Cluster investigations on the self-reformation of perpendicular Earth

1
Cluster investigations on the self-reformation
of perpendicular Earths bow shock
C. Mazelle1, B. Lembège2, A. Morgenthaler3, K.
Meziane4, J.-L. Rauch5, J.-G. Trotignon5,
E.A. Lucek6, I. Dandouras1

• 1CESR, UPS - CNRS, 9 Avenue du Colonel Roche,
  Toulouse, 31400, France
• (christian.mazelle_at_cesr.fr),
• 2 LATMOS / IPSL , CNRS UVSQ, Velizy, France,
• 3LATT, Observatoire Midi-Pyrénées, Univ. of
  Toulouse, France
• 4Physics Department, University of New Brunswick,
  Fredericton, NB, Canada,
• 5LPCE, CNRS, 3A, Avenue de la recherche
  scientifique, France
• 6Space Atmospheric Physics Group, Imperial
  College London, UK.

Cluster 17th workshop, Uppsala, Sweden, May 12-15
2009
2
Outline

• Aim Experimental evidence of shock front
  nonstationarity
• from determination of characteristic
  sub-scales with multi-satellite observations
• previous (pre-Cluster) experimental
  determinations of scales.
• Multi-spacecraft analysis from Cluster. Cases
  studies. Methodology and cautions.
• Statistical analysis of Cluster results.
• Comparison with PIC numerical simulations
  results.
• Comparison with previous experimental results.
• perspective Cross-scale mission, Heliospheric
  shock.

3
Physical characteristics of supercritical
quasi-perpendicular shock
Above a critical value of MA, dispersion is not
sufficient to balance steepening as well as
"resistive" dissipation other ("viscous")
dissipation process by reflected ions mandatory
? characteristics substructures
reflected gyrating ion
Ramp
Overshoot
Foot
4
Non stationarity of supercritical
quasi-perpendicular shock
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
Terrestrial shock geometry
?Bn 90
Lembège et al., 2003
B
2D PIC
MA 5
Time
mp/me400
Earth
B
Q-? (45 - 90)
n
Normalized distance

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

5
Numerical simulations of supercritical
quasi-perpendicular shock
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
Terrestrial shock geometry
?Bn 90
Lembège et al., 2003
B
2D PIC
Overshoot
Foot
Time
Cluster
Earth
B
Q-? (45 - 90)
Ramp
n
c/?pi
Normalized distance

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

6
Outline

• Aim Experimental evidence of shock front
  nonstationarity
• from determination of characteristic
  sub-scales with multi-satellite observations
• previous (pre-Cluster) experimental
  determinations of scales.
• Multi-spacecraft analysis from Cluster. Cases
  studies. Methodology and cautions.
• Statistical analysis of Cluster results.
• Comparison with PIC numerical simulations
  results.
• Comparison with previous experimental results.
• perspective Cross-scale missions, Heliospheric
  shock.

7
Ramp thickness some previous ISEE results

• ISEE
• thicknesses of the laminar (low b) shocks
• 0.4 4.5 c/?pi Russell et al., 1982
• ion inertial length scale
• Supercritical shocks
• ramp thickness
• typically of c/?pi
• Russell and Greenstadt, 1979
• Scudder, 1986

(a)
(b)
Newbury and Russell, GRL, 1996
very thin shock
(b)

(a)
8
Previous study from Cluster data (1)

first examples of some aspects of shock
nonstationarity (or at least variability) were
presented by Horbury et al. 2001
High time resolution is mandatory to reveal the
different sub-structures of the shock even for a
'nearly' perpendicular shock Differ. signat. of
shock crossing shock front variability
what responsible process?
9
Outline

• Aim Experimental evidence of shock front
  nonstationarity
• from determination of characteristic
  sub-scales with multi-satellite observations
• previous (pre-Cluster) experimental
  determinations of scales
• Multi-spacecraft analysis from Cluster. Cases
  studies. Methodology and cautions.
• Statistical analysis of Cluster results.
• Comparison with PIC numerical simulations
  results.
• Comparison with previous experimental results.
• perspective Cross-scale missions, Heliospheric
  shock.

10
Example of analysed shock crossing from Cluster
B (nT)
5 Hz data
11
Methodology
use of high time resolution data
Downstream asymptotic value


ramp
1st overshoot
B (nT)
22 to 64 Hz data
foot
Time (hrs.)
upstream value

• Determination of the limits of the structures in
  time series for each satel. data
• Determine the 'apparent' space width (along each
  sat. traj.)-gt compar. between the 4 s/c.
• Determine the normal velocity of the shock in s/c
  frame (Vshock, Vs/c, angle n - s/c traj.)
• Main goal to determine the real spatial width
  of the structures (ramp, foot, overshoot)
• Careful error determination

along the normal
12
Methodology
use of high time resolution data
Downstream asymptotic value


ramp
1st overshoot
B (nT)
22 to 64 Hz data
foot
Time (hrs.)
upstream value

• Determination of the limits of the structures in
  time series for each satel. data
• For the ramp look for the 'steeper' slope (time
  linear fitting) -gt defines the 'reference
  satellite'
• Determine the 'apparent' space width (along each
  sat. traj.)-gt compar. between the 4 s/c.
• Determine the normal velocity of the shock in s/c
  frame (Vshock, Vs/c, angle n - s/c traj.)
• Main goal to determine the real spatial width
  of the structures (ramp, foot, overshoot)
• Careful error determination

along the normal
13
Methodology
use of high time resolution data
Downstream asymptotic value


ramp
1st overshoot
B (nT)
22 to 64 Hz data
foot
Time (hrs.)
upstream value

• Determination of the limits of the structures in
  time series for each satel. data
• For the ramp look for the 'steeper' slope (time
  linear fitting) -gt defines the 'reference
  satellite'
• Determine the 'apparent' width (along each sat.
  traj.)-gt compar. between the 4 s/c.
• Determine the normal velocity of the shock in s/c
  frame (Vshock, Vs/c, angle n - s/c traj.)
• Main goal to determine the real spatial width
  of the structures (ramp, foot, overshoot)
• Careful error determination

along the normal
14
Methodology
use of high time resolution data
Downstream asymptotic value
Timing method gives shock normal n
and velocity V in s/c frame


ramp
1st overshoot
V
n
B (nT)
foot
22 to 64 Hz data
For ech pair of satellites i and j


Time (hrs.)
upstream value

• Determination 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
  satellite'
• Determine the 'apparent' width (along each sat.
  traj.)-gt compar. between the 4 s/c.
• Determine the normal velocity of the shock in s/c
  frame (Vshock, Vs/c, angle n - s/c traj.)
• Main goal to determine the real spatial width
  of the structures (ramp, foot, overshoot)
• Careful error determination

along the normal
15
Methodology
use of high time resolution data
Downstream asymptotic value
overshoot


ramp
ramp
1st overshoot
B (nT)
B (nT)
foot
22 to 64 Hz data
foot
-1 0 1
Time (hrs.)
upstream value
c/?pi

• Determination 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
  satellite'
• Determine the 'apparent' width (along each sat.
  traj.)-gt compar. between the 4 s/c.
• Determine the normal velocity of the shock in s/c
  frame (Vshock, Vs/c, angle n - s/c traj.)
• Main goal to determine the real spatial width
  of the structures (ramp, foot, overshoot)
• Careful error determination

along the normal
16
Validity criteria for the method (1)
Key points

• 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
• -gt 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 !

17
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)
• -gt determination of Alfvèn velocity -gt MA
• -gt determination of bi

18
Four spacecraft measurements of the
quasi-perpendicular terrestrial bow shock
Horbury et al., JGR, 2002
clean, sharp shock
5 vectors/s
complex, disturbed shock
shock with probable acceleration
19
Four spacecraft measurements of the
quasi-perpendicular terrestrial bow shock
Horbury et al., JGR, 2002
clean, sharp shock
5 vectors/s
complex, disturbed shock
shock with probable acceleration
20
Characteristics of the sample
From 455 shocks 24 shocks with all validated
criteria
Number of occurence
?Bn
?i
MA
majority below 0.1
majority above 84
21
Typical shock crossing
?Bn 89 2
MA4.1
?i0.05
C4
C4
C3
C3
Lramp 5 c/?pe
n
X (km)
Z (km)
C1
C1
C2
C4
C2
C2
Sequence of crossings order
Y (km)
Y (km)
B
S/c positions in (x,n) plane and perpendicular
to n
C1
at ref. time (ramp middle of ref. sat. 4)

• . Very thin ramp some electron inertial lengths
• . Variablilty of ion foot, ramp and overshoot
• thicknesses
• evidence of shock non-stationarity
  and self-reformation

C3
c/?pi
22
Outline

• Aim Experimental evidence of shock front
  nonstationarity
• from determination of characteristic
  sub-scales with multi-satellite observations
• previous (pre-Cluster) experimental
  determinations of scales.
• Multi-spacecraft analysis from Cluster. Cases
  studies. Methodology and cautions.
• Statistical analysis of Cluster results.
• Comparison with PIC numerical simulations
  results.
• Comparison with previous experimental results.
• perspective Cross-scale missions, Heliospheric
  shock.

23
Statistical results (24 shocks 96 crossings)
ramps (1)
Thinnest ramp for each shock
Lramp in


24
Statistical results (24 shocks 96 crossings)
ramps (1)
Thinnest ramp for each shock

• Ramps of the order of a few c/?pe,
• for a large range of ?Bn
• ? electron scale rather than ion
• electron dynamics important

Lramp in


25
Statistical results (24 shocks 96 crossings)
ramps (1)
Thinnest ramp for each shock

• Ramps of the order of a few c/?pe,
• for a large range of ?Bn
• ? electron scale rather than ion
• electron dynamics important
• Change of regime around 85-87
• ? dispersive effects?
• Tend to broaden the ramp?

?
?
Lramp in
critical angle between oblique and
perpendicular shock


for low ? and Mf 1 ?cr 87 (e.g. Balikhin et
al., 1995)
26
Statistical results (24 shocks 96 crossings)
ramps (2)
all ramps
ion inertial length
Larger probability to cross a thin ramp (ltlt
c/?pi) !
27
Statistical results (24 shocks 96 crossings)
ramps (3)
all ramps


Lramp in
Lramp in



no simple trend
trend thickest ramps decrease with MA
only thin ramps close to 90
really perpendicular shocks?
28
Outline

• Aim Experimental evidence of shock front
  nonstationarity
• from determination of characteristic
  sub-scales with multi-satellite observations
• previous (pre-Cluster) experimental
  determinations of scales.
• Multi-spacecraft analysis from Cluster. Cases
  studies. Methodology and cautions.
• Statistical analysis of Cluster results.
• Comparison with PIC numerical simulations
  results.
• Comparison with previous experimental results.
• perspective Cross-scale missions, Heliospheric
  shock.

29
Comparison with 2D PIC simulations






mp/me400
30
Comparison with 2D PIC simulations ramps


mp/me400
31
Statistical Results ion foots (1)





Number of occurence

Lfoot in ? Ci, upstream

• Foot thickness lt Larmor radius as expected
• Mainly low values

32
Ion foots comparison with 2D PIC simulations








mp/me400
Acceleration of the growth of the ion foot both
in amplitude and thickness during one
self-reformation cycle ? higher probability to
cross an ion foot with a small thickness? seems
qualitatively consistent with observations needs
more quantitative investigation
33
Statistical Results ion foots (2)



Red stationary theoretical values Blue
largest observed values

Lfoot in ? Ci, upstream





Shock number

• Comparison of largest observed value with
  'stationary' theoretical value Schwartz et al.,
  1983
• d
  0.648 ?Ci,upstream for ?Bn 90 and ?Vn 0
• ? another signature of shock
  cyclic self-reformation

where
reflected ion turn-around distance Woods, 1969
34
Statistical results (24 shocks 96 crossings)
overshoot
Number of occurence
3








Lovershoot in c/?pi upstream
Majority between 1 and 3 c/?pi as e.g. in Mellott
and Livesey 1987 but also large variability due
to self-reformation of the shock
35
Outline

• Aim Experimental evidence of shock front
  nonstationarity
• from determination of characteristic
  sub-scales with multi-satellite observations
• previous (pre-Cluster) experimental
  determinations of scales.
• Multi-spacecraft analysis from Cluster. Cases
  studies. Methodology and cautions.
• Statistical analysis of Cluster results.
• Comparison with PIC numerical simulations
  results.
• Comparison with previous experimental results.
• perspective Cross-scale missions, Heliospheric
  shock.

36
Previous study from Cluster data (2)
Bale et al., PRL, 2003
macroscopic density transition scale
Shock scale
convective downstream gyroradius
5 Hz data
ion inertial length
Fit of the density profile by an analytical shape
(hyberbolic tangent) No separation
between ramp and foot Typical shock size ion
scales
" This technique captures only the largest
transition scale at the shock" Bale et al.,
2003
Here, different approach ? sub-structures taken
into account
37
Statistical results (24 shocks 96 crossings)
Is the shock front thickness simply dependent on
Mach Number?
Comparaison with results from Bale et al. (2003)




38
Statistical results (24 shocks 96 crossings)
Is the shock front thickness simply dependent on
Mach Number?
Comparaison with results from Bale et al. (2003)

• result seems to depend on the
• sample used.
• no simple dependence

Lrampfoot in c / ?pi
Signature of non stationarity


Magnetosonic Mach number
39
Statistical results (24 shocks 96 crossings)
Is the shock front thickness simply dependent on
Mach Number?
Comparaison with results from Bale et al. (2003)

• result seems to depend on the
• sample used.
• no simple dependence

Lrampfoot in c / ?pi
Signature of non stationarity


Magnetosonic Mach number
40
Previous study from Cluster data (3)
Lobzin et al., GRL, 2007
one case study highly supercritical Q-perp shock
Variability of the shock front with embeded
nonlinear whistler wave trains and "bursty"
quasi-periodic production of reflected ions
proposed as experimental evidence of non
stationarity and self-reformation as described in
Krasnoselskikh et al. 2002
?Bn 81 MA10 ?i0.6

Here, different approach ? accumulation of case
studies (statistics)
41
Other shock sub-structures Electric field spikes
(1)
Walker et al., 2004
42
Other shock sub-structures Electric field spikes
(2)
Histogram of the scale sizes for the spike-like
enhancements
Walker et al., 2004
E-field spikes
c/?pi
43
Other shock sub-structures Electric field spikes
(2)
Histogram of the scale sizes for the spike-like
enhancements
Walker et al., 2004
magnetic ramps
E-field spikes
c/?pi
c/?pi
Similar distribution to that for magnetic ramps
with smaller values
44
Other shock sub-structures Electric field spike
(3)
Dependence of scale size on ?Bn
Walker et al., 2004
E-field spikes
45
Other shock sub-structures Electric field spike
(3)
Dependence of scale size on ?Bn
magnetic ramps

Walker et al., 2004
E-field spikes
Lramp in

Similar trend for only low values close to 90
46
Other shock sub-structures Electric field spike
(4)
Dependence of scale size on upstream Mach number
Walker et al., 2004
E-field spikes
47
Other shock sub-structures Electric field spike
(4)
Dependence of scale size on upstream Mach number
Walker et al., 2004
magnetic ramps

E-field spikes

Similar trend upper limit tend to decrease with
increasing Mach Number
48
Ramp sub-structure
Magnetic ramps often reveal sub-structure
nature?

22 Hz data
Time (hrs.)
49
Ramp sub-structure
Magnetic ramps often reveal sub-structure
nature?
signature due to electric field short scale
structure?

22 Hz data
Time (hrs.)
Need further investigation but electric field
data not always available
50
Outline

• Aim Experimental evidence of shock front
  nonstationarity
• from determination of characteristic
  sub-scales with multi-satellite observations
• previous (pre-Cluster) experimental
  determinations of scales.
• Multi-spacecraft analysis from Cluster. Cases
  studies. Methodology and cautions.
• Statistical analysis of Cluster results.
• Comparison with PIC numerical simulations
  results.
• Comparison with previous experimental results.
• perspective Cross-scale missions, Heliospheric
  shock.

51
Implication for future multi-spacecraft missions
52
Implication for future multi-spacecraft missions

already larger than c/?pe !
53
Termination shock Voyager 2
Burlaga et al., Nature, 2008
Q-perp nature
Complex sub-structure (oscillatory) of the ramp
non uniformity (ripples) / non stationarity?
estimated shock speed 6817 km s-1
ramp thickness c/?pi
but single-s/c determination
MMS10 and ?i0.04 (but without pickup ions)
self-reformation?
54
Conclusions and perspectives

• 1) New results on quasi-perpendicular shocks
• particular cautions with time-series
  (transition -gt real space width)
• Lramp often very thin (electron scale) at
  least for 75 ? ?Bn lt 90
• Lfoot lt ?ci,upstream
• No simple relation between Lramp and ?Bn ,
  Lramp and MA
• between Lfoot and ?Bn
• Signatures of cyclic self-reformation
  (accumul. of reflected ions) as
• predicted by 1D/2D PIC simulations
• --gt accessibility to very thin Lramp (2-6
  c/?pe) varying Lramp
• --gt varying Lfoot in time, varying overshoot
  thickness and amplitude
• --gt in agreement with low to moderate ?i
  (0.02 - 0.6)
• 2) Under progress, necessity
• for increasing the statistics.
• for careful analysis of --gt ion distributions
  (difficulty time resolution)
• --gt associated micro-turbul. in the
  foot/ramp/oversh.
• Mostly thin ramps impact on particle
  acceleration mechanisms

55
END Thank you!
56
Supercritical shock Hybrid simulations
Leroy, 1981
57
First attempts single spacecraft 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.

58
First attempts single spacecraft 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
  Earths bow shock nearly simultaneously. But as
  it was noticed in Russell et al. 1982 the both
  techniques yielded thicknesses of the laminar
  (low b) shocks
• 0.4 4.5 c/wpi 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.

59
Supercritcal quasi-perpendicular shocks

• Among all Earths bow shock crossings subcritical
  shocks are exceptional rather than regular. Under
  the usual solar wind conditions Earths 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/wpe. 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/wpe scale lengths in the shock and they
  conclude that characteristic scales are more
  likely to be c/wpe than c/wpi .
• 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.

60
Four spacecraft measurements of the
quasi-perpendicular terrestrial bow shock
Orientation and motion
Horbury et al., JGR, 2002

• 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 qBN lt 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.

61
ramp thickness determination fitting method
Time (hrs.)
62
Validity of the normal vector n ?
?Bn 89.7 1.5
MA4.5
?i0.04
B (nT)
Bn
Upstream normal component Bn very small Good
consistency with ?Bn 90 and well conserved on
average around the shock ramp
Systematic check for all analysed shock crossings
63
Typical shock crossing (2)
?Bn 88 3
?i0.04
MA3.8
expanding shock

• Very thin ramp evidence of shock reformation
  (reflected ions)

2


3
4
4
?
1
Sequence of crossings order
B (nT)


1
2


3
Satellites positions in (xGSE , n) plane?
At ref. time (ramp middle of ref. sat.)
Lramp 4.5 c/?pe
c / ?pi
64
On the 'danger' of relying only on time series
for shock profiles

MA3.8
?Bn 89
(a)
B (nT)

?i0.04

• which shock is the steepest?

MA3.5
?Bn 89
B (nT)
(b)

?i0.045
Time in hours
65
On the 'danger' of relying only on time series
for shock profiles

MA3.8
?Bn 89
Vshock11 km/s
(a)
B (nT)

• Taking into account the shock velocity
• is of crucial importance
• to avoid misinterpretation
• which shock is the steepest?

?i0.04

MA3.5
?Bn 89
B (nT)
(b)

?i0.045
Vshock78 km/s
Time in hours
66
Which shock is the steepest? answer

(a)



Lramp 4.5 c/?pe
Lramp 11 c/?pe
(b)
Despite the 'appearently steeper' shock in time
series, the real physical width of the ramp is
larger for case (b) than for case (a) because of
the much higher shock velocity.
67
Reformation time
computed as the local gyroperiod in the middle of
the ramp
Number of occurence

Tgyro / Treform
upstream
Typically 2 self-reformation cycles during one
upstream gyroperiod consistent with PIC
simulation results