Gyrofluid Model Research Articles

The toroidal momentum diffusivity in a tokamak plasma: A comparison of fluid and kinetic calculations

Fluid and gyrokinetic calculations of the toroidal momentum diffusivity in a tokamak are compared. The four-moment gyrofluid model predicts the Prandtl number connected with the ion temperature gradient mode reasonably well provided the drift term is kept in the momentum balance. Without the drift term in the momentum balance, some previous gyrofluid models predicted small values of the Prandtl number in the range of experimental observations. It is shown that the drift term enters in the fluid equations through the gyroviscosity. Gyrokinetic calculations of the ion temperature gradient mode with kinetic electrons, and for experimentally relevant parameters yield a Prandtl number in the range 0.7–1.2.

An iterative semi-implicit scheme with robust damping

An efficient, iterative semi-implicit (SI) numerical method for the time integration of stiff wave systems is presented. Physics-based assumptions are used to derive a convergent iterative formulation of the SI scheme which enables the monitoring and control of the error introduced by the SI operator. This iteration essentially turns a semi-implicit method into a fully implicit method. Accuracy, rather than stability, determines the timestep. The scheme is second order accurate and shown to be equivalent to a simple preconditioning method. We show how the diffusion operators can be handled so as to yield the property of robust damping, i.e. dissipating the solution at all values of the parameter DΔt, where D is a diffusion operator and Δt the timestep. The overall scheme remains second order accurate even if the advection and diffusion operators do not commute. In the limit of no physical dissipation, and for a linear test wave problem, the method is shown to be symplectic. The method is tested on the problem of Kinetic Alfvén wave mediated magnetic reconnection. A Fourier (pseudo-spectral) representation is used. A two-field gyrofluid model is used and an efficacious k-space SI operator for this problem is demonstrated. CPU speed-up factors over a CFL-limited explicit algorithm ranging from ∼20 to several hundreds are obtained, while accurately capturing the results of an explicit integration. Possible extension of these results to a real-space (grid) discretization is discussed.

Nonlinear polarization and dissipative correspondence between low-frequency fluid and gyrofluid equations

The correspondence between gyrofluid and low-frequency fluid equations is examined. The lowest-order conservative effects in E×B advection, parallel dynamics, and curvature match trivially. The principal concerns are polarization fluxes, and dissipative parallel viscosity and parallel heat fluxes. The emergence of the polarization heat flux in the fluid model and its contribution to the energy theorem is reviewed. It is shown that gyroviscosity and the polarization fluxes are matched by the finite gyroradius corrections to advection in the long-wavelength limit, provided that the differences between gyrocenter and particle representations are taken into account. The dissipative parallel viscosity is matched by the residual thermal anisotropy in the gyrofluid model in the collision-dominated limit. The dissipative parallel heat flux is matched by the gyrofluid parallel heat flux variables in the collision-dominated limit. Hence, the gyrofluid equations are a complete superset of the low-frequency fluid equations.

Tokamak edge turbulence: background theory and computation

The basic scales of motion and computational requirements for low frequency fluid drift turbulence are summarized in tutorial fashion, with emphasis on the tokamak edge region. Parameters are given by experimental observations, but the computations are otherwise done from first principles. Edge turbulence is fundamentally electromagnetic and nonlinear, not treatable by standard linear or secondary instability analysis. Energetic character is determined by diagnosis of the terms in the energy theorem within the fully developed saturated phase. The spectra of the fluctuations and transport always extend to below the ion gyroradius scale. Direct coupling of pressure fluctuations and E-cross-B eddies through the parallel current is always active. Edge turbulence derives its character from steep gradients, with a parallel/perp scale ratio larger than 100, rather than from collisional effects. Collisionality is neither absent nor strongly dominant for electrons, but very weak for ions. Fluctuations in the axisymmetric component, including the Pfirsch–Schlüter currents, are dynamically integrated into the turbulence. Time scales are one to two orders of magnitude shorter than the ion collision time, hence significant delays occur in the response of heat fluxes and viscosity to temperature gradients and flows. Hence the need for a trans-collisional gyrofluid model to treat cases with comparable ion and electron temperature. Two orders of magnitude in spatial scales and three in time scales are typically involved.

Nonlinear Simulation of Ion Temperature Gradient Driven Drift Wave Turbulence

The gyro-fluid model proposed by Ottaviani et. al is extended, in which the ion parallel flow and ion neoclassical viscosity are taken into account. Using this model, the ion temperature gradient driven drift wave turbulence is analyzed. The saturation level and ion heat flux are evaluated in case with profile relaxation. The multi-scale interaction between the micro-scale and meso-scale is observed in our simulations. The avalanche process and turbulence spreading occur in the nonlinear phase. In addition, the various kind of parallel scheme are tested to improve the performance of the code. We find a way which gives better parallel efficiency of the spectral code.

Edge Turbulence and its Interaction with the Equilibrium

Tokamak edge turbulence is studied using three dimensional computations in a transcollisional, electromagnetic gyrofluid model. All scales between the ion gyroradius and the profile scale length are carried for up to the transport time scale. Profiles and the MHD equilibrium are time dependently self consistent. Basic properties of the turbulence are shown. Zonal flows are compressible and not in equilibrium, though the electric field profile on larger scales is indeed in equilibrium. Edge turbulence in the closed field line region maintains its character whether a scrape-off layer is present, though in the SOL region it becomes interchange like. Divertor separatrix topology is briefly considered, with little effect on the general turbulence/equilibrium situation. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Io‐Jupiter interaction: Alfvén wave propagation and ionospheric Alfvén resonator

A linear, one‐dimensional gyrofluid code has been used to determine the characteristics of propagating Alfvén waves and the ionospheric Alfvén resonator on a Jupiter‐Io flux tube. This model includes electron inertia, electron pressure gradient, and finite ion gyroradius effects, as well as the displacement current correction to prevent the Alfvén velocity from exceeding the speed of light. A quasi‐steady Vlasov code provides realistic density profiles along the flux tube as input parameters for the gyrofluid model. In this paper, we demonstrate that the majority of the wave energy from an initial pulse with a long wavelength (∼0.1 RJ) is unable to reach Jupiter's ionosphere without wave breaking, phase mixing, and/or other nonlinear processes; however, a significant energy flux may be transferred via high‐frequency, small‐wavelength waves to the ionosphere. The waves that reach the ionosphere stimulate an ionospheric Alfvén resonator which is generated between the ionospheric boundary and the first velocity peak of the Alfvén phase speed. The ionospheric density and scale height play important roles to determine the resonant frequency. The eigenfrequency decreases with increasing scale height and with increasing ionospheric density. The fundamental frequency and higher harmonics of the Alfvén resonator are comparable to the observed reoccurring frequency of S bursts between a few and hundreds of Hz. On the basis of this information, we suggest the Alfvén resonator as the likely driver explaining multiple occurrences of S bursts.

Dynamics of large-scale structure and electron transport in tokamak microturbulence simulations

The important issue of whether zonal flows or streamers are preferentially formed in plasma turbulence with electron gyroradius scale is studied based on a gyrofluid model of electron temperature gradient (ETG) driven turbulence. Results from three approaches are presented. It is analytically derived first that the secondary generation of different large-scale structures is determined by the spectral anisotropy of turbulent fluctuation in two-dimensional Charney–Hasegawa–Mima turbulence. This is verified subsequently using three-dimensional simulations of sheared slab ETG turbulence, which show that the magnetic shear governs the pattern selection. It is found that a weak shear favours the enhancement of zonal flows so that the electron transport is strongly suppressed. In contrast, radially elongated streamers are formed nonlinearly in stronger-shear ETG turbulence. Finally, three-dimensional toroidal ETG simulations show that streamers are excited in the linearly stable region along the field (i.e. good curvature region) through a modulation instability after initial saturation of ETG modes. Although the electron transport at the quasi-steady state becomes higher than the initial saturation level, which is dominated by fluctuations with a peaked spectrum, the averaged value is still low at around the gyro-Bohm level. Furthermore, it is shown that the enhanced zonal flows in weak shear ETG turbulence may be limited by a Kelvin–Helmholtz instability. Also, it is found that the electromagnetic effects reduce the generation of zonal flows and reverse the so-called Okawa-scaling of electron transport on the β dependence.

Free-energy conservation in local gyrofluid models

The details of fluctuation free-energy conservation in gyrofluid models are examined. The polarization equation relates E×B flow and eddy energy to combinations of the potential and the density and perpendicular temperature. These determine the combinations which must appear under derivatives in the moment equations so that not only thermal free energy but also its combination with the E×B energy is properly conserved by the parallel and perpendicular compressional effects. The resulting system exhibits the same qualitative energy-transfer properties as corresponding Braginskii or Landau fluid models. One clear result is that the numerical model built on these equations is well behaved for arbitrarily large perpendicular wave number, allowing exploration of two scale phenomena linking dynamics at the ion and electron gyroradii. When the numerical formulation is done in the globally consistent flux tube model, the results with adiabatic electrons are consistent with the “Cyclone base case” results of gyrokinetic models.

Nonlocal nonlinear electrostatic gyrofluid equations: A four-moment model

Extending a previous single-temperature model, an electrostatic gyrofluid model that includes anisotropic temperatures (T‖≠T⊥) and can treat general nonlinear situations is constructed. The model is based on a Lagrangian formulation of gyrofluid dynamics, which leads to an exact energy conservation law. Diamagnetic cancellations are inserted manually in such a way that energy conservation is preserved. Comparison with previous models shows a very good agreement for zero-Larmor-radius terms in the gyrofluid equations of motion.

Nonlocal nonlinear electrostatic gyrofluid equations

Building on Lagrangian field theory methods of fluid dynamics, we construct a set of equations for an electrostatic gyrofluid model which can treat arbitrarily nonlinear situations. Noether’s theorem is used to find the exact energy theorem satisfied by the equations. The exchange of energy between the E×B fluid drift and thermal/kinetic parts of the dynamics is recovered rigorously. Diamagnetic cancellations are inserted manually.

Numerical study of zonal flow dynamics and electron transport in electron temperature gradient driven turbulence

The electron temperature gradient (ETG) driven turbulence in tokamak core plasmas is numerically investigated based on a three-dimensional gyrofluid model with adiabatic ion response. Attentions are focused on the zonal flow dynamics in ETG fluctuations and the resultant electron heat transport. A high electron energy confinement mode is found in the weak magnetic shear region, which is relevant to the self-organization behavior of turbulence through the enhanced zonal flow dynamics rather than the weak shear stabilization of ETG fluctuations. It is demonstrated that the weak shear favors the enhancement of zonal flows in ETG turbulence. The enhanced zonal flows may be plausibly limited by the excitation of a secondary Kelvin–Helmholtz mode, which is observed from the spatial spectral analyses and time-frequency wavelet analyses of turbulent fluctuations. Electromagnetic ETG simulations show that while the Ohkawa’s scaling of anomalous electron transport with beta, χe∝1/βe, is reproduced in the moderate shear plasma, the finite beta effect may reverse the Ohkawa’s scaling in weak shear ETG turbulence due to the reduction of zonal flow generation by the magnetic Reynolds stress. Further, it is shown that the toroidal coupling enhances the zonal flow while it destabilizes the ETG mode. Hence, the electron transport seems insensitive to the toroidicity in the weak shear plasma due to the complex destabilizing and stabilizing competitions.

The toroidicity-induced Alfvén eigenmode structure in DIII-D: Implications of soft x-ray and beam-ion loss data

The internal structure of the toroidicity-induced Alfvén eigenmode (TAE) is studied by comparing soft x-ray profile and beam ion loss data taken during TAE activity in the DIII-D tokamak [W. W. Heidbrink et al., Nucl. Fusion 37, 1411 (1997)] with predictions from theories based on ideal magnetohydrodynamic (MHD), gyrofluid, and gyrokinetic models. The soft x-ray measurements indicate a centrally peaked eigenfunction, a feature which is closest to the gyrokinetic model’s prediction. The beam ion losses are simulated using a guiding center code. In the simulations, the TAE eigenfunction calculated using the ideal MHD model acts as a perturbation to the equilibrium field. The predicted beam ion losses are an order of magnitude less than the observed ∼6%–8% losses at the peak experimental amplitude of δBr/B0≃2–5×10−4.

Full torus Landau fluid calculations of ion temperature gradient-driven turbulence in cylindrical geometry

Results of Landau fluid calculations of ion temperature gradient-driven turbulence in cylindrical geometry but which cover the whole plasma cross section are reported. A simple gyrofluid model is used which evolves in time equations for the ion density or vorticity, the parallel ion velocity, and the ion temperature where Landau damping is included through a linear closure relation. Adiabatic electrons and the electrostatic approximation are assumed to hold. Linear calculations indicate that the growth rates are significantly reduced in the presence of Landau damping. Nonlinear single helicity calculations serve to illustrate the spatial localization of the turbulence in the presence of Landau damping. Finally nonlinear multiple helicity calculations performed with three different values for the ratio of the central ion Larmor radius to the plasma minor radius denoted by ρ* exhibit a definite linear scaling of all the fluctuation characteristics, such as levels, scale sizes, quasilinear profile modifications, self-generated poloidal flows, and heat diffusivities, with ρ*.

Simulation of ion temperature gradient turbulence in tokamaks

Results are presented from non-linear gyrokinetic simulations of toroidal ion temperature gradient turbulence and transport. The ion thermal fluxes are found to have an offset linear dependence on the temperature gradient and are significantly lower than gyrofluid or IFS-PPPL model predictions. A new phenomenon of non-linear effective critical gradients larger than the linear instability threshold gradients is observed and is associated with undamped flux surface averaged shear flows. The non-linear gyrokinetic codes have passed extensive tests, including comparison against independent linear calculations, a series of non-linear convergence tests and a comparison between two independent non-linear gyrokinetic codes. The most realistic simulations to date used actual reconstructed equilibria from experiments and a model for dilution by impurity and beam ions. These simulations highlight the importance of both self-generated and external E × B flow shear as well as the need for still more physics to be included.

Strategies for modifying alpha driven TAE thresholds through q profile and ion temperature control

Two methods are examined for influencing the stability thresholds of alpha driven toroidal Alfven eigenmode (TAE) instabilities in the deuterium-tritium (DT) operation of the Tokamak Fusion Test Reactor (TFTR). These are: (a) increasing the TAE drive through raising the central q(0) value (in order to centre the peak alpha pressure gradient on the lower m gap locations) and (b) decreasing the TAE damping rate on the background ions through cooling perturbations such as helium puffs or injected pellets. A gyrofluid model of the TAE is used that is inherently non-perturbative and includes continuum damping, ion/electron/neutral beam Landau damping, ion FLR effects and a generalized resistivity (finite E1 effects). Specific TFTR DT cases are analysed using a combination of measured and inferred profiles. The resulting predicted thresholds are generally consistent with an observed lack of strong TAE activity. However, it is found that special techniques, especially increasing q(0), can significantly lower the alpha driven TAE threshold into ranges that should be experimentally accessible

On electromagnetic ballooning modes in fluid and kinetic descriptions

A comparison between two-fluid, gyrofluid, and gyrokinetic descriptions of the collisionless ballooning mode in the toroidal limit is performed. The two-fluid model [A. Jarmén et al., Nucl. Fusion 27, 941 (1987)] is reactive and expanded in the finite-Larmor radius (FLR), in contrast to the gyrofluid model [R. E. Waltz et al., Phys. Fluids B 4, 3138 (1992)], which is dissipative and includes finite-Larmor-radius effects to all orders. The eigenmode differential equation is derived and solved numerically. It is shown that the fluid models are able to reproduce the gyrokinetic result, at least qualitatively. The role of the ion magnetic drift resonance and the ion FLR terms, in the fluid and kinetic descriptions, is discussed.

Nonlinear evolution of the toroidal Alfvén instability using a gyrofluid model*

Discrete shear Alfvén modes such as the TAE (toroidal Alfvén eigenmode) are susceptible to destabilization by energetic alpha populations and neutral beams; this can lead to enhanced fast ion losses and degraded heating efficiencies. A gyrofluid model with Landau closure has been developed for understanding both the linear and nonlinear phases of these instabilities. The linear wave–particle resonances necessary to excite Alfvén instabilities are included in a coupled set of fluid equations. This model is used to analyze several nonlinear saturation mechanisms that arise from mode coupling effects. The effects of shear flow velocity generation (through the Reynolds stress) and localized current generation (leading to modifications in the q profile) are specifically examined.

Comparisons of gyrofluid and gyrokinetic simulations*

The gyrokinetic and gyrofluid models show the most promise for large scale simulations of tokamak microturbulence. This paper discusses detailed comparisons of these two complementary approaches. Past comparisons with linear theory have been fairly good, therefore the emphasis here is on nonlinear comparisons. Simulations include simple two-dimensional slab test cases, turbulent three-dimensional slab cases, and toroidal cases, each modeling the nonlinear evolution of the ion temperature gradient instability. There is good agreement in both turbulent and coherent nonlinear slab comparisons in terms of the ion heat flux, both in magnitude and scaling with magnetic shear. However, the nonlinear saturation level for ‖Φ‖ in the slab comparisons shows differences of approximately 40%. Preliminary toroidal comparisons show agreement within 50%, in terms of ion heat flux and saturation level.

Neutral beam driven global Alfvén eigenmodes in the Wendelstein W7-AS stellarator.

During neutral beam injection coherent magnetohydrodynamic activity occurs at low and medium \ensuremath{\beta}, which is driven by the energetic beam particles. As interpretation is given in terms of marginally stable global Alfv\'en eigenmodes (GAE), which are destabilized by Landau resonances of the far circulating particles. Toroidicity induced gaps for toroidal Alfv\'en eigenmodes do not occur in W7-AS but GAE modes in gaps below the shear Alfv\'en continua can be excited. This is investigated within a gyrofluid model, which has been used sucessfully for tokamak cases previously.