Force-free equilibrium magnetic fields in tight-aspect-ratio toroidal configurations are investigated. The study is mainly directed to modelling field configurations in the 'rodomak', a modification to the SPHEX gun-injected spheromak in which a current-carrying rod is inserted along the geometric axis. A family of analytical relaxed states (V x B = p.8, p constant) is presented for a torus of rectangular cross section, with boundary conditions allowing for flux embedded in the walls, representing the gun. Numerically calculated fields in SPHEX geometry, with p profiles relevant to the driven phase of operation, are also given. The dependence of the field conligwations and global quantities such as energy, helicity and toroidal current on the controlling parameters (gun flux, gun current and rod current) and geometry is discussed.
Purpose -The paper sets out to consider the impact of China's reform dynamics on its cooperative economy, examining developments both within the statist sector and beyond, in the variety of experimental quasi-cooperative forms that have grown between plan and market. Design/methodology/approach -The paper employs a field study approach, examining the roles of farmers' specialised cooperatives within the agro-industrial chain with experiments in Shandong province. Findings -The paper shows that reforms in the cooperative economy have been gradual, uneven and jerky, favouring the more entrepreneurial or better-placed farmers to pursue new market opportunities. Creating a sustainable mechanism that would protect and even enhance the interests of the poorer farmers is proving to be far more difficult.Research limitations/implications -The field study of this paper covers just one region. A further investigation of the impact that the reform dynamics has on the cooperative economy is planned for the near future. Practical implications -China's approach has favoured experimentation, encouraging initiatives from below and developing a more flexible relationship between government and farmers. This broad bottom-up approach has allowed for a diversity of models adapted to local conditions, but has also led to uncertainties about the limits of local government influence and the extent of autonomy, with many cooperatives taking an investor-oriented direction. Originality/value -The paper adopts a broader social economy perspective that offers a fruitful way to evaluate developments and consider the future for China's cooperative economy.
It is shown that there is a false assumption hidden in the description of a relaxed state with inhomogeneous boundary conditions as the vector sum of a potential field, satisfying the boundary conditions, and a sum of eigenfunctions of the associated eigenvalue problem expanded by certain coefficients. In particular, although the Jensen and Chu formula ͑1984͒ can provide the correct expansion coefficients, it contains an implicit paradox in its derivation according to a general vector theorem. The same paradox led Chu et al. ͑1999͒ to be concerned about a contradiction obtained by taking the curl of their magnetic field expansion which, if permitted, becomes inconsistent with a current normal to the surface. The assumption that the curl can be commuted across an infinite sum of terms is the mechanism leading to these, apparently paradoxical, conclusions. Two mechanisms for resolving this apparent paradox are possible, one of which will be described in some detail below and the other discussed further in a forthcoming, more theoretical paper ͑Laurence et al., 2000͒. The decomposition of the magnetic field above is valid with convergence in the mean squared sense, but a decomposition of the current needs to be reinterpreted in terms of negative Sobolev spaces. To avoid this, and remain in a more easily managable and familiar setting, we derive the expansion coefficients in a way that involves the commuting of the inverse curl ͑as opposed to the curl͒ and the series. The resulting series converges in a mean square sense. When this is done the calculation can conform to the general vector theorem and a new gaugeinvariant expression for the coefficients is obtained. However the consequence of the non-commutability is nullified in the Jensen and Chu formula, in both simply and multiply connected domains, by the important extra requirement of a boundary condition on the vector potential eigenfunctions; this excludes magnetic field eigenfunctions that carry flux, but there remains a complete set for the expansion and all flux is carried by the potential field. The two formulas are then identical. On a different issue, it is shown that if the general expansion is taken over a half-space, by combining positive and negative eigenvalue terms, then the coefficients are anisotropic, that is they are tensors except when evaluated at the first eigenvalue. A specific example is presented to illustrate the situation and to validate the new method of deriving the coefficients.
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