Poloidal Wavenumber Research Articles

Resistive effects on line-tied magnetohydrodynamic modes in cylindrical geometry

An investigation of the effect of resistivity on the linear stability of line-tied magnetohydrodynamic (MHD) modes is presented in cylindrical geometry, based on the method recently developed in the paper by Evstatiev et al. [Phys. Plasmas 13, 072902 (2006)]. The method uses an expansion of the full solution of the problem in one-dimensional radial eigenfunctions. This method is applied to study sausage modes (m=0, m being the poloidal wavenumber), kink modes (m=1), and m=2 modes. All these modes can be resistively unstable. It is found that m≠0 modes can be unstable below the ideal MHD threshold due to resistive diffusion of the field lines, with growth rates proportional to resistivity. For these resistive modes, there is no indication of tearing, i.e., current sheets or boundary layers due to ideal MHD singularities. That is, resistivity acts globally on the whole plasma column and not in layers. Modes with m=0, on the other hand, can exist as tearing modes if the equilibrium axial magnetic field reverses sign within the plasma.

Detection of Cascading in Drift Wave Turbulence Using Probe Array in Linear Plasmas

Measurement of drift wave turbulence using multi-probe systems was performed in the Large Mirror DeviceUpgrade linear plasma. We compared a drift wave’s two-dimensional (poloidal wave number and frequency) power spectrum with its calculated linear dispersion relation. As a result, a few wave modes that satisfy the linear dispersion relation were observed. Moreover, cascades to non-mode peaks and broadband components that do not satisfy the dispersion relation were identified. c

Recent results on turbulence and MHD activity achieved by reflectometry

Over the last years, owing to hardware progress and the development of new methods, reflectometry has become a common diagnostic on plasma fusion devices. This paper presents some results obtained with reflectometry on transport, turbulence and magnetohydrodynamic (MHD). The emphasis is put on some new results from Tore-Supra. Combining the density profile and fluctuation measurement, it was shown on Tore-Supra that the particle pinch inside the q = 1 surface is close to the neoclassical value in ohmic plasma, while the observed small diffusion is in agreement with a very low level of density fluctuations inside the q = 1 surface. In β scaling experiments, no change in the fluctuation levels was found on Tore-Supra, in agreement with the observation of weak confinement degradation with increasing β. Zonal flows have been detected by Doppler reflectometry in ASDEX-U and with correlation reflectometry in T-10. On Tore-Supra, a fast decrease in the density fluctuation level at high poloidal wavenumbers was measured with Doppler reflectometry, suggesting a minor role of electron temperature gradient driven modes. Various forms of Alfvén eigenmodes (toroidal Alfvén eigenmodes, Alfvén cascades and possibly beta Alfvén eigenmodes) have been detected with reflectometry in TFTR, JET and Tore-Supra. The density fluctuations induced by the mode were found to be higher on the high-field side.

Simulation of fast Alfvén wave interaction with beam ions over a range of cyclotron harmonics in DIII-D tokamak

To study fast Alfvén wave damping on fast ions, a Monte-Carlo code, ORBIT-RF, has been coupled with a 2D full wave code, TORIC4. The ORBIT-RF/TORIC4 combination has been applied to DIII-D experimental conditions to investigate fast wave (FW) heating of injected beam ions over a range of ion cyclotron harmonics. ORBIT-RF using a single dominant toroidal and poloidal Fourier wave number qualitatively reproduces the strong FW–beam interaction at 60 MHz (4ΩD and 5ΩD) and the much weaker interaction at 116 MHz (8ΩD) in DIII-D L-mode plasmas, consistent with experimental observations. The result at 8ΩD differs from linear theory prediction using a Maxwellian to model the fast ion distribution function, suggesting the importance of finite orbit effect, Coulomb collisions for transport across flux surfaces and details of the non-Maxwellian fast ion distribution.

Investigation of geodesic acoustic mode oscillations in the T-10 tokamak

Geodesic acoustic modes (GAMs) were investigated on the T-10 tokamak using heavy ion beam probe, correlation reflectometry and multipin Langmuir probe diagnostics. Regimes with Ohmic heating and with on- and off-axis ECRH were studied. It was shown that GAMs are mainly the potential oscillations. Typically, the power spectrum of the oscillations has the form of a solitary quasi-monochromatic peak with the contrast range 3–5. They are the manifestation of the torsional plasma oscillations with poloidal wavenumber m = 0, called zonal flows. The frequency of GAMs changes in the region of observation and decreases towards the plasma edge. After ECRH switch-on, the frequency increases, correlating with growth in the electron temperature Te. The frequency of the GAMs depends on the local Te as , which is consistent with a theoretical scaling for GAM, where cs is the sound speed within a factor of unity. The GAMs on T-10 are found to have density limit, some magnetic components and an intermittent character. They tend to be more excited near low-q magnetic surfaces.

Theory of turbulence regulation by oscillatory zonal flows

The theory of turbulence regulation by oscillatory zonal flows is presented for passive scalar field models. Zonal flows are assumed to have linear spatial variation of the form U=−xΩ(t)ŷ, where Ω(t) has amplitude Ωm and frequency ωz. The flux and fluctuation levels are found to scale as 1∕∣kyUm∣ and τ*∕∣kyUm∣, respectively, for Ωm>ωz. Here, τ*=τη(ωz∕Ωm)2 is the effective decorrelation time, τη=τ*(Ω=0), Um=xΩm, and ky is the typical poloidal wave number of the turbulence. The effect of stochasticity of oscillatory zonal flows on shear decorrelation is discussed. The results complement the theory of turbulence regulation by low-frequency random zonal flows [E. Kim and P. H. Diamond, Phys. Rev. Lett 91, 075001 (2003)].

Transport phenomena in the edge of Alcator C-Mod plasmas

Two aspects of edge turbulence and transport in Alcator C-Mod are explored. The quasi-coherent mode (QCM), an edge fluctuation present in Enhanced Dα H-mode plasmas, is examined with regard to its role in the enhanced particle transport found in these plasmas, its in/out asymmetry, its poloidal wave number and its radial width and location. It is shown to play a dominant role in the perpendicular particle transport. The QCM is not observed at the inboard midplane, indicating that its amplitude there is significantly smaller than on the outboard side. The peak amplitude of the QCM is found just inside the separatrix, with a radial width ≳5 mm, leading to a non-zero amplitude outside the separatrix and qualitatively consistent with its transport enhancement. Also examined are the characteristics of the intermittent convective transport, associated with the larger scale turbulent structures, also called blobs, and typically occurring in the scrape-off-layer (SOL). These turbulent structures are qualitatively similar in L- and H-mode. When their perpendicular extent, occurrence frequencies and magnitudes are compared, it is found that their size is somewhat smaller in ELMfree H-Mode, while their frequency is similar. A clear difference is seen in the magnitude of these turbulent fluctuations in the far SOL, with ELMfree H-mode showing a smaller perturbation there than L-mode. As the Greenwald density limit is approached (n/nGW ⩾ 0.7), blobs are seen inside the separatrix consistent with the observation that the high cross-field transport region, normally found in the far SOL, penetrates the closed flux surfaces at high n/nGW.

Monte Carlo orbit/full wave simulation of ion cyclotron resonance frequency wave damping on resonant ions in tokamaks

To investigate the experimentally observed interaction between beam ion species and fast Alfvén wave (FW), a Monte Carlo code, ORBIT-RF [V. S. Chan, S. C. Chiu, and Y. A. Omelchenko, Phys. Plasmas 9, 501 (2002)], which solves the time-dependent Hamiltonian guiding center drift equations, has been upgraded to incorporate a steady-state neutral beam ion slowing-down distribution, a quasilinear high harmonic radio frequency diffusion operator and the wave fields from the two-dimensional ion cyclotron resonance frequency full wave code (TORIC4) [M. Brambilla, Plasma Phys. Controlled Fusion 41, 1 (1999)]. Comparison of ORBIT-RF simulation of power absorption with fixed amplitudes of FW fields from TORIC4 power absorption calculation, which assumes Maxwellian plasma distributions, attains agreement within a factor of two. The experimentally measured enhanced neutron rate is reproduced to within 30% from ORBIT-RF simulation using a single dominant toroidal and poloidal wave number.

Microwave Reflectometry of Turbulence Spectrum in Tokamak Plasmas

The recently proposed method of plasma turbulence spectrum measurements in tokamaks by means of conventional microwave reflectometry is extended to the case of arbitrary density profiles. The localization property and sensitivity of the method to the microwave scattering by fluctuations far from the cut-off layer is studied, with a conclusion that, depending on the turbulence spectrum and amplitude, such a scattering can deteriorate the results of spectrum reconstruction. However, if turbulence spectrum parameters do not reveal drastic radial variations, the cumulative effect of fluctuations far from the reflection area is not significant. In such a case the reconstructed poloidal wavenumber spectrum of density fluctuations is well localized near the cut-off layer. The studied cases correspond to the geometry and plasma parameters on the JT-60U tokamak.

Turbulence studies with means of reflectometry at TEXTOR

At TEXTOR, an O-mode heterodyne reflectometer system is installed and operated for the measurement of plasma density fluctuations and turbulence investigations. With two antenna arrays in the equatorial and top positions having two and three horn antennae, respectively, poloidal correlations are investigated under different plasma scenarios. From the amplitude, cross-phase and coherency spectrum, differences in the ohmic and auxiliary heated discharges are investigated. Furthermore the dynamic behaviour of the turbulence is studied in the SOC–IOC transition and in the precursor phase of a disruption. For the latter an increased integrated power spectral density was observed at the X-point of the mode compared with the O-point. Stationary m = 2 mode activity is observed for the first time at TEXTOR by reflectometry. The fluctuation level is calculated for different conditions and rises significantly increasing heating power which is consistent with the L-mode confinement degradation. Correlation measurements yield the measured phase delays which are used to calculate the poloidal phase velocity perpendicular to the magnetic field. In ohmic plasmas the turbulence rotates like a ‘rigid body’ with constant angular velocity inside the q = 2 surface. The rigid body rotation is broken up during tangential neutral beam injection. From the deduced poloidal wavenumber of the turbulence, most likely ion temperature gradient modes are the driving mechanism of the turbulence.

Global full-wave simulation of the Tore-Supra doppler reflectometer

The Tore-Supra doppler reflectometer (TSDR) aims at measuring the poloidal wave number spectrum (k spectrum) of the density fluctuations. To evaluate the performance and the sensitivity of the TSDR, simulations were made using a global full-wave code. The TSDR modeling with a finite-difference time-domain scheme and the required constraints of the simulation are presented, namely the account of the real size of the system: Distance horn plasma and plasma size. Particular attention is given to the role of the radiation pattern both on the TSDR spatial resolution and the wave number detection.

The effect of curvature on quasi-modes in coronal loops

This paper studies quasi-mode oscillations in models of coronal loops that include longitudinal curvature. Using a toroidal coordinate system to incorporate curvature in a basic coronal loop model, the linearized ideal MHD equations are solved for the plasma-β = 0. As a result of the curvature, quasi-modes with different poloidal wave numbers are coupled resulting in modifications of the frequencies. However, for small curvature, only the coupling of quasi-modes with a neighbouring poloidal wave number remains in first order. In addition, the quasi-mode frequencies are unchanged up to first order in the curvature. The imaginary part of the frequency, however, does change in first order, and quasi-modes are slightly more damped in realistically curved coronal loop configurations.

Reconstruction of wavenumber spectra of plasma turbulence in microwave reflectometry

A method of measuring the fluctuation spectrum by means of conventional microwave reflectometry is proposed and discussed. The technique is based on the analytical solution of the full-wave equation of microwave propagation and scattering inside the plasma. The reflectometer signal simulated using the reconstructed fluctuations is consistent with the amplitude spectrum of the experimental reflectometer signal. The accuracy of the method is tested on a number of specially modelled data sets, which correspond to isotropic and anisotropic fluctuations, multiple scattering by strong turbulence, and random motion of turbulent modes. We conclude that for the case of small fluctuations our method allows us to reconstruct the profile of the poloidal wavenumber spectrum with a high accuracy, while the turbulence amplitude is estimated within a 40–60% error bar. Multiple scattering deteriorates both the wavenumber spectrum profile and turbulence amplitude estimations.As an illustration, we apply our technique to estimate the fluctuation level and poloidal wavenumber spectrum shape in the ELM-free H mode plasma of the JT-60U tokamak.

Microwave imaging reflectometry for the measurement of turbulent fluctuations in tokamaks

This paper describes a numerical study of microwave reflectometry for the measurement of turbulent fluctuations in tokamak-like plasmas with a cylindrical geometry. Similarly to what was found previously in plane-stratified plasmas, the results indicate that the characteristics of density fluctuations cannot be uniquely determined from the reflected waves if the latter are allowed to propagate freely to the point of detection, as in standard reflectometry. Again, we find that if the amplitude of fluctuations is below a threshold that is set by the spectrum of poloidal wave numbers, the local characteristics of density fluctuations can be obtained from the phase of the reflected waves when these are collected with a wide aperture antenna, and an image of the cut-off is formed onto an array of phase-sensitive detectors.

Study of electromagnetic microinstabilities in helical systems with the stellarator expansion method

Electromagnetic microinstabilities in helical systems are studied by numerically solving integral eigenmode equations, which are derived from the ion gyrokinetic equation, the quasineutrality equation, the Ampère’s law, and the massless electron approximation. The stellarator expansion technique is used to evaluate finite-beta effects on the guiding-center drift in the helical configuration, where the toroidal plasma shift and the magnetic shear strongly influence the magnetic curvature and accordingly the stability of both magnetohydrodynamics (MHD) and kinetic modes. The kinetic integral equations are shown to reduce to the ideal MHD ballooning mode equation in the fluid limit, from which the Mercier criterion is obtained. For helical geometry like the Large Helical Device (LHD) [Motojima, et al., Nucl. Fusion 43, 1674 (2003)], it is confirmed that, when increasing the beta value, the ion temperature gradient mode is stabilized while the kinetic ballooning mode (KBM) is destabilized due to the unfavorable geodesic curvature resulting from the negative magnetic shear combined with the toroidal plasma shift. Also, dependencies of these kinetic-mode properties on the poloidal wave number and the magnetic shear are investigated. It is found that the KBM-unstable parameter region is narrower than the Mercier-unstable region in the LHD-like configuration.

Density fluctuation measurement at edge and internal transport barriers in JT-60U

A new analytical method using a combination of the O-mode reflectometer and a time-dependent two-dimensional full-wave simulation code has been developed for the quantitative evaluation of density fluctuations in JT-60U. Two statistical parameters of the reflectometer signals, fluctuation index (F) and elongation factor (χ), are introduced as measures of the fluctuation amplitude (γ) and the width of the poloidal wave number spectrum (kθ0). This method is applied to the edge transport barrier (ETB) and internal transport barrier (ITB). At the transition to the ELM free H-mode phase, analysis suggests that the density fluctuation level reduced from 1.9–3.2% to 0.29–0.44%, while the value of kθ0 changed from 1.6–2.0 to 0.77–0.81 cm−1 in the ETB region. On the other hand, the amplitude of the density fluctuation was evaluated as 1.0–2.0% at the ITB region, even after the formation of the box type ITB. Instead, when a pellet was injected into the plasma with a box type ITB as an external perturbation, a remarkable change in the frequency spectrum was observed. Analysis suggests a reduction in the density fluctuation level to 0.4–0.6% after the pellet injection.

Machine modification for active MHD control in RFX

Recent studies on RFP and Tokamak devices call for an active control of the MHD and resistive wall modes to induce plasma mode rotation and to prevent mode phase locking. The results obtained on RFX, where slow rotation of phase locked modes has been induced, support the possibility of extending active MHD mode control through a substantial modification of the device. A new first wall with an integrated system of electric and magnetic transducers has been realised. A close fitting 3 mm thick Cu shell replaces the 65 mm Al shell. A toroidal support structure (TSS) made of stainless steel replaces the shell in supporting all the forces acting on the torus. A system of 192 saddle coils is provided to actively control the MHD modes. This system completely surrounds the toroidal surface and allows the generation of harmonic fields with m=0 and m=1 poloidal wave number and with a toroidal spectrum up to n=24.

Observations of the turbulence in the scrape-off-layer of Alcator C-Mod and comparisons with simulation

The intermittent turbulent transport in the scrape-off-layer (SOL) of Alcator C-Mod [I.H. Hutchinson, R. Boivin, P.T. Bonoli et al., Nucl. Fusion 41, 1391 (2001)] is studied experimentally by imaging with a very high density of spatial measurements. The two-dimensional structure and dynamics of emission from a localized gas puff are observed, and intermittent features (also sometimes called “filaments” or “blobs”) are typically seen. The characteristics of the spatial structure of the turbulence and their relationship to the time-averaged SOL profiles are discussed and compared with those measured on the National Spherical Torus Experiment [M. Ono, S. M. Kaye, Y.-K. M. Pong et al., Nucl. Fusion 40, 557 (2000)]. The experimental observations are compared also with three-dimensional nonlinear numerical simulations of edge turbulence. Radial profiles of the poloidal wave number spectra and the poloidal scale length from the simulations are in reasonable agreement with those obtained from the experimental images, once the response of the optical system is accounted for. The resistive ballooning mode is the dominant linear instability in the simulations. The ballooning character of the turbulence is also consistent with fluctuation measurements made at the inboard and outboard midplane, where normalized fluctuation levels are found to be about 10 times smaller on the inboard side. For discharges near the density limit, turbulent structures are seen on closed flux surfaces.

Laboratory characterization of an imaging reflectometer system

While microwave reflectometry has proven to be a sensitive tool for measuring electron density fluctuations in many circumstances, it has also been shown to have limited viability for core measurements and/or conditions of strong turbulence. To this end, a new instrument based on two-dimensional imaging reflectometry has been developed to measure density fluctuations over an extended plasma region in the TEXTOR tokamak. Laboratory characterization of this instrument has been performed using corrugated reflecting targets as an approximation to plasma reflections including two-dimensional turbulent fluctuations of various magnitude and poloidal wavenumber. Within this approximation, the imaging reflectometer can recover the spectral and spatial characteristics of the reflection layer lost to or otherwise inaccessible to conventional techniques.

Radial wave number spectrum of density fluctuations deduced from reflectometry phase signals

A method to extract the radial wave-number spectrum of the density fluctuations from the reflectometry phase signal is proposed and the main points to interpret it are described. The method buildup is based on reflectometry studies devoted to phase or amplitude variations induced by coherent density perturbations. These previous analyses show that all the wave numbers up to the Bragg detection limit kf (fluctuation radial wave number) <2ko (incident wave number in vacuum) contribute to the phase fluctuations. Although the main part comes from the vicinity of the cutoff layer the entire probing zone contributes to the phase fluctuations. The phase fluctuations dependencies in kf and L (length of the density gradient) found in the literature for the O mode have been estimated for the X mode in the Bragg backscattering zone in the case of extensive density perturbations. The assumptions used to establish this diagnostic and its limitations are discussed. The numerical study with a FDTD code in two-dimensional configuration including the reflectometer antenna shows that the contribution of the poloidal wave number can be neglected in the studied cases. The method used to recover the kf spectrum from the Tore Supra swept reflectometer data and the results are presented.