Proof for small-scale torsional Alfvén waves within the photo voltaic corona

Data description

The information had been obtained on 30 October 2023 by the Cryo-NIRSP instrument situated on the US National Science Foundation’s DKIST. Cryo-NIRSP can observe the corona with a spatial sampling of ~0.12″ (~87 km) alongside the slit and with temporal cadences of 1 s. The Cryo-NIRSP information used listed below are a sit-and-stare statement utilizing the 0.5″-wide spectrograph slit with 0.12″ spatial sampling alongside the slit. The spatial decision of Cryo-NIRSP is optical-limited to round 0.3″ and appears to be seeing-limited to 0.6″ (as a consequence of terrestrial atmospheric seeing)⁵⁰.

The Cryo-NIRSP instrument was tuned to look at the 1,074.7 nm (iron Xiii) coronal emission line and offered high-quality spectra with a spectral dispersion of 4.4 × 10⁻¹² m per pixel and a resolving energy of R ≈ 48,000. The photo voltaic spectrum was noticed between 1,072 nm and 1,076 nm. This line has been extensively used to probe coronal dynamics^{14,17,18,19,53} and is resolved properly by Cryo-NIRSP⁵¹. Supplementary Figs. 1 and a pair of comprise an instance spectrum and the corresponding suits to the road profile, together with a dialogue on the measurement uncertainty related to the Doppler velocities.

The goal area for the observations was centred on (X, Y) = (898″, −568″) (a peak of 0.1R_☉ above the limb). The pointing stability of the information was assessed by analyzing the motions of the fantastic construction, and it appears to be ≲0.1″. Hence, the soundness of the pointing was higher than the estimated seeing decision alongside the slit. Further details about the dataset will be present in ref. ⁵⁰.

Data processing

High-frequency noise within the information can result in points when figuring out the fine-scale coronal constructions. The noise arises from temporal sources (for instance, photon shot noise and variations in seeing) and spatial sources (for instance, detector artefacts). Hence, a two-dimensional Butterworth low-pass filter was utilized each to line depth and Doppler velocity information. The outcomes of the filtering are proven in Fig. 1b,c.

For the road depth information, the noise part has δI < 0.15μB_☉, which was considerably smaller than the coronal emission (I < 15μB_☉), though it may be comparable in magnitude to the additional emission from the overdense constructions. The related spatial energy spectrum for the unfiltered information reveals power-law behaviour all the way down to the noise restrict (Supplementary Fig. 4), indicating the presence of multiscale constructions inside the corona^65,66. The filtering basically smooths the sign over small scales. Hence, there’s a lack of energy for spatial variability beneath scales of two Mm and temporal variability beneath 30 s. Thus, the processed information prohibit measurements to motions of comparatively large-scale (in contrast with the instrument decision) constructions and dynamics.

The Doppler velocity information have an estimated noise stage of δv < 0.1 km s⁻¹ (see Supplementary Information for a dialogue). This worth was considerably decreased after the appliance of the low-pass filter. We notice that the ability spectra proven in Fig. 5c additionally point out the impression of the filtering.

Analysis of kink motions

The location of the fine-scale construction was traced utilizing the NUWT algorithm²¹, which inserts a Gaussian to areas of enhanced brightness (overdense constructions), with the potential to offer subpixel accuracy on the bodily location of the constructions. The evaluation was carried out with unsharp masked line amplitude information, which had additionally been high-pass filtered spatially with a boxcar common filter (9 Mm vast). This filtering eliminated the dominant, large-scale coronal emission and left solely the brightness variations on scales smaller than 9 Mm (Fig. 1b). The places of the height enhanced brightness are traced in time to offer a time sequence of the central location of the overdense constructions. All traced options are proven in Fig. 5a. POS velocities are calculated from the numerical spinoff of the time sequence.

Non-thermal linewidths

Non-thermal linewidths for the dataset used listed below are proven in Extended Data Fig. 5. The non-thermal part of the linewidth ξ is estimated utilizing the next expression⁶⁷:

the place w_I is the instrumental spectral point-spread perform, FWHM is the full-width at half-maximum, T is the ion temperature (assumed right here to be the height line formation temperature for iron Xiii, ~1.6 MK), ok_B is the Boltzmann fixed and M is the ion mass.

The residual Doppler sign

After estimating the places of the flux tubes, we extract the Doppler velocity alerts on the flux-tube centre. Analytical and numerical modelling signifies that this Doppler sign represents the kink mode contribution to the LOS movement. We notice that the Doppler sign may embrace contributions from any non-torsional movement, though different modes (fluting and rotational modes) haven’t been recognized in coronal constructions. The focus right here is on isolating any n = 0 torsional alerts; therefore, within the following: (1) We justify why the sign extracted on the flux-tube centre doesn’t comprise any sign from potential n = 0 modes. (2) We focus on why the residual Doppler sign recognized as a torsional movement doesn’t come up from the kink mode or a higher-azimuthal-order Alfvén mode. (3) We focus on various dynamics that might result in alerts like these of the torsional mode.

Figure 3 shows the rate vectors for the kink mode (Fig. 3a) and the torsional Alfvén modes (Fig. 3b for n = 0 and Supplementary Fig. 7a for n = 1) at their most values (with a temporal dependence of e^iωt, the place ω = 2πf is the angular frequency). The calculations had been carried out for a magnetized cylinder with radius R and a piecewise, discontinuous density profile:

the place ρ_i is the interior density and ρ_e is the exterior density. The magnetic area is homogeneous. Details of tips on how to calculate the mode amplitude profiles (and, therefore, velocity vectors) will be present in Supplementary Information.

For the n = 0 torsional mode, regardless of which LOS is taken with respect to the flux tube, the built-in velocity is at all times zero on the centre of the flux tube, as the rate is at all times perpendicular to the radial course (v_θ solely). One can crudely ahead mannequin the anticipated Doppler velocity sign by taking 〈v_LOS〉, the imply of the rate sign alongside the LOS⁶⁸. The calculation of the imply velocity is weighted with respect to the sq. of the density to imitate the emission measure weighting in an optically skinny plasma (with no photo-excitation). The projected LOS Doppler velocity (Fig. 3d) reveals that the n = 0 torsional mode has zero velocity on the flux-tube centre, whereas the boundaries alternate from blue to purple (a well known consequence⁶⁹). For the n = 1 Alfvén mode, the movement can also be predominantly related to the azimuthal velocities, though there’s a small radial part, which ends up in the seen symmetry throughout the flux-tube axis (Supplementary Fig. 7b).

By distinction, the kink mode has a continuing velocity amplitude throughout the cross part of the cylinder, and the Doppler velocity signature is dependent upon the LOS angle with respect to the course of movement. If movement is perpendicular to the LOS, there isn’t a Doppler sign. There is not any contribution from the interior plasma, and the exterior velocities cancel out as a consequence of symmetry. At different LOS angles relative to the aircraft of the kink movement, the Doppler velocity profile peaks on the flux-tube centre and is close to fixed throughout the cross part. The Doppler velocity sign within the exterior area is 180° out of part with the Doppler sign within the inside area and has a a lot smaller amplitude.

The above dialogue assumes that the coronal constructions will be modelled as discrete, remoted density enhancements, which isn’t sensible. Coronal constructions are thought to transition easily from the interior to exterior plasma. Alfvén waves are recognized to propagate alongside magnetic surfaces. The mannequin used assumes that the coronal constructions will be modelled as a magnetized cylinder and constrains the magnetic surfaces to be inside the cylinder. However, in the true corona, gradients within the magnetic area and modes usually are not strictly confined to the visible boundaries^9,10. In precept, a number of Alfvén modes may exist if a number of magnetic surfaces are current⁷⁰, which may clarify the obvious extension of the torsional Alfvén modes past the visible boundaries of the noticed flux tubes (as famous in the principle textual content).

A easily various density profile from the tube inside to the outside may result in resonance results and excite additional rotational motions (principally n = 1 Alfvén modes) inside the area of easily various density^12,13,71. The presence of those rotational modes within the present observations relies upon upon the damping size of the kink mode, which is proportional to the density distinction (ρ_i/ρ_e)⁷². In common, kink modes in open-field areas appear to be weakly damped¹⁷, which is according to the small density contrasts in coronal holes inferred from imaging observations⁶⁶.

3D MHD simulations

To validate the strategy for extracting torsional wave alerts, we carried out a 3D MHD simulation, specializing in transverse waves in coronal waveguides with a ahead modelling of the anticipated infrared radiation. The simulation set-up intently follows ref. ⁵⁵, with a stratified, density-enhanced magnetic flux tube oriented perpendicular to the photo voltaic floor, which might mimic open coronal constructions⁷³. The tube has a radius of 1 Mm, with each the tube axis and the background magnetic area (~10 G) aligned alongside the z axis. The density distinction was initially set as ρ_i/ρ_e = 3, with inside density ρ_i = 7.5 × 10⁻¹⁵ g cm⁻³. We additionally launched a boundary layer (width = 0.6 Mm) with a hyperbolic tangent profile. The preliminary set-up was relaxed for two,400 s to realize an almost magnetostatic state with out notable background velocity. The simulation area spans from [−3, 3] Mm × [−3, 3] Mm × [0, 150] Mm, with a uniform grid of 256 × 256 × 256 cells. The 3D time-dependent supreme MHD equations had been solved utilizing the PLUTO code⁷⁴. Further particulars of the mannequin will be present in ref. ⁵⁵.

A steady velocity driver was utilized on the decrease boundary of the flux tube to generate each kink and torsional Alfvén waves (Supplementary Fig. 6), as noticed by Cryo-NIRSP. The driver was horizontal (solely within the x–y aircraft) and localized within the flux-tube area, in order that it acted as a superposition of a kink wave driver (with a interval of 300 s) and a torsional Alfvén wave driver (durations various radially from 150 s to 280 s). The velocity amplitudes for each drivers had been set to eight km s⁻¹, and the kink wave driver was aligned alongside the x axis. Further particulars concerning the wave driver are offered in Supplementary Information.

The wave driver generated a combination of kink and torsional Alfvén waves, which, in precept, resembles our observational outcomes. Forward modelling was undertaken with FoMo⁷⁵ to synthesize the iron Xiii 1,074.7-nm spectral profiles for sure LOS angles. We centered on the case when the LOS kinds a forty five° angle to the x axis (course of the kink driver) and lies within the horizontal (x–y) aircraft. The synthesized spectral profiles had been built-in alongside the LOS to offer two-dimensional distributions of the road depth and Doppler velocity. Time versus distance maps had been constructed at z = 50 Mm (Fig. 4a,b).

The time versus distance map for the Doppler velocity primarily displays the attribute of kink waves, with periodic redshifts and blueshifts throughout the flux tube^55,76 (see additionally Fig. 4b). However, after we subtracted the Doppler velocity on the centre place (indicated by the blue dashed line), the residual velocity revealed a typical torsional wave sign (Fig. 4c) that intently resembles Figs. 2b and 3d. In addition, the residual Doppler velocities on the higher and decrease edges of the flux tube clearly have an out-of-phase sample.

The residual Doppler velocity (Fig. 4c) additionally reveals some artefacts arising from the subtraction methodology. The LOS Doppler velocity profile for the kink mode will not be uniform throughout the flux tube (for instance, Fig. 3d). Hence, subtracting the LOS velocity worth on the centre of the flux tube nonetheless leaves some impression of the kink mode. This results in places the place the torsional mode sign appears to increase past the boundaries of the flux tube (for instance, round 200 s, 600 s, 900 s and 1,200 s). There can also be a transparent impression on the residual Doppler velocity sample past the flux tube the place the oscillation will not be current. Hence, the residual Doppler velocity sign is doubtlessly legitimate just for the flux tube upon which it’s calculated. However, the kink modes on a number of neighbouring flux tubes can oscillate coherently²⁰, and in such conditions, the residual Doppler velocity could also be legitimate past the flux tube. Hence, we advise that the residual Doppler velocity alerts from past the flux-tube boundary must be interpreted with warning.

Mode decomposition

The time sequence for the residual Doppler velocity is made up of fluctuations that span a variety of timescales (for instance, Fig. 2c), which is captured by the ability spectrum of the time sequence (Fig. 5c). This is anticipated, because the waves are considered excited by photo voltaic convection, which is a broadband turbulent driver. A earlier evaluation of kink modes revealed that fluctuations with a attribute timescale normally persist for just one to 4 cycles, and these can seem as wave packets, with a number of timescales excited concurrently²¹. This signifies that Fourier evaluation will not be essentially one of the best device for finding out the fluctuations. Hence, we use EMD⁷⁷, a well-established device for analysing non-stationary sequence. Unlike Fourier evaluation, there are not any predefined foundation capabilities. Instead, the strategy defines the IMFs adaptively from the information. The IMFs are domestically outlined to allow a time versus frequency evaluation, which contrasts with the worldwide frequency illustration in Fourier evaluation. We have confirmed that related outcomes will be obtained utilizing frequency filtering with Fourier strategies.

Figure 2f–i and Extended Data Figs. 1–3 present the IMFs derived from the residual Doppler velocities. Only the IMFs related to the longer timescales are proven as a result of the IMFs related to shorter timescales comprise primarily information noise or have amplitudes ≲0.05 km s⁻¹. In some circumstances, a number of the alerts with small amplitudes could also be real. However, we selected to be conservative and centered solely on the motions with bigger amplitudes.

Note that the IMFs can undergo from mode-mixing, so {that a} single IMF might comprise fluctuations on disparate timescales or related fluctuations could also be cut up throughout completely different IMFs. Mode-mixing might be a problem in a number of the examples proven right here, as the ability spectrum of fluctuations reveals that the alerts are broadband and wouldn’t have well-separated timescales. Hence, the strict circumstances we positioned on whether or not a fluctuation constitutes a torsional mode (principal textual content), comparable to equal amplitudes and simultaneous zero crossings, may doubtlessly be relaxed.

Monte Carlo simulations

It is anticipated from earlier work that LOS integration of the radiation via the optically skinny corona results in a discount in wave amplitudes for the kink modes^29,61. Here we develop Monte Carlo simulations to point out that that is additionally the case for the torsional Alfvén mode.

It follows from a consideration of optically skinny radiation that the extent of discount of the LOS Doppler velocity (from the sum of oscillations with a random part) will, in precept, be related for each modes (the derivation is in Supplementary Information). Note that the discount is dependent upon the quantity of ambient plasma in contrast with the quantity of the flux tube. Considering Fig. 3, the LOS averaging of the torsional mode with v = 20 km s⁻¹ for a single cylinder provides v_LOS = 6 km s⁻¹. By distinction, the LOS averaging of the kink mode with v = 20 km s⁻¹ for a single cylinder provides v_LOS = 18 km s⁻¹ (the case in Fig. 3 incurs an extra discount because of the 35° angle with the LOS). Hence, there’s a notable discount within the torsional Alfvén mode amplitude when integrating over the LOS in contrast with the kink mode. This is as a result of the course of the rate vectors inside the flux tube for the torsional mode relies upon upon the spatial location, whereas for the kink mode, the rate vectors all level in the identical course. The lower within the torsional wave amplitude is bigger (on common) than the discount within the kink mode amplitude because of the random polarization angle with respect to the LOS (which ends up in a (1/surd 2) common lower, bringing the 18 km s⁻¹ all the way down to ~13 km s⁻¹).

This was confirmed by a Monte Carlo simulation impressed by strategies used beforehand⁷⁸. We present a dialogue in Supplementary Information as to why we consider this method supplies helpful outcomes, whereas acknowledging the restrictions.

Using analytic fashions of the waves, many (200) oscillating flux tubes had been stacked alongside the LOS, every with a distinct part, interval and amplitude. The part and interval had been drawn from uniform distributions, ({mathscr{U}})(0, 2π) and ({mathscr{U}}(100,,300)), respectively. The amplitude was drawn from log-normal distributions²¹, with the imply amplitude various between 4 km s⁻¹ and 30 km s⁻¹ for various simulations. The LOS velocity was calculated by integrating the velocities over all of the flux tubes and weighting with the emission measure. The emission measure decreased steadily in direction of and away from the central flux tube, with a decay size of 800 Mm. This was carried out for the Alfvén and kink modes individually. As anticipated, we discovered that each modes suffered a considerable discount within the measured Doppler velocity in contrast with the precise mode amplitudes (Extended Data Fig. 4). The LOS velocity of the Alfvén mode was roughly half that of the kink mode, according to the expectations from the one flux tube. The outcomes help our suggestion that precise Alfvén and kink wave amplitudes being equal is conservative.

We ran a second Monte Carlo mannequin to research how the LOS integration influences the emergent Doppler velocity sign past amplitudes. In the simulation, we once more stacked a number of single flux tubes oscillating alongside the LOS, specializing in solely the torsional Alfvén mode. The amplitude and frequency of the torsional Alfvén mode had been outlined by a power-law spectrum. Each flux tube had its personal time sequence that obeyed the identical energy regulation. The Doppler velocity via the flux tubes was averaged alongside a LOS with a weakly decaying emission measure. This will be thought-about a worst-case situation, because the emergent depth was solely weakly weighted to the construction with the biggest emission, as many different constructions alongside the LOS have an identical emissivity.

The emergent Doppler sign was a mix of all Doppler alerts alongside the LOS and didn’t match the Doppler velocity sign on the ‘brightest’ flux tube (Supplementary Fig. 8). The emergent sample is a superposition of the motions. However, there are three essential outcomes from this: (1) The noticed Doppler velocities present a sample of fluctuating purple/blue asymmetry throughout the flux tube. (2) The energy spectrum for the LOS Doppler velocity has the identical spectral slope because the enter spectrum. (3) The energy is decreased by an identical magnitude as in earlier Monte Carlo simulations. Hence, within the worst case, the amplitude fluctuations and the timescales of the LOS Doppler velocity are nonetheless consultant of the behaviour for torsional modes on particular person flux tubes alongside the LOS. In actuality, the emergent Doppler velocity sample might be weighted in direction of the options seen within the line amplitude information, that’s these above the limb. This provides us confidence that the noticed alerts we see are real indications of the torsional movement.

Other sources of purple and blue asymmetry

We additionally acknowledge different potential dynamics within the corona that might result in a purple/blue asymmetry throughout the flux tubes. The purple/blue asymmetry throughout the loop might be as a consequence of transverse kink motions in particular person flux tubes, both within the background or foreground. We count on that a number of constructions will current kink motions alongside the LOS, and there are two circumstances to debate. For kink waves pushed by stochastic, uncorrelated drivers, then the amplitudes, durations and part of two impartial oscillations are unlikely to be the identical. If the waves are correlated, which might happen over spatial scales of 4–8 Mm (refs. ^20,50), they are going to be in part, so there can be no uneven sign. The identical reasoning applies to turbulent motions as properly.

Another origin of the purple/blue asymmetry throughout the loop is because of a background gradient. One choice might be oppositely directed flows from neighbouring loops located alongside the LOS, which can happen if heating occasions are extremely localized. The background gradient case will be dominated out if there are a number of cycles or related magnitude alerts both aspect of the flux-tube centre.