Island Overlap Research Articles

Maintenance of a stable current profile in a reversed field pinch

Externally produced magnetic fields could be used to maintain the reversed field pinch (RFP) plasma confinement configuration in a stable current, j‖∕B, profile. This would be accomplished by the creation of stochastic magnetic field regions in the vicinity of rational surfaces on which the safety factor q is the reciprocal of an integer N. External coils simultaneously excite the poloidal mode number unity and toroidal mode number N and N−1 helical fields from N=Nmax on down in a peristaltic manner, where Nmax is given by the maximal value of q in the RFP plasma. The time scale for turning off and on the N and N−1 pairs of coils is given by τw≡(μo∕η)w2, where w=a∕2Nmax is a typical island half-width required for island overlap, which implies local magnetic stochasticity, and a is the minor radius of the plasma. The total time scale for a complete current spreading cycle is Nmaxτw=(μ0∕η)a2∕4Nmax, which is a factor of Nmax≈10 faster than the time scale for the overall relaxation of the j‖∕B profile.

Hyperresistivity due to viscous tearing mode turbulence

The quasilinear hyperresistivity coefficient DH for flattening a current profile in resistive magnetohydrodynamics is computed. It is found that DH is independent of Δ′, the constant-ψ matching parameter, for viscous tearing modes. This is in contrast to the situation for inertial tearing modes, for which DH scales as Δ′−1∕5. It is reasonable to consider viscous rather than inertial tearing modes for computing DH because inertial tearing modes cross over to the viscous regime near quasilinear saturation, when their growth rates are sufficiently small. This quasilinear calculation of hyperresistivity related to a spectrum of tearing modes invokes the assumption that the tearing layers overlap. This is in contrast to the usual assumption of overlap of magnetic islands, i.e., magnetic field line chaos.

Observations of multiple magnetic islands in the core of a reversed field pinch.

We describe in this Letter the first measurement of multiple islands in the core of a reversed field pinch (RFP). These islands appear with current profile modification leading to magnetic fluctuation reduction in the Madison symmetric torus RFP. Magnetic island widths decrease to an unprecedented level, reducing the overlap of adjacent islands and allowing distinct islands to appear. The structures are observed in multichord measurements of soft-x-ray emissivity. The soft-x-ray data is validated with Poincaré reconstructions of the magnetic field structure in the core.

Simulation of intermittent beam ion loss in a Tokamak Fusion Test Reactor experiment

Recurrent bursts of toroidicity-induced Alfvén eigenmodes (TAE) are studied using a self-consistent simulation model. Bursts of beam ion losses observed in the neutral beam injection experiment at the Tokamak Fusion Test Reactor [K. L. Wong et al., Phys. Rev. Lett. 66, 1874 (1991)] are reproduced using experimental parameters. It is found that synchronized TAE bursts take place at regular time intervals of 2.9 ms, which is close to the experimental value of 2.2 ms. The stored beam energy saturates at about 40% of that of the classical slowing down distribution. The stored beam energy drop associated with each burst has a modulation depth of 10%, which is also close to the inferred experimental value of 7%. Surface of section plots demonstrate that both the resonance overlap of different eigenmodes and the disappearance of KAM surfaces in phase space due to overlap of higher-order islands created by a single eigenmode lead to particle loss. Only co-injected beam ions build up to a significant stored energy even though their distribution is flattened in the plasma center. However, they are not directly lost, as their orbits extend beyond the outer plasma edge when the core plasma leans on a high field side limiter. The saturation amplitude is δB/B∼2×10−2, which is larger than would appear to be compatible with experiment. Physical arguments are presented for why the stored energetic particle response observed in the simulation is still plausible.

Food requirements of breeding king penguins at Heard Island and potential overlap with commercial fisheries

The diet composition of king penguins (Aptenodytes patagonicus) at Heard Island (53°05′S; 73°30′E) was determined from stomach contents of 98 adults captured as they returned to the island throughout 1992. During the two growth seasons, the diet was dominated by the myctophid fish Krefftichthys anderssoni (94% by number, 48% by mass). The paralepidid fish Magnisudis prionosa contributed <1% by numbers but 17% by mass. Mackerel icefish (Champsocephalus gunnari) accounted for 17% by mass of chick diet in late winter, when chicks were malnourished and prone to starvation, although its annual contribution to the penguins' diet was only 3%. Squid was consumed only between April and August; Martialia hyadesi was the commonest squid taken, comprising 40–48% of the winter diet. The remainder of the diet consisted of the squid Moroteuthis ingens and fish other than K. anderssoni. The energy content of the diet mix fed to the chicks varied seasonally being highest during the growth seasons (7.83 ± 0.25 kJ g−1) and lowest in winter (6.58 ± 0.19 kJ g−1). From energetic experiments we estimated that an adult penguin consumed 300 kg of food each, of which its chick received 55 kg during the 1992 season. The chicks received large meals at the beginning of winter (1.2 ± 0.3 kg) and during the middle of the second growth season (1.2 ± 0.3 kg), and their smallest meals in late winter (0.4 ± 0.1 kg). The gross energy required to rear a king penguin chick was estimated to be 724 MJ. The potential impact of commercial fisheries on the breeding activities of king penguins is discussed.

Chaos in trapped particle orbits

An analytic expression is obtained for the threshold for stochastic transport of high energy trapped particles in a tokamak due to toroidal field ripple. The result permits a rapid evaluation of energetic particle loss in general equilibria. The approach to chaos has two limiting cases: island overlap from neighboring precession resonances, and the overlap of bounce resonance webs extending from resonant surfaces. In both cases the physical resonance consists of motion through a distance which is an integer multiple of the field coil separation in one bounce.

Theory of isolated, small-scale magnetic islands in a high temperature tokamak plasma

A theory for the existence of noninteracting small-scale, ‘‘drift’’ magnetic islands in a high temperature tokamak plasma is presented. This situation contrasts with that discussed by Rebut and Hugon [Plasma Phys. Controlled Fusion 33, 1085 (1991)] which involves a background ‘‘sea’’ of magnetic turbulence caused by island overlap. The islands are driven by the effect of finite ion Larmor radius on the particle drifts and they propagate with a velocity comparable to the diamagnetic velocity. In contrast with the work of Smolyakov [Plasma Phys. Controlled Fusion 35, 657 (1993)] collisions are assumed to be rare. Although the saturated island size is independent of the collision frequency in the model discussed here, collisions play a crucial role in determining the frequency of the magnetic islands. An estimate is made of the anomalous heat transport which results from the fluctuations in the electrostatic potential associated with these magnetic islands. The predicted thermal diffusivity has several, but not all, of the characteristics of the Rebut–Lallia–Watkins transport model.

Dimerized island chains in tokamaks

Magnetic surfaces are studied on MHD equilibrium plasmas with non-monotonic safety factor radial profiles, confined on large aspect-ratio tokamaks. Perturbing helical windings lead to twin Poincaré-Bhirkhoff chains and reconnection phenomenon, giving rise to dimerized island chains. Evolution of radial rotation number profiles, together with the radial progression of equilibrium points clarify the reconnection process. Toroidal perturbation leads to reconnection for values of the parameter I h much lower than those in the integrable case. Then, for higher I h , we have overlap of secondary islands and the onset of chaotic regions from the edge region to the center.

Stochastic cyclotron dynamics in the interaction of waves and low-energy particles

In this work we analyse the stochastic dynamics of initially low-energy weakly relativistic particles moving under the action of electrostatic waves. The relevant phase space is not well described by standard nonlinear pendulum approximations, but appropriate resonance studies based on the overlap of primary and secondary islands show that three regions may be identified as the wave vector k of the electrostatic mode is varied: one for small values of k, where the dynamics is always relativistic and regular; one for moderately large values of k, where the dynamics is always relativistic, being regular for small field amplitudes and stochastic for large field amplitudes; and finally one for fairly large values of k, where the dynamics is relativistic and regular for small amplitudes of the wave field and non-relativistic and stochastic for larger values of the amplitude. We perform an extensive numerical analysis in order to check our analytical estimates, and finally we present a brief numerical study of particle diffusion in the relativistic stochastic regime.

The space potential in the tokamak text

A heavy ion beam probe has been used to measure the plasma space potential profiles in the tokamak TEXT [Nucl. Fusion Technol. 1, 479 (1981)]. The Ohmic discharges studied were perturbed by externally produced resonant magnetic fields (an ergodic magnetic limiter or EML). Without these perturbations the plasma central potential is generally consistent with the value calculated from radial ion momentum balance, using experimental values of density and ion temperature and assuming a neoclassical poloidal rotation velocity. Exceptions to the agreement are found when operating with reduced plasma parameters. Possible reasons for this discrepancy are explored, in particular, the effects of intrinsic magnetic field fluctuations, and modifications to the self-consistent radial electric sheath. With the application of the EML fields the edge electric field and potential increase during periods of magnetic island overlap. A test particle calculation of electron transport shows increases in diffusivity also occur during periods of magnetic island overlap. These calculated changes in diffusivity are interpreted in terms of a stochastic layer width, which is itself used to predict a potential change for comparison with the experimental results.

Suppression and control of magnetic islands in toroidal plasmas

Magnetic islands play an important role in determining the stability and confinement properties of toroidal plasmas. The nonlinear growth of magnetic islands can lead to major disruptions in tokamaks. In stellarators, which do not have a continuous symmetry, the formation of magnetic islands can set equilibrium beta limits. In the first part of this paper, a novel method is proposed for reducing drastically the size of m=2, n=1 islands in tokamaks by introducing a beamlet of energetic ions just outside the q=2 surface via parallel neutral-beam injection. A simple physical picture is given of the nonlinear stability of a tearing mode in the Rutherford regime in the presence of energetic ions. This physical picture is supported by a kinetic calculation in the long mean-free-path regime in which the effect of bootstrap currents and resistive interchanges are retained. Estimates show that the energy of the beamlet required for the suppression is a small fraction of that used for Ohmic or beam-heating of the background plasma. In the second part of the paper, an equilibrium beta limit is obtained for stellarators by calculating the island widths induced by finite plasma pressure; the widths are then constrained by the criterion of island overlap. The theory is applied to the Heliotron-E device [Nakamura et al., Phys. Fluids B 2, 2528 (1990)]. It is shown that some aspects of the experimental observations on internal disruptions in Heliotron-E can be interpreted as signatures of the equilibrium beta limit.

Equilibrium beta limits in Heliotron-E due to magnetic island overlap

Widths of magnetic islands induced by finite plasma pressure are computed for the Heliotron-E configuration [Phys. Rev. Lett. 53, 2242 (1984); Nucl. Fusion 27, 895 (1987)]. Equilibrium beta limits are obtained from the criterion of island overlap. Earlier experimental observations on internal disruptions in Heliotron-E are revisited.

Magnetohydrodynamic clump instability

The theory of magnetohydrodynamic (MHD) clump instability in current driven plasma is presented. MHD clump fluctuations are current carrying bundles of correlated magnetic field lines. The instability occurs when turbulent mixing of the mean current density at island overlap produces clumps at a rate faster than their decay as a result of magnetic field line stochasticity. The renormalized dynamical equation describing MHD clump instability is derived from one-fluid MHD equations and conserves the dynamical invariants of the exact equations. The renormalized equation is a nonlinear, turbulent version of the Newcomb equation of linear MHD stability theory and can be cast into the form of a nonlinear MHD energy principle. MHD clump instability is a dynamical route to the Taylor state.

Magnetic field line ergodicity in the edge plasma due to an ion-Bernstein wave antenna

Large ponderomotive forces can occur near a high-power ion cyclotron range of frequency antenna, particularly for the parallel electric field polarization used to excite ion-Bernstein waves (IBW). The magnetohydrodynamic (MHD) equilibrium of the edge plasma is modified by these forces when the antenna is highly localized poloidally and toroidally. Magnetic islands develop at the edge, near q=m/n, where ‖m‖≤πa/wa and ‖n‖≤πR/la, with wa and la the width and length of the antenna, respectively. Ergodicity of the magnetic field lines may be expected when the islands overlap. Numerical calculations of the island overlap criteria are presented for two examples relevant to present day (IBW) heating experiments.

Two- and three-dimensional tokamak equilibrium modifications driven by the localized ponderomotive force of applied radio frequency waves

Large ponderomotive forces can occur when waves in the ion cyclotron range of frequencies (ICRF) are employed in stabilization or high power heating schemes. It is shown that the magnetohydrodynamic (MHD) equilibrium of the plasma is modified when the rf antenna is localized poloidally and/or toroidally. In the two-dimensional (2-D) problem (toroidally uniform rf) a modified Grad–Shafranov equation describes ponderomotively driven magnetic surface distortions. At electric field strengths of interest for ICRF stabilization, the equilibrium distortions are small provided that Lrf≪a and the poloidal localization of the rf is not severe. The character of the 3-D problem (toroidally and poloidally varying rf) depends on whether or not the ponderomotive force couples to rational surfaces within the plasma. For the case without rational surfaces, the 3-D component of the distortions is found to be small by 1/q2a relative to the 2-D component. In the case where rational surfaces are excited, magnetic islands result. A threshold condition for island overlap determines the rf electric field at which ergodicity may be expected. It is shown that ergodicity of the edge plasma may occur for some cases of interest for heating and stabilization, but should be controllable.

Pressure induced islands in three-dimensional toroidal plasma

The production of magnetic islands by plasma pressure in three-dimensional toroidal systems is analyzed. Far from the rational surfaces, a procedure based on linearization in the plasma pressure applies. This yields the solution in terms of δ-function currents at the surface. These currents are found by a nonlinear analysis valid in the vicinity of the island. The result is a set of coupled nonlinear equations determining the island widths. Scaling is found by using the approximation of nearly circular flux surfaces. The results indicate that for typical stellarators, which have a small ratio ιl0/m0 of field line rotational transform to coil rotational transform, the island size in the low-β limit depends dramatically on whether a magnetic well is present. In this case, if a magnetic well is present, islands are insignificant; in contrast, if a magnetic hill is present, island overlap occurs for arbitrarily low pressure.

Three-dimensional magnetohydrodynamic studies of the reversed-field pinch

Three-dimensional computational studies of nonlinear resistive megnetohydrodynamic (MHD) instabilities in the reversed-field pinch are presented. It is found that multihelicity mode coupling effects alter the evolution of m=1 modes from that obtained in single helicity. Strong mode coupling to m=0 and m=2 is observed. Magnetic island overlap results in stochastic field lines. At low values of the pinch parameter this is confined to the central region of the pinch; for larger values the stochasticity may extend to the wall. Healing of outer flux surfaces is observed as modes peak and relax. Decay of positive toroidal flux is retarded. Impact on confinement is discussed, and connection is made with experimental observations.

The structure of magnetic fluctuations in the HBTX-1A reversed field pinch

Arrays of edge magnetic coils and statistical analysis techniques have been used to investigate the magnetic fluctuation structure in the HBTX-1A reversed field pinch. The superficially random fluctuations can in fact be attributed almost entirely to global modes with poloidal mode number m = 0 and 1, provided account is taken of toroidal distortion of the modes. A toroidal array of coils discloses a broad spectrum of toroidal mode numbers with peak at |n| ∼ 10 and significant variation with time and frequency. Cross-correlation establishes that |n| ∼ 10 corresponds to m = 1, a helical mode resonant inside the reversal surface, and also shows the presence of m = 0, n ∼ 0. The time-scales of the fluctuation indicate that the instabilities are probably resistive in character, and the mode amplitudes are such that island overlap and magnetic field ergodization should occur. The energy confinement time due to stochastic transport, estimated from the measured fluctuations, is consistent with that observed experimentally.

Construction of three-dimensional vacuum magnetic fields with dense nested flux surfaces

Toroidal vacuum magnetic fields are analyzed by the surface of section technique and noncanonical perturbation theory. The surface of section analysis shows that it is easy to find magnetic fields with an outermost flux surface of relatively small aspect ratio. However, as the aspect ratio is decreased, so is the rotational transform of the outermost flux surface. A new method of averaging combined with noncanonical perturbation theory shows that island overlap can account roughly for this loss of rotational transform. Finally, it is shown that one can significantly decrease the stochasticity and island structure by making small modifications to the magnetic field.

Stochasticity and reconnection in Hamiltonian systems

A general class of Hamiltonian systems is studied in which neighboring phase-space islands are shifted in phase. This leads to reconnection of Kolmogorov-Arnold-Moser level curves and necessitates a reexamination of the island overlap criterion for the breakdown of adiabatic barriers between island chains. An analytic reconnection threshold is derived from an averaged Hamiltonian and found to agree closely with numerical surfaces of section for a model mapping. Numerous applications to physical problems are indicated.