Dec 1, 1993 - the magnetopause and on the ground [Moskvin and Frank-. 21,365 ..... wood, M. Lockwood, and A. Etemadi, Pressure- driven mag- netopause ...
JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 98, NO. A12, PAGES 21,365-21,371, DECEMBER 1, 1993
Accelerated Motions of the Magnetopauseas a Trigger of the Kelvin-Helmholtz Instability V. Institute of Solar-Terrestrial
V.
MISHIN
Physics, Russian Academy of Sciences, lrkutsk
We consider the effect of accelerated magnetopause motions arising from the arrival of a modified-pressure region of the solar wind with a sharp forefront upon the growth rate of the velocity shear layer instability. The range of values of relative pressure variation on the front
was taken to lie within 1.25-2, and the front thicknessranged from 200 km to 1RE (6400 km). At different phases of motion of the boundary, the instability growth rate can change in either direction as a consequenceof the Rayleigh-Taylor effect, in accordance with a change of sign of acceleration g of the boundary. The typical time during which of the boundary remains inside the
new equilibriumposition (_• I min) is sufficientfor the instability to reach the nonlinearregime. It is possible that plasma flutes on the boundary penetrate the magnetosphere. Generation of disturbances can be modulated by boundary oscillations inside the new equilibrium position. We discussthe interplanetary magnetic field influence upon the growth rate of the MHD instability under consideration. An analysis is made of a number of daytime geophysical conditions which cannot be understood when interpreted in terms of the instability on a stationary boundary. 1. INTRODUCTION
changesabruptly, i.e., plI/pI > 4 (where indicesI and II refer to regionsinsideand outsidethe boundary),and the A magnetohydrodynamic (MHD) boundarylayerof finite flow velocity is minimal [Paschmannet al., 1978]. Since
thicknessis known to form on the magnetosphericboundary as a consequenceof the nonstationary solar wind flow past it. Magnetic field B, flow velocity v, density p, and other parameters of the medium change their values abruptly
the radius of curvature of the geomagneticfield here is also
minimal,R = l0s km [GuterichandShcherbakov, 1971],
the R-T effect here must manifest itself most conspicuously. Its negative character is attributable to the convexity of the [Lundin, 1988]. It will be assumedhere that equihbrium boundary and is associated with two factors' the trapped values of normal vector components v and B are zero. Thus particles travelling in the magnetosphere,and the flow round
we will confine ourselves to the approximation of the bound-
ary layer as a "diffuse"tangential discontinuity(TD). Besides, it will be assumedfor simplicity that the scale of shear of the magnetic field coincides with that of the transition layer of the velocity and density. This is, in principle, justified by the fact that the subsolarmagnetopausewill largely be considered.
It
is known
that
surface
disturbances
on
the convex boundary. The first factor in the theory of plasma traps is usually characterized qualitatively: in the direction of the convexity
of the field lines and the correspondingcentrifugal acceleration of particles go! = P/p, where P is thermal pressure
[Longmire,1962]. An attempt at a quantitativeevaluation
of the R-T effectthrough a direct substitutionof go! into the the TD which are brought about, for example, by pressure equations of isotropic hydrodynamics without taking into and velocity fluctuations, are able to increase as a result of account the stabilization by the magnetic field led Lyatsky the Kelvin-Helmholtzinstability (K-H) in magnetichydroand Safargaleyev [1991]to an overestimatedresult. It would dynamics[Chandrasekhar,1961]. The growth rate of the be more correct to obtain MHD equations in the real geo-
disturbance amplitude in time is characterized by the imaginary part of the frequency, 7 = Imw. Two factors affecting
magnetic field with proper account of the quasi-neutrality
condition.Kovner et al. [1977]wereable to estimatequaninstability are 1) an inflectionpoint in the flow velocityprotitatively the influence of the second factor, the convexity file and 2) a densitygradientdirectedoppositelyto the field of the boundary in the approximation of a cylindrical TD. strength of potential forcesg: g grad p - gp' < 0. Here the prime denotes differentiation with respect to the normal to the boundary. If the first factor is absent and the second is present, then the arising instability is said to be
It was found that long-wavelength disturbances are stable
when m < pH/pz(m is the azimuthalwave number). In this case the effective accelerationge! is due to the inertial
term in the momentum equation:ge! = (vV)v = v2/Ro,
the Rayleigh-Taylor(R-T) or flute instability. A relevant which for the values of v = 100 km s -• and the radius of classical example is the instability of a heavy fluid above a heated, hghter-weight fluid in the gravity field. In the general case of the presence of both factors the influence of the second mechanism upon the value of the K-H growth rate will be henceforth referred to as the R-T effect. In this case,by the positive and negative R-T effects, we mean
an enhancement
and an attenuation
of the K-H
growth rate, respectively. In the vicinity of the subsolarmagnetopausethe density Copyright 1993 by the American Geophysical Union.
curvature of the equatorial cross section of the boundary
R0 = 10Rio= 64,000km is g•! = 0.15km s-2. By the way, such a longwavestabilization was disregarded
by Belmontand Chanteur[1989]when analyzingthe evolution of long-wavelength disturbances).The curvatureeffect on the excitation of the most unstable "short wavelength" disturbancesis, however, small. Therefore, by the magnetospheric boundary K-H instability, it is customary to mean
in the literature the velocityshearinstability (and often, simply the TD velocityinstability). A decadeagosuchan
Paper number 93JA00417.
approachmade it possibleto explain the generationof MHD waves in the range of geomagnetic pulsations observed at
0148--0227/93/93JA-00417505.00
the magnetopauseand on the ground [Moskvinand Frank21,365
21,366
MISHIN: TRIGGER OF MAGNETOSPHERIC BOUNDARYINSTABILITY
Kamenetsky, 1967;Mishin,1981;MishinandMorozov, 1982, for tracing the variation in the K-H instability growth rate 1983; Walker,1981;Miura and Pritchett, 1982]aswell asthe
during magnetopause displacements.
formation of the magnetosphericboundary layer at a sufficient distance from noon and the solar wind momentum
and
2. MAIN
RESULTS
OF THE LINEAR
THEORY
OF THE K-H
INSTABILITY
energyinput into the magnetosphere [Mishin, 1979; Mishin and Matyukhin, 1986]. For other publicationson this topic seea reviewby Lee[1991].
Let us considerthe stability of a transition layer of thickness2d, centered on the plane a:-- 0. It will be assumedthat In spite of the above achievements,such an approach, plasma and magnetic field parameters are homogeneousin however, encountersdifficulties when explaining a number
of effects occurring near the subsolarpoint. Since a stagnation regionexistsin the vicinity of this point [Fairfield, 1976], the growth rate of the velocityshearinstability here must be a minimum or be absent altogether. On the other hand, casesof multiple magnetopausecrossingsby the ISEE
I and 2 satellites[Songet al., 1988]associated with boundary fluctuations do not have a well-defined minimum around noon, while geomagnetic pulsations, even on the contrary, have a dayside maximum, though with a relative minimum
regionsI (x < -d) and II (x > d) and changetheir values inside the layer -d < ß < d (region III). Vectorsv and B can change not only their value but also their direction. The field strength of potential forcesg is directed along the a: axis. Supposethat the system is given a small sinusoidal perturbation which decays with distance from x = 0:
f(x, y, z, t) ~ exp[-nx + i(k•y + k•z - •t)]
(1)
In our study we take into account an arbitrary compressibil-
near noon [Kalisherand Rusakova,1990]. On the contrary, ity (div v y• 0 in the equationof continuity)and the term pg
in the equation of momentum transfer. The linearized system of equations for nondissipativemagnetic hydrodynamics magnetosphere mainly aroundnoon as well [Lundin,1988]. yields a set of differential equations for boundary displaceBesides, it is well known that solar wind dynamic presthe plasma flow from the solar wind appears to penetrate the
surevariationsdrive magnetopause motions[Kaufmanand Konrady,1969]. Thesefacts encouragedSonget al. [1988] to associatethe oscillating magnetopauseobservationswith nonstationary effects, namely bombardment of the magnetosphere by solar wind plasma inhomogeneities. With such an interpretation, each boundary crossingmust have, on the magnetic field profile, the form of a simple step without any additional noise. However, profiles obtained by Song et al.
ment (along the x axis) and for total pressurefluctuations
H• - k2gl•2II• = (p•t• -gp')•- p•t•l&2•'
(2)
Heren2 = k2 - •4/(•c,• - (ka)2c2 ), • = &2_ (lm)2, •c,•2 _ •2_ (kca/c,•)•, wherea is the Alfvenvelocity,
velocity, c,,2 = c2 + a2, and• = [1988]from ISEE I and 2 often exhibit severaladditional c = v/TP/p is thesound peaks (or a superpositionof noise)at the boundarycross- •- kv(z). In the casec -• oo the systemof equations(2) and (3) coincideswith correspondingonesderivedon the ing. Letocite and Roth [1978]offeredthe hypothesisof impul- assumptionof incompressibility[Chandrasekhar,1961]. sive penetration of solar wind plasma clouds into the magnetosphere. It is assumed that as a consequenceof the formation of an indent on the boundary a flute instability must develop, which creates conditions for the ultimate penetration of the inhomogeneity into the magnetosphere. A similar idea as applied to the Venus ionopause was qualitatively
discussed by Russell[1990]. Note that,in orderto provethe supposition about the appearanceof a R-T instability on the
The dispersion dependence calculated by Morozov and
Mishin [1981]from the systemof equations(1)-(3) in the simplestcasek [[ v A_B, v/c 0. If, however, gpt < O, then the K-H in-
equilibrium conditionfor a stationary closedmagnetosphere
(vn =Bn = 0) is the continuityof total pressure-
II = P + B2/2t•0
stability growth rate increases(the R-T effect). Analytic
(7)
results derived by investigating the influence of this effect
andBa/2tt0is magnetic presare givenby Chandrasekhar [1961]for B = 0 in the incom- whereP is thermalpressure
pressible limit c -• oe . The Richardson number J serves sure. Let us now considerwhat will result from a fast change of II in the magnetosheath,which might be associatedwith to characterize the relative importance of the velocity shear an abrupt change in kinetic pressureof the solar wind. In and the R-T effect:
this casethe condition(7) is violated, and an accelerated
J: gpt/(pv?)
(5)
The case J 0 (the negative therangefrom0.1to 10kms-a orhigher[Parkset al.,1979]. R-T effect)it is possibleto constructthe boundaryline of Sinceeffective acceleration ge!= va/Ro'" 0.1km s-a hasa regionsof stable and unstabledisturbancesin the plane (J, stabilizing effect, we will primarily be interestedin displacekd). When J > 0.25, all disturbancesare stable. The influence of the R-T effect upon the K-H instability growth rate is illustrated by the dispersion equation obtained in the
ments with g > ge! . Let us considera one-dimensional model. Let pressurebe a function only of coordinate r di-
to axis x longwavelimit (TD approximation)[Chandrasekhar, 1961]: rectedfrom the Earth to the Sun (corresponding in the geocentricsolar-eclipticcoordinatesystem).We then estimatethe valueof 110= 1I(R0) at the equihbriumposi-
w= a•kv•+ a•kv• 4-i {a•a•(kvH - kv•)2-
-
-
}
The dependenceof the K-H instability growth rate on the wavelength for different values of J studied by Mishin
and Morozov[1982, 1983] is given in Figure 2. One can see that the position of a maximum growth rate with a variation of J changeslittle with respectto the wavelength and,accordingly,to the period. The same dependenceis conservedfor values of J = -0.1. According to our additional calculations, a growth of J in the region J < 0 is accompa-
tion R0 = 10Rls = 64,000 km. The value of magnetic field
strengthwill be takenas Bls = 2Biso/Ro a = 62 nT. Here Bls0 is the strength of geomagneticdipole field on the terrestrial surface on the equator. We take into account the con-
tributionof thermalpressure P. Taking1• = 2ttoP•/Bls •= 0.3-0.4 [Paschmannet al., 1978] we get II0 = 2nPa. A similar value can be derived from the model of compressed
magnetosphericdipole by Mead and Beard [1964]. Since the contribution of P in the inner magnetosphereis small
(/• >1).
0.4 -
II ~
(8)
This permitsus to readily(but to a sufficientaccuracy)estimate the parameters of our interest. Let the arrival of the front, i.e., the boundary of a semispace of increased pressureIi• , be described as an inclined
stepof height(II•-II0) andthickness/•(seeFigure3). Such an approximationis applicableif the front's thicknessalong axis r is much smaller than the inhomogeneity size in axes y and z: l•, I, >>/•. On the other hand, it is clear that the value of/• is bounded below by the value of proton gyrora-
J=-O.25
diusp•i ~ 100 km (whichdeterminesalso2d, the thickness of the subsolarmagnetopause).This, in turn, restrictsthe forceF• = pg• = I- vH01 ~ 1)Ho/•5,underthe action of which the magnetopausewith accelerationg• = F•/aM (a is the meanplasmadensityon the front, M is the proton mass)displacesto the point r = R• . At this point the geomagnetic field pressure compensatesthe original pressure imbalance. Let the value of II• be representedas the prodo
0.1
uct:
= ½n0
ø'øo. '
Idol
Fig. 2. Normalized growth rate as a function of kd for •fferen• values of •ich&r•on number J for incompressible perturbatio•
without magneticfield effect (klB);
½II/½I = 10.
where ½ is someconstant. We substitute this expressioninto
(8) andand:R• = ½-{/to. •n thecase ofweakgradients (½ 1). In the caseof the action of a regionof de- is small comparedwith the duration (r = 2At2) of the "accreasedpressure(•b < 1), this is accompanied, at first, by an tion" of the positiveR-T effect:7-• < r. This conditionis acceleratedexpansionof the magnetosphereand, then, by its
establishment nearthe newequilibrium positionJ•a> R0. 4. THE
MIlD
INSTABILITY
quite well satisfied for all values of •b and f listed in Table 1
(in viewof the condition g _1500km, whoseperiod (>_15 s) correthe stability of a boundary that moves with acceleration g, sponds to the range of Pc2-3 and Pil pulsations. As time it is necessary to include the inertial force Fin - -pg in elapses, the disturbance scale and period appears to grow the momentum equation which takes into account poten-
signof boundaryaccelerationin the dispersionequation(6)
tial forces. As followsfrom equation(6), one shouldthen change the sign of the term describingthe R-T effect. The same change of sign is made when analyzing forces acting on the body being in the acceleratinglift.
as a consequence of both the nonlinearevolution [Belmont and Chanteur,1989]and an increasein the thicknessof the
boundary with distance from noon. 5.
OSCILLATORY
REGIME
OF THE BOUNDARY
INSTABILITY
Mishin [1980]showedthat the value of boundaryaccelerationsobservedby satellitesISEE I and 2 [Parks et al., As has been pointed out in section 4, small oscillations 1979]is sufficientfor enhancingthe growthrate in the entire of the boundary with period T from 2-3 min to _> 10 min range of the K-if instability, with which eventsof multiple boundary traversals can also be associated.The model described in Section 3 is useful for studying the variation in growth rate of the K-if instability in different phasesof motions of the boundary, after a semispacewith modified total pressure interacts with it. The density gradient on the magnetosphericboundary is always directed outward: pt > 0. Therefore the variation of the influence
of R-T
effect
occurs
in accordance
with
a
and amplitude fir _< RE can arise near the position of new equilibrium. When performing suchoscillations,the K-H instability growth rate, though weaker than during the direct arrival of the inhomogeneity front, can periodically change. Namely, it increases when the boundary moves inside the new equilibrium position r < Ra and decreaseswhen the
motion is outsider > R•. (If the regionof decreasedpressure is acting, then the K-H instability intensifiesin a similar
fashion:insidethenewequilibrium positionr < J• but with
changeof sign of g. Thus the motion on the portion R•R2
thedifference that.•a > R0andRa< R0).
(i.e., insidethe positionof new equilibrium)in both direc-
In the daytime on the ground the boundary oscillations must be accompaniedby geomagneticfield fluctuations with
tions corresponds to an enhancement of the K-H instabil-
ity growthrate (a positiveR-T effect)becausein this case, periodgivenby equation(10). Ilumps and troughsof such • = --g < 0 and .•pt < 0 (the signsof g and pt correspond waves must show, respectively an increase and decrease of to the positivesunwarddirectionof axis x(r)). The motion the geomagnetic pulsations involved. on the portion RoR• (alsoin both directions)is characterDifferent types of Sc- and Si-associatedgeomagneticdisized by a positive sign of • and, accordingly,the R-T effect turbanceshave long been studied [Nishida, 1978].Thelat(Opt> 0) leadsto a decrease in growthrate. est simultaneous multisatellite and ground-based observaLet us estimate the extent, to which the growth rate changesdue to the R-T effect. For this purpose, we estimate the Richardsonnumber. Let the shear layer thickness for the velocity and density be the same, and we approximate their profilesby linear functions. From (5) we then
tions have made it possible to establish a relationship of the magnetopause oscillations brought about due to an abrupt change of pressurein the solar wind, with global geomagnetic oscillationsin the magnetosphereand on the ground.
The periodof theseoscillationsusuallyis 8-10 min [Potemra
21,370
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CONCLUSIONS
We have considereda simplemodel describingthe motion of the magnetopause,which was origina]ly located at 10RE from the Earth, under the action of the front of an abrupt changein tota] pressureof the magnetosheathplasma flow. When the pressure varies greatly by factors from 1.25 to 3 on a sca]e of the front less than or about R0, the typica] time of the displacement is 2-3 min, and the distance is about 1-3 RE. The transition from the old equilibrium position to a new one initially occurswith an acceleration, followed by a deceleration. A significant enhancementof the shear instability growth rate occurs on the portion of deceleration,whereasa stabilization takes place during the acceleration. The typical time of enhancement is less than 1 min. The growth time of the disturbances is less than 10 s, and their period is longer than 15 s. The transverse sca]eof the flutes is larger than 1500 km. An enhancement of the K-H instability is a consequenceof the accelerated magnetopausemotions and of the increasein velocity gradient. The above mechanismfor enhancementof the magnetospheric boundary layer instability must be sensitiveto the
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Acknowledgments.The author thanksV.G.Mikhalkovskyfor his assistancein preparing of the English version of the paper. The
Editor
thanks
D. G. Sibeck
assistancein evaluating this paper.
and one other
referee
for their
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(ReceivedJune 28, 1991; revised september 4, 1992;
acceptedFebruary 19, 1993)
