S12-6
Global Vlasov Simulation of a Weakly-Magnetized Small Body
弱磁化小天体のグローバルブラソフシミュレーション
Takayuki Umeda
梅田隆行
Solar-Terrestrial Environment Laboratory, Nagoya UniversityF3-3, Fro-cho, Chikusa-ku, Nagoya 464-8601, Japan
名古屋大学太陽地球環境研究所 〒464-8601 名古屋市千種区不老町 F3-3
The interaction between the solar wind and solar-system bodies, such as planets, satellites and asteroids, is
one of the fundamental global-scale phenomena in space plasma physics. In the present study,
electromagnetic environment around a small dielectric body with a weak intrinsic magnetic field is studied
by means of a five-dimensional full electromagnetic Vlasov simulation with two configuration spaces andthree velocity spaces, which is a challenging task in space plasma physics as well as high performancecomputing. The direct comparison between the low- and high-resolution runs would show importance offully kinetic global simulations.

1. BackgroundThe electromagnetic environment around asolar-system body is formed by the interaction
between the solar wind and the magnetosphere of
the body. The structure and dynamics of theelectromagnetic environment are quite different
depending on the strength of the intrinsic
magnetic field and the condition of the surface ofbodies. The intrinsic magnetic field generally acts
as an obstacle to the solar wind. When theintrinsic magnetic field is weak, ionized gases
from the upper atmosphere of a body act as aconducting obstacle. When a body has neither
intrinsic magnetic field nor atmosphere, on theother hand, the solar-wind plasma directly
impacts the body. The Earth's satellite Moon is
one of examples for a dielectric or insulating
body without intrinsic magnetic field andatmosphere.
The interaction between the solar wind and an
unmagnetized body is one of fundamental“global-scale” issues in space plasma physics.
This process is conventionally studied by meansof MHD simulations where all the plasma
particles are treated by the single-fluid
approximation. On the other hand, one of major
purposes of the present study is to extend theconventional single-fluid simulation to afirst-principle kinetic simulation [1-3], which is achallenging task in space plasma physics as wellas high performance computing.
2. Method
We use a parallel full electromagnetic Vlasovcode [4], which solves the full set of the Maxwellequations (1) and the Vlasov equation (2),1 ∂E∇ × B = μ0 J + c2 ∂t},∇∙𝐁=0𝜌
∇∙𝐄= }

∂B
𝜖0∂t
𝜕𝑓𝑠
𝜕𝑓𝑞
𝜕𝑓+ 𝐯 ∙ 𝜕𝐫𝑠 + 𝑚𝑠 (𝐄 + 𝐯 × 𝐁) ∙ 𝜕𝐯𝑠 = 0𝜕𝑡𝑠

∇×E=-(1)(2)

where 𝐄, 𝐁, 𝐉, ρ, μ0 , 𝜖0 and 𝑐 represent electricfield, magnetic field, current density, charge density,
magnetic permeability, dielectric constant and light
speed, respectively. The Vlasov equation (2)describes the development of the distributionfunctions by the electromagnetic (Lorentz) force,with the collision term in the right-hand side set tozero. The distribution function 𝑓(𝑡, 𝐫, 𝐯) is defined
as a function of time 𝑡, position 𝐫 and velocity 𝐯
with the subscript 𝑠 being the species of singly
charged particles (e.g., 𝑠 = 𝑖 and 𝑒 for ions andelectrons, respectively). The quantities 𝑞𝑠 and 𝑚𝑠
are the charge and the mass of the particle species𝑠. By integrating the Vlasov equation (2) over thevelocity 𝐯 and getting the sum for all the particle
species 𝑠, the continuity equation for charge is
obtained, which describes the necessary condition
for the electric current density,𝜕𝜌+∇∙𝐉=0(3)𝜕𝑡These equations are regarded as the “first principles”
of the collisionless plasma.
Our Vlasov code solves two spatial dimensions(𝑥, 𝑦) and three velocity dimensions (𝑣𝑥 , 𝑣𝑦 , 𝑣𝑧 ).
The numerical schemes are based on a conservative
semi-Lagrangian scheme with several recent
improvements [5-8] and hybrid parallelization with
MPI and OpenMP [9]. The details on the numerical
schemes are found in the literature [4].
The detailed simulation parameters and setting
are described in the literature [3]. Even with thefastest supercomputer in the world in 2014, it isimpossible to use the realistic physical parametersof the Moon in the first-principle full kinetic
simulation. Hence, the radius of the body is“reduced” to be 𝑅𝐿 ~ 2.72 km, which corresponds
to a typical size of an asteroid.
In the present study, we compare two simulation
runs. One is the low-resolution run [3] and the otheris the high-resolution run performed on the Kcomputer. In the low-resolution run, the spatial gridsize is set to be Δ𝑥 = Δ𝑦 = 10𝜆𝐷𝑖 and total of
𝑁𝑥 × 𝑁𝑦 = 160 × 240 grid points are used for thetwo spatial dimensions. In the high-resolution run,the spatial grid size is set to be Δ𝑥 = Δ𝑦 = 𝜆𝐷𝑖
and a total of 𝑁𝑥 × 𝑁𝑦 = 1920 × 2560 grid points
are used for the two spatial dimensions. A total of
𝑁𝑣𝑥 × 𝑁𝑣𝑦 × 𝑁𝑣𝑧 = 60 × 60 × 60 grid points forthe three velocity dimensions in both runs.
3. Result and Summary
Figure 1 shows the temporal development of theion density together with the configuration ofmagnetic field lines as a result of the interaction
between the solar wind and a small dielectric bodywith a weak intrinsic magnetic field. The locationof the inner boundary is indicated by the magenta
circle at the center of the simulation box.
At 𝜔𝑐𝑖 𝑡 = 0, the simulation box is filled with thesolar-wind plasma propagating from the left to theright. The magnetic field lines are open on thenightside while they are closed on the dayside. As
the time elapses, a very-low-density region isformed on the nightside by absorption of thesolar-wind plasma on the surface of the body,which is known as the wake tail. The configurationof magnetic field lines is also modified to follow
the shape of the wake tail. On the dayside, on theother hand, a high-density region is formed whichindicates the formation of a bow-shock-likestructure by the compression of the solar-windplasma by the intrinsic magnetic field of the body.
The profile of ion density shows asymmetry withrespect to the 𝑥 axis. The system reaches almost
the quasi-steady state at 𝜔𝑐𝑖 𝑡 = 6.
One can easily find from the square blocksaround the body that the result in the top panels isobtained by the low-resolution run. It may besurprising, even for those who are familiar withplasma physics, that the difference between the tworuns is very small although there is minor
difference in the magnetic and plasma
environments on the dayside. However, the present
result seems to be consistent with the conventional
understanding of plasma physics that the structureand dynamics of global magnetic fields, which aregenerally described by the MHD equations, are notaffected by electron-scale microphysics.

Fig.1. Development of the ion density together withthe configuration of magnetic field lines as a result of theinteraction between the solar wind and a small dielectricbody with a weak intrinsic magnetic field.

Acknowledgments
The author thanks Yosuke Ito for his assistancein data analysis. This study is supported byMEXT/JSPS Grant-In-Aid 25610144. The highresolution simulation run has been performed on theK computer at the RIKEN AICS as HPCI Systems
Research Projects (hp120092,hp140064,hp140081).
The low-resolution simulation run has been
performed on DELL PowerEdge R815 at STEL, theFujitsu FX1 and HX600 at ITC in Nagoya Univ.
and Fujitsu RX200S6 at RIIT in Kyushu Univ. as acomputational joint research program at STEL, acollaborative research project of computer science
with HPC at Nagoya Univ., and JHPCN programs.
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