress, ϵ is the ExB convection flux, and Γ the particle flux.We also define A T − rP T = S, with S the auxiliary torque source.Moreover, we link neoclassical poloidal damping to friction via coefficients K 1 , K 2 which will depend on collisionality: The exact parametric dependencies of these terms is not important here (made exception for the explicit limit of the linear radial dependence), as this subsection demonstrates the near-axis behavior in a didactical way, not meant to be used for specific calculations.However, we find instructive to show it since it is analytically tractable.The key point is to evidence the linear r dependence where it is important.Solving for the particle flux Γ as a function of the torque source S and the turbulent polarization P T we get: . As such, the actual form of the contribution arising from the external torque S is: For large values of K 1 , one gets: Γ new ≈ S, but for low values of r one gets Γ new ≈ −rK 1 S. Notice that K 1 < 0 (collisional damping/frictions always tend to reduce the plasma flow), as such, the sign of Γ new never changes.

Inclusion of friction and geometry
To obtain the exact expression of the particle flux arising from momentum conservation, we return to equation ( 6) and write it as customary for calculating the neoclassical poloidal flows (again, below everything is intended for a generic impurity species, not for the main ions): where the electric field term vanishes (electrostatic turbulence is poloidally quasi-symmetric, and the equilibrium loop voltage is assumed to be negligible).We split the stress tensor in neoclassical N and turbulence-driven T: We now express the neoclassical term on the left-hand side as: ), where µ is the damping coefficient and U || the impurity parallel flow (and 0 indicates the residual undamped parallel flow arising from the perpendicular gradients [15,31]).Notice that b θ ensures that it is only the poloidal component of U || that gets damped.Also, B = ⟨B⟩.The friction term is written analogously as: ), with ν the cross-species collisionality (e.g.impurity-main ions), and U ||,1 the main ion parallel flow (including the heat flow contribution).In the bananaplateau regime, the left-hand-side damping term dominates, while in the Pfirsch-Schlueter regime, the frictional term is dominant.Note, however, that since we are dealing here with an impurity in the trace limit, both poloidal damping and parallel friction are determined by cross-collisions with the main ion species.As such, neither can be neglected and both have the same order of magnitude.Note also that the auxiliary torque can have a direct effect on the neoclassical flows as well, as it appears in the parallel momentum equation for the main ions.However, we assume that the main ion poloidal damping term dominates over the torque term, thus leading to a negligible modification of the main ion poloidal flow.We now employ this equality for a generic vector where F = RB ϕ .The term in [...] parenthesis can be rewritten expliciting the aspect ratio dependence, since the magnetic field is approximately Typically, the quantity A || also has a similar poloidal variation, leading to a net ε 2 scaling.As such, even at mid-radius, this quantity is small at large local aspect ratio, and it goes to zero as we move towards the core of the plasma (being exactly zero at the magnetic axis).We thus neglect it since we assume that the region of interest is r/a < 0.5.We thus expand the parallel turbulent term Finally, we can compute the parallel flow as: This solution is valid in general.It shows that the presence of turbulence modifies the parallel flow from its neoclassical value, in a way which depends on the relative weight between the injected torque (S), turbulence intensity (P T ) and collisional damping (µ, ν).Now we consider equation ( 4), where we call ε = Ze⟨nE y Rb θ ⟩, we neglect the perpendicular friction term (responsible for classical transport), and of course torque is absent (for the impurity species).Thus: Again we decompose the stress tensor into neoclassical and turbulence-driven: Upon substituting what we found before, we finally arrive at: This allows us to solve for the particle flux as the sum of the following pieces: where: ) We identify these terms as: a 1 is the usual turbulent ExB convection of Larmor-scale density fluctuations, a 2 is the standard neoclassical particle flux, a 3 is the particle flux driven by turbulence-driven perpendicular polarization stress [29], and a 4 is the new particle flux term related to the injected angular momentum torque.Notice that the b θ terms ensure that the flux goes to zero at the magnetic axis, whereas S is still finite.
A couple of interesting limits can be extracted from ( 26): (1) if friction is sub-dominant to poloidal damping, meaning that ν ≪ µb θ , then one finds: a 3 → 0 and a 4 → S, which gives the maximum outward flux achievable by this effect; (2) if friction is dominant, that is ν ≫ µb θ , then: a 3 → −P T and a 4 → 0, which means that the external torque effect goes to 0; (3) for a trace species, which means that poloidal damping and friction are of similar order of magnitude µb θ ∼ ν, and a 4 ≈ S/2.

About quasi-neutrality
It is instructive to discuss the role of ambipolarity in determining the particle fluxes as derived in this work.First of all, for a pure thermal plasma (electron species + 1 main ion species), ambipolarity requires the ions to have the same particle flux as the electrons.However, in the presence of a fast-ions population, additional terms arise in the thermal species equations, that balance the fast ion population dynamics.We recall here the fast ion toroidal momentum balance equation in the case of pure plasma: eΓ fast dΨ dV + ⟨RF ϕ ⟩ fast,D + ⟨RT ϕ ⟩ = 0. Since electrons can be assumed to have negligible inertia, this means that, if we consider this expression for the main ion toroidal momentum balance (we neglect friction with electrons): It is clear that this expression actually means these two separate transport phenomena: In practice, for a pure plasma, the external torque has the effect of modifying the thermal ion particle flux such as to balance the JxB component of the torque.The sum of these must be equal to the electron one, to satisfy ambipolarity.Instead, the total torque is entirely absorbed by the toroidal component of the stress tensor divergence, resulting in an equation for the radial electric field, as well known [32].
For an impure plasma, for example with one main ion species and one (or more) impurity, the fast ion balance is eΓ fast dΨ dV + ⟨RF ϕ ⟩ fast,D + ⟨RF ϕ ⟩ fast,Z + ⟨RT ϕ ⟩ = 0.And the equation ( 27), for each ion species is: Assuming again that the impurity stress tensor is the same as the main ions, but rescaled by the mass density, we have: where we have indicated as S T = ⟨RT ϕ ⟩/ρ M , with ρ M the total ion mass density, the common term.The discrepancy between impurity and electron flux, due to the external torque, will be re-absorbed by the ion flux, which acquires a slightly negative contribution.In practice, while the impurity density profile will become more hollow, the main ion density profile will be slightly more peaked, and the electron density profile will be unchanged.
One could thus conclude that polarization effects on the particle flux are only visible if there are multiple ion species present (including the fast ions), which can share the discrepancy between their particle fluxes and the 'non-polarized' electron particle flux.

Quantitative comparison of the new term to experimental observations
We now compare the contribution of the explicit torque terms against observations from AUG experiments dedicated to studying the behavior of Boron with different heating schemes.In particular we use the case shown in [12], discharge #34400.To this purpose, we have developed a simple code that solves the system of parallel momentum balance equations discussed in [31], where we have added the torque contributions via the fast ions physics, which is coupled to the thermal species via the friction terms.Moreover, in the parallel momentum balance of the thermal ion species we have added the turbulence-driven stress tensor which is supposed to absorb the externally applied torque to satisfy ambipolarity.In practice, we make the same assumptions of equation ( 9), but for the friction terms between fast ions and the thermal species.In this way, we can directly plug in the collisional torque terms T D,Z in the thermal species equations.
After having computed the friction forces, we then calculate the particle flux using the expression (10) for the impurities.In that expression, we do not include the turbulencedriven particle flux contribution Ze⟨nE ϕ R⟩ (and also we do not include the equilibrium-related Ware pinch).Calculated equivalent variation in normalized density gradient of the Boron impurity ∆(R/Ln) as a function of the included effects shown in x-axis.The experimental case is taken from [12], when the maximum torque is obtained, and the radial position is r/a = 0.5.'std': standard neoclassical calculation without torque and turbulence.'case 1': added collisional part of torque and turbulence.'case 2': same as case 2 + JxB torque contribution.'case 3': collisional part of torque only and no turbulence.
We then compare the resulting particle flux with and without including the effect of fast ions and the induced external torque, which will drive a finite turbulence-driven stress as well.The result of this comparison is shown first in figure 1.In this figure, we display the equivalent variation in local normalized density gradient ∆(R/L n ) of the impurities, for the experimental case at maximum injected torque.We then apply the estimate of equation (11) to the same cases shown in [12], where only the JxB component of the torque is used.The result is shown in figure 2. It can be seen that the correction arising from the torque term seems to fit in the discrepancy between the experiment and the values obtained in the previous work.
As could have been previewed from the quantitative dependences of the effect, as shown in equation (11), this additional contribution from the external torque is not dramatically large, nor the dominant component that pushes the impurity normalized density gradient to negative values.However, it is also not negligible, adding on top of the other contributions to push the value into the experimental ballpark.Clearly, this effect is stronger at stronger injected torque normalized to the plasma mass and plasma surface.As such it is expected that, for larger future machines which will operate at lower injected torque per particle, this effect would be negligible.(12).Red stars: experimental value.Green hexagrams: predicted value from [12] plus the new terms.Magenta triangles: predicted value from [12].

Conclusion
Upon using a generic fluid framework to calculate the particle flux of an impurity ion species, we have shown that there is a term arising from the application of auxiliary angular momentum torque.The origin of this term lies in the interaction between parallel impurity inertia, poloidal flow damping and friction, and the link between parallel motion and perpendicular particle flux via the poloidal stress tensor.It is produced by the impact of turbulent toroidal momentum flow on the neoclassical radial flow and, as such, is not present in separate descriptions of neoclassical and turbulent transport.Therefore it has to be added in the modeling of impurity transport where collisional and turbulent transport are computed separately.
The final expression for the particle flux includes known terms (turbulence-driven ExB convection, neoclassical components) and new terms: polarization flux, and torque-driven flux (this work).We have demonstrated that the fluid formalism used here, is consistent with both versions of 2nd order gyrokinetic theory, standard and modified to include the polarization drift directly into the equations of motion.
Comparison between the theory and the experimental observation shows that the effect is sizeable and goes in the right direction.In a future work we plan to apply this model to a broader database.

Adding the torque effect into existing neoclassical codes
In this note, we would like to discuss how to add the terms arising from the external torque effect on the polarization flux to existing neoclassical codes.In particular, adopting the approach followed in section 3. We focus on the fluid moment approach-based codes, although drift-kinetic based codes would be easily adapted too.
First of all, the fast ion species could be simply expressed as a Maxwellian species with a given temperature and density.This is required to compute the frictional terms between fast ions and the thermal ion species.Now, care has to be taken regarding the torque and stress terms: (1) In the parallel momentum balance of the fast ions, the absorbed parallel component of the torque from the neutral beam ⟨BT || ⟩ has to be added.This is important to get the correct value of the frictional forces that will appear, mirrored, in the thermal ions equations; (2) In the parallel momentum balance of the thermal main ions and impurities, the turbulence-driven parallel stress terms have to be included in an ad-hoc way.However, what is important here, is that the total stress summed up over the thermal species equals the absorbed torque; (3) The resulting radial particle flux of the thermal ions maybe better computed using the perpendicular projection, which ensures that the effects related to the external torque are carried over by the neoclassical perpendicular stress tensor, whereas perpendicular friction can be safely ignored.Otherwise, if the particle flux is computed using the toroidal projection of the momentum balance, both toroidal turbulence-driven contribution and frictional terms have to be correctly included.
follow the former work, and estimate the term as (no T pedix): We have ignored the term ∝ −Π : (∇e y ) since it brings in a length scale variation which is longer than the profile scale length from the former term.
We now express Π xy = Mnv E,x v E,y , with v E the turbulenceinduced fluctuating ExB velocity.Using the Fourier modes: Assuming Larmor-scale microturbulence, one can estimate

Appendix B. Consistency with gyrokinetic theory
We consider the consistency of the result found here with gyrokinetic theory [33].In particular, we examine whether the fluid picture derived in the previous sections is consistent with a gyrokinetic treatment of the problem.The setting of such a problem is best cast in a coordinate system which is field-aligned: parallel direction and perpendicular (bi-normal and radial) directions.This natural choice is due to the transformation from particle to gyrocenter (gC) trajectory, where the drifts are all given in terms of (. ..) × B terms, plus parallel motion.This means that the natural comparison would be using the momentum balance projections ( 4) and ( 6).
The gyrokinetic equation can be written in a compact form as ∂ t (JF) + ∇ • ( ṘJF) + ∂/∂v ∥ ( v∥ JF) = JC.J is the Jacobian of the transformation from real space (⃗ x,⃗ v) to gC coordinates ( ⃗ R, v ∥ , µ, ζ), with µ, ζ respectively the particle magnetic moment and the gyroangle, F is the gC distribution function [33,34].C is the collision operator.The particle flux is obtained by considering the following operation: Γ = ⟨ ´f(⃗ x,⃗ v, t)v r d 3 v|∇V|⟩, with v r the radial component of the particle velocity.The phase-space integral is to be done at constant real space position ⃗ x.The customary treatment is to replace f v r with the gC flow density.In this case, the particle flux will be the sum of gC, polarization, and magnetization fluxes [33,35].
We start here from the fluid system analyzed before, and see if we obtain the gyrokinetic equivalent at the most basic level.The derivation shown here follows previous works [36][37][38][39], in a more simplified way.Exact fluid equations of the particle distribution function f up to moments 1, v, vv are (particle mass and charge are assumed to be unity): Upon using 1/B as the expansion parameter, the zero order equations read: with The first order equations are: We do not need to consider higher fluid moments, since we have closed the system here upon expressing Γ 0 , Π 0 from equation (32).Collecting the zero and first order terms of the particle flux we finally have: Notice that the density appearing inside the particle flux expression is the zero order density, that is, it satisfies ∂ t n 0 = −∇ • (n 0 V E ).We now proceed with the manipulation of equation (34).The aim here is to recast the expression of the particle flux Γ in a form which is directly comparable to the single-particle drift motion as obtained from gyrokinetic theory.
In the following we assume the magnetic field to be a vector of constant direction and amplitude.In this case, one can easily see that ∇ • V E = 0, i.e. the ExB flow is incompressible.We also define ϵ = 1/B and b = B/B.Also notice the identity b × First, we recast the divergence of the ExB stress term as: where we have used the zero order relation (32).Next, we apply a well known vector identity: , and we manipulate the last term as: The application of all these operations to equation (34), leads to this modified but equivalent expression for the particle flux: Cast in this form, it can be compared to what one obtains from the gyrokinetic formulation of [34], where the gC motion is given by the drift velocity where The drift velocity can thus be approximated as: Term by term, it is straightforward to see the equivalence of the particle picture with the fluid picture of equation (36).
It is also instructive to look at standard gyrokinetic theory, where the polarization drift does not enter explicitly in Ṙ, which now is given simply by Ṙ = b/B × ∇Ψ, but now the generalized potential is defined as: Ψ = Φ − V 2 E /2 (notice the sign change!).However, with this definition of the gC, one has to add a polarization term explicitly to the fluid density.That is: Now the continuity equation for ions looks like: After lengthy but straightforward vector algebra, using one arrives at: Again, one can recognize in this last equation the same form of equation (34), made exception for the additional last term −nϵb × ∇(V 2 E ) − ϵ 3 |∇ ⊥ Φ| 2 b × ∇n.However, it is easy to see that these two pieces combine into ϵb × ∇(nV 2 E ) = ∇ × (ϵbnV 2 E ), which vanishes under divergence, i.e. ∇ • (∇ × . ..) = 0. Physically, this reflects the fact that standardgyrokinetic and gyrokinetic with polarization drift differ by the magnetization vector bnV 2 E arising from the ExB drift energy, analogously to the magnetization vector bP that links the particle magnetic drifts to the fluid pressure gradient flow.The contribution from magnetization pieces does not produce net transport as expected [35].
Obviously we have obtained this approximate correspondence upon ignoring many other effects like FLR (zero temperature), parallel dynamics, and higher order terms.However, it is striking that we get exactly the same expressions when keeping the lowest-order perpendicular dynamics.This give us confidence that the polarization particle flux should indeed be calculated as in (34), which has the feature of being in conservative form under the divergence.This is similar as what is obtained in [29].

Appendix C. Impurity-main ions difference in toroidal stress tensor
As a last part of the theoretical work, we address the issue of computing rigorously the terms in equation ( 9).There we assumed that the impurity and the main ions have the same stress tensor, when normalized to the mass density.We now consider this assumption in details.First, we can write: For each species, we express the stress tensor radial-toroidal component Π r,ϕ as: Notice that only the diffusive component is included, since in the core of the plasma, for driven rotation, that is the dominant contribution [40].
We consider deuterium D and another impurity j, and thus we have: where we intend the toroidal direction, and T = ⟨RT ϕ ⟩ is the injected torque.We now call A = −χ D U ′ D , and define the quantities ξ = χ j /χ D , δ j = U j − U D , so that: Solving for A: Substituting back into the impurity part, we have: From this last expression, one can see that the impurity stress tensor is self-similar to the one of deuterium (∝ τ with the mass ratio), but deformed via the difference in momentum diffusivities, ξ, and the neoclassical rotation differential δ j .These terms could be calculated via linear gyrokinetic and neoclassical codes.

Figure 1 .
Figure1.Calculated equivalent variation in normalized density gradient of the Boron impurity ∆(R/Ln) as a function of the included effects shown in x-axis.The experimental case is taken from[12], when the maximum torque is obtained, and the radial position is r/a = 0.5.'std': standard neoclassical calculation without torque and turbulence.'case 1': added collisional part of torque and turbulence.'case 2': same as case 2 + JxB torque contribution.'case 3': collisional part of torque only and no turbulence.

Figure 2 .
Figure 2. Comparison of the predicted B normalized density gradient R/Ln(B) with several terms.Blue squares: new term arising from torque(12).Red stars: experimental value.Green hexagrams: predicted value from[12] plus the new terms.Magenta triangles: predicted value from[12].
The bracket parenthesis ⟨. ..⟩ indicate flux-surface average.U r is the radial component of the flow velocity vector U. Ψ is the poloidal magnetic flux and V the plasma volume.Note that |∇Ψ| = |RB θ | (absolute value).As mentioned above, we now project equation (1) along the 3 directions defined by the versors e y , e ϕ , e || .Upon applying the following scalar products Rb θ e y •, Re ϕ •, Rb ϕ e || •, one gets (the species index is only included where needed):