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Turbulent transport driven by trapped electron modes (TEMs) is believed to drive significant heat and particle transport in quasihelically symmetric stellarators. Two three-dimensionally-shaped magnetic configurations with suppressed trapped-electron-mode (TEM)-driven turbulence were generated through optimization that targeted quasihelical symmetry and the available energy of trapped electrons. Initial equilibria have flux surface shapes with a helically rotating negative triangularity (NT) and positive triangularity (PT). In gyrokinetic simulations, TEMs are suppressed in the reduced-TEM NT and PT configurations, showing that negative triangularity does not have the same beneficial turbulence properties over positive triangularity as seen in tokamaks. Heat fluxes from TEMs are also suppressed. Without temperature gradients and with a strong density gradient, the most unstable modes at low $k_y$ were consistent with toroidal universal instabilities (UIs) in the NT case and slab UIs in the PT case. Nonlinear simulations show that UIs drive substantial heat flux in both the NT and PT configurations. A moderate increase in $\beta$ halves the heat flux in the NT configuration, while suppressing the heat flux in the PT geometry. Based on the present work, future optimizations aimed at reducing electrostatic drift wave-driven turbulent transport will need to consider UIs if $\beta$ is sufficiently small.

Using a novel optimization procedure it has been shown that the Helically Symmetric eXperiment (HSX) stellarator can be optimized for reduced trapped-electron-mode (TEM) instability [M.J.~Gerard et al., \textit{Nucl.~Fusion} \textbf{63} (2023) 056004]. Presently, with a set of 563 experimental candidate configurations, gyrokinetic simulations are performed to investigate the efficacy of available energy $E_\mathrm{A}$, quasi-helical symmetry, and flux-surface shaping parameters as metrics for TEM stabilization. It is found that lower values of $E_\mathrm{A}$ correlate with reduced growth rates, but only when separate flux-surface shaping regimes are considered. Moreover, configurations with improved quasi-helical symmetry demonstrate a similar reduction in growth rates and less scatter compared to $E_\mathrm{A}$. Regarding flux-surface shaping, a set of helical shaping parameters is introduced that show increased elongation is strongly correlated with reduced TEM growth rates, however, only when the quasi-helical symmetry is preserved. Using a newly derived velocity-space-averaged TEM resonance operator, these trends are analyzed to provide insights into the physical mechanism of the observed stabilization. For elongation, stabilization is attributed to geometric effects that reduce the destabilizing particle drifts across the magnetic field. Regarding quasi-helical symmetry, the TEM resonance in the maximally resonant trapping well is shown to increase as the quasi-helical symmetry is broken, and breaking quasi-helical symmetry increases the prevalence of highly resonant trapping wells. While these results demonstrate the limitations of using any single metric as a linear TEM proxy, it is shown that quasi-helical symmetry and plasma elongation are highly effective metrics for reducing TEM growth rates in helical equilibria.

Differentially rotating stars and planets transport angular momentum internally due to turbulence at rates that have long been a challenge to predict reliably. We develop a self-consistent saturation theory, using a statistical closure approximation, for hydrodynamic turbulence driven by the axisymmetric Goldreich--Schubert--Fricke (GSF) instability at the stellar equator with radial differential rotation. This instability arises when fast thermal diffusion eliminates the stabilizing effects of buoyancy forces in a system where a stabilizing entropy gradient dominates over the destabilizing angular momentum gradient. Our turbulence closure invokes a dominant three-wave coupling between pairs of linearly unstable eigenmodes and a near-zero frequency, viscously damped eigenmode that features latitudinal jets. We derive turbulent transport rates of momentum and heat, and provide them in analytic forms. Such formulae, free of tunable model parameters, are tested against direct numerical simulations; the comparison shows good agreement. They improve upon prior quasi-linear or ``parasitic saturation" models containing a free parameter. Given model correspondences, we also extend this theory to heat and compositional transport for axisymmetric thermohaline instability-driven turbulence in certain regimes., Comment: Accepted for publication in ApJ

Turbulence in three dimensions ($3$D) supports vortex stretching that has long been known to accomplish energy transfer to small scales. Moreover, net energy transfer from large-scale, forced, unstable flow-gradients to smaller scales is achieved by gradient-flattening instability. Despite such enforcement of energy transfer to small scales, it is shown here not only that the shear-flow-instability-supplied $3$D-fluctuation energy is largely inverse-transferred from the fluctuation to the mean-flow gradient, but that such inverse transfer is more efficient for turbulent fluctuations in $3$D than in two dimensions ($2$D). The transfer is due to linearly stable eigenmodes that are excited nonlinearly. The stable modes, thus, reduce both the nonlinear energy cascade to small scales and the viscous dissipation rate. The vortex-tube stretching is also suppressed. Up-gradient momentum transport by the stable modes counters the instability-driven down-gradient transport, which also is more effective in $3$D than in $2$D ($\mathrm{\approx} 70\% \mathrm{\,\, vs.\,\,}\mathrm{\approx} 50\%$). From unstable modes, these stable modes nonlinearly receive energy via zero-frequency fluctuations that vary only in the direction orthogonal to the plane of $2$D shear flow. The more widely occurring $3$D turbulence is thus inherently different from the commonly studied $2$D turbulence, despite both saturating via stable modes., Comment: To appear in Physics of Fluids

To better understand the interaction of global tearing modes and microturbulence in the Madison Symmetric Torus (MST) reversed-field pinch (RFP), the global gyrokinetic code \textsc{Gene} is modified to describe global tearing mode instability via a shifted Maxwellian distribution consistent with experimental equilibria. The implementation of the shifted Maxwellian is tested and benchmarked by comparisons with different codes and models. Good agreement is obtained in code-code and code-theory comparisons. Linear stability of tearing modes of a non-reversed MST discharge is studied. A collisionality scan is performed to the lowest order unstable modes ($n=5$, $n=6$) and shown to behave consistently with theoretical scaling. The nonlinear evolution is simulated, and saturation is found to arise from mode coupling and transfer of energy from the most unstable tearing mode to small-scale stable modes mediated by the $m=2$ tearing mode. The work described herein lays the foundation for nonlinear simulation and analysis of the interaction of tearing modes and gyroradius-scale instabilities in RFP plasmas.

To comprehensively understand saturation of two-dimensional ($2$D) magnetized Kelvin-Helmholtz-instability-driven turbulence, energy transfer analysis is extended from the traditional interaction between scales to include eigenmode interactions, by using the nonlinear couplings of linear eigenmodes of the ideal instability. While both kinetic and magnetic energies cascade to small scales, a significant fraction of turbulent energy deposited by unstable modes in the fluctuation spectrum is shown to be re-routed to the conjugate-stable modes at the instability scale. They remove energy from the forward cascade at its inception. The remaining cascading energy flux is shown to attenuate exponentially at a small scale, dictated by the large-scale stable modes. Guided by a widely used instability-saturation assumption, a general quasilinear model of instability is tested by retaining all nonlinear interactions except those that couple to the large-scale stable modes. These complex interactions are analytically removed from the magnetohydrodynamic equations using a novel technique. Observations are: an explosive large-scale vortex separation instead of the well-known merger of $2$D, a dramatic enhancement in turbulence level and spectral energy fluxes, and a reduced small-scale dissipation length-scale. These show critical role of the stable modes in instability saturation. Possible reduced-order turbulence models are proposed for fusion and astrophysical plasmas, based on eigenmode-expanded energy transfer analyses., Comment: Selected by the editors of Physics of Plasmas as a Featured article. https://doi.org/10.1063/5.0156560

High-performance fusion plasmas, requiring high pressure $\beta$, are not well-understood in stellarator-type experiments. Here, the effect of $\beta$ on ion-temperature-gradient-driven (ITG) turbulence is studied in Wendelstein 7-X (W7-X), showing that subdominant kinetic ballooning modes (KBMs) are unstable well below the ideal MHD threshold and get strongly excited in the turbulence. By zonal-flow erosion, these sub-threshold KBMs (stKBMs) affect ITG saturation and enable higher heat fluxes. Controlling stKBMs will be essential to allow W7-X and future stellarators to achieve maximum performance., Comment: 16 pages, 5 figures

Visco-resistive magnetohydrodynamic turbulence, driven by a two-dimensional unstable shear layer that is maintained by an imposed body force, is examined by decomposing it into dissipationless linear eigenmodes of the initial profiles. The down-gradient momentum flux, as expected, originates from the large-scale instability. However, continual up-gradient momentum transport by large-scale linearly stable but nonlinearly excited eigenmodes is identified, and found to nearly cancel the down-gradient transport by unstable modes. The stable modes effectuate this by depleting the large-scale turbulent fluctuations via energy transfer to the mean flow. This establishes a physical mechanism underlying the long-known observation that coherent vortices formed from nonlinear saturation of the instability reduce turbulent transport and fluctuations, as such vortices are composed of both the stable and unstable modes, which are nearly equal in their amplitudes. The impact of magnetic fields on the nonlinearly excited stable modes is then quantified. Even when imposing a strong magnetic field that almost completely suppresses the instability, the up-gradient transport by the stable modes is at least two-thirds of the down-gradient transport by the unstable modes, whereas for weaker fields, this fraction reaches up to $98\%$. These effects are persistent with variations in magnetic Prandtl number and forcing strength. Finally, continuum modes are shown to be energetically less important, but essential for capturing the magnetic fluctuations and Maxwell stress. A simple analytical scaling law is derived for their saturated turbulent amplitudes. It predicts the fall-off rate as the inverse of the Fourier wavenumber, a property which is confirmed in numerical simulations., Comment: Accepted for publication in Physics of Plasmas; 43 pages, 15 figures; invited as part of the Special Topic: Papers from the 2022 Sherwood Fusion Theory Conference

Straining of magnetic fields by large-scale shear flow, generally assumed to lead to intensification and generation of small scales, is re-examined in light of the persistent observation of large-scale magnetic fields in astrophysics. It is shown that, in magnetohydrodynamic turbulence, unstable shear flows have the unexpected effect of sequestering magnetic energy at large scales, due to counteracting straining motion of nonlinearly excited large-scale stable eigenmodes. This effect is quantified via dissipation rates, energy transfer rates, and visualizations of magnetic field evolution by artificially removing the stable modes. These analyses show that predictions based upon physics of the linear instability alone miss substantial dynamics, including those of magnetic fluctuations., Comment: Accepted for publication in Physics of Plasmas as a Letter; A simulation movie embedded in Fig. 4

The Kelvin-Helmholtz (KH) instability of a shear layer with an initially-uniform magnetic field in the direction of flow is studied in the framework of 2D incompressible magnetohydrodynamics with finite resistivity and viscosity using direct numerical simulations. The shear layer evolves freely, with no external forcing, and thus broadens in time as turbulent stresses transport momentum across it. As with KH-unstable flows in hydrodynamics, the instability here features a conjugate stable mode for every unstable mode in the absence of dissipation. Stable modes are shown to transport momentum up its gradient, shrinking the layer width whenever they exceed unstable modes in amplitude. In simulations with weak magnetic fields, the linear instability is minimally affected by the magnetic field, but enhanced small-scale fluctuations relative to the hydrodynamic case are observed. These enhanced fluctuations coincide with increased energy dissipation and faster layer broadening, with these features more pronounced in simulations with stronger fields. These trends result from the magnetic field reducing the effects of stable modes relative to the transfer of energy to small scales. As field strength increases, stable modes become less excited and thus transport less momentum against its gradient. Furthermore, the energy that would otherwise transfer back to the driving shear due to stable modes is instead allowed to cascade to small scales, where it is lost to dissipation. Approximations of the turbulent state in terms of a reduced set of modes are explored. While the Reynolds stress is well-described using just two modes per wavenumber at large scales, the Maxwell stress is not., Comment: 39 pages, 17 figures, preprint format

A new optimized quasihelically symmetric configuration is described that has the desir-able properties of improved energetic particle confinement, reduced turbulent transportby 3D shaping, and non-resonant divertor capabilities. The configuration presented in thispaper is explicitly optimized for quasihelical symmetry, energetic particle confinement,neoclassical confinement, and stability near the axis. Post optimization, the configurationwas evaluated for its performance with regard to energetic particle transport, idealmagnetohydrodynamic (MHD) stability at various values of plasma pressure, and iontemperature gradient instability induced turbulent transport. The effect of discrete coilson various confinement figures of merit, including energetic particle confinement, aredetermined by generating single-filament coils for the configuration. Preliminary divertoranalysis shows that coils can be created that do not interfere with expansion of thevessel volume near the regions of outgoing heat flux, thus demonstrating the possibilityof operating a non-resonant divertor., Comment: 13 figures, 3 tables

A linearly unstable, sinusoidal $E \times B$ shear flow is examined in the gyrokinetic framework in both the linear and nonlinear regimes. In the linear regime, it is shown that the eigenmode spectrum is nearly identical to hydrodynamic shear flows, with a conjugate stable mode found at every unstable wavenumber. In the nonlinear regime, turbulent saturation of the instability is examined with and without the inclusion of a driving term that prevents nonlinear flattening of the mean flow, and a scale-independent radiative damping term that suppresses the excitation of conjugate stable modes. A simple fluid model for how momentum transport and partial flattening of the mean flow scale with the driving term is constructed, from which it is shown that, except at high radiative damping, stable modes play an important role in the turbulent state and yield significantly improved quantitative predictions when compared with corresponding models neglecting stable modes., Comment: 34 pages pre-print format, 13 figures, submitted to Physics of Plasmas

Analysis of the saturation of the Kelvin-Helmholtz (KH) instability is undertaken to determine the extent to which the conjugate linearly stable mode plays a role. For a piecewise-continuous mean flow profile with constant shear in a fixed layer, it is shown that the stable mode is nonlinearly excited, providing an injection-scale sink of the fluctuation energy similar to what has been found for gyroradius-scale drift-wave turbulence. Quantitative evaluation of the contribution of the stable mode to the energy balance at the onset of saturation shows that nonlinear energy transfer to the stable mode is as significant as energy transfer to small scales in balancing energy injected into the spectrum by the instability. The effect of the stable mode on momentum transport is quantified by expressing the Reynolds stress in terms of stable and unstable mode amplitudes at saturation, from which it is found that the stable mode can produce a sizable reduction in the momentum flux., Comment: 23 pages, 6 figures, accepted for publication in Physics of Plasmas, June 2017

An analytic equilibrium, the Toroidal Bessel Function Model, is used in conjunction with the gyrokinetic code GYRO to investigate the nature of microinstabilities in a reversed field pinch (RFP) plasma. The effect of the normalized electron plasma pressure ({\beta}) on the characteristics of the microinstabilities is studied. A transition between an ion temperature gradient (ITG) driven mode and a microtearing mode as the dominant instability is found to occur at a {\beta} value of approximately 4.5%. Suppression of the ITG mode occurs as in the tokamak, through coupling to shear Alfven waves, with a critical {\beta} for stability higher than its tokamak equivalent due to a shorter parallel connection length. There is a steep dependence of the microtearing growth rate on temperature gradient suggesting high profile stiffness. There is evidence for a collisionless microtearing mode. The properties of this mode are investigated, and it is found that curvature drift plays an important role in the instability., Comment: 19 pages, 9 figures, submitted to Physics of Plasmas

The forms of electron density structures in kinetic Alfven wave turbulence are studied in connection with scintillation. The focus is on small scales $L \sim 10^8-10^{10}$ cm where the Kinetic Alfv\'en wave (KAW) regime is active in the interstellar medium. MHD turbulence converts to a KAW cascade, starting at 10 times the ion gyroradius and continuing to smaller scales. These scales are inferred to dominate scintillation in the theory of Boldyrev et al. From numerical solutions of a decaying kinetic Alfv\'en wave turbulence model, structure morphology reveals two types of localized structures, filaments and sheets, and shows that they arise in different regimes of resistive and diffusive damping. Minimal resistive damping yields localized current filaments that form out of Gaussian-distributed initial conditions. When resistive damping is large relative to diffusive damping, sheet-like structures form. In the filamentary regime, each filament is associated with a non-localized magnetic and density structure, circularly symmetric in cross section. Density and magnetic fields have Gaussian statistics (as inferred from Gaussian-valued kurtosis) while density gradients are strongly non-Gaussian, more so than current. This enhancement of non-Gaussian statistics in a derivative field is expected since gradient operations enhance small-scale fluctuations. The enhancement of density gradient kurtosis over current kurtosis is not obvious, yet it suggests that modest fluctuation levels in electron density may yield large scintillation events during pulsar signal propagation in the interstellar medium. In the sheet regime the same statistical observations hold, despite the absence of localized filamentary structures. Probability density functions are constructed from statistical ensembles in both regimes, showing clear formation of long, highly non-Gaussian tails.

Because experiment/model comparisons in magnetic confinement fusion have not yet satisfied the requirements for validation as understood broadly, a set of approaches to validating mathematical models and numerical algorithms are recommended as good practices. Previously identified procedures, such as verification, qualification, and analysis of error and uncertainty, remain important. However, particular challenges intrinsic to fusion plasmas and physical measurement therein lead to identification of new or less familiar concepts that are also critical in validation. These include the primacy hierarchy, which tracks the integration of measurable quantities, and sensitivity analysis, which assesses how model output is apportioned to different sources of variation. The use of validation metrics for individual measurements is extended to multiple measurements, with provisions for the primacy hierarchy and sensitivity. This composite validation metric is essential for quantitatively evaluating comparisons with experiments. To mount successful and credible validation in magnetic fusion, a new culture of validation is envisaged., Comment: 27 pages, 1 table, 6 figures

Spatial intermittency in decaying kinetic Alfven wave turbulence is investigated to determine if it produces non Gaussian density fluctuations in the interstellar medium. Non Gaussian density fluctuations have been inferred from pulsar scintillation scaling. Kinetic Alfven wave turbulence characterizes density evolution in magnetic turbulence at scales near the ion gyroradius. It is shown that intense localized current filaments in the tail of an initial Gaussian probability distribution function possess a sheared magnetic field that strongly refracts the random kinetic Alfven waves responsible for turbulent decorrelation. The refraction localizes turbulence to the filament periphery, hence it avoids mixing by the turbulence. As the turbulence decays these long-lived filaments create a non Gaussian tail. A condition related to the shear of the filament field determines which fluctuations become coherent and which decay as random fluctuations. The refraction also creates coherent structures in electron density. These structures are not localized. Their spatial envelope maps into a probability distribution that decays as density to the power -3. The spatial envelope of density yields a Levy distribution in the density gradient., Comment: 31 pages, 3 figures. Replacement contains short additions to Secs. 6 and 7

The role of turbulence in current generation and self-excitation of magnetic fields has been studied in the geometry of a mechanically driven, spherical dynamo experiment, using a three dimensional numerical computation. A simple impeller model drives a flow which can generate a growing magnetic field, depending upon the magnetic Reynolds number, Rm, and the fluid Reynolds number. When the flow is laminar, the dynamo transition is governed by a simple threshold in Rm, above which a growing magnetic eigenmode is observed. The eigenmode is primarily a dipole field tranverse to axis of symmetry of the flow. In saturation the Lorentz force slows the flow such that the magnetic eigenmode becomes marginally stable. For turbulent flow, the dynamo eigenmode is suppressed. The mechanism of suppression is due to a combination of a time varying large-scale field and the presence of fluctuation driven currents which effectively enhance the magnetic diffusivity. For higher Rm a dynamo reappears, however the structure of the magnetic field is often different from the laminar dynamo; it is dominated by a dipolar magnetic field which is aligned with the axis of symmetry of the mean-flow, apparently generated by fluctuation-driven currents. The fluctuation-driven currents have been studied by applying a weak magnetic field to laminar and turbulent flows. The magnetic fields generated by the fluctuations are significant: a dipole moment aligned with the symmetry axis of the mean-flow is generated similar to those observed in the experiment, and both toroidal and poloidal flux expulsion are observed., Comment: 14 pages, 14 figures