Fluctuation Level Research Articles

In-vivo range verification analysis with in-beam PET data for patients treated with proton therapy at CNAO.

Morphological changes that may arise through a treatment course are probably one of the most significant sources of range uncertainty in proton therapy. Non-invasive in-vivo treatment monitoring is useful to increase treatment quality. The INSIDE in-beam Positron Emission Tomography (PET) scanner performs in-vivo range monitoring in proton and carbon therapy treatments at the National Center of Oncological Hadrontherapy (CNAO). It is currently in a clinical trial (ID: NCT03662373) and has acquired in-beam PET data during the treatment of various patients. In this work we analyze the in-beam PET (IB-PET) data of eight patients treated with proton therapy at CNAO. The goal of the analysis is twofold. First, we assess the level of experimental fluctuations in inter-fractional range differences (sensitivity) of the INSIDE PET system by studying patients without morphological changes. Second, we use the obtained results to see whether we can observe anomalously large range variations in patients where morphological changes have occurred. The sensitivity of the INSIDE IB-PET scanner was quantified as the standard deviation of the range difference distributions observed for six patients that did not show morphological changes. Inter-fractional range variations with respect to a reference distribution were estimated using the Most-Likely-Shift (MLS) method. To establish the efficacy of this method, we made a comparison with the Beam’s Eye View (BEV) method. For patients showing no morphological changes in the control CT the average range variation standard deviation was found to be 2.5 mm with the MLS method and 2.3 mm with the BEV method. On the other hand, for patients where some small anatomical changes occurred, we found larger standard deviation values. In these patients we evaluated where anomalous range differences were found and compared them with the CT. We found that the identified regions were mostly in agreement with the morphological changes seen in the CT scan.

Internal noise measures in coarse and fine motion direction discrimination tasks and the correlation with autism traits.

Motion perception is essential for visual guidance of behavior and is known to be limited by both internal additive noise (i.e., a constant level of random fluctuations in neural activity independent of the stimulus) and motion pooling (global integration of local motion signals across space). People with autism spectrum disorder (ASD) display abnormalities in motion processing, which have been linked to both elevated noise and abnormal pooling. However, to date, the impact of a third limit—induced internal noise (internal noise that scales up with increases in external stimulus noise)—has not been investigated in motion perception of any group. Here, we describe an extension on the double-pass paradigm to quantify additive noise and induced noise in a motion paradigm. We also introduce a new way to experimentally estimate motion pooling. We measured the impact of induced noise on direction discrimination, which we ascribe to fluctuations in decision-related variables. Our results are suggestive of higher internal noise in individuals with high ASD traits only on coarse but not fine motion direction discrimination tasks. However, we report no significant correlations between autism traits and additive noise, induced noise, or motion pooling in either task. We conclude that, under some conditions, the internal noise may be higher in individuals with pronounced ASD traits and that the assessment of induced internal noise is a useful way of exploring decision-related limits on motion perception, irrespective of ASD traits.

Large Eddy Simulation of trenched cylindrical film hole with backward compound angles

Many previous studies have demonstrated that both transverse trench and backward cooling injection are capable of improving jet lift-off problem of traditional cylindrical film hole at high blowing ratios. However, few studies have attempted to combine the above two cooling schemes to further alleviate the cooling deterioration phenomenon. This paper proposed two trenched film holes with different backward compound angles of β = 180° and 135°, and used highly-resolved Large Eddy Simulation method to explore in detail their unsteady flow fields including vortex structures, level of velocity fluctuation, turbulent shear stress, distributions of vorticity and adiabatic wall cooling effectiveness. The results are compared against those of the benchmark trenched film hole with β = 0°. The simulations were carried out under blowing ratio of 1.5, and the coolant to mainstream density ratio of 2.0.The detrimental symmetric counter-rotating vortex pair shown in the trench with β = 0° hole is replaced by an asymmetric vortex pair in the trench with β = 180° hole and only a single asymmetric vortex in the trench with β = 135° hole. Inside the trench, hairpin-like vortices are the dominant vortex structures for the trench with backward film holes. In the region downstream of the trench, the trench with β = 180° hole obtains the highest level of three velocity components fluctuations in the central region because of the direct intense collision between the coolant and mainstream, while the trench with β = 135° hole shows a highest level of lateral velocity component fluctuation in the lateral sides and a highest level of turbulent shear stress in the near wall region due to the significantly enhanced lateral motion of the coolant. The trenches with β = 180° and 135° holes significantly improve the lateral uniformity of the coolant, and respectively provide constantly high levels of 0.21 and 0.3 in laterally averaged cooling effectiveness on the wall downstream of the trench, compared to the trench with β = 0° hole with a low level of less than 0.2.

Free‐Electron Tomography of Few‐Cycle Optical Waveforms

AbstractUltrashort light pulses are ubiquitous in modern research but the electromagnetic field of the optical cycles is usually not easy to obtain as a function of time. Field‐resolved pulse characterization requires either a nonlinear‐optical process or auxiliary sampling pulses that are shorter than the waveform under investigation and pulse metrology without at least one of these two prerequisites is often thought to be impossible. Here is reported how the optical field cycles of laser pulses can be characterized with field‐linear sensitivity and no ultrashort probe events. A free‐space electron beam is let to cross with the waveform of interest. Randomly arriving electrons interact by means of their elementary charge with the optical waveform in a linear‐optical way and reveal the optical cycles as the turning points in a time‐integrated deflection histogram. The sensitivity is only limited by the electron‐beam emittance and can reach the level of thermal radiation and vacuum fluctuations. Besides overturning a common belief in optical pulse metrology, the idea also provides practical perspectives for in situ characterization and optimization of optical waveforms in higher‐harmonics experiments, ultrafast transmission electron microscopes, laser‐driven particle accelerators, free‐electron lasers, or generally any experiments with waveform‐controlled laser pulses in a vacuum environment.

Flow transition on the suction surface of a controlled-diffusion compressor blade using a large-eddy simulation

Separation-induced transition is discussed here on a suction surface of a controlled-diffusion compressor blade using a large eddy simulation, where the Reynolds number based on the chord is 210 000. The filtered, incompressible Navier–Stokes and energy equations in the covariant form are solved with second-order spatial and temporal accuracy, where the subgrid stress tensor and temperature flux are assessed by a dynamic model. Flow features are resolved with appreciable accuracy, exhibiting a separation bubble on the suction surface in the vicinity of mid-chord. Excitation of the shear layer is evident with the evolution of Kelvin–Helmholtz (K–H) rolls, depicting amplification of the selective frequency in the first half of the bubble, where the normalized shedding frequency based on the momentum thickness at the point of separation becomes 0.011. The secondary instability appears in the second half of the bubble, which is attributed to the spanwise deformation of K–H rolls. This leads to significant growth of perturbations in the braid region, resulting in breakdown near reattachment. In brief, outer shear layer activities are initiated via inviscid instability, while the near-wall region might be susceptible to the viscous effect in the second half with increasing levels of velocity fluctuations, production, and wall-normal turbulent heat flux. The validity of the universal intermittency curve also bears evidence of a significant viscous effect. The instantaneous temperature contours closely follow the vorticity field, illustrating a strong correlation between species and momentum transport.

Climatology of Equatorial Plasma Bubbles in Ionospheric Connection Explorer/Far-UltraViolet (ICON/FUV) Limb Images

The Far-UltraViolet (FUV) imager onboard the Ionospheric Connection Explorer (ICON) spacecraft provides two-dimensional limb images of oxygen airglow in the nightside low-latitude ionosphere that are used to determine the oxygen ion density. As yet, no FUV limb imager has been used for climatological analyses of Equatorial Plasma Bubbles (EPBs). To examine the potential of ICON/FUV for this purpose, we statistically investigate small-scale (~180 km) fluctuations of oxygen ion density in its limb images. The seasonal-longitudinal variations of the fluctuation level reasonably conform to the EPB statistics in existing literature. To further validate the ICON/FUV data quality, we also inspect climatology of the ambient (unfiltered) nightside oxygen ion density. The ambient density exhibits (1) the well-known zonal wavenumber-4 signatures in the Equatorial Ionization Anomaly (EIA) and (2) off-equatorial enhancement above the Caribbean, both of which agree with previous studies. Merits of ICON/FUV observations over other conventional data sets are discussed in this paper. Furthermore, we suggest possible directions of future work, e.g., synergy between ICON/FUV and the Global-scale Observations of the Limb and Disk (GOLD) mission.

Stochastic Thermodynamics of an Electromagnetic Energy Harvester.

We study the power extracted by an electromagnetic energy harvester driven by broadband vibrations. We describe the system with a linear model, featuring an underdamped stochastic differential equation for an effective mass in a harmonic potential, coupled electromechanically with the current in the circuit. We compare the characteristic curve (power vs. load resistance) obtained in experiments for several values of the vibration amplitude with the analytical results computed from the model. Then, we focus on a more refined analysis, taking into account the temporal correlations of the current signal and the fluctuations of the extracted power over finite times. We find a very good agreement between the analytical predictions and the experimental data, showing that the linear model with effective parameters can describe the real system, even at the fine level of fluctuations. Our results could be useful in the framework of stochastic thermodynamics applied to energy harvesting systems.

Three-dimensional dynamics of fluctuations appearing during pellet ablation process around a pellet in a fusion plasma experiment

Understanding pellet ablation physics is crucial to realizing efficient fueling into a high temperature plasma for the steady state operation of ITER and future fusion reactors. Here we report the first observation of the formation of fluctuation structures in the pellet plasmoid during the pellet ablation process by a fast camera in a medium-sized fusion device, Heliotron J. The fluctuation has a normalized fluctuation level of ~ 15% and propagates around the moving pellet across the magnetic field. By comparing the fluctuation structures with the shape of magnetic field lines calculated with the field line tracing code, we successfully reconstruct the spatio-temporal structure of the fluctuations during the pellet ablation process. The fluctuations are located at the locations displaced toroidally from the pellet and propagate in the cross-field direction around the pellet axis along the field line, indicating a three-dimensional behavior and structure of fluctuations. The fluctuation would be driven by a strong inhomogeneity formed around the pellet and invoke the relaxation of the gradient through a cross-field transport induced by the fluctuations, which could affect the pellet ablation and pellet fueling processes. Such fluctuations can be ubiquitously present at the inhomogeneity formed around a pellet in the pellet ablation process in fusion devices.

New insights on divertor parallel flows,E × B drifts, and fluctuations from in situ, two-dimensional probe measurement in the Tokamak à Configuration Variable

In situ, two-dimensional (2D) Langmuir probe measurements across a large part of the TCV outer divertor are reported in L-mode discharges with and without divertor baffles. This provides detailed insights into time averaged profiles, particle fluxes, and fluctuation behavior in different divertor regimes. The presence of the baffles is shown to substantially increase the divertor neutral pressure for a given upstream density and to facilitate the access to detachment, an effect that increases with plasma current. The detailed, 2D probe measurements allow for a divertor particle balance, including ion flux contributions from parallel flows and E × B drifts. The poloidal flux contribution from the latter is often comparable or even larger than the former, and the divertor parallel flow direction reverses in some conditions, pointing away from the target. In most conditions, the integrated particle flux at the outer target can be predominantly ascribed to ionization along the outer divertor leg, consistent with a closed-box approximation of the divertor. The exception is a strongly detached divertor, achieved here only with baffles, where the total poloidal ion flux even decreases towards the outer target, indicative of significant plasma recombination. The most striking observation from relative density fluctuation measurements along the outer divertor leg is the transition from poloidally uniform fluctuation levels in attached conditions to fluctuations strongly peaking near the X-point when approaching detachment.

Role of secondary long wavelength fluctuation in nonlinear saturation of the kinetic ballooning mode in tokamak plasmas

Nonlinear saturation dynamics of electromagnetic turbulence and associated transport are investigated using a global simulation based on the Landau-fluid model in the finite β tokamak plasmas. The focus is on the kinetic ballooning mode (KBM), while a comparison to the β stabilized ion temperature gradient mode is carried out. Results show that the KBM turbulence creates relatively weaker zonal flows in finite β plasmas. Zonal current could be formed around the low order rational surfaces, but is too narrow and localized to affect the global transport level. It is found that the KBM turbulence is nonlinearly saturated in two sequential stages. The linear KBM instability is first saturated transiently at a low fluctuation level by weak zonal flows. Afterward, robust, linearly stable long wavelength fluctuations are nonlinearly excited and then interact feedback with primary unstable KBM components through the modulation process. As a result, the KBM is finally saturated with a down-shifted wavenumber spectrum. The suppression of turbulent transport by long wavelength fluctuations is identified as mainly resulting from the reduction of KBM turbulence intensity.

How do small changes enable the shift to net-zero? a techno-environmental-economic analysis

With many of the world’s governments committing to achieve net-zero greenhouse gas (GHG) emissions by mid-century, with well-defined milestones along the road, it is important to investigate how each sector can contribute towards achieving this global goal. The manufacturing sector, with its energy-intensive processes, large amounts of wastes, and hazardous and harmful emissions, is one of the main contributors to global GHG emissions, as well as other sustainability aspects, and, thus, it has great potential to contribute substantially to achieve net-zero objectives. This paper presents a techno-environmental-economic analysis of technologies that can play a key, enabling and leading role in the quest towards net-zero. Such technologies typically bring modest improvement in the environmental performance; however, the aim of this paper is to demonstrate how such small changes, when implemented in an industrial setting, can contribute significantly to the collective improvement in the environmental performance. In order to put the potential improvements into perspective, a real case study from the UK aerospace manufacturing sector is conducted. In the case study, metrics measuring potential improvements from the installation of a low-to-medium waste heat recovery system, and the upgrade of electric motors in the shopfloor to more energy efficient ones, are calculated through environmental and economic models. The models are then subject to a series of sensitivity analyses experiments to help understand the impact of different sources of uncertainty on the perceived GHG emissions, and economic and energy savings. The techno-environmental-economic analysis results revealed that these small changes, when implemented in an industrial setting, can indeed bring valuable improvements in the environmental performance of a manufacturing institute. Further, the sensitivity analysis experiments demonstrated how the environmental and economic performances are not adversely affected by different levels of fluctuations in key, likely to fluctuate, input parameters.

MARKET INTEGRATION AND PRICE TRANSMISSION OF CAYENNE PEPPER IN INDONESIA

Cayenne pepper is a horticultural commodity that its production to be seasonal so cause the supply of cayenne pepper to be unavailable in sufficient quantities throughout the year and the price to be fluctuative. Based on this, this study aims to 1) analyze price behavior and the level of price fluctuations of cayenne pepper using the coefficient of variation 2) analyze market integration using Johansen Cointegration Test, and 3) analyze price transmission using the VAR (Vector Auto Regression) / VECM (Vector Error Correction Model). The results show that 1) The price of cayenne pepper is high when production decreases (December-February and July-August), and low when production is abundant (March-June and September-November). The price of cayenne pepper at the consumer and producer levels has a high and unstable level of fluctuation with a coefficient of variation (KV) respectively of 23.9% and 19.7%, 2) The trace statistic value (106.41) > critical value (16.36) so that the producer and consumer markets of cayenne pepper in Indonesia have been vertically integrated, 3) Price transmission is imperfect. In the event of a price shock, at the producer and consumer level, price adjustments will be made towards long-term equilibrium respectively of -0.103 and -0.279.

On Modeling of Spray G ECN Using ROI-Based Eulerian-Lagrangian Simulation

A numerical study has been carried out to understand the effects of Unsteady Reynolds Averaged Navier-Stokes (standard k ― ε and RNG k ― ε model) and large eddy simulations (LES) on a multi-hole gasoline direct injection (GDI) system. The fuel injector considered in this study is the Spray G nozzle from the Engine Combustion Network (ECN). A blob injection model, based on empirical rate of injection (ROI) profile, is considered in this study. The latest data on spray penetrations from Engine Combustion Network is used for model validation along with experimental findings on suction velocity and local droplet diameter. The spray breakup is simulated by using the KH-RT breakup length model. The turbulence model constant Cε1, is tuned to match with the experimental data of liquid and vapor penetrations in simulations while using the standard k-ε turbulence model. On the other hand, the Kelvin-Helmholtz breakup model time constant (B1) and Rayleigh Taylor breakup length constant (Cbl) are tuned for the RNG k ― ε turbulence model. From this work it is observed that by increasing the breakup length model constants (Cbl), the radial dispersion of the spray increases, and the extent of breakup is lowered. The set of optimized model parameters used with RNG k - ε is also used for LES modeling studies with different sub-grid models. The spray penetrations with standard k ― ε turbulence (Cε1=1.44) model are reported underpredicting, and the RNG k ― ε and LES sub-grid models predicted well with the latest and recommended data from ECN. In terms of gas axial velocity comparison, the standard k-ε(Cε1=1.44) simulation setup does not perform as well as the simulation setups using RNG k-ε and LES turbulence models (with breakup parameters: Cbl = 16 and B1 = 32). However, the standard k-ε(Cε1=1.44) simulation setup perform better than the simulation setups using RNG k-ε and LES turbulence models (with breakup parameters: Cbl = 16 and B1 = 32) when it comes to predicting local droplet diameter at 15 mm downstream of the injector tip. A parametric study is also performed considering the geometry of the stepped holes in the computational domain. The rate of injection based simulation is initiated at the end of the smaller hole. The case including the stepped holes led to over-prediction compared to the case with the usual computational domain (i.e., without the stepped holes), in terms of spray penetrations, but exhibited higher levels of fluctuations in the spray morphology. Finally, parametric studies were carried out to understand the relative importance of the individual spray sub-models (breakup, evaporation and collision) and the results are conclusive that for a spray simulation the breakup models are the dominant factors.

Layered complex networks as fluctuation amplifiers

In complex networked systems theory, an important question is how to evaluate the system robustness to external perturbations. With this task in mind, I investigate the propagation of noise in multi-layer networked systems. I find that, for a two layer network, noise originally injected in one layer can be strongly amplified in the other layer, depending on how well-connected are the complex networks in each layer and on how much the eigenmodes of their Laplacian matrices overlap. These results allow to predict potentially harmful conditions for the system and its sub-networks, where the level of fluctuations is important, and how to avoid them. The analytical results are illustrated numerically on various synthetic networks.

On the effects of a separation bubble on fan noise

A scale model of a fan stage operating at approach condition is investigated using wall-resolved Large Eddy Simulations. The configuration is a state-of-the-art fan stage (rotor + stator) of an Ultra High Bypass Ratio turbofan engine, and the computational cost has been reduced by simulating a radial slice of a periodic sector. At low fan speeds and Reynolds number, a recirculation bubble can be observed on the suction side of the fan blades, near the leading edge. This recirculation bubble causes the boundary layer transition to turbulence and is associated with high levels of wall pressure fluctuations. The noise spectra upstream of the fan blades show high frequency tones, which are related to the noise generation mechanism of the recirculation bubble. It has been found that the size, position and pressure disturbances in the bubble depend on the mass flow rate. Furthermore, the frequencies and amplitudes of the high frequency tones in the noise spectra also depend on the mass flow rate. As the mass flow rate decreases, the size of the bubble increases, the bubble shifts towards downstream locations, and high levels of wall pressure fluctuations can be found along the suction side of the fan blade. The acoustic signature of the bubble shifts towards lower frequencies with the mass flow rate, but its amplitude increases. A coherence analysis and a Dynamic Mode Tracking technique are used to confirm that high frequency tones can be generated by the recirculation bubble in the separation region. The flow features at these frequencies suggest that a vortex shedding mechanism is related to the noise radiated from the recirculation bubble.

An Improved Wavelet Threshold Denoising Method for Health Monitoring Data: A Case Study of the Hong Kong-Zhuhai-Macao Bridge Immersed Tunnel

Tunnels generally operate underground or underwater in a complex environment. As a result, the health monitoring system is inevitably affected by various environmental factors, which introduces noise to the system. However, the noise contained in the monitoring sequence may disrupt structural damage identification and health state assessment as the real structural response may be overwhelmed by the noise. To properly eliminate the noise in an objective way, this study proposed an improved wavelet threshold denoising method. Firstly, it adopts a quantitative factor, namely the Sparse Index, to assist the selection of the best wavelet basis in numerous wavelet packages. Then, the decomposition layer and threshold are optimized by a comprehensive evaluation based on a variation coefficient method. At last, the application of the concrete strain health monitoring data of the Hong Kong-Zhuhai-Macao Bridge immersed tunnel verified the effectiveness of the proposed method. It is found that the combination of sym12 and five decomposition layers can obtain the best denoising results within the selected wavelet families and decomposition levels. Moreover, the proposed method achieves good denoising results under different fluctuation levels. Thus, the proposed method is reliable, can solve the problem of optimal parameter selection such as decomposition level and wavelet basis in wavelet denoising, and can be applied in the structural health monitoring of critical infrastructures.

Experimental demonstration: Differential detection of 10 Gbps optical OOK signal over turbulent channel

In this paper, we experimentally demonstrate the bit error rate (BER) performance of fixed threshold differential detection for both 5 Gbps and 10 Gbps On-Off keying (OOK) signals over a turbulent channel, which induces slowly varying intensity fluctuation on the signals. The experimental results are also validated using numerical simulations. This study is significant for the terrestrial free-space optical (FSO) communication system in which the channel induces similar intensity fluctuation due to atmospheric turbulence. To show the efficiency of the differential detection scheme for the OOK signal two proof-of-principle systems are realized. In the first system, the effect of channel-induced intensity variation on the data is realized by passing a differentially coded OOK signal through a Mach–Zehnder modulator (MZM), which is driven by a low frequency sinusoidal electrical signal emulating slowly varying channel fluctuation. The amplitude of the sinusoid has been varied to create different levels of intensity fluctuation, mimicking different turbulent conditions in the optical link. In the second setup, we measure the BER performance of differential detection assisted OOK transmission over a short-range (10)m FSO channel, which is affected by turbulence-induced intensity fluctuations. The different turbulences are created by different heating levels of a blower, which injects hot air across the FSO channel. Experimental results show that in the presence of intensity fluctuation, differential detection of OOK signal using a fixed threshold offers a better BER performance at higher optical to signal ratio (OSNR) values compared to conventional direct detection (DD) scheme. Results obtained from numerical simulation of OOK transmission over turbulent FSO channels also confirm the similar behavior. Both simulation and experimental results show that differential detection can offer a BER of 10−3 at 5 Gbps and 10 Gbps data rates with slow varying channel-induced intensity fluctuation when the DD scheme fails to produce the same BER values.

Numerical study of skewness, kurtosis and variance of droplet-size distribution in air-assisted atomizer

We study the approach to calculating the scale and shape parameters of the Weibull droplet-size distribution by means of the skewness, kurtosis and variance, obtained by simulation of atomization in an air-assisted atomizer with different viscosities, densities, flow rates and surface tensions of atomizing liquid. Such approach evades necessity to determine the shift parameter and to use the approximation of distribution obtained from the calculation by the Weibull distribution. The major obstacle to an application of it in processing the data about the atomization of droplets (obtained through the calculation) is the high level of fluctuations of skewness and kurtosis in case of small averaging time.

Geomagnetic pulsations in 1–4 mHz frequency range (Pc5/Pi3) in the magnetotail at different levels of disturbances in the interplanetary medium

In this paper, we study parameters of geomagnetic pulsations in the 1–4 mHz frequency range (Pc5/Pi3) in the magnetotail, utilizing data obtained by Cluster satellites at different levels of fluctuations in the interplanetary magnetic field (IMF) and the solar wind dynamic pressure in 2016. Particular attention is given to the conditions of “zero” disturbance when amplitudes of fluctuations in the interplanetary medium are smaller compared to their typical values. Both under quiet and disturbed conditions, waves of different spatial scales are recorded, with the occurrence rate of large-scale waves increasing under undisturbed conditions. Amplitudes of the large-scale waves occurring in the magnetotail under low intensity of fluctuations outside the magnetosphere are from few tenths to a few nanoteslas (nT), and their power is approximately equal in longitudinal and transverse components. Presumably, these waves are magnetotail eigen-modes.

Геомагнитные пульсации диапазона 1–4 мГц (Pc5/Pi3) в магнитосферном хвосте при разном уровне возмущенности в межпланетной среде

In this paper, we study parameters of geomagnetic pulsations in the 1–4 mHz frequency range (Pc5/Pi3) in the magnetotail, utilizing data obtained by Cluster satellites at different levels of fluctuations in the interplanetary magnetic field (IMF) and the solar wind dynamic pressure in 2016. Particular attention is given to the conditions of “zero” disturbance when amplitudes of fluctuations in the interplanetary medium are smaller compared to their typical values. Both under quiet and disturbed conditions, waves of different spatial scales are recorded, with the occurrence rate of large-scale waves increasing under undisturbed conditions. Amplitudes of the large-scale waves occurring in the magnetotail under low intensity of fluctuations outside the magnetosphere are from few tenths to a few nanoteslas (nT), and their power is approximately equal in longitudinal and transverse components. Presumably, these waves are magnetotail eigen-modes.