24P151-P
Two-fluid plasma numerical analysis of the interfacial instability induced by acollision-less shock wave
無衝突衝撃波によって誘起される界面不安定の二流体プラズマ数値解析	 
Koichi Mori
森	  浩一	 Nagoya	 University	 Furo, Chikusa-ku, Nagoya 464-8603, Japan名古屋大学 〒464-8603	  名古屋市千種区不老The interactions between a collision-less shock wave and an isolated cylindrical bubble are investigated bysolving the two-fluid equations. A second-order Riemann solver is employed to sharply capture the shock
wave and the contact discontinuity. As a result, filamentation driven by the electron precursor introduces
the turbulence in the plasma ahead of the shock wave.

1. Introduction
Fluid dynamical instability is one of the majormechanisms for building up the turbulence inplasma. The interactions between a shock wave andan isolated bubble, so-called the shock-bubble
interactions have been studied so far with the aim ofunderstanding the compressible turbulence inneutral gas. [1] For plasmas, the effects of themagnetic field on the deformation of a planedensity discontinuity are investigated using themagneto hydrodynamic (MHD) simulation code.
[2] Because the quasi charge neutrality is presumed
in the MHD equations, the shock wave in
un-magnetized plasma is not different from that inneutral gases within the MHD formalism. The
present study deals with the shock-bubble
interaction problems in un-magnetized andcollision-less plasma in order to understand theshock compression in high-temperature plasma andrelevant instability problems. The space-charge willplay a fundamental role. Recently, Shumulak andLoverich solved the one-dimensional Riemannproblem using a full system of two-fluid model ofcollision-less plasma. [3] Similar methods areemployed in the present study.

The numerical fluxes are evaluated using the Roe
scheme. [4] The space and time accuracy isenhanced to the second order by using theMUSCL-Hancock method. [5] The fluid solvers
have been verified by solving the Sod problems andthe shock-bubble interaction problem in the neutralgases. [6] The present fluid solver captures sharply
a shock wave and contact discontinuity within a
few cells. Moreover, the bubble deformation
processes could be reproduced without adding
special treatment to capture the bubble-atmosphereinterface. Major difficulties in solving thetwo-fluid equations are originated from integrating
the stiff source terms. A nine-stepcomposition-method was selected to avoid
unphysical oscillations at moderate grid size.
The computations are conducted for a singly
ionized atomic hydrogen gas. The mass ratio
mi me is kept constant at 1836.0. The half domain
is discretized with 512 × 64 cells. In the initialcondition, the diaphragm and the bubble are located.
The diaphragm separates the high-pressure-room(HPR) and the low-pressure-room (LPR). The
initialvaluesintheLPRare116.
Both( N e , N i , pe , pi ) = (1.0, 1.0, 5.28 × 10 , 9.58 × 10 )2. MethodTwo-fluid equations of collision-less plasmas
are formulated in a normalized form. Here, thesubscript s represents the species either electron (=e) or ion (= i). We have four conservation laws foreach species. Nine equations including the Poisson
equation for the electrostatic field are solved.
The system of the fluid equations is solved in
the conservation form using a finite volume
Riemann solver developed by reference to theupwind method developed in the previous study. [3]the electron and ion fluids are initially at rest in
both rooms. The ratio of the electron temperature to
the ion temperature Te Ti is set at 30.0 uniformly
over the calculation domain for the initial condition
so that typical shock profile is formed. Across theinterface between the bubble and the surrounding
plasma in LPR, the pressure is not changed but thegas density is changed. Both Ni and Ne inside of thebubble are 0.1.

3. Results
Figure 1 shows the density contour of theions and electrons. Because the length scale is close
to the Debye length, the charge neutrality is notalways satisfied. Electron precursor leaches out of
the shock front. Besides, the hot electrons containedin the bubble are spreading out diffusing into thesurrounding plasma. Before the shock front reaches
the bubble, the electrons at the upstream interfaceof the bubble are pushed downstream by theelectron precursor. Moreover, fine filaments appear
at the downstream interface of the bubble. At first,
the filaments extend both in the longitudinal and thelateral directions. At later times, the filaments
extend further only in the longitudinal direction.
The interval between two adjacent filaments almost
equals to the Debye length. The current driven by
the potential gradient at the upstream interface ofthe bubble penetrates the bubble through thedownstream interface breaking up into finefilaments. Filamentation observed in the other PIC
simulation of the collision-less shock wave isthough to be oriented from the two-stream
instabilities. The mushroom shapes appear at one of
the leading tips due to the Rayleigh-Taylor
instability.

Fig.1 Filamentation from the bubble
4. summary
The shock-bubble interactions in un-magnetizedand collision-less plasma are investigated.
Filamentation driven by the electron precursor of
the shock wave is observed. Baroclinic vortex
generation is prevailed by the filamentation
instability in the collision-less plasma.
Acknowledgments
The major part of the study was done while author
was visiting the Institute of Shock Physics, Imperial
College of London. Author is grateful for thevaluable advices from S. Rose and S. Mark. The
visiting research was fund by JSPS excellent young
researcher over sea visits program. 	 

References[1] D. Ranjan, J. Oakley, and R. Bonazza,“Shock-Bubble Interactions,” Annu. Rev. Fluid
Mech. 43, 117 (2011).
[2] J.P.H. Goedbloed, R. Keppens, and S. Poedts,
Advanced Magnetohydrodynamics, Cambridge
University Press, Cambridge, 2010.
[3] U. Shumlak and J. Loverich, “Approximate Riemannsolver for the two-fluid plasma model,” J. Comput.
Phys., 187, 620 (2003)[4] P.L. Roe, “Approximate Riemann Solvers,
Parameter vectors, and Difference Scheme,” J.
Comput. Phys. 43, 357 (1981).
[5] E.F. Toro, Riemann Solvers and Numerical Methods
for Fluid Dynamics, Springer, Berlin, 2009[6] K. Mori, “Blast-bubble interaction for a divergent
case,” to appear.