Dominant Saturation Mechanism Research Articles

Nonlinear dynamics of the reversed shear Alfvén eigenmode in burning plasmas

In a tokamak fusion reactor operated at steady state, the equilibrium magnetic field is likely to have reversed shear in the core region, as the noninductive bootstrap current profile generally peaks off-axis. The reversed shear Alfvén eigenmode (RSAE) as a unique branch of the shear Alfvén wave in this equilibrium, can exist with a broad spectrum in wavenumber and frequency, and be resonantly driven unstable by energetic particles (EP). After briefly discussing the RSAE linear properties in burning plasma condition, we review several key topics of the nonlinear dynamics for the RSAE through both wave-EP resonance and wave-wave coupling channels, and illustrate their potentially important role in reactor-scale fusion plasmas. By means of simplified hybrid MHD-kinetic simulations, the RSAEs are shown to have typically broad phase space resonance structure with both circulating and trapped EP, as results of weak/vanishing magnetic shear and relatively low frequency. Through the route of wave-EP nonlinearity, the dominant saturation mechanism is mainly due to the transported resonant EP radially decoupling with the localized RSAE mode structure, and the resultant EP transport generally has a convective feature. The saturated RSAEs also undergo various nonlinear couplings with other collective oscillations. Two typical routes as parametric decay and modulational instability are studied using nonlinear gyrokinetic theory, and applied to the scenario of spontaneous excitation by a finite amplitude pump RSAE. Multiple RSAEs could naturally couple and induce the spectral energy cascade into a low frequency Alfvénic mode, which may effectively transfer the EP energy to fuel ions via collisionless Landau damping. Moreover, zero frequency zonal field structure could be spontaneously excited by modulation of the pump RSAE envelope, and may also lead to saturation of the pump RSAE by both scattering into stable domain and local distortion of the continuum structure.

Stimulated Raman side and backscatter instabilities of crossed laser beams in plasma

Stimulated Raman scattering (SRS) instability due to two nonlinearly coupled laser beams propagating in homogeneous, thermal plasma is analyzed. The temporal evolution equations for pump, scattered and Langmuir waves excited due to the interaction are set up. Using numerical techniques temporal evolution of amplitudes of pump, scattered and resultant Langmuir waves for side and back SRS instabilities have been obtained in the non-relativistic regime. The temporal growth rate, saturation time and saturation amplitude of waves excited by two crossed (perpendicular) lasers are studied and compared with the single beam case. The model considers pump depletion to be the dominant saturation mechanism. The study points towards a new concept for generation of high amplitude Langmuir waves.

Global gyrokinetic simulation of edge coherent mode in EAST

Linear and nonlinear simulations are carried out for the edge coherent mode (ECM) using the global gyrokinetic code GEM based on the EAST experimental parameters. The linear simulation results show that ECM is an electrostatic mode with dominant toroidal mode number n = 18 and frequency about 48 kHz, and propagates along the direction of electron diamagnetic drift, which are consistent with the experimental results. In addition, the density and electron temperature gradients destabilize the mode, while the collision stabilizes the mode. The nonlinear simulation results show that the saturated particle and heat fluxes induced by ECM are mainly due to the perturbed electrostatic ExB drift, and the fluxes of electrons and ions are almost equal. The ECM drives significant outward particle and heat fluxes, thus greatly promoting the maintenance of the long pulse H-mode. The Fourier decomposition of fluxes and potentials demonstrate that the intermediate-n modes of n = 14, 18 grow fastest in the linear phase, while in the nonlinear saturation phase, the low-n modes such as n = 4, 6 dominate and the fluxes are mainly contributed by the mode of n = 10. It is found that zonal flow is not the dominant saturation mechanism of the turbulence. The inverse spectral cascade of turbulence is inevitably observed in the nonlinear saturation process, indicating that it is a more universal turbulence saturation mechanism. It is also found that radial electric field can greatly reduce the turbulence intensity and transport level. From the analyses of frequency and transport channels, it can be concluded that ECM appears to be the collisionless trapped electron mode.

Effects of equilibrium radial electric field on ion temperature gradient instability in the scrape-off layer of a field-reversed configuration

Linear and nonlinear effects of the equilibrium radial electric field on the ion temperature gradient (ITG) instability in the scrape-off layer (SOL) of a field-reversed configuration have been studied using gyrokinetic particle simulations for a single toroidal mode. Linear simulations with adiabatic electrons find that the flow shear reduces the growth rate and causes a radial tilting of the mode structure on the toroidal plane. Nonlinear simulations find that the flow shear significantly decreases ITG saturation amplitude and ion heat transport in the SOL by reducing both turbulence intensity and eddy size. The turbulence intensity is determined by fluid eddy rotation, which is the dominant saturation mechanism for the SOL ITG instability with a single toroidal mode number. On the other hand, parallel wave-particle decorrelation is the dominant mechanism for the SOL ITG turbulent transport. A random walk model using the guiding center radial excursion as the characteristic length scale and the eddy turnover time as the characteristic time scale fits very well to the scaling of ion heat conductivity with the flow shear.

Measurements of the Growth and Saturation of Electron Weibel Instability in Optical-Field Ionized Plasmas.

The temporal evolution of the magnetic field associated with electron thermal Weibel instability in optical-field ionized plasmas is measured using ultrashort (1.8ps), relativistic (45MeV) electron bunches from a linear accelerator. The self-generated magnetic fields are found to self-organize into a quasistatic structure consistent with a helicoid topology within a few picoseconds and such a structure lasts for tens of picoseconds in underdense plasmas. The measured growth rate agrees well with that predicted by the kinetic theory of plasmas taking into account collisions. Magnetic trapping is identified as the dominant saturation mechanism.

Global gyrokinetic particle simulations of microturbulence in W7-X and LHD stellarators

Global gyrokinetic particle simulations of electrostatic ion temperature gradient (ITG) instability show that the most unstable eigenmode is localized to some magnetic fieldlines or discrete locations on the poloidal plane in the Wendelstein 7-X (W7-X) stellarator due to its mirror-like magnetic fields, which vary strongly in the toroidal direction and induce coupling of more toroidal harmonics (n) to form the linear eigenmode than in the Large Helical Device (LHD) stellarator. Nonlinear electrostatic simulation results show that self-generated zonal flows are the dominant saturation mechanism for the ITG instabilities in both the LHD and W7-X. Furthermore, radial widths of the fluctuation intensity in both the LHD and W7-X are significantly broadened from the linear phase to the nonlinear phase due to turbulence spreading. Finally, nonlinear spectra in the W7-X are dominated by low-n harmonics, which can be generated both by nonlinear toroidal coupling of high-n harmonics and by linear toroidal coupling with large amplitude zonal flows due to the 3D equilibrium magnetic fields.

Measurement of Kinetic-Scale Current Filamentation Dynamics and Associated Magnetic Fields in Interpenetrating Plasmas.

We present the first local, quantitative measurements of ion current filamentation and magnetic field amplification in interpenetrating plasmas, characterizing the dynamics of the ion Weibel instability. The interaction of a pair of laser-generated, counterpropagating, collisionless, supersonic plasma flows is probed using optical Thomson scattering (TS). Analysis of the TS ion-feature revealed anticorrelated modulations in the density of the two ion streams at the spatial scale of the ion skin depth c/ω_{pi}=120 μm, and a correlated modulation in the plasma current. The inferred current profile implies a magnetic field amplitude ∼30±6 T, corresponding to ∼1% of the flow kinetic energy, indicating that magnetic trapping is the dominant saturation mechanism.

Anti-Langmuir decay instability in Langmuir decay instability cascade

Backward stimulated Raman scattering (BSRS) with Langmuir decay instability (LDI) and anti-Langmuir decay instability (ALDI or anti-LDI) has been researched by Vlasov simulation. The decay productions of anti-LDI in LDI cascade and their evolution with time are demonstrated for the first time. The BSRS reflectivity will be decreased largely through LDI cascade and ALDI in the small wave-number region. Different mechanisms to saturate BSRS in CH (or H) and C plasmas have been demonstrated. The dominant saturation mechanism of BSRS in CH (or H) plasmas is LDI cascade and ALDI. However, in C plasmas, due to very weak Landau damping of ion acoustic waves, LDI cascade will promote stimulated Brillouin scattering (SBS) excitation, then SBS will compete with BSRS and saturate BSRS in the later stage. The proportion of the hot electrons is decreased largely through LDI cascade and ALDI. These results give an effective mechanism to suppress BSRS and hot electron generation in the small wave-number region, which are of important significance in the inertial confinement fusion.

Theory of ITG turbulent saturation in stellarators: Identifying mechanisms to reduce turbulent transport

A three-field fluid model that allows for general three-dimensional equilibrium geometry is developed to describe ion temperature gradient turbulent saturation processes in stellarators. The theory relies on the paradigm of nonlinear transfer of energy from unstable to damped modes at comparable wavelength as the dominant saturation mechanism. The unstable-to-damped mode interaction is enabled by a third mode that for dominant energy transfer channels primarily serves as a regulator of the nonlinear energy transfer rate. The identity of the third wave in the interaction defines different scenarios for turbulent saturation with the dominant scenario depending upon the properties of the 3D geometry. The nonlinear energy transfer physics is quantified by the product of a turbulent correlation lifetime and a geometric coupling coefficient. The turbulent correlation time is determined by a three-wave frequency mismatch, which at long wavelength can be calculated from the sum of the linear eigenfrequencies of the three modes. Larger turbulent correlation times denote larger levels of nonlinear energy transfer and hence smaller turbulent transport. The theory provides an analytic prediction for how 3D shaping can be tuned to lower turbulent transport through saturation processes.

Phase conversion dissipation in multicomponent compact stars

We propose a mechanism for the damping of density oscillations in multicomponent compact stars. The mechanism is the periodic conversion between different phases, i.e., the movement of the interface between them, induced by pressure oscillations in the star. The damping grows nonlinearly with the amplitude of the oscillation. We study in detail the case of r-modes in a hybrid star with a sharp interface, and we find that this mechanism is powerful enough to saturate the r-mode at very low saturation amplitude, of order $10^{-10}$, and is therefore likely to be the dominant r-mode saturation mechanism in hybrid stars with a sharp interface.

Nonlinear evolution of the temperature gradient instability in the midlatitude ionosphere

AbstractThe midlatitude Super Dual Auroral Radar Network radars frequently observe quiet time decameter‐scale irregularities in the nightside subauroral ionosphere. Since opposed temperature and density gradients are a persistent feature in the vicinity of the ionospheric projection of the plasmapause, the temperature gradient instability (TGI) is proposed to explain the observed irregularities. In this paper, modeling of the TGI is extended into the kinetic regime appropriate for HF radar frequencies and analyzed as the cause of these irregularities. The nonlinear evolution of the TGI is investigated utilizing gyrokinetic particle‐in‐cell simulation techniques with Monte Carlo collisions for the first time. The purpose of this investigation is to identify the mechanism responsible for the nonlinear saturation as well as the associated anomalous transport. It is found that the saturation amplitude level and the associated diffusion are greatly enhanced as a result of electron collisions. The simulation results indicate that the nonlinear E×Bconvection (trapping) of the electrons is the dominant TGI saturation mechanism. The spatial power spectra of the electrostatic potential and density fluctuations associated with the TGI are also computed, and the results show wave cascading of TGI from kilometer scales into the decameter‐scale regime of the radar observations. This implies that the E region may be responsible for shorting out the F region TGI electric fields before and around sunset. This also suggests that the observed midlatitude decameter‐scale ionospheric irregularities may be produced directly by the TGI or by turbulent cascade from primary longer‐wavelength irregularity structures produced from this instability.

The reduced-description particle-in-cell model for the two plasmon decay instability

Recently, motivated by the resurgent interest in suprathermal electron generation by two plasmon decay (TPD) in direct-drive laser-fusion [J. A. Delettrez et al., Bull. Am. Phys. Soc. 53, 248 (2008); V. A. Smalyuk et al., Phys. Rev. Lett. 100, 185005 (2008); B. Yaakobi et al., Phys. Plasmas 12, 062703 (2005)], the fully kinetic, reduced-description, particle-in-cell (RPIC) methodology has been extended to include TPD. It provides a computationally efficient fully kinetic simulation tool, especially in nonlinear regimes where the Langmuir decay instability (LDI) is a dominant saturation mechanism. This RPIC methodology is an extension of the modeling of laser-plasma instabilities in underdense plasmas reported previously [H. X. Vu, B. Bezzerides, and D. F. DuBois, J. Comput. Phys. 156, 12 (1999)]. The relationship between RPIC and the extended Zakharov model previously used for TPD [D. F. DuBois, D. A. Russell, and H. A. Rose, Phys. Rev. Lett. 74, 3983 (1995)] is explored theoretically and tested in simulations. The modification of the electron velocity distribution—in particular, the generation of hot electrons—as calculated in RPIC leads to weakening of the wave turbulence excited by TPD compared to the Zakharov model predictions but the locations in wave vector space of important spectral features, e.g., arising from the LDI, of the nonlinear wave fluctuations are exactly the same in the two approaches. New results involving two oblique, overlapping laser beams, a common geometrical feature in direct-drive schemes, are presented. The two laser beams can cooperatively excite common primary Langmuir waves which initiate the LDI process.

Role of zonal flows in trapped electron mode turbulence through nonlinear gyrokinetic particle and continuum simulation

Trapped electron mode (TEM) turbulence exhibits a rich variety of collisional and zonal flow physics. This work explores the parametric variation of zonal flows and underlying mechanisms through a series of linear and nonlinear gyrokinetic simulations, using both particle-in-cell and continuum methods. A new stability diagram for electron modes is presented, identifying a critical boundary at ηe=1, separating long and short wavelength TEMs. A novel parity test is used to separate TEMs from electron temperature gradient driven modes. A nonlinear scan of ηe reveals fine scale structure for ηe≳1, consistent with linear expectation. For ηe<1, zonal flows are the dominant saturation mechanism, and TEM transport is insensitive to ηe. For ηe>1, zonal flows are weak, and TEM transport falls inversely with a power law in ηe. The role of zonal flows appears to be connected to linear stability properties. Particle and continuum methods are compared in detail over a range of ηe=d ln Te/d ln ne values from zero to five. Linear growth rate spectra, transport fluxes, fluctuation wavelength spectra, zonal flow shearing spectra, and correlation lengths and times are in close agreement. In addition to identifying the critical parameter ηe for TEM zonal flows, this paper takes a challenging step in code verification, directly comparing very different methods of simulating simultaneous kinetic electron and ion dynamics in TEM turbulence.

Numerical experiments on the drift wave–zonal flow paradigm for nonlinear saturation

This paper confirms that E×B shearing from toroidally symmetric (toroidal mode number n=0) “radial modes” provides the dominant nonlinear saturation mechanism for drift wave (n≠0) turbulence, which in turn nonlinearly drives the modes. In common usage, this is loosely referred to as the “drift wave–zonal flow paradigm” for nonlinear saturation despite the fact that radial modes have several components distinguished in this paper: a residual or zero mean frequency “zonal flow” part and an oscillatory “geodesic acoustic mode” (GAM) part. Linearly, the zonal flows (and GAMs) are weakly damped only by ion-ion collisions, while the GAMs are strongly Landau damped only at low safety factor q. At high q the Hinton–Rosenbluth residual flow from an impulse vanishes and only the weakly damped GAMs remain. With the linear physics and driving rates of the finite-n transport modes unchanged, this paper argues that GAMs are only somewhat less effective than the residual zonal flows in providing the nonlinear saturation, and in some cases E×B shearing from GAMs (or at least the GAM physics) appears to dominate: transport appears to be nearly linear in the GAM frequency. By deleting the drift wave–drift wave nonlinear coupling, it is found that drift wave–radial mode nonlinear coupling triads account for most of the nonlinear saturation. Furthermore, the E×B shear components of the radial modes nonlinearly stabilize the finite-n modes, while the diamagnetic components nonlinearly destabilize them. Finally, from wave number spectral contour plots of the time average nonlinear entropy transfer function (and rates), it is shown that the peak in entropy generation coincides with the peak in transport production, while entropy dissipation (like Landau damping) is spread equally over all n modes (including n=0). Most of these conclusions appear to hold about equally well for all types of drift wave turbulence.

Nonlinear saturation of mirror instability

Mechanism of the nonlinear saturation of compressible electromagnetic mirror instability is studied using the gyrokinetic particle simulation. Phase‐space particle trapping due to the mirror force is found to be the dominant saturation mechanism in the simulation of a single mirror mode with relatively weak drive. At the nonlinear saturation, the phase‐space island of the distribution function is formed. The oscillation frequency of the saturated perturbation amplitude is close to the bounce frequency of the trapped particles, which is comparable to the linear growth rate of the mirror mode. Scaling of the saturation amplitude is consistent with the onset of the particle trapping. With strong instability drive, relaxation toward marginal stability dominates the nonlinear saturation of the mirror instability. Phase‐space trapping, however, persists after the saturation and continues to regulate the nonlinear evolution of the mirror mode.

Nonlinear saturation of collisionless trapped electron mode turbulence: Zonal flows and zonal density

Gyrokinetic δf particle simulation is used to investigate the nonlinear saturation mechanisms in collisionless trapped electron mode (CTEM) turbulence. It is found that the importance of zonal flow is parameter-sensitive and is well characterized by the shearing rate formula. The effect of zonal flow is empirically found to be sensitive to temperature ratio, magnetic shear, and electron temperature gradient. For parameters where zonal flow is found to be unimportant, zonal density (purely radial density perturbations) is generated and expected to be the dominant saturation mechanism. A toroidal mode-coupling theory is presented that agrees with simulation in the initial nonlinear saturation phase. The mode-coupling theory predicts the nonlinear generation of the zonal density and the feedback and saturation of the linearly most unstable mode. Inverse energy cascade is also observed in CTEM turbulence simulations and is reported here.

Nonlinear Saturation of Trapped Electron Modes via Perpendicular Particle Diffusion

In magnetized fusion plasmas, trapped electron mode (TEM) turbulence constitutes, together with ion temperature gradient (ITG) turbulence, the dominant source of anomalous transport on ion scales. While ITG modes are known to saturate via nonlinear zonal flow generation, this mechanism is shown to be of little importance for TEM turbulence in the parameter regime explored here. Instead, a careful analysis of the statistical properties of the ExB nonlinearity in the context of gyrokinetic turbulence simulations reveals that perpendicular particle diffusion is the dominant saturation mechanism. These findings allow for the construction of a rather realistic quasilinear model of TEM induced transport.

Nonlinear damping of zonal modes in anisotropic weakly collisional trapped electron mode turbulence

Comprehensive spectral analysis of a fluid model for trapped electron mode (TEM) turbulence reveals that marginally stable zonal modes at infinitesimal amplitude become robustly damped at finite amplitude. Zonal-mode structure, anisotropy, excitation, and wave number spectra are shown to result from interaction of the zero-frequency drift wave with the density advection nonlinearity. Heuristic dimensional balances, closure theory, and simulations manifest the primacy of the interaction, and yield energy transfer rates, fluctuation levels, spectra and finite-amplitude-induced dissipation. Strong sensitivity to the zero-frequency wave induces a marked spectral energy-transfer anisotropy that preferentially drives zonal modes relative to nonzonal modes. Zonal-mode excitation is accompanied by the nonlinear excitation of a spectrum of damped eigenmodes. The mixing of unstable TEM eigenmodes with the damped spectrum subjects zonal modes to finite-amplitude-induced damping. The combination of anisotropic transfer to zonal wave numbers and their nonlinear damping is shown to make this the dominant saturation mechanism for TEM turbulence.

Physics of saturation of collisionless tearing mode as a function of guide field

The tearing instability has been the subject of much study in space and laboratory plasmas and is thought to play a role as an onset mechanism for reconnection at the magnetotail and at the magnetopause. However, the nonlinear evolution is not completely understood. Here we employ a combination of theory and full particle simulations, including one for realistic mass ratio, to address the saturation of a single, linearly unstable tearing mode as a function of guide field. For the parameter regime considered, we find no agreement with any of the previous theories of tearing mode saturation. There are marked differences in the electron behavior and the associated saturation mechanism in the antiparallel (zero guide field) and finite guide field cases. Electrons become nongyrotropic and also develop temperature anisotropies in the antiparallel case, whereas they remain isotropic in the presence of a guide field. In the antiparallel limit there are two competing saturation mechanisms. One is due to preferential parallel heating of electrons which leads to a saturation amplitude of ws ∼ 2.9 ρe, where ρe is the electron gyroradius in the asymptotic magnetic field and ws is the half‐width of the magnetic island. The other saturation mechanism is electron trapping in the magnetic island which leads to a saturation amplitude comparable to the singular layer thickness (ΔNS). Electron parallel heating is usually the dominant saturation mechanism. However, in the presence of sufficiently fast electron pitch angle scattering, caused either by current‐aligned Weibel instability or turbulence, the second saturation mechanism due to trapping becomes dominant. In the presence of a guide field, the only saturation mechanism is electron trapping which leads to a saturation amplitude of ws ∼ 1.8 ρeG in the strong guide field and ws ∼ (2ρeGΔNS)1/2 in the intermediate regime. Here ρeG is the electron gyroradius in the guide field. The predicted saturation amplitudes for all cases, including the antiparallel regime, are much smaller than the thickness of the magnetopause current layer. Thus a single tearing mode saturates at too small of an amplitude to be of relevance to reconnection in the magnetosphere.

Growth and saturation of toroidal Alfvén eigenmode modes destabilized by ion cyclotron range of frequency produced tails

The linear growth rates of TAE (toroidal Alfvén eigenmode) modes destabilized by ICRF (ion cyclotron range of frequency) heating are calculated over a range of plasma parameters. Nonlinear saturation of a single unstable mode is investigated both analytically and numerically when wave–particle trapping is the dominant saturation mechanism. A numerical code has been developed based on a reduced resonance description of the wave–particle interaction (using a Hamiltonian formalism). A delta-f algorithm was incorporated to allow a low-noise description of mode evolution with particle sources and sinks present. The numerically observed saturation amplitudes correlate well with theoretical predictions to within 20%. Self-excited frequency sweeping resulting from the excitation of many simultaneous wave–particle resonances at different energies is demonstrated and explained as an extension of previous published theory [Berk et al., Phys. Lett. A 234, 213 (1997)].