21st IAEA Fusion Energy Conference

16 - 21 October 2006Chengdu, China

TH/P6-19

Improvement of plasma confinement due to ion and electron heating at the

edge of tokamak

Shurygin R.V., Mavrin A.A.RRC “Kurchatov Institute”, Moscow, Russian Federation

Contact e-mail: alemavrin@yandex.ru

Abstract.

The behavior of turbulent fluxes in the vicinity of a resonace point m/n=qres(x) in the plane wall plasma layer in tokamak is studied by numericaly analyzing the nonlinear MHD equations in four-field electrostatical model. It is shown that the injected auxiliary power into as ion so electron components produces the reduction of the turbulent fluxes, which looks like a L-H transition. Such behavior of the flux is found to be due to the stabilizing effect of the ExB drift velocity, which increases if ion or electron temperature increases.

The behavior of turbulent fluxes in the vicinity of the resonant point m/n = q(xres) in the plane plasma layer (x,y) near tokamak wall is studied numerically.

The simulation was aimed to reveal the dependence of the turbulent particle flux on power heating injected in to ions and electrons inside the edge plasma layer.

Our simulation carries purely theoretical interest. There is not associating with experiment on the present stage of computations.

R

0 x 1

a

Geometry of the problem

In this paper, results are presented from numerical simulations of the parameters of turbulent plasma dynamics in the vicinity of a resonant point m/n=q(xres) in a plane edge plasma layer in a tokamak for modes of the same helicity; this makes it possible to reduce the three-dimensional problem to a two-dimensional one. The simulations were carried out by solving reduced nonlinear two-fluid Braginskii equations. The set of reduced MHD equations, in which it is assumed that pe~pi, should be supplemented with an equation describing a generalized vortex and containing the electron and ion diamagnetic drift velocities.

We use the set of reduced two-fluid Braginskii equations presented in [1-3]. Assuming that the longitudinal ion velocity is zero, u||= 0, and ignoring the thermal current, we can obtain the following set of four-field {, n, pe, pi} nonlinear MHD equations describing the behavior of a collisional plasma at the edge of tokamak. Numerical simulation was based on the system of equations for vorticity, density, ion and electron pressures.

equation for electron pressure:

equation for ion pressure:

),()]()([2

5

2

5)(

2

3 |||| xSWTpepen

Jpp

t

peeieee

ee

e

eE qQV

).(

)]())(

([2

5

2

5)(

2

3 ||||

xS

WTpn

ppep

en

Jpp

t

p

i

eiiiie

ii

ii

i

E

q

QV

Equation for vorticity:

,)(2

|||| WppB

BJ

c

BW

t

Wieci

ci

bVE

equation for density:

,)(|||| nDenpe

Jn

t

ne

QVE

,|||||| en

pJ e

,2

),(B

B

e

cpenW i

bQ

,ie,j,|||| jjjj pp jq

,][

2B

c

BVE ,, ||2

2

2

2

Byx

B

)(3Wei ieei

i

e ppm

m is the exchange term, with ei being the electron-ion

collision frequency. Here, zeB )1(0

0 R

xB , where B0=const is the toroidal

tokamak magnetic field, R0 is the major radius of the torus, x=r/a, r is the radial coordinate and a is the minor radius.

Note that to calculate the electrostatic potential we use the equation for generalized vorticity, which includes both the electric drift and the ion diamagnetic drift.

Using the averaged equation for the radial ion momentum and ignoring the small inertial terms, we arrive at the relationship between the radial electric field, the poloidal ion velocity, and the radial ion pressure gradient:

x

p

enxU

B

xcExV i

oyE

1

)()(

)(

Computations were carried out over the region 0<x<1 for parameters

close of the DIII-D tokamak: R0=170cm, a=67cm, B0=2T, q=4. The width of plane layer is d=3cm.

Se,i are auxiliary heating power injected in the electron or in the ion component. The values Se,i(x) have been imposed as

}]2.0/)2.0[(exp{ 4,0, xqS ieie

0,02 0,03 0,04 0,05 0,06 0,07 0,08 0,09 0,10 0,110,00

0,01

0,02

0,03

0,04

0,05

0,06

0,07

0,08

0,09

0,10

0,11

0,12

0,13

qoi

Q

e

Qi

Fig.1 The dependence of the turbulent flux of particles on the ion heating parameter q0i.

In Fig.1 the dependence of the turbulent particle flux on the ion heating parameter q0i is demonstrated. Our simulation shows that the increase of q0i reduces the turbulent flux of particles flux and of the heat for both ions and electrons. The resulting effect looks like L-H transition.

Such a behavior of the fluxes is found to result from the known stabilizing effect of the shear ExB drift velocity, VE(x), which rises under the ion heating through an increase of the ion diamagnetic velocity.

The equation for the ion pressure, p0i, can easily be obtained by averaging equation for ion pressure over y and by supplementing the right-hand sides of the resulting equations with the necessary dissipative and source terms:

20

20 )(

x

pxS

x

Q

t

p ioii

ii

,

where Qi=<piVx> - turbulent ion heat flux, yVX

.

0,0 0,2 0,4 0,6 0,8 1,00,00

0,02

0,04

0,06

0,08

0,10

dQi/dx<0!

Qi(x) 0.03

0.04 0.05 0.055 0.06 0.07

xFig.2 The radial dependence of the turbulent flux of the ion heat on the ion

heating parameter q0i.

It is obviously that at x=0 area an increase or a decrease of p0i depends on the sign of the derivative of ion flux Qi. In some cases, Qi(0)/ x can change its sign with the increase of q0i. Our simulation shows such change of the sign when got changes from q0i=0.06 to q0i=0.07 (see Fig.2). The value of derivative Qi(0)/ x becomes negative. It implies on occurrence of the additional heating and of increase of p0i due to turbulent ion heat flux. In this case, the positive feedback is produced:

ExBExB

iiii

ExBExBiiii

VV

ppx

Q

x

QVVppSS 0000 ]0

)0(0

)0([

One could say that fluxes sharply decrease with an increase of heating, and the event a L-H transition appears. The simulations show that the mentioned positive feedback takes place only for plasma parameters in very narrow range.

Fig.3 Profiles of the time-averaged ion temperature for different ion heating parameter q0i.

0,0 0,2 0,4 0,6 0,8 1,00

20

40

60

80

100

120

Ti0(x) , eV

q0i = 0.055

q0i = 0.06

q0i = 0.07

X

Fig.4 Profiles of the time-averaged the diamagnetic ion drift velocity for different ion heating parameter q0i.

0,0 0,2 0,4 0,6 0,8 1,0

-16

-14

-12

-10

-8

-6

-4

-2

0

2

X

Vp(x) , km/sec

q0i = 0.055

q0i = 0.06

q0i = 0.07

Fig.5 Profiles of the time-averaged poloidal ion velocity for different ion heating parameter q0i.

0,0 0,2 0,4 0,6 0,8 1,0-1,5

-1,0

-0,5

0,0

0,5

1,0

1,5

2,0

X

U0y

(x)

q0i = 0.055

q0i = 0.06

q0i = 0.07

Fig.6 Profiles of the time-averaged ion pressure for different ion heating parameter q0i.

0,0 0,2 0,4 0,6 0,8 1,00,0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

X

pi0(x)

q0i = 0.055

q0i = 0.06

q0i = 0.07

0,000 0,015 0,030 0,045 0,060 0,075 0,090 0,1050,00

0,01

0,02

0,03

0,04

0,05

0,06

0,07

0,08

0,09

0,10

0,11

0,12

0,13

q0e

Q

e

Qi

Fig.7 The dependence of the turbulent flux of particles on the electron heating parameter q0e.

The similar effect of the transition to the improved confinement was found in the numerical calculations for the case with electron power injected into plasma edge (see Fig.7). It can be explained by two different mechanisms, which can result in suppression of the fluctuations. The first one is an increase of the longitudinal current V||e~J||Te

3/2 due to the increase of the electron temperature. As a consequence, the dissipation power increases, and turbulent particle flux reduces. From the other hand, the electron heating results in the increase of the ion temperature due to Column collisions. In its turn, the growing ion pressure generates the enhanced ExB drift velocity, which provides an extra reduction of the turbulent flux. The analysis show, that the first effect is more important. The power injection with q0e,i ~0.1 in the certain plasma component ( both electrons and ions ) results in the rising of the temperature of the component up to 160-170eV.

0 100 200 300 400 5000,00

0,05

0,10

0,15

0,20

0,25

0,30

0,35

q0i = 0.5

<VE

2>

t , s

Fig.8 Evolution of the turbulent flux of particle and the kinetic energy of the ExB drift when the ion heating parameter q0i at the layer boundary increases from 0 to 0.5 at the time t = 250 µs.

0 100 200 300 400 5000,00

0,05

0,10

0,15

0,20

0,25

0,30

0,35

q0e

= 0.5

t , s

<VE

2>

Fig.9 Evolution of the turbulent flux of particle and the kinetic energy of the ExB drift when the electron heating parameter q0e at the layer boundary increases from 0 to 0.5 at the time t = 250 µs.

Summary

Simulation shows that the additional power heating injected in to ions result in improvement of confinement due to the known stabilizing effect of the shear ExB drift velocity. The resulting effect looks like L-H transition. Also simulation shows that the additional power heating injected in to electrons result in improvement of confinement, but due to other mechanisms. The first one is an increase of the longitudinal current V||e ~ J|| Te

3/2 due to the increase of the electron temperature. As a consequence, the dissipation power increases, and turbulent particle flux reduces. From the other hand, the electron heating results in the increase of the ion temperature due to Column collisions. In its turn, the growing ion pressure generates the enhanced ExB drift velocity, which provides an extra reduction of the turbulent flux. The analysis show, that the first effect is more important.

References

1. Zeiler A., Drake J.F., Rogers B. // Phys.Plasmas.1997.V.4.P.2134.2. Simakov A.N., Catto P. J. // Phys.Plasmas.2003.V.10.P.4744.4. Shurygin R. V. // Fiz. Plazmy 30, 387 (2004) [Plasma Phys. Rep. 30, 353 (2004)].3. Voitsekhovitch I., Garbet X., Benkadda S., Beyer P., Figarella C. // Phys.Plasmas. 2002. V.9. P.4671.