J. Plasma Fusion Res. SERIES. Vol. 5 (2002) 381-385

lon Temperature Gradient Driven Instability, Parallel Flowand Enhanced Reversed Magneic Shear Transition
SEN Sudip
Departinent of Theoretical Physics, The Australian National University, Camberra, ACT 0200, Australia
andDepartment of Mathematics, Hampton University, Hampton, VA 23668, USA(Received: 11 December 2001 / Accepted: 16 April 2002)

Abstrirct
A model for transition to the enhanced reverse shear (ERS) or negative central shear (NCS) mode
triggered in tokamaks is proposed. This model takes into account the linear and quasilinear behaviour of
the ion temperature gradient (ITG) drived perturbation, considered nowadays as the dominant source of
anomal'rus energy losses in the low confinement (L) mode, in the presence of a radially varying parallel
velocitv. Analytic and numerical studies show that when the magnetic shear has the same sign as thesecond derivative of the parallel velocity with respect to the radial coordinate, the linear mode may
becomr: more unstable and turbulent momentum transport increases. On the other hand, when themagnetic shear has the opposite sign to the second derivative of the parallel velocity, the linear mode
may be completely stabilized and turbulent momentum transport reduces.

Keywords:
ITG modes, ERS transition, instability, transport, shear

1. lntroduction
It is belie,ved that the most remarkable story offusion researctr over the past decade is the discovery offurther. This is especially significant because it is now
widely accepted that the negative magnetic shear is not
the only factor needed for the transport reduction in theERS or NCS modes. Some of the clearest evidence
comes from the comparison of the RS (reverse shear)and ERS (enhanced reverse shear) transition data inthe enhanced reverse shear modes (ERS modes) in
Tokamak Fusion Test Reactor (TFTR) [1] and thenegative centriLl magnetic shear modes (NCS modes) in
DIII-D [2]. It ir not often that a system self-organises toa higher energy state with such a large reduction of
turbulence anc. transport when an additional source of
free energy is rLpplied to it [3]. It is usually believed that
the ERS or .\CS configurations can provide thecharacteristics desirable for a fusion reactor [4].

TFIR [5]. It shows that the RS phase is not necessarily
an ERS phase and some other factor is needed toexplain the transition. Theoretical study also indicatesthat the reversal of magnetic shear alone might have alittle effect on the ITG-type microinstabilities [6].
The E x B shear stabilisation mechanism has been

The basic physics involved in the formation oftransport balriers in the ERS or NCS plasma
configurations is therefore fundamental to thedevelopment cf techniques to control such barriers for

proposed to explain the core transport barriers formed inplasmas with negative or reverse magnetic shear
regimes [3]. It is believed that the change in the radial
electric field in the core is produced by a number oftailoring profiles and for improving operating regimes
C orre spondin

I author' se -mail : s en@ nuc leng. ky oto - u. ac.j p381

@2002 by The Japan Society of Plasma
Science and Nuclear Fusion ResearchSen S., Ion Temperature Gradient Driven Instability, Parallel Flow and Enhanced Reversed Magneic Shear Transition

ways, for example, by toroidal flow (voi) [7] and/or byAll other symbols are assumed to have the same mean-pressure gradient (Vpr) [5] and more recently bypoloidal flow (ve;) [8]. However, while this E x B shear

ing as in reference [11] unless otherwise stated explicitly. To model the equilibrium parallel velocity we haveassumed a simple general case of variation with the radial distance,r as:

stabilisation mechanism alone can satisfactorily
explainthe confinement improvement in the edge, it may
not be an obvious explanation for the core confinement

YmYmT
vo(x) = voo *;-,r + -;improvement in the ERS and NCS plasma [9]. For
example, the formation of the ERS mode in TFTR has
been reported [5] at values of To,n (E x B shearingrate), as much as a factor of 3 below 7-u* (the maximumlinear growth rate), while for the suppression of
turbulence-induced transport the condition To, s 2 T^*
needs to be satisfied. It is therefore natural that an
explanation of these experimental results should besought in the effects such optimized magnetic
configurations have on microinstabilities and on theL,t2L,,x- .

Equation (1) is a simple Weber equation. Depending on
the sign of P, we have two types of solution. If P < 0,i.e.,"/" s.s2

QtQ+ Kt

A2

the solution which satisfies the physical boundary condi-tion, i.e., @ -+ 0 at -tr = t- is given by

consequent transport.

|_Q(x\ =0,.^pl- r/2J I pl (" -We identify a mechanism which might be playing
the key role in the reverse shear transition. We show
here when the magnetic shear has the same sign as thesecond derivative of the parallel flow with respect to the^t(2)",)'l

where xo = lQVzlP| The wave therefore does not propagate and is intrinsically undamped.

On the other hand, if P > 0, equation (1) has theradial coordinate, the ion temperature gradient (ITG)
mode may become more unstable and turbulent

solution

momentum transport increases. On the other hand, whenthe magnetic shear has opposite sign to the second

f_t

QG\= Qo"^vl- i rzlllr | (..+*,)'lderivative of the parallel velocity, the ITG mode is.

(3)completely stabilized and turbulent momentum transportThus, we have now a non-localised mode carrying

reduces. This result, therefore, shows that it is therelative sign of the second radial derivative of theequilibrium parallel flow with respect to the magnetic
shear which may be the key factor for the enhanced
reverse shear transition. This is because, a radiallyvarying toroidal flow is routinely observed in the region

energy outward. Because of the convective wave energy

where the transport barrier is formed during thestudied here. We adopt a two-fluid theory in a shearedslab geometry. Following Mattor and Diamond I l] wein the stability of the mode. When the magnetic shearhas the same sign as the parallel flow curvature, i.e., forpositive magnetic shear (2" > 0), parallel flow curvatureacts to destabilise the mode. On the other hand, for thenegative magnetic shear configuration (1" < 0), i.e.,
when the magnetic shear has the opposite sign to thesecond derivative of the parallel flow with respect to thethen write down the linearised eigenvalue equation as:radial coordinate .rr, the parallel flow curvature acts toleakage the perturbation will decay in time in theabsence of any energy source feeding the wave. Thewave is therefore damped.

From the above discussion it is clear that it is theparallel flow curvature which actually plays the key roletransition to the ERS or NCS modes [0].

2. Stability Analysis
The short-wavelength ITG-like perturbation isg.l A-ex+ px'z]=odxtt-

stabilise the mode. Flow curvature now forms anadditional antiwell which pushes the wave function(t)away from the mode rational surface, thereby enhancing

where,/rS
A--b2 l-Q r Ip=zr--Av-- otgz+xt-*.s'a+Kar'O=, K =U:L,^^J *s, o= -9
rknvo'"' {)()+K\-'stabilisation.

The overall stability of the mode may also be
obtained from the dispersion relation

rr= +i,/lPl
A= n2vr(.. ' \/ ' \,,s=!4
r-=lv*L"el.
l.
"1,=ll*J,
rL, , "2-lf\LrzJ

4PI

which can be written more explicitly as:\Lr, l382

Sen S., Ion Temperature Gradient Driven Instability, Parallel Flow and Enhanced Reversed Magneic Shear Transition

4(tz+rr(r-#")lation. For simplicity, we show only the curvature contribution in the figures as our main purposeis to show
the role of the relative sign of the second radial derivative of the equilibrium parallel flow with respect to themagnetic shear. Fig. I exhibits when the magnetic shear

JrQ

has the same sign as the parallel flow curvature, i.e., for
L" > 0, parallel flow curvature acts to destabilize the(t *. oi) e'z +(x rci -r)o =J: {)- rs((2 + K)t(4)'-s(J2+/()mode further. Figure 2, on the other hand, shows that for

Figures I and 2 show the solution of this dispersion re-the negative magnetic shear configuration (2, < 0), i.e.,
when the magnetic shear has the opposite sign to thesecond derivative of the parallel flow with respect to theradial coordinate r, the parallel flow curvature acts tostabilize the mode.

0.5
o.4

3. Ouasilinear Transport
To see what these studies on the linear modestability means to the plasma transport we will now, as

0.3
o.2

an example, derive analytic formulas for the quasilinear

radial flux of momentum. For this we note that thegeneral expression for the perturbed electrostatic

0.1

000.1 0.2 0.3

potential is given by

0.4 0.5 0.6|'

d= Re I r,,do P(x)exp( i ky y - iatk,tt

Fig. 1 Normalized groMh rate Ias a function of normalized kr.Flere, a is dueto L6, Lp= 17 cm and b due'lll.

In the representation of @ we use @6 to characterise
the RMS fluctuation level and 0 (x) the normalised wave

to the absence of flow. The other parameters are
fn= 15cm, s=0.1, K=3, V*= 0.6 Cs, L"= 150 cm.

function. The E x B drift velocitv is;-,Bca6 - .46!

0.220-', 1t,-gla,"t-Bdx'(s)0.215

where B is the toroidal magnetic field and c is the speed

0.210
0.205
o.2oo
0.195
0.190
0.185of light. Now we introduce the two components of the
micro Reynolds stress that measure the radial flux of theperpendicular momentum,

frry= l, vy + vxvy
0.1

0.2

0.3

0.4

0.5

where * stands for complex conjugate. In writing eq.
(6), we leave implicit the summation over all poloidal
mode numbers and all rational surfaces determined by
the toroidal mode number spectra. It is straight forward
to obtain the analytical expression of eq. (6), using eqs.
(3) and (5) as follows:

0.6k,0.30
o.28
0.26
0.22
0.20
0.18

o o = Qol'|

0.1

0.2

0.3

(6)

0.4 0.5 0.6

f,,zt, rtyl(, + *o)(7)which at the reTerence mode rational surface can be sim-k,,

plified toFig. 2 Normalized growth rate /as a function of normalized kr. Here, a is due to L6, Lu2= -17 cm and b
due to the absence of flow.

n,=lQol'$0,ffi
383

(8)

Sen S., Ion Temperature Gradient Driven Instability, Parallel Flow and Enhanced Reversed Magneic Shear Transition

With this simplified formula we are now in a position tosee what effects the parallel flow curvature has on theradial flux with the change of its relative sign with themagnetic shear. We notice that the quantity whichchanges with the relative sign of the parallel flow curva-Table 1

LaL"+140 cm+140 cm-140 cmture and the magnetic shear is P. Table I shows the normalized radial flux as calculated from eq. (8) with various directions of the magnetic shear. For calculating theradial flux we have assumed L, - lO cm, L, - 5 cm, p" 0.1 cm, and VxoolC, - l/10. From this table, one canclearly see that when the shear is positive, i.e., when thesign of the parallel flow curvature is the same as themagnetic shear the radial flux is greater than if there
were no parallel flow curvature (L, - -). On the other
hand, when the shear is negative, i.e., when the sign ofthe parallel flow curvature is in the opposite direction of
the magnetic shear, the radial flux is less than if therewere no parallel flow curvature. It is important to noticethat while the effect of the parallel flow curvature is robust in the case of the linear mode stability its role inthe case of radial flux is rather modest. This apparent
disparity can be attributed to the fact that in the case of
the linear mode stability the contribution of the parallel

5cm

5cm

Flux
0.212
0.1 970.186

Table 2L"

-140 cm-140 cm-140 cm-140 cm-140 cm-140 cm-140 cm

11La

Flux

cm

0.196

9 cm
7 cm
5 cm
3 cm

2cm
1 cm

0.194
0.192
0.186
0.172
0.150
0.101intriguing aspect of Table 2 is that it also shows that as
the scalelength of the parallel flow curvature increases
the radial flux gradually increases before it finallyreaches a state when the flux is no longer sensitive tothe variation of the flow profile.

flow curvature comes through P whereas that in the radial flux comes through {F. Before leaving this section
it is important to mention that one can similarly calcu-4. Conclusion

In conclusion, we present here a model fortransition to the enhanced reverse shear (ERS) or
negative central shear (NCS) mode triggered inlate the radial flux of the parallel momentum 2,,;1 fromthe equation 7t"n= lii, + l)lir, where in the quasilinear
approximation it11= (ekrlL,mia) xQ.Proceeding as before
one can then reach at a conclusion similar to that for theperpendicular counterpart.

tokamaks. Our studies show that when the magnetic
shear has the same sign as the second derivative of theparallel velocity with respect to the radial coordinate,

the linear mode may become more unstable andFinally, having shown the reduction of flux for thenegative magnetics hear in the preceeding sections wewill now address an important question as to what
extent this reduction in the radial flux can be continued
by varying the scalelength of the parallel flow curvature.

turbulent momentum transport increases. On the otherhand, when the magnetic shear has the opposite sign tothe second derivative of the parallel velocity, the linear

mode may be completely stabilized and turbulent
momentum transport. It is also shown that a large
reduction in the momentum transport is possible by

We show thisin Table 2.
The parameters values chosen for calculating thefluxes are the same as assumed in Table 1. It is clear
from Table 2 that, as expected, the radial flux reduces
with the decrease of the scalelength of the parallel flow

suitably tailoring the parallel flow profile. Before
concluding, we note that some related observations werealso made by Li et al. [13] with perpendicular flow
profile for the case of a weak magnetic shear.

curvature, the lower limit of which will be determinedby the threshold of excitation of the Kelvin-Helmholtz(KH) instability. However, it has been noted by Biglari
et al. [12] that in order that KH be excited the condition
Lv (- L,) < Z, needs to be satisfied, where Zr is theReferences[1] F.M. Levinton et al., Phys. Rev. Lett. 75, 4417
( I 995).

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t4l C.B. Forest e/ al., Phys. Rev. Lett. 77,3141 (1996).
t5l E. Maz^rcato et al., Phys. Rev. Lett. 77, 3145

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possible by suitably tailoring the flow profile. Another384

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(2000); E.J. Synakowski et al., Phys. Rev. Lett.78,2972 (1997).
tlll N. Mattor and P.H. Diamond, Phys. Fluids B 31,

t6l S. Sen, II. Nordman, A. Sen and J. Weiland,Physica Scripta 56, 86 (1997).

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Il2lIJ. Biglari et al.,Phys. Plasmas, B 2, 1 (1990).
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