Fyzika tokamaků 1: Úvod, opakování 1

Tokamak PhysicsJan Mlynář

3. Tokamak field equilibrium

Solovjev solution of the Grad-Shafranov equation, Shafranov shift, plasma shape, poloidal beta, vertical field for equilibrium, Pfirsch-Schlüter current

Tokamak Physics 3: Tokamak field equilibrium 2

Grad-Shafranov equation

22

2 2 2002

14 0

2

*

R I R pR R R z

where( )dp

pd

Normalised coordinates

Inverse aspect ratio

0

0

R Rx

az

yaa

R

Unit flux

Dimensionless profiles

1

( ), ( )

1, , ( ), ( ), ( , )x y ( , ), ,x y q ... Shafranov shift, see later

Iterative numerical solutions:In: Out:

Tokamak Physics 3

Soloviev solution

Soloviev solved the special case of the linearised Grad-Shafranov equation:

3: Tokamak field equilibrium

220

02 20 0

1

2 2 4

II FE p p

* 2E FR

Take

i.e. Grad-Shafranov eq.

Analytical solution:

22 2 2 218

2C DR E F D R z

Dimensionless:

2 2 2

22

1 1 1 (2 )2 4

yx x x x

Tokamak Physics 4

Role of dimensionless quantities

3: Tokamak field equilibrium

2 2 2

22

1 1 1 (2 )2 4

yx x x x

2

2

0 1 ok

0 1 elongation4

0 elliptical is a measure of triangularity

0 0 12

y x

x y

y x x

21 1 Shafranov shift

1For example, 0.16

3z

features the up-down symmetry

(but not a HFS-LFS symmetry!!)

Soloviev solution of G-S equation

Tokamak Physics 5

Plasma shape

3: Tokamak field equilibrium

0 ( ) cos

sin

R R r

z

2

( ) cosmm

r S r m

0 cos( arcsin sin )

sin

R R a

z a

2

3

21

4sin

S

rS

r

d

a

Usual form:

elongation:

triangularity:

General form:

i.e. any general shape is decomposed in Fourier series

(and no higher m terms)

Tokamak Physics 6

Poloidal beta

3: Tokamak field equilibrium

2 20 0

8

2pa

pdS dS pdS

B I

01:2a

IB

a

Circular cross-section:

22

20

11

a

p

a

dBr dr

draB

Large aspect ratio

1 :-)a

R

220 02 2

1p

a

B R

B r

2p

explains why tokamaks cannot

reach very high beta

Tokamak Physics 7

Flux shift in circular cross-section

3: Tokamak field equilibrium

0 cos

sin

R R r

z r

0 1( ) ( , )r r

Displaced flux surface:

0 0 0

00

( ) ( cos )

( ) cos

R R r

Rr

220 0

0 020 0 00

22

2

r Bdr p p d

dr rR B

Substituting the Grad-Shafranov equation, integrating…

Tokamak Physics 8

Shafranov shift, vacuum mg. field

internal inductance

3: Tokamak field equilibrium

220 0

0 020 0 00

22

2

r Bdr p p d

dr rR B

separatrix:0 ( ) 0, ( ) 0p a a

0

2p ia

d al

dr R

2

02 2

2a

i

B rdr

la B

0

0 (0) Shafranov shifts

d

dr

Vacuum magnetic field

00

1( )

( ) ( ) 1 cos

12i

p

B aR r

aB a B a

R

l

Tokamak Physics 9

Internal inductance

3: Tokamak field equilibrium

Tokamak Physics 10

Vertical field for equilibrium

3: Tokamak field equilibrium

21

2p p pW L Ip pL I

221

2 2p p

h p p

I LF L I

R R

0

8ln 2

2

insideoutside

ip

lRL R

a

20 8

ln 12 2p i

h

I lRF

a

!

in outp B B hF F F F F

2 22pF p a!

2 pF RI B

0 8 1ln

4 2pI R

BR a

Hoop force

Self-inductance outsideand inside the plasma

Equilibrium:

Tokamak Physics 11

Pfirsch-Schlüter current

3: Tokamak field equilibrium

Total current density

2

p

B

B

j

20 p

B B

j j j

. .

21 : cosP S

a q pj j

R B r

diamagnetic current

Pfirsch-Schlüter current is the

component of the current that is

parallel to the magnetic field

line. It short-cuts the plasma

polarisation which would occur

due to gradB and curvature

drifts.

S

Tokamak Physics 12

Pfirsch-Schlüter current

3: Tokamak field equilibrium

Shafranov shift

.

• In tokamaks, Shafranov shift results from the Grad-Shafranov equation that describes equilibrium. • It is shown that for this shift to appear, vertical field is required. • The vertical field is balanced in plasma by the Pfirsch-Schlüter current. • This current is identical to the current that results due to the shift of the particle trajectory in a toroidal system with field helicity

Tokamak Physics 13

Components of the tokamak field

3: Tokamak field equilibrium

Tokamak Physics 14

Total field, vertical stability

3: Tokamak field equilibrium

Tokamak Physics 15

Tokamak discharge

3: Tokamak field equilibrium