We use a one-dimensional hybrid MHD code to simulate PUI acceleration and ... The simulation box is 4000 c/Ïpi long, subdivided into 8000 grid points.
Hybrid Code Simulations of Pickup Ion Acceleration and Transport David S. Smith and Joe Giacalone Lunar and Planetary Laboratory, University of Arizona {dss, giacalon}@lpl.arizona.edu
Abstract Using a self-consistent, 1-D hybrid MHD simulation of protons and electrons, we examine physics of the acceleration and transport of pickup protons in the solar wind. We find that the streaming instability between the stationary pickup ions and moving solar wind ions does create MHD waves that feed turbulence in the solar wind. The generation, however, is not efficient and the associated pitch angle scattering seems to be inhibited near 90◦ . We find that the pickup ions maintain a significant steady-state anisotropy of levels comparable to that observed by Ulysses (though the PUIs here have much higher abundances).
Background From Gloeckler et al. 1995:
Ulysses observations (Gloeckler et al. 1995) of solar wind proton velocity distributions show an excess of particles with sunward velocities. Some possible explanations are: • Pickup ions have long mean free paths. • Scattering through 90◦ in pitch angle is difficult. • The density of neutral interstellar hydrogen changes rapidly over scales of an AU. • The angle between the field line and the flow direction changes with time and produces an anisotropic source distribution. We chose to employ a previously tested hybrid code (Giacalone 2006) to explore some of the physics of pickup ion acceleration and transport.
Model and Physics PUIs injected uniformly throughout box with zero velocity.
B
B field constant and parallel to SW flow. Solar wind flows in from the left.
Eventually, PUIs accelerated to near SW speed in turbulence generated by streaming instability.
B
We use a one-dimensional hybrid MHD code to simulate PUI acceleration and transport. • The simulation box is 4000 c/ωpi long, subdivided into 8000 grid points. • Solar wind protons are continually injected from one side with velocity 8vA . • Pickup protons are continually injected uniformly throughout the box with no inital velocity relative to the grid (8vA sunward in the SW frame). • The initial magnetic field is constant and in the direction of the flow, with no perpendicular component. • The timestep is 1/100-th of a gyroperiod, and the simulation is run for 2000 gyroperiods.
Results—Simulation Snapshot 1.5
1
0.5
0
-0.5 0
500 nSW nPUI
1000 x (c/ωpi) U/16 By
1500
2000 Bz
An example of output from the simulation for a PUI abundance of 0.01, taken after 2000 ion gyroperiods have elapsed. Note that the bulk speed U has been scaled by 16 to enhance the other curves. Streaming instability-generated waves can be seen forming on the left side of the box and convecting intact for a distance before being dissipated in the turbulence. The PUI density stays small but does increase linearly as the flow shovels up pickup ions. A slight mass loading effect slows the bulk flow (U decreases), but the effect is hard to spot at this low PUI low abundance.
Results—Anisotropies Model Comparison 0
/U
-0.2 -0.4 -0.6 1 0.1 0.01 hybrid code
-0.8 -1 0
1000
2000 x (c/ωpi)
3000
4000
Running the hybrid code with a PUI abundance of 0.01 produces the streaming velocities relative to the wind (hvx i/U ) shown in the magenta curve. The anisotropy approaches -0.6 as the flow exits the box and seems to be nearing steady state. The diffusive streaming flux due to a density gradient (δ = −λ∇ ln f ) should approach zero like −1/(1 + x) in this case, since the PUI density increases linearly across the box. We ran a separate Monte Carlo simulation of particles scattering in pitch angle along field lines to explore the causes of this streaming anisotropy. In the Monte Carlo model, particles are assumed to have a mean free path of 500 c/ωpi and scatter isotropically in pitch angle, but crossing 90◦ fails with a specified probability. This model was inspired by the quasilinear theory prediction of inhibited scattering near 90◦ . Also, this approach is physically similar to the analytical models of Isenberg (1997) and Schwadron (1998). The results of this Monte Carlo simulation are shown in the red, green, and blue points. Red represents no impediment to scattering through 90◦ , while blue means that scatterings that would
take the particle through 90◦ will fail 99% of the time. The new pitch angle in the failed cases is reflected about 90◦ . The red, green, and blue lines are fits to the function δ(x) = −
a + c. 1 + x/b
If this is a purely diffusive effect, a set of parameters with c ' 0 should fit the data well. This means that the PUI velocity distribution will isotropize eventually. If c is nonzero, we have a steady-state anisotropy. Table 1 shows the resulting parameters from a χ2 minimization fit to the three Monte Carlo simulation runs, where P is the probability of getting across the 90◦ “barrier.” P
a
b
c
λ
1
0.888
703
-0.0128
624
0.1 0.561 909 0.01
0.406
694
-0.328 510 -0.492
282
We can see now that the freely scattering case (probability of crossing 90◦ is unity) is purely diffusive, with a very small c. The mean free path (λ = ab) agrees well with the specified value of 500. We conclude that this type of analysis is accurate in extracting the diffusive properties. The blue curve (P = 0.01) does not fit the form of δ(x) specified above very well. We don’t yet understand this. The small-P cases seem to have sizeable steady-state anisotropy which is not a densitygradient effect but is instead produced by inhibiting scattering through 90◦ . More importantly, the hybrid code produces similar results, suggesting that even a self-consistent MHD treatment results in pitch-angle diffusion coefficients that are small near 90◦ for pickup ions in self-generated turbulence.
Results—Power Spectrum and Dµµ 101 hybrid -2.2
k
k P(k) / B0
2
100
10-1
10-2
10
-3
10-3
10-2
10-1
100
k (ωpi/c)
A standard result of quasilinear theory relates the magnetic field power spectrum to the pitch angle diffusion coefficient, Dµµ . Namely, Dµµ =
π kr P (kr ) (1 − µ2 )Ω , 2 B02
where kr ≡ Ω/|µ|v is the resonant wavenumber and Ω is the gyrofrequency. For turbulence that has a power spectrum well-approximated by a power law, this reduces to Dµµ = A|µ|q (1 − µ2 ). The plot above shows the power spectrum of one of the perpendicular B components from the hybrid simulation. This spectrum is quite steep, but still well approximated by a power law. A small bump is located at the gyroradius scale of 8 c/ωpi , presumably due to the ion cyclotron resonances. Using the fitted power spectrum from the hybrid code, we find Dµµ = 5.7 × 10−4 (1 − µ2 )kr (µ)−2.2 . Ω
Future Work Knowing Dµµ from the hybrid code, we can use the technique given by Palmer and Jokipii (1981) to calculate Fokker-Planck coefficients for the PUIs. We have written another Monte Carlo code that scatters the particles by making small changes in pitch angle selected according to the Palmer and Jokipii formula: µ(t + ∆t) = µ(t) + R
p ∂Dµµ 2Dµµ ∆t + ∆t, ∂µ
where R is a random deviate with mean zero and unit variance. In the futere, results from this method will be used to replace the cruder bi-hemisphere approach of our Monte Carlo code. Another possible explanation for the anisotropies in the Ulysses data is an anisotropic source distribution. The figure below from Nem´eth et al. (2000) shows that the observed source distribution is indeed anisotropic. We plan to include a variable source distribution in the hybrid code, which should result in particles being picked up with a distribution of initial pitch angles.
References Giacalone, J., 2006. The hybrid simulation applied to space plasmas. In Numerical Modeling of Space Plasma Flows, ASP Conf. Ser., vol. 359, p. 241. Gloeckler, G., Schwadron, N.A., Fisk, L.A., and Geiss, J., 1995. Weak pitch angle scattering of few MV rigidity ions from measurements of anisotropies in the distribution function of interstellar pickup H+. JGR, 22, 2665. Isenberg, P.A., 1997. A hemispherical model of anisotropic interstellar pickup ions. JGR, 102, 4719. N´emeth, Z., Erd¨os, G., and Balogh, A., 2000. Fluctuations of the heliospheric magnetic field and the anisotropy of pickup ions. Geophys. Res. Lett., 27, 2793. Palmer, I.D., Jokipii, J.R., 1981. Monte-Carlo model of pitch-angle scattering in solar cosmic ray events. In: International Cosmic Ray Conference, vol 3, p. 381–384. Schwadron, N.A., 1998. A model for pickup ion transport in the heliosphere in the limit of uniform hemispheric distributions. JGR, 103, 20643. Winske, D. and Yin, L., 2001. Hybrid codes: Past present and future. Proc. of ISSS-6, p. 1–4.
