Unstable Linear Modes Research Articles

Electron-temperature-gradient-driven ion-scale turbulence in high-performance scenarios in Wendelstein 7-X

Through intercode gyrokinetic numerical simulations, we predict that, for the conditions met during improved performances in the stellarator Wendelstein 7-X, turbulent transport can be dominated by electron-temperature-gradient-driven ion-scale electrostatic turbulence. We find that previously numerically observed large density-gradient-driven turbulence reductions must be attributed to the artificial suppression of the electron temperature gradient. Instead, when electrons have a finite temperature gradient, we observe a moderate turbulence suppression whose quantitative comparison with experimental findings remains challenging. In such partial suppression, the nonlinear dynamics of zonal flows plays a pivotal role as opposed to the underlying most unstable linear modes. Published by the American Physical Society 2024

Linear and nonlinear stability of Rayleigh–Bénard convection with zero-mean modulated heat flux

Linear and nonlinear stability analyses are performed to determine critical Rayleigh numbers ( ${Ra}_{cr}$ ) for a Rayleigh–Bénard convection configuration with an imposed bottom boundary heat flux that varies harmonically in time with zero mean. The ${Ra}_{cr}$ value depends on the non-dimensional frequency $\omega$ of the boundary heat-flux modulation. Floquet theory is used to find ${Ra}_{cr}$ for linear stability, and the energy method is used to find ${Ra}_{cr}$ for two different types of nonlinear stability: strong and asymptotic. The most unstable linear mode alternates between synchronous and subharmonic frequencies at low $\omega$ , with only the latter at large $\omega$ . For a given frequency, the linear stability ${Ra}_{cr}$ is generally higher than the nonlinear stability ${Ra}_{cr}$ , as expected. For large $\omega$ , ${Ra}_{cr} \omega ^{-2}$ approaches an $O(10)$ constant for linear stability but zero for nonlinear stability. Hence the domain for subcritical instability becomes increasingly large with increasing $\omega$ . The same conclusion is reached for decreasing Prandtl number. Changing temperature and/or velocity boundary conditions at the modulated or non-modulated plate leads to the same conclusions. These stability results are confirmed by selected direct numerical simulations of the initial value problem.

Linear instability and nonlinear flow states in a horizontal pipe flow under bottom heating and transverse magnetic field

Linear stabilities of the liquid metal mixed convection in a horizontal pipe under bottom heating and transverse magnetic field are studied through linear global stability analyses. Three branches of the linear stability boundary curves are determined by the eigenvalue computation of the most unstable modes. One branch is located in the region of large Hartmann number and determined by the linear unstable mode which was first revealed by the numerical simulations of Zikanov, Listratov & Sviridov (J. Fluid Mech., vol. 720, 2013, pp. 486–516). This branch curve shows that the global unstable mode exists above a threshold of Hartmann number, which agrees with the experiment of Genin et al. (Temperature fluctuations in a heated horizontal tube affected by transverse magnetic field. In Proc. 8th PAMIR Conf. Fund. Appl. MHD, Borgo, Corsica, France, pp. 37–41, 2011). The other two branch curves determined by two different long-wave unstable modes intersect with each other in the region of small Hartmann number. The critical Grashof number on these two curves increases exponentially with the increase of the Hartmann number. Through energy budget analyses at the critical thresholds of these unstable modes, it is found that, for the unstable mode at large Hartmann numbers, buoyancy is the dominant destabilizing term which demonstrates the hypothetical explanation of Zikanov et al. (2013) who regard natural convection as a destabilization mechanism. It is further revealed that, with respect to the unstable modes on the critical stability curves of small Hartmann numbers, the dominant destabilization comes from the streamwise shear of the basic flow. Finally, within the linear unstable region, fully developed nonlinear flow states of the mixed convection are investigated by direct numerical simulations (DNS) with several sets of selected dimensionless parameters. The spatio-temporal structures of these nonlinear flow states are discussed in detail with comparison with the linear unstable global modes.

Impact of pedestal density gradient and collisionality on ELM dynamics

BOUT++ turbulence simulations are conducted to capture the underlying physics of small ELM characteristics achieved by increasing separatrix density via controlling strike points from vertical to horizontal divertor plates for three EAST discharges. BOUT++ linear simulations show that the most unstable modes change from high-n ideal ballooning modes to intermediate-n peeling–ballooning modes and eventually to peeling–ballooning stable plasmas in the pedestal. Nonlinear simulations show that the fluctuation is saturated at a high level for the lowest separatrix density. The ELM size decreases with increasing separatrix density, until the fraction of this energy lost during the ELM crash becomes less than 1% of the pedestal stored energy, leading to small ELMs. Simulations indicate that small ELMs can be triggered either by the marginally peeling–ballooning instability near the peak pressure gradient position inside the pedestal or by a local instability in the pedestal foot with a larger separatrix density gradient. The pedestal collisionality scan for type-I ELMs with steep pedestal density gradient shows that both linear growth rate and ELM size decrease with increasing collisionality. On the contrary, the pedestal collisionality and pedestal density width scan with a weak pedestal density gradient indicate small ELMs can either be triggered by a high-n ballooning mode or by a low-n peeling mode in a low collisionality region 0.04–0.1. The simulations indicate the weaker the linear unstable modes near marginal stability with small linear growth rate, the lower nonlinearly saturated fluctuation intensity and the smaller turbulence spreading from the linear unstable zone to stable zone in the nonlinear saturation phase, leading to small ELMs.

Dynamic behavior analysis of a mechanical system with two unstable modes coupled to a single nonlinear energy sink

This paper investigates a problem of passive mitigation of vibratory instabilities caused by two unstable modes by means of a single nonlinear energy sink (NES). For this purpose, a linear four-degree-of-freedom (DOF) primary structure having two unstable modes (reproducing the typical dynamic behavior of a friction system) and undergoing, as it is linear, unbounded motions when it is unstable, is coupled to a NES. In this work, the NES involves an essentially cubic restoring force and a linear damping force. We are interested in characterizing analytically the response regimes resulting from the coupling of the two unstable linear modes of the primary structure and the nonlinear mode of the NES. To this end, from a suitable rescaling of the governing equations of the coupled system in which the dynamics of the primary structure is reduced to its unstable modal coordinates, the complexification-averaging method is applied. The resulting averaged system appears to be a fast-slow system with four fast variables and two slow ones related to the two unstable modes of the primary structure. The critical manifold of the averaged dynamics is obtained through the geometric singular perturbation theory and appears as a two-dimensional parametric surface (with respect to two of the four fast variables) which evolves in the whole six-dimensional variable space. The asymptotic analysis reveals that the NES attachment can produce some bounded responses and suggests that the system may have simultaneous stable attractors. Numerical simulations complement the study, highlighting a possible competition between stable attractors and allowing us to investigate their basins of attraction. In each considered situation, a good agreement has been observed between theoretical results and numerical simulations, which validates the proposed asymptotic analysis.

Development of linear unstable modes in supersonic streamwise vortices using a weighted compact nonlinear scheme

Growth processes in supersonic streamwise vortices with linear unstable modes at Mach numbers 2.5 and 5.0 were numerically investigated using a weighted compact nonlinear scheme (WCNS) with three different accuracies. In the evolution of the inviscid linear unstable mode m=−6 as a high-wavenumber mode, the growth rate and eigenfunction profiles of the mode numerically resolved were generally consistent with those obtained with the linear stability theory (LST) during the early transition stage, regardless of the computational accuracy. The numerical scheme used here can capture the growth of three-dimensional linear disturbance in structure. However, the numerical results obtained by nonlinear developments differed according to the accuracy. Among the different accuracies under the same grid resolution, the ninth-order accuracy scheme for the interpolation of primitive variables was able to capture small vortical structures at the downstream in the supersonic flow. In addition, the negative circulation generated and the total disturbance energy indicated that such high accuracy is effective in resolving the developed flow in supersonic vortices, even at moderate grid resolutions.

Unstable modes of oscillations of a tangential discontinuity in partly ionized plasma with charge exchange: long waves

An analytical study of linear unstable modes of oscillations of a tangential velocity discontinuity in partly ionized space plasma with charge exchange is carried out.The long-wave approximation is considered. In this approximation, the scale of perturbations is much larger than a certain characteristic size, which is determined by the charge-exchange cross-section and number density of neutral atoms. It is shown that along with the classical Kelvin – Helmholtz instability, there exists an unstable mode whose increment does not depend explicitly on the relative velocity of the two media separated by the discontinuity. The presence of this mode is entirely determined by the presence of the charge-exchange process.

A Linear Response Framework for Radiative‐Convective Instability

AbstractRadiative‐convective equilibrium is a simple paradigm for the tropical climate, in which radiative cooling balances convective heating in the absence of lateral energy transport. Recent studies have shown that a large‐scale circulation may spontaneously develop from radiative‐convective equilibrium through the interactions among water vapor, radiation, and convection. This potential instability, referred to as radiative‐convective instability, may be posed as a linear stability problem for the water vapor profile by combining a linear response framework with the weak temperature gradient approximation. We design two analytic models of convective linear response to moisture perturbations, which are similar to Betts‐Miller and bulk‐plume convection schemes. We combine these convective responses with either clear‐sky gray or real‐gas radiative responses. In all cases, despite consistent radiative feedbacks, the characteristics of convection dominate the vertical structure of the most unstable linear mode of water vapor perturbations. For Betts‐Miller convection, the stability critically depend on a key parameter: the heating to advection of moisture conversion rate (HAM); warmer atmospheres with higher HAM exhibit more linear instability. In contrast, bulk‐plume convection is stable across temperatures but becomes linearly unstable with a moisture mode peaking in the midtroposphere once combined to radiation, with approximate growth rates of 10 days.

Investigation of the role of pedestal pressure and collisionality on type-I ELM divertor heat loads in DIII-D

A non-dimensional collisionality scan conducted on DIII-D confirms a model for ELM energy densities recently put forward by Eich (2017 Nucl. Mater. Energy 12 84–90), but also reveals key effects that may explain the large scatter typically observed about the scaling. Electron cyclotron heating (ECH) close to the plasma edge was used to raise electron temperatures at the pedestal top and lower collisionality to ITER level, while the power of neutral beam injection (NBI) was decreased during discharges to operate closer to the L–H transition threshold. The scan reveals no explicit pedestal pressure dependence of the ELM energy densities. While collisionality does not play a decisive role, the ratio of heating power to the H-mode threshold power is identified as parameter determining the agreement with the model, with discharges marginally above the threshold showing the largest scatter in the database and exceeding the predicted ELM energy up to twofold. Operation close to the L–H-threshold is accompanied by low ELM frequency and large ELM heat loads. Using linear stability calculations, ELM energy densities are shown to scale inversely with to the most unstable linear mode number before the ELM crash. There are indications that the scatter in the data when compared with the Eich model prediction is caused by including only a limited set out of all quantities considered by linear stability analysis. While further ELM studies near the LH threshold are of great priority, the overall agreement of DIII-D with the Eich model recommends its use in extrapolations towards ITER.

Role of turbulence regime on determining the local density gradient

In this paper we show that the local density gradient in the plasma core depends on the calculated mode-frequency of the most unstable linear mode and reaches a maximum when this frequency is close to zero. Previous theoretical and experimental work on AUG has shown that the ratio of electron to ion temperature, and as such the frequency of the dominant linear gyrokinetic mode, affects the local density gradient close to (Fable et al 2010 Plasma Phys. Control. Fusion 52 015007, Angioni et al 2011 Nucl. Fusion 51 023006). On DIII-D we find that by adding electron cyclotron heating, we modify the dominant unstable linear gyrokinetic mode from an ion temperature gradient (ITG) mode to a trapped electron mode (TEM), which means that the frequency of the dominant mode changes sign (from the ion to the electron direction). Local density peaking around mid-radius increases by 50% right around the cross-over between the ITG and TEM regimes. By comparing how the particle flux changes, through the derivative of the electron density, ne, with respect to time, , we find that the particle flux also exhibits the same trend versus mode frequency. As a result, we find that the changes in local particle transport are inversely proportional to the changes in electron density, indicating that the changes are driven by a change in thermo-diffusive pinch.

The nonlinear Schrödinger equation with generalized nonlinearities and PT-symmetric potentials: Stable solitons, interactions, and excitations.

We investigate the nonlinear Schrödinger (NLS) equation with generalized nonlinearities and complex non-Hermitian potentials and present the novel parity-time-( PT-) symmetric potentials for the NLS equation with power-law nonlinearities supporting some bright solitons. For distinct types of PT-symmetric potentials including Scarf-II, Hermite-Gaussian, and asymptotically periodic potentials, we, respectively, explore the phase transitions for the linear Hamiltonian operators. Moreover, we analytically find stable bright solitons in the generalized NLS equations with several types of PT-symmetric potentials, and their stability is corroborated by the linear stability spectrum and direct wave-propagation simulations. Interactions of two solitons are also explored. More interestingly, we find that the nonlinearity can excite the unstable linear modes (i.e., possessing broken linear PT-symmetric phase) to stable nonlinear modes. The results may excite potential applications in nonlinear optics, Bose-Einstein condensates, and relevant fields.

Linear Stability Prediction of Vortex Structures on High Pressure Turbine Blades

Velocity profiles are extracted from time- and span-averaged direct numerical simulation data, describing the flow over a high-pressure turbine vane linear cascade near engine-scale conditions with reduced inlet disturbance levels. Based on these velocity profiles, local as well as non-local linear stability analysis of the boundary-layer over the suction side of the vane is carried out in order to characterise a linearly unstable region close to the trailing edge. The largest growth rates are found for oblique modes, but those are only slightly more unstable than 2D modes, which describe the locations and frequencies of most unstable modes very well. The frequencies of the most unstable linear modes predict with good accuracy the predominant frequencies found in the direct numerical simulations (DNS) close to the trailing edge.

Thermocapillary modulation of self-rewetting films

Whereas surface tension decreases linearly with temperature for most fluids, here we consider those that exhibit a well-defined minimum. Specifically, our study is motivated by dilute aqueous mixtures of long-chain alcohols, for which surface tension is typically assumed to be a quadratic function of temperature. Utilization of these so-called ‘self-rewetting fluids’ has grown significantly in the past decade, due to observations that heat transfer is enhanced in applications such as film boiling and pulsating heat pipes. With similar applications in mind, we investigate the dynamics of a thin film with quadratic surface tension which is subjected to a temperature modulation in the bounding gas. A model is developed within the framework of the long-wave approximation, and a time-averaged thermocapillary driving force for destabilization is uncovered that results from the nonlinear surface tension. Linear stability analysis of the nonlinear partial differential equation for the film thickness is used to determine the critical conditions at which this driving force destabilizes the film and numerical investigation of this evolution equation reveals that linearly unstable perturbations saturate to regular periodic solutions (when the modulational frequency is set properly). Properties of these flows such as bifurcation at critical points and long-domain flows, where multiple unstable linear modes interact, are also discussed.

Gyrokinetic study of turbulent convection of heavy impurities in tokamak plasmas at comparable ion and electron heat fluxes

In tokamaks, the role of turbulent transport of heavy impurities, relative to that of neoclassical transport, increases with increasing size of the plasma, as clarified by means of general scalings, which use the ITER standard scenario parameters as reference, and by actual results from a selection of discharges from ASDEX Upgrade and JET. This motivates the theoretical investigation of the properties of the turbulent convection of heavy impurities by nonlinear gyrokinetic simulations in the experimentally relevant conditions of comparable ion and electron heat fluxes. These conditions also correspond to an intermediate regime between dominant ion temperature gradient turbulence and trapped electron mode turbulence. At moderate plasma toroidal rotation, the turbulent convection of heavy impurities, computed with nonlinear gyrokinetic simulations, is found to be directed outward, in contrast to that obtained by quasi-linear calculations based on the most unstable linear mode, which is directed inward. In this mixed turbulence regime, with comparable electron and ion heat fluxes, the nonlinear results of the impurity transport can be explained by the coexistence of both ion temperature gradient and trapped electron modes in the turbulent state, both contributing to the turbulent convection and diffusion of the impurity. The impact of toroidal rotation on the turbulent convection is also clarified.

On stable solitons and interactions of the generalized Gross-Pitaevskii equation with PT- and non-PT-symmetric potentials

We report the bright solitons of the generalized Gross-Pitaevskii (GP) equation with some types of physically relevant parity-time- (PT-) and non-PT-symmetric potentials. We find that the constant momentum coefficient Γ can modulate the linear stability and complicated transverse power-flows (not always from the gain toward loss) of nonlinear modes. However, the varying momentum coefficient Γ(x) can modulate both unbroken linear PT-symmetric phases and stability of nonlinear modes. Particularly, the nonlinearity can excite the unstable linear mode (i.e., broken linear PT-symmetric phase) to stable nonlinear modes. Moreover, we also find stable bright solitons in the presence of non-PT-symmetric harmonic-Gaussian potential. The interactions of two bright solitons are also illustrated in PT-symmetric potentials. Finally, we consider nonlinear modes and transverse power-flows in the three-dimensional (3D) GP equation with the generalized PT-symmetric Scarff-II potential.

Hydromagnetic Stability of a Thin Viscoelastic Magnetic Fluid on Coating Flow Using Landau Equation

AbstractThis paper investigates the weakly nonlinear stability of a thin axisymmetric viscoelastic fluid with hydromagnetic effects on coating flow. The governing equation is resolved using long-wave perturbation method as part of an initial value problem for spatial periodic surface waves with the Walter's liquid B type fluid. The most unstable linear mode of a film flow is determined by Ginzburg-Landau equation (GLE). The coefficients of the GLE are calculated numerically from the solution of the corresponding stability problem on coating flow. The effect of a viscoelastic fluid under an applied magnetic field on the nonlinear stability mechanism is studied in terms of the rotation number, Ro, viscoelastic parameter, k, and the Hartmann constant, m. Modeling results indicate that the Ro, k and m parameters strongly affect the film flow. Enhancing the magnetic effects is found to stabilize the film flow when the viscoelastic parameter destabilizes the one in a thin viscoelastic fluid.

Compressibility effects on the structural evolution of transitional high-speed planar wakes

The compressibility effects on the structural evolution of the transitional high-speed planar wake are studied. The relative Mach number ($Ma_{r}$) of the laminar base flow modifies two fundamental features of planar wake transition: (i) the characteristic length scale defined by the most unstable linear mode; and (ii) the domain of influence of the structures within the staggered two-dimensional vortex array. Linear stability results reveal a reduced growth (approximately 30 % reduction up to$Ma_{r}=2.0$) and a quasilinear increase of the wavelength of the most unstable, two-dimensional instability mode (approximately 20 % longer over the same$Ma_{r}$range) with increasing$Ma$. The primary wavelength defines the length scale imposed on the emerging transitional structures; naturally, a longer wavelength results in rollers with a greater streamwise separation and hence also larger circulation. A reduction of the growth rate and an increase of the principal wavelength results in a greater ellipticity of the emerging rollers. Compressibility effects also modify the domain of influence of the transitional structures through an increased cross-wake and inhibited streamwise communication as characteristic paths between rollers are deflected due to local$Ma$ gradients. The reduced streamwise domain of influence impedes roller pairing and, for a sufficiently large relative$Ma$, pairing is completely suppressed. Thus, we observe an increased two-dimensionality with increasing Mach number: directly contrasting the increasing three-dimensional effects in high-speed mixing layers. Temporally evolving direct numerical simulations conducted at$Ma=0.8$and 2.0, for Reynolds numbers up to 3000, support the physical insight gained from linear stability and vortex dynamics studies.

Orientation of X lines in asymmetric magnetic reconnection—Mass ratio dependency

AbstractUsing fully kinetic simulations, we study the X line orientation of magnetic reconnection in an asymmetric configuration. A spatially localized perturbation is employed to induce a single X line, which has sufficient freedom to choose its orientation in three‐dimensional systems. The effect of ion to electron mass ratio is investigated, and the X line appears to bisect the magnetic shear angle across the current sheet in the large mass ratio limit. The orientation can generally be deduced by scanning through the corresponding 2‐D simulations to find the reconnection plane that maximizes the peak reconnection electric field. The deviation from the bisection angle in the lower mass ratio limit is consistent with the orientation shift of the most unstable linear tearing mode in an electron‐scale current sheet.

Multi-dimensional Vlasov simulations and modeling of trapped-electron-driven filamentation of electron plasma wavesa)

Kinetic simulations of two-dimensional finite-amplitude electron plasma waves are performed in a one-wavelength long system. A systematic study of the most unstable linear sideband mode, in particular its growth rate γ and wavenumber ky, is carried out by scanning the amplitude and wavenumber of the initial wave. Simulation results are compared with numerical and analytical solutions to a two-dimensional nonlinear Schrodinger model [H. A. Rose and L. Yin, Phys. Plasmas 15, 042311 (2008)] and to the reduced model by Kruer et al. [Phys. Rev. Lett. 23, 838 (1969)] generalized to two dimensions.

Subcritical versus supercritical transition to turbulence in curved pipes

Transition to turbulence in straight pipes occurs in spite of the linear stability of the laminar Hagen–Poiseuille flow if both the amplitude of flow perturbations and the Reynolds number $\mathit{Re}$ exceed a minimum threshold (subcritical transition). As the pipe curvature increases, centrifugal effects become important, modifying the basic flow as well as the most unstable linear modes. If the curvature (tube-to-coiling diameter $d/D$) is sufficiently large, a Hopf bifurcation (supercritical instability) is encountered before turbulence can be excited (subcritical instability). We trace the instability thresholds in the $\mathit{Re}-d/D$ parameter space in the range $0.01\leqslant d/D\leqslant 0.1$ by means of laser-Doppler velocimetry and determine the point where the subcritical and supercritical instabilities meet. Two different experimental set-ups are used: a closed system where the pipe forms an axisymmetric torus and an open system employing a helical pipe. Implications for the measurement of friction factors in curved pipes are discussed.