Towards the Next-Generation of

Non-Equilibrium Solar Radiative Transfer Models

Chris Osborne

University of Glasgow

Non-LTE Radiative Transfer

Coupling between radiation field and atomic transitions.
Extremely computationally demanding, but necessary for many spectral lines.
Also determines ionisation.

HINODE BFI

The Solar Atmosphere

Information in these narrow wavebands samples different regions of the solar atmosphere.
But not uniquely determined by a single region…

Most Important Quantities

Radiative rates depend on intensity.
Intensity depends on radiative rates.

\begin{align*} J_\nu(\vec{p}) &= \frac{1}{4\pi}\oint_\mathbb{S^2} I_\nu(\vec{p},\hat{\omega}) \mathop{d \hat{\omega} } \\ &\vec{p} \in \mathbb{R}^2, \hat{\omega} \in \mathbb{S}^2\\ \end{align*}

Specific intensity and its first moment.
3 and 5 dimensional functions! (+ wavelength…)
Recursive component solved by fixed-point iteration in the atomic populations.

Current Methods

RT codes designed around the idea of smooth atmospheres…

But ours look like this

Jenkins & Keppens 2021

Carlsson+ 2016

Przybylski+ 2022

Flare Observations

And observations of flares are hardly smooth.

NST/BBSO, Jing et al (2016)

Simplest 2D Chromospheric Model

How are line-profiles affected by 2D transfer effects?
Quiet Sun atmosphere surrounding a flare kernel
- 125, 250, 500 km widths
- Flare-kernel thermodynamics from RADYN.
Time-dependent ionisation
- Charge conservation
No energy feedback into kernel.

125 km Kernel - Hα

Intensity along cuts (red, green, blue), and equivalent plane-parallel model.

500 km Kernel - Ca ɪɪ 8542

Intensity along cuts (red, green, blue), and equivalent plane-parallel model.

Interesting Differences

Line Formation Heights

Even 500 km does not come close to the formation height found in the plane-parallel models.

Radiative Losses

Estimate since this simple model is not energetically consistent.

Motivating Proto-Conclusions

Radiation can propagate energy free from the magnetic field.
2+D radiative transfer for flares really matters.
Formation heights change.
- Losses and gains change too.
Filling factors aren’t uniform in wavelength over a line.

So…

Efficient higher dimensionality treatments are needed.
Plane-parallel can have high error.
A few niggles with the radiation field.

Taking a step back…

Who recognises this?

Cornell University

And how does it connect to this?

VLT (ESO)

Physics!

\begin{align*} (\hat{\omega} \cdot \nabla) L(\vec{p}, \hat{\omega}) = &-\chi(\vec{p}, \hat{\omega}) L(\vec{p}, \hat{\omega}) \\ &+ \eta(\vec{p}, \hat{\omega}) \\ &+ \sigma_s(\vec{p}, \hat{\omega}) \oint_{\mathbb{S}^2} p(\vec{p}, \hat{\omega}^\prime, \hat{\omega}) L(\vec{p}, \hat{\omega}^\prime)\, d\hat{\omega}^\prime \end{align*}

L radiance,
\eta emission coefficient,
\chi absorption coefficient,
\sigma_s scattering coefficient,
p phase function.

Key takeaway: this is a recursive integro-differential equation!

Prominence Albedo

Monte Carlo

Jenkins+ 2024

Ray Effects

Desired Result

Tricky Radiation

In the steady-state solution of radiative transfer coupling is global.
This hinders parallelisation.
Monte Carlo approaches have poor convergence rates.
Is there some middle ground?

Video games are now solving an adjacent problem at reasonable resolutions hundreds of times a second on single workstations…

Aside: Everything Old is New Again

Radiance Cascades

Radiance Cascades: Observations

Linear light source and blocker:

\begin{cases} A < B,\\ \alpha > \beta. \end{cases}

with some small angles approximations…

\begin{cases} \Delta_s < F(D) \propto D\quad&\mathrm{(spatial)}\\ \Delta_\omega < G(1/D) \propto 1/D\quad&\mathrm{(angular)} \end{cases}

We can exploit this!
Represent contributions from different shells separately.

How do we use this?

Encode radiance field as a set of cascades, each describing I_\nu in annuli around \vec{p}.
If penumbra criterion is satisfied, cascade is linearly interpolateable!
- Serves as Nyquist criterion
The cascades sparsely encode the radiation field throughout the domain…
But not in a form we can use directly.
Solution:

Interpolation. But with minimal error.

Interpolation leads to reuse of rays in upper cascades and the effective construction of exponentially more rays than computed directly.

Mipmaps I

We can employ lower resolution proxies for distant rays: mipmaps (recursive spatial averaging) with directionality and adaptive error bounds (variance of log emissivity and opacity), and also skip empty space

Inspired by OpenVDB and longstanding computer graphics practices.

Mipmaps II

Can be efficiently traversed using a hierarchical scheme.
This technique has provided a further order of magnitude speedup on typical models.

Mipmaps III

This approach adapts automatically during the iterative solution!

Ly β COCOPLOT (Donné & Keppens)

Mg ɪɪ k COCOPLOT

Diving into the lines I

Diving into the lines II

Outlook

Simulation structures are typically axis aligned.
Significant complexity arises from looking outside this alignment.
This barely scratches the surface of interdisciplinary computational work, especially for radiation.
- Need to develop common language between fields.
Radiance cascades provide the first meaningful change to solar non-LTE RT in 30 years and it came from outside the field.
- More efficient cascade designs are being developed.

Thanks!

Christopher.Osborne@glasgow.ac.uk