D. Dickinson

Kinetic instabilities that limit β in the edge of a tokamak plasma: a picture of an H-mode pedestal.

Plasma equilibria reconstructed from the Mega-Amp Spherical Tokamak have sufficient resolution to capture plasma evolution during the short period between edge-localized modes (ELMs). Immediately after the ELM, steep gradients in pressure, P, and density, n(e), form pedestals close to the separatrix, and they then expand into the core. Local gyrokinetic analysis over the ELM cycle reveals the dominant microinstabilities at perpendicular wavelengths of the order of the ion Larmor radius. These are kinetic ballooning modes in the pedestal and microtearing modes in the core close to the pedestal top. The evolving growth rate spectra, supported by gyrokinetic analysis using artificial local equilibrium scans, suggest a new physical picture for the formation and arrest of this pedestal.

Towards the construction of a model to describe the inter-ELM evolution of the pedestal on MAST

Pedestal profiles that span the ELM cycle have been obtained and used to test the idea that the pedestal pressure gradient in MAST is limited by the onset of kinetic ballooning modes (KBMs). During the inter-ELM period of a regularly type I ELM-ing discharge on MAST, the pressure pedestal height and width increase together while the pressure gradient increases by only 15% during the ELM cycle. Stability analyses show that the pedestal region over which infinite-n ballooning modes are unstable also broadens during the ELM cycle. To test the relationship between the width of the region that is unstable to n = ∞ ideal magnetohydrodynamic ballooning modes and KBMs the gyrokinetic code, GS2, has been used for microstability analysis of the edge plasma region in MAST. The gyrokinetic simulations find that KBM modes with twisting parity are the dominant microinstabilities in the steep pedestal region, with a transition to tearing parity modes in the shallower pressure gradient core region immediately inside the pedestal top. The region over which KBMs are unstable increases during the ELM cycle, and a good correlation is found between the region where KBMs dominate and the region that is unstable to infinite-n ideal ballooning modes.

Gyrokinetic simulations of spherical tokamaks

This paper reviews transport and confinement in spherical tokamaks (STs) and our current physics understanding of this that is partly based on gyrokinetic simulations. Equilibrium flow shear plays an important role, and we show how this is consistently included in the gyrokinetic framework for flows that greatly exceed the diamagnetic velocity. The key geometry factors that influence the effectiveness of turbulence suppression by flow shear are discussed, and we show that toroidal equilibrium flow shear can sometimes entirely suppress ion scale turbulence in todays STs. Advanced nonlinear simulations of electron temperature gradient (ETG) driven turbulence, including kinetic ion physics, collisions and equilibrium flow shear, support the model that ETG turbulence can explain electron heat transport in many ST discharges.

Structure of micro-instabilities in tokamak plasmas: Stiff transport or plasma eruptions?

Solutions to a model 2D eigenmode equation describing micro-instabilities in tokamak plasmas are presented that demonstrate a sensitivity of the mode structure and stability to plasma profiles. In narrow regions of parameter space, with special plasma profiles, a maximally unstable mode is found that balloons on the outboard side of the tokamak. This corresponds to the conventional picture of a ballooning mode. However, for most profiles, this mode cannot exist, and instead, a more stable mode is found that balloons closer to the top or bottom of the plasma. Good quantitative agreement with a 1D ballooning analysis is found, provided the constraints associated with higher order profile effects, often neglected, are taken into account. A sudden transition from this general mode to the more unstable ballooning mode can occur for a critical flow shear, providing a candidate model for why some experiments observe small plasma eruptions (Edge Localised Modes, or ELMs) in place of large Type I ELMs.

Using a local gyrokinetic code to study global ion temperature gradient modes in tokamaks

In this paper the global eigenmode structures of linear ion temperature gradient (ITG) modes in tokamak plasmas are obtained using a novel technique which combines results from the local gyrokinetic code GS2 with analytical theory to reconstruct global properties. Local gyrokinetic calculations are performed for a range of radial flux surfaces, x, and ballooning phase angles, p, to map out the local complex mode frequency, Ω0(x, p) = ω0(x, p) + iγ0(x, p) for a single toroidal mode number, n. Taylor expanding Ω0 about a reference surface at x = 0, and employing the Fourier-ballooning representation leads to a second order ODE for the amplitude envelope, A(p), which describes how the local results are combined to form the global mode. The equilibrium profiles impact on the variation of Ω0(x, p) and hence influence the global mode structure. The simulations presented here are based upon a global extension to the CYCLONE base case and employ the circular Miller equilibrium model. In an equilibrium with radially varying profiles of a/LT and a/Ln, peaked at x = 0, and with all other equilibrium profiles held constant, including ηi = Ln/LT, Ω0(x, p) is found to have a stationary point. The reconstructed global mode sits at the outboard mid-plane of the tokamak, with global growth rate, γ ~ Max[γ0]. Including the radial variation of other equilibrium profiles like safety factor and magnetic shear, leads to a mode that peaks away from the outboard mid-plane, with a reduced global growth rate. Finally, the influence of toroidal flow shear has also been investigated through the introduction of a Doppler shift, , where Ω is the equilibrium toroidal flow, and a prime denotes the radial derivative. The equilibrium profile variations introduce an asymmetry into the global growth rate spectrum with respect to the sign of , such that the maximum growth rate is achieved with non-zero shearing, consistent with recent global gyrokinetic calculations.

Kinetic microtearing modes and reconnecting modes in strongly magnetised slab plasmas

The problem of the linear microtearing mode in a slab magnetised plasma, and its connection to kinetic reconnecting modes, is addressed. Electrons are described using a novel hybrid fluid-kinetic model that captures electron heating, ions are gyrokinetic. Magnetic reconnection can occur as a result of either electron conductivity and inertia, depending on which one predominates. We eschew the use of an energy dependent collision frequency in the collisional operator model, unlike previous works. A model of the electron conductivity that matches the weakly collisional regime to the exact Landau result at zero collisionality and gives the correct electron isothermal response far from the reconnection region is presented. We identify in the breaking of the constant-

Intermediate-k density and magnetic field fluctuations during inter-ELM pedestal evolution in MAST

A_{\parallel}

Generalised ballooning theory of two-dimensional tokamak modes

approximation the necessary condition for microtearing instability in the collisional regime. Connections with the theory of collisional non-isothermal (or semicollisional) and collisionless tearing-parity electron temperature gradient driven (ETG) modes are elucidated.

The response of toroidal drift modes to profile evolution: a model for small-ELMs in tokamak plasmas?

Measurements of local density and magnetic field fluctuations near the pedestal top, conditionally averaged over the edge localized mode (ELM) cycle, have been made in Mega Amp Spherical Tokamak (MAST). A Doppler backscattering (DBS) system installed at MAST was used to measure intermediate-k density fluctuations at the top of the pedestal. A novel diagnostic technique combining DBS with cross-polarization scattering (CP-DBS) enabled magnetic field fluctuations to also be locally measured at similar wave numbers. Polarization isolation and other effects for CP-DBS are discussed. Both measurements were used in a series of high-β –4.5) MAST plasmas with large type-I ELMs with an period where microtearing modes (MTMs) had been predicted to be unstable in similar conditions (Dickinson 2012 Phys. Rev. Lett. 108 135002). The measured density fluctuation level increased by a factor of about 4 between 2 and 4 ms after the ELM, which was correlated with the recovery of the density profile while the temperature pedestal height continued to increase slowly. Magnetic field fluctuations showed different temporal behaviors, slowly increasing throughout the ELM cycle as the local β increased. Linear GS2 calculations show both MTM and electron temperature gradient (ETG) modes unstable at similar wave numbers as the measurements (although with more overlap between ETG wave numbers and diagnostic spectral resolution) at the top of the pedestal, along with kinetic ballooning modes are unstable lower in the pedestal (at larger wavelengths). The inferred ratio of fluctuation levels from experiment was . The comparable ratios from GS2 were for the MTM and for the ETG. Both the experimental wave number range and the fluctuation ratio are more similar to the linear characteristics of the ETG than the MTM. These results imply that intermediate-k fluctuations due to the ETG play a role in inter-ELM pedestal evolution.

Electromagnetic effects in the stabilization of turbulence by sheared flow

In this work, using solutions from a local gyrokinetic flux-tube code combined with higher order ballooning theory, a new analytical approach is developed to reconstruct the global linear mode structure with associated global mode frequency. In addition to the isolated mode (IM), which usually peaks on the outboard mid-plane, the higher order ballooning theory has also captured other types of less unstable global modes: (a) the weakly asymmetric ballooning theory (WABT) predicts a mixed mode (MM) that undergoes a small poloidal shift away from the outboard mid-plane, (b) a relatively more stable general mode (GM) balloons on the top (or bottom) of the tokamak plasma. In this paper, an analytic approach is developed to combine these disconnected analytical limits into a single generalised ballooning theory. This is used to investigate how an IM behaves under the effect of sheared toroidal flow. For small values of flow an IM initially converts into a MM where the results of WABT are recaptured, and eventually, as the flow increases, the mode asymptotically becomes a GM on the top (or bottom) of the plasma. This may be an ingredient in models for understanding why in some experimental scenarios, instead of large edge localised modes (ELMs), small ELMs are observed. Finally, our theory can have other important consequences, especially for calculations involving Reynolds stress driven intrinsic rotation through the radial asymmetry in the global mode structures. Understanding the intrinsic rotation is significant because external torque in a plasma the size of ITER is expected to be relatively low.