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托卡马克位形优化 (2)

高庆弟

SWIP

核工业西南物理研究院 成都

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Nonlinearity of LH wave absorption

The plasma temperature in HL-2A is much lower than that in future reactor. To establish RS configuration, the LH driven current should be located off-axis where the plasma temperature is even lower, and the plasma absorption of high phase velocity LH waves is too weak to ensure the waves are damped during their first pass.

In the weak electron Landau damping condition LH wave rays make many passes through the wave propagation domain in plasma and undergo numerous reflections at the propagation boundaries. Considering the propagation of lower-hybrid waves in a tokamak. The phase space energy density of the rf field is denoted by U( x, k, t), where x is the position vector, k is the wave vector. U obeys the WKE,

After determining U( x, k), we can calculate physical quantities such as the (time-averaged) absorbed power density,

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THE CYLINDRICAL APPROXIMATIONAppropriate canonical coordinates in tokamak geometry are (x, k)(r, , , kr, m, n). We consider the source S = Pin(2)-2 (r - ro) (kr – kr0), the solution of ( 1) is

The (time-averaged) energy density in the rf parallel electric field,

Here follows from the cold plasma dispersion relation. The damping rate e for electron Landau damping can be written as follows:

)/(/ //// dd

r

rr rv

rrdry

1

1

)(

)()(

where 2

1 )(

)(2r

rr rv

rdr

The absorbed power density,

, with

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The above solution of the WKE can be classified into two distinct para-meter regimes: (i) the multipass regime (for 1), when U tends to be uniform along the entire ray orbit in the (r, k) plane, and (ii) the single-pass regime (for > 5) when nearly all the power is absorbed before the ray reaches the caustic. In the multi-pass regime, the absorption due to electron Landau damping is strongly peaked at the caustic.

Fig. 10 Radial profiles of P and |E//|2

for multipass absorption of a single field harmonic (m = 100, n = 450) in the cylindrical approximation.

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In the tokamak toroidal geometry intrinsic poloidal asymmetry breaks the invariance of m, causing formation of a thick stochastic layer in the ray phase space.

In the LHCD discharges on Tore Supra, a regime with stationary oscillation behavior has been observed because of the nonlinearly coupling effect of both wave-plasma interaction and turbulence suppression by the RS q profile. It is interpreted as that the current density and electron temperature profiles behave as a predator-prey system [Giruzzi, G., et al., Phys. Rev. Lett. 91 (2003) 135001].Fig, 11 Surface-s of section in the (m, )

plane for two parameter sets on Tore Supra.

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LH wave absorption in a quasi-stationary RS plasma I. Simplified dispersion relation

When the WKB approximation is valid, the wave

matrix equation is

0)],,([ 20

2 EkrKkkIkk��

(1)

wher ck /0 , K�

is the dielectric tensor. For a non-

trivial solution,

0)],,([),,( 20

2 krKkkIkkrkD��

(2)

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For LH waves, ceci

kKK yyxx , ce

pexyxyyx iiKK

2

,

izzzz iKK ,//

where

j

jpi

ce

pe2

2,

2

2

1

, 4

2,

2,2

4

2

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3

jTijpi

jTe

ce

pe vv

22

// /1 pe ,

)( //////

////

2

, vkv

fvdvK epe

izz

8

If 1/ kcn , a simplified dispersion relation can be found from the matrix equation by asymptotically expanding

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20 11

~n

onn

nn

,

41

20 11

~n

oEn

EE

With the assumption

2

0

01

n

EEKE

�

, (3)

to the lowest order, 00 EI�

, where nnII��

, so that

000 EnE

9

In first order, there arises the solubility condition:

000 nKn�

This gives the simplified dispersion relation

(electrostatic limit)

02////

24 kkkDr

For cold plasmas )0( ,

The consistency condition (3) is satisfied for 222// / cepen

(~0.5-0.12)

/)/( 222//

2pekk

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LH wave absorption regime

Strong Landau Damping Limit If the LH wave phase velocity is higher than 3.5 times the electron thermal velocity, there are too few velocity-resonant electrons to carry driven current density comparable with the ohmic current density.

n// - Upshift Boundary In the simulated quasi-stationary RS discharges, it turns out

Te01.4kev (Ti02.8kev) with 3191032.2 mne . In such conditions there is a spectral gap between the parallel LHW phase velocity and the electron thermal velocity.

][

5.6////

kevT

ckn

e

11

22// rnn

B

Bn

B

Bn

,

where n is the wave vector component perpendicular to the magnetic field. We are interested in the maximum upshift factor of //n , taking 0rn , which applies at a radial turning point. In tokamak plasmas,

BBnnn /// . By using the cold electrostatic approximation of the dispersion relation,

with x

qq cyl

ˆ , where Ra / .

)ˆ/()/(1

/00////

q

RRnn

pe

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Propagation Domain At the boundary of the propagation domain,

222

//2

nnnn (7) As the tokamak equilibrium is toroidal axisymetric, the toroidal mode number n is conserved. From the definition of n// ,

)(1//

xq

x

n

n

n

n

cyl

(8)

From Eq. (8),

2//22

2//

2

2

)1(ˆ1 n

nq

n

n

n

n

(9)

By using the cold electrostatic approximation of the dispersion relation,

with j

jpi

ce

pe2

2,

2

2

1

]/)/ˆ(1[ˆ

/)/)(ˆ1(1ˆ222

2222

00////

pe

pe

q

qq

R

Rnn

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Fig. 2 Region of LH power absorption (at t=1.0s): electron Landau damping limit (full line); n// -upshift boundary (dotted line); and boundary of the wave propagation domain ( dash line). (a) Ip = 265KA, (b) Ip = 300KA

Fig. 1 Evolution of LH wave driven current profile for the case of the LH spectrum produced with in (a) the Ip = 265kA discharge, and (b) the Ip = 300KA discharge

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Fig.3 Time evolution of the LH wave driven current profile, and the regions of LH power absorption by strong electron Landau damping at two different times: (a) t=1.5s, (b) t=0.8s

The LH wave absorption is bounded in the region defined by the strong Landau-damping limit and the boundary of wave propagation domain. This mechanism of the LH wave absorption causes interplay of the distribution of the LH wave driven current with the modification of the plasma configuration, which constitutes non-linearity in the LH wave deposition.

Due to non-linearity of the LH power absorption, the LH wave deposition position changes spontaneously, generating two distinct quasi-stationary reversed magnetic shear (RS) configura-tions.

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Fig. 4 Time evolution of (a) location of peak of the LH driven current profile, and (b) location of the minimum q in the oscillating RS discharge (full line), the two-phase RS discharge, and the stationary RS discharge (dashed line)

In a NBI heated plasma of Ip = 265kA, BT = 2.8T, and

, by controlling the radiated LH spectrum (PLH = 0

.5MW), quasi-stationary RS discharge [Q. Gao, et al. Nucl Fusion 43 (2003) 982], two-phase RS discharge [Q. Gao, et al. Phys Plasmas 12 (2005) 122507] have been obtained. When = 60 the location of the peak of LH driven current presents oscillation with irregular cyclic.

3191032.2 mne

non-predator-prey oscillation

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(a) Profiles of the ion temperature at t=1.42s (dotted line), and t=1.69s (full line) in the oscillating RS discharge. (b) Ion thermal diffusivity i versus at t=1.42s (dotted line), and t=1.69s (full line) with the corresponding thin lines indicating neo-classical value.

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Oscillations during LH ramp-up in Tore Supra

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• Lotka-Volterra equations

• Used for modelling populations in ecosystems

• 2 coupled nonlinear equations, periodic solutions

• J = predator; T = prey ?

t J J J 1JT

JT

J J JT JT

tT TT 1TJ

TJ

TT TJTJ

Predator-prey systems

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0 J t

1r

r

rr

(T )(J J LH ( j,T ) J bs( j,T ))

nT t

PLH ( J ,T )1r

r

r n( J ,T )Tr

lossescoupling with ions

( J ,T ) c1c2H(s), sshear, resistivity

1

r

r

rr

(J JLH Jbs) J J(1 bT)

PLH (J,T) 1

r

r

r n(J,T)T

r

TT(1 aJ)

dJ

dt J J(1 bT)

dT

dt TT(1 aJ)

Resistive diffusion and heat transport equations:

... have similarities with the Lotka-Volterra equations:

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Oscillations reproduced by CRONOS

coupled resistive and heat diffusion equationshave periodic solutions if:

• jLH(r) j(r)Te(r)

• e is a function of j (e.g., improved confinement for negative shear)

Outputs of the CRONOS code

• resistive diffusion• heat transport• self-cons. equilibrium• jLH(r) j(r)Te(r)

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The oscillatory behavior in the LHCD controlled discharge on HL-2A is induced by nonlinear coupling of the LH power absorption position with the plasma configuration.

According to the analysis by using the wave kinetic equation, in the weak damping regime absorption of the LH waves due to ELD is strongly peaked at caustic.

The peak location of LH driven current is determined by the intersection between the inner boundary (caustic) of the propagation domain and the ELD limit.

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The mechanism of LH power deposition region coupling the q profile and Te profile is the following:

In the tokamak plasma condition, the upper boundary of the LH wave propagation domain is reduced to

In the central plasma region ( < 0.7), is nearly a constant approximately equivalent to 18 because it is mainly dependent on square-root of the plasma density. Thus is a decreasing function of the magnetic geometry factor;

The ELD limit is a decreasing function of Te, and actually it is nearly unchanged in the oscillation since it is inversely proportional to square root of Te.

Whenever the LH driven current moves inwards, decreases due to the safety factor decreasing, the caustic boundary is elevated. The intersection between the caustic boundary and the ELD limit would move inward further

q

qRRnn u )/( 00////

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Fig. 6 Oscillation of the inverse pitch angle of the magnetic field (BT / Bp).

Fig. 5 LH power absorption region in the phasing space (, n//) defined by ELD limit and inner boundary of the wave propagation domain at t=1.42s (dotted line), and t=1.65s (full line) in the oscillating RS discharge.

Variation of the LH deposition region defined by the wave propagation condi-tion and strong Landau damping shows consistency with the plasma oscillation.

In the vicinity of min the oscillating amplitude of the inverse pitch angle of magnetic field (BT / Bp) is quite large. Therefore, it is feasible to measure the oscillation with MSE in experiments.