Ion scales of quasi-perpendicular low-Mach-number interplanetary shocks



[1] A formation of low-Mach-number quasi-perpendicular shocks is expected to be well understood. From theoretical considerations as well as from observations, it follows that the shock ramp thickness would scale with the ion inertial length. We present analysis of 12 subcritical or marginally critical interplanetary shocks that reveals that (1) the ion transition scale determined from direct measurements of plasma moments (speed, temperature, and density) are of the same order as the ramp thickness determined from the magnetic field and (2) the ion transition scale is directly proportional to the ion thermal gyroradius, Rth; it was found to be ≈3.2 Rth in a broad range of solar wind and shock parameters. These results stress a role of the ion kinetics in the shock formation.

1 Introduction

[2] Collisionless shocks play a significant role in the solar wind interaction with the planets, and they are important for understanding physical processes in the vicinity of such astrophysical objects as supernova remnants, plasma jets, binary systems, and ordinary stars. The main process that takes place at the collisionless shock is the redistribution of the energy of the directed plasma motion to the plasma thermalization and acceleration of a part of particles to high energies. These processes depend on the characteristic spatial scales in the transition shock region [e.g., Lee et al., 1986; Scholer et al., 2003].

[3] The energy redistribution at the shock front is related to its structure because interactions with electromagnetic fields and waves act as particle collisions and form a magnetic field shock profile in a collisionless plasma. The most important parameters that classify the shock structure were considered [Formisano, 1977, 1985; Kennel et al., 1985]: the upstream magnetosonic Mach number, MMS (the ratio of the solar wind velocity to the propagation speed of magnetosonic waves), the upstream plasma beta, β (the ratio of the solar wind thermal to magnetic pressures), and θBn (the angle between the magnetic field and shock normal directions). In a low β plasma, the magnetic turbulence is low and a shock is laminar, while the turbulent shock is related to high β.

[4] The shock ramp width is determined by steepest spatial gradients, and this steepening is balanced by dispersion and/or dissipation. A change of the dissipation mechanism and structure is expected between super- (MMS>MC) and subcritical (MMS<MC) shocks where MC is the critical Mach number at which the downstream flow speed is equal to the downstream sound speed [Kennel et al., 1985]. A resistive dissipation alone is enough at low-Mach numbers, while an additional dissipation (e.g., viscosity) is required at higher Mach numbers.

[5] The magnetic field profile of the ramp and the spatial scales over which the field varies have been fairly well documented for low-Mach number quasi-perpendicular shocks [e.g., Russell et al., 1982; Thomsen et al., 1985; Mellott and Livesey, 1987; Newbury et al., 1998; Balikhin et al., 2008; Krasnoselskikh et al., 2013]. Farris et al. [1993] demonstrated a decrease of the shock width with the increasing Mach number in the range of the transition from subcritical to supercritical shocks. Theoretical and kinetic simulations [e.g., Leroy et al., 1982] and many observations suggest that the magnetic ramp occurs on the ion inertial scale, Li=c/ωpi [Papadopoulos, 1985; Matsukiyo and Scholer, 2006]. In low-Mach-number shocks, the width of the magnetic ramp profile was found less than ion convective gyroradius and ions cross the profile in about 0.1 of its gyroperiod [Farris et al., 1993; Gedalin, 1997]. According to Hobara et al. [2010], the ramp width changes by about one order of magnitude from 1.4 to 0.1 Li. A very thin ramp of the order of 0.05 Li was presented by Newbury and Russell [1996] and Walker et al. [1999]. Mellott and Greenstadt [1984] and more recently Hobara et al. [2010] and Mazelle et al. [2010] have shown scales comparable to whistler wavelengths.

[6] At present, the shock thickness studies apply the high-temporal resolution data from magnetic or electric experiments; however, these field profiles bring only indirect evidence on the shock dissipation scales. Measurements of the density transition scale of shocks are rare, mainly because particle instruments usually measure a complete distribution function within several seconds. In Schwartz et al. [2011], major characteristics of the electron distribution function at the bow shock are measured with a 250 ms resolution and reveal directly the time scale of the electron temperature profile. Under the configuration when the magnetic field is roughly aligned with the spin spacecraft axis, the shock motion is slow, and assuming gyrotropy, the authors present the full pitch-angle distribution. They conclude that the electron heating occurs over scales smaller (similarly as in Montgomery et al. [1970] or in Newbury and Russell [1996]) than the convected proton gyroscale (as discussed in Balikhin et al. [1993] and Bale et al. [2003]) and than the ion inertial length [Papadopoulos, 1985; Matsukiyo and Scholer, 2006].

[7] A great majority of the mentioned studies use the planetary bow shocks, whereas interplanetary (IP) shocks are studied experimentally in much less extent. The reason is the bow shock speed in the spacecraft frame can be as low as 10 km/s in fortunate cases [Schwartz et al., 2011], whereas the speeds of IP shocks are hundreds of km/s and thus, the time resolution of measurements would be increased accordingly. On the other hand, the Mach number of IP shocks is often low and such low-Mach bow shocks are encountered rarely [e.g., Formisano, 1977; Mellott, 1985; Balikhin et al., 2008] and investigations of their scales are of an interest. We present the analysis of the ion transition scales at low-Mach quasi-perpendicular IP shocks that is based on direct measurements of ion moments with a high-time resolution (≈31 ms). We show that the best scaling parameter is the proton thermal gyroradius, Rth.

2 Experimental Data

[8] The analysis uses measurements of a BMSW (the Bright Monitor of Solar Wind) solar wind monitor operating onboard the Spektr-R mission. The instrument design and methods of data processing are described by Šafránková et al. [2013]; thus, we repeat only basic principles relevant for the present paper here. The BMSW instrument is placed on the solar panel with the main axis oriented permanently toward the Sun and employs simultaneous measurements of a set of Faraday cups (FCs). The measurement of FC currents is realized with a cadence of ≈31 ms and characteristic plasma moments (bulk velocity, density, and temperature) are determined with a time resolution between 3 s and 31 ms. Three FCs are deflected form the instrument axis and serve for an estimation of the total ion flux and its direction, whereas three other FCs are equipped with retarding grids that facilitate the measurements of a 1-D energy distribution function.

[9] The instrument works in two modes: (1) the retarding voltages of all FCs are swept linearly in a range of 0–3 kV in the sweeping mode and the plasma moments are determined from a Maxwellian fit of the retarding characteristics with the resulting time resolution from 0.5 to 3 s, and (2) the retarding voltages of two FCs are set by feedback loops to the values for which the collector currents of these FCs are depressed to 70% and 30% of the current of that FC without the retarding voltage in the adaptive mode. The plasma moments are determined by a Maxwellian fit to actually measured three points of the energy distribution function with the time resolution of ≈31 ms. We should note that the total ion flux measurements are available always with this high-time resolution, even in the sweeping mode.

[10] Unfortunately, the onboard magnetometer is not in operation and we are forced to use the propagated magnetic field from other spacecraft in the solar wind. Since the IP shocks are structures that can be easily identified, there is no problem with the timing of the data but the quantities computed from our measurements and propagated magnetic field (critical Mach number, proton cyclotron frequency, θBn angle, etc.) suffer with an uncertainty that cannot be exactly determined. For this reason, we process data of all spacecraft observing the same IP shock (usually Advanced Composition Explorer, Wind, Themis, Cluster), compute these parameters from their observations, and use the medians in our statistics. In particular, the shock normal was calculated from the BMSW velocity measurements using the coplanarity theorem, whereas the θBn angle was determined from this normal and median value of the magnetic field from other spacecraft. We should note here that one can use the standard deviations of the magnetic field measured by the spacecraft in different locations as a proxy of the error of quantities determined by this way. This standard deviation does not exceed 7% in upstream and 12% in downstream regions for all investigated events.

3 IP Shock Analysis

[11] In course of the mission, BMSW observed ≈12 IP shocks and a part of them was registered in the adaptive mode, thus with a high-time resolution of plasma moments. Figure 1 presents two extreme examples of the shock observations. The first two panels show the high-time resolution magnetic field from Wind (the propagation time is given at the time scale description) and following panels present the BMSW plasma moments—the velocity, thermal speed, and density. Each panel of the figure contains 3 s of the data centered approximately at the shock ion ramp. In Figures 1a and 1b, one can note typical characteristics of fast-forward IP shocks: the increase of the magnetic field, proton speed, temperature, and density. These changes take place simultaneously, and they are nearly of the same duration. The difference between these two shocks is in shock parameters quantified in the figure captions. Whereas the duration of the 15 March shock ion ramp (Figure 1a) is ≈0.5 s, it is only ≈0.1 s for the 16 June shock (Figure 1b). Recalculation of the ion ramp durations to their widths (a product of the ion ramp duration and shock speed) reveals even more distinct difference because the 15 March shock moved much faster than that on 16 June (compare Figures 1a and 1b).

Figure 1.

Two examples of IP shock observations. At both (a) and (b) panels from top to bottom: the magnetic field strength and components propagated from Wind measured with a time resolution of ≈0.09 s, the solar wind speed (all components in Figure 1b panel), the proton thermal speed, and the proton density (with a resolution of ≈31 ms). IP shock parameters: Figure 1a, 15 March 2012 with MA = 3.8, MMS = 2.4, θBn≈70o, β=1.3, the ramp thickness = 338 km, and the shock speed = 704 km/s; Figure 1b, 16 June 2012 with MA =2.7, MMS = 2.4, θBn≈72o, β=0.2, the ramp thickness = 50.4 km, and the shock speed = 524 km/s. In Figure 1a, a determination of the shock ion ramp is shown (the detail explanation can be found in the text).

[12] The ion ramp width was determined as the time needed for the density to rise from 1.1 of its averaged upstream value to 0.9 of the overshoot height. This procedure is illustrated in Figure 1a (the density, N panel); the corresponding fractions of the density are indicated by the short green horizontal lines and the ion ramp duration is distinguished by the vertical red lines and the red part of the density profile.

[13] The classical theory predicts that the quasi-perpendicular shocks would not exhibit any downstream oscillations [Kennel et al., 1985], but Figure 1 shows clear wavy features in the downstream regions of both shocks. Although θBn and MA of both shocks are similar, the character of these wave trains are strongly different. Whereas a low-frequency modulation can be seen in all parameters (including the magnetic field) in Figure 1a, only temperature and velocity direction are oscillating in Figure 1b. We can note that the temperature oscillations with approximately the same relation between wavelength and shock ramp thickness follow from numerical simulations of Newman et al. [2011]. On the other hand, Figure 1b indicates a wave precursor that cannot be clearly resolved in the magnetic field due to insufficient time resolution. The wavelength of the wave is about equal to the ramp thickness as it is expected for whistler precursors [Mellott, 1985]. Unfortunately, we are not able to classify these waves due to a lack of simultaneous magnetic field observations.

[14] Since the moments in Figure 1 were obtained from Maxwellian fits to three points of the ion distribution function that were actually measured, we will check the reliability of such measurements prior to a conclusion. Figure 2a presents raw data (retarding voltages and FC currents) measured during passage of the IP shock on 29 September 2011. The top panel shows the voltages applied on the retarding grid (HV2 supplying the FC2 was switched off), and corresponding FC currents are shown in the middle panel. The red line belongs to FC2 without the retarding voltage that monitors a full ion flux. One can note that the ion flux fluctuates with a period much shorter than the sweep period and these fluctuations spoil the retarding characteristics. However, all three FCs register exactly the same ion flux; thus, the data of FC0 and FC1 can be normalized with respect to the instantaneous total ion flux (i.e., the FC2 current). The result of this normalization is shown in the last panel.

Figure 2.

An example of the ion distribution evolution across the IP shock. (a) Raw data: HV0,1,2—the voltages applied on the FC retarding grids in Volts, FC0,1,2—currents of FCs in arbitrary units, and FC0/FC2 FC1/FC2 normalized currents. The derivatives of the FC currents as measured just prior to (b) upstream and just after (c) downstream of the shock ramp that was observed at ≈ 01:01:45.5 universal time.

[15] The normalization filters the retarding characteristics and the distribution function that is proportional to the derivative of the retarding characteristics can be reliably determined. The derivatives belonging to the last sweep prior to and the first sweep after the shock ramp are shown in Figures 2b and 2c, respectively. Although these sweeps are separated from the ramp by ≈1 s (≈500 km), they show nearly fully thermalized proton distribution function. According to Balikhin and Wilkinson [1996], the heated downstream distribution is formed inside the ramp due to quasistationary fields and wave-particle interaction.

[16] Based on this example, we can conclude that the ion moments computed in the adaptive mode are reliable and that they can be used for a statistical evaluation. We have collected 12 fast-forward IP shocks with MA ranging from 1.5 to 4 (MMS from 1.1 to 2.5), plasma β from 0.1 to 1.4, and θBn from 50° to 90°. The calculation of the first critical Mach number indicated that all investigated shocks can be classified as subcritical or marginally critical.

[17] As it follows from the introduction, the shock ramp would scale with the proton inertial length but Figure 3a shows that it is probably not the case. We can conclude that the ion ramp width as seen in all parameters is between 1 and 4 proton inertial lengths. Searching for a better scaling parameter, we check the upstream proton gyroradius [Balikhin et al., 2008], Mach number, ion beta, Whistler length [Mellott and Greenstadt, 1984], and other parameters and found that the best organization provides the plot of the ion ramp thickness versus proton thermal gyroradius, Rth (computed from the thermal speed magnitude and magnetic field strength). Since both the temperature and magnetic field rise across the fast-forward shock, the value of Rth is nearly conserved as it can be seen from the plot of the downstream vs upstream thermal gyroradius shown in Figure 3b.

Figure 3.

(a) The ion ramp thickness as a function of the proton inertial length, (b) the dependence of the downstream and upstream thermal gyroradii, and (c) the ion ramp thickness as a function of the proton thermal gyroradius. The dotted line in Figure 3b stands for identity of the gyroradii, and the dashed line in Figure 3c represents the best data fit.

[18] For this reason, we used both thermal gyroradii in Figure 3c. The plot exhibits a nice linear dependence with a slope of ≈3.2 and reliability of the fit equal to 0.77. This result suggests that the mechanism of a formation of the analyzed (subcritical, quasi-perpendicular) IP shocks is connected with the ion gyromotion. Such mechanism was suggested by Balikhin et al. [2008], but the principal difference is that the upstream speed corrected to the θBn angle is replaced with the proton thermal speed in our study. Since the ion reflection is negligible at subcritical shocks, all particles gyrate with their random thermal velocities and thus we did not find any dependence on the magnetic field orientation and upstream bulk speed.

[19] A question whether the upstream or downstream values of the proton thermal gyroradius determine the ion ramp thickness can be hardly answered because (as we pointed above) these values are about equal. Nevertheless, we can note that a plot of the ion ramp thickness versus downstream gyroradius provides a fit with approximately the same slope but the fit reliability increases to 0.87.

4 Conclusion

[20] Our analysis of subcritical quasi-perpendicular IP shocks observed with a high-time resolution revealed the following:

  1. [21] Ion (density, temperature, velocity) scales are nearly identical and of the same order as the ramp thickness determined from the magnetic field.

  2. [22] The ion scales are determined by the proton thermal gyroradius.

  3. [23] The typical width of the ion transition scale is 3.2 proton thermal gyroradii.

[24] These results suggest a leading role of the kinetic processes but the exact mechanism of the shock formation will be a subject of further studies.


[25] We are grateful to D. Burgess and M. Balikhin for useful discussions. The present work was partly supported by the Czech Grant Agency under contracts 205/09/0112 and 209/12/1774, and partly by the Research Plan MSM 0021620860 that is financed by the Ministry of Education of the Czech Republic. O.G. acknowledge support by the Charles University Grant Agency (GAUK 1096213).

[26] The Editor thanks two anonymous reviewers for their assistance in evaluating this paper.