GEOPHYSICALRESEARCHLETTERS,VOL. 22,NO. 20,PAGES2701-2704,OCTOBER15, 1995
A new technique to model plasmoid magnetic field signatures T.K. Nakamura, l T. Nagai, 2Y. Kazama, l K. Amano, l T. Yamamoto, l and S. Kokubun 3 Abstract. A new fitting methodhasbeendevelopedand appliedto the 2. Soft Fitting method plasmoidobservations of the GEOTAIL satellite.This method,which we Classicalfitting methods,such as the least squaremethod, call "softfitting",is regardedas onekind of leastsquaremethod,andcan assume that an analyticalfunction (or someexplicitly-defined determineboththe fittingfunctionandfittingparameters simultaneously. function) can fit observed data. They assume an analytical Its applicationto magneticfield observations by the GEOTAIL satellite functionwith severalunknownparametersas a fitting function, givesan averageshapeof plasmoidBz bipolar structures. The method andtry to determinethe mostprobablevaluesof the parameters can alsoprovideinformationaboutthe velocityand spatialsize of plasusing observeddata. This type of fitting method gives satmoidsfrom the optimalfitting parameters.The resultindicatesthat the isfactory resultswhen an explicit shapeof the fitting function velocityandsizebecomelargeras a plasmoidpropagates tailward.
is provided by a theoretical model or by physical intuition. However, it is not sufficient when the expected shape of observeddata is not known exactly; the Bz bipolar structureof 1. Introduction a plasmoidis a good example.In the presentpaper we introduce Bipolar structuresin the Bz (in the GSM coordinatesystem) a new method of fitting which can determine the fitting funcmagnetic field are frequently observedin the Earth's mag- tion and fitting parameters simultaneouslywhen we have a netotailregionimmediatelyafter substormonsets.Thesebipo- number of similar data sets, as we have for the bipolar struc-
lar structures arereported to be evidence of plasmoids created by near tail reconnection [Baker et al., 1987; Hones et al., 1984; Moldwin and Hughes, 1992; Nagai et al., 1994; Richardson et al., 1987; Scholer, 1986; Slavin et al., 1993]. Each event is superposedon some other structuresor noise, thereforewe need a statisticalaverageto obtaina standardpicture of plasmoids.However, straightforwardaveragingis not appropriatebecausethe scalesin time and amplitudeare different in each event; the observeddurationof bipolar structures varies due to the variation of the size or speedof plasmoids, and amplitudesare different dependingon the plasmoidsize or satellite
tures of substorms.
Suppose wehave n sets ofobserved BzdataBi(t)( i= 1...n) as functions of time
t. We assume that all of the Bz structures
have a similar shape with different scaling factors, i.e., the structurein the i-th event is expressedas
a•Bilt-x•l=F*(t)+Ni(t (1) w,here N,i is noise superposed onthesignal in eachevent; ai andhi arethescaling factors of amplitude andtime,andx?
defines the initial time. Our goal is to find the best values of
parameters (a•, h•, xi , i=l
location.
To overcomethis difficulty in averagingwe employ a new method to obtain an averaged structure of plasmoids. We assumeall the bipolar structureshave similar shapewith different scalingfactors both in time and amplitudeand treat these scalingfactorsas fitting parameters.The fitting parametersare optimizedso as to minimize the deviationbetweeneach event and the averageof all events.With this procedurewe can determine the fitting parametersand the fitting function simultaneously. We call this method "soft fitting" becausethe fitting function can be flexibly determined. We have applied our methodto bipolar structuresobserved by the GEOTAIL satellitemagneticfield experimentto obtaina standardpicture of Bz bipolar structures.
n) andthe shapeof original
function F*(t) simultaneously(hereinafterwe denote true valuesof parametersby asterisks(*)).
If wehavethetruevalues of ( a•, h•, x?), F* (t) canthenbe calculated
as:
F*(t)=-•ai Bi -xi ni=1 h•
(2)
as long as the noise N i vanishesby averaging.On the other hand,if we knowthe originalshapefunction F* (t) thenthe pa-
rameters (a•, h•, x? ) canbe obtained by the conventional
least square method. Wedefine thedeviation D* as
D*(ai,hi,xi)= • dtaiB i t-xi -F*(t) i=1
(3)
hi
for chosen valuesof (ai, hi , xi) anddetermine thebestvalues lInstitute of SpaceandAstronautical Science 2Department of EarthandPlanetary Sciences, TokyoInstituteof Technology 3Solar-Terrestrial Environment Laboratory, NagoyaUniversity
byminimizing D* in thismethod. In our "softfitting"methodwe usechosenvalues(a i, hi ,
xi) instead of thetruevalues (a•, h•, x?)in Equation (2) and substitute into Equation (3). That is, we minimize a func-
tionD(ai,hi,xi) definedas 2
Copyright1995by theAmericanGeophysical Union.
D(a i hi,x i) i=1 • dtaiBz, i t-xi 1i•1 naiz,i-Xi hi n hi
Papernumber95GL02816 0094-8534/95/95GL-02816503.00
(4) 2701
2702
NAKAMURA ET AL.' A NEW TECHNIQUE TO MODEL PLASMOID
insteadof D*. This functionD can give a goodapproximationfor D* nearthe minimumpointof D*, thereforethe parameters (ai, hi , xi) thatminimizeD canbe regarded asthe
lO
optimalparameters (a•, hi, x?). Oncewe obtaintheoptimal parameterswe can calculatethe shapefunction F*(t) from Equation (2). In the actual minimization processwe need one
constraintof •a i =constantto avoid the trivial minimumof ai = 0 for all i. This constraintmeansthat the averageof the
0
10
20
30
40
50
Delay Time (min)
plasmoidshapehas a certain value; not zero. Intuitively speaking, the fitting procedure becomes equivalentto the following steps as long as the function D doesnot convergeto a false minimum point.
N
1) Giveaninitialguessof parameters (a i, hi , xi). 2) Calculatethe averageof Bz with theseparameters. 3) Use this average as a fitting function for the least square methodand find a better set of parameters. 4) Iterate (2) and (3) until D converges. In the actual fitting procedurewe need a function minimization algorithm in 3n dimensionalparameterspaceto find
RescaledDelayTime, (arb. unit)
outtheoptimalparameter setof (a•, ht, xi ) ( i = 1... n). Here
we use Powell's method [Press et al., 1988] for simplicity. A Figure 2. Ten Bz bipolar structuresobservedafter substorm local minimization algorithmlike the Powell'smethodhas the onsets(t=0) are plottedin (a). Our methodrescalesthesedata as disadvantage of often converging to a false minimum. in (b) to obtain an averagepicture. Therefore, we have to choose the initial guess carefully to avoid false minima. Fortunatelywe can give a reasonablygood
initial guessby eye inspectionin our case,and we can confirm ting; we have picked up 61 eventsout of 110 that are clearly the plausibilityof the fitting by checkingthe result plot like appropriatefor fitting. This does not necessarilymean that our Figure2-b. Hencewe do not needto usea moresophisticated method cannot fit other events; more careful use of our method and complicatedglobal minimizationalgorithm.
would make more events available. Here in the first report of our methodwe limit ourselvesonly to the most unambiguous 3. Results eventswithout questionfor fitting. Figure-1 shows the satellite's X position of the bipolar We appliedour methodto Bz bipolarstructures observed by structureobservationsagainstthe time delay (time betweenonthe magnetometer onboardGEOTAIL [Kokubunet al., 1994]. set time and Bz zero crossing).This figure is essentiallythe Nagai et al. [1994] reported53 bipolarstructures associated sameas Figure 5 in [Nagai et al., 1994]; the difference is that with substorm onsets observed during October 1992 we use the Bz crossingtime determinedfrom our fitting. We September 1993.We usea similardatasetobtained for a longer plot ten Bz bipolar structuresin Figure 2-(a) superposed in one period(October1992- February1994).The methodto find the panel as an example. The time interval of each event shouldbe bipolar structuresis the same as in Nagai et al. [1994] , determined so as to be long enough to contain a bipolar however,not all bipolarstructures thereinare suitablefor fitstructure,yet short enough to avoid contaminationfrom other structures.To this end we pick up only well defined single bipolar structures and avoid multiple bipolar events. The 45 interval for each event is determinedby inspection;we have confirmedthat small changesin the time interval do not alter the resultof fitting drasticallyin most of the cases.
40
In each event, the time t = 0 is the substorm onset time
[Nagai et al., 1994], i.e., Bi(O) is the magneticfield at the
substorm onset. By adjusting theparameters (ai , hi , xi ) with
'1o
our method,we can re-scaletheseeventsas shownin Figure 2(b). Averaging these data with Equation (2) gives the most probablepicture of plasmoidBz bipolar structure.Figure 3 is the averageof 61 bipolar structuresobtainedfrom our method;
r"l 5
wehaveminimizedD(ai,hi, xi) in 3 x 61 dimensional spaceto obtainthisfigure.We regardD(ai,hi,xi) to haveconverged
,1s o
i
i
i
when the changeof its value becomesless than 0.1% in a new iteration.
0
100
200
300
Fromtheoptimalparameters of ( a•, h•, 1;i ) we canobtain the duration (size of the structure in time domain) of each
IXl (Re) Figure l. Delay times between substormonset and Bz zero crossing for the bipolar events as a function of GSM X position.
bipolar event, and the delay of its arrival time from the substom onset. Therefore, we can calculate the velocity and spatial size of each plasmoidwhen we assumethesevaluesdo not vary muchduringthe plasmoidpropagation.
NAKAMURA ET AL.' A NEW TECHNIQUETO MODEL PLASMOID
2703
becausethat the locationsof bipolar structureobservationscan be roughlydividedat I Xl = 125 Re due to the satelliteorbit. Resultsare: velocity -390 + 80 km/sec,size -21 + 2 Re in the middle tail (IXI< 125 Re); velocity -610 + 40 km/sec, size -48 + 12 Re in the distant tail (IX I> 125 Re). 1G
Strictly speaking,it is inconsistent in our analysisto think that the velocityof a plasmoidchangesduringits propagation, because we have assumed the velocity to be constant to calculate it from the arrival time. Therefore our estimations
N
-1
Average -z
-3
Time (Arb. Unit)
may be somewhaterroneousespeciallyin the distanttail region. However,the velocityincreasesat mostby a factortwo (390 km/secto 610 km/secin the aboveresult),hencethe obtainedvaluescan give reasonableestimationsfor the averages. We canconcludeat leastqualitativelythat the sizeand velocity of a plasmoid getlargerasit propagates downto thedistanttail region.
Figure 3. The average shape from 61 bipolar events obtainedwith our method.The thick line is the averageand the 4. Summary thin line indicates the width of one standard deviation from the
We have developeda new fitting methodthat can simultaneouslydetermineboth the fitting functionand the fitting 33 Re (see text). parameters. We haveappliedthismethodto Bz bipolarstructuresobservedright after substorm onsets,andobtainedan averagepictureof a plasmoidBz structure. We canalsoestimate Nagai et al. [1994] estimated the averaged plasmoid ve- thevelocityandspatialsizeof eachplasmoid fromtheoptimal locity as 775 km/sec from the coefficient of the linear fitting parameters obtained in ourfittingmethod. Theresultssuggest: on the time-distanceplot (Figure 5 in their paper). We estimate (1) The averagevelocityis -480 km/secandaveragesize
average.One tick mark of the horizontalaxis corresponds to
the velocity in a different way: we assumethat the near tail reconnection line is located at X =-30 Re and calculate the veloc-
ity in each event using the arrival time obtainedfrom the fit-
tingparameters (a•, h•, x?). We define the arrival time of a plasmoidat the zero crossing point of Bz in Figure 3, therefore,the arrival time T in the
resealed unit is definedby thepoint F* (T)= 0 in Equation(1).
is-33
Re.
(2) The deviationin the velocityis muchlargerthanthat in the size, thereforethe differencein the observeddurationis mainlydueto the velocitydifference.
(3) Both the velocityand size increases as a plasmoid travels towards the distant tail region.
The velocity and size obtainedin the presentstudyis
Then the time interval between the onset and the plasmoid somewhatsmaller than those in Nagai et al. [1994] even
arrivalis calculated as Ati =(T-x?)/h* in theactualscalefor thoughtheyuseda similardatasetasours.Thisdifference can the i-thevent. If we determinethe arrival time by the zero beexplained fromthedifference in calculation method applied. crossingpoint of Bz in eachevent,a small amountof noisecan Nagaiet al.. [1994]estimated the velocityfroma linearfit of shift the zero crossing point considerably. However, our methodutilizes the overall shapeof the bipolar structureand is thus more robust to noise.
The averagevelocity of plusmoldsis then calculatedto be -480 _+190 km/sec and the averagespatialsize (the lengththat corresponds to one tick mark in Figure 3) is -33 _+6 Re C_+" indicatesone sigma width). This estimationdoes not depend critically on the assumption for the location of the reconnection line. For example,if we choosethe value of X =25 Re for the reconnectionline the velocity and size become -510
km/sec and -35
e
30
>
20
o
10
z
0
standard deviation of velocities and sizes are 0.40
and 0.17 respectively.This means that the differencein the observeddurationis mainly due to the velocity difference. Nagai et al. [1994]reportedthat the sizes of plasmoidsin the distanttail region are larger than thosein the middle tail region. Now we have obtainedthe velocity and size of each plasmoidfrom our fitting to calculatethe averagedbipolar structure(Figure 3). Hencewe can averagethe velocityand size
0.4 0.8 1.2 1.6
Size (Avr.=1.0)
Re.
Figure 4 is the distributionof velocities and spatial sizes; velocitiesand sizes are normalizedby the averagedvalue. The normalized
o--0.17
20
ß 15
0=0.40
uJ 10
o
5
Z
0 0
' 0.4
' 0.8
' 1.2
'•••• 1.6
2
Size (Avr.=1.0)
in the middle tail and in the distant tail separatelyto check the size/velocity changealong the plasmoid propagation.We es- Figure 4. Histogramsof spatial size (upper panel) and timate the averagevelocity and size for two different regions, velocity(lower panel).Sizesand velocitiesare normalizedby I XI < 125 Re (35 events) and I XI > 125 Re (26 events);this is the average(averagevalue = 1.0).
2704
NAKAMURA ET AL.: A NEW TECHNIQUETO MODEL PLASMOID
the time-distancerelation. They did not assumethe starting We can infer severalexplanations for the velocityand size time and startingpoint of each plasmoid,so the fitting gives a variationalong the plasmoidpropagation(result (3) in the list somewhatunphysicalresult for the startingpoint; its location above).For example,the velocityincreasemightbe the result is sunwardof the Earth. We have assumedthat plasmoidsstart of the decreasein the plasmasheetpressure,andthe increasein from X---30 Re at the substorm onset time. As a result our estithe size might be explainedby the pile up of magneticfield mation gives a somewhatlower estimationfor the velocity and lines during the propagation.Hopefully further analysisof the size.
GEOTAIL datacanrevealthe underlyingphysicsandshedlight
Nagai et al. [1994] calculatedthe averagesize of plasmoids on the plasmoidpropagationmechanism. from the averagevelocity they obtained(775 km/sec) both for the middle tail and distanttail. However, our studyshowsthat References the plasmoidvelocities are also different in thesetwo regions; they becomefasterin the distanttail region.In our analysiswe Baker,D. N., R. C. Anderson,R. D. Zwickl and J. A. Slavin,Average plasma and magnetic field variations in the distant magnetotail can obtain the duration and arrival time of each plasmoid associatedwith near-Earthsubstormeffects,J. Geophys.Res.,92, 71, preciselyfrom the fitting parameters,and we can estimatethe size
based
calculation
on these method
individual
values.
This
difference
is another reason for our smaller
in
size es-
timation in middle tail region. The plasmoid velocity obtained in our calculation is consistent with previous papers [Baker et al., 1987; Richardson,et al., 1987; Slavin et al., 1993], however, the size in our studyis somewhatdifferent from the result reportedby severalauthors[Moldwin and Hughes,1992;Richardson, et al., 1987]. The size reportedby Moldwin and Hughes [1992] is smallerwhile the size obtainedby Richardson, et al., [1987] is larger than our result. Moldwin and Hughes [1992] also concluded that the plasmoid size is almost constant for IXl > 100 Re. There are severalexplanationsfor thesedifferences.One is that the resultsdependon the definition of the plasmoidsize or the event selection criteria as discussedby Moldwin and Hughes [1992]. The differencein methodologymay also cause the discrepanciesbetween the results. Careful examinationof thesepoints could provide insight to the plasmoidpropagation mechanism. However, we need to analyze additional Geotail data, especiallyparticle data, to compareour resultswith the
1987.
Hones,E. W., et and al, Structureof the magnetotailat 220 Re and its responseto geomagneticactivity, Geophys.Res.Lett., 11, 5, 1984. Kokubun,S., T. Yamamoto,M. H. Acufia, K. Hayashi,K. Shiokawaand H. Kawano,The GEOTAIL magneticfield experiment,J. Geomag. Geoelectr., 46, 7, 1994.
Moldwin, M. B. and W. J. Hughes,On the formationand evolutionof plasmoids:A surveyof ISEE 3 geotail data, J. Geophys.Res.,98, 19259, 1992.
Nagai, T., K. Takahashi,H. Kawano,T. Yamamoto,S. Kokubunand A. Nishida,Initial Geotailsurveyof magneticsubstormsignatures in the magnetotail,Geophys.Res.Lett, 21,2991, 1994. Press,W. H., B. P. Flannery, S. A. Teukolsky and W. T. Vetterling, Numerical Recipiesin C, CambridgeUniversity Press,Cambridge, 1988.
Richardson,I. G., S. W. H. Cowley, E. W. Hones and S. J. Bame, Plasmoid-associated energeticion burstsin the deepgeomagnetic tail: Propertiesof Plasmoidsand the postplasmoidplasma sheet, J. Geophys.Res.,92, 9997-10013, 1987. Scholer, M., A review of the ISEE-3 geotail suprathermalion and electronresults,Planet.Space$ci., 34, 915, 1986. Slavin, J. A., M. F. Smith, E. L. Mazur, D. N. Baker, E. W. Hones, T. Iyemori and E. W. Greenstat,ISEE 3 observationsof traveling previousones;this is beyondthe scope9f the presentpaper. compression regionsin the Earth'smagnetotail,J. Geophys.Res.,98,
The purpose of this paper is simply to introduce our new methodand to give a brief report of its first application. We have assumedthat all the observedbipolar structures have similar shapewith different scalingfactorsin our analysis. There is no validation to this assumption;the shapeof observed bipolar structuresmay be different dependingon the satelliteposition,etc. However, the essentialshapeof the Bz structureis expectedto be unchangedat different observation positions(e.g., a bipolar structurecannoteasily becomea trior tetra-polarstructureat a differentposition).Thereforewe can say our averagepicture in Figure 3 can representsubstantial characteristics of plasmoidbipolar structures,althoughits details may not be quantitativelyprecise
15425, 1993.
T. K. Nakamura, Y. Kazama, K. Amano, T. Yamamoto, Institute of
Space and Astronautical Science, 3-1-1 Yoshinodai, Sagamihara, Kanagawa229, Japan (e-mail:
[email protected] [Nakamura]) T. Nagai, Dept. of Earth and PlanetarySciences,Tokyo Instituteof Technology,2-12-10hokayama, Meguro, Tokyo 152, Japan S. Kokubun, Solar-Terrestrial Environment Laboratory, Nagoya University,3-13 Honohara,Toyokawa,Aichi 442, Japan
(Received May 5, 1995;revisedAugust7, 1995; accepted August21, 1995)
