Plasma and Fusion Research: Regular Articles

Volume 14, 3402121 (2019)

Modeling of Heat Pulse Propagation During TESPEL Injection
into the LHD Plasma∗)
Vladimir Yu. SERGEEV1,2) , Alexei Yu. DNESTROVSKIY3) , Vasiliy E. FROLOV1) ,Vladimir D. PUSTOVITOV3,4) and Naoki TAMURA2,5)
1)

2)Peter the Great St. Petersburg Polytechnic University, St.Petersburg, Russia
National Institute for Fusion Science, National Institutes of Natural Sciences, Toki 509-5292, Japan
3)
National Research Center “Kurchatov Institute”, Moscow, Russia
4)
National Research Nuclear University MEPhI, Moscow, Russia
5)
SOKENDAI (The Graduate University for Advanced Studies), Toki 509-5292, Japan(Received 28 December 2018 / Accepted 7 February 2019)

Anomalous transport is a paradigm for analysis of both Power Balance (PB) and Heat Pulse Propagation(HPP) in magnetically confined plasmas including so called non-local transport (NLT) phenomena (core heating
as a reaction on the edge cooling). One of the alternative explanations of the NLT proposes to take into accountthe plasma motion caused by the perturbation of the force balance during HPP. First results of numerical modelingof the electron heat transport by the ASTRA code confirm the viability of this approach.
c 2019 The Japan Society of Plasma Science and Nuclear Fusion ResearchKeywords: non-local transport, plasma perturbation, energy exchange, LHD
DOI: 10.1585/pfr.14.3402121

1. Introduction
Non-local transport (NLT) phenomena (core plasma
heating appearing as a reaction on its cooling at the edge)
attracts attention of the fusion society for decades [1–5].
Up to now there is no convincing explanation of this effect, though a number of models have been proposed, seereviews in [1, 2] and more recent approaches [3, 4]. The
term NLT reflects the inability of simple diﬀusive models
to reproduce the inversion of the heat/cooling pulse, while
their potentials have not yet been exhausted. In such models reviewed in [1, 2] and briefly mentioned in [3–5] (andin seemingly more advanced ones involving the turbulenceetc.) the energy exchange between the plasma and themagnetic field is completely ignored, while it can be essential [2, 6, 7] in tokamaks and stellarators. In such systems,
the plasma-field interaction is the key element to provide
the plasma confinement. Large external forces combinedwith small plasma motion can produce a significant energy
transfer acting as an additional heat source and strongly affecting NLT [2, 6, 7]. This exchange has not been incorporated or even evaluated in [3–5] and similar studies [8, 9].
Here we consider this mechanism and take into accountthe plasma displacement caused by the perturbation of theforce balance during Heat Pulse Propagation (HPP).
The goal of this work is modeling of NLT phenomena
numerically for data observed after TESPEL injection into
low-density LHD plasmas [5]. The results of simulations
author’s e-mail: V.Sergeev@spbstu.ru∗) This article is based on the presentation at the 27th International Toki
Conference (ITC27) & the 13th Asia Pacific Plasma Theory Conference(APPTC2018).

by means of the ASTRA transport code [10, 11] and theiranalysis are presented for the cold edge pulse propagation
with account of the energy transfer [2, 6, 7] between magnetic field and plasma.

2. Formulation of the Problem
We consider such motions that the inertia in the force
balance equation for the plasmadv= −∇p + j × B,
(1)dt
can be neglected. Then the reduced consequence of equation (1) can be used where the pressure expansion is counteracted by the Ampere’s force:mi n∇p + j × B.

(2)

The transition to a new equilibrium state can occur much
faster (on the Alfven time scale, tens of µs) than characteristic times of plasma perpendicular transport.
A local drop in the pressure p due to the pelletproduced cooling at the plasma edge must be accompaniedby the change in the current density δ j. Then a response δBin the magnetic field at a finite distance would be a completely natural event. It reaches the center with some delay,
because the plasma, as a good conductor, prevents immediate penetration of δB into the plasma core. It would berather slow due to the skin eﬀect in a conductor that could
not move. However, the plasma is a mobile gas, not a solidbody. In tokamaks and stellarators, its mass is extremely
small. Therefore, a slightest imbalance of the forces is sufficient for providing a finite velocity and the propagation

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c 2019 The Japan Society of Plasma
Science and Nuclear Fusion Research

Plasma and Fusion Research: Regular Articles

Volume 14, 3402121 (2019)of δB. The standard transport analysis [1, 3–5] postulates
an absence of a plasma perpendicular velocity when theequilibrium is thus perturbed. This is partially justified by
the presence of strong toroidal field preventing large radialdisplacement v·dt in a flux-conserving plasma in tokamaks
and stellarators. During the plasma evolution triggered by
the pellet injection, a relative change in the plasma volume
Vol within a given magnetic surface calculated by equation (1) in [10], ΔVol/Vol, must be, at best, of the order of
10−3 [2], which can be hardly detected in real experimental conditions. However, in [2, 6, 7] it was proved that suchsmall deformations accompanying the plasma equilibrium
perturbations should be taken into account in the energybalance equation for electrons

5
3 ∂nT e+∇ · −χe n∇T e + nT e v = S = j · E+Pe .
2 ∂t
2
(3)
Here, n, T e are the plasma density and electron temperature, χe is the electron heat diﬀusivity, S is the source/sink
term with Pe describing auxiliary heating/cooling. The
plasma velocity v explicitly appears in the convective term,
but it is also hidden in the source term S . Indeed, for fastplasma motions when the magnetic flux conservation approach is only slightly violated, the Ohm’s law
E + v × B = η j,(4)

combined with equation (2) yields
j · E = η j2 + v · ∇p.

(5)Here, along with the Joule heating due to η · j2 (essential
in tokamaks) the additional contribution v · ∇p appears [2].
The arguments presented below prove that both the convective and the source terms with v  0 can strongly aﬀect
the plasma thermal energy balance in LHD.
The plasma velocity v cannot be derived from equation (1) due to small plasma mass density mi · n and imprecise knowledge of other terms there. A validated model
for v evaluation from (4) or similar relations with better
description of j(E) has not yet been developed. Therefore, to make a step for the proof of principle [2] we analyze the possibility of reproducing the main features of the
NLT phenomena during TESPEL injection into the LHD
plasma shot #49708 (Fig. 1 in [5]) by simulations of theelectron energy balance equation (3) with a source term (5)and ad hoc time and space evolution of the radial plasma
velocity v(ρ, t).
In LHD, the NLT phenomena have been observed in awide range of experimental conditions and plasma parameters: in ECR, NBI and NBI+ECR heated plasma, which
demonstrated the independence of NLT on the plasma
heating supply; with diﬀerent sizes of TESPEL and corresponding level of the plasma perturbation at fixed plasma
density prior to the injection [5, 8, 9]; with diﬀerent plasma
densities at fixed TESPEL size that allowed to see the tendency of the transition from the core temperature “reversal” to the diﬀusive picture of the cold pulse propagation

Fig. 1 Profiles of electron density (a), electron heat diﬀusivity(b), electron temperature (c), u(ρ) = 1/q(ρ) and i(ρ) (d),TESPEL ablation rate (e) on the normalized minor radiusρ used in ASTRA simulations. LHD shot #49708.

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Plasma and Fusion Research: Regular Articles

Volume 14, 3402121 (2019)as the plasma density increased [8]. With powerful set of
diagnostics, the LHD data provide a good basis for analysis.
In the absence of the essential net toroidal current in
LHD plasmas we can neglect the Joule heating term η· j2 inthe electron energy balance (3). It is also clear that the cold
pulse propagation and its amplitude reversal in the plasmacore cannot be attributed to the changes in this heating term
related to the current profile alterations [2, 3], which simplifies the analysis.
One of the main aims of the work was to simulate
ΔVol/Vol level at v  0, to compare it with estimates in [2]and see whether the δT e propagation and amplitude can beaﬀected by small radial displacements of the plasma. The
simulation model has to calculate the spatial and temporal evolution of ΔVol/Vol during HPP. For this purpose, theevolution of the magnetic equilibrium during HPP has tobe considered.

3. ModelFrom the latter reason, the ASTRA code (AutomatedSystem for TRansport Analysis) is used here for analysis based on equations (3) and (5) and solving equilibrium equation (2). Details of the algorithm, including thefull scheme summarized in the flow chart are publishedin [10,11]. Some simplifications and assumptions havebeen made prompted by experiments, and some to facilitate computations. The main elements of the model are:(i) Density evolution is not simulated and the steady-state
density profile Ne (ρ = r/a) = const(t) interpolated fromYAG Tomson Scattering (TS) experimental data prior to
TESPEL injection is used (Fig. 1a);(ii) The NBI and ECRH input power profiles simulated inRef. [5] are used for calculation of the steady-state profile of χePB (ρ) (Fig. 1b) to reproduce the interpolation ofelectron temperature profiles (Fig. 1c) measured by TS and2nd harmonic of Electron Cyclotron Emission (ECE) LHD
diagnostics prior to the TESPEL injection [5]. The ion
temperature was not measured in the LHD shot #49708
and is assumed T i (ρ) = T e (ρ)/3 = const(t) according tomeasurements of the core ion temperature in similar LHD
shots [12]. It is assumed that TESPEL injection do not disturb the power heating terms in the plasma energy balance;(iii) The ASTRA7 code is run in the prescribed (fixed)boundary mode with the internal (not SPIDER) equilibrium solver (see [10, 11]. Since the LHD equilibrium is
not implemented in the ASTRA code, we used “equivalent” magnetic equilibrium of tokamak with “eﬀective”
plasma current of about 0.7 MA and with LHD parameters in shot #49708 [5]: major radius at the magnetic axis,
R = 3.5 m; an average minor radius, a = 0.58 m; magnetic field at the axis, Bt = 2.83 T. The “eﬀective” plasma
current value provides a close range of μ(ρ) = 1/q(ρ) in
the tokamak-like simulation model and the LHD rotationaltransform i(ρ) profiles shown in Fig. 1d. The profile of theFig. 2 The measured (ECE –red) and simulated (ASTRA- blue)
electron temperature (a) and ΔVol/Vol (b) at diﬀerent normalized minor radii. The standard diﬀusive model withv ≡ 0 is used here. LHD shot #49708.

rotational transform will aﬀect the distributions of the equilibrium quantities, but cannot reverse the diﬀusion fluxes.
Therefore, if we will succeed in demonstrating the cold
pulse inversion, this should be attributed to other factors.
The “eﬀective” Joule heating η j2 term is not included in
the electron energy balance equations (3) and (5) because
actually it must be small in the current-free LHD plasma;(iv) To simulate the electron temperature drop during
TESPEL injection we use the measured pellet ablation rate
profile Ṅ(ρ) shown in Fig. 1e. According to calculations
in Ref. [13], the density rise and temperature drop formations within the helical domain close to the ablation region
during TESPEL injection are lasting about Δtin j ≈ 2 ms.
The cooling term in equations (3) and (5) is modelled= AṄ(ρ) during Δtin j and the coeﬃcient A is
by PTESPELeadjusted to reproduce the measured electron temperaturedrop after TESPEL injection (see Fig. 2a).

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4. Results of Simulations andDiscussion

At the first step, we performed simulations with v ≡ 0
to test our approach against the results of the standardtransport model with the steady-state values of the electron
heat diﬀusivity χePB (ρ). A comparison of the measured(ECE) electron temperature evolutions with simulated ones
is shown in Fig. 2a. First of all, one can see that the computed profiles reproduce the peripheral drop of the electron temperature just after the TESPEL injection. The subsequent diﬀusive propagation of the cold pulse also looks
quite natural: the time of its arrival increases and the amplitude of the pulse decreases with distance from the ablation
region. It is evident that the diﬀusive model without magnetic field as an energy source is not able to describe therise of core electron temperatures, which is a well-known
theoretical result [1, 2, 5]. Evolution of the relative volume
ΔVol/Vol = (Vol(ρ, t) − Vol(ρ, tin j ))/Vol(ρ, tin j ),inside the deformable magnetic surfaces of three diﬀerent
minor radii is shown in Fig. 2b. It demonstrates rather
weak perturbations of the magnetic equilibrium during
HPP with v ≡ 0.
In Fig. 3a, the simulation results are compared withexperimental data when the adjusted ad hoc plasma velocity v(ρ, t) evolution shown in Fig. 4a, b is taken into account. Note that, because of the model essential assumptions and simplifications, we used rather simple function
for v(ρ, t) = vρ (ρ)·vt (t). The radial dependence described byvρ (ρ) allows to get the electron temperature response maximal in the plasma core (inner half of minor radius). The
temporal factor vt (t) corresponds to the core electron temperature time evolution in experiments. One can see from
Fig. 3a that although the simple function for v(ρ, t), the coreelectron temperature rise was obtained at plasma velocities below 3 m/s. The maximal ΔVol/Vol in the plasma coredoes not exceed 0.4 × 10−3 in agreement with estimates inRef. [2]. As yet, the true profile of the convective velocity is not known mainly because there are no experimental
measurements of the core plasma motion with such velocities. This small amplitude is the most important outcomeof our simulations.
Figures 3a, b show that the temporal behavior of
ΔVol/Vol is similar to that of
ΔT e /T e = [T e (ρ, t) − T e (ρ, tin j )]/T e (ρ, tin j ),in the plasma core. This is revealed more explicitly in
Fig. 5 where ΔT e /T e is shown versus ΔVol/Vol for several
minor radii. There is a linear dependence of ΔT e /T e on
ΔVol/Vol in the plasma core when the plasma velocity istaken into account.
The convective term, (5/2) n · T e · v, and the source
term, v · ∇p, play diﬀerent roles in the electron energy balance equation (3). To see their eﬀect separately, we performed simulations by introducing the convective and thesource terms separately. The results are shown in Fig. 6.

Fig. 3 The measured (ECE –red) and simulated (ASTRA- blue)
electron temperature (a) and ΔVol/Vol (b) at diﬀerent normalized minor radii. The ad hoc v(ρ, t)  0 (shown in
Fig. 4) is used in simulations. LHD shot #49708.

They confirm that the convective term stronger aﬀects theplasma than the source term in the plasma core. This is because the source term, being proportional to the pressure
gradient, cannot be significant near the center where ∇p issmall.
The results demonstrated in Figs. 5 and 6 can be qualitatively interpreted as follows. Due to appearance ofplasma velocity in equations (3) and (5), the electron energy density increases, mainly by means of the convective
term in plasma core, and it causes outward shift of the magnetic surfaces (increase of Vol(ρ)) there as can be expectedfrom equation (2).
It should be noted that increase of the electron energy
content in the inner part of the plasma is rather small incomparison with the magnetic reservoir available. To illustrate this, the time evolution of the simulated toroidal
β and the ratio of the thermal plasma energy variation
ΔWth = Wth (t)−Wth (tin j ) to the total energy of the magnetic

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Fig. 4 The ad hoc v(ρ, t) = vρ (ρ) · vt (t) evolution used in simulations: a) vρ (ρ); b) vt (t). LHD shot #49708.

Fig. 5 Correlation of simulated ΔT e /T e and ΔVol/Vol after
TESPEL injection for LHD shot. #49708.

field Wm within the plasma column are shown in Fig. 7.
Here,


B2t
3
Ne (T e + T i )dW, Wm =
dV.
Wth =
Vol 2
Vol 2μ0
One can see a decrease of ΔWth /Wm just after TESPEL injection due to plasma cooling and then its subsequent rise,
which reflects the core plasma heating. The maximal values of ΔWth /Wm ≈ 0.7 × 10−4 is only 10% of small enoughplasma β ≈ 0.7 × 10−3 . It confirms the necessity of taking into consideration the energy exchange between theplasma and the magnetic field in analysis of the HPP phe-Fig. 6 The measured (ECE − red) and simulated (ASTRA) electron temperature with ad hoc v(ρ, t)  0: a) both terms(blue); b) convective term only (green); c) source term
only (cyan). LHD shot #49708.

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Fig. 7 Evolutions of simulated toroidal plasma β (red) andΔWth /Wm (blue) in LHD shot #49708.

nomena.

5. Summary and Further Plans
This study was motivated by the LHD experimental
results [5, 8] and ideas and estimates of Refs. [2, 6, 7]. The
key element of the concept [2] is that the energy exchange
between the plasma and the magnetic field should be incorporated into the energy balance for HPP such as describedin [3, 5, 8]. Our study is the first attempt to verify suchapproach by comparing the simulation results with LHD
data [5, 8].
Results of simulations without plasma velocity areclose to those in Refs. [5, 8] and again confirm the inability of the standard diﬀusive model to reproduce the behavior of the measured electron temperature such as shown in
Fig. 2. However, similar simulations of the electron heatbalance, but with plasma radial velocity of small enoughmagnitude (meters/second) demonstrate that the temperature data can be reasonably reproduced. Maximal perturbations of the plasma volume ∼ 0.4 × 10−3 are in the rangeof those estimated in [2]. It is clearly diﬃcult to measuresuch small changes in experiment, but their eﬀect is strongon the scale of the energies involved.
We demonstrated a linear correlation between
ΔVol/Vol and ΔT e /T e in the inner part of the plasma whenthe velocity was taken into account in equations (3) and(5). It is shown that the convective term in the electron energy balance (3) dominates in the plasma core producing
the electron energy density increase. This causes outward
shift of the magnetic surfaces (increase of Vol(ρ)) there as
can be expected from the magnetic equilibrium equation.
The increase of the electron energy content in the in-ner half of minor radius is rather small as compared withthe magnetic reservoir available within the plasma. The
maximal values of ΔWth /Wm ≈ 0.7 × 10−4 are one order ofmagnitude below small values of plasma β ≈ 0.7 × 10−3 .
It confirms the conclusion of [2] that the energy exchange
between the plasma and the magnetic field must be taken
into account in analysis of the HPP phenomena.
Simulations of the measured electron density evolution in the LHD shot #49708 has to be done for proof of theprinciple. Experimental and simulated behavior of the ion
temperature evolution would be helpful. The free boundary approach of ASTRA7 might influence the modeling results. The time-dependent magnetic equilibrium for LHD
should be implemented in the ASTRA code. A physics
based model for v(ρ, t) is needed for incorporation into ASTRA simulations. The dependence of NLT phenomena on
the plasma electron density has to be revealed. Simulations of NLT phenomena observed in other devices (bothtokamaks and stellarators) with diﬀerent perturbation techniques are expected.

Acknowledgements
This work is conducted while VS was a visiting
professor at NIFS, and is performed with the supportand under the auspices of the NIFS grant administrative costs (NIFS18ULHH012) and NIFS Collaboration Research program (NIFS18KLPH035). Numerical calculations were performed at the Polytechnic SuperComputer
Center at Peter the Great St. Petersburg Polytechnic University.
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