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Parametric decay instabilityaccompanying electron Bernstein waveheating in MAST

A. Surkov1, G. Cunningham2, A. Gurchenko1, E. Gusakov1,V. Shevchenko2, F. Volpe2

1Ioffe Institute, St.-Petersburg, Russia2EURATOM/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.

2/13

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Outlines

• The EBWH experiment on MAST

• Absolute instability of the UHR wave parametrical reflection

• Threshold power for the fundamental mode of the PDI

• Discussion

• Conclusion

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MAST tokamak

The MAST parameters:

Major radius: R = 0.85 m Plasma current: Imaxp = 2.0 MA

Minor radius: a = 0.65 m Toroidal field: Bmax0 = 0.56 T

Aspect ratio: R/a = 1.3

4/13

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EBW heating system

• f0 = 60 GHz

• 7 beams

• up to 1 MW microwavepower

• steerable focusing mir-rors

5/13

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Experimental results

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6/13

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Theoretical analysis

`uh → `′uh + `lh

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• Slab plasma model

• 1D problem of the para-metric reflection

• “High-frequency” case

ω0 > 2ωce

• Absolute instability

7/13

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Fundamental mode of the decay insta-bility

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Piliya A D and Fedorov V I 1975 Zhurnal Eksperimental’noi i Teo-reticheskoi Fiziki (JETP) 68 987

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In the vicinity of the LHW turning point the equations (see Gusakov Eand Surkov A 2004 Proc. 12th International Congress on PlasmaPhysics, http://hal.ccsd.cnrs.fr/ccsd-00001866/en/) for the daughterwaves potentials

φuh =1

2

[φ1(x)eik1x + c.c.

], φlh =

1

2

[φ2(x)eiκ∗x + c.c.

](1)

can be represented as φ̃′′1 + α1(x− x1)φ̃1 = V12e−iKxφ2

φ′′2 + α2(x− x2)φ2 = V21eiKxφ̃1

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

V12 = − ie

2me

κ2∗E

∗0

ω2ce(5 + µ)`2

T (ω1)k1, V21 =

ie

2me

k1E0

ω2ce`

2T (ω2)

9/13

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Notation:

κ2∗ =

ωpeNlhz

c`T (ω2), k2

∗ =ωce

√1 + Nuh

z2

c`T (ω0)

`2T (ω) =

3

2

(ω2

pe

ω2 − ω2ce

V 2Te

ω2 − 4ω2ce

+ω2

piV2

Ti

ω4

)µ = (k∗/k1)

4, φ̃1 = φ∗1e−i∆kx, ∆k = k1(1 + µ)/(5 + µ),

K = k0 − k1 − κ∗ + ∆k = const,

α1 =[(5 + µ)`2

T (ω1)L(xd)]−1

, α2 =[4`2

T (ω2)L(xd)]−1

,

x1 = xd − `2T (ω1)L(xd)k

21(1 + µ)2

5 + µ, x2 = xd + 2L(xd)

iω′′2ω

2pe

ω2ω2ce

ω2 = ω′2 + iω′′

2

L−1(x) =d ln n

dx+

2ω2ce

ω2pe

d ln B

dx

10/13

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Absolute instability thresholdAccording to Piliya, 1975 the absolute instability threshold for thefundamental mode of the system (1) is determined by the equation

V12V21(α2

2 − α21

)−1/3 ' 0.2

That gives the following expression for the parametric decay, producingthe LH wave with N lh

z ' c/(5VTe), which can be shown to have theminimal threshold

P ∗i

πρ2

[W

cm2

]= 2 · 10−3

[W

cm2/3T1/3GHz1/3eV13/6

]· f

1/30 T̃ 11/12T 5/4

e B1/3

L4/3

where T̃ = Te + 4Ti and ρ is the radius of the heating beam. All theplasma parameters here should be taken in the UHR position. Te ∼ Ti:

P ∗i

πρ2

[W

cm2

]= 0.9 · 10−2

[W

cm2/3T1/3GHz1/3eV13/6

]· f

1/30 T 13/6

e B1/3

L4/3

11/13

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The instability threshold for the MAST

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The experiment parameters:

f0 =ω0

2π= 60 GHz,

Te ∼ Ti = 140 eV,

B = 0.38 T,

L = 3 cm

P ∗i

πρ2≈ 260 W/cm2

ρ = 10 cm⇒P ∗i ≈ 80 kW

12/13

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Discussion

• The threshold power was well exceeded in every heating beam inthe 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 thesuccessful injection of the power exceeding the threshold into theplasma.

• L ∝ f 20 (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.

13/13

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Conclusion

Generation of lower hybrid waves has been observed in EBW heat-ing experiments on MAST for the first time. Theoretical study ofcorresponding decay instability of the pump wave induced backscat-tering was performed. The threshold power was estimated for typicalparameters in MAST. It was shown that the backscattering paramet-ric decay instability can arise only if the pump power exceeds 80 kWin any beam, so there is an indication of substantial coupling of themicrowave power to the UHR in the MAST experiment.

The work was partly funded by the UK Engineering and PhysicalSciences 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 hisresearch.