VOL. 79, NO.7 JOURNAL OF GEOPHYSICAL RESEARCH MARCH 1, \974

A Theory of Long-Period Magnetic Pulsations

Steady State Excitation of Field Line Resonance1.

~

A theory of long-period (Pc 3 to Pc 5) magnetic pulsations is presented based on the ideaof a stady state oscillation of a resona,nt local field line that is excited by a monochromaticsurface wave at the magnetosphere. A coupled wave equation between the shear Alfven waverepresenting the field line oscillation and the surface wave is derived and solved for the dipolecoordinates. The theory gives the frequency, the sense of polarizations, orientation angleof the major axis, and the elliptieity as a function of magnetogphericparameters. It also

clarifies some of the contradicting ideas and observations in relation to the sense of

polarization and excitation mechanism. At lower latitude it is shown that the orientation

angle rather than the sense of rotation is a more critical parameter in finding the directionof wave propagation in the azimuthal coordinate and hence in finding the evidence of Wlweexcitation at the magnetospheric surfaee by the solar wind.

Theories of magnetic pulsations can be categorized basi-cally into two groups. One group of pulsations primarily

concerns those with the excitation mechanism, which is the

'active' aspect of the problem, and the other group of pul-

sations concerns those with the resonance mechanism, which

is the 'passive' aspect of the problem. For the long-peri'od

pulsations ('T ?::: 45 s) the former group considers exeitationby the Kelvin-Helmholtz instability of the outer magneto-

spheric plasmas due to the solar wind [Dungey, 1955;Parker, 1958; Sen, 1963; Southwood, 1968; Boller and

Stolov, 1970J or by the local plasma instabilities such as two-

stream [Ni$hida, 1964; Kimura and Matsumoto, 1968J, drift

wave [Swift, 1967; Hwsegmva, 1971a, bJ, or mirror [Hase-gawa, 1969J instabilities.

The theories that treat the passive aspect of the prob-lem are mostly centered around the idea of the magneto-spheric cavity resonance [Watanabe, 1961; Dungey, 1963;Radoski, 1966J or the resonance of the local field line (inthe limit of a large wave number in the azimuthal direction[Radoski, 1957a; Dungey, 1968; 01'1', 1973J).

Observationally, these long-period continuous pulsationscan be characterized by the following properties.

1. At higher latitude the mostly circularly polarizep pul-sations reverse their sense of polarization near nooll [Sam-

son et aI., 1971J, whereas there is no such systematic re-

versal at lower latitude [Lanzerotti et ai., 1972J.

2. Polarization reversal also occurs at a different latitude[Samson et cd., 1971]. The demarcation line coincides withthe line of peak wave amplitude and linear polarization.

3. Systematic change of the orientation angle in theH-D plane occurs from the second to the first (the first tothe second) quadrant near noon in the northern (southern)hemisphere [Van-Chi et aI., 1958; Lanzerotti et aI., 1972].(As is illustrated in Figure L the tendency is less obvious inthe afternoon sector at Siple.)

4. For each of the events the frequency is independentof latitude, but when it is averaged over many events, thefrequency of the peak amplitude is a decreasing function oflatitude [Sam.son and Rostokel', 1972J.

When the consequences of the theoretical predictions are

~~

Copyright @ 1974 by the American Geophysical Union.

LID CHEN AND AKlRA HASEGAWA

Bell Laboratories, Murray Hill, Nf)w Jersey 07974

compared with these properties of pulsations, it can easilybe seen that none of the foregoing theories are satisfactoryenough to explain the observed phenomena consistently.

For example, although the cavity or the field line reso-nance idea.s can explain either the latitude independent ordependent pulsations by choosing a small or large flZi-muthal wave number, they fail to explain a change of thesense of polarization or the orieI\tatton angle near noon. Nordo they explain why most of the pulsations are ellipticitlly

polarized with its major axis always tilted in the H-D plane,

Le., not aligned with the H or D axis. On 1,110 other hand, the

Kelvin-Helmholtz theory, which is successful in explaining

the switch of the sense of polarization near noon, fails toexplain latitude dependent frequency and the lower-latitudepulsations that do not reveal systematic switches of thesense of polarization.

The local instability theories may apply to some specificcases, but again they are difficult to apply to the majorityof the phenomena that seem to have dependency on aglobal effect such as dawn-dusk asymmetry.

A natural step to take here is therefore to combine the

a~tive lind passive theories and to treat the problem of

c.oupEng between the active and the passive modes.. Becausemany of the high-latitude pulsations seem to switch theirsense of polarization near 1100n, for the act.ive mode itmay be natural to take the Kelvin-Helmholtz mode or asimilar surface mode excited by the solar wind friction.

Whereas fora passive mode it is natural to take the localfield line oscillation by. the shear Alfven wave because ofthe observed localization of the pulsations. A coupling be-tween these modes is being studied independently by So7.dh"

W(lod [1973J, using a straight field line model. We employthe dipole field, which is crucial in deciding the orientationangle of the major axis of the polarization ellipse as will beseen.

METHOD o~' ApPROACH

Here we introduce the basic method of the theoretical

approach to the problem. We consider the oscUlation of the

local field line excited by a surface wavc. The surface WRveis assumed to be monochromatic a.nd is exeited by theKelvin-Helmholtz instability at the magnetospheric bound-

~

CHEN AND HASEGAWA: LoNG-PERIOD MAGNETIC PULSATIONS, 1

0 a

ary or by some other type of friction between the solarwind and the magnetopause. We start with a review ofthe Kelvin-Helmholtz instability. In a unifonn plasma the

surfa,ce wave equation that may be ~xcited by the Kelvin-

Helmholtz instability is given by [e.g., Sen, 196'~1J

\J2[p + (b-B//-Lo)] = 0 (1)

where p is the perturbed plasma pressure, band B are theperturbed and unperturbed magnetic flux density, and J..Lo isthe vacuum magnetic permeability. Because this is a IJaplaceequation, the general solution associated with the surface

perturbation is of the form

exp [i(k.Lx + kllz) - (k.L2 + kIl2?/2 lyjJ

where x and i are parallel tot.he surface and y is perpendicu-lar to the surface. N'ote that the exponentiating distance iscomparable to the wavelength parallel to the surface. Justinside the magnetospheric boundary, x, V, and z correspond

approximately to the a,zimuthal coordinate, the radial (in-

ward) coordinate, and the coordinate parallel to the mag-

netic field, respectively.

Because the magnetospheric boundary is not under a

dynamical equilibrium, a dissipative layer must be assumed

there in general [Eviatar a:nd fiT olf, 1(168]. This assumption

modifies the simple theory of the Kelvin-Helmholtz insta-

bility. However, we assume here that the ordinary resultof the Kelvin-Helmholtz theory still holds if the perturbed

~

1025wavelength is much larger than the proton Larmor radius,which is a typical size of such a dissipative layer.

When incompressibility is assumed for simplicity, the

condition of the instability is well known (see, for example,

Boller and Stolov [1970]) and is given by

(koV.)2 > (~. + -~ JX [N,(koVA),2 + Nm(k'VA)m2j (2)

where subscripts sand m stand for quantities for the solar

wind and the magnetosphere, v, is the velocity of the solar

wind, VA is the Alfven velocity whose direetion is taken to be

parallel to the unperturbed magnetic field, and N is the

unperturbed number density of the plasma. Because IVA I ex:

BN-1I2, the magnetic flux density inside the magnetosphereis much larger than that outside, and N, » N m, (2) can inmost eases be simplified to

kov. , , kl.v, > (koVA)'" = kliVA (2')

where kll is the wave number parallel to the magnetic fieldand VA is t,he magnitude of the Alfven velocity within themagnetosphere. When the compressed geomagnetic field at

the dayside (, ,100)'), VA , , 2 X 10' km/s, a,nd v, , , 4 X

10' krn/s are taken into account, (2/) is satisfied with a large

perpendicular (azimuthal) wave number k, ~ 5 kll) and the

minimum value of lell is decided by the local length of the

field line s by kll = 7r/s.At the threshold 'of the instability the excited frequency

is given by

N, 1UJ = k'v, N+ T "'-' kov, , , "IIVA, m

Therefore it is conceivable that a set of discrete monochro-matic frequencies corresponding to the (2n - 1)7r (n = 1,2, . . .) modes in the standing shear Alfven wave at the mag-netospheric boundary is preferentially excited.

Now we consider how this surface wave can couple to alocal field line oscillatic:>ll deep within the magnetosphere (L

-4). We assume that a onc-flui<.i MIlD equation can suf-

ficiently describe the Alfven wave perturba,tion. With this

assumption we ignore a possible inteJ"action with electro-

static drift. waves. After the electric field is eliminated, the

relevant linearized MHD equations bec~ome

.. 1 1

Pm( = ~ (\1 xb) xB - (\1 xU) xb - \1p

~o ~o

b = '7 x (l; x B)

where ( is the displacement vector defined by

iJl;/iJt = vand v is the perturbed fluid velocity. In (4» Pm and pare

mass density (= miN) and the perturbed plasma pressure.

Because we are interested in the oseiJIation of the local

field line, which is possible only in the shear Alfven waveperturbation, we seek for an incompressible perturbation.

The mass conservation gives

n + N'7' ~ + ~. '7 N = 0

where n is the perturbed number density. From (7) we can seeimmediately that the ordinary incompressibility assumption,n""'" 0, \1ol; , , 0, does not hold because of the nonuniform

plasma density, \1N ;;t!i 0. The incompressible perturbation

~

0H

D

-(3)

(4)

(5)

(6)

(7)

1026 CHEN AND HASEGAWA: LoNG-PERIOD MAGNETIC PUI,SATIONS, 1

(nr-v 0) will have a finite \/, ~ r-v -1<N'~' where1<N r-v \/ (In N); Substituting (5') into (4"),

otherwise, a finite density perturbation n '" f.' \/ N '" .. . 2

(1<N' ~)N results. This fact indicates a coupling between the }.I 0 Pm ~ - (B' \/) ~ = -}.I

shear Alfven wave of our concern and a compressional mode. whereIn association with the variation of the number density,

the plasma pressure also changes. Using the adiabatic C = (b.\/)B -(B'\/assumption and (7), we have for the pressure perturbation ' th b . . . b (5.' ) EWI given y . qua'

p = ---' NT(1<p' ~ + \/. f.) (8) coupling betwe~m the w~v. 'Y wave, representmg the oscil

where T is the plasma, temperature (in energy units) and side is equal to 0), lUU.! the s

1<p '" \/ (In NT), The set of equations (4), (5), (7), and (8) the coupling coefficient C, w'

contains three basic MHD waves: the ion acoustic wave, the nonuniform magnetic field.,

ma,gnetosonic (compressional or isotropic Alfven) wave, and However, if we evaluate

the shear (anisotropic) Alfven wave. (The incompressional sides of (12) using our sma

surface wave of (1), as will be shown, becomes an evanescent hand side is found to invohcompressional mode in the presence of compressibility.) side by a factor of c2. Thi:Because we are looking at a wave having a large perpendicular resonance of the shear Alfvwave numl.)er in consistency with the Kelvin-Helmho1tz unless a particular choice I

perturbations, the magnetosonic wave will have a frequency. reduces the size of these Ia:much higher than the frequency of our interest (which is of The property of this uniquthe order of the shear Alfven wave resonanceof the local shape of the unperturbedfield line). This fact enables us to reduce the coupling between form will be shown in the nthe .shear AIfven wave. and. th .e rest of the mode by using two W . ~;'N . AVE .=.,\UA'l'ION ]

smallparameters Eand {3, where . .. . .. ..

1n this section we elabl

E r-.J kll/kL r-.J KN/kL , "" KB/kL "-' Kp/kL« 1 (9) in the preceding section to

. fora dipole field, WeadoI

(3 =_fT .. 10) /-(,If) used by ~adoskip9(B /211" ( . nates and their resocctlvc

with 1(8 = V In B.'Vith these preparations we introduce here the basic

scheme of the coupling and the theoretical approach to the

problem.

Let us first see how the surface wave (1) is modified owing

to the non.uIljformity. After simple vector operations in (,1)using \1.b =\1. B = 0 we have

\!(/J.oP + b.B)

Henc~if w~ take the divergence of both sides, we have2 . ... : . . . .."

V (p + bo:a/JLo) = 2V'[(b'V)BJ/JLo - Vo(Pml;) (11)

If the plasma is uniforman.d if the iqcompressibility (V 01; =0) is assumed, we recover the wave equation of the surface

wave given In (1). If the plasma density were 'Uniform, we

could assume an incompressibleperturbation, but \l2(p +b.B/ILo) r6 0 if thf3 magnetic fif3ld were nonuniform. Thisfinding shows that the surface wave can couple to the shearAlfven wave through the nonuniform magnetic field. Thecoupling would. vanish. if the. perturbed magnetic field werelinearly pol!trized in the azimuthal direction (because then(b'V)B =0), butbecaus~ the surface wave is circularlypolarized, that cannot be tlw case.

We now 1ook at the modification of the shear AJfven waVeby the surface wave. From (4),

whereas from (5),

b = -}-B(V'O + (B'V)~ - (~'V)B

- Jlo\7[p + (B. bl Jlo)] + C (12

C = (b-V)B -(B-V)[(~'V)B + B(V'~)] (13

vith b given by (5'). Equation 12 illustrates very nicely th,:oupling between the wave. equation of the shear AlfvCIvave, representing the oscillation of the field line (right-han<.ide is equal to 0), am! the surface wave (equation 1) throng!

,he coupling coefficient C. which appear:;; ill conseq uencc of. t h,

lOuuuiform magnetic field.

However, if we evaluate the orders of magnitude of botl,ides of (12) using our smal! parameter:;; e and (3, the rightland side is found to involye terms larger than the left-hanllide by a factor of e-2. This result implies that the field lin'esonance of the shear Alfv~n wave is completely wiped oumless a particular choice of .~ vector is made such that i:educes the siz!;) of these large tenllS on the right-hand siderhe property of this unique ~ vector depends on the actua;hape of the unperturbed magnetic field, and a concret~orm will be shown in the next section.

WAVEEQUA'l'ION IN DIPOLE COORDINA'rEs

In this section Wt1 elaborate on the a,pproach describein the preeedingseetion to derive the coupled wave equatiofor a dipole field. We adopt, here the dipole coordinates (1J-t, If) used by Radoski [1967b) (seeFjgllre 2). These coqrd:nates and their respective scale factors a.re related to thspherical coordinates (7", (), 'P) as follows:

v = sjn2 (Jlr J.! = oos 81r2 <P = <P

h, == (r2/sin.8)(1 + 3 cos2 8)-1/2

h", = Tsin 8 hv = h,h", = M/ B

wberc.¥ is the earth's magnetic dipole moment, and B is t!field strength. the unit vector 1.1 is directed along the field lill~ is ill the azimuthal direction,and v = V x«J is norina) jthe fie~d line and pointing toward the earth (pl'oji~ctecl to tlnorthern (southern) hemisphere, <p and v correspond to

and - tI (+ H) direction). We assume perturbations of tl

followin.g form:

~(J.!, <p, v, t) ~ e-iw'e;m",~(,.I, v)

Here 1m! "-' lie» I, and HJ.I. v) satisfies the MHD equatio

as well as the boundary coriditioT\s. Equation of motion (1

SOLAR WIND

;;//0 . ,"0

Fig. 2. Illustrations of the dipole (v, J.I., <p) coordinate UBin the text.

NT-

(3 =B.212~o

. .

2(b.V)B -/.LoP,,:~ + \! x(b xB)(4/)

The

circularly

(5')

[

CHEN AND HASEGAWA: LoNG-PERIOD MAGNETtc PULSATIONS, 1

= .1 ~

[ JlOP + b"B] + 2K Jl~T!.h, Uv B2 8, B2

2 1 apw Pm~'" = -7 - a -L" Jl

[ W2 1 U ( B U

)];? + Bh" UJl h" UJl ~i

~, a

( B ah , )- hi~hp a}.L hp a}.L'

Thus (17) with KI;' = 0 corresponds to the guided waveequation for the toroidal (~, = 0) or poloidal (~~ = 0) mode.Because we are primarily interested in the coupling problemin the v direction around the equatorial plane, we do notsolve the detailed boundary value problem in the If. direction;instead, we assume that a suit!1blc solution in the J.l. directionis obtained in a WKB form. Then operators [{Ai' whenoperated to HIf., v) become a function of J.l. and v. P!1rticularlyne!1r the equator, where If. "-' 0, [{Ai' can be regarded as a

function of v only. The perturbed magnetic fwld b is relatedto the fluid displacement ~ through (5'), which in the dipolecoordin!1tes becomes

b" = _[ im L + .l a~, - ~, ( J:.2 ah~

)JB h~ h, av h, av

~ = -(\7'~) + ~ ~~~ - KB"~" - 2KB'~'" ft

b~ = 1. o~~ - (~l. Oh~) ~

B h" Oft h~h" 01£ ~

~ = 1. o~, -

( ~ Oh,) ~

B h" oft It,lL" oJ.! '

Equation of state (8) relates the perturbed pressure p to t

I.e. ,

ftop/B2 = ~{3[1<p'~ + -y(\7.nJ (8')

In general, the above set of MHD equations is difficult to

solve and only special cases where the plasma is cold and

the compressional component vanishes have been treated

(see OrT' [19n] and referenees therein).

As was desc,ribed in the preceding section, we approach

this problem by using two small parameters, and (3. First,

we discuss the physical implications of the two sll1.a1l param-

eters from ordering arguments. From (W), (18), and (18')

we obtain the following order estimates:

jJop/B2 "-' {3(Bb"/B2) (21)

j = 'P, v (22)J(A/~i 2Bb~--- r-v -,--kJ.. B2

~

1027

(15) dominates over the left-hand side, and the oscillation ofthe field line becomes 'forced' oscillation rather than 'natural'oscillation. Thus from (14) we can solve bp and p in terms of~ by successive approximations. That is, we put to zerothorder

[B bpi B2]'0) = 0 (2:3)

or from (18) and (18'),

(14)

(15)

(16)

oi = rp) v (17)

Combining (25), (27), and (16), we c~1n see that ~f,(OIf3~/0I; (25) and (27) then become

,...,

(\1'~)() ~ -2KBi;,(O) (25')

J.Lop(O)jB2 ~ -(j[Kp~,(O) + /,('\7-0(0)J (27')

In (25') and (27') we have omitted the subscript jl from}(n and Kp to simplify the notation. Sllbstituting (2.'1), (2.5'),(18)

(18')

(19)

(20)

(26), and (27') into the right-hand side of (15), we arrive atthe wave equation for ~" only:

2 [ (' 2) ].2- a ~~ + ~1-- i. ~- T7 2: - .2- Qhv 1-. Q~

2 2 2 2 L\-A ~ 2

hv Jv h~ KA ~ Jv h, h, Jv hv av

[ 22 ( )+ 4')'fJ m KB 1 ~-h/ KA ~ 2 - 2')'KB

- ---~-~ i. (!:._~!S:A/. Qh; ) rn2J(A,2 J ~h/KA/hv Jv h,z Jv - h~2KA/ ' =

notation. Note, for unif'OITn plasrnas K,~2 = 1(,; (= const) ,then (28) becomes

2(1 a2 rn2)KA'I' h,?' av2- h/ ~, = 0i.e., the shear AlfvE',]} wave (KA/ =: 0) and the compres-sional surface wave (k; + k,/ = 0) are decoupJed. Notealso tb,atin a nonuniform ease as m -+ 00 J KA; = 0 is asolution ,'Of (28). This mode corresponds to R poloidaJ oscil-lation obtajned by Dungey [1968J. However, this mode doesnot have a resonant coupling with the surface wave andhence may not be excited strongly. Equation 28 thus indi-cates the coupling between the surface wave and th(~ shear

Alfven wave given by K,/ =0 due 1.'0 nonunifoffi1ities aswell as field line curvatures.Rou~;hly spea.king, becausem' , (-' » 1 and IC~2/K..,,; ~, 1, the coupling is weak(, (2) except near K..t/ = O. V"! e therefore exp!)ct an cx-.istence of a surface wave away from the resona,nt field linewhere KA/ = 0 and a. shear Alfvpn wave near the resonant

028 CHEN AND HASEGAWA: LoNG-

leld line. Because in the equatorial region the surface wave:xcited by the solar wind has a maximum amplitude andhe coupled wave equation assumes a simplest form, we shall:onfine 'Our treatment in this region and thus reduce the)roblem to one dimension in order to study the nature of:oupling in more detail. Near the equator (p. ~ 0), (28)hen becomes

~2~, + [ -l, dKA 'P 2 + ~J d~.

~v KA 'i' dv v dv

+ [ ~ dKA:" - ";2 KA,22 --~ J ~. = 0 (29)IIKA'i' dv v KA'i' V

1/ LoRE ~ V ~ I/RE

\There

A - ' 2 + ' 4 R 2 2 1. - (Kd.2"1Kn)- ", "II-' m KB ,v dKA 'P 2/ dv

md L. is the L value of the magnetopause. It is underst'Ood

lere that all quantities correspond to values at p. ~- O. In

~articular, KA. and K... are not operators but are functjon5}f v, Although the values of KA: and KA",2 can be rather,lose in low latitudes (iJhjliJj-L, iJB/{fj-L 0:: cos f) ~ 0), we take

~hem to be different. Specifically, we assume K.: ¥= 0 at

K.",2 = O. However, it becomes apparent in later analyses

~hatthe qualitative picture of the coupling is insensitive to~he exact relation between K.",' and K.:. Equation 29plus the appropriate boundary conditions completely de--.ermines ~.. Wave polarization in the tp-vplane thencan be found from the ratio b./b, (see the appendix); i.e.,near the equator, -

J1-'E. = a + iO ~8J;:d8p.b. 81;./ 8p.

~ 1-

[ ~ dl;. + 2v + V2 ~~k.] (30)

mv 1;, dv Bv

Here

ik. == ',.

[81;./ 8E]L .>Co

In the next section we discuss the solution and the associ-ated wave polarization near the resonant field line.

NEAR THE RESONANT FIEW LINE

, When it is assumed that KA.." = Q at v = Yo, then near

the resonant field line we have

KA/ ~ (v ~ vo)( d~:", 2t (31)

, '

Here we usethesubscript 0 to denote quantities evaluated

at v= Vo, and (dKA",'jdv). is taken to benonzero. Substitut-

ing (31) into (29) , we immediately see that the solution has

a singularity at v =Vo. In fact, as wm beshown later, this

singularity is logarithmic. As is- realized also by Southwood[1973], the nature of this singularity is the same as that ofthesingularities encountered in studying wave propagationsin an inhomogeneous medium where the refractive index be-comes infinite [Budden, 196i; Ginzburg, 1967]. The wave

amplitude becomes infinite there, and the energy is ac-cumulated in the vicinity 'Of the pole. This physically un-justifiable result indicates the inappropriateness of treating

d2~. + [ -~ dKAop2 + ~J d~.

dv KA'i' dv V dv

where

Here

CH~1N AND HASEGAWA: LoNG-PERIOD MAGNETIC PULSATIONS, 1

the wa.v~ propa.gation as a steady state process [Ginzburg,1967]. To eliminate this difficulty, in accordance with thecausal relation of the Laplace transformation il1 time, wehave to assume a. small positive imaginary part in (d:

'" = "'r + i"'i Wi > 0 1",';00,1 « 1

K

. 2 ~ [ d(KA. ./)R ] IIO [ ~ - 1 +i1] JAop dll 0 110

(32)

(29)where

(J{,j /)R = He(K,j /)

71 =2wi~r

[ d(KA~)~ ] -1 (33)

VAO va dv a

Note hi ,..., O(W,/Wr) « 1, but in the presence of a loss(such as that due to ionospheric dissipation) I "" is not in-finitely small but rather positive finite to overcompensatethe ~ect of loss; hence, in general, TJ has a finite positivevalue. By defining the new va.riable

(29) beC'Omes, near 8 == 0,

d2~!. + [ 1 + 2

J d~. + ( Go + Do ) ~. = 0

dz z dz z

(34)

Izi « 1

Here

Go =

(30)

~v ~ C2~lv In z(35')

For Izi « Eo, ~'" ~ 1, and ~y can be further approximatedto be

~. ~ C2 In z = C2[~.n + i~/J

= C2[l + i'lr /In \2\] In 12\ (38)

where

CHEN AND HASEGAWA: LoNG-PERIOD MAGNETIC Pur,SATIONS, 1

Meanwhile, we have d~,,/dz ~ a"and from (35')

~:' ~ C2[ (~:'r + i(~:')]

(d~,)[ 1/ + .T,a;. ~ -~2 a""

Equation 30 then gives the ratio b~/b"wave polarization:

In the following three subsections discuss separatelythe values of a and I) as well as the asS'Ociated wave polari-zations for 18/7]1 « I, 18/7]\ ~ 1, and 18/7]1 » 1.

Case 1.. 18/7]1 «1. Here 'fr ~ :!::7r/2, the upper sign

being for 7] > 0 a,nel the lower sign for 7] < O. With 17]1 «

1 we obtain from the above equations

1 .(d~, ) I 1a ~ - m In 17]1 dz "-' -rn7] 1;j.;;T

1 [8 \}i

Jo~m7]lnl7]1 ~-Inhl

Thus the sign of Ci depends on that of in7]; i.e., sign (a) =-sign (m7]). With 7] > 0, a > o for in < 0, and the majoraxis of the ellipse lies inside the first quadrant of the cp-v plane.For rn < 0, a < 0, and the major axis lies inside the secondquadrant. With 7] < 0, the results are opposite. As for thesense of polarization, it depends on the sign of 5, which isindependent of the sign of 7]. For 18/7]1 < 1"/2 In 17]11 (e.g.,exactly at the resonant field line, 8. = 0), sign (5) = - sign(m), Hence wave polarization is all the left-hand side for

m < o and on the right-hand side for m> 0. A little away

from the resonance, where 18/7]1 > 111/2 In 17]11, sign (5) =

-sign (8/ Thus for m < 0, the wave is left-hand-polarized

~

1029

LH

C

LH

C

(39)

m<O

( 40)

( 41)

which decides the Fig. 4. Sketch of the wave amplitude and sense of polar-ization versus radius in the equatorial plane for m < 0 (localmorning). When it is projechod along the field line to theground, it can be viewed as a plot versus latitude. The Cand E stand for circular and elliptical polarization, respectively.

The results are opposite for m > 0 (local afternoon). The

1'0 is the location of the resonant field line. The elliptieityis large (linea,r polarization) near the resonant; field line anddecreases away.

( 42)

( 43)

for s > 0 (closer to the earth) and right-hand-polarized fors < 0 (farther from the earth). For m > 0, the results areopposite. Since 18/0'1 « 1, the ellipticity is large (closeto a linear polarization) with the minimum value, 8M ~ 1 -10/0'1 at a~ "-' l.

Case 2: 18/~1 "-' 1.Here '¥ '" 0(1), ane! we obtain

a ~ 1/21)m In. 1111

b ~ s/21)2m ]n 1111

( 46)

( 47)

Therefore the major axis has the samepreceding case does. Again, the senseindependent of 7/, sign (Ii) = - sign (81m), and has been

discussed in the preceding case. Since 10/ al ~ 1, the ellipticityis smaller than that of the preceding ease with 8M ~ 0.59 at

a2~0.5.

Case 3: Is/1]1 » 1.

properties as theof polarization is

(44)

( 45)

w "-' :!= his I

W :!=7r

Again, the upper sign is for 1) > 0, and the lower sign is for7J < O. Then a and 8 become

1 YIa c::::: ~- - s > 0

ms In Isl s

( 48)l ( 11' YI)a "-' --~- ::!::-~ + - 8 < 0

- ms In Isl In Isl s

i5c::::: l/ms In Isl (49)

Note here that for s < 0, YI/s and::!:: l1'/ln Is! have the same

sign. Hence sign (a) = - sign (YI/m); and the major axis has

the same properties as given in the previous two cases. Also,sign (15) = - sign (ms), which indicates that the wave polari-

zation has properties similar to those of the preceding ease.With la/al » 1 the elliptieity is smaJler (dose to a cireular

polarization) with 8M "-" 0 at i5 ""-' 1. The above results aresummarized in Figures 3 and 4.

Finally, let us assume K..; = 0 at v = 1'0, where IC/ =0, and diseuss the resonanee coupling. It is apparent thatthe (Jl11y changes needed in the above analyses are now

Go= A" , , m2/3 , , ,,-1 (when it is assumed tha.t /3 , ,)

~

.1030 CHEN AN!) HASEGAWA: LONG"

and Dp = - (?n' + 2) ,..., t"', Thus a, ,..., f:"" and a. ,..., (0';in other words, all theaboyc re$ults are expected to hold

for !zl « instead oflzl « <c' as tl~ey do in the. case with

IC: .¥- K,,:, The changes arc therefore only quantitative.

That is, for Izi «co, we have

and for Izi « <,

Ib,,/b.1 ?: Il:~1 « 1

'hus in the former case the waves t(;)nd to incline toward:1e <{J direction. In the latter case, however, the waves have0 particular tendency of inclination. Or,. ulnd 1\.{ atthewi971] compute the resonance frequencies for the poloidalnd toroidal oscillations and find that they are almostlentical (1-3% difference) except for the fundamental

,armonic (30% difference); that is, the latter case may

Ie more common in pmctical situations.

AWAY FaoM THE RESONANT FIELD LINE

In this case, because

(dKA 2/dv) -]~.~ ,r-J V

KA2

me! lC,/JR..; r:-' 1, the wave equation (29) can be ap-Jroxjmatecl to be

d2~, m2

dV2 - ;;-~. ~ 0 (50)

, " '1 1,

Vo < v ':::;- R L R - :::; v < Va, E b E

fhis is a surface wave type equation (i,e., k; + k,; ~ 0).[n fact, it describes the evanescent compressional mode.fhe general solut.ion is

~. ~ Blvl"'! + B2v-lml (51)

rhe values of B, and B, depend on the boundluy concli~

~ions as well as the, actual position of th{', resonant i1eld

line: If we assume that the fields vanish near the earth's

surface, we then have approximately

~. ~ B2v-lml (52)

From (30) and (51) we seethat the wave polarization ismaiI1ly circular and tha,t theseIi,se of polarization dependson sign of m; i.e., the polarization is left-hallded for

m < 0 and is right"handed for 'In > O. These off-resonance

results are also summarized in Figure 4.

SUMMAHY AKD DISCUSSIONS

.. We have shown through general formulations that owing

to nonuniformities as well as field line curvatures, surface

\v8.ves .excited" by the, solar wind at the magneto pause cancouple to thesheal; Alfven waves (guided waves) of theresonant. local field lines inside the.' 1llagnetpsphere. Forsurface waves with fast a.zirm~thal (E-W) variations (smailt:) and in low 13 plasmas the cQupIing that retains thenatural ficld line oscillation can occur only for particularorientation of the ~ vector. (Otherwise, the nature of thefield line oscillation is dominated by the nature of the

CHEN AND HAS~;GA\VA: LONG-PERIOD M"GNETIC Pur,sATIoNS, 1

~xcitjng source.) We then apply this property to derive, coupled wave equation in the dipole coordinates. This

vavC) equation is then solved for the equatorial region.

Chef solution exhibits surface waves away from the resonant

ield line. Near tberesonantfield line, shear Alfven wave>ccurs, aJ.1d the solution has a logarithmic singularity. This

lDphysical property is removed by introducing finite dis-

;ipations. On the basis of the sohJ~ions we then examine,he characteristics of the wave pola.rization, i.e., the tilt of

;hemaj.or axis, the sense of polarization, and the ellipticity.

Phese results are summarized in Figures 3 and 4. Let us

)ow discusS' these theoretical results in thE: light of experi-

nental observations. Since the surfa,(~e wave is ,excited by

lhe solar wind, a natural choice is m <'0 for local morning

md m > 0 for local afternoon.

Tilt of major C1X1:S (F'igttre B). Theory indieates that

the tilt depends on the signs of 'I) arid m. When it is as-

sumed that 'I) >0, the theory then predicts that in localmorning (m < 0) t.he major axis lies in the second (first,)quadrant of the I-I-D plane for most latitudes in the north-ern (sout,hern) hemisphere. This prediction assumes thatthe major axis clirect.i'on cloes not change significantly alongthe direction of the field line. 'This assumption is supportedby conjugacy observation by Lanzerotti et at. [1972]. Fur-

thermore, the tilt is predicted to switch across the local

noo,n. These predictions are Consistent wit.h the low-latitude

conjugate observations of ~anzerotti et aI. [1972) and

Van-Chi ,et 01. [1968]. Samson [1972), using a string

of seven "tations located at different latitudes in, the north-

ern hemisphere (59°-77°N) has also observed similar

,henomena. ' "', ' ,

Let us now disCllSS the physical implications of TJ > 0as suggesteel by the observations. Sinee "-', > 0 to take a,c-

count of di&9ipations, T} >0 then implies that

[d(KA/)R/dv]o> 0

Because

" 22(' 2') R IJ) . 'IJ) 2KA'i' ~-<,j - kilO "'" -<,j - jJ

VA VA

then

fd (IC,,")/d.).

is generally negative except in regions (e.g" the plasmi\.-

pause) where VA decreases with v due to ra,pid density

increase (j.e" faster than EO). The present theory thus

suggests that pulsations tend to occur in regionR ',yheredensity increases faster thanE". '

Recently, Uberoi [1972]ha8 pointed out the clo!1e

analogy between the problem treated here and the prob-

lem of electrostatic oscillations in a cold inhomogeneous

plasma. By tiffing the smallpal'amet,er ~ and looking at

the steady state solution we have ignored in the present

paper the possibility of exciting the weakly dam ped co]-

lective ,(position independent) mode due to a. sudden jump

in v.< as well as the I'elated initial value aspect of this

problem [Scdlacc'!k, 1971J. These considerations are C\.lr-

rendy under study alid are reported in the companion

paper. , , '" ." Wave amplitude, sense, and ellipticity of polarization

(Figure 4). The theory predicts that the wave amplitudepeaks at the resonant field line. The polarization is linear

near the peak, and the ellipticity gradually decreases away

~~

~

CHEN AND HASEGAWA: LONG-PEIUOD MAGNETIC PULSATIONS, 1 103!

reverses not only across local noon but also across theresonant peak, which has diurnal variations, the scnseof polarization thus cannot be expected to exhibit a cleardiurnal pattern. Finally, let us note that the above resultsare also predicted by Southwood [1973] based on similarideas and a straight field line model as well as by AlcC/a:!![1970J based on the coupling of symmetric modes anel the

Hall effects.Latitudinal dependence of fTeqvenc1f. Because the rcoo-

nant frequency increases as the htitude decreases, ourtheory thus satisfactorily explains the observations ofSam.son and Rostokm' [1972]; i,e., for each individua.levent the frequency is the same for alllatitudes (observingthe Kelvin-Hclmholtz surface wave), and, however, thelatitude of the intensity peak decreases as the pulsationfrequency increases (observing the oseillation of the l'oealfielelline), In conclusion, we n,rnaTk here that the theoretical

approach of studying long-period magnetic pulsationo ascoupling between surfaee waves excited by the solar winda,t the magnet'opause and the shear Alfven waves of loealresonant Held lines has yielded results eonsisten\. with manyaspects of thc observational results.

ApPENDIX: GENEHAr, CONSIDERATION OF WAVE POLAHTZA'I'TON

In this appendix we study the characteristics of the wave

polarization given b",/b, = a + 1:8.\Vith the eonvention taken as e-""., Ii", and b, can be

written as

b, = cos U!t b", = Q cas (U!t - A) (AI)

}i'ig. 5. The diurnal variation of the polarization sense forquasimonochromatic micropulsRtions of frequency ~5 mHz[after 'samson el. al., 1971].

,from the peak, Furthermore, the sense of poJarization is

opposite in the two sides of the resonant peak, The theoryalso predicts the reversal of sense of polarization across

local noon, An these theoretical predictions agree with the

obserVlltion of Samson et ai, [1971] (see Figure 5), The

demlll'cation line in Figure 5 then corre..'iponds to theloc.ation of the resonant field lines, Let, us now explain

why in low Ja,titudes no clear dawn-dusk asymmetry is

observed, For [1 station located at low latitudes we can

expect the observed pulsations to be near the resonantfield line where the wave amplitude peaks, Othenvise, thewave signal Inay be too low to be detected, Another indi-cation of excitation of shear Alfven waves is that in a,low latitude most of the observed pulsations have periodsof the order of the local first harmonic shear Alfvenperiod ~40 s (Figure 6), Because the sense of polarization

~

PERIOD, see

Fig. 6. PloL of number events versus period at theconjugate points, Siple and Lac Rebours. It indicatesmost of the events have frequency near the localAlfven frequency (plot provided by, L. J. Lanzerotti).

~

wbere Den = a + i8 From (AI) we obtain

b/ - 2ab~b, + (c/ + f/)b,2 - f/ = 0

By using a new pajr of a,xes (b,/, bv'), whichangle eo in the counterdockwisc direction with(A2) C~Ul be reduced to the standard elliptic equation

b",'2 bv'2(o2/A) + fr/?C) = 1

(A2)

makes an

(b., b,),

(A3)

The angle eo is given by

~ 2a --

tan 200 = 1-=-(;;:2+8")

is decided by the relative value of A and C. It can be shownthat

( -~;) sin 280[(1 i'/? + '1c/] (A6)L1-C=

Hence, for a > 0, A < C, and b/ is the major axis; fora < 0, A > C, and b,' is the major axis. That is, for

a > 0, the maj'or axis lies in the first quadrant of tp-vplane; for a < 0, it is in the second quadrant.

From the above discussions and from (AI) it is easyto see that the sense of polarization is entirely decided

by the sign of I), i .e" left-hand polarization for I) > ()

and right-hand polarization for I) < O. Here the convention

two

that

shear

if A > C or a < O. In terms of a fixed ratio oflB/al wehave the following minimumellipticities. For IB/al « 1,

8M ~ 1 - 151/2

at lal ~ 1; for I a/a I ~ I,

8M ~ 0.59-

at la! ~ 2-1/2; for la/a! » 1,

8M~ 0

at 181 ~1.

Acknowledgments. The authors ackpowledge valuable dis-cussions with H. Fukunishi, L. :r La,nzerotti, and G. Rostoker

and ,correspondence with D. J. Southwood.

* * *

The Editor thanks J. C. Samson and D. J.. Southwood fortheir assista.nce in evaluating this paper.

REFERENCES

Boller, B. R, and H. L. Stolov, Kelvin-Helmholtz instabilityand semiannual variation of geomagnetic activity, J. Geo-phys. Res., 75, 6073, 1970. ' ,

Bu<iden, K. G., Radio Waves in the Ionosphere, pp. 474-481,Cambridge University Press, London, 1961. . '

Dungey,J. W., Electrodynamics of the outer atmosphere, inProceedings of the Ionosphere Conference, p. 229, ThePhysical Society of London, London, 1955. .

Dung[!y, J. W., The Ea"th"s Environment, edited by C. De

Witt, J. Hieblot, and A. Lebeau, p. 505, Gordon and

Breach, New York, 1963.

Dungey,J.W., Hy.clroma.gnetic waves,in Physics of Geo-magnetic. PhenomeM, edited by, S. Matsushita and W.

Campbell, 913 pp., Academic, New York, 1968.

Eviatar, A., and R. A. Wolf, Transfer processes in the mag-netosphere, J. Geophys. Res., 7$, 5161, 1968.

Ginzburg, V. L., The Propagation of Electromagnetic Waves

in Plasmas, pp. 345-349, Pergamon, New York, 19{)7.

Hasegawa, A" Driit min'or instability in the m!\gnetosphere,Phys. Fluids, 12, 2642, 1969. ' ,

Hasegawa, A., Drift wave instability oiacompressional modein ahighfJ plasma,' Phys. Rev. Leu., 27, 11, 1971a.

Hasegawa, A., Drift wave instability at the plasma pause, J.

Geophys. Res., 76, 11, 19719.

CHEN AND HASEGAWA: LONG-PERIOD MAGNETIC PULSATIONS, 1

Kimura., 1., and H. Matsumoto, Hydromagnctic wave instabil-ities in a non-neutral plasma-beam system, Radio 8ci., 3,333, 1968.

Lanzerotti, L. J., A. Hasegawa, and N. A. Tarta.glia, Mor-phology and interpretation of magnetospheric plasma waves

at conjugate points during December solstice, J. Geophys.Res., 77, 6731, 1972.

McClay, J. F., all the resonant modc:.'S of a cavity and thedynamical properties of micro pulsations, Planet. Space Sci.,18, 1673, 1970,u' ,., . ~..'

l'lISh!C1a, A., Theory 01 Irregular magnetic micropulsat,ionsassociated with a. IIlagnetic bay, J. Geophys. Res., 69, 94i,1964.

Orr, D., MagnetIc pulsations within the magnetosphere: Areview, J. At11WS. Terr.Ph1ls., 315,1, 1973.

Off, D., and J. A. D. Matthew, The variation of geomagnetic

micropulsation periods with latitude and the plasmapause,

Planet. Space Sci., 19, 897, 1971.Parker, E. N., Interaction. of solar wind with the geomagnetic

field, Phys. Fluids, 1, 171,1958.Radoski, H. R., Magnetic toroidal resonanCes and vibrating

field lines, J. Geophys. Res., 71,1891,1966.Radoski, H. R., Highly asymmetric MHD resonances: The

guided poloidal mode, J. Geophys. Res., 72, 4026, 1967a.Radoski, H. R., A note on oscillating field lines, .I. Geophys.

Res., 72.418. 1967b.Samson, J. C., Three-dimensional polarization characteris-

tics of high-latitude Pc 5 geomagnetic micropulsations, J,Geophys. Res., 77, 6145,1972.

Samson, J. C., and G.Rostoker, Lat-itude-dependent character-'isticsof high-latitude Pc 4 and Pc 5 micropulsations, J,Geophys. Res., 77, 6133, 1972.

Samson, J. C., J. A. Jacobs, and Q. Rostoker, Latitude-dependent characteristics of long-period geomagnetic mi-cropulsations, J. Geophys. Res., 76, 3675, 1971.

Sedlacek,Z.,Electrostaticoscillations in cold inhomogeneousplasma, 1, Differential equation approach, J. Plasma Phys.,

52~A lQ71 .

S;~,--A:-K~: Stability of hydromagnetic Kelvin-H:elmholtz

discontinuity, Phys. Fluid;;, 6, 1154, 1963. .Southwood, D. J., The hydromat,>;l1etic stability of the magneto-

Spheric bounda.ry, PlaTi-et. Space Sci., 16, 587, 1968.Southwood, D. J., Some. fqatures of. field line resonances in

the l!1agnetosphere,Plq,net. Space Sci., in press, 1973.Swift, D. W., A new ipterpretation of long-period micropulsa-

tions, J. Geophys. Res" 7'2, 4885,1967. ..

Uberoi,C., Alfvenwaves in inhomqgeneous magnetic fields,Phys. Fluids, 16, 1673, 1972, .

Van-Chi. F., B, M. Yahovskiy, A. A. Kovtun, O. M. Raspo-

pay, V. A. Troitskaya., IUld R. Schlich, Investigation of type

Pc 3. geomagnetic pulsations at magnetically conjugatepoints, Geomagn. Aeton" 6, 94, 1968.

Watanabe, T., Geomagnetic days and storm sudden com-mencel!1ents in high latitudes, Sci. Rep. Tohoku Univ.. SeT.6, 13, 62, 1961.

(Received June 22, 1973;accepted November 13, 1973.)