Velocity Perturbations Research Articles

Propagating torsional Alfvén waves in thermally active solar plasma

ABSTRACT The aim of this study is to shed light on the effects connected with thermal misbalance due to non-equal cooling and heating rates induced by density and temperature perturbations in solar active regions hosting either propagating torsional or shear Alfvén waves. A description for the non-linear forces connected with Alfvén waves in non-ideal conditions is provided, based on the second-order thin flux tube approximation. This provides insight into the effects of Alfvén-induced motions on the boundary of thin magnetic structures in thermally active plasmas. The equations describing the process of generating longitudinal velocity perturbations, together with density perturbations by non-linear torsional Alfvén waves, are obtained and solved analytically. It is shown that the phase shift (compared with the ideal case) and the amplitude of the induced longitudinal plasma motions against the period of the mother Alfvén wave are greater for shear Alfvén waves compared with torsional Alfvén waves, although following the same pattern. The difference in the influence of thermal misbalance on the induced velocity perturbations is governed by the plasma-β although its effect is stronger for shear waves. It is deduced that for a harmonic Alfvén driver the induced density perturbations are left uninfluenced by the thermal misbalance.

“Ghost Transient” Corrections to the Southern California GPS Velocity Field from San Andreas Fault Seismic Cycle Models

Abstract Seismic cycle models of the northern and southern San Andreas fault (SAF) were developed to estimate corrections for viscoelastic transients embedded in the present-day Global Positioning System (GPS) velocity field. These models incorporate a viscous shear zone embedded in either a layered or a 3D viscoelastic structure. In the southern SAF model, the 3D effective viscosity structure is based on the Southern California Earthquake Center (SCEC) community rheology model (CRM) and the SCEC community thermal model. Viscoelastic materials are represented with Burgers or Maxwell viscoelastic rheologies, and ranges of rheological parameters and earthquake chronologies are explored. For the northern and southern SAF models, surface velocity perturbations or “ghost transients” (GTs) of up to 3.5–6 mm/yr are obtained for a wide range of conditions. In all cases, the GT velocities are consistent with left-lateral rather than right-lateral motion across the SAF, and subtracting them from the present-day GPS velocity field increases the apparent SAF offset rate. For the southern SAF model, GT velocities are fairly insensitive to the timing of large SAF earthquakes prior to 1857, but they are sensitive to rheology (i.e., temperatures and strain rates assumed for computing CRM effective viscosities). For the northern SAF model, GT velocities are sensitive to both rheology (mantle, lower crust and shear zone effective viscosities, and Burgers Body parameters Δb and RB) and mean earthquake recurrence interval. For the reference northern and southern SAF models provided to U.S. Geological Survey National Seismic Hazard Mapping Project deformation modelers, the maximum GT velocities are 3.73 and 4.84 mm/yr, respectively.

Intelligent detection method for seeding timing in sapphire processing

In sapphire processing by the Kyropoulos method, the detection of seeding timing is the core technology. Existing detection methods are incapable of tracking spoke feature motion on the liquid surface and cannot effectively guide sapphire seeding. Therefore, this paper proposes an innovative visual-perturbation-velocity-fitting method for sapphire seeding timing detection. First, the concept of perturbation velocity, which can reflect the flow state of the melt surface, is presented. Then, the position of spoke feature points is obtained by computer vision and calculated by multi-frame interactive periodic detection. The fitting function model corresponding to the perturbation velocity in different flow states is established. Finally, the seeding timing is determined by the transition of the flow state and the stability of the flow. The experimental results show that the proposed method can improve the seeding success rate from 76.7% to 93.3% and the average successful seeding time from 5.6 h to 4.5 h.

DMSP Poynting Flux: Data Processing and Inter-Spacecraft Comparisons.

Poynting flux (PF) calculated from low Earth orbit spacecraft in situ ion drift and magnetic field measurements is an important measure of energy exchange between the magnetosphere and ionosphere. Defense Meteorological Satellite Program (DMSP) spacecraft provide an extensive back‐catalog of ion drift and magnetic perturbation measurements, from which quasi‐steady PF could be calculated. However, since DMSP are operations‐focused spacecraft, data must be carefully preprocessed for research use. We describe an automated approach for calculating earthward PF focusing on pre‐processing and quality control. We produce a PF data set using nine satellite‐years of DMSP F15, F16, and F18 observations. To validate our process we inter‐compare PF from different spacecraft using more than 2,000 magnetic conjunction events. We find no serious systematic differences. We further describe and apply an equal‐area binning technique to obtain average spatial patterns of PF, magnetic perturbation, electric field and ion drift velocity. We perform our analysis using all components of electric and magnetic field and comment on the adverse consequences of the typical single‐electric‐field‐component DMSP PF approximation on inter‐spacecraft agreement. Including full‐field components significantly increases the relative strength of near‐cusp PF and increases the integrated high‐latitude PF by ∼25%.

WiNDS: An H α Kinematics Survey of Nearby Spiral Galaxies—Vertical Perturbations in Nearby Disk-type Galaxies

We present the Waves in Nearby Disk galaxies Survey (WiNDS) consisting of 40 nearby low-inclination disk galaxies observed through H α high-resolution Fabry–Perot interferometry. WiNDS consists of 12 new galaxy observations and 28 data archived observations obtained from different galaxy surveys. We derive two-dimensional line-of-sight velocity fields that are analyzed to identify the possible presence of vertical velocity flows in the galactic disks of these low-inclination late-type galaxies using velocity residual maps, derived from the subtraction of an axisymmetric rotation model from a rotational velocity map. Large and globally coherent flows in the line-of-sight velocity of nearly face-on galaxies can be associated with large vertical displacement of the disk with respect to its midplane. Our goal is to characterize how frequent vertical perturbations, such as those observed in the Milky Way, arise in the local universe. Our currently available data have allowed us to identify 20% of WiNDS galaxies with strong velocity perturbations that are consistent with vertically perturbed galactic disks.

Parameters of the Radcliffe Wave from Masers, Radio Stars, and T Tauri Stars

The presence of the Radcliffe wave is shown both in the positions and in the vertical velocities of masers and radio stars belonging to the Local Arm. This gives the impression that the structure of the Radcliffe wave is not a wave in the full sense of the word. It is more like a local high-amplitude burst, rapidly fading away. Moreover, this structure has the largest amplitude in the immediate vicinity of the Sun, where the main ``contributors'' are the Gould Belt stars. Based on the spectral analysis of masers, the following estimates of the geometric and kinematic characteristics of the wave were obtained: the largest value of the vertical coordinate $z_{max}=87\pm4$ pc and the wavelength $\lambda=2.8\pm0.1$ kpc, the vertical velocity perturbation amplitude reaches $W_{max}=5.1\pm0.7$ km s$^{-1}$ and the wavelength found from vertical velocities is $\lambda=3.9\pm1.6$ kpc. The Radcliffe wave also manifests itself in the positions of very young stars that have not reached the main sequence stage. We extracted a sample of such stars from the Gaia DR2$\times$AllWISE database and obtained the following estimates from them: $z_{max}=118\pm3$ pc and wavelength $\lambda=2.0\pm0.1$ kpc.

Reliability of the active drag assessment using an isotonic resisted sprint protocol in human swimming

The purpose of the presents study was to investigate the reliability of the active drag (Da) assessment using the velocity perturbation method (VPM) with different external resisted forces. Eight male and eight female swimmers performed 25 m sprints with five isotonic loads (1–2–3–4–5 kg for females; 1–3–5–7–9 kg for males), which were repeated twice on different days. The mean velocity and semi-tethered force were computed for each condition, and the free-swimming maximum velocity was estimated with load-velocity profiling. From the obtained variables, Da at the maximum free-swimming condition was calculated using VPM. Absolute and typical errors and the intra-class correlation (ICC) were calculated to assess test–retest reliability. 95% confidence interval (95% CI) lower bound of ICC was larger than 0.75 in 3, 4 (females only) and 5 kg trials in both sexes (corresponding to 37–60 N additional resistance; all p < 0.001), which also showed small absolute and relative typical errors (≤ 2.7 N and ≤ 4.4%). In both sexes, 1 kg load trial (16–17 N additional resistance) showed the lowest reliability (95% CI of ICC; − 0.25–0.83 in males and 0.07–0.94 in females). These results suggested that a tethered force of 37–60 N should be used to assess Da using VPM.

Molecules with ALMA at Planet-forming Scales (MAPS): A Circumplanetary Disk Candidate in Molecular-line Emission in the AS 209 Disk

We report the discovery of a circumplanetary disk (CPD) candidate embedded in the circumstellar disk of the T Tauri star AS 209 at a radial distance of about 200 au (on-sky separation of 1.″4 from the star at a position angle of 161°), isolated via 13CO J = 2−1 emission. This is the first instance of CPD detection via gaseous emission capable of tracing the overall CPD mass. The CPD is spatially unresolved with a 117 × 82 mas beam and manifests as a point source in 13CO, indicating that its diameter is ≲14 au. The CPD is embedded within an annular gap in the circumstellar disk previously identified using 12CO and near-infrared scattered-light observations and is associated with localized velocity perturbations in 12CO. The coincidence of these features suggests that they have a common origin: an embedded giant planet. We use the 13CO intensity to constrain the CPD gas temperature and mass. We find that the CPD temperature is ≳35 K, higher than the circumstellar disk temperature at the radial location of the CPD, 22 K, suggesting that heating sources localized to the CPD must be present. The CPD gas mass is ≳0.095 M Jup ≃ 30 M ⊕ adopting a standard 13CO abundance. From the nondetection of millimeter continuum emission at the location of the CPD (3σ flux density ≲26.4 μJy), we infer that the CPD dust mass is ≲0.027 M ⊕ ≃ 2.2 lunar masses, indicating a low dust-to-gas mass ratio of ≲9 × 10−4. We discuss the formation mechanism of the CPD-hosting giant planet on a wide orbit in the framework of gravitational instability and pebble accretion.

Impulsively generated kink wave trains in solar coronal slabs

ABSTRACT We numerically follow the response of density-enhanced slabs to impulsive, localized, transverse velocity perturbations by working in the framework of ideal magnetohydrodynamics (MHD). Both linear and non-linear regimes are addressed. Kink wave trains are seen to develop along the examined slabs, sharing the characteristics that more oscillatory patterns emerge with time and that the apparent wavelength increases with distance at a given instant. Two features none the less arise due to non-linearity, one being a density cavity close to the exciter and the other being the appearance of shocks both outside and inside the nominal slab. These features may be relevant for understanding the interaction between magnetic structures and such explosive events as coronal mass ejections. Our numerical findings on kink wave trains in solar coronal slabs are discussed in connection with typical measurements of streamer waves.

A realistic arteriovenous dialysis graft model for hemodynamic simulations.

ObjectiveThe hemodynamic benefit of novel arteriovenous graft (AVG) designs is typically assessed using computational models that assume highly idealized graft configurations and/or simplified boundary conditions representing the peripheral vasculature. The objective of this study is to evaluate whether idealized AVG models are suitable for hemodynamic evaluation of new graft designs, or whether more realistic models are required.MethodsAn idealized and a realistic, clinical imaging based, parametrized AVG geometry were created. Furthermore, two physiological boundary condition models were developed to represent the peripheral vasculature. We assessed how graft geometry (idealized or realistic) and applied boundary condition models of the peripheral vasculature (physiological or distal zero-flow) impacted hemodynamic metrics related to AVG dysfunction.ResultsAnastomotic regions exposed to high WSS (>7, ≤40 Pa), very high WSS (>40 Pa) and highly oscillatory WSS were larger in the simulations using the realistic AVG geometry. The magnitude of velocity perturbations in the venous segment was up to 1.7 times larger in the realistic AVG geometry compared to the idealized one. When applying a (non-physiological zero-flow) boundary condition that neglected blood flow to and from the peripheral vasculature, we observed large regions exposed to highly oscillatory WSS. These regions could not be observed when using either of the newly developed distal boundary condition models.ConclusionHemodynamic metrics related to AVG dysfunction are highly dependent on the geometry and the distal boundary condition model used. Consequently, the hemodynamic benefit of a novel graft design can be misrepresented when using idealized AVG modelling setups.

Velocity perturbations and Reynolds stresses in Holmboe instabilities

The velocity perturbations and Reynolds stresses associated with finite-amplitude Holmboe instabilities are investigated using linear stability analysis, numerical simulations, and laboratory experiments. The rightward and leftward propagating Holmboe instabilities are separated, allowing for a direct comparison of the perturbation fields between the numerical simulations and the linear stability analysis. The decomposition and superposition of the perturbation fields provide insights into the structure and origin of Reynolds stresses in Holmboe instabilities. Shear instabilities in stratified flows introduce a directional preference (anisotropy) in velocity perturbation fields, thereby generating Reynolds stresses. Here, we investigate this anisotropy by comparing pairs of horizontal and vertical velocity perturbations (u′,w′), obtained from the simulations and the laboratory experiment, with predictions from linear stability analysis. For an individual Holmboe mode, both the simulations and linear theory yield elliptical (u′,w′)-pairs that are oriented toward the second and fourth quadrants (u′w′&amp;lt;0), corresponding to the tilted elliptical trajectories of particle movement. Combining the leftward and rightward Holmboe modes yields (u′,w′) ellipses whose orientation and aspect ratio are phase-dependent. When averaged over a full cycle, the joint probability density functions of (u′,w′) in the linear theory and single wavelength simulations exhibit “steering wheel” structures. This steering wheel is smeared out in multiple wavelength simulations and the laboratory experiment due to varying wavelengths, resulting in an elliptical cloud. All of the approaches adopted in the present study yield Reynolds stresses that are comparable to those reported in previous laboratory and field investigations.

Mapping H i 21-cm in the Klemola 31 group at z = 0.029: emission and absorption towards PKS 2020−370

ABSTRACT We present MeerKAT Absorption Line Survey (MALS) observations of the H i gas in the Klemola 31 galaxy group (z = 0.029), located along the line of sight to the radio-loud quasar PKS 2020−370 (z = 1.048). Four galaxies of the group are detected in H i emission, and H i absorption is also detected in front of PKS 2020−370 in Klemola 31A. The emission and absorption are somewhat compensating on the line of sight of the quasar, and the derived column density of the absorption appears underestimated, with respect to the neighbouring emission. A symmetric tilted-ring model of Klemola 31A, assuming the absorbing gas in regular rotation in the plane, yields a rather high spin temperature of 530 K. An alternative interpretation is that the absorbing gas is extra-planar, which will also account for its non-circular motion. The Na i/Ca ii ratio also suggests that the absorbing gas is unrelated to cold H i disc. Two of the galaxies in the Klemola group are interacting with a small companion, and reveal typical tidal tails, and velocity perturbations. Only one of the galaxies, ESO 400−13, reveals a strong H i deficiency, and a characteristic ram-pressure stripping, with a total asymmetry in the distribution of its gas. Since a small galaxy group as Klemola 31 is not expected to host a dense intra-group gas, this galaxy must be crossing the group at a very high velocity, mostly in the sky plane.

Parameters of the Galactic Spiral Density Wave from Masers with Parallax Errors Less Than 10$$\boldsymbol{\%}$$

We have studied the kinematics of Galactic masers and radio stars with measured VLBI trigonometric parallaxes and proper motions. We have considered masers with relative trigonometric parallax errors less than 10\% and determined the Galactic rotation parameters from them. In particular, the linear rotation velocity of the Galaxy at the solar distance $R_0$ has been found to be $244.4\pm4.3$ km s$^{-1}$ (for the adopted $R_0=8.1\pm0.1$ kpc). We have performed a joint and separate spectral analysis of the radial, residual tangential, and vertical maser velocities. For example, from the vertical maser velocities we have estimated the velocity perturbation amplitude $f_W=5.2\pm1.5$ km s$^{-1}$ with the wavelength $\lambda_W=2.6\pm0.7$ kpc, arguing for the influence of the spiral density wave on the vertical stellar velocities. Based on 104 masers within 3 kpc of the Sun, as a result of the joint solution, we have estimated the radial, $f_R=6.7\pm1.1$ km s$^{-1}$, and tangential, $f_\theta=2.6\pm1.2$ km s$^{-1}$, velocity perturbations, the perturbation wavelength $\lambda=2.1\pm0.3$ kpc, and the Sun's phase in the Galactic spiral density wave $\chi_\odot=-148\pm15^\circ$. We have confirmed the presence of the Radcliffe wave in the spatial distribution of masers and radio stars belonging to the Local Arm.

Kinematic Evidence for an Embedded Planet in the IM Lupi Disk

We test the hypothesis that an embedded giant planet in the IM Lupi protostellar disk can produce velocity kinks seen in CO line observations as well as the spiral arms seen in scattered light and continuum emission. We inject planets into 3D hydrodynamics simulations of IM Lupi, generating synthetic observations using Monte Carlo radiative transfer. We find that an embedded planet of 2–3 M Jup can reproduce non-Keplerian velocity perturbations, or “kinks”, in the 12CO J = 2–1 channel maps. Such a planet can also explain the spiral arms seen in 1.25 mm dust continuum emission and 1.6 μm scattered-light images. We show that the wake of the planet can be traced in the observed peak velocity map, which appears to closely follow the morphology expected from our simulations and from analytic models of planet–disk interaction.

Imprints of subducted Palaeo-Tethys oceanic lithosphere on upper-mantle discontinuities and the formation of the Emeishan large igneous province

SUMMARYGenerally, the upwelling of a mantle plume is suggested to have led to the formation of the Emeishan large igneous province (ELIP). However, this idea has been challenged by recent geological and seismological studies. In this investigation, three profiles of velocity perturbations crossing the ELIP are drawn following previous tomographic studies, and common conversion point (CCP) stacking of receiver functions in the ELIP is carried out. A slab-like high-velocity structure is revealed, which may be associated with the lithosphere of the Palaeo-Tethys Ocean (LPTO) subducting toward the northeast. CCP stacking of receiver functions shows that the LPTO generated imprints on the 410- and 660-km discontinuities. A comprehensive analysis suggests that the LPTO may have induced large-scale mantle upwelling, which contributed to the formation of the ELIP.

The Doppler Flip in HD 100546 as a Disk Eruption: The Elephant in the Room of Kinematic Protoplanet Searches

The interpretation of molecular-line data using hydrodynamical simulations of planet–disk interactions fosters new hope for the indirect detection of protoplanets. In a model-independent approach, embedded protoplanets should be found at the roots of abrupt Doppler flips in velocity centroid maps. However, the largest velocity perturbation known for an unwarped disk, in the disk of HD 100546, leads to a conspicuous Doppler flip that coincides with a thick dust ring, in contradiction with an interpretation in terms of a ≳1 M jup body. Here we present new ALMA observations of the 12CO(2–1) kinematics in HD 100546, with a factor of 2 finer angular resolutions. We find that the disk rotation curve is consistent with a central mass 2.1 < M ⋆/M ⊙ < 2.3 and that the blueshifted side of the Doppler flip is due to vertical motions, reminiscent of the disk wind proposed previously from blueshifted SO lines. We tentatively propose a qualitative interpretation in terms of a surface disturbance to the Keplerian flow, i.e., a disk eruption, driven by an embedded outflow launched by a ∼10 M earth body. Another interpretation involves a disk-mass-loading hot spot at the convergence of an envelope accretion streamer.

Comparative study of unsteady flow mechanisms for swept and radially stacked rotors in a dual-stage counter-rotating compressor

Counter-rotating (CR) compressor is a potential configuration to increase the thrust-to-weight ratio of aero-engines. The present study numerically investigated a dual-stage CR compressor by solving unsteady Reynolds-averaged Naiver–Stokes equations. The unsteady flow mechanisms were analyzed and compared for swept and baseline (radially stacked) CR rotors. Both of these rotor configurations were numerically simulated for the entire design speed line. The unsteady flow field near the mid-span and blade tip characterized by tip leakage vortex (TLV) was analyzed in detail for both near peak efficiency (NPE) and near stall (NS) conditions. According to the results, blade loading of the second rotor (R2) was much higher than that of the first rotor (R1). Moreover, the blade loading was higher in the swept rotors in comparison with the baseline CR rotors. In both baseline and swept CR compressors, the potential effect of R2 on the static pressure distribution of R1 was more significant compared to the opposite effect. Under the NS condition, the potential effect was confined in the rear zone of R1 where the potential flow of R2 passed. Despite the obvious impact of wake-induced negative jet on the flow field of R2 blade passage, the static pressure of R2 blade exhibited no evident variation except at 90% spanwise position. Although the blade loading of the swept CR compressor was increased, the potential effect of R2 on R1 was considerably reduced due to growth of axial gap between R1 and R2. Since the blade sweep improved the flow field, the wake passing effect on R2 was more moderate compared to that of the baseline CR compressor. The perturbation velocity magnitude of the swept CR compressor was approximately half of that of the baseline CR compressor. The TLV of R2 was periodically disturbed, and therefore, low axial velocity patches were observed on the TLV trajectory of R2. In addition, the swept CR compressor exhibited smaller low axial velocity patches of TLV due to weaker aerodynamic interaction between R1 and R2.

Turbulence generation by shock interaction with a highly nonuniform medium.

An initially planar shock wave propagating into a medium of nonuniform density will be perturbed, leading to the generation of postshock velocity perturbations. Using numerical simulations we study this phenomenon in the case of highly nonuniform density (order-unity normalized variance, σ_{ρ}/ρ[over ¯]∼1) and strong shocks (shock Mach numbers M[over ¯]_{s}≳10). This leads to a highly disrupted shock and a turbulent postshock flow. We simulate this interaction for a range of shock drives and initial density configurations meant to mimic those which might be presently achieved in experiments. Theoretical considerations lead to scaling relations, which are found to reasonably predict the postshock turbulence properties. The turbulent velocity dispersion and turbulent Mach number are found to depend on the preshock density dispersion and shock speed in a manner consistent with the linear Richtymer-Meshkov instability prediction. We also show a dependence of the turbulence generation on the scale of density perturbations. The postshock pressure and density, which can be substantially reduced relative to the unperturbed case, are found to be reasonably predicted by a simplified analysis that treats the extended shock transition region as a single normal shock.

Direct tracking of reverse-transcriptase speed and template sensitivity: implications for sequencing and analysis of long RNA molecules.

Although reverse-transcriptase (RT) enzymes are critical reagents for research and biotechnology, their mechanical properties are not well understood. In particular, we know little about their relative speed and response to structural obstacles in the template. Commercial retroviral RTs stop at many positions along mixed sequence templates, resulting in truncated cDNA products that complicate downstream analysis. By contrast, group II intron-encoded RTs appear to copy long RNAs with high processivity and minimal stops. However, their speed, consistency and pausing behavior have not been explored. Here, we analyze RT velocity as the enzyme moves through heterogeneous sequences and structures that are embedded within a long noncoding RNA transcript. We observe that heterogeneities in the template are highly disruptive to primer extension by retroviral RTs. However, sequence composition and template structure have negligible effects on behavior of group II intron RTs, such as MarathonRT (MRT). Indeed, MRT copies long RNAs in a single pass, and displays synchronized primer extension at a constant speed of 25 nt/sec. In addition, it passes through stable RNA structural motifs without perturbation of velocity. Taken together, the results demonstrate that consistent, robust translocative behavior is a hallmark of group II intron-encoded RTs, some of which operate at high velocity.

Comparison of flame response characteristics between Non‐premixed and premixed flames under acoustic excitation

A comparative study was conducted on the response characteristics according to the increase in velocity perturbation intensity (u'/u) under acoustic excitation conditions of non-premixed and premixed flames. In addition, the flame dynamic characteristics and flame structure according to acoustic excitation were analyzed, and the response characteristics were compared according to the two flame types through the flame transfer function (FTF) and flame height perturbation analysis. In a non-premixed flame, the number of waveforms (undulations) on the flame surface increased according to the forcing frequency in the flame structure according to the acoustic excitation. On the other hand, as the velocity perturbation intensity increased, the flame height decreased. Thus, in the low forcing frequency range, the magnitude of the flame fluctuation increases, and in the high forcing frequency, the magnitude of the flame fluctuation tends to decrease. Therefore, the Strouhal number and FTF gain are nonlinear properties, and the nonlinear effect depends on the magnitude of the gain and the critical Strouhal number. In a premixed flame, various flame structures are shown according to acoustic excitation. The average flame height does not change much with increasing (u'/u¯). The correlation between the Strouhal number and FTF gain is linear. The flame height perturbation parameters were introduced to examine the flame dynamic characteristics according to the acoustic excitation, and non-premixed and premixed flames were analyzed. As a result, it shows the trend and similarity of the FTF gain, suggesting its potential as a parameter for simple analysis of the heat-release rate. In the correlation between the Strouhal number and FTF gain, the non-premixed flame has a nonlinear characteristic and agrees with the analytical study. On the other hand, the premixed flame is not consistent with analytic studies when the Strouhal number is greater than 1. The reason for various flame structures is to consider parameters such as the unusual flame structure and laminar flame speed by external perturbation.