16 Ultra Low FrequencyWaves in the Magnetosphere

Umberto Villante

Geomagnetic pulsations are the ground manifesta-tion of ultra low frequency hydromagnetic wavespropagating in the magnetosphere. Frequencies typi-cally range between f � mHz and f � Hz; groundamplitudes range from less than . nT to tens orhundreds of nT and generally increase with latitudeup to auroral/cusp regions. The distinct periodicityof most events suggests an interpretation in termsof standing waves reflecting between ionospheres ofopposite hemispheres and hydromagnetic resonanceis the basic process to interpret most aspects ofgeomagnetic pulsations.

The Kelvin–Helmholtz instability at the magne-topause is considered an important energy sourcefor continuous low frequency events ( f � – mHz);an additional contribution might come fromcavity/waveguide modes of the magnetosphere.“Upstream waves” generated by particles reflected fromthe bow shock along interplanetary magnetic fieldlines are important exogenic sources for pulsationsin the mid-frequency band ( f � – mHz). Highfrequency pulsations ( f � . – Hz) are travelingwaves related to ion-cyclotron instabilities occurringwithin the magnetosphere. Irregular pulsationsrepresent transient signals associated with dramaticchanges of the state of the magnetosphere, related tosubstorm manifestations.

The identification of field line resonance pro-cesses represents an important tool for severalaspects of magnetospheric diagnostics: a quantitativedetermination of the set of field line eigenfrequenciescan be used to model the plasma distribution alongthe magnetospheric field lines from equatorial tohigh latitudes, to monitor temporal variations of themagnetospheric plasma concentration and to high-light interesting aspects of plasmasphere/ionospherecoupling.

Contents

16.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 398

16.2 Linear Theory of HydromagneticWaves . . . . . . . . . . . 40016.2.1 The Uniform Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40016.2.2 The Dipole Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 401

16.3 Sources of Geomagnetic Pulsations . . . . . . . . . . . . . . . 40216.3.1 UpstreamWaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40216.3.2 Kelvin–Helmholtz Instability . . . . . . . . . . . . . . . . . . . . . 40316.3.3 Ion-Cyclotron Instability . . . . . . . . . . . . . . . . . . . . . . . . 403

16.4 Effects of the Ionosphereand Field Line Eigenperiods . . . . . . . . . . . . . . . . . . . . . . 404

16.5 Field Line Resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405

16.6 Cavity Resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407

16.7 Low Frequency Pulsations . . . . . . . . . . . . . . . . . . . . . . . 407

16.8 Mid-Frequency Pulsations . . . . . . . . . . . . . . . . . . . . . . . 409

16.9 FLR andMagnetospheric Diagnostics . . . . . . . . . . . . . 412

16.10 Cavity/WaveguideModes . . . . . . . . . . . . . . . . . . . . . . . . 414

16.11 High Frequency Pulsations . . . . . . . . . . . . . . . . . . . . . . . 415

16.12 Irregular Pulsations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417

16.13 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419

Umberto Villante, Ultra Low Frequency Waves in the Magnetosphere.In: Y. Kamide/A. Chian, Handbook of the Solar-Terrestrial Environment. pp. – ()DOI: ./_ © Springer-Verlag Berlin Heidelberg

U. Villante

16.1 Introduction

Geomagnetic (or more simply “magnetic”) pulsationsare the ground manifestation of ultra low frequency(ULF) hydromagnetic waves propagating in the mag-netosphere. Originally termed “micropulsations”, theywere first identified by Celsius (who compared compassmeasurements in Uppsala with auroral fluctuations,1741), by Nervander, 1840’s and by Stewart, 1859.More than fifty years ago, Dungey (1967) arguedthat micropulsations could be interpreted in terms ofAlfvén waves excited on geomagnetic field lines. Infact, their distinct periodicity led Dungey to suggestan interpretation of magnetic pulsations in termsof standing waves reflecting between ionospheres ofopposite hemispheres.

The origin ofmagnetic pulsations is in the interplan-etary medium, in the magnetosphere and, possibly, onthe Sun itself. However, waves detected on the groundare not the samewaves that enter themagnetosphere: in-deed, wave energy is transformed by several processes,and ground signals are electromagnetic waves radiatedfrom currents induced in the ionosphere by the imping-ing hydromagneticwaves. The properties of ground pul-sations also depend on the conductivity of the Earth un-derneath the observer.

Magnetic pulsations typically have frequencies be-tween f � mHz and f � Hz, with highest frequen-cies being determined by the hydrogen gyrofrequencyin the magnetosphere ( f � Hz) and lowest frequenciescorresponding to propagation times across the magne-tosphere (Fig. 16.1). Ground amplitudes range from lessthan . nT (at the highest frequencies) to tens or hun-dreds of nT and generally increase with latitude up toauroral/cusp regions.

As for other areas of geophysics and space physics,the International Geophysical Year (1957–58) stim-ulated a great impetus for research on magneticpulsations. By the early 1970’s, well over 5000 papershad been published on this topic. Since then, pul-sations have also been used as an important tool inmagnetospheric dynamics (“geomagnetic storms” and“substorms”), for determining magnetospheric plasmadensity and for diagnostics of important processes suchas “magnetic reconnection”. Geomagnetic pulsations alsorepresent the source field for electromagnetic inductionstudies of the Earth’s crust, mantle and oceans. ULF

Fig. 16.1. The spectrum of natural signals. A power spectrumrepresenting the natural situation on Earth. ULF waves corre-spond to the lowest frequency band. [Lanzerotti et al., 1990]

geomagnetic signals may occasionally be emitted inassociation with earthquake occurrence.

The International Association of Geomagnetismand Aeronomy (IAGA), classified geomagnetic pulsa-tions into two classes, continuous, Pc, and irregular,Pi. Their further separation in period subclasses doesnot reflect any definite physical difference; rather, froma physical point of view, it would be more reasonable todivide Pc into three distinct frequency bands: low fre-quency ( f � – mHz, see Fig. 16.8a for a typical ex-ample; these waves have wavelengths comparable tothe dimensions of the magnetosphere), mid-frequency( f � – mHz, Fig. 16.6d), and high frequency puls-ations ( f � . – Hz, Fig. 16.10a). However, pulsa-tions sharing the same frequency band often presentdifferent characteristics, reflecting their different origin.Table 16.1 summarizes the classification scheme andmajor energy sources (adapted from Samson, 1991).

The occurrence of pulsations and their charac-teristics depend on the conditions of the solar wind(SW) and on the state of the magnetosphere. Changes

Ultra Low Frequency Waves in the Magnetosphere | Introduction

Table 16.1. The IAGA classification scheme (1964)

[Samson, 1991]

T(s) Frequency Sources

Pc . – High: Ion-cyclotron instabilityPc – . – Hz in magnetosphere.

Pc – Mid: Proton-cyclotron – mHz instability in the SW;

Kelvin–Helmholtz instability.

Pc – Low: Kelvin–Helmholtz instability; – mHz Drift-mirror instability;

Pc – Bounce resonance.Pi – Field aligned current

driven instabilities.Pi – Abrupt changes in convection

in the magnetotail;Flux transfer events.

in the orientation of the interplanetary magnetic field(IMF) can have dramatic effects on the characteristicsof waves seen on the Earth. The morphological andphysical properties of pulsations also depend on thegeomagnetic latitude and longitude (or local time)2.

In IAGA added two new classes (Pc � s,and Pi � s) to the classification scheme. Piinclude fluctuations associated with storm suddencommencements (Psc and Psc), and substorms(Pip, – s; Ps, – min; [Saito, ]). Formore detailed discussions on sources and theo-retical aspects, see [Dungey, ; Hughes, ;Southwood and Hughes, ; Samson, ].

The Z-axis of the geomagnetic coordinate system is par-allel to the magnetic dipole axis, the geographic coordi-nates of which are � .� (colatitude) and � −.� (eastlongitude). The geomagnetic latitude (λ) is measured fromthe geomagnetic equator and is positive northward; thegeomagnetic longitude (ϕ) is measured from the merid-ian that contains the south geographic pole and is posi-tive eastward. The relationship between geomagnetic andgeographic longitude is such that the geomagnetic longi-tude is � � greater than the geographic longitude, exceptnear the poles. The magnetic local time (MLT) is definedas the geomagnetic longitude of the observer minus thegeomagnetic longitude of the Sun expressed in hours plus hr. L is the magnetic shell parameter which identifies thegeocentric equatorial distance of a field line, measured inEarth radii (RE). A related parameter is the invariant lati-tude Λ = cos−(�L)�, which is the latitude where a line offorce intersects the Earth’s surface. For example, for a line

As for other geomagnetic studies, different regionsof interest are usually considered in terms of latitude,or L parameter: the equatorial region (L < ., or Λ <� �), the low latitude region (. K L K ; or � � <Λ < � �), the middle latitude region ( K L K ; or� � < Λ < ��, i.e. up to the expected position ofthe plasmapause 3); the high latitude region (from L � out to the last closed field line); the polar cap, with openfield lines which extend into the tail, or connect to IMFlines. High latitudes encompass several important zonessuch as the auroral oval and the cusp 4. The study ofpulsations in Antarctica is very interesting as Antarcticaextends up to latitudes (corresponding to oceans in thenorthern hemisphere) where local field lines penetrateextreme magnetospheric regions where several gener-ation mechanisms are active (Arnoldy et al., 1988). Atthose latitudes Antarctica also allows geomagnetic mea-surements in a wide longitudinal range.

The classification scheme in terms of L is alsoadopted for magnetospheric studies; in this case Lidentifies the line of force whichmaps to a given groundlatitude: L < and L � usually identify the “inner” and“outer” magnetosphere, respectively. However, sincethe magnetosphere is a highly dynamic system, thestate of the magnetosphere also partially determines themagnetic projection of different regions. Major factorswhich control magnetospheric dynamics are the SWdynamic pressure (ρV

sw, ρ and Vsw being the SW massdensity and flow velocity) and the rate of transport

that extends up to RE in the equatorial plane,Λ is � .�.At high latitudes where field lines are open or non dipolar,L values become meaningless.

The position of the plasmapause (the outer boundary of theplasma population corotating with the Earth) depends onlocal time (and other factors such as geomagnetic activity)and varies roughly between L � (in the dawn sector) andL � (in the dusk sector). Typically the electron densitydrops off by two orders of magnitude across the plasma-pause.

The auroral oval (λ � �–�), more extended equator-ward on the nightside, maps on closed field lines into theplasmasheet.The cusp is a funnel-shaped region separatingclosed field lines extending sunward from those extendingtailward. The dayside cusp is highly confined in latitude;however, its position is dependent on IMF conditions (Λ ��–�), moving equatorward during southward IMF ori-entation.Through the cusp, the magnetosheath plasma hasdirect access to the ionosphere.

U. Villante

of the southward magnetic flux (BsVsw, Bs being thesouthward IMF component).

Dungey (1967) also introduced the concept of hy-dromagnetic resonance, a basic process for interpretingmost aspects of magnetic pulsations, and identified theKelvin–Helmholtz instability (KHI) at the Chapman–Ferraro layer (i.e. the magnetopause ) as an importantenergy source for low frequency events. In addition,“upstream waves” generated by particles reflected fromthe bow shock along IMF lines are considered impor-tant exogenic sources for daytime pulsations in themid-frequency band. High frequency Pc are thoughtto be generated by ion-cyclotron instabilities occurringwithin the magnetosphere. Pi2 pulsations (Fig. 16.11a)represent transient signals associated with dramaticchanges of the state of the magnetosphere; they canonly be extensively treated in the context of substormmanifestation.

Theoretical and experimental aspects of geomag-netic pulsations have been discussed in several booksand review papers (Dungey, 1967; Saito, 1969, 1978; Ja-cobs, 1970; Lanzerotti and Southwood, 1979; Rostoker,1979; Hughes, 1983; Russell and Hoppe, 1983; South-wood and Hughes, 1983; Odera, 1986; Arnoldy et al.,1988; Samson, 1991; Takahashi, 1991, 1998; Allan andPoulter, 1992; Anderson, 1994; Fazakerley and Russell,1994; Hughes, 1994; Le and Russel, 1994; Engebretson,1995; Kivelson, 1995; Villante and Vellante, 1997; Kan-gas et al., 1998; Olson, 1999; McPherron, 2002); pulsa-tions in the magnetosphere have been reviewed by An-derson (1994); hydromagnetic waves upstream of thebow shock have been discussed in a special issue of theJournal of Geophysical Research (June, 1991), by RussellandHoppe (1983), and by Le andRussell (1994);magne-tosheath waves have been examined by Fazakerley andRussell (1994). For more detailed references about as-pects discussed in the next paragraphs, the reader is re-ferred to these reviews.

Following the classical approach to the physics ofmagnetic pulsations, we will summarize basic elementsof hydromagnetic waves and several aspects of themajorprocesses related to the sources of these oscillations.

16.2 Linear Theory of Hydromagnetic Waves

In a plasma imbedded in a magnetic field, B, hydro-magnetic (or “magnetohydrodynamic”, MHD) waves

arise at low frequencies (i.e. lower than both the plasmafrequency, ωps = (nse�εms)�, and the ion gyrofre-quency, Ωi = eB�mi, where ns and ms are the numberdensity and the mass of particles) as a combined effectof mechanical and electromagnetic forces.

16.2.1 The Uniform Field

Following Dungey’s approach (1967) we assume a uni-form fluid, with a density ρ, in a uniform magneticfield B, and consider small amplitude disturbances inthe electric field, velocity, current density, and magneticfield (e, u, j, and b) that vary like ei(kċr−ωt). In theseconditions, basic hydromagnetic equations are thefluid momentum equation, which in a “cold plasma”(i.e. when the thermal pressure can be neglected withrespect to the magnetic pressure) simplifies to

ρ∂u∂t

= j � B � −iωρu = j � B (16.1)

the hydromagnetic form of Ohm’s law

e = −u � B (16.2)

and Maxwell’s equations

∇� e = −∂b∂t

� ωb = k � e (16.3)

∇� b = μj � ik � b = μj (16.4)

in which displacement currents have been neglectedwith respect to the conduction current. Equation (16.1)shows that j is perpendicular to u; e, u, and B are mu-tually perpendicular (Eqs. (16.1) and (16.2)), and thesame is true for j, k, and b (Eqs. (16.3) and (16.4)). Asa consequence (j, e, k), (u, b, k) and (u, b, B) must becoplanar. These conditions present two cases requiringthat either j and e, or u and b, are parallel to k � B.

In the first case (Fig. 16.2a), previous equations pro-vide a dispersion relation such as

ω = �kVa (16.5)

whereVa = B� (μρ)� (16.6)

is the phase velocity (“Alfvén velocity”) which isindependent of the direction of propagation. The corre-sponding wave mode is identified as “fast” mode. It hasgroup velocity parallel to k, propagates isotropically,and energy can be transported in any direction. The

Ultra Low Frequency Waves in the Magnetosphere | Linear Theory of Hydromagnetic Waves

“fast” mode has j ċ B = , so carries no current alongmagnetic field lines. In general, b has a componentparallel to B; so, this mode, which is analogous to anordinary sound wave in a fluid, can trasmit pressurevariations. When this mode propagates perpendicularto B, it is seen as compression and rarefaction of boththe magnetic field and the plasma density.

In the second case (Fig. 16.2b), considering that uand b are both perpendicular to B, we obtain

ω = �kVa cos ϑ (16.7)

where ϑ is the angle between k and B. This is a trans-verse mode (“shear Alfvén mode”) which only bends the

Fig. 16.2a–c.The characteristics of hydromagnetic wave modes.Top panel. The relative orientation of the vector fields: (a) thefast mode; (b) the shear mode. [Dungey, 1967]. Bottom panel.(c) The different wave modes in a dipole field. S is the Poynt-ing vector: it is parallel to B for the toroidal mode (1) andfor the guided poloidal mode (3); it is across B for the cavitymode (2). [Vellante, 1993a]

field lines. b is perpendicular to B and the magneticfield magnitude is constant (to a linear approximation),even in the presence of a wave: the magnetic pressureis constant and the wave is noncompressive. The groupvelocity is parallel to B and the energy is guided alongthe ambientmagnetic field. In general, geomagnetic pul-sations ultimately originate in the magnetosphere eitheras fast or Alfvén modes, or a combination of these twomodes5.

16.2.2 The Dipole Field

In the inner magnetosphere Va � km�s, and pulsa-tion periods of � s or longer are commonly detected.This leads to estimated wavelengths of � – RE, i.e.comparable with the size of the entire magnetosphere.It suggests that the homogeneous plasma approxima-tion is not appropriate for magnetospheric pulsations.If the field is not uniform, previous equations requireadditional terms related to the spatial derivatives of B.These additional terms couple the Alfvén and fast modeand in general it is impossible to separate two distinctwave modes. Interesting simplifications arise when theaxial symmetry ofB (as for a dipole field) is considered.In cylindrical coordinates (r, ϕ, z) an axial symmetric

magnetic field has Bϕ = and∂Br

∂ϕ= ∂Bz

∂ϕ= . In this

case, assuming a longitudinal variation such as ei(mϕ−ωt)for perturbed quantities, it is easy to derive the followingequations (Dungey, 1967; Hughes, 1983)6:

2 ω

V a

+ rB

∇�� �rB∇���35uϕr

6 = mωr

b��B

(16.8)

It is worth noting that in the absence of “cold plasma” con-ditions, two additional compressionalmodes arise: the “fastmagnetoacoustic wave”, with a phase velocity greater thanVa and Cs (Cs = (γp�ρ)� being the sound speed), and the“slowmagnetoacoustic wave”, with a phase velocity less thanVa and Cs . If ϑ = �, phase velocities are Va and Cs , so thatthe motion is a superposition of a pure Alfvén wave anda pure sound wave, both traveling along B. If ϑ = �, thephase velocity is (V

a + Cs )�, and waves propagate per-

pendicularly to B. A similar situation may occur in themagnetosphere, at geocentric distances of � – RE, whereplasma and magnetic pressure may be comparable (Lanze-rotti and Southwood, ).

Interesting aspects of these oscillations emerge in thedipole coordinate system ((Radoski, ), and papers ref-erenced).

U. Villante

2 ω

V a

+ rB∇�� 5∇��rB

63�reϕ� = −iωBr∇b��B

(16.9)

imuϕr

+ rB

∇reϕ = iωb��B

(16.10)

where ∇�� = eB ċ ∇, and ∇ = (eB � ∇)ϕ , eB being theunit vector along B.

This system has eigenperiods corresponding toeigenfunctions which must satisfy certain boundaryconditions. Previous equations can be decoupled intwo limits. If the wave is axisymmetric (m = , whichmeans that the signals are in phase around an entirecircumference) the right hand side of (16.8) vanishes.The left hand side takes the form of a one-dimensionalwave equation with the only spatial derivative along B.It corresponds to a transverse mode (“toroidal mode”with the characteristics of an Alfvén mode, Fig. 16.2c).It can be interpreted as a torsional oscillation of the linesof force, or magnetic shell, which oscillate azimuthally,independently of all others. The magnetic and velocityperturbations (b, u) are azimuthal and the electricfield (e) is normal to the field lines. Such a wave isguided by the magnetic lines of force. In the absenceof ionospheric effects, this mode would correspondto oscillations of the east–west (D) geomagnetic fieldcomponent. In the same limit (16.9) represents a mode(“poloidalmode”, with the characteristics of a fastmode),in which e is azimuthal, u and b are in the meridianplane and the whole cavity oscillates coherently (“cavitymode”). On the Earth’s surface, in the absence of iono-spheric effects, this “field-aligned” mode would affectthe vertical (Z) and north–south (H) components.

The other limiting case corresponds to a disturbanceconfined to a narrow range of longitude (m � 8, whichmeans that adjacent field lines are highly decoupled andperform independent azimuthal oscillations). In thiscase we obtain (16.9) a transverse wave mode whichpropagates along B (“guided poloidal mode”); e is inthe azimuthal direction, while b and u, perpendicularto the field line, lie in the plane of B.

16.3 Sources of Geomagnetic Pulsations

16.3.1 UpstreamWaves

A major source for geomagnetic pulsations in the mid-frequency range is considered the penetration into the

magnetosphere of waves generated in the upstreamingSW (“foreshock region”) by protons reflected by the bowshock along IMF lines. Briefly, such waves are generatedwhen the following resonant condition is matched

ω − k ċVp = Ωp (16.11)

where ω is the wave frequency, Vp, the velocity of re-flected protons (both in the SW frame of reference), andΩp their gyrofrequency. In the spacecraft frame of ref-erence the wave frequency would be:

ωs = ω + k ċV sw (16.12)

Considering that both Vp (� – km�s) andVSW (� – km�s) are usually much larger thanVa (� – km�s in the interplanetary medium),previous equations provide (16.5)

ωs � ΩpVsw

Vp

cos ϑcos ϑ

(16.13)

where ϑ is the angle between k and Vsw and ϑ is theangle between k and Vp. Equation (16.13) reveals thatwave frequency is dependent on IMF magnitude, SWspeed, and the bulk speed of backstreaming ions. As-suming typical values for Vsw, and for the orientationand magnitude of Vp, previous equation gives the nu-merical relation (see also Takahashi et al., 1984; Le andRussell, 1996):

f (mHz) � ( � )B (nT) (16.14)

which, for typical values of IMFmagnitude (� – nT),leads to the prediction of upstream waves mostly in thePc3 regime.

Since these waves propagate slowly, they are con-vected downstream towards the bow shock and, un-der certain conditions, can enter the magnetosphere.Within the magnetosphere, these oscillations may cou-ple to field lines or propagate directly through magneto-spheric cavity resulting in the detection of ground pul-sations.

The orientation of the IMF (and the “cone angle”,θXB, between the IMF and the Earth-Sun line) is alsoan important parameter influencing the structure ofthe bow shock7, the location of the foreshock region When the angle between the IMF and the shock normal islarge (quasi-perpendicular shock), the field is almost paral-

Ultra Low Frequency Waves in the Magnetosphere | Sources of Geomagnetic Pulsations

Fig. 16.3a,b. The generation and penetration of pulsations.(a) The location of the foreshock region in the equatorialplane: Left panel. Upstream waves are generated by pro-tons along the spiral IMF lines. Central panel. For a radialIMF, waves are generated in a foreshock region symmetricaround the subsolar point. Right panel. For a perpendicularIMF, waves are generated in narrow regions close to the bowshock flanks. [Greenstadt et al., 1981; Formisano, 1984]. (b)A schematic representation of the generation and propagationof surface waves on the magnetopause. Kϕ is the azimuthalcomponent of the wave vector. A and B identify two groundstations. Since waves propagate tailward, they would be ob-served to propagate from station A to station B when stationsare located on the morning side, and from B′ to A′ when theyare located on the afternoon side. The polarization of surfacewaves driven by the KHI changes sense near local noon

lel to the shock boundary and the shock appears as a sharpdiscontinuity. When it is small (quasi-parallel shock), thefield is almost perpendicular to the shockboundary and thetransition becomes turbulent. Hence a quasi-parallel shockis also a possible source of ULF waves. Under spiral IMForientation, a quasi-parallel and a quasi-perpendicular bowshock structure are predicted on the dawn and dusk side ofthe magnetosphere, respectively (Fig. .a, top panel).

and wave transmission through the magnetosphere(Fig. 16.3a). Under nominal conditions, the spiralIMF predicts a foreshock region (and a possible wavepenetration) on themorning side of the magnetosphere;convected downstream through magnetosheath, thesewaves, in general, would not reach the magnetopauseeasily. For a radial IMF, a wide and symmetric foreshockregion is predicted around the subsolar point, and wavesare mostly convected toward the magnetosphere. Fora perpendicular IMF, upstream waves would only begenerated in narrow regions close to the bow shockflanks and swept away by the SW flow.

16.3.2 Kelvin–Helmholtz Instability

Surface waves at the magnetopause boundary layer inthe low- and mid-frequency range are expected to ariseas a consequence of the relative motion of the SW andmagnetospheric plasmas. These waves are amplifiedwhen Vsw exceeds a critical value determined by:

(k ċV sw) �5 nsw

+ nM

6

� Vnsw (k ċVa)sw + nM (k ċVa)MW (16.15)

in which n is the number density and subscripts SW andM identify the SW and the magnetosphere, respectively.KHI waves have a phase velocity in the same sense asthe wind which drives them (i.e. westward in the morn-ing and eastward in the afternoon; Fig. 16.3b): as a con-sequence, a phase inversion across the noon meridianshould appear in experimental observations. Beneaththe boundary, a fluid elementwill have an approximatelyelliptical motion (with the rotation being in the oppositesense along the dawn and the dusk flank) similar to thatof the fluid motion associated with a wave on a free sur-face in a pure gravitational field. As the field lines arefrozen into the plasma, they will also rotate generatingelliptical polarized waves which propagate to the Earthvia the field lines.

16.3.3 Ion-Cyclotron Instability

Consider the case of a wave with frequency ω and phasevelocity Va, and a proton with a velocity V�� (i.e. par-allel to the magnetic field) traveling in opposite direc-tions along the magnetic field. In the proton frame of

U. Villante

reference the wave has a Doppler shifted frequency ω′ =ω( + V���Va). Provided the proton has V & , it willbe gyrating around B in a left-handed sense. When ω′and Ωi are the same, the wave electric field and the pro-ton velocity are in resonance. If the wave electric fieldis antiparallel to the ion velocity, the particle will beslowed down and the wave will gain energy. In fact,the distribution of the “pitch angle”, the angle betweenthe particle velocity and the magnetic field, peaks atlarge angles within the magnetosphere. This means thatparticles have more perpendicular than parallel energy:as a net effect waves gain energy and transverse, left-handed, circularly polarized waves are expected to ariseat f � . – Hz (McPherron, 2002).

16.4 Effects of the Ionosphereand Field Line Eigenperiods

The simplest approach to evaluating the effects of theionosphere is to consider it a perfect conductor. So theelectric field vanishes and the energy of transverse wavesguided along the magnetic field is confined to the re-gion of space between ionospheres of opposite hemi-spheres. This approach allows the evaluation of the ex-pected wave periods numerically integrating the cor-responding wave equations. Alternatively, approximatevalues of the eigenperiods Tn for standing oscillationscan be obtained using the WKB approximation, whichis valid when the wavelength is short compared to thescale variation of B and ρ; in particular, for the toroidalmode

Tn = n �

l

dsVa

n = , , . . . (16.16)

where integration is carried out along field lines an-chored on opposite ionospheres. Assuming that the ge-omagnetic is a centred dipole, (16.16) becomes

Tn = πRE

nMμ� cos λ

λ

�

ρ� cos λdλ (16.17)

where M is the dipole moment, and λ is the geomag-netic latitude of the foot of the line of force of a givenshell.

Figure 16.4a shows the classical representation ofthe expected L-dependence of the fundamental period

of the uncoupled toroidal and guided poloidal modeas well as the WKB solution for a simple radial depen-dence of the magnetospheric plasma density. Clearly,the L-variation corresponds to a latitudinal variationat ground magnetometric arrays. Figure 16.4a alsopredicts a continuous spectrum of Tn eigenvalues, witha general tendency for increasing periods with increas-ing latitude (between � – s, with the exception ofa narrow region which corresponds to the ρ cutoff at theplasmapause). Obviously, given the ionosphere height,Eq. (16.17) cannot be applied below L � . – .; a simi-lar conclusion holds for high latitudes, where the dipoleapproximation becomes poor. Eigenfrequencies of thefield lines for more realistic field geometries and moresophisticated density distributions have been evaluatedby several authors: in general, the ρ-dependence makesTn estimates dependent on local time, magnetosphericactivity, the solar cycle, etc.

In addition, a significant portion of the field linelies within the ionosphere at low latitudes: as a conse-quence,mass loading due to ionospheric heavy ions low-ers the expected eigenfrequencies (whose harmonic val-ues are not integer multiples of the fundamental mode)and provides maximum values of the eigenfrequency atλ � � (L � .; Fig. 16.4b). Figure 16.4c shows thestructure of the wave perturbation along the magneticfield line for the two lowest harmonics.

If the ionospherewere a perfect conductivemedium,waves would be confined within the magnetosphere andno signal would be detected on the ground. In reality,the incoming wave drives horizontal current sheetsin the ionosphere because of its finite and anisotropicconductivity and the input signal is not completelyshielded ([Hughes, 1994] and papers referenced). Infact, the ionosphere smears the rapid variations atground level and features with scale length smallerthan � km are strongly damped. In addition, thePedersen current along e generates a magnetic fieldopposite to the wave field, while the Hall currentalong e � B generates a magnetic field perpendicularto the wave field. So, the net effect consists in a �rotation of the original signal (Fig. 16.4d), which isleft-handed in the northern hemisphere and right-handed in the southern hemisphere, when lookingdownward8.

In addition, the excitation of pulsations by rapid changes ofthe ionospheric conductivity induced by solar flare X-ray

Ultra Low Frequency Waves in the Magnetosphere | Field Line Resonance

Fig. 16.4a–d. The effect of the ionosphere. (a) The L-variation of the fundamental period of the toroidal mode, guided poloidalmode and WKB solution for a dipole field, assuming a r− dependence for the plasma density in the plasmasphere (located at� RE), and a r− dependence beyond the plasmapause. Experimental points represent dominant pulsation periods observedat different latitudes. [Villante and Vellante, 1997]. (b) The behaviour of the FLR frequency (dotted line) and Alfvén velocity(solid line); in this case the plasmapause is located at � RE. [Waters et al., 2000]. (c) A schematic representation of the fielddisplacements (dashed lines) in a fundamental and second harmonic mode of FLR and the corresponding perturbation of theelectric andmagnetic fields. (d) A schematic representation of the ionospheric effects on an incident Alfvénwave. [Hughes, 1983]

16.5 Field Line Resonance

Although questioned by some authors, the field line res-onance (FLR) is the principal mechanism and energysource for ground pulsations. Southwood (1974) andChen and Hasegawa (1974) examined a simplemodel inwhich themagnetospheric field line resembles a dampedharmonic oscillator in the presence of a driving force.FLR occurs when the frequency of the incoming (driv-ing) wave is comparable to the field line eigenfrequency.If the incoming wave is monochromatic at f �, the cou-

and EUV fluxes represents an interesting example of gen-eration mechanisms not associated with SW and/or mag-netospheric processes.

pling will be strongest at the closed field line for whichf � is a resonant frequency (i.e. it matches the frequencyof thewave that can stand on that field line). A schematicillustration of this process is shown in Fig. 16.5a (leftpanel) where KHI waves are represented as wiggly linesmoving away from the local noon and line thicknessrepresents wave amplitude. As can be seen, amplitudeprogressively decreases inward, but peaks locally at res-onant L-shells (Kivelson, 1995). In the magnetosphere,the fast and Alfvén mode can interact to generate az-imuthal standing oscillations. In this case the amplitudeof the wave azimuthal component is enhanced; however,as a consequence of the ionospheric rotation, the oc-currence of such resonance processes should lead to anenhancement of the H wave amplitude in ground ob-

U. Villante

Fig. 16.5a–c.Theoretical aspects of the resonance processes. (a) A schematic representation of wave perturbation through the day-sidemagnetosphereproduced by theKHI at themagnetopause (FLR, left panel), and by a compression of themagnetopause nose(cavity mode, right panel). [Kivelson, 1995]. (b) The diurnal and latitudinal variation of the polarization pattern in the northernhemisphere for pulsations with f � mHz; for higher frequencies the entire pattern shifts equatorward. [Samson et al., 1971].(c) The amplitude ratio G( f ) and the phase difference Δϕ between H signals for a pair of stations separated in latitude. fr, isthe resonant frequency at the middle point between stations. [Vellante et al., 2002]

servations. For a broadband driving source such as up-stream waves, a continuum of field lines may resonate,provided the eigenfrequencies are within the frequencyof the source; in addition, multiple harmonics can begenerated on a single shell.

The important aspects of the polarization patternfor FLR were examined considering a field line eigen-frequency that varies across the field lines. Near reso-nance coupling between modes creates a narrow ampli-tudemaximum.As Southwood (1974) argued, the phaseof the radial and azimuthal component are expected tochange in away that thewave polarization changes senseat both the resonant field line and at the amplitude min-imum between the resonant field line and the magne-topause.Waves propagatingwestward (eastward) are ex-pected to have a right-handed (left-handed) polariza-tion poleward of the resonant field line, and left-handed

(right-handed) equatorward of it. In addition, as notedearlier, a polarization switch around local noon is ex-pected for KHI waves. This leads to the prediction ofa polarization pattern which is consistent with the oneobtained at northern auroral latitudes by Samson et al.(1971) (Fig. 16.5b).

Some aspects of FLR can be easily understood interms of forced, damped harmonic transverse oscilla-tions of field lines anchored on opposite hemispheres,driven by incoming fast mode waves. In this simplifiedscheme the field line oscillation can be described as:

bϕ + γbϕ + ωr bϕ = ω

r bz c (sinωdt) (16.18)

where bϕ is the amplitude of the transverse resonancewave, bz is the amplitude of the driving wave at fre-quency ωd, ωr is the resonant frequency, c is a coupling

Ultra Low Frequency Waves in the Magnetosphere | Low Frequency Pulsations

factor between the incoming wave and the field line,and γ is a damping factor (Menk et al., 1994). Assum-ing a broad band source and a low damping, in additionto transient fluctuations at �ωr which decay exponen-tially, it is possible to obtain a steady state solution withamplitude and phase given by

A(ω) = ωr bz c

V�ωr − ω

d� + γω

dW� (16.19)

ϕ(ω) = tan− 2 −γωd

ωr − ω

d3 (16.20)

Clearly, at ωr = ωd, the amplitude assumes maximumvalues and the phase reverses. If data from two stationsclosely spaced in latitude are available, a comparison ofsignal amplitudes (“gradient method”) or phases (“cross-phase method”), can be used for determining the reso-nant frequency. A theoretical expression for the merid-ional structure of the complex amplitude of the H com-ponent in a restricted region around the resonant pointis given by:

H(x , f ) = εHr( f )ε + i [x − xr( f )]

(16.21)

where x is the meridional coordinate, xr( f ) is theresonant point, ε is the resonance width, and Hr( f ) isthe amplitude at the resonant point (Fig. 16.5c; Vellanteet al., 2002 and papers referenced).

16.6 Cavity Resonance

As for toroidal modes, expected periods may alsobe evaluated for poloidal “cavity” modes propagat-ing through the magnetosphere and reflected by itsboundaries. The modern concept of magnetosphericcavity mode was introduced by Kivelson et al. (1984),who proposed a simple box geometry with perfectlyreflecting boundaries, in which the magnetosphererings as a whole at its own eigenfrequencies. At thesame time, Allan et al. (1986) suggested that impulsivestimuli at the magnetopause can set up compressionalcavity resonances which drive FLR within the mag-netosphere. This model was further developed formore realistic conditions. Obviously, the cavity modeeigenperiods are determined by the cavity dimensions.

The presence of several boundaries reflecting signals(the magnetopause, the plasmapause, the ionosphere,etc.) concurs to determine a discrete spectrum ofexpected frequencies. Harrold and Samson (1992),who considered the bow shock as an outer boundary,proposed discrete frequencies at f � ., ., ., .and . mHz. Magnetospheric dimensions, on the otherhand, are continuously changing due to the continuousvariations of the SW and IMF parameters. In addition,given its long tail, the magnetosphere would be betterrepresented as an open-ended waveguide, in whichcompressional modes propagate antisunward andenergy resonates radially between an outer boundary(such as the bow shock or the magnetopause) and an in-ner turning point (Samson et al., 1992). All these aspectsmake estimates (and identification) of magnetosphericcavity/waveguide modes uncertain. Figure 16.5a (rightpanel) shows a schematic representation for possiblecavity resonances driven by impulsive variations ofSW pressure. It is worth noting that as a consequenceof coupling with FLR, the global mode has the char-acteristics of an Alfvén mode near the resonant fieldline, and is similar to a compressional mode away fromthe resonant field line. Multiharmonic cavity modesand toroidal resonances may also be excited when SWpressure pulses impinge the magnetopause.

16.7 Low Frequency Pulsations

During the 1960’s, hydromagnetic fluctuations inthe distant magnetic field were observed by severalspacecraft. Explorer 12 provided a direct observa-tion of magnetic pulsations, both compressional andtransverse, with an approximate period of – min9.Estimates of the field line eigenperiods confirmed theirinterpretation in terms of standing waves. Large am-plitude compressional fluctuations (δB�B � . – .),with periods of � – min, were interpreted in termsof slow modes in which particle pressure and magneticpressure were in anti-phase. Similar fluctuations,

The magnetospheric field exhibits magnetic fluctuations inradial, azimuthal, or parallel (i.e. along B) directions. De-pending onwhich component is dominant, thewave is usu-ally termed “poloidal”, “toroidal”, or “compressional”. Moregenerically, “transverse” waves may include both azimuthaland radial perturbations.

U. Villante

with periods of � min or longer, were observed atL � (indicating a source near the magnetopause)with highest occurrence rates in the morning andafternoon sectors: they were polarized in the meridianplane with comparable compressional and transversalcomponents. At geosynchronous orbit (L � .), waveswere observed to propagate away from noon on bothsides of the magnetosphere, a feature consistent withKHI predictions.

In the meanwhile, evidence from ground observa-tions was also very important: indeed, the same wavetrains were observed simultaneously at the foot of bothlines of force (as expected for guidedmodes), while con-jugate observations were matched cycle for cycle (as ex-pected for standing waves). More in general, the agree-ment between the expected and the dominant periodsobserved at different latitudes during daytime intervals(Fig. 16.4a), as well as the correspondence between con-jugate observations, were considered outstanding argu-ments to interpret Pc3/5 pulsations in terms of oscilla-tions of lines of force which are rooted at magneticallyconjugate points. Simultaneous observations at high lat-itude conjugate stations and at a satellite revealed sym-metric motions of the field line at northern and south-ern end, and indicated the equator as the nodal planeof an odd mode standing wave, with a period close tothe expected for the fundamental mode at those lati-tudes. Nowadays, conjugate phenomena are observedfrom low to high latitudes in a wide frequency range.

However, despite such basic conclusions, the theo-retical elements summarized in the previous paragraphsshould be only considered as rough indicators in com-paring theory and observations (Rostoker, 1979). In-deed, pulsations can rarely be interpreted in terms ofpure toroidal or poloidal modes, the two being invari-ably coupled: ground elliptical polarization, on the otherhand, is clearly indicative of coupling of toroidal andpoloidal modes.

At ground stations, the maximum intensity (up tohundreds of nT) of low frequency pulsations tends to fol-low the approximate position of the auroral oval (Sam-son, 1991). Between λ � −� and λ � −� the low fre-quency power, shows a non monotonic behaviour, witha powerminimum at λ � −�, followed by a further in-crease; moreover, pulsation activity in the auroral zoneand in the polar cap appears decoupled. In the Pc4 range

the maximum wave energy may occasionally occur in-side the plasmapause.

A pronounced morning/afternoon asymmetry(with higher power level in the morning) has beenreported at auroral and cusp latitudes, togetherwith a secondary enhancement near local mid-night. Nighttime enhancement, correlated withsubstorms, is mainly due to more irregular pul-sations sharing the same frequency band. In thissense such enhancement is not indicative of signifi-cant nighttime Pc5 activity. Conversely, in the polarcap, the fluctuation power only maximizes aroundmagnetic local noon, when stations approach thedayside cusp. This feature suggests a minor influ-ence of substorm related events deep in the polarcap.

Pc5 pulsations, due to their frequency, are usuallyobserved from auroral to cusp latitudes. Their oc-currence and intensity are correlated to SW speed, inparticular in the dawn sector. The dependence of thepulsation power on SW speed was stronger at aurorallatitudes than at near cusp latitudes and a thresholdvalue (� – km�s) has been proposed abovewhich SW control of the pulsation power is dominant.Comprehensive auroral surveys revealed statisticalevidence for antisunward propagation and reversalof the polarization sense in latitude and around localnoon (Fig. 16.5b). In the meanwhile, increased activitywas found during times of high SW speeds. In general,local morning fluctuations were attributed to KHI atthe dawn magnetopause (which might be less stablethan the dusk flank), while the less frequent afternoonevents were attributed to corotating SW pressure pulsesimpinging the post-noon magnetopause. At lowerlatitudes (λ � �–�), morning pulsations revealedclockwise polarization and westward propagationsuggesting that these waves are mid-latitude signaturesof SW driven FLR occurring at higher latitudes; con-versely, the polarization characteristics of afternoonevents were interpreted in terms of ground signaturesof compressional cavity modes. Satellite studies alsoshowed that transverse azimuthally polarized waves(correlating to ground observations) predominate inthe morning sector, while compressional events (poorlycorrelated with ground observations) predominate inthe afternoon sector (Anderson, 1994).

Ultra Low Frequency Waves in the Magnetosphere | Mid-Frequency Pulsations

Low frequency events are more rarely observed atlow latitudes, where they are interpreted as signatures ofglobal compressional modes or large scale-cavity reso-nances. They would be difficult to explain in terms ofKHI waves because of the damping rate of the surfacewavemode in the radial direction. Their amplitude (andoccurrence) considerably decreases with decreasing lat-itude; it sharply enhances at the dip equator (i.e. wherethe Z component vanishes) because of the anomalousionospheric conductivity.

Sudden impulses (SI)10 and storm sudden com-mencements (SSC) are often accompanied at middleand high latitudes by long period, often irregular,pulsations which have been interpreted in terms ofFLR; however, this interpretation is in conflict withthe large angles of the polarization axes with respectto the H orientation identified in recent analysis. Theoccurrence of damped Pc4 pulsations that accompaniedmagnetic disturbances has been reported at L � ,exterior to the plasmasphere boundary. An extendedinvestigation of the long period wave response tomagnetic storms from equatorial to cap latitudesrevealed that narrow band Pc5 activity typically occursduring the recovery phase in the dawn-noon sector ateach site; during the main phase the wave activity isbroadband and the strongest power is observed abovef � mHz in the auroral zone, and below f � mHzat middle and low latitudes (Posch et al., 2003). Stormtime magnetospheric pulsations might be generatedby ring current particles as a result of internal plasmainstabilities.

On rare occasions (few times per year), highlymonochromatic, amplitude modulated signals appearin the Pc4 range (termed “giant” or “Pg” pulsations,with a peak to peak amplitude of � – nT at groundlevel and few nT in space). First discussed by Birkeland,these pulsations mostly occur during geomagneticallyquiet conditions, between midnight and noon, withina few degrees of the auroral oval (Rostoker, 1979).Giant pulsations are believed to result from plasmainstabilities within the magnetosphere.

The SI (SSC) itself at ground level consists of a compositesuperposition of several wavelike signals ultimately drivenby impinging SW discontinuities and related to magne-topause and ionospheric currents.The different waveformsin different sectors can be interpreted in terms of the rela-tive importance of different contributions.

16.8 Mid-Frequency Pulsations

Mid-frequency pulsations (with amplitudes ranging be-tween fractions of nT and several nT) are a commonday-time feature of ground observations; they typically reachmaximum amplitudes at the position of the dayside cusp(λ � �) and appear to decline rapidly as the point ofobservations moves to higher latitudes. For many years,the polar cap has been considered to be characterizedby very low activity in the mid-frequency range. How-ever, more recently, several investigations have shownthat this might not be the case (Chugunova et al., 2003).Villante et al. (2002) found peaks of correlation betweenthe Pc4 power and SW speed at dawn and dusk at TerraNova Bay (CGM λ11= −�): this suggests a major roleof KHI at these frequencies.

Although several aspects of the penetration ofupstream wave activity through the magnetopause stillneeds further investigation, it is clear that a significantfraction of the external wave energy enters the mag-netosphere. Attempts to measure the transfer functionof the magnetopause had little success. However, theoccurrence of Pc3 waves at L � . meets highlyfavourable conditions in the morning sector whenSW velocity is high and ground observations reveala close relationship between the amplitude of daytimepulsations and SW velocity. As for lower frequencypulsations, a similar, but somewhat weaker, dependenceon SW speed has also been determined for the energyand occurrence of mid-frequency pulsations (Odera,1986). This feature is also consistent with the generaltendency of the pulsations activity to recur with thesame period of the corotating high velocity SW streams.Figure 16.6a shows the close correlation between theenergy of mid-frequency pulsations and SW velocitydetected at low latitudes (L � ., Yedidia et al., 1991).

As mentioned earlier, the dependence of the pulsa-tion activity on SW speed suggests a major role of KHI

The corrected geomagnetic (CGM) coordinate system hasproven to be an excellent tool in organizing geophysicalphenomena controlled by the Earth’s magnetic field. Fora point in space CGM coordinates are evaluated by tracingthe field line of the International Geomagnetic Field Refer-ence (IGRF) through the specific point to the dipole geo-magnetic equator then returning to the same altitude alongthe dipole field and assigning the obtained dipole coordi-nates as CGM coordinates to the starting point.

U. Villante

Fig. 16.6a–d. The relationships with SW and IMF parameters at L � . (a) A comparison between the daily average power inthe Pc3 band and SW velocity. [Yedidia et al., 1991]. (b) The relationship between the frequency of pulsations and IMF magni-tude. [Villante et al., 1992]. (c) The solar cycle variation of the frequency of the “resonant” mode, fr, of the “upstream” mode, fd,and its predicted value, fu (which is proportional to IMF strength). [Vellante et al., 1993b]. (d) A comparison between the onsetof the pulsation activity (Pc3) and the cone angle. [Vellante et al., 1996]

in the generation of surface waves and/or in the am-plification of already existing waves, as they are con-vected and transmitted through the high latitude mag-netopause. However, since increasing power with in-creasing SW speed is found for most classes of pulsa-tions and in a wide latitudinal range, caution should beadopted before considering this correlation as defini-tive prove of a given source (Anderson, 1994): indeed,a dependence on SW speed might also reflect an effectof magnetospheric compression which makes the wavesource closer to observational points as well as a moreefficient generation and/or transmission process of up-stream waves.

Upstream waves as a source of ground pulsationshave been confirmed by several investigations. Animportant element is the relationship (16.14) betweenIMF strength and the frequency of ground pulsations,with an average coefficient well within the limits oftheoretical predictions (. � ., Fig. 16.6b). The relat-ionship between IMF strength and the frequency ofwaves in the foreshock region was the same as forground pulsations. Le and Russell (1996) and Takahashiet al. (1984) noted the additional influence of the “coneangle” θXB on wave frequency, both in the foreshockregion and in the magnetosphere. Within the magneto-sphere both standing Alfvén waves and compressional

Ultra Low Frequency Waves in the Magnetosphere | Mid-Frequency Pulsations

waves have been identified, and the frequency ofthe compressional fluctuations has also been relatedto IMF strength. Engebretson et al. (1986) reportedharmonically structured pulsations in the outer daysidemagnetosphere in association with upstream waves andwith periods governed by local resonance conditions;in addition, almost monochromatic compressionalfluctuations with periods identical to the those detectedin the SW were occasionally observed. In a similarscenario, two kinds of pulsations would be reasonablyexpected at ground stations (Villante and Vellante,1997): a wide band, irregular wave form with a perioddependent on IMF strength (from upstream wavespropagating through the magnetosphere) and a moreregular wave form with latitude dependent period(from standing oscillations along local field lines). Infact, an analysis conducted at L � . in 1985 showedthat dayside events tend to occur predominantly in twoseparate period ranges (T = � s and T = � s)and these observations were considered consistent withan upstream source spectrum peaked at T , togetherwith a coupling resonant mechanism at T , roughly thefundamental period of the local field line (Vellante et al.,1989). When extended to a longer interval (1985–1994,Fig. 16.6c), the same analysis revealed a clear solar cyclevariation of both dominant periods, consistent withdifferent values of IMF strength (T) and plasmasphericdensity along the field line (T) in different phases:obviously, the two periods may intermingle through thesolar cycle due to the different SW and magnetosphericconditions.

In agreement with model predictions (Fig. 16.3a),IMF orientation also plays a significant role, in thatground pulsations occur more frequently when θXB issmall. Figure 16.6d shows an example of the relationshipbetween the pulsation onset and favourable θXB valuesat L � .. Studies on the transmission of upstreamwaves into the magnetosphere revealed that small θXBprovide quasi-parallel shock conditions at the subsolarbow shock and allow the convection of the turbulentmagnetosheath plasma toward the nose of the mag-netosphere. More in general, combined large Vsw andsmall θXB values were found to enhance significantlythe pulsation occurrence in the dayside magnetosphere.

A careful statistical analysis of the ground polar-ization pattern at low latitudes revealed a polarizationreversal occurring – hours before noon, i.e. consis-

tent with an upstream wave penetration on the morningflanks of the magnetosphere during spiral IMF condi-tions (Fig. 16.3a). During intervals related to radial IMForientation, the polarization reversal was found to oc-cur closer to local noon, as expected for a more sym-metric wave penetration around the subsolar point (Vil-lante et al., 2003). Nevertheless, a greater pulsation oc-currence in the morning sector of the magnetospherewith respect to the afternoon sector was also observedfor IMF orientation far from the spiral (Takahashi et al.,1984).

In general, the relationship between upstream wavesand ground pulsations is better at low than at high lat-itudes: their frequency, on the other hand, is such thatthey preferentially excite FLRs at low and middle lati-tudes. Nevertheless, Villante et al. (2002) found a highercorrelation between the pulsation power and SW speedin the morning, an explicit θXB control and a linear rela-tionship between frequency and IMF magnitude in thePc3 range, at Terra Nova Bay. These features suggestthat the role of upstream waves might also be significantat high latitudes. The occurrence of almost monochro-matic Pc3-4 events in high latitude regions is interpretedeither in terms of higher harmonics of local FLRs, or interms of fast modes propagating earthwards in the equa-torial plane and refracted and diffracted by the chang-ing refractive index of the plasma environment. In addi-tion to the conventional approach which assumes directtransmission of upstream waves from the subsolar mag-netopause, Engebretson et al. (1991) suggested an “iono-spheric transistor” model in which wave transmissionmay also occur as an indirect process involving modu-lation of the dayside Birkeland current12. Observationsat South Pole (CGM λ = −�) provided quantitativeproof of pulsations driven by modulated electron pre-cipitation near the magnetospheric boundary and indi-cated the cusp entry as an important source for pulsationenergy (Olson and Fraser, 1994). At cusp latitudes, theburst-like Pc3/4 signals were highly localized, a resultwhich is consistent with the modulation of precipitat-ing electron beams. At high latitudes, Chugunova et al.

The Birkeland (or field-aligned, FAC) currents, parallel orantiparallel to B, are current systems which, at high lat-itudes, link the SW-magnetosphere system to the iono-sphere.

U. Villante

(2003) identified a peak of activity in themorning sectorwhichmight indicate an additional propagation path viathe magnetotail lobe.

At middle and low latitudes, most aspects of themid-frequency pulsations are interpreted in terms ofresonant phenomena related to the penetration of up-stream fluctuations. Below L � , the occurrence of Pc3waves was considered consistent with compressionalwave modes coupling to shear Alfvén resonances. Mid-frequency pulsations are also detected in the equatorialregion, where they typically show a strong polarizationalong H. Here, in the absence of FLR, these observationsare interpreted either in terms of compressional wavespropagating in the equatorial plane, or in terms ofwaves propagating into the high latitude ionosphere,generating large current oscillations which cause thePc3/4 observations in the equatorial region.

16.9 FLR and Magnetospheric Diagnostics

In addition to what has been previously discussed, veryinteresting results were obtained by radar measure-ments (Walker, 1980): indeed, in agreement with FLRtheory, they showed (Fig. 16.7) that the wave electricfield changed in phase by � � over about � of latitudeand that this corresponded to the half width of the waveamplitude maximum (Hughes, 1994).

Several aspects of magnetospheric research havealso been important in gaining a better understandingof low- and mid-frequency pulsations and their inter-pretation in terms of resonance phenomena. Spatiallylimited polarized pulsations consistent with a secondharmonic standing wave were identified; resonant pro-cesses were observed to provide a significant effect onthe azimuthal component of the magnetospheric field,and clear evidence for series of harmonic structures andfor simultaneous resonant oscillations of a continuumoffield lines was detected at L � – (Fig. 16.8a,b; Taka-hashi and McPherron, 1982; Engebretson et al., 1986).

A vast amount of literature has been published onground signatures of FLR at middle and high latitudes:in particular, the peak of the H component occursat a latitude which is frequency dependent, and istypically accompanied by rapid phase variation. In fact,fr decreased from f � mHz to f � mHz betweenλ � �–�, (Samson et al. 1971; Samson and Rostoker,

1972). Waters et al. (1995) found a FLR at f � mHzat λ � �, decreasing to f � mHz at λ � �.On the other hand, given the variable length of thefield line with the local time and the different plasmacharacteristics in different magnetospheric regions, fora given frequency, the resonance latitude has a localtime dependence and typically shows an arch structurethrough the day.

As noted earlier, important results on polarization( f � mHz) were obtained by Samson et al. (1971).They determined a complex pattern (Fig. 16.5b), inwhich two or more polarization reversals, dependingon latitude (λ � �–�), were observed through theday, with polarization changes occurring approximatelyat noon and across the line of maximum amplitude(where a linear polarization was detected). As men-tioned previously, these results were interpreted interms of surface waves which excite a FLR deep into themagnetosphere on the field line whose eigenfrequencymatches the wave frequency and also create a narrowwave amplitude maximum.

At higher latitudes (Terra Nova Bay; f � – mHz)several reversals of the polarization pattern were

Fig. 16.7.Radar observations of resonance. Latitudinal profilesof the amplitude and phase of the oscillating electric field witha period of � s (STARE radar observations). The solid lineis a model calculation. The narrow peak amplitude and thephase change by � � are both features predicted for FLR.[Walker, 1980]

Ultra Low Frequency Waves in the Magnetosphere | FLR and Magnetospheric Diagnostics

Fig. 16.8a–c. Experimental aspects of the Field Line Resonance. (a) An example of Pc5 pulsations simultaneously observed inthe magnetosphere and on the ground. [Kivelson, 1995]. (b) Spectra of the azimuthal component observed by three satellites.Each spectra observed a series of harmonic peaks. [Takahashi and McPherron, 1984]. (c) H and D power spectral densities andthe corresponding ratio for different time intervals at L � .. The peaks in the ratio identify the resonant frequency and itsharmonics. [Vellante et al., 1993b]

identified through the day, suggesting resonance effectsof lower latitude field lines. It is clear (Fig. 16.5b), thata polar cap station may cross the higher latitude lineof polarization reversal at different local times throughthe day. On the other hand, resonant oscillations ofclosed field lines have been commonly observed atsomewhat lower latitudes (South Pole) during closedmagnetospheric conditions, when the cusp is expectedto be located poleward with respect to the station(Francia et al., 2005).

In general, resonant effects are hardly identifiedfrom a single station because spectral properties tendto reflect the source spectrum. Baransky et al. (1990)proposed the peaks in the ratio between the H and Dspectra to indicate resonant frequency, fr. With sucha technique, up to five harmonics were identified atL � . (Fig. 16.8c). The introduction of the “gradient”and “cross-phase” methods between nearby stationsallowed a significant improvement in the fr identifica-tion and permitted the identification of a continuous fr

U. Villante

variation even in ground measurements. For example,by means of the cross-phase technique at L � . – ,fr was found to increase with decreasing latitude up toL � . and then to decrease at lower latitudes, and thisfeature was considered consistent with the predictedeffects of the mass loading of heavy ions (Fig. 16.4b).Vellante et al. (2002, and papers referenced) adopteddifferent spectral techniques to determine the resonancecharacteristics (L � . – .) during the main phaseof a magnetic storm; fr estimates were found to besignificantly higher than expected, suggesting unusualconditions of the ionosphere-plasmasphere system dur-ing this particular event. In addition, a high frequencyresolution analysis led authors to suggest the possibleoccurrence of FLR driven by cavity/waveguide modes.

These arguments suggest that the clear identifica-tion of FLRs is an important tool for several aspectsof magnetospheric diagnostics. A quantitative deter-mination of the set of field line eigenfrequencies canbe used to model the plasma distribution along thefield lines from equatorial to high latitudes, to monitortemporal variations of the magnetospheric plasmaconcentration, to highlight aspects of the plasmaspherealso in comparison with other methods (Menk et al.,1999; Waters et al., 1996; Takahashi and McPherron,1982). In fact, resonant frequencies are dependent onVa (16.6), a function of the magnetic field and plasmadensity along the field line: since the magnetic fieldis well known, the density can be determined by theobserved resonant frequencies. Over past few years, theeigenfrequency method has been improved in severalaspects. Using this technique a latitudinal magnetomet-ric chain, with a typical pair spacing of – km,is capable of monitoring the radial distribution of themass density from the last closed field lines throughthe plasmapause into the low latitude magnetosphere(Fraser, 2003); moreover, as the chain rotates, it allowsto obtain a map of the plasma mass density fromdawn to dusk. The results of a case event providedexcellent agreement between simultaneous ground andspacecraft observations, particularly in the noon sector(Waters et al., 1996). Interesting results have also beenobtained by Menk et al. (1999) who derived the massdensity profile of the dayside plasmapause (. < L < )with spatial and temporal resolution of �. – . REand � – min. The possibility of determining frby ground measurements is particularly useful at low

latitudes, which are difficult to monitor with spacecraftbecause rapid satellite motion causes spectral broaden-ing and phase shear. Vellante et al. (2004) concludedthat the difference between ground and spacecraft frestimates was consistent with the fast satellite motionthrough the resonant region at L � . – .. They alsoprovided an unprecedented direct confirmation of the� rotation of the polarization ellipse through theionosphere. The boundary between closed and openfield lines is generally identified by means of particlemeasurements. However, since FLR does not occur onopen field lines, it is possible to identify such boundaryas the latitude of the last field line where FLR is detected(Fraser, 2003). Lanzerotti et al. (1999) suggested thatthe demarcation in latitude between the appearance ornot of specific spectral tones may indicate the locationof the dayside magnetopause. Mathiè et al. (1999)identified FLRs on closed field lines at λ � .�, underquiet geomagnetic conditions; they also suggested thatduring perturbed conditions the closed/open boundarymight be located at λ � �–�.

16.10 Cavity/Waveguide Modes

From an experimental point of view the evidence forcavity/waveguide modes is still sparse. Within themagnetosphere, search for cavity modes has mostlyconcentrated on looking for compressional waveswith L-independent frequencies or for enhancementsin the Alfvén continuum at expected cavity modeeigenfrequencies. Rickard and Wright (1995) foundsome correspondence between spacecraft measure-ments and simulations of the magnetic field signalsexpected along the spacecraft trajectory for a waveguidemode. Mann et al. (1999) investigated multisatelliteand ground observations of a tailward propagatingcompressional wave and interpreted the experimentalobservations in terms of a magnetospheric waveguidemode. More recently, Waters et al. (2002) proposeda set of criteria for improving the identification ofcavity modes in spacecraft data which, in addition toamplitude characteristics, include tests for signal phaseinformation.

Several investigations at ground auroral latitudeshave reported evidence for long period waves at the“discrete” frequencies f � ., ., . and .mHz

Ultra Low Frequency Waves in the Magnetosphere | High Frequency Pulsations

(Samson et al., 1992). As noted earlier, such lowfrequencies, also known as cavity mode frequencies(CMS), would involve the bow shock as an outerboundary (Harrold and Samson, 1992). Evidence ofsimilar signals has also been reported at low latitudes. Insome cases the same oscillation modes were observedsimultaneously at low and Antarctic latitudes as wellas in the magnetosphere. These results were tentativelyconsidered consistent with features expected for globalcompressional modes or large scale cavity/waveguideresonances. In this context, the observed variabilityof the “discrete” frequencies might be interpretedconsidering that CMS frequencies represent a set of themost frequently occurring eigenfrequencies; howeverthey are subject to some variability, due to the changingnature of the waveguides. Samson et al. (1995) suggestedthat cavity/waveguide modes of the plasmasphere wereresponsible for low latitude Pc3 as well (L � . – .).In fact, standing Alfvén waves might be excited onlocal field lines by coupling to the waveguide modes; asgroundmagnetometers respond to ionospheric currentsover a range of latitude, the measured power spectraldensity might present a multiharmonic fine envelopestructure centered at the frequencies of the harmonicsof the local standing wave (Takahashi, 1991). Thesearguments were further developed by Waters et al.(2000), who proposed amodel able to reproduce severalfeatures of the low latitude power spectra, indicatingthat the interaction between waveguide and FLR modesmight be important in understanding several aspects oflow latitude observations (Menk et al., 1999).

In this context it is important to mention that sev-eral cases have been presented in which fluctuations inSWdensity and inmagnetospheric fieldwere highly cor-related and often matched some of the CMS frequen-cies (Fig. 16.9a). Kepko and Spence (2002) argued thatfor those events the discrete frequencies were an inher-ent property of the SW and were not related to possiblecavity or waveguide modes. They also speculated a pos-sible solar source related to solar p-modes in the mHzrange. More in general, Francia et al. (1999) identifieda dramatic correlation between continuous variationsof the H component (on time scale of several minutes)and variations of the square root of the SW dynamicpressure (Fig. 16.9b), suggesting that ground measure-ments closely respond to rapid, small amplitude varia-tions of the magnetopause current. In conclusions, be-

Fig. 16.9a,b.Aspects of pulsations at “discrete” frequencies. (a)Two examples of the correlation between the power spectra ofSW dynamic pressure and those of the magnetospheric fieldat geostationary orbit. [Kepko et al., 2002]. (b) An example ofthe strong correlation between continuous variations of SWdynamic pressure and continuous variations of the H compo-nent at low latitudes. [Francia et al., 1999]

fore a definitive interpretation of the “discrete” frequen-cies modes in terms of cavity/waveguide modes can bemade, further investigation is required of spacecraft andground observations.

16.11 High Frequency Pulsations

Unlike lower frequency standing pulsations, highfrequency pulsations are traveling waves which oftenshow spectacular amplitude and frequency modula-tion. This characteristic is represented by the term“pearl necklace” as a reference to a common class ofquasi-periodic sequences of pulsations which appearas structured wave packets and represent the mostcommon Pc1 manifestation at low and middle latitudes.Such “structured” pulsations (which typically appearat conjugate points as a repetitive burst of waves, withmodulated envelopes, Fig. 16.10a) typically range infrequency from . to Hz, with repetition periods

U. Villante

from 100 to s; there is often correlation betweenthe wave period and the repetition period with a pro-portionality factor of � . Pc1 pulsations with a lessregular behaviour (typically identified as “unstructured”pulsations) are the dominant manifestation at higherlatitudes (L L ). An important class is representedby the intervals of pulsations of diminishing period(“IPDP”) which show a typical frequency rise fromf � . to f � Hz in � min; the “hydromagneticchorus” is a mixture of structured and unstructuredpulsations between f � . – . Hz. Actually, thehigh frequency range encompasses a large numberof ground pulsations with different characteristics;several subtypes of high frequency pulsations, basedon their spectral structures, have been classified andeach subtype has a preferential local time occurrence(Fig. 16.10b; Saito, 1969; Fukunishi et al., 1981).High frequency pulsations have maximum amplitude(�. – nT) in the auroral zone and much smalleramplitudes at equatorial latitudes.

The typical tendency of high frequency pulsationsto recur on consecutive days, approximately at the samehours, or to disappear for several days or weeks has beenemphasized in several investigations. In addition, theyappear more frequently during winter months, a fea-ture which is considered consistent with a more effi-cient ionospheric attenuation during the summer. Theirlonger term occurrence appears anticorrelated, at leastat high latitudes, with the sunspot number and severalmechanisms (related to plasmapause position, the pres-ence of heavy ions in the magnetospheric plasma andionospheric waveguides) have been proposed to inter-pret this inverse relationship.

Short period fluctuations with different characteris-tics appear in each phase of geomagnetic storms (Kan-gas et al., 1998). For example, IPDP show strong associ-ation with SI/SSC, in particular in the noon sector andalso tend to occur during the main phase, mostly in theafternoon-evening sector; structured pulsations tend tooccur during the recovery phase. IPDP are also con-nected to substorm activity.

Pc1/2 pulsations, on the other hand, are a commonfeature of the magnetosphere. Since early observations,magnetospheric events have mostly been related tounstructured pulsations (with few exceptions), andfew structured events have been detected beyondgeostationary orbit. A statistical analysis of satellite

Fig. 16.10a,b.Classical aspects of the high frequency pulsations.(a) A classical representation of the alternate appearance ofpearl events in conjugate hemispheres and the correspondingdynamic spectra. [Saito, 1969]. (b) The local time dependenceof the occurrenceof various types of high frequency pulsationsat auroral latitudes. [Kokubun, 1970]

data revealed a main peak of occurrence near themagnetic equator beyond L � and a lower maximumat the plasmapause. Engebretson et al. (2002) identifieda close correspondence between Pc1/2 events in theouter dayside magnetosphere and high latitude obser-vations; they suggested that most events were associatedwith significant compressions of the magnetosphere.However, space/ground comparisons are not straight-forward in that the field line guidance stops at theionosphere, and ground measurements are influencedby the ionospheric waveguide.

The origin of high frequency pulsations is mostlybased on the occurrence of electromagnetic ion-cyclotron instability (EMIC) in the magnetosphere. Inthis sense, the observed predominance of left-handedpolarization as well as the gap of spectral power close tothe helium gyrofrequency are important experimentalaspects consistent with theoretical predictions. Inagreement with theory, high frequency pulsationsappear to be generated in the equatorial plane of

Ultra Low Frequency Waves in the Magnetosphere | Irregular Pulsations

geomagnetic shells between L � – (Anderson et al.,1996); they propagate to Earth along geomagnetic fieldlines. Structured pulsations might originate on fieldlines located near or inside the plasmapause (Fraseret al., 1984). It was suggested, over forty years ago, thatthese events were related to wave packets guided alongfield lines, bouncing from hemisphere to hemispherewith losses being compensated by wave growth at theequator. Satellite observations do not explicitly supportthe bouncing wave packet model; however, alternativegeneration mechanisms still need firm observationalsupport. Unstructured pulsations and Pc1 bursts (occa-sionally extending to Pc2 frequencies) are also observedduring daytime intervals on field lines related to cusp,suggesting an origin related to plasma instabilities nearthe dayside magnetopause.

16.12 Irregular Pulsations

As previously noted, Pi2 events occur as transientand damped signals associated with dramatic changesof the state of the magnetosphere which occur atthe substorm expansive phase (McPherron, 1979;Baumjohann and Glassmeier, 1984). Such irregularevents are observed during nighttime intervals, fromhigh to low latitudes. Their maximum amplitude isdetected at auroral latitudes, close to the region of thesubstorm enhanced westward ionospheric electrojet,while a secondary maximum is detected around theplasmapause. Southwood and Stuart (1979) interpretedthese waves as a transient response to sudden changesin the magnetosphere, acting to communicate andbalance stress between magnetosphere and ionosphere(Allan and Poulter, 1992).

Mid-latitude events are typically very monochro-matic, while high latitude events have much morecomplicated power spectra. Below L � , Pi2 spec-tra contain up to four harmonics and a similarmultiharmonic structure has also been observed inmagnetospheric events. Despite these differences, Pi2tend to have dominant frequencies independent oflatitude. At low latitudes, Pi2 are characterized bya clear initial phase (Fig. 16.11a) and for this reasonthey are often used to identify substorm onset time,although several aspects suggest caution (for example,

the delay time between pulsation onset and the firstauroral brightening).

It is generally accepted that the original source ofPi2 pulsations is the energy andmomentum impulsivelyreleased as the magnetic field of the near Earth tailsuddenly changes from an elongated configurationto a dipolar configuration at substorm onset. Severalobservational aspects are interpreted in terms of thesubstorm current wedge model (SCW, Fig. 16.11b) pro-posed by McPherron (1979) (for the formation of SCWand its association Pi2 pulsations see also Baumjohannand Glassmeier, 1984). In this model the tail current isinterrupted at the substorm onset, current flows alongfield lines into the ionosphere, couples to the westwardelectrojet, and returns to the equatorial plane via anupward FAC. Basically, the most important signaturefor interpreting Pi2 in terms of SCW oscillations is theobserved variation of the orientation of the polarizationaxis with longitude (a feature which also extends to lowlatitudes). Figure 16.11c shows the predicted gradualrotation from northeast (west of the event sourcelongitude), to north/south (close to the centre of thecurrent wedge), and to northwest (east of the eventsource in the northern hemisphere (Lester et al., 1989;Li et al., 1998; and papers referenced). The period ofPi2 pulsations is related to the fundamental eigenperiodof the toroidal mode along the field line where theauroral breakup starts (Fig. 16.11d). As for toroidalPc5/4, which roughly share the same frequency band,this suggests interpreting higher latitude Pi2 waveformsin terms of standing waves reflected between conjugateionospheres: in this case, the rapid damping would bea consequence of the much lower ionospheric conduc-tivity in the nighttime hours. The different waveformsas well as the observation of dayside events suggest ad-ditional Pi2 sources, such as plasmapause surface wavesand cavity resonances of the inner magnetosphereat middle and low latitudes. For example, Yeomanet al. (1990) proposed the superposition of wavesfrom the auroral current system with plasmasphericcavity resonances. Takahashi et al. (1995) conducteda statistical analysis of Pi2 pulsations and found that Pi2were detected in the nightside and primarily at L < ;they also suggested cavity mode resonances excitedin the inner magnetosphere ( < L < ), boundedbelow by the ionosphere and at high altitude by an

U. Villante

Fig. 16.11a–d. Aspects of the Pi2 pulsations. (a) A low latitude Pi2. [Villante et al., 1990]. (b) A representation of theSCW. [McPherron et al. 1973]. (c) The variation of the H and D component due to the substorm manifestation at middle lati-tudes and the Pi2 polarization pattern. [Lester et al., 1984]. (d) The period of Pi2 event vs. the fundamental period of the toroidalmode at the latitude of the auroral breakup. [Kuwashima and Saito, 1981]

Alfvén velocity gradient. Olson (1999), who reviewedtheoretical and experimental aspects, proposed a globalscenario in which the Pi2 signal encompasses a classof pulsations generated by the same event: the onsetof FAC associated with the current disruption in thenear Earth plasmasheet and the impulsive responseof the inner magnetosphere to compressional wavesgenerated at the substorm occurrence or intensification(Fig. 16.12). In this scheme, oscillations in the SCWcurrents produce high and middle latitude Pi2 signals,while compressional waves, traveling inward, stimulateFLR and surface waves at the plasmapause which can beobserved near the plasmapause footprint. In addition,at low latitudes, other resonant and global modes ofthe inner magnetosphere can be observed. Numericalcalculations of cavity quencies gave results consistentwith the low latitude multiharmonic observations.

Higher frequency components of Pi2 waves in theionospheric cavity may produce Pi1 pulsations, whichrepresent an additional typical manifestation of auro-ral and subauroral latitudes; they occur after substormonset and are correlated with pulsating aurora. A com-parison of the results from high latitude stations re-vealed that the significant Pi1 activity associated withsubstorms detected at λ � −� becomes weak at λ �−�. This feature is consistent with the lack of powerenhancement during nighttime hours deep in the polarcaps.

Broad-band bursts of PiB pulsations (Pi1+Pi2,from several Hz to � mHz) are observed at FAConset. Typical burst duration is � – min and – PiBimpulses occur in � – min (Kangas et al., 1998).PiB are also used as high time resolution monitors ofsubstorm development.

Ultra Low Frequency Waves in the Magnetosphere | References

Fig. 16.12.The diagram outlining the flow of energy that man-ifests itself as Pi2 oscillations in the magnetosphere and atground level. [Olson, 1999]

PiC pulsations appear in a narrowband of frequen-cies which is continuous in time, sometimes up toseveral hours. They are generally seen in the morningwith periods of tens of seconds (at the lower frequencyend of Pi1) in correlation with auroral luminosity vari-ations. They have been modeled in terms of a currentsystem of patch of enhanced conductivity in the iono-sphere.

Large amplitude, damped fluctuations (denoted“magnetic impulse events”, MIE, or “traveling currentvortices”, TCV) are also observed from λ � � toλ � �. Most of these transients events consist insingle-cycle pulsations, with periods between � – s, and are possibly associated with flux transferevents (FTE, patchy reconnection events betweenIMF and magnetospheric lines across the daysidemagnetopause) and FAC, or SW pressure pulses.

16.13 Concluding Remarks

As is clear from the arguments above, ULF waves, en-demic within the magnetosphere, are involved in ma-jor manifestations of the magnetospheric dynamics andplay a significant role in the energy transfer from the SW

to the magnetosphere. In the author’s opinion (and ex-perience) some interesting arguments for study in thenear future, among others, are:

− the penetration mechanism of external waves intothe magnetosphere (via the magnetopause nose,flanks and magnetotail lobes), the role of IMFstrength and direction, the wave propagation insidethe magnetosphere, the characteristics of the waveenergy transport, the role of different instabilityprocesses;

− the definite identification of cavity/waveguidemodes, the possible correspondence of “discrete”frequencies modes with simultaneous compres-sional and/or Alfvenic fluctuations at the samefrequencies in the near Earth SW, and their possibleassociation with solar oscillations in the mHzrange;

− additional sources of Pi2 manifestations at middleand low latitudes (plasmapause surface waves, cavityresonances of the inner magnetosphere, etc.);

− the correlation between pulsations and auro-ral manifestations, with a special emphasis onpulsating aurora;

− the role of ULF waves in the energization andtransport of radiation belt particles;

− the improvement of experimental methods formagnetospheric diagnostics, particularly importantat low latitude where spacecraft measurements aregenerally not available.

Acknowledgement. Author is grateful to drs. L. J. Lanze-rotti (New Jersey Inst. Tech.), P. Francia, M. Vellante(University of L’Aquila) for useful comments.

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