The toroidal momentum pinch velocity

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by  A. G. Peeters

The toroidal momentum pinch velocity
A.G. Peeters, C. Angioni, D. Strintzi
Max Planck Institut fuer Plasmaphysik, EURATOM association, Boltzmannstrasse 2 85748 Garching, Germany
7
In this letter a pinch velocity of toroidal momentum is shown to exist for the first time. Using
0
the gyro-kinetic equations in the frame moving with the equilibrium toroidal velocity, it is shown
0
that the physics effect can be elegantly formulated through the “Coriolis” drift. A fluid model is
2
usedtohighlightthemaincouplingmechanismsbetweenthedensityandtemperatureperturbations

n on the one hand and the perturbed parallel flow on the other. Gyro-kinetic calculations are used
a to accurately asses the magnitude of the pinch. The pinch velocity leads to a radial gradient of
J the toroidal velocity profile even in the absence of a torque on the plasma. It is shown to be

2 sizeable in the plasmas of the International Thermonuclear Experimental Reactor (ITER) leading
1 toamoderately peakedrotation profile. Finally,thepinchalso affectstheinterpretation of current
experiments.

]
h
PACSnumbers: 52.25.Fi,52.25.Xz,52.30.Gz,52.35.Qz,52.55.Fa
p
-
m
In a tokamak the total toroidal angular momentum rotation), µ the magnetic moment, m (e) the particle
as is a conserved quantity in the absence of an external mass (charge), and u∗0 = u0+vkb. For the background
l source. Transport phenomena determine the rotation velocity (u0) we assume a constant rigid body toroidal
p profile which is of interest because a radial gradient in rotationwithangularfrequencyΩ(thisisanequilibrium
.
s the toroidal rotation is connected with an ExB shearing solution see, for instance, Refs. [26, 27, 28])
c
that can stabilise micro-instabilities [1, 2, 3] and, hence,
si improveconfinement. Furthermore,atoroidalrotationof u0 =Ω×X=R2Ω∇ϕ, (3)
y
sufficientmagnitudecanstabilisethe resistivewallmode
h where ϕ is the toroidal angle. We briefly outline the
[4,5,6]. Inpresentdayexperimentsthe rotationisoften
p
derivation of the final equations here. More details can
[ dsuelttesrmfroinmedthbeyntheeuttroarlobideaalmtohrqeauteinogn.tShuecphlaasmtoarqthuaetwriel-l be found in [29]. The backgroundvelocity u0 will be as-
1 sumed smaller than the thermal velocity, and only the
belargelyabsentinareactoranditisgenerallyassumed
v thattherotation,andhenceitspositiveinfluence,willbe terms linear in u0 will be retained. This eliminates the
7
centrifugalforces but retains the Coriolis force. Further-
4 small. The novel pinch velocity described in this letter,
more the low beta approximation is used for the equi-
1 however, may generate a sizeable toroidal velocity gra-
librium magnetic field (i.e. b·∇b ≈ ∇ B/B where ⊥
1 dient in the confinement region even in the absence of a ⊥
0 torque. indicates the component perpendicular to the magnetic
7 field). With these assumptions
We will focus onthe IonTemperature Gradient(ITG)
0
mode, which is expected to be the dominant instability
cs/ governingthe ion heat channel in a reactor plasma. The u∗0·∇u∗0 ≈vk2∇B⊥B +2vkΩ×b. (4)
equations are formulated using the gyro-kinetic frame-
i
s work [7, 8, 9, 10], which has been proven success- Using the definition of B∗ (see Ref. [26]) and expanding
y
ful in explaining many observed transport phenomena up to first order in the normalised Larmor radius ρ∗ =
h
p [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22]. Because ρ/R, where R is the major radius, one obtains
: of the rotation, the background electric field cannot be
Xiv othrdeedreerdivsamtiaolnl [i2s3a,s2e4t,o2f5e,q2u6a],tiaonndsftohretshtearttiminegepvooilnuttifoonr B∗ =B+ ωB∇×u∗0 =B(cid:20)b+ 2ωΩ + ωvk B×B2∇B(cid:21) (5)
c c c
oftheguidingcentreXandtheparallel(tothemagnetic
r
a field) velocity component (vk) in the co-moving system and Bk∗ = b·B∗ = B(1+2Ωk/ωc) (ωc = eB/m is the
(with background velocity u0) obtained from Ref. [26] gyro-frequency). Expandingnowtheequationsofmotion
dX b retaining only terms up to first order in ρ∗ yields
=v b+ ×(e∇φ+µ∇B+mu∗·∇u∗), (1)
dt k eBk∗ 0 0 dX =v b+b×∇φ+vk2+v⊥2/2B×∇B+2vkΩ (6)
dt k B ω B2 ω ⊥
dv B∗ c c
k =− ·(e∇φ+µ∇B+mu∗·∇u∗). (2)
dt mB∗ 0 0 Thetermsinthisequationarefromlefttoright,thepar-
k allelmotion(v b),theExBvelocityv ,thecombination
k E
Here b = B/B is the unit vector in the direction of the ofcurvatureandgrad-Bdriftv , andanadditionalterm
d
magnetic field (B), φ is the perturbed gyro-averagedpo- proportional to Ω . An interpretation of this term can
⊥
tential (i.e. the part not connected with the background be found if one uses the standard expression for a drift
2
velocity (v ) due to a force (F) perpendicular to the where F is the Maxwell distribution. Note that trans-
D M
magnetic field v =F×B/eB2. Substituting the Cori- lation symmetry in the z-direction is assumed, eliminat-
D
olis force F =2mv×Ω, and taking for the velocity (v) ing the parallel dynamics. Building moments of these
c
the lowest order (parallel) velocity one obtains equations neglecting the heat fluxes (this a clear simpli-
fication, see for instance [34, 35, 36, 37]), and taking the
F ×B 2v
v = c = kΩ (7) space and time dependence of the perturbed quantities
dc eB2 ωc ⊥ as exp[ikθz−iωt], one arrivesat the following equations
for the perturbed density (n) normalised to the back-
The last term in Eq. (6) is therefore the Coriolis drift.
ground density (n0), the perturbed parallel velocity (w)
Expanding the terms in the equation for the parallel ve-
normalisedwith the thermal velocity, and the perturbed
locity to first order in ρ∗ one can derive
iontemperature(T)normalisedwiththebackgroundion
dv dX dX temperature (T0)
k
mv =−e ·∇φ−µ ·∇B (8)
k dt dt dt
R
ωn+2(n+T)+4uw= −2 φ, (12)
where dX/dt is given by Eq. (6). The derived equations (cid:20)LN (cid:21)
are similar to the non-rotating system, with the differ-
ence being the additional Coriolis drift. It follows that
ωw+4w+2un+2uT =[u′−2u]φ, (13)
thisCoriolisdriftappearsinacompletelysymmetricway
compared with the curvature and grad-B drift.
In this letter the approximation that assumes circular 4 14 8 R 4
ωT + n+ T + uw = − φ. (14)
surfacesandsmallinverseaspectratio(ǫ)isused. Inthis
3 3 3 (cid:20)L 3(cid:21)
T
case the Coriolis drift adds to the curvature and grad-B
drift Here R/LN ≡−R∇n0/n0, R/LT ≡−R∇T0/T0, the po-
v2+2v RΩ+v2/2 tentialφis normalisedto T0/e,andthe frequency isnor-
v +v ≈ k k ⊥ e , (9) malisedwiththedriftfrequencyωD =−kθT0/eBR. The
d dc z
ωcR Coriolisdrift(alltermsproportionaltou)introducesthe
perturbed velocity in the equations for the perturbed
where e is in the direction of the symmetry axis of the
z density, and temperature. However, since u ≪ 1 the
tokamak. The linear gyro-kinetic equation is solved us-
influence of the Coriolis drift on the “pure” ITG (with
ing the ballooning transform[30]. The equations,except
u = 0) is relatively small. The Coriolis drift generates
fromthe Coriolisdrift arestandardandcanbe foundin,
a coupling between w and the density, temperature as
forinstance,Ref.[31]. Inthefollowingu′ ≡−R∇RΩ/v
th well as potential fluctuations. Note that for u = 0 the
and u ≡ RΩ/v . Unless explicitly stated otherwise all
th perturbed velocity is directly related to the gradient u′,
quantitieswillbemadedimensionlessusingthemajorra-
resulting in a purely diffusive flux. For finite rotation
dius R, the thermalvelocity v ≡ 2T/m , and the ion
th i (u6=0)the ITGwillgenerateaperturbedparallelveloc-
mass mi. Densities will be normalipsed with the electron ity w, which is then transported by the perturbed ExB
density. Thetoroidalmomentumfluxisapproximatedby
velocity. Iftheperturbedtemperatureiskepttheexpres-
the flux of parallel momentum (Γ ) which is sometimes
φ sions for the momentum flux become rather lengthy and
normalised with the total ion heat flow (Q )
i are, therefore, reported elsewhere [29]. Retaining only
the coupling with the perturbed density and potential,
1
(Γφ,Qi)=(cid:28)vEZ d3v(cid:18)mvk,2mv2(cid:19)f(cid:29), (10) and assuming an adiabatic electron response (n = φ/τ
with τ = Te/T0 being the electron to ion temperature
ratio) one can derive
wheref isthe(fluctuating)distributionfunctionandthe
brackets denote the flux surface average.
1 2+2τ
Before turning to the gyro-kinetic calculations, first Γ = k ρIm[φ†w]=χ u′− u , (15)
φ θ φ
4 (cid:20) τ (cid:21)
the implications of the Coriolis drift will be investigated
using a simple fluid model (more extended models have
with
been published in Refs. [32, 33]). A (low field side) slab
likegeometrywillbeassumedwithallplasmaparameters 1 γ 2
χ =− k ρ |φ| . (16)
being a function of the x-coordinate only. The magnetic φ 4 θ (ω +4)2+γ2
R
field is B = Be , ∇B = −B/Re , The model can be
y x
Here, the dagger denotes the complex conjugate, ω is
build by taking moments of the gyro-kinetic equation in R
(X,v ,v ) coordinates the real part of the frequency, and γ the growth rate
k ⊥
of the mode. Note that χ is positive since ω (γ) are
φ R
∂f +(v +v )·∇f =−v ·∇F −eFM(v +v )·∇φ, normalised to ωD = −kT0/eBR. The second term be-
d dc E M d dc tween the square brackets of Eq. (15) represents an in-
∂t T
(11) ward pinch of the toroidal velocity (the word pinch is
3
used here because the flux is proportional to u, unlike more negative, i.e. the pinch always enhances the abso-
off-diagonal contributions that are due to pressure and lute value of the velocity gradientin agreementwith the
temperature gradients [38, 39]) If one assume no torque, results from the fluid theory. Fig. 1 also shows that the
i.e. Γ = 0 it can be seen that the pinch can lead to a pinch decreases with k ρ . It is noted here that also χ
φ θ i φ
sizeablegradientlengthR/L ≡R∇u/u=4(forτ =1). in becomes smaller for smaller k ρ [39].
u θ i
Thepeakingisinroughlythesamerangeastheexpected
0
density peaking [40].
0.5 −1
0.3
−2
0.3 0.28
Qi 0.26 γR / vth χV / φφ−3
Γ(R / 2 L) / φT−00..11′u012−−−1100....5050 −0.1 0kkθθu ρρii ==0 .001..25 R−−54
−0.3 −1 0.5 −6
−2 1.0 1.5 kθ ρi = 0.8 0 1 R2/LN, q, 3s3, 20ε, 6kθ4 ρi 5 6
−0.5 −0.1 0 u 0.1
−0.15 −0.1 −0.05 0 0.05 0.1 0.15
u
FIG. 2: RVφ/χφ as a function of various parameters: R/LN
(x),3sˆ(+),q (o), and 20ǫ (diamonds), and 6kθρi (squares)
FIG. 1: (R/2LT)Γφ/Qi as a function of u for three values of
kθρi 0.5(o),0.2(squares),and0.8(diamonds). Thetopright Fig.2showsthenormalisedpinchvelocityRV /χ asa
graphshowsthegrowth rateasafunctionofuandthedown φ φ
function of various parameters. The magnetic shear and
left graph the contour lines of (R/2LT)Γφ/Qi as a function
ofuandu′,bothforkθρi =0.5. Inthelattergraphthethick the density gradient have a rather large impact. Note
line denotes zero momentum flux, i.e. the stationary point that both due to sˆ, as well as due to q, R/LN and ǫ, the
for zero torque pinch velocity is expected to be small in the inner core,
but sizeable in the confinement region.
Fig. 1 shows the parallel momentum flux as a func- The novel pinch velocity described in this letter has
tion of the toroidalvelocityu obtained from linear gyro- several important consequences. It can explain a gradi-
kinetic calculations using the LINART code [41] (in ent of the toroidal velocity in the confinement region of
which unlike Eq. (11) the parallel dynamics is kept) the plasma without momentum input. A spin up of the
for three different values of the poloidal wave vector plasmacolumnwithouttorquehasindeedbeenobserved
(k ρ = 0.2, 0.5, and 0.8). The parameters of each of [51, 52, 53, 54, 55, 56]. Although a consistent descrip-
θ i
thegyro-kineticcalculationsinthisletterarethoseofthe tionorderingthedifferentobservationsisstilllacking,the
Waltz standard case [42]: q = 2, magnetic shear sˆ= 1, calculations of this letter show that the pinch velocity is
ǫ = 0.1, R/L = 3, R/L = 9, τ = 1, u = u′ = 0. expected to play an important role. This finite gradient
N T
In the presented scans one of these parameters is var- without torque is especially important for a tokamak re-
ied while keeping the others fixed. Since the flux from actor in which the torque will be relatively small. From
Fig. 1 is linear in the velocity, a constant pinch velocity thecalculationsshownabove,andfortypicalparameters
exists in agreement with the fluid model. The influence in the confinement region of a reactor plasma, one ob-
of the toroidal velocity on the growth rate is small. The tains a gradient length R/Lu = u′/u in the range 2-4
bottom left graph shows the contour lines of the flux as representing a moderate peaking of the toroidal velocity
a function of u and u′. The fact that the contour lines profile similar to that of the density. Unfortunately, the
are straight means that the momentum flux is a linear current calculation only yields the normalised toroidal
combinationofthe diffusive part(∝χ u′)andthe pinch velocitygradient. Inordertodeterminethevelocitygra-
φ
velocity (V u) dientone wouldneed to know the edge rotationvelocity.
φ
This situation is similar to that of the ion temperature
Γ =[χ u′+V u] (17) [50].
φ φ φ
The existence of a pinch can resolve the discrepancy
The diagonal part has been calculated previously using between the calculated χ and the experimentally ob-
φ
fluid [43, 44, 45, 46, 47] as well as gyro-kinetic theory tained effective diffusivity (χeff = Γφ/u′). The latter is
[48,49]. Thepinchvelocityisnegative(inward)forposi- oftenfoundtodecreasewithincreasingminorradiusand
tiveusuchthatitenhancesthegradient. Itchangessign tobesmallerthanthetheoreticalvalueofχ intheouter
φ
with u such that for negative velocities it will make u′ regionof the plasma [57, 58, 59]. The pinch indeed leeds
4
to a decrease of χeff Thecalculationsinthis lettershowthatthe secondterm
inthebracketscanbeoftheorder-1,leadingtoχeff <χi.
RV 1
φ
χeff =χφ 1+ . (18)
(cid:20) χ R/L (cid:21)
φ u
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