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The multiscale dynamics of sparks and lightning
Ute Ebert
CWI Amsterdam and TU Eindhoven
http://homepages.cwi.nl/~ebert/
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The multiscale dynamics of sparks and lightning
Puzzles in lightning
Physical mechanisms
Computations and Analysis
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Lightning: • ca. 45 flashes/second worldwide,
• major source of O3 and NOx.
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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
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Lightning – is it possible at all?
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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?
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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
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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?
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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:
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Air, +28 kV on 40 mm,
exposure 0<t<300 ns
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Air, +28 kV on 40 mm,
exposure 46<t<47 ns
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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]
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Self-organized plasma reactor dots
produce O*, X-rays(?), …
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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?
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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
— — — — — —
+ + + + + +
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:
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e-
A
A+
E
A*
Streamer mechanism + + + + + +
— — — — — —
++
+ --
-
Echarge layer
++
--
-+
+
--
-+
+-
--
++
--
-+
-+
--
+-
-+
--
++ +
++
----
- --
--
+
+++
IonizedRegion
Nonionized Region
E
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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
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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
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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]
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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
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L
Charge distribution and (electro-)dynamics different from single streamer!
Anode
Cathode
Direction of propagation
Periodic array of negative streamers in 2D:
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LAnode
Cathode
Direction of propagation
Periodic array of negative streamers in 2D:
Thin front structure, almost a moving boundary
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Moving Ionization Boundaries
Ideal conductor
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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
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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.
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Hele-Shaw Flow
HoleHole
GlycerolGlycerol
Colored Colored WaterWater
Radial Symmetry
Channel configuration Saffman-Taylor finger
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An array of streamers (2D, fluid-model):L
Saffman-Taylor finger with λ=½!
Mathematics of selection?
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From few channels to more.
DBM
L
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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
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Elves, sprites, jets correlated
with lightning strokes
Predicted 1925,observed since 1989.
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Sprite dischargeabove a thundercloud
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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.
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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.