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Based on the analysis of a large number of high-fidelity nonlinear gyrokinetic simulations, we propose a novel strategy to improve confinement in 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 MAST and a preliminary equilibrium from SMART., Comment: 7 pages, 5 figures

Strong $E\times B$ 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., Comment: 43 pages, 23 figures

Because negative triangularity plasma scenarios remain in L-mode, they do not require external heating systems that exceed the H-mode power threshold. Operating with less heating has the potential to improve performance as heating generally degrades confinement in tokamaks. Using simple zero dimensional power balance and standard empirical scaling laws for confinement, we analyze the impact of external heating on several different reactor-relevant devices (i.e. SPARC, MANTA, ITER and DEMO). We compare the nominal externally heated scenarios against equivalent negative tringularity cases without external heating. For most of these devices, the Ohmically heated negative triangularity versions achieve better performance, particularly for devices with high magnetic field and/or high fusion gain.

Using local nonlinear gyrokinetic simulations, we demonstrate that turbulent eddies can extend along magnetic field lines for hundreds of poloidal turns in tokamaks with weak or zero magnetic shear $\hat{s}$. We observe that this parallel eddy length scales inversely with magnetic shear and at $\hat{s}=0$ is limited by the thermal speed of electrons $v_{th,e}$. We examine the consequences of these "ultra long" eddies on turbulent transport, in particular, how field line topology mediates strong parallel self-interaction. Our investigation reveals that, through this process, field line topology can strongly affect transport. It can cause transitions between different turbulent instabilities and in some cases triple the logarithmic gradient needed to drive a given amount of heat flux. We also identify a novel "eddy squeezing" effect, which reduces the perpendicular size of eddies and their ability to transport energy, thus representing a novel approach to improve confinement. Finally, we investigate the triggering mechanism of Internal Transport Barriers (ITBs) using low magnetic shear simulations, shedding light on why ITBs are often easier to trigger where the safety factor has a low-order rational value.

In this work, we study the impact of aspect ratio A = R0 /r (the ratio of major radius R0 to minor radius r) on the confinement benefits of Negative Triangularity (NT) plasma shaping. We use high-fidelity flux tube gyrokinetic GENE simulations and consider several different scenarios: four of them inspired by TCV experimental data, a scenario inspired by DIII-D experimental data and a scenario expected in the new SMART spherical tokamak. The present study reveals a surprising and non-trivial dependence. NT improves confinement at any value of A for ITG turbulence, while for TEM turbulence confinement is improved only in the case of large and conventional aspect ratios. Additionally, through a detailed study of a large aspect ratio case with pure ITG drive, we develop an intuitive physical picture that explains the beneficial effect of NT at large and conventional aspect ratios. This picture does not hold in TEM-dominated regimes, where a complex synergistic effect of many factors is found. Finally, we performed the first linear gyrokinetic simulations of SMART, finding that both NT and PT scenarios are dominated by micro-tearing-mode (MTM) turbulence and that NT is more susceptible to MTMs at tight aspect ratio. However, we found that a regime where ITG dominates in SMART can be found, and in this regime NT is more linearly stable.

In tokamak experiments, sufficiently strong $E\times B$ flow shear reduces turbulent transport, thereby improving the prospects for fusion power plants. It is therefore of great importance to efficiently explore parameter space to find where strong plasma flow can be achieved. To this end, we propose a new, physically motivated quasi-linear model for estimating momentum transport from turbulence in the presence of toroidal flow shear and plasma shaping. The method gives good estimates of momentum transport for up-down asymmetric geometries as well as low magnetic shear and tight aspect ratio. The results are benchmarked with high-fidelity nonlinear GENE simulations, demonstrating that it provides a fast and accurate estimate of momentum transport., Comment: 34 pages, 17 figures

In this work, we modify the standard flux tube simulation domain to include arbitrary ion gyroradius-scale variation in the radial profile of the safety factor. To determine how to appropriately include such a modification, we add a strong ion gyroradius-scale source (inspired by electron cyclotron current drive) to the Fokker-Planck equation, then perform a multi-scale analysis that distinguishes the fast electrons driven by the source from the slow bulk thermal electrons. This allows us to systematically derive the needed changes to the gyrokinetic model. We find new terms that adjust the ion and electron parallel streaming to be along the modified field lines. These terms have been successfully implemented in a gyrokinetic code (while retaining the typical Fourier representation), which enables flux tube studies of non-monotonic safety factor profiles and the associated profile shearing. As an illustrative example, we investigate tokamaks with positive versus negative triangularity plasma shaping and find that the importance of profile shearing is not significantly affected by the change in shape., Comment: 33 pages, 4 figures

Non-adiabatic electron response leads to significant changes in Ion Temperature Gradient (ITG) eigenmodes, leading in particular to fine-structures that are significantly extended along the magnetic field lines at corresponding Mode Rational Surfaces (MRSs). These eigenmodes can nonlinearly interact with themselves to drive zonal flows via the so-called self-interaction mechanism. In this paper, the effect of collisions on these processes are studied. In presence of non-adiabatic electrons, the linear growth rate of ITG eigenmodes decreases with increasing collisionality. Detailed velocity space analysis of the distribution function shows that this results from collisions leading to a more adiabatic-like response of electrons away from MRSs. In linear simulations, collisions are furthermore found to broaden the radial width of the fine-structures, which translates to narrower tails of the eigenmode in extended ballooning space. The characteristic parallel scale length associated to these tails is shown to scale with the mean free path of electrons. In nonlinear turbulence simulations accounting for physically relevant values of collisionality, the fine structures located at MRSs, together with the associated drive of zonal flows via self-interaction, are shown to persist and play a significant role.

Local gyrokinetic simulations use a field-aligned domain that twists due to the magnetic shear of the background magnetic equilibrium. However, if the magnetic shear is strong and/or the domain is long, the twist can become so extreme that it fails to properly resolve the turbulence. In this work, we derive and implement the "non-twisting flux tube," a local simulation domain that remains rectangular at all parallel locations. Convergence and runtime tests indicate that it can calculate the heat flux more efficiently than the conventional flux tube. For one test case, it was 30 times less computationally expensive and we found no case for which it was more expensive. It is most advantageous when the magnetic shear is high and the domain includes at least two regions of turbulent drive (e.g. stellarator simulations, pedestal simulations, tokamak simulations with several poloidal turns). Additionally, it more accurately models the inboard midplane when the magnetic shear is large. Lastly, we show how the non-twisting flux tube can be generalized to allow further optimization and control of the simulation domain., Comment: 49 pages, 13 figures

Self-interaction is the process by which a microturbulence eigenmode that is extended along the direction parallel to the magnetic field interacts with itself non-linearly. This effect is particularly significant in gyrokinetic simulations accounting for kinetic passing electron dynamics. Self-interaction is known to generate stationary $E\times B$ zonal flow shear layers at radial locations near low order mode rational surfaces [Weikl et. al., Phys. Plasmas 25, 072305 (2018)]. We find however that it also plays a significant role in generating fluctuating zonal flows, which is critical to regulating transport throughout the radial extent. Unlike the usual picture of zonal flow drive where microturbulence eigenmodes coherently amplify the flow, the self-interaction drive of zonal flows from these eigenmodes are uncorrelated with each other. It is shown that the associated shearing rate of the fluctuating zonal flows therefore reduces as more toroidal modes are resolved in the simulation. In flux-tube simulations accounting for the full toroidal domain, such an increase in the density of toroidal modes corresponds to an increase in the system size, leading to a finite system size effect that is distinct from the well-known profile shearing effect.

Local linear gyrokinetic simulations show that electron temperature gradient (ETG) instabilities are the fastest growing modes for $k_y \rho_i \gtrsim 0.1$ in the steep gradient region for a JET pedestal discharge (92174) where the electron temperature gradient is steeper than the ion temperature gradient. Here, $k_y$ is the wavenumber in the direction perpendicular to both the magnetic field and the radial direction, and $\rho_i$ is the ion gyroradius. At $k_y \rho_i \gtrsim 1$, the fastest growing mode is often a novel type of toroidal ETG instability. This toroidal ETG mode is driven at scales as large as $k_y \rho_i \sim (\rho_i/\rho_e) L_{Te} / R_0 \sim 1$ and at a sufficiently large radial wavenumber that electron finite Larmor radius effects become important; that is, $K_x \rho_e \sim 1$, where $K_x$ is the effective radial wavenumber. Here, $\rho_e$ is the electron gyroradius, $R_0$ is the major radius of the last closed flux surface, and $1/L_{Te}$ is an inverse length proportional to the logarithmic gradient of the equilibrium electron temperature. The fastest growing toroidal ETG modes are often driven far away from the outboard midplane. In this equilibrium, ion temperature gradient instability is subdominant at all scales and kinetic ballooning modes are shown to be suppressed by $\mathbf{ E} \times \mathbf{ B} $ shear. ETG modes are very resilient to $\mathbf{ E} \times \mathbf{ B}$ shear. Heuristic quasilinear arguments suggest that the novel toroidal ETG instability is important for transport., Comment: 67 pages, 29 figures, accepted in Nuclear Fusion 2020

In this work, we highlight an issue that may reduce the accuracy of many local nonlinear gyrokinetic simulations - turbulent self-interaction through the parallel boundary condition. Given a sufficiently long parallel correlation length, individual turbulent eddies can span the full domain and "bite their own tails," thereby altering their statistical properties. Such self-interaction is only modeled accurately when the simulation domain corresponds to a full flux surface, otherwise it is artificially strong. For Cyclone Base Case parameters and typical domain sizes, we find that this mechanism modifies the heat flux by roughly 40% and it can be even more important. The effect is largest when using kinetic electrons, low magnetic shear, and strong turbulence drive (i.e. steep background gradients). It is found that parallel self-interaction can be eliminated by increasing the parallel length and/or the binormal width of the simulation domain until convergence is achieved., Comment: 19 pages, 9 figures

Specific energy (i.e. energy per unit mass) is one of the most fundamental and consequential properties of a fuel source. In this work, a systematic study of measured fusion cross-sections is performed to determine which reactions are potentially feasible and identify the fuel cycle that maximizes specific energy. This reveals that, by using normal hydrogen to breed deuterium via neutron capture, the conventional catalyzed D-D fusion fuel cycle can attain a specific energy greater than anything else. Simply surrounding a catalyzed D-D reactor with water enables deuterium fuel, the dominant stockpile of energy on Earth, to produce as much as 65% more energy. Lastly, the impact on space propulsion is considered, revealing that an effective exhaust velocity exceeding that of deuterium-helium-3 is theoretically possible., Comment: 22 pages, 6 figures

The cold ion limit of the local gyrokinetic model is rigorously taken to produce a nonlinear system of fluid equations that includes background flow shear. No fluid closure is required. By considering a simple slab geometry with magnetic drifts, but no magnetic shear, these fluid equations reduce to the Charney-Hasegawa-Mima model in the presence of flow shear. Analytic solutions to this model are found to study the impact of ExB flow shear on the stability of a single Parallel Velocity Gradient (PVG) driven mode. Additionally, the model is used to investigate the effect of background ExB flow shear on the basic three-mode nonlinear coupling, which reveals differences between zonal and non-zonal modes. These analytic results agree with gyrokinetic simulations and can serve to benchmark the numerical implementation of flow shear and nonlinear coupling., Comment: 34 pages, 12 figures

Breaking the up-down symmetry of the tokamak poloidal cross-section can significantly increase the spontaneous rotation due to turbulent momentum transport. In this work, we optimize the shape of flux surfaces with both tilted elongation and tilted triangularity in order to maximize this drive of intrinsic rotation. Nonlinear gyrokinetic simulations demonstrate that adding optimally-tilted triangularity can double the momentum transport of a tilted elliptical shape. This work indicates that tilting the elongation and triangularity in an ITER-like device can reduce the energy transport and drive intrinsic rotation with an Alfv\'{e}n Mach number on the order of $1\%$. This rotation is four times larger than the rotation expected in ITER and is sufficient to stabilize MHD instabilities. It is shown that this optimal shape can be created using the shaping coils of several experiments., Comment: 16 pages, 5 figures

Introducing up-down asymmetry into the tokamak magnetic equilibria appears to be a feasible method to drive fast intrinsic toroidal rotation in future large devices. In this paper we investigate how the intrinsic momentum transport generated by up-down asymmetric shaping scales with the mode number of the shaping effects. Making use the gyrokinetic tilting symmetry (Ball et al (2016) Plasma Phys. Control. Fusion 58 045023), we study the effect of envelopes created by the beating of different high-order shaping effects. This reveals that the presence of an envelope can change the scaling of the momentum flux from exponentially small in the limit of large shaping mode number to just polynomially small. This enhancement of the momentum transport requires the envelope to be both up-down asymmetric and have a spatial scale on the order of the minor radius., Comment: 20 pages, 5 figures

Bulk toroidal rotation has proven capable of stabilising both dangerous MHD modes and turbulence. In this thesis, we explore a method to drive rotation in large tokamaks: up-down asymmetry in the magnetic equilibrium. We seek to maximise this rotation by finding optimal up-down asymmetric flux surface shapes. First, we use the ideal MHD model to show that low order external shaping (e.g. elongation) is best for creating up-down asymmetric flux surfaces throughout the device. Then, we calculate realistic up-down asymmetric equilibria for input into nonlinear gyrokinetic turbulence analysis. Analytic gyrokinetics shows that, in the limit of fast shaping effects, a poloidal tilt of the flux surface shaping has little effect on turbulent transport. Since up-down symmetric surfaces do not transport momentum, this invariance to tilt implies that devices with mirror symmetry about any line in the poloidal plane will drive minimal rotation. Accordingly, further analytic investigation suggests that non-mirror symmetric flux surfaces with envelopes created by the beating of fast shaping effects may create significantly stronger momentum transport. Guided by these analytic results, we carry out local nonlinear gyrokinetic simulations of non-mirror symmetric flux surfaces created with the lowest possible shaping effects. First, we consider tilted elliptical flux surfaces with a Shafranov shift and find little increase in the momentum transport when the effect of the pressure profile on the equilibrium is included. We then simulate flux surfaces with independently-tilted elongation and triangularity. These two-mode configurations show a $60\%$ increase over configurations with just elongation or triangularity. A rough analytic estimate indicates that the optimal two-mode configuration can drive rotation with an on-axis Alfven Mach number of $1.5 \%$ in an ITER-like machine., Comment: 150 pages, 35 figures, PhD thesis, University of Oxford, Trinity term 2016

Tokamaks with up-down asymmetric poloidal cross-sections spontaneously rotate due to turbulent transport of momentum. In this work, we investigate the effect of the Shafranov shift on this intrinsic rotation, primarily by analyzing tokamaks with tilted elliptical flux surfaces. By expanding the Grad-Shafranov equation in the large aspect ratio limit we calculate the magnitude and direction of the Shafranov shift in tilted elliptical tokamaks. The results show that, while the Shafranov shift becomes up-down asymmetric and depends strongly on the tilt angle of the flux surfaces, it is insensitive to the shape of the current and pressure profiles (when the geometry, total plasma current, and average pressure gradient are kept fixed). Next, local nonlinear gyrokinetic simulations of these MHD equilibria are performed with GS2, which reveal that the Shafranov shift can significantly enhance the momentum transport. However, to be consistent, the effect of $\beta'$ (i.e. the radial gradient of $\beta$) on the magnetic equilibrium was also included, which was found to significantly reduce momentum transport. Including these two competing effects broadens the rotation profile, but leaves the on-axis value of the rotation roughly unchanged. Consequently, the shape of the $\beta$ profile has a significant effect on the rotation profile of an up-down asymmetric tokamak., Comment: 40 pages, 18 figures

Breaking the up-down symmetry of tokamaks removes a constraint limiting intrinsic momentum transport, and hence toroidal rotation, to be small. Using gyrokinetic theory, we study the effect of different up-down asymmetric flux surface shapes on the turbulent transport of momentum. This is done by perturbatively expanding the gyrokinetic equation in large flux surface shaping mode number. It is found that the momentum flux generated by shaping that lacks mirror symmetry (which is necessarily up-down asymmetric) has a power law scaling with the shaping mode number. However, the momentum flux generated by mirror symmetric flux surface shaping (even if it is up-down asymmetric) decays exponentially with large shaping mode number. These scalings are consistent with nonlinear local gyrokinetic simulations and indicate that low mode number shaping effects (e.g. elongation, triangularity) are optimal for creating rotation. Additionally it suggests that breaking the mirror symmetry of flux surfaces may generate significantly more toroidal rotation., Comment: 39 pages, 4 figures

A poloidal tilting symmetry of the local nonlinear $\delta f$ gyrokinetic model is demonstrated analytically and verified numerically. This symmetry shows that poloidally rotating all the flux surface shaping effects with large poloidal mode number by a single tilt angle has an exponentially small effect on the transport properties of a tokamak. This is shown using a generalization of the Miller local equilibrium model to specify an arbitrary flux surface geometry. With this geometry specification we find that, when performing an expansion in large flux surface shaping mode number, the governing equations of gyrokinetics are symmetric in the poloidal tilt of the high order shaping effects. This allows us to take the fluxes from a single configuration and calculate the fluxes in any configuration that can be produced by tilting the large mode number shaping effects. This creates a distinction between tokamaks with mirror symmetric flux surfaces and tokamaks without mirror symmetry, which is expected to have important consequences for generating toroidal rotation using up-down asymmetry., Comment: 31 pages, 8 figures

Using analytic calculations, the effects of the edge flux surface shape and the toroidal current profile on the penetration of flux surface shaping are investigated in a tokamak. It is shown that the penetration of shaping is determined by the poloidal variation of the poloidal magnetic field on the surface. This fact is used to investigate how different flux surface shapes penetrate from the edge. Then, a technique to separate the effects of magnetic pressure and tension in the Grad-Shafranov equation is presented and used to calculate radial profiles of strong elongation for nearly constant current profiles. Lastly, it is shown that more hollow toroidal current profiles are significantly better at conveying shaping from the edge to the core., Comment: 11 pages, 13 figures

Recent work demonstrated that breaking the up-down symmetry of tokamak flux surfaces removes a constraint that limits intrinsic momentum transport, and hence toroidal rotation, to be small. We show, through MHD analysis, that ellipticity is most effective at introducing up-down asymmetry throughout the plasma. We detail an extension to GS2, a local $\delta f$ gyrokinetic code that self-consistently calculates momentum transport, to permit up-down asymmetric configurations. Tokamaks with tilted elliptical poloidal cross-sections were simulated to determine nonlinear momentum transport. The results, which are consistent with experiment in magnitude, suggest that a toroidal velocity gradient, $(\partial u_{\zeta i} / \partial \rho) / v_{th i}$, of 5% of the temperature gradient, $(\partial T_{i} / \partial \rho) / T_{i}$, is sustainable. Here $v_{th i}$ is the ion thermal speed, $u_{\zeta i}$ is the ion toroidal mean flow, $\rho$ is the minor radial coordinate normalized to the tokamak minor radius, and $T_{i}$ is the ion temperature. Since other intrinsic momentum transport mechanisms scale poorly to larger machines, these results indicate that up-down asymmetry is the most feasible method to generate the current experimentally-measured rotation levels in reactor-sized devices.

A local magnetic equilibrium solution is sought around the magnetic axis in order to identify the key parameters defining the magnetic-surface's up-down asymmetry in the core of tokamak plasmas. The asymmetry is found to be determined essentially by the ratio of the toroidal current density flowing on axis to the fraction of the external field's odd perturbation that manages to propagate from the plasma boundary into the core. The predictions are tested and illustrated first with an analytical Solovev equilibrium and then using experimentally relevant numerical equilibria. Hollow current-density distributions, and hence reverse magnetic shear, are seen to be crucial to bring into the core asymmetry values that are usually found only near the plasma edge., Comment: 6 pages, 2 figures, submitted for publication