Fishbones in JET Plasmas with High ICRH Driven Fast Ions ... · Fishbones in JET Plasmas with High ICRH Driven Fast Ions Energy Content. ... the initial frequency of the fishbone

F. Nabais, D. Borba, M. Mantsinen, M.F.F. Nave, S.E. Sharapovand JET EFDA contributors

EFDA–JET–PR(04)65

Fishbones in JET Plasmas withHigh ICRH Driven Fast Ions

Energy Content

.

Fishbones in JET Plasmas withHigh ICRH Driven Fast Ions

Energy ContentF. Nabais1, D. Borba1, M. Mantsinen2, M.F.F. Nave1, S.E. Sharapov3

and JET EFDA contributors*

1Association EURATOM/IST, Av. Rovisco Pais, 1049-001 Lisboa, Portugal2Association EURATOM/TEKES, Helsinki University of Technology, Espoo, Finland

3 Association EURATOM/UKAEA Fusion Association, Culham Science Centre, Abingdon,Oxfordshire, OX14 3DB

* See annex of J. Pamela et al, “Overview of Recent JET Results and Future Perspectives”,Fusion Energy 2002 (Proc.19 th IAEA Fusion Energy Conference, Lyon (2002).

Preprint of Paper to be submitted for publication inPhysics of Plasmas

“This document is intended for publication in the open literature. It is made available on theunderstanding that it may not be further circulated and extracts or references may not be publishedprior to publication of the original when applicable, or without the consent of the Publications Officer,EFDA, Culham Science Centre, Abingdon, Oxon, OX14 3DB, UK.”

“Enquiries about Copyright and reproduction should be addressed to the Publications Officer, EFDA,Culham Science Centre, Abingdon, Oxon, OX14 3DB, UK.”

1

ABSTRACT

JET ICRH-only discharges with low density plasmas and high fast ions energy contents provided a

scenario where fishbones behaviour has been observed to be related with sawtooth activity: Crashes

of monster sawteeth abruptly changed the type of observed fishbones from low frequency fishbones

[3] to high frequency fishbones [2]. During periods between crashes, the type of observed fishbones

gradually changed in the opposite way. Two new fishbones regimes have been observed in

intermediate stages: Fishbones bursts covering both high and low frequencies and low amplitude

bursts of both types occurring simultaneously. Both sawtooth and fishbones behaviour has been

explained using a variational formalism.

1. INTRODUCTION

Bursts of MHD activity with the shape of “fishbones” have first been observed in 1983, on the

tokamak PDX during the injection of high power neutral beams [1]. These bursts were caused by a

rotating mode with toroidal mode number n=1 and dominant poloidal mode number m=1, which

caused the poloidal magnetic field fluctuations to oscillate in that peculiar way. The mode was

associated with the loss of energetic ions, which were expelled from the plasma core during the

bursts of MHD activity. This may not only reduce the auxiliary heating efficiency, but also the

maximum achievable β (kinetic pressure / magnetic pressure).

Towards the middle and the end of the eighties, two models were proposed to explain this

newfound MHD activity. Both assumed that the instability was caused by the presence in the plasma

of trapped fast particles precessing toroidally with frequency Dω , which for deeply trapped particles,

is given by

EqmrRω0

ωD= ,, (1)

where E is the particle energy, m the mass of the particle, r the minor plasma radius, R the major

plasma radius, q the safety factor, mZeB /0 =ω the gyrofrequency, and Ze is the particle charge. In

the first model, proposed by Chen, White and Rosenbluth [2], the inclusion of the fast particles

energy functional in the dispersion relation would create a new branch on its solution, which becomes

unstable when the fast particles beta hβ increases above a critical value. In this case, the mode is

created with the same frequency of the average precessional drift frequency Dω and it is

destabilized by resonant interaction with the trapped fast ions. The source of energy that drives the

mode unstable is then the spatial gradient of the fast ions distribution function. In the second model,

proposed by Coppi and Porcelli [3, 4], the trapped energetic ions destabilise an already existing

mode, which propagates in the thermal ion diamagnetic direction, and that was rendered marginally

stable by diamagnetic effects. The resonant interaction between the trapped particles and this mode

taps the source of energy for the instability, which is related to the pressure gradient of the plasma

bulk. In this model, the initial frequency of the fishbone instability is around the ion diamagnetic

2

frequency of thermal ions,

ω ≈ ω*i= 1

ZieniBr

dPi

dr- (2)

taken at the radius where the helicity of the perturbation matches that of the magnetic field, 1rr =

where 1r is the radius of the q=1 surface. Zie, ni, and Pi are the plasma thermal ions charge, density,

and pressure respectively. The two models for the fishbone instability were later found to correspond

to different limiting situations. The high frequency fishbones occurred for high values of the fast

ions’ beta hβ while the low frequency fishbones occurred for low values of, both regimes being

separated by a stable window. The two regimes merge when the diamagnetic frequency increases

to a sufficiently high value [5, 6]. In this paper, the higher frequency fishbones will be designed by

precessional (drift) fishbones and the lower frequency fishbones will be designated by diamagnetic

fishbones.

In JET plasmas diamagnetic fishbones are commonly observed; However, in ICRH-only

experiments carried out with low plasma densities and high contents of fast ions energy, MHD

activity in the range 45-75kHz, identified as precessional fishbones, was also observed [7]. In later

experiments, it was observed that the type of fishbones that appear in this kind of discharges were

related to the sawtooth stability. When sawteeth were unstable with frequent crashes precessional

fishbones were observed, but when sawteeth were stabilized and the crashes ceased, precessional

fishbones were gradually replaced by diamagnetic fishbones. Sawteeth crashes have already been

observed to affect fishbones activity. Indeed, it was reported that a sawtooth crash can temporarily

suppress the fishbone activity [8]. In JET experiments with high fast ion energy content another

effect was observed. Monster sawteeth crashes also suppress fishbones temporarily, but when

fishbone activity is resumed the type of observed fishbones changes from diamagnetic (before the

crash) to precessional (after the crash). The remainder of this paper is organized as follows. In Sec.

2 the existing theory is briefly reviewed, pointing out separately diamagnetic and fast particles

effects on fishbone activity. In Sec.3 the stability domains of sawteeth and fishbones activity,

including fast particle, diamagnetic and resistive effects are determined. In Sec.4 experimental

results showing the evolution of fishbones behaviour along a monster sawtooth cycle is presented.

The types of orbits of the ions resonating with the high frequency fishbones are shown in Sec.5.

The analysis and explanation of the experimental observations is done in Sec.6. Finally, conclusions

are presented in Sec.7.

2. THEORY REVIEW

2.1 DISPERSION RELATION

The common approach to analyse the stability of the internal kink mode taking into account the

effects of finite Larmor radius, resistivity and fast particles is by means of a variational formalism

3

[2], [9-13]. The plasma is assumed to be composed by a thermal background component treated

with resistive or ideal MHD and a hot component composed by the fast ions population, which is

dealt with a gyrokinetic formalism. Under these circumstances and assuming a large aspect ratio

circular cross-section, the internal kink mode behaviour is described by the following dispersion

relation [11-13].

δWMHD + δWHOT - = 0,8iΓ [(Λ3/2 +5)/4] [ω(ω - ω*i)]

/2

Λ9/4Γ[(Λ3/2-1)/4]ωΑ (3)

where,

Λ=i[ω (ω − ω*e)(ω - ω*i)]

1/3^

γR

, (4)

γR = S-1/3ωA is the resistive growth rate, S is the magnetic Reynolds number, ωA is the Alfven

frequency, is the electron diamagnetic frequency

ω*e =1

ene Br

dPe

dr, (5)

Pe and ne are the electron pressure and density respectively and

1 dTe

dr.ω*e= ω*e + 0.71^

eBr (6)

The Euler gamma functions in equation (3) come from the inertial layer and are evaluated at the

q=1 surface. is the usual isotropic functional for ideal internal kink modes [14] and

δWHOT =23/2mπ2

B2d(λB2/B0)

dE E5/2 K2 ω (δ/δ E + ω*/ωD)F2^

Kb(ωD -ω)

(7)

is the kinetic contribution coming from the fast ions distribution function F. Here, is the normalised

magnetic momentum, is a differential operator associated with the fast ion diamagnetic drift

frequency and K2 and Kb are elliptic functions arising from bounce averaging [10, 13].

4

2.2 DIAMAGNETIC EFFECTS

In the case of an ideal mode, and not considering diamagnetic or fast ions effects, the solution of

dispersion relation is

γI = - ωA δWMHD.. (8)

defines then the growth rate of the ideal internal kink mode associated with the minimized variational

energy. Including ion diamagnetics effects but keeping out resistive and fast ions effects, the

dispersion relation (3) becomes

γI + i[ω (ω-ω*i)]1/2 = 0, (9)

which has the analytical solutions,

ω =ω*i

2 2

ω*i– ( (2

-γI

1/2

. (10)

The behaviour of the solutions, represented in Fig.1, depends then on two frequencies, γI and ω*i.

When the diamagnetic frequency is zero, the ideal internal kink mode is a pure growing mode.

Diamagnetic effects tend to reduce the mode growth rate and at the same time the mode acquires a

real frequency. For values of i∗ω above a critical value that depends on the ideal growth rate

( Ii γω 2>∗ ), the internal kink mode is stabilized and two different branches become marginally

stable. The low frequency branch is the kink branch, which is now marginally stable, and the high

frequency branch is the ion branch. When including viscous effects as those provided by a resonant

interaction with trapped fast ions, the ion branch becomes unstable producing fishbone bursts while

the kink branch is damped. Nevertheless, the kink branch may also be unstable if resistive effects

are significant.

2.3 TRAPPED FAST IONS EFFECTS

The existence of the precessional (fishbone) branch is determined by the inclusion of the term

HOTWδ in the dispersion relation and can be predicted without diamagnetic or resistive effects. To

solve the dispersion relation when a fast ion population is present, it is necessary to specify the

distribution function. This is usually done for the case of a slowing down distribution in energy,

which is simpler than the Maxwellian case and produces qualitatively the same results. The numerical

solutions of the ideal dispersion relation with i∗ω = 0 are presented in [10]. For each set of parameters

(γI, βh) there are two different unstable solutions. The low frequency solution corresponds to the

5

kink branch and the high frequency solution corresponds to a new branch, the fishbone branch. The

growth rate of this mode goes to as goes to zero, but it becomes unstable for values of above a

critical value. This branch of the dispersion relation is an entirely new branch created by the inclusion

of the term , while on the contrary, the diamagnetic fishbones are caused by an already existing

branch, which can become unstable in the presence of fast ions.

2.4 FISHBONE REGIMES

As seen in sections II.2 and II.3, the existence of fishbones depends critically on the diamagnetic

frequency ω*i and on the fast ions beta βh. Diamagnetic fishbones can only be observed for values

of ω*i above a critical value while precessional fishbones can only be observed if βh is above a

critical value. For moderate values of ω*i (above the critical value) two different regimes of fishbones

can exist for different values of βh separated by a stable window [4, 5]. The diamagnetic fishbones

regime appears for low values of βh, while the precessional fishbones regime appears for high

values of βh. When increasing ω*i the stable window on βh narrows and for sufficiently high values

of ω*i it disappears. At this point the two fishbones regimes coalesce. The ideal unperturbed growth

rate γI plays here a role of a parameter and for increasing values of γI lower values of ω*i are

required in order to achieve the coalescence of the two fishbones regimes.

3. STABILITY DOMAINS

Considering a fast ion population produced by ICRF heating, the distribution in energy can be

approximated to a Maxwellian and, for on-axis heating, the population can be approximately

characterized by a single value of the normalized magnetic momentum λ ≡ µB0/E = 1,

F(E, λ, r) = n(r) δ(λ - 1)e -E/THOT . (11)

Here THOT is the temperature characterizing the Maxwellian distribution in energy. Introducing

this distribution function in the dispersion relation (3) and taking the average value of ωD on r, the

dispersion relation in the ideal limit can be written as [13]

- -i -γI

ωD ωD

ωωD

ωωD

ωωD

ωωD

ω*i

ωD

ω1_2

3_2

1_2

4_3

1_2

+ +βh

ε Z = 0 , (12)

and the threshold condition ω = real, i.e. the condition for which the stability of the mode changes,

is given by

6

-γ I = ωD

ωωD

ωωD

ωωD

ωωD

ω*i

ωD

ωωD

ω1_2

3_2

3_2

1_2

3_4

1_2

++ ReZ = 0 ,

-

(13)

with the corresponding value of given by,

-ωD

ωωD

ωωD

ω*i

ωD

ω1_2

5_2

3_

4

-

βh =εωA

ωDπ1/2

eω/ωD . (14)

If ω*i is of the same order of magnitude as <ωD>, then equation (13) has two solutions provided that

γI < γM, where γM is the maximum value that the right hand side of equation (13) can have, and βh

is a monotonic function of w.

If βh1 and βh2 are the values of βh corresponding to the solutions of equation (13) with βh1 < βh2,

then when the critical value βh1 is reached two different possibilities exist depending on the ratio

γI / ω*i. If γI / ω*i > 1/2, then this solution corresponds to the stabilization of the kink branch. If

γI / ω*i < 1/2 then βh1 is the threshold value for the stabilization of the ion branch.

The second threshold given by equations (13) and (14), βh2, corresponds to the threshold for the

fishbone branch destabilization. For βh > βh2 the precessional fishbone branch is unstable. In the

case of ω*i <<<ωD> the function βh is no longer monotonic and, when increasing βh the precessional

fishbone branch can be destabilized before the kink branch be stabilized.

If resistive effects are included, for values of γi sufficiently low depending on the Reynolds

magnetic number, the kink branch loses its ideal character and the resistive kink branch is unstable.

For plasmas containing high energy fast ions, finite orbit width effects over sawtooth stabilization

must also be taken account. It is known that an increase in the size of the trapped particles interacting

with sawteeth may reduce the efficiency of its stabilization [15]. Thus, if finite orbit width effects

are added, for sufficiently high values of THOT the stabilizing effect of the fast ions over the kink

branch may be lost and this branch may become unstable again. Since βh depends on THOT, there

must be a threshold value in βh above which the kink branch becomes unstable again. Note that the

same value of may be reached by increasing the fast particles density or the fast ions temperature

. If the threshold in is reached increasing the kink branch destabilization is not expected. So, when

considering finite orbit width effects, the threshold in should be replaced by a threshold in . The

resulting stability diagram is presented in Fig.2.

In region I the kink branch is unstable. Fast particles effects are too weak to stabilize the mode

( 1hh ββ < ) and the diamagnetic effects are also weak when compared with the ideal unperturbed

growth rate 21>∗iI ωγ . In region II the kink branch is stabilized by diamagnetic effects, but the

ion branch is now destabilized by fast ions. In region III the ion branch is still unstable and the kink

branch becomes also unstable due to resistive effects. In region IV the ion branch is stabilized

7

1hh ββ > , but the kink branch remains unstable due to resistive effects. In region V the precessional

branch is also destabilized by fast particles since the condition 2hh ββ > was reached. In region VI,

Iγ is high enough for resistive effects to be weak and the kink branch is stabilized by fast particles

effects. In region VII, if hβ is increased due to an increase in the fast ions’ temperature the kink

branch becomes unstable due to finite orbit width effects. If the increase in is due to an increase in

the fast ions’ density’ the kink branch remains stable. In region VIII, where , the low frequency

branch and the high frequency branch coalesce. For the precessional branch coalesces with the

kink branch (case represented in Fig.2) and for the precessional branch coalesces with the ion

branch. In both cases there is always one and only one unstable branch. The value of decreases

when increasing. In region IX all branches are stable.

4. EXPERIMENTAL RESULTS

In recent ICRH-only discharges on the JET tokamak with low plasma densities and high fast ion

energy contents, the fishbone behaviour was observed to change during the cycle of a monster

sawtooth.

Figure 3 shows the spectrogram of MHD activity for Pulse No:54300 and just the same sawteeth

and fishbone behaviour was observed in the subsequent Pulse No:54301, No:54305 and No:54306.

In these pulses, carried out under identical conditions, the regime of short sawtooth crashes appeared

always accompanied by precessional fishbones. As soon as the sawteeth were stabilized and the

frequent crashes stopped, precessional fishbones were gradually replaced by diamagnetic fishbones

until only diamagnetic fishbones existed.

Figure 4 shows a zoom where the temporal evolution of the fishbone regimes during a single

monster sawteeth is presented for Pulse No:54301.

At the beginning of the sawtooth free period, around t=9.2s, only precessional fishbones can be

observed. Later in the discharge precessional fishbones are replaced by “hybrid fishbones”, which

can be clearly seen around t=9.6s. The hybrid fishbones are a newly observed type of fishbones

that have characteristics of both the precessional and diamagnetic fishbones ranging all the way

from high to low frequencies. The amplitude of the hybrid fishbones bursts eventually starts

decreasing and they begin chirping down for a smaller range of frequencies. At some point, they

become precessional fishbones and at the same time diamagnetic fishbones appear in the low

frequencies. A new stage was reached where low amplitude bursts of both types of fishbones can be

observed simultaneously. This occurs just before t=9.8s. The precessional fishbones progressively

disappear until only diamagnetic fishbones remain. A monster sawtooth crash occurs just after

t=9.91s and the fishbone activity is suppressed, but after a short time (t=10.0s) precessional fishbones

reappear. The complete cycle is represented in Fig.5.

5. RESONANT IONS

One surprising feature of the precessional fishbones is their unexpectedly high frequencies. If the

mode was being destabilized by a population of trapped particles with a temperature around 1 MeV,

8

then one would expect averaged precessional drift frequencies below 50 kHz while the observed

MDH activity was in the range 50-70 kHz. To identify which particles were destabilizing the fishbone

branch, the CASTOR-K code [16] was used to calculate the resonant transference of energy between

a mode with a frequency of 70 kHz and an on-axis ICRH driven fast ions population with MeV,

which is around the value estimated for these pulses. The CASTOR-K used the eigenfunction

calculated by the MISHKA code [17] and simulations were carried out for a vast range of conditions.

Figure 6 shows the energy exchanged between the mode and a fast ion population peaked in the

centre and characterized by (on axis heating), MeV. As seen in this figure, the particles resonating

with the mode are not only trapped particles but also particles with potato orbits (Pϕ < 0). These

particles have a much higher precessional drift frequency ωD than the trapped particles, and also

contribute to the average precessional frequency of resonant particles <ωD>. Thus, this can explain

why the observed mode frequency is higher than the predicted by the average precessional drift of

trapped particles.

6. STABILITY ANALYSIS

6.1 INTRODUCTION

In the variational approach presented in section III, fishbone and sawtooth stability depends basically

on three parameters ( ihI ∗ωβγ ,, ), if resistive effects are not taken account. Resistive effects are

only important if the unperturbed mode is close to the marginal stability ( Iγ close to zero). For the

ideal internal kink mode, any change in the fishbones or sawtooth behaviour must be due to a

change in any of the three parameters ( ihI ∗ωβγ ,, ). Quantitative comparisons between the stability

diagram of Fig.2 and experimental results are very difficult. On one hand, the approximations used

in the dispersion relation and in the distribution function, and especially the fact that the radial

dependence of was not taken account, make the borders of the regions in the stability diagram

quite uncertain. Also the large size of the orbits may contribute to the uncertainty in the location of

the lines corresponding to and. On the other hand there is also an amount of uncertainty in the

values of ( ihI ∗ωβγ ,, ) calculated with experimental data. Thus, a qualitative analysis is more adequate

than trying to fit the calculated values of ( ihI ∗ωβγ ,, ) within the regions of the stability diagram.

6.2 PRECESSIONAL FISHBONES

In Pulse Nos:54300 to No:54305 small period sawteeth are always observed along with precessional

fishbones. This occurs when the plasma density is below a certain threshold [7], which, since the

plasma density is inversely proportional to the fast ions temperature [18], corresponds also to a

threshold in. Assuming that the occurrence of frequent sawteeth crashes are due to finite orbits

effects [19], then this regime corresponds to the region VII in Fig.2. When the plasma density

increases and the fast ions’ temperature decreases, sawteeth are stabilized by the fast particles but

precessional fishbones continue to be observed. This transition corresponds to crossing the pink

line in Fig.2 and moving from region VII to region VI. The growth rate of the mode causing

precessional fishbones has been calculated numerically [20].

9

6.3 HYBRID FISHBONES

During sawteeth free periods the q=1 surface expands as result of magnetic diffusion and the ideal

growth rate increases. At the same time the absence of crashes allows the background ions’ pressure

profile to peak. The diamagnetic frequency is mainly determined by the radial gradient of the

background ions’ pressure, so this means that when sawteeth are stabilized by fast ion effects, both’

and’ increase. It is possible to estimate the increase in the diamagnetic frequency through the sawtooth

free period. In similar pulses (No:47575 and No:47576, taken as reference), the frequency in the

laboratory frame ƒ*iLAB = ω*i/2π + ƒrot for the regime where frequent sawteeth crashes and

precessional fishbones were observed was calculated [7] and a value below 3kHz was obtained.

Since the initial frequency of diamagnetic fishbones is around ω*i, it can be observed from the

MHD spectrograms that ƒ*iLAB increases from a value below 3kHz at the beginning of the sawteeth

free phase to more than 10kHz when diamagnetic fishbones are first observed. After the appearance

of diamagnetic fishbones continues to increase reaching typically values around 20kHz when the

sawteeth free period is ended by a monster crash. This dramatical increase in the diamagnetic

frequency drives the whole system. The effect over the stability diagram of Fig.2 is that the blue

line (γI = ω*i/2) drifts upward and at the same time the dark green line (γI = γM) drifts downward.

The condition γI = ω*i/2 was reached first but the system remains in the region VI of Fig.2 and only

precessional fishbones are still observed. After that, the diamagnetic frequency continues to increase

and at some point the condition γI > γM is also reached. When this happens, the fishbone branch of

the dispersion relation coalesces with the ion branch (see Fig.8) and the region VIII.a in the stability

diagram of Fig.7 is reached.

The ion-fishbone branch is always unstable and, for a given value of ω*i, behaves like the fishbone

branch for high values of βh and like the ion branch for lower values of βh. If βh is high enough when

a fishbone burst is triggered, it begins as a precessional burst. During the burst fast particles are

expelled from the plasma core and βh decreases significantly. Thus, it is possible that βh reaches values

small enough for the mode behaviour change to that of a diamagnetic fishbone, as indicated in Fig.8.

If the condition MI γγ > was reached before the condition 2iI ∗< ωγ , then the unstable branch

would be the kink-fishbone branch. In this case, the precessional burst and consequent decrease in

βh would have caused the mode to behave like the unstable kink and a sawtooth crash would have

been observed.

Figure 9 shows the temporal evolution of Bθ for the hybrid fishbones. The hybrid burst is initiated

with fast oscillations in Bθ, at frequencies characteristic of the precessional fishbones. The amplitude

of the oscillations reaches a maximum and begins decreasing, but resonant fast particles continue

to be expelled from the core and continues decreasing. At some point is low enough for a new

source of energy to be tapped. This new source of energy, related with the bulk ions, allows the

amplitude of the oscillations to grow again but now they have the characteristics of a diamagnetic

fishbones with much slower oscillations in Bθ.

In Fig.10 it is shown that each fishbone event where the frequency chirps down from around

80kHz to 10kHz, is related to a single burst. The frequency of occurrence of hybrid fishbone bursts

~

~

~

10

is nearly the same as the precessional fishbones, which is not surprising since the trigger condition

is the same. On the other hand, the amplitude of the hybrid bursts is usually higher than that of the

precessional or diamagnetic fishbones. The reason for this is that the energy related to the bulk ions

is normally tapped before the amplitude of the precessional part of the hybrid fishbone reaches its

maximum. Thus the energy tapped from the bulk ions is added to the already existing energy

coming from the fast particles allowing higher amplitudes to be reached.

6.4 COEXISTENCE OF BOTH TYPES OF FISHBONES

A further increase in the diamagnetic frequency i∗ω and in the ideal growth rate Iγ causes the ion-

fishbone branch to gradually lose its precessional behaviour. At this intermediate point where none

of the behaviours is dominant, both types of fishbones can be triggered independently, but they

both can only reach small amplitudes. Small amplitude precessional fishbones do not cause

diamagnetic energy to be tapped as in the previous case and so both types of fishbones exist

independently.

Figure 11 shows the transition from the hybrid fishbone regime to the regime where both fishbone

types coexist. The amplitude of the precessional part of the hybrid bursts become so small that no

longer triggers the diamagnetic burst. At the same time, diamagnetic bursts appear in the low

frequencies. The events are not correlated and near t=9.06s it can be observed that three precessional

bursts occur during a single diamagnetic burst. Even after t=9.1s a closer analysis of the Bθ signal

show the presence of very small amplitude precessional bursts.

6.5 DIAMAGNETIC FISHBONES

The small amplitude precessional fishbones eventually disappear and only diamagnetic fishbones

remain. The reason for this is once again the increases in Iγ and i∗ω , which causes the ion-fishbone

branch to behave as the ion branch. It has also been suggested that Alfven eigenmodes may expel

fast particles from the plasma core [21] and in this case there would be a decrease in hβ which

would further push the mode behaviour towards the diamagnetic side. Since the branch is unstable,

the usual diamagnetic fishbone behaviour is observed.

Diamagnetic bursts are less frequent than the precessional ones and, as seen in Fig.11, both the

amplitude of the bursts and the initial mode frequency, which is around the diamagnetic frequency

ω*i, increase in time.

6. 6 MONSTER SAWTOOTH CRASH

Diamagnetic fishbone bursts cease when a monster sawtooth crash occurs. The usual explanation

for the occurrence of monster sawteeth crashes is related with an increase of the radius of the q=1

surface r1 [22]. As the safety factor on the axis decreases and increases during the sawtooth free

period, according to the Bussac model [14], the energy required to stabilise the m=1 oscillations

increases as r13. At a certain point, the stabilizing effect provided by the fast particles is no longer

sufficient to stabilize the mode and a crash occurs. An alternative explanation recently proposed for the

~

11

occurrence of monster crashes is based on the possibility that Alfven instabilities would cause a depletion

of stabilizing ions from the core of the plasma [21]. In terms of the stability diagram of Fig.7, for the

crash to occur the region VIII.c must be accessed, which requires the condition γI/ω*i > 1/2 to be

verified. This means that the depletion of resonant ions from the plasma core alone cannot explain

the crash, since the stabilising effect of the finite diamagnetic frequency could not be overcome.

The crash must then be caused by a rapid increase in the ideal growth rate γI related to the expansion

of the q=1 surface. However, it is possible that the number resonating fast particles is small before

some crashes occur, as shown in Fig.12.

The diamagnetic bursts are suppressed almost one tenth of a second before the monster crash

occur, which suggests that the number of fast particles in the plasma core must effectively be small.

Nevertheless the suppression of diamagnetic fishbones just before the monster crash is not generally

observed. Some time after the crash the fast particle population is restored and precessional fishbones

are observed. Thus, the instability causing the fast particles to be expelled from the plasma core, if

there is any, must also have been suppressed by the sawtooth crash.

CONCLUSIONS

JET ICRH-only discharges with low density plasmas and high fast ion energy content provided a

new scenario where the stability of the internal kink mode in the presence of highly energetic ICRH

driven ions can be analysed. The qualitative behaviour of fishbones and sawteeth has been

successfully explained using a variational formalism. In this variational approach, the stability of

the ideal internal kink mode depends basically on three parameters ( ihI ∗ωβγ ,, ). A stability diagram

indicating the stability of each branch of the dispersion relation was drawn as a function of these

parameters. Precessional fishbones were observed in a regime of high hβ and low i∗ω . During

sawteeth free periods, precessional fishbones were observed to be gradually replaced by diamagnetic

fishbones. This is due to an increase in the diamagnetic frequency i∗ω and in the ideal growth rate

γI that causes the ion branch to coalesce with the fishbone (precessional) branch. The ion-fishbone

branch can behave as the ion branch or the fishbone branch depending on the parameters ( ihI ∗ωβγ ,, ).

The increase in i∗ω and γI, that occurs during sateeth free periods, push the behaviour towards the

ion branch type. When the precessional behaviour is slightly dominant, hybrid bursts that chirp

down both ranges of frequencies are observed. When none of the behaviours is dominant, low

amplitude bursts of both types occur simultaneously and independently. When the diamagnetic

behaviour is dominant, diamagnetic bursts are observed. A monster sawtooth crash flattens the

bulk ion profile and reduce the r1 radius, restoring low values of ω*i and γ1, and causing precessional

fishbones to occur. The fishbone cycle is then restarted by a monster sawtooth crash.

Aside from particles with banana orbits, the resonant ions include a minority of ions with potato

orbits that precess toroidally at higher frequencies ωD. This may account for the unexpectedly high

mode frequencies observed experimentally.

12

ACKNOWLEDGEMENTS

This work, supported by the European Communities and “Instituto Superior Técnico” under the

Contract of Association between EURATOM and IST, has been carried out within the framework

of the European Fusion Development Agreement. Financial support was also received

from”“Fundação para a Ciência e Tecnologia” in the frame of the Contract of Associated Laboratory.

The views and opinions expressed herein do not necessarily reflect those of the European

Commission, IST and FCT.

REFERENCES

[1]. K. McGuire et al. Phys. Rev. Lett. 50 891 (1983)

[2]. L. Chen, R. White and M. Rosenbluth, Phys. Rev. Lett. 52 1122 (1984)

[3]. B. Coppi and F. Porcelli, Phys. Rev. Lett. 57, 2272 (1986)

[4]. B. Coppi S. Migliuolo and F. Porcelli, Phys. Fluids–31, 1630 (1988)

[5]. Y. Zhang, H. Berk and S. Mahajan, Nucl. Fusion 29, 848 (1989)

[6]. B. Coppi et al. Phys. Rev. Lett. 63, 2733 (1989)

[7]. M. Mantsinen et al. Plasma Phys. Control. Fusion 42, 1291, (2000)

[8]. T. Krass et al. Nuclear Fusion 38, 807 (1998)

[9]. R. White et al. Phys. Fluids 26, 2958 (1983)

[10]. R. White, L.Chen, F. Romanelli and R. Hay, Phys. Fluids–28, 278 (1985)

[11]. R. White, P. Rutherford and P. Colestock, Phys. Rev. Lett. 60, 2038 (1988)

[12]. R. White, M. Bussac and F. Romanelli, Phys. Rev. Lett. 62, 539 (1989)

[13]. R. White, F. Romanelli and M. Bussac, Phys. Fluids B 2, 745 (1990)

[14]. M. Bussac et al. Phys. Rev. Lett. 35, 1638 (1975)

[15]. F. Porcelli–et al. Phys. Fluids B–4, 3017 (1992)

[16]. D. Borba and W. Kerner, J. Computational Physics 153, 101 (1999)

[17]. A.B. Mikhailovskii–et al. Plasma Physics Reports–10, 844 (1997)

[18]. T.H. Stix, Nuclear Fusion 15, 737 (1975)

[19]. F. Nabais et al. Submitted to Plasma Phys. Control. Fusion

[20]. N.N. Gorelenkov et al. EPS 2003, St. Petersburg, Russia (2003)

[21]. S. Barnabei–et al. Phys. Rev. Lett. 84, 1212 (2000)

[22]. C.K. Phillips et al. Phys. Fluids B 4 (7) 2155 (1992)

13

JG05

.37.

2c

VIII-C

I-S

IX-StableII-D

VII-S.P

VI-P

V-S,PIV-S

6

4

2

0

-2

-4

8

2 4 6 80 10ω*i (a.u.)

JG05

.37-

1c

Kink branchIon branch

ω*i = 2γI

γI

ω,γ

(a.u

.)

Figure 1: Solutions of the dispersion relation (9). Thegrowth rate (dashed lines) and real frequency (solid lines)of the mode are presented as function of the diamagneticfrequency …*i while the ideal growth rate γI plays therole of a parameter.

Figure 2: Stability diagram for the internal kink mode.In the regions labelled with S the kink branch responsiblefor sawteeth is unstable, in the regions labelled with Dthe ion branch responsible for diamagnetic fishbones isunstable and in the regions labelled with P the fishbonebranch responsible for precessional fishbones is unstable.For γI >γM the low frequency and high frequency branchescoalesce (region labelled with C).

Figure 3: Spectrogram of MHD activity for Pulse No:54300. Bothprecessional and diamagnetic fishbones activity are indicated with arrows.

1018

1

0

0

00 0

ND

e (m

-3 )

Time (s)

JG05

.

0

20

40

60

80

100

120

10 12 148 16Time (s)

JG05

.37.

3c

Diamagnetic Fishbones

Freq

uenc

y (k

Hz)

Pulse No. 54300 channel: 002 Amplitude

Precessional Fishbones

GiantSawteethCrashes

14

Figure 5: The fishbones cycle around a monster sawtoothperiod.

Figure 6: Energy exchanged between the fishbone branchof the internal kink mode with a frequency of 70 kHz anda population of on-axis ICRH driven fast ions with atemperature of 1MeV.

Figure 4: The fishbones cycle around the sawtooth crash in Pulse No: 54301.

Fre

quen

cy (

kHz)

0

20

40

60

80

100

120

9.6 9.8 10.0 10.29.49.2Time (s)

JG05

.37-

4c

Pulse No: 54301 channel: 002 Amplitude

Hybrid fishbones Both fishbonestypes coexist

Diamagnetic fishbones

Precessional fishbones Gaint sawtooth crash

Giantsawtooth

crash

Precessionalfishbones

Hybridfishbones

Both types of fishbones

Diamagneticfishbones

JG05

.37-

5c

0.015

0.010

0.005

0

-0.005

0.020

1000 20000 3000Energy (keV)

JG05

.37-

6c

0.000278840 / fish orbits 5

Nor

mal

ized

tor

oida

l can

onic

al m

omen

tum

15

Figure 8: Schematic diagram of the solution of thedispersion relation including diamagnetic and fast ioneffects. The smaller arrow indicates the evolution of themode behaviour during a hybrid burst, as fast ions areexpelled from the plasma core and hβ decreases.

Figure 9: Temporal evolution of Bθ for the hybrid fishbonein Pulse No:54300.

~

VIII.c- C:SP

VIII.b- C:DP

VIII.a- C:DP

VI-P

JG05

.37-

7c

Figure 7: Stability diagram for the internal kink mode.In the regions labelled with P the fishbone branchresponsible for precessional fishbones is unstable, in theregions labelled with C:DP the ion branch responsiblefor diamagnetic fishbones coalesces with the precessionalbranch and in the regions labelled with C:SP the kinkbranch responsible for sawteeth coalesce with theprecessional branch. The point X marks the state of theplasma when hybrid fishbones are observed.

JG05

.37-

8c

CoalescentIon-Fishbone

branch

precessionalbehaviour

Fishbonebranch

diamagneticbehaviour

_h decreasesduring the burst

Ionbranch

-3000

-2000

-1000

0

1000

2000

3000

12.551 12.553 12.555Time (s)

JG05

.37-

9c

16

4000

2000

0

-2000

-4000

9.54 9.58 9.629.50Time (s)

JG05

.37-

10c

120

100

80

60

40

20

09.54 9.58 9.629.50

Time (s)

JG05

.37-

11c

Freq

uenc

y (k

Hz)

Pulse No: 54301 channel: 002 Amplitude

100

80

60

0

120

8.9 9.1 9.3 9.5Time (s)

JG05

.37-

13c

Pulse No: 54300 channel: 002 Amplitude

40

20

Freq

uncy

(kH

z)

-4000

-2000

0

2000

4000

6000

8.9 9.1 9.3 9.5Time (s)

JG05

.37-

12c

~Figure 10: Temporal evolution of Bθ for the hybrid fishbones and correspondent MHD spectrogram.

Figure 11: Temporal evolution of Bθ and MHD spectrogram showing the transition to the diamagnetic fishbone regime.~

17

Pulse No: 54300 Channel: 002 Amplitude

0

40

100

80

60

20

120

9.4 9.5 9.69.3 9.7

Freq

uenc

y (k

Hz)

Time (s)

JG05

.37-

15c

Figure 12: Fishbone behaviour around a monster sawtooth crash.