Nonlinear Ohm’s Law: Plasma Heating by Strong Electric Fields

and the Ionization Balance in PPDs

1

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)