0 5 100

5

10

15

critical gradients(II)i second (II)

e

i-ITB

valuesused in

ourcriterion

zp - deviation

z0 - distance between

critical gradients

valuesused in

JETcriterion

zp

z0

First (I) and

electrons

ions

hea

t fl

uxe

s i,

e (a

.u.)

-aT'/T

0,0 0,2 0,4 0,6 0,8 1,00

10

20

A=5, =1, =0, T-10A=3, =1.6, =0.3, JETA=1.4, =1.75, =0.325, MAST

T-10

JET

MAST

Tc=

-RT

c'/T

c

0,0 0,2 0,4 0,6 0,8 1,00

1

2

3

4

5 Critical gradients,normalised to minor radius

MAST, A=1.5

JET, A=3

T-10, A=5

Cri

tica

l gra

die

nt

aT

c'/T

c

0,0 0,2 0,4 0,6 0,8 1,00

2

4

6

8

10

12

14First critical gradient

DIII-D, #89943, t=1.7 sec

-R/L

Te=

-RT

c'/T

c

r/a

0.0 0.2 0.4 0.6 0.8 1.00

2

4

6

qexp

DIII-D #89943t=1.7 s

qcalc

q

0.0 0.2 0.4 0.6 0.8

0

2

4

6

Shear

Z0i

DIII-D, #89943, t=1.7 sec

0.0 0.2 0.4 0.6 0.8

0

2

4

6

Shear

Z0i

1. The first critical gradient

We find the first critical gradient by the canonical profiles theory. The canonical profile for the function = 1/q (denoted below as с) can be found by the solution of the Euler equation for the free plasma energy functional [1]

  2G c2/ + (/2) / ((1/ V) (VGc)) = Cc/V.(1)

Here: index S means the plasma boundary,

iс = 1/V /(G Vс) is the dimensionless current density,

V is the plasma volume, V= V/,

G = R2<(grad )2/r2> is the metric coefficient.

The solution of Eq. (1) and the constants С and are determined by the following four boundary conditions:

c(0) = 0 ~ 1, c(0) = 0, c(max) = S,

X [ic/(2G c)]S = G(a)1/2 S /0 (2)

The first dimensionless critical gradients for the temperature and density are following:

Tc = R/LTc -RTc/Tc = - 2/3 R ic/ic,

nc -Rnc/n = - 1/3 R ic/ic. (3)