Title: 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
2Abstract
- 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.
31.- 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.
5Figure 1 Cross section of the tokamak magnetic
surface showing the reference curves for the
coordinates used in the text.
62.- 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)
7and following the procedure as in the linear case
we found where Now as
demonstrated elsewhere in our basis
coordinates
(3)
(4)
83.) 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.
9On 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)
10Where 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)
11Now Sin(q) is defined as
(10)
and we can rewrite (9) in the from
(11)
12where 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)
13when 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)
14if 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 .
155. 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
29
(18)
where is defined by
(19)
16With 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)
17Now putting in terms of and
, we have
(22)
and
(23)
where
(24)
18and 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
206.- 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)
21and
(30)
On the other hand we have also following
definitions
(31)
22In 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.
23Fig.2 Toroidal density current ellipticity k and
triangularits along the major radius through the
minor magnetic
24Fig.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.
25Fig.4 Dimensionless poloidal magnetic field
around a magnetic surfaces with and without hole
for different ellipticisties.
26Fig.5 Dimensionless poloidal magnetic field
around a magnetic surfaces with and without hole
for different tringularities.
27REFERENCES 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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