S. R. Grosshauser

Toroidal momentum transport in a tokamak caused by symmetry breaking parallel derivatives

A new mechanism for toroidal momentum transport in a tokamak is investigated using the gyro-kinetic model. First, an analytic model is developed through the use of the ballooning transform. The terms that generate the momentum transport are then connected with the poloidal derivative of the ballooning envelope, which are one order smaller in the normalised Larmor radius, compared with the derivative of the eikonal. The mechanism, therefore, does not introduce an inhomogeneity in the radial direction, in contrast with the effect of profile shearing. Numerical simulations of the linear ion temperature gradient mode with adiabatic electrons, retaining the finite ρ* effects in the E × B velocity, the drift, and the gyro-average, are presented. The momentum flux is found to be linear in the normalised Larmor radius (ρ*) but is, nevertheless, generating a sizeable counter-current rotation. The total momentum flux scales linear with the aspect ratio of the considered magnetic surface, and increases with increasi...

Toroidal momentum transport in a tokamak due to profile shearing

The effect of profile shearing on toroidal momentum transport is studied in linear and non-linear gyro-kinetic simulations. Retaining the radial dependence of both plasma and geometry parameters leads to a momentum flux that has contributions both linear in the logarithmic gradients of density and temperature, as well as contributions linear in the derivatives of the logarithmic gradients. The effect of the turbulence intensity gradient on momentum transport is found to be small for the studied parameters. Linear simulations at fixed normalized toroidal wave number predict a weak dependence of the momentum flux on the normalized Larmor radius ρ*=ρ/R. Non-linear simulations, however, at sufficiently small ρ* show a linear scaling of the momentum flux with ρ*. The obtained stationary rotation gradients are in the range of, although perhaps smaller than, current experiments. For a reactor plasma, however, a rather small rotation gradient should result from profile shearing.

Comparison of gradient and flux driven gyro-kinetic turbulent transport

Flux and gradient driven ion temperature gradient turbulence in tokamak geometry and for Cyclone base case parameters are compared in the local limit using the same underlying gyro-kinetic turbulence model. The gradient driven turbulence described using the flux tube model with periodic boundary conditions has a finite ion heat flux Qi≈10n0T0ρ*2vth, where n0 (T0) is the background density (temperature), ρ*=ρ/R is the normalized Larmor radius, R is the major radius of the device, and vth is the ion thermal velocity at the nonlinear threshold of the temperature gradient length for turbulence generation. Consequently, the gradient driven local transport model is unable to accurately describe heat fluxes below Qi 10n0T0ρ*2vth, and at higher heat fluxes, the statistics of the turbulence is ...

Gradient-driven flux-tube simulations of ion temperature gradient turbulence close to the non-linear threshold

It is shown that Ion Temperature Gradient turbulence close to the threshold exhibits a long time behaviour, with smaller heat fluxes at later times. This reduction is connected with the slow growth of long wave length zonal flows, and consequently, the numerical dissipation on these flows must be sufficiently small. Close to the nonlinear threshold for turbulence generation, a relatively small dissipation can maintain a turbulent state with a sizeable heat flux, through the damping of the zonal flow. Lowering the dissipation causes the turbulence, for temperature gradients close to the threshold, to be subdued. The heat flux then does not go smoothly to zero when the threshold is approached from above. Rather, a finite minimum heat flux is obtained below which no fully developed turbulent state exists. The threshold value of the temperature gradient length at which this finite heat flux is obtained is up to 30% larger compared with the threshold value obtained by extrapolating the heat flux to zero, and t...

Ion temperature gradient turbulence close to the finite heat flux threshold

The dependence of the heat flux on the temperature gradient length in collisionless ion temperature gradient turbulence has recently been revisited. It has been found that the heat flux is discontinuous at a finite heat flux threshold larger than the (Dimits) interpolated threshold. In this paper, the influence of collisions on the heat flux close to the threshold is investigated. It is found that up to relatively high collision frequencies, relevant to the modern day experiments, a discontinuous behaviour of the heat flux as a function of the gradient length persists. Collisions, however, do lead to a reduction in the gradient length at which the discontinuity is observed. Below the finite heat flux threshold, a state of low turbulence with a vanishing small heat flux persists, which can drive the zonal flow against the collisional dissipation. This state is characterised by the fully developed staircases in the radial ExB shearing profile. Increasing the collision frequency at a fixed gradient length le...

On the tertiary instability formalism of zonal flows in magnetized plasmas

This paper investigates the so-called tertiary instabilities driven by the zonal flow in gyro-kinetic tokamak core turbulence. The Kelvin Helmholtz instability is first considered within a 2D fluid model and a threshold in the zonal flow wave vector kZF>kZF,c for instability is found. This critical scale is related to the breaking of the rotational symmetry by flux-surfaces, which is incorporated into the modified adiabatic electron response. The stability of undamped Rosenbluth-Hinton zonal flows is then investigated in gyro-kinetic simulations. Absolute instability, in the sense that the threshold zonal flow amplitude tends towards zero, is found above a zonal flow wave vector kZF,cρi≈1.3 ( ρi is the ion thermal Larmor radius), which is comparable to the 2D fluid results. Large scale zonal flows with kZF<kZF,c are unstable for sufficiently large amplitudes with increasing trend for an increasing radial scale. However, the critical E × B-shearing rate associated with the stability boundary ωE×B,c exceeds typical values connected to the pure flow state at marginal stability by more than an order of magnitude, which therefore lies deeply in the stable parameter region. Furthermore, the impact of zonal temperature perturbations on the tertiary instability is examined. Although temperature perturbations favor instability, the realistic values of gradient-driven gyro-kinetic simulations still lie deeply in the stable parameter regime. Therefore, the relevance of the tertiary instability as a saturation mechanism to the zonal flow amplitude is questioned, as most of the zonal flow intensity is concentrated in modes satisfying kZF≪kZF,c as well as ωE×B≪ωE×B,c.

Turbulence spreading in gyro-kinetic theory

In this letter a new operative definition for the turbulence intensity in connection with magnetized plasmas is given. In contrast to previous definitions the new definition satisfies a Fisher–Kolmogorov–Petrovskii–Piskunov type equation. Furthermore, explicit expressions for the turbulence intensity and the turbulence intensity flux, that allow for the first time direct numerical evaluation, are derived. A carefully designed numerical experiment for the case of a tokamak is performed to study the impact of turbulence spreading. The effective turbulence diffusion coefficient is measured to be smaller than the heat conduction coefficient and the turbulence spreading length is found to be of the order of the turbulence correlation length. The results show that turbulence spreading can play a role in the non-local flux gradient relation, or in the scaling of transport coefficients with the normalized Larmor radius, only over lengths scale of the order of the turbulence correlation length. A new turbulence convection mechanism, due to the drift connected with the magnetic field inhomogeneities, is described. The convective flux integrates to zero under the flux surface average unless there is an up–down asymmetry in the tubulence intensity. The latter asymmetry can be generated through a radial inhomogeneity or plasma rotation. It is shown that the turbulence convection can lead to a spreading of the order of the correlation length.

Influence of centrifugal effects on particle and momentum transport in National Spherical Torus Experiment

This paper studies the effect of rotation on microinstabilities under experimentally relevant conditions in the spherical tokamak National Spherical Torus Experiment (NSTX). The focus is specifically on the centrifugal force effects on the impurity and momentum transport in the core ( r/a=0.7) of an H-mode plasma. Due to relatively high beta, the linear simulations predict the presence of both microtearing mode (MTM) and hybrid ion temperature gradient-kinetic ballooning mode (ITG-KBM) electromagnetic instabilities. Rotation effects on both MTM and ITG-KBM growth rates and mode frequencies are found to be small for the experimental values. However, they do influence the quasi-linear particle and momentum fluxes predicted by ITG-KBM (MTM contributes only to electron heat flux). The gradient of the intrinsic carbon impurity in the source-free core region is predicted to be locally hollow, strengthened by centrifugal effects. This result is consistent with experimental measurements and contradicts neoclassic...

Damping of zonal modes through turbulent momentum transport

It is shown that the radial transport of parallel momentum provides a damping mechanism for the zonal flow relevant for plasma turbulence close to the nonlinear threshold. The damping mechanism is confirmed by a “Rosenbluth-Hinton” test with a model radial momentum diffusion, in which the decay rate of the residual potential is found to be proportional to the model diffusion coefficient and in good agreement with the analytical result. Nonlinear simulations show that, when momentum transport is suppressed, stronger long wavelength zonal flow shearing occurs. The suppression of momentum transport then allows for the development of fully developed staircase structures in the E × B shear, which can suppress turbulence completely for a finite time window. No impact on shorter wavelength zonal flows is observed, in contrast to the analytical prediction which suggests a high damping rate. The latter result raises the question of the relevance the residual zonal flow plays in turbulence saturation.

The radial propagation of turbulence in gyro-kinetic toroidal systems

In this paper, a conservation equation is derived for the radially dependent entropy in toroidal geometry using the local approximation of the gyro-kinetic equation. This naturally leads to an operative definition for the turbulence intensity. It is shown that the conservation equation can be split into two contributions, one describing the dynamics of the zonal modes and one for the non-zonal modes. In essence the paper provides an operative tool for both analytic as well as numeric studies of the radial propagation of turbulence in tokamak plasmas.