S8-7
Hydrodynamical Instabilities in Strongly Magnetized Plasma
強磁場下での流体不安定性	 
Takayoshi Sano
佐野孝好
Institute of Laser Engineering, Osaka University, Suita, Osaka 565-0871, Japan
大阪大学レーザーエネルギー学研究センター	  〒565-0871	  大阪府吹田市山田丘2-6
In this paper, the effects of a magnetic field on hydrodynamical interfacial instabilities, such as theRichtmyer-Meshkov and Rayleigh-Taylor instability, is reviewed briefly. Especially, the suppression
mechanism of the unstable growth and the critical condition for the field strength are discussed.

1. Backgrounds
Turbulent mixing driven by interfacial
instabilities often plays a crucial role in many
research fields such as inertial confinement
fusion (ICF), astrophysical phenomena, andinterplanetary plasmas [1,2]. Space plasmas
are commonly magnetized and the dynamical
effects of a magnetic field cannot be negligible.
Recently, inclusion of an external magnetic
field is considered intensely for the laboratory
experiments [3]. Therefore, it is an urgent needto understand the magnetohydrodynamical(MHD) evolutions of the interfacial instabilities
in terms of these applications.
Introduction of a magnetic field brings two
important consequences into the turbulent
mixing process, which are the amplification ofan ambient field [4] and the suppression of theunstable motions [5]. In this paper, we will pay
attention to the latter effect mainly.
2. Suppression by a Magnetic Field
Richtmyer-Meshkov instability (RMI) is oneof the interfacial instabilities, which occurs
when an incident shock strikes a corrugatedcontact discontinuity separating two fluidswith different densities. Because of thecorrugation of the interface, the surface profilesof the transmitted and reflected waves (shockor rarefaction) are also rippled. The RMI isdriven by the vorticity left by these rippled
shocks at the interface and in the fluids. The
fluctuations of the interface grow linearly withtime and the growth velocity vlin can bedetermined by the Mach number of the incident
shock, density ratio at the interface, andcorrugation amplitude [2,6].
When a uniform magnetic field parallel tothe shock velocity exists initially, the RMI canbe stabilized as a result of the extraction ofvorticity from the interface. A useful formula

describing a critical condition for MHD RMI
has been derived based on the results of direct
numerical simulations [5]. The stability
condition is given by the Alfven number A =vA/vlin > 0.1 where vA is the Alfven speed (seeFig. 1). Interestingly, this criterion is
independent of any parameters for the initialconditions. Then the critical field strength tosuppress the RMI can be written as Bcrit = 0.1(4π ρ )1/2vlin ≒ 0.01 ρ 1/2Ui, where Ui is theincident shock velocity. The RMI growthvelocity is usually an order of magnitude
smaller than Ui, so a relation vlin = 0.1Ui isassumed for this estimation.
Rayleigh-Taylor instability (RTI) can be alsostabilized by a strong magnetic field [7]. The
magnetic tension force suppresses the RTIgrowth for the fluctuations shorter than thecritical wavelength λcrit = 2πvA2/(At g), where
At is the Atwood number and g is thegravitational acceleration (see Fig. 2). If thecritical wavelength is longer than the systemsize L, no growth of the RTI can be expected.
Then the critical strength for RTI is given by
Bcrit = (2ρAt g L)1/2.
These criteria are quite useful for the design ofexperimental setup controlling the growth of theinstabilities. For the case of ICF condition, thecritical strength is roughly of the order of 0.1-1 kT.

3. Other Effects of a Magnetic Field
It have been demonstrated that a magnetic
field can be amplified efficiently through thestretching motions caused by the interfacialinstabilities. For the case of RMI, themaximum field strength is more than twoorders of magnitude higher than the initial size,
and it appears associated with the interface aswell as the bulk vorticity left by the rippledtransmitted shock [4]. The amplified field could

modify largely the nonlinear evolutions of
turbulent mixing.
Heat conduction is also affected by the presenceof a magnetic field. The thermal conductivity inmagnetized plasmas is anisotropic with respect tothe direction of the magnetic field. Rapid
temperature diffusion could govern the dynamics of
RM or RT unstable system [8]. Then it would be aninteresting future work to study the effect of theanisotropic diffusion by numerical simulations andalso by laser plasma experiments.

Fig.2. Dispersion relation of the linear stability for
MHD RTI, in which n2 > 0 denotes exponential unstable
growth of the perturbations. Because of the suppression
by magnetic tension force, fluctuations with the shorter
wavelength can be stabilized.

Fig.1. Nonlinear stability of the RMI performed by
two-dimensional MHD simulations for single-modeanalysis. The circles denote the unstable models thatexhibit the nonlinear growth of RMI in the numericalsimulations, whereas the crosses stand for the models
which are stabilized by the ambient magnetic field. The
critical condition for RMI growth can be given by theAlfven number A < 0.1, and which is independent of theMach number, density ratio, corrugation amplitude, andeven the direction of the initial magnetic field.

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