arXiv subjects

Mijie Shi

Publications and source records attributed to Mijie Shi.

At least 19 recordsLinked to original sources

arXiv2026

Three-dimensional kink modes in solar coronal slabs: group velocities and their implications for impulsively excited waves

Little attention has been paid to group velocities of three-dimensional (3D) MHD waves in solar coronal seismology. This study aims to present a rather comprehensive examination on the group velocities of trapped 3D kink modes in coronal slabs, emphasizing the connection of mode analysis to both mode characterization and impulsively excited 3D kink waves. We work in linear, ideal, pressureless MHD, and take the equilibrium slab to be symmetrically structured only in one transverse direction. The dispersion relation is numerically solved, with the results understood by making in-depth analytical progress. We address both the transverse fundamental and its first overtone. We develop a three-subgroup scheme for categorizing 3D kink modes on the plane spanned by the axial and out-of-plane wavenumbers. The group ($\vec{v}_{\rm gr}$) and phase velocities ($\vec{v}_{\rm ph}$) sit on the same side of the equilibrium magnetic field ($\vec{B}_0$) for the ``$\vec{B}_0$-same-side A'' and ``$\vec{B}_0$-same-side F'' subgroups, which are further discriminated by the directional similarity of $\vec{v}_{\rm gr}$ and $\vec{B}_0$. The ``$\vec{B}_0$-straddling'' subgroup is peculiar in that $\vec{v}_{\rm gr}$ and $\vec{v}_{\rm ph}$ lie astride $\vec{B}_0$, a feature that cannot be found for waves in unbounded uniform media in pressureless MHD. This ``$\vec{B}_0$-straddling'' subgroup pertains to both the fundamental and its overtones. We further place our results in the context of impulsive waves, employing the method of stationary phase to predict the large-time wavefront morphology in the plane of symmetry of the equilibrium slab. Wavefronts directed toward $\vec{B}_0$ derive exclusively from ``$\vec{B}_0$-straddling'' modes, and are confined to narrow sectors.

astro-ph.SR↗

arXiv2026

The Effect of Gravitational Stratification on Kink Oscillations in Curved Coronal Loops

Kink oscillation frequency is a key parameter for coronal seismology. It is still unclear how gravitational stratification affects the kink frequency in curved coronal loops. This work aims to investigate the effect of gravitational stratification on the frequency of kink oscillations in curved coronal loops and discuss their seismological potential. We conduct numerical computations within the ideal MHD framework to study different kink polarizations and harmonics in a curved, gravitationally stratified coronal loop. The oscillation frequencies derived from the Lagrangian displacement are compared with the WKB approximation. For the vertically polarized fundamental mode, the oscillation frequency deviates from the WKB approximation by about 18\% in the current numerical setup. Nevertheless, the oscillation frequency closely matches the local Alfv\'en frequency near the loop apex. On the other hand, the frequency of the horizontally polarized fundamental mode exhibits only a 7\% deviation in our current model from the WKB approximation and closely matches the local Alfv\'en frequency near one quarter of the loop. For the first overtones, the frequencies for both polarizations can be well described by the WKB approximation. The frequency of vertically polarized fundamental kink modes can be predicted by the local Alfv\'en frequency near the loop apex. In contrast, the WKB approximation remains highly reliable for estimating the frequency of horizontally polarized fundamental modes and first overtones, which is also well described by the local Alfv\'en frequency near one quarter of the loop. These results therefore pave the way for spatially dependent coronal seismology, enabling, e.g., the probing of magnetic field strength at different locations along a coronal loop.

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arXiv2025

QFP Waves Driven by the Tuning-Fork Effect during Magnetic Reconnecion

Through three-dimensional MHD simulations, we have uncovered a kind of fast coronal wave originating from both ends of a current sheet (CS) during a solar eruption. These waves are observed to appear near the top and bottom ends of the reconnection-related CS. The simulations demonstrate the presence of termination shock regions above the two ends of the CS. As the reconnection outflows escape from the vertical CS and encounter these termination shocks, they undergo partial reflection, redirecting towards the CS terminal fork walls. The identified waves propagate rapidly at a speed of approximately 1400 km/s with a period of just 2 s. Concurrently, the time-evolution of intensity within a small region of the CS terminal fork structures, exhibits a similar oscillation period of 2 s. All these evidence supports the notion that these QFP (Quasi-periodic Fast-Propagating) waves were excited by tuning fork effects within the CS system. Essentially, the rapid reconnection outflows are reflected by the terminal shocks, striking the fork walls at the CS ends. Moreover, parts of the oscillations along the tuning fork handle are transformed into thermal energy, accumulating in the CS center and elevating the temperature. This is the first time to report such QFP waves resulting from tuning fork effects within the CS during a solar eruption. These waves are anticipated to manifest closely following the propagation of CMEs and adjacent to the related post-flare loops in observations, with partial confirmation in current observations.

astro-ph.SR↗

arXiv2025

Multi-episodes of remote brightenings driven by a coronal EUV jet on the Sun

Remote brightening (RB) is compact brightening at footpoints of magnetic loops, which are remotely-connecting to and confining an eruption in the solar atmosphere. Here, we report on observations of an RB resulting from an EUV jet with a speed of about 90\,km\,s$^{-1}$. The loops connecting the RB and the jet have an apparent length of about 59\,Mm. Intriguingly, the RB exhibits at least two episodes of brightenings, as characterised by two peaks in its lightcurve. The energies, which sustain the first and second peaks of the RB, are $6.3\times10^{26}$\,erg and $8.4\times10^{26}$\,erg, respectively, and take a significant proportion of the total energy. The first peak of the RB brightenings coincides with the jet's peak with a time delay of 12 seconds, while the second peak lags behind by 108 seconds. Besides the flows of the ejecta, we have identified two additional flows originating from the eruption site. One is relatively cool with a temperature of $log_{10}T/K=5.8-6.1$ and has a speed of about $275\pm15$\,km\,s$^{-1}$. The other is hot with a temperature of $log_{10}T/K=7.0-7.3$ and travels much faster with a speed of about 750$\pm$\,70\,km\,s$^{-1}$. We attribute the second peak of RB directly to this hot flow, which our numerical experiments suggest is the result of a slow shock wave. Considering the minimal time delay between the first peak of RB and the eruption, we infer this first episode is due to heating by nonthermal electrons. Our research demonstrates that the dynamics in an RB can offer vital insights into the nature of the corresponding eruption and help understand how the energy is distributed throughout the solar atmosphere.

astro-ph.SR↗

arXiv2024

Temporal evolution of axially standing kink motions in solar coronal slabs: An eigenfunction expansion approach

We aim to provide more insights into the applicability to solar coronal seismology of the much-studied discrete leaky modes (DLMs) in classic analyses. Under linear ideal pressureless MHD, we examine two-dimensional (2D) axial fundamental kink motions that arise when localized velocity exciters impact some symmetric slab equilibria. Continuous structuring is allowed for. A 1D initial value problem (IVP) is formulated in conjunction with an eigenvalue problem (EVP) for laterally open systems, with no strict boundary conditions (BCs) at infinity. The IVP is solved by eigenfunction expansion, allowing a clear distinction between the contributions from proper eigenmodes and improper continuum eigenmodes. Example solutions are offered for parameters typical of active region loops. Our solutions show that the system evolves towards long periodicities due to proper eigenmodes (of order the axial Alfven time), whereas the interference of the improper continuum may lead to short periodicities initially (of order the lateral Alfven time). Specializing to the slab axis, we demonstrate that the proper contribution strengthens with the density contrast, but may occasionally be stronger for less steep density profiles. Short periodicities are not guaranteed in the improper contribution, the details of the initial exciter being key. When identifiable, these periodicities tend to agree with the oscillation frequencies expected for DLMs, despite the differences in the BCs between our EVP and classic analyses. The eigenfunction expansion approach enables all qualitative features to be interpreted as the interplay between the initial exciter and some response function, the latter solely determined by the equilibria. Classic theories for DLMs can find seismological applications, with time-dependent studies offering additional ways for constraining initial exciters.

astro-ph.SR↗

arXiv2024

Damped kink motions in a system of two solar coronal tubes with elliptic cross-sections

This study is motivated by observations of coordinated transverse displacements in neighboring solar active region loops, addressing specifically how the behavior of kink motions in straight two-tube equilibria is impacted by tube interactions and tube cross-sectional shapes.We work with linear, ideal, pressureless magnetohydrodynamics. Axially standing kink motions are examined as an initial value problem for transversely structured equilibria involving two identical, field-aligned, density-enhanced tubes with elliptic cross-sections (elliptic tubes). Continuously nonuniform layers are implemented around both tube boundaries. We numerically follow the system response to external velocity drivers, largely focusing on the quasi-mode stage of internal flows to derive the pertinent periods and damping times. The periods and damping times we derive for two-circular-tube setups justify available modal results found with the T-matrix approach. Regardless of cross-sectional shapes, our nonuniform layers feature the development of small-scale shears and energy accumulation around Alfven resonances, indicative of resonant absorption and phase-mixing. As with two-circular-tube systems, our configurational symmetries make it still possible to classify lower-order kink motions by the polarization and symmetric properties of the internal flows; hence such mode labels as $S_x$ and $A_x$. However, the periods and damping times for two-elliptic-tube setups further depend on cross-sectional aspect ratios, with $A_x$ motions occasionally damped less rapidly than $S_x$ motions. We find uncertainties up to $\sim 20\%$ ($\sim 50\%$) for the axial Alfven time (the inhomogeneity lengthscale) if the periods (damping times) computed for two-elliptic-tube setups are seismologically inverted with canonical theories for isolated circular tubes.

astro-ph.SR↗

arXiv2024

Horizontally Polarized Kink Oscillations Supported by Solar Coronal Loops in an Asymmetric Environment

Kink oscillations are ubiquitously observed in solar coronal loops, their understanding being crucial in the contexts of coronal seismology and atmospheric heating. We study kink modes supported by a straight coronal loop embeded in an asymmetric environment using three-dimensional magnetohydrodynamic (MHD) simulations. We implement the asymmetric effect by setting different exterior densities below and above the loop interior, and initiate the simulation using a kink-like velocity perturbation perpendicular to the loop plane, mimicking the frequently measured horizontally polarized kink modes. We find that the external velocity fields show fan blade structures propagating in the azimuthal direction as a result of the successive excitation of higher azimuthal Fourier modes. Resonant absorption and phase mixing can still occur despite an asymmetric environment, leading to the development of small scales at loop boundaries. These small scales nonetheless develop asymmetrically at the upper and lower boundaries due to the different gradients of the Alfven speed. These findings enrich our understanding of kink modes in coronal loops embedded within an asymmetric environment, providing insights helpful for future high-resolution observations.

astro-ph.SR↗

arXiv2024

Small-amplitude Compressible Magnetohydrodynamic Turbulence Modulated by Collisionless Damping in Earth's Magnetosheath: Observation Matches Theory

Plasma turbulence is a ubiquitous dynamical process that transfers energy across many spatial and temporal scales and affects energetic particle transport. Recent advances in the understanding of compressible magnetohydrodynamic (MHD) turbulence demonstrate the important role of damping in shaping energy distributions on small scales, yet its observational evidence is still lacking. This study provides the first observational evidence of substantial collisionless damping (CD) modulation on small-amplitude compressible MHD turbulence cascade in Earth's magnetosheath using four Cluster spacecraft. Based on an improved compressible MHD decomposition algorithm, turbulence is decomposed into three eigenmodes: incompressible Alfvén modes, and compressible slow and fast (magnetosonic) modes. Our observations demonstrate that CD enhances the anisotropy of compressible MHD modes because CD has a strong dependence on wave propagation angle. The wavenumber distributions of slow modes are mainly stretched perpendicular to the background magnetic field ($\mathbf{B_0}$) and weakly modulated by CD. In contrast, fast modes are subjected to a more significant CD modulation. Fast modes exhibit a weak, scale-independent anisotropy above the CD truncation scale. Below the CD truncation scale, the anisotropy of fast modes enhances as wavenumbers increase. As a result, fast mode fractions in the total energy of compressible modes decrease with the increase of perpendicular wavenumber (to $\mathbf{B_0}$) or wave propagation angle. Our findings reveal how the turbulence cascade is shaped by CD and its consequences to anisotropies in the space environment.

astro-ph.SR↗

arXiv2023

Excitation of Multi-periodic Kink Motions in Solar Flare Loops: Possible Application to Quasi-periodic Pulsations

Magnetohydrodynamic (MHD) waves are often invoked to interpret quasi-periodic pulsations (QPPs) in solar flares. We study the response of a straight flare loop to a kink-like velocity perturbation using three-dimensional MHD simulations and forward model the microwave emissions using the fast gyrosynchrotron code. Kink motions with two periodicities are simultaneously generated,with the long-period component P_L = 57s being attributed to the radial fundamental kink mode and the short-period component P_S=5.8s to the first leaky kink mode. Forward modeling results show that the two-periodic oscillations are detectable in the microwave intensities for some lines of sight. Increasing the beam size to (1")^2 does not wipe out the microwave oscillations. We propose that the first leaky kink mode is a promising candidate mechanism to account for short-period QPPs. Radio telescopes with high spatial resolutions can help distinguish between this new mechanism with such customary interpretations as sausage modes.

astro-ph.SR↗

arXiv2022

Standing Sausage Perturbations in Solar Coronal Slabs with Continuous Transverse Density Profiles: cutoff wavenumbers, evanescent eigenmodes, and oscillatory continuum

The lack of observed sausage perturbations in solar active region loops is customarily attributed to the relevance of cutoff axial wavenumbers and the consequent absence of trapped modes (called ``evanescent eigenmodes'' here). However, some recent eigenvalue problem studies yield that cutoff wavenumbers may disappear for those equilibria where the external density varies sufficiently slowly, thereby casting doubt on the rarity of candidate sausage perturbations. We examine the responses of straight, transversely structured, coronal slabs to small-amplitude sausage-type perturbations that excite axial fundamentals by solving the pertinent initial value problem with eigensolutions for a closed domain. The density variation in the slab exterior is dictated by some steepness parameter $μ$, and cutoff wavenumbers are theoretically expected to be present (absent) when $μ\ge 2$ ($μ< 2$). However, our numerical results show no qualitative difference in the system evolution when $μ$ varies, despite the differences in the modal behavior. Only oscillatory eigenmodes are permitted when $μ\ge 2$. Our discrete eigenspectrum becomes increasingly closely spaced when the domain broadens, and an oscillatory continuum results for a truly open system. Oscillatory eigenmodes remain allowed and dominate the system evolution when $μ<2$. We show that the irrelevance of cutoff wavenumbers does not mean that all fast waves are evanescent. Rather, it means that an increasing number of evanescent eigenmodes emerge when the domain size increases. We conclude that sausage perturbations remain difficult to detect even for the waveguide formulated here.

astro-ph.SR↗

arXiv2022

Three-Dimensional Propagation of Kink Wave Trains in Solar Coronal Slabs

Impulsively excited wave trains are of considerable interest in solar coronal seismology. To our knowledge, however, it remains to examine the three-dimensional (3D) dispersive propagation of impulsive kink waves in straight, field-aligned, symmetric, low-beta, slab equilibria that are structured only in one transverse direction. We offer a study here, starting with an analysis of linear oblique kink modes from an eigenvalue problem perspective. Two features are numerically found for continuous and step structuring alike, one being that the group and phase velocities may lie on opposite sides of the equilibrium magnetic field ($\vec{B}_0$), and the other being that the group trajectories extend only to a limited angle from $\vec{B}_0$. We justify these features by making analytical progress for the step structuring. More importantly, we demonstrate by a 3D time-dependent simulation that these features show up in the intricate interference patterns of kink wave trains that arise from a localized initial perturbation. In a plane perpendicular to the direction of inhomogeneity, the large-time slab-guided patterns are confined to a narrow sector about $\vec{B}_0$, with some wavefronts propagating toward $\vec{B}_0$. We conclude that the phase and group diagrams lay the necessary framework for understanding the complicated time-dependent behavior of impulsive waves.

astro-ph.SR↗

arXiv2022

Oblique Quasi-Kink Modes in Solar Coronal Slabs Embedded in an Asymmetric Magnetic Environment: Resonant Damping, Phase and Group Diagrams

There has been considerable interest in magnetoacoustic waves in static, straight, field-aligned, one-dimensional equilibria where the exteriors of a magnetic slab are different between the two sides. We focus on trapped, transverse fundamental, oblique quasi-kink modes in pressureless setups where the density varies continuously from a uniform interior (with density $ρ_{\rm i}$) to a uniform exterior on either side (with density $ρ_{\rm L}$ or $ρ_{\rm R}$), assuming $ρ_{\rm L}\leρ_{\rm R}\leρ_{\rm i}$. The continuous structuring and oblique propagation make our study new relative to pertinent studies, and lead to wave damping via the Alfv$\acute{\rm e}$n resonance. We compute resonantly damped quasi-kink modes as resistive eigenmodes, and isolate the effects of system asymmetry by varying $ρ_{\rm i}/ρ_{\rm R}$ from the ``Fully Symmetric'' ($ρ_{\rm i}/ρ_{\rm R}=ρ_{\rm i}/ρ_{\rm L}$) to the ``Fully Asymmetric'' limit ($ρ_{\rm i}/ρ_{\rm R}=1$). We find that the damping rates possess a nonmonotonic $ρ_{\rm i}/ρ_{\rm R}$-dependence as a result of the difference between the two Alfv$\acute{\rm e}$n continua, and resonant absorption occurs only in one continuum when $ρ_{\rm i}/ρ_{\rm R}$ is below some threshold. We also find that the system asymmetry results in two qualitatively different regimes for the phase and group diagrams. The phase and group trajectories lie essentially on the same side (different sides) relative to the equilibrium magnetic field when the configuration is not far from a ``Fully Asymmetric'' (``Fully Symmetric'') one. Our numerical results are understood by making analytical progress in the thin-boundary limit, and discussed for imaging observations of axial standing modes and impulsively excited wavetrains.

astro-ph.SR↗

arXiv2022

Influence of fine structures on gyrosynchrotron emission of flare loops modulated by sausage modes

Sausage modes are one leading mechanism for interpreting short period quasi-periodic pulsations (QPPs) of solar flares. Forward modeling their radio emission is crucial for identifying sausage modes observationally and for understanding their connections with QPPs. Using the numerical output from three-dimensional magnetohydrodynamic (MHD) simulations, we forward model the gyrosynchrotron (GS) emission of flare loops modulated by sausage modes and examine the influence of loop fine structures. The temporal evolution of the emission intensity is analyzed for an oblique line of sight crossing the loop center. We find that the low- and high-frequency intensities oscillate in-phase at the period of sausage modes for models with or without fine structures. For low-frequency emissions where the optically thick regime arises, the modulation magnitude of the intensity is dramatically reduced by the fine structures at some viewing angles. On the contrary, for high-frequency emissions where the optically thin regime holds, the effect of fine structures or viewing angle is marginal. Our results show that the periodic intensity variations of sausage modes are not wiped out by the fine structures, and sausage modes remains a promising candidate mechanism for QPPs even when flare loops are fine-structured.

astro-ph.SR↗

arXiv2022

Impulsively Generated Kink Wave Trains in Solar Coronal Slabs

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 nonlinear 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 nonetheless arise due to nonlinearity, 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.

astro-ph.SR↗

arXiv2021

Analysis of Magnetohydrodynamic Perturbations in Radial-field Solar Wind from Parker Solar Probe Observations

We report analysis of sub-Alfvénic magnetohydrodynamic (MHD) perturbations in the low-\b{eta} radial-field solar wind using the Parker Solar Probe spacecraft data from 31 October to 12 November 2018. We calculate wave vectors using the singular value decomposition method and separate the MHD perturbations into three types of linear eigenmodes (Alfvén, fast, and slow modes) to explore the properties of the sub-Alfvénic perturbations and the role of compressible perturbations in solar wind heating. The MHD perturbations there show a high degree of Alfvénicity in the radial-field solar wind, with the energy fraction of Alfvén modes dominating (~45%-83%) over those of fast modes (~16%-43%) and slow modes (~1%-19%). We present a detailed analysis of a representative event on 10 November 2018. Observations show that fast modes dominate magnetic compressibility, whereas slow modes dominate density compressibility. The energy damping rate of compressible modes is comparable to the heating rate, suggesting the collisionless damping of compressible modes could be significant for solar wind heating. These results are valuable for further studies of the imbalanced turbulence near the Sun and possible heating effects of compressible modes at MHD scales in low-\b{eta} plasma.

astro-ph.SR↗

arXiv2021

Fast Sausage Oscillations in Coronal Loops with Fine Structures

Fast sausage modes (FSMs) in flare loops have long been invoked to account for rapid quasi-periodic pulsations (QPPs) with periods of order seconds in flare lightcurves. However, most theories of FSMs in solar coronal cylinders assume a perfectly axisymmetric equilibrium, an idealized configuration apparently far from reality. In particular, it remains to examine whether FSMs exist in coronal cylinders with fine structures. Working in the framework of ideal magnetohydrodynamics (MHD), we numerically follow the response to an axisymmetric perturbation of a coronal cylinder for which a considerable number of randomly distributed fine structures are superposed on an axisymmetric background. The parameters for the background component are largely motivated by the recent IRIS identification of a candidate FSM in Fe XXI 1354 Å~observations. We find that the composite cylinder rapidly settles to an oscillatory behavior largely compatible with a canonical trapped FSM. This happens despite that kink-like motions develop in the fine structures. We further synthesize the Fe XXI 1354 Å~emissions, finding that the transverse Alfvén time characterizes the periodicities in the intensity, Doppler shift, and Doppler width signals. Distinct from the case without fine structuring, a non-vanishing Doppler shift is seen even at the apex. We conclude that density-enhanced equilibria need not be strictly axisymmetric to host FSM-like motions in general, and FSMs remain a candidate interpretation for rapid QPPs in solar flares.

astro-ph.SR↗

arXiv2021

Observations of an Electron-cold Ion Component Reconnection at the Edge of an Ion-scale Antiparallel Reconnection at the Dayside Magnetopause

Solar wind parameters play a dominant role in reconnection rate, which controls the solar wind-magnetosphere coupling efficiency at Earth's magnetopause. Besides, low-energy ions from the ionosphere, frequently detected on the magnetospheric side of the magnetopause, also affect magnetic reconnection. However, the specific role of low-energy ions in reconnection is still an open question under active discussion. In the present work, we report in situ observations of a multiscale, multi-type magnetopause reconnection in the presence of low-energy ions using NASA's Magnetospheric Multiscale data on 11 September 2015. This study divides ions into cold and hot populations. The observations can be interpreted as a secondary reconnection dominated by electrons and cold ions located at the edge of an ion-scale reconnection. This analysis demonstrates a dominant role of cold ions in the secondary reconnection without hot ions' response. Cold ions and electrons are accelerated and heated by the secondary process. The case study provides observational evidence for the simultaneous operation of antiparallel and component reconnection. Our results imply that the pre-accelerated and heated cold ions and electrons in the secondary reconnection may participate in the primary ion-scale reconnection affecting the solar wind-magnetopause coupling and the complicated magnetic field topology affect the reconnection rate.

physics.space-ph↗

arXiv2021

Forward Modeling of Simulated Transverse Oscillations in Coronal Loops and the Influence of Background Emission

We simulate transverse oscillations in radiatively cooling coronal loops and forward-model their spectroscopic and imaging signatures, paying attention to the influence of background emission. The transverse oscillations are driven at one footpoint by a periodic velocity driver. A standing kink wave is subsequently formed and the loop cross-section is deformed due to the Kelvin-Helmholtz instability, resulting in energy dissipation and heating at small scales. Besides the transverse motions, a long-period longitudinal flow is also generated due to the ponderomotive force induced slow wave. We then transform the simulated straight loop to a semi-torus loop and forward-model their spectrometer and imaging emissions, mimicking observations of Hinode/EIS and SDO/AIA. We find that the oscillation amplitudes of the intensity are different at different slit positions, but are roughly the same in different spectral lines or channels. X-t diagrams of both the Doppler velocity and the Doppler width show periodic signals. We also find that the background emission dramatically decreases the Doppler velocity, making the estimated kinetic energy two orders of magnitude smaller than the real value. Our results show that background subtraction can help recover the real oscillation velocity. These results are helpful for further understanding transverse oscillations in coronal loops and their observational signatures. However, they cast doubt on the spectroscopically estimated energy content of transverse waves using the Doppler velocity.

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