Surface Current Model Research Articles

Effect of rigid toroidal rotation on the stability of a high-beta tokamak to external kink modes

The effect of rigid toroidal rotation on the stability of a tokamak to external kink modes is examined. For simplicity a surface current model is assumed. For a high-β tokamak it is shown that to leading order the equations governing stability in the presence of rotation are identical to those for a static plasma with β replaced by β+S, where β is the plasma beta and S is a parameter which provides a measure of the rotation rate. For a circular cross-section tokamak the critical beta for stability to external kink modes in the presence of rigid toroidal rotation is given, to leading order in the inverse aspect ratio ε, by β=0.21ε−S. For an elliptical cross-section tokamak the largest critical beta is obtained for a vertical elongation of 2.2 and is given, to leading order in ε, by β=0.37ε−S. The lower limit on the kink safety factor q* or the Mercier q increases with increase in S. Quantitative estimates of this increase can be obtained from the stability boundaries for the static problem.

The introduction of surface resistance in the three-dimensional finite-difference method in frequency domain

A full-wave treatment of lossy three-dimensional structures using the finite-difference method in the frequency domain is presented. This accounts for both dielectric and conductor losses. By the introduction of a surface resistance, the effect of conductor losses and surface roughness can be modeled very efficiently. The modifications of the finite-difference-frequency-domain (FDFD) algorithm are presented. Comparisons between the conventional approach using elementary cells with finite conductivity and this new discretization method with surface current cells are given, and the advantages and limitations of the surface current model are shown.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Electric surface current model for the analysis of the radiation resistance of a spherical circular microstrip antenna

AbstractAn approach to the analysis of a spherical circular microstrip antenna which is applicable to a relatively thick substrate is presented. Based on the spectral‐domain Green's function formulation, the effect of the dielectric substrate as well as the spherical ground are incorporated. Numerical results for the radiation resistance agree well with the results using the generalized transmission line model (GTLM), which is accurate for a thin dielectric substrate only. © 1993 John Wiley &amp; sons, Inc.

Nonlinear kink mode dynamics in circular and noncircular pinches

It is shown that the global (free-boundary) m=1 kink instability of the ideal, magnetohydrodynamic (MHD) sharp boundary (surface current) pinch is stabilized by nonlinear effects, provided Be≲1 and βp&amp;lt;1, where βp=1+B2e−B2i and Bi and Be denote, respectively, the internal and external axial magnetic fields of the pinch, normalized to the poloidal magnetic field. The stabilization has to do with the bending of the interior, ‘‘frozen’’ field lines and associated volume currents induced in the pinch, and does not occur in a pure surface current model, which neglects these currents and only conserves the total magnetic flux through the pinch. It is suggested that the global, helical m=1 structures observed in various pinch experiments may have to do with the stabilizing mechanism above. The nonlinear stability has been calculated by means of a new approach to the bifurcated equilibria of the helical m=1 mode, and the method should also be useful in connection with other nonlinear, ideal MHD phenomena. The regime of nonlinear stability above corresponds to intermediate or short wavelengths of the marginal mode (ka≳1). In the opposite, long-wavelength regime, the ideal MHD model and the pure surface current model give similar results, predicting nonlinear instability for, e.g., the nearly marginal Kruskal–Shafranov mode in tokamaks, in agreement with previous theories. Effects of mode rotation as well as of a noncircular cross section of the pinch, modeling the Extrap [Fusion Technol. 16, 7 (1989)] configuration, have also been considered, extending the results of a previous, linear investigation [Phys. Fluids B 2, 1601 (1990)] to the nonlinear regime.

Far field analysis of spherical-circular microstrip antennas by electric surface current models

The far field radiation patterns of a circular disk microstrip antenna mounted on a sphere are studied theoretically. The radiator is replaced by an assumed surface current distribution. The effects of the dielectric layer and the metallic sphere are rigorously taken into account by the spectral domain Green's function formulation. In the spectral domain represented by the vector Legendre series, the Green's function matrix is diagonal. The results are verified by comparison with results obtained from the cavity model theory. The cavity model agrees asymptotically with the present method for the case of a thin dielectric substrate.

Patch antennas on a spherical body

The far field radiation patterns of wrap-around, circular disk and annular ring microstrip patch antennas mounted on a sphere are studied theoretically. The patches are replaced by equivalent magnetic currents based on cavity model theory. The radiation fields are then determined using the modal expansion technique in spherical co-ordinates. Results for a wraparound antenna of a half-wavelength in width, for the broadside TM11 mode of a circular disk antenna and for the TM11 and TM12 modes of an annular ring antenna are given. The accuracy of the simple theory is confirmed by a more rigorous but complicated theoretical approach based on electric surface current model.

Viscous damping of Free Boundary Models in a Non-Circular z-Pinch

The MHD stability properties of a slightly non-circular z-pinch are studied, taking classical viscosity into account. A surface current model is used, and the perturbations are assumed to be incompressible with infinite axial wavelength.It is found that the only unstable mode is that with m = 1, m being the poloidal mode number. This mode is unstable for all values of the viscosity coefficient μ. Although the growth rate can be made arbitrarily small, even a moderate reduction requires a very large of μ.Clearly, the observed stability of Extrap, where the non-circularity is due to currents in a number of external conductors, can not be explained by viscosity alone. The required viscosity is about six orders of magnitude larger than the observed classical one.

Electric surface current model for the analysis of microstrip antennas on cylindrical bodies

An approach to the analysis of microstrip antennas on cylindrical bodies is presented. The printed radiator is replaced by as assumed surface current distribution, and the fields are solved taking into account the presence of the dielectric layer and the metallic cylinder. Calculation takes place in the Fourier domain. The far field, calculated asymptotically from this solution, is used to get the radiation patterns of the wraparound antenna for any dielectric and the half-wavelength patch for \epsilon_{r} = 1 .

Electric surface current model for the analysis of microstrip antennas with application to rectangular elements

An approach to the analysis of microstrip antennas which is applicable also to relatively thick substrates using the relevant Green's function is presented. The Green's function is derived and closed form expressions for various antenna characteristics which explicitly take into account the presence of the dielectric material are obtained in terms of the electric surface current density. For rectangular microstrip elements near resonance the current distribution is approximated using lossless transmission line analysis, thus enabling the complete evaluation of the characteristics of the element near resonance. The results obtained in this approach for the radiation resistance, surface wave resistance, radiation pattern, directivity, and bandwidth are presented in a detailed set of graphs for a representative set of parameters.

MHD Stability Analysis of Axisymmetric Surface Current Model Tokamaks Close to the Spheromak Regime

In the toroidal coordinates, a stability analysis is presented for very low-aspect-ratio tokamaks with circular cross section which is described by a surface current model (SCM) of axisymmetric equilibria. The energy principle determining the stability of plasma is treated without any expansion of aspect ratio. Numerical results show that, owing to the occurrence of the non-axisymmetric ( n =1) unstable modes, there exists no MHD-stable ideal SCM spheromak characterized by zero external toroidal vacuum field. Instead, a stable spheromak-type plasma which comes to the ideal SCM spheromak is provided by the configuration with a very weak external toroidal field. Close to the spheromak regime (1.0<aspect ratio\( \lesssim \)1.1), the minimum safety factor and the critical β-values increase monotonically with aspect ratio decreasing from a large value, and curves of β p versus β in the marginal stability approach to an ideal SCM spheromak line β p =β.

Ideal magnetohydrodynamic stability limits of a high-β tokamak with superimposed helical fields

The stability of a high-β tokamak with a superimposed single helicity stellarator field is investigated in the context of the sharp boundary surface current model. The analysis treats external ideal magnetohydrodynamic modes including both pressure-driven ballooning effects and current-driven kink instabilities. The results show that the addition of an l = 2 field is unfavorable, lowering the allowable Ohmic heating current and decreasing the maximum stable β. The addition of an l = 3 field, however, is favorable. Both the allowable Ohmic heating current and maximum stable β are increased as the l = 3 amplitude increases.

Combined n = 0 and n ≠ 0 MHD stability analysis of axisymmetric surface current model equilibria

For a surface current model of axisymmetric equilibria with constant plasma pressure, the energy principle is used to perform a combined stability analysis of axisymmetric and nonaxisymmetric MHD modes in the absence of wall stabilization. An improper treatment of the n ≠ 0 modes at low aspect ratio A found in the literature is pointed out and corrected. In the regime 3.8 ⪅ A < ∞, for fixed values of the aspect ratio and β poloidal, β is maximized by optimizing the plasma shape, while the requirements of full MHD stability are observed. In the regime A ⪅ 3.8 no maximum of β is found by shape optimization, and therefore a special selection of typical shapes is considered; specifically, the stability of configurations with circular, elliptical, doublet- and D-shapes is studied. For circular and D-shapes, critical β-values are calculated down to the extreme case of A almost equal to one. In addition, surface-current-model (SCM) spheromaks are considered, which turn out to be MHD-unstable when there is no wall stabilization, the type of situation to which this paper is restricted.

AC losses in multifilamentary Nb&amp;lt;inf&amp;gt;3&amp;lt;/inf&amp;gt;Sn conductors

We have studied losses at 4.2K in a number of multifilamentary Nb and Nb 3 Sn wires fabricated by the bronze route. Losses due to alternating current and to transverse alternating field showed a fourth power dependence on current or field which is incompatible with the critical state model, even when the observed field dependence of the bulk critical-current density is taken into account. In order to reveal the source of the low losses in our materials more clearly, we have studied losses with the field parallel to the filament surface. As expected, the results show that losses in Nb filaments are low because of the presence of the Meissner state. However, superficially similar results for Nb 3 Sn filaments cannot be interpreted as evidence for an anomalous lower critical field; a more likely explanation is the presence of a surface current. Although a simple surface current model cannot account in detail for the transverse field losses, one incorporating losses due to surface roughness, field orientation effects and the field dependence of the surface current can probably provide an adequate description.

Feedback stabilization of axisymmetric MHD instabilities in tokamaks

A stability principle is derived and applied to simple tokamak configurations in order to study feedback stabilization of unstable, vertically elongated tokamak plasmas. For practical applicability it is assumed that the fast instabilities are slowed down by passive conductors so that only slow motions have to be considered. Numerical results are presented for a surface current model of plasma with one conjugate pair of axisymmetric feedback loops. Stabilization is possible, except in a limited region of loop positions. The optimum loop position in the region with possible stabilization is determined.

Stability of tokamaks with respect to slip motions

With the help of the energ y principle in tokamaks, a class of perturbations is studied, which, if unstable, cannot be stabilized by the toroidal main field. On the assumption of usual tokamak ordering and in the limit of infinite aspect ratio, these perturbations are shown to be minimizing among all axisymmetric perturbations. In the case of finite aspect ratio, a detailed stability analysis is carried out by using a constant-pressure surface-current model with elliptic, triangular or rectangular plasma cross-section. Definite stabilization by toroidal effects and by ßp is demonstrated.

Emission of Neutrons from Muonic Lead and Bismuth

The emission of prompt neutrons from muonic bismuth observed experimentally is believed to be caused by the radiationless excitation of the target nucleus above the neutron separation energy by muonic transitions of the type np → 1s[Formula: see text]. The work of Zaretskii and Novikov for muon induced fission in uranium is extended to the case of muonic lead and bismuth. The ratio of rates of radiationless excitation to the radiative transition probabilities is calculated by the surface current model of Sliv to check a random phase approximation used in the calculation by Zaretskii. The neutron spectrum obtained using this nuclear branching in the cascade calculation is given for lead and bismuth.

Use of the M&amp;S Configuration in Theta Pinches

The M &amp; S configuration gives equilibrium for a toroidal plasma without net current in the azimuthal direction and without rotational transform. Explicit solutions are obtained only for the restricted cases β=0, and β ≦ 1 in the surface current model, but configurations with continuous pressure profile are proved to exist. The use of the M &amp; S configuration to confine a toroidal theta pinch plasma leads to problems in the establishment of the equilibrium and in particular in its stability. The stability of the M &amp; S configuration is investigated for the simplified case of the corrugated linear theta pinch. LIMPUS. This configuration is theoretically unstable with respect to m ≧ 1 flutes. Mechanisms to suppress these modes are based on finite gyroradii stabilisation for m ≧ 2. and on the conducting wall for m = 1, the latter case being limited to very high β. No m 2 instabilities are observed experimentally, although m = 1 instabilities with growth times agreeing with MHD theory are detected. An extrapolation of these results to an M &amp; S configuration gives τm&amp;s/τD≈ (ξ0/τD)1/2 where τD is the average minor radius, ξ0 an initial deviation from the M &amp; S equilibrium position, and τ M&amp;S and τD are the M &amp; S confinement time and the toroidal drift time, respectively. M&amp;S experiments with theta pinch plasmas near 10 eV, τm&amp;S is extended by a factor 3 - 4 beyond τD , which is in agreement with the above relation. This suggest firstly, that the M &amp; S configurations investigated are an adequate approximation to the required equi­librium, and, secondly, that the MHD instabilities substantially limit the confinement time. Simple models show that oscillating axial currents, a steady axial motion of the LIMPUS mag­netic field configuration or a standing wave like oscillation can suppress the instabilities mentioned. Experiments on such dynamic stabilisation are in progress.

Internal conversion coefficients for magnetic multipoles forz=39

Results of new internal conversion calculations for two values of gamma-energy equal to 0.05 mc2 and 0.15 mc2 are presented and compared with earlier works ofRose, andSliv andBand in the case of Y87. Matrix-elements for the surface current model and the no-penetration model are presented. Differences were found between present calculations and the earlier works fork equal to 0.05 mc2 and in particular for theK shell. It is suggested that these differences are partly due to the screening model used and possibly also due to the mesh-size employed. New calculations are based on the non-relativistic Hartree-Fock potential in the Slater exchange approximation. The earlier work was based on the Thomas-Fermi-Dirac model. Our calculated eigenvalues agree extremely well with the experimental electron binding energies. A large number of points (2500) were used to minimize any errors due to mesh-size effects.