cluster investigations on the self-reformation of perpendicular earth’s bow shock

Cluster investigations on the self-reformation of perpendicular Earth’s bow

shock

Cluster 17th workshop, Uppsala, Sweden, May 12-15 2009Cluster 17th workshop, Uppsala, Sweden, May 12-15 2009

11CCESRESR, , UPS - UPS - CNRS, 9 Avenue du Colonel CNRS, 9 Avenue du Colonel Roche, Toulouse, 31400, France Roche, Toulouse, 31400, France ([email protected]),([email protected]),

2 2 LATMOS / IPSL , CNRS UVSQ, Velizy, France, LATMOS / IPSL , CNRS UVSQ, Velizy, France, 33LATT, Observatoire Midi-Pyrénées, Univ. of Toulouse, FranceLATT, Observatoire Midi-Pyrénées, Univ. of Toulouse, France44Physics Department, University of New Brunswick, Fredericton, NB, Physics Department, University of New Brunswick, Fredericton, NB, Canada,Canada,55LPCE, LPCE, CNRS, 3A, Avenue de la recherche scientifique,CNRS, 3A, Avenue de la recherche scientifique, FranceFrance66Space & Atmospheric Physics Group, Imperial College London, UK.Space & Atmospheric Physics Group, Imperial College London, UK.

C. MazelleC. Mazelle11, B. Lembège, B. Lembège22, A. Morgenthaler, A. Morgenthaler33, K. Meziane, K. Meziane44, , J.-L. RauchJ.-L. Rauch55, ,

J.-G. TrotignonJ.-G. Trotignon55, , E.A. LucekE.A. Lucek66, I. Dandouras, I. Dandouras11

OutlineOutline

Aim: Experimental evidence of Aim: Experimental evidence of shock front nonstationarityshock front nonstationarity

from determination of from determination of characteristic sub-scalescharacteristic sub-scales with multi-satellite observationswith multi-satellite observations

previous (pre-Cluster) experimental determinations of previous (pre-Cluster) experimental determinations of scales.scales.

Multi-spacecraft analysis from Cluster. Cases studies. Multi-spacecraft analysis from Cluster. Cases studies. Methodology and cautions.Methodology and cautions.

Statistical analysis of Cluster results.Statistical analysis of Cluster results.

Comparison with PIC numerical simulations results. Comparison with PIC numerical simulations results.

Comparison with previous experimental results.Comparison with previous experimental results.

perspective: Cross-scale mission, Heliospheric shock.perspective: Cross-scale mission, Heliospheric shock.

Physical characteristics of supercritical quasi-Physical characteristics of supercritical quasi-perpendicular shockperpendicular shock

OvershootFoot

Ramp

reflected gyrating ionreflected gyrating ion

Above a critical value of MAbove a critical value of MAA, dispersion is not sufficient to balance steepening as well as, dispersion is not sufficient to balance steepening as well as "resistive" dissipation: other ("viscous") dissipation process by reflected ions mandatory"resistive" dissipation: other ("viscous") dissipation process by reflected ions mandatory

characteristics substructures:characteristics substructures:

Non stationarity of supercritical quasi-perpendicular shockNon stationarity of supercritical quasi-perpendicular shock

PIC simul.: Shock non stationary -> Cyclic "shock front self-reformation".PIC simul.: Shock non stationary -> Cyclic "shock front self-reformation". Different proposed mechanisms of non stationarity Different proposed mechanisms of non stationarity signaturessignatures : variation of the characteristic structures (foot, ramp, overshoot). : variation of the characteristic structures (foot, ramp, overshoot).

Terrestrial shock geometry

Tim

e

PIC Numerical simulations:1D: Biskamp and Welter, 1972; Lembège and Dawson, 1987;

Hada et al., 2004; Schöler and Matsukyo, 2004; ….2D: Lembège and Savoini, 1992; Lembège et al., 2003 …

BB

n Q-(45° - 90°)

Bn

EartEarthh

Normalized distance

BB

[Lembège et al., 2003][Lembège et al., 2003] BnBn= 90°= 90°2D PIC

MMAA= 5= 5

mmpp/m/mee=400=400

Numerical simulations of supercritical quasi-perpendicular Numerical simulations of supercritical quasi-perpendicular shockshock

PIC simul.: Shock non stationary -> Cyclic "shock front self-reformation".PIC simul.: Shock non stationary -> Cyclic "shock front self-reformation". Different proposed mechanisms of non stationarity Different proposed mechanisms of non stationarity signaturessignatures : variation of the characteristic structures (foot, ramp, overshoot). : variation of the characteristic structures (foot, ramp, overshoot).

Terrestrial shock geometry

Tim

e

PIC Numerical simulations:1D: Biskamp and Welter, 1972; Lembège and Dawson, 1987;

Hada et al., 2004; Schöler and Matsukyo, 2004; ….2D: Lembège and Savoini, 1992; Lembège et al., 2003 …

BB

n Q-(45° - 90°)

Bn

EartEarthh

Normalized distance

FootOvershoot

BB

[Lembège et al., 2003][Lembège et al., 2003] BnBn= 90°= 90°

Ramp

c/ωpi

Cluster

2D PIC

OutlineOutline

Aim: Experimental evidence of shock front nonstationarity Aim: Experimental evidence of shock front nonstationarity

from determination of from determination of characteristic sub-scales characteristic sub-scales with multi-satellite observationswith multi-satellite observations






perspective: Cross-scale missions, Heliospheric shock.perspective: Cross-scale missions, Heliospheric shock.

Ramp thickness: some previous ISEE Ramp thickness: some previous ISEE resultsresults

[Newbury and Russell, [Newbury and Russell, GRLGRL, 1996], 1996]

very thin shockvery thin shock

ISEE:ISEE:

thicknesses of the laminar (low ) shocks :0.4 – 4.5 c/ωpi [Russell et al., 1982]ion inertial length scale

Supercritical shocks:Supercritical shocks:

ramp thicknessramp thicknesstypically of ~ typically of ~ c/ωpi

[Russell and Greenstadt, 1979;

Scudder, 1986]

(a)(a)

(b)(b)

(a)(a)(b)(b)

Previous study from Cluster data (1)Previous study from Cluster data (1)

High time resolution is mandatoryHigh time resolution is mandatory to reveal the different sub-structures of the shock to reveal the different sub-structures of the shock even for a 'nearly' perpendicular shockeven for a 'nearly' perpendicular shockDiffer. signat. of shock crossing shock front variability: what responsible process?Differ. signat. of shock crossing shock front variability: what responsible process?

first examples of some aspects first examples of some aspects of shock nonstationarityof shock nonstationarity(or at least (or at least variabilityvariability))were presented by were presented by Horbury et al. [2001]Horbury et al. [2001]::

OutlineOutline



previous (pre-Cluster) experimental determinations of previous (pre-Cluster) experimental determinations of scalesscales






Example of analysed shock crossing from Example of analysed shock crossing from ClusterCluster

B (nT)B (nT)

5 Hz data5 Hz data

MethodologyMethodology

Determination of the limits of the structures in time series for each satel. dataDetermination of the limits of the structures in time series for each satel. data

Determine the 'apparent' space width (along each sat. traj.)-> compar. between the 4 s/c.Determine the 'apparent' space width (along each sat. traj.)-> compar. between the 4 s/c. Determine the normal velocity of the shock in s/c frame (VDetermine the normal velocity of the shock in s/c frame (Vshockshock, V, Vs/cs/c, angle , angle n n - s/c traj.)- s/c traj.) Main goal: to determine the Main goal: to determine the reareal spatial width of the structuresl spatial width of the structures (ramp, foot, overshoot) (ramp, foot, overshoot) Careful error determinationCareful error determination

Time (hrs.)

B (

nT)

Downstream asymptotic valueDownstream asymptotic value

upstream valueupstream value

22 to 64 Hz data22 to 64 Hz datafootfoot

rampramp11stst overshoot overshoot

along the normalalong the normal

use of high time resolution datause of high time resolution data


Determination of the limits of the structures in time series for each satel. dataDetermination of the limits of the structures in time series for each satel. data For the ramp: look for the 'steeper' slope (time linear fitting) -> defines the 'reference For the ramp: look for the 'steeper' slope (time linear fitting) -> defines the 'reference

satellite'satellite' Determine the 'apparent' space width (along each sat. traj.)-> compar. between the 4 s/c.Determine the 'apparent' space width (along each sat. traj.)-> compar. between the 4 s/c. Determine the normal velocity of the shock in s/c frame (VDetermine the normal velocity of the shock in s/c frame (Vshockshock, V, Vs/cs/c, angle , angle n n - s/c traj.)- s/c traj.) Main goal: to determine the Main goal: to determine the reareal spatial width of the structuresl spatial width of the structures (ramp, foot, overshoot) (ramp, foot, overshoot) Careful error determinationCareful error determination

Time (hrs.)

B (

nT)








Determination of the limits of the structures in time series for each satel. dataDetermination of the limits of the structures in time series for each satel. data For the ramp: look for the 'steeper' slope (time linear fitting) -> defines the 'reference For the ramp: look for the 'steeper' slope (time linear fitting) -> defines the 'reference

satellite'satellite' Determine the 'apparent' width (along each sat. traj.)-> compar. between the 4 s/c.Determine the 'apparent' width (along each sat. traj.)-> compar. between the 4 s/c. Determine the normal velocity of the shock in s/c frame (VDetermine the normal velocity of the shock in s/c frame (Vshockshock, V, Vs/cs/c, angle , angle n n - s/c traj.)- s/c traj.) Main goal: to determine the Main goal: to determine the reareal spatial width of the structuresl spatial width of the structures (ramp, foot, overshoot) (ramp, foot, overshoot) Careful error determinationCareful error determination

Time (hrs.)

B (

nT)








Determination of the limits of the structures in time series for each satel. dataDetermination of the limits of the structures in time series for each satel. data For the ramp: look for the 'steeper' slope (time linear fitting) : defines the 'reference For the ramp: look for the 'steeper' slope (time linear fitting) : defines the 'reference


Time (hrs.)

B (

nT)



Timing method: gives shock normal n and velocity V in s/c frame

For ech pair of satellites i and j :

nn

footfoot


22 to 64 Hz data22 to 64 Hz data



VV


Determination of the limits of the structures in time series for each satel. dataDetermination of the limits of the structures in time series for each satel. data For the ramp: look for the 'steeper' slope (time linear fitting) : defines the 'reference For the ramp: look for the 'steeper' slope (time linear fitting) : defines the 'reference


Time (hrs.)

B (

nT)


upstream valueupstream valueB

(nT

)

overshootovershoot

footfoot

rampramp

-1 0 1-1 0 1c/ω

pi

footfoot


22 to 64 Hz data22 to 64 Hz data



Validity criteria for the method (1)Validity criteria for the method (1)

Criterion 1: careful determination of the Bn

- determination of the 'mean' normal seen by the 4-spacecraft set (timing correlation analysis). - check the conservation of normal magnetic field component Bn.

- check the mean upstream magnetic field vector seen by each satellite: -> estimate of B0 for the tetrahedron and associated error.

Criterion 2: careful conversion of temporal scales (time series of the shock crossings) to real spatial scales- take into account the shock velocity in each s/c frame- relative orientations of the s/c trajectories w.r.t. the shock normal: determination of the width along the normal.

A long 'temporal' scale can lead to 'real' narrow ramp width … !

Key points:Key points:

Validity criteria for the method (2)Validity criteria for the method (2)

Criterion 3: careful determination of the upstream parameters

solar wind ion density and temperature

caution: not reliable when Cluster CIS in magnetospheric mode.

Use of ACE data and Cluster/WHISPER (plasma frequency) data.

caution: He++/H+ ratio (to avoid ~20 % error in mass density)

-> determination of Alfvèn velocity -> MA

-> determination of i

Four spacecraft measurements of the quasi-perpendicular terrestrial bow shock: [Horbury et al., JGR, 2002][Horbury et al., JGR, 2002]

5 vectors/s5 vectors/sclean, sharp shockclean, sharp shock

complex, disturbed shockcomplex, disturbed shock

shock with probable accelerationshock with probable acceleration

Characteristics of the sampleCharacteristics of the sampleN

um

ber

of

occ

ure

nce

BnBn iiMMAA

From 455 shocks:From 455 shocks:24 shocks with all validated criteria24 shocks with all validated criteria

majority above 84°majority above 84° majority below 0.1majority below 0.1

|B|

Typical shock crossing Typical shock crossing

.. Very thin ramp Very thin ramp: some : some electron electron inertial inertial lengthslengths

.. Variablilty of ion foot, ramp and Variablilty of ion foot, ramp and overshoot overshoot

thicknessesthicknesses evidence of shock non-stationarity evidence of shock non-stationarity and and self-reformationself-reformation

C4

C2

C1

C3

S/c positions in (x,n) plane and perpendicular to n

Y’ (km)

C4

C2

C1

C3

C2

C3

C1

C4

X’

(km

)

Y’ (km)

Z’

(km

)

n

BnBn= 89° = 89° ±± 2° 2°Sequen

ce o

f crossin

gs o

rder

Sequen

ce o

f crossin

gs o

rder

at ref. time (ramp middle of ref. sat. 4)at ref. time (ramp middle of ref. sat. 4)

LLrampramp= 5 = 5 c/c/pepe

MMAA=4.=4.11

ii=0.05=0.05

c/ωpi

OutlineOutline









Statistical results (24 shocks = 96 crossings): Statistical results (24 shocks = 96 crossings): ramps (1)ramps (1)

Thinnest rampfor each shock

L ram

p in



L ram

p in

• Ramps of the order of a few c/ωpe,

for a large range of BnBn

electron scale rather than ion electron dynamics important



L ram

p in

• Ramps of the order of a few c/ωpe,

for a large range of BnBn

electron scale rather than ion electron dynamics important

• Change of regime around 85-87° dispersive effects? Tend to broaden the ramp?

????

critical angle between ‘oblique’ and ‘ perpendicular’ shockcritical angle between ‘oblique’ and ‘ perpendicular’ shock

for lowfor low and Mand Mff ~1 ~1 crcr = 87°= 87° (e.g. Balikhin et al., 1995)

Ai

ecr M

M

mcos

Larger probability to cross a thin ramp (<< Larger probability to cross a thin ramp (<< c/ωpi) !!

ion inertial lengthion inertial length

all rampsall ramps

Statistical results (24 shocks = 96 crossings): ramps (2)Statistical results (24 shocks = 96 crossings): ramps (2)

Statistical results (24 shocks = 96 crossings): ramps (3)Statistical results (24 shocks = 96 crossings): ramps (3)

all rampsall ramps

only thin ramps close to 90°only thin ramps close to 90°

no simple trendno simple trend

trend: thickest ramps trend: thickest ramps decrease with Mdecrease with MAA

really perpendicular shocks?really perpendicular shocks?

L ram

p in

L ram

p in

OutlineOutline






Comparison with PIC numerical simulations results.Comparison with PIC numerical simulations results.



Comparison with 2D PIC simulationsComparison with 2D PIC simulations

mmpp/m/mee=400=400

Comparison with 2D PIC simulations: Comparison with 2D PIC simulations: rampsramps

mmpp/m/mee=400=400

Statistical Results: ion foots (1)Statistical Results: ion foots (1)

Foot thickness < Larmor radius as expectedFoot thickness < Larmor radius as expected

Mainly low valuesMainly low values

Lfoot in Ci, upstream

Num

ber

of

occ

ure

nce

Ion foots: comparison with 2D PIC Ion foots: comparison with 2D PIC simulationssimulations

mmpp/m/mee=400=400

Acceleration of the Acceleration of the growth growth of the ion foot of the ion foot both in amplitude and both in amplitude and thicknessthickness during one during one self-reformation cycleself-reformation cycle

higher probabilityhigher probabilityto cross an ion foot to cross an ion foot with a small thickness?with a small thickness?

seems qualitatively seems qualitatively consistent with consistent with observationsobservations

needs more quantitative needs more quantitative investigationinvestigation

Statistical Results: ion foots (2)Statistical Results: ion foots (2)

Comparison of largest observed value with 'stationary' theoretical Comparison of largest observed value with 'stationary' theoretical value [Schwartz et al., 1983] :value [Schwartz et al., 1983] :

d = 0.648 d = 0.648 Ci,upstreamCi,upstream for for BnBn = 90° and = 90° and VnVn =0°=0°

another signature of shock cyclic self-reformationanother signature of shock cyclic self-reformation

Red : stationary theoretical valuesBlue : largest observed values

Shock number

Lfo

ot in

C

i, up

stre

am

wherewhere

reflected ion turn-around distancereflected ion turn-around distance [Woods, 1969] [Woods, 1969]

Statistical results (24 shocks = 96 crossings): overshootStatistical results (24 shocks = 96 crossings): overshoot

Num

ber

of

occ

ure

nce

Lovershoot in c/ωpi upstream

33

Majority between 1 and 3 Majority between 1 and 3 c/ωpi as e.g. in as e.g. in Mellott and LiveseyMellott and Livesey [1987] [1987]

but also large variability due to self-reformation of the shockbut also large variability due to self-reformation of the shock

OutlineOutline









Previous study from Cluster data (2) Previous study from Cluster data (2) [Bale [Bale et alet al., PRL, 2003]., PRL, 2003]

Fit of the density profile by an analytical shape (hyberbolic tangent)Fit of the density profile by an analytical shape (hyberbolic tangent) No separation between ramp and footNo separation between ramp and foot

Typical shock size: ion scalesTypical shock size: ion scales

ionion inertial inertiallengthlength

convectiveconvectivedownstreamdownstreamgyroradiusgyroradius

ShockShockscalescale::

Here, different approachHere, different approach sub-structures taken into account sub-structures taken into account

"" This technique captures This technique captures only the largest transition scaleonly the largest transition scale at the shock at the shock"" [Bale [Bale et alet al., 2003]., 2003]

macroscopic density transition scale

5 Hz data5 Hz data

Comparaison with results from Bale et al. (2003)

Statistical results (24 shocks = 96 Statistical results (24 shocks = 96 crossings)crossings)

Is the shock front thickness simply dependent on Mach Number?Is the shock front thickness simply dependent on Mach Number?


result seems to depend on the sample used.

no simple dependence

L ram

p+

foot i

n c

/ ω

pi

Magnetosonic Mach Magnetosonic Mach numbernumber

Signature of non stationaritySignature of non stationarity





result seems to depend on the sample used.

no simple dependence

L ram

p+

foot i

n c

/ ω

pi

Magnetosonic Mach Magnetosonic Mach numbernumber


Signature of non stationaritySignature of non stationarity

Previous study from Cluster data Previous study from Cluster data (3)(3)

[Lobzin [Lobzin et alet al., GRL, 2007]., GRL, 2007]

Here, different approachHere, different approach accumulation of case studies (statistics) accumulation of case studies (statistics)

BnBn= 81° = 81° MMAA=10 =10

ii=0.6=0.6

one case study: highly supercritical Q-perp shockone case study: highly supercritical Q-perp shock

Variability of the shock front with embeded Variability of the shock front with embeded nonlinear whistler wave trains nonlinear whistler wave trains and "bursty" quasi-periodic production of and "bursty" quasi-periodic production of reflected ions reflected ions proposed as experimental evidence of proposed as experimental evidence of non stationarity and self-reformation asnon stationarity and self-reformation asdescribed described in Krasnoselskikh et al. [2002]

Other shock sub-structures: Electric field Other shock sub-structures: Electric field spikes (1)spikes (1)

[Walker [Walker et alet al., 2004]., 2004]


Other shock sub-structures: Electric field Other shock sub-structures: Electric field spikes (2) spikes (2)

Histogram of the scale sizes for the spike-like enhancements

E-field spikesE-field spikes

c/ωpi


Histogram of the scale sizes for the spike-like enhancements

c/ωpic/ωpi

Similar distribution to that for magnetic ramps with smaller valuesSimilar distribution to that for magnetic ramps with smaller values

magnetic rampsmagnetic rampsE-field spikesE-field spikes

Other shock sub-structures: Electric field Other shock sub-structures: Electric field spikes (2) spikes (2)

Other shock sub-structures: Electric field Other shock sub-structures: Electric field spike (3) spike (3)


Dependence of scale size on Bn



L ram

p in

magnetic rampsmagnetic ramps


Dependence of scale size on Bn


Similar trend for only low values close to 90°Similar trend for only low values close to 90°



Dependence of scale size on upstream Mach number





magnetic rampsmagnetic ramps

Similar trend: upper limit tend to decrease with increasing Mach NumberSimilar trend: upper limit tend to decrease with increasing Mach Number

Dependence of scale size on upstream Mach number

Ramp sub-structureRamp sub-structure

Time (hrs.)Time (hrs.)

Magnetic ramps often reveal sub-structure : nature?Magnetic ramps often reveal sub-structure : nature?

22 Hz data22 Hz data

Ramp sub-structureRamp sub-structure


signature due to electric field signature due to electric field short scale structure?short scale structure?

Magnetic ramps often reveal sub-structure : nature?Magnetic ramps often reveal sub-structure : nature?

Need further investigation but electric field data not always available Need further investigation but electric field data not always available

22 Hz data22 Hz data

OutlineOutline









Implication for future multi-spacecraft Implication for future multi-spacecraft missionsmissions

Implication for future multi-spacecraft Implication for future multi-spacecraft missionsmissions

already larger than already larger than c/ωpe ! !

Termination shock: Voyager 2Termination shock: Voyager 2 [Burlaga et al., Nature, 2008][Burlaga et al., Nature, 2008]

estimated shock speed: 68±17 km s-1 ramp thickness ~ ~ c/ωpi

but single-s/c determination…

MMMSMS~10 ~10 andand i~0.04 i~0.04 ((but without pickup ionsbut without pickup ions)) self-reformation? self-reformation?

Q-perp natureQ-perp nature

Complex sub-structure (oscillatory) of the ramp: Complex sub-structure (oscillatory) of the ramp: non uniformity (ripples) / non uniformity (ripples) / non stationaritynon stationarity??

Conclusions and perspectivesConclusions and perspectives1) 1) New results on quasi-perpendicular shocksNew results on quasi-perpendicular shocks

* particular cautions with time-series (transition -> real space width)* particular cautions with time-series (transition -> real space width)

* * LLrampramp often very thin (electron scale)often very thin (electron scale) at least for 75° at least for 75° BnBn < 90° < 90°

* L* Lfoot foot < < ci,upstreamci,upstream

* No simple relation between L* No simple relation between Lrampramp and and BnBn , L , Lrampramp and M and MA A

between Lbetween Lfootfoot and and BnBn

* Signatures of cyclic self-reformation (accumul. of reflected ions) as* Signatures of cyclic self-reformation (accumul. of reflected ions) as predicted by 1D/2D PIC simulations:predicted by 1D/2D PIC simulations:

--> accessibility to very thin L--> accessibility to very thin Lrampramp (2-6 c/ (2-6 c/pepe) + varying L) + varying Lrampramp

--> varying L--> varying Lfootfoot in time, varying overshoot thickness and amplitude in time, varying overshoot thickness and amplitude

--> in agreement with low to moderate --> in agreement with low to moderate ii (0.02 - 0.6) (0.02 - 0.6)

2) 2) Under progress, necessity:Under progress, necessity: for increasing the statistics.for increasing the statistics. for careful analysis of: --> for careful analysis of: --> ion distributionsion distributions (difficulty: time resolution) (difficulty: time resolution)

--> associated micro-turbul. in the foot/ramp/oversh.--> associated micro-turbul. in the foot/ramp/oversh.

Mostly thin ramps: impact on particle acceleration mechanisms Mostly thin ramps: impact on particle acceleration mechanisms

ENDEND

Thank you!Thank you!

Supercritical shock: Hybrid simulationsSupercritical shock: Hybrid simulations

[Leroy, 1981][Leroy, 1981]

First attempts: single spacecraft First attempts: single spacecraft determination (1)determination (1)

To distinguish directly between spatial and temporal variations at least for some temporal and spatial scale range, and thus to determine spatial scales of structures became really possible only after the ISEE-l,2 launch.

However, already in pre ISEE era some indirect methods were elaborated to define spatial scales. The precision and reliability of these method were very low, but at least some of them gave results which are in agreement with later results obtained by ISEE.

The first attempt to estimate shock scale was made in Holzer et al. [1966] where results of magnetic field measurements obtained from OGO-1 were presented. The proposed method was used for Explorer 12 data in Kaufmann [1967], and for OGO-1 in Heppner et al. [1967]. It was assumed that the bow shock motion can be represented by zigzag line. Estimates of the amplitude of this line can be made on the basis of distance between first and last bow shock crossings.

Then the velocity can be estimated in terms of this amplitude and a number of crossings. In spite of the fact that this is a very strong assumption about shock motion, which seems not to be very reliable, estimates of the shock velocity Vsh ~ 10 km/s were quite reasonable.

First attempts: single spacecraft First attempts: single spacecraft determination (2)determination (2)

The second method which was used in pre-ISEE era was based on two nearly simultaneous encounters of bow shock by two satellites OGO-5 and Heos-1 which were quite distant one from another [Greenstadt et al., 1975].

The shock velocity was estimated in assumption that the shock surface is a coherent surface. This assumption was checked on the basis of OGO-5 and Heos-1 measurements during the bow shock crossing [Greenstadt et al., 1972].

Such method cannot be applicable to numerous bow shocks, due to the small probability that two different satellites occasionally will cross Earth’s bow shock nearly simultaneously. But as it was noticed in Russell et al. [1982] the both techniques yielded thicknesses of the laminar (low ) shocks

0.4 – 4.5 c/pi thus ion inertial length scale

which were in a good agreement with those obtained later from two ISEE satellites.

The decrease of the thickness of the shock as approaching 90° have been only qualitatively shown.

Supercritcal quasi-perpendicular shocksSupercritcal quasi-perpendicular shocks Among all Earth’s bow shock crossings subcritical shocks are exceptional rather than

regular. Under the usual solar wind conditions Earth’s bow shock is in supercritical regime.

It has been theoretically speculated that an exactly perpendicular shock behaves like the soliton wave solution from classic cold plasma theory when some additional dissipation is provided to transform it into a fast magnetosonic waves [Karpman, 1964; Tidman and Krall, 1971]. This would lead to a much thinner ramp of the order of c/pe. Further theoretical studies predicted also such small scale for supercritical shocks [e.g. Galeev et al., 1976, 1989; Krasnoselskikh et al., 1985, 2002 ]

In Friedricks et al. [1967] it was noted that the presence of bursts of electric field fluctuations in the regions of steep slopes of |B| can be a strong argument in favor of the presence of c/pe scale lengths in the shock and they conclude that characteristic scales are more likely to be ~ c/pe than ~ c/pi .

But only after results obtained by ISEE magnetometer, it became possible to determine directly the size of the ramp. The major issue is the accuracy of the shock velocity determination.

Russell and Greenstadt [1979] fit exponential curves to supercritical quasi-perpendicular shock crossing and obtaines thicknesses of the order of ~0.4 c/wpi. Scudder [1986] got ~0.3 c/wpi for a single shock crossing.

Four spacecraft measurements of the quasi-perpendicular terrestrial bow shock: Orientation and motion

Measurements of the magnetic field at the four Cluster spacecraft, typically separated by 600 km, during bow shock crossings allow the orientation and motion of this structure to be estimated.

Results from 48 clean and steady quasiperpendicular crossings during 2000 and 2001, covering local times from 0600 to 1700, reveal the bow shock normal to be remarkably stable, under a wide range of steady upstream conditions.

Nearly 80% of normals lay within 10° of those of two bow shock models, suggesting that the timing method is accurate to around 10°, and possibly better, and therefore that four spacecraft timings are a useful estimator of the orientation and motion of quasiperpendicular bow shocks.

In contrast, only 19% of magnetic coplanarity vectors were within 10° of the model normal. The mean deviation of the coplanarity vector from the timing-derived normal for shocks with BN < 70° was 22° ± 4°.

Typical shock velocities were 35 km.s-1, although the fastest measured shock was traveling outbound at nearly 150 km.s-1 and 48% have a velocity less than 10 km.s-1.

[Horbury et al., JGR, 2002][Horbury et al., JGR, 2002]

ramp thickness determination: fitting ramp thickness determination: fitting method method


Validity of the normal vector Validity of the normal vector n ?n ?

|B|

(nT)

Bn

BnBn= 89.7° = 89.7° ±± 1.5° 1.5° MMAA=4.=4.55

ii=0.04=0.04

Upstream normal component BUpstream normal component Bnn very small very small

Good consistency with Good consistency with BnBn~ 90°~ 90°and well conserved on average around the shock and well conserved on average around the shock

ramp ramp

Systematic check for all analysed shock crossingsSystematic check for all analysed shock crossings

Typical shock crossing (2)Typical shock crossing (2)

Very thin rampVery thin ramp: evidence of : evidence of shock reformation shock reformation (reflected ions)(reflected ions)

Satellites positions in (xGSE , n) plane

c / ωpi

B (

nT)

22

1144

33

LLrampramp= 4.5 c/= 4.5 c/pepe

BnBn= 88° ± 3°= 88° ± 3°

At ref. time (ramp middle of ref. sat.)At ref. time (ramp middle of ref. sat.)

Sequen

ce o

f crossin

gs o

rder

Sequen

ce o

f crossin

gs o

rder

expanding shockexpanding shock

22

44

11

33

??

MMAA=3.=3.88

ii=0.04=0.04

On the 'danger' of relying only on time series for shock On the 'danger' of relying only on time series for shock profiles…profiles…

which shock which shock is the is the steepest?steepest?

B (

nT)

Time in hours

B (

nT)

BnBn= 89°= 89°

BnBn= 89°= 89°

MMAA=3.8=3.8

MMAA=3.5=3.5

ii=0.045=0.045

ii=0.04=0.04

(a)(a)

(b)(b)

On the 'danger' of relying only on time series for shock On the 'danger' of relying only on time series for shock profiles…profiles…

Taking into Taking into account the account the shock velocity shock velocity

is of crucial is of crucial importance importance

to avoid to avoid misinterpretatimisinterpretation:on:

which shock is which shock is the steepest?the steepest?

B (

nT)

Time in hours

B (

nT)

VVshockshock=11 =11 km/skm/s

VVshockshock=78 =78 km/skm/s

BnBn= 89°= 89°

BnBn= 89°= 89°

MMAA=3.8=3.8

MMAA=3.5=3.5

ii=0.045=0.045

ii=0.04=0.04

(a)(a)

(b)(b)

LLrampramp= 11 c/= 11 c/pepe

Despite the 'appearently steeper' shock in time series, the realDespite the 'appearently steeper' shock in time series, the realphysical width of the ramp is larger for case (b) than for case (a) physical width of the ramp is larger for case (b) than for case (a) because of thebecause of the much higher shock velocity. much higher shock velocity.

Which shock is the steepest?: Which shock is the steepest?:

answeranswer

LLrampramp= 4.5 = 4.5 c/c/pepe

(a)(a)

(b)(b)

Reformation timeReformation time

TTgyrogyro / T / Treformreform

Num

ber

of

occ

ure

nce

Typically 2 self-reformation cycles during one upstream gyroperiodTypically 2 self-reformation cycles during one upstream gyroperiod

consistent with PIC simulation resultsconsistent with PIC simulation results

computed as the local gyroperiod in the computed as the local gyroperiod in the middle of the rampmiddle of the ramp

upstreamupstream

cluster investigations on the self-reformation of perpendicular earth’s bow shock

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