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. *) email@example.com
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. , 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.
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 ellipticity[1,2,3] 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.
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.
Figure 1: Cross section of the tokamak magnetic surface showing the reference curves for the coordinates used in the text.
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)
and following the procedure as in the linear case we found where Now as demonstrated elsewhere in our basis coordinates (3) (4)
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.
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 Ampere’s law in the linear MHD approximation we get (5) Now considering stationary equilibrium (6) we obtain (7)
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)
Now Sin(q)is defined as (10) and we can rewrite (9) in the from (11)
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)
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)
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 .
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  (18) where is defined by (19)
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)
Now putting in terms of and , we have (22) and (23) where (24)
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  we achieve
(27) When the form of is not know, and can be determined
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 al[V.Ya] (28) where (29)
and (30) On the other hand we have also following definitions (31)
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.
Fig.2 Toroidal density current ellipticity k and triangularits along the major radius through the minor magnetic
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.
Fig.4 Dimensionless poloidal magnetic field around a magnetic surfaces with and without hole for different ellipticisties.
Fig.5 Dimensionless poloidal magnetic field around a magnetic surfaces with and without hole for different tringularities.
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