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 object only if it looses angular momentum. One needs a way to transport angular momentum outward to have accretion: «angular momentum transport problem» Turbulence produces a «turbulent viscosity» ν t = α c s H 2
The magnetorotational instability (MRI) A B Field line A MRI is an efficient mechanism which seems α ∼ 10 − 3 —10 − 1 to produce B Need a relatively weak field (sub-thermal) Ideal MHD instability, modified by nonideal effects Balbus & Hawley 1991, Balbus 2003 3
Simulation example Simulation parameters: Re=1000, Pm=1, β =1000 3D map of v y (azimuthal velocity) 0.1 It works! 0.08 � 0.06 Is it the end of the 0.04 0.02 story? 0 2 4 6 8 10 12 14 t (orbits) 4
Protoplanetary discs Protoplanetary discs are far from being in the ideal MHD regime: very low ionisation fraction ∼ 10 − 13 3 non-ideal effects Ohmic resistivity (electrons-neutrals collisions) Hall effect (electrons-ions drift) Ambipolar diffusion (electrons-neutral drift) ⌘ 1 / 2 ⇣ v A O n ⌘ − 1 ⇣ = 8 × 10 17 cm − 3 H c s ⌘ − 1 / 2 ⇣ ⌘ 1 / 2 ⇣ v A A n T ⇣ ⌘ = 9 × 10 12 cm − 3 10 3 K H c s Hall dominates for «intermediate» densities 5
Non-ideal protoplanetary discs 0.1 AU Density at z = 4 h, e ff ective disk Midplane 1) Ambipolar temperature 10 3 temperature, 2) Hall density 3) Ohmic 1 AU 10 2 1) Ohmic T (K) 2) Hall 10 AU 3) Ambipolar 1) Hall 10 1 2) Ohmic 3) Ambipolar 10 2 AU 1) Hall 2) Ambipolar 10 0 3) Ohmic 10 –17 10 –15 10 –13 10 –11 10 –9 10 –7 10 –5 ρ (g cm –3 ) (Armitage 2011) 6
Hall effect basics Fully ionised plasmas Equation of motion for electrons dt = − e ( E + u e × B ) − 1 d u e r P e − ν ei m e ( u e − u i ) m e n e Long timescale compared to electrons gyro-frequency U ∼ u i Introduce currents and average bulk velocity J = − en e ( u e − u i ) U ∼ u i Ohm’s Law: 1 1 E = − U ⇥ B + J ⇥ B − r P e + η J en e en e Electron Ideal MHD Hall effect Ohmic resistivity pressure Whistler waves: ⇣ ⌘⇣ ⌘ c ∂ t δ b = − k × δ b k · B 0 4 π en e 7
MRI in the Hall regime Linear stability analysis Introduce two dimensionless numbers Λ η = v 2 Λ H = en e B A ρ c Ω η Ω Ohmic Elsasser number Hall Elsasser number Growth rate of the most unstable MRI mode 10 0.6 8 0.5 6 0.4 Λ − 1 η 0.3 4 0.2 2 0.1 0 0 5 10 Λ − 1 H MRI is more unstable with Hall and Ω · B > 0 8
Literature: Sano & Stone (2002) 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 e ff ect 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, Re M � v 2 A = � � (where v A is the Alfve ´n speed, � is the magnetic di ff usivity, and � is the angular frequency), is initially larger than a critical value Re M ; crit , the MRI evolves into MHD turbulence in which angular momen- tum is transported e ffi ciently by the Maxwell stress. If Re M < Re M ; 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 e ff ect 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 v 2 A z = � � , where v A z 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 v 2 A z = � � e 1, and this critical value is independent of the strength and geometry of the magnetic field or the size of the Hall term. If the magnetic field strength in an accretion disk can be estimated observationally and the magnetic Reynolds number v 2 A = � � is larger than about 30, this would imply that the MRI is operat- ing in the disk. Subject headings: accretion, accretion disks — di ff usion — instabilities — MHD — turbulence On-line material: color figures Fig. 5. —Saturation level of the Maxwell stress as a function of the Hall parameter X 0 for the models with � 0 ¼ 800, 3200, and 12,800. The magnetic Reynolds number is Re M 0 ¼ 1 for all the models. Hall effect «does nothing» 9
Literature: Wardle & Salmeron 2012 Hall diffusion and the magnetorotational instability in protoplanetary discs Mark Wardle 1 ⋆ and Raquel Salmeron 2 1 Department of Physics & Astronomy and Research Centre for Astronomy, Astrophysics & Astrophotonics, Macquarie University, Sydney, NSW 2109, Australia 2 Planetary 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 otherwise be possible. However, simulations incorporating Hall and Ohm diffusion give the 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 of Hall diffusion on the MRI and the implications for the extent of magnetohydrodynamic (MHD) turbulence in protoplanetary discs, where Hall diffusion dominates over a large range of radii. Wardle & Salmeron 2012 Simulations did not explore the right regime 10
The incompressible shearing box model Separate the mean shear from the fluctuations: u = − q Ω x e y + v z y H x Shearing box equations: = 0 r · v ∂ t v − q Ω x ∂ y v + v · r v = − r P + B · r B − 2 Ω ⇥ v + q Ω v x e y + ν ∆ v = r ⇥ ( v ⇥ B − x H J ⇥ B ) − q Ω B x e y + η ∆ B ∂ t B − q Ω x ∂ y B 11
Boundary conditions 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 12
Spectral methods for shearing boxes Shearing wave decomposition Courtesy T. Heinemann 13
The Snoopy code a spectral method for sheared flows 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. 14
Testing whistler waves with Snoopy Falle (2003) «Explicit Hall-MHD codes are Kunz & Lesur (2013): stable for high order schemes unconditionally unstable» Whistler branch 10 2 10 1 ω / ω H 10 0 s e v a w n é v f l A 10 − 1 10 − 1 10 0 10 1 k ℓ H Nyquist frequency Whistler waves are well captured down to the grid scale Stable explicit scheme (RK3) 15
Hall-MRI: turbulent viscosity Does Hall-MRI look like «ideal» MRI? Sano & Stone 2002 ℓ H sgn( B z ) 10 0 0 0 . 5 1 1 . 5 2 10 − 2 10 − 1 10 − 4 10 − 2 10 − 6 10 − 3 α 10 − 8 α Λ − 1 10 − 4 H = 0 ν t = α c s H 10 − 10 Λ − 1 Λ − 1 10 − 5 H = 2 H = 32 10 − 12 Λ − 1 Λ − 1 10 − 6 H = 16 H = 100 10 − 7 0 100 200 300 400 500 600 0 20 40 60 80 100 Λ − 1 t H Although a powerful instability is present, Hall-MRI simulations have a very low level of turbulent transport 16
Hall-MRI: turbulent viscosity Does Hall-MRI look like «ideal» MRI? 10 0 Z B 3(I1,H1- 9) 10 − 1 Z B 1(I1,H1- 6) 10 − 2 Z B 10(I2,H2- 5) Z B 3(I2,H10- 16) 10 − 3 α 10 − 4 10 − 5 10 − 6 10 − 7 0 1 2 3 ℓ H sgn( B z ) ◆ 1 / 2 ✓ ⇢ ✓ m i c 2 ◆ 1 / 2 Transport is controlled by ` H ≡ 4 ⇡ e 2 n i ⇢ i 17
Hall-MRI animation: Bz MRI+Ohmic+Hall MRI+Ohmic resistivity 18
Zonal field structures in Hall-dominated discs 0 . 15 4 2 0 . 15 � B z � − B 0 0 . 1 0 . 1 2 1 = r = r 0 . 05 0 . 05 0 x 0 x 0 0 − 2 − 1 − 0 . 05 − 0 . 05 � B z � − B 0 − 0 . 1 − 0 . 1 0 200 400 600 0 500 1000 1500 2000 t t Self Organisation! 19
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