Abstract
A 2D Riemann problem is designed to study the development and dynamics of the slow shocks that are thought to form at the boundaries of reconnection exhausts. Simulations are carried out for varying ratios of normal magnetic field to the transverse upstream magnetic field (i.e., propagation angle with respect to the upstream magnetic field). When the angle is sufficiently oblique, the simulations reveal a large firehosesense temperatureanisotropy in the downstream region, accompanied by a transition from a coplanar slow shock to a noncoplanar rotational mode. In the downstream region the firehose stability parameter tends to plateau at 0.25. This balance arises from the competition between counterstreaming ions, which drive ɛ down, and the scattering due to ion inertial scale waves, which are driven unstable by the downstream rotational wave. At very oblique propagating angles, 2D turbulence also develops in the downstream region.
This work was supported in part by NASA Grant Nos. NNX08AV87G, NNX09A102G and NNN06AA01C. Computations were carried out at the National Energy Research Scientific Computing Center.
I. INTRODUCTION
II. SIMULATION MODELS AND DETAILS
III. GENERAL FEATURES OF THE 75° CASE
IV. THE SOURCE OF TEMPERATUREANISOTROPY: ALFVÉNIC COUNTERSTREAMING IONS
V. TEMPERATUREANISOTROPY VS. PROPAGATION ANGLES
VI. THE DOWNSTREAM TURBULENTWAVES AND PARTICLE SCATTERING
VII. SUMMARY AND DISCUSSION
Key Topics
 Plasma waves
 67.0
 Magnetic fields
 24.0
 Anisotropy
 22.0
 Magnetohydrodynamic waves
 22.0
 Magnetohydrodynamics
 16.0
Figures
(Color online) The exhaust from steady reconnection in a PIC simulation. Panel (a): The outofplane electron current density J _{ ey }; Panel (b): , where positive values have been set to 0. The colored region is firehose unstable; Panel (c): The magnitude of B _{ x } showing the development of 2D turbulence; Panel (d): A cut of ɛ at x/d _{ i } = –35 (the vertical line in (b)). The horizontal lines demarcate ɛ = 0.25 and ɛ = 0.
(Color online) The exhaust from steady reconnection in a PIC simulation. Panel (a): The outofplane electron current density J _{ ey }; Panel (b): , where positive values have been set to 0. The colored region is firehose unstable; Panel (c): The magnitude of B _{ x } showing the development of 2D turbulence; Panel (d): A cut of ɛ at x/d _{ i } = –35 (the vertical line in (b)). The horizontal lines demarcate ɛ = 0.25 and ɛ = 0.
(Color online) The evolution of a system with θ_{ BN } = 75° (Run f). Panel (a): The evolution of B from time 0–200/Ω_{ ci }. A pair of fast rarefactions (FR) propagate out from the symmetry line, followed by a pair of slow shocks (SS). Each curve has been shifted so that it intersects the vertical axis at the given time. The time between the yellow curves is 100/Ω_{ ci }; Panel (b): The predicted FR and switchoff slow shock (SSS) from ideal MHD theory; Panel (c): The same as (a) but with the vertical axis measuring B; Panel (d): The evolution of B _{ z } from time 0–200/Ω_{ ci }.
(Color online) The evolution of a system with θ_{ BN } = 75° (Run f). Panel (a): The evolution of B from time 0–200/Ω_{ ci }. A pair of fast rarefactions (FR) propagate out from the symmetry line, followed by a pair of slow shocks (SS). Each curve has been shifted so that it intersects the vertical axis at the given time. The time between the yellow curves is 100/Ω_{ ci }; Panel (b): The predicted FR and switchoff slow shock (SSS) from ideal MHD theory; Panel (c): The same as (a) but with the vertical axis measuring B; Panel (d): The evolution of B _{ z } from time 0–200/Ω_{ ci }.
(Color online) Parameters from the run with θ_{ BN } = 75° (Run f) at time 200/Ω_{ ci }. Panel (a): Temperature anisotropy ɛ and xdirection heat flux Q _{ x }; Panel (b): Magnetic field components; Panel (c): Parallel and perpendicular temperatures (the offdiagonal components T _{ ixy }, T _{ ixz }, T _{ iyz } are plotted together in green, denoted as T _{off}, and are small); Panel (d): Total plasma pressure components and P _{ x } + B ^{2}/2μ _{0}. Panel (e): The plasma β and local θ_{ BN } = cos^{–1}(B _{ x }/B); Panel (f): Plasma density. The dotted curves in each panel are the predicted magnitude and position of the switchoff slow shocks (SSS) from isotropic MHD for B _{ z } in (b), T in (c), P in (d), β in (e), and n in (f).
(Color online) Parameters from the run with θ_{ BN } = 75° (Run f) at time 200/Ω_{ ci }. Panel (a): Temperature anisotropy ɛ and xdirection heat flux Q _{ x }; Panel (b): Magnetic field components; Panel (c): Parallel and perpendicular temperatures (the offdiagonal components T _{ ixy }, T _{ ixz }, T _{ iyz } are plotted together in green, denoted as T _{off}, and are small); Panel (d): Total plasma pressure components and P _{ x } + B ^{2}/2μ _{0}. Panel (e): The plasma β and local θ_{ BN } = cos^{–1}(B _{ x }/B); Panel (f): Plasma density. The dotted curves in each panel are the predicted magnitude and position of the switchoff slow shocks (SSS) from isotropic MHD for B _{ z } in (b), T in (c), P in (d), β in (e), and n in (f).
(Color online) The phase space of the run with θ_{ BN } = 75° (Run f) at time 200/Ω_{ ci }. From top to bottom the left column shows the ion distribution in: V _{ z } – x space, where the backstreaming ions from the discontinuities are clearly seen; V _{ y } – x space; V _{ x } – x space. The right column is the electron distribution in V _{ z } – x space, V _{ y } – x space and V _{ x } – x space. The white dashed lines indicate the locations of the velocity distributions shown in Fig. 5. The color bar is normalized to the maximum value in each panel.
(Color online) The phase space of the run with θ_{ BN } = 75° (Run f) at time 200/Ω_{ ci }. From top to bottom the left column shows the ion distribution in: V _{ z } – x space, where the backstreaming ions from the discontinuities are clearly seen; V _{ y } – x space; V _{ x } – x space. The right column is the electron distribution in V _{ z } – x space, V _{ y } – x space and V _{ x } – x space. The white dashed lines indicate the locations of the velocity distributions shown in Fig. 5. The color bar is normalized to the maximum value in each panel.
(Color online) The ion velocity distributions measured at locations 409.1–410.1d _{ i }, 415–416d _{ i } and 430–431d _{ i } of Fig. 4 (the white dashed lines). From top to bottom are V _{ z } – V _{ x }, V _{ z } – V _{ y } and V _{ x } – V _{ y } distributions. The distributions are color coded and the white contours help identify different ion parcels. The local magnetic field is denoted by blue arrowed lines beginning at origin. The axis scales, when cut by a factor of 2, also measure the magnitude of the field. Ions that stream along the magnetic field are clearly seen at these locations.
(Color online) The ion velocity distributions measured at locations 409.1–410.1d _{ i }, 415–416d _{ i } and 430–431d _{ i } of Fig. 4 (the white dashed lines). From top to bottom are V _{ z } – V _{ x }, V _{ z } – V _{ y } and V _{ x } – V _{ y } distributions. The distributions are color coded and the white contours help identify different ion parcels. The local magnetic field is denoted by blue arrowed lines beginning at origin. The axis scales, when cut by a factor of 2, also measure the magnitude of the field. Ions that stream along the magnetic field are clearly seen at these locations.
(Color online) From top to bottom are runs with θ_{ BN } = 30° (Run a) at 100/Ω_{ ci }, 45° (Run b) at 200/Ω_{ ci }, 52° (Run c) at 100/Ω_{ ci }, 60° (Run d) at 250/Ω_{ ci }, 75° (Run f) at 400/Ω_{ ci }, and 83° (Run k) at 700/Ω_{ ci }. The first column shows the temperature anisotropy, and the second column the magnetic field components as a function of x. The third column displays hodograms taken from the right half of the simulation domains. The dotted curves in the second column are the predicted magnitudes and positions of switchoff slow shocks (SSS) and fast rarefactions (FR) from isotropic MHD theory for B_{z}.
(Color online) From top to bottom are runs with θ_{ BN } = 30° (Run a) at 100/Ω_{ ci }, 45° (Run b) at 200/Ω_{ ci }, 52° (Run c) at 100/Ω_{ ci }, 60° (Run d) at 250/Ω_{ ci }, 75° (Run f) at 400/Ω_{ ci }, and 83° (Run k) at 700/Ω_{ ci }. The first column shows the temperature anisotropy, and the second column the magnetic field components as a function of x. The third column displays hodograms taken from the right half of the simulation domains. The dotted curves in the second column are the predicted magnitudes and positions of switchoff slow shocks (SSS) and fast rarefactions (FR) from isotropic MHD theory for B_{z}.
(Color online) Evolution of ɛ for the case with θ_{ BN } = 60° (Run d) for equally spaced times between 100 – 500/Ω_{ ci } from lighter grey to darker grey in (a), the θ_{ BN } = 75° case (Run f) for time 100 – 500/Ω_{ ci } in (b), and the θ_{ BN } = 83° case (Run k) for time 100 – 700/Ω_{ ci } in (c). The bottom is a plot of B _{ x } for the θ_{ BN } = 83° case at time 700/Ω_{ ci } showing the 2D turbulence that develops.
(Color online) Evolution of ɛ for the case with θ_{ BN } = 60° (Run d) for equally spaced times between 100 – 500/Ω_{ ci } from lighter grey to darker grey in (a), the θ_{ BN } = 75° case (Run f) for time 100 – 500/Ω_{ ci } in (b), and the θ_{ BN } = 83° case (Run k) for time 100 – 700/Ω_{ ci } in (c). The bottom is a plot of B _{ x } for the θ_{ BN } = 83° case at time 700/Ω_{ ci } showing the 2D turbulence that develops.
(Color online) The ɛ distributions of runs θ_{ BN } = 60° (Run d) at 500/Ω_{ ci }, 75° (Run f) at 200/Ω_{ ci }, 83° (Run k) at 700/Ω_{ ci }. (The 83° case is shifted to the right by 204.8d _{ i }.)
(Color online) The ɛ distributions of runs θ_{ BN } = 60° (Run d) at 500/Ω_{ ci }, 75° (Run f) at 200/Ω_{ ci }, 83° (Run k) at 700/Ω_{ ci }. (The 83° case is shifted to the right by 204.8d _{ i }.)
(Color online) Panel (a): The evolution of ɛ, B _{ z } and B _{ y } for equally spaced times between 50–500/Ω_{ ci } (from left to right) in the θ_{ BN } = 75°, w _{ i } = 10d _{ i } case (Run g). The downstream largerscale rotational wave breaks into waves of wavelength ∼ 6d _{ i }. Panel (b): A blowup of the downstream B _{ y } at time 450/Ω_{ ci }.
(Color online) Panel (a): The evolution of ɛ, B _{ z } and B _{ y } for equally spaced times between 50–500/Ω_{ ci } (from left to right) in the θ_{ BN } = 75°, w _{ i } = 10d _{ i } case (Run g). The downstream largerscale rotational wave breaks into waves of wavelength ∼ 6d _{ i }. Panel (b): A blowup of the downstream B _{ y } at time 450/Ω_{ ci }.
(Color online) The evolution of B _{ y }, B _{ z } and B for equally spaced times between 0–100/Ω_{ ci }. The red curve indicates the time 100/Ω_{ ci }. Panel (a): Run 1 with both initial streaming ions and modulated rotational parent wave. Panel (b): The same as panel (a) without the initial spatial modulation (Run 3). Panel (c): The same as panel (a) without initial beams (Run 5). Panel (d): The same as panel (c) without initial polarization (Run 6).
(Color online) The evolution of B _{ y }, B _{ z } and B for equally spaced times between 0–100/Ω_{ ci }. The red curve indicates the time 100/Ω_{ ci }. Panel (a): Run 1 with both initial streaming ions and modulated rotational parent wave. Panel (b): The same as panel (a) without the initial spatial modulation (Run 3). Panel (c): The same as panel (a) without initial beams (Run 5). Panel (d): The same as panel (c) without initial polarization (Run 6).
The evolution of for equally spaced times between 0–100/Ω_{ ci } (from lighter grey to darker grey) of Run 1 (Fig. 10(a)). The temperature anisotropy of the ions is reduced, which indicates particle scattering is taking place.
The evolution of for equally spaced times between 0–100/Ω_{ ci } (from lighter grey to darker grey) of Run 1 (Fig. 10(a)). The temperature anisotropy of the ions is reduced, which indicates particle scattering is taking place.
Tables
Parameters and results of shock simulations.
Parameters and results of shock simulations.
Parameters and results.
Parameters and results.
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