Water bag modelling of a multispecies plasmaP. Morel,1, a) E. Gravier,1 N. Besse,1 R. Klein,1 A. Ghizzo,1 P. Bertrand,1 C. Bourdelle,2 and X. Garbet21)Institut Jean Lamour, UMR 7198 CNRS-Universite Henri Poincare, F-54506 Vandoeuvre-les-Nancy Cedex,France2)Association EURATOM-CEA, CEA/DSM/IRFM, CEA Cadarache, Saint-Paul-lez-Durance,France.

(Dated: 11 February 2011)

We report in the present paper a new modelling method to study multiple species dynamics in magnetizedplasmas. Such a method is based on the gyro water bag modelling, which consists in using a multi step-likedistribution function along the velocity direction parallel to the magnetic field. The choice of a water bagrepresentation allows an elegant link between kinetic and fluid descriptions of a plasma. The gyro water bagmodel has been recently adapted to the context of strongly magnetized plasmas. We present its extensionto the case of multi ion species magnetized plasmas : each ion species being modeled via a multi waterbag distribution function. The water bag modelization will be discussed in details, under the simplificationof a cylindrical geometry that is convenient for linear plasma devices. As an illustration, results obtainedin the linear framework for Ion Temperature Gradient instabilities are presented, that are shown to agreequalitatively with older works.

I. INTRODUCTION

By definition, any plasma is a multi species one, sim-ply regarding a pure case with electrons and just one ionpopulation. For the sake of simplicity, most of the stud-ies have been achieved by focusing on a given species,assuming for example the other(s) adiabatic.From a numerical point of view, the increase of the

computational power makes possible to study the cou-pling between different species. The limiting factor isgiven by the mass ratio between the different species totake into account. Even if the mass ratio between speciesis not so high the coupling between species increase thenumerical effort, so that any alternative is welcome todecrease the dimension of a given simulation. The waterbag approach has recently been demonstrated to providesuch a decrease, and has motivated the present work.From an experimental point of view strongly magne-

tized plasmas always include many species due to par-ticles eroded from experiment’s wall (mainly Tungstenand/or Carbon), that can be carried until the plasmacore. In the extensively investigated case of a fusionreactor, one has to deal with Tritium and Deuteriumfor achieving the fusion reaction, and with the Heliumproduced. Independently of their origin, such impuritiescould cause a non negligible loss in confinement proper-ties and their effects on the plasma stability remains tobe fully understood.In this paper, we describe magnetically confined plas-

mas, characterized by negligible collision rate and highmagnetic field. The toroidal version of the water bagmodel being under development, we present here resultsobtained in cylindrical geometry. Such a geometry is es-

a)pmorel@ulb.ac.be; Physique Statistique et Plasmas, UniversiteLibre de Bruxelles, CP 231, Campus Plaine, Boulevard du Triom-phe, 1050 Brussels, Belgium.

pecially suitable for linear plasma devices1–5, that offera simplified plasma geometry with an homogeneous ax-isymmetric magnetic field. Moreover, such a simplifica-tion allows to recover major features of the Ion Tempera-ture Gradient instability that bases our investigations inthe present work.Two kinds of theoretical descriptions can be used in

plasma physics : the fluid and the kinetic ones. De-spite lot of progresses have been made until recent yearsby the use of fluid numerical solvers for studying multi-species plasmas6,7, a more accurate description of micro-instabilities and anomalous transport requires the use ofnonlinear gyro-kinetic codes8–11. In these works, the keyquestion is about the modification of particles and heatnonlinear fluxes due to the presence of impurities andcontrolled by the density and temperature radial gradi-ents. Older works concentrated on the linear responseof a multi-species plasma, especially on finding the crit-ical density and temperature gradients destabilizing theplasma12, and trying to give an expression for the fluxesfrom quasi-linear theory13,14.Despite considerable progresses during last decades,

gyrokinetic solvers require a huge numerical effort, andtheir extension to multiple species study increases sucha difficulty. One can too point the fact that the operat-ing cost of a numerical simulation of a realistic plasmabecomes comparable to the one of a real experiment.Some alternative ways to describe kinetic effects with asimplified formulation are also useful, that could allowa less requiring numerical effort. Such an effort in de-picting multi-species plasma can be decreased by the useof quasi-linear solvers15 or gyro-fluid transport models16.The present work present another interesting alternative,based on the water bag representation.The gyro-water-bag equations are an exact reformula-

tion of the Vlasov Poisson system, in the case of a specialform of the distribution function : the water bag whichtakes the form of a multi step-like distribution functionalong the velocity direction (each step representing a

2

bag). The interest of the water bag formulation consistsin the discrete summation over the bags replacing theusual continuous distribution function17–21. The paral-lel velocity dependency and the partial derivative alongthis direction do no more appear in equations and arereplaced with the coupling between each bag dynamicsensured by a discrete summation in the quasi neutral-ity equation. The Vlasov Poisson system can be refor-mulated as a multi fluid one, with an exact adiabaticclosure22,23.Based on recent papers23–26, that ensure the water bag

model to be an interesting approach to depict kinetic fea-tures, our present work first consists in the generalizationof the water bag concept to the case of a multispeciesplasma, in the framework of a cylindrical geometry (sec-tion II). An interesting result is that the water bag modeleasily offers an analytical expression of the linear stabil-ity threshold, in place of the usual Fried-Conte plasmadispersion function (section III). As illustrations, linearstability frontiers are drawn in the plane of the tempera-ture and density gradients of main ions. We then discussthe stability dependence on the nature, the ratio and theradial density and temperature profile of impurities (sec-tion IV).

II. A WATER BAG APPROACH TO MODELIMPURITIES

In the collisionless limit, each species s of a multi-species plasma is described with a Vlasov equation act-ing on its guiding centers distribution function fs =fs(R, v‖, t), with R defining the guiding center position,v‖ being the velocity coordinate along the magnetic fieldline. We consider a cylindrical geometry, assuming a con-stant magnetic field B oriented along the azimuthal axiscalled z. We assume for simplicity that the Larmor radiusrLs = v⊥Ms/qsB and the associated magnetic momentµ = Msv

2⊥/2B are constant for each species (particles of

mass Ms, charge qs = Zse (with e the elementary charge)and perpendicular velocity v⊥), meaning that all parti-cles of a species have the same gyro-motion. So do weomit in the following the usual summation over differentvalues of µ for a given species (we refer the reader toreference26 for a water bag analysis of this problem). Foreach species s, the Vlasov equation then reads :

∂tfs + 〈vE(r, t)〉s .∇⊥fs + v‖∂zfs+Zs

Ms〈E(r, t)〉s ∂v‖fs = 0 (1)

We note here that by using gyro-average (denoted< . >s) and considering a constant perpendicular veloc-ity associated to gyro-motion, only the parallel velocityremains an independent variable.The quasi-neutrality equation ensures the coupling be-

tween the different ion species and electrons, that are

moreover assumed adiabatic with a low electrostatic en-ergy eφ << Te :

ne0

(1 +

eφ(r, t)

Te

)=

∑s

Zs

(∇⊥.

[Msns0

qsB2∇⊥φ(r, t)

]

+

∫ +∞

−∞

⟨fs(R, v‖, t)

⟩sdv‖

)(2)

φ, E, and vE are respectively electrostatic potentialand field, and the corresponding electric drift vE =−∇φ × B/B2. Each ion species s is defined with itscharge qs = Zse, mass Ms and equilibrium density ns0.Te is the electron temperature, e their charge and ne0

their equilibrium density. Finite Larmor Radius effectsare contained in the gyroaveraged distribution function⟨fs(R, v‖, t)

⟩s, and in the first term of the left hand side

of the quasi neutrality equation (2), commonly known asthe polarisation effect.We now represent each species with a water bag dis-

tribution function along the velocity parallel to the mag-netic field, given by the sum of Ns Heaviside step func-tions Υ :

fs =

Ns∑

j=1

Asj

[Υ(v‖ − v−sj(R, t)

)−Υ(v‖ − v+sj(R, t)

)]

(3)Each bag j is enclosed by positive and negative velocity

contours v±sj and has an height Asj (Fig. 1), while thetotal number of bags Ns is free for any species. Moreoverand accordingly to the conservation property of phasespace measured by the distribution function, the heightAsj of each bag is a constant of motion. By insertingmulti-water-bag distributions (3) into equations (1, 2),we obtain :

∂tv±sj(R, t) + 〈vE(r, t)〉s .∇⊥v±sj(R, t)

+v±sj(R, t)∂zv±sj(R, t) =

qsMs

〈E(r, t)〉s(4)

ne0

(1 +

eφ(r, t)

Te

)=

∑s

Zs

(∇⊥.

[Msns0

qsB2∇⊥φ(r, t)

]

+

Ns∑

j=1

Asj

⟨v+sj(R, t)− v−sj(R, t)

⟩s

,

(5)

We can rewrite these equations by defining for eachspecies a density nsj = Asj(v

+sj − v−sj), an average ve-

locity usj = (v+sj + v−sj)/2, and a partial pressure psj =

Msn3sj/12A

2sj relatively to each bag :

3

∂tnsj + 〈vE〉s ∇⊥nsj + ∂zusjnsj = 0 (6)

∂tusj + 〈vE〉s ∇⊥usj + usj∂zusj =

qsMs

〈E〉s −1

Msnsj∂zpsj (7)

ne0

[1 +

eφ

Te

]=

∑s

Zs

[∇⊥.

(Msns0

qsB2∇⊥φ

)

+

Ns∑

j=1

〈nsj〉s

(8)

We recover the hydrodynamic formulation of the waterbag equations, providing a continuity (6) and a Euler (7)equation for each bag with an exact closure given by thepartial pressures psj = Msn

3sj/12A

2sj . Such a formulation

is a typical feature of our modelling choice of a water bagdistribution function22.

III. LINEAR ANALYSIS

In the present section, we derive the expression of thedielectric function ε(ω) of a multispecies plasma, aimingfor obtaining a linear stability threshold. We assume anhomogeneous equilibrium along the magnetic axis (z di-rection) and the orthoradial coordinate (θ), without anyequilibrium electric potential. Contours are expressedwith an even equilibrium ±asj(r) and a fluctuating partw±

sj . Fluctuating quantities are expanded on a Fourierbasis for θ and z :

v±sj = ±asj(r) + w±sj(r) e

i(mθ+k‖z−ωt) (9)

φ = 0 + δφ(r) ei(mθ+k‖z−ωt), (10)

where m is the orthoradial mode number and k‖ is theparallel wave vector.For simplicity, we neglect finite gyroradius effects

by considering for each species rLs = 0, we then obtainthe linearized gyro-water-bag equations :

(ω ∓ k‖asj)w±sj =

[k‖

qsMs

∓ m

rBdrasj

]δφ (11)

ene0

Teδφ =

∑s

Zs

Ns∑

j=1

Asj(w+sj − w−

sj)

(12)

The gyro-water-bag multispecies dielectric function isobtained by eliminating contour fluctuations (w±

js) given

by (11) into quasi-neutrality equation (12) :

ε(ω) = 1− Te

ene0

∑s

Zsns0

qsMs

Ns∑

j=1

αsj

k2‖ − ωkθκsj/Ωcs

ω2 − k2‖a2sj

(13)Where we introduce the bag relative density αsj =

2asjAsj/ns0 and the radial derivative for each equilib-rium contour κsj = dr ln asj . More usual parametersare the poloidal wave vector kθ = m/r (r being the ra-dial point considered) and the cyclotron frequency Ωcs =qsB/Ms. Introducing the parameters Z?

s = ZsTe/Ts andv2Ts = Ts/Ms, the dielectric plasma function can be writ-ten :

ε(ω) = 1−∑s

Z?

s

Zsns0

ne0v2Ts

Ns∑

j=1

αsj

k2‖ − ωkθκsj/Ωcs

ω2 − k2‖a2sj

(14)The linear stability threshold is defined with ε(ω) = 0

and dωε(ω) = 024. By applying these conditions to thedielectric plasma function (14), we obtain :

∑s

Z?s

Zsns0

ne0v2Ts

Ns∑

j=1

αsj

k2‖ − ωkθκsj/Ωcs

ω2 − k2‖a2sj

= 1 (15)

∑s

Z?s

Zsns0

ne0v2Ts

Ns∑

j=1

αsj

(ω2 + k2‖a2sj)kθκsj/Ωcs − 2ωk2‖

(ω2 − k2‖a2sj)

2= 0

(16)We can notice that the calculation of such a thresh-

old is a good illustration of the ability of the water bagmodel, because in the general case of multiple continu-ous distribution functions, no analytic expression of thethreshold can be derived. In our case, the threshold isaccessible very simply by using an ω parametrization ofprevious equations (15, 16).

IV. LINEAR STABILITY

We will now focus on the case of a two speciesplasma (indiced D for fixed Deuterium and s for varyingspecies in the following). In such a two species case, thelinear threshold of stability (15, 16) is a surface in a spaceof four dimensions defined by the density (κnD, κns) andtemperature (κTD, κTs) gradients of each species. Forrepresentation facilities, we assume the same tempera-ture profile for each species. The density gradients arelinked with respect to the definition of the effective chargeZeff (introducing its radial derivative κZeff

) :

Zeff =Z2DnD0 + Z2

sns0

ZDnD0 + Zsns0(17)

κZeff=

ZDnD0 Zsns0(ZD − Zs) (κnD − κns)

(ZDnD0 + Zsns0)2 (Z2DnD0 + Z2

sns0)(18)

4

and we will consider two different cases:

? The first one is to consider a flat impurity pro-file, with an arbitrary density profile for the otherspecies. The effective charge profile κZeff

is then :

κZeff=

ZDnD0 Zsns0(ZD − Zs)

(ZDnD0 + Zsns0)2 (Z2DnD0 + Z2

sns0)κnD (19)

? The second case consists in forcing the impurities infollowing main ions (κns = ξκnD), while the effec-tive charge radial profile is given by the definitionof κZeff

(18).

We apply in the following the results of linear analysis(14, 15, 16) to three practical cases of mixed plasmas.The main species is chosen to be Deuterium while theother one is detailed below :

? To ensure fusion reactions at low temperature, Tri-tium must be present in a Tokamak. Its properties(ZT = 1 etMT ' 3mp) are closed to the Deuteriumreference.

? Carbon (MC ' 12mp, Zc = 6) often constitutesTokamak wall elements. It gives an interesting in-termediary value of charge and mass values.

? To study the effect of heavy particles the Tungstenis chosen (MW ' 184mp, ZW ' 40)27, which is agood candidate for long lifetime ITER divertor andfirst wall28.

A. Water bag parameters

The water bag model replaces an imaginary pole witha finite number of real resonances that are associatedto discontinuities of the water bag distribution functionalong the velocity coordinate21,30,31. Such discontinu-ities correspond to values ω = ±k‖asj , as can be seen inthe expression of the dielectric plasma function (14). Asshown in Fig. (1), we superpose discontinuities linked toeach species by choosing the same interval between twobags ∆a.Such a superposition allows us to treat easily the real

poles of the dielectric plasma function, that can be ex-pressed as a rational function of two polynoms. Theroots are degenerate in the velocity interval where thetwo species are defined.Practically, regarding the velocity grids relative to each

species, we use constant and equal interval lengths :∆aD = ∆as = ∆a. For each species s (including Deu-terium), the cutoff velocity aNss is chosen to be equal tofive thermal velocities :

∀j ≤ Ns, ajs =

(j +

1

2

)∆as

Ns : 5vTs =

(Ns +

1

2

)∆as.

−5 −4 −3 −2 −1 0 1 2 3 4 50

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

parallel velocity (along magnetic field, vTD

units)

multi−water−bag (Deuterium)multi−water−bag (Carbon)

equilibrium contoursa

jD and a

jC

FIG. 1. Maxwellian multi-water-bag distribution functionsplotted against the parallel velocity direction (case of Deu-terium (D) and Carbon (C), vTD units).

The velocity equilibrium contours asj being fixed, thebag densities αsj as well as the logarithmic gradients ofequilibrium velocities κsj remain to be given. As detailedin a previous paper29, we used a moment sense equiva-lence with respect to maxwellian distribution functions :

fs =ns0√2πv2Ts

e−v2‖/v

2Ts , (20)

that allows us to link the unknown water bag parametersαsj , κsj to the maxwellian moments.In the following we will present stability diagrams in

the plane of density gradient (κnD) and temperature gra-dient (κTD) of the Deuterium. As usual in the study oftemperature gradient modes, the unstable areas are theones containing second or fourth quarters of the stabilityplane. Such diagrams present a structure relative to ourchoice of water bag distribution functions. For detailedexplanations about the specificity of a water bag repre-sentation, we refer the reader to previous articles23,24.The important information we need to precise is that

such water bag stability diagrams present a lobe-likestructure, where each lobe is related to a given paral-lel velocity population. The fastest particles are locatednear (κnD = 0, κTD = 0) couples, while the slow onesfollow the η → 2 asymptote in the case of a pure plasma(η being the ratio between temperature and density gra-dients of main ions η = κTD/κnD).It is important to notice that these two areas can be

associated to fluid (for the η branch) and kinetic parts(region close to zero) of the plasma response24. By con-sidering the case of a one-bag plasma with one ion species,the ratio between temperature and density gradients isfixed such that η is strictly equal to 2. In that case no in-stability can develop. On the contrary, the linear stabilityfrontier of a single ion plasma described by a multi-water-bag maxwellian equilibrium recovers the η = 2 value asan asymptote for the large density and temperature gra-dients limit.

5

In the numerical results presented in the following,length are expressed in k‖ units, velocities are normal-ized with respect to the Deuterium thermal velocitiy :v = v/VTD, and frequencies are consequently expressedin k‖/vTD units. Since there is no kθ mode selectionin the zero Larmor radius approximation, we can takekθrLD = 1 without a loss of generality.

B. Flat density profile of impurities

By considering a flat density profile, we simplify to di-lution effects the impact of impurities on the linear sta-bility threshold.

a)

−20 −15 −10 −5 0 5 10 15 20−40

−30

−20

−10

0

10

20

30

40

density gradient κnD

in k|| units

tem

pera

ture

gra

dien

t κT in

k|| u

nits

nT0

= 0

nT0

= 0.1

nT0

= 1

STABLE

UNSTABLE

η1 ∼ 0.90

η0.1

∼ 1.8

b)

−20 −15 −10 −5 0 5 10 15 200

0.2

0.4

0.6

0.8

1

1.2

1.4

density gradient κnD

in k|| units

linea

r gr

owth

rat

e γ

in k

|| vT

D u

nits

n

T0 = 0

nT0

= 0.1

nT0

= 1

κT = −10 k

||

FIG. 2. Impact of relative density of impurities on the linearthreshold of stability (a) in the plane of Deuterium gradi-ents (κnD, κT in k‖ units), and on the linear growth rate γ(b) along the density gradient κnD for a given value of thetemperature gradient κT = −10k‖. Case of a Deuterium- Tritium plasma (ND = 30, NT = 24, amaxD = 5vTD,amaxT = 4.08vTD, κnT = 0).

In the case of Tritium, we are considering two pop-ulations with comparable density values. We plot inFig. 2 (a) the linear stability thresholds of a mixed D-T plasma, for different values of Tritium relative density.

We choose to use a relatively large bag number (ND = 30NC = 24), in order to neglect the water bag typical lobe-like structure24. We report in Fig. 2 (b) the dependenceof the linear growth rate (γ in k‖vTD units) on the densitygradient of main Deuterium ions κnD for a fixed value oftemperature gradient κT .The mixed Deuterium-Tritium plasma exhibits two

kinds of response (see upper part of Fig. 2). We candistinguish stabilized areas related to small values of thetemperature profile, linked to high velocities classes ofparticles. But the plasma is more affected in the fluidbranch, associated to slow particles, where the Tritiumeffect is to destabilize the plasma. Regarding the lineargrowth rate (lower part of Fig. 2), we observe quanti-tatively two antagonist effects: γ decreases around κnD

close to zero (kinetic branch), while the unstable domainincreases for higher absolute values of the density gradi-ent (fluid branch).These effects become perceptible from values of the

relative Tritium density equal to 10% of the main Deu-terium density.Next, we represent in Fig. 3 the linear stability thresh-

old (a) and the associated linear growth rate (b) ofa plasma with Deuterium and a relatively high popu-lation of Carbon ions (10% of the Deuterium density,Zeff = 2.875). Effects ever observed in the case of Deu-terium Tritium plasma are recovered. The linearly unsta-ble area is extended by adding Carbon to the Deuteriumplasma, proportionally to the relative Carbon density(nC0 = 0.1nD0 shows a stability diagram comparable tonT0 = ND0).Moreover the more unstable values of the linear growth

rate located around flat Deuterium density profile, aredecreased by more than 20%. That is very high whencompared to the case of Tritium impurities (even withnT0 = nD0) that showed very similar values of growthrates.Tungsten exhibits very high values of charge and mass.

To describe its effect on the linear stability threshold ofa Deuterium plasma, we choose to deal with a very highnumber of bags in order to have a satisfying description ofthe tungsten equilibrium distribution function, accordingto the large mass ratio (ND = 500, NW = 52).We draw in Fig. 4 the linear stability threshold (a)

and the linear growth rate (b) of a Deuterium-Tungstenplasma. A typical effect is that the addition of Tungstengives rise to a secondary lobe-like structure, distinct fromthe Deuterium relevant one. Such a separation is due tothe fact that the distribution function of the Tungstenpopulation is very different from the Deuterium one, es-pecially the thermal velocities.In the lower part of Fig. 4, we use a constant tem-

perature gradient κT = −20k‖. That value is assumeddifferent from previous cases (Figs. 3, 2) in order to bein the second lobe relative to the Tungsten ions. Weobserve that the growth rate related to the second lobetakes very low values (less than 10% of the maximumvalue), while the maximum of growth rate very slowly

6

a)

−20 −15 −10 −5 0 5 10 15 20−40

−30

−20

−10

0

10

20

30

40

density gradient κnD

in k|| units

tem

pera

ture

gra

dien

t κT in

k|| u

nits

n

C0 = 0, Z

eff = 1

nC0

= 0.1, Zeff

= 2.875

UNSTABLE

STABLE

η0.1

∼ 0.93

b)

−20 −15 −10 −5 0 5 10 15 200

0.2

0.4

0.6

0.8

1

1.2

1.4

linea

r gr

owth

rat

e γ

in k

|| vT

D u

nits

density gradient κnD

in k|| units

nC0

= 0

nC0

= 0.1κ

T = −10 k

||

FIG. 3. Impact of relative density of impurities on the linearthreshold of stability (a) in the plane of Deuterium gradients(κn, κT ), and on the linear growth rate γ (b) along the densitygradient κnD for a given value of the temperature gradientκT = −10k‖. Case of a Deuterium - Carbon plasma (ND =30, NC = 12, amaxD = 5vTD, amaxC = 2.04vTD, κnC = 0).

decreases around κnD = 0 values.Whatever the species we consider, we obtain two dif-

ferent responses of the plasma, depending on the valuesof density and temperature profiles :

? The first one, very low, is a stabilization for smallvalues of density gradients (kinetic branch).

? The second effect is an important destabilizationof the mixed plasma, in regions with high values oftemperature and density profiles (fluid branch).

These two effects are in agreement with12, where au-thors used the assumption of a flat effective charge pro-file. The increase of impurity relative ratio, and/or theircharge and mass leads to a stabilizing effect in the flatdensity limit (κnD, κTD ' 0.0 in our stability diagrams).On the other hand, the low frequency branch, corre-sponding to the fluid branch presents a destabilizationof the plasma, that is proportional to the relative density

a)

−20 −15 −10 −5 0 5 10 15 20−40

−30

−20

−10

0

10

20

30

40

density gradient κnD

in k|| units

dens

ity g

radi

ent κ

T in

k|| u

nits

nW0

= 1e−4, Zeff

= 1.16

nW0

= 1 e−3, Zeff

= 2.5

UNSTABLE

STABLE

η0.001

∼ 1.5

b)

−20 −15 −10 −5 0 5 10 15 200

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

density gradient κnD

in k|| units

linea

r gr

owth

rat

e γ

in k

|| vT

D u

nits

nW0

= 1. 10−3

nW0

= 1. 10−4

κT = −20 k

||

FIG. 4. Impact of relative density of impurities on the linearthreshold of stability (a) in the plane of Deuterium gradients(κn, κT ) and on the linear growth rate γ (b) along the densitygradient κnD for a given value of the temperature gradientκT = −20k‖. Case of a Deuterium - Tungsten plasma (ND =500, NW = 52, amaxD = 5vTD, amaxW = 0.52vTD, κnW = 0).

and/or charge and mass of impurities. Such a result hasbeen observed by other authors14.

C. Effect of density gradient of impurities

We focus here on the case of a plasma composed ofDeuterium and Carbon. We study the effect of the radialprofile of impurities on the linear stability of the plasmain the plane of the main ions parameters (κnD and κTD).The temperature profiles are equalled (κTC = κTD), andwe connect the density gradients in any point of the sta-bility diagram with a fixed coefficient:

κnC = ξκnD (21)

Under such a condition, the effective charge profile(18), is easily linked to the density gradient of Deuterium

7

a)

−20 −15 −10 −5 0 5 10 15 20−40

−30

−20

−10

0

10

20

30

40

density gradient κnD

in k|| units

tem

pera

ture

gra

dien

t κT in

k|| u

nits

κnC

= κnD

κnC

= 0.1 κnD

κnC

= 10 κnD

η10

∼ 13.2

η0.1

∼ 1

STABLE

UNSTABLE

b)

−20 −15 −10 −5 0 5 10 15 200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

density gradient κnD

in k|| units

linea

r gr

owth

rat

e γ

in k

|| vT

D u

nits

κ

nc = κ

nD

κnc

= 0.1 κnD

κnc

= 10 κnD

κT = −10 k

||

FIG. 5. Impact of density peaking of impurities on the linearthreshold of stability (a) in the plane of Deuterium gradients(κn, κT ), and on the linear growth rate γ (b) along the densitygradient κnD for a given value of the temperature gradientκT = −10k‖. Case of a Deuterium - Carbon plasma (ND =30, NC = 12, amaxD = 5vTD, amaxC = 2.04vTD, nC0 = 0.1,Zeff = 2.9).

ions :

κZeff=

ZDnD0 ZCnC0(ZD − ZC)(1− ξ)κnD

(ZDnD0 + ZCnC0)2 (Z2DnD0 + Z2

CnC0)(22)

We represent in Fig. 5 the linear stability diagramof a plasma mixing Deuterium and Carbon. The ra-tio between the density of Deuterium and the densityof Carbon is constant. We choose a relatively high valuenC0 = 0.1nD0, in order to enlight the effect of the Carbondensity peaking.We observe that the stability threshold strongly de-

pends on the ratio between radial density gradients ofDeuterium and Carbon. In the case of a Carbon den-sity gradient lower than the Deuterium one (blue curve,κnC = 0.1κnD), the fluid asymptot (η0.1 ∼ 1) corre-sponding to large values of density and temperature gra-dients is lower than the classical value (η = 2), and theunstable area globally increases.

On the contrary, if we consider an impurity profilemore peaked than the majority ions (red curve, κnC =10κnD), we observe an exactly opposite effect. The un-stable area is much less extended, and the asymptoticvalue of the η ratio is clearly greater than the usual value(η10 ∼ 13.2).Representing the linear growth rate in Fig. 5 (b) en-

lightens these facts : the density peaking of impurity de-creases the width of the unstable area, while it increasesits maximum value. In other terms, the density peakingof the impurities stabilizes the slowly moving particlesof the plasma associated to the fluid description, anddestabilizes the fast ones. Such results are in qualitativeagreement with older works12,14.

V. DISCUSSION

Linear stability of a multi species plasma has beendiscussed within the framework of a cylindrical plasmacolumn (i.e. without magnetic curvature and/or mag-netic gradient effect). An interesting feature of the waterbag linear analysis is that there is no ambiguity aboutthe linear growth rates, because the water bag dielectricfunction presents at the most a unique couple of complexconjugate roots.The stability threshold has been shown to depend on

the relative densities of species. The coupling inducedby the impurities can destabilize the plasma. The am-plitude of destabilization is proportional to the mass ofimpurity considered. In the case of a flat density profileof impurities, increasing relative density of heavy ionsdestabilizes the plasma in the fluid branch, while it sta-bilizes the plasma in the central area associated to smalldensity gradients (kinetic branch).If the density profile is not flat, we have chosen to

link it to the main ions density profile by using a linearrelation. The relative peaking of impurities can stabilize(in the case of impurities more peaked than the mainions) or destabilize (with impurities more flat than mainions) the fluid branch of the ITG linear stability diagram.The two antagonist effects shown with flat density byplaying on the density ratios are recovered : the peakingof impurity density profile plays a role comparable to theincrease of their relative density or mass.The next step for the gyro water bag model is its

extension to toroidal geometry, that is currently underinvestigation32. Another interesting feature would beto take into account kinetic electrons with a water bagmodel, the main difficulty would be the large discrepancyin thermal velocities due to the ion to electron mass ratio.But such a difficulty is not due to our particular choiceof a water bag representation.

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