$$ \def\ab{\boldsymbol{a}} \def\bb{\boldsymbol{b}} \def\cb{\boldsymbol{c}} \def\db{\boldsymbol{d}} \def\eb{\boldsymbol{e}} \def\fb{\boldsymbol{f}} \def\gb{\boldsymbol{g}} \def\hb{\boldsymbol{h}} \def\kb{\boldsymbol{k}} \def\nb{\boldsymbol{n}} \def\pb{\boldsymbol{p}} \def\qb{\boldsymbol{q}} \def\rb{\boldsymbol{r}} \def\tb{\boldsymbol{t}} \def\ub{\boldsymbol{u}} \def\vb{\boldsymbol{v}} \def\xb{\boldsymbol{x}} \def\yb{\boldsymbol{y}} \def\zb{\boldsymbol{z}} \def\Ab{\boldsymbol{A}} \def\Bb{\boldsymbol{B}} \def\Eb{\boldsymbol{E}} \def\Fb{\boldsymbol{F}} \def\Jb{\boldsymbol{J}} \def\Ub{\boldsymbol{U}} \def\xib{\boldsymbol{\xi}} \def\evx{\boldsymbol{e}_x} \def\evy{\boldsymbol{e}_y} \def\evz{\boldsymbol{e}_z} \def\evr{\boldsymbol{e}_r} \def\evt{\boldsymbol{e}_\theta} \def\evp{\boldsymbol{e}_r} \def\evf{\boldsymbol{e}_\phi} \def\evb{\boldsymbol{e}_\parallel} \def\omb{\boldsymbol{\omega}} \def\dA{\;\mathrm{d}\Ab} \def\dS{\;\mathrm{d}\boldsymbol{S}} \def\dV{\;\mathrm{d}V} \def\dl{\mathrm{d}\boldsymbol{l}} \def\bfzero{\boldsymbol{0}} \def\Rey{\mathrm{Re}} \def\Real{\mathbb{R}} \def\grad{\boldsymbol\nabla} \newcommand{\dds}[1]{\frac{\mathrm{d}{#1}}{\mathrm{d}s}} \newcommand{\ddy}[2]{\frac{\partial{#1}}{\partial{#2}}} \newcommand{\ddt}[1]{\frac{\mathrm{d}{#1}}{\mathrm{d}t}} \newcommand{\DDt}[1]{\frac{\mathrm{D}{#1}}{\mathrm{D}t}} $$
Magneto-frictional modelling
My thesis with Duncan Mackay (St Andrews) developed a global model non-potential model for the large-scale mean field in the solar corona, using the magneto-frictional (MF) approach.
Introduction
In MF we retain the MHD induction equation, but replace the momentum equation by a fictional relaxation velocity: \[\begin{align*} \ddy{\Bb}{t} &= \nabla\times(\ub\times\Bb),\\ \nu\ub &= \Jb\times\Bb,\\ \Jb &= \frac{1}{\mu_0}\nabla\times\Bb,\\ \nabla\cdot\Bb &= 0. \end{align*}\] This causes the coronal magnetic field \(\Bb\) to relax toward a force-free state (\(\Jb\times\Bb=\bfzero\)). Our global coronal model makes two additions:
The inner boundary (solar surface) is driven by imposing an electric field. This models flux emergence and surface flux transport. The coronal \(\Bb\) evolves quasi-statically in response.
Near the outer boundary, \(\ub\) is modified to include the effect of a solar wind outflow. This gives the characteristic streamer shapes, and removes sensitivity to the precise boundary height.
For an overview, see my talk Magneto-frictional modelling of the solar corona (given to a Lorentz Centre workshop in 2021). A more complete description of the model is given in [1], or a more mathematical viewpoint in [2].
For a comparison of different non-potential modelling approaches, see the work of our ISSI team [3], or our Living Review [4].
Selected papers
[5], [6], [7] – the original study from my thesis where I used the MF model to explain the origin of the hemispheric pattern of filament chirality. I’m most proud of the (less cited) third paper, where I explain why the model gives results consistent with observations.
[8] – I believe this was the first study to self-consistently model both the formation and eruption of coronal magnetic flux ropes, without imposing them at particular locations. A follow up paper compared the eruption locations to real CMEs [9] – basically we found that the model produces fewer (but larger) ejections, with less strong clustering of eruptions in active regions. This is to be expected given the model was driven with synoptic data. In these original studies, we defined flux ropes based on the vertical magnetic pressure and tension forces. This is good at finding flux rope cores, but less good at finding their full extents, as needed to measure their flux or helicity content. This problem was later solved with postdoc Chris Lowder [10], where we defined ropes instead by thresholding of field line helicity.
[11] - we showed how the MF model enhances the open solar flux compared to traditional potential field equilibria. The enhancement is time dependent, and arises from (i) quasi-steady inflation of \(\Bb\) by electric currents and (ii) bursty enhancement through eruptions. We originally attributed the latter to flux ropes, but from more recent work with postdoc Prantika Bhowmik we now know that these eruptions also come from overlying arcades, like “streamer blowouts” [12].
[2] – I really like this paper: it sounds unlikely but I found that looking at a one-dimensional test case can really help to understand how MF behaves compared to full MHD.
[1] – my most recent version of the model, where I use “local inductive” electric fields to insert HMI SHARP regions with controllable twist. I focused on quantitative outputs in order to compare different runs (and, in future, different time-evolving models). Here I avoided identifying flux ropes, instead finding a simpler measure of eruptivity using the second time-derivative of the open flux.