Raquel Salmeron (1) and Mark Wardle (2) (1) Research School of Astronomy and Astrophysics, The Australian National University (2) Department of Physics, Macquarie University which leads to This system is linearized about an initial state where the fluid is in Keplerian rotation and the magnetic field is vertical. The equations are solved by ‘shooting’ from the midplane to the surface of the disk while simultaneously adjusting the growth rate of the perturbations and E ` at the midplane. [1] Balbus S. A., Hawley J. F., 1991, ApJ, 376, 214. [2] Salmeron R. & Wardle M., 2005, MNRAS, 361, 45. [3] Wardle M. & Ng C., 1999, MNRAS, 303, 239 [4] Salmeron R. & Wardle M., 2003, MNRAS, 345, 992 The magnetorotational instability (MRI, [1]) transports angular momentum radially outwards in protostellar disks through the distortion of the magnetic field lines that connect fluid elements. However, the low ionization fraction of the gas limits the ability of the field to couple to the fluid and the disk magnetic response is strongly dependent on the conductivity and its spatial variation. We present the vertical structure and linear growth of the MRI in a weakly ionized, stratified protostellar disk, assuming dust grains are well mixed with the fluid. All grains have the same radius (a = 0.1, 1 or 3 μm) and constitute 1% of the total mass of the gas. Solutions are obtained at representative radii for different strengths of the initially vertical magnetic field, configurations of the conductivity tensor and grain sizes. The method [2] incorporates a realistic ionization profile, produced by cosmic rays, X- rays and radioactive decay. The contributions of ambipolar diffusion, the Hall conductivity and Ohmic resistivity are all included. a Ratio of the Alfven speed to the sound speed at the midplane Hall conductivity Magnetic coupling at the midplane Pedersen conductivity -Explore the effect of dust settling and a realistic grain size distribution on the solutions. -Investigate the implications of the disk structure to planet formation and migration Left panel: Perturbations grow for B < 8 G (1 AU), B < 800 mG (5 AU) and B < 250 mG (10 AU). For a range of B, max is of order the ideal-MHD rate (0.75 ) [1, 2]. Right Panel:Hall conductivity modifies the growth rate of global unstable modes at 1 AU for all magnetic field strengths that support MRI [2]. Left panel: Growth rate of the most unstable modes of the MRI for R = 1, 5 and 10 AU as a function of the strength of the magnetic field for the minimum-solar nebula disk and including the ionization provided by cosmic rays, X-rays and radioactive decay. Right panel: Growth rates of the fastest growing modes at 1 AU as a function of the strength of the magnetic field for different configurations of the conductivity tensor Left panels: Full conductivity modes have higher wavenumber, and grow closer to the midplane, than modes in the ambipolar diffusion limit. Right panels: When dust grains are present, both the wavenumber and growth rate of the perturbations are reduced and the dead zone is more extended. Structure of the most unstable modes of the MRI for the minimum-mass solar nebula model as a function of the magnetic field strength (top right corner of each panel). The maximum growth rate ( max ) is shown in bottom right corners. Solid lines display B r and dashed ones correspond to B . Left panel: At R = 1 AU for different configurations of the conductivity tensor. Right panel: R = 10 AU. The left column shows the case where dust grains have settled, while the right one displays results when 0.1μm grains are present. (a) Effect of dust grains (R = 5 AU) Structure of the most unstable MRI modes at R = 5 AU as a function of the strength of the magnetic field and for different grain sizes. The growth rate is indicated in the bottom right corner of each panel whereas the field strength appears at the top right corner. ∂ρ ∂t + ∇· (ρv)=0 ∂ v ∂t +(v ·∇)v − 2Ωv φ ˆ r + 1 2 Ωv r ˆ φ − v 2 K r ˆ r + c 2 s ρ ∇ρ + ∇Φ − J × B cρ =0 ∂ B ∂t = ∇× (v × B) − c∇× E − 3 2 ΩB r ˆ φ J = σ E + σ H ˆ B × E ⊥ + σ P E ⊥ E = J σ + σ H σ 2 ⊥ J × B B − σ P σ 2 ⊥ − 1 σ (J × B) × B B 2 J = c 4π ∇× B Left panel: No grains are present. For z/H < 2.5 the fluid is in the Hall conductivity regime (| H | > P ). At higher z, ambipolar diffusion dominates. Near the surface | H | << P [2]. Right Panel: 0.1 μm grains are present. All terms drop drastically and | H | shows ‘spikes’ at the z where it changes sign (in response to particular species becoming decoupled to the field [3]). H >0 when 0 < z/H < 1.6 and 2.5 < z/H < 4.1. The behaviour of the charged species at selected heights is sketched below. Components of the conductivity tensor ( ||, | H| and P), magnetic coupling ( ) and | H|/ ⊥ as a function of height for R = 10 AU and B = 10 mG. Left panel: Dust grains are assumed to be settled. Right panel: A population of 0.1 μm grains is well mixed with the gas. (1a) z/H = 1, Electrons and 1/3 of the ions stick to the grains. Their contributions to H are similar, with a small positive excess, so H >0. (1b) z/H = 2, Ions are the dominant positively charged species, while the negative charges are still on the grains, so H <0 in this region. (1c) z/H = 3, Ions and electrons are the main charge carriers, the former dominate the contribution to H, which is > 0. In all cases, whenever < | H |/ ⊥ , Hall diffusion modifies the structure and growth of MRI unstable modes [4]. (2) Structure of the perturbations (1) The conductivity tensor (b) Conductivity regime (b1) R = 5 AU, 3 μm grains (b2) R = 10 AU, 0.1 μm grains (3) Growth rates (a) No grains, at different radii (b) Effect of dust grains (R = 5 AU) As the grain size is reduced, the central dead zone is more extended and the wavenumber of the perturbations diminishes. When 0.1 μm grains are present, the disk is inactive for z/H < 2. Even for a = 3 μm, the modes are different from the no-grains solutions. Hall diffusion modifies the most unstable MRI modes for all magnetic field strengths of interest, diminishing the extent of the central dead zone and increasing the wavelength of the perturbations. Structure of the most unstable MRI modes for R = 5 AU and assuming 3 μm grains are present, as a function of the strength of the magnetic field and for different configurations of the conductivity tensor. The growth rate is indicated in the bottom right corner of each panel whereas the field strength appears at the top right corner. When dust grains are present, MRI modes grow for B < 80 mG (a=1 μm) and B < 15 mG (0.1 μm), down from B < 800 mG when grains are assumed to be settled (see section 3a, above). For 1 mG < B < 5 mG, the fastest growing modes are obtained when 0.1 μm grains are present. These modes peak at higher z where the local growth rate is higher (see case 2a, centre). Growth rate of the most unstable modes of the MRI for R = 5 AU as a function of the strength of the magnetic field and two choices of the single-size grain population (a = 0.1 and 1 μm). For comparison, the no-grain case is also shown. q = q 0 + δq (z )e iωt Perturbations Field-aligned conductivity 0.1 H H P P H H (1a) (1b) (1c)
Welcome message from author
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
Raquel Salmeron(1) and Mark Wardle(2)
(1) Research School of Astronomy and Astrophysics, The Australian National University(2) Department of Physics, Macquarie University
which leads to
This system is linearized about an initial state where the fluid is in Keplerian rotation and the magnetic field is vertical. The equations are solved by ‘shooting’ from the midplane to the surface of the disk
while simultaneously adjusting the growth rate of the perturbations and E ` at the midplane.
[1] Balbus S. A., Hawley J. F., 1991, ApJ, 376, 214.
[2] Salmeron R. & Wardle M., 2005, MNRAS, 361, 45.
[3] Wardle M. & Ng C., 1999, MNRAS, 303, 239
[4] Salmeron R. & Wardle M., 2003, MNRAS, 345, 992
The magnetorotational instability (MRI, [1]) transports angular
momentum radially outwards in protostellar disks through the
distortion of the magnetic field lines that connect fluid elements.
However, the low ionization fraction of the gas limits the ability of the
field to couple to the fluid and the disk magnetic response is strongly
dependent on the conductivity and its spatial variation.
We present the vertical structure and linear growth of the MRI in a weakly ionized, stratified protostellar disk, assuming dust grains are well mixed with the fluid. All grains have the same radius (a = 0.1, 1 or 3 μm) and constitute 1% of the total mass of the gas. Solutions are obtained at representative radii for different strengths of the initially vertical magnetic field, configurations of the conductivity tensor and grain sizes.
The method [2] incorporates a realistic ionization profile, produced by cosmic rays, X-rays and radioactive decay. The contributions of ambipolar diffusion, the Hall conductivity and Ohmic resistivity are all included.
a Ratio of the Alfven speed to the sound speed at the midplane
Hall conductivityMagnetic coupling at the midplane
Pedersen conductivity
-Explore the effect of dust settling and a realistic grain size distribution on the solutions.-Investigate the implications of the disk structure to planet formation and migration
Left panel: Perturbations grow for B < 8 G (1 AU), B < 800 mG (5 AU) and B < 250 mG (10 AU). For a range of B, max is of order
the ideal-MHD rate (0.75 ) [1, 2].
Right Panel:Hall conductivity modifies the growth rate of global unstable modes at 1 AU for all magnetic field strengths that support MRI [2].
Left panel: Growth rate of the most unstable modes of the MRI for R = 1, 5 and 10 AU as a function of the strength of the magnetic field for the minimum-solar nebula disk and including the ionization provided by cosmic rays, X-rays and radioactive decay. Right panel: Growth rates of the fastest growing modes at 1 AU as a function of the strength of the magnetic field for different configurations of the conductivity tensor
Left panels: Full conductivity modes have higher wavenumber, and grow closer to the midplane, than modes in the ambipolar diffusion limit.Right panels: When dust grains are present, both the wavenumber and growth rate of the perturbations are reduced and the dead zone is more extended.
Structure of the most unstable modes of the MRI for the minimum-mass solar nebula model as a function of the magnetic field strength (top right corner of each panel). The maximum growth rate ( max) is shown in bottom right corners. Solid lines display Br and dashed ones correspond to B . Left panel:
At R = 1 AU for different configurations of the conductivity tensor. Right panel: R = 10 AU. The left column shows the case where dust grains have settled, while the right one displays results when 0.1μm grains are present.
(a) Effect of dust grains (R = 5 AU)
Structure of the most unstable MRI modes at R = 5 AU as a function of the strength of the magnetic field and for different grain sizes. The growth rate is indicated in the bottom right corner of each panel whereas the field strength appears at the top right corner.
∂ρ
∂t+ ∇ · (ρv) = 0
∂v
∂t+ (v · ∇)v − 2Ωvφr̂ +
1
2Ωvrφ̂
−v2
K
rr̂ +
c2
s
ρ∇ρ + ∇Φ −
J × B
cρ= 0
∂B
∂t= ∇× (v × B) − c∇× E
′−
3
2ΩBrφ̂
J = σ‖E′
‖+ σH
ˆB × E
′
⊥+ σPE
′
⊥
E′ =
J
σ‖
+σH
σ2
⊥
J × B
B−
(σP
σ2
⊥
−1
σ‖
)(J × B) × B
B2
J =c
4π∇× B
Left panel: No grains are present. For z/H < 2.5 the fluid is in the
Hall conductivity regime (| H| > P). At higher z, ambipolar
diffusion dominates. Near the surface | H| << P [2].
Right Panel: 0.1 μm grains are present. All terms drop
drastically and | H| shows ‘spikes’ at the z where it changes sign
(in response to particular species becoming decoupled to the field
[3]). H>0 when 0 < z/H < 1.6 and 2.5 < z/H < 4.1. The behaviour
of the charged species at selected heights is sketched below.
Components of the conductivity tensor ( ||, | H| and P), magnetic coupling ( ) and | H|/ ⊥ as a function of height for R = 10 AU and B = 10 mG.
Left panel: Dust grains are assumed to be settled. Right panel: A population of 0.1 μm grains is well mixed with the gas.
(1a) z/H = 1, Electrons and 1/3 of the ions stick to the grains. Their contributions to H are similar, with a small positive excess, so H>0.(1b) z/H = 2, Ions are the dominant positively charged species, while the negative charges are still on the grains, so H <0 in this region.(1c) z/H = 3, Ions and electrons are the main charge carriers, the former dominate the contribution to H, which is > 0.
In all cases, whenever < | H|/ ⊥, Hall diffusion modifies the
structure and growth of MRI unstable modes [4].
(2) Structure of the
perturbations(1) The conductivity tensor
(b) Conductivity regime
(b1) R = 5 AU, 3 μm grains(b2) R = 10 AU, 0.1 μm grains
(3) Growth rates
(a) No grains, at different radii
(b) Effect of dust grains (R = 5 AU)
As the grain size is reduced, the central dead zone is more extended and the wavenumber of the perturbations diminishes. When 0.1 μm grains are present, the disk is inactive for z/H < 2. Even for a = 3 μm, the modes are different from the no-grains solutions.
Hall diffusion modifies the most unstable MRI modes for all magnetic field strengths of interest, diminishing the extent of the central dead zone and increasing the wavelength of the perturbations.
Structure of the most unstable MRI modes for R = 5 AU and assuming 3 μm grains are present, as a function of the strength of the magnetic field and for different configurations of the conductivity tensor. The growth rate is indicated in the bottom right corner of each panel whereas the field strength appears at the top right corner.
When dust grains are present, MRI modes grow for B < 80 mG (a=1 μm) and B < 15 mG (0.1 μm), down from B < 800 mG when grains are assumed to be settled (see section 3a, above).
For 1 mG < B < 5 mG, the fastest growing modes are obtained when 0.1 μm grains are present. These modes peak at higher z where the local growth rate is higher (see case 2a, centre).
Growth rate of the most unstable modes of the MRI for R = 5 AU as a function of the strength of the magnetic field and two choices of the single-size grain population (a = 0.1 and 1 μm). For comparison, the no-grain case is also shown.
q = q0 + δq(z)eiωtPerturbations
Field-aligned conductivity
0.1
H
H
P
P
H
H
(1a)
(1b)
(1c)
Raquel Salmeron(1) and Mark Wardle(2)
(1) Research School of Astronomy and Astrophysics, The Australian National University(2) Department of Physics, Macquarie University
which leads to
This system is linearized about an initial state where the fluid is in Keplerian rotation and the magnetic field is vertical. The equations are solved by ‘shooting’ from the midplane to the surface of the disk
while simultaneously adjusting the growth rate of the perturbations and E ` at the midplane.
[1] Balbus S. A., Hawley J. F., 1991, ApJ, 376, 214.
[2] Salmeron R. & Wardle M., 2005, MNRAS, 361, 45.
[3] Wardle M. & Ng C., 1999, MNRAS, 303, 239
[4] Salmeron R. & Wardle M., 2003, MNRAS, 345, 992
The magnetorotational instability (MRI, [1]) transports angular
momentum radially outwards in protostellar disks through the
distortion of the magnetic field lines that connect fluid elements.
However, the low ionization fraction of the gas limits the ability of the
field to couple to the fluid and the disk magnetic response is strongly
dependent on the conductivity and its spatial variation.
We present the vertical structure and linear growth of the MRI in a weakly ionized, stratified protostellar disk, assuming dust grains are well mixed with the fluid. All grains have the same radius (a = 0.1, 1 or 3 μm) and constitute 1% of the total mass of the gas. Solutions are obtained at representative radii for different strengths of the initially vertical magnetic field, configurations of the conductivity tensor and grain sizes.
The method [2] incorporates a realistic ionization profile, produced by cosmic rays, X-rays and radioactive decay. The contributions of ambipolar diffusion, the Hall conductivity and Ohmic resistivity are all included.
a Ratio of the Alfven speed to the sound speed at the midplane
Hall conductivityMagnetic coupling at the midplane
Pedersen conductivity
-Explore the effect of dust settling and a realistic grain size distribution on the solutions.-Investigate the implications of the disk structure to planet formation and migration
Left panel: Perturbations grow for B < 8 G (1 AU), B < 800 mG (5 AU) and B < 250 mG (10 AU). For a range of B, max is of order
the ideal-MHD rate (0.75 ) [1, 2].
Right Panel:Hall conductivity modifies the growth rate of global unstable modes at 1 AU for all magnetic field strengths that support MRI [2].
Left panel: Growth rate of the most unstable modes of the MRI for R = 1, 5 and 10 AU as a function of the strength of the magnetic field for the minimum-solar nebula disk and including the ionization provided by cosmic rays, X-rays and radioactive decay. Right panel: Growth rates of the fastest growing modes at 1 AU as a function of the strength of the magnetic field for different configurations of the conductivity tensor
Left panels: Full conductivity modes have higher wavenumber, and grow closer to the midplane, than modes in the ambipolar diffusion limit.Right panels: When dust grains are present, both the wavenumber and growth rate of the perturbations are reduced and the dead zone is more extended.
Structure of the most unstable modes of the MRI for the minimum-mass solar nebula model as a function of the magnetic field strength (top right corner of each panel). The maximum growth rate ( max) is shown in bottom right corners. Solid lines display Br and dashed ones
correspond to B . Left panel: At R = 1 AU for different configurations of the conductivity tensor. Right panel: R = 10 AU. The left column
shows the case where dust grains have settled, while the right one displays results when 0.1μm grains are present.
(a) Effect of dust grains (R = 5 AU)
Structure of the most unstable MRI modes at R = 5 AU as a function of the strength of the magnetic field and for different grain sizes. The growth rate is indicated in the bottom right corner of each panel whereas the field strength appears at the top right corner.
∂ρ
∂t+ ∇ · (ρv) = 0
∂v
∂t+ (v · ∇)v − 2Ωvφr̂ +
1
2Ωvrφ̂
−v2
K
rr̂ +
c2
s
ρ∇ρ + ∇Φ −
J × B
cρ= 0
∂B
∂t= ∇× (v × B) − c∇× E
′−
3
2ΩBrφ̂
J = σ‖E′
‖+ σH
ˆB × E
′
⊥+ σPE
′
⊥
E′ =
J
σ‖
+σH
σ2
⊥
J × B
B−
(σP
σ2
⊥
−1
σ‖
)(J × B) × B
B2
J =c
4π∇× B
Left panel: No grains are present. For z/H < 2.5 the fluid is in the
Hall conductivity regime (| H| > P). At higher z, ambipolar
diffusion dominates. Near the surface | H| << P [2].
Right Panel: 0.1 μm grains are present. All terms drop
drastically and | H| shows ‘spikes’ at the z where it changes sign
(in response to particular species becoming decoupled to the field
[3]). H>0 when 0 < z/H < 1.6 and 2.5 < z/H < 4.1. The behaviour
of the charged species at selected heights is sketched below.
Components of the conductivity tensor ( ||, | H| and P), magnetic coupling ( ) and | H|/ ⊥ as a function of height for R = 10 AU and
B = 10 mG. Left panel: Dust grains are assumed to be settled. Right panel: A population of 0.1 μm grains is well mixed with the gas.
(1a) z/H = 1, Electrons and 1/3 of the ions stick to the grains. Their contributions to H are similar, with a small positive excess, so H>0.(1b) z/H = 2, Ions are the dominant positively charged species, while the negative charges are still on the grains, so H <0 in this region.(1c) z/H = 3, Ions and electrons are the main charge carriers, the former dominate the contribution to H, which is > 0.
In all cases, whenever < | H|/ ⊥, Hall diffusion modifies the
structure and growth of MRI unstable modes [4].
(2) Structure of the
perturbations(1) The conductivity tensor
(b) Conductivity regime
(b1) R = 5 AU, 3 μm grains(b2) R = 10 AU, 0.1 μm grains
(3) Growth rates
(a) No grains, at different radii
(b) Effect of dust grains (R = 5 AU)
As the grain size is reduced, the central dead zone is more extended and the wavenumber of the perturbations diminishes. When 0.1 μm grains are present, the disk is inactive for z/H < 2. Even for a = 3 μm, the modes are different from the no-grains solutions.
Hall diffusion modifies the most unstable MRI modes for all magnetic field strengths of interest, diminishing the extent of the central dead zone and increasing the wavelength of the perturbations.
Structure of the most unstable MRI modes for R = 5 AU and assuming 3 μm grains are present, as a function of the strength of the magnetic field and for different configurations of the conductivity tensor. The growth rate is indicated in the bottom right corner of each panel whereas the field strength appears at the top right corner.
When dust grains are present, MRI modes grow for B < 80 mG (a=1 μm) and B < 15 mG (0.1 μm), down from B < 800 mG when grains are assumed to be settled (see section 3a, above).
For 1 mG < B < 5 mG, the fastest growing modes are obtained when 0.1 μm grains are present. These modes peak at higher z where the local growth rate is higher (see case 2a, centre).
Growth rate of the most unstable modes of the MRI for R = 5 AU as a function of the strength of the magnetic field and two choices of the single-size grain population (a = 0.1 and 1 μm). For comparison, the no-grain case is also shown.
q = q0 + δq(z)eiωtPerturbations
Field-aligned conductivity
0.1
(1a)
(1b)
(1c)
H
H
P
P
H
H
Related Documents
