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S.E.Sharapov, Lecture 4, Australian National University, Canberra, 7-9 July 2010
TOROIDAL ALFVÉN EIGENMODES
S.E. Sharapov
Euratom/CCFE Fusion Association, Culham Science Centre, Abingdon, Oxfordshire OX14 3DB, UK
S.E.Sharapov, Lecture 4, Australian National University, Canberra, 7-9 July 2010
OUTLINE OF LECTURE 4
• Toroidicity induced frequency gaps and Toroidal Alfvén Eigenmodes • TAE observations on JET
• Doppler shift
• n-dependence of the TAE fast ion drive
• Bursting Alfvén instabilities on MAST
• Summary
S.E.Sharapov, Lecture 4, Australian National University, Canberra, 7-9 July 2010
GLOBAL ALFVÉN EIGENMODE IN CYLINDRICAL PLASMA
• The existence of weakly-damped GAE in cylindrical plasma is determined by the extremum point of ( )rVrkr AA )()( =ω :
0)(
0==rr
A
dr
rdω, i.e. dr
dV
Vdr
dk
k
A
A
11−=
5
4
3
2
10.5 1.00
ω A
r / a
JG01
.459
-10c
m = -3
m = -2
m = -1
• How the function ( )rVrkr AA )()( =ω looks like in toroidal geometry? Could some extremum
points exist there?
S.E.Sharapov, Lecture 4, Australian National University, Canberra, 7-9 July 2010
THE PARALLEL WAVE-VECTOR IN TORUS In a torus, the wave solutions are quantized in toroidal and poloidal directions:
( ) ( ) ( ) ( ) ..expexp,,, ccimrintitrm
m +−+−= ∑ ϑφζωζϑφ
n is the number of wavelengths in toroidal direction and m is the number of wavelengths in poloidal direction • The parallel wave-vector for the m -th harmonic of a mode with toroidal mode number n is
−=
)(
1)(
rq
mn
Rrk
m .
• This function is determined by the safety factor PT RBrBrq /)( = . • Depending on the value of )(rq with respect to rational nm / , the parallel wave-vector can
be zero, positive, or negative
S.E.Sharapov, Lecture 4, Australian National University, Canberra, 7-9 July 2010
TOROIDICITY INDUCED COUPLING - 1
• In torus, two cylindrical SA modes with m and m+1 mode numbers (same n number) have the same frequency at radial point satisfying
)()()()( 1 rVrkrVrk AmAm +−==ω
• Toroidicity induced “gap” is formed in the SA continuous spectrum at this frequency • This extremum point in the continuum may cause Toroidal Alfvén Eigenmode to exist
(a) (b)
S.E.Sharapov, Lecture 4, Australian National University, Canberra, 7-9 July 2010
TOROIDICITY INDUCED COUPLING - 2
Typical structure of Alfvén continuum (and SM continuum) in toroidal geometry. Two discrete
TAE eigenfrequencies exist inside the TAE gap.
S.E.Sharapov, Lecture 4, Australian National University, Canberra, 7-9 July 2010
PROPERTIES OF TOROIDAL ALFVÉN EIGENMODES - 1
• In contrast to GAE, Toroidal Alfén Eigenmode consists of two poloidal harmonics minimum
• Similarly to GAE in cylinder, TAE frequency does not satisfy local Alfvén resonance condition in the region of TAE localization, )(rATAE ωω ≠ → TAE does not experience
strong continuum damping
1.2
1.0
0.8
0.6
0.4
0.2
1.4
00.2 0.4 0.6 0.8 1.00
Re
(rVr)
s=√ψ/ψs
JG01
.459
-12c
r
m = 1
m = 2
m = 3
m = 40
Radial dependence of the Fourier harmonics of TAE
S.E.Sharapov, Lecture 4, Australian National University, Canberra, 7-9 July 2010
PROPERTIES OF TOROIDAL ALFVÉN EIGENMODES - 2
• Start from the TAE-gap condition )()()()( 1 rVrkrVrk AmAm +−==ω (*)
↓
)()(1rkrk
mm +−=
↓
+−−=
−
)(
11
)(
1
00 rq
mn
Rrq
mn
R
↓
n
mrq
2/1)(
+=
• TAE gap is localised at ( ) nmrq /2/1)( 0 += determined by the SA values of n and m
• Substitute this value of safety factor in the starting equation (*) to obtain
( )( )00
00
2 rqR
rVA=ω
S.E.Sharapov, Lecture 4, Australian National University, Canberra, 7-9 July 2010
PROPERTIES OF TOROIDAL ALFVÉN EIGENMODES - 3
• Typical JET parameters are:
TB 30 ≅ ; 319105 −×= mni ; Di mm =
↓ smVA /106.6 6×≅
• For typical value of the safety factor q=1, the frequency of TAE mode should be
16
0 sec10 −≅ω
kHzf 1602/00 ≅≡ πω
• Add fast ions to plasma and see…
S.E.Sharapov, Lecture 4, Australian National University, Canberra, 7-9 July 2010
HOW TO DETECT SHEAR ALFVÉN WAVES?
• Shear Alfvén waves have the following perturbed electric and magnetic fields:
0=Bδ , 0=Eδ
[ ]( ) ..exp cctmnir
E mr +−−
∂∂
−= ωϑζφδ [ ]( ) ..exp cctmnir
imE m +−−= ωϑζφδ ϑ
[ ]( ) ..exp cctmnir
imckB mr +−−⋅−= ωϑζφ
ωδ [ ]( ) ..exp cctmni
r
ckB m +−−
∂∂⋅−= ωϑζφ
ωδ ϑ
• Oscillatory perturbed poloidal magnetic field, ϑδB , is the easiest to detect with magnetic pick-up coils (Mirnov coils) just outside the plasma
S.E.Sharapov, Lecture 4, Australian National University, Canberra, 7-9 July 2010
MAGNETIC PICK-UP COILS
JG08.412-2c
H301
H302
H303
H304
H305
JET cross-section showing the position and directivity of five Mirnov coils separated toroidally
• The coils measure edgeedge BB
tϑϑ δωδ ⋅≅
∂∂
and are VERY sensitive! Thanks to
high values of 16 sec10 −≅ω perturbed
fields 8
0 10/ −≅BBedgeϑδ are measured
• Sampling rate 1 MHz allows
measurements of AE up to 500 kHz to be made
• The coils are calibrated, i.e. give
the same amplitude and phase response to the same test signal
S.E.Sharapov, Lecture 4, Australian National University, Canberra, 7-9 July 2010
TAE IN JET DISCHARGE WITH ICRH-ACCELERATED FAST IONS
16.0 16.2 16.4 16.6 16.8 17.0
Time (s)
0
100
200
300
400
500
Freq
uenc
y (k
Hz)
Pulse No. 40399 amplitude
JG01
.459
-21c
S.E.Sharapov, Lecture 4, Australian National University, Canberra, 7-9 July 2010
FREQUENCIES OF ALFVÉN EIGENMODES AGREE WITH THEORY
16.0 16.2 16.4 16.6 16.8 17.0
Time (s)
0
100
200
300
400
500
Freq
uenc
y (k
Hz)
Pulse No. 40399 amplitude
JG01
.459
-21c
Spectrogram of AE activity excited by ICRH ions
0 0.2 0.4 0.6 0.8 1.0
s=√ψ/ψs
0
100
200
300
400
500
Freq
uenc
y (k
Hz) EAE gap
TAE gap
JG97
.366
/25c
Pulse No. 40399
Continuous spectrum gaps for n=1…6 as functions of radius
S.E.Sharapov, Lecture 4, Australian National University, Canberra, 7-9 July 2010
MANY TAEs CAN BE EXCITED AT ONCE
S.E.Sharapov, Lecture 4, Australian National University, Canberra, 7-9 July 2010
TAE ARE SEEN AT HIGHER FREQUENCY IF PLASMA ROTATES
S.E.Sharapov, Lecture 4, Australian National University, Canberra, 7-9 July 2010
TOROIDAL MODE NUMBER MAY BE FOUND FROM PHASE DATA
• In tokamak, toroidal mode number can be found from the phase shift measured between two (or more) Mirnov coils at different toroidal angles
JG07.470-1
c
Bω
*et0, ω0, ϕ2
t0, ω0, ϕ1
ω*i
ϕ2
ϕ1
α
Bϕ
R0
r0
B × ∇ B
Beams Ions
Electrons
q = 2
ϕ θ
Bθ
J
Sinusoidal signals measured at different toroidal angles 21 , ϕϕ at the same time and
same frequency are shifted in phase by α.
S.E.Sharapov, Lecture 4, Australian National University, Canberra, 7-9 July 2010
TOROIDAL MODE NUMBERS FROM PHASE SPECTROGRAM
Fre
quen
cy (
Hz)
(x1
05 )
3.8
3.6
3.4
3.2
3.012.50 12.52 12.54 12.56 12.58 12.60
Time (s)
35
30
25
20
15
10
5
0JG99
.404
/6c
n + 18 (36 low |dB|, 37 badfit, 38 non-int 39 |n|>17)4.0
n = 11
n = 10
n = 9
n = 8
n = 7
n = 6
S.E.Sharapov, Lecture 4, Australian National University, Canberra, 7-9 July 2010
GRADUAL INCREASE OF ICRH POWER
• Power waveforms of ICRH driving TAE and NBI (NBI provides damping)
S.E.Sharapov, Lecture 4, Australian National University, Canberra, 7-9 July 2010
ICRH-DRIVEN TAE DURING THE INCREASE OF ICRH POWER
S.E.Sharapov, Lecture 4, Australian National University, Canberra, 7-9 July 2010
EXPERIMENTALLY OBSERVED TAE – THE QUESTIONS TO ASK
• TAE frequencies are well above the expected 160 kHz. Why? Doppler shift • TAE modes with 96 −=n are excited easily, while TAE modes with lower or
higher n ’s need a higher pressure of fast ions. Why? Stability depends on n • The TAE spectral lines broaden and split after some time. Why? Nonlinear
amplitude modulation becomes important (Lecture 5A)
S.E.Sharapov, Lecture 4, Australian National University, Canberra, 7-9 July 2010
DOPPLER SHIFT OF THE MODE FREQUENCY
• Uni-directional NBI on JET drives a significant toroidal plasma rotation (up to 40 kHz)
Geometry of NBI injection system on JET (view from the top of the machine)
• Frequencies of waves with mode number n in laboratory reference frame, LAB
nf , and in the
plasma, 0
nf , are related through the Doppler shift )(rnf rot :
)(0 rnfff rotn
LAB
n +=
S.E.Sharapov, Lecture 4, Australian National University, Canberra, 7-9 July 2010
TAE DRIVE BY FAST IONS – 1
• For TAE, parallel electric field is zero. How fast ions transfer their power to TAE via perpendicular electric field?
• TAE is attached to magnetic flux surface, while the resonant ions experience drift across the flux surfaces and the TAE mode structure
S.E.Sharapov, Lecture 4, Australian National University, Canberra, 7-9 July 2010
TAE DRIVE BY FAST IONS - 2
• When a resonant ion moves radially across TAE from point A to point B, the mode and
the ion exchange energy φ∆e as Figure shows.
• The maximum of φ∆e corresponds to the orbit size, orbit∆ , equal to the mode width,
/TAE TAEr m∆ ≈ . This is why energetic ion drive has a maximum at /TAE orbitm nq r≈ ≈ ∆ .
S.E.Sharapov, Lecture 4, Australian National University, Canberra, 7-9 July 2010
MAIN TAE DAMPING EFFECTS • Thermal ion Landau damping due to 3/Ai
VV = resonance with thermal (or beam) ions does not depend on n:
( )( ) ( ) ii
i q βλλλλβπωγ
9/1,exp2114
3
2
3
22
33
22/1
=−++−≅
• Electron Landau damping and electron “collisional” damping for ideal MHD
TAE does not depend on n either
↓ • Threshold condition for TAE excitation,
eifast γγγ +>
depends on n via fast ion drive alone
S.E.Sharapov, Lecture 4, Australian National University, Canberra, 7-9 July 2010
THE OBSERVED TAE MAY BE CONSIDERED AS THE TAE STABILITY DIAGRAM
S.E.Sharapov, Lecture 4, Australian National University, Canberra, 7-9 July 2010
ALFVÉN NBI-DRIVEN INSTABILITIES ON START AND MAST LOOK VERY DIFFERENT THAN ICRH-DRIVEN MODES ON JET
• Tight aspect ratio (R0 /a ∼∼∼∼ 1.2÷÷÷÷1.8) limits the value of magnetic field at level BT ∼∼∼∼ 0.15÷÷÷÷0.6 in present-day STs ⇒ Alfvén velocity in ST is very low
•
VA = BT / (4ππππnimi)1/2≅≅≅≅ 106 ms-1 (MAST)
• Even a relatively low-energy NBI, e.g. 30 keV hydrogen NBI is super-Alfvénic
VNBI ≅≅≅≅ 2.4××××106 ms-1 > VA ,
The super-Alfvénic NBI can excite Alfvén waves via the fundamental resonance VNBI = VA as in the case of fusion alphas.
S.E.Sharapov, Lecture 4, Australian National University, Canberra, 7-9 July 2010
NBI-DRIVEN AEs ARE DIFFERENT THAN ICRH-DRIVEN
• AEs are seen in bursts not as steady-state modes
40 50 60 70 80 90 100 110 120 t,ms
f, kHz 300
200
100
0
S.E.Sharapov, Lecture 4, Australian National University, Canberra, 7-9 July 2010
NBI-DRIVEN AEs on MAST
S.E.Sharapov, Lecture 4, Australian National University, Canberra, 7-9 July 2010
S.E.Sharapov, Lecture 4, Australian National University, Canberra, 7-9 July 2010
COMPARING TAE DRIVEN BY ICRH AND NBI
• Why there is a dramatic difference in TAEs excited by ICRH and by NBI? • Which regime of TAE excitation, the steady-state (JET-like) or the bursting
(MAST-like) will be relevant for alpha-driven TAEs on ITER?
S.E.Sharapov, Lecture 4, Australian National University, Canberra, 7-9 July 2010
SUMMARY
• TAEs are localised at ( ) nmrq /2/1)( 0 += and their frequency range is given by ( )( )00
0
02 rqR
rVA=ω
• For typical JET parameters TB 30 ≅ ; 319105 −×= mni ; Di mm = , JET has
smVA /106.6 6×≅ ; kHzf 1602/00 ≅≡ πω
• Mirnov coils are good for both amplitude and phase measurements of TAE with frequency below half of the sampling rate
• ICRH-driven TAEs are common on JET. These are seen as steady-state modes with frequencies satisfying
)(0 rnfff rotn
LAB
n +=
• NBI-driven TAEs are common on MAST. These are seen as bursting modes
with sweeping frequency
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