Fyzika tokamaků 1: Úvod, opakování 1

Tokamak PhysicsJan Mlynář

2. Magnetic field, Grad-Shafranov Equation

Basic quantities, equilibrium, field line Hamiltonian, rotational transform, axisymmetric tokamak, q profiles, Grad-Shafranov equation.

Tokamak Physics 2: Mg. field, Grad-Shafranov equation 2

Revision of basic quantities

Magnetic field (magnetic induction) B

Magnetic flux d B S

0 0 0

0

0

t

t

EB j

BE

B

E

Ampère’s law

Faraday’s law

"the curl operator"

Maxwell’sequations

0 : ( )G B A B A A

A .... Magnetic vector potentialG .... Gauge (~ particular choice)

Faraday’s law G

t t t

A AE

Tokamak Physics 2: Mg. field, Grad-Shafranov equation 3

Field line, equilibrium

Magnetic field line

yx z

x y z

dldl dldds

ds B B B

xB

“nested surfaces” Equilibrium:

0 0, 0p p p F j B B j

Axisymmetry nested mg. flux surfaces

Magnetic field lines and j lie on the magnetic fluxsurfaces (but can not overlap otherwise the pressure gradient would be zero!)

Suppose that never vanishes

Tokamak Physics 2: Mg. field, Grad-Shafranov equation 4

Mg field in arbitrary coordinates

2 2G

AAll functions of x, t

1

2

B A

/ 2B B

coordinates , ,

Jacobian of the transformation , , x

1

2 B

x x x

Magnetic field lines:d d d

ds

B B B

Tokamak Physics 2: Mg. field, Grad-Shafranov equation 5

Magnetic field Hamiltonian

is the magnetic flux in the direction:2

d d d B S B

From the equations of magnetic field lines:

d

d

d

d

is Hamiltonian, generalised momentum,

generalised coordinate and generalised time

Tokamak Physics 2: Mg. field, Grad-Shafranov equation 6

Rotational transform, q

Safety factor

Integraton

Transformation

( ) & ( )

( , , ) x x

gives complete topology

If canonical transformation leading to (axisymmetry), then ( )

0 0,d

d

dq

d

rotational transform /

Tokamak Physics 2: Mg. field, Grad-Shafranov equation 7

Axisymmetric tokamak

/ 2 / 2 B

p B B T TRB B B

1

R

( ) 0 B

d d d B l l

2

Bd d d d dl

RB

B l l

2

Bdq dl

d RB

for circular cross-sectionrB

qRB

Tokamak Physics 2: Mg. field, Grad-Shafranov equation 8

Poloidal coordinates

Field line is straight if 1

0

l Bq dl

RB

Tokamak Physics 2: Mg. field, Grad-Shafranov equation 9

q profiles

Ampère’s law 02 ( )prB I r

Circular plasma:

2 20

( ) 2( )

r

r

I rj j r rdr

r r

0

2

r

Bq

R j

0

0

a

a

q j

q j

in particular

model: 2

0 20

1 1aqrj j

a q

divertor: 95instead, ( 0.95 )aq q r a

Tokamak Physics 2: Mg. field, Grad-Shafranov equation 10

R, , z coordinates

, , , cylindrical coordinates R zB B R z

( )10 0R zRB B

R R z

B

0

(R, z) 2

1

2

1 1

2

2

R

p

z

T

A R

AB

z R z BRA

BR R R RI

B BR

0 1

2 2

T p

I

B

B B

Tokamak Physics 2: Mg. field, Grad-Shafranov equation 11

Grad-Shafranov equation

0

(a) 0 0

(b) 0 ( & ) 0

p

p pp

z R R zRB RBp p

pz R R z

j B

B

j B j

We shall work in cylindrical coordinates and assume axisymmetric field

p as well as RB are functions of only.

0 ( )2

IRB I I

(c) component?R(1)z z

pj B j B

R

00

1 1 ( )

2z

B Ij

R R R

B j

Tokamak Physics 2: Mg. field, Grad-Shafranov equation 12

Grad-Shafranov equation2 *

20 0 0

1 1 1 1

2 2R zB B

jz R R z R R R R

2* 2

2 2

1 1elliptic operator R R

R R R z R

*

0

0

1 ( ) ( )0

2 2

& 2 , ,2

z

z

I pB B

R R R R

I dpRB B p

R R d

* 2 2 20 04 0I I R p

two arbitrary profiles I(), p() ; boundary condition

From (1):

const.a

Notice: The form on the title slide (copy from Wesson) is different as many authors

use a different definition of flux, while here we defined 1

2d

B S d B S

Tokamak Physics 2: Mg. field, Grad-Shafranov equation 13

Grad-Shafranov equation

Something to think about:Why is it not similar to a magnetic dipole field?

Next lecture:

Solovjev solution of the Grad-Shafranov equation, Shafranov shift, plasma shape, poloidal beta, flux shift in the circular cross-section, vacuum magnetic field, vertical field for equilibrium, Pfirsch-Schlüter current

*02 2 ' 'R j j Rp B I

2 ' 'Rp I j e B