R. Gatto

Nonlinear stability and instability in collisionless trapped electron mode turbulence

A two-field model for collisionless trapped electron mode turbulence has both finite amplitude-induced stability and instability, depending on wave number. Effects usually identified with nonlinear plasma instability (self-trapping, kinetics, 3D mode structure, magnetic shear) are absent. Nonlinear stability and instability reside in E×B advection of density. It drives modes of a purely damped branch of the dispersion relation to finite amplitude and changes the rate at which free energy is released into the turbulence by shifting the density-potential cross phase. Analysis shows that modes of the purely damped branch cannot be ignored in saturation, and that the linear growth rate is a poor indicator of driving at finite amplitude, invalidating mixing length and quasilinear approximations. Using statistical closure theory, the nonlinear eigenmode and growth rate are determined from the saturation level of modes on all branches, stable and unstable, and the nonlinear cross phase that governs finite-amplit...

Nonlinear damping of zonal modes in anisotropic weakly collisional trapped electron mode turbulence

Comprehensive spectral analysis of a fluid model for trapped electron mode (TEM) turbulence reveals that marginally stable zonal modes at infinitesimal amplitude become robustly damped at finite amplitude. Zonal-mode structure, anisotropy, excitation, and wave number spectra are shown to result from interaction of the zero-frequency drift wave with the density advection nonlinearity. Heuristic dimensional balances, closure theory, and simulations manifest the primacy of the interaction, and yield energy transfer rates, fluctuation levels, spectra and finite-amplitude-induced dissipation. Strong sensitivity to the zero-frequency wave induces a marked spectral energy-transfer anisotropy that preferentially drives zonal modes relative to nonzonal modes. Zonal-mode excitation is accompanied by the nonlinear excitation of a spectrum of damped eigenmodes. The mixing of unstable TEM eigenmodes with the damped spectrum subjects zonal modes to finite-amplitude-induced damping. The combination of anisotropic transf...

Nonlinear inward particle flux component in trapped electron mode turbulence

Trapped electron turbulence is shown to have a significant inward particle flux component associated with nonlinear deviations of the density-potential cross correlation from the quasilinear value. The cross correlation is altered because the density advection nonlinearity mixes a linearly stable eigenmode with the eigenmode of the instability. The full nonlinear flux is evaluated by solving spectrum balance equations in a complete basis spanning the fluctuation space. An ordered expansion for small collisionality, perpendicular wave number, and temperature/density-gradient instability threshold parameter enables an analytic solution for a weakly driven regime. The solution quantifies the role of zonal modes on transport via their saturation of the turbulence under intensely anisotropic transfer. The inward transport is neither diffusive nor convective, but is driven by temperature gradient and enhanced by flat density gradients. It is slightly smaller than the outwardly directed flux associated with the growing eigenmode, making the flux a small fraction of the quasilinear value.

Self-consistent electron transport in tokamaks

Electron particle, momentum, and energy fluxes in axisymmetric toroidal devices are derived from a version of the action-angle collision operator that includes both diffusion and drag in action-space [D. A. Hitchcock, R. D. Hazeltine, and S. M. Mahajan, Phys. Fluids 26, 2603 (1983); H. E. Mynick, J. Plasma Phys. 39, 303 (1988)]. A general result of the theory is that any contribution to transport originating directly from the toroidal frequency of the particle motion is constrained to be zero when the electron temperature is equal to the ion temperature. In particular, this constraint applies to those components of the particle and energy fluxes that are proportional to the magnetic shear, independent of the underlying turbulence and of whether the particles are trapped or untrapped. All the total fluxes describing collisionless transport of passing electrons in steady-state magnetic turbulence contain contributions proportional to the conventional thermodynamic drives, which are always outward, and contr...

Anomalous ion heating from ambipolar-constrained magnetic fluctuation-induced transport

A kinetic theory for the anomalous heating of ions from energy stored in magnetic turbulence is presented. Imposing self-consistency through the constitutive relations between particle distributions and fields, a turbulent Kirchhoff’s Law is derived that expresses a direct connection between rates of ion heating and electron thermal transport. This connection arises from the kinematics of electron motion along turbulent fields, which results in granular structures in the electron distribution. The drag exerted on these structures through emission into collective modes mediates an effective ambipolar constraint on transport. Resonant damping of the collective modes by ions produces the heating. In collisionless plasmas the rate of ion damping controls the rate of emission, and hence the ambipolar-constrained electron heat flux. The heating rate is calculated for both a resonant and nonresonant magnetic fluctuation spectrum and compared with observations. The theoretical heating rate is sufficient to accoun...

Magnetic relaxation and hyper-resistivity during helicity injection

Transport theory is applied to magnetic helicity injection into plasmas with toroidal geometry. Magnetic relaxation during helicity injection can be described as hyper-resistive diffusion of the current. By using the generalized Balescu–Lenard extension of quasi-linear transport theory, it is shown that hyper-resistive diffusion is generally slow compared with heat transport. It follows that magnetic relaxation due to such turbulence tends to flatten the temperature profile, as observed in reversed-field pinches. Given flattened temperature profiles, Taylors minimum principle for magnetic relaxation is usefully reformulated as minimum dissipation, yielding circuit equations for electrostatic helicity injection in laboratory devices such as spheromaks and tokamaks. A favorable heat pinch could benefit helicity injection into tokamaks. These results are also relevant to natural phenomena involving the generation of fields by magnetic relaxation.

Temperature dependence of pinches in tokamaks

The issue of turbulent pinches induced by inhomogeneities in the magnetic field confining tokamak plasmas is studied in the framework of the self-consistent action-angle transport theory. It is found that, in magnetic turbulence, electron particle and energy fluxes proportional to the magnetic shear and originating from the toroidal frequency of the particle motion are different from zero only if the electron temperature is different from the ion temperature, independent of whether the electrons are trapped or untrapped. Restricting to passing electrons, it is shown that both the magnitude and the sign of these fluxes depend crucially on the ratio Ti/Te.

Turbulent sources in axisymmetric plasmas

Successful operation of tokamaks and other magnetic confinement schemes of fusion interest rely on the tailoring of the parallel momentum/current density and temperature profiles via resonant absorption of externally injected waves. Similarly, it is to be expected that a turbulent spectrum of waves, internally generated to free the energy stored in the gradients of the equilibrium profiles, could transfer locally momentum and energy to the particle degree of freedom of the plasma. Turbulent sources stem out nicely from the action-angle transport formalism, as a detailed derivation of the general transport law from the collision operator (which includes both the diffusion and the friction coefficients) in action-space shows. The special case of magnetic turbulence is considered, and explicit expressions for the electron parallel momentum and energy sources are presented. An interesting feature of the sources resides in their dependence on the first and second powers of the safety factor derivative, a dependence that is often found in turbulent fluxes as well. One term in the energy source depends, in a determinant way, also on the relative magnitude of the electron and ion temperature. This dependence, an output of the retention of the friction term in the collision operator, leads to an energy flow that is always directed from the hotter to the cooler species, a desirable property that is missed when a quasilinear approach is employed.

Turbulent contributions to Ohm's law in axisymmetric magnetized plasmas

The effect of magnetic turbulence in shaping the current density in axisymmetric magnetized plasmas is analyzed using a turbulent extension of Ohms law derived from the self-consistent action-angle transport theory. Besides the well-known hyper-resistive (helicity-conserving) contribution, the generalized Ohms law contains an anomalous resistivity term and a turbulent bootstrap-like term proportional to the current density derivative. The numerical solution of the equation for equilibrium and turbulence profiles characteristic of conventional and advanced scenarios shows that, through the “turbulent bootstrap” effect and anomalous resistivity, power and parallel current can be generated which are a sizable portion (about 20%–25%) of the corresponding effects associated with the neoclassical bootstrap effect. The degree of alignment of the turbulence peak and the pressure gradient plays an important role in defining the steady-state regime. In a fully bootstrapped tokamak, the hyper-resistivity is essent...

Current Density Equation in Turbulent Magnetized Plasmas

A turbulent extension of Ohm’s law, derived from the self-consistent action-angle transport theory, is presented. The equation describes the steady-state profile of the current density in axisymmetric magnetized plasmas in the presence of magnetic turbulence. The hyper-resistive, helicity-conserving contribution, usually derived in the framework of magneto-hydro-dynamics, is recovered, and the hyper-resistivity is defined. Additionally, the generalized Ohm’s law contains an anomalous resistivity term, and a term proportional to the current density derivative. For given thermodynamic profiles, the numerical solution of the equation shows that turbulent contributions, besides regularizing the current density profile in the central region, lead to an increase of the total plasma current. This “turbulent bootstrap” effect provides a possible explanation to discrepancies recently observed between experimental current profiles and neoclassical predictions.