A Fokker–Planck Framework for Studying the Diffusion of Radio Burst Waves in the Solar Corona

Abstract

Electromagnetic wave scattering off density inhomogeneities in the solar corona is an important process which determines both the apparent source size and the time profile of radio bursts observed at 1 AU. Here we model the scattering process using a Fokker-Planck equation and apply this formalism to several regimes of interest. In the first regime the density fluctuations are considered quasi-static and diffusion in wavevector space is dominated by angular diffusion on the surface of a constant energy sphere. In the small-angle ("pencil beam") approximation, this diffusion further occurs over a small solid angle in wavevector space. The second regime corresponds to a much later time, by which scattering has rendered the photon distribution near-isotropic resulting in a spatial diffusion of the radiation. The third regime involves time-dependent fluctuations and, therefore, Fermi acceleration of photons. Combined, these results provide a comprehensive theoretical framework within which to understand several important features of propagation of radio burst waves in the solar corona: emitted photons are accelerated in a relatively small inner region and then diffuse outwards to larger distances. En route, angular diffusion results both in source sizes which are substantially larger than the intrinsic source, and in observed intensity-versus-time profiles that are asymmetric, with a sharp rise and an exponential decay. Both of these features are consistent with observations of solar radio bursts.

Anisotropic Radio-wave Scattering and the Interpretation of Solar Radio Emission Observations

“…While there have been a number of Monte Carlo simulations developed to describe wave scattering (mostly for isotropic density fluctuations), these do not all agree. Therefore, the present work addresses this important issue both by extending the isotropic plasma treatment of Bian et al (2019) into the anisotropic scattering domain and by improving the previous descriptions by Steinberg et al (1971), Arzner & Magun (1999), and Thejappa & MacDowall (2008). The description presented captures both multiple scattering of radio waves in anisotropic small-scale turbulence and refraction of waves in the presence of large-scale plasma inhomogeneity.…”

“…The propagation of radio waves in a turbulent medium can be effectively described using a kinetic approach (e.g., Mangeney & Veltri 1979;Arzner & Magun 1999;Bian et al 2019). This approach describes the evolution of radio waves, in an inhomogeneous plasma with quasi-static density fluctuations, in the geometrical optics approximation (Tatarskii 1961;Ishimaru 1978), i.e., when the scale length for variation of the wavelength λ due to inhomogeneity is much smaller than the wavelength itself:…”

“…The diffusion tensor D ij appropriate to scattering (cf. Arzner & Magun 1999;Bian et al 2019) is given by…”

“…where n = n is the average plasma density, taken to be a slowly varying function of position. Note that Equations (5) and (6) include a scaling of (2π) 3 in the definition of the spectral density S(q), consistent with the treatment of Arzner & Magun (1999), but not with the scaling used by Bian et al (2019).…”

A Review of Recent Solar Type III Imaging Spectroscopy

“…Krupar et al (2018) using the STEREO spacecraft and Krupar et al (2020) using Parker Solar Probe found that from the arrival time, the exponential decay times observed at low frequencies from spacecraft are able to be explained through the scattering of radio waves by density inhomogeneities. Bian et al (2019) modeled the scattering process using a Fokker-Planck equation and were able to reproduce the time profile but not the inverse frequency dependence of the decay time, which they concluded was down to the exclusion of a large-scale refractive term. Kontar et al (2019) recently extended the work of Bian et al (2019) but treated the scattering in the anisotropic domain, with the dominant effect being perpendicular to the heliospheric radial domain (Kontar et al, 2017).…”