Heating the Solar CoronaThomas Howson

Supervised by Ineke De Moortel

Contributions from Patrick Antolin, Jenny O’Hara and Paolo Pagano

Contents1. Introduction - Energy Required for Coronal Heating - Reconnection Heating - Wave Heating

2. Thermal Conduction and Optically Thin Radiation in LARE3D - Implementation - Maintaining Equilibrium

3. Numerical Experiments - Braiding - Waves - Searching for the Kelvin-Helmholtz Instability

4. Future Work - Comparing Energy Input and Heating

Energy Losses

In order to maintain the coronal temperature, energy lost from the solar atmosphere must be replaced.

Understanding mass transfer is also essential.

Corona at ~106 K

Space

Lower Layers at ~104 KConduction

Radiation

Solar Wind

Energy BudgetDifferent coronal regions have different energy requirements (Withbroe & Noyes, 1977).

Coronal Hole ~ 800 Wm-2

Active Region ~ 10000 Wm-2

Quiet Sun ~ 300 Wm-2

Heating MechanismsConvective buffeting of the photosphere is thought to be the source of the energy required to heat the corona.

Coronal magnetic field is rooted in the photosphere and so footpoint motions induce a Poynting flux into the upper atmosphere.

Movement of existing magnetic field.

Emergence of new field.

S =1

µ0

Z

SE⇥B · dS =

1

µ0

Z

S��Bk · vk

�B? +

�Bk ·Bk

�v?dS.

Heating MechanismsThe nature of the convective buffeting splits the resultant heating into two broad groups.

Motions slower than the local Alfvén speed tend to produce braiding of magnetic field.

Faster motions can produce waves that may propagate into the atmosphere.

Both of these effects could cause heating and may be relevant in different coronal regions.

Convective Driving

Photosphere

Reconnection HeatingNanoflare heating.

Slow convective driving can braid the magnetic field around itself.

Can create large gradients in the magnetic field leading to reconnection and heating.

Is sufficient energy built up before release?

Field line dominated conduction requires energy release across entire loop cross section.

(a) (b)

Initial Conditions for magnetic field and current density (isosurface showing 25% of max value) for braiding experiment in Pontin et al., (2011).

Wave HeatingCoronal wave dissipation rates are far too slow: - need to build up large gradients in the magnetic field.

Gradients in Alfvén speed i.e. density profile can lead to phase mixing and enhanced dissipation.

Observations of waves being damped does not imply heating - mode conversion?

Significant reflection in lower atmosphere.

Destroys the required density profile? (Cargill et al., 2016)

Alfven Wave Phase Mixing• Shear Alfven waves become quickly out of phase as

they propagate along the field lines with large (perpendicular) gradients in the Alfven speed

• Small length scales are generated – dissipation is enhanced.

• Eventually, all the wave energy is dissipated.

Perpendicular gradients in vA can lead to propagating waves becoming out of phase. - generates the small length scales and hence enhanced dissipation.

My ProjectNumerical experiments using the code LARE3D.

Modelling and comparing heating mechanisms in the corona.

How much heating do we get from a certain amount of energy input?

Over what timescales do different heating mechanisms release energy?

What are the effects of thermal conduction and optically thin radiation on different heating mechanisms.

Is energy released at a sufficient rate to compensate for cooling?

In what ways are numerical results code dependent?

What can we infer from forward modelling?

Heat Transfer in LARE3DLARE3D includes routines for thermal conduction and optically thin radiation.

Thermal conduction calculated with the method of Successive Over Relaxation (hopefully this will converge).

Following K. Tam’s PhD Thesis (2014) an additional restriction is included to ensure heat changes by no more than 1% during any time step: - this may occur at the peak of radiative losses.

LARE3D offers the inclusion of an additional/background heating function allowing for an equilibrium to be maintained.

Thermal ConductionIn the presence of magnetic field, conduction along field lines dominates.

We require the form of thermal conduction to reduce to isotropic conduction in the presence of magnetic null points

LARE3D implements

⇢@✏

@t= r · q,

where�q = k

✓1

B2min +B2

◆(B ·rT )B+ k

B2min

B2min +B2

rT,

k = 10�11T52Wm�1K�1,

and Bmin is some small but finite number.

Optically Thin RadiationIn the corona this is typically dominated by thermal conduction.

Can be significant in regions of enhanced density.

Modelled by

where

is a piecewise continuous loss function with and being functions of temperature (Klimchuk et al., 2008).

⇢@✏

@t= �Lr,

Lr = n2e�T

↵.

� ↵

Maintaining EquilibriumConsider the change in temperature in a particular grid cell given by

Tn+1 = Tn � Lc � Lr

Conductive loss/gain Radiative loss

In order to maintain an equilibrium, we must include a background heating term.

May be spatially and temporally dependent - tracking required to heat oscillating loops

+QH .

Background Heating

Maintaining EquilibriumInstead of calculating this background heating term, we follow Kuan’s method and use

This is easier to implement.

Is it valid? It only affects regions with minimal heating. Could change physics?

Tn+1= max

�Tn � Lc � Lr, T

0�.

Experiment 1Setup

- Coronal model (no lower layers).

- Two initially straight flux tubes allowed to relax towards a numerical equilibrium.

- During this phase, the density and temperature are held constant.

- Velocity driver imposed on the upper and lower boundaries causes braiding.

- η only non-zero away from the driving boundaries.

O’Hara (2016)

Jenny’s Experiment Setup

Normalisation -

512 x 512 x 256 grid modelling a 13 box.

Thermal conduction and radiation turned on during the driving - temperature restricted to be at least the initial value.

B0 = 0.01G

L0 = 7.5⇥ 107m t0 ⇡ 34 sT0 ⇡ 5.8⇥ 108 K

⇢0 = 1.67⇥ 10�11 kg m�3 v0 ⇡ 2.2⇥ 106 m s�1

Volume Integrated Energy: Heat Transfer Vs No Heat Transfer

With Conduction & Radiation Without Conduction & Radiation

Temperature along field lines traced from driven foot points

Temperature along field lines traced from simulation centre

Experiment 2- Uniform magnetic field

parallel to loop.

- Dense, ‘cold loop’.

- Density transition in ‘shell’ region.

- Small driver imposed on lower boundary.

Core

Shell

External

Wave Experiment Setup

- Following the setup of Pascoe et al. (2010).

- Uniform magnetic field.

- High density flux tube with linear transition region.

- Equilibrium maintained using ‘cold loop’

- β = 0.02

- 256 x 256 x 128 grid points with a box of dimensions 4 x 4 x 20.

Wave ExperimentImpose a velocity driver at lower boundary for 1 period (10 normalised seconds)

Periodic boundaries.

‘Tiny’ - not much energy put into system so very little heating.

No conduction or radiation due to insignificant heating.

How do we adapt to put more energy into the system?

Wave Propagation

Experiment 3Global Kink Mode

Initially straight flux tube

‘Cold Loop’

Initial velocity induces a standing wave

Looking for Kelvin-Helmholtz Instability

Patrick’s ExperimentInitial Set-up

‘Cold’ flux tube in uniform magnetic field.

Non-uniform grid.

β = 0.02

Do we observe the Kelvin-Helmholtz instability?

Potential alternative heating mechanism.

Patrick’s Run

- Evidence of Kelvin-Helmholtz Instability

- Not observed with all density profiles

- What are the consequences for wave dissipation?

- Are results comparable with those of other codes?

Density Evolution at Loop Apex

Evidence of Mode Conversion

Time Evolution of Mode Conversion

Paolo’s run in VAC:

Upcoming WorkBraiding- Comparison of results with and without thermal conduction and radiation.

Waves - Increased energy input. - Turn thermal conduction and optically thin radiation on. - How does heating compare with the braiding experiment given some energy input?- How do results compare with the VAC code?

Patrick’s Experiment- Code comparison - Is the Kelvin-Helmholtz instability physical?

This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 647214).