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The phenomenon of beam focusing of microwaves in a plasma near a turning-point caustic is discussed in the context of the analytical solution to the Gaussian beam-tracing equations in the 2D linear-layer problem. The location of maximum beam focusing and the beam width at that location are studied in terms of the beam initial conditions. The analytic solution is used to study the effect of this focusing on Doppler backscattering (DBS). We find that the filter function that characterises the scattering intensity contributions along the beam path through the plasma is inversely proportional to the beam width, predicting enhanced scattering contributions from the beam focusing region. We show that the DBS signal enhancement for small incident angles between the beam path and the density gradient is due to beam focusing and not due to forward scattering. The analytic beam model is used to predict the measurement of the $k_y$ density-fluctuation wavenumber power spectrum via DBS, showing that the spectral exponent of the turbulent, intermediate-to-high $k_y$ density-fluctuation spectrum might be quantitatively measurable via DBS, but not the spectral peak corresponding to the driving scale of the turbulent cascade.

We report the first experimental detection of a zero-frequency fluctuation that is pumped by an Alfv\'en mode in a magnetically confined plasma. Core-localized bidirectional Alfv\'en modes of frequency inside the toroidicity-induced gap (and its harmonics) exhibit three-wave coupling interactions with a zero-frequency fluctuation. The observation of the zero-frequency fluctuation is consistent with theoretical and numerical predictions of zonal modes pumped by Alfv\'en modes, and is correlated with an increase in the deep core ion temperature, temperature gradient, and confinement factor $H_{89,P}$. Despite the energetic particle transport induced by the Alfv\'en eigenmodes, the generation of a zero-frequency fluctuation that can suppress the turbulence leads to an overall improvement of confinement.

A solid target in contact with a plasma charges (negatively) to reflect the more mobile species (electrons) and thus keep the bulk plasma quasineutral. To shield the bulk plasma from the charged target, there is an oppositely (positively) charged sheath with a sharp electrostatic potential variation on the Debye length scale $\lambda_{\rm D}$. In magnetised plasmas where the magnetic field is inclined at an oblique angle $\alpha$ with the target, some of the sheath potential variation occurs also on the ion sound gyroradius length scale $\rho_{\rm S} \cos \alpha$, caused by finite ion gyro-orbit distortion and losses. We consider a collisionless and steady-state magnetised plasma sheath whose thickness $l_{\rm ms} \sim \max (\lambda_{\rm D}, \rho_{\rm S} \cos \alpha)$ is smaller than the characteristic length scale $L$ of spatial fluctuations in the bulk plasma, such that the limit $l_{\rm ms} / L \rightarrow 0$ is appropriate. Spatial structures are assumed to be magnetic field-aligned. In the case of small magnetic field angle $ \alpha \sim \delta \equiv \rho_{\rm S} / L \ll 1$, electric fields tangential to the target transport ions towards the target via ExB drifts at a rate comparable to the one from parallel streaming. A generalised form of the kinetic Bohm-Chodura criterion at the sheath entrance is derived by requiring that the sheath electric field have a monotonic spatial decay far from the target. The criterion depends on tangential gradients of potential and ion distribution function, with additional nontrivial conditions., Comment: 79 pages, 3 figures

We present a new neoclassical transport model for large aspect ratio tokamaks where the gradient scale lengths are of the size of the poloidal gyroradius. Previous work on neoclassical transport across transport barriers assumed large density and potential gradients but a small temperature gradient, or neglected the gradient of the mean parallel flow. Using large aspect ratio and low collisionality expansions, we relax these restrictive assumptions. We define a new set of variables based on conserved quantities, which simplifies the drift kinetic equation whilst keeping strong gradients, and derive equations describing the transport of particles, parallel momentum and energy by ions in the banana regime. The poloidally varying parts of density and electric potential are included. Studying contributions from both passing and trapped particles, we show that the resulting transport is dominated by trapped particles. We find that a non-zero neoclassical particle flux requires parallel momentum input which could be provided through interaction with turbulence or impurities. We derive upper and lower bounds for the energy flux across a transport barrier in both temperature and density and present example profiles and fluxes., Comment: 52 pages, 13 figures, submitted to Journal of Plasma Physics

The Doppler backscattering (DBS) diagnostic, also referred to as Doppler reflectometry, measures turbulent density fluctuations of intermediate length scales. However, when the beam's wavevector is not properly aligned perpendicular to the magnetic field, the backscattered power is attenuated. In previous work, we used beam tracing and reciprocity to derive this mismatch attenuation quantitatively. In this paper, we applied our model, in the small but finite mismatch limit, to a several new cases. We compared our predictions with multiple O-mode channels for the first time. We then identified a $\sim 3^{\circ}$ error in the MAST Q-band's quasioptics, showing that our model is useful for commissioning DBS diagnostics. For both O- and X-mode, we compared experimental data with our model's predictions at multiple times during the shots, unlike our previous work, where only a single time was analysed. Finally, we analysed other contributions to the backscattered signal, evaluating how much they affect our measurements of mismatch attenuation, giving comparisons with data from both MAST and MAST-U. This paper's detailed study systematically validates and demonstrates the usefulness of our model for quantitatively interpreting DBS data from spherical tokamaks.

We use the beam model of Doppler backscattering (DBS), which was previously derived from beam tracing and the reciprocity theorem, to shed light on mismatch attenuation. This attenuation of the backscattered signal occurs when the wavevector of the probe beam's electric field is not in the plane perpendicular to the magnetic field. Correcting for this effect is important for determining the amplitude of the actual density fluctuations. Previous preliminary comparisons between the model and Mega-Ampere Spherical Tokamak (MAST) plasmas were promising. In this work, we quantitatively account for this effect on DIII-D, a conventional tokamak. We compare the predicted and measured mismatch attenuation in various DIII-D, MAST, and MAST-U plasmas, showing that the beam model is applicable in a wide variety of situations. Finally, we performed a preliminary parameter sweep and found that the mismatch tolerance can be improved by optimising the probe beam's width and curvature at launch. This is potentially a design consideration for new DBS systems.

New orbit averaged equations for low collisionality neoclassical fluxes in large aspect ratio stellarators with mirror ratios close to unity are derived. The equations retain finite orbit width effects by employing the second adiabatic invariant $J$ as a velocity space coordinate and they have been implemented in the orbit-averaged neoclassical code KNOSOS. The equations are used to study the $1/\nu$ regime and the lower collisionality regimes. For generic large aspect ratio stellarators with mirror ratios close to unity, as the collision frequency decreases, the $1/\nu$ regime transitions directly into the $\nu$ regime, without passing through a $\sqrt{\nu}$ regime. An explicit formula for the neoclassical fluxes in the $\nu$ regime is obtained. The formula includes the effect of particles that transition between different types of wells. While these transitions produce stochastic scattering independent of the value of the collision frequency in velocity space, the diffusion in real space remains proportional to the collision frequency. The $\sqrt{\nu}$ regime is only recovered in large aspect ratio stellarators close to omnigeneity: large aspect ratio stellarators with large mirror ratios and optimized large aspect ratio stellarators with mirror ratios close to unity. Neoclassical transport in large aspect ratio stellarators with large mirror ratios can be calculated with the orbit-averaged equations derived by \cite{calvo17}. In these stellarators, the $\sqrt{\nu}$ regime exists in the collisionality interval $\epsilon |\ln \epsilon| \ll \nu_{ii} R a/\rho_i v_{ti} \ll 1/\epsilon$. In optimized large aspect ratio stellarators with mirror ratios close to unity, the $\sqrt{\nu}$ regime occurs in an interval of collisionality that depends on the deviation from omnigeneity $\delta$: $\delta^2 |\ln \delta | \ll \nu_{ii} R a/\rho_i v_{ti} \ll 1$., Comment: Accepted in the Journal of Plasma Physics. 79 pages, 15 figures

We use beam tracing -- implemented with a newly-written code, Scotty -- and the reciprocity theorem to derive a model for the linear backscattered power of the Doppler Backscattering (DBS) diagnostic. Our model works for both the O-mode and X-mode in tokamak geometry (and certain regimes of stellarators). We present the analytical derivation of our model and its implications on the DBS signal localisation and the wavenumber resolution. To determine these two quantities, we find that it is the curvature of the field lines and the magnetic shear that are important, rather than the curvature of the cut-off surface. We also provide an explicit formula for the hitherto poorly-understood quantitative effect of the mismatch angle. Consequently, one can use this model to correct for the attenuation due to mismatch, avoiding the need for empirical optimisation. This is especially important in spherical tokamaks, since the magnetic pitch angle is large and varies both spatially and temporally., Comment: This is the version that passed peer review. No major changes, but many improvements to writing style

The Debye sheath is known to vanish completely in magnetised plasmas for a sufficiently small electron gyroradius and small angle between the magnetic field and the wall. This angle depends on the current onto the wall. When the Debye sheath vanishes, there is still a potential drop between the wall and the plasma across the magnetic presheath. The magnetic field angle corresponding to the predicted sheath collapse is shown to be much smaller than previous estimates, scaling with the electron-ion mass ratio and not with the square root of the mass ratio. This is shown to be a consequence of the kinetic electron and finite ion orbit width effects, which are not captured by fluid models. The wall potential with respect to the bulk plasma at which the Debye sheath vanishes is calculated. Above this wall potential, it is possible that the Debye sheath will invert., Comment: 59 pages, 17 figures

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

The magnetic presheath is a boundary layer occurring when magnetized plasma is in contact with a wall and the angle $\alpha$ between the wall and the magnetic field $\vec{B}$ is oblique. Here, we consider the fusion-relevant case of a shallow-angle, $\alpha \ll 1$, electron-repelling sheath, with the electron density given by a Boltzmann distribution, valid for $\alpha / \sqrt{\tau+1} \gg \sqrt{m_{\text{e}}/m_{\text{i}}}$, where $m_{\text{e}}$ is the electron mass, $m_{\text{i}}$ is the ion mass, $\tau = T_{\text{i}}/ZT_{\text{e}}$, $T_{\text{e}}$ is the electron temperature, $T_{\text{i}}$ is the ion temperature, and $Z$ is the ionic charge state. The thickness of the magnetic presheath is of the order of a few ion sound Larmor radii $\rho_{\text{s}} = \sqrt{m_{\text{i}} \left(ZT_{\text{e}} + T_{\text{i}} \right) } / ZeB$, where $e$ is the proton charge and $B = |\vec{B}|$ is the magnitude of the magnetic field. We study the dependence on $\tau $ of the electrostatic potential and ion distribution function in the magnetic presheath by using a set of prescribed ion distribution functions at the magnetic presheath entrance, parameterized by $\tau$. The kinetic model is shown to be asymptotically equivalent to Chodura's fluid model at small ion temperature, $\tau \ll 1$, for $|\ln \alpha| > 3|\ln \tau | \gg 1$. In this limit, despite the fact that fluid equations give a reasonable approximation to the potential, ion gyro-orbits acquire a spatial extent that occupies a large portion of the magnetic presheath. At large ion temperature, $\tau \gg 1$, relevant because $T_{\text{i}}$ is measured to be a few times larger than $T_{\text{e}}$ near divertor targets of fusion devices, ions reach the Debye sheath entrance (and subsequently the wall) at a shallow angle whose size is given by $\sqrt{\alpha}$ or $1/\sqrt{\tau}$, depending on which is largest., Comment: 44 pages, 8 figures

Main ions influence impurity dynamics through a variety of mechanisms; in particular, via impurity-ion collisions. To lowest order in an expansion in the main ion mass over the impurity mass, the impurity-ion collision operator only depends on the component of the main ion distribution that is odd in the parallel velocity. These lowest order terms give the parallel friction of the impurities with the main ions, which is typically assumed to be the main cause of collisional impurity transport. Next-order terms in the mass ratio expansion of the impurity-ion collision operator, proportional to the component of the main ion distribution that is even in the parallel velocity, are usually neglected. However, in stellarators, the even component of the main ion distribution can be very large. In this article, such next-order terms in the mass ratio expansion of the impurity-ion collision operator are retained, and analytical expressions for the neoclassical radial flux of trace impurities are calculated in the Pfirsch-Schl\"uter, plateau and $1/\nu$ regimes. The new terms provide a drive for impurity transport that is physically very different from parallel friction: they are associated to anisotropy in the pressure of the main ions, which translates into impurity pressure anisotropy. It is argued that main ion pressure anisotropy must be taken into account for a correct description of impurity transport in certain realistic stellarator plasmas. Examples are given by numerically evaluating the analytical expressions for the impurity flux., Comment: 40 pages, 4 figures. Published version

Differential rotation is induced in tokamak plasmas when an underlying symmetry of the governing gyrokinetic-Maxwell system of equations is broken. One such symmetry-breaking mechanism is considered here: the turbulent acceleration of particles along the mean magnetic field. This effect, often referred to as the `parallel nonlinearity', has been implemented in the $\delta f$ gyrokinetic code $\texttt{stella}$ and used to study the dependence of turbulent momentum transport on the plasma size and on the strength of the turbulence drive. For JET-like parameters with a wide range of driving temperature gradients, the momentum transport induced by the inclusion of turbulent acceleration is similar to or smaller than the ratio of the ion Larmor radius to the plasma minor radius. This low level of momentum transport is explained by demonstrating an additional symmetry that prohibits momentum transport when the turbulence is driven far above marginal stability., Comment: 15 pages, 3 figures, updated to reflect published version + additional references that were initially overlooked

Using a kinetic model for the ions and adiabatic electrons, we solve a steady state, electron-repelling magnetic presheath in which a uniform magnetic field makes a small angle $\alpha \ll 1$ (in radians) with the wall. The presheath characteristic thickness is the typical ion gyroradius $\rho_{\text{i}}$. The Debye length $\lambda_{\text{D}}$ and the collisional mean free path of an ion $\lambda_{\text{mfp}}$ satisfy the ordering $\lambda_{\text{D}} \ll \rho_{\text{i}} \ll \alpha \lambda_{\text{mfp}}$, so a quasineutral and collisionless model is used. We assume that the electrostatic potential is a function only of distance from the wall, and it varies over the scale $\rho_{\text{i}}$. Using the expansion in $\alpha \ll 1$, we derive an analytical expression for the ion density that only depends on the ion distribution function at the entrance of the magnetic presheath and the electrostatic potential profile. Importantly, we have added the crucial contribution of the orbits in the region near the wall. By imposing the quasineutrality equation, we derive a condition that the ion distribution function must satisfy at the magnetic presheath entrance --- the kinetic equivalent of the Chodura condition. Using an ion distribution function at the entrance of the magnetic presheath that satisfies the kinetic Chodura condition, we find numerical solutions for the self-consistent electrostatic potential, ion density and flow across the magnetic presheath for several values of $\alpha$. Our numerical results also include the distribution of ion velocities at the Debye sheath entrance. We find that at small values of $\alpha$ there are substantially fewer ions travelling with a large normal component of the velocity into the wall., Comment: 69 pages, 20 figures

The component of the neoclassical electrostatic potential that is non-constant on the magnetic surface, that we denote by $\tilde\varphi$, can affect radial transport of highly charged impurities, and this has motivated its inclusion in some modern neoclassical codes. The number of neoclassical simulations in which $\tilde\varphi$ is calculated is still scarce, partly because they are usually demanding in terms of computational resources, especially at low collisionality. In this paper the size, the scaling with collisionality and with aspect ratio, and the structure of $\tilde\varphi$ on the magnetic surface are analytically derived in the $1/\nu$, $\sqrt{\nu}$ and superbanana-plateau regimes of stellarators close to omnigeneity; i. e. stellarators that have been optimized for neoclassical transport. It is found that the largest $\tilde\varphi$ that the neoclassical equations admit scales linearly with the inverse aspect ratio and with the size of the deviation from omnigeneity. Using a model for a perturbed omnigeneous configuration, the analytical results are verified and illustrated with calculations by the code KNOSOS. The techniques, results and numerical tools employed in this paper can be applied to neoclassical transport problems in tokamaks with broken axisymmetry., Comment: 30 pages, 12 figures, 1 table. Published version

The control of impurity accumulation is one of the main challenges for future stellarator fusion reactors. The standard argument to explain this accumulation relies on the, in principle, large inward pinch in the neoclassical impurity flux caused by the typically negative radial electric field in stellarators. This simplified interpretation was proven to be flawed by Helander et al. [Phys. Rev. Lett. 118, 155002 (2017)], who showed that in a relevant regime (low-collisionality main ions and collisional impurities) the radial electric field does not drive impurity transport. In that reference, the effect of the component of the electric field that is tangent to the magnetic surface was not included. In this Letter, an analytical calculation of the neoclassical radial impurity flux incorporating such effect is given, showing that it can be very strong for highly charged impurities and that, once it is taken into account, the dependence of the impurity flux on the radial electric field reappears. Realistic examples are provided in which the inclusion of the tangential electric field leads to impurity expulsion., Comment: 10 pages, 1 figure. Published version

The collisionless axisymmetric zonal flow residual calculation for a tokamak plasma is generalized to include electromagnetic perturbations. We formulate and solve the complete initial value zonal flow problem by retaining the fully self-consistent axisymmetric spatial perturbations in the electric and magnetic fields. Simple expressions for the electrostatic, shear and compressional magnetic residual responses are derived that provide a fully electromagnetic test of the zonal flow residual in gyrokinetic codes. Unlike the electrostatic potential, the parallel vector potential and the parallel magnetic field perturbations need not relax to flux functions for all possible initial conditions.

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

Due to their capability to reduce turbulent transport in magnetized plasmas, understanding the dynamics of zonal flows is an important problem in the fusion programme. Since the pioneering work by Rosenbluth and Hinton in axisymmetric tokamaks, it is known that studying the linear and collisionless relaxation of zonal flow perturbations gives valuable information and physical insight. Recently, the problem has been investigated in stellarators and it has been found that in these devices the relaxation process exhibits a characteristic feature: a damped oscillation. The frequency of this oscillation might be a relevant parameter in the regulation of turbulent transport, and therefore its efficient and accurate calculation is important. Although an analytical expression can be derived for the frequency, its numerical evaluation is not simple and has not been exploited systematically so far. Here, a numerical method for its evaluation is considered, and the results are compared with those obtained by calculating the frequency from gyrokinetic simulations. This "semianalytical" approach for the determination of the zonal-flow frequency reveals accurate and faster than the one based on gyrokinetic simulations., Comment: 30 pages, 14 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

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. For sufficiently low collision frequencies (below the values that define the $1/\nu$ 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 preserve radial locality in the drift-kinetic equation for collisionalities below the $1/\nu$ regime. Then, it is shown that neoclassical transport is determined by two layers in phase space. One of the layers corresponds to the $\sqrt{\nu}$ regime and the other to the 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 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 $\sqrt{\nu}$ 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., Comment: 36 pages. Version to be published in Plasma Physics and Controlled Fusion

We develop a gyrokinetic treatment for ions in the magnetic presheath, close to the plasma-wall boundary. We focus on magnetic presheaths with a small magnetic field to wall angle, $\alpha \ll 1$ (in radians). Characteristic lengths perpendicular to the wall in such a magnetic presheath scale with the typical ion Larmor orbit size, $\rho_{\text{i}}$. The smallest scale length associated with variations parallel to the wall is taken to be across the magnetic field, and ordered $l = \rho_{\text{i}} / \delta$, where $ \delta \ll 1$ is assumed. The scale lengths along the magnetic field line are assumed so long that variations associated with this direction are neglected. These orderings are consistent with what we expect close to the divertor target of a tokamak. We allow for a strong component of the electric field $\vec{E}$ in the direction normal to the electron repelling wall, with strong variation in the same direction. The large change of the electric field over an ion Larmor radius distorts the orbit so that it is not circular. We solve for the lowest order orbits by identifying coordinates, which consist of constants of integration, an adiabatic invariant and a gyrophase, associated with periodic ion motion in the system with $\alpha = \delta = 0$. By using these new coordinates as variables in the limit $\alpha \sim \delta \ll 1$, we obtain a generalized ion gyrokinetic equation. We find another quantity that is conserved to first order and use this to simplify the gyrokinetic equation, solving it in the case of a collisionless magnetic presheath. Assuming a Boltzmann response for the electrons, a form of the quasineutrality equation that exploits the change of variables is derived. The gyrokinetic and quasineutrality equations give the ion distribution function and electrostatic potential in the magnetic presheath if the entrance boundary condition is specified., Comment: 55 pages including appendices and references

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

Intrinsic toroidal rotation in a tokamak can be driven by turbulent momentum transport due to neoclassical flow effects breaking a symmetry of turbulence. In this paper we categorize the contributions due to neoclassical effects to the turbulent momentum transport, and evaluate each contribution using gyrokinetic simulations. We find that the relative importance of each contribution changes with collisionality. For low collisionality, the dominant contributions come from neoclassical particle and parallel flows. For moderate collisionality, there are non- negligible contributions due to neoclassical poloidal electric field and poloidal gradients of density and temperature, which are not important for low collisionality., Comment: 28 pages, 5 figures, and 7 tables

The radial flux of toroidal angular momentum is needed to determine tokamak intrinsic rotation profiles. Its computation requires knowledge of the gyrokinetic distribution functions and turbulent electrostatic potential to second-order in $\epsilon = \rho/L$, where $\rho$ is the ion Larmor radius and $L$ is the variation length of the magnetic field. In this article, a complete set of equations to calculate the radial transport of toroidal angular momentum in any tokamak is presented. In particular, the $O(\epsilon^2)$ equations for the turbulent components of the distribution functions and electrostatic potential are given for the first time without assuming that the poloidal magnetic field over the magnetic field strength is small., Comment: 39 pages. Version to be published in Plasma Physics and Controlled Fusion

A stellarator is said to be omnigeneous if all particles have vanishing average radial drifts. In omnigeneous stellarators, particles are perfectly confined in the absence of turbulence and collisions, whereas in non-omnigeneous configurations, particle can drift large radial distances. One of the consequences of omnigeneity is that the unfavorable inverse scaling with collisionality of the stellarator neoclassical fluxes disappears. In the pioneering and influential article [Cary~J~R and Shasharina~S~G 1997 {\it Phys. Plasmas} {\bf 4} 3323], the conditions that the magnetic field of a stellarator must satisfy to be omnigeneous are derived. However, reference [Cary~J~R and Shasharina~S~G 1997 {\it Phys. Plasmas} {\bf 4} 3323] only considered omnigeneous stellarators in which all the minima of the magnetic field strength on a flux surface must have the same value. The same is assumed for the maxima. We show that omnigenenous magnetic fields can have local minima and maxima with different values. Thus, the parameter space in which omnigeneous stellarators are possible is larger than previously expected. The analysis presented in this article is only valid for orbits with vanishing radial width, and in principle it is not applicable to energetic particles. However, one would expect that improving neoclassical confinement would improve energetic particle confinement., Comment: 10 pages, 5 figures

A set of flux tube gyrokinetic equations that includes the effect of the spatial variation of the density, temperature and rotation gradients on the turbulence is derived. This new set of equations uses periodic boundary conditions. In the limit $l_\bot/L \ll 1$, where $l_\bot$ is the characteristic perpendicular length of turbulent structures and $L$ is the characteristic size of the device, this new set of flux tube gyrokinetic equations is shown to be equivalent to the traditional global $\delta f$ gyrokinetic equations to an order higher in $l_\bot/L$ than the usual flux tube formulations., Comment: 21 pages

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

Self-consistent equations for intrinsic rotation in tokamaks with small poloidal magnetic field $B_p$ compared to the total magnetic field $B$ are derived. The model gives the momentum redistribution due to turbulence, collisional transport and energy injection. Intrinsic rotation is determined by the balance between the momentum redistribution and the turbulent diffusion and convection. Two different turbulence regimes are considered: turbulence with characteristic perpendicular lengths of the order of the ion gyroradius, $\rho_i$, and turbulence with characteristic lengths of the order of the poloidal gyroradius, $(B/B_p) \rho_i$. Intrinsic rotation driven by gyroradius scale turbulence is mainly due to the effect of neoclassical corrections and of finite orbit widths on turbulent momentum transport, whereas for the intrinsic rotation driven by poloidal gyroradius scale turbulence, the slow variation of turbulence characteristics in the radial and poloidal directions and the turbulent particle acceleration can be become as important as the neoclassical and finite orbit width effects. The magnetic drift is shown to be indispensable for the intrinsic rotation driven by the slow variation of turbulence characteristics and the turbulent particle acceleration. The equations are written in a form conducive to implementation in a flux tube code, and the effect of the radial variation of the turbulence is included in a novel way that does not require a global gyrokinetic formalism., Comment: 88 pages, 4 figures

Quasisymmetric stellarators are a type of optimized stellarators for which flows are undamped to lowest order in an expansion in the normalized Larmor radius. However, perfect quasisymmetry is impossible. Since large flows may be desirable as a means to reduce turbulent transport, it is important to know when a stellarator can be considered to be sufficiently close to quasisymmetry. The answer to this question depends strongly on the size of the spatial gradients of the deviation from quasisymmetry and on the collisionality regime. Recently, formal criteria for closeness to quasisymmetry have been derived in a variety of situations. In particular, the case of deviations with large gradients was solved in the $1/\nu$ regime. Denoting by $\alpha$ a parameter that gives the size of the deviation from quasisymmetry, it was proven that particle fluxes do not scale with $\alpha^{3/2}$, as typically claimed, but with $\alpha$. It was also shown that ripple wells are not necessarily the main cause of transport. This paper reviews those works and presents a new result in another collisionality regime, in which particles trapped in ripple wells are collisional and the rest are collisionless., Comment: 14 pages, 2 figures. To appear in Plasma Physics and Controlled Fusion

Plasma flow is damped in stellarators because they are not intrinsically ambipolar, unlike tokamaks, in which the flux-surface averaged radial electric current vanishes for any value of the radial electric field. Only quasisymmetric stellarators are intrinsically ambipolar, but exact quasisymmetry is impossible to achieve in non-axisymmetric toroidal configurations. By calculating the violation of intrinsic ambipolarity due to deviations from quasisymmetry, one can derive criteria to assess when a stellarator can be considered quasisymmetric in practice, i.e. when the flow damping is weak enough. Let us denote by $\alpha$ a small parameter that controls the size of a perturbation to an exactly quasisymmetric magnetic field. Recently, it has been shown that if the gradient of the perturbation is sufficiently small, the flux-surface averaged radial electric current scales as $\alpha^2$ for any value of the collisionality. It was also argued that when the gradient of the perturbation is large, the quadratic scaling is replaced by a more unfavorable one. In this paper, perturbations with large gradients are rigorously treated. In particular, it is proven that for low collisionality a perturbation with large gradient yields, at best, an $O(|\alpha|)$ deviation from quasisymmetry. Heuristic estimations in the literature incorrectly predicted an $O(|\alpha|^{3/2})$ deviation., Comment: 24 pages, 2 figures. To appear in Plasma Physics and Controlled Fusion

A generic non-symmetric magnetic field does not confine magnetized charged particles for long times due to secular magnetic drifts. Stellarator magnetic fields should be omnigeneous (that is, designed such that the secular drifts vanish), but perfect omnigeneity is technically impossible. There always are small deviations from omnigeneity that necessarily have large gradients. The amplification of the energy flux caused by a deviation of size $\epsilon$ is calculated and it is shown that the scaling with $\epsilon$ of the amplification factor can be as large as linear. In opposition to common wisdom, most of the transport is not due to particles trapped in ripple wells, but to the perturbed motion of particles trapped in the omnigeneous magnetic wells around their bounce points., Comment: 6 pages, 2 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.

Conventional radially-local neoclassical calculations become inadequate if the radial gradient scale lengths of the H-mode pedestal become as small as the poloidal ion gyroradius. Here, we describe a radially global $\delta f$ continuum code that generalizes neoclassical calculations to allow stronger gradients. As with conventional neoclassical calculations, the formulation is time-independent and requires only the solution of a single sparse linear system. We demonstrate precise agreement with an asymptotic analytic solution of the radially global kinetic equation in the appropriate limits of aspect ratio and collisionality. This agreement depends crucially on accurate treatment of finite orbit width effects.

Gyrokinetic simulations have greatly improved our theoretical understanding of turbulent transport in fusion devices. Most gyrokinetic models in use are delta-f simulations in which the slowly varying radial profiles of density and temperature are assumed to be constant for turbulence saturation times, and only the turbulent electromagnetic fluctuations are calculated. New massive simulations are being built to self-consistently determine the radial profiles of density and temperature. However, these new codes have failed to realize that modern gyrokinetic formulations, composed of a gyrokinetic Fokker-Planck equation and a gyrokinetic quasineutrality equation, are only valid for delta-f simulations that do not reach the longer transport time scales necessary to evolve radial profiles. In tokamaks, due to axisymmetry, the evolution of the axisymmetric radial electric field is a challenging problem requiring substantial modifications to gyrokinetic treatments. In this thesis, I study the effect of turbulence on the global electric field and plasma flows. By studying the current conservation equation, or vorticity equation, I prove that the long wavelength, axisymmetric flow must remain neoclassical and I show that the tokamak is intrinsically ambipolar, i.e., the radial current is zero to a very high order for any long wavelength radial electric field. Intrinsic ambipolarity is the origin of the problems with the modern gyrokinetic approach since the lower order gyrokinetic quasineutrality (if properly evaluated) is effectively independent of the radial electric field. I propose a new gyrokinetic formalism to solve for the global radial electric field., Comment: MIT Thesis, 202 pages, 11 figures

The induced electric field in a tokamak drives a parallel electron current flow. In an inhomogeneous, finite beta plasma, when this electron flow is comparable to the ion thermal speed, the Alfven mode wave solutions of the electromagnetic gyrokinetic equation can become nearly purely growing kink modes. Using the new "low-flow" version of the gyrokinetic code GS2 developed for momentum transport studies [Barnes et al 2013 Phys. Rev. Lett. 111, 055005], we are able to model the effect of the induced parallel electric field on the electron distribution to study the destabilizing influence of current on stability. We identify high mode number kink modes in GS2 simulations and make comparisons to analytical theory in sheared magnetic geometry. We demonstrate reassuring agreement with analytical results both in terms of parametric dependences of mode frequencies and growth rates, and regarding the radial mode structure., Comment: Revised version of the manuscript, to be submitted to PPCF

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

Rotation is favorable for confinement, but a stellarator can rotate at high speeds if and only if it is sufficiently close to quasisymmetry. This article investigates how close it needs to be. For a magnetic field $\mathbf{B} = \mathbf{B}_0 + \alpha \mathbf{B}_1$, where $\mathbf{B}_0$ is quasisymmetric, $\alpha\mathbf{B}_1$ is a deviation from quasisymmetry, and $\alpha\ll 1$, the stellarator can rotate at high velocities if $\alpha < \epsilon^{1/2}$, with $\epsilon$ the ion Larmor radius over the characteristic variation length of $\mathbf{B}_0$. The cases in which this result may break down are discussed. If the stellarator is sufficiently quasisymmetric in the above sense, the rotation profile, and equivalently, the long-wavelength radial electric field, are not set neoclassically; instead, they can be affected by turbulent transport. Their computation requires the $O(\epsilon^2)$ pieces of both the turbulent and the long-wavelength components of the distribution function. This article contains the first step towards a formulation to calculate the rotation profile by providing the equations determining the long-wavelength components of the $O(\epsilon^2)$ pieces., Comment: 59 pages, 2 figures. arXiv admin note: text overlap with arXiv:1204.1509

The ion toroidal rotation in a tokamak consists of an $E\times B$ flow due to the radial electric field and a diamagnetic flow due to the radial pressure gradient. The turbulent pinch of toroidal angular momentum due to the Coriolis force studied in previous work is only applicable to the $E\times B$ flow. In this Letter, the momentum pinch for the rotation generated by the radial pressure gradient is calculated and is compared with the Coriolis pinch. This distinction is important for subsonic flows or the flow in the pedestal where the two types of flows are similar in size and opposite in direction. In the edge, the different pinches due to the opposite rotations can result in intrinsic momentum transport that gives significant rotation peaking., Comment: 5 pages and 3 figures

Using a two-dimensional fluid description, we investigate the nonlinear radial-longitudinal dynamics of intense beams in storage rings and cyclotrons. With a multiscale analysis separating the time scale associated with the betatron motion and the slower time scale associated with space-charge effects, we show that the longitudinal-radial vortex motion can be understood in the frame moving with the charged beam as the nonlinear advection of the beam by the $\mathbf{E}\times\mathbf{B}$ velocity field, where $\mathbf{E}$ is the electric field due to the space charge and $\mathbf{B}$ is the external magnetic field. This interpretation provides simple explanations for the stability of round beams and for the development of spiral halos in elongated beams. By numerically solving the nonlinear advection equation for the beam density, we find that it is also in quantitative agreement with results obtained in PIC simulations., Comment: 23 pages, 7 figures

The injection of lower hybrid waves for current drive into a tokamak affects the profile of intrinsic rotation. In this article, the momentum deposition by the lower hybrid wave on the electrons is studied. Due to the increase in the poloidal momentum of the wave as it propagates into the tokamak, the parallel momentum of the wave increases considerably. The change of the perpendicular momentum of the wave is such that the toroidal angular momentum of the wave is conserved. If the perpendicular momentum transfer via electron Landau damping is ignored, the transfer of the toroidal angular momentum to the plasma will be larger than the injected toroidal angular momentum. A proper quasilinear treatment proves that both perpendicular and parallel momentum are transferred to the electrons. The toroidal angular momentum of the electrons is then transferred to the ions via different mechanisms for the parallel and perpendicular momentum. The perpendicular momentum is transferred to ions through an outward radial electron pinch, while the parallel momentum is transferred through collisions., Comment: 22 pages, 4 figures

Recently, the electrostatic gyrokinetic Hamiltonian and change of coordinates have been computed to order $\epsilon^2$ in general magnetic geometry. Here $\epsilon$ is the gyrokinetic expansion parameter, the gyroradius over the macroscopic scale length. Starting from these results, the long-wavelength limit of the gyrokinetic Fokker-Planck and quasineutrality equations is taken for tokamak geometry. Employing the set of equations derived in the present article, it is possible to calculate the long-wavelength components of the distribution functions and of the poloidal electric field to order $\epsilon^2$. These higher-order pieces contain both neoclassical and turbulent contributions, and constitute one of the necessary ingredients (the other is given by the short-wavelength components up to second order) that will eventually enter a complete model for the radial transport of toroidal angular momentum in a tokamak in the low flow ordering. Finally, we provide an explicit and detailed proof that the system consisting of second-order gyrokinetic Fokker-Planck and quasineutrality equations leaves the long-wavelength radial electric field undetermined; that is, the turbulent tokamak is intrinsically ambipolar., Comment: 70 pages. Typos in equations (63), (90), (91), (92) and (129) corrected

The generation of intrinsic rotation by turbulence and neoclassical effects in tokamaks is considered. To obtain the complex dependences observed in experiments, it is necessary to have a model of the radial flux of momentum that redistributes the momentum within the tokamak in the absence of a preexisting velocity. When the lowest order gyrokinetic formulation is used, a symmetry of the model precludes this possibility, making small effects in the gyroradius over scale length expansion necessary. These effects that are usually small become important for momentum transport because the symmetry of the lowest order gyrokinetic formulation leads to the cancellation of the lowest order momentum flux. The accuracy to which the gyrokinetic equation needs to be obtained to retain all the physically relevant effects is discussed.

A low flow, $\delta f$ gyrokinetic formulation to obtain the intrinsic rotation profiles is presented. The momentum conservation equation in the low flow ordering contains new terms, neglected in previous first principles formulations, that may explain the intrinsic rotation observed in tokamaks in the absence of external sources of momentum. The intrinsic rotation profile depends on the density and temperature profiles and on the up-down asymmetry., Comment: 20 pages

Two symmetries of the local nonlinear delta-f gyrokinetic system of equations in tokamaks in the high flow regime are presented. The turbulent transport of toroidal angular momentum changes sign under an up-down reflection of the tokamak and a sign change of both the rotation and the rotation shear. Thus, the turbulent transport of toroidal angular momentum must vanish for up-down symmetric tokamaks in the absence of both rotation and rotation shear. This has important implications for the modeling of spontaneous rotation., Comment: 15 pages, 2 figures

Gyrokinetic theory is based on an asymptotic expansion in the small parameter $\epsilon$, 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 $\epsilon^2$. In particular, a new expression for the complete second-order 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 $\epsilon^2$ and a Poisson equation accurate to order $\epsilon$. The final expressions are explicit and can be implemented into any simulation without further computations., Comment: 55 pages. Version with typo in equation (135) corrected. The second term in the second line of (135) was missing the subindex that indicates that only the perpendicular component of the gradient enters this term