GEOPHYSICAL RESEARCHLETTERS, VOL. 23, NO. 22, PAGES 3251-3254,NOVEMBER 1, 1996
A comparisonof self-consistentkinetic and quasi-MHD
simulations:Application to a dipolarizing field reversal
SandraC. Chapmanand ChrisG. Mouikis•
Space and AstrophysicsGroup, University of Warwick, UK
Self consistentone dimensionalhybrid code
an isotropic plasma. Subsequentlythe magnetic rever-
(particleions, massless
electronfluid) simulationsare
sal relaxes(dipolarizes)as there is insufficientparal-
used to examine the possiblestructure and evolution of
a dipolarizing field reversal in the near earth geotail.
Here, we run the simulations in both the ion kinetic
lel plasma pressure to maintain equilibrium. Hybrid
scales,and the quasi- MHD limit, where all ion trajectories are still fully resolvedbut the ion moments are
taken, and the electromagneticfields are advancedon
spatial scaleslarger than the gyroradiiof the ions. One
simulation selfconsistentlyincludesthe ion current and
densityresolvedon ion Larmor scaleswhereasthe other
doesnot; both simulationshave sufficientbandwidth to
well resolveall the low frequencywave modesexpected
is also seenin hybrid simulationsof rotational disconti-
Abstract.
codesimulationsof this system[Richardsonand Chapman, 1994]showthat ionsare accelerated
givinga non
(hybrid) limit, where the ion momentsand the elec- isotropicion pressuretensorand a crosstail (GSE y
tromagnetic fields are all well resolvedon ion Larmor directed)magneticfield componentis generatedwhich
nuities[e.g.Lin andLee,1993].
2.
Simulation
Details
The hybrid schemeusedhere [Terasawaet al., 1986]
representsthe ion f(v) with an ensembleof computational particles that follow trajectories prescribedby
fromkinetic(in the hybridcase)or 2 fluid (in the quasi- the Lorentz force law; the changein positions and veMHD case)theory. It is shownthat the evolutionof locities of these computational particles are resolvedon
the electromagneticfield and ion pressuretensor are
markedlydifferent in the two cases.The resultssuggest
that the selfconsistentquasi- MHD solution obtained in
this way cannot be obtainedfrom a coarsegrainingof
the full kinetic (hybrid) selfconsistent
solution.
scales determined by the simulation timestep At. At
each timestep the ion number density and current density n•, J• are obtained by taking moments of the computational particle ensembleonto the simulation grid
and are henceresolvedat the grid spacingAx (and for
numerical stability a Courant condition is also satisfied
i.e. Ax > cat where c is the velocity of the fastest
1.
wave mode in the simulation). The fields are then
Introduction
Selfconsistent simulations are now widely used to
study magnetosphericboundary layers and structures.
The set of equations used, and hence the solution obtained, dependson the a priori choiceof the spario-temporal scalesto be represented.Another approach,used
for example in the geotail, is to integrate large numbers
advanced on the simulation grid using quasi- neutrality, the "massless"electron fluid momentum equation
where the inertial term has been neglected(and with
P• = nkT•, T• constant),and Maxwellequationsat low
frequency(displacementcurrent term neglected). At
the next timestepthe fieldson the grid are then extrapolated to the particle positions to advance the particle
of singleparticle trajectoriesin prescribedfields that trajectories.
The two simulation
runs shown here have identical
are obtainedfrom either globalempiricalmodels[e.g.
Ashour-Abdallaet al., 1994]or from MHD simulations bulk plasmaparametersand time resolution(in terms
[e.g. Joyceet al., 1995]. This hasthe advantagethat of the ion Larmor period) differingonly in the spatial
the region where ions are acceleratedis connectedto a resolution of the simulation grid. In the hybrid case,
global system, but the disadvantageof non selfconsis- Ax << R• the thermal ion gyroradiusin the backtency in the sensethat the the bulk plasma moments ground field: the fields are advancedusingcurrentsre(numberdensityn, ioncurrentJi, ionpressure
7vi)of solvedon scales<< R• and the ionsexperiencechanges
the evolvingion f(v) do not contribute.tothe fieldson in the fields as they move severalAx in a singlegyraion Larmor scales. Here we examine this a priori as- tion about the field. In the MHD scalecase,Ax > R•:
sumption: that solutionsof this type can be adequately the field advance does not include spatial variations in
obtained without resolvingselfconsistencyon ion Lar- the current on ion Larmor scalesand the ions experimor scales. The simple example used here is localized ence little changein the fields as they remain within a
"dipolarization" in the near earth geotail. The initial singlegrid cell Ax as they gyrate about the field, only
condition of the simulation is out of equilibrium: the being carried from one cell to the next by field aligned
magnetic field reversessmoothly and symmetricallyin or drift motion of the guiding centre. In both cases
the bandwidth
of the simulation
is sufficient to well re-
solvethe expectedlow frequencymodes(e.g. KraussVarbanet al. [1994])which are not identicalin the
1Now at Max-Planck-Institut fiir Aeronomie, Germany
kinetic and the quasi- MHD limits. The physical pa-
Copyright
1996by theAmericanGeophysical
Union.
rametersfor both simulationsare: v•,•fto• = 0.1V•,•,
(V•,• is the Alfven speedin •) /3• = 0.1, T• = 0.1T•,
n- 0.2cm-•, B• = 20nTand B• = 2nT. The code
Papernumber96GL02845.
0094-8534/96/96GL-02845505.00
3251
21252
CHAPMAN
AND MOUIKIS' KINETIC
AND QUASI- MHD SIMULATIONS
weakinflow~ 0.1Vax
I
_____
•• /
z
W•V•S
.....
.........
Direction of
variation
waves
1
we• •flow - 0.1V•
Figure 1. Geometry of the simulation.
parameters
are: Az -- 1/10Rgi(O.03C/Wpi)
in the hy- Rgi (329.6C/Wpi)
for the MHD scalerun. The parambrid caseand Az = 2Rgi(0.65C/Wpi)
in the MHD scale eters are shownfor t = 9.8rgi, sufficientlylate in the
case;At = 1/1630rgiin both cases;the simulation simulations to correspondto the "post escape"phase
andChapman,
1994]
box has 4096 grid cells in the hybrid caseand 512 in in the hybridsolution[Richardson
when the ions escapefrom the reversaland can be seen
to be moving towards the boundaries.
Electromagnetic Fields: From early times in the
a bandwidthof 0.055 < k < 111.6(C/Wpi)
-1 (hybrid
the MHD scale case; 100 computationalparticlesper
cell are used in both cases and the simulations
have
case)and0.019< k < 4.8(c/wpi)
-1 (MHD scale)with hybrid solutiona bipolar Bu componentevolvesas the
0.1< w < 800fl•• in bothcases.
The initial and boundaryconditionsare identical,the
ion distribution is initially isotropic, has weak inflow
towards the centre z = z0 and is spatially uniform,
Bx = Bxotanh((z-zo)/L),(L = 1.95c/COpi)
By = 0, the
electric field then evolvesselfconsistently.The boundaries are wave and particle absorbing. The geometry
of the one dimensionalsimulationis shownin Figure 1.
All scalarsand vectorcomponentsare retainedbut vary
in the onespacedirection(•) directedacrossthe reversal, and time. The field threading the reversalat t = 0,
Bz, is constant(V. B = 0) in this onedimensional
geometry. All wavesthen propagatein q-•, and the VB 2
magnetic pressureacts along q-• to contain the plasma
in the weak field region at the centre of the reversal,
whereasthe (B. V)B curvatureforceacts perpendicular to the direction
of variation
and accelerates the
Time=9.8e+00Tgi
Vx(l)
Vx(r)
20000-
-200020000-
-2000200-
Ez
-20-
5o - "---"'-"--
Ey
-5200-
Bx,By
-20
0.4-
0.2-
nl,nr
01000-
(J^B)x
-1001000-
(J^B)y
-10010000-
(J^B)z
-1000500-
plasma. The electric field in the simulation frame is:
Viz
r, - -v i ^ B
J AB
ne
V. 7•e
ne
I
0-
-500-
Cei
2-210-
for singleion and electronfluids in the hybrid limit and
-1020-
(2)
by a term of order J A B/he in the hybrid and MHD
cases.We will seehoweverthat the principal differences
in the two solutionsare characterisedby differencesin
vi A B, whichis largecompared
with J A B/he.
Results
The bulk parametersand ion phasespacefor the two
simulationsare shownin Figure 2 (hybrid) and Figure 3 (MHD scale) plotted versusdirectionof variation
across the reversal
z.
In both
cases the central
0-
-20-
(Vi,V)x
for MHD. In the MHD limit v - vi in (2) soE differs
0-
(Vi^BJ^B/ne)y
(Vi^B,J^B/ne)z
3.
0-
(Vi^B,J^B/ne)x
ne
E - -v ^ B +
HYBRID
(Vi,V)y
2•-200010000-
-1000-
-3638.4 (-22.3,-7.2)
0.0(0.1•,0.0)
3638.4,22.3,7.2)
z/km (rgi,c/wpi)
Figure 2. From top to bottom: V• phasespacefor left
and right enteringions(kms-•) (linesindicatewhere
f(v) is takenfor Figures4, 5), Ez, Ey (mVm-•), B•
(solid)and By (dashed)(nT), numberdensityof the
left (solid)andthe right(dashed)enteringions(cm-3),
normalized
J^B components
(.103),z component
ofthe
ionbulkvelocity
(kms-•), V i ^ B (solid
lines)andJ ^
(mV/m), bulkvelocity
512 cells of the simulation box are showncorrespond- B/he (dashedlines)components
ing to 46 Rgi (14.4c/wpi)for the hybridrun and 1026 x andy components
V• (solid)andV (dashed)(kms-•).
CHAPMAN AND MOUIKIS: KINETIC AND QUASI- MHD SIMULATIONS
Time=9.8e+000Tgi
Vx(l)
_2oo•O.
Vx(r)
.2oo•Oi
magneticpressure/ioninertial length). In both cases
this term is significantonly in the central reversalregion so that it doesnot act on the ionsoncethey have
escapedfrom this regionin the hybrid case. The z com-
"MHD"
ponentthencorresponds
to VB 2 in (1) andactsto con-
2000.
tain the plasmain the reversaland the x (and in the
hybrid case,alsoy) componentscorrespondto the cur-
2.
Ez
0.
-2.
5-
Ey
vature force that acceleratesthe plasma. Although the
magnitude of J A B is about 20 times smaller in the
o-520-
Bx,By
MHD scalecase,in physicalunits the width of the central reversalat this time is about 20 times larger, hence
the rate of delivery of momentum from the relaxing B
field to to the plasma over the entire simulation box is
o-20
0.4-
nl,nr
o.2-
(J^B)x
0-
the same in both simulations.
Ion Distribution
Functions:
-5O.t'
(J^B)y
o-
The difference in the
ion dynamics in the two casescan be seen in the top
41.1tOO-
0^B)z
3253
0T= 0 Tgi
500'
Viz
o .....
2
2'
• 2
• 2
(Vi^B,J^B/ne)x o
-210'
(Vi^B,J^B/ne)y o-to10'
-2
-2
(Vi^B,J^B/ne)z o-
-2
-to2000-
(Vi,V)x
0
-2
2
-2
Vperpl I Va,x
0
-2
2
0
2
0
2
Vperp21 Va,x
o
T= 9.8 Tgi
t000-
(Vi,V)y
o
2
-t000-
-83747.3 •
-5!3.0,164.8)
0.0((•.0,0.0)
z7km(rgi,c/wpi)
2
• 2
83747.3
(5!3.0,164.8)
o
Figure 3. As for Figure 2 but for the MHD scalecase.
The double dashed bar indicates the full size of the hybrid casesimulationbox, the singledashedbar indicates
the size of the region shownin Figure 2.
-2
0
-2
2
0
2
-2
Vperp2/ Va,x
•perpl I •,x
Vperp2/ Va,x
T- 9.8 Tgi
2
magnetic
fieldrotatesoutofthex, z planeacross
there-
-2
versal. The MHD scalesolutiondoesnot developsignifi-
-2
cantBy i.e. the magneticfieldremainsin the x, z plane.
In both casesthe B= componentreversessharply in a
central region where the density is strongly enhanced
and is slightly depressedin a regionjust outside. This
0
Vz / Va,x
-2
-2
2
0
2
_
-2
Vy / Va,x
0
2
Vy I Va,x
T- 9.8 Tgi
• 2
depression
in B=justoutside
thereversal
isphysically
different
inthetwo
solutions,
however.
Inthehybrid
case,B• is depressedas the local densityis increasedby
the ions escapingfrom the reversal. In the MHD scale
case, the local density is decreasedin this region, accompaniedby an increasein the vz componentof bulk
-2
0
2
-2
Vperpl I Va,x
0
2
0
Vperp2/ Va,x
2
Vperp21Va,x
T= 9.8 Tgi
velocityinto the reversaland an accompanying
signature in E• - -vzB•. This compressional
modehas•an
2
2
,I,
analogue
in the hybridsolutionin the formof a fast
•.o
modewavepacketwhichis generatedat early times and
which has moved out of the simulation box. A bipolar
E• (withcorresponding
(vi A B)•) evolves
in thehybrid
caseover a region occupiedby the escapingions. The
E• component(not shown)is within the noiselevelsof
-2
-2
0
Vz / Va,x
2
-2
0
Vy / Va,x
2
-2
0
2
Vy I Va,x
the simulations
in both cases.The termsvi A B and Figure 4. Ion f(v) in the reversalcentrefor the MHD
J A B/ne are plottedon the samescalesfor comparison scalecase(A, B) and hybrid scalecase(C, D). Vpar,
and at this stagein the simulations
the vi A B term Vperpl, Vperp2(vii, Vñl,vñ2)forma righthandedordominates,that is, the J A B/ne term in equation(1) thogonalcartesiancoordinatesystem(see text). Vx,
does not account for the differencesin E in the hybrid
and MHD
scale solutions.
Bulk Plasma Acceleration:
The componentsof
J ^ B are also shownon Figures2-3 (normalizedto x
Vy, Vz correspondto GSE. Va,x is the Alfven speedin
the B= field. The logarithmic grey scale rangesfrom
! computationalparticleper pixel (light grey) to .-. 50
computationalparticlesper pixel (black).
3254
CHAPMAN AND MOUIKIS: KINETIC AND QUASI- MHD SIMULATIONS
rection.In the hybridcase(C, D) the ion f(v) is heated
morestronglyalongthe fieldand hasa crossfielddrift
T= 0 T•i
2'
,• 2
• 2
in both x and y which is higher also .• 0.5- 1V•. Just
outsideof the reversal(Figure5) in the MHD scalecase
(A, B) the ion f(v) is almostunchanged
fromthe input
-2
-•2
-2
0
-2
-2
2
0
distribution, it has the same temperature and a cross
-2
2
0
Vperp2/ Va,x
Vperpl / Va,x
2
vPerp2
] Va,x
T= 9.8 Tgi
field drift, in +z (this distributionis takenin the region
wherethere is significantEy). In the hybrid case(C,
D) the escapingionscan be clearlyseen,movingout of
the reversalalong the field with up to .• 2V•. In a sufficiently large simulation box this unstable distribution
would ultimately thermalize.
-2
0
2
Vperpl/ Va•
-2
0
2
Vperp2/ Va•
-2
0
2
Vperp2/ Va,x
4.
Conclusions
For a simple dipolarizing reversal, we have found
that important features of the hybrid solution cannot
be "coarsegrained" to yield the MHD scale solution.
T= 9.8 Tgi
It should be stressed that
the details
of these results
will differ for different systems, or in more realistic
geometries(allowingquantitiesto vary in three space
coordinates). Neverthelessthis exampledemonstrates
the importanceof resolvingselfconsistently
on ion Larmor scales a plasma configuration that energizesthe
ions. This has relevance in particular for recent studies of ion energizationand predictionsof the ion distri-
T= 9.8 Tgi
butionfunctionin the geotailthat involveintegrating
large numbersof test particle trajectories in prescribed
-2
0
2
-2
VperpiI ¾a,x
0
2
-2
Vperp21
Va,x
0
2
Vperp21
Va,x
fieldsobtainedon MHD scales(i.e. largerthan ion gyroscales),either from phenomenological
field modelsor
from the output of MHD codes.
T= 9.8 T•i
•
""'
-2
•o
-2
-2
0
Vz I Va,x
2
Acknowledgments.
This work was conductedwhilst
C.G.M. was supported by PPARC and S.C.C. was in part
supportedby a Nuffield FoundationResearchFellowship.
References
-2
-2
0
2
Vy I Va,x
-2
0
2
Vy I Va,x
Ashour-Abdalla, M., L. M. Zelenyi, V. Peroomian and R.
L. Richard, Consequencesof magnetotail ion dynamics,
J. Geophys. Res., 99, 14891, 1994
Figure 5. Ion f(v) from a regionoutsidethe reversal
for the MHD scalecase(A, B) and the hybridscalecase Joyce, G., J. Chen, S. Slinker, D. L. Holland, and J. B.
Harold, Particle energization near an X line in the mag(C, D), sameformatasFigUre4.
netotail basedon global MHD fields, J. Geophys.Res.,
100, 19167, 1995
panel of Figures2 and 3 whichshowsvx of individ- Krauss-Varban, D., N. Omidi, and K. B. Quest, Mode propual computationalparticlesversusz. Distribution functions have been plotted for the regionsmarked with the
verticallines,inside(Figure4) andoutside(Figure5)
of the reversal. These show projections of the distri-
butionin the wit,vii planeandthe vñ],vl2 plane,at
t = 0 andat the timeshown
in Figures
2 and3 (with
viidirectedalongthelocalB, viz liesin thex, z plane
andvii, vii, vl2 forma righthandedcartesian
coordi-
erties of low-frequencywave: Kinetic theory versusHall-
MHD, J. Geophys. Res., 99, 5987, 1994
Lin, Y., and L. C. Lee, Structure of the Dayside Reconnection Layer in Resistive MHD and hybrid Models, J.
Geophys. Res., 98, 3919, 1993
RiChardson,A., and S.C. Chapman, Self consistentonedimensional hybrid code simulations of a relaxing field
reversal, J. Geophys.Res., 99, 17391, 1994
Terasawa, T., M. Hoshino, J. I. Sakai, and T. Hada, Decay Instability of Finite-Amplitude Circularly Polarized
nate system). The t • 0 distributionsare alsoshown
Plottedin vx,vy,vz space,corresponding
to the approximately GSE coordinatesystemof the simulation(see
Figure 1). •om Figure 4 we seethat insidethe reversal the MHD scalesolution(A,B) is characterised
by
Alfven
Waves:
A Numerical
Simulation
of Stimulated
Brillouin Scattering, J. Geophys. Res., 91, 4171, 1986
Space and AstrophysicsGroup, Univ. of Warwick, UK
an isotropicion distributionthroughoutthe run which
is heated to maximum thermal speed.• V• and which (ReceivedApril 11, 1996; revisedJuly 9, 1996;
has a cross field bulk motion of .• 0.5V•
in the +x di-
acceptedSeptember11, 1996.)