Proceedings of ITC/ISHW2007

Transport analysis of dynamics for the plasma proﬁles in helicaldevices
Shinichiro TODA1) and Kimitaka ITOH1)
1)

National Institute for Fusion Science, Toki-shi, Gifu, 509-5292, Japan

The plasma dynamics and structure are studied by use of the one-dimensional theoretical model for theanomalous transport diﬀusivities. In this analysis, the high collisional Pﬁrsch-Schlüter collisional regime isexamined and the anomalous particle diﬀusivity is employed. The reduction of the anomalous particle diﬀusivity
and the steep gradient in the density proﬁle can be obtained. This prediction may be the theoretical explanation
for the Internal Diﬀusion Barrier observed in Super Dense Core plasmas of Large Helical Device.
Keywords: helical toroidal plasmas, turbulent transport, transport barrier, improvement mode, radial electric ﬁeld,
particle conﬁnement
1Introduction

The turbulence-driven transport and transport barriers are
the key issues in fusion research. Main eﬀorts have been
focused at the understanding of improved conﬁnement
modes (such as the H-mode). Among various improved
conﬁnement modes, two main kinds of transport barriersare observed in helical experimental results. At ﬁrst, thetransport barrier because of the radial transition of Er andthe improvement of the electron conﬁnement (’electron internal transport barrier’, e-ITB) were observed inside of
the transition point for Er . The transport analysis with theeﬀect of zonal ﬂows could predict an e-ITB in helical plasmas [1]. A comparison with the experimental results wasmade [2]. As the second kind of transport barriers, an internal diﬀusion barrier (IDB) is recently observed with thehigh gradient of the density in a super dense core (SDC)
plasma in LHD when a series of pellets is injected [3]. In acore region, the obtained high density and the temperature
are around 5×1020 m−3 and 0.85keV, respectively. A transport study of a SDC plasma is an urgent and critical issue inhelical conﬁnement plasmas. We solve the temporal transport equations for the density, the radial electric ﬁeld andthe temperature to study the dynamics of SDC plasmas ina cylindrical conﬁguration. The additional source proﬁleof the density which corresponds to that in the case of thepellet injection in LHD experiments is studied in the transport equations. The analytical form used here for the neoclassical particle and heat ﬂux related with the helical ripple trapped particles in the high collisional regime is given
in ref. [4]. The radial electric ﬁled is assumed to be determined by the ambipolar condition which is constituted
with the neoclassical particle ﬂux. The proﬁles of the density and temperature are determined from the neoclassicaland anomalous transport. A transport model for anomalous
diﬀusivities is adapted to describe the turbulent plasma. A
dynamics for a self-consistent solution for plasma proﬁles
author’s e-mail: toda@nifs.ac.jp

is examined and the barrier of the particle transport at theinner radial point in the density proﬁle can be reproduced.
That can explain a SDC plasma which is accompanied withan IDB in LHD. The proﬁle of the thermal diﬀusivity is examined when a high-density plasma which corresponds toa SDC plasma is obtained as a calculation result.

2 Model Equations
The model equations used here are shown. The onedimensional transport model is employed. The cylindricalcoordinate is used and r-axis is taken in the radial cylindrical plasma in this article. The region 0 < ρ < 1 isconsidered, where a is the minor radius and ρ = r/a. The
expression for the radial neoclassical ﬂux associated withhelical-ripple trapped particles is given in [4] which covers
the Pﬁrsch-Schlüter collisional regime, because the plasmastate of a SDC corresponds to the high collisional regime.
The total particle ﬂux Γt is written as Γt = Γna −Da n′ , whereΓna is the neoclassical ﬂux associated with the helicalripple trapped particles, and the prime denotes the radialderivative. Here, Da is the turbulent (anomalous) particle
diﬀusivity. The energy ﬂux related with the neoclassicalripple transport, Qnaj is obtained like the neoclassical particle ﬂux. The total heat ﬂux Qtj for the species j is written′′as Qtj = Qna
j − nχa T j − 5Da n T j /2, where χa is the anomana
lous heat diﬀusivity and Q j is the neoclassical contribution from the Pﬁrsch-Schlüter regime. A theoretical modelfor the anomalous heat conductivity χa is adopted and isexplained later. The temporal equation for the density is1 ∂∂n=−(rΓt ) + S n + S p ,∂tr ∂r(1)where the term S n represents the particle source and theparameter S p is the term to simulate the experimental procedure of the pellet injection. The detail about this term
S p will be explained later. The equation for the electron

Proceedings of ITC/ISHW2007
temperature is given as3∂
1 ∂me n(nT e ) = −(rQte ) −(T e − T i ) + Phe ,(2)2 ∂tr ∂rmi τewhere the term τe denotes the electron collision time. Theterm Phe represents the absorbed power due to the ECRH
heating. The temporal equation for the ion temperature is3∂
1 ∂me n(nT i ) = −(rQti ) +(T e − T i ) + Phi . (3)2 ∂tr ∂rmi τe
The term Phi represents the absorbed power of ions. Theequation for the radial electric ﬁeld in a nonaxisymmetricsystem is expressed by the ambipolar condition as∑
Z j Γna(4)j = 0.
jIn this article, we examine the plasma which consists ofthe electron and the hydrogen ion. Therefore, the amna
bipolar condition, Eq. (4) can be rewritten as Γnai = Γe .
It is well known that the neoclassical transport is dominant when the radial electric ﬁeld is formed in helical plasmas [5]. The analysis including the temporal equation ofEr [6] which includes the eﬀect of the electric ﬁeld diffusion is left for the future study. The source proﬁles arechosen here as follows. The particle source S n is set to be
S n = S 0 exp((r − a)/L0 ), where L0 is set to be 0.1m. This
proﬁle represents the peaking at the plasma edge of theparticle source due to the ionization eﬀect. The intensity,
S 0 , governs the average density, and is taken as a control
parameter to specify the density in this article. The radial
proﬁles of the electron and ion heating terms, Phe and Phi ,are assumed to be proportional to exp(−(r/(0.5a)2 ) for thesake of the analytic insight.
3

Model for Anomalous Transport Diﬀusivities andBoundary Conditions

We adopt the model for the turbulent heat diﬀusivity χabased on the theory of the self-sustained turbulence dueto the ballooning mode and the interchange mode, bothdriven by the current diﬀusivity [7, 8]. The anomaloustransport coeﬃcient for the temperatures is given as χa =3
χ0 /(1 + Gω2E1 ) (χ0 = F(s, α)α 2 c2 vA /(ω2pe qR), where ω peis the electron plasma frequency. The factor F(s, α) is thefunction of the magnetic shear s and the normalized pressure gradient α, deﬁned by s = rq′ /q and α = −q2 Rβ′ . For
the ballooning mode turbulence (in the system with a magnetic well), we employ the anomalous thermal conductivity χa,BM . The details about the coeﬃcients F(s, α), G andthe factor ωE1 , which stands for the poloidal E × B rotation
frequency, are given in [8] in the ballooning mode turbulence. In the case of the interchange mode turbulence forthe system of the magnetic hill [7], we adopt the anomalous thermal conductivity χa,I M . The details about the coeﬃcients F, G and the factor ωE1 in the case of the interchange mode were given in [7]. The greater one of thesetwo diﬀusivities is adopted as χa = max(χa,BM , χa,I M ). The
value for the anomalous diﬀusivities of the particle is set
as Da = χa to examine the radial variation in the proﬁle ofthe particle diﬀusivity Da when the steep radial gradient in
the density proﬁle can be obtained as a calculation result.
The equations of density, temperature and electricﬁeld (1)-(4) are solved, with the prescribed source proﬁles,under the appropriate boundary conditions. We ﬁx theboundary condition at the center of the plasma (ρ = 0) suchthat n′ = T e′ = T i′ = 0. The boundary conditions at the edge(ρ = 1), with respect to the density and temperature, aregiven by specifying the gradient scale lengths. We employ
these conditions, −n/n′ = 0.1m, −T e /T e′ = −T i /T i′ = 0.1m
in this article. The machine parameters which are similar
to those of LHD are set to be R = 3.6m, a = 0.6m, B = 3T,ℓ = 2 and m = 10. In this case, we set the safety factor
and the √helical ripple coeﬃcient as q = 1/(0.4 + 1.2ρ2 ) andϵh = 2 1 − (2/(mq(0)) − 1I2 (mr/R), respectively. Here,q(0) is the value of the safety factor at ρ = 0 and I2 is thesecond-order modiﬁed Bessel function.
4

Dynamical Response after Pellet Injection

Using these parameters and boundary conditions, the onedimensional transport analysis for the LHD-like plasma
has been performed and the proﬁles of n, T e , T i and Er
are solved using equations (1), (2), (3) and (4). We adapt
a theoretical model for the anomalous transport diﬀusivities driven by the current diﬀusivity as a candidate. Anexample is taken from the plasma which is sustained by
the ECRH. At ﬁrst we obtain self-consistent steady proﬁles of n, T e , T i and Er for the given source proﬁle. Inorder to set the line-averaged density to be around n̄ ≅1 × 1020 m−3 , the line-averaged temperature of electrons
to be around T̄ e ≅ 0.4keV and the line-averaged temperature of ions to be around T̄ i ≅ 0.4keV, the absorbed
power of electrons is set to be 1MW and the coeﬃcient
S 0 is taken as 7 × 1020 m−3 s−1 , for the choice of the above
mentioned anomalous transport coeﬃcients, where the absorbed power of ions is taken as 0kW. The radial electricﬁeld takes the negative value in the entire radial region.
Next, we use these obtained proﬁles of n, T e , T i and Er as
an initial condition, i.e., we begin the new calculation fromthe proﬁle of the density as n̄ ≅ 1×1020 m−3 s−1 and the negative Er in the entire region. In this parameter region examined here, both typical energy and particle conﬁnement
times are about 1s. To simulate the experimental procedure of the pellet injection, we add the parameter S p of theparticle source in Eq. (1). This parameter S p has a distribution as S p = S p0 exp(−(r/r p )2 ) and is set to have a value
from the initial time t = 0 to 1ms. We set the half widthof the proﬁle S p as r p = 0.2a. In other words, we set as
S p0 = 1 × 1023 m−3 s−1 for 0 < t ≤ 1ms and S p0 = 0m−3 s−1
for t > 1ms. We show the dynamics of the plasma radial
proﬁles n (Fig. 1), T e (Fig. 2), T i (Fig. 3) and Er (Fig. 4)
at the times 0ms, 1ms and 10ms. The proﬁles labelled byProceedings of ITC/ISHW2007
0ms represent the initial conditions used in the calculationin this article (with the dotted lines). As the time goes on,
the density increases and the positive electric ﬁeld appears.
In Fig. 1, the rapid change of the gradient in the density
proﬁle at t = 10ms can be found at ρ = ρT (≅ 0.2). Theparameter ρT represents the location of the transition fromthe positive Er to the negative Er at 10ms in Fig. 4. In
Fig. 4, we can show the steep gradient of the radial electricﬁeld at the transition point ρT . (The proﬁle of the radial
electric ﬁeld is determined by the ambipolar radial electricﬁeld and the single solution of the radial electric ﬁeld forthe ambipolar condition can be obtained at a radial point.)
The gradient of the radial electric ﬁeld is strong enough tosuppress the anomalous transport: |Er′ | ≅ 1 × 105 Vm−2 atρ = ρT . Therefore, the improvement near the transition
point can be obtained. In Figs. 2 and 3, the temperatures
are found to decrease toward the plasma center when thevalue of the density rapidly increases. After the time 1ms,
the density in the core region continues to increase because
there is an inward neoclassical transport because the positive gradient of the temperature proﬁles. If the time passes
over the conﬁnement time, the plasma proﬁles takes initial
ones. At t = 10ms, the clear barrier with respect to theparticle transport in the density proﬁle can be obtained inFig. 1.
A proﬁle in the region ρ < 0.5 of the anomalous transport diﬀusivity is shown in Fig. 5. In the region ρ < ρT ,the reduction of the anomalous particle diﬀusivity can befound, because the value of the anomalous transport diﬀusivity is proportional to the square of the safety factor q andthe temperatures of electrons and ions decrease toward theplasma center at 10ms. A clear reduction of the anomalousparticle diﬀusivity is found at the transition point ρ = ρT
due to the strong gradient of Er compared with the region
ρ > ρT . In the region ρ < 0.3 at the time 10ms, the neoclassical particle diﬀusivity DNEO (= −Γna /n′ ) is found to
be much smaller than the absolute value of the eﬀective
anomalous particle diﬀusivity and is |DNEO | < 0.01m2 s−1
with the dotted line. In the core region, the slightly negative value for the eﬀective neoclassical particle diﬀusivity
DNEO and the particle neoclassical pinch are obtained at
the time 10ms. In Fig. 6, the dashed line shows the radial proﬁle of the anomalous particle diﬀusivity Da , thesolid line shows the radial proﬁle of the sum of neoclassical and anomalous particle diﬀusivity Dtot (= DNEO + Da )at the time 10ms, respectively The clear reduction canbe obtained in the sum of the neoclassical and anomalous transport. Therefore, we can obtain the clear barrier
with respect to the particle transport in Fig. 1 (n proﬁleat t = 10ms). It is emphasized that the barrier formation starts to occur after the end of the particle fuelling.
It takes a few ms for the establishment of a barrier seenin Fig. 5. It is found that the anomalous particle transport has a much more important role on the improvementof the particle conﬁnement in the plasma core region than
the neoclassical particle transport in the parameter region

examined here. If we set much smaller value for S p0 ,S p0 = 1 × 1022 m−3 s−1 than that (S p0 = 1 × 1023 m−3 s−1 )in this calculation, we can not ﬁnd the steepening in thedensity gradient, because the gradient of Er after the pellet
fuelling (1ms) is not strong enough. It is shown that there
is a threshold value for S p to form the barrier with respect
to the particle conﬁnement in the density radial proﬁle.
5Summary and Discussions

We have studied the strong reduction of the particle transport when the dynamics and the radial structure of proﬁles of the density, the temperatures and the electric ﬁeld
are examined in toroidal helical plasmas. The analysis isperformed by use of the one-dimensional transport equations. These transport equations includes the contributions
from the neoclassical transport and the anomalous transport driven by the current diﬀusivity. The neoclassical theory for the particle and heat ﬂux in the the Pﬁrsch-Schlüterregime is adapted. The radial proﬁle of the electric ﬁeld isdetermined by the ambipolar condition. The clear changeof the gradient in the density proﬁle and the reduction of
the anomalous particle transport in the core region after theparticle fuelling, if the value of the particle fuelling (thepellet size in the experiment) exceeds the threshold, can beshown when the temporal evolution of the plasma proﬁles
are examined. This theoretical prediction may explain theInternal Diﬀusion Barrier (IDB) observed in LHD plasmas.
It is predicted that the additional particle source S p has athreshold value to obtain the reduction of the anomaloustransport diﬀusivity. Furthermore, this threshold value of
S p0 strongly depends on the shape of the distribution, e.g.
the parameter r p which determines the half-width length of
the additional particle source. To study the threshold valueof S p0 to obtain the clear barrier of the particle transport,
the parameter survey of the calculations results is needed.
These are left for future studies.
The authors would like to acknowledge
Prof. A. Fukuyama for the helpful suggestions relatedwith the numerical methods.
[1] S. Toda, K. Itoh, A. Fujisawa, S. -I. Itoh, M. Yagi, A.
Fukuyama, P. H. Diamond and K. Ida, Nucl. Fusion 47 914(2007).
[2] K. Itoh, S. Toda, A. Fujisawa, S. -I. Itoh, M. Yagi, P. H.
Diamond and K. Ida, Phys. Plasma 14 020702 (2007).
[3] N. Ohyabu et al., Phys. Rev. Lett. 97 055002 (2006).
[4] K. C. Shaing and J. D. Callen, Phys. Fluids 26 3315 (1983).
[5] K. Ida, Plasma Phys. Control. Fusion 40 1429 (1998).
[6] D. E. Hastings, Phys. Fluids 28 334 (1985).
[7] K. Itoh, S. -I. Itoh, A. Fukuyama, H. Sanuki and M. Yagi,Plasma Phys. Control. Fusion 36 123 (1994).
[8] K. Itoh, S. -I. Itoh, A. Fukuyama, M. Yagi and M. Azumi,Plasma Phys. Control. Fusion 36 279 (1994).

Proceedings of ITC/ISHW2007

Fig. 1 The dynamics of the density radial proﬁles. The dottedline, the dashed line and the solid line show the states at
the times 0ms, 1ms and 10ms, respectively.

Fig. 2 The proﬁles of the electron temperature at the times 0ms,1ms and 10ms. Because the density rapidly increases in
the core region, the electron temperature is found to decrease in that region (ρ < ρT ).

Fig. 3 The proﬁles of the ion temperature at the times 0ms, 1ms
and 10ms.

Fig. 4 The dynamics of the radial electric ﬁeld. At the time
10ms, the radial transition of Er from the positive value
to the negative value can be obtained.

Fig. 5 The radial proﬁles of the anomalous particle diﬀusivity
Da in the region 0 < ρ < 0.5 at the times 0ms (with thedotted line), 1ms (with the dashed line) and 10ms (withthe solid line). We can show the clear reduction of theanomalous particle diﬀusivity at the time 10ms in the region ρ < ρT .

Fig. 6 The radial proﬁles of the particle diﬀusivity in the region
0 < ρ < 0.5 at the times 10ms. We can show the strong
reduction in the sum of the neoclassical particle diﬀusivity and the anomalous particle diﬀusivity at the time 10msin the region ρ < ρT with the solid line.