Presentation Puerta

8/8/2019 Presentation Puerta

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Poloidal Magnetic Field Topology for

Tokamaks with Current Holes

Julio Puerta, Pablo Martn and Enrique Castro

Departamento de Fsica, Universidad Simn Bolvar,

Apdo. 89000,Caracas 1080A, Venezuela.

USBLABORATORIO DE FSICA DE PLASMA

*) [email protected]


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The appearance of hole currents [1-3] in tokamaks seems to be very

important in plasma confinement and on-set of instabilities, and this

paper is devoted to study the topology changes of poloidal magnetic

fields in tokamaks. In order to determine these fields different models

for current profiles can be considered. It seems to us, that one of the

best analytic description is given by V. Yavorskij et. al. [3], which has

been chosen for the calculations here performed. Suitable analytic

equations for the family of magnetic field surfaces with triangularityand Shafranov shift are written down here. The topology of the

magnetic field determines the amount of trapped particles in the

generalized mirror type magnetic field configurations [4,5]. Here it is

found that the number of maximums and minimums of Bp depends

mainly on triangularity, but the pattern is also depending of the

existence or not of hole currents. Our calculations allow to compare

the topology of configurations of similar parameters, but with and

without hole currents. These differences are study for configurations

with equal ellipticity but changing the triangularity parameters.

Positive and negative triangularities are considered and compared

between them.

Abstract


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1.- INTRODUCTION

Linear treatment of equilibrium in Tokomaks is in ourknowledge well developed by Russian authors to get the famousGrad-Shafranov equations. Now, several types of heating or beaminjection and rf heating induce toroidal and poloidal plasma flowsand indeed non-linear terms become important. In the low velocity

approximation in axis-simmetry Tokamaks, a theory of non-linearequilibrium has been developed a new kind of Grad-Shafranov Vequation including triangularity and ellipticity[1,2,3]

In general it is very difficult the non-linear treatment due tothe appearance of two complex differential equations like Grad-

Shafranov and Bernoulli types. Now considering the H-modeoperation when turbulence and vorticity are very low [7,8] it isjustifiable to treat the non-linear situation as a first approximationin the low vorticity limit, in order to calculate the poloidalmagnetic field topology in Tokamaks in the hollow current limitand compare for the case no hole current exist.


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Now is useful to point out that we use the orthogonal set of naturalcoordinates as defined elsewhere [fig.1] to make the calculation ofthe poloidal magnetic field. As is it well known, this coordinate

system form a natural basis for better development of transporttheory and stability theory due to the fact, that one of thecoordinates lies in the magnetic surface and the another one, isorthogonal and therefore, in equilibrium, parallel to the pressuregradient.


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Figure 1: Cross section of the tokamak magnetic surface showing thereference curves for the coordinates used in the text.


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2.- Theory

The non-linear MHD equations for equilibrium is

In this equation only the main term of the pressure tensor has beenconsidered. Using this equations and the vorticity defined by,

21c

v v = v v j B pV V v rrr r r

v[ ! v

r r

(1)

(2)


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and following the procedure as in the linear case we found

where

Now as demonstrated elsewhere in our basis coordinates

* 2 ( )

4

Fc

Rx

H ]

] VT H( !

* 2 2

4

2

*

2

( ) ( )( ) ( )

4

( )

cc I

I

z

T

] ]] V ]

T ] ]x x

( ! x x

x x x

(!

x x x

%

( )I RBN] !

( ) ( )F F] W!% %

(3)

(4)


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3.) Poloidal Magnetic Field

Now it is well known that ellipticity and triangularity are

important parameters for tokamaks plasmas because their

affect in general the efficiency of this facilities. Here thetechnique is prescribing and in order to

calculate the flux function using the G-S equation. In our

case we consider the magnetic field as given and

calculating all parameters using the knowledge on the

surfaces.

( )I ]


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On the other hand the analytical form of the along the middle

line through the minor axis is also given in terms ofa

VP !

4.) Poloidal current density equations. Using Amperes law

in the linear MHD approximation we get

(5)

(6)pJC

!v1

Now considering stationary equilibrium

we obtain

(7)NN JtJJ p !

4B J

c

T v !

r r


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Where it is well known, where is no component of orthogonal

to the magnetic surfaces.In the study state equilibrium we have

0J !

r

(8)

and considering axisymmetry we get

0p pJ Sin

J Js R

J

W

K

UO

x ! x %

(9)


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Now Sin(Uis defined as

RSin

s WU

x ! x %

(10)

and we can rewrite (9) in the from

p Pp

J J RJs R s

W

W

Ox x x x %(11)


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where we used here the notation of the new coordinates defined

in previous paper. Equation (11) can be also writes

1

p

p

RJR

J s

W

W

O ! x x %

(12)

integrating (12) along and arbitrary magnetic surface yield

11 1 1

0

exp[ ] ( )

sp p

p

R J R J J ds s

R RW WO Q! | (13)


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if we consider the reference curve

1 1

4I

p

s p

s

BB J

cN

TO

W

x ! x %

(16)

equation (11) and (12) can be formerly solved and written in theform

11

0

exp

sp

p s

R BB ds

RO W

!

- (17)

this equation allows us the calculation of for any prints without the

poloidal flux-function .


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5. Calculation without hole

Now in order to show something interesting numerical result, we

choose elliptic surfaces with shift and triangularity. The toroidalcurrent density along the central line (z = 0) is [29]

0

12 ; = ; =1, 2, 3, 4....

1

j

j

a

N RR R

W W

RW

!

-

%

%(18)

where is defined by

1 mR RW ! % (19)


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With is each point on the - reference curve which here

coincide with the outer point in each magnetic surface, and

is the radius of the minor magnetic surfaces

W

mR

1R

2 2 00

2 20 0

0

, cos2

, 24

m

aTR R a

aE Tz E a sin sin sin

P U P P U P

P U P U P U U

! (

! (20)

where

21 0, 0 mR R a

a

P U P P

VP

! ! (

!

0 mR R a( ! 2

1 0mR R aW P P! | (%

(21)


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Now putting in terms of and , we have sQ P U

2 2 2

0 0 0

32 2 22 2

0 0

4 2 cos(3 ),

2 cos( ) cos(2 ) 4 1 cos(2 sin ( )s

o

E T T R

a E T T

P P UP U

P U P U P U U

!

-

(22)

and

0

, exp , d

U

Q P U Q P U U

!

- (23)

where

'2

,1

m m m

m Q P U P

U Px x

! x x

(24)


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and defined by the slope of the magnetic field line

z

mr

P

P

U

U

xx

! xx

(25)

Now from (20) and (21), we determine in the form

0

1 3 22 2

0

4 1

2s

o

TR Z Z R

E a T R Z

U UU U UU

U U

PO

P P

! ! -

(26)

and therefore ( along the reference line) can becalculated if is prescribed for this line. In fact, using

equation (16) we obtain a differential equation that can be

solved for and combining this result with the value of

calculated elsewhere [19] we achieve

1p

Bp

B

JN

JN pB


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1

1 1,

exp[ ( , ) ] ,, , ,

p

p

B R Rd

B R R

P U P PQ P U U Q P U

P U P U P U! !

When the form of is not know, and can be determined

(27)

pB JN


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In figure 3 it is shown the dimensionless poloidal field for , with

and without hole. It is good to see, that in the case with a hole

current profile a deeper depression in the poloidal magnetic field

profile appear grater than for the case without the hole. That meansa better confinement will be achieved. Similar behavior is observed

in figures 4 and 5, but in the case of the figure 4 a better

confinement is achieved with the hole current profile when the

ellipticity goes higher, that shows the importance of this parameters.


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Fig.2 Toroidal density current ellipticity k and triangularits

along the major radius through the minor magnetic


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Fig.3 Dimensionless poloidal magnetic field around with hole

and with out a magnetic surfaces. The value = r correspond to

the inners point of the magnetic surface and = 0 is the outward

point.


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Fig.4 Dimensionless poloidal magnetic field around a

magnetic surfaces with and without hole for different

ellipticisties.


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Fig.5 Dimensionless poloidal magnetic field around a magnetic

surfaces with and without hole for different tringularities.


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REFERENCES

1.- G. T. A. Huysmans, T. C. Hender, N. C. Hawkes, and X.

Litaudon, Phys. Rev. Lett. 87 (2001) 245002-1.

2.- T. Ozeki and JT-60 team, Plasma Phys. Control Fusion 45

(2

003) 6453.- V. Yavorskij, V. Goloborodko, K. Schoepf, S.E. Sharapov,

C.D. Challis, S. Reznikand D. Stork, Nucl. Fusion 43 (2003)1077

4.- N. I. Grishanov, C. A. Acevedo, and A. S. de Assis, PlasmaPhys. Controlled Fusion 41 (1999) 1791

5.- P. Martn, M. G. Haines and E. Castro, Phys. Plasmas 12(2005) 082506


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