Proceedings of ITC/ISHW2007

Inter-linkage of transports and its bridging mechanism
Katsumi IDA
National Institute for Fusion Sciences, Toki, Gifu 509-5292, Japan(Received / Accepted )The physical mechanisms determining the diﬀusive and non-diﬀusive terms of particle, momentum
and heat transport are described. The non-diﬀusive term in the particle transport, which causes inward
pinch or outward flow is driven by the temperature gradient and the magnetic field curvature. The
non-diﬀusive term in the momentum transport, which drives spontaneous toroidal rotation, is foundto be sensitive to the sign of the electric field. In heat transport, there is no clear non-diﬀusive term
observed. The temperature and temperature gradient dependences of the diﬀusive terms are discussed.
Keywords: Plasma Transport Inter-linkage
DOI: 10.1585/pfr.1.001
The existence of the non-diﬀusive term of transport has been recognized in particle transport (andimpurity transport) as the Ware pinch by the toroidal
electric field[1] or convection by ion temperature gradient driven turbulence[2] . The peaked profile ofthe electron density observed in the steady-state withgas puﬀ fueling clearly shows the significant contribution of the non-diﬀusive inward particle flux, because
the particle source is localized near the plasma periphery. The inward pinch observed in a stellarator
where there is no toroidal electric field, suggests that
turbulence driven convection can easily overcome thethermodiﬀusion[3]. The Non-diﬀusive term in the momentum transport was found experimentally[4] and it
appears as an spontaneous toroidal rotation[5]. The
significant contribution of the non-diﬀusive term to
the transport shows a strong inter-linkage of the transport between particle, (poloidal and toroidal) momentum and (ion and electron) heat transport. In
the plasmas with a transport barrier, where a strong
gradient is produce, the inter-linkage of the transport becomes more significant. (A good example isa large spontaneous toroidal rotation at the internaltransport barrier[6, 7]). This is because these nondiﬀusive terms are correlated to the gradients of otherplasma parameters (ion temperature gradient for particle transport or potential gradient for momentum
transport). Therefore these non-diﬀusive term areconsidered to be oﬀ-diagonal terms of the so-called
transport matrix. Recent turbulence transport theory explores the physics bridging mechanism for theoﬀ-diagonal terms of the transport matrix[8]. For example, the theoretical model of symmetry breakingof turbulence by radial electric field shear[9] could explain the strong correlation between non-diﬀusive momentum flux and ion pressure gradients (and also raauthor’s e-mail: xxxyyyzzz@u-plasma.ac.jp

dial electric field) observed in experiments. In thispaper, non-diﬀusive terms of particle and momentum
transport and diﬀusive terms of heat transport are discussed based on experiments in stellarators and tokamaks.
It is well known that the non-diﬀusive term hasa significant contribution to the radial flux of particles and it causes the inward flux which is necessary to achieve a peaked density profile with a particle source localized near the plasma edge. Due tothe role of spontaneous toroidal rotation in MHD stability, the non-diﬀusive term of momentum transport
has become highlighted in recent momentum transport studies. On the other hand, the non-diﬀusive
term of heat transport has not been clearly identified,although there were experimental results that suggest
the heat pinch. Figure 1 shows the physics mechanism
determining the diﬀusive term and non-diﬀusive termsof the radial flux of a particle, momentum and energy transport. There are various physics mechanisms
contributing to the non-diﬀusive term in the particle
transport. The inward pinch driven by the toroidal
electric field is well known as the Ware pinch in tokamaks. However, the inward pinch observed in steadystate plasmas in tokamaks and in stellarators can not
be explained by the Ware pinch, because there is no
toroidal electric field in these experiments. In stellarator plasmas, the thermodynamic force driven by theelectron temperature gradient contributes to significant outward flux. This term has been confirmed experimentally in stellarators. It is well known that momentum and energy transport have convection terms
due to the radial flux of particles. In general, theseconvective terms are relatively small. It is interesting that the radial electric field and the magnetic fieldcurvature aﬀects both the non-diﬀusive terms of particle transport and momentum transport, while the radial electric field and its shear aﬀect both the diﬀusiveProceedings of ITC/ISHW2007

Fig. 1 Physics mechanism determining the diﬀusive term and non-diﬀusive terms of the radial flux of particle, momentum
and energy transport.

term of momentum transport and energy transport.
Therefore particle, momentum and heat transport arecoupled to each other through these physics elements.
The diﬀusive term of particle transport has been
derived from the time dependent transport analysis
of a discharge with gas puﬀ modulation. Figure 2 (a)shows the temperature dependence of the diﬀusion coeﬃcient evaluated in LHD. The diﬀusion coeﬃcientmeasured is larger than the prediction by neoclassical theory by one order of magnitude. Both in thecore and edge, the diﬀusion coeﬃcient has the electron
temperature dependence of α ≈ 1, where D ∝ Teα ,which is weaker than that of the neoclassical prediction (α ≈ 2.6 ). Figure 2(b) shows the relation between the normalized temperature and density gradients experimentally obtained in Wendelstein 7AS and
LHD. Since both the density gradient and temperature gradient contribute to the outward flux, there
should be an inward flux to sustain the steady state
density profile where Γ ≈ 0. The particle flux can be
written as Γ = −ne (D11 ∇ne /nee + D12 ∇Te /T e − u).
Where D11 and D12 are the on- and oﬀ-diagonal elements of the transport matrix and the velocity u is aconvective contribution. The ratio of these coeﬃcientderived from the slope of the data plotted in Fig2(a)are D12 /D11 ≈ 1.4 and u/D11 ≈ −12.5 m−1 , respectively. The significant contribution of the inward termof u/D11 overcomes the outward term of D12 /D11 of

the non-diﬀusive term as well as the diﬀusive termand sustains the peaked density profile in steady state.
Because various physics mechanisms contribute to thenon-diﬀusive term of particle transport, it is still open
to question which physics mechanisms become dominant. A more comprehensive understanding should besought from the comparison of density profiles in stellarators (typically flat or hollow) with these in tokamaks (typically peaked).
Similar experiments have been done in steadystate fully noninductive current plasmas in tokamaks
to eliminate the contribution of the Ware pinch due
to toroidal electric field to drive ohmic current. Figure 3 shows the relations between ∇ne /ne , ∇Te /Teand ∇q/q in the Tore Supra tokamak. As seen in
Fig.3(a) and Fig.3(b), there are two distinguished regions (ρ < 0.3 and 0.35 < ρ < 0.6 ) that show different slopes. The direction of the thermodiﬀusion
flux is found to change when moving from the outer
to the inner plasma. The sign of the thermodiﬀusion
pinch is determined by the slope. This experimentshows that thermo-diﬀusion can be inward (ρ < 0.3 )or outward (0.35 < ρ < 0.6) depending on the plasma
radius, where the unstable mode (ITG or ETG) is expected to be diﬀerent because of the change in theratio of ∇Te /∇Ti . In the region of ρ = 0.35 − 0.6,
where the thermo-diﬀusion flux is outward, a physics
mechanism driving the inward pinch is necessary inProceedings of ITC/ISHW2007

Fig. 2 (a) Temperature dependence of the diﬀusion coeﬃcient in the core (0.4 < ρ < 0.7) and edge (ρ > 0.7) experimentally
evaluated and neoclassically predicted in LHD[10] and (b) average density versus temperature gradient scale length
estimated between r = 2 and 8 cm for three densities. For Γ ≈ 0 it follows that D12 /D11 ≈ 1.4 and u/D11 ≈ −12.5
m−1 [3] and (b) the relation of normalized temperature and density gradient. The core and edge values are estimatedfrom the average value in the core (0.4 < ρ < 0.7) and edge (0.7 < ρ < 1.0) at Rax = 3.6m, Bt = 2.75, 2.8T.

order to sustain the sharp density gradient in steadystate. A clear relation between ∇ne /ne and ∇q/q in
Fig.3(c) strongly suggests that the magnetic field curvature drives the inward pinch. It is interesting thatthe sign of the magnetic field curvature driven flux instellarators is opposite to that in tokamak and outward flux is expected. It is still open to question what
the physics mechanism is causing the inward flux near
the plasma edge in sterllarators, where there is no
Ware pinch and both the thermo-diﬀusion flux andthe flux driven by the magnetic field curvature are
expected to be outward because of ∇Te /Te < 0 and∇q/q < 0. Because the impurity has a higher ion
charge, Z, impurity transport is more sensitive to theradial electric field than bulk particle transport. The
positive radial electric field in the electron root tends
to drive outward flux and suppress the accumulationof impurities and radiation collapse[15, 16]. It is a crucial issue to achieve poor impurity confinement (impurity exhaust) and good energy confinement, simultaneously. In stellarators, poor impurity confinement
is observed in plasmas with good energy transport, forexample the HDH mode in Wendelstein 7-AS[17] andthe electron ITB and high ion temperature mode in
LHD.
Since the damping of the plasma toroidal velocity owing to parallel viscosity is significant near theplasma periphery in stellarators[12], toroidal rotationnear the plasma edge is determined by the radial electric field, while the core toroidal rotation is deter-

mined by the momentum from the tangential NBI.
The momentum flux including the non-diﬀusive term
due to the radial electric field can be expressed as
Pφ = −mn(μ⊥ ∇Vφ −ξEr /Bθ )[13]. In order to investigate the eﬀect of the radial electric field on the toroidal
rotation, a density scan is made near the threshold of
the transition from the ion root (negative Er ) to electron root (positive Er ). In LHD, the transition fromthe ion root to the electron root takes place at the low
density where the plasma is well into the collisionlessregime μ∗e < 0.1, because of the non-ambipolar particle flux as predicted by neoclassical theory. As seenin Fig.4, the plasma rotates more in the counter direction (anti parallel to equivalent plasma current) for
positive Er , while it rotates more in the co direction
for negative Er . The relation between the toroidal rotation and the radial electric field is plotted in Fig4(c).
This result shows that the positive radial electric field
drives the toroidal rotation in the counter direction,
where the toroidal rotation contributes to the negative radial electric field. It should be emphasized thatthe direction of the toroidal flow is anti-parallel to thedirection of the < Er × Bθ > drift. This is in contrast
to the spontaneous flow in a tokamak, where the direction of the toroidal flow is parallel to the directionof the < Er × Bθ > drift, because the toroidal viscosity is nearly zero due to the toroidal symmetry. The
spontaneous toroidal flow becomes more significant in
the plasma with an internal transport barrier, wherelarge electric fields are observed. In CHS the sponta-

Proceedings of ITC/ISHW2007

Fig. 3 ∇ne /ne ∇Te /Te from a set of seven discharges: (a) for r/a ≤ 0.3, Te /Ti = 2 ± 0.4, ∇Te /Te = 3.8-4.8; (b) for
0.35 ≤ r/a ≤ 0.6, Te /Ti = 1.2 ± 0.4, ∇Te /Te = 0.7-3.5. Corresponding variation of ∇q/q is displayed at the top.
(c) ∇ne /ne versus ∇q/q within 0.3 ≤ r/a ≤ 0.6 from a set of seven discharges, keeping |∇Te /Te | at the values of
6m−1 ± 10 % and ∇Te /Te = 0.6 ± 0.3 [11].

neous toroidal flow overcomes the toroidal flow driven
by co-NBI and the plasma rotates in the counter direction associated with the formation of a large positive Er . In contrast, in JT-60U, the plasma rotatesin the counter direction at the ITB region where astrong negative radial electric field is produced. The
significant diﬀerence between tokamaks and heliotron
devices is the relationship between the poloidal field
direction and the direction of the dominant symmetry. The pitch angle of the dominant symmetry iseven larger than that of the averaged poloidal field ina Heliotron device, while it is zero due to the toroidal
symmetry in a tokamaks.
In a stellarator turbulence in the plasma as well
as non-ambipolar flux can drive flows perpendicular(E × B) and parallel (spontaneous toroidal flow ) to
the magnetic field though Reynolds stress. The experiments to investigate the turbulence driven E × B
flow and the spontaneous toroidal flow have been donein the vicinity of the last closed flux surface (LCFS)
in JET and in TJ-II. The radial-perpendicular component of the production term has been investigatedin the LCFS vicinity in the JET tokamak as seen in
Fig.5(a). Figure 5(b) shows the radial structure of thecross correlation between parallel and radial fluctuating velocities in the proximity of the LCFS in the TJ-II
stellarator. The level of cross correlation < vM|| > increases for plasma densities above the threshold value
to generate E × B sheared flows. The appearanceof gradients on < vM|| > is due to both radial variations in the level of fluctuations and in the cross-phasecoherence between fluctuating radial and parallel velocities. It has been found that the energy transferedfrom dc flows to turbulence, directly related to themomentum flux and the radial gradient in the perpendicular flow, can be both positive and negative inthe proximity of sheared flows. Furthermore, the energy transfer rate is comparable with the mean flowof kinetic energy normalized to the correlation timeof the turbulence, implying that this energy transfer
is significant. These results show that turbulence canact as an energy sink and energy source for the mean
flow near the shear layer.
There is no clear evidence of a non-diﬀusive termof the energy transport except for a few experiments
in plasmas with ECH[21]. In most cases the diﬀusive
term only can explain the energy transport observed inexperiments. The diﬀusive term of the energy transport is evaluated as thermal diﬀusivity (the ratio ofheat flux normalized by density to the temperature
gradient). In general, the heat flux depends on thetemperature and the temperature gradient and it isnon-linear. Therefore the thermal diﬀusivity has atemperature dependence and also a temperature gradient dependence in the non-linear regime. Figure 6
shows the ∇Te dependence of χe (obtained from thestatic analysis i.e. the power balance analysis) in JT60U, Wendelstein 7-AS and LHD. In Fig.6, the ther3/2
mal diﬀusivity is normalized by Te , which is predicted by Gyro-Bohm transport, in order to eliminate
the Te dependence of χe and make the ∇Te dependence clear. The ∇Te is normalized by R/Te , Because
the ratio of the major radius to the scale length of thetemperature is expected to give the threshold of turbulence in the critical temperature gradient transportmodel. As seen in Fig.6, there is no clear critical gra-Proceedings of ITC/ISHW2007

Fig. 4 Radial profiles of (a) toroidal and (b) poloidal rotation velocities and (c) relation between toroidal and poloidal
rotation velocities in LHD.

Fig. 5 Radial profile of the (a) radial-perpendicular Reynolds stress component in the plasma boundary region in the JET
tokamak and the (b) radial-parallel Reynolds stress component in the TJ-II plasma boundary region at diﬀerent
plasma densities. Parallel velocity is quantified by the parallel Mach number [14, 18].

dient scale length observed in LHD, although the existence of a critical gradient scale length is suggested in
Wendelstein 7-AS and JT-60U. The thermal diﬀusivity χe shows a Gyro-Bohm type Te dependence when
the temperature gradient is small enough, while it depends mostly on the scale length above a critical valueof the temperature gradient. This implies that there
is a strong nonlinear mechanism in transport. One of
the candidates for the mechanism causing strong nonlinearity is non-linear flow damping of the zonal flow
in a tokamak plasma. In the LHD heliotron, there isno critical value of the temperature gradient, because
the transition from L-mode to an electron ITB takeplace at a lower temperature gradient[22]. The transition from L-mode to the electron ITB is due to thechange in the temperature dependence of α from positive values (∼ 3/2 as predicted by gyro-Bohm scal-ing) to negative values (∼ −3/2), where α is defined
as Q/ne ∝ Teα ∇Te [23]. The temperature dependence
tends to decrease (weak temperature dependence) as
the electron density is increased. This change in thetemperature dependence of the thermal diﬀusivity is
consistent with the change in the power degradation of
the energy confinement time. It should be noted that
the L-mode scaling of the global energy confinement
time expressed by tE ∝ (ne /P )3/5 is equivalent to thetemperature dependence of the thermal diﬀusivity ex3/2
pressed by χ ∝ Te . Therefore the weak temperaturedependence (α = 1/2) appears as a tendency of thesaturation of the energy confinement time at higher
1/3
density as ne . The temperature dependence factor
α becomes ∼ 0.5 at lower temperatures below the critical value (in the more collisional regime[24] ) or in theplasma with pellet injection[25]Proceedings of ITC/ISHW2007

Fig. 6 ∇Te and Te dependence of χe at diﬀerent normalized radii in (a)JT-60U, (b) Wendelstein 7AS and (c) LHD NBI
plasmas. χe is normalized by the gyro-Bohm Te dependence and ∇Te is normalized by (R/Te ), where R is the major
radius and LT = ∇Te /Te in (a) and (c)[19, 20].

In summary, the non-diﬀusive term driven by thethermo-diﬀusion due to the temperature gradient isexperimentally identified in stellarators and tokamaks.
In stellarators, thermo-diﬀusion always drives outward
flow, while that in tokamak drives inward or outward
flow depending on the unstable mode of the turbulence. The inward pinch due to the magnetic fieldcurvature is found in steady state fully non-inductive
current plasmas in tokamaks. However, this does not
explain the inward pinch observed in the heliotron
plasma, where the non-diﬀusive term due to the magnetic field curvature is expected to be outward. In
momentum transport, the significant contribution of
the non-diﬀusive term is experimentally observed andis identified as spontaneous flow. The physics mechanisms driving the spontaneous flow, which is parallelto the magnetic field, are investigated. Both the mean
flow perpendicular and parallel to the magnetic field
are driven by turbulence as well as by non-ambipolar
flux. The mean flow parallel to the magnetic field(spontaneous toroidal flow) is in the direction of thetoroidal flow anti-parallel to the < Er × Bθ > drift,
which is in contrast to the spontaneous flow in a tokamak, where the direction of the toroidal flow is parallel to the direction of the < Er × Bθ > drift. Finallythe parameter dependence of the diﬀusive term of theenergy transport is discussed. The thermal diﬀusivity shows a Gyro-Bohm type Te dependence when thetemperature gradient is small. When the temperature gradient exceeds the threshold of the scale length,
the thermal diﬀusivity increases rapidly if the plasma
stays in L-mode. The threshold of the scale length isobserved in tokamaks and Wendelstein 7-AS plasmas,
but not in the LHD plasma, where the transition from
L-mode to the electron ITB mode take place before thethreshold.

The author would like to thank Drs.K.Tanaka
and M.Yoshinuma (NIFS) for fruitful discussions inorganizing this paper. The author also would liketo thank Drs.C.Hidalgo (CIMAT) and U.Stroth(Kiel
Univ.), G.T.Hoang(CEA) for useful discussions on theinterpretation of the experimental data in their devices.
[1] A. A. Ware, Phys. Rev. Lett. 25, 15 (1970).
[2] P.W.Terry, Phys. Fluids B 1, 1932 (1989).
[3] U.Stroth, et al., Phys. Rev. Lett. 82, 928 (1999).
[4] K.Ida et al., Phys Rev Lett 74, 1990 (1995).
[5] B. Coppi Nucl. Fusion 42, 1 (2002).
[6] K.Ida et al., Phys Rev Lett 86 3040 (2001).
[7] Y.Sakamoto et al., Nucl Fusion 41 865 (2001).
[8] S.-I.Itoh Phys Fluids B 4 796 (1992).
[9] O.D.Gurcan , P.H.Diamond, et al., Phys Plasmas 14,042306 (2007).
[10] K.Tanaka, et al., Nucl. Fusion, 46, 110 (2006).
[11] G.T.Hoang, et al., Phys. Rev. Lett. 93, 135003(2004).
[12] K.Ida et al., Phys Rev Lett. 67 58 (1991).
[13] K.Ida et al., Plasma Phys. Control. Fusion, 44, 361(2002)[14] C.Hidalgo et. al., Plasma Phys. Control. Fusion, 48,
S169 (2006)[15] Y.Nakamura, et. al., Nucl. Fusion 43 219 (2003).
[16] K.Ida et al., Nucl. Fusion 45 391 (2005).
[17] K. McCormick, et. al., Phys. Rev. Lett. 89, 015001(2002).
[18] B.Goncalves, et al., Phys. Rev. Lett. 96, 145001(2006).
[19] S.Inagaki, H.Takenaga, K.Ida, et. al., Nucl. Fusion
46, 133 (2006).
[20] F.Ryter et. al., Plasma Phys. Control. Fusion, 48,
B453 (2006)[21] T.C.Luce, et. Al., Phys. Rev. Lett. 68, 52 (1992).
[22] K.Ida et. al., Phys Rev Lett 91 085003 (2003).
[23] K.Ida et. al., Phys Plasmas 11 2551 (2004).
[24] J. Miyazawa et al., Plasma Phys. Controlled Fusion
47 801 (2005).
[25] K.Ida et. al., Phys Rev Lett 96 1250063 (2006).