Stable Eigenmodes Research Articles

Real time detection of multiple stable MHD eigenmode growth rates towards kink/tearing modes avoidance in DIII-D tokamak plasmas

Real time detection of time evolving growth rates of multiple stable magnetohydrodynamic (MHD) eigenmodes has been achieved in DIII-D tokamak experiments via multi-mode three-dimensional (3D) active MHD spectroscopy. The measured evolution of the multi-modes’ growth rates is in good accordance with the variation of the plasma β N . Using experimental equilibria, resistive MARS-F simulations found the two least stable modes to have comparable growth rates to those experimentally measured. Real time and offline calculations of the modes’ growth rates show comparable results and indicate that cleaner system input and output signals will improve the accuracy of the real time stability detection. Moreover, the shortest real time updating time window of multi-mode eigenvalues can be about 2 ms in DIII-D experiments. This real time monitoring of stable, macroscopic kink and tearing modes thus provides an effective tool for avoidance of the most common causes of tokamak disruption.

Three-dimensional shear-flow instability saturation via stable modes

Turbulence in three dimensions (3D) supports vortex stretching that has long been known to accomplish energy transfer to small scales. Moreover, net energy transfer from large-scale, forced, unstable flow-gradients to smaller scales is achieved by gradient-flattening instability. Despite such enforcement of energy transfer to small scales, it is shown here that the shear-flow-instability-supplied 3D-fluctuation energy is largely inverse-transferred from the fluctuation to the mean-flow gradient, and such inverse transfer is more efficient for turbulent fluctuations in 3D than in two dimensions (2D). The transfer is due to linearly stable eigenmodes that are excited nonlinearly. The stable modes, thus, reduce both the nonlinear energy cascade to small scales and the viscous dissipation rate. The vortex-tube stretching is also suppressed. Up-gradient momentum transport by the stable modes counters the instability-driven down-gradient transport, which also is more effective in 3D than in 2D (≈70% vs ≈50%). From unstable modes, these stable modes nonlinearly receive energy via zero-frequency fluctuations that vary only in the direction orthogonal to the plane of 2D shear flow. The more widely occurring 3D turbulence is thus inherently different from the commonly studied 2D turbulence, despite both saturating via stable modes.

Linear stability analysis of wake vortices by a spectral method using mapped Legendre functions

A spectral method using associated Legendre functions with algebraic mapping is developed for a linear stability analysis of wake vortices. These functions serve as Galerkin basis functions, capturing correct analyticity and boundary conditions for vortices in an unbounded domain. The incompressible Euler or Navier–Stokes equations linearised on a swirling flow are transformed into a standard matrix eigenvalue problem of toroidal and poloidal streamfunctions, solving perturbation velocity eigenmodes with their complex growth rate as eigenvalues. This reduces the problem size for computation and distributes collocation points adjustably clustered around the vortex core. Based on this method, strong swirling$q$vortices with linear perturbation wavenumbers of order unity are examined. Without viscosity, neutrally stable eigenmodes associated with the continuous eigenvalue spectrum having critical-layer singularities are successfully resolved. The inviscid critical-layer eigenmodes numerically tend to appear in pairs, implying their singular degeneracy. With viscosity, the spectra pertaining to physical regularisation of critical layers stretch out toward an area, referring to potential eigenmodes with wavepackets found by Mao & Sherwin (J. Fluid Mech., vol. 681, 2011, pp. 1–23). However, the potential eigenmodes exhibit no spatial similarity to the inviscid critical-layer eigenmodes, doubting that they truly represent the viscous remnants of the inviscid critical-layer eigenmodes. Instead, two distinct continuous curves in the numerical spectra are identified for the first time, named the viscous critical-layer spectrum, where the similarity is noticeable. Moreover, the viscous critical-layer eigenmodes are resolved in conformity with the$Re^{-1/3}$scaling law. The onset of the two curves is believed to be caused by viscosity breaking the singular degeneracy.

Near-cancellation of up- and down-gradient momentum transport in forced magnetized shear-flow turbulence

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

Hydroacoustic interaction between draft tube and penstock eigenmodes under Francis turbine full load instability

At high load, Francis Turbines may experience self-sustained pressure surge leading to significant power swing and pressure fluctuations along the waterway. The physical mechanism initiating this instability phenomenon has been the subject of much research. The development of the axisymmetric cavitating vortex rope at the runner outlet modifies the hydroacoustic properties of the draft tube waterway. Very low wave speed due to high cavitation volume combined with a high swirling number initiates the unstable axial pulsations of the cavitating vortex rope which frequency corresponds to a penstock’s eigenfrequency. The 15 MW power plant of Monceaux-la-Virole in France, composed of two units fed by a single penstock, experiences such full-load surge. On-site tests have been carried out to analyze the envelope of pressure fluctuations along the penstock once instability occurs. Combined with a 1D SIMSEN model of the power plant, these measurements have allowed to enhance the understanding of this instability phenomenon. To achieve this, an advanced draft tube modelling taking into account distributed wave speed, convective terms and divergent geometry is used and frequency analysis is carried out. Unstable draft tube eigenmodes and stable penstock eigenmodes are predicted. The key draft tube model parameters such as wave speed and second viscosity are calibrated to set the draft tube eigenmode frequency to the unstable measured frequency for different operating points. This frequency analysis concludes that high load instability occurs when a matching between the draft tube and the penstock eigenfrequencies is experienced. Moreover, it is shown that the unstable draft tube eigenmode is able to interact with different order penstock eigenmodes as function of the operating point of the unit.

The onset of instability in a hydromagnetic channel flow of Casson fluid: the accurate solutions

We study the temporal stability of linear two-dimensional disturbances of plane Poiseuille flow of a Casson fluid in a rigid parallel channel under the influence of a uniform magnetic field. When the disturbance is taken in normal mode it transforms the perturbed equations to a fourth-order eigenvalue problem which is then solved by the spectral method. We first obtain sufficient conditions for the hydrodynamic stability in the channel by analyzing the nature of the growth rate and the Reynolds number. For the chosen set of parameters, the eigenspectra of the flow exhibit the familiar Y-shaped structure with three distinct modes depending on the phase speed approach. The eigenfunctions corresponding to the least stable and most unstable eigenmodes have the symmetric and antisymmetric modes. For a certain range of Casson parameter β and the magnetic number M, instability always appears on the wall mode for which the eigenfunction is antisymmetric about the channel center-line. The neutral stability curves that correspond to various values of β and M are found to be an extension of the continuation of the Tollmien-Schlichting instability that is noticed for a Newtonian channel flow. The regions of stability are enlarged for small β and large M.

Mechanism for sequestering magnetic energy at large scales in shear-flow turbulence

Straining of magnetic fields by large-scale shear flow, which is generally assumed to lead to intensification and generation of small scales, is reexamined in light of the persistent observation of large-scale magnetic fields in astrophysics. It is shown that, in magnetohydrodynamic turbulence, unstable shear flows have the unexpected effect of sequestering magnetic energy at large scales due to counteracting straining motion of nonlinearly excited large-scale stable eigenmodes. This effect is quantified via dissipation rates, energy transfer rates, and visualizations of magnetic field evolution by artificially removing the stable modes. These analyses show that predictions based upon physics of the linear instability alone miss substantial dynamics, including those of magnetic fluctuations.

Resolvent-based modelling of coherent structures in a turbulent jet flame using a passive flame approach

Motivated by the recent success in finding coherent structures in turbulent flows and describing their noise emission using the Resolvent Analysis (RA), the method is applied to a turbulent jet flame at Re=15,000 and a corresponding non-reacting flow to investigate their axisymmetric dynamics. The RA, which in this study assumes a passive flame and neglects compressibility effects, is based on the governing equations linearized around the temporal mean state. It is validated against Spectral Proper Orthogonal Decomposition (SPOD) obtained from time-resolved snapshots. Both the temporal mean state and the snapshots are obtained by Large Eddy Simulation (LES). The SPOD reveals that an axisymmetric, convective Kelvin–Helmholtz (KH) instability is the dominant hydrodynamic mechanism in the jet flame within a narrow frequency band and incorporates up to 40% of the turbulent kinetic energy. Results show that the RA is capable of reproducing the mode shapes seen in the SPOD. The RA furthermore allows to address the origin of these hydrodynamic structures: While the corresponding KH mode in a non-reacting turbulent jet flow is most sensitive to perturbations in the nozzle boundary layer, the same dominant mode in the turbulent Bunsen flame is most receptive to perturbations in the region between the nozzle edge and the annular pilot burner. The results suggest that the strong density gradients in this region initiate perturbations in the baroclinic torque, which are feeding the KH mode. Finally, a linear stability analysis proves that the high sensitivity of the KH structure is due to resonance with stable linear eigenmodes, which explains its high energy content. By applying the RA to a turbulent reacting flow, this study opens up a new pathway in analyzing the role of hydrodynamic structures in reacting flows.

Influence of freestream turbulence on the flow over a wall roughness

We demonstrate that free-stream turbulence has a strong impact on the dynamics of flow past a cylindrical roughness element. Previous works have shown that, depending on operating conditions, the leading flow instability can either be varicose (symmetric) or sinuous (antisymmetric). In both cases, when the flow is excited by broadband frequency forcing, dynamic mode decomposition extracts only varicose coherent structures, even though optimal response analysis predicts strong amplification of sinuous disturbances with frequency near the marginally stable sinuous eigenmode. We show that this is due to the flow essentially being an amplifier of varicose perturbations rather than a resonator.

Predicting the critical gradient of ITG turbulence in fusion plasmas

The quasilinear mixing-length approach to efficient prediction of transport in fusion devices is improved to account for the ‘Dimits’ upshift between linear and nonlinear critical pressure gradients in zonal-flow-saturated turbulence regimes. This modification uses the frequency mismatch between modes interacting turbulently to track changes in saturation efficiency. Near criticality, energy is transferred exclusively to stable eigenmodes, rapidly increasing the efficacy of the nonlinearity. The modified quasilinear model is able to predict below-threshold turbulent ion-temperature-gradient-driven transport accurately and also yields significantly improved predictions for trapped-electron-mode transport.

A categorization of marginally stable thermoacoustic modes based on phasor diagrams

Phasor diagrams are utilized to plot and analyze thermoacoustic modes in a duct with ideal closed/open acoustic boundary conditions that contains a velocity-sensitive flame. Inspection of phasor diagrams that represent fluctuations of velocity, pressure, heat release rate and characteristic wave amplitudes at the flame elicits characteristic features of marginally stable, intrinsic thermoacoustic (ITA) modes: the sign of velocity fluctuations and the sign of the gradient of pressure fluctuations change across the flame. These sign changes result from a reversal of direction of the velocity phasor across the flame, affected by unsteady heat release exactly out-of-phase with respect to upstream velocity fluctuations and of sufficient strength. Unlike alternative methods proposed for the identification of ITA modes, the proposed categorization does not involve a parameter sweep, but relies on an analogy with the structure of ITA modes in an anechoic environment. The phasor diagram also elucidates that continuous transitions from acoustic to ITA modes and vice versa can be achieved by an increase or decrease of the gain of the flame transfer function. The representation of acoustic wave propagation in terms of phasors facilitates the formulation of a compact dispersion relation of the thermoacoustic configuration under consideration. The value of the transitional gain of the flame transfer function, where acoustic/ITA transitions occur, may be deduced easily from the dispersion relation. A plot of solution branches of acoustic and ITA modes illustrates how the respective mode transitions, and how they can be related to the well-known quarter wave modes of a closed/open resonator. It is observed that under variation of transfer function gain and phase, marginally stable eigenmodes may occur at almost any frequency, not only at frequencies that correspond to the quarter-wave modes.

Removing instabilities in the hierarchical equations of motion: Exact and approximate projection approaches.

The hierarchical equations of motion (HEOM) provide a numerically exact approach for computing the reduced dynamics of a quantum system linearly coupled to a bath. We have found that HEOM contains temperature-dependent instabilities that grow exponentially in time. In the case of continuous-bath models, these instabilities may be delayed to later times by increasing the hierarchy dimension; however, for systems coupled to discrete, nondispersive modes, increasing the hierarchy dimension does little to alleviate the problem. We show that these instabilities can also be removed completely at a potentially much lower cost via projection onto the space of stable eigenmodes; furthermore, we find that for discrete-bath models at zero temperature, the remaining projected dynamics computed with few hierarchy levels are essentially identical to the exact dynamics that otherwise might require an intractably large number of hierarchy levels for convergence. Recognizing that computation of the eigenmodes might be prohibitive, e.g., for large or strongly coupled models, we present a Prony filtration algorithm that may be useful as an alternative for accomplishing this projection when diagonalization is too costly. We present results demonstrating the efficacy of HEOM projected via diagonalization and Prony filtration. We also discuss issues associated with the non-normality of HEOM.

Stellarator microinstabilities and turbulence at low magnetic shear

Gyrokinetic simulations of drift waves in low-magnetic-shear stellarators reveal that simulation domains comprised of multiple turns can be required to properly resolve critical mode structures important in saturation dynamics. Marginally stable eigenmodes important in saturation of ion temperature gradient modes and trapped electron modes in the Helically Symmetric Experiment (HSX) stellarator are observed to have two scales, with the envelope scale determined by the properties of the local magnetic shear and an inner scale determined by the interplay between the local shear and magnetic field-line curvature. Properly resolving these modes removes spurious growth rates that arise for extended modes in zero-magnetic-shear approximations, enabling use of a zero-magnetic-shear technique with smaller simulation domains and attendant cost savings. Analysis of subdominant modes in trapped electron mode (TEM)-driven turbulence reveals that the extended marginally stable modes play an important role in the nonlinear dynamics, and suggests that the properties induced by low magnetic shear may be exploited to provide another route for turbulence saturation.

Global stability analysis of axisymmetric boundary layer over a circular cylinder

This paper presents a linear global stability analysis of the incompressible axisymmetric boundary layer on a circular cylinder. The base flow is parallel to the axis of the cylinder at inflow boundary. The pressure gradient is zero in the streamwise direction. The base flow velocity profile is fully non-parallel and non-similar in nature. The boundary layer grows continuously in the spatial directions. Linearized Navier–Stokes (LNS) equations are derived for the disturbance flow quantities in the cylindrical polar coordinates. The LNS equations along with homogeneous boundary conditions forms a generalized eigenvalues problem. Since the base flow is axisymmetric, the disturbances are periodic in azimuthal direction. Chebyshev spectral collocation method and Arnoldi’s iterative algorithm is used for the solution of the general eigenvalues problem. The global temporal modes are computed for the range of Reynolds numbers and different azimuthal wave numbers. The largest imaginary part of the computed eigenmodes is negative, and hence, the flow is temporally stable. The spatial structure of the eigenmodes shows that the disturbance amplitudes grow in size and magnitude while they are moving towards downstream. The global modes of axisymmetric boundary layer are more stable than that of 2D flat-plate boundary layer at low Reynolds number. However, at higher Reynolds number they approach 2D flat-plate boundary layer. Thus, the damping effect of transverse curvature is significant at low Reynolds number. The wave-like nature of the disturbance amplitudes is found in the streamwise direction for the least stable eigenmodes.

The spectrum of multi-region-relaxed magnetohydrodynamic modes in topologically toroidal geometry

A general formulation of the problem of calculating the spectrum of stable and unstable eigenmodes of linearized perturbations about a magnetically confined toroidal plasma is presented. The analysis is based on a new hydromagnetic dynamical model, multi-region relaxed magnetohydrodynamics, which models the plasma magnetic-field system as consisting of multiple regions, containing compressible Euler fluid and Taylor-relaxed magnetic field, separated by interfaces in the form of flexible MHD current sheets. This is illustrated using a first-principles analysis of a two-region slab geometry, with periodic boundary conditions to model the outer regions of typical tokamak or reversed-field pinch plasmas. The lowest and second-lowest eigenvalues in plasmas unstable to tearing and kink-tearing modes are calculated. Near-marginal stability, the lowest mode obtained using the incompressible approximation to the kinetic energy normalization of the present study is shown to correspond to the eigenvalues found in previous studies where all mass was artificially loaded onto the interfaces.

On the role of corotation radius in the lowT/Wdynamical instability of differentially rotating stars

We investigate the nature of so-called low $T/W$ dynamical instability in a differentially rotating star by focusing on the role played by the corotation radius of the unstable oscillation modes. An one dimensional model of linear perturbation, which neglects dependence of variables on the coordinate along the rotational axis of the star, is solved to obtain stable and unstable eigenmodes. A linear eigenmode having a corotation radius, at which azimuthal pattern speed of the mode coincides with the stellar angular velocity, is categorized to either a complex (growing or damping) mode or a purely real mode belonging to a continuous spectrum of frequency. We compute canonical angular momentum and its flux to study eigenmodes with corotation radius. In a dynamically unstable mode, sound wave transports its angular momentum in such a way that the absolute value of the angular momentum is increased on both sides of the corotation radius. We further evaluate growth of amplitude of reflected sound wave incident to a corotation point and find that the over-reflection of the wave and the trapping of it between the corotation radius and the surface of the star may qualitatively explain dependences of eigenfrequencies on the stellar differential rotation. The results suggest that the low $T/W$ instability may be caused by over-reflection of sound waves trapped mainly between the surface of the star and a corotation radius.

Self-similar slip instability on interfaces with rate- and state-dependent friction.

We examine the development of a frictional instability, with diverging sliding rate, at the interface of elastic bodies in contact. Evolution of friction is determined by a slip rate and state dependence. Following Viesca (2016 Phys. Rev. E93, 060202(R). (doi:10.1103/PhysRevE.93.060202)), we show through an appropriate change of variable, the existence of blow-up solutions that are fixed points of a dynamical system. The solutions show self-similarity of the simple variety: separable dependence of time and space. For an interface with uniform frictional properties, there is a single-problem parameter. We examine the linear stability of these fixed points, as this problem parameter is varied. Specifically, we consider two archetypical elastic settings of the slip surface, in which interactions between points on the surface are either local or non-local. We show that, independent of the nature of elastic interactions, the fixed-points lose stability in the same matter as the parameter is increased towards a limit value: an apparently infinite sequence of Hopf bifurcations. However, for any value of the parameter, the nonlinear development of the instability is attraction, if not asymptotic convergence, towards these fixed points, owing to the existence of stable eigenmodes. For comparison, we perform numerical solutions of the original evolution equations and find precise agreement with the results of the analysis.

Dispersive-to-nondispersive transition and phase-velocity transient for linear waves in plane wake and channel flows.

In this study we analyze the phase and group velocity of three-dimensional linear traveling waves in two sheared flows: the plane channel and the wake flows. This was carried out by varying the wave number over a large interval of values at a given Reynolds number inside the ranges 20-100, 1000-8000, for the wake and channel flow, respectively. Evidence is given about the possible presence of both dispersive and nondispersive effects which are associated with the long and short ranges of wavelength. We solved the Orr-Sommerfeld and Squire eigenvalue problem and observed the least stable mode. It is evident that, at low wave numbers, the least stable eigenmodes in the left branch of the spectrum behave in a dispersive manner. By contrast, if the wave number is above a specific threshold, a sharp dispersive-to-nondispersive transition can be observed. Beyond this transition, the dominant mode belongs to the right branch of the spectrum. The transient behavior of the phase velocity of small three-dimensional traveling waves was also considered. Having chosen the initial conditions, we then show that the shape of the transient highly depends on the transition wavelength threshold value. We show that the phase velocity can oscillate with a frequency which is equal to the frequency width of the eigenvalue spectrum. Furthermore, evidence of intermediate self-similarity is given for the perturbation field.

Non-actively controlled double-inverted-pendulum-like dynamics can minimize center of mass acceleration during human quiet standing.

Multiple joint movements during human quiet standing exhibit characteristic inter-joint coordination, shortly referred to as reciprocal relationship, in which angular acceleration of the hip joint is linearly and negatively correlated with that of the ankle joint (antiphase coordination) and, moreover, acceleration of the center of mass (CoM) of the double-inverted-pendulum (DIP) model of the human body is close to zero constantly. A question considered in this study is whether the reciprocal relationship is established by active neural control of the posture, or rather it is a biomechanical consequence of non-actively controlled body dynamics. To answer this question, we consider a DIP model of quiet standing, and show that the reciprocal relationship always holds by Newton's second law applied to the DIP model with human anthropometric dimensions, regardless of passive and active joint torque patterns acting on the ankle and hip joints. We then show that characteristic frequencies included in experimental sway trajectories with the reciprocal relationship match with harmonics of the eigenfrequency of the stable antiphase eigenmode of the non-actively controlled DIP-like unstable body dynamics. The results suggest that non-actively controlled DIP-like mechanical dynamics is a major cause of the minimization of the CoM acceleration during quiet standing, which is consistent with a type of control strategy that allows switching off active neural control intermittently for suitable periods of time during quiet standing.

Experimental tests of linear and nonlinear three-dimensional equilibrium models in DIII-D

DIII-D experiments using new detailed magnetic diagnostics show that linear, ideal magnetohydrodynamics (MHD) theory quantitatively describes the magnetic structure (as measured externally) of three-dimensional (3D) equilibria resulting from applied fields with toroidal mode number n = 1, while a nonlinear solution to ideal MHD force balance, using the VMEC code, requires the inclusion of n ≥ 1 to achieve similar agreement. These tests are carried out near ITER baseline parameters, providing a validated basis on which to exploit 3D fields for plasma control development. Scans of the applied poloidal spectrum and edge safety factor confirm that low-pressure, n = 1 non-axisymmetric tokamak equilibria are determined by a single, dominant, stable eigenmode. However, at higher beta, near the ideal kink mode stability limit in the absence of a conducting wall, the qualitative features of the 3D structure are observed to vary in a way that is not captured by ideal MHD.