A Survey of Venus Shock Crossings Dominated by Kinematic Relaxation

Collisionless shocks are one of the most effective particle accelerators in the known universe. Even low Mach number shocks could have a significant role in particle heating and acceleration. Theory suggests that kinematic collisionless relaxation, the process whereby a downstream nongyroptopic ion population becomes thermalized through collisionless gyrophase mixing, is the dominant energy redistribution mechanism in quasi‐perpendicular, low Mach number, and low β shocks. However, there have only been a limited number of observations of these shocks using in situ measurements at Venus, Earth and in interplanetary space. This paper presents the results of the first detailed study using in situ measurements, of the effect of fundamental parameters on the formation of these shocks. All low Mach number shocks occurring during the magnetic cloud phase of an interplanetary coronal mass ejection are identified in Venus Express magnetic field data over the duration of the mission. From the 92 shock crossings identified, 38 show clear evidence of kinematic relaxation. It is shown that kinematic relaxation is dominant at Venus when the angle between the local shock normal and upstream magnetic field is greater 50° and the Alfvén Mach number is less than 1.4. These shocks are also observed across a range of solar‐zenith‐angles indicating that it is likely that any location on the Venus bow shock could form such a structure. Venus Express plasma measurements are used to verify the parameters estimated from the magnetic field and indicate the importance of heavy ions, including potential pickup O+.


Introduction
Understanding collisionless shocks is important for many astrophysical processes. They are key providers of particle acceleration, both within the heliosphere and further afield. It is important to understand the physics of all types of collisionless shocks. Even low Mach number shocks could have a significant role in particle acceleration and heating (Ryu et al., 2003). Within the heliosphere they have a key role in planetary interaction with the solar wind (Russell, 1985) and it is only here that they can be directly observed using in situ measurements. These direct observations therefore have important consequences for understanding astrophysical collisionless shocks. Their study within the heliosphere using direct observation, is also crucially important for many remote astrophysical objects, as radiation generated at collisionless shocks often provides the only observational data about the environment in the vicinity of these objects.
In a collisionless plasma a shock forms when a magnetosonic flow encounters an object and is subsequently decelerated to a submagnetosonic speed. When the flow is decelerated across the shock the upstream kinetic energy is redistributed through various processes into thermalization of the bulk of the plasma flow and acceleration to high energies of a fraction of the particles. The processes that lead to this energy redistribution vary depending on the shock parameters, most importantly the Mach number, the ratio of the upstream plasma kinetic to magnetic pressure ( ) and the angle between the upstream magnetic field and normal to the shock surface ( B, n ). Understanding the role of the different processes that lead to the energy re-distribution is one of the most important tasks in collisionless shock physics.
One subset of shocks is those where the shock is quasi-perpendicular (i.e., B,n ≳ 45 • ). These shocks usually have a well structured magnetic field profile (Burgess et al., 1989;Kennel et al., 1985;Mellott, 1985;Scudder et al., 1986). As the Mach number increase the magnetic field compression across the shock increases. The number of ions which are reflected and appear ahead of the shock ramp also increases and these have a significant effect on the structure of the shock (e.g., Scudder et al., 1986). The shock ramp can be defined as the region in the shock transition which has the steepest increase in the magnetic field. After crossing the shock ramp, the initially reflected ions form a superthermal population in the downstream region.
Under the accepted theory of low Mach number quasi-perpendicular shocks the magnetic field transition across the shock is smooth and ends in the downstream region without an overshoot or downstream oscillations (Kennel et al., 1985;Mellott, 1985). This is the case for both resistive and dispersive shocks. In contradiction to this accepted theory, Balikhin et al. (2008) observed low Mach number quasi-perpendicular shock crossings detected in the Venus Express magnetic field data that had a noticeable overshoot and/or a significant downstream oscillations. They proposed that the source of this overshoot/oscillations was the kinematic relaxation of downstream ions with a nongyrotropic velocity distribution. This was supported by theory and numerical analysis based on earlier work for downstream gyrating ion populations (Gedalin, 1996(Gedalin, , 1997Zilbersher et al., 1998). Balikhin et al. (2008) referred to such shocks as "kinematic shocks" in much the same manner as dispersive or resistive shocks are named after the predominant energy redistribution process. The nongyrotropy of the downstream ion distributions leads to a spatially dependent ion pressure, which due to the requirement for pressure balance, creates a spatially dependent and out-of-phase magnetic pressure (Gedalin, 2015). This is the cause for the overshoot and oscillations in the magnetic field observed by Balikhin et al. (2008). If the upstream ions have a noncold distribution, spatial gyrophase mixing in the downstream regions leads to mixing of the plasma and consequently a decay in the magnitude of the oscillations. The rate of mixing is related to . When is low the relaxation length is large, so that a set of well-defined coherent oscillations can appear downstream of low Mach number shocks (Gedalin, 2015;Gedalin et al., 2015). The mixing of the directly transmitted and initially reflected ions at high Mach number shocks usually leads to significantly different downstream structure, often devoid of coherent oscillations (Gedalin, 2016;Ofman & Gedalin, 2013). Since the influence of reflected ions is expected to be small, low Mach number and low shocks provide the best opportunity to observe and study the process of kinematic relaxation. This was confirmed by theoretical analysis and hybrid simulations conducted by Ofman et al. (2009).
Since this initial discovery by Balikhin et al. (2008), several other studies have focused on such shocks. This includes observation of low Mach number interplanetary shocks with such a structure in the magnetic field (Kajdič et al., 2012;Russell et al., 2009). Nonsimultaneous magnetic field and plasma data was used to investigate interplanetary shocks with this structure (Goncharov et al., 2014). However, this did not directly confirm the antiphase oscillations in the magnetic and ion pressure predicted by theory. Recently, Pope et al. (2019) used the first observations of such shocks at the Earth to confirm this theory with direct simultaneous measurement of antiphase oscillations in the magnetic and ion pressure. This study confirmed kinematic relaxation as the dominant process for energy redistribution in these quasi-perpendicular low mach number shocks. It also showed the role of the different ion species (proton and particles in this case) in forming the downstream distribution.
Despite recent work, little is understood about the exact range of upstream conditions under which shocks dominant by kinematic relaxation form. Pope et al. (2019) confirmed the most likely opportunity to observe such shocks at planets is during the magnetic cloud phase of an interplanetary coronal mass ejection (ICME). The magnetic cloud embedded within an ICME has the low Mach number and environment predicted by theory as the most likely conditions under which kinematic relaxation can be observed. In addition, such shocks usually form at high altitude instances of the planetary bow shock, caused by the low Mach number solar wind.  analyzed the location of the Venus bow shock during the planets interaction with a strong ICME. These shocks included those studied by Balikhin et al. (2008) and occurred at a much higher altitude than the nominal value determined using data from a similar time period . Suitable Earth observation satellites do not regularly explore such comparatively high altitudes. In contrast, extraterrestrial innerplanetary exploration spacecraft, such as Venus Express, often have a much higher apoapsis and short orbital duration allowing this region to be sampled regularly and thus increasing the likelihood of observing such shocks when the solar wind conditions are suitable. In this paper the magnetic clouds identified by Vech et al. (2015) over the duration of the Venus Express mission are used as the search space for low Mach number shocks which show evidence of an overshoot and/or downstream oscillations.

10.1029/2020JA028256
This paper identifies all instances of such shocks during these intervals. Following identification of these shocks, their structure are analyzed in terms of their dependence on key parameters such as Alfvén Mach number (M A ) and B, n . Their location in terms of solar zenith angle (SZA) and altitude are also investigated to determine if there is any bias for these shocks to form in a more subsolar or flank location. The results show a clear range of upstream conditions under which these shocks occur. They also show that they can occur from the subsolar through to flank locations at Venus.

Data
In this study magnetic field and plasma data measured by the MAG (Zhang et al., 2007) and ASPERA  instruments onboard Venus Express was used. The primary data set is the 1 Hz magnetic field measurements, which provides complete orbital coverage apart from during occasional planned or unplanned spacecraft operations. Using these data, the initial search space for each planet was reduced by only considering the time periods when each planet was subject to an ICME with a clear and strong magnetic cloud. Previous studies have shown that these provide the most favorable conditions for observation of kinematic relaxation Pope et al., 2019). The time intervals when Venus experiences the clear magnetic cloud phase of an ICME during the orbital mission lifetime of Venus Express (2006)(2007)(2008)(2009)(2010)(2011)(2012)(2013)(2014) has previously been identified by Vech et al. (2015). This study identified six such intervals and these are investigated for the presence of shocks showing evidence of kinematic relaxation. This approach does not necessarily lead to the identification of all such shocks at Venus observed by Venus Express, but it should identify the majority and greatly speeds up the process of identification. Within these intervals, shock crossings were searched for which had (1) low magnetic compression (B d /B u ) across the shock (≾1.5), indicative of a very low Mach number; (2) low-frequency downstream oscillations polarized along the shock normal direction which onset immediately after the shock ramp; and (3) multiple crossings of this shock structure. Criteria number 3 is not strictly required as initial evidence of observation of a shock dominated by kinematic relaxation, but it can aid in their identification. The reason is that the Venus bow shock altitude is primarily driven by the solar wind Mach number (Russell et al., 1988). Since the shocks of interest occur during very low Mach numbers, even small changes can lead to significant changes in bow shock location. In particular, in a very low Mach number magnetic cloud the proton density is usually very small. For example, it is close to unity for the kinematic shock observed at the Earth (Pope et al., 2019). In this case even a small absolute change in proton density can lead to a significant relative change in M A . Multiple bow shock crossings by Venus Express due to a dynamic bow shock moving back and forth across the slower moving spacecraft are therefore often a signature of the very low Mach number conditions of interest. Following identification of the shocks in the magnetic field data, ASPERA proton number density, temperature, and velocity and oxygen number density at 192 s sample intervals with good/excellent quality flags was used to further study the upstream conditions. The plasma data does not provide complete orbital coverage, so data were not available for all shock crossings identified.
Due to the reliance on single spacecraft measurements with low sample rate and noncontinuous plasma measurements, the shock normaln for each of the shocks studied is determined from the magnetic field data using both minimum variance analysis (MVA) and the coplanarity theorem (CP). Both of these methods can be subject to errors due to factors such as a small number of data samples across the shock ramp and the presence of non-shock-related structures in then upstream and downstream regions. However, if carefully implemented they can be used to determine a reasonably accurate estimate of the shock normal. As an example, in Pope et al. (2019) there was <4 • between the MVA and CP normal and that calculated using the double coplanarity theorem which requires both the magnetic field and ion velocity data. Due to the often abnormally high altitude of the observed shocks, the shock normal calculated using a model bow shock and observed SZA was not considered. The shock normal is used to determine the angle between the average upstream magnetic field and shock normal direction B, n .
When suitable plasma data were available, the Alfvén Mach number was calculated directly as M A = v u, x /v A , the ratio of the upstream flow velocity v u, x in the shock normal direction in the shock rest frame, to the upstream Alfvén speed v A = B 2 u ∕ n i,u m i (B u is the average upstream magnetic field magnitude, n i, u is the upstream ion density, and m i is the ion mass). In the calculation of the Alfvén velocity the subscript i is the ion species. Where possible, single (proton) and multifluid (proton and oxygen) v A were calculated and the importance of including heavy ions investigated. The Alfvén Mach number for all of the detected  Balikhin et al. (2008). The green and blue shaded regions in panels (a)-(f) indicate new groups of very low Mach number shocks identified in this study. For clarity of presentation, the limits of both the time and magnetic field axes are set independently for each panel (a-f). very weak shocks was estimated directly from the magnetic field data using M A,B ≈ √ R(R + 1)∕2, where R = B d ∕B u is the ratio of the downstream and upstream magnetic field magnitudes. This is valid for a cold perpendicular shock . Pope et al. (2019) recently showed good agreement between this estimate and the directly calculated value of M A for the very low Mach number shock in which kinematic relaxation dominates, which was observed at the Earth. The magnetosonic Mach number was not considered, since in an ICME magnetic cloud the Alfvén speed is usually significantly larger than the sound speed. This is caused by the abnormally large (for the solar wind) magnetic field and the low proton temperature and often low proton densities (Burlaga et al., 1981;Leamon, 2002) The upstream i,u = p i ∕p B = 2 0 p u ∕B 2 u (the ratio of kinetic to magnetic pressure) is calculated when suitable plasma data is available. The total kinetic pressure in the solar wind is estimated using the available proton density and temperature data as p u = n p, u k b (1.16T p, u + 1.55 × 10 5 ) (Burlaga & Ogilvie, 1970). Finally, the SZA and altitude A of the observed shocks are determined. These are compared to model shock altitudes A m at solar minimum for the observed SZA , that is, when the bow shock is on average at its most compressed.

Observations
Six intervals during which Venus Express observes the magnetic cloud phase of an ICME, as identified by Vech et al. (2015), are shown in Figure 1a-f. For each of these six intervals, Venus Express observes at least one group of multiple crossings of very low Mach number shock. These groups of shock crossings are marked by the colored regions in Figures 1a-1f. The profile of the magnetic field magnitude and the shock parameters (mainly derived from the magnetic field measurements) for all of these shock crossings on each of these 6 days are contained in the supporting information in Figures S1-S6 and Tables S1-S6, respectively. A subset of these which show certain interesting characteristics, or for which plasma data are available, are included in Figures 2-4 and Table 1 and discussed in the following subsections.

10-11 September 2006
The shocks previously studied by Balikhin et al. (2008) occurred on 10 September 2006 during an ICME with a prolonged magnetic cloud, which commenced at around 18:20 UT on 10 September and continued throughout the following day, spanning an entire Venus Express orbit. The magnetic field measured by Venus Express during this period is shown in Figure 1a. The red shaded region marks the seven shock crossings which were studied by Balikhin et al. (2008) and occurred on the inbound trajectory of Venus Express, but at an abnormally high altitude. The spacecraft then detects two more sets of shocking crossings on the outbound and the next inbound trajectory. These are marked by the green and blue regions in Figure 1a. Figure S1 shows the profile in the magnetic field data, and Table S1 contains the magnetic field derived parameters for all of these individual shock crossings. Figure 2 shows the three shock crossings in the second group and eight shock crossings in the third group (i.e., the green and blue shaded regions in Figure 1a). Table 1 contains the associated shock parameters mainly derived from the magnetic field data. All of the shocks in the second group are quasi-perpendicular with B,n = 72-75 • . This is based on the MVA derived shock normals as they give consistent values across the three closely spaced shock crossings. When there are no data gaps present, all three shock crossings show a structure with an overshoot and downstream oscillations of varying magnitude. They also all have a very low Alfvén Mach number, estimated to be in the range 1.24-1.37. The range of Mach numbers and B, n for these shock crossings is very similar to the shock crossings previously studied by Balikhin et al. (2008), that is, the first group of shock crossings ( B, n = 76-87 • and M A = 1.19-1.23). They are also very similar to the very low     Pope et al. (2019). All of the shock crossings in the third group (blue region) are also quasi-perpendicular, but with values closer to 45 • and even lower Alfvén Mach number in the range 1.06-1.09. All of these shocks, apart from Shock 3f, also show a structure with an overshoot and downstream oscillations of varying magnitude. These two additional groups of shock crossings (apart from 3f) on 11 September 2006, can be categorized as being very low Mach number shocks dominated by kinematic relaxation. The original group of shocks investigated by Balikhin et al. (2008) occurred on the nightside flank (SZA = 112-116 • ) and at abnormally high altitude of 8.6-9.4 R V , compared to a model value of 2.79-2.94 R V at solar minimum. In contrast the second group of shocks extends observations to the dayside (SZA = 40-49 • ) and as a consequence a much lower altitude (2.78-3.63 R V ). This is still much higher than the nominal bow shock location of 1.44-1.52 R V . The third group of shocks covers the dayside terminator region (81-89 • ) and very high altitude of 9.72-11.4 R V . This is 5.0 to 5.4 times greater than the nominal bow shock altitude.
No suitable plasma data was available for the original group of shock crossings studied by Balikhin et al. (2008). However, for both of the second and third group of shock crossings, plasma data are available. This can be used to help verify the parameters derived from the magnetic field data and also provide additional insight. For the second group of shocks, ASPERA plasma data is available for the duration of the required period. The quality flag of this data is only excellent for the first shock crossing in the sequence and falls to satisfactory for the subsequent two crossings. This is sufficient to determine the upstream conditions between the first two shocks. Due to the low density and temperature and high magnetic field magnitude when compared to nominal solar wind conditions at 0.72 AU, is very low (0.02-0.03) and the proton-only Alfvén velocity is high (457 km/s). The upstream parameters used to calculate these values are given in Table S7. Calculating the proton-only Alfvén Mach number using the minimum variance shock normal and assuming a shock velocity of 0, leads to M A = 0.74 for the first shock. The assumption of zero shock velocity is incorrect, since for the shock to pass back and forth across the spacecraft over a short period of time requires a nonzero shock velocity. However, this value of M A is not particularly sensitive to assuming a shock velocity of 0. For example, a shock velocity of 20 km/s directed outwards (away from Venus) gives only a marginally higher M A = 0.78. The value of M A < 1 could be due to measurement errors associated with the low number density. Considering the excellent data flag, another reason could be a significant heavy ion component, which can act to lower the Alfvén velocity. Pope et al. (2019) have previously shown the contribution that a heavy ion component in the solar wind in the form of particles can play for kinematic shocks. At Venus it is well-known that pickup ions can be abundant, in particular O + (e.g., Barabash, Fedorov, et al., 2007;Fedorov et al., 2011). The plasma data do not indicate a significant heavy ion component, but an O + number density of 15% that of the protons would sufficiently lower the Alfvén velocity to give the required M A = 1.37 estimated for the first shock using the magnetic field.
For the third group of shock crossings (blue shaded region), the plasma data is available for a short time interval approximately 3 hr after the last crossing. Vech et al. (2015) previously calculated the magnetosonic Mach number for this magnetic cloud as 1.12. It is unclear which interval of plasma data was used to determine this value. Using the proton and magnetic field data measured 3 hours after the last shock, gives M A = 1.41. The upstream parameters used to calculate this value are given in Table S7. Alternatively, using the same proton data but the higher magnetic field magnitude at the shock crossing locations, gives M A = 1.03. This is consistent with the Alfvén Mach numbers of 1.06-1.09 estimated for these distant shocks using the magnetic field data. The calculation of M A = 1.03 at the shock location assumes that the proton velocity and density is comparable to that measured 3 hr later and does not take into account the normal incidence frame transformation. However, it does provide an indication of the very low Mach number nature of the magnetic cloud around the time of these shocks. The value of is 0.08 if the plasma data measured 3 hr after the third group is used. It falls to 0.04 if the magnetic field data at the last crossing in the third group is used. The values of calculated for both groups is consistent with the assumption of a very low and with < 0.1 during the shock observations at the Earth (Pope et al., 2019).

16-17 October 2011
Another prolonged magnetic cloud occurred during an ICME that arrived at Venus on 16 October 2011. The magnetic cloud commenced at some point while Venus Express was within the induced magnetosphere of Venus. The spacecraft subsequently detected a group of thirteen shocking crossings on its outbound and then seven shock crossings on its inbound trajectory during the magnetic cloud. The first group occurred POPE 8 of 16 on the flank (SZA 96-123 • ). All of these shock crossings exhibit a very low magnetic compression and are marked by the green shaded region in Figure 1b. The second group occurred on the day side and are marked by the blue shaded region in Figure 1b. All of the shock crossings in the second group have a very low magnetic compression and downstream oscillations of varying magnitude. The important magnetic field derived parameters for these shocks are shown in Table 1 and their magnetic profiles are shown in more detail in Figure 3.
The first group of shock crossings start on the night side flank at an altitude slightly above the model altitude and progress to close to the terminator at an altitude much higher (just over five times higher) than the model value. The earlier crossings in the sequence start off as quasi-parallel shocks, with both the minimum variance and coplanarity normal agreeing reasonably well and proceed to become quasi-perpendicular shocks. The minimum variance and coplanarity normal do not agree as well later in the sequence, with the most likely reason the presence of numerous local rotations in the solar wind magnetic field. The transition from quasi-parallel to quasi-perpendicular shock is consistent with the observed slow rotation of the field in the magnetic cloud over this interval. Only crossing 1j, 1l, and 1m are definitively quasi-perpendicular (both shock normals are >60 • ) and have a sufficiently long period in the downstream region to observe the presence of an overshoot and some indication of downstream oscillations due to kinematic relaxation. However, the interval of data present in the downstream region is still reasonably short, so that in some instances less than a single period of slowly oscillating field can be observed, for example, Shock 1j. The most interesting shock crossings in terms of observation of kinematic relaxation are those in Group 2. These consist of seven closely spaced shock crossings much closer to the subsolar point. They occurred at SZA 50-60 • and at an altitude 1.7-2.4 times higher than the model altitude. This further adds to the SZA at which this type of shock has been observed. All of these shock crossings are well defined as quasi-perpendicular and with generally good agreement between the two shock normal directions. The coplanarity normal in particular does not deviate much across all seven crossings. As the planet is approached the estimated Mach number gradually increases from 1.24 to 1.56, as would be expected for a more compressed induced magnetosphere. All of these shock crossings have well-defined downstream oscillations, apart from crossing 2b which has a data gap across the ramp and into the downstream region. The absolute magnitude of the downstream oscillations is approximately the same for all of these shock crossings (2.2 ± 0.6 nT), but relative to the size of the shock ramp their magnitude decreases from 0.3 to 0.1.
For the first group there is no simultaneous plasma data. However, with the second group there is simultaneous plasma and magnetic field measurements. This allows the upstream conditions to be resolved. Within this second group there are two longer solar wind intervals with excellent quality flag (the time intervals between Shock Crossings 2b and 2c and Crossings 2f and 2g). The plasma data upstream of Shock 2a is not used due to the lower quality flag and the interval between 2d and 2e is too short compared to the sample time of the plasma data to extract any meaningful measurements. These plasma data intervals are used together with the simultaneous upstream magnetic field averages for Shock Crossings 2c, 2f, and 2g to calculated some of the important shock parameters. Shock crossing 2b is not considered due to the data gap. The upstream parameters used to calculate the values are given in Table S8.
All three shocks are low with values of 0.08 for Shock 2c and 0.1 for Shocks 2f and 2g. The Alfvén Mach numbers are calculated using a shock velocity of 0 km/s and the coplanarity shock normals. This gives values of 1.28, 1.22, and 1.44 for Shock Crossings 2c, 2f, and 2g, respectively, which is 7%, 11%, and 9% smaller than the Mach numbers estimated using the magnetic field. The spacecraft velocity projected onto the shock normal is 3-5 km/s, indicating that a relatively small shock velocity would be required to create the oscillatory back and forth motion across the spacecraft trajectory. Including a shock velocity of ±20 km/s leads to a small 4-6 % change in the estimated Mach number, indicating that other factors might also contribute to the difference. Other than uncertainties in the measurements used and the contribution from the shock velocity, another reason for this difference would be the presence of heavy ions. The Venus Express plasma data indicates a very small and fluctuating component of O + (<3% by number compared to the proton density). Taking into account 1-2% O + increases the Alfvén Mach numbers to the values of 1.37, 1.36, and 1.56 estimated from the magnetic field. This indicates the sensitivity of very low Mach number calculations to the presence of even a small amount of heavy ions and the role that they might play.

5 November 2011
On 5 November 2011 Venus interacts with an ICME that leads to the detection of a bow shock with a large magnetic compression of 3.44 on the inbound trajectory of Venus Express. This strong shock and associated effect of the ICME on the Venus interaction region has been studied by Dimmock et al. (2018) using a combination of observations and hybrid simulations. Within the ICME a large magnetic cloud with a peak measured magnetic field of 50 nT commences at some point while the spacecraft is in the magnetosheath or induced magnetosphere and continues until the end of the day. During the magnetic cloud phase Venus Express crosses the bow shock seven times at SZA 123-117 • and altitude 7.79-9.11 R V . These shock crossings are marked by the green shaded region in Figure 1c. Figure S3 shows the profile in the magnetic field data and Table S3 contains the magnetic field derived parameters for the individual shock crossings. All of these shock crossings have a low magnetic compression leading to estimated Alfvén Mach numbers of 1.15-1.26. The altitude of all of the shock crossings are considerably above the model altitude at the respective SZA, with a tendency for this difference to increase as the Mach number reduces. All of the shock normals derived by the coplanarity theorem lie within 40 • (60% lie within 20 • ) of each other. However, despite this reasonably good agreement, B, n ranges between 11 and 67 • due to numerous short and long interval rotations in the magnetic field during the magnetic cloud. Three of these shocks have B, n ≥ 50 • and two of these (1a and 1d) show clear evidence of an overshoot and downstream oscillations of varying magnitude. No plasma data is present at or near the interval in which these crossings occur.

19 November 2011
On 19 November 2011 Venus interacts with an ICME which reaches Venus while Venus Express is in the solar wind. The magnetic cloud phase commences shortly before the spacecraft crosses the Venus bow shock on its inbound trajectory and continues until the end of the day. The spacecraft detects a sequence of 23 shock crossings associated with a very dynamic shock motion over an interval of 1 hr and 40 min on its outbound trajectory. These shock crossings (Shock Group 1) are indicated by the green shaded region in Figure 1d. Figure S4 shows the profile in the magnetic field data and Table S4 contains the magnetic field derived parameters for the individual shock crossings. These crossings occur from SZA 130-121 • and at an altitude of 6.75-8.67 R V . The magnetic compression gradually drops across the interval, such that the estimated Alfvén Mach number falls from 1.7 to 1.3. All of the shocks are quasi-perpendicular for both shock normal calculations with B, n ranging from 61-81 • (average of the two values for each shock crossing). The shock crossings have a tendency to become more perpendicular through the interval. When the downstream interval for each crossing is sufficiently long to allow this region to be investigated, clear downstream oscillations only become evident later in the sequence from Shock 1n onward. No plasma data with good/excellent quality flags is available for this interval.
Venus Express subsequently encounters six additional shock crossings between 19:00 and 21:15 UT. These shock crossings (Group 2) are indicated by the blue shaded region in Figure 1d. The important magnetic field derived parameters for these shocks are shown in Table 1, and their magnetic profiles are shown in more detail in Figure 4. They occur over an interval near the peak magnetic field in the cloud and from SZA 101-96 • and altitude 11.97-11.92 R V . The estimated Mach numbers of these shock crossings is very low and in the range 1.01-1.06. To our knowledge, these are the lowest Mach number shocks that have been observed at Venus. The altitude of all of the shock crossings on 19 November 2011 are considerably above the model altitude at the respective SZA, with a tendency for this difference to increase as the Mach number reduces. The ratio of observed to nominal model altitude for the lowest Mach number shocks in Group 2 is just over 5, which is consistent to the other days when the Mach number is just above 1. The shock normals indicate that they are all oblique to quasi-parallel with a tendency to become more parallel through the interval, apart from crossing 2b which is quasi-perpendicular. The change in shock normal for this shock crossing is due to a small rotation in the field in the downstream region just before the spacecraft crosses the shock back into the magnetic cloud. The quasi-perpendicular Shock Crossing 2b has B,n = 66 • (average of the two shock normal calculations) and an estimated Mach number of 1.06. It is also evident that the downstream region contains a clear sequence of oscillations, indicating the presence of kinematic relaxation. For the last three shock crossings in this sequence good/excellent quality proton data is available. Using the observed upstream proton measurements just after the last shock and taking into account 10% O + (the quality flag for O + fluctuates such that the number density varies significantly between 0 and approximately 80%) the Alfvén Mach number of the bulk flow is estimated as 1.01 and as 0.04. These are only approximate values due to the low sample rate of the data and errors associated with determining such low and marginal values.
However, they do serve as an indication of the very low Mach number and nature of the magnetic cloud. The upstream parameters used to calculate these values are given in Table S9.

16 June 2012
On 15 June 2012 Venus Express detected the leading shock of an ICME while in the solar wind. The magnetic cloud phase commenced at about 19:30UT and continued into the following day. Venus Express detected 14 shock crossings on its outbound trajectory from SZA 137-125 • and altitude 5.43-7.83 R V . These shock crossings are indicated by the green shaded region in Figure 1e. Figure S5 shows the profile in the magnetic field data and Table S5 contains the magnetic field derived parameters for the individual shock crossings. All of these shocks are quasi-perpendicular with all but one B,n = 72-84 • (average values from the two shock normal calculations) and 60% have B, n ≥ 80. The one exception (Shock Crossing 1j) has B,n = 65 • . As such, these are the closest to "perpendicular" very low Mach number shocks observed in this study. All of these shocks have a very low estimated Mach number, which starts at 1.44 and 1.47 for the first two shocks and gradually falls through the interval to 1.24 for the last shock. The altitude of all of the shock crossings are above the model altitude at the respective SZA, with a tendency for this difference to increase as the Mach number reduces. Most of these shocks show evidence of downstream oscillations or an overshoot, but some show evidence of neither. However, the sampling interval is relatively long with respect to the time required to cross the shock ramp, which might inhibit the possibility of clearly observing any downstream oscillations. Plasma data are available for most of this interval, but excellent/good quality proton data are only available in the region upstream of the first shock. For this interval and using the upper limit of measured oxygen contribution of 4% by number (the oxygen data varies from bad to good in this region), M A = 1.45 for the bulk solar wind flow. This Mach number agrees very well with 1.44 estimated from the magnetic compression of the first shock. The value of = 0.20 is notably larger when compared to the shock crossings observed on the other days. The upstream parameters used to calculate these values are given in Table S10.

28 October 2013
On 27 October 2013, just before midday, Venus Express detects the leading shock of an ICME. The ICME continues into 28 October and a magnetic cloud phase commences around the time that the spacecraft crosses the Venus bow shock on its inbound trajectory. The magnetic cloud phase continues until the end of the day, but the magnetic field gradually weakens to a magnitude representative of the undisturbed solar wind at 0.72 AU. On the outbound trajectory, Venus Express encounters three closely spaced bow shock crossings at SZA 114 and altitude 4.20-4.26 R V . These shock crossings are indicated by the green shaded region in Figure 1f. Figure S6 shows the profile in the magnetic field data and Table S6 contains the magnetic field derived parameters for the individual shock crossings. These crossings have a reasonably low magnetic compression leading to estimated Alfvén Mach numbers of 1.70-1.85. The altitude of all of the shock crossings are approximately 50% higher than the model altitude at the respective SZA. The shock normal determined for the three shock crossings using both methods agree reasonable well, such that B,n = 53-62 • . This indicates that the shocks are quasi-perpendicular, but toward the lower end of the possible range of B, n . The spacecraft spent sufficient time in the downstream region of only the first crossing to detect a clear set of downstream oscillations. For this first shock crossing there is some indication of downstream oscillations, but they are of low magnitude and only last two to three wave periods. The magnetic field in the magnetic cloud is not particularly large at the time of the shock crossings. This would reduce the Alfvén velocity and raise . Unfortunately, the proton data that is available has low quality flags, preventing a direct calculation of these values.

Statistical Results, Analysis, and Discussion
When plasma data are available for these shock crossings, the calculated Alfvén Mach number is generally in good agreement with that estimated from the magnetic field data. This gives confidence in using the estimated Mach number for further analysis of these shocks. It also indicates that , the ratio of kinetic to magnetic pressure, is low for these shocks. The calculated values are ≤ 0.1 for all but one value, which is = 0.2. Figure 5 plots B, n (average of the coplanarity and minimum variance derived values) against the estimated Alfvén Mach number. Each shock is plotted as a symbol which represents one of four conditions (downstream oscillations present, no downstream oscillations present, overshoot present and potential downstream oscillations, overshoot only). Figure 5 shows that only quasi-perpendicular very low Mach The transition from this region to more quasi-parallel and higher Mach numbers is indicated by the shocks with potentially some evidence of downstream oscillations, or with only an overshoot. The existence of these transition regions provides good evidence that kinematic relaxation is only clearly observable for very low Mach number quasi-perpendicular shocks. Evidence of just an overshoot for these shocks is likely to be due to the formation of less than one period of downstream oscillations, that is, a situation in which the oscillations are quickly damped. A higher damping rate for smaller values of B, n would be consistent with theory (Gedalin, 2015). However, a longer wave train is expected for higher magnetic compression. The reason for the clear transition observed in Figure 5 could be that the upstream is greater for these shocks. Since cannot be consistently measured for all of the shocks in this study, its effect on this transition cannot be verified.
Additional evidence of this transition from lower to higher Mach numbers can be seen in some of the sequences of shock crossings on each of the days. All of the shock crossings in Shock Group 2 on 17 October 2011 are quasi-perpendicular with very similar B, n , but the Mach number increases throughout the sequence. As seen from Figure 3, both the magnitude of the oscillations with respect to the ramp magnitude and the number of observable periods of the oscillations decreases throughout the sequence. When it is available the plasma data indicates that does not change significantly across these shocks (for Shock 2c = 0.08, Shock 2f = 0.10, and Shock 2g = 0.10). This indicates that the observed changes are due to the increase in Mach number and are not due to changes in . A second example is Shock Group 2 on 19 November 2011. All of these shocks have a similar very low Mach number and all are quasi-parallel, apart from Shock 2b. As seen from Figure 4, it is only the quasi-perpendicular Shock 2b that has downstream oscillations present.
When calculating the Alfvén Mach number from the plasma data, it was found that the contribution from heavy ions in the form of O + needed to be included to get a match to the values estimated from the magnetic field data. Pope et al. (2019) showed the role of heavy ions in forming the downstream distribution. In the case of the Earth, particles in the magnetic cloud were the source of heavy ions. It was shown that they contribute to the pressure balance by providing both a fixed and oscillating component to the dynamic pressure. At Venus there will still be particles present in the magnetic cloud and likely of a similar amount to that investigated by Pope et al. (2019). However, pickup ions are also likely to have a similar effect and ASPERA data do indicate a potentially significant population of O + . Due to the dependence of the wavelength of these oscillations on the ion gyrofrequency, the observed period will be 1/16 that of the protons. Therefore, it is unlikely that such a long period will be observable in the data due to insufficient time spent in the downstream region. The largest number of oscillations observed due to protons is 9, that is, approximately half the wavelength of the O + oscillations. The two most promising candidates for observation of oscillations due to O + are Shock 3c during 10-11 September 2006 and Shock 1n during 16 June 2012. Both of these shocks have eight to nine clear downstream oscillations which are superimposed onto a much slower change. The frequency of this slower change would be consistent with approximately 1/16 that of the higher frequency proton oscillations. It should be noted that the altitude of the shocks in this study are higher than what is normally considered as suitable for significant levels of pickup ions to be present. Wei et al. (2017) has investigated Venus pickup ions at higher altitudes and found associated O + above the nominal bow shock altitude. The period studied in Wei et al. (2017) would include VEX orbits for some of the shocks in this study, indicating that higher altitude pickup ions should be possible. There is insufficient data to reliably determine whether the contribution of O + to the shocks, is more significant in the hemisphere associated with a higher abundance of O + and due to the positive solar wind motional electric field. In terms of potential mechanisms leading to their presence, one notable feature is that the very low solar wind density during these observations could lead to escape of exosphere oxygen to higher altitudes than normal, before ionization by impact and charge exchange processes. The very low Mach number shocks identified in Venus Express data cover SZAs from 40 • through to 137 • . Evidence of kinematic relaxation for quasi-perpendicular shock geometry is also observed throughout this range of SZA. This indicates that when the solar wind conditions are suitable, any location on the Venus bow shock can form a structure in which kinematic relaxation is the dominant energy redistribution mechanism. Figure 6a shows the estimated Alfvén Mach number plotted against the ratio of the observed to solar minimum model bow shock altitude. The observations are plotted as stars and have been binned according to SZA, which is indicated by the different colors. It shows the tendency for the bow shock to move to a higher altitude as the Mach number falls, which is consistent with previous studies (Chai et al., 2014;Russell et al., 1988). However, the effect is more pronounced when M A ≲ 1.4. This is below the range previously studied. Above this value the altitude increase is less than a factor of two, but increases to greater than five as M A = 1 is approached. Chai et al. (2014) fit a conic section model to shocks identified in VEX data at solar minimum. This showed a linear relationship between terminator altitude and the upstream magnetic field magnitude, which would be consistent with a nonlinear relationship with the Alfvén Mach number. The very low Mach number shocks in this study have B u ≈ 20-50 nT, which places them toward the upper range and above the values investigated by Chai et al. (2014). Extrapolation of the trend identified by Chai et al. (2014) to the upper end of the range studied here, gives a terminator shock altitude of approximately 3R V , significantly below the upper values observed in this study. Chai et al. (2014) also showed that the shock altitude did not depend on solar wind velocity or dynamic pressure. However, this does not mean that the plasma density, included as part of the dynamic pressure, does not have a role in determining the shock altitude. Since the Alfvén Mach number is a function of the square of the magnetic field magnitude and the inverse of the plasma density, the very small density values observed in this study can significantly increase the Alfvén Mach number and could help explain the observed trend. Unfortunately for the shocks observed, there is an insufficient number of accurate upstream density measurements to accurately verify this.
The grouping of the colors of the stars in Figure 6a occurs due to the tendency to observe a group of multiple shock crossings across a small time interval. Despite this grouping, it is evident that the SZA does not appear to have a noticeable affect on the abnormal increase in shock altitude as the Mach number decreases. For example, the shocks observed at SZA ≤ 60 • and SZA > 120 • had a similar range of Mach numbers but show broadly similar increases in altitude. The shocks observed at 100 • < SZA ≤ 120 • are split into two groups with lower and higher Mach numbers and their relative increase in altitude is also consistent with the general trend. Figure 6b shows the same data but plots the Alfvén Mach number against the SZA, with the data binned according to the ratio of the observed to solar minimum model bow shock altitude. This also shows the tendency for the altitude ratio to increase as the Mach number falls. It also highlights that the highest relative altitudes are observed closer to the equator. However, the split either side of the terminator for the 1.5 < A/A m ≤ 2.5 data (light blue stars) indicates that this feature might be due to the limited observations. In fact, the highly elliptical polar orbit of VEX will bias observations of high altitude bow shock crossings to the polar terminator regions.
To determine if the general trend of increasing relative altitude as the Mach number falls is representative of all SZA, the conic section bow shock model  given by Equation 1 has been fitted to the data at different Mach numbers. The simple conic section bow shock model used by  has been chosen as it only requires two parameters (eccentricity and terminator altitude L T ) and thus can be estimated using a minimum of two data points. The use of a small number of data points can compromise the accuracy of the estimated model due to the uncertainty in the measurements. However, the objective here is to assess the general trend of the shock location as the Mach number changes, not to determine a high-accuracy bow shock model for future predictions. For each model calculation, a minimum of two data points were selected which spanned a Mach number range of 0.02. To ensure that the vertex of the conic section is at the subsolar point, a mixture of dayside and flank shock crossings were selected. The resulting seven Mach numbers chosen for the model (1.07, 1.09, 1.24, 1.34, 1.37, 1.41, and 1.55) arise due to the available data and these restrictions on data selection. The full set of data used to determine the models, together with the resulting model parameters and a plot of the relative locations of the models in relation to Venus as included in supplementary material Tables S11 and S12 and Figure S7, respectively.
The ratio of the Mach number-dependent model altitudes to the model altitudes at solar minimum is plotted in both Figures 6a and 6b. In Figure 6a the model altitudes are plotted as lines, with the colors corresponding to the SZA bins, that is, 50 • to 130 • at 20 • intervals. These lines are closely spaced and show the same overall trend, indicating that the relationship between relative bow shock altitude and Mach number is not a function of SZA. In Figure 6b the relative bow shock altitude is indicated by the contour plot, which is consistent with the observations which overlay it. This contour plot indicates that assuming the bow shock can be described by a conic section, as it can be a solar minimum, the relative altitude increase as the solar wind Mach number becomes very small is not significantly affected by the SZA, that is, similar increases in shock altitude are seen across the observed SZA from 40 • through to 140 • . At solar minimum  found that the conic section model was a good fit up to a SZA of 117 • . For higher SZA a Mach cone was found to fit the data better. This might be the reason that the conic section models start to show a small deviation from the general trend above approximately 120 • in Figure 6. However, the deviation is small and the overall trend is still in line with that seen at smaller SZA.

Conclusion
Very low Mach number quasi-perpendicular shocks in which the main energy redistribution mechanism is through the kinematic relaxation of nongyrotropic downstream ion populations have been previously observed in limited studies at Venus , Earth (Pope et al., 2019), and in interplanetary space (Goncharov et al., 2014;Kajdič et al., 2012;Russell et al., 2009). In this study a thorough survey of Venus Express data during suitable solar wind conditions (i.e., the magnetic cloud phase of an ICME) is conducted to identify the majority of such shocks observed during the entire Venus Express mission. These shocks are then analyzed in terms of the fundamental parameters. The main results of this study can be summarized as follows:

10.1029/2020JA028256
Mach numbers is indicated by the shocks with potentially some evidence of downstream oscillations or only an overshoot. Evidence of just an overshoot is likely to be due to the formation of less than one period of downstream oscillations, that is, a situation in which the oscillations are quickly damped. 4. When Venus Express plasma data were available, it generally supports the estimates of fundamental parameters made using the magnetic field alone, in particular the Alfvén Mach number. Since the proton density is usually very low in magnetic clouds, certain calculations such as the Alfvén velocity, are very sensitive to the inclusion of the effect of heavy ions. In this case the contribution from O + , most likely occurring as pickup ions, needed to be included to provide good agreement. 5. The shock crossings which show kinematic relaxation, are observed across a range of SZAs from 40 • through to 130 • . This indicates that it is likely that all locations of the Venus bow shock can form a shock structure in which kinematic relaxation is the dominant energy redistribution mechanism. 6. The altitude of the observed shocks are generally considerably higher than the Venus model bow shock at solar minimum. The increase in shock altitude is correlated with a reduction in Alfvén Mach number. This is consistent with previous results (Russell et al., 1988). However, the increase is much more pronounced for M A ≲ 1.4, which is below the range studied by Russell et al. (1988) and does not appear to be affected by the solar zenith angle of the shock location.