Discharge initiation and plasma column formation

in aspect ratio A=2 tokamak.

 

R.R. Khayrutdinov1 E.A. Azizov1, A.D. Barkalov1, G.G.Gladush1, I.L.Tajibaeva2, Ph.W.West3

 

1Troitsk, Moscow Reg., Russia2NNC, RK

3 General Atomics, USA

Introduction• To describe a current ramp-up process at tokamaks it isimportant to know place where plasma is initiated, conditions for breakdown and dynamics of plasma current rise. This is necessaryto create conditions for better plasma confinement, for minimization of breakdown voltage and to decrease a volt-second consumption. At the same time a study of physics of breakdown and plasma initiation is not complete yet. • After avalanche and creation of “quasi-neutral” plasma it isnecessary to jump through the “ionization-radiation” barrier. This is connected with phenomena that the initially plasma volumeis less than the vacuum vessel one and the plasma density also ismuch more less than neutral density. So at the beginning a “burn-through” of neutrals should take place.

Plasma initiation consists of several parts:

1. Creation of maximum poloidal flux value at the plasma region by increasing of current in the inductor to the maximum available value.

2. Programmed change of currents in the inductor and PF coils to producea required External Voltage at the plasma region (due to eddy currents at the vacuum vessel and passive structure there is delay at appearance of desired voltage at the plasma region).

3.  Formation of Field Null and Programmed Voltage in the plasma region at time of breakdown with taking into account of eddy currents.

4.   Avalanche Phase of initial plasma current formation (most part of plasma current flows along the opened magnetic surfaces) and density rise.

5.  Quasi-neutral plasma phase, plasma current increase, formation of closed magnetic surfaces.

6.  Formation of initial plasma for ramp-up phase (all neutrals are “burned-through” and magnetic surfaces inside plasma are closed).

• Phases 1-3 are well studied theoretically and experimentally and are almost standard for the all existing tokamaks and ITER. Phases 4-6are usually replaced by the only one phase, in which: • There is no plasma 2D equilibrium to be solved. Plasma is considered to be a circular; • Approximate equations for circular plasma are used to calculate magnetic field to be necessary to keep plasma at prescribed position; • 0D equations are solved for the plasma and neutrals density, electron and ion temperature and empirical equations are applied forImpurity; • Circuit equations are solved for PF coils, eddy and plasma currents.

Plasma is represented as a single filament

Today’s status of plasma initiation model

In developed 2D model of plasma initiation the each 4-6 phases are considered separately:

Avalanche phase:

- plasma region and initial values of plasma current and density are calculated in 2D region for each grid point with use of Avalanche Model;

- 2D plasma equilibrium is calculated taking into account eddy currents;

-the most part of plasma current flows along the open field lines;

Quasi-neutral phase:

-Plasma is considered to at “quasi-static” 2D equilibrium;

-Time dependent equations for PF coils and eddy currents are self-consistently solved together with plasma free boundary equilibrium problem (plasma current flows along the open and closed field lines);

-Plasma current density and total plasma current are calculated by use of 2-D diffusion of magnetic field and Grad-Shafranov equations.

-As a result plasma current density and total plasma current are calculated;

- Set of 0-D non-stationary equations are solved for neutral densities, electron and ion temperatures and for impurity species;

- as a result, initial plasma is obtained for ramp-up phase (all neutrals are “burned-through” and the magnetic surfaces inside plasma becomes closed).

• Plasma start-up at tokamaks can be divided into three phases:

• Avalanche phase, a breakdown phase and plasma current ramp-up phase.

• During the avalanche phase, electrons multiply until the ionization rate becomes 0.1.

• During breakdown phase, electric resistance due to Coulomb collisions becomes important and almost all hydrogen atoms are ionized, emitting bright light.

• Entering the ramp-up phase, joule heating increases Te, and p becomes low to obtain Ip ramp-up.

Model description

Determination of plasma initiation region (avalanche phase).

Townsend’s criteria for avalanche breakdown:

(1)

– first Townsend’s ionization,

=0.0010.1 – coefficient of secondary emission, and integral is taken along magnetic field line.

Magnetic lines, for which condition (1) is correct, will determine boundary of region for avalanche and plasma initiation.

,

11ln

dl

6 0 7 0 8 0 9 0 1 0 0 1 1 0 1 2 0 1 3 0 1 4 0

- 6 0

- 5 0

- 4 0

- 3 0

- 2 0

- 1 0

0

1 0

2 0

R , cм

Z ,с м

1

2

1 0

2 0

As an example on the Fig., the regions of plasma initiation are presented for KTM tokamak, which were calculated for different values of hydrogen pressure p and at fixed value of loop voltage. It is seen that with increasing pressure the avalanche region is increased too.

Boundary of avalanche regions in the КТМ tokamak for different values of pressure: p =1, 2, 10, 20 мP.

0-D equations for quasineutral plasma

016.02

3016.0)(

2

3

E

eeionOHee

TnPPPTn

dt

d

016.02

3016.0)(

2

3

E

iecxie

TnPPTn

dt

d

p

eie0

20e 10n

Snndt

dn

pie020

p

pe0V 10 VSnn

Vn

dt

dnV

Vp - volume of plasma region, Vv - vacuum chamber volume,

Te и Ti - electrons and ions temperatures,

POH - ohmic heating power,

P - equilibration power between electrons and ions,

Pion - neutral gas ionization losses,

Pcx - charge exchange losses, Е, Р - confinement times

EQUATIONS FOR MAGNETIC FIELD DIFFUSION

The projection of the general Ohm’s along the magnetic field is given by:

j

adj

ppttpptt jj )(

d

R

FRd

R

F

R

Rdtt

1

422 200

222

dRd

R

Rd pppp

Rd

pFRd

R

F

d

dF

RpR

Rdj tt 0

02

02 222

4

R

d

d

dF 220 F

4

R

dFRd

RR

FRdj pp

24

1

22

/4

12

2

1

42 2020

20

Rd

jR

dFRdpF

R

dF

d

dFadd

R

dF

2

0

Rd

Rd

FF

RdRd

pRd

F

0

20 2

14

2

1

kjkj

kpl jG

kext

kplk

kkjjk

j

Bd

dF

d

dFA )()(

22

tjRZRRR

02

2

2

2 1

d

dF

RpRjt

202

24

Let’s denote by index k each magnetic surface

then using relation that total

is a sum of - plasma flux

and - external flux

pl

ext

kkext

kplk

Grad-Shafranov equation:

Plasma current, electrons temperature, minor radius and feedback current behavior during breakdown simulation

Algorithm of solution • As first step in the iteration scheme the 2D free boundary equilibrium equation on rectangular grid is solved at given external PF and passive structure currents self-consistently withcircuit equations for these currents. Plasma can flow at open and closed field lines.• As second step, after finding equilibrium configuration,averaging is carried out to find all metric coefficients andcoefficients for matrix equation. Value for electron temperature used tocalculate plasma conductivity is taken from results of solution of 0D setof transport equations. As a result plasma current density and totalplasma current Ip are calculated.• As third step set of 0D equations is solved. Value of Ip is used to calculate Ohm’s heating of electrons. Average value of minorplasma radius a is calculated as Sp/, where Sp is area occupied byplasma. Plasma pressure profile pin this case is equal to zero. As aresult of solution new value of electron temperature is obtained, whichis used in the step 2.

• Steps 1, 2 and 3 are iterated until required accuracy is achieved.• Then next time step follows.• Test simulation for developed model has been carried

out for KTM tokamak, which includes all three stages of

plasma breakdown.• Examples of magnetic configuration for KTM tokamak at

the beginning of initiation and at the end of breakdown are presented below at Figs.

Poloidal configuration of КТМ tokamak at the beginning of plasma initiation, plasma current is Ip=3kA.

Poloidal configuration of КТМ tokamak at the end of plasma breakdown, plasma current is Ip=5

0kA.

Conclusion. 1.Plasma initiation model with 2D plasma equilibrium solver has been

developed. 2. All necessary equations have been derived.3. Preliminary stage with use of 2D equilibrium and 0D transport is

programmed and tested. 4. Next stage will consist of in developing NEW code which will be

integration of TRANSMAK [1] code and this 2D breakdown model [2].

1. V.A. Beliakov, V.I. Vasiliev, K.I. Lobanov, L.P. Makarova, Л.П. A.B. Mineev, VII Int. Conf. of Thermonuclear Reactor Engineering Problems, 28 - 31 Oct. 2002 г. St.-Petersburg, p. 178.

2. E.A. Azizov, A.D. Barkalov, G.G. Gladush, R.R. Khayrutdinov. Int. Conf. and School on Plasma Phys. and Controlled Fusion, Alushta, Ukraine, Sept. 16 – 21, 2002.