Abstract. We present a new recursive procedure to find a full f electrostatic gyrokinetic equation correct to first order in an expansion of gyroradius over magnetic field characteristic length. The procedure provides new insights into the limitations of the gyrokinetic quasineutrality equation. We find that the ion distribution function must be known at least to second order in gyroradius over characteristic length to calculate the long wavelength components of the electrostatic potential selfconsistently. Moreover, using the example of a steady-state θ-pinch, we prove that the quasineutrality equation fails to provide the axisymmetric piece of the potential even with a distribution function correct to second order. We also show that second order accuracy is enough if a more convenient moment equation is used instead of the quasineutrality equation. These results indicate that the gyrokinetic quasineutrality equation is not the most effective procedure to find the electrostatic potential if the long wavelength components are to be retained in the analysis.

Scaling laws for ion temperature gradient driven turbulence in magnetized toroidal plasmas are derived and compared with direct numerical simulations. Predicted dependences of turbulence fluctuation amplitudes, spatial scales, and resulting heat fluxes on temperature gradient and magnetic field line pitch are found to agree with numerical results in both the driving and inertial ranges. Evidence is provided to support the critical balance conjecture that parallel streaming and nonlinear perpendicular decorrelation times are comparable at all spatial scales, leading to a scaling relationship between parallel and perpendicular spatial scales. This indicates that even strongly magnetized plasma turbulence is intrinsically three-dimensional.Introduction. Microscale turbulence is a ubiquitous feature of the plasmas used for magnetic confinement fusion. It is driven by kinetic instabilities feeding predominantly off a strong mean gradient in the ion temperature, and it is responsible for the majority of particle and heat transport observed in experiment. As with neutral fluid and magnetohydrodynamic turbulence, exact analytical results for kinetic plasma turbulence are rare, and numerical simulations are costly. Phenomenological scaling laws are thus useful for guiding simulation and providing gross predictions of plasma behavior in a multi-dimensional parameter space.Experimental, numerical, and analytical results (cf. [1-4]) have long been used to predict the dependence of turbulent fluxes on the mean plasma gradients and on the magnetic field configuration. However, scalings based on empirical observations provide limited physical insight, and the theoretical predictions, which are predominantly based on linear or quasilinear arguments, are not sufficiently detailed to be easily falsifiable. A more detailed examination of the properties of kinetic plasma turbulence has been conducted for scales smaller than the ion Larmor radius [5][6][7], but it is the ion temperature gradient (ITG) driven turbulence above the Larmor scale that is most relevant for heat transport in fusion devices (cf. [8]). Recent advances in plasma fluctuation measurements [9, 10] have provided turbulence spectra in this scale range; direct numerical simulations have also calculated spectra [11] and studied energy injection, transfer, and dissipation [12,13].In this Letter, we propose a phenomenological scaling theory of ITG turbulence. A number of simple, physically-motivated conjectures about the nature of this turbulence are formulated and applied to obtain fluctuation spectra from the driving scale to the ion Larmor scale. We then derive predictions for the dependence of heat flux on plasma current and ion temperature gradient. Numerical results are presented to support our predictions and justify our conjectures.Gyrokinetic turbulence. Plasma fluctuations in a strong mean magnetic field are anisotropic with respect to the mean field direction and have typical frequencies

Toroidal momentum transport mechanisms are reviewed and put in a broader perspective. The generation of a finite momentum flux is closely related to the breaking of symmetry (parity) along the field. The symmetry argument allows for the systematic identification of possible transport mechanisms. Those that appear to lowest order in the normalized Larmor radius (the diagonal part, Coriolis pinch, E ×B shearing, particle flux, and up-down asymmetric equilibria) are reasonably well understood. At higher order, expected to be of importance in the plasma edge, the theory is still under development.

Abstract. In general, the orbit-averaged radial magnetic drift of trapped particles in stellarators is non-zero due to the three-dimensional nature of the magnetic field. Stellarators in which the orbit-averaged radial magnetic drift vanishes are called omnigeneous, and they exhibit neoclassical transport levels comparable to those of axisymmetric tokamaks. However, the effect of deviations from omnigeneity cannot be neglected in practice, and it is more deleterious at small collisionalities. For sufficiently low collision frequencies (below the values that define the 1/ν regime), the components of the drifts tangential to the flux surface become relevant. This article focuses on the study of such collisionality regimes in stellarators close to omnigeneity when the gradient of the non-omnigeneous perturbation is small. First, it is proven that closeness to omnigeneity is required to actually preserve radial locality in the drift-kinetic equation for collisionalities below the 1/ν regime. Then, using the derived radially local equation, it is shown that neoclassical transport is determined by two layers located at different regions of phase space. One of the layers corresponds to the so-called √ ν regime and the other to the so-called superbanana-plateau regime. The importance of the superbanana-plateau layer for the calculation of the tangential electric field is emphasized, as well as the relevance of the latter for neoclassical transport in the collisionality regimes considered in this paper. In particular, the role of the tangential electric field is essential for the emergence of a new subregime of superbanana-plateau transport when the radial electric field is small. A formula for the ion energy flux that includes the √ ν regime and the superbanana-plateau regime is given. The energy flux scales with the square of the size of the deviation from omnigeneity. Finally, it is explained why below a certain collisionality value the formulation presented in this article ceases to be valid.

Nonlinear gyrokinetic simulations have been conducted to investigate turbulent transport in tokamak plasmas with rotational shear. At sufficiently large flow shears, linear instabilities are suppressed, but transiently growing modes drive subcritical turbulence whose amplitude increases with flow shear. This leads to a local minimum in the heat flux, indicating an optimal E × B shear value for plasma confinement. Local maxima in the momentum fluxes are also observed, allowing for the possibility of bifurcations in the E × B shear. The sensitive dependence of heat flux on temperature gradient is relaxed for large flow shear values, with the critical temperature gradient increasing at lower flow shear values. The turbulent Prandtl number is found to be largely independent of temperature and flow gradients, with a value close to unity.Introduction. Experimental measurements in magnetic confinement fusion devices indicate that sheared mean E × B flows can significantly reduce and sometimes fully suppress turbulent particle, momentum, and heat fluxes [1,2]. Since these turbulent fluxes determine mean plasma density and temperature profiles, their reduction leads to a local increase in the profile gradients. This increase can be dramatic: transport barriers in both the plasma core and edge have been measured with radial extents on the order of only tens of ion Larmor radii [3]. The associated increase in core density and temperature results in increased fusion power. Thus, understanding how shear flow layers develop and what effect they have on turbulent fluxes is both physically interesting and practically useful.This Letter reports a numerical study of the influence of sheared toroidal rotation on turbulent heat and momentum transport in tokamak plasmas. Two main effects of sheared toroidal rotation were identified in previous numerical work [4][5][6][7][8]: suppression of turbulent transport by shear in the perpendicular (to the mean magnetic field) velocity and linear destabilization due to the parallel velocity gradient (PVG). While the former observation indicates that a finite flow shear improves plasma confinement, the latter raises the question of whether more shear is always beneficial. Below we report that the PVG-driven linear instability [9] is stabilized at sufficiently large flow shear values, consistent with fluid theory in slab geometry [10]. Correspondingly, fluxes decrease with increasing flow shear as the linear stabilization point is approached. However, beyond this point, transiently growing modes driven by the PVG give rise to subcritical turbulence. The fluxes associated with this turbulence increase with flow shear. This implies an optimal flow shear for each temperature gradient; the fact that the minimum heat flux value is finite indicates that there is a maximum attainable temperature gradient that can be maintained for a given heat flux. Additionally, the observed presence of maxima in the momentum fluxes admits the possibility of bifurcations in the flow shear (and thus the temperature gr...

Abstract.We derive a self-consistent equation for the turbulent transport of toroidal angular momentum in tokamaks in the low flow ordering that only requires solving gyrokinetic Fokker-Planck and quasineutrality equations correct to second order in an expansion on the gyroradius over scale length. We also show that according to our orderings the long wavelength toroidal rotation and the long wavelength radial electric field satisfy the neoclassical relation that gives the toroidal rotation as a function of the radial electric field and the radial gradients of pressure and temperature. Thus, the radial electric field can be solved for once the toroidal rotation is calculated from the transport of toroidal angular momentum. Unfortunately, even though this methodology only requires a gyrokinetic model correct to second order in gyroradius over scale length, current gyrokinetic simulations are only valid to first order. To overcome this difficulty, we exploit the smallish ratio B p /B, where B is the total magnetic field and B p is its poloidal component. When B p /B is small, the usual first order gyrokinetic equation provides solutions that are accurate enough to employ for our expression for the transport of toroidal angular momentum. We show that current δf and full f simulations only need small corrections to achieve this accuracy. Full f simulations, however, are still unable to determine the long wavelength, radial electric field from the quasineutrality equation.

Divertor detachment may be essential to reduce heat loads to magnetic fusion tokamak reactor divertor surfaces. Yet in experiments it is difficult to control the extent of the detached, low pressure, plasma region. At maximum extent the front edge of the detached region reaches the X-point and can lead to degradation of core plasma properties. We define the ‘detachment window’ in a given position control variable C (for example, the upstream plasma density) as the range in C within which the front location can be stably held at any position from the target to the X-point; increased detachment window corresponds to better control. We extend a 1D analytic model [] to determine the detachment window for the following control variables: the upstream plasma density, the impurity concentration and the power entering the scrape-off layer (SOL). We find that variations in magnetic configuration can have strong effects; increasing the ratio of the total magnetic field at the X-point to that at the target, , (total flux expansion, as in the super-x divertor configuration) strongly increases the detachment window for all control variables studied, thus strongly improving detachment front control and the capability of the divertor plasma to passively accommodate transients while still staying detached. Increasing flux tube length and thus volume in the divertor, through poloidal flux expansion (as in the snowflake or x-divertor configurations) or length of the divertor, also increases the detachment window, but less than the total flux expansion does. The sensitivity of the detachment front location, zh, to each control variable, C, defined as , depends on the magnetic configuration. The size of the radiating volume and the total divertor radiation increase and , respectively, but not by increasing divertor poloidal flux expansion or field line length. We believe this model is applicable more generally to any thermal fronts in flux tubes with varying magnetic field, and similar sources and sinks, such as detachment fronts in stellarator divertors and solar prominences in coronal loops.

Gyrokinetic theory is based on an asymptotic expansion in the small parameter ǫ, defined as the ratio of the gyroradius and the characteristic length of variation of the magnetic field. In this article, this ordering is strictly implemented to compute the electrostatic gyrokinetic phase-space Lagrangian in general magnetic geometry to order ǫ 2 . In particular, a new expression for the complete secondorder gyrokinetic Hamiltonian is provided, showing that in a rigorous treatment of gyrokinetic theory magnetic geometry and turbulence cannot be dealt with independently. The new phase-space gyrokinetic Lagrangian gives a Vlasov equation accurate to order ǫ 2 and a Poisson equation accurate to order ǫ. The final expressions are explicit and can be implemented into any simulation without further computations.