Basic tokamak variables

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I ntroduction to Nuclear Fusion

Prof. Dr. Yong-Su Na

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Basic tokamak variables

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• High <n

e

>/n

_GW

• High b

_N

• High H

98

(y,2)

• Pulse length

s bar keVs

m T

n 

_E

 3  10

²¹ ^³

 5 

→ Stability, confinement issue

Objectives of the Tokamak Operation

3

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Plasma

(Plasma pressure)

Plasma Equilibrium, Stability and Transport

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fishbones (q=1)

I_p (MA)

P_NBI (MW)

b_N

H₉₈ (y,2)

<n_e>/n_GW D_a

4xl_i

#14521

• No sawteeth, good confinement, and b

_N

~ 3.5, T

_i

~ T

_e

, <n

_e

>/n

_GW

~ 0.88, averaged over 3.6 seconds (~ 50 

_E

).

Objectives of the Tokamak Operation

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6

fishbones (q=1)

P_nbi (MW)

b_N

H₉₈ (y,2)

<n_e>/n_GW 4xl_i

~ 18 MW / m²  6 MW / m²

outer divertor

inner divertor

Small ELMs

(type II)

I_p (MA) ^#14521

D_a

Objectives of the Tokamak Operation

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Tokamak

http://www.splung.com/content/sid/5/page/fusion

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• Cylindrical and local coordinates for a tokamak

- Major radius: R₀, minor radius: a - Aspect ratio: Λ = R0/a ~ 3-5

- Inverse aspect ratio: ε = a/R0 ~ 0.2~0.35

ex) KSTAR: Λ = 3.6, ε = 0.28, ITER: Λ = 3.1, ε = 0.32

Basic Tokamak Variables

A. A. Harms et al, “Principles of Fusion Energy”, World Scientific (2000)

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Basic Tokamak Variables

x

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Basic Tokamak Variables

• VEST (Versatile Experiment Spherical Torus)

- Basic research on a compact, high b ST (Spherical Torus) with elongated chamber in partial solenoid configuration - Study on advanced tokamak scenario including

innovative start-up, non-inductive H&CD, high performance, and innovative divertor concept, etc

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Basic Tokamak Variables

• VEST (Versatile Experiment Spherical Torus)

- Basic research on a compact, high b ST (Spherical Torus) with elongated chamber in partial solenoid configuration - Study on advanced tokamak scenario including

innovative start-up, non-inductive H&CD, high performance, and innovative divertor concept, etc

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- Elongation: κ - Triangularity: δ

Basic Tokamak Variables

• Plasma equilibrium parameters

κ ↑

δ ↑

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• Elongation

a

 b

 a

d c

2  



• Triangularity

Basic Tokamak Variables

• Plasma equilibrium parameters

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Strike point

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• Separation of plasma from wall by a limiter and a divertor

Basic Tokamak Variables

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Major Radius, R₀ Minor Radius, a Plasma Current, I_P Elongation, _x Triangularity, _x Toroidal Field, B₀ Pulse Length

Fuel

Parameters

1.8 m 0.5 m 2.0 MA 2.0 0.8 3.5 T 300 s H, D

KSTAR

6.2 m 2.0 m 15 MA 1.85 0.5 5.3 T 500 s D, T

ITER

- Plasma shape

Basic Tokamak Variables

• Plasma equilibrium parameters

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Basic Tokamak Variables

• Normalized beta – stability limit

p t t

N

I

b aB b 

0 2

/ 2  b

t

B

p 

 

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Basic Tokamak Variables

• Energy confinement time

power heating

applied

energy stored







 



in in

E

P

W t

P W

 W

- To predict the performance of future devices, the energy confinement time is one of the most important parameter.

- Since tokamak transport is anomalous, empirical scaling laws for energy confinement are necessary.

- Empirical scaling laws: regression analysis from available experimental database.

a

a

a a

a

 



_th^fit_,_E

 CI

^I

B

^B

P

^P

n

ⁿ

M

^M

R

^R

in engineering variables





_th_,_E

~ a

²

/

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Basic Tokamak Variables

78 . 0 58 . 0 97 . 1 19 . 0 41 . 0 69 . 0 15 . 0 93 . 0 )

2 (

98 ^y,

0 . 0562 I B P n M R

a

IPB

th,E

 

 

^

• Energy confinement time enhancement factor

98( ,2) ,

) 2 , (

98 IPB y

E th

E

H

y



 

• Energy confinement time

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Basic Tokamak Variables

• Greenwald density

a

2

n

_GW

I

^p

 

- As the limit is approached, the plasma becomes increasingly susceptible to disruption and data become sparser.

M. Greenwald et al, NF 28 199 (1988): one of the most cited paper in NF Martin Greenwald, PPCF 44 R27 (2002)

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• Safety factor q = number of toroidal orbits per poloidal orbit

Magnetic field lines Magnetic flux surfaces

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



B

B   





number of toroidal windings number of poloidal windings

 q









B B R

 r

 2

- Rotational transform:

Δθ? when ΔΦ = 2π











 

 

 rB

RB

R B

r B

2 2 2



 

 









B rd B

Rd 



Basic Tokamak Variables

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q-profile

3

0 0.5 0.95

Normalised radius 1

5

2

Advanced scenario Baseline scenario

q₉₅

4 Hybrid scenario

Basic Tokamak Variables

• Safety factor q = number of toroidal orbits per poloidal orbit

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dr dq q r r

s ( ) 

- Measuring the change in pitch angle of a magnetic field line from one flux surface to the next

- Playing an important role in stabilizing MHD instabilities, particularly those driven by the pressure gradient:

A perturbation aligned with B(r) will,

at a point with increased minor radial distance r+dr, encounter field lines at a different angle which again will vary as the perturbation

grows to another distance r+dr’.

Any helically resonant instabilities are thus radially localised.

Basic Tokamak Variables

• Magnetic shear

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→ Force balance

→ Ampere‘s law

→ Closed magnetic field lines

B J p  







J B  



0







 0



 B 

• Plasma Equilibrium

kinetic pressure

balanced by JxB (Lorentz) force

, 2

nqB B v

_D _p

p



_

    

, 2

,

B

p v B

q n v

q n

J 

_i _i _D _i



_e _e _D _e

  

 

Diamagnetic current

 0



 p J 

 0



 p B 

- If B is applied, plasma equilibrium can be built by itself due to induction of diamagnetic current.

induced by the pressure gradient:

causing a decrease in B → diamagnetism

B

ion

←∇p x

B J p  







Magnetic Flux Surface

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 0



 p B 

p B

J     

- In a tokamak configurations with confined plasmas the magnetic lines lie on a set of nested toroidal surfaces called flux surfaces.

- Pressure is constant along a magnetic field line.

- Magnetic lines lie in surfaces of constant pressure.

- Flux surfaces are surfaces of constant pressure.

- The current lines lie on surfaces of constant pressure.

 0



 p J 

Magnetic line

Magnetic axis

Magnetic Flux Surface

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http://blog.daum.net/eco205sky/12854349 25 http://www.freewebs.com/weatherexplorer/apps/blog/show/286262

http://www.econym.demon.co.uk/isotut/isobars.htm

Isobar: p = const.

Magnetic Flux Surface

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Spherical tokamaks

Start operation

Strongly shaped Divertor

High field

Superconducting Compression DT operation

Tokamak