H G Eriksson

Effect of combined triangularity and ellipticity on the stability limit of the ideal internal kink mode in a tokamak

The sawtooth period and amplitude in tokamaks is known to depend strongly on the shape of the q = 1 surface [1]. Furthermore, this shape dependence seems to correlate, at least to some extent, with the stability limit of the ideal, internal kink mode. The strong dependence on shaping, especially on ellipticity e and triangularity δ, of the stability limit of this mode is well-known from numerical computations [2]. The strongly destabilizing effect of ellipticity alone, especially at small ∆ q = 1 − q0, has recently also been given an analytical explanation by including terms of order e 2 e, where e is the inverse aspect ratio, in a perturbation expansion of the potential energy δW of the ideal m = n = 1 mode [3]. In the case of a parabolic current profile near the axis, and for small values of ∆ q and r1, a normalized form of this contribution to δW is given by [3]

Stability properties of a toroidal z-pinch in an external magnetic multipole field

MHD stability of m = 1, axisymmetric, external modes of a toroidal z-pinch immersed in an external multipole field (Extrap configuration) is studied. The description includes the effects of a weak toroidicity, a non-circular plasma cross-section and the influence of induced currents in the external conductors. It is found that the non-circularity of the plasma cross-section always has a destabilizing effect but that the m = 1 mode can be stabilized by the external feedback if the non-circularity is small.

Nonlinear theory of the long-wavelength external m=1 mode in screw pinches and tokamaks

The nonlinear evolution of the long-wavelength external m=1 (kink) mode of a general, straight screw pinch is investigated analytically by means of bifurcation theory. The investigation covers both the ordinary screw pinch ordering of parameters, Bz approximately Bphi , as well as the tokamak limit, Bz>>Bphi . Conducting wall effects are included in both cases. In the case of the ordinary screw pinch, general expressions, valid up to second order in the axial wavenumber, are derived for the coefficients D1 and D3 in the mode evolution equation eta n+D1 eta +D3 eta 3=0. Generally, the nonlinear stability near a linear stability boundary is found to depend on the current profile, but not on the axial field profile. Furthermore, peaked current profiles are found to be more stable than broad profiles, but for complete nonlinear stability a conducting wall is in general required. In the tokamak case, a general expression for D3, covering all previously calculated values of this coefficient (for particular profiles), is derived. Near the first marginal point qa=1/b2 (b is the wall radius normalized to the plasma radius), the shape of the current profile is found to be of decisive importance for the nonlinear evolution of the m=1 mode. In general, sufficiently peaked profiles lead to nonlinear stability, whereas broad profiles lead to explosive growth of the mode. Near the second marginal point qa=1, D3 is shown to have a singularity either of order (1-qa)-1, or of order (1-qa)-2, depending on whether Ja not=0 or Ja=0, respectively, where Ja is the current density at the plasma edge.

Axisymmetric stability of a toroidal pinch with non-circular cross-section

Presents an analytic ideal MHD normal-mode analysis of the axisymmetric, m=1, free-boundary mode. The ordering Btheta /Bp approximately O(1) is considered and an incompressible perturbation is used which mainly has implications for the horizontal mode. The analysis is based on an expansion in the inverse aspect ratio and on a shaping parameter, characterizing the effect of a weakly non-circular (up-down symmetric) cross-section. A closed analytical expression, giving the leading order contributions to omega 2 due to toroidicity, non-circularity and the coupling between toroidicity and non-circularity, is obtained in terms of a number of simple parameters, describing the effects of non-circularity and diffuse current and field profiles. When the inverse aspect ratio and the non-circularity are approximately of the same order, the toroidicity can give a contribution to omega 2 which is comparable in magnitude with the contribution due to non-circularity present in a straight pinch. In particular, the relative effect of toroidicity increases with peaking of the current profile.

Nonlinear dynamics of an elliptic magnetic stagnation line

The nonlinear evolution of the kink instability of a plasma with an elliptic magnetic stagnation line is studied by means of an amplitude expansion of the ideal magnetohydrodynamic (MHD) equations. A cylindrically symmetric plasma with circular field lines is used to model the magnetic field geometry close to the stagnation line. Due to the symmetry with respect to ±z, the linear stability problem of such a system has a two-folded degeneracy, with equal eigenvalues for helical kink perturbations with positive and negative polarization. It is shown that, near marginal stability, the nonlinear evolution of the instability can be described in terms of a two-dimensional potential U(X,Y), where X and Y represent the amplitudes of the perturbations with positive and negative helical polarization. The potential U(X,Y) is found to be nonlinearly stabilizing for all values of the polarization. Furthermore, in addition to the equilibrium point (X,Y)=(0,0), the nonlinear potential has eight equilibrium points in the...

Nonlinear stability analysis of external hydromagnetic modes in a tokamak

This paper deals with the nonlinear stability properties of external, ideal magnetohydrodynamic (MHD) modes with helical symmetry in a tokamak. Based on an analysis of bifurcated equilibria in cylindrical geometry, the investigation is concerned with non-resonant, nearly marginal modes with poloidal mode numbers . The analysis is to a large extent analytical and based on an expansion in the helical mode amplitude. The final results, however, are obtained numerically. A systematic investigation of the dependence of the nonlinear effect on poloidal mode number, current profile and wall distance is performed, including current profiles which are peaked off-axis. The results show that helical modes with with realistic wall distances are, in principle, nonlinearly stable, i.e. the bifurcation is supercritical, for all studied current profiles. The stabilizing effect increases with m and depends strongly on the mode number. For high m, the nonlinear effect is significant even at very small mode amplitude and, consequently, the amplitudes of helical states with high m are very small. Also, the stabilizing effect depends strongly on the current profile and increases when the profile is peaked. However, if too much current is flowing in a region near the plasma edge, the bifurcation is still supercritical but the amplitudes of the bifurcated equilibria are very large, thus indicating a situation where nonlinear stability would not be obtained in practice. This finding may be of interest in connection with high-performance tokamaks operating with a large fraction of bootstrap current. The requirement of reduced current density near the plasma edge is most stringent for the m = 2 mode.

Estimates of plasma equilibria from external magnetic measurements

Abstract An algorithm based on an asymptotic expansion of the magnetic field from a toroidal plasma with weakly noncircular cross section together with a least-squares minimization scheme, has been developed for rapid determination of several plasma parameters from external measurements of the radial and tangential components of the B field, using poloidal probe coil arrays. The accuracy of the method and its robustness against random and certain physically relevant systematic errors is tested by extensive numerical simulations. The method works well; it should be possible to determine the plasma position and β p + l i 2 based on measurements with 8–10% accuracy. With high enough accuracy in the measurements it is also possible to determine the noncircularity of the cross section and, assuming a linear Grad-Shafranov equation, an estimate (Bessel function approximation) of the current profile and hence of the separate values of βp and li may then also be obtained.

On the effect of passive feedback in the toroidal Extrap configuration

The ideal MHD stability properties of the axisymmetric free-boundary m = 1 mode of toroidal Extrap equilibria with large aspect ratio and weakly non-circular cross-section are investigated. The effect of passive feedback due to induced currents in the external conductors, represented as ideal current loops, and in a surrounding conducting wall, is taken into account in an approximate fashion by using the inductances of the circuits. The dependence of the effect of the feedback on different parameters such as the plasma radius, the radius of the wall and the horizontal position of the equilibrium plasma is studied. It is found that the effect of feedback becomes significant in the Extrap configuration when the plasma radius is not too small. The axisymmetric mode is thereby stabilized over a parameter regime which is roughly corresponding to the experimental situation of the Extrap T1 device.