Strong E× B plasma flow shear is beneficial for reducing turbulent transport. However, traditional methods of driving flow shear do not scale well to large devices such as future fusion power plants. In this paper, we use a large number of nonlinear gyrokinetic simulations to study a novel approach to increase flow shear: decreasing the momentum diffusivity to make the plasma “easier to push”. We first use an idealized circular geometry and find that one can obtain low momentum diffusivity at tight aspect ratio, low safety factor, high magnetic shear and low temperature gradient. This is the so-called Low Momentum Diffusivity (LMD) regime. To drive intrinsic momentum flux, we then tilt the flux surface, making it up-down asymmetric. In the LMD regime, this intrinsic momentum flux drives strong flow shear that can significantly reduce the heat flux and increase the critical temperature gradient. We also consider the actual experimental geometry of the MAST tokamak to illustrate that this strategy can be practical and create experimentally significant flow shear. Lastly, a preliminary prediction for the SMART tokamak is made.

A quasi-linear reduced transport model is developed from a database of high- $\beta$ electromagnetic nonlinear gyrokinetic simulations performed with spherical tokamak for energy production (STEP) relevant parameters. The quasi-linear model is fully electromagnetic and accounts for the effect of equilibrium flow shear using a novel approach. Its flux predictions are shown to agree quantitatively with predictions from local nonlinear gyrokinetic simulations across a broad range of STEP-relevant local equilibria. This reduced transport model is implemented in the T3D transport solver that is used to perform the first flux-driven simulations for STEP to account for transport from hybrid kinetic ballooning mode turbulence, which dominates over a wide region of the core plasma. Nonlinear gyrokinetic simulations of the final transport steady state from T3D return turbulent fluxes that are consistent with the reduced model, indicating that the quasi-linear model may also be appropriate for describing the transport steady state. Within the assumption considered here, our simulations support the existence of a transport steady state in STEP with a fusion power comparable to that in the burning flat top of the conceptual design, but do not demonstrate how this state can be accessed.

Electromagnetic microtearing modes (MTMs) have been observed in many different spherical tokamak (ST) regimes. Understanding how these and other electromagnetic modes nonlinearly saturate is likely critical in understanding the confinement of a high beta ST. Equilibrium E x B sheared flows have sometimes been found to significantly suppress low beta ion scale transport in both gyrokinetic simulations and in experiment. This work aims to understand the conditions under which E x B sheared flow impacts on the saturation of MTM simulations, as there have been examples where it does (Guttenfelder et al 2012 Phys. Plasmas 19 056119) and does not (Doerk et al 2012 Phys. Plasmas 19 055907) have a considerable effect. Two experimental regimes are examined from MAST and NSTX, on surfaces that have unstable MTMs. The MTM driven transport on a local flux surface in MAST is shown to be more resilient to suppression via E x B shear, compared to the case from NSTX where the MTM transport is found to be significantly suppressed. This difference in the response to flow shear is explained through the impact of magnetic shear, s<^>, on the MTM linear growth rate dependence on ballooning angle, theta 0. At low s<^>, the growth rate depends weakly on theta 0, but at higher s<^>, the MTM growth rate peaks at theta 0 = 0, with regions of stability at higher theta 0. Equilibrium E x B sheared flows act to advect the theta 0 of a mode in time, providing a mechanism to reduce the linear drive and suppress the transport from modes where the growth rate is strongly peaked in theta 0 (Roach et al 2009 Plasma Phys. Control. Fusion 51 124020). Bicoherence analysis demonstrates that with the inclusion of E x B shear there is a more effective coupling between the linearly driven drift-waves responsible for transport and zonal modes, which enhances damping. The dependence of gamma MTM on theta 0 is in qualitative agreement with a recent theory (Hardman et al Plasma Phys. Control. Fusion 65 045011) at low beta when q similar to 1, but the agreement worsens at higher q where the theory breaks down. At higher s<^>, MTMs drive more stochastic transport due a stronger overlap of magnetic islands centred on neighbouring rational surfaces, but equilibrium E x B shear acts to mitigate this. This is especially critical towards the plasma edge where s<^> can be larger and where the total stored energy in the plasma is more sensitive to the local gradients. This work highlights the important role of the safety factor profile in determining the impact of equilibrium E x B shear on the saturation level of MTM turbulence.

Based on the analysis of a large number of high-fidelity nonlinear gyrokinetic simulations, we propose a novel strategy to improve confinement in spherical tokamak plasmas by combining up-down asymmetric flux surface shaping with the Low Momentum Diffusivity (LMD) regime. We show that the intrinsic momentum flux driven by up-down asymmetry creates strong flow shear in the LMD regime that can significantly reduce energy transport, increasing the critical gradient by up to 25%. In contrast to traditional methods for generating flow shear, such as neutral beam injection, this approach requires no external momentum source and is expected to scale well to large fusion devices. The experimental applicability of this strategy in spherical tokamaks is addressed via simulations by considering actual equilibria from Mega Ampere Spherical Tokamak and a preliminary equilibrium from SMART.

Recent results from MAST Upgrade are presented, emphasising understanding the capabilities of this new device and deepening understanding of key physics issues for the operation of ITER and the design of future fusion power plants. The impact of MHD instabilities on fast ion confinement have been studied, including the first observation of fast ion losses correlated with Compressional and Global Alfvén Eigenmodes. High-performance plasma scenarios have been developed by tailoring the early plasma current ramp phase to avoid internal reconnection events, resulting in a more monotonic q profile with low central shear. The impact of m / n = 3/2, 2/1 and 1/1 modes on thermal plasma confinement and rotation profiles has been quantified, and scenarios optimised to avoid them have transiently reached values of normalised beta approaching 4.2. In pedestal and ELM physics, a maximum pedestal top temperature of ∼350 eV has been achieved, exceeding the value achieved on MAST at similar heating power. Mitigation of type-I ELMs with n = 1 RMPs has been observed. Studies of plasma exhaust have concentrated on comparing conventional and Super-X divertor configurations, while X-point target, X-divertor and snowflake configurations have been developed and studied in parallel. In L-mode discharges, the separatrix density required to detach the outer divertors is approximately a factor 2 lower in the Super-X than the conventional configuration, in agreement with simulations. Detailed analysis of spectroscopy data from studies of the Super-X configuration reveal the importance of including plasma-molecule interactions and D _2 Fulcher band emission to properly quantify the rates of ionisation, plasma-molecule interactions and volumetric recombination processes governing divertor detachment. In H-mode with conventional and Super-X configurations, the outer divertors are attached in the former and detached in the latter with no impact on core or pedestal confinement.

In this work, we present first-of-their-kind nonlinear local gyrokinetic simulations of electromagnetic turbulence at mid-radius in the burning plasma phase of the conceptual high-$\beta$, reactor-scale, tight-aspect-ratio tokamak STEP (Spherical Tokamak for Energy Production). A prior linear analysis in D. Kennedy et al. 2023 Nucl. Fusion 63 126061 reveals the presence of unstable hybrid kinetic ballooning modes, where inclusion of the compressional magnetic field fluctuation, $\delta B_{\parallel}$, is crucial, and subdominant microtearing modes are found at binormal scales approaching the ion-Larmor radius. Local nonlinear gyrokinetic simulations on the selected surface in the central core region suggest that hybrid kinetic ballooning modes can drive large turbulent transport, and that there is negligible turbulent transport from subdominant microtearing modes when hybrid kinetic ballooning modes are artificially suppressed (through the omission of $\delta B_{\parallel}$). Nonlinear simulations that include perpendicular equilibrium flow shear can saturate at lower fluxes that are more consistent with the available sources in STEP. This analysis suggests that hybrid kinetic ballooning modes could play an important role in setting the turbulent transport in STEP, and possible mechanisms to mitigate turbulent transport are discussed. Increasing the safety factor or the pressure gradient strongly reduces turbulent transport from hybrid kinetic ballooning modes in the cases considered here. Challenges of simulating electromagnetic turbulence in this high-$\beta$ regime are highlighted. In particular the observation of radially extended turbulent structures in the absence of equilibrium flow shear motivates future advanced global gyrokinetic simulations that include $\delta B_\parallel$.

This study presents an analytical investigation of energetic particle-induced geodesic acoustic modes (EGAMs) within a gyro-kinetic model, incorporating finite-orbit-width (FOW) effects up to the second order. The inclusion of second-order FOW effects introduces two distinct types of energetic particle-wave resonances, occurring at omega = omega(h)(t) and omega = 2 omega(h)(t) , respectively, where omega th denotes the transit frequency of energetic particles (EPs). It is found that two unstable EGAM branches coexist: a low frequency branch (LFB) characterized by 0 < omega(LFB )< omega(h)(t,max) , and a high frequency branch (HFB) marked by omega(h )(t,max)<omega(HFB )< 2 omega(h)(t,max). The instability of LFB primarily arises from the resonance omega = omega(h)(t) , mainly introduced by first-order FOW effects. As a result, the instability of LFB always exists regardless of the presence or absence of second-order FOW effects, and is barely modified by these effects. In contrast, the instability of HFB is exclusively attributed to the resonance omega = 2 omega(h)(t) induced by second-order FOW effects. Consequently, the HFB exhibits instability in the presence of these effects.

Plasma elongation effects on energetic particle-induced geodesic acoustic modes (EGAMs) are theoretically investigated by using gyro-kinetic equations and the Miller local equilibrium model. Including an arbitrary elongation kappa and a finite radial derivative s kappa=r partial differential r kappa/kappa , a general EGAM dispersion relation is obtained for an arbitrary energetic particle (EP) distribution. In particular, we obtain analytical EGAM dispersion relations for both the double-shifted Maxwellian distribution and the standard slowing-down distribution of EPs. In both cases, the frequency of the unstable EGAM branch decreases slowly with increasing elongation, while its growth rate decreases rapidly with kappa when the ratio of the EP to the bulk ion density, nh/ni > 0.1 . These trends agree well with previous GENE and ORB5 simulations (Di Siena et al 2018 Nucl. Fusion 58 106014), but differ significantly from the elongation effects on geodesic acoustic modes (GAMs) (Gao et al 2009 Nucl. Fusion 49 045014). The portion of the EGAM dispersion relation accounting for the first-order finite-orbit-width shows greater sensitivity to frequency compared to that of GAM, which explains the smaller variations in the frequency of EGAM as kappa changes. When the EP number is small (typically, nh/ni approximate to 5% ) in the double-shifted Maxwellian case, the growth rate of EGAMs first increases with the increasing elongation and then decreases, while it monotonically increases with kappa in the slowing-down case. Furthermore, the effects of s kappa on EGAMs are similar to the elongation kappa effects but weaker.

STEP is a spherical tokamak prototype power plant that is being designed to demonstrate net electric power. The design phase involves the exploitation of plasma models to optimise fusion performance subject to satisfying various physics and engineering constraints. A modelling workflow, including integrated core plasma modelling, MHD stability analysis, SOL and pedestal modelling, coil set and free boundary equilibrium solvers, and whole plant design, has been developed to specify the design parameters and to develop viable scenarios. The integrated core plasma model JETTO is used to develop individual flat-top operating points that satisfy imposed criteria for fusion power performance within operational constraints. Key plasma parameters such as normalised beta, Greenwald density fraction, auxiliary power and radiated power have been scanned to scope the operational space and to derive a collection of candidate non-inductive flat-top points. The assumed auxiliary heating and current drive is either from electron cyclotron systems only or a combination of electron cyclotron and electron Bernstein waves. At present stages of transport modelling, there is a large uncertainty in overall confinement for relevant parameter regimes. For each of the two auxiliary heating and current drive systems scenarios, two candidate flat-top points have been developed based on different confinement assumptions, totalling to four operating points. A lower confinement assumption generally suggests operating points in high-density, high auxiliary power regimes, whereas higher confinement would allow access to a broader parameter regime in density and power while maintaining target fusion power performance.

Spherical tokamaks have many desirable features that make them an attractive choice for a future fusion power plant. Power-plant viability is intrinsically related to plasma heat and particle confinement, and this is often determined by the level of micro-instability-driven turbulence. Accurate calculation of the properties of turbulent microinstabilities is therefore critical for tokamak design; however, the evaluation of these properties is computationally expensive. The considerable number of geometric and thermodynamic parameters and the high resolutions required to accurately resolve these instabilities make repeated use of direct numerical simulations in integrated modeling workflows extremely computationally challenging and create the need for fast, accurate, reduced-order models. This paper outlines the development of a data-driven reduced-order model, often termed a surrogate model for the properties of micro-tearing modes (MTMs) across a spherical tokamak reactor-relevant parameter space utilizing Gaussian process regression and classification, techniques from machine learning. These two components are used in an active learning loop to maximize the efficiency of data acquisition, thus minimizing computational cost. The high-fidelity gyrokinetic code GS2 is used to calculate the linear properties of the MTMs: the mode growth rate, frequency, and normalized electron heat flux, and core components of a quasi-linear transport model. Data cross-validation and direct validation on unseen data are used to ascertain the performance of the resulting surrogate models.

The results of a gyrokinetic analysis of turbulent transport driven by the electron temperature gradient (ETG) in the MAST pedestal are presented. Local nonlinear simulations from the gyrokinetic code, GENE, show that heat flux produced by ETG turbulence is 10%–30% of the total applied heating power in the upper pedestal and pedestal top during both the pre-edge-localized mode (ELM) (80%–99% inter-ELM period) and post-ELM (0%–20%) periods. Increasing strongly with the ETG, the ETG transport appears to be stiff. Considering radiation losses, ion thermal transport, and the strong sensitivity of the transport to the ETG, we propose that ETG transport is a plausible mechanism mediating the inter-ELM temperature profile on MAST. Cognizant of the possibility that sharp variations in the pedestal profiles may violate the local approximation, we conducted global nonlinear simulations; the results are in good agreement with local simulations except near the pedestal top, where extended radial structures and high transport levels (far beyond experimental) develop in the local simulations. We quantify and parameterize the discrepancy between local and global simulations by calculating the ratio of the radial correlation length to a length scale representative of the profile curvature. When this ratio is sufficiently small, local and global simulations agree as expected.

This paper discusses the importance of parallel perturbations of the magnetic-field in gyrokinetic simulations of electromagnetic instabilities and turbulence at mid-radius in the burning plasma phase of the conceptual high-beta, reactor-scale, tight-aspect-ratio tokamak STEP. Previous studies have revealed the presence of unstable hybrid kinetic ballooning modes (hKBMs) and subdominant microtearing modes at binormal scales approaching the ion Larmor radius. In this STEP plasma it was found that the hKBM requires the inclusion of parallel magnetic-field perturbations to be linearly unstable. Here, the extent to which the inclusion of fluctuations in the parallel magnetic-field can be relaxed is explored through gyrokinetic simulations. In particular, the frequently used MHD approximation (dropping delta B parallel to and setting the del B drift frequency equal to the curvature drift frequency) is discussed and simulations explore whether this approximation is useful for modelling STEP plasmas. It is shown that the MHD approximation can reproduce some of the linear properties of the full STEP gyrokinetic system, but is too stable at low ky and nonlinear simulations using the MHD approximation result in very different transport states. It is demonstrated that the MHD approximation is challenged by the high beta ' values in STEP, and that the approximation improves considerably at lower beta ' . Furthermore, it is shown that the sensitivity of STEP to delta B parallel to fluctuations is primarily because the plasma sits close to marginality and it is shown that in slightly more strongly driven conditions the hKBM is unstable without delta B parallel to. Crucially, it is demonstrated that the state of large transport typically predicted by local electromagnetic gyrokinetic simulations of STEP plasmas is not solely due to delta B parallel to physics.

In magnetic confinement fusion devices, the ratio of the plasma pressure to the magnetic field energy, β , can become sufficiently large that electromagnetic microinstabilities become unstable, driving turbulence that distorts or reconnects the equilibrium magnetic field. In this paper, a theory is proposed for electromagnetic, electron-driven linear instabilities that have current layers localised to mode-rational surfaces and binormal wavelengths comparable to the ion gyroradius. The model retains axisymmetric toroidal geometry with arbitrary shaping, and consists of orbit-averaged equations for the mode-rational surface layer, with a ballooning space kinetic matching condition for passing electrons. The matching condition connects the current layer to the large scale electromagnetic fluctuations, and is derived in the limit that β is comparable to the square root of the electron-to-ion-mass ratio. Electromagnetic fluctuations only enter through the matching condition, allowing for the identification of an effective β that includes the effects of equilibrium flux surface shaping. The scaling predictions made by the asymptotic theory are tested with comparisons to results from linear simulations of micro-tearing and electrostatic microinstabilities in MAST discharge #6252, showing excellent agreement. In particular, it is demonstrated that the effective β can explain the dependence of the local micro-tearing mode (MTM) growth rate on the ballooning parameter θ 0 –possibly providing a route to optimise local flux surfaces for reduced MTM-driven transport.

First nonlinear gyrokinetic simulations of microtearing modes in the core of a MAST case are performed on two surfaces of the high-collisionality discharge used in Valovič et al (2011 Nucl. Fusion 51 073045) to obtain the favorable energy confinement scaling with collisionality, τ E ∝ ν ∗ − 1 . On the considered surfaces microtearing modes dominate linearly at binormal length scales of the order of the ion Larmor radius. While the effect of electron collision frequency is moderate in linear simulations, a strong dependence on this parameter is found in nonlinear simulations at r / a = 0.5 , where r and a are the surface and tokamak minor radius, respectively. The dynamics of magnetic islands generated by microtearing modes is analysed, showing that the radial extent of the stochastic region caused by islands overlapping plays an important role in determining the saturation level of the microtearing mode driven heat flux. Local nonlinear gyrokinetic simulations show that the microtearing mode driven heat flux, Q e M T M , is largely dominated by magnetic flutter and depends strongly on the magnetic shear, s ˆ . Comparing two surfaces, r / a = 0.5 and r / a = 0.6 , reveals that Q e M T M is negligible at r / a = 0.5 ( s ˆ = 0.34 ), with the electron temperature gradient (ETG) driven heat flux, Q e E T G , comparable to the experimental electron heat flux, Q e e x p , while Q e M T M is significantly larger and comparable to Q e E T G and Q e e x p at r / a = 0.6 ( s ˆ = 1.1 ). Microtearing modes cause more experimentally significant transport in higher s ˆ regions and may influence (together with ETG modes) the observed scaling of energy confinement time with collisionality (Valovič et al 2011 Nucl. Fusion 51 073045).

Geodesic acoustic modes (GAMs) represent the oscillating counterpart of zonal flow in tokamak plasma and can affect transport due to their interaction with turbulence eddies. GAMs have been observed in many experiments and modelled under different conditions, but because of their variety of characteristics, we do not yet have a complete picture of their dynamics. It has been demonstrated that optical methods can be efficiently used to describe and predict several characteristics of the GAM radial structures that can be interpreted as 'waves' propagating in the space-time. We exploit complex eikonal theories to investigate the behavior of GAMs that are commonly observed in experiments, and find that their periodic modulation and intermittency can be explained by the properties of the equilibrium temperature profile. Theoretical results obtained in this work are supported by gyrokinetic simulations for several equilibria. Implications for existence criteria and GAM dynamics in different plasma equilibrium conditions are discussed, with particular attention to the edge plasma in low and high confinement modes.

A predictive model for the electron temperature profile of the H-mode pedestal is described, and its results are compared with the pedestal structure of JET-ILW plasmas. The model is based on a scaling for the gyro-Bohm normalized, turbulent electron heat flux qe/qe,gB resulting from electron temperature gradient (ETG) turbulence, derived from results of nonlinear gyrokinetic (GK) calculations for the steep gradient region. By using the local temperature gradient scale length LTe in the normalization, the dependence of qe/qe,gB on the normalized gradients R/LTe and R/Lne can be represented by a unified scaling with the parameter ηe=Lne/LTe, to which the linear stability of ETG turbulence is sensitive when the density gradient is sufficiently steep. For a prescribed density profile, the value of R/LTe determined from this scaling, required to maintain a constant electron heat flux qe across the pedestal, is used to calculate the temperature profile. Reasonable agreement with measurements is found for different cases, the model providing an explanation of the relative widths and shifts of the Te and ne profiles, as well as highlighting the importance of the separatrix boundary conditions. Other cases showing disagreement indicate conditions where other branches of turbulence might dominate. This article is part of a discussion meeting issue ‘H-mode transition and pedestal studies in fusion plasmas’.

We present herein the results of a linear gyrokinetic analysis of electromagnetic microinstabilites in the conceptual high − β , reactor-scale, tight-aspect-ratio tokamak Spherical Tokamak for Energy Production, https://step.ukaea.uk . We examine a range of flux surfaces between the deep core and the pedestal top for two candidate flat-top operating points of the prototype device. Local linear gyrokinetic analysis is performed to determine the type of microinstabilities that arise under these reactor-relevant conditions. We find that the equilibria are dominated at ion binormal scales by a hybrid version of the kinetic ballooning mode (KBM) instability that has significant linear drive contributions from the ion temperature gradient and from trapped electrons, while collisional microtearing modes (MTMs) are sub-dominantly also unstable at similar binormal scales. The hybrid -KBM and MTM exhibit very different radial scales. We study the sensitivity of these instabilities to physics parameters, and discuss potential mechanisms for mitigating them. The results of this investigation are compared to a small set of similar conceptual reactor designs in the literature. A detailed benchmark of the linear results is performed using three gyrokinetic codes; alongside extensive resolution testing and sensitivity to numerical parameters providing confidence in the results of our calculations, and paving the way for detailed nonlinear studies in a companion article.

Spherical tokamaks (STs) have been shown to possess properties desirable for a fusion power plant such as achieving high plasma ? and having increased vertical stability. To understand the confinement properties that might be expected in the conceptual design for a high $\beta$ ST fusion reactor, a 1GW ST plasma equilibrium was analysed using local linear gyrokinetics to determine the type of micro-instabilities that arise. Kinetic ballooning modes (KBMs) and micro-tearing modes (MTMs) are found to be the dominant instabilities. The parametric dependence of these linear modes was determined and from the insights gained, the equilibrium was tuned to find a regime marginally stable to all micro-instabilities at $\theta_0$ = 0:0. This work identifies the most important micro-instabilities expected to generate turbulent transport in high $\beta$ STs. The impact of such modes must be faithfully captured in first principles based reduced models of anomalous transport that are needed for predictive simulations.

We present the results of GENE gyrokinetic calculations based on a series of JET–ITER-like-wall (ILW) type I ELMy H-mode discharges operating with similar experimental inputs but at different levels of power and gas fuelling. We show that turbulence due to electron-temperature-gradient (ETGs) modes produces a significant amount of heat flux in four JET–ILW discharges, and, when combined with neoclassical simulations, is able to reproduce the experimental heat flux for the two low gas pulses. The simulations plausibly reproduce the high-gas heat fluxes as well, although power balance analysis is complicated by short ELM cycles. By independently varying the normalised temperature gradients (ωTe) and normalised density gradients (ωne) around their experimental values, we demonstrate that it is the ratio of these two quantities ηe=ωTe/ωne that determines the location of the peak in the ETG growth rate and heat flux spectra. The heat flux increases rapidly as η e increases above the experimental point, suggesting that ETGs limit the temperature gradient in these pulses. When quantities are normalised using the minor radius, only increases in ωTe produce appreciable increases in the ETG growth rates, as well as the largest increases in turbulent heat flux which follow scalings similar to that of critical balance theory. However, when the heat flux is normalised to the electron gyro-Bohm heat flux using the temperature gradient scale length LTe , it follows a linear trend in correspondence with previous work by different authors.