The multiscale dynamics of sparks and lightning

Ute Ebert

CWI Amsterdam and TU Eindhoven

http://homepages.cwi.nl/~ebert/

The multiscale dynamics of sparks and lightning

Puzzles in lightning

Physical mechanisms

Computations and Analysis

Lightning: • ca. 45 flashes/second worldwide,

• major source of O3 and NOx.

Sparks and lightning evolve in three stages:

1. Charge separated -> voltage builds up

2. Streamer/leader: conducting channels grow

3. Short circuit: Ohmic heating, visible stroke

Lightning – is it possible at all?

100 MV on 10 km = 10 kV/m …

electric breakdown of air requires 30 kV/cm

100 MV

10 km

-> average field 100 MV/10 km = 100 V/cm

A field paradox?

100 MV on 10 km = 10 kV/m …

electric breakdown of air requires 30 kV/cm

100 MV

10 km

-> average field 100 MV/10 km = 100 V/cm

Highest field measured inside thundercloud 3 000 V/cm

A field paradox?

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e-

A

A+

— — — — — —

+ + + + + +Free electrons, if present, drift and diffuse in local E-field –

like a ball jumping down a slope.

Collisions with neutral molecules:

Impact ionization -> electron gain

Attachment to O2 -> electron loss

Electron number gain larger than loss above ~30 000 V/cm

(in air at 1 bar and 300 K)

E

100 MV on 10 km = 10 kV/m …

electric breakdown of air requires 30 kV/cm

100 MV

10 km

-> average field 100 MV/10 km = 100 V/cm

Highest field measured inside thundercloud 3 000 V/cm

Electric breakdown of air requires ~30 000 V/cm

Hammer, nail and wall: field focussing!

A field paradox?

Movie of Lightning leader [G.M. McHarg, US Air Force Academy, summer 2007]

shows how the lightning leader searches its way to the ground.

The total duration of the movie is only 3.5 milliseconds, time steps are 5 microseconds.

Not the total channel is illuminated, but only the actively propagating tip.

In this tip electrical forces are focused, similarly to the focusing of mechanical forces in the tip of the nail.

But the “lightning nail” is not pre-fabricated, but self-organized. We later will see how.

Similar glowing tips are seen on smaller scale in the lab:

Air, +28 kV on 40 mm,

exposure 0<t<300 ns

Air, +28 kV on 40 mm,

exposure 46<t<47 ns

exposure: 1 ns(46 ns < t < 47 ns)

10 ns(50 ns < t < 60 ns)

50 ns(50 ns < t < 100 ns)

300 ns(0 ns < t < 300 ns)

Air, 1 bar, +28 kV pulse on point above, 40 mm gap to plate below

[Ebert et al., PSST 06, Briels et al., J Phys D 2006]

Self-organized plasma reactor dots

produce O*, X-rays(?), …

Terrestrial Gamma-Ray Flashes, > 50/day[discovered 1994, here RHESSI satellite data 2006]

correlated with lightning strokes

There are puzzles in cosmology, but do we understand our own earth?

12 stage 2.4MV Marx generator

Hypothesis:

Enhanced field region at streamer tip

= electron accelerator

-> Bremsstrahlung

-> gamma-rays

Gamma-ray bursts now also observed in MV-lab discharges

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e—

A

A+

E

— — — — — —

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Fast processes in the ionization front (in pure N2 or Ar for simplicity):

10-9 m:

10-6 m:

Electrons drift and diffuse in local E-field.

Elastic, inelastic and ionizing collisions with neutral molecules.

Degree of ionization < 10-4.

Fluid approximation with

Impact ionization e— + A → 2 e— + A+

Ohm’s law j ~ ne E

Coulomb’s law n+— ne = div E

→ Minimal streamer model for electron density σ, ion density ρ and electric field E:

@t¾ = Dr 2¾+ r ¢(E¾) + ¾f (jE j);

@t½ = ¾f (jE j);

¡ r 2Á = ½¡ ¾; E = ¡ r Á;

f (x) = jxj e¡ 1=jxj ; D = 0:1:

e-

A

A+

E

A*

Streamer mechanism + + + + + +

— — — — — —

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Echarge layer

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IonizedRegion

Nonionized Region

E

Propagating streamer

r (mm) r (mm)r (mm) r (mm)

z (m

m)

Negative electrons ne

Net charge n+ - ne

Electric field

Positive ions n+

Strong local field enhancement

The multiscale challenge:

Solve Poisson equation everywhere.

Solve densities in ionized region.

Resolve steep density gradients with high accuracy.

Do not exceed computational memory.

[Montijn et al., 2006, Luque et al., 2008]

z

r

electrons

r

z

net charge

Numerical decoupling of domains and moving local grid refinement

Whole computational domain

Grids for densitiesGrids for Poisson

equation

Coupling of the computational grids

σ, ρ

E

¢x=4¢x=2

¢x=1¢x=1/2¢x=1/4¢x=1/8

[C. Montijn et al., J. Comp. Phys. 2006, Phys. Rev. E 2006]

2 interacting streamers in 3D:

Surfaces of equal electron density

Quasispectral method for the Poisson equation

[Luque et al., PRL 08, Research Highlight Nature 08]

Electrostatic repulsion versus attraction through photoionization

L

Charge distribution and (electro-)dynamics different from single streamer!

Anode

Cathode

Direction of propagation

Periodic array of negative streamers in 2D:

LAnode

Cathode

Direction of propagation

Periodic array of negative streamers in 2D:

Thin front structure, almost a moving boundary

Moving Ionization Boundaries

Ideal conductor

Coordinates around body uncharged body in an external field

The electric potential φ around a conducting body

(solutions of ¢φ = 0 with boundary conditions)

Electric field = slope of φ = - r φ

Moving Ionization Boundaries

Ideal conductor

Air-oil-flow (between glass plates) mathematically equivalent:

Viscous oil: v = -rp, incompressible r∙v = 0

=> r2p = 0 in oil

v = -rp on interface

Nonviscous air: p = const.

Hele-Shaw Flow

HoleHole

GlycerolGlycerol

Colored Colored WaterWater

Radial Symmetry

Channel configuration Saffman-Taylor finger

An array of streamers (2D, fluid-model):L

Saffman-Taylor finger with λ=½!

Mathematics of selection?

From few channels to more.

DBM

L

Physics/electroengineering.: Streamer discharges: experiments and applications

5 ns 5 µs

Nonlinear Dynamics: Fronts and interfaces, model reduction

geophysics: Sprite discharges

Computational Science: adaptive grids, hybrid (MC-continuous)

Spark formationin Nature and Technology

Elves, sprites, jets correlated

with lightning strokes

Predicted 1925,observed since 1989.

Sprite dischargeabove a thundercloud

4 cm

Telescopic images of sprite discharges [Gerken et al., Geophys. Res. Lett. 2000]

4 cm1 bar Approximate similarity

between different gas densities,

better than theory predicts.

Artikelen voor allgemeen publiek zijn te vinden op http://homepages.cwi.nl/~ebert/PublPubl.html, b.v.

Bliksem boven bliksem over reuzenachtige sprite ontladingen boven onweerswolken

of

Vroege Vonken onder de virtuele microscoop over simulaties van streamer ontladingen.