•o•as,n o• G•.o•¾s•c^n R•.s•.^ac•, S•^c•. P•¾s•cs
Von. 75, No. 31, Nov•.•i•.• 1, 1970
Kelvin-Helmholtz Instability and the SemiannualVariation of
GeomagneticActivity
BRUCE R. BOLLER AND HAROLD L. STOLOV
Department o• Physics
The City College o• The City University o• New York
New York, New York 10031
Kelvin-Helmholtz instability along the flanks of the magnetosphereexhibits a semiannual
variation, instability maximums occurring at the equinoxesand instability minimums at the
solstices.It is suggestedthat Kelvin-Helmholtz instabilities, generated at the magnetopause,
initiate the modulation of geomagnetic disturbance detected as the semiannual variation of
geomagnetic activity. The Kelvin-Helmholtz explanation predicts a universal time variation
of geomagneticdisturbance.This prediction is confirmed by several analysesof the geomagnetic activity indices. Since the physical mechanism suggestedfor the semiannual variatioh
of geomagneticactivity dependson the tilt of the dipole axis, the 'equinoctial' hypothesisis
supported.
annual variation to the changeof the tilt of the
The semiannual variation of geomagnetic
activity is a well-established
phenomenon,having maximums near the equinoxesand minimums near the solstices(range •15 7). There
dipoleaxis,the times of greateractivity occur-
ring when the dipole is perpendicular to the
solar wind flow. The maximum activity occurs,
have been two views as to the cause of this
therefore, twice yearly at the equinoxes.The
semiannual variation:
exact physical mechanismis not specified.
There has been a continuingcontroversyover
1. Cortie's[1912] 'axial' hypothesisexplained
that the geomagneticvariation wes associated the correct explanationof the semiannualvariawith changesof.the earth'sheliographiclatitude tion of geomagneticactivity, since the nature
(maximum north heliographic latitude of 7.2ø of the geomagneticdata makes it difficult to
on September 7 and maximum south helio- identify the dates of the maximums. Priester
graphic latitude of 7.2ø on March 6). At the and Cattani [1962] favor the axial hypothesis
times of these maximums,the earth would be in their analysis of Barrels' [1940] monthly
more favorably situated with respect to the geomagneticactivity data, whereas Barrels
solar streams coming from the active regions [t932, 1963] and recentlyShapiro [1969] favor
between 10 and 20 deg north and south helio- the equinoctialhypothesisin their analysisof
graphic latitude, thereby resulting in increased the geomagnetic
data.Mishin,et al. [1961]and
geomagneticactivity at thesetimes. The axial Mishina [!967] invoke a more complicated
hypothesisdependsbasicallyupon two things: interplay of several factors in explainingtheir
that the solar streams cause geomagneticac- geomagneticanalyses.
The lack of a physical mechanismto adetivity, and that the solar streamsemitted radiquately explain the cause of the semiannual
ally are not diffusedvery much in the interplanetary mediumby the time they get to the variationhasbeenthe major problem.A knowlorbit of the earth.
2. Mclntosh's [1959] 'equinoctial' hypothesis, on the other hand, attributes the semi• Presented at the symposium on Instabilities
in the Magnetosphere of the IAGA General Scientific Assembly, Madrid, September 1-12, 1969.
Copyright (•) 1970 by the American GeophysicalUnion.
edgeof the mechanismwill identify the parametersresponsible
for the phenomenon
and therefore make possiblethe understandingof the
semiannualvariation of geomagneticactivity.
K½,nvIN-H½,nM I-IOnTZ HYPOTHESIS
It is the purposeof this paper to associate
the semiannualvariation of geomagneticactiv-
ity with the variation of Kelvin-Helmholtzin6073
BOLLER
6074
AND
STOLOV
stabilityat the boundaryof the magnetosphere.The symbols
B andp stand.
for magnetic
field
intensityand massdensity,respectively.
It has
Helmholtz instability are due to the seasonal been assumed that the wave vector of the
changesof the orientationof the earth'smag- perturbation
is in thelJ direction.The presence
netic dipolewith respectto the solarwind flow. of parallelconiponents
of magneticfield with
It is suggested
that instabilitiesgeneratedat respect
to thesolarwindstreamat theboundary
the magnetopause
may initiatethe geomagnetic createa stabilizinginfluence.If the magnetic
disturbancedetectedby surfacemagneticob- fieldswereeithernonexistent
or perpendicular
servatoriesas the semiannualvariationof mag- to the solarstream,a situationof complete
It will be shown tha• the variations of Kelvin.
netic disturbance.
Frictional or viscous-like interactions be-
instability
wouldresultforwavevectors
alonglJ.
lishing the initial conditionsin both cases.
titative estimateof the Kelvin-He!mholtzinsta-
Parallelcomponents
of magneticfieldstherefore
tweenthe solarwind and the magnetospherehavea stabilizing
influence
on the boundary.
haye been studied quantitativelyby Ax/ord The angle• is a functionof threequantities'
[!964] and Parker [1958] on the basis of a thepointonthemagnetopause,
thetimeof year,
sound-waverefractionmechanism
and particle andthetimeofday.Some
regions
ofthemagnetoscatteringby inhomogeneities
of the magnetic pause are thereforemore susceptibleto this
field, respectively.The Ke!vin-Helmholtzphe- instabilitythan othersand, in addition,they
nomenoncould possiblybe the processestab- exhibita time variation.An illustrativequanThe seasonaldistributionof magnetosphericbility criterionis givenin theappendix.
substormsmay well contribute to the semiIt is clear from the Ke!vin-Helmholtz in.
annual variation of geomagneticactivity. equality that interplanetaryfields will conAkaso/u [!968, p. 224] citesten theoriesthat tributeto variations
of instability.
However,
have been suggested
to explainthe magneto- sinceseasonal
variationsof the interplanetary
spheric substorm.In two of these theories, fieldhavenotbeenobserved,
anyinterplanetary
Kelvin-Helmholtzinstabilitymight againserve field influenceon a semiannualvariation of inas the intermediate between the solar wind and
stability will be averagedout when several
geomagneticdisturbance.It should not be assumed that the Kelvin-Helmholtz mechanism
years' data are considered.Short-term varia-
tions of the interplanetarymagneticfield and
is exclusive,
but ratl•erin competition
with their influences
on geomagnetic
activity, as
other disturbance
phenomena.
observed
by Fairfield[1967],Schattenand
The Ke!vin-Helmholtz instability of the Wilcox•!967], and Wilcoxet al. [!967] and
explainedin terms of interconnectionbetween
boundarybetweentwo magnetohydrodynamic
incompressiblefluids in relative motion has interplanetaryand geomagnetic
field lines in
beenstudiedby Chandrasekhar
[1961]. A num- the.mannerof Dungey[1961],are by nomeans
ber of important:contributions
to the problem excludedby the presentdiscussion.
havebeenmaq•e
.by Ax/ord[1960,1962],Se•
[1963,1964],Fejer [1963,1964],Talwar[1964,
1965], and Southwood[!968]. For a goodreview of this subjectsee Dung.
ey [1968] and
Getwin [t968]. As a first approximation,
the
instabilitycriteriondeveloped
by Chandrasekhar
can be appliedto the magnetopause,
giving
u>
4•rpxp•
+ s],cos
We will now associate the semiannual varia-
tion of geomagnetic
activity with the seasonal
variations of the angle •k•, in the KelvinHelmholtz theory. A sketch of the magnetosphereof the earth is shownin Figure !. The
Y-Z plane cuts • dawn-dusk cross section
through
the flanksof the magnetosphere
and
passesthroughthe centerof the earth, which
is at the orig!nof the coordinate
system,
The
solarwind is in the X direction.The dipoleis
whereI standsfor interplanetary
valuesoutside shownhere'to be alongthe Z axisand perpenthe magnetosphere
(magnetosheath
values)and dicularto the solarwind.This particularorienM standsfor magnetospheric
values.The symbol tation of the dipole with respectto the solar
U is the streamspeedof the solarwind at the wind will occurtwice a day during the periods
magnetopause,
and • is the angle betweenthe that the earth'srotationaxis is perpendicular
localstreamvelocityI/and the magnetic
fields. to the solar wind, i.e. during the equinoxes.
KELVIN-HELMHOLTZ INSTABILITY
6075
z
y •
SUN
Fig.1. Sketch
ofthemagnetosphere
withY-Zplane
through
theflanks
[afterWalters,
1966,
Fig. 1].
An appropriate
rotationof thiscoordinate
sys- favoringinstability.This valueof •kMoccursat
tem about the X axis (solar wind direction) 1030 and 2230 UT (sunriseand sunsetat the
pole,respectively).
For this
canalwaysbe madesothat the dipoleis con- northgeomagnetic
field
tained in the X-Z plane and the X-Y plane case,it is apparentthat magnetospheric
linescannotexert stabilizinginfluences
in the1I
intersects
the magnetopause
alongthe flanks.
Figure2 is a cross-sectional
view of the direction.
The least unstable situation at the flanks
magnetosphere
in theX-Y planelooking
in the
during
theequinoxes
occurs
when•kM= 78.5øor
negative
Z direction
for the orientation
of the
dipole of Figure 1. Inside the magnetopause101.5ø. These valuesof •k• occurat 0430 and
all alongthe dawnand duskflanksthe geo- 1630 UT (midnight and noon at the north
pole,respectively).
The condition
magnetic
fieldlinesarein the positive
Z direc- geomagnetic
tion. A local coordinatesystemX', yt, Z• (X •
in the streamingdirection,Y' perpendicular
to
for instabilitynow becomes
U• > pt
q-__p.p•
[B•cos
• •p•q_0.0397B•2]
the magnetopause,
Z• = Z) canbe placedany4z'p•p•u
wherealongthe dawnand duskflanks.Such
,• coordinate
systemis shownat the duskflank Obviously,
thecomponents
of fieldlinesalong
U
in Figure 2. Clearlythe angle½• is 90ø at are a stabilizinginfluence.
For the mostunstablesituationat the winter
the flanksin the planeof symmetry,and the
and summersolstices,•k• is 78ø and 102ø,
inequality becomes
respectively,
78ø occurring
at 1630UT and
U2> pt
q-p• B•2cos
• •p•
4z'p•p•u
102ø occurring
at 0430UT. The inequalitycan
then be written as
6076
BOLLER
AND
STOLOV
Figure 3 shows the stable and unstable re-
U2• pt
-I-pM
[B•2cos2
• -I-0.0431BM
2] gionsof Usplottedagainst•
4z'prp•
alone.The values
of • duringthe equinoxes
can correspond
to
The mostunstableconfiguration
at the solstices unstableconditionswith relativelylowervalues
nearlycorresponds
to the moststableconfigura- of U than at the solstices.
If the otherquantition at. the equinoxes.
ties in the inequalityhave no significantseaThe least unstableconfiguration
during the sonal dependence,
the instability at the flanks
winter and su•er solstices
occurswhen•
is dependent
solelyuponthe anglebetweenthe
is 55ø and 125ø, respectively,
the formerangle dipole and the sun-earth line, as in the
occurringat 0430UT and the latter occurring equinoctialhypothesis.However,any seasonal
at 1630 UT. The instability conditionthen be-
changesin the intensity of the solar streams
comes
(assuggested
by the axialhypothesis)
will have
their Kelvin-Helmholtz effects.,Thesehave not
been observed.
U2•>m
']-p•[B•.cøs
•.• _]_
0.329B•.-[
47rP•PM
PREDICTIONOF KELVIN-HELMHOLTZ
There are other regions where the solar
streamingis perpendicular
to the geomagnetic
HYPOTHESIS
field.However,theseregionsare exclusively
at
The parametersof the solar wind are known
the front of the magnetosphere,
wherethe low to be quite variable.Thesespatialvariations
valueof solarstreaming,as well as the absence in the solarwind are convectedthroughthe
of any seasonal
dependence
in this vicinity, bowshockandpassthroughthemagnetosheath.
serveto excludeit from the presentdiscussion. At anygiventimetheseparameters
maychange
SOLAR
STREAM
e
WIND
LINES
MAGNETOPAUS!
0-•
GO
Oo
oo
EARTH
Fig. 2.
Sectional
viewof themagnetosphere
in theX-Y planewithlocalcoordinate
system
at the dusk flank.
KELVIN-HELMHOLTZ
INSTABILITY
6077
2 Cos2 'h+",,
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B
s
o
T
A
!1
"JL_
125ø
78o78.ø5
101ø.5 102 ø
180 ø
Stableandunstableregionsof U'2plottedagainst
so as to alter the relative magnitudesof the
be determinedfrom the instability conditionas
left- andright-hand
sidesof theinstability
con- in Table1, whereD andI, referto thedensity
term, respectively.
dition.Whenthe magnetospheric
termis small, factor and interplanetary
the probability
for instability
to occurwill be Therefore,if the instabilityof the magneto-
effective,
theKelvinhigh.If themagnetospheric
termis large,the pauseis geomagnetically
Helmholtz
explanation
predicts
a
universal-time
probability
for instability
to occurwouldbe
activity.Illustrative
correspondingly
low.Thetimesduringthe day variationof geomagnetic
of instability
asa funcof the maximumand minimumprobabilities
for plotsof theprobability
instabilityare seasonally
dependent
and may tion of universal time are shown in Figure 4
TABLE
1.
Diurnal Variationof Kelvin-Helmho!tzInstability
SummerSolstice
Equinoxes
Max = 1030, 2230 UT
•k• = 90 o, 90 o
U 2 > D[lt]
Max
= 0430 UT
•p•r = 102ø
U2 > D[!t + 0.04B• 2]
Min = 0430, 1630 UT
Min
•p• = 78.5 ø, 101.5ø
U 2 > D[I, + 0.04BM2]
•p•r = 125ø
= 1630UT
U2 > D[lt + 0.33B•r2]
D, densityfactor;It, interplanetary
term.
Winter Solstice
Max
= !630 UT
,p• = 78 ø
U2 > D[It + O.04B• 2]
Min
•
= 0430 UT
= 55ø
U2 • D[!• -{--0.33B• •']
6078
BOLLER
DECEMBER SOLSTICE
AND STOLOV
EQUINOXES
,JUNE
SOLSTICE
-90 ø
o
of:
-78.5,10l
.,.
102 ø
o •
s6;5 '•
113ø.5
125 ø
.....
o
6
I
J2
18
24UT
, •.
0
,
6
.
.i
12
!
.
18
24UT
0
6
12
18
24UT
Fig.4. Plotof universal-time
variation
of Kelv•n-Helmholtz
instability.
fortheDecember
solstice,
theequinoxes,
and priorireason
fora linear
assumption,
it repretheJunesolstice.
Wehaveassumed
herethat sents
the simplest
working
hypothesis.
It is
theprobability
of instability
isapproximately
a clearfromTable1 that the flankshavethe
linearfunction
of thedifference
between
the highest
probability
ofinstability
attheequinoxes
left-hand
sideandthe right-hand
sideof the andin addition
havea smaller
diurnalvaria~
instability
condition.
Although
thereis no a tionthanthe solstices.
Theexpected
diurnal
JUNE
MINUS
DECEMBER
A44
i
I
i
6
12
18
24UT
.
0
6
PREDICTED
!
!
!
12
.
.
18
24UT
McINTOSH
b
44.
+2
i
--4
.
0
,
.
6
[
.
12
MAYAUD
C
. i [
18
i
24UT
24UT
NICHOLSON,
WULF
Fig.5. Predicted
JuneminusDecember
universal-time
variation
of geomagnetic
disturbancecompared
with data analyses
of (b) Mcintosh[1959],(c) Mayaud[1967],and (d)
Nicholson and Wul• [1955].
KEI,VIN-HELMHOLTZ
variationat, the solstices
is approximately
7
INSTABILITY
6079
firrnedby severalanalyses
of availablegeomag-
timesgreaterthanat the equinoxes,
andthe netic data.
maximumprobabilityof instabilityat the
]-)ATA ANALYSIS: CONFIRMATION OF
solstices
isapproximately
equalto theminimum
KELVIN-HELM HOLTZ HYPOTHESIS
probability
of instability
at the equinoxes,
as
The universal-timevariationpredictedby the
Table I and Figure4 show.The ordinateof
Kelvin-Helmholtz
hypothesisshould be reFigure5a is thedifference
between
theprobvealed
in
the
geomagnetic
data on the removal
abilityofinstability
at theJunesolstice
andthe
probability
of instability
at the Decemberof the local-time effect. The changesof geosolstice
butislabeled
asthedifference
betweenmagneticactivity with local time are well
theexpected
disturbances
(A disturbance),
s.ince known and have little seasonal variability.
theinstability
condition
of tl•emagnetopause
is Therefore,the removalof the local-timevariaby taking
taken to be geomagnetically
.effective.
Figure tion may be simplyaccomplished
6apresents
a similarplot,buttheordinate
now differencesbetween seasons.The June minus
represents
the difference
between
the sumsof December disturbancedifferenceleads to the
theexpected
disturbances
at theequinoxes
and doublingof the predictedvariationand is illu-
the sumsof the expecteddisturbances
at the strated in Figure 5a. The maximumoccursat
solstices.
The phaseandamplitude
of the pre- 0430 UT and the minimum at 1630 UT. The
betweenMarch and September
would
dicted curves of Figures 5a and 6a are' con- difference
EQUINOXES
MINUS SOLSTICES
T
I
0
i
6
•
!
6
12
PREDICTED
o
0
I•.
MAYAUD
C
18
24UT
0
6
12
McINTOSH
!8
24UT
0
6
12
!8
24UT
b
!8
P.4UT
NICHOLSON, WULF
d
Fig.6. Predicted
equinoxes
minus
solstices
universal-time
variation
ofgeomagnetic
dis,
turbance
compared
withdataanalyses
of (b)McI•tosh
[1959],
(c)Mayaud
[1967],
and(d)
Nicholson and WulJ [1955].
6080
BOLLER
AND
STOLOV
obviouslyeliminate both the predictedeffect universal time. An excellent verification of the
and the local time variation. Therefore, the predictedphaseis obtainedin all three cases,
sum of the equinoxesminus the sum of the and the relative amplitudesare essentiallyas
solsticesresultsin a doublingof the predicted expected.The fact that the agreementwith the
equinoctialvariation seenin Figure 6a. Maxi- Mayaud data is lessexact can be attributed to
mums occur at 1030 and 2230 UT and minithe longer data interval taken about the key
mums at 0430 and 1630 UT.
days.
In order to verify further the Kelvin-HelmThe predictedvariationis compared
with several analysesof geomagnetic
activity in Figures holtz mechanism,supportedby the geomagnetic
5 and 6. Figures5b and6b present.
the indicated analysisof others,the data from four observadifferencesof the average daily variation of tories closer to the auroral zone are studied:
the K index of 12 observatoriesduring 1950- Murmansk (63.5øN, 126.2øE), Wellen (61.8øN,
1955 on selecteddays of Kp sum •20 reported 237.1øE), Kerguelen (56.5øS, 127.8øE), and
by Mcintosh [1959]. Using data from all stations,the averagevalue of K is obtainedseparately for the monthsof June,December,March,
and Septemberfor each3-hour interval of uni-
Maquarie Island (60.7øS,243.0øE), where geomagnetic coordinates are indicated. The K indices analyzed are taken from three weeks be-
fore to three weeksafter the key dates of the
equinoxesand solsticesfrom December1, 1959,
The eight 3-hour intervals are then averaged to October 12, 1964. The indices are. lin.earized
for each of these months to obtain the average as in the method of Bartels by forming ak, the
monthlyK. The differencebetweenthe monthly equivalent amplitude, for each station and then
average and the individual 3-hour monthly converting these into the range in y. These
averagesgives the 3-hour deviationsfrom the values are averagedseparatelyfor the two stamean fer each month. This method is generally tions in the north and in the south at each seaapplied in determining values used in Figures son for each 3-hour period of universal time
5 and 6. Figures 5c and 6c present the dif- The indicated differences are then taken and
ferences of the average daily variation of plotted in Figures 7a and 7c. Compositesfor
Mayaud's index (am) computedfor sixteensta- all four stationsare shownin Figures 7b and
tions for three selected years (1959, 1961, 7d. As before, the phase verification is excel1964), including only 3-hour intervals with lent, and, in addition, the relative amplitudes
Kp •_ 2o [Mayaud, 1967]. Mayaud's amindex are now clearer for the near auroral stations.
is derived from his Km index in the same manAnalysis of the K indices from eight polarner as a,, is derived from Kp. The index Km cap observatoriesare performed for the same
is formed from the K indices of ten stations in
time periodand, as expected,reveal no seasonal
versal time over the indicated interval of years.
the north and six stations in the south between
Kelvin-Helmholtz
45ø and 55ø geomagneticlatitude, after certain modificationsare made so that amis representative of an equivalent amplitude in the
unit •. Mayaud's June solsticeconsistsof the
polar cap representsquite a different regime
of geomagneticactivity.
monthsMay, June,July, and Aug.ust;his December solstice consists of the months Novem-
ber, December,January, and February. Mayaud's equinoxesconsistof the months March,
April, September, and October. Figures 5d
effect. It
is clear that
the
SATELLITE EVIDENCE Or WAVES AND
INSTABILITIES
Several satellite and related measurements
are available that support the proposed instability of the magnetopause:
and 6d present the indicated differencesof the
1. Heppner et al. [1967], with OgoA maglatitude,
average daily variation of the K index for six netometersbetweenñ15 ø geomagnetic
observatories,includingall days of 1940-1946 find that the flanks of the magnetosphereare
calculatedby Nicholsonand WulJ [1955]. highly unstable.
These curves were formed from Nicholson and
2. Anderson et al. [1968] associate Imp
Wulf's individual monthly curves (June, De- 2 electron-fluxchangeshaving periods between
cember, March, and September) of the mean 3 and 20 min with hydromagneticdisturbances
deviation of average K values as a function of propagating on the magnetopause.
KELVIN-HELMHOLTZ INSTABILITY
JUNE
6081
DECEMBER
MINUS
7'
,
+50
0
-50
-I00
-30
0
-60
-50
-I00
-90
i
12
18
24 UT
EQUINOXES
MINUS
SOLSTICES
7'
7'
+1oo
+50
+60
o
ALL
4-$0
4-1oo
+50
o
o
;;
i
i
!
!
c
o
!
'!
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i
,s
d
Fig. 7. Variation of geomagneticactivity with universaltime for two stationsin the north
and two stationsin the southfor JuneminusDecember,and equinoxes
minussolstices,
including composites.
3. McDiarmid
and Wilson [1968] with
dencelocalizesthe sourceof thesehydromag-
Alouette2 find a loweringof the high-latitude neticwavesto the magnetopause.
electronboundarydue to a strongercoupling
6. Kaufman and Konradi [1969] with Exbetweenthe solarwind and the magnetospl•ere plorer 12 find large:amplitude distortions
when the angle between the. sun-earth line traveling alongthe flanks toward the tail. These
and the geomagnetic
axisis near 90ø.
distortions
are frequentlytriggeredby a change
4. Nagata et al. [1963] detectmagnetically in the propertiesof the solar wind.
conjugatetransversegiant micropulsations
havSUMMARY
ing left-handed polarizations at dawn and
right-handedpolarizationsat dusk.ThesemicroKelvin-Helmholtzinstabilityalongthe-flanks
pulsations(pc 5) suggestthe presenceof sur- of the magnetosphereexhibits a semiannual
facewaveson the magnetopause
[Atki•son and variation, with instability maximums at the
Watanabe, 1966].
equinoxes.and instability minimums at the
5. Du•gey and Southwood[1969] with Ex- solstices.Kelvin-Helmholtzinstabilities,genplorer 33 find hydromagnetic
wavesnear the erated at the magnetopause,
initiate the modumagnetopause
with polarizationsin agreement lation of geomagneticdisturbancedetectedas
with data presentedby Nagata et al. [1963]. the semiannualvariation of geomagneticacThey find polarizations
just outsidethe mag- tivity. The Kelvin-Helmholtzexplanationprenetopause opposite to those inside. This evidictsa universal-time
variationof geomagnetic
6082
BOLLER AND STOLOV
TABLE 2. Data from Explorer 34
parameter
Crossing1
(Kp = 3o)
Crossing2
(Kp = 1 - )
B•
ou
,:i,•
B[
o•.
iI:,•r
XsE
Ysr
ZsE
Rat
U,•
n•
45?
0ø
315ø
40y
10ø
310"
6.96
0.21
-6.17
9.30
539 km/sec
2.3/cm 8
30?
20"
240 ø
15y
-45 ø
210"
5.07
-12.71
-7,40
15.56
382 km/sec
2.0/cm 8
disturbancethat is verified by the data. Since
the physicalmechanismsuggested
dependson
the tilt of the dipoleaxis,the 'equinoctial'hypothesisis supported.
APPENDIX
ILLUSTRATIVE
QUANTITATIVE
ESTIMATES
OFTHE
KELVIN-HELMHOLTZ INSTABILITY CRITERION
bow shock.Both crossingswere outbound,so
that the hourly average values of solar wind
speed(U•) and numberdensity(n•) measured
are thoseobtainedjust after the satellitepassed
throughthe bowshock.Table2 liststhesemeasured parameters.The magneticfield measurementsand satelliteposition(in earth radii) at
the crossingpoint are given in solar ecliptic
coordinates[Ness et al., 1964].
The measuredparametersare not in a form
that couldbe used in the instability condition.
In particular, the values of the solar wind
streamspeed(U) and massdensities(pt, p•)
are neededat the satellitecrossingpoint. These
valuescan be obtainedby usingSpreiteret al.'s
[1968] gasdynamiccurves (e.g., for a freestreamMach numberof 5). They displaycontours of the ratio of magnetosheathto free-
streamvaluesof thesequantitiesfor the dayside
magnetosheath.The free-stream values (U,
and n•) determin• the stream speed and
the mass density at the crossingpoint. The
magnetospheremass density is assumedto be
the sameas the magnetosheath
massdensityin
the absenceof interior plasmameasurements,
The directionof the solar streamingat the
The magnitudesof the left- and right-hand
sides of the instability criterion will be estimated'for disturbedand quiet conditions
of the
mag.n.
etosphere.Explorer 34 data are used for
tWOcrossings
of the magnetosphere
boundary.
The first crossingoccurredon September14,
satellitecrossing
pointmustbe foundso that
the componentsof the magneticfields in this
direction can be determined. The direction of
the solar streamingis uniquelydeterminedby
the intersectionof the plane containingthe
satellitecrossingpoint and the sun-earthline
1967,at 0618 UT duringa moderatelydis- and the planeformedby the magnetosheath
and
turbedperiod(Kp - 3o). The secondcrossing magnetospheric
magneticfields at the crossing
occurredon November18, 1967,at 0940 UT
point.The quantities
B, cos•, andB• cos•
during a quiet period (Kp - 1-). Each are then found.
crossingsuppliedmagneticfield measurements Table 3 contains the calculated values of the
on either side of the magnetopause.
Reliable parametersto be used in the instability conplasmadata were availableonly outsidethe dition for both crossings.
The instabilityconTABLE
3.
Calculated
Values
,.
Quantity
U, cm/sec
or: o•, g/cm•
B) cos•, gauss
B• cos•r, gauss
?, cm'/sec'
•.r
,• 0MIBm'
cos'
•l'q-B•'cos
•], gauss
zcmS/g
4•rPlPM
Instability conditionfulfilled
Crossing
i (•Kp-- 30)
2.16
1.23
--7.09
--1.63
4.66
X 10?
>< 10-•s
>< 10-•
>< 10-4
>< 1014
4,08 >< 1014
Yes
Crossing2 (Kp := 1- )
2.29
7.68
8,.39
2.12
5.25
X 10 ?
>< 10-•4
>< 10-s
>< 10-4
>< 10•4
1.08 >< 101•
No
KELVIN-HELMHOLTZ
dition is fulfilled for the more disturbed cross-
INSTABILITY
60'83
the auroral zones, Phys. Rev. Le.tt., 6, 47-48,
1961.
ing point, but not for the quiet crossingpoint.
Dungey, J. W., Waves and particles in the magApplication of the gasdynamic curves of
netosphere,in Physics o• the Magnetosphere,
Spreiter et al. [1968] to determineconditions
edited by R. L. Carovillano, J. F. M cClay, and
H. R. Radoski, pp. 218-259, D. Reidel, Dordat the flanks yield a similar result. The two
casesstudied here are merely for illustration.
A detailed statistical study of the KelvinHelmholtz inequality and its significancefor
the onset of geomagneticactivity is currently
in progress.
Acknowledgment. This researchwas supported
in part by the National Aeronautics and Space
Administration
wish to thank
under contract NsG
Drs. D. H. Fairfield
197-62. We
and K. W.
Ogilvie for their kind assistancein making available the magnetic field and plasma data. We also
wish to thank Dr. Robert Jastrow for his hospitality at the Institute for Space Studies.
The Editor
wishes to thank G. V. A tkinson,
R. A. Wolf, and another referee for their assistance in evaluating this paper.
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