Fyzika tokamaků 1: Úvod, opakování 1

Tokamak PhysicsJan Mlynář

6. Transport: Theory and experiments

Bootstrap current, revision of transport, dimensional analysis, Buckingham theorem, dimensionless quantities, scale invariance of equations, Bohm and Gyro-Bohm scaling,identity and similarity experiments, etc.

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Bootstrap current

7: Transport: Theory and experiment

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Bootstrap current

7: Transport: Theory and experiment

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Bootstrap current

7: Transport: Theory and experiment

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Bootstrap current

"I was still a couple of miles above the clouds when it broke, and with such violence I fell to the ground that I found myself stunned, and in a hole nine fathoms under the grass, when I recovered, hardly knowing how to get out again. Looking down, I observed that I had on a pair of boots with exceptionally sturdy straps. Grasping them firmly, I pulled with all my might. Soon I had hoist myself to the top and stepped out on terra firma without further ado." Campaigns and Adventures of Baron Munchausen (Baron Prášil in Czech), 1786.

7: Transport: Theory and experiment

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Revision of transport

7: Transport: Theory and experiment

What can be done if “first principles” (models based on plasma theory)

won’t fit the data?

- dimensional analysis

- scale invariance

-33 1 3 [ Wm ]

2 2e e

e e he ei e e e

T ndn T P R P r D T

dt r r r r

-33 1 3 [ Wm ]

2 2i i

i i hi cx ei i i i

T ndn T P L P r D T

dt r r r r

radiation conductive loss convective loss

charge exchange

electrons:

ions:

Neoclassical transport

However, in experiments:

3/2 2. . ( . .) 2

( )P S G S L ei

nD q r

B T

2p

E

B

nT

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Dimensional analysis

7: Transport: Theory and experiment

Courtesy of C C Petty fromthe General Atomics, USA

Buckingham - theorem

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Buckingham theorem

7: Transport: Theory and experiment

Boris Kadomtsev 1928-1998

Kadomtsev applied the theorem on tokamaks

in 1975. Published in a brief and simple article.

Choice of dimensional quantities defining state of the system:

BT, Bp, T, n, mi , R, a

+ „fundamental“: me , e, m0 (c)

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Dimensional analysis

7: Transport: Theory and experiment

But there are only 7 quatnitites + 3 ratios:

eT

p i

mBaq

R B m

4 independent quantities BT, T, n, a

4 dimensionless independent parameters required

to describe the tokamak plasma

Typical choice:

02

3/2

* normalised Larmor radius

2 beta

* normalised collisionalityv

* normalised Debye length

Li

T b

D

r

anT

B

Rq

a

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Dimensional analysis

Physical limitations: Does

the set Q1, ... Qn describe

the system completely?

7: Transport: Theory and experiment

1 2

1 2 1 2

( , , ) 0

( , , , , , ) 0n

p p p n

F Q Q Q

F r r r

e.g. following the Buckingham recipe:

independentdimensionless

ratios

1 2 3 4 *

1where 1 0 1

2

a b c d

a b c d

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Scale invariance

7: Transport: Theory and experiment

Complementary to dimensional analyses• take equations that describe the system• find transformations for which the equations are invariant

e.g. the Vlasov (collisionless) equation + quasineutrality:

3

( ) 0

0

j jj

j

j jj

f ef f

t m

e f d

vv E v B

v

3 invariant transformations:

1 2

1 1

(1)

(2) v v t

(3)

j jf f

t E E B B

t t x x E E B B

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Scale invariance

7: Transport: Theory and experiment

From Kadomtsev, we expect: p q r sE n B T a

3 2 31v

3n fd nT m fd v v

substituting to

2 2

(1) 01

(2) 2 1

(3) 0

r

E

pT

r qB a B

s q

Physical limitations:does the system of equations describe the system correctly ?

Collisional high-beta equation is more realistic than Vlasov equation

results in 54

1( )E F Ba

B

For constant n, T that is, constant

i.e. “identity experiments” 5 54 4const. (only)EBa a

*, *,

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Combining scale invariance and dimensional analysis

7: Transport: Theory and experiment

4 dimensional independent parameters

n, B, T, a 4 dimensionless independent parameters

e.g. **, *, ,D

(notice: all doubts concerning completeness of this choice remain valid)

* * *r s q pE n B T a

It can be speculated that * has no influence on transport, i.e. = 0

a constraint on r, s, q, p known as the “Kadomtsev condition”:

1( *, , *)E F

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Identity and similarity experiments

7: Transport: Theory and experiment

Identical & “ratios” (see below) * *, ,

“Identity experiments” are based on the same idea as

the “wind-tunnel” experiments (remember the exercise

with the Reynolds number, from Feynman part II)

Two fixed parameters, and one changing (usually the *)

Identity experiments:

Similarity experiments:

However, some authors tend to use “identity” for equal dimensional values.

Notice that the ratios must be fixed as well!

, , , , , ,eeff

i s

vTq a Z M

T c

M …. Mach number… this can cause difficulties due to different plasma rotation in different facilities !

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Identity and similarity experiments

7: Transport: Theory and experiment

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Bohm and Gyro-Bohm scaling

7: Transport: Theory and experiment

? Scaling with * ?

1. Bohm scaling

also known as “Bohm-like diffusion”

i.e. independent of *

fluctuations are driven by gradient lengths

2

( , *)

where 16

E B

Be

F

eBa

T

2. Gyro-Bohm scaling( , *)

*B

E F

2

3

* Bohm scaling

* gyro-Bohm scaling

E

E

B

B

Experiment decides!

Notice: „Global scaling“ need not be realistic.

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Bohm and Gyro-Bohm scaling

7: Transport: Theory and experiment

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Pedestal profiles – no atomic physics

7: Transport: Theory and experiment

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Hidden physics

7: Transport: Theory and experiment

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Regression analysis

7: Transport: Theory and experiment

can help to distinguish the transport processes, e.g.

breaks the zonal flows

But stabilises TEM (~ bananas)

Important : driven by electrostatic or magnetic turbulences?

It is significant, as the former option may mean

higher than predicted Q in ITER.

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Role of scale experiments

7: Transport: Theory and experiment

( *)E ii

( )E( )E

( )E

ei

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Promise for ITER ?

7: Transport: Theory and experiment

Extrapolation of scale invariance has large error ITER has to be built

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Conclusions

7: Transport: Theory and experiment

To remember

Experiments can clarify whether certain processes are

important for plasma dynamics, via their scaling

properties (e.g. atomic physics proved irrelevant for the

edge dynamics)

On the other hand, sometimes the experiments do not

scale as expected

this means there is some hidden physics

(e.g. influence of ripple)

Multiparameter regression analysis is not consistent

with separated scaling experiments

extrapolations have large error and

tend to be conservative (crossfingers !!)

1)

2)

3)