Poloidal Magnetic Field Topology for Tokamaks with Current Holes

1
Poloidal Magnetic Field Topology for Tokamaks
with Current Holes
USB LABORATORIO DE FÍSICA DE PLASMA

•  
• Julio Puerta, Pablo Martín and Enrique Castro
• Departamento de Física, Universidad Simón
  Bolívar, Apdo. 89000,
• Caracas 1080A, Venezuela.

) jpuerta_at_usb.ve
2
Abstract

• 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 triangularity and
  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.

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1.- INTRODUCTION

• Linear treatment of equilibrium in Tokomaks is
  in our knowledge well developed by Russian
  authors to get the famous Grad-Shafranov
  equations. Now, several types of heating or beam
  injection and rf heating induce toroidal and
  poloidal plasma flows and indeed non-linear
  terms become important. In the low velocity
  approximation in axis-simmetry Tokamaks, a theory
  of non-linear equilibrium has been developed a
  new kind of Grad-Shafranov V equation including
  triangularity and ellipticity1,2,3
•  
• In general it is very difficult the non-linear
  treatment due to the appearance of two complex
  differential equations like Grad-Shafranov and
  Bernoulli types. Now considering the H-mode
  operation when turbulence and vorticity are very
  low 7,8 it is justifiable to treat the
  non-linear situation as a first approximation in
  the low vorticity limit, in order to calculate
  the poloidal magnetic field topology in Tokamaks
  in the hollow current limit and compare for the
  case no hole current exist.

4

• Now is useful to point out that we use the
  orthogonal set of natural coordinates as defined
  elsewhere fig.1 to make the calculation of the
  poloidal magnetic field. As is it well known,
  this coordinate system form a natural basis for
  better development of transport theory and
  stability theory due to the fact, that one of the
  coordinates lies in the magnetic surface and the
  another one, is orthogonal and therefore, in
  equilibrium, parallel to the pressure gradient.

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Figure 1 Cross section of the tokamak magnetic
surface showing the reference 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 been
considered. Using this equations and the
vorticity defined by,
(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
(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 the
  technique 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.

 

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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 of
4.) Poloidal current density equations. Using
Amperes law in the linear MHD approximation we
get
(5)
Now considering stationary equilibrium
(6)
we obtain
(7)
10

Where it is well known, where is no component of
orthogonal to the magnetic surfaces. In the study
state equilibrium we have
(8)
and considering axisymmetry we get
(9)
11

Now Sin(q) is defined as
(10)
and we can rewrite (9) in the from
(11)
12

where we used here the notation of the new
coordinates defined in previous paper. Equation
(11) can be also writes
(12)
integrating (12) along and arbitrary magnetic
surface yield
(13)
13

when in the radius of each point in the
G-reference curve. Now considering equation
(14)
and from the Ampere Law, it is easy to obtain
(15)
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if we consider the reference curve

(16)
equation (11) and (12) can be formerly solved and
written in the form
(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 toroidal
current density along the central line (z 0) is
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(18)
where is defined by
(19)
16
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

(20)
where
(21)
17
Now putting in terms of and
, we have

(22)
and
(23)
where
(24)
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and defined by the slope of the magnetic field
line
(25)
Now from (20) and (21), we determine in the form
(26)
and therefore ( along the reference
line) can be calculated 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
19

(27)
When the form of is not know, and can
be determined
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6.- Calculation with hole   For the case we have
a have a hollow current profile we use for the
toroidal density current the model proposed by V.
Yavorskij et alV.Ya

(28)
where
(29)
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and
(30)
On the other hand we have also following
definitions
(31)
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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 means a 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 (2003)
645 3.- V. Yavorskij, V. Goloborodko, K.
Schoepf, S.E. Sharapov, C.D. Challis, S. Reznik
and D. Stork, Nucl. Fusion 43 (2003) 1077 4.-
N. I. Grishanov, C. A. Acevedo, and A. S. de
Assis, Plasma Phys. Controlled Fusion 41 (1999)
1791 5.- P. Martín, M. G. Haines and E. Castro,
Phys. Plasmas 12 (2005) 082506
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