Large-mode-number magnetohydrodynamic instability driven by sheared flows in a tokamak plasma with reversed central shear

0.51 MB·English

by  Andreas Bierwage

Large-mode-number magnetohydrodynamic instability driven by sheared flows
in a tokamak plasma with reversed central shear
Andreas Bierwage,1,∗ Qingquan Yu,1 and Sibylle Gu¨nter1
1Max-Planck-Institut fu¨r Plasmaphysik, EURATOM Association, D-85748 Garching, Germany
(Dated: February 2, 2008)
7 Theeffectofanarrowsub-Alfv´enicshearflowlayerneartheminimumqmin ofthetokamaksafety
0 factor profile in a configuration with reversed central shear is analyzed. Sufficiently strong velocity
0 shear gives rise to a broad spectrum of fast growing Kelvin-Helmholtz (KH)-like ideal magnetohy-
2 drodynamic (MHD) modes with dominant mode numbers m,n ∼ 10. Nonlinear simulations with
finiteresistivityshowmagnetic reconnection nearripples caused byKH-likevortices, theformation
n
ofturbulentstructures,andaflatteningoftheflowprofile. TheKHmodesarecomparedtodouble
a
J tearing modes (DTM) which dominate at lower shearing rates. The possible application of these
results in tokamaks with internal transport barrier is discussed.
2
1

Shear flows can give rise to the well-known Kelvin- 3
] (a)
asm-ph HntgteoiaeonmlnsKmisiocHhnioIn(,lMtaztmrHheiaenDgstn)pthaeleastbrysiiecmslfitoteapyrmrewe(sKs,thshiHuhcorhaIse)seawatrwonhehdiccephorame,mritapniigtcenumtceelaaatirngwclnyitceshotshouuehfispaycerleedl.d1prwtoliRdibintyelhee-- safety factor q2222....22468 000001....2468current density jz0
0 0.2 0.4 0.6 0.8 1
l MHD instabilities. A system where such coupling is ex- r
p 0.02 0.05
ics. ptthoekecatpemdraeksteoncpotlnasfityugudanrya,tiimisopnto,hrewtahrneictvherroissleed,a-asshntedraornwgh(RiccSahn)mdaioddtvaiatvenactfeeoddr ωquency ϑ00.01 (b) 0 vorticity uz0
ys achieving self-sustained nuclear fusion conditions. It has ar fre −0.05
h a non-monotonic toroidal current profile [Fig.1(a)] and ngul
the magnetic shear, s = (r/q)dq/dr, changes its sign at a 0 −0.1
p 0 0.2 0.4 0.6 0.8 1
[ some radius rmin. Here r is the radial coordinate (mi- radius r

nor radius of the torus), and q(r) is the safety factor
2 profile measuring the field line pitch [Fig.1(a)]. In the FIG. 1: Equilibrium profiles of (a) the safety factor q and
v
region with negative shear (r < r ) pressure-driven current density jz0, and (b) the angular frequency ωϑ0 and
6 min
9 ideal MHD ballooning modes are stable, so that higher vorticityuz0. qs =2resonancesareindicatedbydottedlines.
1 pressures can be achieved in the core. Strongly sheared
7 poloidal flows may further improve confinement in RS gradientmaydestabilizeDTMs.17,18 Thus,adynamicin-
0 tokamaks. Thereisevidencethatsuchzonalflowsplayan teraction between DTMs and shear flows is expected.5
6
important role in the formation of an internal transport
InthisLetter,wedescribeincylindricalgeometryhow
0
barrier (ITB) near r .2,3 MHD instabilities are often
/ min a broad spectrum of linear DTMs is stabilized by in-
s observed in ITB discharges.4,5 While there is evidence
creasing velocity shear and show that, above a certain
c
that certain ITB equilibria with strong pressure gradi-
i threshold, fast growing electromagnetic KH-like modes
s ent in the region of weakly positive shear may be stable
with dominant high mode numbers m 10 appear. The
y
to ideal MHD ballooning modes,6 low shear around q ∼
h min linear instability of both DTMs and KH-like modes is
may allow pressure-driven infernal modes to develop.7,8
p analyzed and first nonlinear simulation results are pre-
: Recently, non-curvature-driveninstabilities have also at- sented for a case with finite resistivity. The latter shows
v tractedattention.9,10 Thedestabilizationofanidealelec-
i magnetic reconnection,the formationofturbulent struc-
X trostatic KHI in a region with low magnetic shear was
tures, and a flattening of the flow profile.
demonstrated in Ref. 11. KHI coupled to parallel mag-
r For simplicity, we use the well-known reduced magne-
a netic perturbations for s=0 was considered in Ref. 12.
tohydrodynamic(RMHD) modelincylindricalgeometry
(r,ϑ,z).19 Inthelimitofzeropressure,theRMHDequa-
The scenario considered in this Letter consists of
tions for the magnetic flux ψ and the vorticity u are
a radially localized shear flow layer near q in an z
min
equilibrium which is also unstable to a broad spec- ∂ ψ+v ∇ψ = ηj (1)
t z
t(rDuTmMo).f13c,1u4rreTnht-idsrivmeanyreoscisctuivrewdhoeunbelveerteaqrmining mpaosdseess ρm(∂tuz+v·∇· uz) = −B·∇jz+ν∇2⊥uz. (2)
through a low-rational value.4,5 Shear flow is known to Here, ρ is the mass density, η the resistivity and ν the
m
stabilize DTMs through its tendency to decouple the viscosity. Currentdensity j and vorticity u are related
z z
resonances.15,16 Inthenonlinearregime,lockingbetween to ψ and φ through µ j = 2ψ and u = 2φ, re-
0 z ⊥ z ⊥
−∇ ∇
finite-size magnetic islands and flattening of the velocity spectively. In the following, time t is measured in units
2
0.015 0.01 10−2
(a) (2,1) (b) (2,1) (c) m =14
(6,3) 0.009 (6,3) m =10 peak
(12,6) 0.008 (12,6) peak
∆fm 2π∆f
γh rate lin 0.01 ωncy /mlin00..000067 γh rate lin10−3 mpeak=6
linear growt 0.005 m=12 m=6 m=2 linear freque 000...000000345 linear growt mpeak=4
G=0.00, d=0.02
0.002 G=0.75, d=0.02
0.001 10−4 G=1.50, d=0.02
G=0.80, d=0.01 m =18 m =26
0 0 max max
0 0.5 1 1.5 2 0 0.5 1 1.5 2 0 5 10 15 20 25 30
core rotation parameter G core rotation parameter G poloidal mode number m
FIG. 2: (a) Linear growth rate γlin and (b) rotation frequency ωlin/m for the modes (m,n) = (2,1), (6,3) and (12,6), in
dependence of the core rotation parameter G. The effective and specific shearing rates, ∆fm and ∆f, are plotted as dash-
dotted lines. In (c) spectra of linear growth rates γlin(m) are shown for three cases with d=0.02: G=0 stationary, G=0.75
stabilized, G=1.5 destabilized. A spectrum obtained with a narrower shear layer is also shown (d=0.01, G=0.8).
of the poloidal Alfv´en time τHp = √µ0ρma/B0 (with For the shear-flow-drivenmodes described in this Let-
′ ′
B being the strong axial magnetic field) and the radial ter the vorticity gradient term ρ (im/r)v u (u =
0 m r z0 z0
coordinate r is normalized by the minor radius of the du /dr) in Eq. (2) is crucial. This term iseknown to be
z0
plasma,a. Resistivedissipationis measuredby themag- theoriginofKHI.1Inaccordancewiththecriteriongiven
netic Reynolds number S =τ /τ (with τ =a2µ /η by Eq. (23) in Ref. 10, the presence of resonant surfaces
Hp η Hp η 0
being the resistive diffusion time) and viscous damping is found to be crucialfor instability. We find that low-m
by the kinematic Reynolds number Re = a2/ντ . modesevenrequiretwo nearbyresonancesandthatmul-
Hp Hp
The perturbed fields are Fourier transformed as h = tiplebranchesofunstablemodes(idealandresistive)can
e be destabilized simultaneously.
Pm,nehm,n(r)exp(imϑ−inz/R0)(withR0beingthema-
jor radius of the tokamak), where m/n=q =2 (single- Results obtained by solving the linearized RMHD
s
helicity perturbation). The model and numerical meth- equations as an initial-value problem are presented in
ods are described in detail in Refs. 14 and 20. Figs.2and3. InFig.2(a)thelineargrowthratesγlinand
The equilibrium configurationused is shownin Fig. 1. in Fig. 2(b) the linear frequencies ωlin of several modes
Thesafetyfactorhastwoq =2resonantsurfaceslocated areplotted asfunctions ofG. Spectra ofunstable modes
s
a small distance D = r r = 0.066 apart (r = areshownin Fig.2(c) forseveralvaluesofG. The linear
12 s2 s1 s1
0.381,r =0.447). Themag−neticshearatthesesurfaces mode structures are shown in Fig. 3. The data were ob-
s2
iss1 = 0.10ands2 =0.12. Thepoloidalvelocityprofile tainedwithSHp =107 andReHp =1011, sotheviscosity
v has−a shear layer near r and is given by effect is negligible.14
ϑ0 min
ωϑ0 =vϑ0/r =Ω0G(cid:2)C1−tan−1((r−r0)/d)(cid:3)/C0, (3) GT=h0e (γnlion(flGo)wc)u,rsvoesoninlyFGig. 2(0ai)sacroenssyidmermedet.riScinacreoutnhde
≥
with C1 = tan−1[(1 r0)/d] and C0 = C1 distanceD12 issmall,theresonancescouplestronglyand
tan−1[( r )/d]. The stan−dard case for ω is shown i−n a broadspectrum of resistiveDTMs with dominanthigh
0 ϑ0
Fig. 1(b−). The shear layeris located near the inner reso- poloidal and toroidal mode numbers m and n are found
nance (r = 0.4) and has a width (2d = 0.04) compara- forG=0(cf. Refs.13,14). Intherange0<G.1.0,the
0
ble to D . The parameter G controls core rotation and effectoftheshearflowonDTMsisprimarilyastabilizing
12
thustheshearingratebetweentheresonances. Wedefine one. However,it canbe seen in Fig. 2(a) that this stabi-
the specific shearing rate as ∆f = (ω ω )/2π [with lizationmaystagnateandevenreverse. Forinstance,the
02 01
ω =ω (r )]. The effective shearing ra−te for the mode growthrate of (m,n)=(12,6)mode rises briefly around
0i ϑ0 si
(m,n) is ∆f =m∆f. Since the equilibrium flow is not G 0.2 before descending further. The origin of this
m ∼
includedself-consistently,onlysub-Alfv´enicvelocitiesare behavior is not yet understood. The growth rates reach
considered (Ω0 = 0.01, G 2). The characteristics of a minimum around Gmin =1.0 and increase for G>1.0
the shear flow are simil|ar|t≤o those reported in Ref. 2. almost linearly with G. When a smaller velocity gradi-
Both the static MHD equilibrium and the shear flow are ent( G/d)isused,orwhentheinter-resonancedistance
∝
separatelyunstable: the former to resistive DTMs14 and D12 isincreased,alargervalueisobtainedforGmin (and
the latter to the ideal KHI:1 the Rayleigh criterion for vice versa). Without resonant surfaces (qmin > 2) no
KHI is fulfilled [u′ (r ) = 0] and the existence of an in- unstable modes with helicity m/n=2 are found.
z0 0
stability threshold m , is demonstrated shortly. The frequencies ω (G) in Fig. 2(b) undergo a shift
max lin
3
around G Gmin. For G < Gmin the rotation at the magnitude phase
outer reson≈ance determines the frequency of the mode. 1 2 1 1
(a) (b)
This is typical for DTMs (since typically s1 < s2) and ψ| (m,n)= φ|
indicatesthatthecurrentdrivedominatesinthisregime. | (10,5) 1 | 0
0.5 0.5 (m,n)=
ForG>G the frequencyequalsthatatthe inner res-
min (10,5)
0 −1
onance, which is closer to the shear layer. Hence, G = π π
Gmin is identified as the transition point between pre- α / ψ α / φ
0 −1 0 −2
dominantly current-driven and shear-flow-driven modes. 0.3 0.4 0.5 0.3 0.4 0.5
1 2 1 2
With a shear layer closer to the outer resonance there is (c) (d)
no frequency shift. ψ| (m,n)(=c) φ| (m,n)=
| (20,10) 1 | (20,10) 1
A comparison between the growth rates and the ef-
0.5 0.5
fective shearing rates in Fig. 2(a) shows that γ (G) in-
lin 0 0
π π
tchreaatsseosmweitohftGheactoulepalsintgasbeftawsteeanst∆hefmre.sonTahniscessu,gwgheiscths α / ψ α / φ
0 −1 0 −1
0.3 0.4 0.5 0.3 0.4 0.5
is important for DTMs, may pertain even for G>Gmin. radius r radius r
For the (2,1) mode we have observedthis effect by com-
paring the results of the RS configuration with results FIG. 3: Linear eigenmode structures for (m,n)=(10,5) [(a)
obtained with a monotonic q profile with a single qs =2 and (b)] and (20,10) [(c) and (d)]. The flow parameters are
resonantsurfaceandotherwisesimilarconditions: inthe r0 = 0.4, d = 0.02 and G = 1.50. Flux ψe and potential φe
formercase(RS)the shear-flow-driven(2,1)modeisun- are decomposed into magnitude and complex phases as ψ =
iα
stable [cf. Fig. 2(a)], in the latter case (monotonic q) it |ψ|e ψ. Vertical dotted lines indicate resonant surfaces.
is stable. In contrast,higher-m modes such as (6,3) and
(12,6) have similar growth rates in both cases, so for
thesethecouplingbetweentheresonancesdoesnotseem range 107 S 1015 reveal that shear-flow-driven
Hp
to play a crucial role. modes (G >≤G )≤on the dominant branch grow inde-
min
The full spectra of linear growth rates γlin(m) are pendently of SHp. The ideal character of the instability
shown in Fig. 2(c). For the same shear layer width as is reflected in the linear eigenmode structures shown in
in Figs. 2(a) and (b) (half width d = 0.02), the spectra Fig. 3(a) and (b) for (m,n)=(10,5): the flux perturba-
are plotted for three values of G: no shear flow (G=0), tion switches signs at the resonances, as can be inferred
stabilized (G = 0.75), and destabilized (G = 1.5). The from ψ (r ) = 0 and the phase shifts ∆α(r ) = π
dominance of modes with m > 2 modes can be clearly |e| si si ±
[Fig. 3(a)]. The potential φ [Fig. 3(b)] is non-zero at the
seen. The DTM spectrum (G = 0) has m = 6 as e
peak resonancesand its complex phase exhibits a jump at the
its fastest growing mode and the last unstable mode is ′
inflection point of the velocity profile, where u = 0.
m = 18. In the stabilized case (G = 0.75) m z0
max peak Resistivity independent modes are also observed with a
has decreased to 4. In addition, a second branch of
single resonant surface and otherwise similar conditions.
weakly unstable modes is present, with m =18. For
peak
The typical structure of modes on the secondary
G=0.75 the frequencies of modes on both branches are
branchisshowninFig.3(c)and(d)for(m,n)=(20,10).
determinedbytherotationoftheouterresonance. Inthe
Theperturbationislocalizedbetweentheshearlayer(r )
destabilized case (G = 1.5), the modes on the dominant 0
branch around mpeak = 10 grow rapidly and their fre- and the outer resonance (rs2). At r = rs2, the flux ψe is
quenciesareup-shifted, implying rotationwiththe inner non-zero and the potential φ has a discontinuity, sug-
e
resonance. Thegrowthratesonthesecondarybranchare gestingatearing-typeinstability. This mode growsmost
almost the same as for G=0.75 and the frequencies are rapidly for SHp 107, while the growth rate is reduced
∼
still determined by the rotation of the outer resonance. for larger and lower resistivity.
In Fig. 2(c) a case with a smaller shear layer width Furthercalculationsshowthatthegrowthratesonthe
d = 0.01 is shown as well. Compared to the d = 0.02 dominant branch are rather insensitive to the exact lo-
case, here the destabilization of shear-flow-drivenmodes cation of the shear layer between the resonances. This
has a lowerthreshold G and the spectrum is broader: indicates that shear-flow-driven modes have a non-local
min
m = 14 and m = 26. Indeed, it is found that character,inthesensethatitisnotsufficienttoconsider
peak max
mmax is controlled by the parameter d, while G seems values of ωϑ0 at the resonances only (cf. Ref. 10).
to be irrelevant. This is similar to KHI which possesses Finally, nonlinear simulation results are presented in
an upper threshold for the wave number kϑ = m/r of Fig. 4 (parameters settings: G = 1.5, d = 0.02, SHp =
unstablemodes:1forγ >0itisrequiredthatkϑd.Ø(1). 107, ReHp = 5 108, single helicity qs = 2, 128 modes,
Here,ford=0.02andr0 =0.4the dominantbranchhas dealiased). At ×the time of the first snapshot, Fig. 4(a),
mmax = 18 [Fig. 2(c)], so that kϑd= 0.9. 1. Modes on thenonlinearmodecouplinghasreducedthevelocitygra-
the secondary branch do not seem to be subject to the dient around r =0.4, while the profile in the surround-
0
same threshold. ings is steepened. The helical flux contours show islands
Calculations with magnetic Reynolds numbers in the around the inner resonance, with a dominant m = 2
4
ωangular freq. ϑ0.01 (a) (b) 0.01ωngular freq. ϑ tsphtlaaeTgsphmereiaismorlbaiekrsseyeislrytsviaevttdioutyrbmaertaaitgothnnheeemtrfliceatcthrhtaeeancnnoiintnsugmnrebocifuntiltoethnhneceeiflesoaewdrfflueypecrntosto.ofinlleHfii.nnoeiwater-
0 t=0 t=0 0 a
safety factor q2.021 t=1200 t=1770 22.01afety factor q eccovulbeersarrer,ernvhttehodegwrrfleamadatuitreceenhnttinweaagnocdohinftcshttoheanebvtriefllilibootwyucittdepysrriogvafiritnaledagisegmunigvetg,echenasanttndsiimstimhte.asist,Tntthhhoeeet
1.99 ψ(t=1200) ψ(t=1770) 1.99s current-gradient-driveninstability (i.e., DTM) may play
* *
a more significant role in later stages. The study of the
0.8 0.8
long-time evolution of this system requires turbulence
π π
ϑngle /0.6 0.6ϑngle / smimoduelaltainondsnwumhiecrhicgaol cboedyeounsdedthheersec.ope of the physical
a a
oidal 0.4 0.4oidal In summary, we have analyzed the effect of sheared
pol pol flows in a RS tokamak configuration when qmin is below
0.2 0.2 but close to a low-order rational value q = m/n and a
s
pairofnearbyresonantsurfaceshasformed. The growth
0 0
0.35 0.4 0.45 0.35 0.4 0.45 rates of DTMs are reduced, although the spectrum re-
radius r radius r
mains peaked at m>2 even with strong shear flow. For
sufficientlystrongvelocityshearDTMsarereplacedbya
FIG.4: Snapshotsofnonlineardynamicsat(a)t=1200 and
broadspectrumofshear-flowdrivenKH-likemodes with
(b) t = 1770. Top: ωϑ profiles. Dotted lines indicate the
original resonant radii. Middle: q profiles. Bottom: helical dominant high m 10. In the nonlinear regime and
∼
magnetic fluxcontours [ψ∗ =ψ+r2/(2qs)] in theϑ-r plane. with finite resistivity,magnetic reconnectionoccurs near
ripples caused by KH-like vortices. The tendency to de-
velop turbulent structures in the inter-resonance region
topology and high-m ripples. In the X-point regions, andflattening ofthe velocityprofileisobserved. Further
poloidally localized KHI-like vortices (only one shown calculationswithexperimentaldataasinputarerequired
due toπ-periodicity)canbe observed. Subsequently,the inordertoidentify theroleofKH-likemodesinITBdis-
m=2 islands undergo fragmentation and the structures charges. Note also that since high-m modes are involved
become increasinglyturbulent, ascanbe seenin the sec- thesingle-fluidMHDpicturemaybeinaccurate. Ourre-
ond snapshot in Fig. 4(b). sults motivate further investigations with more realistic
Note that the shear-flow-induced perturbations ex- models to understand the properties and effects of KH-
tendovertheentireinter-resonanceregion(andbeyond). like modes in tokamak plasma with ITB. Since high-m
Thus, one may expect a coupling similar to that in modes are involved, we expect that similar instabilities
DTMs. Indeed,Fig.4(b)showsislandformationonboth may be found for other values of qs suchas 3/2or7/3.14
resonanceswhile the centralflux surfaces are still intact. A. B. thanks the Max-Planck-Institut fu¨r Plasma-
Furthermore, in Fig. 4(b) a flattening of the flow profile physik in Garching for its support and hospitality,
in a wide region can be observed, while q is still well and gratefully acknowledges valuable discussions with
min
below q = 2. This shows that when resistivity is low, S. Hamaguchi, Z. Lin, E. Strumberger and T. Tatsuno.
s
∗ Electronic address: [email protected]; Present address: (2005).
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