J. Plasma Fusion Res. SERIES, Vol. 8 (2009)

Fast Heat Pulse Propagation by Turbulence Spreading
Volker Naulin, Jens Juul Rasmussen, Paola Mantica1) , Diego del-Castillo-Negrete2), and JET-EFDA
contributors3)
Association EURATOM - Risø DTU, Risø National Laboratory for Sustainable Energy, Technical University of Denmark,
P.O. 49, DK-4000 Roskilde, Denmark
1)
Istituto di Fisica del Plasma, EURATOM/ENEA-CNR Association, 20125, Milano, Italy
2)
Oak Ridge National Laboratory Oak Ridge TN, 37831-6169, USA
3)
JET-EFDA, Culham Science Centre, OX14 3DB, Abingdon, UK(Received: 1 September 2008 / Accepted: 17 November 2008)

The propagation of a cold pulse initiated by edge cooling in JET is compared to propagation of the heat wave
originating from a modulation of the heating source roughly at mid radius. It is found that the propagation of thecold pulse is by far faster than what could be predicted on the basis of the heat wave propagation, and within local
transport models no suﬃcient explanation for this behaviour can be found. Recently, modelling of the cold pulse
propagation using non-local eﬀects and a transport equation that uses fractional derivatives has been successfully
applied to model the eﬀect [1]. Here we discuss a model in which the non-locality is introduced by the processof turbulence spreading. Transport models with turbulence spreading have been proposed in [2] and conditions
under which the perturbation in the turbulence profile could travel at the required high speed from edge to thecore have been established [3]. Here we report on recent results in the modelling of cold pulse propagation using
turbulence spreading transport models.
Keywords: nonlocal transport, turbulence, model, cold pulse, turbulence spreading

1. Introduction

JG08.203-1c

7Temperature
Growthrate
6
Turbulence
Gradient
5
Source

4

3

4
3

2

Amp

T (keV), heating

Experiments using modulation of the sources of heatand more recently momentum are routinely carried out at
the JET tokamak and most of the major magnetic confinement fusion devices. Modulation has proved to be a powerful tool for validating and disseminating various physicsbased transport models. For using modulation techniques
as a tool in transport investigations, it is usually assumed
that the transport properties of the plasma during transient
events such as modulation are the same, or at least very
close to, the transport properties in steady state. However,it has been found that the transient propagation of a cold
pulse initiated by a local cooling near the plasma edge ismuch faster than the propagation of the heat modulation
wave under same plasma conditions [4] . This poses a serious challenge to modelling. Specifically, if the propagation speed of a perturbation is characterised by the local
properties of the plasma only and does thus depend solely
on the local values of temperature, density and their gradients, it seems impossible to explain both fast and slow
propagation of the perturbation at the same local plasma
conditions within such local models. This is reflected in
the observation that for parameters for which the modulation experiments are well described by a local model suchas the critical gradient model (CGM) [4, 5], the model fails
to capture the fast propagation of the cold pulse. Alternatively it is possible to model the fast response of the plasma
to the cold pulse with the CGM, but at the expense of being
unable to reproduce the modulation results. More recently
the use of fractional diﬀusion based transport models [1]has been suggested to introduce the necessary non-locality

2
1

1

0
0
0

0.1

0.2

0.3

0.4

ρ

0.5

0.6

0.7

0.8

-1
0.9

Fig. 1 Profiles of temperature (red) (in keV) and source profile (blue)(left scale). Right scale: turbulence intensity(cyan), κT (magenta) (κc = 1.8) and growthrate (green).
(ρ = r/a).

to be able to describe both heat wave propagation and cold
pulse. Indeed within such a non-local model both fast andslow propagations could be reproduced, but it remains unclear which physical process is responsible for the non locality and under which conditions a non-local response of
the plasma is triggered.

2. Turbulence Spreading Transport Model
We propose an alternative approach that accounts forturbulence spreading [6, 7] as a step towards a physics
based model including non locality and describing boththe cold pulse transient and the modulation part of experiments. In the Turbulence Spreading Transport Model(TSTM) the local intensity of the turbulent fluctuations in-author’s e-mail: volker.naulin@risoe.dk

55©2009 by The Japan Society of Plasma
Science and Nuclear Fusion Research

V. Naulin et al., Fast Heat Pulse Propagation by Turbulence Spreading�� �� �the flux as qh = ξ T̃ 2 v2r . We use the growth-rate γ
to estimate the cross coherence ξ, with the assumption thatonce a significant growthrate is achieved the correlation
saturates with value one, thus we prescribe ξ = tanh(γ).
The relative temperature fluctuation level is assumed pro-stead of the growthrate is used to determine the fluxes of
the transported quantities. The turbulence itself becomes atransported quantity and can spread into linearly stable regions of the plasma raising the transport there to high levels. Consequently the model describes asymmetric radial
spreading of the turbulence, up-gradient transport and front
propagation [7]. Clearly the turbulence introduces a possibility to trigger non-linear plasma responses, mediated bychanges in the turbulence level, which in this model canoriginate from spatially separated regions.
In detail we consider a 1D model in cylindrical symmetry for the profiles of turbulent energy E and electron
temperature T . For simplicity we assume that the density
profile is frozen and flat and choose normalisations thatbring our model close to the CGM model in the absence of
turbulence spreading (D0 = 0):

50

3.5
3.0

40
30

Phase

A (keV)

2.5
2.0
1.5

20

0
0

1
∂t E = ∂r r [D0 E∂r E] + γE − (γ0 + βE)E, (1)r��3
5/2 3/2 ∇T+ S (r, t). (2)∂t T = −∇ · qh + ∇ · χ0 T q
2T

0.5
0.2 0.4 0.6 0.8 1.0
ρ

0
0

0.2

0.4

ρ

0.6

0.8

JG08.203-2c

1.0
10

1.0

Fig. 2 Fourier-analysis of the heat modulation in comparison
with JET pulse 55809 (points), numerical results from
TSTM as lines. Amplitude a) and phase b) evolution of
fundamental (black) and third harmonic (blue).

with for positive growth rates
���−R∂r T e
γ=λ− κc ,
Te√
portional� � to that of the velocity fluctuations, T̃ /T = C E,
i.e., T̃ 2 = C 2 �E��T �2 , where C is a parameter absorbing the spatial scale of the turbulence somewhat analogous
to standard mixing length arguments [7]. In general sucha system will approach a temperature profile close to themarginally stable one, which thus appears as a stiﬀ profile,with the degree of stiﬀness measuring by which amount theheatflux has to be increased to obtain a rise in the temperature gradient. For situations in which turbulence spreading
is not important the stiﬀness mainly determined by γ i.e.,the capability of the system to generate anomalous transport. In situations where the eﬀects of turbulence spreading become dominant, the stiﬀness can take values that are
determined by the local increase in the turbulence level,a clear consequence of the non locality of the model. It
should be noted that within the present model in regions
where the turbulence is damped the flux will be negative,
thus the transport is up-gradient, with the cross coherence ξ
being negative as intrinsic reason for up-gradient transport.
As turbulence is able to penetrate into the stable regions of
the domain, it contributes to steepening the temperature
gradient and increasing the local thermal energy, naturally
at the expense of the turbulent energy. It should be stressedthat the net transport is always down-gradient, and only afraction of the turbulence energy will contribute to this upgradient transport.
It should be noted that for zero turbulence spreading D0 = 0 the model approaches the CGM model(e.g.,
Ref. [5] given by:
� �
�
�
−R∇T
∇T
3
− κc + χ0∂t T = ∇ · q3/2 T 5/2 χ s
2
T
T
H
(3)

and for subcritical regions
� ��−R∂r T e− κc .
γ = −0.1λ −Te
The low damping rate in the subcritical region is introduced to describe the stabilisation of modes in the absenceof a better model for growth and damping rates. Here it ismainly responsible for a smallish up-gradient transport as
has also been observed in direct numerical simulations [7].
In detail the heatflux from turbulence is calculated as
qh = CET tanh(γ).
We motivate the model as follows: The diﬀusion of the
turbulence is taken to be non-linear and proportional to thelevel of turbulence, with D0 constant. The energy input
rate is proportional to the growth rate of the underlying instability γ. Additionally the turbulent energy has a weakdamping (γ0 ) and a non-linear saturation (β) described by
the last term in Eq. (1). The growth rate is given by thesquare-root of the deviation of the temperature gradient√from a critical value κc , i.e., γ = λ [κT − κc ], where λis a parameter and κT ≡ |R∂r T |/T (∂r T < 0 for standard
profiles) and is used in the present form to mimic trappedelectron mode behavior. The temperature T evolves due toa spatially dependent source S (r), the diﬀusivity, χ�0 , and�the divergence of the radial turbulent heat flux, q = T̃ vr ,where T̃ are the temperature fluctuations and� �vr is the fluctuating radial velocity component (E ≈ v2r ). Assuming
a finite cross coherence ξ between T̃ and vr , we express

56

V. Naulin et al., Fast Heat Pulse Propagation by Turbulence Spreading

Te (keV)

4.6

ρ = 0.11

ρ = 0.11

4.4

3.5

ρ = 0.11

4.5

4.4

3.3
4.0

4.3

3.4

Te (keV)ρ = 0.3

ρ = 0.3

ρ = 0.3

3.0

2.9
3.2
2.8

2.8
2.7

3.0

1.5

1.5

1.4

1.6

1.4

1.3

Te (keV)ρ = 0.87

ρ = 0.87

ρ = 0.87
0.6

0.7

0.5

0.6

0.8

0.6
10.10

10.12

10.14
Time (s)

10.16

10.18

1.00

1.02

1.04
Time (s)

1.06

1.08

1.99

2.01

2.02
2.03
Time (s)

2.04

2.05

2.06

JG08.203-9c

Te (keV)

1.6

1.4

ρ = 0.60

ρ = 0.60

ρ = 0.60

Fig. 3 Cold pulse in experiment JET pulse 55809 (left) and simulations with CGM (middle) and TSTM (right), vertical lines indicate startof cold pulse.

where the index H indicates the Heaviside function of thebracket multiplied to the bracket. We will now apply the
TSTM to describe transient heat transport phenomena withparticular attention to the heat modulation and cold pulse
experiments in JET [4] and compare as well to CGM results.

to the high value of turbulence spreading. The profile of1000

3. Simulation results

10
1
0.1

The system Eq. (1) and Eq. (2) is solved numerically,using a third order stiﬄy stable time stepping scheme, witha spatial resolution of 1000 grid points. With the source
profile modeling an oﬀ-axis heating profile we have performed simulations over a wide range of parameters. For
selected parameters, we observe polarity reversal, i.e., theedge cooling results in a transient temperature increase at
the center as first observed in TEXT [8]. We concentrateon modeling the perturbative transport experiments in JET
by Mantica et al. [4]. Specifically we have considered shot
No. 55809. We use the source profile (oﬀ-axis ICH + NBI
heating) from this experiment. The results shown for the
TSTM are for parameters C = 0.5, λ = 1.7 and D0 = 35.
The growth rate was chosen as large as possible and is atpresent limited by numerical instabilities, D0 determines
the speed of the turbulence response, resulting in a rather
flat turbulence profile at the value given. C is the principal value to fit the modulation behavior. Saturation of the
turbulence is chosen at β = 1 and χ0 = 0.6 taken close
to the values used in CGM modeling. Simulations startfrom noise and are run to steady state, reached after typically 0.5 to 1 second, with cold pulse triggered at 2.5 seconds and modulation started at 4 seconds, after decay ofthe cold pulse perturbation. In Fig. 1 the source profile andtemperature profile in the saturated state are depicted (lefty-scale). The temperature profile is a fair match to the experimental one. Furthermore, in the same figure the profileof the turbulent energy is show, which is mainly flat due

0.01
0

1

2

∇T/T

3

4

5

JG08.203-4C

qn T5/2

100

Fig. 4 Flux gradient relationship for TSTM during modulation(red) and during cold pulse (green). The vertical line indicates the critical gradient. Note the logarithmic scale.
Note strong non-linearity in response to the cold pulse.

the temperature gradient κT shows that the plasma is stable
until a radius ρ = 0.3, where the heatflux is observed to be
negative, i.e., up-gradient. Outside of ρ = 0.3 the gradientexceeds the critical one (κT < κc ). Figure Fig. 2 shows theresults from the modulation part of the simulations. While
the amplitude of the first and third harmonic are well reproduced, only the phase for the first harmonic fits well.
The phase of the third harmonic could agree better. Figure
Fig. 3 shows electron temperature traces from experimentand modeling by CGM and TSTM respectively. Parametervalues taken from the best achieved fit to the modulation
data. Temperature values are clearly reproduced better in
the CGM model, but it should be mentioned that the maineﬀort of the fitting was aimed at providing a good fit forthe modulation and cold pulse data. In the TSTM modeling the drop in electron temperature occurs rather quickly
and we measure a 30eV drop at after 11ms, compared with
4ms for the experiment and 22ms for the CGM. It is worthwhile to note that the innermost trace for the TSTM model57

V. Naulin et al., Fast Heat Pulse Propagation by Turbulence Spreading
4shows a slight increase in temperature just after the cold
pulse is triggered, that is a reversal in the polarity of thepulse. This is due to the turbulence profile adjusting on avery short timescale (see Figure 5), leading transiently to
up-gradient transport.

T

11.0

0
+1ms
+2ms
+3ms
+4ms
+5ms
+7ms
+10ms
+15ms

3

2

10.5
10.0
9.5

8

8.5

6

8.0

E

4. Nonlocality and stiﬀness

5
4
3
0

7.5
JG08.203-7c

To investigate non locality in the TSTM, we first consider the relationship between normalised heat flux and local gradient. For the CGM, which features local transport
only, we find for
� �
�
�
−R∇T
∇T
3 3/2 5/2
− κc + χ0(4)χsqh = q T
2
T
T
H

9.0

7

Turbulence
0.2

0.4

X

0.6

7.0

0.8

Fig. 5 Temperature (top) and Fig. 6
turbulence
intensity
field (bottom) after cold
pulse initiation.

so that if we plot the heatflux normalised with q3/2 T 5/2 wefind straight line segments with the slope ξ being the stiﬀness of the model. Clearly there is a transition from low
stiﬀness χ0 to high anomalous stiﬀness once the criticalgradient is exceeded. The whole dynamics of the model isdetermined by these two parameters. As both modulationand cold pulse experience the same local stiﬀness and the
propagation speed of a perturbation is roughly proportional
to the stiﬀness, the model cannot accommodate for two different perturbation propagation velocities, reflected in the
20 ms delay for the cold pulse arrival. For the TSTM thesituation is fundamentally diﬀerent, as can be seen from
Figure 4. As long as the change in the plasma is slow andthe turbulence is everywhere close to its saturated value,
we find a behaviour that is very similar to the CGM model
behavior, namely for every radial position a near linear dependence between flux and gradient, so that the concept ofa well defined local stiﬀness is fulfilled. The actual values
for the stiﬀness should not be to far away from the ones
described in the CGM model, as the modulation part of theexperiment was well described within the CGM. Actually
the observation that for slow or weak perturbations the nonlocality is small explains why in most practical situations
local models are indeed very successful in modeling transport. In case of the large and rapid perturbation of the cold
pulse, however, the situation changes and the trajectories in
the flux gradient diagram become more complex. Specifically, we see that during the cold pulse the transport rises
sharply even before the gradient changes. This is due tothe fast propagation of turbulence intensity from the edge
to the core of the plasma. An increased level of turbulence
then increases transport and this changes the local gradient, see Figure 5. Initially the turbulence increase toward
the edge, when the cold pulse is triggered the turbulence
increases fastly at the edge and then propagates inwards
on a fast time scale. There is no well defined slope of the
trajectories in this case, showing that the concept of a local
stiﬀness parameter is invalid in this situation. Fig. 6 shows
the nonlocal response after the cold pulse in more detail.
The trajectory evolves clockwise and shows an increase intransport as gradient decreases, a behavior not compatible
with local transport models.

1.96

1.98

2.00
T/T

2.02

JG08.203-8c

Temperature

qh T5/21

2.04

Flux gradient relation
after cold pulse at r =0.4.
The trajectory
evolves clockwise.

Interesting in this context is the behavior in the very
center, where the gradient is below critical initially. The
increase in turbulence here leads for a short time to an increase in temperature, with the flux going down, e.g. an
inversion of the cold pulse, before the increased gradient
reverses the heat flux and the temperature decays.

5. Conclusion
We have shown that the TSTM can reproduce both fastcold pulse propagation and slow modulation heat wave expansion at the same parameters. Even though the fittingof the temperature profile could be better, here we convey
the two points that for diﬀerent responses of the plasma to
diﬀerent perturbations (modulation and cold pulse) non locality is a necessity, which appears naturally in the TSTM.
The strong non locality is introduced to the system throughthe propagation of turbulence intensity, which can give riseto increased transport without a change in the local temperature gradients. For these situations the concept of stiﬀness
does not hold as the dependency between flux and gradientis non-local and complex. There are clear ways to improvethe model further, but in addition non locality produces
clear signatures which could be tested experimentally, so
it would be interesting to check in experiment if non-local
eﬀects in the flux/gradient relationship can be found.
[1] D. del-Castillo-Negrete, P. Mantica, V. Naulin, et al., Nucl.
Fusion 48, 075009, (2008)[2] V. Naulin, A.H. Nielsen, J.J. Rasmussen, Phys. Plasmas
12 (122306 (2005), J.J. Rasmussen et al., 33rd EPS Conf.
on Plasma Phys. Rome, June 2006 ECA Vol.30I,P-1.076(2006)[3] X. Garbet, Y. Sarazin, F. Imbeaux, et al. , Phys. Plasmas 14,
122305 (2007)[4] P. Mantica et al., (Proc. 19thIAEA Conf. Lyon, 2002)
EX/P1-04, IAEA 2002
[5] X. Garbet, P. Mantica, F. Ryter et al., Plasma Phys. Controlled Fusion 46, 1351 (2004); ibid 47, 957 (2005).
[6] Z. Lin and T. S. Hahm, Phys. Plasmas 11, 1099, (2004).
[7] V. Naulin, A.H. Nielsen and J. Juul Rasmussen, Phys. Plasmas 12 122306 (2005).
[8] K.W. Gentle et al., Phys. Rev. Lett. 74, 3620, (1995).

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