Advances in Space Research 33 (2004) 742–746

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Investigations of MHD wave coupling in a 3-D numericalmodel: effects of temperature gradients

D.H. Lee a,*, R.L. Lysak b, Y. Song b

a Department of Astronomy and Space Science and Institute of Natural Sciences, Kyung Hee University, Yongin, Kyunggi 449-701, South Koreab School of Physics and Astronomy, University of Minnesota, Minneapoli, MN 55455, USA

Abstract

The inhomogeneity of the plasma pressure becomes important near the boundary between cold and hot plasmas. The Alfven

speed undergoes significant variations in such regions in order to satisfy the total pressure balance. We have developed a new three-

dimensional (3-D) MHD wave model that allows for finite plasma pressure. Based on this time-dependent model, we study the

resonant absorption properties when temperature gradients are significant. We consider two models, one corresponding to the inner

edge of the plasma sheet while the second models the plasma sheet boundary layer. We discuss how both transverse and com-

pressional waves are affected by inhomogeneous pressure and temperature by examining the wave spectra. Our results show that

localized field-aligned currents are strongly excited at the boundary between the plasma sheet and its surrounding regions.

� 2003 COSPAR. Published by Elsevier Ltd. All rights reserved.

Keywords: Magnetotail; Plasma sheet; MHD wave; Field-aligned current

1. Introduction

The magnetotail is a non-uniform region which

consists of plasmas with a wide variation of tempera-

tures. To maintain pressure balance equilibrium, the

density and magnetic field strength should vary rapidly

near the boundary region between the two different

plasmas. This inhomogeneity leads to a variation of

magnetohydrodynamic wave speeds, which leads to acoupling between the different MHD wave modes

(Chen and Hasegawa, 1974). Previous studies on the

MHD wave properties near the magnetotail (Goertz

and Smith, 1989; Allan and Wright, 2000; Lee et al.,

2001) have discussed the mode conversion between fast

and shear Alfven modes in a simplified model. Some

features of auroral emission are also known to be as-

sociated with these shear Alfven waves (Samson et al.,1992). However, no numerical studies have been at-

tempted so far in a self-consistent MHD wave model

* Corresponding author: Tel.: +82-31-201-2449; fax: +82-31-204-

7082.

E-mail address: dhlee@khu.ac.kr (D.H. Lee).

0273-1177/$30 � 2003 COSPAR. Published by Elsevier Ltd. All rights reser

doi:10.1016/S0273-1177(03)00635-5

which includes both hot and cold plasmas in themagnetotail. The fast mode is greatly affected where the

temperature increases, while the shear Alfven waves are

not. Therefore, it is important to investigate the details

of MHD waves in a reasonable model, which reflects

realistic plasma temperature of the plasma sheet.

Recent observations from Polar (Wygant et al.,

2000) have found strong shear Alfven wave turbulence

at the plasma sheet boundary layer near 5 RE (RE is theEarth radius) with sufficient Poynting flux to power the

aurora. Coordinated observations with Polar and

Geotail (Angelopoulos et al., 2002) show even larger

Poynting fluxes at Geotail at 18 RE, with a significant

fluctuation in the total magnetic field indicative of

compressional waves. These observations suggest that

the coupling of MHD waves in the magnetotail is an

important aspect of magnetotail dynamics.In this study, we present the initial results from a

new numerical model for MHD wave propagation in

hot plasmas. We consider the boundary regions of the

plasma sheet and show how both fast and shear modes

can couple and produce field-aligned currents near

these regions.

ved.

D.H. Lee et al. / Advances in Space Research 33 (2004) 742–746 743

2. Model

In this study, we consider two boundary regions in

the magnetotail: (A) the equatorial region between the

inner magnetosphere and the plasma sheet and (B) theplasma sheet boundary layer (PSBL). Figs. 1 and 2 show

the physical properties in each region. For A, the vari-

ation of plasma parameters in the equatorial plasma

sheet, plotted in Fig. 1 as a function of the radial dis-

tance (x), is based on an empirical profile (Moore et al.,

1987). The magnetic field is assumed to be along the

north-south direction (̂z). For B, the PSBL in Fig. 2 is

based on Lui (1987). In this case, the inhomogeneityis assumed to be in the z direction, and the magnetic field

is parallel to x. In both cases, we assume a one-dimen-

sional equilibrium plasma, including the strong gradi-

ents of physical quantities near the boundaries.

One major difference between A (Fig. 1) and B

(Fig. 2) lies in the relative amplitude of wave speeds. At

the equatorial plasma sheet (A), the fast mode speed in

the hot plasma sheet becomes much greater than the

Fig. 1. The background equilibrium plasmas assumed in model A: (a) the mag

pressure PB and thermal plasma pressure PP, and (d) the fast mode speed CF

Fig. 2. The background equilibrium plasma parameters assumed in the mode

central plasma sheet (CPS), the plasma sheet boundary layer (PSBL), and th

Alfven speed since the sound speed is significantly larger

than the Alfven speed. At the PSBL (B), the Alfven

speed is close to the fast mode since the sound speed is

almost negligible, and the speed profiles become similar

to those for cold plasmas. Thus, the case of B can be

approximated by a cold plasma model if we focus onlyon the wave speeds.

In our numerical model, the linearized Maxwell and

MHD wave equations are solved in a time-dependent

manner by a leapfrog scheme:

~r�~E1 ¼ �ot~B1; ~r�~B1 ¼ l0~J 1; ð1Þ

~E1 ¼ �~v1 �~B0; ð2Þ

q0ot~v1 ¼ �~rp1 þ~J 1 �~B0 þ~J 0 �~B1; ð3Þ

otq1 þ ~r � ðq0~v1Þ ¼ 0; ð4Þ

p1 ¼cq0

q0

q1; ð5Þ

netic field B and density q, (b) the temperature T and b, (c) the magnetic

, the Alfven speed VA, and the sound speed CS.

l B, plotted in the same format as Fig. 1. The regions representing the

e tail lobe are indicated on the figure.

744 D.H. Lee et al. / Advances in Space Research 33 (2004) 742–746

where P0ð~xÞ, ~B0~x, and q0ð~xÞ represent the background

pressure, magnetic field, and density, respectively.

The simulation box is assumed to be 15 RE in the

parallel direction (the field-aligned direction), 20 RE in

the east–west direction, and 10 RE in the radial direction.

The perfectly reflecting boundary conditions are adop-ted at all boundaries. An impulsive source of compres-

sional waves with a size of about 3 RE � 3 RE � 3 RE, is

assumed at the center of the plasma sheet in both

models. Thus, the source region is located at x ¼ 14 in

Fig. 1 and z ¼ 0 in Fig. 2. We start the simulation with

this radially inward propagating impulse in the mag-

netotail, which are assumed to be symmetric in latitude.

Thus only odd harmonic transverse modes will appearin the wave spectra shown.

Fig. 3. The power spectra of the magnetic fields, the perturbed velocities, a

dependent distribution of the parallel current density Jk is shown in the bot

3. Numerical results

Figs. 3 and 4 show the spectra of the magnetic fields

and velocities as well as the current densities for each

case. In these figures, for convenience, the �inhomoge-neity� direction corresponds to the coordinate x in A and

z in B, respectively. To avoid confusion here, we use

�parallel� for the field-aligned direction, and �east–west�for the other coordinate.

It is well known (e.g., Lee et al., 2000) that in this

geometry transverse waves or shear modes are repre-

sented by the east–west component of ~B and ~v in the

spectrum of an inhomogeneous cold plasma, althoughthese components may be mixed with compressional

spectral power. In addition, compressional waves are

nd the current densities in the equatorial plasma sheet (A). The time-

tom panel at the right column.

Fig. 4. The power spectra of magnetic fields, the perturbed velocities, and the current densities near the plasma sheet boundary layer (B) in the same

format as Fig. 3.

D.H. Lee et al. / Advances in Space Research 33 (2004) 742–746 745

represented by the parallel or radial component of~B and~v. In Figs. 3 and 4, this feature is confirmed well. The

radial and parallel components show compressional

modes that have spectral peaks with frequencies inde-

pendent of position, indicating a global mode structure.

The east–west components show continuous modes with

position-dependent frequencies indicative of transverse

waves across the boundaries in both cases.It should be noted that there are some differences

between A (Fig. 3) and B (Fig. 4). It is evident that the

transverse modes in Fig. 4 have more distinct continu-

ous spectral features than those in Fig. 3. Fig. 4 is

similar to the spectra in previous cold plasma studies

(Lee et al., 2000). This correspondence is understandable

in the sense that the wave speed profile in B is similar to

a cold plasma as discussed above. The fact that the

continuous bands in A are less distinct than in B indi-

cates that transverse waves in hot plasmas have different

spectral characteristics when the fast mode speed and

the Alfven speed become clearly differentiated. There-

fore, our results suggest that the identification of

transverse modes in ~B and~v components may not be so

straightforward in such cases.

The current densities are also shown in Figs. 3 and 4.Since field-aligned currents can be carried only by

transverse waves, it should be the case that the Jkstructures represent pure shear Alfven waves. Both Figs.

3 and 4 show strong localized Jk at the boundary regionswhere the Alfven speed gradient is strong. The bottom

panel in the right column in Figs. 3 and 4 shows the

time evolution of Jk. During the initial period of

0 < t < 800 s, it is evident that Jk is first excited at the

746 D.H. Lee et al. / Advances in Space Research 33 (2004) 742–746

sharp boundary, and then it is gradually extended to the

neighboring magnetic shells, which have less strong

wave speed gradients.

4. Discussion and summary

We have developed a 3-D MHD wave code in hot

plasmas, which allows magnetic field, density and tem-

perature gradients. Note that the model does not include

any aspects of the dipolar geometry of the inner mag-

netosphere, and so not all features will be reproduced.

Nevertheless, box models of this type have been com-

monly used to gain insight into the behavior of wavecoupling. In the more distant tail geometry, or if we are

looking only in local regions in the tail, this model is a

reasonable approximation to the realistic situation.

Our initial results show that impulsive field-aligned

currents are strongly excited near the PSBL and the

boundary between the magnetospheric cold plasma and

the hot plasma sheet at the equator. In these regions, the

Alfven speed undergoes a rapid variation, and thus in-tensive shear Alfven waves can be excited through mode

conversion.

In these three dimensional, inhomogeneous plasmas,

it is evident that all components of the wave field are

coupled, and the spectral feature can be different among

various kinds of hot plasmas. However, the use of Jk asa diagnostic of shear mode waves provides clear evi-

dence of their presence at the gradients. The east–westcomponents of the perturbed magnetic field and veloc-

ities show less distinctive spectral feature of transverse

waves when the Alfven speed becomes significantly dif-

ferent from the fast mode speed.

In a realistic magnetosphere, there can be field-

aligned variations in all physical quantities, which are

neglected in this study. Since our numerical model al-

lows a full 3-D calculation, more complicated equilib-rium background plasmas can be included in the model

in a straightforward manner. This subject remains as

future work.

Acknowledgements

Work at Kyung Hee University was supported by the

Korea Science and Engineering Foundation Grant R14-

2002-043-01000-0. Work at University of Minnesota

was supported by NASA Grant NAG5-11868. Super-computing resources were provided by the Minnesota

Supercomputer Institute.

References

Allan, W., Wright, A.N. Magnetotail waveguide: fast and Alfven

waves in the plasma sheet boundary layer. J. Geophys. Res. 105,

317, 2000.

Angelopoulos, V., Chapman, J.A., Mozer, F.S., et al. Plasma sheet

electromagnetic power generation and its dissipation along auroral

field lines. J. Geophys. Res. 107 (A8), 1181, 2002, doi:10.1029/

2001JA900136.

Chen, L., Hasegawa, A. Plasma heating by spatial resonance of Alfven

wave. Phys. Fluids 17, 1399, 1974.

Goertz, C.K., Smith, R.A. The thermal catastrophe model of

substroms. J. Geophys. Res. 94, 6581, 1989.

Lee, D.H., Lysak, R.L., Song, Y. Field line resonances in a non-

axisymmetric magnetic field. J. Geophys. Res. 105, 10703, 2000.

Lee, D.H., Lysak, R.L., Song, Y. Generation of field-aligned currents

in the near-earth magnetotail. Geophys. Res. Lett. 28, 1883, 2001.

Lui, A.T.Y. Road map to magnetotail domains, in: Lui, A.T.Y. (Ed.),

Magnetotail Physics. The John Hopkins University Press, Balti-

more, London, pp. 3–10, 1987.

Moore, T.E., Gallagher, D.L., Horwitz, J.L., Comfort, R.H. MHD

wave breaking in the outer plasmasphere. Geophys. Res. Lett. 14,

1007, 1987.

Samson, J.C., Wallis, D.D., Hughes, T.J., et al. Substorm intensifica-

tions and field line resonances in the nightside magnetosphere. J.

Geophys. Res. 97, 8495, 1992.

Wygant, J.R., Keiling, A., Cattell, C.A., et al. Polar comparisons of

intense electric fields and Poynting flux hear and within the plasma

sheet-lobe boundary to UVI images: an energy source for the

aurora. J. Geophys. Res. 105, 18675, 2000.