E.Peluso1, T.Craciunescu2, A.Murari3 , M.Gelfusa1, E.Lerche3,4 , L.Garzotti3, M.Lungaroni1 and

JET Contributors§,1 University of Rome “Tor Vergata”, Via del Politecnico 1, 00133 Rome, Italy

2 National Institute for Laser, Plasma and Radiation Physics, Magurele-Bucharest, Romania

3 Consorzio RFX (CNR, ENEA, INFN, Universita’ di Padova, Acciaierie Venete SpA),Corso Stati Uniti 4, 35127 Padova, Italy

3 Euratom/CCFE Fusion Association, Culham Science Centre, Abingdon, United Kingdom

4 LPP-ERM/KMS, Association EUROFUSION-Belgian State, T EC partner, Brussels, Belgium

See the author list of “Overview of the JET preparation for Deuterium-Tritium Operation” by E. Joffrin et al. to be published in Nuclear Fusion Special issue: overview and summary

reports from the 27th Fusion Energy Conference (Ahmedabad, India, 22-27 October 2018)

THE CONCEPT OF CAUSALITY HORIZON

AND ITS APPLICATION TO SYNCHRONIZATION

EXPERIMENTS IN TOKAMAKS

Outline

• About Synchronization

• Causality Horizon, definition

• Sawteeth triggering with ICRH modulation:

– Overview

– Results

• ELMs pacing via Pellets:– Overview

– Results

• Conclusions

Outline

• About Synchronization

• Causality Horizon, definition

• Sawteeth triggering with ICRH modulation:

– Overview

– Results

• ELMs pacing via Pellets:– Overview

– Results

• Conclusions

Synchronization (of dynamical systems)

• Studying the behaviour of two or more dynamical systems,

“synchronization” occurs when a relationship of coupling can be

defined. Consequently, depending on the type of coupling, different

synchronizations can be defined:

• COMPLETE SYNCHRONIZATION: it occurs when the systems evolve in the

same way in time.

• PHASE SYNCHRONIZATION: it occurs when the phase difference is bounded

(not their amplitude)

• LAG SYNCHRONIZATION, which implies the existence of an asymptotic bound

between the output of one system and the time-delayed output of a second one

• INTERMITTENT LAG SYNCHRONIZATION, which is equivalent to a lag

synchronization interrupted by bursts of non-synchronous behavior.

• ALMOST SYNCHRONISATION results in the asymptotic boundedness of the

difference between a subset of the variables of one system and the

corresponding subset of variables of the other system.

• ….

Synchronization (of dynamical systems)

We are interested at the so called:

GENERALIZED SYNCHRONIZATION:it refers to completely different systems where the dynamical variables of

one system (the response system) are determined by the other system

(the drive system)

To assess and quantify the actual

synchronization of relevant dynamical systems

in fusion, different methods have been used to

assess what we have called causality horizon.

Outline

• About Synchronization

• Causality Horizon, definition

• Sawteeth triggering with ICRH modulation:

– Overview

– Results

• ELMs pacing via Pellets:– Overview

– Results

• Conclusions

Definition of the «Causality horizon»

• Let us consider two examples:

a) Sawteeth triggering with ICRH

(top) what’s the average time

interval between the ICRH

modulation and the saweteeth

crashes?

b) ELMs pacing (bottom) how many

ELMs have been triggered in the

most efficient coupling time?

• No theoretical model can

establish or quantify the

synchronization between

experimental data.

• There is an actual need of

applying a statistical criterion.

Examples reported to show the raw signals and highlighteninghow a statistical approach is actually needed

Definition of the «Causality horizon»

• Considering the definitions of causality and synchronization, it is

possible to formalize the intuitive concept of «causality horizon»

(CH) as:

– the CH is the maximum time interval into which two physical

quantities are coupled (i.e syncronized) and in which one observable

can be thougth as the «drive mechanism» of the second observable.

• In our field, two main typolgies of experiments have been studied

quite recently to asses the above mechanism:

• the ELMs pacing with Pellets [A. Murari et al 2016 Nucl. Fusion 56 076008]

• the Sawteeth pacing with ICRH modulation [A. Murari et al 2017 Nucl. Fusion

57 126057]

Different methods have been applied to quantify the CH

will be described.

• Considering the JRP, the time of the “maximum causal influence” can be

found by varying the delay between the time series.

• Maximal values of the JRP’ quantification factors (e.g. Entropy of Diagonal

Lines) have been considered as indicators of synchronisation.

Definition of «causality horizon» for JRP

0 20 40 60 80 1000.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

CC

M c

orr

ela

tion c

oeffic

ient

Time series delay (ms)

(Y causally influences X)

(X causally influences Y)

• For CCM the intersection of the curves describing the variation of the

Pearson correlation coefficients 𝜌(𝑋 → 𝑌) (X is the “causal driver” of Y)

and 𝜌(𝑌 → 𝑋) (Y is the “causal driver” of X) is used to determine the

“causal horizon”.

“Causality horizon”

Definition of «causality horizon» for CCM

Definition of «causality horizon» for TE

TE example for Pellets pacing:𝑐𝑎𝑢𝑠𝑎𝑙 ℎ𝑜𝑟𝑖𝑧𝑜𝑛 = (2.7 ±2)ms

TE example for Swteeth pacing:𝑐𝑎𝑢𝑠𝑎𝑙 ℎ𝑜𝑟𝑖𝑧𝑜𝑛 = (67 ±9)ms

• Studying the Transfer Entropy (TE), the CH has been defined as the time interval in which the TE falls at around the 95% of its normalized maximum.

Outline

• About Synchronization

• Causality Horizon, definition

• Sawteeth triggering with ICRH modulation:

– Overview

– Results

• ELMs pacing via Pellets:– Overview

– Results

• Conclusions

Sawtheeth pacing with ICRH modulation: Overview

49 50 51 52 53 54 55 56 57 580

2

4

Pic

rf (

MW

)

89820

49 50 51 52 53 54 55 56 57 581

2

3

4

5

Te

0 (ke

V)

49 50 51 52 53 54 55 56 57 580

0.05

0.1

MH

D

N=1N=2

49 50 51 52 53 54 55 56 57 580.5

1

1.5

2

Wm

hd (

MJ)

time(s)

3.5keV

1.5MW

4MW

X[H]=2%)

2Hz RF (400ms ON, 100ms OFF) B=2.75T;

Ip=2MA, f=42MHz (0p0p)

• In these experiments the

population of fast ions is

reduced by notches in the

RF power, so reducing their

stabilising effect and

triggering the sawtooth

crash [E. Lerche et al 2017

Nucl. Fusion 57 036027]. One

of the main issues is that

sawteeth are quasiperiodic

and therefore, in case of

pulsed actions, after a while

an sawtooth is going to

occur.

• Due to the lack of a physical model able to fully explain the occurrence of

sawteeth by first principle, the efficiency of the modulation can be

assessed only statistically determining the causality horizon, i.e the time

interval in which is this likely that the sawteeth have been really triggered

by the external intervention.

• If they are not too large, sawteeth are not particularly

detrimental to plasma performance.

• They can cause a moderate confinement degradation but they

can be even beneficial for the expulsion of helium ash or

impurities from the plasma core in the reactor.

• On the other hand, if the sawteeth crashes become too large

they have the potential to trigger more deleterious instabilities,

such as neo-classical tearing modes (NTM’s)

• Such NTM modes typically have a very negative effect on

plasma confinement and they can even induce disruptions.

Sawtheeth pacing with ICRH modulation: Overview

• With regard to the plasma conditions relevant to the data analysed, a

sawtooth crash occurs when two conditions are fulfilled:

1. the magnetic shear s at the q=1 surface reaches a certain threshold

𝑠𝑐𝑟𝑖𝑡2. the normalised potential energy functional 𝛿 𝑊 associated with the

m=1 mode, which is responsible for the sawtooth crash, is smaller

than a certain value proportional to the normalised Larmor radius of

the ions.

• Therefore when 𝛿 𝑊 is smaller than this value a sawtooth crash will occur

when 𝑠 > 𝑠𝑐𝑟𝑖𝑡.

• The normalised potential energy functional presents an additive component

which is associated with the fast ion population in the plasma. The primary

effect of ICRH is to modify the fast ion population and therefore to act on

the 𝛿 𝑊 term, i.e inducing and reducing the fast ion content and thereby

inducing a sawtooth crash.

Sawtheeth pacing with ICRH modulation: Overview

Outline

• Introduction to Causality and Synchronization

• Causality Horizon, definition

• Sawteeth triggering with ICRH modulation:

– Overview

– Results

• ELMs pacing via Pellets:– Overview

– Results

• Conclusions

Results: Sawtheeth pacing with ICRH

Pulsenumber

Regime

CCM Causality horizon

[ms]

JRPCausalityhorizon

[ms]

TECausalityhorizon

[ms]

Triggering[%]

Slowing down time

of the ions [ms]

89822 L 51 [53,52] 63±5 80 50 ±10

89826 L 52 54 87±3 71 50 ±10

90005 H 69 72 67±9 42 80 ±20

90006 H 98 [95,93] 75±5; 94±10 72 80 ±20

• The three indicators give similar estimates of the “causality horizon” and are

in excellent agreement with the slowing down time of the ions.

• The only exception is the pulse 89826 which the TE did not provide a clear

maximum.

The triggering percentages are calculated from the ratio of the number of sawteeth triggered by the RF

notches within the average delay value computed using the TE and the CCM

Outline

• Introduction to Causality and Synchronization

• Causality Horizon, definition

• Sawteeth triggering with ICRH modulation:

– Overview

– Results

• ELMs pacing via Pellets:– Overview

– Results

• Conclusions

ELMs Pacing with Pellets: Overview

• ELMs are particularly problematic H-mode plasma instabilities

causing a significant reduction of the energy confinement by

compromising the edge transport barrier.

• They can be basically described as expulsions of plasma,

tipically in a sub millisecond time scale. For this reason they

have cause serious consequences for the plasma facing

components, the divertor in primis.

• For ITER a form of active ELM control is considered essential.

DEMO is expected to run in ELM-free scenario.

• Therefore, to support the development of reactor-relevant

scenarios, in many machines various ELM pacing techniques

have been tested.

• One of the most promising is based on triggering them with

pellets.

ELMs Pacing with Pellets: Overview

• The objective of the aforementioned technique and of the dedicated

experiments is to control, by triggering it, the frequency of ELMs with

pellets.

• In this way, the ELMs pacing could be adjusted so that the gradients at the

edge would not have time to increase excessively between subsequent

ELMs.

• Consequently it is expected that the expulsions of plasma due to ELMs

can be kept to manageable levels, not being able to damage the plasma-

facing components.

Outline

• About Synchronization

• Causality Horizon, definition

• Sawteeth triggering with ICRH modulation:

– Overview

– Results

• ELMs pacing via Pellets:– Overview

– Results

• Conclusions

Results: ELMs Pacing with Pellets

Pulse

JRPCausalityhorizon

[ms]

% of triggering

TECausalityhorizon

[ms]

%of triggering

Δt [ms]%

of triggering

84688 1.8 9 1.26 3 2 9

84690 3.4 25 3.22 25 2 17

84693 3.2 25 3.51 28 2 17

82885 1.5 6 1.54 6 2 7

82886 3.3 15 3.78 17 2 6

82887 4.2 24 4.06 23 2 11

82889 4.5 21 4.48 21 2 4

84696 3.5 9 3.53 9 2 2

The triggering percentages are calculated from the ratio of the number of ELMs triggered by pellets, divided

by the total number of pellets reaching the plasma for each shot. The column of 2ms refers to the usually used triggering time

• TE and JRP provides similar results for the estimations of the “causality horizon”.

• Discharges exhibit different behaviours, so the assumption of a single time intervalof 2ms is not supported by the analysis [A. Murari et al 2016 Nucl. Fusion 56 076008]

Outline

• Introduction to Causality and Synchronization

• Causality Horizon, definition

• Sawteeth triggering with ICRH modulation:

– Overview

– Results

• ELMs pacing via Pellets:– Overview

– Results

• Conclusions

Conclusions• TE, CCM and JRP are powerful tools to identify “causal horizons” in

physical relevant signals.

• They have been applied to two different typolgies of experiments, but

being fully general, can be applied also in other contexts.

1. Application to ELMs pacing with pellets:– The choice of a fixed triggering time is not supported and furthermore its use

could not be sufficient to fully understand the pellets’ capability of triggering

ELMs.

– Indeed while the ELMs frequency can be due also to the plasma’s parameters

like the density profile, that pellets actually modify, the tools applied here are

based only on the statistical relations between the measured signals. In other

words the methodology allows assessing the actual capability of triggering

ELMs with Pellets on a shot to shot basis, paving the way for a better

understanding and disentagling of any correlated effect.

2. Application to Sawteeth pacing with ICRH:– When properly optimized, high triggering efficiency can be achieved not only in

L mode, but also in H mode.

– The causality horizon is coherent with the ion relaxation time, supporting the

interpretation that the ICRH acts on the fast ions to destabilize the sawteeth.

Thanks for Your

Attention!