The 12 th Asia Pacific Physics Conference of AAPPS Chiba, Japan, July 14-19, 2013 Ryusuke NUMATA 1,*) , N. F. Loureiro 2) 1) Graduate School of Simulation Studies, University of Hyogo 2) Instituto de Plasmas e Fusão Nuclear, Instituto Superior Técnico *) Email: [email protected] Web: http://www.rnumata.org Electron and Ion Heating during Magnetic Reconnection in Weakly Collisional Plasmas D1-PWe-07 This work was supported by JSPS KAKENHI Grant Number 24740373. This work was carried out using the HELIOS supercomputer system at Computational Simulation Centre of International Fusion Energy Research Centre (IFERC-CSC), Aomori, Japan, under the Broader Approach collaboration between Euratom and Japan, implemented by Fusion for Energy and JAEA. Acknowlegements Introduction Simulation Setup Conclusion References Simulation results ● Magnetic reconnection is ubiquitous in fusion and astrophysical plasmas, which allows topological change of field lines, and convert field energy into plasma flow and heat. We focus on plasma heating—how much energy is transformed into the thermal energy? To address thermodynamic properties of plasmas, inter particle collisions are especially important even though plasmas are considered to be collision-free. ● In weakly collisional plasmas, kinetic effects, such as Landau damping or finite Larmor radius effects may leads phase mixing to create fine structures in velocity space [1]. Those structures suffer strong collisional dissipation. As long as collisions are sufficiently infrequent, the rate of energy dissipation does not depend on collisions. ● In this work, we present gyrokinetic simulations of magnetic reconnection using AstroGK [2], and show energy dissipation due to collisions are strongly enhanced by phase mixing process, which contributes to background plasma heating. Simulations are performed using AstroGK [2] in doubly periodic slab domain. We follow a setup for linear tearing instability study [3] We assume uniform background (∇n 0 = ∇T 0 ∇ = B 0 = 0), and ∂/∂z = 0. Parameters are k y a = 0.8, ∆'a = 23.2, m i /m e = 100, β e = 0.01, T 0i /T 0e = 1, and ρ i /a = 0.25. For these parameters, other kinetic scales are ρ i = d e = 0.1d i = 10ρ e . Initial cond.: shifted Maxwellian electron (finite u ||,e ), non-shifted Maxwellian ion (u ||,i =0) → Electron flow (amplitude and profile) is chosen to give AstroGK accurately reproduces the Spitzer resistivity, for which the electron-ion collision frequency (ν ei ) and the resistivity (η) are related by η/µ 0 = 0.380ν ei d e 2 . The resistivity is recast in terms of the Lundquist number S = 2.63 (ν ei τ A ) -1 (d e a) -2 where τ A = a/V A , V A is the Alfvén velocity corresponding to B y max . We fix collision frequencies ν ee τ A =ν ei τ A =ν ii τ A =8×10 -5 . (collisionless limit is achieved [4]) A ∥ eq = A ∥ 0 eq cosh 2 (( x L x / 2 )/ a ) S ( x ) ( is to make periodic) S ( x ) [1] W. Dorland and G. Hammett, Phys. Fluids B 5, 812 (1993). [2] R. Numata et al., J. Comput. Phys. 229, 9347 (2010); ibid. 245, 493 (2013). [3] R. Numata et al., Phys. Plasmas 18, 112106 (2011). [4] R. Numata and N. F. Loureiro, Bull. Am. Phys. Soc. 57 (2012), CP9-59. [5] G. G. Howes et al., Astrophys. J. 651, 590 (2006). [6] A. A. Schekochihin et al., Astrophys J. Suppl. 182, 310 (2009). [7] N. F. Loureiro et al., arXiv:1301.0338 (2013). [8] A. Zocco and A. A. Schekochihin, Phys. Plasmas 18, 102309 (2011). Heating diagnostics ● We have performed gyrokinetic simulations to investigate plasma heating during magnetic reconnection in strongly magnetized, weakly collisional plasmas. ● In the collisionless limit where the macroscopic behavior of plasmas is insensitive to collisions, linear parallel and nonlinear perpendicular phase mixing create fin structures in velocity space leading to strong energy dissipation. ● We have shown low-β plasmas. The electron heating occurs after the dynamical reconnection phase, and continues for longer time. Perpendicular phase mixing follows parallel phase mixing as it is a nonlinear process. ● The low-β result is consistent with the previous study using a hybrid fluid/kinetic model [7,8]. ● We have also shown a relatively high-β case. Ion heating rate becomes larger as β increases although it is still small for β=0.1. W = ∑ s E s p + E ⊥ m + E ∥ m = ∫ [ ∑ s ∫ T 0s δ f s 2 2 f 0s d v + ∣ ∇ ⊥ A ∥ ∣ 2 2 µ 0 + ∣ δ B ∥ ∣ 2 2 µ 0 ] d r D s = ∫ 〈 T 0s h s f 0s ( ∂ h ∂ t ) coll 〉 r d v d r > 0 To estimate plasma heating, we measure the collisional energy dissipation rate Without collisions, the gyrokinetic equation conserves the generalized energy consisting of the particle part E p and the magnetic field part E m where is the perturbation of the distribution function, and h s is the non-Boltzmann part obeying the gyrokinetic equation, and the generalized energy is dissipated by collisions as . The collisional dissipation increases the entropy (related to the first term of the generalized energy), and is turned into heat [5,6]. δ f s =( q s ϕ/ T 0s ) f 0s + h s dW / dt = ∑ s D s 0 0.05 0.1 0.15 0.2 Reconnection rate [E X /(V A B y max ] 0 0.2 0.4 0.6 0.8 1 1.2 Energy [E m (t=0)] total mag. perp. mag. para. part. ion part. electron 10 -7 10 -6 10 -5 10 -4 10 -3 0 5 10 15 20 25 30 35 40 Dissipation rate [E m (t=0)/τ A ] Times [τ A ] ion electron Figure 1: Time evolutions of the reconnection rate (top), The energy components (middle), and the dissipation rate of Ions and electrons (bottom). Reconnection rate is measued by the electric field at the X point. The peak reconnection is about 0.2 achieving the fast reconnection During magnetic reconnection, the magnetic energy is converted to the particle's energy reversibly. Collisionally dissipated energy is about 1% of the initial magnetic energy after the dynamical phase (~25τ A ). The energy dissipation starts to grow rapidly when the maximum reconnection rate is achieved. It stays large long after the dynamical phase, and an appreciable amount is lost in the later time. The ion dissipation is negligibly small compared with the electron's for the low-β case. Velocity space structure In the earlier time of the nonlinear stage, the distribution function only has gradients in the parallel direction indicating parallel phase mixing. Later, the perpendicular FLR phase mixing follows to create structures in the perpendicular direction. Figure 2: Magnetic flux and electron temperature (left), and electron distribution function structure in velocity space (right). Time: 10 [τ A ], x/d e =23.5, y/d e =15.7 -5 -4 -3 -2 -1 0 1 2 3 4 5 v || [v th,e ] 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 v ⊥ [v th,e ] -0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08 [ε n 0e /(v th,e √ π29 3 ] Time: 20 [τ A ], x/d e =26.1, y/d e =15.7 -5 -4 -3 -2 -1 0 1 2 3 4 5 v || [v th,e ] 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 v ⊥ [v th,e ] -0.15 -0.1 -0.05 0 0.05 0.1 0.15 [ε n 0e /(v th,e √ π29 3 ] High-β case 0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0 5 10 15 20 25 30 D i /D e Time [τ A ] β e =0.01 β e =0.1 Figure 3: Ratio of ion to electron dissipation rate for different β. In high-β plasmas, compressible fluctuations will be excited which are strongly damped collisionlessly. This may open up another dissipation channel where phase mixing of ions ends up with ion heating. Comparison of the ratio of energy dissipation of ions to electrons for different β shows that ion dissipation increases with increasing β. Ion heating may be relevant for much higher β.
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The 12th Asia Pacific Physics Conference of AAPPSChiba, Japan, July 14-19, 2013
Ryusuke NUMATA1,*), N. F. Loureiro2)
1) Graduate School of Simulation Studies, University of Hyogo2) Instituto de Plasmas e Fusão Nuclear, Instituto Superior Técnico*) Email: [email protected] Web: http://www.rnumata.org
Electron and Ion Heating during Magnetic Reconnectionin Weakly Collisional Plasmas
D1-PWe-07
This work was supported by JSPS KAKENHI Grant Number 24740373. This work was carried out using the HELIOS supercomputer system at Computational Simulation Centre of International Fusion Energy Research Centre (IFERC-CSC), Aomori, Japan, under the Broader Approach collaboration between Euratom and Japan, implemented by Fusion for Energy and JAEA.
Acknowlegements
Introduction
Simulation Setup
Conclusion
References
Simulation results●Magnetic reconnection is ubiquitous in fusion and astrophysical plasmas, which allows topological change of field lines, and convert field energy into plasma flow and heat. We focus on plasma heating—how much energy is transformed into the thermal energy? To address thermodynamic properties of plasmas, inter particle collisions are especially important even though plasmas are considered to be collision-free.
●In weakly collisional plasmas, kinetic effects, such as Landau damping or finite Larmor radius effects may leads phase mixing to create fine structures in velocity space [1]. Those structures suffer strong collisional dissipation. As long as collisions are sufficiently infrequent, the rate of energy dissipation does not depend on collisions.
●In this work, we present gyrokinetic simulations of magnetic reconnection using AstroGK [2], and show energy dissipation due to collisions are strongly enhanced by phase mixing process, which contributes to background plasma heating.
Simulations are performed using AstroGK [2] in doubly periodic slab domain.We follow a setup for linear tearing instability study [3]We assume uniform background (∇n
0 = ∇T
0∇ = B
0 = 0), and ∂/∂z = 0.
Parameters are kya = 0.8, ∆'a = 23.2, m
i/m
e = 100, β
e = 0.01, T
0i/T
0e = 1, and ρ
i/a = 0.25. For
these parameters, other kinetic scales are ρi = d
e = 0.1d
i = 10ρ
e.
Initial cond.: shifted Maxwellian electron (finite u||,e
), non-shifted Maxwellian ion (u||,i=0)
→ Electron flow (amplitude and profile) is chosen to give
AstroGK accurately reproduces the Spitzer resistivity, for which the electron-ion
collision frequency (νei) and the resistivity (η) are related by η/µ
0 = 0.380ν
eid
e
2.
The resistivity is recast in terms of the Lundquist number S = 2.63 (νeiτ
A)-1 (d
ea)-2
where τA = a/V
A, V
A is the Alfvén velocity corresponding to B
y
max.
We fix collision frequencies νeeτ
A=ν
eiτ
A=ν
iiτ
A=8×10-5. (collisionless limit is achieved [4])
A∥eq=
A∥0eq
cosh2((x�Lx/2)/a)S(x) ( is to make periodic)S(x)
[1] W. Dorland and G. Hammett, Phys. Fluids B 5, 812 (1993).[2] R. Numata et al., J. Comput. Phys. 229, 9347 (2010); ibid. 245, 493 (2013).[3] R. Numata et al., Phys. Plasmas 18, 112106 (2011).[4] R. Numata and N. F. Loureiro, Bull. Am. Phys. Soc. 57 (2012), CP9-59.[5] G. G. Howes et al., Astrophys. J. 651, 590 (2006).[6] A. A. Schekochihin et al., Astrophys J. Suppl. 182, 310 (2009).[7] N. F. Loureiro et al., arXiv:1301.0338 (2013).[8] A. Zocco and A. A. Schekochihin, Phys. Plasmas 18, 102309 (2011).
Heating diagnostics
●We have performed gyrokinetic simulations to investigate plasma heating during magnetic reconnection in strongly magnetized, weakly collisional plasmas.
●In the collisionless limit where the macroscopic behavior of plasmas is insensitive to collisions, linear parallel and nonlinear perpendicular phase mixing create fin structures in velocity space leading to strong energy dissipation.
●We have shown low-β plasmas. The electron heating occurs after the dynamical reconnection phase, and continues for longer time. Perpendicular phase mixing follows parallel phase mixing as it is a nonlinear process.
●The low-β result is consistent with the previous study using a hybrid fluid/kinetic model [7,8].
●We have also shown a relatively high-β case. Ion heating rate becomes larger as β increases although it is still small for β=0.1.
W=∑sEs
p+E⊥m+E∥
m=∫[∑s∫T0sδ f s
2
2 f 0s
dv+∣∇⊥ A∥∣
2
2µ0
+∣δ B∥∣
2
2µ0]d r
Ds=�∫⟨T0shs
f 0s(∂h∂ t )coll
⟩r
d v d r>0
To estimate plasma heating, we measure the collisional energy dissipation rate
Without collisions, the gyrokinetic equation conserves the generalized energy consisting of the particle part Ep and the magnetic field part Em
where is the perturbation of the distribution function, and hs is the
non-Boltzmann part obeying the gyrokinetic equation, and the generalized energy is dissipated by collisions as . The collisional dissipation increases the entropy (related to the first term of the generalized energy), and is turned into heat [5,6].
δ f s=�(qsϕ/T0s) f 0s+hs
dW/dt=�∑sDs
0
0.05
0.1
0.15
0.2
Rec
onne
ctio
n ra
te[E
X/(
VA
Bym
ax]
0
0.2
0.4
0.6
0.8
1
1.2
Ene
rgy
[Em
(t=
0)]
total
mag. perp.
mag. para.
part. ionpart. electron
10-7
10-6
10-5
10-4
10-3
0 5 10 15 20 25 30 35 40
Dis
sipa
tion
rate
[Em
(t=
0)/τ A
]
Times [τA]
ion
electron
Figure 1: Time evolutions of the reconnection rate (top),The energy components (middle), and the dissipation rate of Ions and electrons (bottom).
Reconnection rate is measued by the electric field at the X point. The peak reconnection is about 0.2 achieving the fast reconnection
During magnetic reconnection, the magnetic energy is converted to the particle's energy reversibly.
Collisionally dissipated energy is about 1% of the initial magnetic energy after the dynamical phase (~25τ
A).
The energy dissipation starts to grow rapidly when the maximum reconnection rate is achieved. It stays large long after the dynamical phase, and an appreciable amount is lost in the later time.
The ion dissipation is negligibly small compared with the electron's for the low-β case.
Velocity space structure
In the earlier time of the nonlinear stage, the distribution function only has gradients in the parallel direction indicating parallel phase mixing. Later, the perpendicular FLR phase mixing follows to create structures in the perpendicular direction.
Figure 2: Magnetic flux and electron temperature (left), and electron distribution function structure in velocity space (right).
Time: 10 [τA], x/de=23.5, y/de=15.7
-5 -4 -3 -2 -1 0 1 2 3 4 5
v|| [vth,e]
0 0.5
1 1.5
2 2.5
3 3.5
4 4.5
v ⊥ [
v th,
e]
-0.08-0.06-0.04-0.02 0 0.02 0.04 0.06 0.08
[ε n
0e/(
v th,
e√ π
)3 ]
Time: 20 [τA], x/de=26.1, y/de=15.7
-5 -4 -3 -2 -1 0 1 2 3 4 5
v|| [vth,e]
0 0.5
1 1.5
2 2.5
3 3.5
4 4.5
v ⊥ [
v th,
e]
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
[ε n
0e/(
v th,
e√ π
)3 ]
High-β case
0 0.002 0.004 0.006 0.008 0.01
0.012 0.014 0.016 0.018
0 5 10 15 20 25 30
Di/D
e
Time [τA]
βe=0.01βe=0.1
Figure 3: Ratio of ion to electron dissipation rate for different β.
In high-β plasmas, compressible fluctuations will be excited which are strongly damped collisionlessly. This may open up another dissipation channel where phase mixing of ions ends up with ion heating.
Comparison of the ratio of energy dissipation of ions to electrons for different β shows that ion dissipation increases with increasing β.
Ion heating may be relevant for much higher β.
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