Introduction Methodology Results Summary Appendix References
Thesis Results (Part 1) Ballistic Relaxation, Perturbation, Resistivity Samuel A. Lazerson University of Alaska, Geophysical Institute
April 1, 2008
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Overview Motivation The Question Plasma Parameters
Overview
Introduction The DENISIS Code Results of Ballistic Relaxation Results of Perturbation Results of Resistivity Remarks
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Overview Motivation The Question Plasma Parameters
Motivation Chondrules are the millimeter sized spherical inclusions found in chondrites (meteorites). The process by which they formed in largely unknown. They are discussed as the first solids in the solar system. They are the transitional material between dust and meter sized stones. They present a geological record of the conditions present in the protosolar nebula.
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The Scientific Question Can magnetic reconnection in a dusty plasma explain the heating necessary for chondrule properties? 4.5 By old
Stereotypical Chondrite (Sears, 2004)
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The Scientific Question Can magnetic reconnection in a dusty plasma explain the heating necessary for chondrule properties? 4.5 By old nm to mm size
Stereotypical Chondrite (Sears, 2004)
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The Scientific Question Can magnetic reconnection in a dusty plasma explain the heating necessary for chondrule properties? 4.5 By old nm to mm size Heating rates in the range of 2000 − 5000K /hr
Stereotypical Chondrite (Sears, 2004)
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The Scientific Question Can magnetic reconnection in a dusty plasma explain the heating necessary for chondrule properties? 4.5 By old nm to mm size Heating rates in the range of 2000 − 5000K /hr Multiple heating events are recorded. Stereotypical Chondrite (Sears, 2004)
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Introduction Methodology Results Summary Appendix References
Overview Motivation The Question Plasma Parameters
The Scientific Question Can magnetic reconnection in a dusty plasma explain the heating necessary for chondrule properties? 4.5 By old nm to mm size Heating rates in the range of 2000 − 5000K /hr Multiple heating events are recorded. Exposed to a magnetic field on the order of 1[G ]. Lazerson (
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Stereotypical Chondrite (Sears, 2004)
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Overview Motivation The Question Plasma Parameters
Plasma Parameters Number Density Charge Number Mass Temperature Plasma Frequency Cyclotron Frequency Neutral Collision Frequency Debye Length Dust-Electron Collision Frequency Ion-Dust Collision Frequency Electon-Ion Collision Frequency Inertial Length Scales Magnetization Plasma Parameter Plasma Beta Coulomb Coupling The following assumptions are made: σn
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nk Zk mk Tk ωpk ωck νkn λDk νde νid νei c ωpk c ωck
Dust 0.1 10, 000 1x10−16 500 0.017 0.0016 0.0004 49 7.59x10−17 1.20x10−8 3.78x10−30 17.6x109
Ions 1001 1 1.67x10−27 500 41.7 9600 102 49
Electrons 1 1 9.11x10−31 500 56.4 17.6x106 4352 1543
Neutrals 1x109
Units m−3
1.67x10−27 500
5.3x106
kg K s −1 −1 s s −1 m s −1 −1 s s −1 m
7.2x106
187x109
31, 000
17
m
Λ 1 β 1 Γc 1 −11 = 5x10 m−2 , vTn = 2030 m/s, B = 10−4 T , and rd = 10−6 m.
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Organization DENISIS Simulation Parameters Initial Conditions
The Numerical Experiment I plan to conduct the following studies using the DENISIS 4-Fluid Code Ballistic Relaxation Perturbation Resistivity Collision Frequencies Adiabatic Index Aerodynamic Heating
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Organization DENISIS Simulation Parameters Initial Conditions
The DENISIS Code The DENISIS (Dust Electron Neutral Ion Self-consistent Integration Scheme) code has proven useful in studying dusty plasmas in the space environment. (Schr¨ oer et al., 1998). Fluid Code Dust, Ion and Neutral Continuity Equations Dust and Neutral Equations of Motion Dust, Ion, Electron and Neutral Energy Equations Ohms’s Law (intertialess Ion EOM) Electron Density (Quasineutrality)
Leap-Frog and Dufort Frankel Integration Schemes 3-D Nonuniform Cartesian Grid Lazerson (
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Simulation Parameters Normalizations B-Field Time Smallest Grid Scale Normalized Values Dust Mass Density Ion Mass Density Neutral Mass Density Current Sheet Thickness Grid Dimensions NX = 49 NY = 49 NZ = 15 Collision Frequencies Dust-Neutral Ion-Neutral Electron-Neutral
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= = =
0.1 G 180 s 12.5 × 106 m
= = = =
1.0 0.1 1.0 0.2 x ∈ [−10, 10] y ∈ [−2, 2] z ∈ [0, 10]
= = =
0.026 1000 0.00
Mass Density Length Velocity
= = =
1 × 10−17 kg 500 × 106 m 3 × 106 m/s
Dust Charge Number Ion Charge Number Dust Mass Ion Mass
= = = =
10 1 1.00 0.01
Equidistant Non-Equidistant Equidistant Dust-Electron Ion-Dust Ion-Electron
∆Xmin = 0.41 ∆Ymin = 0.0125 ∆Zmin = 0.67 = = =
0.0000001 0.00128 0.00
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Organization DENISIS Simulation Parameters Initial Conditions
Initial Condition
The simulation begins with a Harris-like current sheet (Harris, 1962)
~ = B0 tanh B
y ˆ d x
ρ = ρ0 + ρpeak cosh21(y /d) p = p0 + ppeak cosh21(y /d)
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Ballistic Relaxation Perturbation Resistivity
No Relaxation The initial condition is not a good equilibrium
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Ballistic Relaxation Perturbation Resistivity
Canonical Relaxation Velocities set to zero a maximums in kinetic energy (Hesse et al., 1993)
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Manual Relaxation Velocities set to zero at specified points
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Ballistic Relaxation Perturbation Resistivity
Relaxed Magnetic Field
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Relaxed Dust Density
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Relaxed Dust Internal Energy
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Ballistic Relaxation Perturbation Resistivity
The Sweet-Parker Mode
The lack of magnetic field dynamic durring ballistic relaxation allows a reconnective mode to be specified in terms of the magnetic field profile. A perturbation similar to that described in the Sweet-Parker model of reconnection is used.
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Ballistic Relaxation Perturbation Resistivity
Sweet-Parker Perturbation First first attempt at a perturbation was to simply assume inflow and outflow regions with no spatial variance.
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Ballistic Relaxation Perturbation Resistivity
Sweet-Parker Mass Flux
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Sweet-Parker Compression
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Ballistic Relaxation Perturbation Resistivity
Sweet-Parker Dust Mass
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Ballistic Relaxation Perturbation Resistivity
Sweet-Parker Internal Energy
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Ballistic Relaxation Perturbation Resistivity
Sweet-Parker Magnetic Energy
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Ballistic Relaxation Perturbation Resistivity
Modified Sweet-Parker Perturbation An attempt at an improvement was made by assuming that the outflow velocity was the local Alfv´ en velocity and computing a associated inflow mass flux.
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Ballistic Relaxation Perturbation Resistivity
Modified Sweet-Parker Compression
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Ballistic Relaxation Perturbation Resistivity
Modified Sweet-Parker Dust Mass
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Ballistic Relaxation Perturbation Resistivity
Modified Sweet-Parker Internal Energy
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Modified Sweet-Parker Magnetic Energy
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Velocity Profile Modified Sweet-Parker
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Ballistic Relaxation Perturbation Resistivity
Effects of Global Resistivity To evaluate the effects of resistivity on the process of reconnection various simulations were run with global resistivities ranging from those relevant to the proto-solar nebula and 2 order of magnitude lower.
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Introduction Methodology Results Summary Appendix References
Ballistic Relaxation Perturbation Resistivity
Effects of Global Resistivity
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Introduction Methodology Results Summary Appendix References
Ballistic Relaxation Perturbation Resistivity
Effects of Global Resistivity
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Ballistic Relaxation Perturbation Resistivity
Effects of Global Resistivity
(a) η = 0.0001
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(b) η = 0.0005
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Effects of Global Resistivity
(c) η = 0.08
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(d) η = f (νin )
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Summary
The following were explored: The Ballistic Relaxation General Evolution of the Current Sheet Choice of Relaxation Method
The Effects of Reconnective Mode The Effects of Global Resistivity
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Summary
The following were explored: The Ballistic Relaxation The Effects of Reconnective Mode Transition from Current Sheet to Steady-State Reconnection Types of Reconnective Modes
The Effects of Global Resistivity
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Summary The following were explored: The Ballistic Relaxation The Effects of Reconnective Mode The Effects of Global Resistivity Sweep of Global Resistivities Effect on Magnetic Field Configuration Effect on Magnetic Energy Effect on Reconnective Mode Parameter Dependance
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Future Work
The following studies are being conducted Effect of Collision Frequency Effect of Adiabatic Index Aerodynamic Heating
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The End
The grassland meteorite showing embedded condrules. (Wikipedia-Chondrule)
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The Protosolar Nebula DENISIS Equations
The Protosolar Nebula
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The Protosolar Nebula DENISIS Equations
Continuity Equations ∂ρd = −∇ · (ρd v~d ) ∂t ∂ρi = −∇ · (ρi v~i ) ∂t ∂ρn = −∇ · (ρn v~n ) ∂t Z d ρd ρi − ρe = m e mi md
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The Protosolar Nebula DENISIS Equations
Momentum Equations
∂(ρd v~d ) ∂t
=
−∇ ·(ρd v~d v~d) − ∇ (pe + pi + pd ) ~ ×B ~ + 1 ∇×B 4π
−νdn ρd (v~d − v~n ) − νin ρi (~ vi − v~n ) ∂(ρn v~n ) ∂t
=
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−∇ · (ρn v~n v~n ) − ∇pn +νdn ρd (v~d − v~n ) + νin ρi (~ vi − v~n )
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Energy Equations
1 ∂pi γi −1 ∂t
− γi 1−1 ∇ · (pi v~i ) − pi ∇ · v~i d n ρi νin (~ vi − v~n )2 + mdm+m ρ ν (~ v − v~ )2 + mim+m n i i id i d ρi νid kB Ti kB Ti kB Td i νin −2 mρi +m γi −1 − 2 mi +md γi − γd −1 n Ion Energy Equation shown for reference. Similar equations exist for each of the 4 species. =
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The Protosolar Nebula DENISIS Equations
Induction Equation
∂B ∂t
=
− me i ∇ × + me i ∇ × − me i ∇ ×
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∇pi mi ~ +m Zd ∇ × ρρdi v~d × B d ρi ~ ∇×B ~ − η∇2 B ~ ×B ρh i i { nndi Zd − nndi νid + Zd νin v~d
− νin v~n }
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References
1
D. Sears. The Origin of Chondrules and Chondrites. Cambridge Planetary Science, Cambridge (2004).
2
A. Schr¨ oer, G. T. Birk and A. Kopp. Comp. Phys. Comm. 112 (1998).
3
E. G. Harris. Il Nuovo Cimento. 23 (1962).
4
M. Hesse and J. Birn. J. Geo. Res. 98 (1993).
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