Staircase resiliency in a fluctuating cellular array.

Inhomogeneous mixing by stationary convective cells set in a fixed array is a particularly simple route to layering. Layered profile structures, or staircases, have been observed in many systems, including drift-wave turbulence in magnetic confinement devices. The simplest type of staircase occurs in passive-scalar advection, due to the existence and interplay of two disparate timescales, the cell turn-over (τ_{H}), and the cell diffusion (τ_{D}) time. In this simple system, we study the resiliency of the staircase structure in the presence of global transverse shear and weak vortex scattering. The fixed cellular array is then generalized to a fluctuating vortex array in a series of numerical experiments. The focus is on regimes of low-modest effective Reynolds numbers, as found in magnetic fusion devices. By systematically perturbing the elements of the vortex array, we learn that staircases form and are resilient (although steps become less regular, due to cell mergers) over a broad range of Reynolds numbers. The criteria for resiliency are (a) τ_{D}≫τ_{H} and (b) a sufficiently high profile curvature (κ≥1.5). We learn that scalar concentration travels along regions of shear, thus staircase barriers form first, and scalar concentration "homogenizes" in vortices later. The scattering of vortices induces a lower effective speed of scalar concentration front propagation. The paths are those of the least time. We observe that if background diffusion is kept fixed, the cell geometric properties can be used to derive an approximation for the effective diffusivity of the scalar. The effective diffusivity of the fluctuating vortex array does not deviate significantly from that of the fixed cellular array.

Generation of momentum transport in weakly turbulent ß-plane magnetohydrodynamics.

Magnetohydrodynamic turbulence on a ß plane with an in-plane mean field, a system which serves as a simple model for the solar tachocline, is investigated analytically and computationally. We first derive two useful analytic constraints: We express the mean turbulent cross-helicity in terms of the mean turbulent magnetic energy, and then show that (for weak turbulence) the time-averaged momentum transport in the system can be expressed in terms of the cross-helicity spectrum. We then complete a closure of the system using weak turbulence theory, appropriately extended to a system with multiple interacting eigenmodes. We use this closure to perturbatively solve for the spectra at lowest order in the Rossby parameter ß and thereby show that the momentum transport in the system is O(ß^{2}), thus quantifying the transition away from Alfvénized turbulence. Finally, we verify our theoretical results by performing direct numerical simulations of the system over a broad range of ß.

Turbulence model reduction by deep learning.

A central problem of turbulence theory is to produce a predictive model for turbulent fluxes. These have profound implications for virtually all aspects of the turbulence dynamics. In magnetic confinement devices, drift-wave turbulence produces anomalous fluxes via cross-correlations between fluctuations. In this work, we introduce an alternative, data-driven method for parametrizing these fluxes. The method uses deep supervised learning to infer a reduced mean-field model from a set of numerical simulations. We apply the method to a simple drift-wave turbulence system and find a significant new effect which couples the particle flux to the local gradient of vorticity. Notably, here, this effect is much stronger than the oft-invoked shear suppression effect. We also recover the result via a simple calculation. The vorticity gradient effect tends to modulate the density profile. In addition, our method recovers a model for spontaneous zonal flow generation by negative viscosity, stabilized by nonlinear and hyperviscous terms. We highlight the important role of symmetry to implementation of the alternative method.

Curvature of Radial Electric Field Aggravates Edge Magnetohydrodynamics Mode in Toroidally Confined Plasmas.

We show that the radial electric field (E_{r}) plays a dual role in edge magnetohydrodynamics (MHD) activity. While E_{r} shear (first spatial derivative of E_{r}) dephases radial velocity and displacement, and so is stabilizing, a new finding here is that E_{r} curvature (second spatial derivative of E_{r}) tends to synchronize the radial velocity and displacement, and so destabilizes MHD. As a highlighted result, we analytically demonstrate that E_{r} curvature can destabilize an otherwise stable kink mode, and so form a joint vortex-kink mode. The synergetic effects of E_{r} shear and E_{r} curvature in edge MHD extend the familiar E×B shearing paradigm. This theory thus explains the experimental findings that a deeper E×B well may aggravate edge MHD, and so trigger the formation of the edge harmonic oscillation. A simple criterion linking E_{r} structure and the edge MHD activity is derived.

Spontaneous transport barriers quench turbulent resistivity in two-dimensional magnetohydrodynamics.

This Rapid Communication identifies the physical mechanism for the quench of turbulent resistivity in two-dimensional magnetohydrodynamics. Without an imposed, ordered magnetic field, a multiscale, blob-and-barrier structure of magnetic potential forms spontaneously. Magnetic energy is concentrated in thin, linear barriers, located at the interstices between blobs. The barriers quench the transport and kinematic decay of magnetic energy. The local transport bifurcation underlying barrier formation is linked to the inverse cascade of 〈A^{2}〉 and negative resistivity, which induce local bistability. For small-scale forcing, spontaneous layering of the magnetic potential occurs, with barriers located at the interstices between layers. This structure is effectively a magnetic staircase.

Tracing the Pathway from Drift-Wave Turbulence with Broken Symmetry to the Production of Sheared Axial Mean Flow.

This study traces the emergence of sheared axial flow from collisional drift-wave turbulence with broken symmetry in a linear plasma device-the controlled shear decorrelation experiment. As the density profile steepens, the axial Reynolds stress develops and drives a radially sheared axial flow that is parallel to the magnetic field. Results show that the nondiffusive piece of the Reynolds stress is driven by the density gradient, results from spectral asymmetry of the turbulence, and, thus, is dynamical in origin. Taken together, these findings constitute the first simultaneous demonstration of the causal link between the density gradient, turbulence, and stress with broken spectral symmetry and the mean axial flow.

How turbulence fronts induce plasma spin-up.

A calculation which describes the spin-up of toroidal plasmas by the radial propagation of turbulence fronts with broken parallel symmetry is presented. The associated flux of parallel momentum is calculated by using a two-scale direct-interaction approximation in the weak turbulence limit. We show that fluctuation momentum spreads faster than mean flow momentum. Specifically, the turbulent flux of wave momentum is stronger than the momentum pinch. The scattering of fluctuation momentum can induce edge-core coupling of toroidal flows, as observed in experiments.

Formation and evolution of target patterns in Cahn-Hilliard flows.

We study the evolution of the concentration field in a single eddy in the two-dimensional (2D) Cahn-Hilliard system to better understand scalar mixing processes in that system. This study extends investigations of the classic studies of flux expulsion in 2D magnetohydrodynamics and homogenization of potential vorticity in 2D fluids. Simulation results show that there are three stages in the evolution: (A) formation of a "jelly roll" pattern, for which the concentration field is constant along spirals; (B) a change in isoconcentration contour topology; and (C) formation of a target pattern, for which the isoconcentration contours follow concentric annuli. In the final target pattern stage, the isoconcentration bands align with stream lines. The results indicate that the target pattern is a metastable state. The band merger process continues on a time scale exponentially long relative to the eddy turnover time. The band merger process resembles step merger in drift-ZF staircases; this is characteristic of the long-time evolution of phase-separated patterns described by the Cahn-Hilliard equation.

How mesoscopic staircases condense to macroscopic barriers in confined plasma turbulence.

This Rapid Communication sets forth the mechanism by which mesoscale staircase structures condense to form macroscopic states of enhanced confinement. Density, vorticity, and turbulent potential enstrophy are the variables for this model. Formation of the staircase structures is due to inhomogeneous mixing of (generalized) potential vorticity (PV). Such mixing results in the local sharpening of density and vorticity gradients. When PV gradients steepen, the density staircase structure develops into a lattice of mesoscale "jumps" and "steps," which are, respectively, regions of local gradient steepening and flattening. The jumps then merge and migrate in radius, leading to the emergence of a new macroscale profile structure, so indicating that profile self-organization is a global process, which may be described by a local, but nonlinear model. This work predicts and demonstrates how mesoscale condensation of staircases leads to global states of enhanced confinement.

Zonal Flow Patterns: How Toroidal Coupling Induces Phase Jumps and Shear Layers.

A new, frequency modulation mechanism for zonal flow pattern formation is presented. The model predicts the probability distribution function of the flow strength as well as the evolution of the characteristic spatial scale. Magnetic toroidicity-induced global phase dynamics is shown to determine the spatial structure of the flow. A key result is the observation that global phase patterning can lead to zonal flow formation in the absence of turbulence inhomogeneity.

Logarithmic discretization and systematic derivation of shell models in two-dimensional turbulence.

A detailed systematic derivation of a logarithmically discretized model for two-dimensional turbulence is given, starting from the basic fluid equations and proceeding with a particular form of discretization of the wave-number space. We show that it is possible to keep all or a subset of the interactions, either local or disparate scale, and recover various limiting forms of shell models used in plasma and geophysical turbulence studies. The method makes no use of the conservation laws even though it respects the underlying conservation properties of the fluid equations. It gives a family of models ranging from shell models with nonlocal interactions to anisotropic shell models depending on the way the shells are constructed. Numerical integration of the model shows that energy and enstrophy equipartition seem to dominate over the dual cascade, which is a common problem of two-dimensional shell models.

Synchronization of Geodesic Acoustic Modes and Magnetic Fluctuations in Toroidal Plasmas.

The synchronization of geodesic acoustic modes (GAMs) and magnetic fluctuations is identified in the edge plasmas of the HL-2A tokamak. Mesoscale electric fluctuations (MSEFs) having components of a dominant GAM, and m/n=6/2 potential fluctuations are found at the same frequency as that of the magnetic fluctuations of m/n=6/2 (m and n are poloidal and toroidal mode numbers, respectively). The temporal evolutions of the MSEFs and the magnetic fluctuations clearly show the frequency entrainment and the phase lock between the GAM and the m/n=6/2 magnetic fluctuations. The results indicate that GAMs and magnetic fluctuations can transfer energy through nonlinear synchronization. Such nonlinear synchronization may also contribute to low-frequency zonal flow formation, reduction of turbulence level, and thus confinement regime transitions.

On the interplay between neoclassical tearing modes and nonlocal transport in toroidal plasmas.

This Letter presents the first observation on the interplay between nonlocal transport and neoclassical tearing modes (NTMs) during transient nonlocal heat transport events in the HL-2A tokamak. The nonlocality is triggered by edge cooling and large-scale, inward propagating avalanches. These lead to a locally enhanced pressure gradient at the q = 3/2 (or 2/1) rational surface and hence the onset of the NTM in relatively low ß plasmas (ßN < 1). The NTM, in return, regulates the nonlocal transport by truncation of avalanches by local sheared toroidal flows which develop near the magnetic island. These findings have direct implications for understanding the dynamic interaction between turbulence and large-scale mode structures in fusion plasmas.

Nonperturbative mean-field theory for minimum enstrophy relaxation.

The dual cascade of enstrophy and energy in quasi-two-dimensional turbulence strongly suggests that a viscous but otherwise potential vorticity (PV) conserving system decays selectively toward a state of minimum potential enstrophy. We derive a nonperturbative mean field theory for the dynamics of minimum enstrophy relaxation by constructing an expression for PV flux during the relaxation process. The theory is used to elucidate the structure of anisotropic flows emerging from the selective decay process. This structural analysis of PV flux is based on the requirements that the mean flux of PV dissipates total potential enstrophy but conserves total fluid kinetic energy. Our results show that the structure of PV flux has the form of a sum of a positive definite hyperviscous and a negative or positive viscous transport of PV. Transport parameters depend on zonal flow and turbulence intensity. Turbulence spreading is shown to be related to PV mixing via the link of turbulence energy flux to PV flux. In the relaxed state, the ratio of the PV gradient to zonal flow velocity is homogenized. This homogenized quantity sets a constraint on the amplitudes of PV and zonal flow in the relaxed state. A characteristic scale is defined by the homogenized quantity and is related to a variant of the Rhines scale. This relaxation model predicts a relaxed state with a structure which is consistent with PV staircases, namely, the proportionality between mean PV gradient and zonal flow strength.

From phase locking to phase slips: a mechanism for a quiescent H mode.

We demonstrate that E×B shear, V_{E×B}^{'}, governs the dynamics of the cross phase of the peeling-ballooning-(PB-)mode-driven heat flux, and so determines the evolution from the edge-localized (ELMy) H mode to the quiescent (Q) H mode. A physics-based scaling of the critical E×B shearing rate (V_{E×B,cr}^{'}) for accessing the QH mode is predicted. The ELMy H mode to the QH-mode evolution is shown to follow from the conversion from a phase locked state to a phase slip state. In the phase locked state, PB modes are pumped continuously, so bursts occur. In the slip state, the PB activity is a coherent oscillation. Stronger E×B shearing implies a higher phase slip frequency. This finding predicts a new state of cross phase dynamics and shows a new way to understand the physics mechanism for ELMy to the QH-mode evolution.

Finding the elusive E×B staircase in magnetized plasmas.

Turbulence in hot magnetized plasmas is shown to generate permeable localized transport barriers that globally organize into the so-called "ExB staircase" [G. Dif-Pradalier et al., Phys. Rev. E, 82, 025401(R) (2010)]. Its domain of existence and dependence with key plasma parameters is discussed theoretically. Based on these predictions, staircases are observed experimentally in the Tore Supra tokamak by means of high-resolution fast-sweeping X-mode reflectometry. This observation strongly emphasizes the critical role of mesoscale self-organization in plasma turbulence and may have far-reaching consequences for turbulent transport models and their validation.

Elasticity in drift-wave-zonal-flow turbulence.

We present a theory of turbulent elasticity, a property of drift-wave-zonal-flow (DW-ZF) turbulence, which follows from the time delay in the response of DWs to ZF shears. An emergent dimensionless parameter |〈v〉'|/Δωk is found to be a measure of the degree of Fickian flux-gradient relation breaking, where |〈v〉'| is the ZF shearing rate and Δωk is the turbulence decorrelation rate. For |〈v〉'|/Δωk>1, we show that the ZF evolution equation is converted from a diffusion equation, usually assumed, to a telegraph equation, i.e., the turbulent momentum transport changes from a diffusive process to wavelike propagation. This scenario corresponds to a state very close to the marginal instability of the DW-ZF system, e.g., the Dimits shift regime. The frequency of the ZF wave is ΩZF=±Î³d1/2γmodu1/2, where γd is the ZF friction coefficient and γmodu is the net ZF growth rate for the case of the Fickian flux-gradient relation. This insight provides a natural framework for understanding temporally periodic ZF structures in the Dimits shift regime and in the transition from low confined mode to high confined mode in confined plasmas.

Effects of magnetic shear on toroidal rotation in tokamak plasmas with lower hybrid current drive.

Application of lower hybrid (LH) current drive in tokamak plasmas can induce both co- and countercurrent directed changes in toroidal rotation, depending on the core q profile. For discharges with q(0) <1, rotation increments in the countercurrent direction are observed. If the LH-driven current is sufficient to suppress sawteeth and increase q(0) above unity, the core toroidal rotation change is in the cocurrent direction. This change in sign of the rotation increment is consistent with a change in sign of the residual stress (the divergence of which constitutes an intrinsic torque that drives the flow) through its dependence on magnetic shear.

Gyrokinetic theory of turbulent acceleration of parallel rotation in tokamak plasmas.

A mechanism for turbulent acceleration of parallel rotation is discovered using gyrokinetic theory. This new turbulent acceleration term cannot be written as a divergence of parallel Reynolds stress. Therefore, turbulent acceleration acts as a local source or sink of parallel rotation. The physics of turbulent acceleration is intrinsically different from the Reynolds stress. For symmetry breaking by positive intensity gradient, a positive turbulent acceleration, i.e., cocurrent rotation, is predicted. The turbulent acceleration is independent of mean rotation and mean rotation gradient, and so constitutes a new candidate for the origin of spontaneous rotation. A quasilinear estimate for ion temperature gradient turbulence shows that the turbulent acceleration of parallel rotation is explicitly linked to the ion temperature gradient scale length and temperature ratio Ti0/Te0. Methods for testing the effects of turbulent parallel acceleration by gyrokinetic simulation and experiment are proposed.

Physics of stimulated L→H transitions.

We report on model studies of stimulated L→H transitions. These studies use a novel reduced mesoscale model. Studies reveal that L→H transitions can be triggered by particle injection into a subcritical state (i.e., P