Poloidal Mode Research Articles

Effect of Toroidal Current on Rotational Transform Profile by MHD Activity Measurement in Heliotron J

The effect of toroidal current on the rotational transform was investigated in Heliotron J by measuring magnetohydrodynamic (MHD) activities at two configurations with a rotational transform (ι/2π) close to 0.5. The resonant m/n = 2/1 mode was observed in an ECH + co-NBI plasma in the configuration with ι/2π = 0.48 at ρ = 0.7. Here, m and n are the poloidal and toroidal mode numbers, respectively. ρ is the normalized minor radius. The rotational transform is increased presumably due to the toroidal current. The location of the ι/2π = 0.50 rational surface was determined to be ρ = 0.8-0.9 by soft X-ray (SX) fluctuations related to the MHD mode. An equilibrium calculation considering the toroidal current showed that the increase in the rotational transform due to the toroidal current was consistent with experimental results. The resonant mode structure was also investigated in an ECH + counter-NBI plasma at the ι/2π = 0.50 configuration. The location of the ι/2π = 0.50 rational surface, as determined by SX signals, did not change significantly compared with that obtained under a vacuum configuration. There is no significant difference between the rotational transform profile that considers toroidal currents by equilibrium calculation and that of the vacuum configuration. These results suggest that the change in the rotational transform profile caused by the toroidal current was small, owing to the balance between the bootstrap current and counter-flowing NB driven current.

Density Collapse and Fluctuation Observed in Poloidally Rotating Plasma on TU-Heliac

The ohmic heated plasma in TU-Heliac was biased by a hot-cathode to drive E × B poloidal rotation. Coincident measurements of the line density and ion saturation currents revealed the existence of density collapse accompanied by fluctuation in the poloidally rotating plasma. The radial density profiles just before and after the collapse were estimated. The steep density gradient vanished with the density collapse. The power spectra of the fluctuation were calculated from the ion saturation current obtained with a high-speed triple probe. The frequency of fluctuation was compared to the E × B poloidal rotation frequency, where the fluctuation appeared with density collapse. The fluctuation frequency was 2 to 3 times the rotation frequency. This suggested that the poloidal mode number of fluctuation was m = 2o r 3.

Temporal Evolution of the Pressure Profile and Mode Behavior during Internal Reconnection Events in the MAST Spherical Tokamak

Mode behaviors including non-linear development and pressure profiles during internal reconnection events are clarified in the MAST spherical tokamak. In a q0 > 1 discharge, the tearing mode is a trigger; however, non-linearity of modes is not confirmed. On the other hand, in a q0 < 1 discharge, the harmonics of a m/n = 4/1 mode of ∼22 kHz are confirmed. Method for identification of poloidal mode in ST configuration is also given.

Three-dimensional features of GAM zonal flows in the HL-2A tokamak

A novel design of the three-step Langmuir probe (TSLP) array has been developed to investigate the zonal flow (ZF) physics in the HL-2A tokamak. Three TSLP arrays are applied to measure the three-dimensional (3D) features of ZFs. They are separated by 65 mm in the poloidal and 800 mm in the toroidal directions, respectively. The 3D properties of the geodesic acoustic mode (GAM) ZFs are presented. The poloidal and toroidal modes of the radial electric fields of the GAM perturbations are simultaneously determined in the HL-2A tokamak for the first time. The modes have narrow radial wave numbers (krρi = 0.03–0.07) and short radial scale lengths (2.4–4.2 cm). High coherence of both the GAM and the ambient turbulence separated by toroidal 22.5° along a magnetic field line is observed, which contrasts with the high coherence of the GAM and the low coherence of the ambient turbulence apart from the field line. The nonlinear three wave coupling between the turbulent fluctuations and the ZFs is a plausible mechanism for flow generation. The skewness and kurtosis spectra of the probability distribution function of the potential perturbations are contrasted with the corresponding bicoherence for the first time, which support the three wave coupling mechanism.

Unified linear response function for zonal flows with full finite orbit effects

A new formulation of the linear response function of electrostatic potential to nonlinear drive (due to turbulence) is presented in this paper; zonal flows play important roles in the self-regulation of turbulence and their basic physics are contained in response functions. Two branches of zonal flows [stationary zonal flow and geodesic acoustic mode (GAM)] are known to exist in the low and high frequency ranges. However, they have been analyzed separately using different approximations due the difference in their frequency ranges. This paper visits this problem and gives a unified expression of the response function by taking full account of finite orbit effects. The drift kinetic equation is integrated along particle orbits by expanding them in Fourier series. Thus, a separate handling of passing and trapped particles is facilitated revealing some important aspects of zonal flows: (1) neoclassical poloidal mode coupling due to finite orbit effects, (2) enhancement of the nonuniform potential field due to reduced parallel transport, and (3) the presence of two propagation bands of GAM as the quadratic dispersion relation is solved.

Resistive internal kink modes in a differentially rotating cylindrical plasma

The resistive internal kink modes in a differentially rotating cylindrical plasma column are studied. It is shown that the Velikhov effect, which causes the magnetorotational instability in astrophysics, contributes to the magnetic hill/well and thereby enhances or suppresses these modes, depending on the character of radial profile of the rotation frequency. It is pointed out that, in the case of unfavorable rotation frequency profile, such a rotation-induced magnetic hill can exceed the hill effect due to the plasma pressure gradient. Under this condition, there appears a new variety of resistive-interchange modes, which are referred to as rotational modes. On the other hand, for a favorable rotation frequency profile, the Velikhov effect suppresses the resistive-interchange modes. These results concern the m&amp;gt;1 modes, where m is the poloidal mode number. In the case of perturbations with m=1, the favorable rotation frequency profile leads to decreasing the growth rate of the reconnecting mode.

Extending the collisional fluid equations into the long mean-free-path regime in toroidal plasmas. IV. Banana regime

In Part I [Phys. Fluids B 2, 1190 (1990)] and Part II [Phys. Plasmas 12, 082508 (2005)], it was emphasized that the equilibrium plasma viscous forces when applied for the magnetohydrodynamic (MHD) modes are only rigorously valid at the mode rational surface where m−nq=0. Here, m is the poloidal mode number, n is the toroidal mode number, and q is the safety factor. This important fact has been demonstrated explicitly by calculating the viscous forces in the plateau regime in Parts I and II. Here, the effective viscous forces in the banana regime are calculated for MHD modes by solving the linear drift kinetic equation that is driven by the plasma flows first derived in Part I. At the mode rational surface, the equilibrium plasma viscous forces are reproduced. However, it is found that away from the mode rational surface, the viscous forces for MHD modes decrease, a behavior similar to that observed in the viscous forces for the plateau regime. The proper form of the momentum equation that is appropriate for the modeling of the MHD modes is also discussed.

Effect of perturbations on magnetic fields as a possible problem for implementing toroidal magnetic configurations with a zero rotational transform

Approved reliable methods and a unified quasi-resonant approach are used to reproduce the results characterizing the effect of dangerous perturbations on three particular types of toroidal magnetic configurations with a zero rotational transform (ι/2π = 0). An analysis is made of the perturbations whose geometry and amplitudes correspond to imperfections and defects in the assembly of the actual magnetic systems of conventional stellarators. The effect of perturbations on magnetic configurations with ι/2π = 0 is of the same quasi-resonant nature as that for systems with ι/2π ≠ 0 and a zero magnetic shear. However, in contrast to shearless configurations with a nonzero rotational transform, the most dangerous perturbations are those with n = 0 and m = 1, 2, 3, … (where m is the poloidal mode number and n is the toroidal mode number) and the effect of harmonics with the smallest poloidal and toroidal numbers is more destructive and nonlocal even for the amplitudes corresponding to imperfections and defects in the assembly of conventional stellarators. Such an extremely high sensitivity of magnetic systems with ι/2π = 0 to the perturbations under consideration may be a serious obstacle for their practical implementation.

Building magnetic barriers in tokamaks

The second order perturbation method of creating invariant manifold inside chaos in Hamiltonian systems (Ciraolo G et al 2004 J. Phys. A: Math. Gen. 37 3589; Ciraolo G et al 2004 Phys. Rev. E 69 056213; Vittot M 2004 Phys. A: Math. Gen. 37 6337; Chandre C et al 2005 Phys. Rev. Lett. 94 074101) is applied to tokamaks to build magnetic barriers inside the region of magnetic chaos created by resonant magnetic perturbations. A symplectic map in magnetic coordinates is derived for calculation of magnetic field line trajectories in toroidal magnetic confinement configurations for fusion plasmas. A class of safety factor, q, profiles is chosen to represent the Ohmically heated tokamaks (OHT), the DIII-D and the ASDEX UG devices. For OHT, the equilibrium poloidal flux is calculated from the standard safety factor profile. Magnetic perturbation in OHT are poloidal mode number m = 3 and toroidal mode number n = 2 and m = 2, n = 1. A magnetic barrier is created at about midway between these two resonant magnetic surfaces. The barrier reduces the diffusion of magnetic field lines roughly by half. The barrier is fortified by adding up to third order magnetic perturbation for the amplitude of the tearing modes, δ < 2.25 × 10−3. For δ > 2.25 × 10−3, the barrier is not sustainable. A close-by noble barrier is found to be much more robust. For the DIII-D, equilibrium poloidal flux is calculated from the data for shot 115467 at 3000 ms. The magnetic perturbation chosen in the case of the DIII-D q-profile has mode numbers (m, n) equal to (3,1) and (4,1). A magnetic barrier is created inside chaos for the DIII-D q-profile by adding the second order magnetic perturbation to equilibrium poloidal flux. The robustness of this barrier in the DIII-D against topological noise is studied. For the ASDEX UG q-profile, (3,2) and (4,3) modes are chosen and a barrier is created about midway by the second order magnetic perturbation.

Stabilization of neoclassical tearing modes by electron cyclotron current drive in JT-60U

Results of active control of neoclassical tearing modes (NTMs) by electron cyclotron current drive (ECCD) in JT-60U are described. Growth of an NTM with poloidal mode number m = 3 and toroidal mode number n = 2 has been suppressed by ECCD inside the sawtooth inversion radius in the co-direction, showing the possibility of the coexistence of sawtooth oscillations and a small-amplitude m/n = 3/2 NTM without large confinement degradation. Stabilization of an m/n = 2/1 NTM by ECCD at the mode rational surface has been demonstrated with a small ratio of the current density driven by the electron cyclotron (EC) wave to the local bootstrap current density (∼ 0.5). In addition, dependence of the stabilization effect on ECCD location has been investigated in detail. It has been found that an m/n = 2/1 NTM can be completely stabilized with the misalignment of the ECCD location less than about half of the full island width, and that the m/n = 2/1 NTM is destabilized with the misalignment comparable to the full island width. Time-dependent, self-consistent simulation of magnetic island evolution using the TOPICS code has shown that the stabilization and destabilization of an m/n = 2/1 NTM are well reproduced with the same set of coefficients of the modified Rutherford equation. The TOPICS simulation has also clarified that EC wave power required for complete stabilization can be significantly reduced by narrowing the ECCD deposition width.

Observation of instabilities during density limit experiments in the Hefei Tokamak-7

During a recent campaign at the Hefei Tokamak-7 (HT-7) [M. Asif et al., Phys. Plasmas 12, 082502 (2005)], experiments were performed with a moveable graphite limiter to investigate the influence of the horizontal plasma position on the density limit. It has been found that in HT-7 density limits differ by the radiative instability in the plasma boundary and various magnetohydrodynamic (MHD) instabilities, which precedes the disruption. Control of the impurity content, edge, and recycling properties prevents the growth of the multifaceted asymmetric radiation from the edge (MARFE) and makes it possible to exceed the Greenwald limit in HT-7. It has been observed that the m=2, MHD component dominates during almost the whole discharge duration, when the density limit is approached at values of the edge safety factor 3&amp;lt;q(a)&amp;lt;4. The observed sawtooth oscillation shows that oscillation with m=1 and n=1, where m and n are the poloidal and toroidal mode numbers, respectively, triggers m=2 and n=1 in sawtooth collapse, which finally causes disruption in the HT-7 tokamak.

Comparison between resistive and collisionless double tearing modes for nearby resonant surfaces

The linear instability and nonlinear dynamics of collisional (resistive) and collisionless (due to electron inertia) double tearing modes (DTMs) are compared with the use of a reduced cylindrical model of a tokamak plasma. We focus on cases where two q = 2 resonant surfaces are located a small distance apart. It is found that regardless of the magnetic reconnection mechanism, resistivity or electron inertia, the fastest growing linear eigenmodes may have high poloidal mode numbers m ∼ 10. The spectrum of unstable modes tends to be broader in the collisionless case. In the nonlinear regime, it is shown that in both cases fast growing high-m DTMs lead to an annular collapse involving small magnetic island structures. In addition, collisionless DTMs exhibit multiple reconnection cycles due to reversibility of collisionless reconnection and strong E × B flows. Collisionless reconnection leads to a saturated stable state, while in the collisional case resistive decay keeps the system weakly dynamic by driving it back towards the unstable equilibrium maintained by a source term.

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.

Quasilinear description of radiofrequency-induced radial diffusion

We derive the bounce-averaged quasilinear (QL) operator in axisymmetric toroidal geometry using the standard QL formalism. When specialized to resonant ion cyclotron harmonic interactions, this operator predicts the same radiofrequency-induced radial diffusion as the orbit-averaged operator (Eriksson and Helander 1994 Phys. Plasmas 1 308) obtained using action-angle variables. By assuming the wave field known as a superposition of toroidal and poloidal Fourier modes, the QL diffusion coefficients are written in a form which can be directly evaluated using the output of a spectral full-wave solver of Maxwell equations in toroidal plasmas.

The origin of convective structures in the scrape-off layer of linear magnetic fusion devices investigated by fast imaging

A fast imaging camera is used to unveil the spatio-temporal properties of radially convective events in the CSDX linear plasma device [M. J. Burin et al., Phys. Plasmas, 12, 052320 (2005)]. The exposure time is set to 1μs and the time between frames to 10μs. The time series from a Langmuir probe and from a pixel in the 50000-frame movie are compared and cross-correlated. Excellent agreement between the two diagnostics is found for spatial scales greater than 2.5mm. The fluctuations inside the main plasma column are found to change between different poloidal mode numbers as a function of time. Accordingly, the power spectra determined in these linear devices reflect the sum over these modes. Outside the main plasma column, avaloids are observed to remain attached to the main plasma, hence their behavior does not become independent of the dynamics inside the main plasma column. Avaloid properties, assessed from imaging, agree with Langmuir probes done on various devices, except that the radial length is found to be much larger than previously determined because the blob-shape assumption is not valid. The link between fluctuations inside and outside the main plasma column indicates that the nonlinear evolution of the m=1 poloidal mode number is responsible for the creation of avaloids.

Quantum Chaos? Genericity and Nongenericity in the MHD Spectrum of Nonaxisymmetric Toroidal Plasmas

The eigenmode spectrum is a fundamental starting point for the analysis of plasma stability and the onset of turbulence, but the characterization of the spectrum even for the simplest plasma model, ideal magnetohydrodynamics (MHD), is not fully understood. This is especially true in configurations with no continuous geometric symmetry, such as in a real tokamak when the discrete nature of the external magnetic field coils is taken into account, or the alternative fusion concept, the stellarator, where axisymmetry is deliberately broken to provide a nonzero winding number (rotational transform) on each invariant torus of the magnetic field:line dynamics (assumed for present purposes to be an integrable Hamiltonian system). Quantum (wave) chaos theory provides tools for characterizing the spectrum statistically, from the regular spectrum of the separable case (integrable semiclassical dynamics) to that where the semiclassical ray dynamics is so chaotic that no simple classification of the individual eigenvalues is possible (quantum chaos). The MHD spectrum exhibits certain nongeneric properties, which we show, using a toy model, to be understable from the numb er-theoretic properties of the asymptotic spectrum in the limit of large toroidal and poloidal mode (quantum) numbers when only a single radial mode number is retained. Much more realistically, using the ideal MHD code CAS3D, we have constructed a data set of several hundred growth-rate eigenvalues for an interchange-unstable three-dimensional stellarator equilibrium with a rather flat, nonmonotonic rotational transform profile. A statistical analysis of eigenvalue spacings shows evidence of generic quantum chaos, which we attribute to the mixing effect of having a large number of radial mode numbers.

Magnetic Diagnostics of Magnetic Island in LHD

Characteristics of magnetic islands are investigated by magnetic diagnostics in the L arge Helical Device (LHD). The structure of the magnetic island with m/n = 1/ 1( where,m and n are poloidal and toroidal mode number, respectively) can be estimated from the perturbed magnetic field appearing when a magnetic island changes. To measure the toroidal profile of the perturbed magnetic field δb1 originating from the plasma, a toroidal array of magnetic flux loops is set up in the LHD. The toroidal profile of δb1 is then spatially Fourier decomposed to determine the amplitude of the n = 1c omponent, δb n=1 1 and its phase, φn=1 whic hc orrespond the change of the island width and the toroidal position of the X-point of the island, respectively. Therefore, the information about the magnetic island structure can be obtained from δb n=1 1 and φn=1 .I ncase the island width becomes larger than the seed island, measurements show that δb n=1 1 is non-zero and φn=1 is temporally constant. A non-zero δb n=1 1 can also be observed when the island width becomes smaller than the seed island. In this case, the angle φn=1 shifts by about π[rad] compared with the increasing case and the δb n=1 1 is limited to a certain value whic hc orresponding to the magnetic field suppressing the seed island.

Quasi-single helicity state by a small positive pulse of toroidal magnetic field in TPE-RX reversed field pinch experiment

By applying a small positive pulse (ΔBta) in toroidal magnetic field, the quasi-single helicity (QSH) state can be obtained with a controllable and reproducible manner in a reversed-field pinch (RFP) experiment on the large RFP machine, TPE-RX [Y. Yagi et al., Fusion Eng. Des. 45, 421 (1999)]. The QSH state in RFP is one of the states where the improved confinement can be observed, and is important for development toward the pure single helicity (SH) state. In the SH state, the dynamo-action for sustaining the RFP configuration will be driven by a single helical mode and its harmonics, and the anomalous plasma loss can be avoided which is caused by the multi-helicity dynamo action in ordinary RFPs. In the operating condition presented here, the reversal of toroidal magnetic field (Bta) is maintained at a shallow value (∼−1mT) for a certain period (∼20ms) after the setting up of the RFP configuration and then the positive ΔBta (≲5mT magnitude and ∼2ms width) is applied to the Bta, which is usually negative during the sustaining phase of RFP. Just after applying the pulse, the m∕n=1∕6 mode (m and n being the poloidal and toroidal Fourier mode numbers, respectively) grows dominantly and the configuration goes into QSH state. This QSH state can be sustained for a long period (up to ∼45ms) almost until the end of discharge by applying a delayed reversal of Bta with appropriate timing and magnitude. The setting up of the QSH states shows a reproducibility of almost 100% with the same timing corresponding to the applied positive pulse. This observation can confirm the interpretation in the former report [Y. Hirano et al., Phys. Plasmas 12, 112501 (2005)], in which it is claimed that the QSH state is obtained when a small positive pulse in toroidal magnetic field spontaneously appears.

Large-scale structures and turbulence saturation in gyrofluid ETG simulations

The dynamics of secondary fluctuations with poloidal long wavelength and low frequency in electron-temperature-gradient (ETG) turbulence saturation is considered in sheared slab geometry. A simplified modeling analysis shows that this kind of large-scale structure including streamers may saturate primary ETG instability through poloidal mode coupling. Spectral analyses in three-dimensional ETG simulations clearly exhibit the secondary excitation and the probable relevancy to KTG saturation. The result may implicate a lower fluctuation level in ETG turbulence.

Effects of the magnetic field model and wave polarisation on the estimation of proton number densities in the magnetosphere using field line resonances

The cold, core plasma mass density in the Earth's magnetosphere may be deduced from the resonant behaviour of ultra-low frequency (ULF; 1–100 mHz), magnetohydrodynamic (MHD) waves. Ground-based magnetometers are the most widely used instruments for recording the signature of ULF wave activity in the magnetosphere. For a suitable model of the background magnetic field and a functional form for the variation of the proton number density with radial distance, the resonant frequencies of ULF waves provide estimates of the equatorial plasma mass density. At high latitudes, the magnetic field model becomes critical when estimating the plasma mass density from FLR data. We show that a dipole field model is generally inadequate for latitudes greater than ∼65° geomagnetic compared with models that are keyed to magnetic activity, interplanetary magnetic field and solar wind properties. Furthermore, the method often relies on the detection of the fundamental ULF resonance, which changes frequency depending on the polarisation of the oscillation. Using idealised toroidal and poloidal oscillation modes, the range of the derived densities as the ULF wave polarisation changes is of the same order as changing the density function from a constant value throughout the magnetosphere to assuming constant Alfven speed in a dipole geometry.