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
OUTLINE
 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
 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) ~ 4.5-5.0
 Two 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.
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.
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.
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.
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.
The time evolution of the electron temperature after off-axis ECRH
switch-off. Mark the very slow change of central temperature.
8
MATHEMATICAL MODEL
Equation for the electron temperature for transient process
after ECRH switch-on:
3
1  
T  1 
nT  
rnueT   POH  Q  PEC ,
 rn e

2 t
r r 
r  r r
T
( r  0, t )  0, T ( r  1, t )  T0 , t  tS
(1)
r
T ( r, t  tS )  T S ( r ), 0  r  1.
Equation for the steady state electron temperature T S (r )
3  S S 1   S S T S  1 
S
 rn  e
 
n T 
rn S ueS T S  POH
 QS ,
2 t
r r 
r  r r


T S
( r  0, t )  0, T S ( r  1, t )  T0 ,
r


(2)
9
In Eq. (1) and Eq. (2) the total heat flux W consists of the diffusive
and the convective terms:
W   n e
T
 nu e T ,
r
(3)
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:
~
~
s
s
n(r, t )  n (r)  n (r, t ),T (r, t )  T (r)  T (r, t ). (4)
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.
~
~
n
T

,
S
S
n
T
~
n
 1.
S
n
(5)
10
Finally we can write the equation for variation temperature
which describes the transient process after ECRH switch-on:
~
3  S ~ 1   S T  1 
~
 rn  e
 
n T 
rn S ueT  PEC 
2 t
r r 
r  r r

P
OH




S
  Q  Q S  
 POH
S
 1 
1   S

T
S
S
S
S




rn




rn
u

u
T
,
e
e
e
e


r r 
r  r r
~
T
~
( r  0, t )  0, T ( r  1, t )  0, t  t S
r
~
T ( r, t  t S )  0, 0  r  1.


( 6)
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.
11
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:
 e ( r, t )  
S
e (r) 
S
~
 e ( r ), ue ( r, t )  ue ( r )  u~e ( r ),
t  tS ~
t  tS
0,
0,
~
 e (r)   ~
, ue ( r )   ~
.
  e ( r ), t  tS
ue ( r ), t  tS
( 7)
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:
~
3  S ~ 1   S S ~ T  1 
~
S
S
~
n T 
rn (  e   e )  
rn (ue  ue )T 

2 t
r r 
r  r r

 

 PEC  POH  Q   P~  Pu~ ,
~
T
~
( r  0, t )  0, T ( r  1, t )  0, t  tS
r
~
T ( r, t  tS )  0, 0  r  1.
1   S ~ T S
 rn  e
P~ 
r r 
r
(8)

1 
, Pu~ 
rn S u~eT S .
r r



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.
13
STATEMENT 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.
The discrepancy functional can be written as:


 
M N
2
1M N
~
k
k 2
J   i T ( ri , tk )  f i /   i f i , (9)
2 k 1 i 1
k 1 i 1
where i are weight factors, which are selected in accordance with the
error of the measurement for every channel.
The inverse problem is formulated as follows:
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).
14
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 lownoise experimental data. In this report we consider the simplified
version of the inverse problem.
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.
15
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). 16
Fig.1a, Т-10, #32918.
Time dependences of the electron temperature variation after on-axis
ECRH switch-on at different radii.
17
Fig.1b, Т-10, #32918.
Time dependences of the variation electron temperature after on-axis
ECRH switch-on at different radii.
18
Fig.1c, Т-10, #32918.
Time dependences of the variation electron temperature after on-axis
ECRH switch-on at different radii.
19
Fig.2-3, Т-10, #32918.
Solution of the inverse problem: e is the heat diffusivity; ue is the jump of the
convective velocity; Pec is ECRH profile; Poh is the change of the Ohmic heating;
Pu is the heat sink which it is depended from the jump of the convective velocity. 20
Fig.0, Т-10, #32917.
Radial profiles of the electron temperature variation after off-axis ECRH
switch-off at different time instants.
21
Fig.1a, Т-10, #32917.
Time dependences of the variation electron temperature after off-axis
ECRH switch-off at different radii.
22
Fig.1b, Т-10, #32917.
Time dependences of the variation electron temperature after off-axis
ECRH switch-off at different radii.
23
Fig.1c, Т-10, #32917.
Time dependences of the variation electron temperature after off-axis
ECRH switch-off at different radii.
24
Fig.2-3, Т-10, #32917.
Solution of the inverse problem: e is the heat diffusivity; ue is the jump of the
convective velocity; Pec is ECRH profile; Poh is the change of the Ohmic heating;
Pu is the heat source which it is depended from the jump of the convective velocity.25
Signal D.
Time dependences of the experimental signal D after ECRH switchon (t=560 ms, t=750 ms) and ECRH switch-off (t=820 ms, t=950 ms).
26
CONCLUSION
The transient process after ECRH switch-on (-off) shows
the following characteristic features:
 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 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