VOL. 79, NO.7

A Theory of Long-Period Magn~tic Pulsations

2. Impulse Excitation of Surface Eigenmode

A theory of long-period magnetic pulsations (Pca to Pc 5) is presented as an initial value

problem to explain impulse-excited pulsations. By using a. one-dimensional model a wave

equation that sbows a. coupling between a surfaGewave and a shear Alfven wave is derived.By solving this equation on the basis of initial value approach we conclude that there is a

continuous spectrum with damping proportional to inverse power of time. and that there are

weakly, damped discrete eigenmodes (surface eigenmodes) due to sharp variations in theplasma parameters. The. frequency IAI. and the damping rate "I of the surface eigenmode aregiven approximately by IAI.= ku[(B,2 + BII2)/p.,(p, + pUH'/:I and "1/"" = Ik!/V(ln v..") I re-spectively, where v..(:=B/(p<>p)l/O) is the Alfen speed, ku and k" are wave numbers paralleland perpendicular to the magnetic field, and subscripts I and II refer to quantities associatedwith each side of the surface. The result is used to explain recent observations of plasmapause-associated magnetic pulsations as well as magnetic pulsations excited by sudden commence-ments and sudden impulses near the magnetopause.

In the companion paper [Chen and Hasegawa, 1974]we presented a theory of long-period magnetic pulsationsbased on the idea of a steady state reSonance coupling be-tween a monochrolDatic surface wave excit~d at magneto-pause due to Kelvin-Helmholtz instability (and/or other

mechanisms) and a shear Alfven wave associated with

local field line oscillations. We have shown that this theory

can unify many contradictory observations and predict,among other things, the dawn-dusk asymmetries in the

sense of polarization as well as the tilt of the major axis

of the polarization ellipse. However,recentIy, Lanzerottiet ul.. [1973], using latitudinal data near the plasmapause,4ave observed weakly damped coherent oscillations atL ~ 3.2, which are. accompanied by impulselike perturba-tions at higher latitudes, L~ 4.4 and 4.0. The steady stateapproach, which shows latitude independent frequency,obviously cannot explain these observations. A, natur~l

step to take here is to apply an initial value .approachand study effects of sharp changes in magnetospheric plasmaparameters. The present paper. describes this approach.

First, we use a one-dimensional model and ideal MHD

equations to derive a coupled wave. equation. We then

obtai!,). the solution of this equation by constructing itsGreen function. For this purpose we note the analogy ofthepresent problem to that of an electrostatic oscillation in

a nonuniform cold plasma. From. this. analogy we canconcillde the existence of a continuous spectrum (non~

collective modes) and a discret~ spectrum ( collective

modes). Since the non collective modes damp away as

inverse power of time oWing to phase mixing, we con-

centrate on the collective modes, where damping can berather weak if sharp discontinuities in the plasma param-eters exist. We then calculate the osciHationfrequencyand damping rate in terms of magnetospheric parameters.Finally, we. apply these theoretical results to the recentobservations [Lanzerotti etal,., 1973] of plasmapause-associated magnetic plllsationsas well as magneticpulsa-tions excited' by the. sudden commencements and suddenimpulses at the magnetopanse [Saito and Matsushita,

Copyright @ 1974 by the American Geophysical Union.

MARCH I. 1974JOURNAL OF GEOPHYSICAL RESEARCH

LID CHEN AND AKIRA HASEGAWA

Bell Laboratories, Murray Hill, New Jersey 01974

1967] and show how the magnetospheric pa.rameters can

be inferred by using the observed frequency and dampingrate in time and in space.

MODEL AND THE WAVE EQUATION

We aSSllme the plasma to be an ideal MHD fluid with itsequilibrium properties (density p, pressure P, and confining

magnetic field B = Bz) varying only in the y direction. (In

the magnetosphere, x and y rough\y correspond to W-E and

radiaUy inward directions, respectively.) Here P and B

satisfy the equiliJ:>rium condition d/dy (P + B 2/2IL 0) = O.(If we include the effect of the curved field line, this conditionis modified,. In particular, one can have a situation in whichboth P and B increase' (or decrease) with radius.) Lineariz-

ing the standard set of MHD equations, we have the equation

of motion as

'.. 2 " ' " ' .

ILoP~'- (B. V) ~ = -ILoVP - B(B.V)(V.~) (1)

Here ~ is the fluid displacement \Tector,8~/at = v; p = p +B.b/ILo is the perturbed total pressure; b, the perturbedmagnetic field, is expressed in terms of ~ as

b = (B.V)~ ~ B(V.~) - (dB/dy) ~yz (2)

For a compressible ~uid the adiabatic equation of state

and the continuity equation relate the perturbed p and ~:

p= -~ydP/dy - 'YP(V.~) (3)

(For an incompressible fluid (V.~ = 0), one has a similarequation [Uberoi, 1972]. However, as is shown in the com-

panion paper [Chen and Hasegawa, 1974], the nonuniformitydoes not generally allow incompressible perturbation.)

We then take. the Laplace time ,and the Fourier space trans-forms of the perturbed quantities; i.e.,

~(y, kll' k;'1 w) =L'" dt L: dx L"'G) dz ~(x, y, z, t)

. ex? [i(wt - k;..x -kllz) 1m w 2: 0

:.Lx-kuz) 1m 00 2: 0

ikl. L: dku ~(y, ku,k.L' 00)

. exp [i(kl.x + kHz - oot)J

~(x, y, It, t) = (2~)2 1. cU.J i: dkJ. i:

1033

1034 CHJ;:N' AND HASJ;:OAWA: LONG-PJ;:RIOD MAGNJ;:TIC PULSATIONS, 2

where the integration path c runs parallel to the real axis which asymptotically correEof the CJJ plane above all singularities of ~, Using the above tions with position depende)

representations and (2) and (3) in (1), we obtain, after portional to the inverse powe

straightforward calculations, thE! following coupled wave Second, there also ex;

equation in ~": (position independent) wij[ 2 .. damping is weak if E has aha)-1:.-. eaB 2 2 d~" + E~" = S(y, CJJ) (4) When we,apply ~he abo~

dy E - aB k 1. dyJ . one sees themterestmgposs)

-- collective modes due to thE

Sedlacek

ANALOGY Wl'l'H ELECTROSTA'l'IC OSCILLATIONS IN

NONUNIFORM: CoLD PLASMAS

The oscillation, spectrum of It. n,on,llIuform cold plasma

b.as been studied by many authors. These oscillations canbe described by a Wave equatioll similar to (4'), ~u beingreplaced by the potential1/! and ~ being replaced by ~' =

wi.~ CJ)p2{y) , Note that the boundary conditions ~u(1/!) ~ 0

as y ~ :!:: ro are the same. Furthermore, since the matching

conditions are also the sj:t.me, i.e., ~y(1/!) is .continuous and

. ~ - .aB2.. .dEu ( . d1/;)p = -tkj. E. = - E ~ aB2k2 e dy e'E = -e' dy

is continuous, we can expect the Green functions to haveidentical strt1ct'lr~. From this analogy we can draw thefollowing qualitative pictures.. [Sedl6/:ek, 1971a].

First, there exists acontinuous spectrum,

lei I mill <.IA2 5 <.125 max WA2} <.I/(y) = kn2B2//looP

~(y)

~

Bo

EI

E (y) = W2fJ-op - kl12 82

Fig. 1. Profile of .(y) used in the text.

which asymptotically corresponds to noncollective oscilla.tions with position dependent frequency and damping pro.portional to the inverse power of time.

Second, there also exist collective eigenosc~lati'Ons(position independent) with exponential dampings. The

damping is weak if E has sharp discontinuities.

When we apply the above results to the magnetosphere,one sees the interesting possibility of exciting weakly dampedcollective modes due to the sharp jump in density and/ormagnetic field at the magnetopause, plasmapause, and per-haps at the edge of the ring current, In the absence ofsuch a jump the eigenmode in the magnetosphere is adiffused damped mode corresponding to the non collectiveoscillati'on, and a monochromatic oscillation is excited onlyby a monochromatic source function as discussed in tlle

companion paper. In the next section we a.dopt a simple

model for 'f (y) and calculate the eigenmmillation freqt\ency

and the damping rate of the collective modes. The analysis

is closely parallel to that of Sedlacek [1971 a] ,

CALCULATIONS OF THE COLLECTIVE MODES

The collective modes come in through the Green function,which can be written as

G(y, y';<») = ri[1h(y,<»)t/;.(y',<»)H(y' - y)

+t/;.(y, CJJ)t/;l(y', CJJ)H(y - V')] (5)

where dependences on k. and kn are understood. Here His the unit step function, .;, and .". are two homogeneoussolutions. to (4') satisfying boundary conditions at y = -00

and y= + 00, respectively, J is the conjunct of .p,and .p.,

J(kil kuic.J) = E(y)[d:; 1/11 _~~11/l2J (6)

and J is independent of y andy'.Let us take .£(y) to be (see Figure 1)

E(Y) = EJ = ClJ2p.OPI '-kU2BJ2 Y ~ -a

E(Y) = By + 11 --a;S; Y ~ a (7)

E('II) = En =ClJ2JJ.nOTT ~ k/BTT2 11 ;::. a

where8 == «II -(,)/2a and." = «II + (,)}2, the sub-scripts I and II indicating each side of the surface.(We have also analyzed this problem with (y) nowhereconstant. As might be expected, the results are similar aslong as !k.L! » 1, where Lis the scale length of dY)variation at Iyl = a+.) We a\soassume that for Iyl s a,OJ,' monotoI1ically decreases with y. Then J becomes

J(kJ. kn:",) = I>Y.YTID(ki. kg;",) (8)

where Y1 = k.El/8, Yn .= k.En/8, and D, the 'dispersionfunction,' is

D(kJ., kM;Cd) = -[Io(Yu) + It(Yu)](Ko(YI) + K1(YI)]

t [T(V\ - T(V\Hv(v \ - vlv \1 In\

I L£U\£'I £l\£'/JL£~U\-UI --'\-Un. \-,

Here land Kare modified Bessel functions, and D is arriultivalued function .due lothe logarithmic terms in K.This gives rise to branch cuts and therefore Riemann sheets(Figure 2). The complete solution ~.(y, t) can be obtainedby performing inverse Laplace transform and integrating

y

,~':Il!)

CHEN AND HASEGAWA: LON.a-PERIOD MAGNETIC PULSATIONS, 2

over the initial conditions; Le"

My, t) = ;... I dy' I au, e-'Uy, t) = ;'/1" J dy' [ cU" e-i"'G(y, y'j ",)S'(y', "') (10)

In performing the (0 integration one closes the integrationpath by extending it into the lower half of the (J) plane. Thetwo major contributions are from integrations along thebranch cuts (on the real axis of the 0) plane) of G, which

corresponds to the continuous spectruIIl, and from simple

poles of the analytic continuation of D into the lowerhalf of the (J) plane, which corresponds to the collectivemodes. This finding is similar to the theory of Landaudamping in warm plasma oscillations [Sedlacek, 1971b].We are only interested in the least-damped modeS, whichexist in the 11. = ::t1 Riemann sheets when Ik,aj « 1.Assuming jk,aj « 1 or, equivalently, IV,!. IYII! « 1, wehave the following 'dispersion relation' on the 11. = ::t1sheet:

My, t) = 2~ J

From (11) we obtain the oscillation frequency <Or and thedamping .rate 'Y :

CAJr ~ [ kIl2(BI2 -+ BII2)]I/2 (12)

/LO(PI + PIT)

'Y ~ _?! (k1.2a) ./LOPIPn(V,tI2 ;- V.W2]- (13)CAJr 2 . (PI + PIT)(BI + BII )

Note that .owi[)g to the difference in ( between this case

and the electrostatic case a simple substitution of <Up

in electrostatic case by kliVA does not lead to (12) and (13).On the n = -1 sheet we have <0/ = -<Or and y' = 'Y.

-., -Note also that (J)AnM :S ofJJ: :S. (J)AI".At the plasmapause, pn ».PI and B,2 :::::: Bn2; we then

have

.. 1/2

tJJr .~ 2 kUV,UI

'Y/tJJr '"-J. -(7r/4) I k.L/K IHere KCl == 2a.~ [d In p/dyl.=o]-' is the scale length ofdensity variation at the plasma-pause.

At the magneto pause, however, pu » PI and Bu" « BtHere subscripts I and II stand for the earthside and thesunside, respectively. We then have

.(/Jr ~ knBI!(/LOPII)'/2 (16)

~/CJJr~. -(7r/2) Ikj./,,'I (17)

Here (K't' is the scale length of density a.nd magnetic fieldvlfriationat the magnetopause.

Physically, this collective mode corresponds to the eigen-mode of the surface wave excited at the boundary due tothe discontinuity in iO. The sharper the discontinuity is,i.e., the smaller IkJKlis, the weaker the damping is. Thenumber of the collective modes is approximately equal ,tothe number of discontinuities, [Harston, 1964]. As theprofile becomessDlooth, thesemodes become heavily damped,

and one if'; left only with the continuous spectrum. Finally,as can be expected from a surface wave, let us note thatthe collective mode has a peak at the jump and decaysaway exponentially as exp [ -lk"ylJ.

sdg

\035

w PLANE

(10)

-/U(+)e

Fig. 2. The branch cuts and the least-damped roots ofthe dispersion function on its n .= +1 Riemann sheet; C isthe integration path for the inverse Laplace transform.

SUMMARY AND DISCUSSION

In the previous sections we have presented a theory oflong-period micropulsations as an initial value problem.

Using a one-dimensional model and the Laplace transform,

we derive a wave equation that shows the coupling be-

tween the surface wave (i.e., the. evanescent compressionalAlfven wave) and the shear Alfven wave. We then notethe analogy of this problem to that of electrostatic oscilla-tions in nonuniform cold plasmas. From this analogy we

can conclude that there exists first, a continuous spectrum

{ell minooA2 ~OO2 ~ maxooA2\

which corresponds to noncollective modes with, asymptoti-

cally, position dependent frequency OOA(Y) and damping

proportional to inverse power of time,' and second, a discretespectrum, which corresponds to collective modes (surfaceeigenmodes) with position independent frequency andexponential damping. The damping can be rather weak if theplasma' parameters have sharp variations. ExpreSsions ofthe frequency oo~and damping rate'Y of the surrace eigen-

modes are then derived in terms of plasma parameters. As we

have shown in the companion paper, the dynamics of MHD

plasmas in a dipole magnetic field can be described by acoupled wave equation with a structure similar to the oneused here. Therefore we can expect that' the fundamentalphysics obtained here, i.e., the continuous spectrum and thesurface eigenmodes, also exists in the more complicated andrealistic case. There are, however, other effects due to thepresence of the ionosphere-earth boundary. With the bound-

ary located at Y = c, the bot~ndary condition then becomes

~y = 0 at Y = - oo,cinstead of at y = ::I:: 00 .An additional

damping introduced by this effect is negligible, however,if kJ.c» I, because 'YI(J) due to this effect may be estimatedto be exp [- k.Lc], even if the ionosphere is totally absorbing.Theother effect isa damping due to the ionospheric dissipa-tions associated with the motion of the field line in theionosphere.' The order' of magnitude of this damping rate' call

be roughly estimated by'Y '" PIW. Here P is the dissipated

power, and Wis the total wave energy. If we take the skin

depth of the ionosphere and the ground X as the dissipative

layer, 'Yloo then may be deduced. to be ---All, where l is thelengtl1 of the field line. Consequently, damping due to thiseffect is also negligible. .

Let us now discuss the above theoretical results withregard to recent observations of the plasmapause..:associatedmicropulsations mentioned earlier (Lanzerotti et al.,

(11)

(14)

(15)

CHEN AND HASEGAWA: LONG-PERIOD MAGNETIC PULSATIONS, 21036

1973]. Four typical events are sbown in Figure 3. Thedamped oscillations observed at L = 3.2 are assumed tobe the excitations of weakly damped surface eigenmodesdue to a sharp density jump. The disappearance of suchoscillations at higher latitudes (L = 4.0 and 4.4) is con-sistent with the theoretical result that. the surface eigen-mode has its peak magnitude at the jump and decaysaway exponentially with ascRle of k.-1. These observationsthen suggest that during these events the plasmapause waslocated near L= 3.2. Away from the plasmapause (e.g.,at L = 4.0), only modes. corresponding to the continuousspectrum were excited and, consequently, damped awayowing to spatial phase mixing. Furthermore, using theob;;erved osciIlationfrequency «0 ,...J 0.07 rad/s), dampingrate (y/<o ,..., 0.()8), and maximum decay length (i.e.,

k.,m'!O ,..., 1/0.8Ila) in (14) and (15), we obtain for the

fundamental mode V.m ~650 km/s and Kmu-1 ~ 0.05Ra.

These results are consistent with satellite observations

[Burton et 01., 1970; Chappellet mo, 1970].

The surface eigenrriode may also explain the long-perioddamped type pulsations often ass()ciatcd with sudden com-mencements and sudden impulses; i.e., Psc 4 and Psc 5[Saito and MatsU8hita, 1967]. Although detailed infor-mation is lacking, we can roughly check the frequencyrange. Near the magnetopause (L ::::::8), from (16) and

the balance of momentum flux (B."/2p,o = pn V 82), the

eigenmode frequency is (if the Doppler shift by the solarm;~;1 ~~+;~~ ;~ ~n~lnn+n;l\HUm UWONU'Y "~,,,WVW"'

"'. .~ 21/2kn V 8 (18)

Here VB is the solar wind speed. Taking VB ~ 400 km/s,

we then have for the fundamental mode a period T ~ 400

s,. which corresponds to Psc 5 predominantly observed. inauroral zone. Finally, we emphasize the significance ofusing n}icropulsation data as a possible diagnostic tool

fOf the dynamics of the plasmapause and,. perhaps, the

magnetopause and solar wind, and we note that damped

Alfven waves attributed as surface eigenmodes have alsobeen observed in laboratory theta pinch experiments[Grossmann and Tataronis, 1973].

Acknowledgments. The authors acknowledge valuable dis-

cussions with H. Fukunishi and L. J. Lanzerotti.

'" '" '"

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

REFERENC:ES

Ba,'Ston, E. M., Electrostatic oscillations in inhomogeneouscold plasmas, Ann. Phys., 29, 282, 1964.

Burton, R K, C: T. Russell, and C. R. Chappell, The Alfven

velocity in the magnetosphere and its relationship to ELF

emissions, J. Geophy. Res., 75, 5582, 1970.Chappell, C. R, K K. Harris, and G. W. Sharp, A study

of the influence of magnetic activity on the location ofthe plasmapause as measured by ago 5, J. Geophys. Res.,75, 50, 1970.

Chen, L., and A. Hasegawa. A theory of long-period magneticpulsations, 1,Steady state excitation of field line resonance,J. Geophys. Res., 79, this issue, 1974.

Grossmann, W., and J. Tataronis, Decay of MHD wavesby phase mixing, 2, The theta-pinch in cylindrical geom-etry, Z. Phys., 261, 217,1973.

CHEN AND HASEGAWA: LONG-PERIOD MAGNETIC PULSATIONS, 2

Lanzerotti, L. J., H. Fukunishi, A. Hasegawa, and L. Chen,Exeitation of the plasrnapause at ULF frequencies, Phys.Rev. LeU., 31, 624, 1973.

Saito, T., and S. Matsushita, Geomagnetic pulsations as-sociated with sudden commencements and sudden im-pulses, Planet. Space Sci., 15, 573, 1967.

Sedhicek, Z., Electrostatic oscillations in cold inhomogeneousplasma, 1, Differential equation approach, J. Plasma Phys.,

6, 239, 1971a.

1037

Sedlacek, Z., Electrostatic oscillations in cold inhomogeneousplasma, 2, Integral equation approach, J. Plfls7no Phys.,6, 187, 1971b.

Uberoi, C., AUven waves in inhomogeneous magnetic fields,Phys. Fluids, 15, 1673, 1972.

(Received July 26, 1973;accepted November 13, 1973.)