turing instability of electrical discharges · 2017. 4. 20. · schoenbach, 2004 formation of the...
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Turing instability of electrical discharges
Sh. Amiranashvili
August 27, 2007
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Institute of Applied Physics at Münster
Johannes LavenWeifeng ShangLars StollenwerkHans-Georg PurwinsHendrik BödekerSvetlana Gurevich
100 % Experiment
100 % Theory
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Outline
1 Experimental data
2 Turing instability and RD patterns
3 Applications to the glow discharge
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Part I. Experimental data
+-
There are more things between cathode and anode than aredreamt in your philosophy.
H. Raether
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Rubens, 1940
Formation of the anode spots in the glow discharge mode.
-+
Spherical cathode R = 26.7 cm;Spherical anode r = 5.1 cm;Nitrogen p ∼ 1 Torr;Current I ∼ 500 mA.
A homogeneous anode glow is unstable and evolves to thesymmetric arrangements of current spots.
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Anode spots (Rubens, 1940)
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Schoenbach, 2004
Formation of the cathode spots in the glow discharge mode.
1 Cathode d = 100 µm;2 Spacer 250 µm;3 Hollow anode D = 1.5 mm;4 Xenon p ∼ 200 Torr.
A homogeneous cathode glow is unstable and evolves to thesymmetric arrangements of current spots.
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Cathode spots (Schoenbach, 2004)
Theory: Benilov1986, 2007
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Cathode vs anode spots, Nasuno, 2003
Nitrogen, d ∼ 100 µm, p ∼ 40 Torr, sandwich-like geometry.
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Maxwell time
Metalτm → 0
Dielectric
τm →∞
High-ohmic barrier
τm = ε0ρ ∼ τi
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Purwins group at Münster
Semiconductor
Electrode
Current spots
Glass
Camera
Light
Us
R
Gas
CCD
Electrode
Gas, e.g.: N2 0.5 mm 100 Torr
Cathode: GaAs 0.5 mm 107 Ω cm
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Par t II. RD systems
un-1Jn un
Jn+1 un+1
dun
dt= Qn + Jn - Jn+1
Qn = f (un)
Jn ∼ un−1 − un
Jn+1 ∼ un − un+1
A discrete RD equation
dun
dt= f (un) + un−1 − 2un + un+1
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One component RD systems
Fisher, Kolmogorov 1937
∂tu = D∂2x u + f (u)
e.g.,
f (u) = u − u3
Local evolution
∂tu(x , t) = f (u)
u= 0 u= 1u= -1 u -2 -1 1 2
-2
-1
1
2
3
4
x
u
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Free energy
Stable
Stable
Connected stable states;Fronts propagation;Phase transitions, flames, etc.
Free energy always reduces
∂tu = −δFδu
F[u] =
∫ [D(∂xu)2
2+ F (u)
]dnx
withF ′(u) = −f (u) ∂tF < 0
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“The chemical basis of morphogenesis”
Turing, 1952
∂tu = Du∂2x u + f (u, v)
∂tv = Dv∂2x v + g(u, v)
Reaction part
dudt
= f (u, v) anddvdt
= g(u, v)
Problem
How stability of a stationary state is affected by diffusion?
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Prey-Predator Model
Linearized uniform system
dudt
= αu − βv
dvdt
= γu − δv
Stability conditions
α < δ
αδ < βγ
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Slow preys, fast predators
Linearized full system
∂u∂t
= Du∂2x u + αu − βv
∂v∂t
= Dv∂2x v + γu − δv
The most important instabilitycondition
Du < Dv
(to be revised).
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Stability analysis
Linearized full system is considered for u ∼ eikx , v ∼ eikx
dudt
= αu − βv
dvdt
= γu − δv
duk
dt= (α− Duk2)uk − βvk
dvk
dt= γuk − (δ + Dv k2)vk
Stability conditions
α < δ
αδ < βγ
α− Duk2 < δ + Duk2
(α− Duk2)(δ + Duk2) < βγ
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Turing instability
αδ < βγ, but (α− Duk2)(δ + Dv k2) > βγ
maxk
[αδ + (αDv − δDu)k2 − (Duk2)(Dv k2)
]> βγ
A necessary condition
αDv > δDuk0
k
Α∆
ΒΓ
A finite space scale
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Resume
Parametersα, β, γ, δ, Du, Dv
Citeria
α < δ “Sp” criterion
αδ < βγ “Det” criterion
αδ + DuDv k20 > βγ No simple interpretation
αDv > δDu Slow preys, fast predators
The most dangerous mode: k20 =
αDv − δDu
2DuDv
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An example
Equations
∂tu = D∂2x u + αu − βv
∂tv = ∂2x v + u − v
Criteria
D < α < 1
α < β < α +(α− D)2
4D
0.25 0.5 0.75 1 1.25 1.5 1.75 2
0.25
0.5
0.75
1
1.25
1.5
1.75
2
Α
Β
D = 0.9
0.25 0.5 0.75 1 1.25 1.5 1.75 2
0.25
0.5
0.75
1
1.25
1.5
1.75
2
0.25 0.5 0.75 1 1.25 1.5 1.75 2
0.25
0.5
0.75
1
1.25
1.5
1.75
2
Α
Β
D = 0.1
0.25 0.5 0.75 1 1.25 1.5 1.75 2
0.25
0.5
0.75
1
1.25
1.5
1.75
2
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Nonlinear stage of instability
∂tu = D∂2x u + αu − βv −u3︸︷︷︸
stab.
∂tv = ∂2x v + u − v
Two space dimensions
Spatial structureDifferent symmetriesSpace scaleDissipative solitons
Turing pattern
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Part III. Applications
To which extent the concept ofTuring patterns can be used to ex-plain self-organization of glows?
+-
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Key processes
Particle driftIonizationGamma process
E.g., 106 avalanchespermanently coexistin the gas gap.
E
A seed electron Ionization Gamma process
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Transport equations
Continuity equation∂tn +∇Γ = Q
+ -+
+
+
+
+
+
+
++
+
+
++
+
+
+
+
+
+
+
+
++
+
--
-
-
-
-
-
-
0 d zd+dc
j jc
E Ec
+ -
bi E -beE
j=j
A
j=j
C
j=j
B
Flux model
Γ = drift + diffusion
Source model
Q = ionization
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Physical ideas
E
Average velocity
vdrift = b E
Particle flux
Γ = n vdrift − D∇n
Impact ionization(dndt
)ionization
= α(E) Γe
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Mathematical model for gas
Particles
∂tne +∇Γe = Qe
∂tni +∇Γi = Qi
Source
Qe = α(E) Γe
Qi = α(E) Γe
Fluxes
Γe = −nebeE− De∇ne
Γi = +nibiE− Di∇ni
Field
E = −∇ϕ
ε0∇2ϕ = −q(ni − ne)
Drift dominates over diffusion!
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Reduction idea
Separation of scales:
d = 0.1–0.5 mm and D ∼ 1 mm
Cathode
Anode
A locally 1D solution.A continuous set of 1Dsolutions is required.
CVC
Ugas = U(I)
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Procedure
General formulation:
L0(∂z)u = L1(∂x , ∂y , ∂t )u
1D solution
L0(∂z)u = 0 → u = ueq(z,C)
Perturbation
u = ueq(z,C(x , y , t)
)+ u(x , y , z, t)
Compatibility
L′0(∂z)u = L1(∂x , ∂y , ∂t )ueq︸ ︷︷ ︸A singular equation
→ 〈 Projector |L1ueq(z,C)〉 = 0
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Reduced equations
+ -+
+
+
+
+
+
+
++
+
+
++
+
+
+
+
+
+
+
+
++
+
--
-
-
-
-
-
-
0 d zd+dc
j jc
E Ec
+ -
bi E -beEj=j
A
j=j
C
j=j
B
“Prey”
jz(x , y , t)
“Predator”
δU = Ugas(x , y , t)−Ub
∂t jz = Debi
be∇2jz +
Cγ
τi
(α′bδU +
α′′bτ2i d
24ε20j2z
)jz
∂tδU =d2
c3τc∇2δU +
Us − Ub − δUτc
− jzc
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Normalized form
A two-component reaction-diffusion system:
“current” u(x , y , t)“voltage” v(x , y , t)
τ∂tu = L2∇2u + (v + ηu2)u
∂tv = ∇2v + s − u − v
E.g., for the transient oscillations (current vs time)
10 20 30 40
0.2
0.4
0.6
0.82 4 6 8 10
0.2
0.4
0.6
0.8
1.2
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Results
0.1 0.2 0.3 0.4
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Tow
nsen
dm
ode
Glo
wm
ode
Η
LAn example for τ = 1 and s = 1
∂tu = L2∇2u + (v + ηu2)u
∂tv = ∇2v + 1− u − v
MechanismInstability thresholdSpatial scaleInitial dynamics