Growth Rate Spectrum Research Articles

Improved flux-surface parameterization through constrained nonlinear optimization

Parameterization of magnetic flux-surfaces is often used for magnetohydrodynamic stability analysis and microturbulence modeling in tokamaks. Shape parameters for such local parameterization of a (numerical) equilibrium are traditionally computed analytically using geometrically derived quantities. However, often the shape is approximated by the average of values for different sections of the flux-surface contour or a truncated series, which does not guarantee an optimal fit. Here, instead nonlinear least squares optimization is used to compute these parameters, with a weighted sum of squared error cost function that is robust to outliers. This method results in a lower total absolute error for both the parameterization of the flux-surface contour and the poloidal magnetic field density than current methods for several parameterizations based on the well-known “Miller geometry.” Furthermore, rapid convergence of shape parameters is achieved, no approximate geometric measurements of the contour are needed, and the method is applicable to any analytical shape parameterization. Validation with local, linear gyrokinetic simulations using these optimized shape parameters showed reduced root mean square errors in both the growth rate and frequency spectra when compared with simulations based on numerical equilibria. In particular, the popular Turnbull–Miller parameterization benefits from this approach, extending its usability closer toward the last-closed flux-surface for cases with minor up-down asymmetry.

Modulational instability in vector exciton-polariton condensates with photonic spin-orbit coupling

The paper demonstrates the existence of modulational instability (MI) in nonlinear exciton-polariton condensates (EPCs). A set of coupled two-dimensional (2D) driven-dissipative Gross-Pitaevskii equations describing the spin-up and spin-down polaritons, containing the photonic spin-orbit (SO) coupling and the rate equations describing the density of incoherent excitonic reservoir density, is reduced to a system of two coupled 2D complex Ginzburg-Landau (2DCGL) equations, with saturable nonlinearities for complex polariton order parameters, thanks to the adiabatic approximation and small density fluctuations approximation. The analytical approach relies on the linear stability analysis of continuous waves (CWs) to derive an expression for the growth rate and conduct a parametric study of MI. The effect of different parameters on the growth rate spectrum is discussed with an emphasis on the photonic SO coupling and the pumping power. The predictions are verified against a direct simulation of the 2DCGL equations, and excellent agreement is found. The emergence of solitonic clusters manifests the evolution of MI. Phase diagrams on a MI spectrum for the CWs are presented against the magnetic field ($B$) and SO coupling strength ($\ensuremath{\sigma}$), based on which the dynamical behaviors of the emerging structures are debated, along with their response to changing $\ensuremath{\sigma}$ and $B$. The results suggest that the photonic SO coupling and the magnetic field constitute efficient tools for nonlinear mode selection and characterization via the MI process and are equally important in any eventual experimental realization of such a process in the studied EPC system.

Dephasing and phase-locking: Dual role of radial electric field in edge MHD dynamics of toroidally confined plasmas

We carry out several numerical simulations to illustrate how the radial electric field (Er) impacts the edge magnetohydrodynamic (MHD) instabilities. The analyses reveal that Er-shear (Er′, here the prime denotes the derivative with respect to the radial direction) tends to stabilize the kink/Peeling–Ballooning modes by dephasing the perturbed radial velocity (ṽr) and displacement (ξ̃r). However, Er-curvature (Er″) tends to destabilize the kink/peeling modes by inducing a phase lock between ṽr and ξ̃r. More specifically, the ratio between them could be measured to quantify their relative competition strength. Consequently, the shape of Er is crucial to the shape of linear growth rate spectrum γ(n) (here n is the toroidal mode number), which further determines the nonlinear dynamics. On the one hand, relatively larger Er-curvature causes narrower γ(n), leading to larger nonlinear energy loss fraction. On the other hand, relatively larger Er-shear has the opposite effect.

The role of ETG modes in JET–ILW pedestals with varying levels of power and fuelling

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

Effect of triangularity on ion-temperature-gradient-driven turbulence

The linear and nonlinear properties of ion-temperature-gradient-driven turbulence with adiabatic electrons are modeled for axisymmetric configurations for a broad range of triangularities δ, both negative and positive. Peak linear growth rates decrease with negative δ but increase and shift toward a finite radial wavenumber kx with positive δ. The growth-rate spectrum broadens as a function of kx with negative δ and significantly narrows with positive δ. The effect of triangularity on linear instability properties can be explained through its impact on magnetic polarization and curvature. Nonlinear heat flux is weakly dependent on triangularity for |δ|≤0.5, decreasing significantly with extreme δ, regardless of sign. Zonal modes play an important role in nonlinear saturation in the configurations studied, and artificially suppressing zonal modes increased nonlinear heat flux by a factor of about four for negative δ, increasing with positive δ by almost a factor of 20. Proxies for zonal-flow damping and drive suggest that zonal flows are enhanced with increasing positive δ.

Non-universality of the mass function: dependence on the growth rate and power spectrum shape

ABSTRACT The abundance of dark matter haloes is one of the key probes of the growth of structure and expansion history of the Universe. Theoretical predictions for this quantity usually assume that, when expressed in a certain form, it depends only on the mass variance of the linear density field. However, cosmological simulations have revealed that this assumption breaks, leading to 10–20 per cent systematic effects. In this paper, we employ a specially designed suite of simulations to further investigate this problem. Specifically, we carry out cosmological N-body simulations where we systematically vary growth history at a fixed linear density field, or vary the power spectrum shape at a fixed growth history. We show that the halo mass function generically depends on these quantities, thus showing a clear signal of non-universality. Most of this effect can be traced back to the way in which the same linear fluctuation grows differently into the non-linear regime depending on details of its assembly history. With these results, we propose a parameterization with explicit dependence on the linear growth rate and power spectrum shape. Using an independent suite of simulations, we show that this fitting function accurately captures the mass function of haloes over cosmologies spanning a vast parameter space, including massive neutrinos and dynamical dark energy. Finally, we employ this tool to improve the accuracy of so-called cosmology-rescaling methods and show they can deliver 2 per cent accurate predictions for the halo mass function over the whole range of currently viable cosmologies.

Influence of plasma shaping effect on peeling-ballooning instability

The influence of plasma shaping effect on peeling-ballooning mode is studied linearly and nonlinearly using the three-field module of BOUT++. Without considering the variations of pressure pedestal and bootstrap current induced by shaping, our simulations demonstrate that shaping effect on magnetic surfaces also influences peeling-ballooning instability significantly. The shape of growth rate spectrum tends to shift to a lower toroidal mode number () with the increase of elongation (), and the growth rates become smaller for all modes with the increase of triangularity () in the linear simulation. Further analysis demonstrates that peeling-ballooning mode is stabilized mainly by the increased magnetic shear () for elongation and the decreased normalized pressure gradient () for triangularity. In the nonlinear simulation, the elongation can lead to a larger pedestal collapse while the triangularity plays an opposing role. That is because the dominance of low mode that arises with the increase of and decrease of can lead to a larger energy loss. The results indicate that the effect of triangularity on the magnetic surface is quite important since it leads to ELM mitigation without any variations of pressure pedetal and bootstrap current, which provide an innovative mechanism to explain the ELM mitigation phenomenon due to plasma shaping.

Predicting nonresonant pressure-driven MHD modes in equilibria with low magnetic shear

Nonresonant internal modes can be difficult to anticipate as there is no resonant surface in the plasma. However, equilibria that are unstable to multiple nonresonant magnetohydrodynamic (MHD) modes may be more prone to global loss of confinement since these instabilities generate spatially extended linear displacements, potentially enhancing magnetic field line chaos via nonlinear interactions. Here, we successfully predict the unstable nonresonant pressure-driven modes for equilibria with zero shear in the plasma core, irrational q on axis, and a central pressure gradient, which is consistent with pre-crash profiles in sawtoothing tokamak plasmas in the large-aspect-ratio limit. A criterion for identifying nonresonant modes most likely to be unstable is developed from the convergents of the continued fraction representation of q0. A higher-order analysis of the standard Energy Principle reveals the conditions under which these modes are expected to dominate. Linear growth rate spectra, as a function of toroidal mode number (up to n = 30), calculated using the initial-value extended-MHD code, M3D-C1, recover the characteristic dependence observed for ideal infernal modes. Nonresonant modes have also been invoked in some ideal sawtooth crash models. This work provides a mechanism to predict the mode numbers of infernal modes and, potentially, the width of some post-sawtooth-crash profiles.

Peeling-ballooning stability of tokamak plasmas with applied 3D magnetic fields

The poloidal harmonics of the toroidal normal modes of an unstable axisymmetric tokamak plasma are employed as basis functions for the minimisation of the 3D energy functional. This approach presents a natural extension of the perturbative method considered in Anastopoulos Tzanis et al (2019 Nucl. Fusion 59 126028). This variational formulation is applied to the stability of tokamak plasmas subject to external non-axisymmetric magnetic fields. A comparison of the variational and perturbative methods shows that for D-shaped, high βN plasmas, the coupling of normal modes becomes strong at experimentally relevant applied 3D fields, leading to violation of the assumptions that justify a perturbative analysis. The variational analysis employed here addresses strong coupling, minimising energy with respect to both toroidal and poloidal Fourier coefficients. In general, it is observed that ballooning unstable modes are further destabilised by the applied 3D fields and field-aligned localisation of the perturbation takes place, as local ballooning theory suggests. For D-shaped high βN plasmas, relevant to experimental cases, it is observed that the existence of intermediate n unstable peeling-ballooning modes, where a maximum in the growth rate spectrum typically occurs, leads to a destabilising synergistic coupling that strongly degrades the stability of the 3D system.

Stabilization of Short Wavelength Resistive Ballooning Modes by Ion-to-Electron Temperature and Gradient Ratios in Tokamak Edge Plasmas**Supported by the National Key R&D Program of China (Grant Nos. 2017YFE0300405 and 2017YFE0301200) and the National Natural Science Foundation of China (Grant Nos. 11775067, 11775069, 11875019, and 11805058).

We numerically investigate the effects of ion-to-electron temperature ratio Ti/Te and temperature gradient ratio ηi/ηe on resistive ballooning modes (RBMs) under tokamak edge plasma conditions. The results show that the growth rates of the RBMs exhibit the characteristic of a quite broad poloidal wavenumber spectrum in the cold ion limit. The growth rate spectrum becomes narrower and the peak of the spectrum shifts from the short to long wavelength side with increasing Ti/Te and ηi/ηe. The electron temperature gradient has a very weak effect on the stability of RBMs. However, the ion-to-electron temperature ratio and the temperature gradient ratio have strong stabilizing effects on short-wavelength RBMs, while they have relatively weak effects on long-wavelength RBMs.

Edge state selection by modulating E × B shearing profile in toroidally confined plasmas

We investigate how the E × B shearing profile impacts the energy transport induced by the MHD mode in the edge of H-mode plasmas. By enhancing the outer layer (ψnor ≈ 0.9–1) shear, the growth rate spectrum γ(n) gets narrowed with the fastest growth rate having a lower toroidal mode number. In the nonlinear stage, the MHD fluctuations tend to reach a coherent, oscillating state, which is analogous to a quiescent(Q) H mode state with edge-harmonic-oscillations. By enhancing the inner layer (ψnor ≈ 0.7–0.8) shear, γ(n) shifts to lower n, and its width does not change much. In the nonlinear stage, the edge fluctuations tend to stay in an incoherent, oscillating state, which is analogous to a turbulent QH-mode state. We also explore the underlying physics of the results and show that they are closely related to the different cross-phase dynamics of the edge MHD fluctuations induced by the variation of the edge E × B shearing profile.

Theory of the Drift-Wave Instability at Arbitrary Collisionality

A numerically efficient framework that takes into account the effect of the Coulomb collision operator at arbitrary collisionalities is introduced. Such a model is based on the expansion of the distribution function on a Hermite-Laguerre polynomial basis to study the effects of collisions on magnetized plasma instabilities at arbitrary mean-free path. Focusing on the drift-wave instability, we show that our framework allows retrieving established collisional and collisionless limits. At the intermediate collisionalities relevant for present and future magnetic nuclear fusion devices, deviations with respect to collision operators used in state-of-the-art turbulence simulation codes show the need for retaining the full Coulomb operator in order to obtain both the correct instability growth rate and eigenmode spectrum, which, for example, may significantly impact quantitative predictions of transport. The exponential convergence of the spectral representation that we propose makes the representation of the velocity space dependence, including the full collision operator, more efficient than standard finite difference methods.

Study on edge localized mode during plasma vertical swing in HL-2A tokamak

An experiment on edge localized modes (ELMs) during the vertical swing is performed in HL-2A tokamak. ELM mitigation is observed when the plasma moves upward. The plasma column expands accompanied by a decrease in both lower triangularity and the absolute value of local magnetic shear at the outer mid-plane during the upward moving. The critical pressure gradient in pedestal is reduced for the mitigating case according to the linear simulation based on peeling-ballooning theory. The physics of the ELM crash process is also explored through nonlinear simulation. For the mitigating case, small current drive for low-n modes and a weak ion diamagnetic effect on high-n modes lead to a higher dominant mode and a wider growth rate spectrum, which result in weak turbulence transport and a small ELM size.

Linear analyses of peeling-ballooning modes in high beta pedestal plasmas

We present the linear simulations of edge plasma instabilities using the 3-field peeling-ballooning model and gyro-Landau-fluid model under the BOUT++ framework. A series of realistic equilibria of shifted circular geometry are generated by a global equilibrium solver CORSICA, where the Shafranov shift, elongation effects, and bootstrap current are included. The linear growth rate spectrum of the peeling-ballooning modes is shown in a wide range of pressure gradient and parallel current density in the pedestal region. The results show that the bootstrap current stabilizes high beta ballooning modes. The simulations with different fractions of bootstrap current indicate a trend for the existence of the high beta peeling-ballooning mode stability region. Taking the kinetic effects into account, the linear simulations of kinetic peeling-ballooning mode using the gyro-Landau-fluid model show that this region can be accessed.

Non-ideal effects on ballooning mode stability in the presence of resonant magnetic perturbations

The ideal ballooning mode model in the presence of the externally applied resonant magnetic perturbation (RMP), developed by Bird and Hegna [Nucl. Fusion 53, 013004 (2013)], is extended to include the non-ideal effects, i.e., the finite resistivity and diamagnetic drift effects. Using the eigenvalue equation with the non-ideal effects, a comprehensive analysis is carried out to elucidate the impact of the RMP on ballooning mode stability. Finite resistivity is shown to reduce the line bending stabilization in low to intermediate toroidal mode number (n), resulting in the more peaked growth rate spectrum with respect to n. It is shown that the combination of finite resistivity and the RMP-induced local shear modulation have a strong influence on line bending stabilization, leading to an interesting threshold behavior of the ballooning mode stability. This signifies the importance of the RMP to the line bending stabilization coupled with finite resistivity, as well as the local shear modulation.

Impact of bootstrap current and Landau-fluid closure on ELM crashes and transport

Results presented here are from 6-field Landau-Fluid simulations using shifted circular cross-section tokamak equilibria on BOUT++ framework. Linear benchmark results imply that the collisional and collisionless Landau resonance closures make a little difference on linear growth rate spectra which are quite close to the results with the flux limited Spitzer-Härm parallel flux. Both linear and nonlinear simulations show that the plasma current profile plays dual roles on the peeling-ballooning modes that it can drive the low-n peeling modes and stabilize the high-n ballooning modes. For fixed total pressure and current, as the pedestal current decreases due to the bootstrap current which becomes smaller when the density (collisionality) increases, the operational point is shifted downwards vertically in the Jped – α diagram, resulting in threshold changes of different modes. The bootstrap current can slightly increase radial turbulence spreading range and enhance the energy and particle transports by increasing the perturbed amplitude and broadening cross-phase frequency distribution.

A comprehensive numerical study of immiscible and miscible viscous fingers during chemical enhanced oil recovery

Miscible and immiscible viscous fingering in porous media are undesirable for many engineering applications such as chemical enhanced oil recovery, contaminant transport, and liquid chromatography. For chemical enhanced oil recovery, polymer is injected to prevent immiscible viscous fingering between aqueous and oil phases. Polymer solution weakens the growth of immiscible viscous fingers but it may actuate miscible viscous fingering behind the polymer bank when polymer flooding is followed by water flooding. Due to the adsorption of polymer and presence of connate water, two saturation shocks are most commonly formed during polymer flooding. Therefore during typical polymer flooding two immiscible viscous fingering can occur around two saturation shocks. We have performed a Fourier analysis of the saturation and polymer concentration contours at water-oil interface and concentration front after the onset of viscous fingering to obtain the amplitude spectra for different instabilities. We also have calculated root mean square (RMS) of fingers and its variation with time. The Fourier analysis and calculation of RMS at different times are very useful to quantify the evolution of growth rate spectrum for immiscible and miscible instabilities. We have shown that due to adsorption the width of polymer bank decrease very fast for low injection concentration. In this case polymer bank disappear and polymer flooding becomes ineffective. But for high injection concentration the miscible viscous fingers grow very fast and reach the oil bank. This can affect the efficiency of oil displacement.

A model for generation of high wavenumber fluctuations by external magnetic field perturbations in edge pedestal plasmas

We study the impact of external magnetic perturbations on the stability of ballooning modes. A unique feature of our analysis is the two-step parametric process [Chaturvedi and Kaw, J. Geophys. Res. 81, 3257 (1976)], which enables us to calculate contributions from all the modes with high toroidal mode numbers. The analysis shows that the externally applied magnetic field perturbations can modify the linear dispersion characteristics of the ballooning mode. Specifically, the growth rate spectrum of the ballooning modes becomes broader in poloidal wavenumber (kθ) space, implying the generation of high-k fluctuations. The increase of high-k fluctuations (micro-turbulence) can lead to the mitigation of an edge localized mode crash by increasing turbulent transport in the pedestal. In addition to this, a new nonlinear instability is found even below the threshold of the ballooning mode instability when the amplitude of magnetic perturbation is sufficiently large (i.e., δB/B0≥1.0×10−4). A discussion is given of the implication of this new finding.

Excitation of THz hybrid modes in an elliptical dielectric rod waveguide with a cold collisionless unmagnetized plasma column by an annular electron beam

The dispersion relation of electromagnetic waves propagating in an elliptical plasma waveguide with a cold collisionless unmagnetized plasma column and a dielectric rod is studied analytically. The frequency spectrum of the hybrid waves and the growth rate for excitation of the waves by a thin annular relativistic elliptical electron beam (TAREEB) is obtained. The effects of relative permittivity constant of dielectric rod, geometrical dimensions, plasma frequency, accelerating voltage, and current density of TAREEB on the growth rate and frequency spectra of the waveguide will be investigated.

The role of zonal flows in the saturation of multi-scale gyrokinetic turbulence

The 2D spectrum of the saturated electric potential from gyrokinetic turbulence simulations that include both ion and electron scales (multi-scale) in axisymmetric tokamak geometry is analyzed. The paradigm that the turbulence is saturated when the zonal (axisymmetic) ExB flow shearing rate competes with linear growth is shown to not apply to the electron scale turbulence. Instead, it is the mixing rate by the zonal ExB velocity spectrum with the turbulent distribution function that competes with linear growth. A model of this mechanism is shown to be able to capture the suppression of electron-scale turbulence by ion-scale turbulence and the threshold for the increase in electron scale turbulence when the ion-scale turbulence is reduced. The model computes the strength of the zonal flow velocity and the saturated potential spectrum from the linear growth rate spectrum. The model for the saturated electric potential spectrum is applied to a quasilinear transport model and shown to accurately reproduce the electron and ion energy fluxes of the non-linear gyrokinetic multi-scale simulations. The zonal flow mixing saturation model is also shown to reproduce the non-linear upshift in the critical temperature gradient caused by zonal flows in ion-scale gyrokinetic simulations.