A new System of co-ordinates is found and a method developed to determine the toroidal equilibrium of plasmas with arbitrary current distribution and plasma cross-section. The method depends on knowledge of the equilibrium of a straight plasma column of similar cross-section and similar current distribution. A large aspect ratio is assumed. By successive approximations, better solutions can be obtained. An explicit formula is presented for the poloidal flux of a nearly circular plasma. This can be written in terms of a function related to the asymmetry of the poloidal field due to toroidality. The method works provided that there is only one magnetic axis.
We do not believe that the minimum energy theory [l] is applicable to driven plasmas. We assume that relaxation occurs towards a state of minimum entmpy production rate. As the equilibrium Ohm's law cannot be satisfied in peaked-current density states, the final state must be steadily turbulent. What is new about our theory is, fundamentally: (i) the pressure balance is considered as a relevant constraint upon the relaxed state and this is expressed as an integral relation; (ii) the energy dissipation by tuhulence is assumed to be minimal and (iii) anisotropic resistivity is regarded as the main cause of poloidal current. The numerical calculations of magnetic field and current density reproduce typical observations in tokamaks.
An averaging method is developcd in order to dctermine anslytically the magnetic stwcturc of a system with symmetry broken by small pcrturbations. In particular, Puincare maps are obtained for toroidal hclical fields using the typical parameters of the Brazilian Tokamak TBR-I. The technique is fairly general as the small parameter uscd in the construction of this lheory is the original symmetry-breaking perturbation pardmcler; it is applicable in analyses of ficlds in more compact devices or with cross-sections with any shape as well. Unless the unperturbed system is doubly symmetric (circular cylinder) a single helical perturbation made (PE, n) can excite many other modes. Thc coupling between any two modes arises in a very natural way.
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