22E04
Fluid modeling of Short Large Amplitude Magnetic Structures (SLAMS) near
the earth's bowshock
SLAMSの流体モデル	 Tohru Hada1, Yasuhiro Nariyuki2 and Marty Lee3
羽田	  亨1, 成行泰裕2，マーティ・リー3
1. Interdisciplinary Graduate School of Engineering Sciences, Kyushu University
6-1 Kasuga Koen, Kasuga, 816-8580, Japan
2. Department of Electrical Engineering, Kochi National College of Technology,
200-1 Mononobeotsu, Nangoku, Kochi, 783-8508 Japan
3. Department of Physics, University of New Hampshire,
105 Main Street, Durham, New Hampshire 03824 USA
1. 九州大学大学院総合理工学研究院、816-8580 福岡県春日市春日公園６−１2. 高知工業高等専門学校電気情報工学科、783-8508 高知県南国市物部乙200-1
3. ニューハンプシャー大学物理学部、30824 アメリカ合衆国ニューハンプシャー州ダーラム市
The so-called Short Large Amplitude Magnetic Structures (SLAMS) are one of the most outstanding features frequently
observed upstream of the earth's bowshock. Although the properties of these structures have been studied extensively
using spacecraft observations, the mechanism leading to their formation still remains unclear. Since the SLAMS growin a region with a gradient of energetic ions, the ion heat flux is likely to be the main energy source for their growth. Wemodel their evolution using the Landau-fluid type framework, by including a nonlocal interaction between the ion heat
flux and the magnetic envelope modulation. Numerical simulations show that, in the presence of inverse Landau
interaction, a series of magnetic pulsations similar to the SLAMS grow rapidly. The growth is nonlinear and singular.

1. Introduction
From in-situ spacecraft observations we havelearned that the collisionless quasi-parallel shocks
are associated with a wide variety of wave activity
and an abundance of supra-thermal particles.
Among them are the short large amplitude magneticstructures, or “SLAMS” [1], which are also oftengrouped with longer structures and referred tocollectively as “pulsations” [2,3].
Figure 1 gives a few examples of the SLAMs
detected by four-spacecraft Cluster experiment [4].
Plotted is the total magnetic field amplitude versus
time, and different colors correspond to different
spacecraft. Three panels show examples for threedifferent spacecraft separation scales (shown at theupper left corner of each panel). These observations
suggest that the overall extent of the SLAMs is afew hundred to a few thousand kilometers, i.e.,much larger than the typical ion Larmor radius, andthat the structure can grow within a very shortperiod of the order of seconds, i.e., about the sameorder as (or even shorter than) the ion gyro-period.
The rapid growth and the apparently“nonlinear” shape of the SLAMs suggest that they
are generated by some external energy injection,rather than internal plasma instabilities. From earlydays of observations it was inferred that foreshock
ULF waves (MHD waves often found in a region
upstream of the bowshock) provide the seeds of theSLAMs[1,3]. Also, a superimposed epoch analysis

of suprathermal ion data around the SLAMs
clarified that the background energetic ion flux,
presumably of the bowshock origin, is responsible
for feeding energy to the SLAMs growth [5].
Hybrid simulation (super-particle ions + a massless
electron fluid) of a quasi-parallel shock and its
upstream region suggests that the foreshock ULF
waves are amplified as they interact with energetic
ions backstreaming from the shock, and they
nonlinearly steepen and form a pulsation-like wave
packet [6].

Fig. 1: Examples of SLAMs observed by Cluster

2. Fluid Model
In order to study the growth and nonlinear
evolution of the SLAMs from the fluid point ofview, we employ a Landau-fluid type formulation[7,8],"u"u " p "S " b 2(1)! + !u +++ ( )=0"t"z "z "z "z 8#!
! p!q(2)( + u )( # ) +=0!t!z "!z"v"v B "b(3)! + !u # 0=0"t"z 4$ "z!b!v!b!u(4)+u+ b " B0=0!t!z!z!zdistribution function).
Figure 2 shows time evolution of seed magnetic
fluctuations (upper panel) when they are immersed
in a plasma with normal (red)/zero (black)/inverse(blue) Landau interactions. Energetic ions from theshock are able to grow the initial magnetic
fluctuations to a very large amplitude. In addition,
as Figure 3 shows, the growth is intrinsicallynonlinear, and is associated with a (almost) blow-up
at a certain finite timescale.

where all the notations are standard, except that b is
the transverse magnetic field, u and v are thelongitudinal and the transverse plasma velocity
components, and that S and q are the nonlocal
momentum and heat flux defined via their Fouriercounterparts,ik(5)ûk|k|
ik ˆ(6)q̂k = ! 2n0 vt "Tk .
|k|The transverse dynamics (such as propagation of
Alfven waves) in a medium specified by thelongitudinal variables is described by (3) and (4),while (1) and (2) describe the longitudinal
dynamics modified by the ponderomotive force andthe nonlocal flux contributions. The latter isindependent of the spatial gradient scale (factork/|k| in (5) and (6)), characteristic of the Landau
type interaction.
If the interaction between two oppositely
propagating waves (or structures) can be ignored,(1)-(5) may be written in a way similar to theDNLS equation,Ŝk = ! 2mn0 vt µ!b 1 !
v ! b+(U NL b) + e "=0!t 2 !z
2# !z 22A2(7)where vA is the Alfven speed, Ω is the ion
gyrofrequency, e is the unit vector along the z-axis.
The nonlinear response due to finite amplitude
magnetic field is given byv
U NL = A (c1 (| b |2 ! | b |2 ) + c2 H(| b |2 ! | b |2 )) (8)2where the bar indicates the average, and H stands
for the Hilbert transformation. The first and thesecond terms in the r.h.s. of (8) represent the fluid
and the Landau response, and their magnitude isspecified via coefficients c1 and c2, which arefurther determined by the plasma states (such as theFig.2. Time evolution of finite amplitude magnetic
fluctuations in a plasma with Landau interations

Fig. 3. Nonlinear growth of the magnetic field
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