J. Plasma Fusion Res. SERIES, Vol.2 (1999) 505-508

Self-Organization in the Two-Dimensional Kelvin-Helmholtz
lnstability at the Magnetopause
MIURA Akira
Department of Earth and Planetary Physics, University of Tolcyo, Tokyo I13-0033, Japan(Received: 8 December 1998 / Accepted: 17 February 1999)

Abstract
A two-dimensional (2-D) magnetohydrodynamic (MHD) simulation of the Kelvin-Helmholtz (K-H)
instability is performed in the plane transverse to the magnetic field for a random initial velocity
perturbation. The relative constancy of the total kinetic energy and a rapid decay of the enstrophy by theselective dissipation with the numerical viscosity indicate that the successive pairings of vortices
occurring in the nonlinear stage of the K-H instability is a self-organization process. The spectral
distributions of the integrated kinetic energy, the enstrophy, and the integrated magnetic energy show
power law distributions at the medium subrange of the wavenumber in the well developed nonlinearstage.

Keywords:
self-organization, Kelvin-Helmholtz instability, sheared plasma flow, MHD, magnetopause

1. Introduction
The shear in the flow velocity is a ubiquitousfeature in space and astrophysical plasmas and it isvelocity perturbation and the qualitative feature of theself-organization in the 2-D K-H instability is rather
insensitive to the form of the initial seed oerturbation.

susceptible to the K-H instability. The importance of thevelocity shear in the fusion plasma confinement has also
been recognized recently. The K-H instability occurring

2. Simulation Model
A 2-D MHD simulation of the K-H instability isat the terrestrial magnetopause is a prototype example ofthe K-H instability occurring in the MHD sheared flows

performed for an initial velocity profile of vor@) = (Vsl2) f,l - tanh (x/a)l and for a convective fast mode Mach
number of 0.35 [1-2]. The initial uniform magnetic field
is perpendicular to the simulation plane (x-y plane).

[]. The nonlinear evolution of the K-H instability isresponsible for the transport of momentum and energy

across the velocity shear layer []. The pairing ofinstability is a well-known nonlinear effect []. It has
been shown by a 2-D MHD simulation starting from a
coherent initial seed perturbation that the successive
pairings of vortices occurring in the nonlinear stage ofthe K-H instability is a self-organization process [2].

Time t is normalized by 2alVs, where 2a is the initial
thickness of the velocity shear layer and V6 is the total
jump of the flow velocity across the velocity shear layer.
The length of the periodic simulation box in the ydirection is equal to 8 times as long as the wavelengthof the linearly fastest growing mode (i+c1a = 15.7a) I3l.
In Miura [2] a coherent perturbation, which was a sumThe purpose of this brief report is to show that the selforganization in the K-H instability, i.e., the successive

subharmonics, was given as an initial seed perturbationvortices occurring in the nonlinear stage of theof the linearly fastest growing mode and its

pairings of vortices, also arises for a random initial@1999 by The Japan Society of Plasma
Science and Nuclear Fusion Research505

Miura A., Self-Organization in the Two-Dimensional Kelvin-Helmholtz Instability at the Magnetopause

with the peak amplitude of vr equal to 0.005Ve and theterm, the baroclinic term, and the viscous term [l-2], weconclude that the rapid decay in the enstrophy shown in
Figure 2 must be due to the viscous dissipation. Noticethat although the present simulation is done for the ideal
MHD without any explicit physical viscosity, small
implicit numerical and artificial viscosities are included
to prevent spurious mesh oscillations. Therefore, theenstrophy, which is the invariant in the 2-D, inviscid,
and incompressible case, decreases with time due to theselective dissipation [4] by the numerical viscosity. The
relative constancy of the total kinetic energy shown in
Figure I and the selective decay of the enstrophy shown

temporal evolution of the instability was investigated. Inorder to see the dependence of the self-organization on
the initial perturbation a random velocity perturbation
with its peak amplitude of v, equal to 0.00005V6 is given
as an initial seed perturbation in the present run. A twostep Lax-Wendroff scheme with an artificial viscosity isused in the present simulation and the number of grid
points is equal to 400 x 400.

3. Simulation Results
Figure I shows temporal evolutions of the totalenergy (dotted curve), total internal energy (dashedcurve), total magnetic energy (dot-dash curve), and thetotal kinetic energy (solid curve) integrated in the wholesimulation region. Each total energy is normalized by

the evolution of the K-H instability in the present 2-D
MHD transverse configuration is a self-organization

0.0628a2p6, where p6 is the initial uniform pressure. It is

process [5].

in Figure 2 are similar to the results obtained for acoherent initial seed perturbation [2] and indicate that

Figure 3 shows the spectral amplitudes of theobvious from this figure that the total kinetic energy
remains almost constant during the simulation run in thepresent 2-D MHD transverse configuration.
Figure 2 shows temporal evolutions of the totalsquare vorticity (enstrophy) integrated in the wholesimulation region (solid curve), the contribution to thekinetic energy integrated across x versus thewavenumber in the y direction at T = 0 (a) and Z = 600(b), where T = tVo/(2a), the spectral amplitudes of theenstrophy versus the wavenumber at T = O (c) and I =
600 (d), and the spectral amplitudes of the magnetic
energy integrated across.r versus the wavenumber at T
= 0 (e) and Z = 600 (0. The spectral amplitudes of theintegrated kinetic energy are normalizedby 2aps.The
spectral amplitudes of the enstrophy are normalized bychange of the total square vorticity due to thecompressibility (dotted curve) and the baroclinic
contribution to the change of the enstrophy (dashedcurve), which are normalized by Vo2. It is obviously seenin this figure that the enstrophy decreases rapidly with(V0/2a)2. The spectral amplitudes of the integrated

time with a small oscillating component, which is due tothe compressibility. It is also seen in this figure that thebaroclinic term is negligible. Since the time change ofthe enstrophy is equal to the sum of the compressible

magnetic energy are normalized by apo-tBrz, where Bs is
the initial uniform magnetic field strength. In this figure
ft-1n is the wavenumber equal to kFcM/8, where ftp6s isthe wavenumber in the y direction of the linearly fastest^45inoJss.i so'E ,'0 450000
CLa!

400000GOc{(oqotr350000...,..'.'..,\Tota|.'.-..

300000o 200000
co150000
tto
o 100000t''o 5(X)00

c9zo

250000

0$tt

3ro
at

E5

Eo-5

-\Kinetic

0100 200 300 4(x) 500 600

200 300 400 500 600

Time ( x2alVo)

Time ( x2alVo)
Fig. 2 Temporal evolutions of the total square vorticity(enstrophy), the contribution of the compressible

Fig. 1 Temporal evolutions of the total energy, totalinternal energy, total magnetic energy, and theterm to the change of the enstrophy, and thetotal kinetic energy in the whole simulation

contribution of the baroclinic term to the changeof the enstrophy.

region.
s06

Miura A., Self-Organization in the Two-Dimensional Kelvin-Helmholtz Instability at the Magnetopause(c)I.gdb.n-gy

I
0.t
0.01

0.mt
0.@1t€

tE4
tEo
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tE-tE{ttE{l1tot@

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Fig.3 The spectral amplitudes of the total kinetic energy versus the wavenumber at f = 0 (a) and f = 600 (b). theenstrophy at f= 0 (c) and f= 600 (d), and the total magnetic energy at f= 0 (e) and f= 600 (f).

growing mode. At T = 0, a random incompressible

as predicted by the linear theory [3], but as avelocity perturbation with a small amplitude was addedconsequence of successive pairings of vortices, a large

to the background flow velocity. Figures 3(a) and 3(c)show that the spectral distributions of the integrated
kinetic energy and enstroph! at T = 0 are almost flat
(white noise). However, at T = 600, the spectral

isolated vortex is formed inside the simulation region atZ = 600. Figures 4(b) and (d) show electric current
vectors at T = 70 and Z = 600. It is seen in these panels
that an eddy current is associated with each flow vortex.

distributions of the integrated kinetic energy, theThese eddy currents appear because of thecompressibility, i.e., the eddy current is equal to curl ofenstrophy, and the integrated magnetic energy become
almost continuous and are well represented by powerlaw distributions at klk^in 1 90. Irregular noise
components at large ft values (klk^in> 90) at T = 600
appeared because of the artificial viscosity added to thesystem. The spectral peaks at k = k^in at ? = 600 of theintegrated kinetic energy, the enstrophy, and theintegrated magnetic energy occurred because of theinverse cascade. The power law exponents of theintegrated kinetic energy, the enstrophy, and theintegrated magnetic energy at medium subrange of thewavenumber from klk^in = l0 to 5O at T = 600 are equal
to -4.61, -2.58, and -4.60, respectively. Since theenstrophy is an integral of the square of curl of v, thethe z component of the magnetic field perturbation (fastmagnetosonic component of the magnetic field). These
cuffents are mainly carried by the inertia cunent [1-2].

4. Conclusion
By a 2-D MHD simulation of the K-H instability
starting from a random velocity perturbation it has been
established that the successive pairings of vortices
occurring in the nonlinear stage of the instability is aself-organization process and the qualitative feature of
the self-organization process is rather insensitive to theform of the initial seed perturbation. The present result,if applied to the spatial growth of the instability along
the magnetopause, suggests that small-scale vortices
excited at the dayside magnetopause by the instability
evolve into global scale vortices in the tail of themagnetosphere by the successive pairings of vortices.
Therefore, the self-organization process is responsible

observed fact that the power law exponent of theenstrophy is nearly equal to the power law exponent of
the total kinetic energy plus 2 is reasonable. Notice thatthe selective dissipation of the enstrophy and the relative
constancy of the total kinetic energy are caused by thelarge difference of the power law exponents between thefor the creation of a large scale order (vortex) in the tailof the magnetosphere.

spectra of the integrated kinetic energy and theenstrophy.
Figures 4(a) and (c) show contours of vorticity at Z

= 70 and 7= 600. AtT =7O eieht vortices are observed507

Miura A., Self-Organization in the Two-Dimensional Kelvin-Helmholtz Instability at the Magnetopause
Vorticity

T=70Q(a)(c)

Vorticity
40

30

30

20

20't010S,Ox

J0.10-t0-20-20.30{0 I020

T=70/o-4060

Y('a)
Curtent

100(b)t20(d)

40403030m

20

t0

10

€.0x.90x-.t0.10-20-20-30-30.10 E
0'f"'i'.

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. , i \ -/,,
'r.-'i.40

60m

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80

Y('a)

Fig.4 (a) Contours of the vorticity atT=70. (b) Electric currentvectors atT=70. (c) Contours of the vorticity at f = 600.
(d) Electric current vectors at f = 600.

References

Acknowledgements

tll A. Miura, Phys. Plasmas 4,2871 (1997).

This work has been supported by Grants-in-aid for
Scientific Research 09640529 and, in part, by RASC ofKyoto university, ISAS and CRL as a joint research
project. The computation for this work was performed at
the computer center of the University of Tokyo.

12) A. Miura, J. Geophys. Res. 104, 395 (1999).
t3l A. Miura and P.L. Pritchett, J. Geophys. Res. 87,743r (1982).

t4l D. Montgomery, L. Turner and G. Vahala, Phys.
Fluids, 21,757 (1978).
t5l A.Hasegawa, Adv. Phys.34, I (1985).

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