Hall effect in protoplanetary discs Geoffroy Lesur (IPAG, Grenoble, - - PowerPoint PPT Presentation
Hall effect in protoplanetary discs Geoffroy Lesur (IPAG, Grenoble, France) Matthew W. Kunz (Univ. of Princeton, USA) An accretion problem... Accretion discs are known to form around young stars and compact objects Gas can fall on the central
Geoffroy Lesur (IPAG, Grenoble, France) Matthew W. Kunz (Univ. of Princeton, USA)
Accretion discs are known to form around young stars and compact objects Gas can fall on the central object only if it looses angular momentum. One needs a way to transport angular momentum
«angular momentum transport problem» Turbulence produces a «turbulent viscosity»
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Balbus & Hawley 1991, Balbus 2003
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MRI is an efficient mechanism which seems to produce Need a relatively weak field (sub-thermal) Ideal MHD instability, modified by nonideal effects
Simulation parameters: Re=1000, Pm=1, β=1000 3D map of vy (azimuthal velocity)
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2 4 6 8 10 12 14 0.02 0.04 0.06 0.08 0.1
t (orbits)
Protoplanetary discs are far from being in the ideal MHD regime: very low ionisation fraction 3 non-ideal effects Ohmic resistivity (electrons-neutrals collisions) Hall effect (electrons-ions drift) Ambipolar diffusion (electrons-neutral drift)
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O H = ⇣ n 8 × 1017 cm−3 ⌘1/2⇣vA cs ⌘−1 A H = ⇣ n 9 × 1012 cm−3 ⌘−1/2⇣ T 103 K ⌘1/2⇣vA cs ⌘
∼ 10−13
1) Ambipolar 2) Hall 3) Ohmic 1) Hall 2) Ambipolar 3) Ohmic 1) Hall 2) Ohmic 3) Ambipolar 1) Ohmic 2) Hall 3) Ambipolar 0.1 AU 1 AU 10 AU 102 AU Midplane temperature, density Density at z = 4 h, effective disk temperature 10 –17 10 –15 10 –13 10 –11
ρ (g cm–3) T (K)
10 –9 10 –7 10 –5 10 3 10 2 10 1 10 0
(Armitage 2011)
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me due dt = −e(E + ue × B) − 1 ne rPe − νeime(ue − ui)
Equation of motion for electrons Introduce currents and average bulk velocity Ohm’s Law: Whistler waves:
Long timescale compared to electrons gyro-frequency
U ∼ ui
Ideal MHD Hall effect Electron pressure Ohmic resistivity
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E = −U ⇥ B + 1 ene J ⇥ B − 1 ene rPe + ηJ
U ∼ ui
Λ −1
H
Λ −1
η
5 10 2 4 6 8 10 0.1 0.2 0.3 0.4 0.5 0.6
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Introduce two dimensionless numbers Growth rate of the most unstable MRI mode MRI is more unstable with Hall and
Λη = v2
A
ηΩ
ΛH = eneB ρcΩ
Ohmic Elsasser number Hall Elsasser number
Ω · B > 0
THE EFFECT OF THE HALL TERM ON THE NONLINEAR EVOLUTION OF THE MAGNETOROTATIONAL INSTABILITY. II. SATURATION LEVEL AND CRITICAL MAGNETIC REYNOLDS NUMBER Takayoshi Sano and James M. Stone
Department of Astronomy, University of Maryland, College Park, MD 20742-2421; sano@astro.umd.edu Received 2002 March 31; accepted 2002 May 22
ABSTRACT The nonlinear evolution of the magnetorotational instability (MRI) in weakly ionized accretion disks, including the effect of the Hall term and ohmic dissipation, is investigated using local three-dimensional MHD simulations and various initial magnetic field geometries. When the magnetic Reynolds number, ReM v2
A= (where vA is the Alfve
´n speed, is the magnetic diffusivity, and is the angular frequency), is initially larger than a critical value ReM;crit, the MRI evolves into MHD turbulence in which angular momen- tum is transported efficiently by the Maxwell stress. If ReM < ReM;crit, however, ohmic dissipation suppresses the MRI, and the stress is reduced by several orders of magnitude. The critical value is in the range of 1–30 depending on the initial field configuration. The Hall effect does not modify the critical magnetic Reynolds number by much but enhances the saturation level of the Maxwell stress by a factor of a few. We show that the saturation level of the MRI is characterized by v2
Az=, where vAz is the Alfve
´n speed in the nonlinear regime along the vertical component of the field. The condition for turbulence and significant transport is given by v2
Az=e1, and this critical value is independent of the strength and geometry of the magnetic field
and the magnetic Reynolds number v2
A= is larger than about 30, this would imply that the MRI is operat-
ing in the disk. Subject headings: accretion, accretion disks — diffusion — instabilities — MHD — turbulence On-line material: color figures
parameter X0 for the models with 0 ¼ 800, 3200, and 12,800. The magnetic Reynolds number is ReM0 ¼ 1 for all the models. 9
Wardle & Salmeron 2012
Hall diffusion and the magnetorotational instability in protoplanetary discs
Mark Wardle1⋆ and Raquel Salmeron2
1Department of Physics & Astronomy and Research Centre for Astronomy, Astrophysics & Astrophotonics, Macquarie University,
Sydney, NSW 2109, Australia
2Planetary Science Institute, Research School of Astronomy & Astrophysics and Research School of Earth Sciences, Australian National University,
Canberra, ACT 2611, Australia Accepted 2011 October 15. Received 2011 October 13; in original form 2011 March 18
ABSTRACT
The destabilizing effect of Hall diffusion in a weakly ionized Keplerian disc allows the magnetorotational instability (MRI) to occur for much lower ionization levels than would
impression that the consequences of this for the non-linear saturated state are not as significant as suggested by the linear instability. Close inspection reveals that this is not actually the case as the simulations have not yet probed the Hall-dominated regime. Here we revisit the effect
(MHD) turbulence in protoplanetary discs, where Hall diffusion dominates over a large range
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Separate the mean shear from the fluctuations: Shearing box equations:
H
x y z
r · v = ∂tv − qΩx∂yv + v · rv = −rP + B · rB − 2Ω ⇥ v +qΩvxey + ν∆v ∂tB − qΩx∂yB = r ⇥ (v ⇥ B − xHJ ⇥ B) − qΩBxey + η∆B
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Courtesy T. Heinemann
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Use shear-periodic boundary conditions= «shearing-sheet» Allows one to use a sheared Fourier Basis periodic in y and z (non stratified box)
Courtesy T. Heinemann
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MHD equations solved in a co-moving sheared frame Compute non linear terms using the pseudo spectral method 3rd order low storage Runge-Kutta integrator Non-ideal effects: Ohmic, Hall, ambipolar (coming soon), Braginskii Available online http://ipag.osug.fr/~glesur/snoopy.html Advantages: Shearing waves are computed exactly (natural basis) Exponential convergence Magnetic flux conserved to machine precision Sheared frame & incompressible approximation: no CFL constrain due to the background sheared flow/sound speed.
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Whistler waves are well captured down to the grid scale Stable explicit scheme (RK3)
kℓ H ω/ωH 10 − 1 10 0 10 1 10 − 1 10 0 10 1 10 2
Whistler branch A l f v é n w a v e s
Nyquist frequency Falle (2003) «Explicit Hall-MHD codes are unconditionally unstable»
Kunz & Lesur (2013): stable for high order schemes
Λ − 1
H
α 20 40 60 80 100 10 − 7 10 − 6 10 − 5 10 − 4 10 − 3 10 − 2 10 − 1 ℓ H sgn(B z)
1
2
Does Hall-MRI look like «ideal» MRI? νt = αcsH
t α 100 200 300 400 500 600 10 − 12 10 − 10 10 − 8 10 − 6 10 − 4 10 − 2 10 0 Λ − 1
H = 0
Λ − 1
H = 2
Λ − 1
H = 16
Λ − 1
H = 32
Λ − 1
H = 100
Sano & Stone 2002
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Does Hall-MRI look like «ideal» MRI?
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ℓ H sgn(B z) α 1 2 3 10 − 7 10 − 6 10 − 5 10 − 4 10 − 3 10 − 2 10 − 1 10 0
Z B 3(I1,H1- 9) Z B 1(I1,H1- 6) Z B 10(I2,H2- 5) Z B 3(I2,H10- 16)
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t x 500 1000 1500 2000 −1 1 2 −0. 1 −0. 05
B z − B 0 t x 200 400 600 −2 2 4 −0. 1 −0. 05
B z − B 0
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= r = r
Consider the induction equation with Hall only Assume: Antidiffusive if !
with
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∂thBzi ⇠ 1 ene ∂xhJ ⇥ Biy ⇠
4πene ∂xhB · rByi ⇠
4πene ∂2
xhBxByi
⇠ cΩ2H2ρ0 ene ∂2
xα(hBzi)
h·i = ZZ dy dz
∂tδBz = Q ⇣ ∂α ∂Bz ⌘
B0∂2 xδBz
hBzi = B0 + δBz
⇣ ∂α ∂Bz ⌘
B0 < 0
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B z M x y × 100 10 − 4 10 − 3 10 − 2 10 − 1 −1 −0. 8 −0. 6 −0. 4 −0. 2
−α
t x 1 2 3 4 5 −1 1 2
B z − B 0
Induction Vorticity
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∂tB = r ⇥ ⇣ v ⇥ B − J ⇥ B ene ⌘
Canonical vorticity behaves like magnetic field in ideal MHD Field line redistribution implies a redistribution of vorticity in the flow
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t x 500 1000 1500 2000 −1 1 2 −0. 6 −0. 4 −0. 2
ωz − 2A t x 500 1000 1500 2000 −1 1 2 −0. 1 −0. 05
B z − B 0
long-lived zonal flows are associated to Hall-MRI Good for planet formation?
Hall MRI does not saturate like ideal MRI Turbulent transport reduced by 2-3 orders of magnitude Production of zonal fields Mean field theory captures this behaviour Zonal flows produced by zonal field regions: dust trapping regions? Open questions: Stratification, compressibility? Vertical ionisation profile?
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References: Sano & Stone (2002), ApJ, 577, 534–553 Wardle & Salmeron (2012), MNRAS, 422, 2737–2755 Kunz & Lesur (2013), MNRAS, accepted