J. Plasma
J.
Plasma Fusion
Fusion Res.
Res. SERIES,SERIES,Vol.
Vol.6 6(2004)(2004)283–286
000–000

Radial and Poloidal Structure Coupled with Shear Viscosity under
the Existence of a Large Flow Shear in Tokamaks
KASUYA Naohiro, ITOH Kimitaka and TAKASE Yuichi1
National Institute for Fusion Science, Toki 509-5292, Japan
1
University of Tokyo, Bunkyo-ku 113-0033, Japan(Received: 9 December 2003 / Accepted: 13 March 2004)

Abstract
A large poloidal flow is generated in an improved confinement state in tokamaks. Poloidal shock has been
predicted theoretically when a poloidal flow velocity increases. An extension of the model to include the poloidal
structure is made to obtain a self-sustained two dimensional structure. Its special cases with no shear viscosity(decoupled) and large shear viscosity are solved.
Keywords:
poloidal shock, two dimensional structure, shear viscosity, E × B flow shear, tokamak

1. Introduction

exists. In this paper a self-sustained two dimensional structure
model with shear viscosity (µ) is proposed. The existence ofthe diamagnetic flow can also weaken the shock structure[13 ]. In our approach, shear viscosity is taken into accountto consider radial structural coupling. Interaction between
quantities on different flux surfaces through shear viscosity
reveals a two dimensional structure in tokamaks. Special caseswith µ = 0 and µ >> 1 are solved. These calculations give atypical shock structure and a structure determined by shear
viscosity, respectively.

The H-mode [1 ] offers a great advantage for improving
plasma confinement in tokamaks. A large poloidal flow isgenerated by a large radial electric field in the tokamak edge
region in H-mode [2 ], and a shear flow is responsible forimproving confinement by suppressing anomalous transport[3 ]. Many studies have been carried out to understand theformation mechanism of a steep structure in the radialdirection [4-6].
In the tokamak edge region in H-mode, the poloidal flow
increases to Mp ~ 1, where Mp is the poloidal Mach number.
Appearance of a poloidal shock structure in the potential anddensity profile has been predicted theoretically with such afast poloidal flow [7,8 ]. A measurement in the CCT tokamakshows a poloidal density variation [9 ]. The existence of asteep poloidal structure induces a deviation of constantdensity contours from magnetic surfaces [10 ], and makes alarge poloidal electric field, which gives a large E × B flowin the radial direction [11 ]. However, transport analyses areoften carried out with flux surface averaged quantities, i.e.
the poloidal structures are neglected. Neoclassical calculation
with low order Fourier expansion has been carried out toinclude poloidal asymmetry [12 ], but it can take account of
only mild poloidal variations. There remains a key question
whether poloidal shocks are formed or not in H-mode
plasmas. If shocks are established, an extension of theprevious model to include the poloidal structure should beneeded for a quantitative transport analysis in H-mode.
A two dimensional structure with coupled radial andpoloidal variations must be considered, if a shock really

2. Two dimensional structure
2.1 Basic equations
To construct model equations for the structure withevaluation of self-sustained two dimensional effects, we takethe same assumption as Ref. [7]. Poloidal variations of thedensity and the electrostatic potential are considered, but thatof the temperature is neglected. Electrons are isothermal, ionsare adiabatic, and ni = ne ≡ n is taken, where ni and ne are theion and electron density, respectively. Extending the modelin Ref. [7], radial flow and shear viscosity are taken intoaccount. By these terms, radial and poloidal structures arecoupled with each other. The structures are governed by themomentum balance equation,

mi nd
Vi = J × B − ∇( pi + pe ) − ∇ ⋅ π i ,dt(1)where Vi is the flow velocity, J is the plasma current, pi andpe are the ion and electron pressure, π i is the viscosity tensorof ions, and mi is the ion mass. The viscosity of electrons is©2004 by The Japan Society of Plasma
Science and Nuclear Fusion Research

Corresponding author’s e-mail: kasuya@nifs.ac.jp1283

Kasuya N. et al., Radial and Poloidal Structure Coupled with Shear Viscosity under the Existence of a Large Flow Shear in Tokamaks

neglected because it is smaller by a factor of the order ofme /mi . In tokamaks the toroidal symmetry is satisfied, sothe parallel component and averaged poloidal component ofthe momentum balance are given to be−nIBp ∂Φ ∂ ⎡ 1 ⎛ KB ⎞ 2 ⎤ Bp ∂ ⎡ 1 ⎛ KB ⎞ 2 ⎤⎢⎥+⎢⎥KB2 r ∂θ ∂ψ ⎣ 2 ⎝ n ⎠ ⎦ r ∂θ ⎣ 2 ⎝ n ⎠ ⎦+IBp Bς ∂Φ ∂ ⎡ I ∂Φ ⎤ KBp Bς ∂ ⎡ I ∂Φ ⎤⎥⎢⎥−⎢nr ∂θ ⎢⎣ Bς ∂ψ ⎥⎦B2 r ∂θ ∂ψ ⎢⎣ Bς ∂ψ ⎥⎦⎞Bp ∂ ⎛ Pe
5 Pi23nln N +=−⎟⎜⎟
2 n5 3mi r ∂θ ⎜⎝ n⎠−()(neoclassical process [14 ], and the shear viscosity, given byanomalous process [11], respectively. The radial flow is taken
into account, so ∂Φ/∂θ terms are involved in the left side ofEqs. (2) and (3). The Boltzmann relationn = n exp, (2)2.2 Decoupled model)

A decoupled limit neglecting the radial flow velocity(Vr /Vp << 1) and the shear viscosity (µ = 0) is considered toshow what kind of structure is formed when large Er shear
exists. This is the case solved in Ref. [7]. When the radialflow velocity can be neglected, variable K becomes a functionof ψ only. The model equation (2) for the parallel component
can be reduced to benIBp ∂Φ ∂ ⎡ 1 ⎛ KBp ⎞ 2⎤ Bp ∂ ⎡ 1 ⎛ KBp ⎞ 2⎤⎢ ⎜⎢ ⎜⎟⎥⎟ ⎥+KB2 r ∂θ ∂ψ ⎢⎣ 2 ⎝ n ⎠ ⎥⎦ r ∂θ ⎢⎣ 2 ⎝ n ⎠ ⎥⎦=Bp ⋅ ∇ ⋅ π i

1 JBp Bς
1−minmi1−min),= ε [( M + 2C ) cos θ + D sin θ ]

, (3)

(2 ∂χ
D+ 1 − Mp 2 χ + 2 A′ χ 2 − χ 23 ∂θ(bulkBp ⋅ ∇ ⋅ π in(4)is adopted here to determine variables, where Ti is the iontemperature, f̄ and ∆ f represent the spatial average andperturbed parts of quantity f, respectively. The variable thatmust be determined from Eqs. (2)-(4) are K, Φ and n, whichhave radial and poloidal variations.

11B⋅ ∇ ⋅π i
B⋅ ∇ ⋅π i
−bulk
shear
mi n
mi n−e∆ΦTi)

(5)

2p—
where χ = ln(n/n̄), Mp = KB0/n̄vtiCr , D = (4 √π /3)[IpsKB0/(n̄vtiCr2)], C = I 2(∂Φ/∂ψ)2/2vti2B02Cr2), Cr2 = (5/3 + Te /Ti)/2, A′= Mp2/2 + 5/(36Cr2), vti is the thermal velocity of ions, and Ips

shear

where Φ is the electrostatic potential, K = nVp /Bp, I = R2 B·∇ς, (∇ · π i)bulk and (∇ · π i)shear are the bulk viscosity, given by

10.75
0.5
0.25
0-0.25-0.50a[cm0-5Fig. 1r--1]-0.75

χ profile when a large E × B shear flow exists, where a is the minor radius and θ = 0 is defined as the low field side midplane.
This is the case of a decoupled model where the radial profile of Mp is given as a parameter to solve the poloidal structure.
This figure corresponds to the density perturbation profile.

2284

Kasuya N. et al., Radial and Poloidal Structure Coupled with Shear Viscosity under the Existence of a Large Flow Shear in Tokamaks

is the function defined in Ref. [7]. For obtaining Eq. (5), n is—
replaced by the variable χ, and assumed to be χ ~ O(√ε ).
Only terms up to O(ε) are kept in Eq. (5). The variable χ isrelated to a potential perturbation ∆Φ by Eq. (4). Thisequation gives a shock solution when Mp ~ 1. Parameter Mp
corresponds to the poloidal Mach number. When the Er profilein the radial direction is given to be the solitary structure,
which is formed in electrode biasing H-mode [4], and thepoloidal flow is determined by E × B flow, a two dimensional
potential profile is obtained. We call the region where thesolitary structure exists shear region. The χ profile is shown–in Fig. 1. A large magnitude of the perturbation ∆Φ ~ Φ withpoloidal variation is shown. A composition of the radialsolitary and the poloidal shock structure gives the total profileof Φ. Their contours correspond to the streamline of theplasma. A radial flow arises in the vicinity of the midpoint of
the shear region. This is due to the existence of the shock.
The shock position differs when Mp is different, so the large
variation of Er like in the solitary structure generates a flow
pointing to the radial direction. This result contradicts to theassumption that Vr can be neglected, coming from decouplingin the radial direction, so radial coupling must be taken intoaccount.

−µ+⎤χ2 ⎞r B0 ∂ 2 ⎡ ⎛− E⎥1−χ+M⎢p⎜⎟
22 ⎠vti Cr Bp ∂r ⎢⎣ ⎝⎥⎦2∂2 χ∂χ∂χD(1 − χ ) 2 + 1 − Mp 2+ 4 A′χ3∂θ∂θ∂θ()⎛ ⎧⎪µ B0 ⎡ 2 ∂ 2Mp= ε ⎜ ⎨D −, (6)⎢2 r
rvti Cr Bp ⎣∂r 2⎝ ⎩⎪⎤ ⎪⎫∂Mp + E − Mp + E ⎥ ⎬ cos θ+ 2r∂r⎦ ⎪⎭⎞− Mp Mp + E sin θ ⎟⎠where E = I(∂Φ /∂ψ)/(vtiB0Cr). Simplified case with Mp given(() ())

by the solitary equation and strong toroidal damping E = Mp
is considered here. The χ profile is shown in Fig. 2. The
boundary condition is χ = 0 at the boundary of the shear
region. This condition corresponds to no perturbation caused
by a poloidal flow outside the shear region. The profile shownin Fig. 2 is for a case with large shear viscosity (µ = 70 [m2/s], which is much larger than the values estimated fromexperiments in the TEXTOR tokamak µ = 10–2 ~ 100 [m2/s][15 ]) to obtain a characteristic structure formed by shear
viscosity. In this case the first term of the left hand side ofEq. (6) is dominant, so an approximated solution is given as

2.3 Viscosity coupling model
When shear viscosity is taken into account, the steep
structures can be smoothed and there is possibility to sustain
the assumption Vr /Vp << 1. In this system Eq. (2) becomesχ ( r, θ )~ε vti Cr D2 + 4 Mp2 Bp
2 µ rMp

B00.025
0.02
0.015
0.01
0.005
0-0.005-0.01-0.015-0.02-0.025(r − a) (r − a + d ) sin(θ + θα )0-5r-a[cm]0Fig. 2 χ profile for a large viscosity case with µ = 70 [m2/s]. The maximum amplitude is much smaller than that in Fig. 1.

3285, (7)Kasuya N. et al., Radial and Poloidal Structure Coupled with Shear Viscosity under the Existence of a Large Flow Shear in Tokamaks

where tan θα = –D/2Mp2. Smoothed structure is obtained, anda poloidal variation is sufficient small (∆n/n̄ < 4%) to satisfy
the assumption Vr /Vp << 1. The maximum value of χ is
proportional to 1/µ. The peak position depends on Mp, andthe peak in Fig. 2 appears at that of Mp near the boundary.
This structure results from the strong radial coupling with theshear viscosity and the strong boundary constraint χ = 0.

the shock is possible to affect to shorten the transition time[11].
The poloidal shock structure arises from toroidicity.
Another cause of formation of a poloidal structure is adivertor configuration. A potential hill near the divertor Xpoint is observed in DIII-D [16 ]. This effect is also importantfor poloidal structural formation mechanism and must be
taken into account.

3. Conclusion
Acknowledgements

The two dimensional potential structure, which plays animportant role in a tokamak improved confinement state, isstudied in this paper. An extension of the model to include
the poloidal structure is carried out. A poloidal shock structure
has been predicted when the poloidal Mach number Mp ~ 1.
We propose two dimensional coupling model equations. A
decoupled limit, which corresponds to the model in Ref. [7],shows a significant modification of the potential profile by
the appearance of the poloidal shock, and generation of radialflow. The decoupled limit model is not self-consistent, andshear viscosity is taken into account to smooth the structure.
A large shear viscosity case is analyzed, and a wellsuppressed perturbation profile is obtained. The structure isaffected by those near the boundary of the shear region
because of the strong radial coupling with the shear viscosity
and the strong boundary constraint. Only the limiting casesof poloidal shock and shear viscosity dominant, respectively,
are considered here, and an intermediate case that the poloidal
shock and shear viscosity effect are comparable is left for afuture work. The intermediate case is what is expected to berealized in tokamak experiments. Figures 1 and 2 indicate thatboth the magnitude and the peak position of the density
perturbation depend strongly on the magnitude of µ .
Therefore, a measurement of the poloidal density profile canbe used to estimate µ. Even the large shear viscosity caseshows a little but finite poloidal variation, so a detailed
measurement of the poloidal density profile gives important
information on the structural formation mechanism.
Previous transport analyses based on flux surface
averaged quantities have shown good qualitative agreementwith experiments, and a clear shock structure has not been
observed yet in experiments. An effect of the poloidalstructure might be negligible as in the large shear viscosity
case. On the other hand in a transient phase in L-H transition

This work is partly supported by the Grant-in-Aid for
Scientific Research of MEXT Japan and by the collaboration
programme of National Institute for Fusion Science.

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