Experimental observations and modelling of intrinsic rotation reversals in tokamaks

0.19 MB·

by  Y. Camenen

Experimental observations and modelling of
intrinsic rotation reversals in tokamaks
7
1 Y. Camenen1, C. Angioni2, A. Bortolon3, B.P. Duval4,
0 E. Fable2, W.A. Hornsby2, R.M. McDermott2, D.H. Na5,
2 Y-S. Na5, A.G. Peeters6 and J.E. Rice7

n 1 CNRS,Aix-MarseilleUniv.,PIIMUMR7345,Marseille,France
a 2 MaxPlanckInstitutfu¨rPlasmaphysik,Garching,Germany
J 3 PrincetonPlasmaPhysicsLaboratory,Princeton,USA
7 4 EPFL,SwissPlasmaCenter(SPC),Lausanne, Switzerland
2 5 Departement ofNuclearEngineering,Seoul NationalUniversity,Seoul,Korea
6 PhysicsDepartment,UniversityofBayreuth, Bayreuth,Germany
] 7 PSFC,MIT,Cambridge,Massachusetts, USA
h
p
Abstract. The progress made in understanding spontaneous toroidal rotation
-
m reversals in tokamaks is reviewed and current ideas to solve this ten-year-old
puzzleareexplored. Thepaperincludesasummarialsynthesisoftheexperimental
s observations inAUG,C-Mod,KSTAR,MASTandTCVtokamaks, reasonswhy
a
turbulent momentum transport is thought to be responsible for the reversals, a
l
p reviewofthetheoryofturbulentmomentumtransportandsuggestionsforfuture
. investigations.
s
c
i
s
y
h
p
[

1
v
5
9
0
8
0
.
1
0
7
1
:
v
i
X
r
a
2
1. Introduction approaches to its resolution.
The definitions and conventions adopted in this
Duringthe1980s,itwasshownthatstationarytoroidal paper areintroducedinSec.2, followedbya summary
flows that reach up to 20% of the thermal velocity of the experimental observations in Sec. 3. The
can develop in tokamak plasmas in the absence of constraints these observations put on the theory and,
externally applied torque (a summary of these early in particular, the reasons why turbulent momentum
observations is available in Table 1 of [1]). This transport is thought to be responsible for rotation
phenomenon, dubbed intrinsic rotation, has practical reversals are discussed in Sec. 4. The theory of
implications for future low torque devices like ITER turbulent momentum transport is briefly reviewed in
owing to the potential stabilising impact of plasma Sec. 5 before summarising the current status of the
flows onturbulence anddeleterious MHD instabilities. modelling activities in Sec. 6. Finally, in Sec. 7 future
Following initial measurements, experiments were work and open issues are discussed.
performed to explore the physics of intrinsic rotation.
The observations have been regularly summarised
2. Definitions and conventions
in review articles [2, 3, 4, 5, 6]. Momentum
transport, reconnection events (sawteeth and ELMs),
Throughout this paper intrinsic rotation refers to
non-axisymmetric magnetic fields (from magnetic
the toroidal rotation that develops in the absence of
perturbation coils, error fields or large MHD modes),
externally applied torque. The toroidal rotation is
orbit losses and interactions with neutrals were
noted v and its direction is given with respect to
ϕ
all observed to impact intrinsic rotation. The
the plasma current: v > 0 for co-current rotation
ϕ
underlying physics is surprisingly rich, involving
and v <0 for counter-current rotation. The rotation
ϕ
many competing mechanisms that determine the final
profile is saidto be peaked (hollow)when v increases
ϕ
rotation profile. These include collisional momentum
(decreases) from the edge to the magnetic axis. Note
transport, fluctuation-induced Reynolds and Maxwell
thatotherdefinitionsexistinthelitterature,wherefor
stresses (turbulent transport), charge exchange with
instance |v | is used in place of v to define peaked
ϕ ϕ
neutrals,J ×δB torqueinducedbyresonantmagnetic
k and hollow profiles. The present definition is deemed
perturbations, J × B torque induced by non-
r more appropriate to describe profiles that cross v =
ϕ
axisymmetric magnetic fields (neoclassical toroidal
0. Three regions are distinguished in the rotation
viscosity NTV), ionisation currents and orbit losses.
profile following [13, 14, 15]: the sawtooth region 0 ≤
A convenient set of transport equations has been r/a . rinv/a, with rinv the sawtooth inversion radius
proposed in [7] to describe the evolution of toroidal
and a the plasma minor radius, the gradient region
flows in tokamak plasmas resulting from these various rinv/a . r/a . 0.8 (typically) and the edge region
mechanisms in a consistent fluid moment framework. 0.8 . r/a ≤ 1. The separation between the gradient
This approach highlights the complexity of intrinsic
and edge regions relies on the different dependencies
rotationinherenttothenumberofmechanismsatplay.
ofthe toroidalrotationgradientonplasmaparameters
The present paper focuses on a specific puzzle
in these two regions. The plasma core includes the
withinintrinsicrotation: spontaneoustoroidalrotation
sawtooth and gradient regions.
reversals. This intriguing phenomenon was reported
Toroidal rotation reversals are defined as a large
10 years ago on the TCV tokamak where the core change (& 100%) of the intrinsic toroidal rotation
toroidal rotation was observed to flip from counter-
gradient over the whole gradient region triggered
current to co-current when a threshold in density was by minor changes (. 20%) in the control plasma
exceeded in Ohmic L-mode plasmas [8]. Rotation
parameters. It is important to notice that this
reversals have since been demonstrated in C-Mod [9],
definition does not require a change in the direction
AUG[10],MAST[11]andKSTAR[12]. Inparallel,the
of the central toroidal rotation, nor of the toroidal
theoryofintrinsicrotationhasundergoneconsiderable
rotation gradient, which is somehow at odds with the
development and many possible physical mechanisms
reversal qualifier. Toroidal rotation bifurcation would
have been identified. In spite of this progress,
be a more accuratedescriptionofthis phenomenology.
understanding toroidal rotation reversals still eludes
For consistency with the past literature, however,
us and predicting the direction of the core rotation in
we retain toroidal rotation reversal and broaden its
OhmicL-modesremainsachallenge. Toroidalrotation
definitiontoincludecaseswherethesignofthetoroidal
reversals do not directly affect plasma performance,
rotation and/or of its gradient do not change.
but they represent a critical test for the theory of
intrinsicrotation. Thepurposeofthepresentworkisto
survey the observations and the theoreticalframework
with the goal of presenting the current understanding
of this research and explaining current ideas and
3
3. Experimental observations (Type II.a reversals), in AUG and TCV diverted
plasmasatveryhighdensity (Type II.b reversals)
The experimental observations of toroidal rotation andinMASTlowcurrentandlowdensitydiverted
reversals collected over the last ten years in AUG plasmas (Type II.c reversals).
[10,16],C-Mod[17,18,19,20,21],MAST[11],KSTAR
AUG:Fig. 4in[16]andFig. 7in[10],TCV:Fig. 1in[8]
[12] and TCV [8, 13, 14, 15, 22] are summarised in
Note that the distinction made above is purely
this section. In some cases, direct reference is made to
phenomenological and does not exclude that all
figures in the published works.
reversals be manifestations of the same physical
mechanismobservedindifferentplasmaconditions. In
3.1. Measurements
particular, in Ohmic plasmas, T , T , n and q are
e i e
(i) Toroidal rotation reversals were reported observ- strongly coupled and a unique threshold identified as
ing impurity ions (boron, carbon and argon)with a combination of these parameters may be traversed
a variety of diagnostics: X-ray imaging crystal several times in a density ramp, with a trajectory
spectrocsopy (XICS) in C-Mod and KSTAR (ar- possibly dependent on the operation mode (limited
gon),chargeexchangerecombinationspectroscopy versus diverted for instance).
(CXRS) with a diagnostic neutral beam in TCV
(carbon), CXRS using short pulses of a heating 3.4. Initial and final states
neutral beam in AUG (boron) and KSTAR (car-
Typicalpre-andpost-reversaltoroidalrotationprofiles
bon).
areshowninFig.1forAUG,C-ModandTCVplasmas.
(ii) Reversals have also been inferred from Doppler
back-scattering measurements of the perpendicu- (i) In the sawtooth region, the toroidal rotation
larvelocityofelectrondensityfluctuationsinAUG profile is mostly flat with a bulge in the co-
andMAST(assumingadominantE×B velocity). currentdirection,independentofthereversalstate
(measurement integrated over several sawtooth
(iii) The measurements were mostly performed in
cycles)
Ohmic L-modes, but reversals were also reported
in the presence of ion cyclotron heating [21] and (ii) In the gradient region, the toroidal rotation
electron cyclotron heating [15], still in L-mode. gradient has a wide range of values and often a
different sign before and after the reversal.
3.2. Triggers (iii) In the edge region, the toroidal rotation profiles
are similar just before and just after the reversal.
(i) Toroidal rotation reversals have been triggered
(iv) In C-Mod, the modificationof the rotationprofile
by density ramps, plasma current-ramps, toroidal
in Type I reversals occurs in the region q .3/2.
magnetic field ramps, impurity injection and by
C-Mod: Fig16in[20]
switching on/off electron cyclotron heating, see
e.g. [6, 15].
3.5. Dynamics
(ii) Thereversalsappeartobehighlyreproducibleand
weakly sensitive to machine conditioning [13]. The dynamics of reversals have been investigated
during density ramp experiments for Type I reversals
3.3. Reversal direction in C-Mod and Type II.a reversals in TCV. A similar
behaviour is reported in the two cases.
The reversal direction is discussed here as a function
(i) In C-Mod and TCV, the reversal process appears
of increasing density, as density ramps are the most
as a clear break in slope of the toroidal rotation
common trigger of reversals.
response to an increase in density.
(i) Type I. Co-current to counter-current reversals
C-Mod: Fig. 1in[17],TCV:Fig. 5.3in[15]
(or more precisely bifurcations from peaked/flat
(ii) After the reversal process commences, the tem-
to hollow profiles in the gradient region) have
poral dynamics of the central toroidal rotation is
been observed in AUG, C-Mod, MAST and TCV
rather well described by an exponential fit of the
(diverted configuration) and in KSTAR (limited
form exp[−t/τrev]. The characteristic time of the
configuration) above a critical density.
reversal τ is comparable to, or longer than, the
rev
AUG:Fig. 4in[10],C-ModFig. 13in[17],KSTAR:Fig.15
energy confinement time. In TCV, for the Type
in[12],MAST:Fig. 4in[11],TCV:Fig. 6in[13].
II.areversalsshowninFig. 5.3,5.4and5.6of[15],
(ii) Type II. Counter-current to co-current reversals
τrev ranges from 40 to 120ms.
(transition from hollow to peaked profiles) have
beenobservedinTCVlimitedplasmasforq95 .3
4

20 C−mod a) 20 AUG #28387 c) 20 TCV #36469, #36461 e)
15 15 15
10 10 10
5 5 5
v [km/s]φ −50 v [km/s]φ −50 v [km/s]φ −50
−10 −10 −10
−15 n =10.5 −15 n =1.36 −15 n =4.3
el19 el19 el19
−20 nel19=11.4 −20 nel19=1.66 −20 nel19=4.8

0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1
r/a r/a r/a

30 C−mod b) 20 TCV #36003, #36004 d) 20 TCV #36463 f)
25 15 15
20 10 10
15 5 5
v [km/s]φ 150 v [km/s]φ −50 v [km/s]φ −50
0 −10 −10
−5 n =6.5 −15 n =3.9 −15 n =6.1
el19 el19 el19
−10 nel19=8.3 −20 nel19=4.5 −20 nel19=6.3

0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1
r/a r/a r/a
Figure 1. Typical pre/post reversal toroidal rotation profiles measured in C-Mod (adapted from Fig. 13 and 15 in [17]), AUG
(adapted fromFig4 in[10])and TCV (adapted fromFig. 6.3and 8.1in[15]). The lineaveraged density foreach case is givenin
unitsof1019m−3 andtheinversionradiusisindicatedbyadashedline.
a)C-Mod,TypeIreversal,lowersinglenull(B×∇B upward),Ip=1.05MA,BT =5.4T,q95=3.2.
b)C-Mod,TypeIreversal,uppersinglenull(B×∇B upward),Ip=0.8MA,BT =5.4T,q95=4.7.
c)AUG,TypeIreversal,lowersinglenull(B×∇B downward),Ip=0.5MA,BT =1.5T,q95=4.9.
d)TCV,TypeII.areversal,limitedconfiguration, Ip=0.34MA,BT =1.45T,q95=2.7.
e)TCV,TypeIreversal,lowersinglenull(B×∇B downward),Ip=0.26MA,BT =1.45T,q95=3.6.
f)TCV,TypeII.breversal,lowersinglenull(B×∇B downward), Ip=0.26MA,BT =1.45T,q95=3.6.
(iii) When the reversalis triggeredby a density ramp, 3.6. Critical density for the reversal
thetimescaleofthereversalisindependentofthe
(i) The density threshold for Type I reversals
density ramp-rate.
increaseswithincreasingplasmacurrent(C-Mod).
C-Mod: Fig. 2in[17],TCV:Fig. 5.4in[15]
This was also indirectly observed in experiments
(iv) Even with small increases of the line averaged
wheretheplasmacurrentwasscannedatconstant
density, stabilising the profiles between the initial
density (TCV).
and final states of a reversal has never been
C-Mod: Fig. 10 in[17], TCV: Fig. 6.5 in [15] and Fig. 1
demonstrated. Fig. 6 in [13] shows the typical
in[22].
gap between the two stationary states.
(ii) The density threshold for Type I reversals
(v) During the reversal, there is a transient evolution
decreases with increasing toroidal magnetic field.
of the edge rotation (0.8 . r/a . 1) in the
C-Mod: Fig. 12in[17].
directionoppositetothatofthecore(edgerecoil).
(iii) In C-Mod, the I and B dependencies of the
The edge rotation then relaxes to its pre-reversal p T
density threshold can be unified by a critical
value.
density proportional to 1/q95. For the cases
C-Mod: Fig. 19,20in[17],TCV:Fig. 5.7and5.8in[15]
investigated, a critical collisionality of the form
(vi) The reversal is, itself, reversible, with some νrev ∝ neZeff/Te2 works equally well as the factor
hysteresis in density > 10%. There may also be 2
Zeff/Te was nearly constant at the reversal [20].
a hysteresis in the plasma current, but this is not
In AUG, a critical collisionality better unifies the
as clear due to a relatively slow current diffusion
data than a critical density, see Fig 7 in [10].
time.
(iv) The density threshold for Type II.a reversals
C-Mod: Fig. 4and7in[17],TCV:Fig. 5.5in[15]
(backwards with respect to Type I reversals)
decreases with increasing plasma current and,
therefore, has an opposite dependence on I
p
5
compared to Type I reversals. a (1,1) sawtooth precursor, whose amplitude in-
TCV:Fig. 5.6in[15]. creases with the plasma density is often detected
(v) The density threshold for Type II.a reversals by magnetic probes. As for the sawteeth,no clear
increases with increasing ECH power. correlation is found between the MHD activity
and the rotation direction, with co- and counter-
TCV:Fig. 5.17and5.18in[15].
rotation observed for similar MHD spectrograms
(vi) The scaling of the density threshold for Type II.b
[14, 15]
and Type II.c reversals with respect to plasmas
parameters is, to date, unexplored.
3.10. LOC/SOC transition and turbulence changes
3.7. Poloidal rotation (i) Type I reversals generally occur close to, but
not necessarily at, the transition from linear
Poloidal rotation profiles before and after Type II.a
Ohmic confinement (LOC) to saturated Ohmic
reversals have been measured in TCV.
confinement (SOC) [10, 12, 16, 17, 18, 19, 20, 25]
(i) The pre- and post- reversal profiles are similar and to the non-local heat/cold pulse cut-off [20].
and no large excursions of poloidal rotation are
(ii) The prediction of the turbulence regime from
observedduringthereversal(onthemeasurement
linear stability calculation is a delicate exercise
timescale).
due to the sensitivity of the TEM/ITG transition
TCV:Fig5.9in[15]andFig8in[13]. to input parameters that are difficult to measure
(ii) No departure from the neoclassicaltheory predic- precisely (temperature, density and rotation
tion is observed within the measurement uncer- gradients,collisionality,magnetic shear,etc.) and
tainties (.2km/s) [23]. to the choice of the collision operator in the
numericalsimulations [26]. In addition, the linear
3.8. Plasma shape stabilitydoesnotnecessarilyreflectthenon-linear
state. Experimentally, the characterisation of the
(i) In TCV, Type II.a reversalsvanished for negative
turbulence regime from temperature and density
triangularity: the rotation profile is peaked even
fluctuation measurements is not straigtforward
at low density (with the edge counter-current
either. These caveats in mind, linear stability
rotating) andshifts rigidly towardsmore counter-
calculations [16, 27, 10] for AUG and C-Mod
currentrotationwhenthedensityisincreased[14].
plasmas indicate that toroidal rotation reversals
(ii) In KSTAR, no sharp evolution of the toroidal often occur close to the boundary between the
rotation is observed when ramping the plasma TEM and ITG instabilities and fluctuations
density in low elongation limited plasmas. The measurementsonC-Mod[17,20,27]showchanges
toroidal rotation remains counter-current and in the fluctuation spectra across toroidal rotation
displays a mild U-curve behaviour with density reversals. These changes in the turbulence
[12]. characteristics do not appear, however, to trigger
thereversalastheycanoccurinaregionwherethe
3.9. Sawteeth and MHD activity toroidal rotation gradient is not modified [27] or
therotationprofileexperienceareversalwithouta
(i) Toroidal rotation reversals are often observed in
change of the predicted dominant instability [10].
plasmas that also display sawtooth phenomena.
However,thepresenceofsawteethappearstobea
3.11. Dependence of the toroidal rotation gradient on
consequence of the constrained operationalspace.
plasmas parameters
Reversals have been triggered in plasmas exhibit-
ing a wide variety of sawtooth characteristicsand (i) Multi-variable regressions performed for a large
nocorrelationhasbeenestablishedbetweenthere- database of AUG Ohmic L-modes [10] and for a
versals and the sawtooth frequency, amplitude or reduced set of AUG and TCV Ohmic L-modes
inversionradius[14]. InAUG,hollowrotationpro- [28], show that the toroidal rotation gradient
files with the core rotating in the counter-current is mostly correlated to the normalised density
direction(highdensitybranchofTypeIreversals) gradient R/Ln and effective collisionality νeff.
were observed in the absence of detectable saw- Larger R/L increases the hollowness of the
n
teeth [24]. rotation profile. Large variations of the toroidal
(ii) Low amplitude MHD activity is sometimes ob- rotationgradient(includingachangeofsign)can,
served during toroidal rotation reversal experi- nevertheless,beobservedatnearlyconstantR/Ln
ments. For instance, in TCV, a (2,1) mode and [25].
6
(ii) Interestingly, in AUG the strong dependence of Sec.3.5). Theedgerecoilcannoteasilybeproducedby
the toroidal rotation gradient on R/L is also alocalisedtorqueordampingterm: thiswouldrequire
n
observed for a wide operational range including us to invoke several radially localised contributions of
OhmicandelectroncyclotronheatedL-modesand opposite directions and different temporal behaviour.
H-modes[16]. Stronglyhollowrotationprofilesare In contrast, a sudden (i.e. faster that the momentum
only observed at large R/L values. confinement time) change of momentum transport in
n
theplasmacoreatconstantedgemomentumtransport
4. Why is momentum transport considered as producesanedgerecoilasaconsequenceofmomentum
the key to explain the reversals? conservation. The profile evolves on a time scale
dictated by momentum diffusion, i.e. τrev ∼ a2/χϕ
The mechanisms invoked to explain toroidal rotation with χϕ the momentum diffusivity.
reversalsneed to be consistentwith allthe experimen-
tal observations summarised in Sec. 3. This includes 5. Momentum transport theory
theobserveddynamics(timescale,edgerecoil,hystere-
sis), the parametric dependencies of the critical den- 5.1. Momentum conservation and momentum flux
sityandtheconstancyofthepre/post-reversalrotation
Assuming momentum transport to be the only
profiles in the edge region.
mechanism at play, the toroidal rotation in the core
The main mechanisms expected to impact the
of an axisymmetric tokamak is governed by the
intrinsic rotation profile in the gradient region,
redistribution of toroidal angular momentum, e.g. [7]:
where the reversal takes place, are the neoclassical
arenddistrtiubrubtuiolenn)t, thmeonmeeonctlausmsicatlratonrsopiodratl v(imscoomsiteyntduume ∂∂tXhnsmsRvϕ,si+ V1′∂∂r [V′Πϕ]=0 (1)
s
to field ripple or a strong MHD mode (damping
with n , m and v the density, mass and toroidal
towards a diamagnetic level offset), a torque due s s ϕ,s
velocity of species s, respectively, R the local major
to resonant non-axisymmetric fields (locking to the
radius, h.i the flux surface average, r a radial
wall) and sawteeth (momentum redistribution and
coordinate (flux surface label), V the flux surface
possibly transient torque). Sawteeth and strong MHD
volume, V′ = ∂V/∂r (radial derivative) and Π =
modes certainly affect the intrinsic rotation profile, ϕ
hΠ ·∇ri the flux surface averaged radial component
but acausallink betweensawteeth/MHDandtoroidal ϕ
of the toroidal momentum density flux. Eq. (1)
rotation reversal has yet to be established. As
is obtained from the flux surface average of the
described in Sec. 3.9, toroidal rotation reversals are
momentum conservation equation and simply states
observed with little to no MHD activity and for a
that,intheabsenceofsourcesandsinks,theevolution
varietyofMHDspectrogramsandsawtoothbehaviour,
ofthe toroidalangularmomentumdensityisdrivenby
independent of the reversal state, The magnitude of
the divergence of the momentum flux Π .
the toroidal rotation in the pre- and post- reversal ϕ
states is often larger than a diamagnetic level rotation Up to first order in ρ∗ = ρi/R0, the species flow
entering Eq. (1) lies within a flux surface and is given
and, therefore, somewhat incompatible with NTV
bythesumoftheparallelstreamingalongthemagnetic
or resonant non-axysimmetric fields as the dominant
field lines, of the E ×B drift and of the diamagnetic
process. Inaddition,rotationreversalsareobservedin
flow
tokamaks with very low ripple like KSTAR but not
in tokamaks with high ripple like Tore Supra [29]. vs =vkb+vE+vdia+O(ρ2∗) (2)
In Tore Supra, a slightly hollow profile of counter-
Here,ρi =mivthi/(eB0)isthemainionLarmorradius,
current rotation is measured in Ohmic L-modes that
R0 and B0 are a reference major radius and magnetic
is satisfactorily described by NTV theory. Close
field, respectively, and vthi = p2Ti/mi is the thermal
to the LOC/SOC transition, a small departure from
velocity. The parallel flow can be split into three
NroTtaVtiopnredreicvteirosnaslibsuotbsfearrvefdro,mremsiignnisicfiecnatntolfyaatffoercotiidnagl components vk =vkE+vkdia+vkθ so that vkEb+vE and
vdiab+vdia are purely toroidalwhereas the remaining
the rotation profile: when NTV dominates, toroidal k
rotation reversals are hampered [29]. Summing up contribution to the total flow, vkθb, has finite poloidal
and toroidal components. The two purely toroidal
these various considerations, momentum transport is
flows are given by:
the only viable candidate left to explain rotation
reversals. An additional strong argument in this ∂Φ
direction is brought by the transient acceleration of vkEb+vE =RωΦeϕ =−R∂ψeϕ (3)
the plasma edge in the direction opposite to that of 1 ∂p
the core observed during a reversal (edge recoil, see vkdiab+vdia =Rωp,seϕ =−RZ en ∂ψseϕ (4)
s s
7
with Φ the electrostatic potential, ψ the poloidal rotation to sustain a gradient. A residual stress
magnetic flux, e the unit vector in the toroidal contributionis,therefore,requiredtodescriberotation
ϕ
direction and Z and p the species charge number profiles crossing zero, as observed in Fig. 1 for the
s s
and pressure, respectively. The parallel flow vθ impurity rotationv or in [10] for the E×B angular
k ϕ,s
is constrained by neoclassical physics. Combining frequency ωΦ.
Eqs. (2), (3) and (4), yields the customary expression In the core of an axisymmetric tokamak, the
of the toroidal flow [30]: neoclassical momentum flux is typically an order of
magnitude smaller than the gyro-Bohm momentum
B
t
vϕ,s =RωΦ+Rωp,s+vθ,s (5) flux [31] and negligible compared to the turbulent
B
p
momentum flux. The following discussion, therefore,
The first term, related to the E × B flow, is the
focuses on turbulent momentum transport. The main
lowest order contribution. It is species independent
mechanisms are briefly outlined in the framework of
and can assume arbitrarily large values in an
gyrokinetic theory, emphasising their potential link
axisymmetric tokamak. The lowest order momentum
with rotation reversals. For a more comprehensive
transport theory is formulated with respect to ωΦ. description of the theory of turbulent momentum
The next contributions, related to the diamagnetic
transport and further references to the original work,
and poloidal flows, are first order in ρ for a
∗ the reader is referred to published reviews [32, 33, 34,
neoclassical level poloidal flow and roughly scale as
35].
12ρ∗BBptvthiR/LTi, with R/LTi = −R0∂lnTi/∂r the
normalised temperature gradient. For ρ = 1/600,
∗ 5.2. Lowest order contributions
B /B = 10 and R/L = 6, the first order toroidal
t p Ti
flow in Eq. (5) is about 0.05vthi and, therefore, not To lowest order, with respect to the gyrokinetic
negligible compared to the total toroidal flow, which ordering (local limit, ρ∗ → 0), five mechanisms that
is often less than 0.2vthi for intrinsic rotation. When can generate a momentum flux are described. The
dealingwithintrinsicrotation,the distinctionbetween parallel [36] and perpendicular [37] components of the
the total toroidal flow v , that is the measured toroidal flow shear give rise to a diagonal flux. For
ϕ,s
quantity, and the lowest order toroidal flow RωΦ positive magnetic shear, these two contributions have
therefore needs to be taken into account. opposite sign and the perpendicular component of the
The momentum flux entering the transport toroidal flow shear acts to reduce toroidal momentum
equation, Eq. (1), is now decomposed into diagonal diffusivity [38]. The pinch also has two contributions:
(diffusive), pinch (convective) and residual stress the Coriolis pinch [39] and the momentum carried by
components: any particle flux. In the stationary state, the second
contribution vanishes if no particle source remains.
Πϕ =nmR0vthi[χϕu′+R0Vϕu+Cϕ] (6) Finally, the only contribution to the lowest order
Here, the decomposition is performed with respect residual stress arises from the up-down asymmetry of
to the lowest order flow, i.e. the diagonal part the magnetic flux surfaces CFS [40].
ϕ
componentswithrespecttou′ =−R0/vthi∂ωΦ/∂rand The ratio of the toroidal momentum diffusivity
the pinch components with respect to u = R0ωΦ/vthi. and ion heat diffusivity, the Prandtl number, is
In the expression above, nm = n m is the species typically predicted as Pr = χ /χ ∼ 0.7 [32], but
P s s ϕ i
averaged mass density and the momentum transport values in the range 0.4 to 1.5 are possible depending
coefficients have also been species averaged using on the plasma parameters [38]. The Coriolis pinch
A = n m A / n m where A represents the is generally directed inward and acts to increase the
P s s s P s s
momentumdiffusivityχ ,pinchvelocityV orresidual absolutevalue ofthe rotation. The pinchtodiffusivity
φ ϕ
stresscoefficientCϕ. Instationarystate,Πϕ =0sothe ratio R0Vϕ/χϕ typically ranges from -1 to -4 with a
intrinsic rotation profile is determined by the balance markeddependenceonthenormaliseddensitygradient
between the diagonal flux, which tends to flatten the R/L . The Coriolis pinch tends to be smaller in the
n
profile, and the non-diagonal flux (pinch and residual TEMregime [41]andcanbe directedoutwardcloseto
stress) which tends to sustain a finite gradient. The the kinetic ballooning mode threshold [42]. The ratio
sign and magnitude of the resulting rotation gradient of the residual stress from the flux surface asymmetry
is dictatedby the ratioofthe pinchandresidualstress to the momentumdiffusivity is typically |CFS/χ |.1
ϕ ϕ
components to the momentum diffusivity: near the edge where the flux surface shaping is the
highest and |CFS/χ | . 0.3 in the core [40, 43].
u′ =−R0Vϕu− Cϕ (7) The sign of CFϕS isϕdetermined by the flux surface
χϕ χϕ ϕ
asymmetryandthedirectionofthemagneticfield. The
The fundamental difference between the pinch and
momentum diffusivity, pinch and up-down asymmetry
residual stress is that only the pinch requires a finite
residual stress have all been identified experimentally
8
and found to be in fair agreement with lowest order the rotation gradient is limited to the edge region [58]
gyrokinetic theory predictions [44, 45, 46, 47]. As the andthereforenotdirectlyrelevantfortoroidalrotation
intrinsic rotation gradient in the vicinity of u = 0 reversals.
is typically between −1.5 . u′ . 1.5, including up- Of course, for all first order contributions, C /χ
ϕ ϕ
down symmetric plasmas for which CFS = 0, intrinsic is expected to depend on ρ . This dependence is
ϕ ∗
rotation can clearly not be described by the lowest linear in ρ for (iii) and (v) according to analytical
∗
order theory: it lacks residual stress contributions. calculations, but may be more complicated for the
othercontributions. Forinstance,ithasbeenshownin
5.3. First order contributions global non-linear simulations that for profile shearing,
C /χ is first linear in ρ but saturates at high ρ
ϕ ϕ ∗ ∗
To next order in ρ , new contributions to the residual
∗ values [51]. Overall, the ρ scaling of residual stress is
∗
stress arise from:
still debated and deserves further investigation. What
(i) the impact of a poloidally inhomogeneous turbu- is certainly true, however,is that the exponent on any
lence on the parallel symmetry [48] ρ∗ scaling is between 0 and 1 and could quite possibly
depend on the plasma parameters.
(ii) profile shearing, i.e. the shear in the drifts and
parallel motion due to first and second order
derivatives of the magnetic equilibrium, density 5.4. Numerical simulations
and temperature profiles [49, 50, 51]
All the contributions listed above combine to generate
(iii) the generic impact of a radially inhomogeneous the total residual stress. Unfortunately, many tend
turbulence on the parallel symmetry [52, 53] to be of comparable magnitude, at least from simple
(iv) theimpactofaradiallyinhomogeneousturbulence scaling arguments, with various signs, making the
on passing ions with different orbit shifts [54] prediction of their sum a delicate exercise. A
quantitative prediction of intrinsic rotation therefore
(v) the deviation of the equilibrium distribution
requires numerical simulations. The lowest order
function from a Maxwellian, i.e. the impact of
momentum flux can be computed in gyrokinetic δf
the neoclassical equilibrium [55], which includes
flux-tube codes provided that the background E ×B
the pressure gradient contributions to the E×B
toroidalflowandanarbitraryfluxsurfacegeometryare
shear [56].
included. The first order contribution (i) can also be
For the parameter dependencies, all contributions computed within the flux-tube approach but requires
that rely on coupling by parallel compression between the inclusion of higher order parallel derivatives. The
density and parallel velocity fluctuations increase contributions (ii) to (iv) require a radially global
in magnitude with R/Ln. This is the case for approach. Contribution (v) can be treated in a δf
contributions (i-iii) and a part of (v). The flux-tube codeby addingthe neoclassicalcorrectionto
dependence already appears in reduced fluid models the background Maxwellian distribution function. It
whenconsideringagenericparallelsymmetrybreaking, can also be treatedby solving the coupled neoclassical
see e.g. [57]. Another robust feature of residual and turbulent problem. The second option requires
stress is that its magnitude tends to be smaller for an accurate collision operator and is considerably
TEM dominated turbulence than for ITG dominated more computationally expensive as the simulations
turbulence, typically by a factor & 2, consistently must cover several ion-ion collision times to reach
with the more symmetric mode structure with respect a stationary state with respect to the neoclassical
to the horizontal midplane obtained in the TEM physics. This method includes, however, the impact
regime. The residual stress contributions related to of turbulence on neoclassical physics, which may also
the radial inhomogeneity of turbulence [53, 54], to the be relevant for momentum transport [59]. Finally, all
shear in the perturbed E × B drift advection of the simulations that aim for a quantitative prediction of
background [49] and to the neoclassical equilibrium themomentumfluxmusttreattheelectronskinetically
[55]allstronglydependonthesecondorderderivatives asanadiabaticelectronapproximationhasadramatic
of the temperature and/or density profiles. This impact on the parallel symmetry [60].
dependence can extend as far as to change their The first principle prediction of the intrinsic
sign. The neoclassical equilibrium residual stress also rotation profile resulting from momentum transport
stronglydependson,andcanchangesignwith,theion- including the interplay between neoclassical and
ioncollisionality. Thecontributionrelatedtotheshear turbulence physics represents a formidable challenge.
of the parallel motion and of the curvature and ∇B Only one example of a global simulation with kinetic
drift [50] does not depend on second orderderivatives. electrons (at reduced ion to electron mass ratio)
Interestingly, contribution (iv) alone strongly depends including all the lowest and first order contributions
on the radial position of the X-point. Its impact on to momentum transport and evolving the rotation
9
profile over a confinement time has been reported magnitude to the up-down asymmetry residual stress
[61]. It remains an extremely important result as it andtheCoriolispinch. Thepredictedtoroidalrotation
demonstrates that a rotation profile reaching 0.15vthi gradient was up to an order of magnitude smaller
can be sustained by the internal redistribution of than measured, demonstrating the need to invoke
momentum by turbulence, see Fig 10 in [61], lending other contributions to explain the measurements. The
further support to the interpretation of intrinsic impactofprofileshearingwasinvestigatedforaDIII-D
rotation in this framework. It also confirms the plasmainwhichthetoroidalrotationwasnearlyzeroed
criticalroleofincludingkineticelectronsasthetoroidal out by counter-current neutral beam injection [49]. In
rotationwasshowntodevelopintheoppositedirection these conditions, the Coriolis pinch is negligible and
when the electrons were described adiabatically. the momentuminputbalancesthe residualstress. The
At some point, the issue was raised as to whether simulations were performed in the non-linear regime,
the conventional gyrokinetic ordering was sufficient including all the lowest order contributions to the
to properly describe turbulent momentum transport, momentumfluxandtheprofileshearingresidualstress
in particular in full-f global simulations [62]. It was (fromfirstandsecondorderderivatives). Aroundmid-
first proven in the context of the gyrokinetic field radius, the predicted momentum flux was comparable
theory that momentum conservation is guaranteed in magnitude to experiment, but could differ, even in
to any order provided the approximations are made sign,dependingonthe chosensecondorderderivatives
at the level of the Hamiltonian [63] and it was of the temperature and density profiles. Further
then demonstrated numerically that the conventional studies are required to better quantify the relevance
gyrokineticorderingissufficienttodescribemomentum of this contribution. Finally, the impact of a generic
transportinthelongwavelengthapproximationthatis symmetry breaking term was explored for the AUG
valid for ITG and TEM turbulence [64]. databases of [10] and [16] by imposing a finite
ballooning angle shift θ0 in the linear gyrokinetic
6. On-going modelling activities simulations. A unique θ0 value was chosen for all the
cases in the TEM regime and another for those in the
At present, numerical simulations as reported in [61] ITGregime. After these twoad-hocvaluesarechosen,
are far too costly to be systematically compared the experimental toroidal rotation gradient around
to experimental observations. Modelling activities mid-radiusissurprisinglywellreproducedbythequasi-
therefore focus on the residual stress contributions linear prediction across the whole database capturing
separately in the hope that one of these contributions the strong R/Ln dependence of the rotation gradient.
dominates the others in magnitude. The collisionality This suggests that the residual stress mechanism
dependence of the neoclassical equilibrium residual sustaining the intrinsic rotation profile relies on the
stress CNC is appealing and was invoked to explain coupling,byparallelcompression,betweendensityand
ϕ
Type I rotation reversals in MAST [11]. A simplified parallel velocity fluctuations, as this coupling directly
qualitative model was used to predict the reversal engenders a R/Ln dependence of residual stress.
state. According to this model, the rotation profile is
predicted to be peakedinthe plasma regionwhere the 7. Summary and discussion
collisionalityislowerthanathresholdvalueandhollow
in the region above, with the transition between the Basedontheexperimentalobservationsgatheredinthe
two regions moving radially inward across a density last ten years in AUG, C-Mod, MAST, KSTAR and
ramp. While the prediction of the reversal state TCV and the developments in the theory of intrinsic
was reasonably successful, the model did not appear rotation,turbulentmomentumtransportappearstobe
compatiblewiththeexperimentalobservationthatthe the most likely candidate to explain toroidal rotation
rotation profile is strongly modified at the critical reversals in the core of Ohmic L-modes.
densitybutrelativelyindependentofthedensitybefore Concerning the stationary rotation profiles, the lowest
and after the reversal. More recently, the impact order contributions in the turbulent momentum flux
of CNC has been investigated for the AUG database (the diagonal part, the Coriolis pinch and the up-
ϕ
assembled in [10] and covering pre- and post- reversal down asymmetry residual stress) cannot account
profiles. The focus of this work was the prediction for the experimental observations and higher order
of the toroidal rotation gradient around mid-radius contributions to the residual stress are required. The
with the modelling based on a quasi-linear approach first order contributions are now identified [35] and in
supported by a few non-linear simulations [65]. The the process of being tested against the experimental
gyrokinetic simulations included all the lowest order observations. Combining one of these first order
termsandtheresidualstressdrivenbytheneoclassical contributions, the neoclassical equilibrium residual
equilibrium. The latter appeared comparable in stress,withthelowestordercontributionswasrecently
10
demonstrated to be insufficient to reproduce the of collisionality (it occurs at a different collisionality
experimentallymeasuredintrinsicrotationgradientfor for the different wavevectors). For hypothesis (ii), a
a database of AUG Ohmic L-modes [65]. The focus bifurcationinmomentum transportcouldbe triggered
is now moving to another first order contribution, if the momentum diffusivity becomes locally negative,
profile shearing residualstress,which was shownto be i.e. the momentum flux locally decreases as the
sufficiently largeto reproducethe requiredmomentum rotation gradient increases. This could, in principle,
flux in a zeroed-rotation DIII-D case [49]. One occur if the contribution to the toroidal momentum
difficulty of this validation exercise arises from the diffusivityfromtheperpendiculardynamicsovercomes
dependence of several residual stress contributions on the parallel one, which requires large values of ǫ/q.
the second derivatives of the temperature and density Whether such a mechanism can be at play for
profiles, which are unlikely to be ever sufficiently well realistic plasma conditions remains, however, to be
measured experimentally. There are two main ways demonstrated. Hypothesis (iii) can probably be
to minimise the impact of this issue: either perform dismissed as no plasma parameter except the toroidal
the modelling at multiple radial positions as in [65] to rotation has so far been observed to strongly vary at
account for the constraint engendered by the value of the reversal and this despite an exhaustive search.
thefirstderivativesor,whenpossible,useasufficiently To summarise, in spite of considerable progress, there
complete simulation to compare the magnitude of the is, to date, no modelling that quantitatively predicts
different residual stress terms. At present, it remains the core intrinsic rotation gradient over a large scale
unclear whether a first order contribution dominates databaseencompassingpre-andpost-reversalprofiles,
for specific experimental conditions. nor the dynamics of a reversal. Possible routes to
For the reversals dynamics, the rotation is progress are suggested below.
observed to be a very sharp function of the plasma
(i) Is there a single or several reversals? Is there
density at the reversal, at least in C-Mod and TCV.
a common threshold on a local parameter that
Such a sharp variation could be the signature of:
unifies the different types of reversals?
(i) acontinuousbutsharpdependenceofCϕ/χϕ ona Important parameters for turbulent transport
plasmaparameterthatvariesinthedensityramp, are the normalised gradients R/L , R/L and
Te Ti
e.g. density, collisionality, Te/Ti, etc. R/Ln, the safety factor, the magnetic shear,
(ii) abifurcationinmomentumtransporttriggeredby Te/Ti,thecollisionality,thelocalplasmaβandthe
the density increase magnetic equilibrium (elongation, triangularity,
etc.).
(iii) amoderatedependenceofC /χ onaplasmapa-
ϕ ϕ
rameter that exhibits a strong variation (continu- (ii) In the same vein, further characterisation of the
ous or bifurcation) close to the critical density scalings of the threshold(s) in terms of local
plasma parameters, in particular for Type II
Conceptually, a bifurcation is very different from a
reversals would be helpful. Here, an important
continuous transition as it requires unstable states
issue is whether a critical collisionality better
and some direct feedback of the rotation profile
unifies thedatathanacriticaldensity. Isthatthe
on momentum transport. Some features of the
case in all devices? Dedicated experiments with
reversals in C-mod and TCV (the insensitivity of the
electron cyclotron heating power ramps may help
dynamics to the density ramp rate, the hysteresis and
decouple density and collisionality effects.
the gap observed in the stationary profiles) suggest
a bifurcation rather than a continuous transition. (iii) Does the strong correlation between the intrinsic
This aspect would deserve further investigation, in rotation gradient and R/Ln observed in AUG
particular in AUG and KSTAR, as the choice between hold across C-Mod, KSTAR and the full TCV
a bifurcation and a continuous transition not only databases? This would hint at a residual stress
impacts the way data should be handled in multi- mechanism that relies on parallel compression
variable regressions (one or two sets?) but also and would merits examination over as wide an
providesastrongconstraintonthetheoreticalsolution. operational range as possible.
From a theoretical perspective, a mechanism that (iv) An interesting observation from C-Mod is that
supports hypothesis (i) is not directly offered by the region where the toroidal rotation reverses
current theories, since the dependence of turbulent is typically restricted to q . 3/2. In TCV,
transport on plasma parameters is predicted to be Type II.a reversals were only observed for a
rather mild in general. A change of sign of one sufficiently low q95 value and in KSTAR no
of the residual stress components at the TEM/ITG reversals are observed for low elongation high q95
transition[52,50]couldbe invokedbutthe TEM/ITG plasmas, which may or not be related. From the
transition is, itself, not a particularly sharp function theory standpoint, the toroidal projection of the