Excitation and steepening of ion‐acoustic wavesin the ionospheric Alfvén resonator

D. Sydorenko,1 R. Rankin,1 and K. Kabin1,2

Received 9 March 2010; revised 30 June 2010; accepted 5 August 2010; published 18 November 2010.

[1] A nonlinear two‐dimensional fluid model describing excitation of the ionosphericAlfvén resonator by a shear Alfvén wave coming from the magnetosphere is developed.Initially, the plasma is in an equilibrium defined by a balance between the gravity,electric field, and pressure gradient forces. This equilibrium is perturbed when a standingAlfvén wave is excited in the resonator. The nonlinear Lorentz force of the wave createsconverging and diverging plasma flows along the geomagnetic field, thus producingcompressions and rarefactions in the plasma density. Simulation reveals that densityperturbations evolve into ion‐acoustic shock waves in a process similar to the nonlinearsteepening of sound waves in neutral gases. A shock associated with compression ofhydrogen ions propagates faster than a shock associated with compression of oxygenions. One‐dimensional shock‐capturing Poisson simulation reveals that the shocksappear as double layers at first, but then they decay into ion‐acoustic wave packets. Thedrop of potential across each shock is negligible at any stage of shock’s development,making these shocks unfavorable for auroral electron acceleration.

Citation: Sydorenko, D., R. Rankin, and K. Kabin (2010), Excitation and steepening of ion‐acoustic waves in the ionosphericAlfvén resonator, J. Geophys. Res., 115, A11212, doi:10.1029/2010JA015448.

1. Introduction

[2] The interest to nonlinear structures in the Earth mag-netosphere, such as shock waves, solitons, and double layers,is related with their particle acceleration effect [Mozer et al.,1980]. The present paper addresses a well known mechanismof formation of such structures, namely the nonlinear wavesteepening. Formation of shock waves due to nonlinearsteepening of compressive MHD waves, for example, is awell known phenomena in the solar corona, see the review ofWarmuth [2007, and the references therein]. Erkaev et al.[2001] demonstrated that a pressure pulse produced by avolcanic eruption on Io creates slow MHD waves whichpropagate toward Jupiter and grow due to the nonlinearsteepening. The nonlinear steepening of obliquely propa-gating inertial Alfvén waves in the near‐Earth plasma isstudied theoretically and numerically and compared withspacecraft observations by Seyler et al. [1995]. The nonlinearsteepening of ion‐acoustic waves propagating parallel to thegeomagnetic field, however, is largely overlooked in thestudies of the low‐altitude magnetosphere.[3] The reason why the nonlinear steepening of ion‐

acoustic waves is usually discarded [Prakash, 1997] may bethat this process is not possible due to strong Landaudamping if the electron temperature is about the ion tem-

perature [Andersen et al., 1967]. Therefore, the nonlinearsteepening is an unlikely mechanism for altitudes below2000 km, where the electron and ion temperatures are of thesame order, 2000–4000 K. At altitudes above 2000 km,however, the electron temperature may be of the order of afew eV (about 20000–30000 K) [Kletzing et al., 1998] and, ifthe ion temperature stays in the range of a few thousand K,the nonlinear steepening of ion acoustic waves becomespossible.[4] Various satellites observed double layers and solitary

waves in the magnetosphere at different altitudes: Freja at1700 km [Dovner et al., 1994], FAST at 4000 km [Ergunet al., 1998], S3‐3 at 6000 km [Temerin et al., 1982],Polar at 7000 km [Dombeck et al., 2001], Viking at 9700 km[Bostrom et al., 1988]. Such structures are usually accom-panied by intense upward ion beams and may be associatedwith nonlinear electrostatic ion cyclotron or ion acousticwaves. An ion‐acoustic double layer, for example, forms ifthere is an intense current or a flow of electrons relative toions with the speed comparable with the electron thermalspeed [see Hudson and Mozer, 1978, and the referencestherein]. Although such rapid flows are typical in the auroralacceleration region (altitude of 1–2 Earth radii), they areusually not registered at lower altitudes. The nonlinear struc-tures at these altitudes still may appear as a result of nonlinearevolution of an intense ion‐acoustic wave, which occurs evenwithout having the intense parallel electric current.[5] Montgomery [1967] pointed out that propagation of

an ion‐acoustic wave is mathematically similar to a non-linear sound wave in a neutral gas. A sound wave steepensand, under certain conditions, develops into a discontinuity

1Department of Physics, University of Alberta, Edmonton, Alberta,Canada.

2Now at Department of Physics, Royal Military College of Canada,Kingston, Ontario, Canada

Copyright 2010 by the American Geophysical Union.0148‐0227/10/2010JA015448

JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 115, A11212, doi:10.1029/2010JA015448, 2010

A11212 1 of 12

thus becoming a shock wave [e.g., Landau and Lifshitz,1987]. Steepening of an ion‐acoustic wave into a shockis observed both in laboratory plasmas [Andersen et al.,1967] and numerical simulations [White et al., 1974; Kozlovand McKinstrie, 2002].[6] In low‐altitude magnetosphere, Sydorenko et al.

[2008] showed that an intense initial density perturbationcan be formed by the nonlinear Lorentz force (ponder-omotive effect) of a standing Alfvén wave in the ionosphericAlfvén resonator (IAR) [Polyakov and Rapoport, 1981]. If ahigher harmonics of the resonator is excited, the densityperturbationsmay be formed at altitudes up to about 4000 km,where the top boundary of the resonator is situated. Thepresent paper shows that such perturbations evolve intocompressional shock waves. The shock waves propagateupward along the geomagnetic field with the speed equal tothe speed of fast or slow ion‐acoustic waves in a plasmawith multiple warm ion species [Fried et al., 1971; Kozlovand McKinstrie, 2002]. Formation of the shock waves in theIAR is demonstrated using a two‐dimensional multifluidquasineutral model of low‐altitude auroral flux tubes. Thismodel, capable of describing large‐scale dynamics of thenear‐Earth plasma, provides coarse spatial resolution and isnot suited for description of sharp discontinuities, which iswhy these simulations have to be stopped at an early non-linear stage of shock’s development. A much more detailedpicture of shock’s evolution is obtained with the help of aone‐dimensional shock‐capturing multifluid electrostaticcode. Simulation of a relatively short system, where theelectron Debye length is well resolved by the grid, revealsthat the ion acoustic shock wave at first appears as a doublelayer and then decays into a short intense ion‐acoustic wavepacket, in qualitative agrement with Hirose et al. [1978] andVerheest [1989].

[7] The paper is organized as follows. Section 2 describesthe two‐dimensional numerical model of the near‐Earthmagnetospheric plasma. In section 3, the nonlinear steep-ening of ion‐acoustic wave structures in magnetospheric si-mulations is discussed. Issues related with insufficient spatialresolution and numerical artefacts of the magnetosphericmodel are clarified in section 4 by means of shock‐capturingone‐dimensional simulation. The one‐dimensional modelsolves the Poisson equation and resolves the electron Debyelength. A summary of the results is given in section 5.

2. Model Description

2.1. Alfvén Wave Electromagnetic Field and PlasmaDynamics Equations

[8] A two‐dimensional (2‐D) model of low‐altitudeauroral flux tubes discussed below is developed from theIAR model described by Sydorenko et al. [2008]. Thepresent model studies nonlinear plasma dynamics inducedby Alfvén waves in the near‐Earth magnetosphere. It isassumed that the plasma is azimuthally symmetric and thegeomagnetic field is dipole. The model uses a 2‐D uniformstructured grid in dipole coordinates. Directions along andacross the geomagnetic field in the meridional plane areresolved. The northern and southern boundaries of thesimulated area coincide with geomagnetic field lines (curvesAB and CD in Figure 1), the bottom boundary correspondsto a perfectly conducting ionosphere (curve BC in Figure 1),and the top end is open to Alfvén waves (curve AD inFigure 1). The boundary conditions are the same as ofSydorenko et al. [2008], however, the electromagnetic fieldand the plasma dynamics equations are essentially different.[9] The present model considers a purely transverse Alfvén

wave:

@E2

@t¼ �V 2

A;eff

h1h3

@h3B3

@�1;

@B3

@t¼ � 1

h1h2

@h2E2

@�1;

ð1Þ

where the effective Alfvén speed is

V�2A;eff ¼ c�2 þ

X�¼O;H

�0n�m�

B2E

;

c is the light speed, BE is the geomagnetic field,ma and na arethe mass and the number density of ion species a, subscriptsa = O and a = H denote oxygen and hydrogen ions. Thedipole coordinates x1,2,3 are x1 = cosϑ/r2, x2 = sin2ϑ/r, andx3 = −’, where {r, ϑ, ’} are the ordinary spherical co-ordinates. The metric factors h1,2,3 are h2 = r2/(sinϑ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ 3 cos2 #

p), h3 = r sinϑ, and h1 = h2h3. Subscript 1

corresponds to vector components directed along the geo-magnetic field line (in the x1 direction), subscript 2 corre-sponds to vector components normal to the geomagneticfield line in the meridional plane (in the x2 direction), andsubscript 3 corresponds to azimuthally directed vectorcomponents (in the x3 direction). Below, all vector com-ponents directed along the geomagnetic field are referred toas the parallel ones. The wave equations (1) are coupled tothe plasma motion through the dynamic equations describedbelow.

Figure 1. Schematic of the simulated area. Corners A, B,C, and D of the area in the real configuration space corre-spond to the corners A′, B′, C′, and D′ of the rectangle inthe dipole coordinate space, respectively.

SYDORENKO ET AL.: EXCITATION AND STEEPENING OF ION‐ACOUSTIC WAVES A11212A11212

2 of 12

[10] Ion velocity across the geomagnetic field (in themeridional plane) corresponds to the inertial ion current:

u�;2 ¼ m�

eB2E

@E2

@t; � ¼ O;H ; ð2Þ

where e is the elementary charge, e > 0. The azimuthal ionvelocity ua,3 = −E2/BE is the velocity of drift in crossedwave electric field E2 and geomagnetic field BE. The parallelion velocity is

@u�;1@t

¼ � ðu� � rÞu�½ �1þe

m�ðE1 þ u�;2B3Þ

� 1

m�n�h1

@

@�1n�T� þ g1; � ¼ O;H ; ð3Þ

where Ta is the ion temperature, g1 is the parallel acceler-ation due to the gravity force, and the parallel component ofthe convective velocity derivative is

ðu� � rÞu�½ �1 ¼u�;1h1

@u�;1@�1

þ u�;2h2

@u�;1@�2

þ u�;1u�;22h21h2

@h21@�2

� u2�;22h1h22

@h22@�1

� u2�;32h1h23

@h23@�1

:

[11] The azimuthal electron velocity is the same as that ofions, ue,3 = −E2/BE. Electron motion across the geomagneticfield in the meridional plane is omitted, ue,2 = 0. The parallelelectron velocity is

@ue;1@t

¼ � ðue � rÞue½ �1�e

meE1 � 1

meneh1

@

@�1neTe þ g1; ð4Þ

where me, ne, and Te are the electron mass, number density,and temperature, respectively, and the parallel component ofthe convective velocity derivative is

ðue � rÞue½ �1¼ue;1h1

@ue;1@�1

� u2e;32h1h23

@h23@�1

:

The ion densities are calculated via the continuity equations

� @n�@t

¼ 1

h1h2h3

@ðn�u�;1h2h3Þ@�1

�þ @ðn�u�;2h1h3Þ

@�2

�; � ¼ O;H ;

ð5Þ

while the electron density is obtained from the quasineu-trality condition

ne ¼ nO þ nH : ð6Þ[12] For the sake of simplicity, it is assumed that local

values of temperatures of all species do not change withtime, i.e., ∂Ta/∂t = 0, where a = e, O, H.

2.2. Quasineutral Parallel Electric Field

[13] In order to satisfy condition (6), the divergence of thetotal electric current must be zero, which results in

@

@�1ðJ1h2h3Þ ¼ � @

@�2ðJ2h1h3Þ; ð7Þ

where J1 =P

�¼e;O;Hqanaua,1 is the parallel electric current,

J2 =P

�¼O;Hqanaua,2 is the transverse electric current, qe = −e

and qO,H = e are the electron and ion charges. The transverse

current is defined by the ion dynamics of the Alfvén wave.With J2 known, equation (7) becomes an equation for theparallel current J1.[14] The parallel electric field, which creates the parallel

current satisfying condition (7), can be obtained by differ-entiating (7) over time and using equations of ion andelectron motion, (3) and (4):

E1

X�¼e;O;H

n�e2

m�¼

X�¼e;O;H

q�1

h1

@

@�1

n�T�m�

� ��

�u�;1@n�@t

þ n� ðu� � rÞu�½ �1�

�X

�¼O;H

n�e2

m�u�;2B3 þ @J1

@t: ð8Þ

[15] Note, using r · nuu = u(r · nu) + n(u · r)u and thecontinuity equation, the second and the third terms in theright‐hand side of (8) reduce to

�u�;1@n�@t

þ n� ðu� � rÞu�½ �1¼ r � ðn�u�u�Þ½ �1;

which in case of one‐dimensional motion is proportional tothe parallel gradient of the ram pressure, rmanaua,1

2 .

2.3. Initial Equilibrium State

[16] The presence of the thermal pressure and gravityforces in the equations of parallel motion (3) and (4) re-quires an equilibrium state as an initial condition for thesimulation. Such an equilibrium is produced as follows.First, the oxygen and ion densities are assumed to be:

nOðrÞjt¼0 ¼ n0O exp � r � RE � hI

l1 þ l2 tanhðr � RE � hI Þ

l3

2664

3775;

nH ðrÞjt¼0 ¼ n0HRE þ hI

r

� �p

;

ð9Þ

where nO0 = 3.2 × 1011 m−3, nH

0 = 7 × 108 m−3, RE = 6380 kmis the Earth radius, hI = 400 km is the altitude of the topionospheric layer, l1 = 130 km, l2 = 325 km, l3 = 2400 km,and p = 3. Functions (9) approximate densities providedby the International Reference Ionosphere model (avail-able at http://omniweb.gsfc.nasa.gov/vitmo/iri_vitmo.html).The ion density profiles are shown in Figure 2a. Withdensities (9) and the dipole geomagnetic field (shown inFigure 2b), the Alfvén speed has a nonmonotonic profileshown in Figure 2c. Note that according to this profile, theIAR is formed between the ionosphere and the altitude ofabout 4000–5000 km, where the upward gradient of theAlfvén speed is maximal.[17] Second, the oxygen ion temperature is set constant

TO = 0.2eV throughout the whole simulation area. Thisassumption allows to find the parallel equilibrium electric fieldfrom the equation of momentum balance for oxygen ions:

E1 ¼ TOenOh1

@nO@�1

� mO

eg1:

SYDORENKO ET AL.: EXCITATION AND STEEPENING OF ION‐ACOUSTIC WAVES A11212A11212

3 of 12

The profile of the parallel electric field in the equilibrium isshown in Figure 2e.[18] Finally, hydrogen ion and electron temperatures are

prescribed at the ionospheric boundary x1 = x1,I: TH (x1,I) =0.2eV and Te (x1,I) = 0.1eV. In the rest of the system, thesetemperatures are determined by the corresponding momen-tum balance equations:

@

@�1nHTH ¼ h1nH ðeE1 þ mHg1Þ;

@

@�1neTe ¼ h1neð�eE1 þ meg1Þ:

[19] The electron and ion temperature profiles are shown inFigure 2d. Note that hydrogen and electron temperaturesexhibit monotonic growth similar to temperature profilesused, e.g., from Chaston et al. [2004]. It should be noted,however, that some data show a much slower electron tem-

perature growth (within 1–5eV) up to altitudes of 8000 km[Kletzing et al., 1998]. Besides, kinetic model of equilibriumof anisotropic plasmas developed by Ergun et al. [2000]predicts a stepwise increase of the electron temperature ataltitudes of several thousand kilometers instead of the smoothgrowth. So, the question of how realistic are the temperatureprofiles obtained above remains open.

2.4. Discussion of the Numerical Scheme

[20] The continuity equation for ions (5) is rewritten forthe density logarithm ln(na) and is solved using a semi‐Lagrangian method described by, e.g., Staniforth and Cote[1991]. The advection is described by a three‐time‐levelscheme, i.e., the values of density in the nodes of the grid attime tn+1 are calculated using the nodal density values attime tn−1 and velocity values at time tn, superscripts n andn ± 1 denote time tn and tn±1 = tn ± Dt when a variablewith the superscript was calculated (Dt is the time step).The 2‐D‐interpolation of the ion density in space is per-formed using cubic Hermite splines with limiters asdescribed by Carlson and Fritsch [1985], which preservesmonotonicity of the density profile. The use of the logarithmof the density ensures its positivity.[21] The three‐time‐level advection provides the advanced

densities nan+1 before finding the advanced transverse ion

velocities ua,2n+1.With known na

n+1, equations for the transverseelectromagnetic field (1) and transverse ion dynamics (2) inimplicit finite difference form produce a three‐diagonal sys-tem of linear equations with respect to the updated wavemagnetic field B3

n+1. After the system is solved, the advancedtransverse electric field E2

n+1 and ion velocity ua,2n+1 are readily

calculated.[22] The advanced parallel current J1

n+1 is calculated fromthe finite difference equivalent of (7) when the updatedtransverse current J2

n+1 =P

�¼O;Hqana

n+1ua,2n+1 is found. With

known J1n+1 and na

n+1, the updated parallel quasineutralelectric field En+1 is found from the implicit finite differenceform of (8). Finally, the parallel flow velocities are updatedusing the implicit finite difference form of (3) and (4).[23] The implicit quasineutral semi‐Lagrangian algorithm

described above can be relatively easily transferred to non-uniform grids. No sign of numerical instability appears insimulations covering 4minutes of themagnetospheric plasmaevolution. The semi‐Lagrangian advection implemented inthe present model, however, is not conservative and showsnoticeable numerical diffusion, which affects position ofthe front and reduces the amplitude of nonlinear waves[Priestley, 1993]. Moreover, even though advection itselfdoes not introduce spurious maxima or minima into thedensity profiles of the plasma species, the parallel electriccurrent calculated as a difference between the electron andion fluxes may contain spurious oscillations.[24] In order to avoid nonphysical effects introduced by

the numerical method, simulations are carried out with dif-ferent spatial resolution (grid cell size) and the convergencebetween the results of these simulations is monitored. Asimulation is stopped once it develops oscillatory structureswith the spatial scale proportional to the size of the grid cell.Usually this happens soon after the beginning of the non-linear stage, as described in section 3. Furthermore, theconclusions made on the basis of simulations with the 2‐D

Figure 2. Initial profiles of (a) ion number density, (b) geo-magnetic field, (c) Alfvén speed, (d) temperature, and (e) par-allel electric field. In Figure 2a, curves 1 and 2 are for oxygenand hydrogen, respectively, markers show values obtainedwith the online International Reference Ionosphere model(available at http://omniweb.gsfc.nasa.gov/vitmo/iri_vitmo.html). In Figure 2d, curves 1, 2, and 3 are for oxygen, hydro-gen, and electrons, respectively. The ionospheric boundary isat altitude of 400 km.

SYDORENKO ET AL.: EXCITATION AND STEEPENING OF ION‐ACOUSTIC WAVES A11212A11212

4 of 12

IAR model are verified in section 4 using a one‐dimensionalshock‐capturing model.

2.5. Simulation Parameters

[25] The simulation area is defined by positions of threecorner points: the southern ionospheric corner (C in Figure 1)is at latitude 69.68° and altitude 400 km, the northern iono-spheric corner (B in Figure 1) is at latitude 70°, the southernhigh‐altitude corner (D in Figure 1) is at latitude 58°. Thewidth of the ionospheric boundary (segment BC in Figure 1)is 39.2 km, the width of the high‐altitude boundary (segmentAD in Figure 1) is 148.7 km, the distance between the mid-points of these boundaries calculated along the geomagneticfield line is 9356 km. The computational grid has 60 cellsin the x2‐direction and from 400 to 1600 cells in the x1‐

direction. In case of 400 cells, the grid cell size in thedirection parallel to the geomagnetic field varies from6944 m at the low‐altitude end to 94544 m at the high‐alti-tude end. The time step Dt = 0.00267s is the maximal valuesatisfying an equivalent of Courant’s condition for Alfvénwaves Dt < h1Dx1/VA,eff everywhere in the simulation area.In case of 1600 cells, the parallel grid cell size varies from1736 m to 23635 m, the time step is Dt = 0.00067s.

3. Nonlinear Ion‐Acoustic Waves in QuasineutralLarge‐Scale Simulations of MagnetosphericPlasmas

3.1. Formation of Density Perturbation in the IAR

[26] A simulation is performed where a downward prop-agating Alfvén wave packet enters the system through thehigh‐altitude open‐end boundary. The wave packet consistsof 20 wave periods with the frequency 0.515 Hz corre-sponding to the first IAR harmonic. The wave packetenvelope is similar to the one used by Sydorenko et al.[2008], with a maximal transverse electric field amplitudeE2+ = 0.2V/m, as shown in Figure 3a.[27] The transverse profile of the wave packet is Gaussian,

with a maximum in the middle of the high‐altitude boundary.Note, since the density perturbations of interest are thestrongest where the driving Alfvén wave is the strongest, thepresent paper considers only profiles along the middle geo-magnetic field line of the simulation domain. Also, startingfrom Figure 3, distance L1 along the geomagnetic field linecalculated upward from the ionospheric end is used as theposition coordinate for the profiles unless otherwise stated,as from Sydorenko et al. [2008]. The reason for this choice isthat such distance is required for calculation of the speed ofwave propagation along the geomagnetic field.[28] The wave packet excites the first harmonic of the IAR.

The nonlinear Lorentz (ponderomotive) force of the standingwave (see the profile in Figure 3b) creates plasma flowsconverging toward a point with L1 ≈ 500 km and divergingaway from a point with L1 ≈ 1250 km (see Figure 3c). Afterjust 40 seconds, these flows produce significant large‐scalemodification of the plasma density, as shown in Figure 3d.Sydorenko et al. [2008] suggested that such process may beresponsible for formation of density cavities in the low‐altitude magnetosphere. The plasma from Sydorenko et al.[2008] was cold. The present model accounts for thermaleffects and reveals a much more complex plasma dynamics.[29] The aforementioned density modification increases

the gradient of the electron pressure at L1 ≈ 550 km, whichresults in the local increase of the parallel electric field, inagreement with equation (8). This electric field perturbationgrows with time and acquires the form of a negative (directedupward) spike of the parallel electric field propagatingupward (see evolution of the spike with label S (“slow”) inFigure 4).[30] The nonlinear Lorentz force of the Alfvén wave

packet does not depend on the ion mass, while the force ofthe parallel electric field does. Since hydrogen ions are muchlighter than the oxygen ones, there is a significant differencebetween the motion of the two ion species when the Alfvénwave decays (compare velocity profiles for hydrogen andoxygen ions in Figure 5a). As a result, modification of the

Figure 3. Formation of density perturbation when the firstIAR harmonic is excited. (a) Electric field of the incomingwave versus time in the middle of the magnetosphericboundary. Profiles of (b) average acceleration due to thenonlinear force, (c) ion velocity along the geomagneticfield, and (d) relative plasma density perturbation. Positivevalues in Figures 3b and 3c are directed downward, towardthe ionosphere L1 = 0. Averaging in Figure 3b is per-formed over the wave period starting at t = 18.4s. Ionvelocity in Figure 3c and relative density perturbation in Fig-ure 3d are obtained at t = 39.6s. In Figure 3c, the hydrogenand the oxygen curves have labels H and O, respectively.The relative density perturbation in Figure 3d is calculatedas ne(t)/ne(0) − 1, where ne(t) is the plasma density at time tand ne(0) is the initial plasma density. The amplitude of thedriving Alfvén wave packet is E2

+ = 0.2V/m. The numericalgrid has 400 cells in the parallel direction.

SYDORENKO ET AL.: EXCITATION AND STEEPENING OF ION‐ACOUSTIC WAVES A11212A11212

5 of 12

hydrogen ion density profile is different from that of theoxygen ions (compare profiles in Figure 5b). In particular,by t ≈ 80s, hydrogen ion flows converging toward L1 ≈1000 km amplify the hydrogen ion density gradient (and,due to quasineutrality, the electron density gradient) at L1 ≈1150 km and produce another electric field perturbation(labeled F (“fast”) in Figures 4 and 5c). This perturbationpropagates upward at a much higher speed then the otherone.[31] Profiles of the aforementioned wave structures ob-

tained in simulations with different spatial resolution in theparallel direction are very close to each other during thefirst 100 seconds, as it is shown in Figure 6a. After thistime, the value of the parallel electric field maximum inspikes S and F grows faster in a simulation with a finergrid. This is a reasonable behavior because limitation ofthe gradient by a coarse grid is an unwanted numericaleffect. However, by t ≈ 120s, noticeable oscillatory wakesappear behind both spikes (oscillations following spike Fare seen in Figure 4d). The wavelength of these oscilla-tions is about 8 grid cells regardless of what the cell size is

(see Figure 6b), which allows to conclude that they are ofnumerical rather than physical origin. Because of this, thesimulation is stopped at t = 120s.

3.2. Dispersion Equation of Ion‐Acoustic Waves

[32] The nature of perturbations described above becomesclear if one compares their dispersion with the dispersion ofion‐acoustic waves. The ion‐acoustic dispersion equationcan be obtained as a result of a standard procedure, where aharmonic perturbation is substituted into linearized equa-tions of motion (3) and (4), ion continuity equations (5), andequation for the quasineutral parallel electric field (8). It isfurther assumed that (1) the quasineutrality (6) holds forelectron and ion density perturbations, (2) there is no motionin transverse directions, (3) contributions of curvature andtemperature gradients can be neglected. It is important toaccount for the flows of electrons and ions. Omitting thelengthy but straightforward algebra, the dispersion relationof ion‐acoustic waves is

� vph þ 2ue;1 ¼X

�¼O;H

n�me

m�ne

2ue;1 þ vph v2ph þ u�;1vph � u2e;1 � u2�;1 � 1 vphðvph � u�;1Þ � v2�;th

; ð10Þ

where vph is the wave phase velocity in the laboratory frame,ue,1, uH,1, and uO,1 are the electron, hydrogen and oxygenflow velocities in the laboratory frame, vH,th

2 = TH/mH andFigure 4. Temporal evolution of the parallel electric fieldprofile in the simulation with the amplitude of the drivingAlfvén wave packet E2

+ = 0.2V/m. (a–d) Values obtainedat times ta = 81.5s, tb = 91.2s, tc = 100.9s, and td =110.6s. The oxygen (slow) ion‐acoustic wave structurehas label S. The hydrogen (fast) ion‐acoustic wave structurehas label F. Wave structures S and F correspond to the wavestructures with the same labels in Figure 4a of Figure 8.Positive field values are directed downward, toward theionosphere at L1 = 0. The numerical grid has 1600 cells inthe parallel direction.

Figure 5. Difference between formation of the slow andthe fast wave. Profiles of the (a) parallel ion flow velocity,(b) relative ion density perturbation, and (c) parallel electricfield. In Figures 5a and 5b, the hydrogen and the oxygencurves have labels H and O, respectively. In Figure 5c,the slow and the fast ion‐acoustic wave structures have la-bels S and F, respectively. Figures 5a–5c are obtained attime t = 91.2s. The amplitude of the driving Alfvén wavepacket is E2

+ = 0.2V/m. The numerical grid has 1600 cellsin the parallel direction.

SYDORENKO ET AL.: EXCITATION AND STEEPENING OF ION‐ACOUSTIC WAVES A11212A11212

6 of 12

vO,th2 = TO/mO are thermal speeds of hydrogen and oxygenions, the aforementioned velocities are normalized by theelectron thermal velocity ve,th = (Te/me)

1/2. In general,equation (10) has 5 roots. In the absence of the flows, atrivial root vph = 0 corresponds to a stationary state. Inaddition, two pairs of symmetric (positive and negative)roots describe slow and fast ion‐acoustic waves propagatingin both directions [Fried et al., 1971;Kozlov and McKinstrie,2002]. Below, the roots of (10) are referred to usingascending order, starting with the largest in magnitude neg-ative root.[33] Fast plasma flows may significantly modify disper-

sion of ion‐acoustic waves if the velocities of these flows(Figure 7d) are comparable with local thermal speeds(Figure 7e). In Figure 7a, profiles of phase velocities aregiven for the initial equilibrium state. With time, profiles ofhydrogen and oxygen ion number densities become stronglyperturbed (see Figure 7c) and rapid flows develop (seeFigure 7d) due to both the nonlinear Lorentz force of theAlfvén wave and the aforementioned waves with intenseparallel electric fields. As a result, local values of the phasevelocities of ion‐acoustic waves deviate strongly from theirinitial values (compare Figure 7b with Figure 7a).[34] The dispersion of waves obtained in the simulation

can be checked by comparing the wave propagation veloc-ities with theoretical values given by equation (10). Notethat the roots of (10) change constantly in space and time. InFigure 8a, the electric field along the middle geomagneticfield line is plotted as a function of coordinate L1 and time t.Two wave structures selected for comparison are labeled Sand F. In Figure 8b, trajectories of two test particles areplotted in the L1 − t phase plane. Test particle F propagateswith the local phase velocity of the upward propagating fastion‐acoustic wave [the first root of the dispersion equation(10)]. Test particle S propagates with the local phasevelocity of the upward propagating slow ion‐acoustic wave[the second root of the dispersion equation (10)]. There is anexcellent agreement between the trajectories of the testparticles and the wave structures. Therefore, the wavestructures observed in the simulation and shown in Figure 4can be conclusively identified as the ion‐acoustic waves.

3.3. Nonlinear Wave Steepening

[35] A detailed evolution of the parallel electric fieldspike, which has label S in Figure 4 and Figure 8a, is shownin Figure 9. The spike moves upward with the velocity ofthe slow ion‐acoustic wave. Below, this structure is referredto as an oxygen shock wave (the term “shock” is usedbecause of the reasons described below). The oxygen shockwave is a compressional perturbation of the oxygen iondensity, that is the density behind the shock is higher thanthe density in front of the shock. With time, the parallelspatial scale of the density perturbation decreases (seeFigure 9a), which increases the density gradient and theparallel electric field (see Figure 9b). This process resemblesformation of a shock wave due to the nonlinear steepeningFigure 6. Parallel electric field profiles in magnetospheric

simulations with E2+ = 0.2V/m, where the numerical grid has

400 (curves 1), 800 (curves 2), and 1600 (curves 3) cells inthe parallel direction. (a) Profiles are obtained at t = 100.9s.(b) Profiles are obtained at t = 110.6s. Markers in Figure 6brepresent electric field in the nodes of the grid.

Figure 7. Effect of electron and ion flows and densitymodification on the dispersion of ion‐acoustic waves. (a)Initial profiles of ion‐acoustic phase velocities (roots ofequation (10)). Final profiles of (b) phase velocities, (c)densities, (d) flow velocities obtained at time t = 108.6s.(e) Profiles of thermal speed. In Figures 7c–7e, curves forelectrons, hydrogen ions, and oxygen ions have labels e, H,and O, respectively. In Figures 7a, 7b, and 7d, positivevelocity values are directed downward, toward the iono-sphere at L1 = 0. The amplitude of the driving Alfvén wavepacket is E2

+ = 0.2V/m. The numerical grid has 1600 cells inthe parallel direction.

SYDORENKO ET AL.: EXCITATION AND STEEPENING OF ION‐ACOUSTIC WAVES A11212A11212

7 of 12

of a sound wave in gas dynamics [Landau and Lifshitz,1987]. Note that in front of the oxygen shock wave, theoxygen ion flow is directed toward the wave, as shown inFigure 9c. Therefore, in the frame moving with the oxygenshock wave, the oxygen ion flow before the shock issupersonic, while behind the wave the flow is subsonic. Thisis a characteristic property of a shock wave [Landau andLifshitz, 1987].[36] Consider another wave structure, which has label F in

Figure 4 and Figure 8a. This is a compressional perturbationof hydrogen ions (see Figure 10a). It propagates with thephase velocity of the fast ion‐acoustic wave. Below, thisstructure is referred to as the hydrogen shock wave. Detailedevolution of the hydrogen shock wave is similar to the evo-lution of the oxygen one (compare Figure 10 with Figure 9).The difference is that it is the hydrogen ion flow which issupersonic before and subsonic after the hydrogen shockwave (see Figure 10c).[37] Due to the presence of dynamically changing flows,

at some locations and times the roots of the dispersionequation (10) for the ion‐acoustic waves may becomecomplex. However, no correlation between the complex

roots and the transformation of initially smooth densityperturbations into steep shock waves was found. Also, therelative drift of electrons and ions is much slower than theelectron thermal speed. In fact, in the absence of transversecurrents, the quasineutrality condition ensures that the totalelectron current is zero. Thus, formation of the shock wavesin the present model is not related to the ion‐ion two‐streaminstability [Wahlund et al., 1992] and the current driveninstability [Quon and Wong, 1976; Sato and Okuda, 1981;Foster et al., 1988; Rietveld et al., 1991].[38] Since ion‐acoustic waves in the simulation exhibit

nonlinear behavior, it is instructive to investigate the effectof the wave amplitude. Test runs with different values of theamplitude of the driving Alfvén wave reveal that, on onehand, the speed of wave propagation weakly depends on thewave amplitude. On the other hand, if the wave packetamplitude is below some threshold value (0.1V/m for theparameters of the present simulation), shock waves do notdevelop during 240 seconds of system evolution, as shownin Figures 11a (E2

+ = 0.05V/m) and 11b (E2+ = 0.1V/m).

Simulations with E2+ = 0.12V/m (Figure 11c) and E2

+ =0.14V/m (Figure 11d), where the shock waves are observed,are stopped earlier than at 240 seconds, once the oscillatorywakes appear.[39] The nonlinear wave steepening may not occur

because of the Landau damping if the electron and iontemperatures are of the same order, which is usually the casein the real magnetosphere for altitudes below 2000 km.

Figure 8. Verification of dispersion of ion‐acoustic wavepackets observed in simulation with the driving Alfvénwave packet’s amplitude E2

+ = 0.2V/m and the numericalgrid with 1600 cells in the parallel direction. (a) The colormap represents evolution of the parallel electric field profilein time (the vertical axis). Perturbations S and F are the slowand the fast ion‐acoustic wave structures labeled S and F inFigure 4. (b) Trajectories of test particles in “coordinate‐time” phase space. The velocities of test particles S and Fare equal to the local phase velocities of the slow and fastion‐acoustic wave propagating upward, respectively. In bothFigures 8a and 8b, the ionospheric boundary is at L1 = 0.

Figure 9. Steepening of a slow ion‐acoustic wave. Profilesof (a) oxygen ion density, (b) parallel electric field, and (c)parallel oxygen ion flow velocity in the laboratory frame.Curves 1, 2, 3, and 4 are obtained at times t1 = 97s, t2 =102.8s, t3 = 108.6s, and t4 = 114.4s, respectively. In Figures9b and 9c, negative values are directed upward, away fromthe ionosphere at L1 = 0. Markers denote values in the nodesof the numerical grid. The driving Alfvén wave packet’samplitude is E2

+ = 0.2V/m, the grid has 1600 cells in the par-allel direction.

SYDORENKO ET AL.: EXCITATION AND STEEPENING OF ION‐ACOUSTIC WAVES A11212A11212

8 of 12

Simulations shown in the present paper are carried out withthe first IAR harmonic, which excites the density perturba-tions at a relatively low altitude, about 1000 km. The firstharmonic is chosen because, first, it produces a convergingplasma flow in one location only, and, therefore, only twoshock waves appear which do not overlap and are easy totrace. Second, the numerical grid provides a much betterresolution at lower altitudes. If a higher IAR harmonic isexcited, it will produce converging plasma flows in multiplelocations, including the altitude range from 2000 km toabout 4000 km (the top IAR boundary). In the presentmodel, the difference between shocks excited at lower andhigher altitudes will be mostly quantitative. Nevertheless, itis important to keep in mind that IAR harmonics with highernumbers are better candidates to produce density perturba-tions which may evolve into nonlinear shock waves.

4. Nonlinear Ion‐Acoustic Waves inOne‐Dimensional Simulations Resolvingthe Electron Debye Length

[40] In the simulation described in the previous section,the parallel grid size is of the order of 2–3 km at altitudesof 1–2 thousand kilometers, which exceeds the local valueof the electron Debye length (0.1 m–1 m) by several ordersof magnitude. On one hand, such coarse resolution is suffi-cient to describe large‐scale dynamics of the magnetosphericplasma, on the other hand, it limits the parallel electric fieldin the shocks significantly. Moreover, the numerical scheme

of the 2‐D IAR model is not the best for description of shockwaves, as discussed above.[41] A correct description of the process of nonlinear

ion‐acoustic wave steepening must (1) resolve the Debyelength, (2) calculate the electrostatic potential from thePoisson equation, and (3) describe sharp discontinuitieswithout introducing spurious oscillations. In order to dem-onstrate the nonlinear evolution of an ion‐acoustic waveunobscured by the limitations of the large‐scale quasineutralmodel, a simple one‐dimensional (1‐D) multifluid numericalmodel is applied, as described below.[42] The model considers a uniform 1‐D plasma consist-

ing of electrons, hydrogen and oxygen ions. The plasma isperiodical with the period length H. The model includesequations for the density, momentum, and energy for thethree species in the conservative form (a = e, O, H):

@n�@t

þ @

@xn�v�ð Þ ¼ 0; ð11Þ

@n�v�@t

þ @n�v2�@x

¼ � n�q�m�

@F@x

� 1

m�

@

@xn�T�ð Þ; ð12Þ

@n�T�@t

þ @n�T�v�@x

¼ � 2

3n�T�

@v�@x

; ð13ÞFigure 10. Steepening of a fast ion‐acoustic wave. Profilesof (a) hydrogen ion density, (b) parallel electric field, and (c)parallel hydrogen ion flow velocity in the laboratory frame.Curves 1, 2, 3, and 4 are obtained at times t1 = 97s, t2 =100.9s, t3 = 104.8s, and t4 = 108.6s, respectively. In Figures10b and 10c, negative values are directed upward, awayfrom the ionosphere at L1 = 0. Markers denote values inthe nodes of the numerical grid. The driving Alfvén wavepacket’s amplitude is E2

+ = 0.2V/m, the grid has 1600 cellsin the parallel direction.

Figure 11. Final parallel electric field profiles for differentvalues of the amplitude of the driving Alfvén wave packet.(a) Values obtained at time ta = 237.5s with the wave ampli-tude E2,a

+ = 0.05V/m. (b) Values obtained at tb = 237.5s withE2,b+ = 0.1V/m. (c) Values obtained at tc = 201.9s with E2,b

+ =0.12V/m. (d) Values at td = 166.3s with E2,c

+ = 0.14V/m.Positive field values are directed downward, toward the ion-osphere at L1 = 0. The numerical grid has 400 cells in theparallel direction.

SYDORENKO ET AL.: EXCITATION AND STEEPENING OF ION‐ACOUSTIC WAVES A11212A11212

9 of 12

and the Poisson equation

@2F@x2

¼ � 1

"0

X�¼e;O;H

q�n�;

where x is the spatial coordinate, v is the velocity, F is theelectrostatic potential, the rest of the variables is the same asin the 2‐D model described in section 2. Equation (13)corresponds to adiabatic compression of monatomic idealgas with g = 5/3. Equations (11)–(13) are solved using aflux‐corrected transport (FCT) algorithm proposed by Borisand Book [1973], which is a proven method to solve shockproblems. The numerical scheme considers the density,momentum, velocity, temperature, and potential on the samegrid and at the same moments of time. As a result, timeadvancing includes iterations. The numerical scheme isdescribed and studied in section II.A of Rambo and Denavit[1991]. This algorithm is stable and the iterations converge

when wpeDt < 1 and Dx >Dt(gTe/me)1/2. Unfortunately, this

model is numerically very expensive and cannot be appliedto large simulation domains stretching for millions of Debyelengths.[43] A test simulation is carried out with the following

initial plasma parameters: ne = 3.09 × 109 m−3, nH = 3.9 ×108 m−3, nO = 2.7 × 109 m−3, Te = 5.6eV, TH = 1.2eV, TO =0.2eV. These values are close to the parameters of plasma inthe magnetospheric model at L1 ≈ 1400 km, where the fastion‐acoustic wave (the one with label F in Figure 4 andFigure 8) becomes significant. The 1‐D system of lengthH = 316.5 m is divided into 32000 cells of sizeDx = lDe/32 ≈ 9.89 × 10−3 m, where lDe = ("0Te/nee

2)1/2 is the electronDebye length. The time step is Dt = wpe

−1/(21/2 32) ≈ 7.046 ×10−9s, where wpe

2 = nee2/"0me is the electron plasma fre-

quency. In order to initiate ion‐acoustic waves, all plasmacomponents have an initial velocity perturbation as shown inFigure 12a.[44] The initial velocity perturbation compresses plasma

in the middle of the system, like the nonlinear Lorentz forcedoes in the IAR. The compressed area emits two pairs ofion‐acoustic waves propagating symmetrically rightwardand leftward: slow waves associated with the compressionof oxygen (labeled S+ and S− in Figures 12b and 12d), andfast waves associated with the compression of hydrogen(labeled F+ and F− in Figures 12c and 12d). These waves aresimilar to perturbations S and F in the simulation of themagnetospheric plasma (see Figures 4 and 8a). The wavessteepen and become oxygen (slow) and hydrogen (fast)shocks.[45] Initially, the electric field in the shocks has a shape of

a unipolar spike, as in a double layer. The electric field isdirected along the direction of the shock propagation. Thedouble layer pattern, however, does not hold for long. Soonoscillatory trails start growing behind the shock fronts andthe amplitude of the spikes gradually decays. The maximalelectric field in the oxygen shock S+, registered while thedouble‐layer pattern is dominant, is 0.75V/m at time t =0.0123s, the width of the double‐layer electric fieldspike (calculated at the half of spike’s amplitude) is about1.5 meters or 5lDe. The hydrogen shock F+ is weaker andnarrower, with the maximal electric field of about 0.42V/mat time t = 0.0031s and the width of 0.7 meters or 2.5lDe.There are several reasons to believe that oscillatory wakesin the 1‐D model are physical. First, simulations withdifferent grid resolution demonstrate clear convergence ifthe grid cell size is Dx < lDe/16 (see Figure 13). Second,similar oscillatory wakes are reported both in experiments[e.g., Taylor et al., 1970] and simulations [e.g., White et al.,1974; Kozlov and McKinstrie, 2002]. The reason for thesewakes is the dispersion of ion‐acoustic waves. Eventually, aunipolar spike transforms into awave packet with asymmetricenvelope, in qualitative agreement with Hirose et al. [1978].[46] It is interesting that the nonlinear waves in the

magnetospheric simulation exhibit similar behavior at muchlower parallel electric fields due to excessive numericaldiffusion of the advection algorithm.[47] The double layers observed in the simulations above

appear as a transitional phenomenon only and do notbecome stationary, which does not contradict availabletheories. Verheest [1989] concluded that an ion‐acousticdouble layer in a plasma with multiple ion species is

Figure 12. Simulation of nonlinear ion‐acoustic waves withPoisson equation. (a) Profile of initial perturbation of flowvelocity. Profiles of (b) oxygen ion density, (c) hydrogenion density, (d) electric field, and (e) electrostatic potential.Labels S± and F± denote oxygen (slow) and hydrogen (fast)ion‐acoustic waves propagating in the positive and negativex‐directions, respectively. Figures 12b–12e are obtained attime t = 0.00648s. The numerical grid has 32000 cells.

SYDORENKO ET AL.: EXCITATION AND STEEPENING OF ION‐ACOUSTIC WAVES A11212A11212

10 of 12

impossible if all ion species are positive and there is onlyone electron population, which exactly matches the prop-erties of the models discussed in the present paper. In orderto obtain the double layer as a stationary propagatingstructure, one must either include both hot and cold electronpopulations [Reddy and Lakhina, 1991] or negativelycharged ions [Mishra et al., 2002].[48] Finally, the drop of the electrostatic potential across

the whole area of an oxygen or a hydrogen shock wave isnegligible both when the shock looks like a double layer(see the potential profile in the oxygen shocks S± in Figure12f) and when it looks like a short wave packet (see thepotential profile in the hydrogen shocks F± in Figure 12f).Therefore, the ion‐acoustic shock waves are not effective forauroral particle acceleration.

5. Summary

[49] The nonlinear Lorentz force of a standing Alfvénwave in the ionospheric Alfvén resonator (IAR) compressesand rarefies the plasma along the geomagnetic field. Withfinite electron temperature, the density modification am-plifies the parallel electric field in proportion to the electronpressure gradient. Ions with different mass respond differ-ently to this electric field, which produces spatially sepa-rated local enhancements of light (hydrogen) and heavy(oxygen) ions. These perturbations propagate as ion‐acoustic waves, and their amplitude grows due to the pro-cess of nonlinear wave steepening similar to shock tubes.[50] An oxygen (hydrogen) density perturbation evolves

into an oxygen (hydrogen) compressional shock wave: theoxygen (hydrogen) ion density is bigger behind the shockand it changes in a jump‐like manner across the shock front,the shock propagates upward with the speed of a slow (fast)ion‐acoustic wave. In the shock’s frame, the incomingoxygen (hydrogen) flow in front of the shock is supersonic,behind the shock it is subsonic.[51] Initially, the ion‐acoustic shock waves in the IAR

were obtained in numerical simulation with a 2‐D multifluidmodel of low‐altitude auroral flux tubes. This model de-

scribes large‐scale motion of near‐Earth plasma on a coarsegrid using an algorithm which is not the best for descriptionof sharp discontinuities. Simulations with the 2‐D modelhave to be stopped at an early stage of the nonlinearsteepening process in order to prevent the appearance ofnumerical artifacts. Therefore, a 1‐D Poisson shock‐cap-turing multifluid code was applied to simulate a relativelyshort (1000lDe) plasma system with fine resolution (lDe/32)and provide a detailed picture of evolution of the shockwaves.[52] The 1‐D Poisson simulation confirms the formation

of hydrogen and oxygen ion‐acoustic shock waves from adensity perturbation caused by converging plasma flows. Itis found that a shock wave starts as a double layer, with aunidirectional electric field. At this stage, the maximalelectric field for the selected plasma parameters is about0.4–0.7V/m and the width of the shock is 2–5lDe. Thedouble layer, however, decays into an intense ion‐acousticwave packet with an asymmetric, rapidly increasing andgradually decreasing envelope. The shocks appeared to benot suitable for particle acceleration neither when they looklike double layers, nor when they transform into the wavepackets.[53] Finally, a perturbation growing into an ion‐acoustic

shock wave may be created not only by a standing Alfvénwave in the IAR, but by other mechanisms. There are sat-ellite observations of intense ion‐acoustic wave packets inthe plasma sheet, well outside the IAR area [Cattell et al.,1998]. The nonlinear steepening will occur if, first, theperturbation is sufficiently strong. In the IAR model, forexample, shock waves form when the amplitude of thetransverse electric field of the driving Alfvén wave packet isno less than 0.12V/m, which is a high but not an unrealisticvalue. The second requirement is that the electron temper-ature must be much higher than the ion temperature, whichprobably limits application of this mechanism to altitudesabove 2000 km. Inside the IAR, density perturbations atthese altitudes may be produced by the resonator harmonicswith higher numbers.

[54] Acknowledgments. The present study was supported by theCanadian Space Agency (CSA) and the National Sciences and EngineeringResearch Council of Canada (NSERC).[55] Robert Lysak thanks Joachim Vogt and another reviewer for their

assistance in evaluating this paper.

ReferencesAndersen, H. K., N. D’Angelo, P. Michelsen, and P. Nielsen (1967), Inves-tigation of Landau‐damping effects on shock formation, Phys. Rev. Lett.,19, 149–151.

Boris, J. P., and D. L. Book (1973), Flux‐corrected transport. I. SHASTA, afluid transport algorithm that works, J. Comput. Phys., 11, 38–69.

Bostrom, R., G. Gustaffson, B. Holback, G. Holmgren, and H. Koskinen(1988), Characteristics of solitary waves and weak double layers in themagnetospheric plasma, Phys. Rev. Lett., 61, 82–85.

Carlson, R. E., and F. N. Fritsch (1985), Monotone piecewise bicubic inter-polation, SIAM J. Numer. Anal., 22, 386–400.

Cattell, C., J. Wygant, J. Dombeck, F. S. Mozer, M. Temerin, and C. T.Russell (1998), Observations of large amplitude parallel electric fieldwave packets at the plasma sheet boundary, Geophys. Res. Lett., 25,857–860.

Chaston, C. C., J. W. Bonnell, C. W. Carlson, J. P. McFadden, R. E. Ergun,R. J. Strangeway, and E. J. Lund (2004), Auroral ion acceleration in dis-persive Alfvén waves, J. Geophys. Res., 109, A04205, doi:10.1029/2003JA010053.

Figure 13. Convergence of Poisson simulations with dif-ferent grid cell size. Detailed pictures of the electric fieldin the rightward‐going (a) slow wave and (b) fast wave.These waves are labeled S+ and F+ in Figure 12, respec-tively. Curves 1, 2, 3, and 4 are obtained in simulations wherethe numerical grid has 4000, 8000, 16000, and 32000 cells,respectively, at time t = 0.00648s.

SYDORENKO ET AL.: EXCITATION AND STEEPENING OF ION‐ACOUSTIC WAVES A11212A11212

11 of 12

Dombeck, J., C. Cattell, J. Crumley, W. K. Peterson, H. L. Collin, and C.Kletzing (2001), Observed trends in auroral zone ion mode solitary wavestructure characteristics using data from Polar, J. Geophys. Res., 106,19,013–19,021.

Dovner, P. O., A. I. Eriksson, R. Bostrom, and B. Holback (1994), Frejamultiprobe observations of electrostatic solitary structures, Geophys.Res. Lett., 21, 1827–1830.

Ergun, R. E., et al. (1998), FAST satellite wave observations of electricfield structures in the auroral zone, Geophys. Res. Lett., 25, 2025–2028.

Ergun, R. E., C.W. Carlson, J. P.MacFadden, F. S.Mozer, and R. J. Strangeway(2000), Parallel electric fields in discrete arcs, Geophys. Res. Lett., 27,4053–4056.

Erkaev, N. V., V. S. Semenov, V. A. Shaidurov, D. Langmayr, H. K. Biernat,and H. O. Rucker (2001), Propagation of nonlinear slow waves producedby pressure pulses along the Io flux tube, Adv. Space Res., 28, 1481–1488.

Foster, J. C., C. del Pozo, K. Groves, and J.‐P. St.‐Maurice (1988),Radar observations of the onset of current driven instabilities in thetopside ionosphere, Geophys. Res. Lett., 15, 160–163, doi:10.1029/GL015i002p00160.

Fried, B. D., R. B. White, and T. K. Samec (1971), Ion acoustic waves in amulti‐ion plasma, Phys. Fluids, 14, 2388–2392.

Hirose, A., O. Ishihara, S. Watanabe, and H. Tanaca (1978), Response ofion acoustic waves to an impulse, Plasma Phys., 20, 1179–1184.

Hudson, M. K., and F. S. Mozer (1978), Electrostatic shocks, double layers,and anomalous resistivity in the magnetosphere, Geophys. Res. Lett., 5,131–134, doi:10.1029/GL005i002p00131.

Kletzing, C. A., F. S. Mozer, and R. B. Torbert (1998), Electron tempera-ture and density at high latitude, J. Geophys. Res., 103, 14,837–14,845.

Kozlov, M. V., and C. J. McKinstrie (2002), Sound waves in two‐ion plas-mas, Phys. Plasmas, 9, 3783–3793.

Landau, L. D., and E. M. Lifshitz (1987), Fluid Mechanics, Course Theor.Phys., 2nd ed., Pergamon Press, New York.

Mishra, M. K., A. K. Arora, and R. S. Chhabra (2002), Ion‐acoustic com-pressive and rarefactive double layers in a warm multicomponent plasmawith negative ions, Phys. Rev. E, 66, 046402.

Montgomery, D. (1967), Evolution of a nonlinear ion acoustic wave, Phys.Rev. Lett., 19, 1465–1466.

Mozer, F. S., C. A. Cattell, M. K. Hudson, R. L. Lysak, M. Temerin, andR. B. Torbert (1980), Satellite measurements and theories of low alti-tude auroral particle acceleration, Space Sci. Rev., 27, 155–213.

Polyakov, S. V., and C. O. Rapoport (1981), Ionospheric alfven resonator,Geomagn. Aeron., 21, 816.

Prakash, M. (1997), Excitation of ion‐acoustic turbulence and its role inauroral processes at Freja altitude, Phys. Chem. Earth, 22, 773–776.

Priestley, A. (1993), A quasi‐conservative version of the semi‐Lagrangianadvection scheme, Mon. Weather Rev., 121, 621–629.

Quon, B. H., and A. Y. Wong (1976), Formation of potential double layersin plasmas, Phys. Rev. Lett., 37, 1393–1396.

Rambo, P. W., and J. Denavit (1991), Fluid and field algorithms for time‐implicit plasma simulation, J. Comput. Phys., 92, 185–212.

Reddy, R. V., and G. S. Lakhina (1991), Ion acoustic double layers andsolitons in auroral plasma, Planet. Space. Sci., 39, 1343–1350.

Rietveld, M. T., P. N. Collis, and J.‐P. St.‐Maurice (1991), Naturallyenhanced ion acoustic waves in the auroral ionosphere observed withthe EISCAT 933‐MHz radar, J. Geophys. Res., 96, 19,291–19,305.

Sato, T., and H. Okuda (1981), Numerical simulations on ion acousticdouble layers, J. Geophys. Res., 86, 3357–3368.

Seyler, C. E., J.‐E. Wahlund, and B. Holback (1995), Theory and simu-lation of low‐frequency plasma waves and comparison to Freja satelliteobservations, J. Geophys. Res., 100, 21,453–21,472, doi:10.1029/95JA02052.

Staniforth, A., and J. Cote (1991), Semi‐Lagrangian integration schemes foratmospheric models—A review, Mon. Weather Rev., 119, 2206–2223.

Sydorenko, D., R. Rankin, and K. Kabin (2008), Nonlinear effects in theionospheric Alfvén resonator, J. Geophys. Res., 113, A10206,doi:10.1029/2008JA013579.

Taylor, R. J., D. R. Baker, and H. Ikezi (1970), Observation of collisionlesselectrostatic shocks, Phys. Rev. Lett., 24, 206–209.

Temerin, M., K. Cerny, W. Lotko, and W. S. Mozer (1982), Observationsof double layers and solitary waves in the auroral plasma, Phys. Rev.Lett., 48, 1175–1179.

Verheest, F. (1989), Ion‐acoustic double layers in multi‐species plasmasmaintained by negative ions, J. Plasma Phys., 42, 395–406.

Wahlund, J.‐E., F. R. E. Forme, H. J. Opgenoorth, M. A. L. Persson, E. V.Mishin, and A. S. Volokitin (1992), Scattering of electromagnetic wavesfrom a plasma: enhanced ion acoustic fluctuations due to ion‐ion two‐stream instabilities, Geophys. Res. Lett., 19, 1919–1922.

Warmuth, A. (2007), Large‐scale waves and shocks in the solar corona,Lect. Notes Phys., 725, 107–138.

White, R. B., B. D. Fried, and F. V. Coroniti (1974), Formation of ion‐acoustic shocks, Phys. Rev. Lett., 17, 211–214.

K. Kabin, Department of Physics, Royal Military College of Canada,PO Box 17000 STN Forces, Kingston, ON K7K7B4, Canada.(konstantin.kabin@rmc.ca)R. Rankin and D. Sydorenko, Department of Physics, University of

Alberta, Room 238 CEB, 11322 ‐ 89 Ave., Edmonton, AB T6G 2G7,Canada. (rrankin@ualberta.ca; sydorenk@ualberta.ca)

SYDORENKO ET AL.: EXCITATION AND STEEPENING OF ION‐ACOUSTIC WAVES A11212A11212

12 of 12