Nonlinear Gyrokinetic Equation Research Articles

Overview of FTU results

New FTU ohmic discharges with a liquid lithium limiter at IP = 0.7–0.75 MA, BT = 7 T and ne0 ⩾ 5 × 1020 m−3 confirm the spontaneous transition to an enhanced confinement regime, 1.3–1.4 times ITER-97-L, when the density peaking factor is above a threshold value of 1.7–1.8. The improved confinement derives from a reduction of electron thermal conductivity (χe) as density increases, while ion thermal conductivity (χi) remains close to neoclassical values. Linear microstability reveals the importance of lithium in triggering a turbulent inward flux for electrons and deuterium by changing the growth rates and phase of the ion-driven turbulence, while lithium flux is always directed outwards. A particle diffusion coefficient, D ∼ 0.07 m2 s−1, and an inward pinch velocity, V ∼ 0.27 m s−1, in qualitative agreement with Bohm–gyro-Bohm predictions are inferred in pellet fuelled lithized discharges. Radio frequency heated plasmas benefit from cleaner plasmas with edge optimized conditions. Lower hybrid waves penetration and current drive effects are clearly demonstrated at and above ITER densities thanks to a good control of edge parameters obtained by plasma operations with the external poloidal limiter, lithized walls and pellet fuelling. The electron cyclotron (EC) heating system is extensively exploited in FTU for contributing to ITER-relevant issues such as MHD control: sawtooth crash is actively controlled and density limit disruptions are avoided by central and off-axis deposition of 0.3 MW of EC power at 140 GHz. Fourier analysis shows that the density drop and the temperature rise, stimulated by modulated EC power in low collisionality plasmas are synchronous, implying that the heating method is the common cause of both the electron heating and the density drop. Perpendicularly injected electron cyclotron resonance heating is demonstrated to be more efficient than the obliquely injected one, reducing the minimum electric field required at breakdown by a factor of 3. Theoretical activity further develops the model to interpret high-frequency fishbones on FTU and other experiments as well as to characterize beta-induced Alfvén eigenmodes induced by magnetic islands in ohmic discharges. The theoretical framework of the general fishbone-like dispersion relation is used for implementing an extended version of the HMGC hybrid MHD gyrokinetic code. The upgraded version of HMGC will be able to handle fully compressible non-linear gyrokinetic equations and 3D MHD.

Exact momentum conservation laws for the gyrokinetic Vlasov-Poisson equations

The exact momentum conservation laws for the nonlinear gyrokinetic Vlasov-Poisson equations are derived by applying the Noether method on the gyrokinetic variational principle [A. J. Brizard, Phys. Plasmas 7, 4816 (2000)]. From the gyrokinetic Noether canonical-momentum equation derived by the Noether method, the gyrokinetic parallel momentum equation and other gyrokinetic Vlasov-moment equations are obtained. In addition, an exact gyrokinetic toroidal angular-momentum conservation law is derived in axisymmetric tokamak geometry, where the transport of parallel-toroidal momentum is related to the radial gyrocenter polarization, which includes contributions from the guiding-center and gyrocenter transformations.

Kinetic theory of the turbulent energy pinch in tokamak plasmas

The turbulent energy fluxes, including up-gradient ‘energy pinch’ effects, are derived using the nonlinear bounce-kinetic equation for trapped electrons and the nonlinear gyrokinetic equation for ions in toroidal geometry. The quasi-universal type of inward turbulent equipartition (TEP) energy pinch is recovered for both ions and trapped electrons, with different field dependence coefficients due to toroidal effects. A contribution from the density gradient to an outward convective energy flux is also obtained. The direction of the total energy convection is primarily determined by the competition between the TEP energy pinch and the outward density gradient driven energy convection. The magnetic shear dependence of the electron energy pinch is discussed. The energy pinches can provide possible explanations for some puzzling experimental observations.

Nonlinear gyrokinetics: a powerful tool for the description of microturbulence in magnetized plasmas

Gyrokinetics is the description of low-frequency dynamics in magnetized plasmas. In magnetic-confinement fusion, it provides the most fundamental basis for numerical simulations of microturbulence; there are astrophysical applications as well. In this tutorial, a sketch of the derivation of the novel dynamical system comprising the nonlinear gyrokinetic (GK) equation (GKE) and the coupled electrostatic GK Poisson equation will be given by using modern Lagrangian and Lie perturbation methods. No background in plasma physics is required in order to appreciate the logical development. The GKE describes the evolution of an ensemble of gyrocenters moving in a weakly inhomogeneous background magnetic field and in the presence of electromagnetic perturbations with wavelength of the order of the ion gyroradius. Gyrocenters move with effective drifts, which may be obtained by an averaging procedure that systematically, order by order, removes gyrophase dependence. To that end, the use of the Lagrangian differential one-form as well as the content and advantages of Lie perturbation theory will be explained. The electromagnetic fields follow via Maxwell's equations from the charge and current density of the particles. Particle and gyrocenter densities differ by an important polarization effect. That is calculated formally by a ‘pull-back’ (a concept from differential geometry) of the gyrocenter distribution to the laboratory coordinate system. A natural truncation then leads to the closed GK dynamical system. Important properties such as GK energy conservation and fluctuation noise will be mentioned briefly, as will the possibility (and difficulties) of deriving nonlinear gyrofluid equations suitable for rapid numerical solution—although it is probably best to directly simulate the GKE. By the end of the tutorial, students should appreciate the GKE as an extremely powerful tool and will be prepared for later lectures describing its applications to physical problems.

Exact energy conservation laws for full and truncated nonlinear gyrokinetic equations

The exact global energy conservation laws for the full and truncated versions of the nonlinear electromagnetic gyrokinetic equations in general magnetic geometry are presented. In each version, the relation between polarization and magnetization effects in the gyrokinetic Poisson and Ampère equations and the quadratic ponderomotive gyrocenter Hamiltonian is emphasized.

Symmetry breaking effects of toroidicity on toroidal momentum transport

A derivation of symmetry breaking toroidicity effects on toroidal momentum transport has been made from the stress tensor. This effect is usually stronger than the symmetry breaking caused by the flowshear on the eigenfunction. The model obtained generalizes a recent derivation of diagonal transport elements from the stress tensor to convective elements of turbulent equipartition or thermoelectric types. This makes it possible to interpret the same type of effects previously obtained from a phase space conserving nonlinear gyrokinetic equation.

Fully electromagnetic nonlinear gyrokinetic equations for tokamak edge turbulence

An energy conserving set of the fully electromagnetic nonlinear gyrokinetic Vlasov equation and Maxwell’s equations, which is applicable to both L-mode turbulence with large amplitude and H-mode turbulence in the presence of high E×B shear has been derived. The phase-space action variational Lie perturbation method ensures the preservation of the conservation laws of the underlying Vlasov–Maxwell system. Generalized ordering takes ρi⪡ρθi∼LE∼Lp⪡R [here ρi is the thermal ion Larmor radius and ρθi=B∕(Bθρi)], as typically observed in the tokamak H-mode edge, with LE and Lp being the radial electric field and pressure gradient lengths. k⊥ρi∼1 is assumed for generality, and the relative fluctuation amplitudes eδϕ∕Ti∼δB∕B are kept up to the second order. Extending the electrostatic theory in the presence of high E×B shear [Hahm, Phys. Plasmas 3, 4658 (1996)], contributions of electromagnetic fluctuations to the particle charge density and current are explicitly evaluated via pullback transformation from the gyro...

Arbitrary poloidal gyroradius effects in tokamak pedestals and transport barriers

A technique is developed and applied for analyzing pedestal and internal transport barrier (ITB) regions in a tokamak by formulating a special version of gyrokinetics. In contrast to typical gyrokinetic treatments, canonical angular momentum is taken as the gyrokinetic radial variable rather than the radial guiding center location. Such an approach allows strong radial plasma gradients to be treated, while retaining zonal flow and neoclassical (including orbit squeezing) behavior and the effects of turbulence. The new, nonlinear gyrokinetic variables are constructed to higher order than is typically the case. The nonlinear gyrokinetic equation obtained is capable of handling such problems as collisional zonal flow damping with radial wavelengths comparable to the ion poloidal gyroradius, as well as zonal flow and neoclassical transport in the pedestal or ITB. This choice of gyrokinetic variables allows the toroidally rotating Maxwellian solution of the isothermal tokamak limit to be recovered. More importantly, we prove that a physically acceptable solution for the lowest order ion distribution function in the banana regime anywhere in a tokamak and, in particular, in the pedestal must be nearly this same isothermal Maxwellian solution. That is, the ion temperature variation scale must be much greater than the poloidal ion gyroradius. Consequently, in the banana regime the background radial ion temperature profile cannot have a pedestal similar to that of plasma density.

Turbulent equipartition theory of toroidal momentum pinch

The mode-independent part of the magnetic curvature driven turbulent convective (TurCo) pinch of the angular momentum density [Hahm et al., Phys. Plasmas 14, 072302 (2007)], which was originally derived from the gyrokinetic equation, can be interpreted in terms of the turbulent equipartition (TEP) theory. It is shown that the previous results can be obtained from the local conservation of “magnetically weighted angular momentum density,” nmiU∥R∕B2, and its homogenization due to turbulent flows. It is also demonstrated that the magnetic curvature modification of the parallel acceleration in the nonlinear gyrokinetic equation in the laboratory frame, which was shown to be responsible for the TEP part of the TurCo pinch of angular momentum density in the previous work, is closely related to the Coriolis drift coupling to the perturbed electric field. In addition, the origin of the diffusive flux in the rotating frame is highlighted. Finally, it is illustrated that there should be a difference in scalings between the momentum pinch originated from inherently toroidal effects and that coming from other mechanisms that exist in a simpler geometry.

Nonlinear gyrokinetic theory of toroidal momentum pinch

The turbulent convective flux of the toroidal angular momentum density is derived using the nonlinear toroidal gyrokinetic equation which conserves phase space density and energy [T. S. Hahm, Phys. Fluids, 31, 2670 (1988)]. A novel pinch mechanism is identified which originates from the symmetry breaking due to the magnetic field curvature. A net parallel momentum transfer from the waves to the ion guiding centers is possible when the fluctuation intensity varies on the flux surface, resulting in imperfect cancellation of the curvature drift contribution to the parallel acceleration. This mechanism is inherently a toroidal effect, and complements the k‖ symmetry breaking mechanism due to the mean E×B shear [O. Gurcan et al., Phys. Plasmas 14, 042306 (2007)] which exists in a simpler geometry. In the absence of ion thermal effects, this pinch velocity of the angular momentum density can also be understood as a manifestation of a tendency to homogenize the profile of “magnetically weighted angular momentum density,” nmiR2ω‖∕B2. This part of the pinch flux is mode-independent (whether it is trapped electron mode or ion temperature gradient mode driven), and radially inward for fluctuations peaked at the low-B-field side, with a pinch velocity typically, VAngTEP∼−2χϕ∕R0. Ion thermal effects introduce an additional radial pinch flux from the coupling with the curvature and grad-B drifts. This curvature driven thermal pinch can be inward or outward, depending on the mode-propagation direction. Explicit formulas in general toroidal geometry are presented.

Foundations of nonlinear gyrokinetic theory

Nonlinear gyrokinetic equations play a fundamental role in our understanding of the long-time behavior of strongly magnetized plasmas. The foundations of modern nonlinear gyrokinetic the- ory are based on three important pillars: (1) a gyrokinetic Vlasov equation written in terms of a gyrocenter Hamiltonian with quadratic low-frequency ponderomotive-like terms; (2) a set of gyrokinetic Maxwell (Poisson-Ampere) equations written in terms of the gyrocenter Vlasov dis- tribution that contain low-frequency polarization (Poisson) and magnetization (Ampere) terms derived from the quadratic nonlinearities in the gyrocenter Hamiltonian; and (3) an exact energy conservationlaw for the gyrokineticVlasov-Maxwell equations that includes all the relevant linear and nonlinear coupling terms. The foundations of nonlinear gyrokinetic theory are reviewed with an emphasis on the rigorous applications of Lagrangian and Hamiltonian Lie-transform perturba- tion methods used in the variationalderivationof nonlineargyrokineticVlasov-Maxwell equations. The physical motivations and applications of the nonlinear gyrokinetic equations, which describe the turbulent evolution of low-frequency electromagnetic fluctuations in a nonuniform magnetized plasmas with arbitrary magnetic geometry, are also discussed.

Astrophysical Gyrokinetics: Basic Equations and Linear Theory

Magnetohydrodynamic (MHD) turbulence is encountered in a wide variety of astrophysical plasmas, including accretion disks, the solar wind, and the interstellar and intracluster medium. On small scales, this turbulence is often expected to consist of highly anisotropic fluctuations with frequencies small compared to the ion cyclotron frequency. For a number of applications, the small scales are also collisionless, so a kinetic treatment of the turbulence is necessary. We show that this anisotropic turbulence is well described by a low frequency expansion of the kinetic theory called gyrokinetics. This paper is the first in a series to examine turbulent astrophysical plasmas in the gyrokinetic limit. We derive and explain the nonlinear gyrokinetic equations and explore the linear properties of gyrokinetics as a prelude to nonlinear simulations. The linear dispersion relation for gyrokinetics is obtained and its solutions are compared to those of hot-plasma kinetic theory. These results are used to validate the performance of the gyrokinetic simulation code {\tt GS2} in the parameter regimes relevant for astrophysical plasmas. New results on global energy conservation in gyrokinetics are also derived. We briefly outline several of the problems to be addressed by future nonlinear simulations, including particle heating by turbulence in hot accretion flows and in the solar wind, the magnetic and electric field power spectra in the solar wind, and the origin of small-scale density fluctuations in the interstellar medium.

Clebsch-type coordinates for nonlinear gyrokinetics in generic toroidal configurations

The nonlinear gyrokinetic equations are frequently used as a basis for simulations of small-scale turbulence in magnetized toroidal plasmas. In this context, field-aligned coordinates are usually employed in order to minimize the number of necessary grid points. The present work proposes a system of Clebsch-type coordinates which does not depend on the existence of flux surfaces. The construction and use of these coordinates is explained, and the corresponding formulation of the nonlinear gyrokinetic equations is accomplished. This setup paves the way toward the investigation of nonaxisymmetric toroidal geometries, also in the region of magnetic islands as well as inside the ergodic layer where flux surfaces cease to exist. For testing purposes, in the axisymmetric, large aspect ratio case, the well-known ŝ-α expressions are recovered for closed flux surfaces. Moreover, geometric data for a specific stellarator configuration are computed and discussed.

Relevance of the parallel nonlinearity in gyrokinetic simulations of tokamak plasmas

The influence of the parallel nonlinearity on transport in gyrokinetic simulations is assessed for values of ρ* which are typical of current experiments. Here, ρ*=ρs∕a is the ratio of gyroradius, ρs, to plasma minor radius, a. The conclusion, derived from simulations with both GYRO [J. Candy and R. E. Waltz, J. Comput. Phys., 186, 585 (2003)] and GEM [Y. Chen and S. E. Parker J. Comput. Phys., 189, 463 (2003)] is that no measurable effect of the parallel nonlinearity is apparent for ρ*<0.012. This result is consistent with scaling arguments, which suggest that the parallel nonlinearity should be O(ρ*) smaller than the E×B nonlinearity. Indeed, for the plasma parameters under consideration, the magnitude of the parallel nonlinearity is a factor of 8ρ* smaller (for 0.000 75<ρ*<0.012) than the other retained terms in the nonlinear gyrokinetic equation.

Non-linear zonal dynamics of drift and drift-Alfvén turbulence in tokamak plasmas

The present work addresses the issue of identifying the major non-linear physics processes which may regulate drift and drift-Alfvén turbulence using a weak turbulence approach. Within this framework, on the basis of the non-linear gyrokinetic equation for both electrons and ions, an analytic theory is presented for non-linear zonal dynamics described in terms of two axisymmetric potentials, δϕz and δA|| z, which spatially depend only on a (magnetic) flux co-ordinate. Spontaneous excitation of zonal flows by electrostatic drift microinstabilities is demonstrated both analytically and by direct 3-D gyrokinetic simulations. Direct comparisons indicate good agreement between analytic expressions of the zonal flow growth rate and numerical simulation results for ion temperature gradientdriven modes. Analogously, it is shown that zonal flows may bespontaneously excited by drift-Alfvén turbulence, in the form of modulational instability of the radial envelope of the mode as well, whereas in general, excitations of zonal currents are possible but have little feedback on the turbulence itself.

Nonlinear relativistic gyrokinetic Vlasov-Maxwell equations

A set of self-consistent nonlinear gyrokinetic equations is derived for relativistic charged particles in a general nonuniform magnetized plasma. Full electromagnetic-field fluctuations are considered with spatial and temporal scales given by the low-frequency gyrokinetic ordering. Self-consistency is obtained by combining the nonlinear relativistic gyrokinetic Vlasov equation with the low-frequency Maxwell equations in which charge densities and current densities are expressed in terms of moments of the gyrokinetic Vlasov distribution. For these self-consistent gyrokinetic equations, a low-frequency energy conservation law is also derived.

Theory of plasma transport induced by low‐frequency hydromagnetic waves

On the basis of the quasi‐linear limit of the nonlinear gyrokinetic equations we have derived self‐consistent, comprehensive theoretical results on plasma transport induced by low‐frequency hydromagnetic waves. Effects such as finite ion Larmor radii, realistic magnetic field geometries, general plasma equilibria, magnetic trapping, and arbitrary wave polarizations are included in the present analysis. Corresponding transport equations for flux tube‐averaged radial fluxes of particle, energy, and parallel momentum as well as local heating and parallel flow generation rates are obtained. The resultant transport coefficients exhibit the appealing physical pictures of exchanges of energies and momenta between resonant plasma particles and hydromagnetic wave quasi‐particles. The present results also indicate that radial transport generally consists of convective components in addition to the usual diffusive ones. Applications to the particle transport at the dayside magnetopause as well as the nonlinear saturation of drift Alfvén‐ballooning mode instabilities [Chen and Hasegawa, 1991] are also presented.

Nonlinear electromagnetic gyrokinetic equation for plasmas with large mean flows

A new nonlinear electromagnetic gyrokinetic equation is derived for plasmas with large flow velocities on the order of the ion thermal speed. The gyrokinetic equation derived here retains a collision term and is given in the form which is valid for general magnetic geometries including the slab, cylindrical and toroidal configurations. The source term for the anomalous viscosity arising through the Reynolds stress is identified in the gyrokinetic equation. For the toroidally rotating plasma, particle, energy and momentum balance equations as well as the detailed definitions of the anomalous transport fluxes and the anomalous entropy production are shown. The quasilinear anomalous transport matrix connecting the conjugate pairs of the anomalous fluxes and the forces satisfies the Onsager symmetry.

Nonlinear gyrokinetic equations for turbulence in core transport barriers

An energy-conserving set of the nonlinear electrostatic gyrokinetic Vlasov and Poisson equations is derived for the first time in the presence of equilibrium E×B velocity uE∼vTi, via phase-space Lagrangian Lie-perturbation theory. In this general formulation, only the basic small parameter ε with ω/Ω∼k∥/k⊥∼ε and δn/n0∼1/k⊥L∼ε, is used, while no device-specific expansion has been made. Here, L is the equilibrium scale length. For application to microturbulence in tokamak core transport barriers, an additional small ordering parameter δB≡Bθ/B≪1 is utilized. This leads to a useful form of the nonlinear gyrokinetic system which is applicable to a realistic situation in which the gradient lengths of the equilibrium radial electric field and pressure are of the same order as the ion poloidal gyroradius. The ordering for fluctuations is also modified to δn/n0∼εδB≪1/k⊥L∼δB for a better description of sub-mixing-length level fluctuations. uE/vTi∼δB and ρθi∼Lp put the pressure-gradient contribution to Er and the toroidal-flow contribution to Er at the same order. δB∼ε is shown to be a maximal ordering for studying the E×B flow shear suppression of turbulence.

Toroidal gyrofluid equations for simulations of tokamak turbulence

A set of nonlinear gyrofluid equations for simulations of tokamak turbulence are derived by taking moments of the nonlinear toroidal gyrokinetic equation. The moment hierarchy is closed with approximations that model the kinetic effects of parallel Landau damping, toroidal drift resonances, and finite Larmor radius effects. These equations generalize the work of Dorland and Hammett [Phys. Fluids B 5, 812 (1993)] to toroidal geometry by including essential toroidal effects. The closures for phase mixing from toroidal ∇B and curvature drifts take the basic form presented in Waltz et al. [Phys. Fluids B 4, 3138 (1992)], but here a more rigorous procedure is used, including an extension to higher moments, which provides significantly improved accuracy. In addition, trapped ion effects and collisions are incorporated. This reduced set of nonlinear equations accurately models most of the physics considered important for ion dynamics in core tokamak turbulence, and is simple enough to be used in high resolution direct numerical simulations.