THE BEHAVIOUR OF THE HEAT CONDUCTIVITY COEFFICIENT AND THE HEAT CONVECTIVE VELOCITY AFTER ECRH SWITCH-ON (-OFF)

IN T-10

THE BEHAVIOUR OF THE HEAT CONDUCTIVITY COEFFICIENT AND THE HEAT CONVECTIVE VELOCITY AFTER ECRH SWITCH-ON (-OFF)

IN T-10

V. F. Andreev, Yu. N. Dnestrovskij,

K. A. Razumova, A. V. Sushkov

Nuclear Fusion Institute, RRC “Kurchatov Institute”,

Moscow, Russia.

1

O U T L I N EO U T L I N E

Introduction.Experimental results with ECRH switch-on

or switch-off in T-10 tokamak.Mathematical model.Numerical method of the solution.Results of the calculation. Conclusion.

2

The conditions of the experiment

The conditions of the experiment

Total plasma current - Ip= 175 [kA].Average density - <n>=1.4 [1019 1/m3].Toroidal magnetic field - Bz=2.35 [T].Safety factor q(a) ~ 4.5-5.0Two gyrotrons (140 GHz) in ~0.4 - to

suppress saw-tooth oscillations.One gyrotron (130 GHz) in the center of

plasma.

3

Fig.2, T-10, #32920.Fig.2, T-10, #32920.

The radial profiles of the electron temperature after on-axis ECRH switch-on (two off-axis gyrotrons suppress saw-tooth oscillations).The radial profiles of the electron temperature after on-axis ECRH switch-on (two off-axis gyrotrons suppress saw-tooth oscillations). 4

Fig.2, T-10, #32917.Fig.2, T-10, #32917.

The radial profiles of the electron temperature after on-axis ECRH switch-on (two off-axis gyrotrons suppress saw-tooth oscillations).The radial profiles of the electron temperature after on-axis ECRH switch-on (two off-axis gyrotrons suppress saw-tooth oscillations). 5

Fig.1, T-10, #32917.Fig.1, T-10, #32917.

The time evolution of the temperature after on-axis ECRH switch-on at different radii (two off-axis gyrotrons suppress saw-tooth oscillations).The time evolution of the temperature after on-axis ECRH switch-on at different radii (two off-axis gyrotrons suppress saw-tooth oscillations).6

Fig.2, T-10, #32917.Fig.2, T-10, #32917.

The evolution of the electron temperature profiles after off-axis ECRH switch-off. Mark the very slow change of central temperature.

The evolution of the electron temperature profiles after off-axis ECRH switch-off. Mark the very slow change of central temperature. 7

Fig.1, T-10, #32917.Fig.1, T-10, #32917.

The time evolution of the electron temperature after off-axis ECRH switch-off. Mark the very slow change of central temperature.The time evolution of the electron temperature after off-axis ECRH switch-off. Mark the very slow change of central temperature. 8

MATHEMATICAL MODELMATHEMATICAL MODEL

,),1( ,0),0(

(2) ,11

2

3

0TtrTtrr

T

QPTurnrrr

Trn

rrTn

t

SS

SSOH

SSe

SS

Se

SSS

)(rT S

.10 ),(),(

(1) ,),1( ,0),0(

,11

2

3

0

rrTttrT

ttTtrTtrr

T

PQPTrnurrr

Trn

rrnT

t

SS

S

ECOHee

9

Equation for the steady state electron temperature

Equation for the electron temperature for transient process after ECRH switch-on:

In Eq. (1) and Eq. (2) the total heat flux W consists of the diffusive and the convective terms:

(3) ,Tnur

TnW ee

10

We represent the electron density n(r,t) and the temperature T(r,t) for transient process as a sum of the steady state values and theirs variations:

)4( .),(~

)(),(),,(~)(),( trTrsTtrTtrnrsntrn After ECRH switch-on the main power is deposited into the electrons, therefore the relative variation of the temperature is grater than the relative variation of the density. Here we consider the transient process after ECRH switch-on (10-20 ms), therefore we can neglect the variation of the electron density.

)5( .1~

,~~

SSS n

n

T

T

n

n

.10 ,0),(~

,0),1(~

,0),0(~

)6( ,11

~1~

1~2

3

rttrT

tttrTtrr

T

Tuurnrrr

Trn

rr

QQPP

PTurnrrr

Trn

rrTn

t

S

S

SSee

SS

See

S

SSOHOH

ECeS

eSS

Finally we can write the equation for variation temperature which describes the transient process after ECRH switch-on:

11

In Eq. (6) transport coefficients and ECRH power profile are unknown. Therefore we should define the specific model for each of them before formulation of the inverse problem for reconstruction of transport coefficients.

NON-LOCAL MODEL OF TRANSPORT COEFFICIENTS

NON-LOCAL MODEL OF TRANSPORT COEFFICIENTS

We suppose that transport coefficients are non-local functions of plasma parameters. This means that transport coefficients can change in any point of the plasma radius even if local values of the density and the temperature are not changed in the given point. We represent the heat diffusivity and the convective velocity for transient process after ECRH switch-on in the following form:

)7( . ),(~

,0)(~ ,

),(~ ,0

)(~

),(~)(),( ),(~)(),(

Se

Se

Se

Se

eSeee

See

ttru

ttru

ttr

ttr

rurutrurrtr

In other words we assume that the variation of the heat diffusivity and the convective velocity are changed faster than characteristic time of the transient process. 12

In this case the equation for variation electron temperature has following form:

13

The Eq. (8) contains terms which are determined steady state temperature and the variation of transport coefficients, but its are not depended on the variation of the electron temperature. Note, that the Eq. (8) is not linearized equation.

.~1 ,~1

.10 ,0),(~

)8( ,0),1(~

,0),0(~

,

~)~(

1~

)~(1~

2

3

~~

~~

Se

Su

S

eS

S

S

uOHEC

eSe

Se

Se

SS

Turnrr

Pr

Trn

rrP

rttrT

tttrTtrr

T

PPQPP

Tuurnrrr

Trn

rrTn

t

STATEMENT OF THE INVERSE PROBLEMSTATEMENT OF THE INVERSE PROBLEM

If we know experimental values of temperature variations, measured in N radial points ri (i=1,…, N) and in M temporal points tk (k=1,…, M), and also

we know some integral plasma parameters: major and minor plasma radii, the total plasma current, Ohmic and ECRH power.

)9( ,/),(~

2

1

1 1

2

1 1

2

M

k

N

i

kii

M

k

N

i

kikii fftrTJ

where i are weight factors, which are selected in accordance with the

error of the measurement for every channel.

We should find the ECRH power profile PEC, the heat diffusivity , the convective velocity u, variations of the heat diffusivity and the convective velocity, so the solution of the equation (8) provides the minimum of the functional (9).

The inverse problem is formulated as follows:The inverse problem is formulated as follows:

The discrepancy functional can be written as:The discrepancy functional can be written as:

14

1) We assume that the total heat flux W in the steady state has diffusive component only, i.e. u() =0.

2) We assume that, during transient process after ECRH is switched on or off, only the convective velocity changes, i.e. ()=0 and u()0. Therefore we have only three unknown functions.

3) We set () and u() as polynomial functions.

4) We set ECR power profile as gaussian one, the center of the gaussian function and half-width w as unknown parameters.

5) Note we usually use 1000-2000 experimental points for the electron temperature.

Thus, it is necessary to reconstruct five unknown function, namely, (), u(), their variations (), u() and ECR power profile Pec() It is a fairly difficult task because it requires low-noise experimental data. In this report we consider the simplified version of the inverse problem.

15

Fig.0, Т-10, #32918.Fig.0, Т-10, #32918.

The radial profiles of the temperature variation after on-axis ECRH switch-on (two off-axis gyrotrons suppress saw-tooth oscillations).The radial profiles of the temperature variation after on-axis ECRH switch-on (two off-axis gyrotrons suppress saw-tooth oscillations). 16

Fig.1a, Т-10, #32918.Fig.1a, Т-10, #32918.

Time dependences of the electron temperature variation after on-axis ECRH switch-on at different radii.

Time dependences of the electron temperature variation after on-axis ECRH switch-on at different radii. 17

Fig.1b, Т-10, #32918.Fig.1b, Т-10, #32918.

Time dependences of the variation electron temperature after on-axis ECRH switch-on at different radii.

Time dependences of the variation electron temperature after on-axis ECRH switch-on at different radii. 18

Fig.1c, Т-10, #32918.Fig.1c, Т-10, #32918.

Time dependences of the variation electron temperature after on-axis ECRH switch-on at different radii.

Time dependences of the variation electron temperature after on-axis ECRH switch-on at different radii. 19

Fig.2-3, Т-10, #32918.Fig.2-3, Т-10, #32918.

Solution of the inverse problem: Solution of the inverse problem: ee is the heat diffusivity; u is the heat diffusivity; uee is the jump of the is the jump of the

convective velocity; Pec is ECRH profile; convective velocity; Pec is ECRH profile; Poh is the change of the Ohmic heating; Poh is the change of the Ohmic heating; Pu is the heat sink which it is depended from the jump of the convective velocity.Pu is the heat sink which it is depended from the jump of the convective velocity.20

Fig.0, Т-10, #32917.Fig.0, Т-10, #32917.

Radial profiles of the electron temperature variation after off-axis ECRH switch-off at different time instants.

Radial profiles of the electron temperature variation after off-axis ECRH switch-off at different time instants. 21

Fig.1a, Т-10, #32917.Fig.1a, Т-10, #32917.

Time dependences of the variation electron temperature after off-axis ECRH switch-off at different radii.

Time dependences of the variation electron temperature after off-axis ECRH switch-off at different radii. 22

Fig.1b, Т-10, #32917.Fig.1b, Т-10, #32917.

Time dependences of the variation electron temperature after off-axis ECRH switch-off at different radii.

Time dependences of the variation electron temperature after off-axis ECRH switch-off at different radii. 23

Fig.1c, Т-10, #32917.Fig.1c, Т-10, #32917.

Time dependences of the variation electron temperature after off-axis ECRH switch-off at different radii.

Time dependences of the variation electron temperature after off-axis ECRH switch-off at different radii. 24

Fig.2-3, Т-10, #32917.Fig.2-3, Т-10, #32917.

Solution of the inverse problem: Solution of the inverse problem: ee is the heat diffusivity; u is the heat diffusivity; uee is the jump of the is the jump of the

convective velocity; Pec is ECRH profile; convective velocity; Pec is ECRH profile; Poh is the change of the Ohmic heating; Poh is the change of the Ohmic heating; Pu is the heat source which it is depended from the jump of the convective velocity.Pu is the heat source which it is depended from the jump of the convective velocity.25

Signal D.Signal D.

Time dependences of the experimental signal DTime dependences of the experimental signal D after ECRH switch- after ECRH switch-on (t=560 ms, t=750 ms) and ECRH switch-off (t=820 ms, t=950 ms).on (t=560 ms, t=750 ms) and ECRH switch-off (t=820 ms, t=950 ms).26

CONCLUSIONCONCLUSION

The variation of the electron temperature is localized, when on-axis ECRH follows after off-axis ECRH.

The delay of the change of the central electron temperature after off-axis ECRH switch-off on the level of 20 ms is seen.

There is a zone with very low heat diffusivity coefficient ~0.02-0.05 м2/s. The size of this zone is close to 3.5-4.0 см.

The jump of the convective velocity allows us to describe the experimental data with sufficient accuracy. But so far we can not affirm strongly that these experiments should be explained by the change of the convective velocity only.

The transient process after ECRH switch-on (-off) shows the following characteristic features:

The work was supported by Nuclear Science and Technology Department of Minatom RF and Russian Basic Research Foundation Grant N00-15-96536 and N01-02-17483.

27