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Compact toroidal magnetized plasmas are an important part of the world’s magnetic fusion and plasma science efforts. These devices can play an integral role in the development of magnetic fusion as a viable commercial energy source, and in our understanding of plasma instabilities, particle and energy transport, and magnetic field transport. In this paper, we are developing a numerical program to study the magnetic dynamo or relaxation of CT’s characterized by arbitrary tight aspect ratio (major to minor radii of tokamak) and arbitrary cross-sections (Multi-pinch and D-Shaped). The lowest ZFE’s has been calculated through the Taylor’s relaxed state (force-free) toroidal plasmas equation. For ZFE’s, we use the toroidal flux vanishing boundary condition along the whole boundary of tokamaks. Several runs of the program for various wave numbers showed that ZFE was very insensitive to the choice of wave numbers. Besides, the CT’s poloidal magnetic field topologies are well represented. It was very interesting to check our methods for the cases when aspect ratio tends to unity (zero tokamak whole). A good fulfillment of the boundary condition is achieved.

Compact toroidal magnetized plasmas became recently an important part of the world’s magnetic fusion and plasma science efforts. These devices can play an integral role in the development of magnetic fusion as a viable commercial energy source, and in our understanding of plasma instabilities, particle and energy transport, and magnetic field transport.

Compact toroid (CT) configuration is of simpler construction than the conventional tokamak and has important advantages due to the novel physics properties of its low aspect ratio. It could afford smaller, less-expensive fusion power plants, so it is becoming increasingly popular. There are several worldwide types of compact toroids as spheromaks: field reversed configuration (FRC), and other compact toroidal devices which had achieved remarkable successful results via fusion plasma parameters.

These devices can play an integral role in the development of magnetic fusion as a viable commercial energy source, and in our understanding of plasma instabilities, particle and energy transport, and magnetic field transport, but still there is a need to develop an experimental programs to study some of these phenomena with emphasis on transport, the relaxation state, non-classical ion acceleration and magnetic reconnection [

For reversed field pinch and spheromak configuration, magnetic helicity has been believed to play an important role as a global invariant in the process of self-organization and relaxation for magnetized plasma in CT. Accordingly, we have to go back to J. B. Taylor model [

to study and investigate the relaxed state (force-free) for compact to kamaks. The theory has been successfully applied to many fusion related plasmas, especially CT’s.

The steady state equilibrium of CT is characterized by

Compact tori are typically characterized by low

Equation (1) describes the relaxed toroidal plasmas to a force-free configuration of minimum energy. It was a representation of the convential Beltrami equilibrium, which also restricted to only force-free equilibrium.

If

which defines current limitation in tooidal systems.

Solutions of (1) are perceived to describe the gross features of FRC, spheromak configurations, current limitation in toroidal plasmas and others. Two parameters are determining the relaxed state for toroidal system with

a perfectly conducting boundary. Firstly, is the magnetic helicity

toroidal flux

To select the correct minimum energy solution of Equations (1) and (2), one can consider the eigenvalues of this system [

The aim of present work is to reply to the questions which may be arise about the applicability of collocation methods [

We calculated the lowest eigenvalues

In order to determine the minimum eigenvalues, we apply to equations (1) and (2) a method of solving equations of this type―collocation method [

According to Taylor [

(i) The boundary condition (which is applied to all cases):

(ii) The zero field condition:

(iii) The zero flux condition:

For a non-axisymmetric mode the boundary condition (i) alone determine a zero-flux eigenvalue, the condition

For axisymmetric mode, the boundary condition (i) alone is not sufficient to determine an eigenvalue. The zero flux condition (iii) must be explicitly imposed to determine axisymmetric eigen-modes. The vanishing field condition (ii) is really an independent problem. It determines its own set of eigenvalues (the zero-field eigenvalues). The connection with the zero-flux problem arises because some (but not all) zero0field eigenvalues, defined by (ii), exactly coincide with some (but not all) zero-flux eigenvalues defined by (iii).

The meridional cross-section in the r, z plane of the toroidal metallic vessel wall shall be described by an arbitrary curve

From (1), the following expressions linking the magnetic field components are obtained:

As the toroidally non-axisymmetric states are not so evident in the experiments, we concentrate mainly on axisymmetric or

Solution of Equations (3) gives the magnetic field components for axisymmetric toroidal plasmas as:

where,

It should be mentioned here that, for the case of a straight cylinder (or for infinite aspect ratio of toroidal plasma), the coefficient

Inserting (5) into (2), the zero flux boundary condition for an axisymmetric container of finite aspect ratio and arbitrary cross-section reads [

Plasma is an inhomogeneous anisotropic nonlinear medium and modeling of plasma devices and the solutions of the associated mathematical problems are not a simple task. Many plasma machines are of toroidal geometry, hence their associated physical processes are described by partial differential equations with space dependent coefficients. Often these equations can not be separated in ordinary differential equations and one has to take refuge with numerical methods. The large computer CODES have been developed and worked quite well. However these codes showed disadvantages via small and compact tokamaks (characterized by great curvature or small aspect ratio). Should we arrive that a partial differential equations is not separable in a special coordinate system, then they will be solved in a coordinate system in which it is separable. This offers the great advantages that the boundary curves can be described by another coordinate systems or by a set of collocation points and that this description is absolutely independent of the coordinate system in which the partial differential equations is separable.

The collocation method [

For tokamak with multi-pinch cross section, the meridional curve is given by:

Thus,

This shows that, by increasing aspect ratio α, the zero flux eigenvalue is slightly increases.

All our results have been calculated for a multi-pinch cross section given by the curve (8). This curve, along which the boundary condition is fulfilled, must also be described by the zero flux boundary condition (7). For C = 0, this gives the poloidal magnetic field lines along the container. Besides, for C has different values, this gives the poloidal magnetic flux lines inside the container. The toroidal shift (Shafranov’s shift) is clearly seen and depends on the aspect ratio, as per

The collocation points

Besides,

Let us consider now a tokamak with D-Shaped cross section with dimensions: R = 1.2, a = 0.6, α = R/a = 2, and collocation points given in table 2.

20 | 5 | 2 | 1.5 | 1.2 | 1 | α = R/a |
---|---|---|---|---|---|---|

2.05119 | 2.05064 | 2.04733 | 2.04409 | 2.03967 | 2.0338 | μ |

1.80 | 1.76 | 1.48 | 1.20 | 0.92 | 0.60 | 0.60 | 0.60 | r_{i} |
---|---|---|---|---|---|---|---|---|

0.01 | 0.28 | 0.73 | 0.91 | 0.99 | 0.99 | 0.44 | 0.01 | z_{i} |

For different arbitrary aspect ratio α, mode N = 4, and k-values (k = 4.3999, 3.42212, 2.44434, 1.46657), we obtained the zero flux eigenvalues as per table 3.

Applying the zero flux condition (7), for C = 0, and different values of C, we get the following poloidal magnetic field picture inside the D-shaped cross-sections for arbitrary aspect ratio (

Besides,

The paper presents a method for determining the ZFE’s of relaxed plasma states in compact tori of Multi-pinch and D-Shaped cross-sections.

If plasma is approximately force-free and approximately in equilibrium, then its magnetic field is determined by the simple force-free equation

As shown in

It is also shown that the numerical method used here, works quite well for (CT) with tight aspect ratio (

The toroidal shift (Shafranov’s shift) is clearly seen and depends on the aspect ratio, as per

Recently, the problem studied in this paper, Taylor/Beltrami relaxation, is still finding a great interest [

20 | 10 | 5 | 2 | 1.1 | 1.05 | α = R/a |
---|---|---|---|---|---|---|

4.58023 | 4.57964 | 4.57714 | 4.55571 | 4.43098 | 4.40478 | μ |

Low aspect ratio plasmas in devices such as the mega ampere spherical tokamak (MAST) are characterized by strong toroidicity, strong shaping and self fields, low magnetic field, high beta, large plasma flow and high intrinsic E × B flow shear. These characteristics have important effects on plasma behaviour, provide a stringent test of theories and scaling laws and offer new insight into underlying physical processes, often through the amplification of effects present in conventional tokamaks (e.g. impact of fuelling source and magnetic geometry on H-mode access) [

Strong plasma shaping can lead to high steady state confinement and stable beta which ensure ignited burn capability in this plasma. Normal conducting toroidal field coils that dissipate power become acceptable under these conditions. High self-driven current reduces external current drive and recirculating power.

An interesting point is arised to consider in future the coupling magnetic and fluid aspects of plasma (i.e., for

Sherif MohamedKhalil,Noura AliAlomayrah, (2016) Zero Flux Eigenvalues (ZFE’s) of Tight Aspect Ratio Tokamaks. World Journal of Nuclear Science and Technology,06,15-22. doi: 10.4236/wjnst.2016.61002