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.
The ergodic magnetic limiter is a device designed to generate a cold boundary layer of chaotic magnetic field lines at the peripheral region of a tokamak, with the main purpose of reducing the deleterious effects of the plasma-wall interaction. In the TCABR tokamak an ergodic limiter was constructed and recently installed inside the vacuum chamber. We developed a theoretical model for the action of an ergodic magnetic limiter in a large aspect-ratio tokamak taking into account the finite width of the limiter. The theoretical results are in good agreement with measurements of the vacuum magnetic field created by the limiter. Poincaré maps of field line flow are computed to reveal the resulting magnetic field line structure due to the ergodic limiter and show that the operation of the ergodic limiter in the TCABR tokamak is feasible and results in a chaotic boundary layer for limiter currents of about 6% of the plasma current.
The paper provides an empirical analysis of the macroeconomic factors that enhance revenue gap in South Africa using the multivariate cointegration techniques for the period 1965 to 2012. The results from the cointegration analysis indicate that the revenue gap in South Africa is negatively associated with the level of imports while positively related to external debt and underground economy. The former finding is consistent with the notion that imports are subjected to more taxation than domestic activities because of certain features of international trade that tend to make tax evasion difficult. On the other hand, the positive relationship between external debt and tax gap shows that the South African government relies upon external debt to finance its budget deficit resulting from missing revenues. Furthermore, the observed negative effect of the post-apartheid dummy confirms that the tax policy reforms that South Africa introduced following the liberation in 1994 have led to a reduction in missing revenues. The results from the Granger causality test also show that there is a unidirectional causality running from imports and underground economy to revenue gap, while revenue gap on the other hand is found to Granger-cause national income and external debt in South Africa.
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.
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