Anomalous diffusion, clustering, and pinch of impurities in plasma edge turbulence

0.54 MB·English

by  M. Priego

Anomalous diffusion, clustering, and pinch of impurities in plasma edge turbulence
∗
M. Priego, O. E. Garcia, V. Naulin, and J. Juul Rasmussen
Association EURATOM–Risø National Laboratory, OPL-128 Risø, DK-4000 Roskilde, Denmark
(Dated: January 7, 2005)
The turbulent transport of impurity particles in plasma edge turbulence is investigated. The
impurities are modeled as a passive fluid advected bythe electric and polarization drifts, while the
ambientplasmaturbulenceismodeledusingthetwo-dimensionalHasegawa–Wakataniparadigmfor
5
resistivedrift-waveturbulence. Thefeaturesoftheturbulenttransportofimpuritiesareinvestigated
0
by numerical simulations using a novel code that applies semi-Lagrangian pseudospectral schemes.
0
2 The diffusive character of the turbulent transport of ideal impurities is demonstrated by relative-
diffusionanalysisoftheevolutionofimpuritypuffs. Additionaleffectsappearforinertialimpurities
n
asaconsequenceofcompressibility. First,thedensityofinertialimpuritiesisfoundtocorrelatewith
a
thevorticityoftheelectricdriftvelocity,thatis,impuritiesclusterinvorticesofapreciseorientation
J
determinedbythechargeof theimpurityparticles. Second,aradial pinchscaling linearly with the
7
mass–charge ratio of the impurities is discovered. Theoretical explanation for these observations is

obtained by analysis of the model equations.
]
h
p
- I. INTRODUCTION netic fieldis relatedto thatofthe bulk plasma,since the
m motionperpendiculartothemagneticfieldofallcharged
s particlesintheplasmaisdominatedbyonesamevelocity
Oneofthepersistentproblemsinthedevelopmentofa
a field, the electric drift velocity. In this sense, impurities
l magneticfusionreactoristhedegradationofplasmacon-
p and bulk plasma are partly subject to a common mech-
finement caused by turbulent transport. In spite of the
. anism of turbulent transport, which relates the problem
s strongconfiningmagneticfield,highlevelsofenergyand
c of confinement to that of controlling the impurity con-
charged-particletransportacrossmagneticfield lines are
i tent. However, the distinctive nature of impurity parti-
s observedinavarietyofexperimentalconfigurations,such
y cles, notably their mass and electric charge, as well as
astokamaksandstellarators. Thistransporthasreceived
h their different influence on the plasma allow for dispari-
the adjective “anomalous,” as its features cannot be ex-
p ties between the transport of impurities and that of the
[ plained by the classical or neoclassical collisional theo-
ries. Instead,anomaloustransportisgenerallyattributed bulk plasma. In fact, recent experiments not only re-
1 tosmall-scale,electrostaticplasmaturbulence.1,2 Inpar- veal disparities in the transport of impurities and bulk
v plasma,4 but also in that of different impurity species.5
ticular, the transport in the plasma edge is believed to
9 Insuchexperimentalinvestigationsimpuritytransportis
3 be largelycontrolledby drift-waveturbulence, whichap-
usuallycharacterizedintermsofeffectivediffusioncoeffi-
0 pears naturally in this region as a result of the strong
cients and drift velocities,the latter alsoknownas pinch
1 radialpressuregradient. Thedrift-wavetransportmech-
velocities.1,6Thesetransportcoefficientsaretypicallyde-
0 anism has been the subject of much theoretical and ex-
5 terminedbyanalysisofthetemporalevolutionofinjected
perimental research over the past several decades, and
0 wasrecentlyreviewedbyHorton.2 Nevertheless,muchof impurity puffs. While the explanation for the diffusive
/ character of turbulent transport goes back to the work
s this transportphenomenonisyettobe fully understood.
c of Taylor and Richardson in the 1920s,7,8 the origin of
i Anotherimportantprobleminthequestforcontrolled the anomalouspinchvelocitiesisstillamatter ofintense
s
y thermonuclear fusion that is linked to turbulence con- research.6 Moreover, the inward pinch observed in lab-
h cernstheimpuritycontentoftheplasma. Theproperties oratory experiments tends to concentrate the impurities
p offusionplasmasarestronglyinfluencedbythe presence inthecenteroftheplasmacolumn,1 preciselywherethey
v: of impurities. For instance, impurities enhance radiative are least wanted, and thus constitutes one of the major
i energylossesanddilute the hydrogenfuelwithinthehot unresolved transport problems in fusion plasma physics.
X plasmacenter,atthesametimebeingthemainagentsin
r the erosion and deposition processes that affect plasma- Inthiscontributionweinvestigatetheturbulenttrans-
a facing components.3 Consequently, the understanding of portofimpuritiesinagenericmodelforplasmaedgetur-
the transportmechanismsthatcontrolthe impuritycon- bulence. Specifically, the investigations are based on the
tent in the plasma is of crucial importance for the per- simple and extensively studied model for resistive drift-
formance and safe operation of future fusion devices. wave turbulence proposed by Hasegawa and Wakatani.9
For the impurities, we use a consistent passive-fluid
The transport of impurity particles across the mag-
model that takes into account impurity-particle inertia,
that is to say, it contemplates the polarization drift.
Thisenablesthemodeltoexhibitdifferenttransportfea-
∗AlsoatDepartmentofMathematics,TechnicalUniversityofDen- tures depending on the mass and charge of the impuri-
ties, as observed in laboratory experiments. In particu-
markandDepartamentodeFluidosyCalor,UniversidadPontificia
Comillas–ICAI;E-mailaddress: [email protected] lar, the polarization drift brings compressibility into the
2
model, giving rise to subtle yet qualitatively important The HW model assumes an unperturbed, uniform
effects. The investigations are carried out by numerical magnetic field in the z-direction, that is, B = B ˆz. The
0
simulations of the model equations using a newly devel- x- and y-directions correspond respectively to the radial
oped code that applies semi-Lagrangian pseudospectral and the poloidal directions in a slab at the edge of the
schemes. We proceed separately for ideal (i.e., massless) confined plasma. The electrons are assumed isothermal
impurities and inertial ones. In the case of ideal impuri- with temperature T , whereas the ions are considered
e
ties, we focus on the phenomenon of turbulent diffusion, cold. The equilibrium plasma density n is assumed to
0
which is studied from the relative-diffusion perspective. behomogeneousinthepoloidaldirectionandtodecrease
Forinertialimpurities,weconcentrateontheneweffects in the radial direction in such a way that the equilib-
that arise from compressibility. First, the density of in- rium density length scale L ≡n /|dn /dx| is constant.
n 0 0
ertial impurities is found to correlate with the vorticity Hence, in this thin-layer approximation the second term
of the electric drift velocity, that is, impurities cluster in on the left-hand side of Eq. (1a) actually represents the
vorticesofapreciseorientationdeterminedbythecharge advectionoftheinhomogeneousequilibriumplasmaden-
oftheimpurityparticles. Thisbehaviorisadirectconse- sity by the electric drift.
quence of compression by the polarization drift and tur- The quantities in Eqs.(1) have been made dimension-
bulent mixing. Second, we discover a radial pinch that less using the gyro-Bohmnormalization:
scales linearly with the mass–chargeratio of the impuri-
x y c n L eϕL
ties. An understanding of this phenomenon is obtained s n n
→x, →y, t →t, →n, →ϕ.
by analysis of the evolution equation for the global im- ρs ρs Ln n0 ρs Te ρs
purity flux.
Thispaperisorganizedasfollows. InSec.IIthemodel Here, cs ≡ (Te/mi)1/2 is the sound speed and ρs ≡
for resistive drift-wave turbulence and the passive-fluid (miTe)1/2/eB0 isthethedriftscale,thatis,theiongyro-
model for the impurities are described. The diffusive ef- radius at the sound speed. The parameter C coupling
fectofturbulenceisdemonstratedinSec.IIIforthecase Eqs. (1a) and (1b) is known as the adiabaticity parame-
of ideal impurities. The effects that arise from impurity ter, and is defined as
inertia, the clustering and the anomalous radial pinch,
arerespectivelypresentedinSecs.IVandV. Theresults C ≡ Te kq2 .
aresummarizedanddiscussedinSec.VI. Lastly,theAp- e2ηqn0cs/Ln
pendix is devoted to a brief description of the numerical
schemes applied in the simulations. Here, ηq is the parallel resistivity and kq is the single
parallel wave number that is considered in the reduc-
tion to a 2D model. In the strongly collisional limit
C → 0, Eq. (1b) reduces to the 2D Navier–Stokes vor-
II. MODEL EQUATIONS
ticity equation. In this so-calledhydrodynamic limit the
equations become decoupled and the plasma density is
In this section we provide a brief presentation of the
passively advected by the electric drift velocity. In the
resistive drift-wave paradigm used for modeling the tur-
weakly collisional limit C → ∞, the adiabatic electron
bulenceattheplasmaedge. Wealsodescribethepassive-
response yields Boltzmann-distributed density perturba-
fluidmodelforthe impurityspecies,whichtakesintoac-
tions, n ≈ ϕ, and the HW model reduces to the well-
count impurity-particle inertia.
known Hasegawa–Mimaequation.10
The diffusive terms in Eqs. (1a) and (1b) correspond
respectively to the collisional diffusion of electrons and
A. Resistive drift-wave turbulence
the ion viscosity. These dissipative processes are essen-
tial for the stabilization of resistive drift modes as well
As a basic model for plasma edge turbulence we
as for the feasibility of numerically simulating the HW
consider the two-dimensional (2D) Hasegawa–Wakatani
model. Linear analysis of the HW model reveals the ex-
(HW) model for resistive drift-wave turbulence:9
istence ofunstable modes for 0<C <∞, with allmodes
D (n−x)=C(ϕ−n)+µ ∇2n, (1a) beingunstableintheabsenceofdissipation.11,12 Itisthe
t n ⊥
combination of this linear instability with nonlinear in-
Dtω =C(ϕ−n)+µω∇2⊥ω, (1b) teractionthroughthe advective terms and dissipationat
smallscalesthatleadstoquasistationaryturbulentstates
where within the HW model. In particular, when initialized
with a broadband set of small-amplitude, random-phase
ω ≡∇2ϕ, (1c)
⊥ modes, the system evolves in two stages: first, a linear
Dt ≡∂t+ˆz×∇⊥ϕ·∇⊥. (1d) phaseinwhichthe linearlyunstablemodesgrowandthe
dampedones decay;second,a phase ofnonlinear satura-
Here,ndenotesthefluctuatingcomponentoftheplasma tion leading to a quasistationary turbulent state whose
density, ϕis the electrostaticpotential, andω is the vor- statistics solely depend on the parameters of the HW
ticity of the electric drift velocity vE =ˆz×∇⊥ϕ. model.
3
Two relevant nonlinear invariants in the HW model turbulence model, the flow velocity of the impurities is
are the total energy E and the generalized enstrophy U, hence given by
defined as
vθ =vE +vp =ˆz×∇⊥ϕ−ζDt∇⊥ϕ,
1
E ≡ (n2+|∇⊥ϕ|2)dx,
2Z where
1 mθ e ρs
U ≡ (n−ω)2dx. ζ ≡ .
2Z qθ miLn
The evolution of these quantities within the HW model Here,mθ andqθ arerespectivelythemassandtheelectric
is governed by charge of the impurity particles. The impurity model
resulting from the continuity equation is thus given by
dE
dt =Γn−Γc−DE, Dtθ−∇⊥· θζDt∇⊥ϕ =µθ∇2⊥θ, (2)
(cid:0) (cid:1)
dU =Γ −DU, where the term on the right-hand side arises from colli-
n
dt sional diffusion.
The polarization drift, which represents the effect of
where
impurity-particle inertia, constitutes a higher-order cor-
rection to the electric drift velocity. The influence of
Γn ≡− n∂yϕdx, the former on the dynamics of impurities is determined
Z
by the dimensionless parameter ζ, which varies with the
Γ ≡C (n−ϕ)2dx, mass–charge ratio of the impurity particles relative to
c
Z that of the bulk-plasma ions. For ideal impurities ζ = 0
and the impurities are solely advected by the electric
DE ≡ (µn|∇⊥n|2+µω|∇⊥ω|2)dx, drift. For inertial impurities ζ has a nonzero value and
Z
the polarization drift can play a role in the dynamics.
DU ≡ ∇⊥(n−ω)·∇⊥(µnn−µωω)dx. The value ofζ is inanycasesmall,since the lengthscale
Z quotientρ /L isbyassumptionsmall. Nevertheless,the
s n
polarization drift adds important qualitative features to
Hence, Γ is the only source in the system, and corre-
n the dynamics of impurities. In the case of static, uni-
sponds to energy extracted from the equilibrium density
form magnetic field the electric drift is incompressible,
gradient. Conversely, Γ is a sink corresponding to re-
c whereasthepolarizationdriftiscompressible. Hence,the
sistivedissipationthroughtheparallelcurrent,whileDE
inclusion of the polarization drift in the impurity model
andDU aresinksarisingfromcollisionalityandviscosity.
allows for genuinely compressible effects, such as those
Beingthesolesourceofenergyandgeneralizedenstrophy
described in Secs. IV and V.
source within the HW model, Γ is in practice positive
n
definite.11 This factis usedinSec.Vto identify a source
for radial drift of impurities.
III. DIFFUSION OF IDEAL IMPURITIES
Theoretical studies of the diffusive effect of plasma
B. Impurity passive-fluid model
edge turbulence on passively advected impurities have
previously relied on analysis of the absolute dispersion
Impurities are here regardedas a passive species, that of test particles.13,14 The presentapproachis necessarily
is, they are advected by the turbulence but do not have
different,astheimpuritiesarenotmodeledhereasparti-
any influence on it. This is a reasonable approximation
cles but as a fluid. Inthis section we investigatethe tur-
for the situation in which the density of impurity par-
bulent diffusion of ideal impurities by relative-diffusion
ticles is much lower than that of the bulk plasma. The analysis of the evolution of impurity puffs.15,16
modelingofthedynamicsofimpuritiesisconsistentwith
that of the turbulence, in that it also follows from the
drift-ordered fluid approach and the assumption of un- A. Relative diffusion
perturbed, uniform magnetic field.
The impurities are characterized by a density θ and
Weconsiderthereleaseofapuffofidealimpuritiesinto
a flow velocity v . Just as in the case of the ions in
θ the turbulent field, which is a saturated turbulent state
the bulk plasma, the impurities are assumed cold, with
described by the HW model. The amount of impurities
their motion parallelto the magnetic field likewise being
released is
neglected. Therefore, the motion of the impurity fluid
comprises only the electric drift v and the polarization
E Q≡ θdx,
drift vp. Under the same normalization applied to the Z
4
and the position c of the center of mass of the impurity evolutionofS canbecharacterizedbyeffectivediffusiv-
ii
puff at any time instant is given by ities D such that
i
1 dS
c≡ xθdx. ii =2D . (3)
QZ dt i
We define the relative position vector x′ ≡ x−c, which These diffusivities solely depend on the statistical prop-
indicates position in a reference frame moving with the erties of the turbulent velocity field—more precisely,
center of mass of the puff. The average evolution of im- they are related to the mean squared velocities and the
purity puffs can then be characterized by the following Lagrangianrelative time scale.
ensemble-averageddispersion tensors: The last phase, dominated by turbulent diffusion, is
mostrelevantforus,asiteventuallydominatesthetrans-
1 port of impurities. In fact, the effective diffusivities D
Σ ≡ x x θdx , i
ij i j
Q(cid:28)Z (cid:29) that characterize the phase of turbulent diffusion are
equivalentintheirdefinitiontothoseemployedintheex-
S ≡ 1 x′x′θdx , perimentalmodelingofanomalousimpuritytransport.1,6
ij Q(cid:28)Z i j (cid:29) Becauseinthis laststagethe diagonalelements S grow
ii
linearlyaccordingtoEq.(3),theeffectivediffusivitiesD
M ≡hc c i. i
ij i j can easily be obtained from the asymptotic evolution of
Here, the components of vectors and tensors have been Sii. This is the procedure used here for estimating the
indicated by subindices and the angular brackets stand effectivediffusivitiesoftheturbulentfieldactingonideal
for the ensemble average,that is, the averageover many impurities. Inprinciple,anyinitialpuffscouldbeusedfor
releases. Theabsolute-dispersiontensorΣ measuresthe the determination of the diffusivities. From the discus-
ij
dispersion with respect to the origin of the fixed frame, sionabove,largeinitialpuffs rapidly experiencethe final
whereas the relative-dispersion tensor S measures the phase of turbulent diffusion, and thus provide the effec-
ij
dispersionrelativeto the centerofmass. The tensorM tivediffusivitiesshortlyaftertheirrelease. However,care
ij
quantifies the “meandering” of the center of mass. The mustbe takeninnumericalsimulationsthatthe puffsdo
dispersion tensors are related by not reach the boundaries of the simulation domain, as
this would result in underestimates for the diffusivities.
Σ =S +M , In order to determine the effective diffusivity in one di-
ij ij ij
rection, we take Gaussian stripes in the perpendicular
which means that absolute dispersion consists of rela- direction as initial puffs and use periodic boundary con-
tive dispersion and meandering of the center of mass. ditions. In this way, the puffs are as large as possible in
Relative-diffusionanalysisfocusesontheevolutionofSij, onedirectionwhile the conclusionsofthe simulationsre-
that is, on the spreading of the puff relative to its center main valid. Moreover, because Gaussians are similarity
of mass. Specifically, it is usually the diagonal elements solutions to the diffusion equation, the evolution of S
ii
Sii that are of interest once the reference axes are ade- would be perfectly linear if the effect of turbulence were
quately selected, in our case those corresponding to the purely diffusive from the start.
radialand poloidal directions. In view of the above rela-
tion, the meandering of the center of mass is eliminated
in relative-diffusion analysis, which thus leads to better B. Simulation results
statistics in comparison with those of absolute diffusion.
Theevolutionofthedispersiontensorsdependsonthe
The HW model for plasma edge turbulence, Eqs. (1),
initial size and shape of the puffs, the statistical prop-
and the model for ideal impurities, Eq. (2) with ζ =
erties of the turbulent velocity field, and the collisional
0, were simultaneously solved numerically using the
diffusivity µ . Under the assumption that µ is small,
θ θ schemes described in the Appendix. The simulations
the evolution of S for initially concentrated puffs fol-
ii were carried out on a doubly periodic square domain of
lows three stages: first, a phase dominated by collisional
side length 40 using 512×512 grid points. The viscosity
diffusion that lasts while the puff is much smaller than
µ and the collisionaldiffusivities µ and µ were set to
ω n θ
the turbulent structures and is characterizedby
the common value 0.02 in all the simulations presented
inthis paper. We consideredfour differentvaluesfor the
dS
ii
≈2µθ; adiabaticity parameter C in the HW model: 0.5, 1, 2,
dt
and4. As notedinthe precedingsection,lowvalues ofC
second, a transitional phase in which the size of the puff correspond to hydrodynamic-like regimes, whereas high
is comparableto thatofthe turbulentstructuresandS valuescorrespondtonearlyadiabaticregimes. Inthisre-
ii
typicallyevolvefasterthanlinearlywithtime(seeRef.15 spect,weregardC =1asatransitionalvalue,representa-
for further discussions); and last, a phase dominated by tiveofanintermediateregime. Aninitialturbulentstate
turbulentdiffusionthatstartswhenthepuffislargecom- was achieved by letting a uniform distribution of small-
pared to the turbulent structures and during which the amplitude, random-phase modes evolve for a sufficiently
5
long time in order to reach a quasistationary saturated
turbulentregime. Gaussianstripeshavingstandarddevi-
ation2wereusedasinitialimpuritypuffs. Foreachvalue
of the adiabaticity parameter and each direction, radial
and poloidal, 20 releases of such stripes were simulated,
and the dispersion tensors were subsequently averaged.
In Fig. 1 we present snapshots of the evolution of two
perpendicularstripesofimpuritiesreleasedintothesame
saturated turbulent field with C = 1. The anisotropy in
the diffusion of impurities is evident in the figure, as the (a) (b)
poloidal spread of the puff in Fig. 1(e) is significantly
larger than the corresponding radial spread in Fig. 1(f).
Hence, diffusion acts in this case faster in the poloidal
direction than in the radial direction. The degree of
anisotropy is in fact controlled by the adiabaticity pa-
rameter,the turbulence ranging from isotropicin hydro-
dynamic limit, C →0, to strongly anisotropic in the adi-
abatic limit, C →∞.13
The evolutionof the diagonalelements S of the rela-
ii
tivedispersiontensorforthefourvaluesoftheadiabatic- (c) (d)
ity parameter is shown in Fig. 2. For the three smallest
values of C, it is possible to distinguish the transitional
and turbulent phases of growth in the plot for relative
radial dispersion. In the plot corresponding to poloidal
dispersionthereisnocleardistinction,astheevolutionof
S very rapidly becomes approximately linear. For the
22
sidelengthofoursimulationdomain,dispersioncanstart
becomingunderestimatedwhenS reachvaluesofabout
ii
65. Nevertheless, for the three smallest values of C the
linear trend in S characterizing the phase of turbulent
ii
diffusion is establishedbefore sucha point is reached. In
(e) (f)
contrast, the eventually linear regime is neither appar-
ent in the radialnor the poloidaldispersionplots for the
FIG. 1: (Color online) Evolution of Gaussian stripes of ideal
highest value of the adiabaticity parameter, C =4. This
impurities in a HW saturated turbulent state with C = 1:
different behavior is attributed to the weakerturbulence
radial stripe at (a) t = 0, (c) t = 6, and (e) t = 12; and
athighervaluesofCandtheparticulartransportfeatures
poloidal stripe at (b) t=0, (d) t=6, and (f) t=12.
of the adiabatic limit, the Hasegawa–Mimaequation.17
EstimatesfortheeffectivediffusivitiesD oftheturbu- TABLE I: Effective diffusivities in the radial direction D1
i
lenceforthe threesmallestvaluesofC areobtainedfrom and the poloidal direction D2 of HW saturated turbulence
for different values of C.
linear fits to the linear portions of the curves in Fig. 2.
Theestimatedvaluesfortheradial,D1,andpoloidal,D2, C D1 D2
effectivediffusivitiesarepresentedinTableI. Thevalues 0.5 1.12 2.32
obtainedherearequalitativelysimilartothosecalculated
1 0.65 2.26
in Ref. 13 by means of a test-particle, absolute-diffusion
2 0.36 2.63
approach. Inparticular,thedecreaseofD withC aswell
1
as the rise of D with C for C ≥ 1 are here reproduced.
2
We notethatthe accuracyofthe presentapproachcould
be improved by simulating the evolution of the impurity IV. CLUSTERING OF INERTIAL IMPURITIES
puffs for longer times on a larger domain. The number
of puff releasescould as well be increased in order to en- The importance ofinertia in the advectionofparticles
sureconvergenceto the ensemble averages. We alsonote by turbulent flows is well known in fluid dynamics.18 As
that different values for µ and µ were used in the lat- inmagnetizedplasmas,thetransportofinertialparticles
ω n
terreference,withcollisionaldiffusionofimpuritiesbeing by fluids shows compressible features even when the ad-
discarded because of their test-particle modeling. vecting flow is incompressible. Particle inertia has been
6
90 transitional value 1, while we contemplated various val-
80 uesfortheparameterζ,rangingfrom−0.01to0.05. The
70 impurity density field θ wasinitialized with the constant
value θ = 1, and the initial turbulent fields were taken
60 0
from a HW saturated turbulent state.
50
2 In Fig. 3 we present side by side snapshots of the evo-
2
S 40 lution of the vorticity ω of the electric drift velocity and
30 the impurity density field for the case ζ = 0.01. Merely
5 time units after the simulation is started, there is al-
20
ready indication of a correlation between vorticity and
10
impurity density. Att=50,there is visuallylittle differ-
0 encebetweenthetwofields. Figure4showsascatterplot
70 ofvorticityandrelativeimpurity densityθ/θ0 att=100
for three different values of ζ. The plot clearly suggests
60
an approximate linear relation given by
50
θ/θ ≈1+Kω,
0
40
1
1 with the regression coefficient K depending on ζ. The
S
30 least-squares estimates of the coefficient K are plotted
as a function of the parameter ζ in Fig. 5. This figure in
20
turn indicates a linear relation between K and ζ in the
10 form K ≈ 0.82ζ. In summary, we find that the density
of inertial impurities eventually becomes linearly corre-
0
0 5 10 15 20 lated with the vorticity, the regression coefficient being
t proportional to the parameter ζ.
FIG.2: EvolutionoftherelativeradialdispersionS11andthe
relative poloidal dispersion S22 of ideal impurity Gaussian B. Turbulent mixing and clustering
stripes released in HW saturated turbulence with C = 0.5
(solid), C = 1 (dashed), C = 2 (dotted), and C = 4 (dash-
The behavior just described can easily be explained
dotted).
using an argument of turbulent mixing, or turbulent
equipartition.20,21,22 Performing basic algebraic manip-
ulations on Eq. (2) while accounting for the particular
showntoresultinclusteringinvorticalstructures,andits
definition of the Lagrangian derivative D in Eq. (1d),
role in phenomena such as planetary formationhas been t
suggested.19 While the dynamics ofimpurity particlesin we mayrewrite the impurity modelequationin the form
magnetizedplasmasare notgovernedby the sameforces
amsaignnefltuizideds,pinlaesrmtiaaseinntearssimthieladrywnaaymaiscsthoefCimopriuorliistifeosrcine Dt(lnθ−ζω)=ζ∇⊥lnθ·Dt∇⊥ϕ+ µθθ ∇2⊥θ, (4)
entersthoseofheavyimpuritiesinrotatingfluids. Hence,
where the left-hand side describes advectionby the elec-
corresponding clustering effects for inertial impurities in
tricdriftandcompressionbythepolarizationdrift,while
magnetized plasmas should be expected. In this section
the right hand-side describes advection by the polar-
wediscoverandexplainthecorrelationbetweenimpurity
ization drift as well as collisional diffusion. We will
densityandvorticitythatresultsfromthecompressibility
now argue that the right-hand side of this equation is
introducedbythepolarizationdriftinthecaseofinertial
small, which implies that lnθ −ζω is approximately a
impurities. In order to most clearly reveal this effect we
Lagrangian invariant, or in other words, that lnθ −ζω
considerahomogeneousinitialdistributionofimpurities,
is nearly conserved along the trajectories defined by the
since sucha distribution would remainunchangedin the
electric drift velocity. We split the impurity density θ
incompressible, that is, ideal-impurity case.
intoitsmeanvalueθ andthefluctuationsθ arisingfrom
0 1
compressibility. Based on the computational results, we
further postulate that θ /θ = O(ζ). Selectively apply-
A. Simulation results 1 0
ing the split θ ≡θ +θ to Eq. (4), we obtain
0 1
Simulations of the HW model for plasma edge turbu- θ µ
lence, Eqs. (1), and the impurity model, Eq. (2), were Dt(lnθ−ζω)=ζ∇⊥ln(cid:18)1+ θ1(cid:19)·Dt∇⊥ϕ+ θθ ∇2⊥θ1.
0
again performed on a doubly periodic square domain of
side length 40 using 512×512 grid points. The adia- Assuming that the normalizations used in the impurity
baticity parameter C in the HW model was set to the modeladequatelyrepresentthescalesoffluctuations,the
7
1.5
1.4
1.3
1.2
0 1.1
θ
/
θ 1
0.9
0.8
0.7
0.6
(a) (b) −8 −6 −4 −2 0 2 4 6 8
ω
FIG.4: (Coloronline)Scatterplotofvorticityω andrelative
density θ/θ0 of inertial impurities in a HW saturated turbu-
lent state with C = 1 for ζ = 0.05 (red/steepest), ζ = 0.01
(green), and ζ =0.002 (blue/flattest).
0.05
0.04
0.03
(c) (d)
K 0.02
0.01
0
−0.01
−0.01 0 0.01 0.02 0.03 0.04 0.05
ζ
FIG.5: Least-squaresestimatesofthecoefficientK asafunc-
tion of ζ for the impurity-density–vorticity linear regression
θ/θ0 ≈1+Kω inaHWsaturatedturbulentstatewithC =1.
(e) (f)
The fittedline (dashed) corresponds to K =0.82ζ.
FIG. 3: (Color online) Evolution of the vorticity ω and the
density θ of inertial impurities (ζ =0.01) in a HW saturated
be recast into the simpler form
turbulent state with C = 1: (a) ω and (b) θ at t = 0, (c) ω
and(d)θatt=5,and(e)ωand(f)θatt=50. Greencorre-
θ/θ ≈1+ζω.
sponds to mean values, while red and bluerepresent positive 0
and negative fluctuations respectively.
The present analysis thus predicts a linear relation be-
tween the impurity density and the vorticity, the regres-
sion coefficient being the parameter ζ. Hence, the the-
right-handside of this last equation is of order ζ2+ζµ ,
θ oretical argument explains the effect discovered in the
whereasthefluctuatingcomponentofthequantityinside
simulations up to a slight mismatchin the regressionco-
the Lagrangianderivative on left-hand side is of order ζ.
efficient. We emphasize that the particular features of
Hence, lnθ−ζω variesslowlyalongthe electricdrifttra-
the HW turbulence model were not used in the previous
jectories, thus constituting an approximate Lagrangian
argument,the conclusionsrelyingsolelyonthe modeling
invariant.
of the dynamics of impurities and the argument of tur-
Turbulent mixing homogenizes Lagrangian invariants, bulent mixing. It follows that the clustering of inertial
andthustendstodrivesystemstostatesofso-calledtur- impurities in vortical structures is a generic effect, in-
bulentequipartition. Theapplicationofthisargumentto dependent of the underlying instability mechanisms and
the approximate invariant here yields the specific characteristics of the turbulence.
Thefactthatthedensityofinertialimpuritiesbecomes
lnθ−ζω ≈const. directly correlated with the vorticity implies that impu-
rities of positive charge cluster in positive vortices, the
Because the amount of impurities is conserved and it is opposite taking place for negatively charged impurities.
assumed that θ /θ =O(ζ), the previous expression can As previously introduced, this effect is similar to the ag-
1 0
8
0.005
gregation of heavy particles in anticyclonic vortices that
cantakeplaceinrotatingfluidsasaresultoftheCoriolis
force.18 Incontrastwithcompressionbythe polarization 0
drift, which yields D lnθ ∼ ζD ω, compression by the
Coriolis force enters tthe dynamicts of heavy impurities in ˆx−0.005
·
the form D lnθ ∼ −2Ωω, with Ω being the overall rota- θ
t V−0.010
tion. Hence, aggregation in the rotating-fluid case does
not necessarily lead to a linear relation between impu-
−0.015
rity density and vorticity, but can go on further to form
highly concentrated clusters in the cores of anticyclonic
−0.020
vortices. For this reason, it has for example been sug- 0 25 50 75 100 125 150
gestedas a mechanism, in combinationwith gravitation, t
forplanetaryformationinrotatingastrophysicaldisks.19
FIG.6: EvolutionoftheradialdriftvelocityVθ·xˆofinertial
impurities in a HW saturated turbulent state with C =1 for
V. PINCH OF INERTIAL IMPURITIES ζ =−0.01(solid),ζ =0.005(dashed),ζ =0.02(dotted),and
ζ =0.05 (dash-dotted).
In this section we finally investigate the possible role
of inertia in the radially inward pinch that concentrates 0.001
impuritiesinthecenteroftheplasmacolumninmagnetic
0
confinement devices.1,5 As a measure of the overall drift
of impurities, we consider the global impurity flux −0.001
ˆx
Γθ ≡ΓEθ +Γpθ ≡Z θvEdx+Z θvpdx. ·Vθ−0.002
−0.003
Here,wehavedistinguishedthe componentsrespectively
−0.004
resulting from advection by the electric drift and by the
polarization drift. In terms of these global fluxes, we −0.005
define the impuritydrift velocityV andits components −0.01 0 0.01 0.02 0.03 0.04 0.05
θ
VE and Vp by means of ζ
θ θ
V ≡VE +Vp ≡Q−1ΓE +Q−1Γp, FIG. 7: Time-averaged radial drift velocity Vθ·xˆ of inertial
θ θ θ θ θ
impurities as a function of ζ in a HW saturated turbulent
where once againQ≡ θdx. For a homogeneous distri- with C =1. The fitted line (dashed) corresponds toVθ·xˆ=
bution of impurities inRa periodic domain, both ΓE and −0.090ζ.
θ
Γp are zero. Hence, ideal impurities cannot experience a
θ
net drift when their initial distribution is homogeneous,
since such a distribution remains unchanged in the ideal a continuous radially inward drift, as observed in exper-
case. In contrast, inertial impurities may experience a imental investigations of impurity transport.1,5 In Fig. 7
drift even if their initial distribution is uniform, given we show the time-averaged radial drift velocities Vθ ·xˆ
thatthelatterisalteredbycompressibleeffects,asshown in the saturated regimes, computed using the values of
in the preceding section. the drifts between t = 25 and t = 150. The variation of
thetime-averagedradialdriftwithζ isremarkablylinear,
well fitted by V ·xˆ = −0.090ζ. Therefore, the present
θ
A. Simulation results radial drift scales linearly with the mass–charge ratio of
the impurity particles.
WebeginbymonitoringtheradialdriftvelocityV ·xˆ The reason for this peculiar pinch effect is not clear
θ
in the simulations referred to in Sec. IV. These were at the outset. An analysis of the components VE and
θ
initializedwithhomogeneousdistributionsofinertialim- Vp of the drift velocity in the simulations reveals that
θ
purities in a HW saturated turbulent state with C = 1. the contribution of the electric drift is by far dominant.
In Fig. 6 we present the evolution of the radial drift ve- Hence, while the net drift arises as a result of the polar-
locity for four different values of ζ. In every case, the izationdrift, the mainroleofthe latterisnottoglobally
evolution of the radial drift comprises a strong transient advect impurities, but rather to distribute them in such
burst, after which the drift enters a saturated quasista- a way that they experience net transport by the electric
tionaryregime. Moreover,theradialdriftvelocityisseen drift. From a naive point of view, it could be thought
tohaveadefinitesignoppositetothatofζ,thatistosay, that the correlation between impurity density and vor-
oppositetothetypeofchargeoftheimpurityparticles. It ticity discovered in Sec. IV is responsible for the radial
follows that positively charged impurities are subject to drift. Indeed, transport of trapped particles by coher-
9
ent vortices is for instance known to play an important We will now resort to the particular HW turbulence
role in transport processes in rotating fluids.18 However, model to show that Λ constitutes a source for global
2
because an incompressible flow in a 2D periodic domain radial flux in the present case. Substitution of the HW
cannotgloballytransportitsvorticity,apurelylinearre- vorticity equation in the above expression for Λ yields
2
lationbetweenimpuritydensityandvorticitywouldyield
no drift at all. Thus, the explanation for the radial drift Λ = ζθ C(ϕ−n)+µ ∇2ω v dx
2 ω ⊥ E
of inertial impurities requires deeper thought. Z
(cid:2) (cid:3)
=−ζCθ nv dx+O(ζ2+ζµ ).
0 E ω
Z
B. Net transport and pinch
Projectingthisequationontotheradialdirectionwefind
Intrigued by the initial burst and the subsequent sat-
uration of the radial impurity drift velocity, we analyze Λ2·xˆ=−ζCθ0 n∂yϕdx+O(ζ2+ζµω)
Z
the evolution of the global impurity flux. In particular,
wefocusonthe ΓE component,giventhatthenettrans- =−ζCθ0Γn+O(ζ2+ζµω).
θ
portofimpurities isdominatedbythe electricdrift. Ina
periodicdomain,the evolutionofΓE under theimpurity Here,wehaveidentifiedthe solesourceΓn ofenergyand
θ generalizedenstrophywithinthe HWmodel. Since Γ is
model in Eq. (2) is governedby n
inpracticepositivedefinite (see,e.g.,TableIinRef.11),
dΓE it follows that Λ2·xˆ has a definite sign opposite to that
θ =Λ1+Λ2+Λ3+Λ4, of ζ, thus acting as a source for the global radial flux
dt
ofinertialimpurities. Asexplainedbefore,Λ isinitially
2
where theonlynonzerocontributiontodΓE/dt,thusexplaining
θ
the transient bursts observed in Fig. 6. More generally,
Λ ≡ θD v dx, this term sustains the radial transport of impurities in
1 t E
Z the turbulent states described by the HW model.
From the above estimates, we are led to expect that
Λ ≡ ζθv D ωdx,
2 E t Λ is responsible for the saturation of the radial impu-
Z 1
rity drift seen in Fig. 6. The reason for this saturation
Λ3 ≡ ζ(∇⊥θ·Dt∇⊥ϕ)vEdx, becomes clear when we account for the correlation that
Z develops between impurity density and vorticity. The
argumentof turbulent equipartition used in Sec. IV pre-
Λ4 ≡ µθvE∇2⊥θdx. dictedaneventualrelationθ /θ ≈ζωbetweentheimpu-
Z 1 0
rity density fluctuations and the vorticity. Considering
These four contributions correspond respectively to evo- this relation and the fact that vorticity is not globally
lutionoftheelectricdriftvelocity,compressionbythepo- transported by the electric drift we obtain
larization drift, advection by the polarization drift, and
collisional diffusion of impurities. In order to reveal the
Λ ≈ ζθ ωD v dx
1 0 t E
magnitude of the different terms, we again split the im- Z
puritydensityintoitsmeanvalueθ0 andthefluctuations d
=ζθ ωv dx−ζθ v D ωdx
θ1, keeping in mind that θ1/θ0 = O(ζ). Selectively ap- 0dtZ E 0Z E t
plying the split θ ≡ θ +θ to the above definitions we
0 1
obtain =− ζθ0vEDtωdx≈−Λ2
Z
Λ1 = θ1DtvEdx=O(ζ), whenever turbulent equipartition holds true. Conse-
Z
quently, when such a state is reached, the two terms
dominating the evolution of the radial drift approxi-
Λ = ζθv D ωdx=O(ζ),
2 E t
Z mately cancel, and the radial drift enters a quasista-
tionary regime. The time it takes for this regime to be
Λ3 = ζ(∇⊥θ1·Dt∇⊥ϕ)vEdx=O(ζ2), reachedisindependentofζ, asthe developmentofcorre-
Z
lation is exclusively dependent on the turbulent mixing
Λ = µ v ∇2θ dx=O(ζµ ). bytheelectricdrift. Infact,theevolutionofθ1isapprox-
4 θ E ⊥ 1 θ
Z imatelylinearwithζ withinshorttime intervals,sincein
line with the discussion in Sec. IV it follows that
According to these estimates, Λ and Λ dominate the
1 2
dynamics of the global impurity flux. In the case of uni- D θ =ζθ D ω+O(ζ2+ζµ ).
t 1 0 t θ
form impurity density, Λ is the only nonvanishing con-
2
tribution to dΓE/dt, being therefore responsible for the Giventhatsolelythedensityfluctuationsyieldnettrans-
θ
initial development of drift velocities. port of impurities, the evolution of the impurity drift is
10
likewise approximately linear with ζ. This explains the charge ratio of the impurity particles and is inward for
apparent linear scaling of the curves in Fig. 6 and the positively charged impurities. This anomalous pinch re-
linearity of the time-averagedradial drift in Fig. 7. liesontheparticularfeaturesoftheturbulencedescribed
Whiletheargumentspresentedheredonotyieldapre- by the HW model. In contrast with other known turbu-
dictionforthequasistationaryleveloftheradialimpurity lent pinches,6 the present pinch is neither caused by a
drift, they do identify the mechanism for its onset and specificmagneticfieldgeometrynorbytemperaturegra-
likewise explain its scaling with the parameter ζ. We dients,giventhatourmodelsassumeauniformmagnetic
emphasize that the former mechanism relies on the spe- field and cold impurities. Consequently, our finding of a
cific properties of the turbulence described by the HW net radial drift due to impurity-particle inertia consti-
model. Nevertheless, an analogue pinch effect would ap- tutes a new contribution towards the understanding of
pearforotherdrivinginstabilitiesprovidedthattheterm the anomalous pinch of impurities in magnetic confine-
Λ remains a source for a definite impurity drift. ment devices.
2
VI. CONCLUSION Acknowledgments
We have studied the transportof impurity particles in The first author is grateful to M. P. Sørensen for co-
plasmaedgeturbulence, basingourinvestigationsonthe supervising his M.Sc. thesis, which constitutes the basis
paradigmaticHasegawa–Wakatani(HW)modelforresis- for this publication. O.E.G. was sponsored by the Re-
tive drift-wave turbulence and a consistent passive-fluid search Council of Norway.
model for the impurities. The latter model takes into
account impurity-particle inertia, which enters the im-
purity flow velocity in the form of the polarization drift.
APPENDIX: NUMERICAL SCHEMES
Inertiahasindeedbeenfoundtoplayasignificantrolein
theturbulenttransportofimpurities,sinceitgivesriseto
The numerical simulations presented in this paper
subtle yet qualitatively important compressible effects.
were performed using a semi-Lagrangian (SL) pseudo-
The model equationshave beeninvestigatedboththe-
spectralcodedevelopedbythe firstauthor.23 Inessence,
oretically and computationally. The numerical simula-
SL schemes combine the very stable Lagrangian treat-
tions were performed using the semi-Lagrangian (SL)
ment of advection with the convenience of a fixed spa-
pseudospectral code briefly described in the Appendix.
tialdiscretization.24,25Thisisachievedbyintegratingthe
Although the emphasis of this paper is not on the nu-
partialdifferentialequationsinquestionalongadifferent
merical methods employed, we point out the suitability
family of advective trajectories in each time step, pre-
ofSLschemesfortheadvection-dominatedproblemsthat
cisely those that by the end ofthe time step traversethe
appear in plasma turbulence. In particular, we highlight
fixed points on which the spatial discretization is based.
thesuperiorstabilityoftheSLsemi-implicitschemeused
We now outline how the aforementioned code combines
here for the HW model, which is known to constitute a
SL method and the pseudospectral discretization to nu-
numerically challenging nonlinear problem.
merically solve the HW model and the impurity passive-
Inthecaseofidealimpurities,wehavefocusedontheir
fluid model described in Sec. II.
turbulent diffusion by the electric drift. This effect has
We discretize time using a constant time step ∆t,
been demonstrated by relative-diffusion analysis of the
which yields the sequence {t } of time instants related
evolution of impurity puffs. The effective diffusivities m
byt =t +∆t. Likewise,weusetheFourierpseudo-
of the turbulent advecting field have been calculated for m+1 m
spectral method to discretize the doubly periodic rect-
several values of the adiabaticity parameter in the HW
angular simulation domain.24 This results in a regular
model. The resulting turbulent diffusivities are in quali-
grid {x } of collocation points. In our application of
tative agreement with those obtained in an earlier study i
based on an absolute-diffusion, test-particle approach.13 the SL method we consider the (incompressible) electric
In the case of inertial impurities, we have discovered drift velocity vE =ˆz×∇⊥ϕ as the sole advecting field.
Hence,theadvectivetrajectoryX(x ,t ;t)thatpasses
thattheirdensityeventuallycorrelateswiththevorticity i m+1
through the grid point x at time t is given by
of the electric drift velocity. This clustering effect scales i m+1
linearly with the mass–charge ratio of the impurity par-
d
ticles and results from compression by the polarization X(x ,t ;t)=v X(x ,t ;t),t ,
i m+1 E i m+1
dt
drift incombinationwith turbulentmixing. The cluster- (cid:0) (cid:1)
ing of impurities in vortices of definite sign is a generic X(x ,t ;t )=x .
i m+1 m+1 i
effectinmagnetized-plasmaturbulence,inthesensethat
itisindependentofthespecificcharacteristicsofthetur- At each time step, we firstly determine the so-called up-
bulence. stream points {X(x ,t ;t ) ≡ x −α }. These are
i m+1 m i i
Finally, we have also found an overall radial drift of numerically calculated using the following second-order
inertial impurities that scales linearly with the mass– scheme, which combines the implicit midpoint rule and