In gyrokinetic theory, the quadratic nonlinearity is known to play an important role in the dynamics by redistributing (in a conservative fashion) the free energy between the various active scales. In the present study, the free energy transfer is analyzed for the case of ion temperature gradient driven turbulence. It is shown that it shares many properties with the energy transfer in fluid turbulence. In particular, one finds a forward (from large to small scales), extremely local, and self-similar cascade of free energy in the plane perpendicular to the background magnetic field. These findings shed light on some fundamental properties of plasma turbulence, and encourage the development of large eddy simulation techniques for gyrokinetics.Fully developed turbulence is fundamentally linked to a conservative transfer of (free) energy in wavenumber space from drive to dissipation scales [1]. While the respective cascade dynamics for simple fluids (described by the Navier-Stokes equation) has been the subject of countless studies and is fairly well understood, the situation is quite different for turbulent plasmas, both at large scales (compared to the gyroradii of the particles) -described in the context of magnetohydrodynamics -and, in particular, at small scales -described by the gyrokinetic equations [2]. The latter case, in which one deals with a gyrocenter distribution function in three spatial dimensions as well as two velocity space dimensions (the third velocity space coordinate can be removed analytically in a low-frequency ordering), shall be the focus of the present work.In three-dimensional Navier-Stokes turbulence, the kinetic energy is conserved by the convective nonlinearity. It is usually assumed to be injected into the system at the largest scales through mechanical forcing, and to be dissipated at the smallest scales by viscous effects. The role of the nonlinearity is then to transfer the kinetic energy from the large scales to the small ones in what is usually referred to as a cascade process. In the gyrokinetic formalism, on the other hand, the free energy acts as the quadratic conserved quantity (see, e.g., Ref.[3] and various references therein). It is usually injected into the system at large scales via the background density and temperature gradients, and expected to be dissipated at small (space and/or velocity space) scales. It is anticipated that one role of the nonlinear term in gyrokinetic turbulence is to transfer the free energy from the largest perpendicular scales to the smallest ones [4][5][6], but a definitive investigation of the free energy transfer dynamics in a self-driven, three-dimensional system (which is the standard case for magnetically confined plasmas) is still lacking and shall be provided for the first time in the present Letter.Our study is based on numerical solutions of the nonlinear gyrokinetic equations obtained by means of the Gene code [7][8][9]. Although Gene is able to treat an arbitrary number of fully gyrokinetic particle species as well as general toroi...

Free energy plays an important role in gyrokinetic theory since it is known to be a nonlinear invariant. Its evolution equations are derived and analyzed for the case of ion temperature gradient driven turbulence, using the formalism adopted in the Gene code. In particular, the ion temperature gradient drive, the collisional dissipation as well as entropy/electrostatic energy transfer channels represented by linear curvature and parallel terms are analyzed in detail.

The Large Eddy Simulation (LES) approach is adapted to the study of plasma microturbulence in a fully three-dimensional gyrokinetic system. Ion temperature gradient driven turbulence is studied with the GENE code for both a standard resolution and a reduced resolution with a model for the sub-grid scale turbulence. A simple dissipative model for representing the effect of the sub-grid scales on the resolved scales is proposed and tested. Once calibrated, the model appears to be able to reproduce most of the features of the free energy spectra for various values of the ion temperature gradient.

Particle simulations on a flat-topped somewhat underdense (typically n0/nc = 0.6) plasma slab by Nikolic [Phys. Rev. E 66, 036404 (2002)] were seen to give transient stimulated scattering behavior with frequency shift [omega0 - omegas(approximately omegap)] considerably less than the plasma frequency omegap. This has been linked to the electron acoustic wave (EAW) and the scattering was thus seen as another example of stimulated electron acoustic scattering inferred by Montgomery [Phys. Rev. Lett. 87, 155001 (2001)] from experiments on low-density plasmas. Montgomery had noted the difficulty of how one could have a very narrow observed scattering from a wave whose damping was at least initially very high. Our Vlasov-Maxwell simulations for such somewhat underdense (n0/nc> or = 0.25) plasmas show that the simulation resonance was in fact determined by the beating of the pump with a new "radiating pseudocavity" electromagnetic mode for the slab at a frequency close to omegap with relatively low loss. This allows the initial narrow-band excitation of the kinetic electrostatic electron nonlinear (KEEN) waves (the nonlinear "cousins" of EAWs) at a well-defined frequency (omegaK approximately omega0 - omegap < omegap) which is not necessarily the value given by the EAW dispersion relation. (The KEEN wave characteristics have been discussed by Afeyan [33rd AAAC (2003), #238, IFSA 2003].) The consideration of such a mechanism is relevant to moderately underdense hot plasmas.

Large Eddy Simulations (LES) of gyrokinetic plasma turbulence are investigated as interesting candidates to decrease the computational cost. A dynamic procedure is implemented in the GENE code, allowing for dynamic optimization of the free parameters of the LES models (setting the amplitudes of dissipative terms). Employing such LES methods, one recovers the free energy and heat flux spectra obtained from highly resolved Direct Numerical Simulations (DNS). Systematic comparisons are performed for different values of the temperature gradient and magnetic shear, parameters which are of prime importance in Ion Temperature Gradient (ITG) driven turbulence. Moreover, the degree of anisotropy of the problem, that can vary with parameters, can be adapted dynamically by the method that shows Gyrokinetic Large Eddy Simulation (GyroLES) to be a serious candidate to reduce numerical cost of gyrokinetic solvers. I. MOTIVATION AND CONTEXTIn the area of fluid turbulence, theories are usually based on the notion of an inertial range in which the energy cascades from larger scales to (somewhat) smaller scales mediated by the quadratic nonlinearity. The role of the smallest scales is then to dissipate energy in the so-called dissipative range. In numerical simulations, this picture has led to the development of Large Eddy Simulation (LES) techniques that are based on the idea that neglecting the small scales can be compensated by introducing a dissipative model for the eddy viscosity 1 .A Direct Numerical Simulation (DNS) is supposed to retain all the scales from the injection range down to the dissipative range. This requires an enormous numerical effort in the case of high Reynolds number flows. On the contrary, a LES coarsens the simulation grid and only retains the largest scales (which are problem-dependent), while the small scales (which are assumed to be universal) are replaced by a model. In Fourier space, such a coarsening can be seen as the action of a low-pass filter. Since the scale range is truncated, the dissipation scales can not be reached, and the modeling basically consists of the introduction of artificial dissipation mechanisms. From a more mathematical viewpoint, one notes that the filtering operation does not commute with the nonlinear term that transfers energy from largest to smallest scales, and the major problem of LES consists in finding a satisfying closure for representing the influence of the unresolved scales.Recent gyrokinetic studies have shown that Ion Temperature Gradient (ITG) driven turbulence exhibits a direct and local cascade of a nonlinear invariant, namely the free energy. 2 Such a cascade is analogous to the kinetic energy cascade in three dimensional NavierStokes turbulence. The important difference is that the a) pmorel@ulb.ac.be quadratic conserved quantity in fluid dynamics is the kinetic energy, while it is the free energy in gyrokinetics. The latter quantity is the sum of both the perturbed entropy and the electrostatic energy. Transfers between entropy and electrostatic energy ar...