The determination of upstream parameters entails a steady asymptotic upstream interplanetary magnetic field (IMF) B0 associated with each shock crossing by the four spacecraft. For highly perpendicular shocks, however, it is often possible to define an unperturbed upstream field on a timescale of around a few minutes. For each satellite, we must define an upstream time interval displaying a magnetic field quite steady both in components and magnitude. For each spacecraft s/ci, where s/ci represents for spacecraft i among the Cluster quartet (where i=1 to 4), an average upstream field B0i is computed. Its magnitude is shown in the lower horizontal green line in Fig. 3a for one example of a single spacecraft shock crossing. Again, an initial estimated time interval is obtained from the visual inspection of the time series. The time duration cannot be too long since the usual stability timescale of the IMF due to the intrinsic solar wind variability (turbulence and directional discontinuities) is rarely more than a few tens of minutes (e.g., Burlaga, 1971), which we can take as an upper limit Δtu. In order to be physically reliable, Δtu cannot be too short. One reasonable lower limit Δtl is the time interval during which a solar wind parcel travels a distance of one ion inertial length (about 100 km usually) that is about 1 min typically, considering the mean spacecraft velocities are all unchanged within a reasonable range. In this purpose, the six relative angles θij measured between upstream fields B0i and B0j for s/ci and s/cj with (i,j=1 to 4) are computed. The upstream vector B0i measured by satellite i must be as close as possible to the vector B0j measured by another satellite j. If all angles θij are less than 5∘ (arbitrary value chosen herein from empirical considerations), then the procedure is pursued; otherwise, the upstream time intervals (where these θij values are calculated) are automatically modified by slightly shifting and reducing them until the above constraint on θij is satisfied. The shock event is rejected from the present analysis when the adjustment fails or is not satisfactory.

When possible, the use of a larger time interval improves the statistics. Using the initial estimate of Δt and constraining it to be between Δtl and Δtu, the best upstream interval corresponds to the lowest values of the standard deviations σ for both the magnitude and components of the field. Although this can be easily verified by visual inspection, it allows us to ensure the absence of any solar wind disturbance. The following step deals with the determination of a unique unperturbed ambient IMF B0 for every shock crossing by the four-spacecraft constellation. First, it is necessary to compare the individual upstream fields B0i. This determination of B0 is of crucial importance for the present study since it impacts both the value of the shock Alfvén Mach number through the field magnitude used and the determination of the angle θBn0 between B0 and the global shock normal n0. This is obviously the first possible source of error in the determination of both the upstream Alfvén velocity, the Alfvén and magnetosonic Mach numbers, and the angle θBn0.

For each satellite, the standard deviations, σ, obtained above for the magnetic field components and magnitude allow for the determination of the time when the spacecraft penetrates inside the magnetic foot. We determine, for each component and magnitude, the time when it is above a 3σ threshold level and close to the initially estimated time of the entry into the foot obtained by visual inspection (ta dashed blue line in Fig. 3a). Then, we select the “external” limit of this entry (t1 in Fig. 3a) among these four times as the closest to the external estimate ta, and we select the “internal limit” (t2 in Fig. 3a) as the closest to the internal estimate tb (dashed blue lines in Fig. 3a). The derived times, t1 and t2, then define the error bar of the foot entry in the time domain. For the present case, the visual estimate tb nearly coincides with the time t2 automatically determined. Moreover, the possible presence of solar wind or foreshock transients as well as a detached whistler wave packet must be avoided. The last case occurs quite often even in the nearly perpendicular configuration and can easily lead to a wrong determination. Also, solar wind transients are always possible, while foreshock transients may only be an issue for a more “oblique” configuration which is not addressed in the present study. Nevertheless, it is necessary to make a cautious post-visual inspection.

Similarly, the initial estimated times of the other microstructures are labeled with letters, and the final times determined from the present methodology are labeled with numbers. The times tc and td are the initial “visual” estimates for the exit from the foot and entry into the ramp, respectively. They will be only used for initializing the automated algorithm described below. A major caveat in many previous attempts to compute the ramp thickness in the literature is to include in the ramp the part of the magnetic field profile up to the maximum of the first overshoot. To avoid this and to define at least the first overshoot well, we need to determine a downstream (asymptotic) level which would be required, for instance, for checking the Rankine–Hugoniot conditions. For that purpose, it is necessary to define a downstream interval in the magnetosheath far enough from the consecutive overshoots or undershoots displaying a quasi-steady magnetic field. This choice is highly important and not always straightforward in the case of very close consecutive crossings (multiple crossings) where only an average over the region of the sequence of overshoots or undershoots will be estimated. When a “clean” time interval downstream from this overshoot or undershoot interval is available, the average values obtained for B magnitude and components can be compared to the values inside the overshoot or undershoot interval. If these values are in good agreement, the downstream interval is appropriate for the determination of the asymptotic value. It is then used to give an initial estimate (te, tf) of the final time interval (t5, t6) for the exit from the ramp or entry into the overshoot by using ±5 % of the computed asymptotic value (which is arbitrary but appears to be a good empirical compromise). It should be mentioned that a 1σ level can also be determined here, but it will usually be large in this region (local microturbulence in the magnetosheath) and not appropriate for the present purpose. The two times (te, tf) are again only used for initializing the next automated step.

From the initial time estimates (tc, td) and (te, tf), the determination of the final times for the entry into and exit from the ramp, (t3, t4) and (t5, t6), respectively, is made with an automatic algorithm. In order to characterize a steep gradient, we perform a linear fit of the data points between tc and tf. The time range for the regression can vary on both sides. The process is initialized by the largest possible time interval with t3=tc and t6=tf. Then, the entry time can vary between tc and td, while the exit time can vary between te and tf. The choice of a linear fit mainly excludes any “pollution” of the ramp region by a part of the extended foot (around the upstream edge of ramp) and by downstream fluctuations around the overshoot (around the downstream edge of the ramp). The resulting time interval is associated with the best linear regression coefficient, and the bounding times (tc and tf) explored during the automatic procedure are kept to estimate the error bars. In other words, we impose t3=tc and t6=tf. The automatic procedure only defines t4 and t5 (internal limits of the ramp). Then, the crossing of the downstream asymptotic value closer to the first estimate defines the exit from the overshoot (nearly equivalent to the middle of the gradient between the first overshoot and the first undershoot, not shown in detail here). The times t7 and t8 for the exit from the first overshoot are defined in the same way as the entry (Fig. 3a). The time tg shown by a vertical dashed blue line is – as for the other microstructures – simply the initial rough “visual” estimate and is just shown for reference for a later visual check.

For every shock crossing by the Cluster quartet, we make the same procedure for each satellite. The satellite that is associated with the steeper ramp from this analysis is used to arbitrarily define what is called the “reference satellite” for a shock crossing event. Then, we determine the middle time tramp,i of the ramp crossing time interval for each individual satellite i and the “reference time” for a shock crossing is simply this middle time tref for the arbitrarily chosen reference satellite. Next, we determine the shock velocity and normal from the four different middle times tramp,i and the four spacecraft positions in the GSE (Geocentric Solar Ecliptic ) frame by using the classical “timing method” (e.g., Schwartz, 1998; Horbury et al., 2001, 2002). This method works if the separation between the spacecraft is not too large (otherwise the hypothesis of a planar structure breaks down) and provided that the four spacecraft are not coplanar (otherwise the method fails) and that the shape of the tetrahedron is not very elongated (which is not the case for the present study). This method provides a global normal vector n0 valid only at the length scale of the spacecraft tetrahedron (i.e., the largest inter-distance between two satellites) and at the temporal scale defined by the time interval between the first crossing and the last crossing among the four satellites. The error bars for the entry or exit times are used to determine an error bar for the normal vector. It must be mentioned that each local normal vector ni for the crossing by s/ci can differ from the global n0 because of the small-scale turbulence of the front (shock front ripples) but cannot be determined experimentally as reliably as n0 by any classical single-spacecraft method. Moreover, as will be discussed later, we intend here to compare the values determined experimentally with simulation results for which the normal vector n0 and thus the θBn0 angle are globally defined for a shock front both averaged spatially perpendicular to the normal and in time (see Sect. 4.4). This will also conversely justify the averaging along the shock front made in the simulations (that will be described in Sect. 4.4). The local effect due to the variation of the local normal direction ni with respect to B0 (local curvature effect due to the shock rippling at a lower scale) is not analyzed in the present work.

For all the shock front microstructures, the above analysis gives an “apparent” temporal extension along each satellite path and allows us to compare them between the four satellites. In every individual spacecraft frame, we compute an individual shock velocity vector. Since the spacecraft velocities in the Earth frame are typically around a few kilometers per second (km/s) as also found for many slowly moving shocks in the same frame, it is essential to also take these small spacecraft velocities into account. The major aim of the present analysis is to reconstruct a local spatial profile along the global shock normal n0 as shown in Fig. 3b. For this purpose, the angle between each spacecraft i trajectory and n0 has been used. Then, it is possible to measure the local physical thicknesses along n0 of the microstructures of the shock front and compare them between the different satellites. It must be mentioned that the shock velocity determined by the four-spacecraft timing method here is an averaged velocity within the time interval separating the first and the last shock crossings. Then, it is supposed to be constant during this time interval and homogeneous over the spatial scale of the tetrahedron. One cannot preclude that the shock velocity can vary inside the considered time interval and that the shock motion is subject to some acceleration inward and/or outward. However, this possible effect can be assumed as weak if the temporal profiles of the shock crossing are quite similar in time series taken at low temporal resolution (such as in Fig. 1a). In contrast, when a sudden shock front acceleration or deceleration (due to a solar wind fluctuation) occurs between the crossings by two satellites, there is a clear signature in the time series (see in Fig. 4, derived from Fig. 1 in Horbury et al., 2002). Figure 4b and c are typical cases of observations which are eliminated from our present analysis. We use the same shock velocity in the Earth's frame for the four crossings to derive the spatial widths along the global normal n0 for any case like in Fig. 4a.

Then, we can compare the obtained thicknesses of the shock microstructures with physical length values predicted either by theory or from numerical simulation results. It must be mentioned that using only time series can lead to wrong comparative classification of the observed crossings. A large angle between the spacecraft velocity vector and the shock front normal can produce a long crossing duration for a real very small physical thickness and will also produce a small shock speed. In summary, a particular precaution is necessary to avoid conclusions deduced exclusively from the temporal profiles, especially in the selection process which is often based solely on the steepness of the temporal gradient in the literature.

In order to have a qualitative check of the determination of the global normal n0, the time series of the component Bn of the instantaneous magnetic field along this normal is systematically examined. First, this normal component is supposed to be conserved on both sides of the shock front. Second, for a nearly perpendicular shock, this component is expected to vanish (exactly zero for 90∘ shock) upstream from the shock front and inside the ramp, since the vector of the magnetic field is supposed to be strictly along the shock front. Figure 5 displays the temporal evolution of Bn for a shock front crossing on 31 March 2001 around 17:38 UT for one of the four satellites (magenta lower curve) in comparison with the total magnitude B (black upper curve). For this shock, the θBn angle is about 86∘. If one takes into account the fluctuations observed in and around the ramp, the conservation in the front is well respected, and the upstream value is also very low as expected for a nearly perpendicular case. The downstream evolution in the overshoot is also consistent with expected oscillations around the average downstream value (e.g., Kennel et al., 1985).

Herein, the spatial ramp width is identified within the shock front by a linear fit defined in the steepest part of the magnetic field gradient (Fig. 11b). However, the procedure to define this width slightly differs from that used for Cluster experimental data in Sect. 2. Presently, its lower bound (dashed line B) is defined where the B amplitude departs clearly (due to the foot region) from this linear fit. On the other hand, its upper bound (dashed line C) is defined where the B profile departs clearly (due to the proximity of the overshoot) from this linear fit. We do not use the overshoot as the upper bound since it is polluted by reflected ions as these gyrate and penetrate downstream after being once reflected at the ramp (Leroy et al., 1982); then this would overestimate the ramp width. On the other hand, let us note that using the mean downstream B amplitude as a reference (as done for experimental data analysis in Sect. 3) may be a source of inaccuracy when applied to the present numerical simulation results. Indeed, the size of the downstream region is not long enough to deduce a precise estimate of the averaged downstream B amplitude. In the B amplitude of Fig. 11b, these lower and upper bounds are, respectively, located at x̃=3755 and 3745 which provides a ramp thickness L̃ramp=10. Repeating the same measurement procedure at different times within the self-reformation cycle allows us to prove that the ramp thickness stays always very narrow around a (time-averaged) value of L̃ramp=3c̃/ω̃pe (Fig. 11c); the width of the ramp is almost independent of time and of the strong fluctuations of the maximum B amplitude at the overshoot (not shown herein), which is mainly driven by reflected ions (Leroy et al., 1982). Since one has to compare statistical results of experimental observations with simulations, it is important to clarify the following points: (i) considering a perpendicular shock represents a reference case in the sense that the ramp is expected to be the thinnest for 90∘, as compared with oblique cases (lower than 90∘), since indeed whistler wave emission (precursor) in oblique shocks tends to extend the thickness; (ii) herein, the concerned simulation is based on a relatively low β̃i value (= 0.10) or equivalently a high ratio ṽshock/ṽthi (= 31.66), which fits with the range of experimental conditions where data sets of shock crossing have been chosen (Fig. 8c–d); (iii) over a full self-reformation cyclic time period, the shock front covers a distance of Δx̃=597 (or equivalently 9.8c̃/ω̃pi). In the case study shown in Fig. 6c, the satellites inter-distances along the normal are, respectively, 1.4c/ωpi for C4–C2, 1.6c/ωpi for C2–C1, and 2.6c/ωpi for C1–C3, which makes the total distance for the shock crossing of 5.6c/ωpi; (iv) shock curvature effects and front rippling of the shock front have been neglected herein, in order to approach the experimental conditions where the inter-distance between satellites is small. The y averaging of the simulated planar shock is in agreement with the hypothesis of considering locally a planar shock front in experimental data for estimating the relative speed of each satellite versus the shock velocity. Then one can extract at present the width of the fine structures (foot and ramp) of the shock front mainly along the shock normal with a good accuracy. Moreover, in contrast with the methodology used for experimental data (Sect. 2), it is not necessary to use herein two different levels of identification (“visual” and “automatic”) for defining the upper and lower bounds of each microstructure (ramp and foot), since profiles have been already smoothed out enough by the y averaging.

The variation of the ramp width versus time reported in Fig. 11c shows that three successive time ranges can be defined where, respectively, the width decreases (range i defined by t̃<3456), very slightly decreases (range ii defined by 3456<t̃<4400), and increases (range iii defined by t̃>4400) as represented by pink, blue, and yellow areas, respectively. Then, the width stays almost constant during a large time range (ii). Within this range, the width reaches a lower-bound value around 3c/ωpe (dashed–dotted red line), which can be used as a reference scale in the comparison with experimental data instead of comparing exact scales measured at a given crossing time of the shock. In order to analyze the whole-time variation of the ramp width in detail, it can be associated with that of the foot and with ion phase space as described in Sect. 4.4.2.

Figure 11d shows the time variation of the foot width issued from the same PIC simulation. The upstream edge of the foot (vertical dashed line A in Fig. 11b) is defined as the location where the magnetic field has increased by 6.67 % over its upstream value (Burgess et al., 1999; Yang et al., 2009, 2011). The value of 6.67 % has been determined as corresponding to the maximum amplitude in the magnetic field fluctuations measured upstream (which is sensitive to the particle number per cell). Then, the amplitude of the upstream magnetic field turbulence is below the value (1 + 6.67 %) ×B0 in our simulations.

The time variation of the foot width can be detailed with the help of the three time intervals defined for the ramp thickness in Sect. 4.4.1 and is associated with the ramp as follows. In range (i), the foot width increases in anticorrelation with the ramp variation; at that time, reflected ions are accelerated at the ramp and only initialize their gyromotion (not shown herein). The range (ii) (where the ramp width slightly decreases) includes three successive phases for the foot dynamics: one (ii-a) where the foot width is constant (3456<t̃<3984), another (ii-b) where the foot width strongly increases (3984<t̃<4176), and a last one (ii-c) where the foot width is still increasing. But, in addition, a secondary foot can be identified upstream (blue curve for t̃>4176); a double foot appears as detailed below. Finally, in range (iii) (t̃>4416), the foot width reaches a maximum value (Lfoot/ρci,us) = 0.6, which remains constant at later times while the ramp thickness starts increasing.

A detailed analysis of the ion phase space (not shown here) at these corresponding time intervals allows us to interpret more precisely the time variation of both ramp and foot thicknesses. In range (i) (i.e., t̃<3456), the ramp (named ramp-1) is freshly formed from the steepening of the foot resulting from the previous self-reformation and the foot in panel (d) is only due to the old reflected ions (previous self-reformation) still located upstream of the new ramp-1 while accomplishing their full gyration (before being transmitted downstream); this steepening corresponds to the decrease of the ramp width (range i in panel c). In range (ii-a), ramp-1 starts to accelerate and reflect new upstream ions, which add to old reflected ions which did not reach the downstream region yet. Then, the local foot (named foot-1) is composed of two (old and new) reflected ion populations (reflected at two successive reformation cycles). During this acceleration phase, the energy of macroscopic fields at the ramp is transferred to newly reflected ions, which smooths the local steepening, and the ramp-1 width slightly decreases only. Within the range (ii-b), all old reflected ions have reached the downstream region; only freshly reflected ions are present upstream of the ramp, continue their gyromotion, and contribute to foot-1. Consequently, the foot-1 width strongly increases as mainly carried by these newly reflected ions. Correspondingly, the width of ramp-1 still slightly decreases. But, more and more new ions are reflected and accumulate in time, and a local new foot (named foot-2 ) builds up. This ion accumulation is large enough so that the upstream edge of this new foot-2 steepens and becomes a new ramp (named ramp-2, not shown in panel d) which starts reflecting new ions; let us mention that the width of ramp-2 cannot be precisely identified before t̃<4368 by using the best linear fit (as in Fig. 11b) since this part of the front microstructure is still mainly dominated by the growing foot (steepening is not strong enough in the upstream edge of foot-2 within this time range). This new foot-2 (blue curve in panel d) co-exists with the old foot-1 (red curve) within the same time interval t̃>4176. Simultaneously, the amplitude of the new overshoot (not shown here) associated with the new ramp-2 increases, and the field gradient between the old and the new overshoot becomes weaker. Therefore, the ramp-1 width increases at a later time (time range iii). For t̃>4608, the old ramp-1 is totally absorbed into the shock profile (between the old and the new overshoots), and its width cannot be identified precisely.

Then, the shock front may acquire a multi-peak profile including traces of old and new successive ramps (not to be confused with emission of large amplitude whistler waves emitted from the main ramp). The width of the whole shock front is the summation of the ramp and the feet contribution and can be estimated from the upper edge of the new foot-2 to the lower edge of the old ramp in the front. Present results provide maximum value measured at the final stage of self-reformation cycle (L̃shock,max=176.3=2.93c̃/ω̃pi at t̃=4608). Let us stress that this value represents an upper bound of the shock front width and not a common measurement of the front width itself.

Most studies based on the analysis of experimental data issued from ISEE-1 and ISEE-2 satellites indicate a scaling of the ramp width within the ion inertial length. In a very detailed analysis of a single shock crossing, Scudder et al. (1986) have reported the ramp thickness to be 0.2c/ωpi, a measurement which was based on an exponential fit of the foot–ramp transition. By using the same fit technique, Russell and Greenstadt (1979) have measured a slightly thicker ramp of about 0.4c/ωpi in an initial study of ISEE bow shock observations. Later on, Newbury and Russell (1996) analyzed seven nearly perpendicular shocks (θBn>80∘) and found a range of ramp widths (based on a linear fit to the ramp transition) from 0.5–0.8 c/ωpi. However, in one case the ramp width was particularly thin 0.05c/ωpi (= 2c/ωpe). To the knowledge of the authors, it was the first time that such a thin ramp had been observed associated with the terrestrial shock. All the observations lead to the following statements: a.

In contrast with the tentative explanations proposed at that time (summarized in Newbury et al., 1998), this very thin ramp (2c/ωpe) takes a full meaning herein and is in quite good agreement with the present statistical results. It represents quite well a potential signature of the self-reformation process which allows the ramp to have a very small thickness (a few c/ωpe).

At the ISEE-1 and ISEE-2 time, the ramp was usually characterized by two different scales: (i) a large scale (or global ramp width) within which the main transition from upstream to downstream magnetic fields takes place and (ii) a thinner subramp scale which contains steep jumps in the magnetic field with amplitudes sometimes comparable to the overall change in the magnetic field at the shock. One conclusion (Newbury et al., 1998) was that both scales are characteristic of the quasi-stationary shock profile (which – at that time – was considered stationary within the ion gyroperiod). More precisely, Newbury et al. (1998) have analyzed a set of 20 shock crossings and found that the width of the overall ramp transition is typically within the range 0.5–1.5 c/ωpi, which is quite large as compared with our present measurements. This difference can be easily explained by the fact that the authors have defined the ramp thickness based on a linear fit of the total magnetic field profile from the foot to the overshoot, which enlarges the estimate of the ramp thickness, as compared with the present method used in Sects. 3 and 4.

Cluster data and present statistical results allow us to correct this view within the frame of the intrinsic shock front self-reformation. The so-called “large scale” of the ramp presents a large width variability (extending from a part of the ion inertia length to a few electron inertia lengths as displayed in Fig. 9a), which was not observed at that time for the reason mentioned above. But, simultaneously, the so-called “small scales” can result from instability developing with the foot region as the ECDI, which represents Electron Cyclotron Drift Instability (Muschietti and Lembège, 2006, 2013) or MTSI (Scholer et al., 2003; Matsukyio and Scholer, 2003; Muschietti and Lembège, 2017), for instance, and can propagate (convection effects). In the present work, the use of a linear fit applied within the ramp region only (as described in Sect. 3) allows us to average these local small-scale fluctuations and to measure the ramp width with minimizing errors. Analyzing small-scale field fluctuations (as those also observed within the ramp in Figs. 3 and 5) requires a further investigation which will be presented elsewhere.

From the spacecraft floating potential measured on board the four Cluster satellites, Bale et al. (2003) have determined the electron plasma density and studied the macroscopic density transition scale at 98 crossings of the quasi-perpendicular terrestrial bow shock. This first tentative clue for measuring the shock transition width from Cluster data was attractive but contains several limitations for the following reason: a hyperbolic tangent function has been fitted to each density transition in order to capture the main shock transition but no microstructures of the front – as foot, ramp, or overshoot – have been identified and scaled. Such a fit partially includes parts of the shock front outside the ramp itself so that it cannot apply to determine the ramp thickness with acceptable accuracy since both ramp and foot structures are not separated. This technique takes into account only the widest transition scale at the shock front, which overestimates the ramp width and restricts drastically its application. Then, the conclusions stating that, for high Mach number shocks (including the range MA = 4–5 as that considered in the present work), only the convected gyroradius is the preferred scale for the shock density transition are strongly questionable since the real ramp is much thinner. This resulting scale corresponds to the scale of the foot rather than that of the ramp, and no conclusive statements can be obtained.

On the other hand, in the light of Cluster data, looking for an estimate of the shock transition presents, nowadays, a limited interest, since recent results show that terrestrial shock is rather nonstationary and a “unique” typical spatial scaling must be replaced by some interval or bounds of variation (lower bound and upper bound) within which the spatial scales of the fine structures can vary.

Figure 12 helps us to better compare our results with those by Bale et al. (2003), which have been reproduced in Fig. 12a, b, and d. The same representation as in Fig. 12b and d has been used for Fig. 12c and e to plot the front thickness computed as the sum of the ramp and the foot thicknesses from our methodology versus the magnetosonic Mach number Mms. The pink area in Fig. 12b illustrates the range of Mms explored from the present study. Figure 12c reveals that in this range of Mms values no clear dependence considering the large error bars (as on above Fig. 12b) can be identified from the present study. It would be obviously impossible here to conclude that the front thickness simply increases with Mms and conversely the best fit would conclude here that there is a trend to decrease. Our results are therefore in strong contradiction with conclusions of Bale et al. (2003). Moreover, if we consider another Mms range of similar extension as the one shown in Fig. 12c from our study and highlighted by the green area in Fig. 12b, a similar representation (not shown here) would conclude to exactly the opposite to what the dotted line in Fig. 12b shows, considering again the large error bars. This illustrates again the impossibility to make a simple scaling of the front thickness by a single parameter as the magnetosonic Mach number. Figure 12e shows that the front thickness normalized by the downstream gyroradius is more or less consistent with the results of Bale et al. (2003) shown in Fig. 12d. There is a trend there to decrease with Mms increase. As mentioned in the present study, the use of the so-called “density transition scale” by Bale et al. (2003) shown in Fig. 12a is inappropriate since it mixes the time variability of the ramp and the ion foot. They also mention a stationary shock from their Fig. 1. But, this figure does not show the highest temporal resolution of the magnetic field data but only corresponds to five values per second. For many shock crossings used in the present study, the profiles displayed at 5 Hz look very similar, whereas the highest-time-resolution profiles clearly reveal different features (as illustrated in our Fig. 6).

Krasnoselskikh et al. (2002) have proposed another mechanism as a possible source of nonstationarity of the shock front (so-called gradient catastrophe), where the shock appears as a balance between nonlinear and dispersive effects only. In this mechanism, the transition to nonstationarity is characterized by the lack of the phase standing whistler wave train inside the shock front. This process takes place for oblique quasi-perpendicular shock (for a direction θBn not too far from 90∘) as the Alfvén Mach MA is above a certain critical Mach number given by Mnw=(1/2)1/2(Mi/me)1/2cos⁡θBn; this condition can be equivalently expressed for a fixed MA as the direction θBn is below a critical angular value. In this case, the nonlinear steepening of the wave train cannot be balanced by the dispersion effects alone. However, as mentioned in Sect. 1, this theoretical mechanism has severe limitations since (i) it is based on a 1D model only and (ii) it excludes any dissipative effect. Indeed, reflected ions (which play a key role in supercritical Mach regime) are neglected, and their impact on the shock dynamics – and in particular on the temporal fluctuations of the shock front microstructures which is our center of interest in the present paper – is totally absent. Because of the above severe limitations, this process is still a source of controversy and is not considered herein.

Nevertheless, a further study made by Lobzin et al. (2007) has proposed the gradient catastrophe process as being at the origin of the shock front nonstationarity observed by Cluster on 24 January 2001. While the front nonstationarity is quite clear in the experimental data which show quite different shock front profiles of the magnetic field crossed by the four satellites of Cluster, the proposed interpretation stays quite questionable for the following reasons: (i) the overall analysis is based on one shock crossing only (no statistical analysis has been performed), (ii) no procedure detailing the transfer from time to spatial scaling of the shock front (in particular focused separately on the ramp or foot structures) has been indicated, so one ignores the percentages of errors underlying the spatial field profiles measurements which are not shown; (iii) no mapping and/or basic information have been provided on the 3D disposal and inter-distance of the satellites, so one ignores whether the four satellites cross the same shock front within the same or different successive self-reformation cycles; (iv) only one measurement of the ion density showing some fluctuations has been presented which is not enough for confirming that it could be due to the gradient catastrophe process. Moreover, note that such similar density fluctuations have been also measured quantitatively versus time in previous numerical simulations (Lembège and Savoini, 1992; Yang et al., 2009) but have been attributed as being due to the self-reformation process fed by the accumulation of reflected ions and not to the gradient catastrophe process. Similar self-reformations have been also obtained as supported by the MTSI for an obliquely propagating shock (Comisel et al., 2011); (v) a deeper analysis shows that the upstream ion temperature used in this paper is not the correct one since the CIS instrument was in magnetospheric mode (see Sect. 2), so the moment calculations provide a large overestimate of the solar wind proton core temperature. Indeed, the determination of the upstream ion thermal velocity requires a delicate procedure and is of central interest since the stationarity or nonstationarity of the shock front is strongly dependent on the ratio vsh/vthi (Scholer et al., 2003) or similarly on the ratio βi (defined as the ion kinetic energy over magnetic energy) (Hada et al., 2003). When using the solar proton temperature from the ACE satellite (4.5 eV thermal energy), the ion beta found is 0.6 instead of the value of 2.0 reported by Lobzin et al. (2007).

The same data of Cluster measured on 24 January 2001 has been re-examined in detail by Comisel et al. (2011) and re-interpreted quite differently by using PIC simulations in plasma conditions and Mach number regimes chosen to be appropriate with experimental conditions. Basically, the authors have shown that the self-reformation is quite well recovered but is not controlled by the gradient catastrophe process (dispersive effects only) mentioned above but rather by the self-reformation mechanism fed up by the MTSI which is an indirect consequence of the ion reflection (dissipative effects which have been neglected in Krasnoselskikh et al., 2002). Indeed, for slightly oblique shocks (out of 90∘), the MTSI develops between the incoming electrons and incoming ions, with a growth rate large enough to develop during the gyration of the reflected ions (Scholer and Matsukiyo, 2004). These authors have shown that for a nonrealistic mass ratio, the self-reformation is still controlled by the collection of reflected ions at the upstream edge of the foot as in Lembège and Savoini (1992), since the MTSI has a weak growth rate and has no impact on the nonstationary dynamics of the shock front. In contrast, for realistic mass ratio, the growth rate is much larger, and MTSI can be easily excited, which leads to ion phase mixing and thermalization. Then, the ion pressure increases considerably locally somewhere within the foot itself (and not at the edge of the foot) with associated building up and overtaking of the magnetic field. A new shock ramp builds up within the foot which initiates to reflect new incoming ions and a cyclic reformation is initialized. The cyclic period is lower than the one associated with the self-reformation due to the accumulation of reflected ions (as in Lembège et Savoini, 1992), since the process takes place before ions accomplish their full gyration. More precisely, by using the appropriate conditions of the 24 January 2001 shock crossing in the PIC simulations, Comisel et al. (2011) have shown that the waves triggered by this instability are of whistler type and have phase velocities directed downstream in the shock frame, and the associated Poynting flux (and wave group velocity) is also directed in the downstream direction. However, as these approach the density and magnetic field overshoot, the waves are refracted; as a consequence, the Poynting flux changes direction and is directed upstream in the shock frame. In summary, there is no indication of a phase standing linear or nonlinear whistler precursor produced by the shock ramp itself in this process. This scenario does not support the scenario that proposed by Krasnoselskikh et al. (2002), where a nonlinear whistler precursor is required to initialize the gradient catastrophe process as a source of the front nonstationarity.

A statistical study based on 77 crossings of the terrestrial bow shock (30 by THEMIS (Time History of Events and Macroscale Interactions during Substorms) and 47 by Cluster) has been performed by Hobara et al. (2010) in order to analyze the ramp scale and the associated gradient width. Their study claims that the spatial scale of the ramp decreases as the shock Alfvén Mach number increases. However, detailed comparison with our results is difficult (almost impossible) at the present time because of important missing information. (i) No detailed information is provided on the procedure used for extracting the spatial widths of the ramp from the temporal experimental data as detailed in present Sect. 2; (ii) the duration of the ramp crossing has been defined as a time interval between the upstream edge of the ramp and the maximum of overshoot. However, let us note that the upstream edge is not defined precisely (see the precaution to account for in our Sect. 3), and the use of the overshoot can lead to an overestimate of the ramp width as compared to the method used herein (Sect. 3) since it is partially polluted by reflected ions; (iii) statistical errors are indicated in their Figs. 2 and 3 only for a range of Mach numbers and not on individual measurements; (iv) no statistics are shown versus the direction (θBn) of the shock front normal; as a consequence, one ignores how and whether the measurements of the ramp or foot bounds are polluted by the whistler (either linear or nonlinear), whose emission strongly depends on the angle θBn; (v) no statistics or information is indicated on the ion βi value (which has a strong impact on the shock front self-reformation) (Hada et al., 2003); (vi) the authors use two different statistics: one based on the ramp spatial scale (their Fig. 3) and the other based on magnetic ramp spatial gradient scale (their Fig. 4). Moreover, they used a simple theoretical argument (based on dispersive whistler) to stress that the wavelength Lw of the linear or nonlinear whistler varies as Lw∼1/MA. But the fact that both statistics show that the width of the measured structure decreases as MA increases does not imply that dispersion determines the size of the magnetic ramp even for supercritical shocks (as discussed later on); (vii) at least, only a very limited number of measurements (whatever the MA value is within the concerned range MA = 2–12 in Hobara et al., 2010) indicate that the ramp width succeeds to reach a few electron inertial lengths (equal to or lower than 5c̃/ω̃pe, in their Figs. 2 and 3), which is in strong contrast with our statistical results of Fig. 9a. The lack of statistics versus θBn and βi (above points iv and v) and the absence of precise information on the methodology used in the statistics of Hobara et al. (2010) (as listed above) could explain this difference.

Moreover, Hobara et al. (2010) deduce that their measurements (in particular the decrease of the ramp thickness as MA increases) show that the major process responsible for the shock front nonstationarity is the gradient catastrophe process proposed by Krassnoselskikh et al. (2002) and Lobzin et al. (2007). However, three main facts contradict this statement: (i) the first is arguments describing the severe limitations of the theoretical model (Krassnoselskikh et al., 2002) and the revised analysis of Comisel et al. (2011) as already described above in the present section, (ii) Hobara et al. (2010) use the overshoot (observed in the set of their experimental measurements) as an upper bound to define the ramp width, but the overshoot of supercritical shocks is a clear signature of the reflected ions as these gyrate downstream after being once reflected at the ramp (Leroy et al., 1982). Then, a contradiction occurs since dissipation effects (strongly carried by the reflected ions) have been fully neglected in the model of gradient catastrophe which cannot support their statements; (iii) a similar variation of the ramp thickness as MA increases has also been observed in the self-reformation process in PIC simulations (not shown here), which means that this variation cannot be used as a unique signature of the gradient catastrophe process. At least, in order to support their statements, Hobara et al. (2010) recall that PIC simulations evidencing the self-reformation due to the accumulation of reflected ions at a foot distance from the ramp (as in Lembège and Savoini, 1992) as well as due to the MTSI (as in Matsukyio and Scholer, 2006 use low ωpe/ωce ratio, which overestimates the electric field to magnetic field ratio (in practice, using a realistic ωpe/ωce ratio is very difficult at present time for computational constraints). If were true, this argument would suggest that the electric field would be too large and that the self-reformation process would result from an artifact. However, no quantitative result has been obtained until now to support such a statement. In contrast, let us remind the reader that (i) the parametric analysis performed by Lembège et al. (2009) clearly shows that both hybrid and PIC simulation retrieve the same self-reformation process of the shock front in similar plasma conditions and Mach regime, provided that the space grid is small enough (i.e., high resolution) in hybrid simulations as shown initially by Hellinger et al. (2002); this last result would suggest that the field gradients at the ramp are important and not the absolute fields amplitude; (ii) electron scales are neglected in common hybrid simulations (as those used in Hellinger et al., 2007; Lembège et al., 2009, and references therein), which are independent of the ratio ωpe/ωce. This suggests that the ratio ωpe/ωce has no direct impact on the self-reformation. Note that the ramp width may be estimated in terms of percentages of ion inertial length in hybrid simulations and both in terms of electron and ion inertial lengths in PIC simulations.

In summary, the measurements of Hobara et al. (2010) do not support the model describing the ramp as an evolving nonlinear whistler wave; neither does the mechanism of the gradient catastrophe process of Krassnoselskikh et al. (2002) as a source of the front nonstationarity.

Schwartz et al. (2011) have determined the scale of the electron temperature gradient via electron distributions measured in situ by the Cluster spacecraft and have identified two different scales: when the Mach number is supercritical (MA=3.8) and when the shock propagation is oblique (θBn=83∘). First, the authors found that half of the electron heating coincides with a narrow layer covering several electron inertia lengths (c/ωpe) and found an inflated electron distribution due to primarily the electron filling in or the entrapment in regions of phase space that would not be accessible. Second, the remaining solar wind maximum is only present in the smoother increase of the field which precedes the ramp; this beam is totally destroyed by the time this electron scale ramp is crossed. The authors found that this scale is comparable to that deduced from the theoretical limiting case of a wave capable of phase standing in the incident flow and concluded that this suggests that supercritical shocks steepen to this whistler limit since dissipation processes are insufficient to broaden the transition further.

However, several elements of information are missing: (i) the authors mentioned to have used a technique to convert the time series of data to distance along the shock normal similar to that of Schwartz (1998), i.e., the procedure to determine the normal and the velocity of the shock from multi-spacecraft measurements (as in Sect. 2), but did not provide detailed information on the procedure itself, associated errors estimate, or on the definition of the ramp itself (their Fig. 1); (ii) the results are limited to one shock crossing (one spacecraft only), and some attempts for other shock crossings are mentioned, but no statistical results have been shown or summarized; (iii) the observed results cannot lead to the conclusions that dissipation processes are insufficient to broaden the transition further, since once again the key dissipation processes driven by the (reflected) ions are not mentioned or analyzed in the study. Of course, dispersive effects must be considered in the global balance of nonlinear steepening of the front by both dispersive or dissipative effects, but this does not mean that the shock front dynamics are controlled by dispersive effects. Moreover, other sources of field fluctuations may take place in the shock front (such as formation of multi-peaks (signatures of old or new self-reformation (Sect. 4.4.2) and/or whistler waves excited by the MTSI within the front) without invoking any dispersive effect.

Very recently, Yang et al. (2020) claimed to have identified shock front self-reformation with the help of high-resolution Magnetospheric MultiScale (MMS) satellite data. This study using measured ion phase space together with B profiles clearly shows the importance of dissipation effects carried by the reflected ions. However, the comparison with the results of the present study proves to be quite difficult due to the lack of precise information. While the MMS data show clearly that the crossed shock is nonstationary, a clear proof of the self-reformation is still questionable for the following reasons: a.

The whole study is based on one shock crossing only and eventually restricted to a comparison between two satellites since three of them show very similar profiles and appeared to be in a plane nearly parallel to the shock front. No statistical results are shown or even summarized, which could have stressed the ion vortex formation over different steps of its formation.