Ion Sound Larmor Radius Research Articles

Isotope effects on energy transport in the core of ASDEX-Upgrade tokamak plasmas: Turbulence measurements and model validation

Design and operation of future tokamak fusion reactors using a deuterium–tritium 50:50 mix requires a solid understanding of how energy confinement properties change with ion mass. This study looks at how turbulence and energy transport change in L-mode plasmas in the ASDEX Upgrade tokamak when changing ion species between hydrogen and deuterium. For this purpose, both experimental turbulence measurements and modeling are employed. Local measurements of ion-scale (with wavevector of fluctuations perpendicular to the B-field k⊥< 2 cm−1, k⊥ρs< 0.2, where ρs is the ion sound Larmor radius using the deuterium ion mass) electron temperature fluctuations have been performed in the outer core (normalized toroidal flux ρTor=0.65−0.8) using a multi-channel correlation electron cyclotron emission diagnostic. Lower root mean square perpendicular fluctuation amplitudes and radial correlation lengths have been measured in hydrogen vs deuterium. Measurements of the cross-phase angle between a normal-incidence reflectometer and an ECE signal were made to infer the cross-phase angle between density and temperature fluctuations. The magnitude of the cross-phase angle was found larger (more out-of-phase) in hydrogen than in deuterium. TRANSP power balance simulations show a larger ion heat flux in hydrogen where the electron-ion heat exchange term is found to play an important role. These experimental observations were used as the basis of a validation study of both quasilinear gyrofluid trapped gyro-Landau fluid-SAT2 and nonlinear gyrokinetic GENE codes. Linear solvers indicate that, at long wavelengths (k⊥ρs<1), energy transport in the deuterium discharge is dominated by a mixed ion-temperature-gradient (ITG) and trapped-electron mode turbulence while in hydrogen transport is exclusively and more strongly driven by ITG turbulence. The Ricci validation metric has been used to quantify the agreement between experiments and simulations taking into account both experimental and simulation uncertainties as well as four different observables across different levels of the primacy hierarchy.

Stability of a weakly collisional plasma with runaway electrons

We investigate the problem of the tearing stability of a post-disruption weakly collisional plasma where the current is completely carried by runaway electrons. We adopt here a two fluid model which takes into account also ion sound Larmor radius and electron inertia effects in the description of the reconnection process. In the past, it has been demonstrated in [Helander et al. Phys. Plasmas 14, 12, (2007)] that in the purely resistive regime the presence of runaway electrons in plasma has a significant effect on the saturated magnetic island width. In particular, runaway electrons generated during disruption can cause an increase of 50% in the saturated magnetic island width with respect to the case with no runaway electrons. These results were obtained adopting a periodic equilibrium magnetic field that limited the analysis to small size saturated magnetic islands. Here we present our results to overcome this limitation adopting a non-periodic Harris’ type equilibrium magnetic field. Preliminary results on the effects of the ion sound Larmor radius effects will also be presented.

Microscopic scales of linear tearing modes: a tutorial on boundary layer theory for magnetic reconnection

We revise in detail and in a pedagogical way the analysis of the boundary layer theory of warm tearing modes in slab, reduced magnetohydrodynamics (MHD), when magnetic reconnection is driven by electron inertia and/or resistivity, and ion-sound Larmor radius effects are included. By comparison with the numerical solution of the corresponding eigenvalue problem, we interpret these results by means of a heuristic approach, which in the warm-electron regime, we show to be in general not feasible without knowledge of the scaling of the gradient of the magnetic flux function, differently from what happens in the cold-electron regimes. We put in evidence for a non-trivial relation between the first derivative of the magnetic flux function and of the velocity parallel to the neutral line, evaluated in its proximity, by thus providing insight to the multiple boundary layer analysis that Pegoraro & Schep (Plasma Phys. Control. Fusion, vol. 28, 1986, p. 647) first showed to be required in warm-tearing regimes. In this way, we also suggest and justify a general operational definition of the reconnecting layer width and we discuss the linear appearance of microscopic scales related to the gradients of the eigenfunctions of the tearing modes.

Moving toward mass-conserving simulations of plasma turbulence and kinetic neutrals in the tokamak boundary with the GBS code

A model for the simulation of the tokamak boundary is described, which is mass-conserving to leading order in ρs/R0 (where ρs is the ion sound Larmor radius and R0 is the tokamak major radius at the magnetic axis). The model integrates a three-dimensional plasma turbulence model, based on the two-fluid drift-reduced Braginskii equations, a neutral model, based on the discretization of the Boltzmann equation integrated along the neutral path, and proper boundary conditions at the vessel walls. The implementation of the model in the GBS code [Ricci et al., Plasma Phys. Control. Fusion 54, 124047 (2012)] is described and a technical proof of mass conservation is obtained. The simulation results highlight the impact of mass conservation on the plasma and neutral profiles, and the formation of the plasma density profile is explained in terms of plasma and neutral particle fluxes.

Excitation of the axisymmetric Alfvén eigenmodes by micro-turbulence

Electrostatic, short-wavelength (kθρs>1, with ρs being the ion sound Larmor radius) micro-turbulence, which is present at the plasma edge due to its intrinsic immunity to equilibrium E × B shear suppression, can drive global, axisymmetric [n=0,m=±1, with n(m) being the toroidal (poloidal) mode number] Alfvén eigenmode (GAE) with the inertial layer located at the plasma edge, i.e., in the turbulent zone. The excitation mechanism is the modulation instability induced by turbulent Reynolds stress. The growth rate of this instability can overcome GAE continuum damping.

A gyrokinetic model for the plasma periphery of tokamak devices

A gyrokinetic model is presented that can properly describe large and small amplitude electromagnetic fluctuations occurring on scale lengths ranging from the electron Larmor radius to the equilibrium perpendicular pressure gradient scale length, and the arbitrarily large deviations from thermal equilibrium that are present in the plasma periphery of tokamak devices. The formulation of the gyrokinetic model is based on a second-order accurate description of the single charged particle dynamics, derived from Lie perturbation theory, where the fast particle gyromotion is decoupled from the slow drifts assuming that the ratio of the ion sound Larmor radius to the perpendicular equilibrium pressure scale length is small. The collective behaviour of the plasma is obtained by a gyrokinetic Boltzmann equation that describes the evolution of the gyroaveraged distribution function. The collisional effects are included by a nonlinear gyrokinetic Dougherty collision operator. The gyrokinetic model is then developed into a set of coupled fluid equations referred to as the gyrokinetic moment hierarchy. To obtain this hierarchy, the gyroaveraged distribution function is expanded onto a Hermite–Laguerre velocity-space polynomial basis. Then, the gyrokinetic equation is projected onto the same basis yielding the spatial and temporal evolution of the Hermite–Laguerre expansion coefficients. A closed set of fluid equations for the lowest-order coefficients is presented. The Hermite–Laguerre projection is performed accurately at arbitrary perpendicular wavenumber values. Finally, the self-consistent evolution of the electromagnetic fields is described by a set of gyrokinetic Maxwell equations derived from a variational principle where the velocity integrals are explicitly evaluated.

Interaction of magnetic islands with turbulent electron temperature fluctuations in DIII-D and in GENE nonlinear gyrokinetic simulations

We present localized measurements of interactions between neoclassical tearing modes (NTMs) and turbulent electron temperature fluctuations () in the expected kθρs = 0–0.5 wave-number range of the ion temperature gradient (ITG) instability measured via correlation electron cyclotron emission (kθ and ρs are the poloidal wave-number and ion-sound Larmor radius, respectively). Comparison to GENE gyrokinetic simulations, shows qualitative agreement with being reduced (increased) inside (outside) of the islands due to modified local gradients, and being higher (lower) in the region outside the island where lower (higher) flow shear is expected due to radially asymmetric island shape. These results are consistent with previously reported modifications of long and intermediate wavelength turbulent density fluctuations and cross-field electron thermal diffusivity reduction at the O-point of magnetic islands. Interestingly, a 30% increase of is detected at the NTM rational surface when the island width is a few times ρs, which is replicated by GENE through zonal flow damping due to the destroyed magnetic field topology of the NTM rational surface.

Direction of cascades in a magnetofluid model with electron skin depth and ion sound Larmor radius scales

The direction of cascades in a two-dimensional model that takes electron inertia and ion sound Larmor radius into account is studied, resulting in analytical expressions for the absolute equilibrium states of the energy and helicities. These states suggest that typically both the energy and magnetic helicity at scales shorter than the electron skin depth have a direct cascade, while at large scales the helicity has an inverse cascade as established earlier for reduced magnetohydrodynamics (MHD). The calculations imply that the introduction of gyro-effects allows for the existence of negative temperature (conjugate to energy) states and the condensation of energy to the large scales. Comparisons between two- and three-dimensional extended MHD models (MHD with two-fluid effects) show qualitative agreement between the two.

Marginal stability, characteristic frequencies, and growth rates of gradient drift modes in partially magnetized plasmas with finite electron temperature

The detailed analysis of stability of azimuthal oscillations in partially magnetized plasmas with crossed electric and magnetic fields is presented. The instabilities are driven by the transverse electron current which, in general, is due to a combination of E×B and electron diamagnetic drifts. Marginal stability boundary is determined for a wide range of the equilibrium plasma parameters. It is shown that in some regimes near the instability threshold, only the low-frequency long-wavelength oscillations are unstable, while the short-wavelength high-frequency modes are stabilized by the finite Larmor radius effects. Without such stabilization, the high-frequency modes have much larger growth rates and dominate. A new regime of the instability driven exclusively by the magnetic field gradient is identified. Such instability takes place in the region of the weak electric field and for relatively large gradients of plasma density (ρs/ln>1, where ρs is the ion-sound Larmor radius and ln is the scale length of plasma density inhomogeneity).

The onset of ion heating during magnetic reconnection with a strong guide field

The onset of the acceleration of ions during magnetic reconnection is explored via particle-in-cell simulations in the limit of a strong ambient guide field that self-consistently and simultaneously follow the motions of protons and α particles. Heating parallel to the local magnetic field during reconnection with a guide field is strongly reduced compared with the reconnection of anti-parallel magnetic fields. The dominant heating of thermal ions during guide field reconnection results from pickup behavior of ions during their entry into reconnection exhausts and dominantly produces heating perpendicular rather than parallel to the local magnetic field. Pickup behavior requires that the ion transit time across the exhaust boundary (with a transverse scale of the order of the ion sound Larmor radius) be short compared with the ion cyclotron period. This translates into a threshold in the strength of reconnecting magnetic field that favors the heating of ions with high mass-to-charge. A simulation with a broad initial current layer produces a reconnecting system in which the amplitude of the reconnecting magnetic field just upstream of the dissipation region increases with time as reconnection proceeds. The sharp onset of perpendicular heating when the pickup threshold is crossed is documented. A comparison of the time variation of the parallel and perpendicular ion heating with that predicted based on the strength of the reconnecting field establishes the scaling of ion heating with ambient parameters both below and above the pickup threshold. The relevance to observations of ion heating in the solar corona is discussed.

Density fluctuation measurements by far-forward collective scattering in the MST reversed-field pinch

The multichannel polarimeter-interferometer system on the MST reversed-field pinch can be utilized to measure far-forward collective scattering from electron density fluctuations. The collective scattering system has 11 viewing chords with ∼8 cm spacing. The source is a 432 μm (694 GHz) far infrared laser and the scattered power is measured using a heterodyne detection scheme. Collective scattering provides a line-integrated measurement of fluctuations within the divergence of the probe beam covering wavenumber range: k(⊥) < 1.3 cm(-1), corresponding k(⊥)ρ(s) < 1.3 (ρ(s) is the ion-sound Larmor radius), the region of primary interest for turbulent fluctuation-induced transport. The perpendicular wavenumber consists of toroidal, poloidal, and radial contributions, which vary with chord position. Coherent modes associated with tearing instabilities and neutral-beam driven fast particles are observed along with broadband turbulence at frequencies up to 500 kHz. Changes in frequency are consistent with a Doppler shift due to parallel plasma flow.

Seed island formation by forced magnetic reconnection

Neoclassical tearing modes observed in experiments often grow from seed magnetic islands induced by triggers like sawteeth. The formation of seed islands is studied in this paper using both the reduced MHD and two-fluid equations, with the trigger being modelled by externally applied resonant magnetic perturbations. In the linear phase the growth rate of the driven mode is found to be the same as that of the trigger. A slowly growing trigger drives a tearing mode, while a fast one drives a kink-like mode, which becomes a tearing mode later when the trigger's growth slows down. A finite ion sound Larmor radius (ion Larmor radius by using electron temperature) and electron inertia are found to lead to a larger seed island for a given external perturbation. The electron diamagnetic drift and plasma rotation, if increasing the relative rotation between the trigger and the driven mode, decrease the seed island width.

Turbulence in the TORE SUPRA Tokamak: Measurements and Validation of Nonlinear Simulations

Turbulence measurements in TORE SUPRA tokamak plasmas have been quantitatively compared to predictions by nonlinear gyrokinetic simulations. For the first time, numerical results simultaneously match within experimental uncertainty (a) the magnitude of effective heat diffusivity, (b) rms values of density fluctuations, and (c) wave-number spectra in both the directions perpendicular to the magnetic field. Moreover, the nonlinear simulations help to revise as an instrumental effect the apparent experimental evidence of strong turbulence anisotropy at spatial scales of the order of ion-sound Larmor radius.

Studies of turbulence and transport in Alcator C-Mod ohmic plasmas with phase contrast imaging and comparisons with gyrokinetic simulations

Recent advances in gyrokinetic simulation have allowed for quantitative predictions of core turbulence and associated transport. However, numerical codes must be tested against experimental results in both turbulence and transport. In this paper, we present recent results from ohmic plasmas in the Alcator C-Mod tokamak using phase contrast imaging (PCI) diagnostic, which is capable of measuring density fluctuations with wave numbers up to 55 cm−1. The experiments were carried out over the range of densities covering the ‘neo-Alcator’ (linear confinement time scaling with density, electron transport dominates) to the ‘saturated ohmic’ regime. We have also simulated these plasmas with the gyrokinetic code GYRO and compared numerical predictions with experimentally measured turbulence through a synthetic PCI diagnostic method. The key role played by the ion temperature gradient (ITG) turbulence has been verified, including measurements of turbulent wave propagation in the ion diamagnetic direction. It is found that the intensity of density fluctuations increases with density, in agreement between simulation and experiments. The absolute fluctuation intensity agrees with the simulation within experimental error (±60%). In the saturated ohmic regime, the simulated ion and electron thermal diffusivities also agree with experiments after varying the ion temperature gradient within experimental uncertainty. However, in the linear ohmic regime, GYRO does not agree well with experiments, showing significantly larger ion thermal transport and smaller electron thermal transport. Our study shows that although the short wavelength turbulence in the electron temperature gradient (ETG) range is unstable in the linear ohmic regime, the nonlinear simulation with kθρs up to 4 does not raise the electron thermal diffusivity to the experimental level, where kθ is the poloidal wavenumber and ρs is the ion-sound Larmor radius. At the present time, it is not known whether even shorter wavelength turbulence would account for the measured electron transport.

Clarifications to the limitations of the s-α equilibrium model for gyrokinetic computations of turbulence

In the context of gyrokinetic flux-tube simulations of microturbulence in magnetized toroidal plasmas, different treatments of the magnetic equilibrium are examined. Considering the Cyclone DIII-D base case parameter set [Dimits et al., Phys. Plasmas 7, 969 (2000)], significant differences in the linear growth rates, the linear and nonlinear critical temperature gradients, and the nonlinear ion heat diffusivities are observed between results obtained using either an s-α or a magnetohydrodynamic (MHD) equilibrium. Similar disagreements have been reported previously [Redd et al., Phys. Plasmas 6, 1162 (1999)]. In this paper it is shown that these differences result primarily from the approximation made in the standard implementation of the s-α model, in which the straight field line angle is identified to the poloidal angle, leading to inconsistencies of order ε (ε=a/R is the inverse aspect ratio, a the minor radius and R the major radius). An equilibrium model with concentric, circular flux surfaces and a correct treatment of the straight field line angle gives results very close to those using a finite ε, low β MHD equilibrium. Such detailed investigation of the equilibrium implementation is of particular interest when comparing flux tube and global codes. It is indeed shown here that previously reported agreements between local and global simulations in fact result from the order ε inconsistencies in the s-α model, coincidentally compensating finite ρ∗ effects in the global calculations, where ρ∗=ρs/a with ρs the ion sound Larmor radius. True convergence between local and global simulations is finally obtained by correct treatment of the geometry in both cases, and considering the appropriate ρ∗→0 limit in the latter case.

Kinetic theory of electromagnetic geodesic acoustic modes

The electromagnetic effects on geodesic acoustic modes due to electron parallel motion are investigated by employing the kinetic theory and appropriate expansion of the electron distribution function. The existence of several electromagnetic modes involving average geodesic curvature with different frequencies and polarization are revealed. It is shown that there are two Alfven side-band modes with different polarizations which also have different values of adiabatic indices due to average geodesic compressibility. The dispersion corrections to the eigen-frequencies of geodesic acoustic and related electromagnetic modes are investigated. It is shown that the dispersion due to electron dynamics is different depending on whether the electron response is adiabatic or hydrodynamic, i.e. depends on the ratio of the parameter, . It is noted that for m = n = 0 only the hydrodynamic regime is possible, . Principal electron effects leading to the dispersion involve a finite ion-sound Larmor radius , electron inertia and a finite electron magnetic drift frequency, .

Nonlinear acceleration of the electron inertia-dominated magnetohydrodynamic modes due to electron parallel compressibility

The behavior of the collisionless magnetohydrodynamics modes is investigated by the gyrokinetic particle simulation in a cylindrical tokamak plasma in the parameter region where the effects of electron inertia and electron parallel compressibility are competitive for magnetic reconnection. Although the linear growth of the m=1 internal kink-tearing mode is dominated by the electron inertia, it is found that the growth rate can be nonlinearly accelerated due to the electron parallel compressibility proportional to the ion sound Larmor radius ρs. It is also found that, as decreasing the electron skin depth δe, the maximum growth rate before the internal collapse saturates independently of the microscopic scales such as δe and ρs. The acceleration of growth rate is also observed in the nonlinear phase of the m=2 double tearing mode.

Growth and stabilization of drift-tearing modesin weakly collisional plasmas

In the limit where the electron drift-wave frequency exceeds theelectron-ion collision frequency, drift-tearing modes are found to growwith a linear growth rate independent of resistivity and proportional tothe product of the electron inertial skin depth and the ion sound Larmorradius. The stabilization of these modes in collisionless andsemi-collisional regimes is investigated. The stabilization mechanism isrelated to the coupling and propagation of drift-acoustic perturbationsaway from the reconnecting mode-rational surface. Analytic and numericalsolutions of the four-field reduced fluid model in the slab geometryapproximation with constant electron temperature and negligible iontemperature are presented. The actual stability threshold can occur atvalues of the normalized tearing mode stability parameter Δ' ashigh as 102.\\pacs{52.35.P}}\\fnm{1}{Permanent address: Istituto Nazionale Fisica della Materia, Department of Energetics, Politecnico di Torino, Italy

Hamiltonian magnetic reconnection

Magnetic reconnection in two dimensional (2D), collisionless, non-dissipative regimes is investigated analytically and numerically by means of a finite difference code in the nonlinear regime where the island size becomes macroscopic. The cross-shaped structure of the reconnection region, originally reported by Cafaro et al (1998 Phys. Rev. Lett. 80 20) is analysed as a function of the ratio between the ion sound Larmor radius and the inertial skin depth. This cross shape structure is found to survive in the presence of weak dissipation. Further insight on the quasi-explosive behaviour of the magnetic island width as a function of time and on the spatial structure of the perturbed current density is provided. We confirm that the amount of reconnected flux becomes of order unity on the time scale of the inverse linear growth rate. Results in the collisionless limit are interpreted on the basis of the Hamiltonian properties of the adopted collisionless, 2D, fluid model. Thus, collisionless reconnection is a fast, non-steady-state process, fundamentally different from 2D resistive magnetic reconnection, of which the Sweet-Parker model is the classic paradigm.