The Physical Consequences of Under-resolution for N-dimensional Models

of the Solar Atmosphere

Dr. Stephen BradshawRice University

Prof. Peter CargillImperial College London

&University of St. Andrews

1. Questions

What are the consequences of under-resolving the solar atmosphere in numerical models of coronal heating?

What resolution can we get away with for particular regimes? (E.g. quiet Sun, active regions, flares)

What can be done to help mitigate or overcome some of these challenges?

See: Bradshaw, S. J., & Cargill, P. J. 2013, ApJ, 770, 12

2. What challenges do the models face?

When the corona is heated in-situ the temperature gradient of the cooler, lower atmosphere is forced to steepen to help get rid of the excess thermal energy

Take a 4 MK active region loop in hydrostatic equilibrium

The coefficient of thermal conduction scales as T5/2

A shallow gradient is sufficient to transport excess energy in the corona

few thousand kilometers (CORONA)

We are interested in understanding and quantifying the consequences of inadequate spatial resolution. Under what conditions can one claim that their models are physically realistic?

few thousand meters (TRANSITION REGION)

Not so extreme in the quiet Sun, vastly more extreme in flares ( meters in the TR)

The timescale for a thermal conduction front to propagate across a grid cell is

An explicit code must time-step on at least this timescale (usually with a safety factor < 1.0)

An implicit code need not, but then fails to resolve the principle timescale of the system

can vary between 0.1 s in the quiet Sun corona to < 10-7 s in flares

Three possible solutions:

Use a fixed, non-uniform grid and hope the gradients don’t propagate too far

Use a stretching algorithm that allows the high resolution region to move with the gradient; keeps the number of cells constant but only one high resolution region

Employ a fully adaptive grid that locally increases the density of grid cells wherever needed such that multiple steep gradients can be simultaneously resolved

3. Numerical experiments

RL = 0: 400 km

RL = 2: 100 km

RL = 4: 25 km

RL = 6: 6.25 km

RL = 8: 1.56 km

RL = 10: 390 m

RL = 12: 98 m

84 runs in total

GROUP 3

GROUP 6

GROUP 12

4. Summary of key findings

Under-resolving the transition region can lead to a gross underestimate of the coronal density, with attendant implications for EM and spectral modeling

The enthalpy flux into the corona is drastically underestimated, with consequences for understanding energy transport through the Sun’s atmosphere

The temperature depends on the grid resolution only during the cooling phase because of the interplay between thermal conduction and radiation, which depend on the density

Not due to numerical errors violating the conservation laws. We checked:

(conserved to better than 0.1% for entire run)

These findings imply that energy is NOT ‘bottled up’ in the corona when the transition region is under-resolved (under-resolved cases cool more quickly)

5. What happens to the energy?

Group 3 at t = 50 s

RL = 0 (solid)

RL = 6(dotted)

RL = 12(dashed)

In a well-resolved transition region the heat flux progresses in a series of steps

At each step, some energy is radiated, some goes into driving an enthalpy flux (ablation) and the remainder continues on

In a poorly resolved transition region the energy ‘falls over’ the cliff and more is radiated away by the chromosphere than drives ablation

6. What can we conclude?

What are the consequences of under-resolving the solar atmosphere in numerical models of coronal heating?

What resolution can we get away with for particular regimes?

What can be done to help mitigate or overcome some of these challenges?

See: Bradshaw, S. J., & Cargill, P. J. 2013, ApJ, 770, 12

Active region cores: RL = 4 – 6 (6.25 – 25 km) for 75%; RL = 6 – 8 (1.56 – 6.25 km) for 90%

Flare models: RL > 8 (< 1.56 km)

Grossly underestimate the coronal density and upward energy transport; led to the wrong conclusion concerning the source of the coronal plasma (neglect transition region contribution); get the energy balance wrong

Quiet Sun models: RL = 2 (100 km) for 75% accuracy; RL = 4 (25 km) for 90%