Cluster investigations on the self-reformation of perpendicular Earth’s bow shock Cluster 17th workshop, Uppsala, Sweden, May CESR, UPS - CNRS, 9 Avenue du Colonel Roche, Toulouse, 31400, France 2 LATMOS / IPSL, CNRS UVSQ, Velizy, France, 3 LATT, Observatoire Midi-Pyrénées, Univ. of Toulouse, France 4 Physics Department, University of New Brunswick, Fredericton, NB, Canada, 5 LPCE, CNRS, 3A, Avenue de la recherche scientifique, France 6 Space & Atmospheric Physics Group, Imperial College London, UK. C. Mazelle 1, B. Lembège 2, A. Morgenthaler 3, K. Meziane 4, J.-L. Rauch 5, J.-G. Trotignon 5, E.A. Lucek 6, I. Dandouras 1

Outline Aim: Experimental evidence of shock front nonstationarity from determination of characteristic sub-scales with multi-satellite observations 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.

Physical characteristics of supercritical quasi-perpendicular shock OvershootFoot Ramp reflected gyrating ion Above a critical value of M A, 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 shock  PIC simul.: Shock non stationary -> Cyclic "shock front self-reformation".  Different proposed mechanisms of non stationarity  signatures : variation of the characteristic structures (foot, ramp, overshoot). Terrestrial shock geometry Time 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 … B n Q-  (45° - 90°) Earth Normalized distance B [Lembège et al., 2003]  Bn = 90° 2D PIC M A = 5 m p /m e =400

Numerical simulations of supercritical quasi-perpendicular shock  PIC simul.: Shock non stationary -> Cyclic "shock front self-reformation".  Different proposed mechanisms of non stationarity  signatures : variation of the characteristic structures (foot, ramp, overshoot). Terrestrial shock geometry Time 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 … B n Q-  (45° - 90°) Earth Normalized distance Foot Overshoot B [Lembège et al., 2003]  Bn = 90° Ramp c/ω pi Cluster 2D PIC

Outline Aim: Experimental evidence of shock front nonstationarity from determination of characteristic sub-scales with multi-satellite observations 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.

Ramp thickness: some previous ISEE results [Newbury and Russell, GRL, 1996] very thin shock ISEE: thicknesses of the laminar (low  ) shocks : 0.4 – 4.5 c/ω pi [Russell et al., 1982] ion inertial length scale Supercritical shocks: ramp thickness typically of ~ typically of ~ c/ω pi [Russell and Greenstadt, 1979; Scudder, 1986] (a) (b) (a) (b)

Previous study from Cluster data (1) 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? first examples of some aspects of shock nonstationarity (or at least variability) were presented by Horbury et al. [2001]:

Outline Aim: Experimental evidence of shock front nonstationarity from determination of characteristic sub-scales with multi-satellite observations 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.

Example of analysed shock crossing from Cluster B (nT) 5 Hz data

Methodology  Determination 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 normal velocity of the shock in s/c frame (V shock, V s/c, angle n - s/c traj.)  Main goal: to determine the real spatial width of the structures (ramp, foot, overshoot)  Careful error determination Time (hrs.) B (nT) Downstream asymptotic value upstream value 22 to 64 Hz data foot ramp 1 st overshoot along the normal use of high time resolution data

Methodology  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' space width (along each sat. traj.)-> compar. between the 4 s/c.  Determine the normal velocity of the shock in s/c frame (V shock, V s/c, angle n - s/c traj.)  Main goal: to determine the real spatial width of the structures (ramp, foot, overshoot)  Careful error determination Time (hrs.) B (nT) Downstream asymptotic value upstream value 22 to 64 Hz data foot ramp 1 st overshoot along the normal use of high time resolution data

Methodology  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.)-> compar. between the 4 s/c.  Determine the normal velocity of the shock in s/c frame (V shock, V s/c, angle n - s/c traj.)  Main goal: to determine the real spatial width of the structures (ramp, foot, overshoot)  Careful error determination Time (hrs.) B (nT) Downstream asymptotic value upstream value 22 to 64 Hz data foot ramp 1 st overshoot along the normal use of high time resolution data

Methodology  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.)-> compar. between the 4 s/c.  Determine the normal velocity of the shock in s/c frame (V shock, V s/c, angle n - s/c traj.)  Main goal: to determine the real spatial width of the structures (ramp, foot, overshoot)  Careful error determination Time (hrs.) B (nT) Downstream asymptotic value upstream value Timing method: gives shock normal n and velocity V in s/c frame For ech pair of satellites i and j : n foot ramp 1 st overshoot 22 to 64 Hz data along the normal use of high time resolution data V

Methodology  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.)-> compar. between the 4 s/c.  Determine the normal velocity of the shock in s/c frame (V shock, V s/c, angle n - s/c traj.)  Main goal: to determine the real spatial width of the structures (ramp, foot, overshoot)  Careful error determination Time (hrs.) B (nT) Downstream asymptotic value upstream value B (nT) overshoot foot ramp c/ω pi foot ramp 1 st overshoot 22 to 64 Hz data along the normal use of high time resolution data

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 B n. - check the mean upstream magnetic field vector seen by each satellite: -> estimate of B 0 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:

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 -> M A -> determination of  i

Four spacecraft measurements of the quasi-perpendicular terrestrial bow shock: [Horbury et al., JGR, 2002] 5 vectors/s clean, sharp shock complex, disturbed shock shock with probable acceleration

Four spacecraft measurements of the quasi-perpendicular terrestrial bow shock: [Horbury et al., JGR, 2002] 5 vectors/s clean, sharp shock complex, disturbed shock shock with probable acceleration

Characteristics of the sample Number of occurence  Bn iiii MAMAMAMA From 455 shocks: 24 shocks with all validated criteria majority above 84° majority below 0.1

|B| Typical shock crossing. Very thin ramp: some electron inertial lengths. Variablilty of ion foot, ramp and overshoot thicknesses thicknesses  evidence of shock non-stationarity and self-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  Bn = 89° ± 2° Sequence of crossings order at ref. time (ramp middle of ref. sat. 4) L ramp = 5 c/  pe M A =4.1  i =0.05 c/ ω pi

Outline Aim: Experimental evidence of shock front nonstationarity from determination of characteristic sub-scales with multi-satellite observations 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.

Statistical results (24 shocks = 96 crossings): ramps (1) Thinnest ramp for each shock L ramp in

Statistical results (24 shocks = 96 crossings): ramps (1) Thinnest ramp for each shock L ramp in Ramps of the order of a few c/ω pe,  Bn for a large range of  Bn  electron scale rather than ion electron dynamics important

Statistical results (24 shocks = 96 crossings): ramps (1) Thinnest ramp for each shock L ramp in Ramps of the order of a few c/ω pe,  Bn 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? ? ? critical angle between ‘oblique’ and ‘ perpendicular’ shock for low  and M f ~1  cr = 87° for low  and M f ~1  cr = 87° (e.g. Balikhin et al., 1995)

Larger probability to cross a thin ramp (<< ! Larger probability to cross a thin ramp (<< c/ω pi ) ! ion inertial length all ramps Statistical results (24 shocks = 96 crossings): ramps (2)

Statistical results (24 shocks = 96 crossings): ramps (3) all ramps only thin ramps close to 90° no simple trend trend: thickest ramps decrease with M A decrease with M A really perpendicular shocks? L ramp in

Outline Aim: Experimental evidence of shock front nonstationarity from determination of characteristic sub-scales with multi-satellite observations 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.

Comparison with 2D PIC simulations m p /m e =400

Comparison with 2D PIC simulations: ramps m p /m e =400

Statistical Results: ion foots (1)  Foot thickness < Larmor radius as expected  Mainly low values L foot in  Ci, upstream Number of occurence

Ion foots: comparison with 2D PIC simulations m p /m e =400 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

Statistical Results: ion foots (2) Comparison of largest observed value with 'stationary' theoretical value [Schwartz et al., 1983] : d =  Ci,upstream for  Bn = 90° and  Vn =0° d =  Ci,upstream for  Bn = 90° and  Vn =0°  another signature of shock cyclic self-reformation  another signature of shock cyclic self-reformation Red : stationary theoretical values Blue : largest observed values Shock number L foot in  Ci, upstream where reflected ion turn-around distance [Woods, 1969]

Statistical results (24 shocks = 96 crossings): overshoot Number of occurence L overshoot in c/ω pi upstream 3 Majority between 1 and 3 as e.g. in Mellott and Livesey [1987] Majority between 1 and 3 c/ω pi as e.g. in Mellott and Livesey [1987] but also large variability due to self-reformation of the shock

Outline Aim: Experimental evidence of shock front nonstationarity from determination of characteristic sub-scales with multi-satellite observations 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.

Previous study from Cluster data (2) [Bale et al., PRL, 2003] Fit of the density profile by an analytical shape (hyberbolic tangent) No separation between ramp and foot No separation between ramp and foot Typical shock size: ion scales ion inertial length convectivedownstreamgyroradius Shock scale: Here, different approach  sub-structures taken into account " This technique captures only the largest transition scale at the shock" [Bale et al., 2003] macroscopic density transition scale 5 Hz data

Comparaison with results from Bale et al. (2003) 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 L ramp+foot in c / ω pi Magnetosonic Mach number Signature of non stationarity Statistical results (24 shocks = 96 crossings) Is the shock front thickness simply dependent on Mach Number?

Statistical results (24 shocks = 96 crossings) Comparaison with results from Bale et al. (2003)   result seems to depend on the sample used.   no simple dependence L ramp+foot in c / ω pi Magnetosonic Mach number Is the shock front thickness simply dependent on Mach Number? Signature of non stationarity

Previous study from Cluster data (3) [Lobzin et al., GRL, 2007] Here, different approach  accumulation of case studies (statistics)  Bn = 81° M A =10  i =0.6 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 described in Krasnoselskikh et al. [2002]

Other shock sub-structures: Electric field spikes (1) [Walker et al., 2004]

Other shock sub-structures: Electric field spikes (2) Histogram of the scale sizes for the spike-like enhancements E-field spikes c/ω pi

[Walker et al., 2004] Histogram of the scale sizes for the spike-like enhancements c/ω pi Similar distribution to that for magnetic ramps with smaller values magnetic ramps E-field spikes Other shock sub-structures: Electric field spikes (2)

Other shock sub-structures: Electric field spike (3) E-field spikes Dependence of scale size on  Bn [Walker et al., 2004]

Other shock sub-structures: Electric field spike (3) L ramp in magnetic ramps E-field spikes Dependence of scale size on  Bn [Walker et al., 2004] Similar trend for only low values close to 90°

Other shock sub-structures: Electric field spike (4) [Walker et al., 2004] Dependence of scale size on upstream Mach number E-field spikes

Other shock sub-structures: Electric field spike (4) [Walker et al., 2004] E-field spikes magnetic ramps Similar trend: upper limit tend to decrease with increasing Mach Number Dependence of scale size on upstream Mach number

Ramp sub-structure Time (hrs.) Magnetic ramps often reveal sub-structure : nature? 22 Hz data

Ramp sub-structure Time (hrs.) signature due to electric field short scale structure? Magnetic ramps often reveal sub-structure : nature? Need further investigation but electric field data not always available 22 Hz data

Outline Aim: Experimental evidence of shock front nonstationarity from determination of characteristic sub-scales with multi-satellite observations 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.

Implication for future multi-spacecraft missions

already larger than ! already larger than c/ω pe !

Termination shock: Voyager 2 [Burlaga et al., Nature, 2008] ~ estimated shock speed: 68±17 km s -1 ramp thickness ~ c/ω pi but single-s/c determination… M MS ~10 and  i~0.04 (but without pickup ions) self-reformation? Q-perp nature Complex sub-structure (oscillatory) of the ramp: non uniformity (ripples) / non stationarity?

Conclusions and perspectives 1) New results on quasi-perpendicular shocks * particular cautions with time-series (transition -> real space width) * L ramp often very thin (electron scale) at least for 75°   Bn < 90° * L foot <  ci,upstream * No simple relation between L ramp and  Bn, L ramp and M A between L foot and  Bn * 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 ramp (2-6 c/  pe ) + varying L ramp --> varying L foot in time, varying overshoot thickness and amplitude --> in agreement with low to moderate  i ( ) 2) Under progress, necessity:  for increasing the statistics.  for careful analysis of: --> ion distributions (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

END Thank you!

Supercritical shock: Hybrid simulations [Leroy, 1981]

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.

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 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 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/w pi. Scudder [1986] got ~0.3 c/w pi 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]

ramp thickness determination: fitting method Time (hrs.)

Validity of the normal vector n ? |B| (nT) BnBn  Bn = 89.7° ± 1.5° M A =4.5  i =0.04 Upstream normal component B n very small Good consistency with  Bn ~ 90° and well conserved on average around the shock ramp Systematic check for all analysed shock crossings

Typical shock crossing (2) Very thin ramp: evidence of shock reformation (reflected ions) Satellites positions in (x GSE, n) plane‏ c / ω pi B (nT) L ramp = 4.5 c/  pe  Bn = 88° ± 3° At ref. time (ramp middle of ref. sat.) Sequence of crossings order expanding shock ? M A =3.8  i =0.04

On the 'danger' of relying only on time series for shock profiles… which shock is the steepest? which shock is the steepest? B (nT) Time in hours B (nT)  Bn = 89° M A =3.8 M A =3.5  i =0.045  i =0.04 (a) (b)

On the 'danger' of relying only on time series for shock profiles… Taking into account the shock velocity Taking into account the shock velocity is of crucial importance is of crucial importance to avoid misinterpretation: to avoid misinterpretation: which shock is the steepest? which shock is the steepest? B (nT) Time in hours B (nT) V shock =11 km/s V shock =78 km/s  Bn = 89° M A =3.8 M A =3.5  i =0.045  i =0.04 (a) (b)

L ramp = 11 c/  pe 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. Which shock is the steepest?: answer L ramp = 4.5 c/  pe (a) (b)

Reformation time T gyro / T reform Number of occurence Typically 2 self-reformation cycles during one upstream gyroperiod consistent with PIC simulation results computed as the local gyroperiod in the middle of the ramp upstream