nonlinear ohm’s law - indico€¦ · nonlinear ohm’s law:! plasma heating by strong electric...
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Nonlinear Ohm’s Law: Plasma Heating by Strong Electric Fields
and the Ionization Balance in PPDs
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Satoshi Okuzumi (Tokyo Institute of Technology)
in collaboration with Shu-ichiro Inutsuka (Nagoya University)
Ref: Okuzumi & Inutsuka, submitted (arXiv:1407.8110)
Disk Ionization and MRI
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Neutral Gas
H2+
+
+ -
-- -
HCO+
e-e-
e-
H3+
Mg+
Grains
Star
Disk
cosmic ray nonthermalionization
gas-phaserecombination IonsElectrons
stellar X-ray
?
● Ionization by external high-energy sources ● Recombination in gas phase and in “solid phase” ● MRI turbulence if gas is sufficiently ionized ● How MRI turbulence changes ionization state?
Electron Heating in Weakly Ionized Gas
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−E
e− H2
H2
H2
H2H2
H2H2
Lifshitz & Pitaevskii, Physical kinetics (1981) Golant et al., Fundamentals of plasma physics (1980)
ℓe
Electrons cannot move straight because they frequently collide with neutrals. ⇒ Random velocity >> drift velocity
Critical E-Field Strength
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ℓe=(σennn)−1: electron m.f.p. T: neutral gas temperature
Electrons are significantly heated when E is higher than
E-field Strength in MRI Turbulence
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(Muranushi, Okuzumi, & Inutsuka 2012)
(≳1 for MRI to be active)
● By using Ohm’s law E′ = (4πη/c2)J, we obtain
(~100–1000 for fully saturated turbulence)
● MHD simulations show RMS current density is insensitive to ohmic resistivity:
Criterion for Electron Heating in MRI Turbulence
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● RMS strength of comoving electric field in MRI turbulence
If 1≲ Λz ≲100, MRI-induced E-fields heat up free electrons (up to ~ 1 eV)
(see also Inutsuka & Sano 2005)
(Okuzumi & Inutsuka, 2014; based on Muranushi, Okuzumi, & Inutsuka 2012)
● Critical E-field strength for electron heating
Ionization Model with Plasma Heating
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- fe : Davydov–Druyvesteyn - fi : Offset Maxwellian ● Analytic functions of E ● Inelastic energy losses neglected
sticking to grain
gas-phase rec.
impact ionization
Okuzumi & Inutsuka (2014)
Model Parameters
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Ionization Balance
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Model B (d/g=10−4)
Electron heating ➡ Electron–grain collision freq. ↑ ➡ Electron abundance ↓, Grain charge ↑ (if grain charging dominates over gas-phase rec.)
electron heating
Model C (d/g=10−2)
electron heating
J–E Relation
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Model B (d/g=10−4) Model C (d/g=10−2)
J decreases with increasing E until Ji takes over Je
➡ N-shaped J–E curve!
Effect of Impact Ionization
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● “Electric Discharge” at ⟨𝜖e⟩ ≈ 3eV (cf. IP =15eV)
● For grain-rich cases, the discharge current becomes triple-valued (S-shaped J-E curve)
Nature of Multiple Equilibria
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(STABLE) (UNSTABLE) (STABLE)
1.7 1.7
0.66 11
11
430 430
430430
0.006
4.8×107 4.8×1071.7×108
2.2×108 2.2×10 8
impact impact
● Low State (stable) (external ionization) = (grain charging)
● Middle State (unstable) (impact ionization) = (grain charging)
● High State (stable) (impact ionization) = (gas-phase recombination)
L
M
H
Nature of Multiple Equilibria
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L
M
H
Negative Differential Resistance (NDR)
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NDR Destabilizes E-Field!
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Maxwell-Ampère Eq.
● Equilibrium:
● Perturbation:
In the long-wavelength limit (just for simplicity),
(Ampère’s law)
If σdiff < 0 (NDR), δE grows.
displacement current
Quasi-steady Ampère’s law is no longer valid!
Nonlinear Ohm’s Law: Summary
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“N”-curve“S”-curve
Linear Ohm Nonlinear Ohm
Application to PPDs
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Example: MMSN, d/g = 0.01, a = 0.1µm(Mori & Okuzumi, in prep.)
● At < 60 AU, Λ falls below 1 in nonlinear regime (self-regulated MRI?)
● Caveat: non-ohmic effects (gyromotion of i & e) not included here
J [e
su c
m−2
s−1 ]
J [e
su c
m−2
s−1 ]
⬇Λ<1 ⬇Λ
<1
J=JMRI J=JMRI
Eʹ′ [esu cm−2] Eʹ′ [esu cm−2]
Conclusions
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● A ≪generalized≫ nonlinear Ohm’s law (including AD & Hall drift) is also coming soon!
● MRI-induced electric fields heat up electrons in PPDs (in particular when 1<Λ<100).
● Under electron heating, the conductivity decreases with increasing E. Even the electric current J decreases (“N-shaped” J–E curve).
● Discharge current can have an unstable intermediate branch (“S-shaped” J–E curve) when dust is abundant.
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Effect of Inelastic Energy Losses
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-Pℓ: elastic energy loss -PR: rotational excitation -PV: vibrational excitation -PI: ionization - Pℓ + PR + PV + PI = 1
Engelhardt & Phelps (1963)
elastic only (Pℓ=1) w/ inelastic loss
With inelastic losses, the electron kinetic energy is
Toward a Generalized Nonlinear Ohm’s Law
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Gyromotion suppresses heating by E perp. to B:
● Ωc,α: gyrofrequency of species α ● ts,α: stopping time of species α
(Golant et al. 1980)