Transverse Velocity Perturbations Research Articles

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.

Quasi-periodic spicule-like cool jets driven by Alfvén pulses

ABSTRACT We perform a 2.5D magnetohydrodynamic simulation to gain a comprehensive understanding of the formation of spicule-like cool jets caused by initial transverse velocity pulses akin to Alfvén pulses in the solar chromosphere. We invoke multiple velocity (Vz) pulses between 1.5 and 2.0 Mm in the solar atmosphere, which create the initial transverse velocity perturbations. These pulses transfer energy non-linearly to the field-aligned perturbations via the ponderomotive force. This physical process further creates magnetoacoustic shocks followed by quasi-periodic plasma motions in the solar atmosphere. The field-aligned magnetoacoustic shocks move upwards, which subsequently causes the quasi-periodic rise and fall of chromospheric plasma into the overlying corona as thin and cool spicule-like jets. The magnitude of the initial applied transverse velocity pulses is taken in the range of 50–90 km s−1. These pulses are found to be strong enough to generate spicule-like jets. We analyse the evolution, kinematics and energetics of these spicule-like jets. We find that the transported mass flux and kinetic energy density are substantial in the local solar corona. These mass motions generate in situ quasi-periodic oscillations on the scale of ≃ 4.0 min above the transition region.

Investigation of the spectral characteristics of a thermoviscous fluid flow in an annular channel

The flow of a thermoviscous model fluid in an annular channel with a given temperature field is considered. The problem of the stability of the flow of a thermoviscous fluid is solved on the basis of the generalized equation by the spectral method of expansion in Chebyshev polynomials of the first kind. The effect of taking into account the exponential dependence of the fluid viscosity on temperature and channel geometry on the spectral characteristics of the equation of hydrodynamic stability of an incompressible fluid flow in a flat channel for various wall temperatures is studied. Spectral patterns of eigenvalues of the generalized equation are constructed. The spectral characteristics determine the structure of the eigenfunctions and the critical parameters of the flow of a thermoviscous fluid. In this case, the eigenfunctions demonstrate the behavior of transverse velocity perturbations, their possible growth or decay with time. It is shown that the structure of the spectra largely depends both on the properties of the liquid, determined by the index of the functional dependence of viscosity, and on the geometry of the channel. It has been established that for small values of the thermoviscosity parameter, the spectrum is comparable to the spectrum for an isothermal fluid flow in a flat channel, however, as it increases, the number of eigenvalues and their density increase, that is, there are more points at which the problem has nonzero amplitudes of transverse velocity perturbations. The stability of a thermoviscous fluid flow depends on the presence of an eigenvalue with a positive imaginary part among the entire set of found eigenvalues for fixed parameters of the Reynolds number and wave number. It is shown that, at fixed values of the Reynolds number and wave number, the flow can become unstable with an increase in the thermoviscosity parameter.

Forward modelling of MHD waves in braided magnetic fields

Aims. We investigate synthetic observational signatures generated from numerical models of transverse waves propagating in complex (braided) magnetic fields. Methods. We consider two simulations with different levels of magnetic field braiding and impose periodic, transverse velocity perturbations at the lower boundary. As the waves reflect off the top boundary, a complex pattern of wave interference occurs. We applied the forward modelling code FoMo and analysed the synthetic emission data. We examined the line intensity, Doppler shifts, and kinetic energy along several line-of-sight (LOS) angles. Results. The Doppler shift perturbations clearly show the presence of the transverse (Alfvénic) waves. However, in the total intensity, and running difference, the waves are less easily observed for more complex magnetic fields and may be indistinguishable from background noise. Depending on the LOS angle, the observable signatures of the waves reflect some of the magnetic field braiding, particularly when multiple emission lines are available, although it is not possible to deduce the actual level of complexity. In the more braided simulation, signatures of phase mixing can be identified. We highlight possible ambiguities in the interpretation of the wave modes based on the synthetic emission signatures. Conclusions. Most of the observables discussed in this article behave in the manner expected, given knowledge of the evolution of the parameters in the 3D simulations. Nevertheless, some intriguing observational signatures are present. Identifying regions of magnetic field complexity is somewhat possible when waves are present; although, even then, simultaneous spectroscopic imaging from different lines is important in order to identify these locations. Care needs to be taken when interpreting intensity and Doppler velocity signatures as torsional motions, as is done in our setup. These types of signatures are a consequence of the complex nature of the magnetic field, rather than real torsional waves. Finally, we investigate the kinetic energy, which was estimated from the Doppler velocities and is highly dependent on the polarisation of the wave, the complexity of the background field, and the LOS angles.

On Alfvén wave propagation along a circle on dipolar coordinates

The multipolar representation of the magnetic field has, for the lowest-order term, a magnetic dipole that dominates the far field. Thus the far-field representation of the magnetic field of the Earth, Sun and other celestial bodies is a dipole. Since these bodies consist of or are surrounded by plasma, which can support Alfvén waves, their propagation along dipole magnetic field lines is considered using a new coordinate system: dipolar coordinates. The present paper introduces multipolar coordinates, which are an example of conformal coordinates; conformal coordinates are orthogonal with equal scale factors, and can be extended from the plane to space, for instance as cylindrical or spherical dipolar coordinates. The application considered is to Alfvén waves propagating along a circle, that is a magnetic field line of a dipole, with transverse velocity and magnetic field perturbations; the various forms of the wave equation are linear second-order differential equations, with variable coefficients, specified by a background magnetic field, which is force free. The absence of a background magnetic force leads to a mean state of hydrostatic equilibrium, specified by the balance of gravity against the pressure gradient, for a perfect gas or incompressible liquid. The wave equation is simplified to a Gaussian hypergeometric type in the case of zero frequency, otherwise, for non-zero frequency, an extended Gaussian hypergeometric equation is obtained. The solution of the latter specifies the magnetic field perturbation spectrum, and also, via a polarisation relation, the velocity perturbation spectrum; both are plotted, over half a circle, for three values of the dimensionless frequency.

Magnetohydrodynamic waves in braided magnetic fields

Aims. We investigate the propagation of transverse magnetohydrodynamic (MHD) wave fronts through a coronal plasma containing a braided magnetic field. Methods. We performed a series of three dimensional MHD simulations in which a small amplitude, transverse velocity perturbation is introduced into a complex magnetic field. We analysed the deformation of the wave fronts as the perturbation propagates through the braided magnetic structures and explore the nature of Alfvénic wave phase mixing in this regime. We considered the effects of viscous dissipation in a weakly non-ideal plasma and evaluate the effects of field complexity on wave energy dissipation. Results. Spatial gradients in the local Alfvén speed and variations in the length of magnetic field lines ensure that small scales form throughout the propagating wave front due to phase mixing. Additionally, the presence of complex, intricate current sheets associated with the background field locally modifies the polarisation of the wave front. The combination of these two effects enhances the rate of viscous dissipation, particularly in more complex field configurations. Unlike in classical phase mixing configurations, the greater spatial extent of Alfvén speed gradients ensures that wave energy is deposited over a larger cross-section of the magnetic structure. Further, the complexity of the background magnetic field ensures that small gradients in a wave driver can map to large gradients within the coronal plasma. Conclusions. The phase mixing of transverse MHD waves in a complex magnetic field will progress throughout the braided volume. As a result, in a non-ideal regime wave energy will be dissipated over a greater cross-section than in classical phase mixing models. The formation rate of small spatial scales in a propagating wave front is a function of the complexity of the background magnetic field. As such, if the coronal field is sufficiently complex it remains plausible that phase mixing induced wave heating can contribute to maintaining the observed temperatures. Furthermore, the weak compressibility of the transverse wave and the observed phase mixing pattern may provide seismological information about the nature of the background plasma.

Influence of viscosity temperature dependence on the spectral characteristics of the thermoviscous liquids flow stability equation

The flow of a viscous model fluid in a flat channel with a non-uniform temperature field is considered. The problem of the stability of a thermoviscous fluid is solved on the basis of the derived generalized Orr-Sommerfeld equation by the spectral decomposition method in Chebyshev polynomials. The effect of taking into account the linear and exponential dependences of the fluid viscosity on temperature on the spectral characteristics of the hydrodynamic stability equation for an incompressible fluid in a flat channel with given different wall temperatures is investigated. Analytically obtained profiles of the flow rate of a thermovisible fluid. The spectral pictures of the eigenvalues of the generalized Orr-Sommerfeld equation are constructed. It is shown that the structure of the spectra largely depends on the properties of the liquid, which are determined by the viscosity functional dependence index. It has been established that for small values of the thermoviscosity parameter the spectrum compares the spectrum for isothermal fluid flow, however, as it increases, the number of eigenvalues and their density increase, that is, there are more points at which the problem has a nontrivial solution. The stability of the flow of a thermoviscous fluid depends on the presence of an eigenvalue with a positive imaginary part among the entire set of eigenvalues found with fixed Reynolds number and wavenumber parameters. It is shown that with a fixed Reynolds number and a wave number with an increase in the thermoviscosity parameter, the flow becomes unstable. The spectral characteristics determine the structure of the eigenfunctions and the critical parameters of the flow of a thermally viscous fluid. The eigenfunctions constructed in the subsequent works show the behavior of transverse-velocity perturbations, their possible growth or decay over time.

Research of eigenfuctions perturbation of the transverse component velocity thermoviscous liquids flow

The viscous model fluid flow in a plane channel with a linear temperature profile is considered. The problem of the thermoviscous fluid flow stability is solved on the basis of the previously obtained generalized Orr–Sommerfeld equation by the spectral method of decomposition into Chebyshev polynomials. We study the effect of taking into account the linear and exponential dependences of the viscosity of a liquid on temperature on the eigenfunctions of the hydrodynamic stability equation and on perturbations of the transverse velocity of an incompressible fluid in a plane channel when various wall temperatures are specified. Eigenfunctions are found numerically for two eigenvalues of the linear and exponential dependence of viscosity on temperature. Presented pictures of their own functions. The eigenfunctions demonstrate the behavior of the transverse velocity perturbations, their possible growth or attenuation over time. For the given eigenfunctions, perturbations of the transverse flow velocity of a thermoviscous fluid are obtained. It is shown that taking the temperature dependence of viscosity into account affects the eigenfunctions of the equations of hydrodynamic stability and perturbations of the transverse flow velocity. Perturbations of the transverse velocity significantly affect the hydrodynamic instability of the fluid flow. The results show that when considering the unstable eigenvalue over time, the velocity perturbations begin to grow, which leads to turbulence of the flow. The maximum values of the eigenfunctions and perturbations of the transverse velocities are shifted to the hot wall. It is seen that for an unstable eigenvalue, the perturbations of the transverse flow velocity increase over time, and for a stable one, they decay.

Periods and Damping Rates of Fast Sausage Oscillations in Multishelled Coronal Loops

Standing sausage modes are important in interpreting quasi-periodic pulsations in the lightcurves of solar flares. Their periods and damping times play an important role in seismologically diagnosing key parameters like the magnetic field strength in regions where flare energy is released. Usually such applications are based on theoretical results neglecting unresolved fine structures in magnetized loops. However, the existence of fine structuring is suggested on both theoretical and observational grounds. Adopting the framework of cold magnetohydrodynamics (MHD), we model coronal loops as magnetized cylinders with a transverse equilibrium density profile comprising a monolithic part and a modulation due to fine structuring in the form of concentric shells. The equation governing the transverse velocity perturbation is solved with an initial-value-problem approach, and the effects of fine structuring on the periods $P$ and damping times $\tau$ of global, leaky, standing sausage modes are examined. A parameter study shows that fine structuring, be it periodically or randomly distributed, brings changes of only a few percent to $P$ and $\tau$ when there are more than about ten shells. The monolithic part, its steepness in particular, plays a far more important role in determining $P$ and $\tau$. We conclude that when measured values of $P$ and $\tau$ of sausage modes are used for seismological purposes, it is justified to use theoretical results where the effects due to fine structuring are neglected.

Dust-acoustic solitary and rogue waves in a Thomas-Fermi degenerate dusty plasma

The formation and propagation of dust-acoustic (DA) solitary and rogue waves are studied in a non-relativistic degenerate Thomas-Fermi thermal dusty plasma incorporating transverse velocity perturbation effects. The electrons and ions are described by the Thomas-Fermi density distributions, whereas the dust grains are taken as dynamic and classical. By using the reductive perturbation technique, the cylindrical Kadomtsev-Petviashvili (CKP) equation is derived, which is then transformed into a Korteweg-deVries (KdV) equation by using appropriate variable transformations. The latter admits a solitary wave solution. However, when the carrier waves frequency is much smaller than the dust plasma frequency, the DA waves evolve into the nonlinear modulation instability, generating modulated wave packets in the form of Rogue waves. For the study of DA-rogue waves, the KdV equation is transformed into a self-focusing nonlinear Schrodinger equation. The variation of dust temperature and the electron density affects the nonlinearity and dispersion coefficients which suppress the amplitudes of the DA solitary and rogue waves. The present results aim to describe the nonlinear electrostatic excitations in astrophysical degenerate dense plasma.

Weakly nonlinear stability analysis of magnetohydrodynamic channel flow using an efficient numerical approach

We analyze weakly nonlinear stability of a flow of viscous conducting liquid driven by pressure gradient in the channel between two parallel walls subject to a transverse magnetic field. Using a non-standard numerical approach, we compute the linear growth rate correction and the first Landau coefficient, which in a sufficiently strong magnetic field vary with the Hartmann number as \documentclass[12pt]{minimal}\begin{document}$\mu _{1}\sim (0.814-\mathrm{i}19.8)\times 10^{-3}\textit {Ha}$\end{document}μ1∼(0.814−i19.8)×10−3Ha and \documentclass[12pt]{minimal}\begin{document}$\mu _{2}\sim (2.73-\mathrm{i}1.50)\times 10^{-5}\textit {Ha}^{-4}.$\end{document}μ2∼(2.73−i1.50)×10−5Ha−4. These coefficients describe a subcritical transverse velocity perturbation with the equilibrium amplitude \documentclass[12pt]{minimal}\begin{document}$|A|^{2}=\Re [\mu _{1}]/\Re [\mu _{2}](\textit {Re}_{c}-\textit {Re})\sim 29.8\textit {Ha}^{5}(\textit {Re}_{c}-\textit {Re}),$\end{document}|A|2=ℜ[μ1]/ℜ[μ2](Rec−Re)∼29.8Ha5(Rec−Re), which exists at Reynolds numbers below the linear stability threshold \documentclass[12pt]{minimal}\begin{document}$\textit {Re}_{c}\sim 4.83\times 10^{4}\textit {Ha}.$\end{document}Rec∼4.83×104Ha. We find that the flow remains subcritically unstable regardless of the magnetic field strength. Our method for computing Landau coefficients differs from the standard one by the application of the solvability condition to the discretized rather than continuous problem. This allows us to bypass both the solution of the adjoint problem and the subsequent evaluation of the integrals defining the inner products, which results in a significant simplification of the method.

Damping of kink waves by mode coupling

Context. Recent observations of the corona reveal ubiquitous transverse velocity perturbations that undergo strong damping as they propagate. These can be understood in terms of propagating kink waves that undergo mode coupling in inhomogeneous regions.Aims. The use of these propagating waves as a seismological tool for the investigation of the solar corona depends upon an accurate understanding of how the mode coupling behaviour is determined by local plasma parameters. Our previous work suggests the exponential spatial damping profile provides a poor description of the behaviour of strongly damped kink waves. We aim to investigate the spatial damping profile in detail and provide a guide to the approximations most suitable for performing seismological inversions.Methods. We propose a general spatial damping profile based on analytical results that accounts for the initial Gaussian stage of damped kink waves as well as the asymptotic exponential stage considered by previous authors. The applicability of this profile is demonstrated by a full parametric study of the relevant physical parameters. The implication of this profile for seismological inversions is investigated.Results. The Gaussian damping profile is found to be most suitable for application as a seismological tool for observations of oscillations in loops with a low density contrast. This profile also provides accurate estimates for data in which only a few wavelengths or periods are observed.

Klein-Gordon equations for horizontal transverse oscillations in two-dimensional coronal loops

We present a theory for hydromagnetic waves in an axi-symmetric background magnetic field in which the wave equations for the horizontal transverse magnetic field and velocity perturbations can be transformed into Klein-Gordon (KG) equations. For harmonic time variations, the KG equations become a set of ordinary differential equations that can be solved along any individual field line, subject to boundary conditions at the two ends. The solutions provide the spatial (latitudinal) profiles of the transverse magnetic field and velocity oscillations, especially in the horizontal direction, along the field line. In particular, we examine the KG solutions for two background field geometries: a local dipole field line, and a stretched global dipole field line which may approximate coronal loop geometries in the solar corona. The results yield the oscillation frequencies in agreement with observations (periods on the order of minutes), and the spatial profiles which are characteristic of a propagating type near the center of the loop and a possible evanescent type towards the footpoints of the loop. The latter solution arises when the oscillation frequency is less than a critical cut-off frequency which varies spatially along the loop. The oscillation amplitude is also affected by an adiabatic growth/decay factor along the loop. We discuss the implications of our results and future applications to coronal loop oscillations.

Three-dimensional ion-acoustic wave packet in magnetoplasmas with superthermal electrons

Three-dimensional modulated electrostatic ion-acoustic wave packet in magnetized plasma with electrons obeying kappa distribution is investigated. The standard reductive perturbation theory is used to derive the corresponding three-dimensional nonlinear Schrödinger (3D-NLS) equation which governs the dynamics, as well as the slow modulation of the ion-acoustic wave packets. Both the dispersive and the nonlinear coefficients of the 3D-NLS equation are modified by the inclusion of the transverse velocity perturbations, as well as the external magnetic field. New instability regimes arise when the ion to electron temperature ratio σi, the spectral index κ (i.e. measuring deviation from Maxwellian equilibrium), and the ion gyrofrequency ωci are presented. Importantly, the latter establishes new instability regimes that are not presented in an unmagnetized plasma.

PROPAGATING COUPLED ALFVÉN AND KINK OSCILLATIONS IN AN ARBITRARY INHOMOGENEOUS CORONA

Observations have revealed ubiquitous transverse velocity perturbation waves propagating in the solar corona. We perform three-dimensional numerical simulations of footpoint-driven transverse waves propagating in a low β plasma. We consider the cases of distorted cylindrical flux tubes and a randomly generated inhomogeneous medium. When density structuring is present, mode coupling in inhomogeneous regions leads to the coupling of the kink mode to the Alfven mode. The decay of the propagating kink wave is observed as energy is transferred to the local Alfven mode. In all cases considered, modest changes in density were capable of efficiently converting energy from the driving footpoint motion to localized Alfven modes. We have demonstrated that mode coupling efficiently couples propagating kink perturbations to Alfven modes in an arbitrary inhomogeneous medium. This has the consequence that transverse footpoint motions at the base of the corona will deposit energy to Alfven modes in the corona.

Acoustic–convective mode conversion in an aerofoil cascade

When an acoustic wave impinges on an aerofoil cascade, a convective vorticity mode is generated giving rise to transverse velocity perturbations. This mode conversion process is investigated to explain the flow dynamics observed when swirlers are submitted to incident acoustic disturbances. The phenomenon is first studied in the case of a two-dimensional aerofoil cascade using a model derived from an actuator disk theory. The model is simplified to deal with low-Mach-number flows. The velocity field on the downstream side of the cascade features two components, an axial perturbation associated with the transmitted acoustic wave and a transverse disturbance corresponding to the vorticity wave generated at the cascade trailing edge. The model provides the amplitude of both components and defines their phase shift. Numerical simulations are carried out in a second stage to validate this model in the case of a cascade operating at a low Reynolds number Rec = 2700 based on the chord length. Space–time diagrams of velocity perturbations deduced from these simulations are used to retrieve the two types of modes. Experiments are then carried out in the case of an axial swirler placed in a cylindrical duct and submitted to plane acoustic waves emitted on the upstream side of the swirler. The amplitude and phase of the two velocity components measured in the axial and azimuthal directions are found to be in good agreement with theoretical estimates and with numerical calculations. This analysis is motivated by combustion dynamics observed in flames stabilized by aerodynamic swirlers in continuous combustors.

Coupled Alfvén and kink oscillations in an inhomogeneous corona

AbstractWe perform 3D numerical simulations of footpoint-driven transverse waves propagating in a low β plasma. The presence of inhomogeneities in the density profile leads to the coupling of the driven kink mode to Alfvén modes by resonant absorption. The decay of the propagating kink wave as energy is transferred to the local Alfvén mode is in good agreement with a modified interpretation of the analytical expression derived for standing kink modes. This coupling may account for the damping of transverse velocity perturbation waves which have recently been observed to be ubiquitous in the solar corona.

COUPLED ALFVÉN AND KINK OSCILLATIONS IN CORONAL LOOPS

Observations have revealed ubiquitous transverse velocity perturbation waves propagating in the solar corona. However, there is ongoing discussion regarding their interpretation as kink or Alfven waves. To investigate the nature of transverse waves propagating in the solar corona and their potential for use as a coronal diagnostic in MHD seismology, we perform three-dimensional numerical simulations of footpoint-driven transverse waves propagating in a low β plasma. We consider the cases of both a uniform medium and one with loop-like density structure and perform a parametric study for our structuring parameters. When density structuring is present, resonant absorption in inhomogeneous layers leads to the coupling of the kink mode to the Alfven mode. The decay of the propagating kink wave as energy is transferred to the local Alfven mode is in good agreement with a modified interpretation of the analysis of Ruderman & Roberts for standing kink modes. Numerical simulations support the most general interpretation of the observed loop oscillations as a coupling of the kink and Alfven modes. This coupling may account for the observed predominance of outward wave power in longer coronal loops since the observed damping length is comparable to our estimate based on an assumption of resonant absorption as the damping mechanism.

Wave propagation in multiple flux tubes and chromospheric heating

AbstractThis investigation is a continuation of earlier work on the dynamics of the magnetic network. In a previous calculation (Hasan et al. 2005), we examined the response of a single flux tube to transverse motions of its footpoints. We now extend this analysis to a more realistic model of the network consisting of multiple flux tubes. We apply a transverse velocity perturbation uniformly along the lower boundary located at the base of the photosphere. Our 2-D MHD simulations enable us to study the complex wave pattern due to waves generated in the individual tubes as well as their interaction with those emanating from adjacent tubes. Our results show that the dominant heating of the chromosphere occurs due to slow magnetoacoustic waves in a region that is close to the central region of the flux tube.

High‐resolution investigation of shear wave anisotropy in D″ beneath the Cocos Plate

Broadband shear wave signals from 16 deep South American earthquakes are used in a high‐resolution analysis of lowermost mantle structure beneath the Cocos Plate. Shear wave splitting of 162 ScS phases is generally compatible with vertical transverse isotropy (VTI) in the D″ region, with little path length or spatial dependence across the 200 × 800 km2 area sampled. Assuming a 250 km thick D″ layer, the VTI has an average value of 0.63%. ScS‐S differential travel time anomalies for the same data indicate a lateral gradient in positive D″ transverse component velocity perturbations, increasing northward. S‐wave triplication arrivals from a shear velocity discontinuity at the top of D″ show splitting similar to the ScS phases, but with smaller magnitude. Vertical resolution remains limited, but the data are consistent with VTI material being concentrated in just the upper portion or at the top and bottom of D″.