This document discusses parametric decay instability accompanying electron Bernstein wave heating in the MAST tokamak. It presents experimental results showing lower hybrid waves generated during EBW heating experiments, a theoretical analysis of the instability threshold, and estimates that the threshold power was exceeded in the MAST experiments, indicating substantial coupling of microwave power. The work provides evidence that parametric decay instability can explain backscattering observed for pump powers above 80 kW.

1. 1/13
Parametric decay instability
accompanying electron Bernstein wave
heating in MAST
A. Surkov1, G. Cunningham2, A. Gurchenko1, E. Gusakov1,
V. Shevchenko2, F. Volpe2
1 Ioffe Institute, St.-Petersburg, Russia
2 EURATOM/UKAEA Fusion Association, Culham Science Centre, UK
e-mail: a.surkov@mail.ioffe.ru
6th International Workshop ”Strong microwaves in plasmas”, July 25 - August 1, 2005.

2. Outlines
2/13
• The EBWH experiment on MAST
• Absolute instability of the UHR wave parametrical reflection
• Threshold power for the fundamental mode of the PDI
• Discussion
• Conclusion

3. MAST tokamak
3/13
The MAST parameters:
max
Major radius: R = 0.85 m Plasma current: Ip = 2.0 MA
max
Minor radius: a = 0.65 m Toroidal field: B0 = 0.56 T
Aspect ratio: R/a = 1.3

4. EBW heating system
4/13
• f0 = 60 GHz
• 7 beams
• up to 1 MW microwave
power
• steerable focusing mir-
rors

5. Experimental results
Shot #11420
1
Power, MW
RF Power 5/13
0
3
Intensity, a.u. <n l>, 1020m-2
2
Plasma Density
1
e
0
2
4 D‫؁‬
Shot #11420 1
3
LHE signal, mW
0.255 s 0
0.285 s 4
2
0.290 s 101 MHz
LH signal, mW
3
1 2
1
0 0
100 200 300 400 500 0,00 0,05 0,10 0,15 0,20 0,25 0,30
Frequency, MHz Time, s

6. Theoretical analysis
uh → uh + lh
6/13
k UHR(ω0)
k0 • Slab plasma model
k2
• 1D problem of the para-
LH
metric reflection
• “High-frequency” case
x ω0 > 2ωce
UHR(ω1) k1
• Absolute instability

7. Fundamental mode of the decay insta-
bility
7/13
k k0-k1 k
k0-k1
k2
k2
xd1 xd2
xd1 x
x x1 x2 xd2
Piliya A D and Fedorov V I 1975 Zhurnal Eksperimental’noi i Teo-
reticheskoi Fiziki (JETP) 68 987

8. In the vicinity of the LHW turning point the equations (see Gusakov E
and Surkov A 2004 Proc. 12th International Congress on Plasma
Physics, http://hal.ccsd.cnrs.fr/ccsd-00001866/en/) for the daughter
waves potentials
1 1 8/13
φuh = φ1(x)eik1x + c.c. , φlh = φ2(x)eiκ∗x + c.c. (1)
2 2
can be represented as

˜ ˜
 φ1 + α1(x − x1)φ1 = V12e−iKxφ2
2 2 21
˜1
 φ + α (x − x )φ = V eiKxφ
2
2
x1 denotes a virtual turning point for the UH wave.
x2 gives the real turning point of the LH wave.
The interaction coefficients take the form
2 ∗
ie κ∗ E0 ie k1E0
V12 = − , V21 =
2me ωce(5 + µ) 2 (ω1)k1
2
T 2me ωce 2 (ω2)
2
T

9. Notation:
ωpeNzlh ωce 1 + Nzuh2
2 2
κ∗ = , k∗ =
c T (ω2) c T (ω0)
2
2 3 ωpe VT2e 2
ωpiVT2i 9/13
T (ω) = +
2 ω 2 − ωce ω 2 − 4ωce
2 2 ω4
µ = (k∗/k1)4, ˜
φ1 = φ∗e−i∆kx, ∆k = k1(1 + µ)/(5 + µ),
1
K = k0 − k1 − κ∗ + ∆k = const,
−1 −1
α1 = (5 + µ) 2 (ω1)L(xd)
T , α2 = 4 2 (ω2)L(xd)
T ,
2
2 2 (1 + µ)2 iω2 ωpe
x1 = xd − T (ω1)L(xd)k1 , x2 = xd + 2L(xd) 2
5+µ ω2ωce
ω2 = ω2 + iω2
2
−1 d ln n 2ωce d ln B
L (x) = + 2
dx ωpe dx

10. Absolute instability threshold
According to Piliya, 1975 the absolute instability threshold for the
fundamental mode of the system (1) is determined by the equation
10/13
2 2 −1/3
V12V21 α2 − α1 0.2
That gives the following expression for the parametric decay, producing
the LH wave with Nzlh c/(5VT e), which can be shown to have the
minimal threshold
1/3 ˜
Pi∗ W W f0 T 11/12Te5/4B 1/3
= 2 · 10−3 ·
πρ 2 cm2
cm2/3T1/3GHz1/3eV13/6 L4/3
˜
where T = Te + 4Ti and ρ is the radius of the heating beam. All the
plasma parameters here should be taken in the UHR position. Te ∼ Ti:
1/3
Pi∗ W −2 W f0 Te13/6B 1/3
= 0.9 · 10 ·
πρ2 cm2 cm2/3T1/3GHz1/3eV13/6 L4/3

11. The instability threshold for the MAST
1,2
1,0
ne (10 m ), Te (keV)
11/13
0,8
The experiment parameters:
0,6
-3
0,4
ω0
20
f0 = = 60 GHz,
0,2 shot #11420, 270 ms 2π
0,0
Te ∼ Ti = 140 eV,
0,2 0,4 0,6 0,8 1,0 1,2 1,4
Major Radius, m
B = 0.38 T,
100 Fpe L = 3 cm
shot #11420, 270 ms Fce
Fuh
80 Fuc
Flc
Pi∗
Frequency, GHz
60 ≈ 260 W/cm2
πρ2
40
20 ρ = 10 cm ⇒ Pi∗ ≈ 80 kW
0
0,2 0,4 0,6 0,8 1,0 1,2 1,4 1,6
Major Radius, m

12. Discussion
• The threshold power was well exceeded in every heating beam in 12/13
the recent MAST experiments.
• Pi is less than the heating beam power due to the conversion effi-
ciency in the UHR, which is less than 100%.
• The decay instability excitation can be considered as a proof of the
successful injection of the power exceeding the threshold into the
plasma.
2
• L ∝ f0 (linear density profile) ⇒ to optimize the EBWH experi-
ment the decrease of the heating frequency can be recommended.
It leads additionally to the broadening of the transparency window
(see Pilya A D and Tregubova E N 2005 Plasma Phys. Control.
Fusion 47 143), hence simplifying the design of an experiment.

13. Conclusion
Generation of lower hybrid waves has been observed in EBW heat- 13/13
ing experiments on MAST for the first time. Theoretical study of
corresponding decay instability of the pump wave induced backscat-
tering was performed. The threshold power was estimated for typical
parameters in MAST. It was shown that the backscattering paramet-
ric decay instability can arise only if the pump power exceeds 80 kW
in any beam, so there is an indication of substantial coupling of the
microwave power to the UHR in the MAST experiment.
The work was partly funded by the UK Engineering and Physical
Sciences Research Council, by EURATOM, by the RFBR (project no.
04-02-16404) and by the scientific school support program 2159.2003.2.
A.V. Surkov is thankful to the Dynasty foundation for supporting his
research.