D. B. Jess^1,2, V. E. Reznikova³, R. S. I. Ryans¹, D. J. Christian², P. H. Keys^1,4, M. Mathioudakis¹, D. H. Mackay⁵, S. Krishna Prasad¹, D. Banerjee⁶, S. D. T. Grant¹, S. Yau¹ & C. Diamond^1
1 Astrophysics Research Centre, School of Mathematics and Physics, Queen’s University Belfast, UK
² Department of Physics and Astronomy, California State University Northridge, USA
³ Center for Mathematical Plasma Astrophysics, Department of Mathematics, KU Leuven, Belgium
⁴ Solar Physics and Space Plasma Research Centre (SP²RC), University of Sheffield, UK
⁵ School of Mathematics and Statistics, University of St Andrews, UK
⁶ Indian Institute of Astrophysics, Koramangala, Bangalore, India

The high local temperatures and weak magnetic fields expected in the corona make measurements of coronal magnetism particularly challenging, often requiring observing times exceeding one hour to build up sufficient signal¹. However, a wide variety of propagating coronal wave phenomena in the immediate vicinity of active regions have been observed with periodicities and amplitudes governed by the photospheric p-mode spectrum². They often display characteristics synonymous with upwardly propagating magneto-acoustic slow-mode waves, which travel along the magnetic field lines with phase speeds equal to the local tube speed. Can the ubiquitous presence of upwardly propagating slow-mode waves in coronal structures provide us with information regarding the magnetic fields in which they are embedded?

Figure 1 | The upper panel displays a time-distance diagram of the azimuthally averaged intensities, following the application of a 4-min Fourier filter, propagating radially away from the umbral barycenter. A solid green line highlights a line-of-best-fit used to calculate the period-dependent phase speeds, which are displayed in the lower panel following the Fourier filtering of the time series, and plotted as a function of oscillatory period. Longer period waves (which preferentially occur further from the sunspot umbra) propagate with reduced phase speeds.

To answer this question³ we utilized SDO observations to study active region NOAA 11366 on 2011 December 10. The region was relatively inactive during the 75 minute observing sequence, with no large-scale eruptive phenomena. Examination of the temperature (T_e) maps (obtained using a differential emission measure, DEM, technique⁴ with AIA images) revealed that a large portion of the surrounding coronal plasma was relatively cool (~0.7 MK). A wealth of propagating oscillatory motion was visible, best in AIA’s 171 Å channel, within the confines of the coronal fans. To better characterize the presence of wave phenomena we Fourier-filtered the image sequence and re-generated new time series that had been decomposed into frequency bins, each separated by 30 s, and corresponding to periodicities of 1–11 minutes. Examination of the filtered time series clearly revealed oscillatory signatures covering a wide range of frequencies, with longer periodicities preferentially occurring at increasing distances from the sunspot umbra, and propagating with diminishing phase speeds (Figure 1). The dominant periodicity was then defined as the periodicity that had the most relative power within each AIA pixel, with dominant periodicities ~3 minutes found within the sunspot umbra, increasing rapidly within the penumbral locations, and eventually plateauing at periodicities ~13 minutes further along the coronal fan structures.

Figure 2 | A 171 Å image acquired by the STEREO–A spacecraft, where white dashed lines display heliographic co-ordinates with each vertical and horizontal line separated by 10°. The solid white box highlights the region extracted for measurements in panels (d). (b) The STEREO–A image having been ran through a high-pass filter to reveal small-scale coronal structuring. (c) The resulting filtered image (i.e. the addition of panels (a) and (b)) detailing the improved contrast associated with fine-scale coronal loops and fans. (d) A zoom-in to the region highlighted by the white boxes in panels (a–c) and rotated so the solar limb is horizontal. The white arrow indicates the direction towards the geosynchronous orbit of the SDO spacecraft.

Importantly, since the inclination angles of the coronal fans with respect to SDO’s line-of-sight are likely to change with distance along the structure, our measured phase speeds of the propagating waves will have been underestimated to a certain degree, particularly for locations close to the umbral barycenter where the magnetic field inclination was minimal. We utilized observations from the leading STEREO-A spacecraft, which was able to see the features from a side-on perspective (Figure 2). We then computed the spatial variations of the structures with atmospheric height, and hence derived the inclination angles of the features with respect to SDO’s line-of-sight (aided by a coronal field model extrapolated from an HMI vector magnetogram). From this, we calculated the true phase speeds of the waves, finding tube speeds, c_t, of approximately 185 km s^-1 towards the center of the umbra, decreasing radially within several Mm to values of approximately 25 km s^-1. Since the sound speed, c_s, is dependent upon the local plasma temperature, we used the T_e maps to derive the spatial distribution of the sound speeds. With the distributions of c_s and c_t known, we estimated the spatial distribution of the Alfvén speeds, v_A, through the relationship, v_A = c_sc_t / (c_s² – c_t²)^1/2.

Figure 3 | Upper left: The measured tube speed (c_t; black line with grey error contours) following compensation from inclination angle effects to provide the true tube speed irrespective of angle ambiguities, the temperature-dependent sound speed (c_s; blue line with light blue error contours) calculated from the local plasma temperature, and the subsequently derived Alfvén speed (v_A; black line with red error contours) displayed as a function of distance from the underlying umbral barycenter. Lower left: The magnitude of the magnetic field strength, B, plotted as a function of radial distance from the umbral barycenter, where the vertical dashed lines represent the umbral and penumbral boundaries established from photospheric SDO/HMI continuum images at approximately 3.8 Mm and 10.1 Mm, respectively, from the center of the sunspot umbra. Right panels: Co-spatial images providing a two-dimensional representation of the magnitude of the photospheric magnetic field strength (lower; displayed on a log-scale and saturated between 30 and 3000 G to aid clarity), the EUV AIA 171 Å intensity (middle lower), the electron densities (middle upper; saturated between log(n_e) = 9.3 and log(n_e) = 9.6 to assist visualization) and the reconstructed magnitude of the coronal magnetic field strength (upper; displayed on a log-scale and saturated between 0.3 and 30 G to aid clarity). White and black crosses indicate the position of the umbral barycenter, which are connected between atmospheric heights using a dashed white line.

The derived Alfvén speeds, displayed alongside the measured tube speeds and estimated sound speeds in Figure 3, are highest (1500 km s^-1) towards the center of the sunspot umbra, dropping to their lowest values (25 km s^-1) at the outermost extremity of the active region. The large values found towards the sunspot core are consistent with previous coronal fast-mode wave observations related to magnetically confined structures⁵.

The Alfvén speed depends on two key parameters: the magnitude of the magnetic field strength, B, and the local plasma density, ρ. We are able to extract spatially-resolved plasma densities from the EM and T_e maps that were derived from the application of DEM techniques, resulting in a range of densities on the order of (2.1–5.6)×10^-12 kg m^-3. Thus, with the Alfvén speeds and local plasma densities known, we derived the magnitude of the magnetic field at each radial location. The resulting values for B range from 32 ± 5 G at the center of the underlying sunspot, rapidly decreasing to 1 G over a lateral distance of approximately 7000 km (Figure 3).

Importantly, we show for the first time a novel way of harnessing the omnipresent nature of propagating slow-mode waves in the corona to more-accurately constrain the magnetic field topology as a function of spatial location, with the potential to provide coronal B-field maps with cadences as high as 1 minute (i.e., the period corresponding to the most rapid frequency detected). This would provide more than an order-of-magnitude improvement in temporal resolution compared with deep exposures of coronal Stokes profiles. Furthermore, since magnetic reconnection phenomena are often observed in the vicinity of active regions, a high-cadence approach to monitoring the spatial variations of the coronal magnetic field would be critical when attempting to understand the pre-cursor events leading to solar flares.

References

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[3] Jess, D. B., Reznikova, V. E., Ryans, R. S. I., et al., 2015, Nat. Phys., doi:10.1038/nphys3544
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