Magnetic Energy: Transport and Dissipation

Magnetic Energy:

Transport and Dissipation

Åke Nordlund Niels Bohr Institute

University of Copenhagen

Spineto Summer School, June 2012 SWIFF WP3: Coupling at the Sun

The Magnetic Energy Equation

To discuss and understand the transport and

dissipation of magnetic energy, it is fundamentally useful to consider the equation that controls the change of magnetic energy with time:

The three quantities on the right have names after

historic persons: The Poynting flux, the Lorentz work, and the Jould dissipation.

So, let’s deriv it!

Induction eqn:

Magnetic energy:

Vector identity:

Electic current:

Electric field:

All together:

Collect terms:

or

where

Time evolution of magnetic energy in sub-surface magnetoconvection

starting with zero field

Poynting flux

 Blue = up, red = down

Evolution of Poynting flux

Energy equation parts, early

Time evolution

Time evolution detail

Surface region

Magnetic dissipation in

a horizontal slice just below the surface, with

24 km grid size

Ditto, with 12 km grid size

Late time, high res balance

Zoom into small ROI

 Work, dissipation, sum of the two:

Ditto, smoothed

 Box car averaged over 5x5x5 cells

Magnetic energy density evolution

Dependence on numerical resolutoion

Very long time evolution

The same, as averages

Magnetic and kinetic energy density

Transition to 2x numerical resolution

Surface detail, 24 – 12 km transition

Small ROIs, in dissipation

Now up into the

Chromosphere & Corona!

Solar Chromosphere & Corona Conditions:

 Which of the MHD approximations are valid?

 Enough charged particles to make a plasma?

 Certainly! Typical Debye lengths are a few mm!

 Slow enough motions relative to the speed of light?

 Well ”yes, most of the time!”

 Frequent enough collisions for thermal behavior?

 No!! Mean-free-paths similar to loop lengths

 Coulomb collisions dominate

 Have interesting run-away behavior!

The one most important macro-scale kinetic effect:

 Particle acceleration!

 Might be a recursive accerelation (ping-pong) effect

 But more likely it is a direct acceleration effect!

 Mean free path comparable to structure scales

 Coulomb cross sections decrease with energy!

 Minority population can have run-away

 Need particle-in-cell codes to investigate!

Computational Costs

 MHD codes, cost per cell

 On Intel Nehalem, IBM Power-6, and similar CPUs

 Costs of the order 2-10 microseconds per cell-update

 Depending on equation-of-state, radiative transfer, …

 Particle-in-cell (PIC) codes

 On similar CPUs

 Costs of the order 1 microsecond per particle-update

 Need 10-50 particles per cell

 Naïve bottom line: only about 5-10 times slower

 However...; physical scales!!

Masson et al. 2009

Flare ribbon

FROM OBSERVATIONS TO PIC

MHD

simulation

?

PIC

simulation

06:27 UT

TRACE and RHESSI observations

Image: H. Reid

12-25 keV (>50%) 25-100 keV (>50%)

A dense electrons beam is accelerated by the E- field in the reconnection region and travels along the fan magnetic field lines. The impact on the photosphere creates X- rays and UV emission.

INITIAL EXTRAPOLATED BFIELD

Potential extrapolation, side view

Potential extrapolation, top view

3D nullpoint reconnection

nullpoi nt

magnetic field lines

Sequence of modeling

 SOHO cut-out, potential field extrapolation, MHD model, PIC-cutout, particles accelerated

APPLIED DRIVER

The elliptical driver applies a sub-Alfvénic velocity of about 20 km/s to the lower boundary after about 150 sec of relaxation time.

Flux emergence on 16^th Nov ‘02

SPINE/FAN MOTION AND J//B

About 5h 30min. real time

RESISTIVE MHD SIMULATION

Code: Stagger MHD code (6^th order space, 3^th order in time)

Computational box size: ^{62 x 175 x}

100 [Mm]

Stretched mesh^, max. resolution: 80 [km]

Boundaries: ^closed

SPINE/FAN MOTION AND J//B

About 5h 30min. real time

Initial average vertical structure

Overview of MHD experiments

Stretched mesh, grid size = f(index)

Magnetic field line connectivity

Electric currents in the current sheet

Relative displacement of the spines

Current density in the fan plane

Perpendicular and parallel electric fields

Evolution with time of

magnetic energy and magnetic dissipation

RESULTS MHD

Current density in horizontal slice Current parallel B

Magnetic dissipation (blue surface) and magnetic field lines near the null

Rate of change of magnetic energy, and rate of magnetic dissipation

Why no flare?

1. Numerical resolution is not the reason, since

 Runs with different resolution are very similar

 All runs behave as expected from scaling arguments

2. Kinetic effects (non-MHD behavior) would not helpt – there is simply not enough free energy available for a C-class flare

 Free energy is proportional to the angle difference across the current sheet, which we show scale with the drive speed, indendent of resistivity …

 … and the solar boundary motions were much slower!

MHD experiment passing over to PIC (Particle-in-Cell) experiment

 MHD snapshot provides

 A cut-out of the ”dome” with uniform mesh

 Initial conditions (including electric currents)

 Boundary conditions (vector B-field at boundaries)

 PIC setup

 Box size covers the ”dome” structure

 Stops about 1.3 Mm above the solar surface

 Mesh size 70, 35, and 17.5 (!) km

 The 17.5 resolution case has more mesh points (~ 3 billion) than any MHD model so far

Particle-in-cell simulations

 Field equations (Maxwell):

 Equations of motion (Newton):

Large scale PIC-simulations

 Copenhagen PhotonPlasma Code

 MPI-parallelized PIC code

 Includes particle-interactions

 Most relevant in the corona: Coulomb scattering

 Modular: also Compton scattering, pair creation & annihilation, ...

 Could also do collisional, time dependent ionization this way!

 Scales to thousands of CPUs / cores

 Pleiades (NASA/Ames); ~47.000 cores, tested up to 4096 cores

 Uses about 1 core-microseconds per particle-update

 (95% efficiency from 8 to 4096 cores)

PIC Scaling

Gyroradius:

Debye length:

PIC Scaling

L_Debye ~ 0.5 dx

L_skin ~ 6 - 8 dx > L_Debye

v_drift < v_thermal_ion ~ v_sound < v_Alfven <

v_thermal_electron < c

100 Mm

175 Mm

PIC runs at JUGENE (Jülich)

 The largest run, 5L

 Grid: 2518x1438x923 (3.3 billion cells)

 Particles: 135 billion

 Ran for 36h on 262.144 (2^18) cores on JUGENE

 about 10 million core-hours

Scaling on JUGENE

 Both the MHD code and the PIC code scale to the full size of the machine (2¹⁸ cores)

Initial PIC state: 44x25x16 Mm box

Projection planes

VERTICAL

PROJECTION

HORIZONTAL

PROJECTION

CURRENT DENSITY IN THE NULL PLANES

Impact area of electrons

Comparison with TRACE and RHESSI observations

Image: H.Reid

12-25 keV (>50%) 25-100 keV (>50%)

A dense electrons beam is accelerated by E//B in the reconnection region and travels down the

inner spine and along the fan magnetic field lines.

The impact on the

photosphere creates X- rays and UV emission.

J//B maximum

Energy distribution

Different simulations

Energy histogram, sampled from N_tot=1.35 10¹¹

The dN/dE power-law index of -1.75 Implies that the electric

current is mainly carried by the low energy electrons, while energy is mainly carried by the high energy ones.

The histograms here show dN/dlnE = E dN/dE, so the dN/dE

distribution index is steeper by one unit; i.e. has slope approximately -1.75

PARTICLE POSITIONS

IN THE POWER-LAW TAIL

PARTICLE POSITIONS

IN THE POWER-LAW TAIL

Location of accelerated particles

Inner spine Outer spine

Fan plane

Trace of accelerated particles

Losers, in low energy part of tail

X - Z Plane

X - Y Plane

Tracing particles in the power-law tail...

MOVIE WITH EVOLVING BACKGROUND

add movie of accelerated particles from lower part of power tail, their trajectories, edotv, and energies

add screenshot of the failed ones.( their trajectories, edotv, and energies)

Lower energy power law particles occur in essentially the same region

Total Current Density

Total electric

current density in slices through the null point

neighborhood

Electric current contributions

Parallel (accelerating) electric field

Note the scale! stabilizes

in a few seconds

Summary

 Need to consider kinetic effects in context!

 micro-scales; modified dissipation details

 meso-scales; dissipation hierarchy

 macro-scales; particle acceleration!

 New tool: Particle-In-Cell (PIC) simulations

 can now handle resolutions ~ MHD a few years ago

 careful scalings need to be applied

 radiation can be included (Monte Carlo RT)

 time dependent ionization is possible

 synthetic observations over a broad range (gamma ray, X- ray, visual, radio, ...) possible ..

Take home: myths to kill!

 Ideal MHD

 School book magnetic fields

 Sweet-Parker current sheets

 Scalings cannot be done

 No collision -> no magnetic feld aligned E