First, we estimate the background magnetic field intensity Bbg using a low-pass Butterworth filter as in for Venus and for Mars. We adopt a 2 min window (fband=1/120 Hz), a passband ripple of 2 dB and a stopband attenuation of 20 dB.

Criterion 1 ensures that the structure is compressional with an absolute B-field fluctuation of: 2Δ|B|Bbg=|B|-BbgBbg≥0.15; that is, magnetic field fluctuations are larger than or equal to 15% of the background field. To obtain criteria 2–4, we then perform a moving minimum variance analysis noted MVA; see with a 15 s moving window, which is about the width of the larger MM structures found at Mars , and a shift of 1 s. From the MVA, we define two angles: ΘmaxV (ΦminV) is the angle between the maximum (minimum) variance eigenvector direction and that of the background magnetic field. A small (large) enough angle denotes also a compressional wave with propagation in the direction perpendicular to B , as expected and shown in the kinetic MM simulations of . We use also the ratio of the maximum-to-intermediate eigenvalues λmax/λint (intermediate-to-minimum eigenvalues λint/λmin) to further constrain the shape of the variance ellipsoid into a cigar-shaped one quasi-linear polarisation; seeand references therein.

At this stage and using Criteria 1–4, the total number of detected 1 s duration MM-like events in the 1 November 2014–7 February 2021 MAVEN dataset was 2 046 533. Two examples of B-field-only detections are presented in Fig.  (higher solar activity conditions) and Fig.  (lower solar activity) as events in green and grey. We will come back to discussing them in Sect. .

The detection criteria 1–4 of Table  may not exclude structures which may closely resemble MMs but, in effect, are not.

Some structures, appearing as highly compressional, can be generated upstream of and in and around the bow shock surface; they testify to the variability and complex structure of the Martian shock. These non-MM structures include upstream waves such as proton cyclotron waves (PCWs) and ULF foreshock waves seefor observations and simulations. Because PCWs are linked to pickup ion processes, they are usually observed at Mars when the exosphere is particularly extended and neutral hydrogen densities significantly increase, that is, around Ls=270∘ (just after perihelion, Lsph=251∘) and during the dust storm season for Ls=[135–225∘] , in the whole upstream region. PCWs are non-compressive in nature, which is ruled out by Criteria 1–2. Foreshock waves, on the other hand, are linked to back-streaming ions in the quasi-parallel shock. Both types of upstream waves are generated with a circular polarisation, characteristics which are filtered out by our Criteria 3–4 ensuring quasi-linear polarisation. However, due to relative phase velocities with respect to the solar wind plasma, they may become steepened in the vicinity of the shock seefor comets and Mars, respectively, their nonlinear growth stage tending towards more elliptical and quasi-linear polarisations, and with a rotation of the magnetic field across the structure in the steepened edge of the structure . These nonlinearly evolved waves may sometimes be captured as MM-like by our B-field-only detection Criteria 1–4.

Other types of structures, such as the so-called “fast-mode” type waves, can have similar compressional and polarisation characteristics as MMs; they occur deeper in the magnetosheath and in the magnetosphere. For example, fast-mode magnetosonic waves with quasi-linear polarisations have been observed at Mars downstream of the IMB , in a manner reminiscent of similar waves found downstream of the magnetic pileup boundary at comet 1P/Halley , each time with MM waves in B–N antiphase situated just upstream of the boundary. In contrast, the magnetosonic-type waves found by close to the ionosphere and generated at the IMB by solar-wind-driven pressure pulses in the foreshock region are usually elliptically polarised. However, as before, certain associated steepened wave packets with strong fluctuations (Δ|B|/Bbg≥0.1) could also display a more quasi-linear polarisation and be potentially kept as MM-like candidates if we only apply Criteria 1–4 of Table .

In all previous cases, magnetic field intensity and plasma density will typically be in phase, as opposed to the expected MM behaviour . However, this information is sometimes neither available at the desired (high) time resolution nor practical to derive as in large statistical surveys. We suggest here two mitigation strategies for our B-field-only investigation: (1) making sure that the magnetic field across the structures does not rotate more than about ±10∘, as theoretically predicted for MMs and in agreement with past observations , and (2) restricting the detections to magnetosheath conditions only and excluding the region around the bow shock to avoid foreshock transients and upstream waves.

Strategy (1) constrains the detected events to behaviours more reminiscent of MMs: we apply Criterion 5 of Table  which ensures that the magnetic field does not rotate significantly across a MM-like structure. From the 1 s magnetic field vector in MSO coordinates, thereby ignoring higher-frequency features which are not at the same scale as the ion-scale candidate mirror-mode structures, magnetic azimuth and elevation angles are defined as 3az=arctan⁡By/Bxandel=arctan⁡Bz/Bx2+By2. First, within our original database of candidate MM-like detections, we define so-called “detection periods” containing all events consecutively detected, with two periods separated by a minimum of 30 s between one another. We discard periods of isolated singular events, defined as a detected event lasting no more than 1 s within a period. In this way, some of these discarded isolated events (DIEs) are not part of the usual quasi-periodic train of MM structures, which would result in more than one detection within a period. They are instead more reminiscent of the so-called “linear magnetic holes” (LMHs), in the original definition of , structures that we wish to filter out from the database. In contrast, if multiple detections occur consecutively within a whole period, these events may represent in reality either a large MM-like structure or a train of shorter MM-like structures, depending on the length of the period they belong to. The particular value of 30 s between detection periods was chosen empirically as double the length of the longest MM structures found at Mars or Venus see, for example. We then estimate how much azimuth and elevation angles fluctuate at the detected position of the candidate event by calculating their running standard deviation 〈σ(az,el)〉 over a 2 min sliding interval, keeping only those events where 〈σ(az,el)〉 is less than 10∘ for each angle see.

Complementarily, strategy (2) may make use of the position of the bow shock crossing in the spacecraft data and ignore the detected events in a range of radial distances around it (or equivalently, in a range of durations around the time of the crossing). For this part, we use the automatic bow shock detection technique explained in , who used a predictor–corrector algorithm based on magnetic field measurements only at Mars. Their list of automatically detected crossings is freely available . It discriminates between (i) crossings from the solar wind into the magnetosheath and from the magnetosheath into the solar wind and (ii) quasi-perpendicular (noted “Q-⟂” in the following) and quasi-parallel (noted “Q-||') bow shock crossings. The latter aspect is useful for investigating how the shock configuration impacts the spatial distribution of MM-like structures (see Sect. ). It is important to note here that the technique of is biased towards the detection of Q-⟂ crossings as it is based on the classic B-field signature of the Q-⟂ shock, with clearly defined foot, ramp and overshoot. Between 2014 and 2021 of MAVEN operations, recorded a total of 11 967 Q-⟂ and 2962 Q-∥ “clear” crossings, discarding 1,615 crossings where the shock signatures were difficult to automatically ascertain. In addition, knowing where the shock is can be used to automatically ensure that the structures are located in the magnetosheath. In turn, this naturally fulfils Criterion 6 in Table , where the background magnetic field Bbg needs to be 2 times higher than the average, but highly variable, interplanetary magnetic field Bimf.

In practice, knowing when the bow shock crossing took place, we first identify in spacecraft ephemerides the portion of the orbit in the solar wind as opposed to inside the bow shock structure. Then, we remove all MM-like events that are closer than 0.075RM (∼250 km) to the shock position, assuming that those are shock substructures, situated immediately upstream, at or downstream of the shock itself. This value was chosen empirically from a subset of orbits with magnetic fluctuations that were incorrectly captured by the original detection algorithm (see for an illustration Fig. ).

Table  displays the number of MM-like events detected when incrementally applying the criteria of Table , together with the two mitigation strategies discussed above. From the original ∼2×106 MM-like events detected in Sect. , strategy 1 (using Criteria 5 and ignoring 23822 isolated events) removed 85% of the total number of events: we end up with only 283792 MM-like detections. Criterion 5 is the criterion that removes most of those events due to the harsh constraint on the stability of the magnetic field angular fluctuations within trains of MM-like events. It shows moreover that Criterion 5 is of utmost importance to remove from the database candidates that are not linearly polarised. Applying strategy 2 to filter out shock substructures and solar wind magnetic holes (based on Criteria 6 and an automatic estimate of the position of the shock) removes another 107751 events, resulting in a total of 176041 MM-like events in our final database.

Figure  (high solar activity conditions) and Fig.  (low solar activity) show examples of MM-like detections, with total magnetic field and Criteria 1–4 of Table . Detections matching Criteria 1–4 in coincidence are shown as vertical grey and green lines, with the green lines marking the final detections with the removal of false positives (Criteria 1–4 and 5–6 in coincidence).

The events of 8 September 2016 (Fig. ) occur at and in the wake of a predominantly Q-⟂ bow shock crossing, with angle θBn between the normal to the shock surface and the average interplanetary magnetic field (IMF) direction of about 80∘. Comparison of magnetic field and density variations is displayed in the bottom panel, with magnetic field (left axis, blue) and density (right axis, red) fluctuations being in antiphase when ΔB/|B| and ΔN/N have opposite signs. For the detections in grey in the ramp of the shock as initially captured by Criteria 1–4 but without removal of false positives, B and N appear mostly in phase at the cruder resolution of the SWIA instrument (here 8 s, with ΔB/|B| and ΔN/N having most of the time the same sign in the bottom panel). Together with the position within the shock ramp, this implies a false positive detection: further applying Criteria 5–6, they are correctly removed from the final database, leaving only the events in green. The events in green display a mix of B–N in-phase and antiphase behaviours (bottom panel), with some of the shorter detected events appearing in phase, a characteristic more reminiscent of fast mode-type (e.g. magnetosonic) waves; this is however difficult to unambiguously ascertain owing to SWIA's 8 s resolution which cannot capture these shorter magnetic field structures. Around 00:19:30 UT, some oscillations are clearly in antiphase, whereas before and after they seem to become in phase again. Closer to 00:30 UT, B–N fluctuations appear again in antiphase but are not captured by the detection algorithm, due to constraints on the linearity of the structures. This points further to the necessity of a more in-depth case study as in to ascertain the nature of the fluctuations captured by our algorithm and shows the limitations of a B-field-only algorithm to detect MM structures. From a general point of view, the temperature anisotropies responsible for the generation of MMs are expected to take place in the wake of a Q-⟂ shock see. Consequently, we expect such time series, together with those discussed in (24 December 2014 around 11:25 UT, also at high solar activity) or (26 December 2014 around 15:00 UT in their Fig. 1), to harbour “textbook” examples of MM occurrences.

In Fig.  (23 March 2019), detections occur in the solar wind region around 19:15 UT (not shown) and in the magnetosheath towards the IMB around 18:38 UT, behind a predominantly Q-|| shock (θBn≈25∘, noted “Q-||” in the following). Multiple crossings of the shock from the magnetosheath into the solar wind take place. The automatic estimate of the shock's location following pinpoints the last great magnetic field enhancement around 19:05 UT, whereas the magnetosheath-to-solar wind position is visually closer to 18:52 UT: this is due to the fact that automatic estimate choosing the last occurrence of a shock-like structure when crossing from the magnetosheath to the solar wind. In the solar wind after 19:05 UT, magnetic field and density seem mostly in phase: two “linear magnetic hole” (LMH) candidates (not shown) are first captured by Criteria 1–4. After they are removed from the database using Criteria 5–6, only MM-like events which are roughly in B–N antiphase (around 18:39 UT) are kept.

That said, other events deeper in the magnetosheath (around 18:33 and 18:36 UT) and part of the database in grey are also removed because they are isolated events: the second of these events around 18:36 UT presents a clear B–N anticorrelation. This shows that although the detection method using Criteria 1–6 appears quite apt at detecting regions where MM-like events are present and at removing events that are clearly not MMs, it may also ignore promising candidates (especially around 18:30–18:34 UT). Conversely, as already mentioned in Fig. , we expect the method to also keep events that are likely not MMs although situated in the magnetosheath but with B and N in phase. For illustration, isolated 1 s events, discarded in the final database, amount to 23822 additional events, which make about 14% of the final total number of events (see Table ). As a consequence, on this criterion only (isolated event), we estimate that the frequency of true MM-like detections in our final database could be underestimated by about 10%.

Another aspect is that several individual structures as part of trains of MMs are likely to have been missed by our B-field-only method; see, for example, in Fig. , around 18:35 UT where clear B–N anticorrelations are not captured even by the original detection method. This is an inherent caveat of any automatic detection method based on somewhat arbitrary thresholds and a possibly flawed estimate of the background field seewho used upper and lower quartiles of B to overcome some of this difficulty. Choosing laxer detection criteria would increase the likelihood of capturing such structures; however this would also be at the expense of increasing false positive detections.

Finally, we only consider in our study so-called “linear” MM-like structures; theoretically, single MMs are purely growing modes, which an elliptical polarisation would prevent. However, as shown in studies made in the Earth's magnetosheath , MMs can also be elliptically polarised in certain conditions. A discrimination between all polarisations states is difficult to achieve with B-field-only criteria and is outside of the scope of our study. Therefore, our method may miss a certain (but difficult to estimate) amount of structures which are MMs but are non-linearly polarised in nature.

Several tests have been made on reduced datasets to find the sweet spot of all criteria gathered in Table  see also the discussion in to keep as many confirmed MM-like structures as possible whilst avoiding other compressional structures that are not MMs. This points overall to the difficulty of characterising such structures reliably with magnetometer-only data. In all cases, after applying our mitigation strategies, the number of events that are captured is thus most likely a lower estimate of the frequency of linearly and non-linearly polarised MM-like structures present around Mars. As shown in Figs.  and , one way to mitigate these aspects is to check for the antiphase behaviour between B and N (at 4–8 s resolution), at the expense of the smaller MM-like structures. This is outside the scope of our current work which aims at evaluating the B-field-only method of detection; using the full plasma instrumentation on board MAVEN is thus left to a future study, in a way similar to the recent study of .

First, we apply the B-field-only criteria of Table  in coincidence to the whole MAVEN dataset, from November 2014 to February 2021. This yields the timing and duration of MM-like structures, noted tstruct and ΔTstruct, respectively. It is important to remark here that a full dip or peak of any given MM-like structure is usually longer than the total sum of 1 s detections for that structure. This arises from the fact that all the criteria of Table  must be simultaneously met to validate this as a detection. Consequently, ΔTstruct can only be an underestimate of the total duration of a structure. Following , we evaluate such an underestimate to at least 50%, based on visual comparisons in the subset of events presented in Fig. , Fig.  and in , who, with the same detection algorithm, captured 33 s out of a total of 77 s of visually identified MM structures.

Second, we define a spatial grid for the data accumulation. We start with spacecraft coordinates expressed in the Mars Solar Orbital (MSO) coordinate system (sometimes called “Sun-state” coordinates), where the XMSO axis points towards the Sun from the centre of Mars, and ZMSO points towards Mars' north pole and is perpendicular to the orbital plane defined as the XMSO–YMSO plane passing through the centre of Mars, with YMSO completing the right-hand triad. We then transform this system into aberrated MSO coordinates by rotating the XMSO–YMSO plane 4∘ anticlockwise around the ZMSO axis in order to account for the apparent “aberration” of the orbital motion of Mars with respect to the average direction of the solar wind . The new aberrated coordinate system, aligned with the average apparent solar wind direction, is noted XMSO′, YMSO′ and ZMSO′ (although ZMSO′=ZMSO by construction). For brevity, the +XMSO′ (-XMSO′) direction is sometimes referred to as “subsolar” (“antisolar”) direction in the following, although, strictly speaking, they should be referred to as “subsolar-wind” and “antisolar-wind” directions.

A two-dimensional cylindrical coordinate grid can then be created, with the abscissa i along the XMSO′ axis and the ordinate j defined as YMSO′2+ZMSO′2. The grid resolution is chosen so that, on average, the residence time of the spacecraft in each grid cell is large enough in all cells covered by the spacecraft's orbit. For example, for Mars (average planetary radius RM=3389.5 km) and applied to MAVEN 2014–2021 ephemerides, a resolution of ΔR=0.05RM ensures that the spacecraft residence time in each (i,j) cell is of the order of 500–1000 s (8–16 min) per grid cell for a typical Mars year of 687 Earth days; for a resolution of ΔR=0.1RM, average residence times are of the order of 5×104 s (∼14 h). The latter resolution of ΔR=0.1RM is thus chosen to maximise the statistical significance of the MM-like detections. Examples of spacecraft residence times with this grid resolution are shown in Fig. , left panels, for several consecutive Mars years.

We can now define the probability P of detecting MM-like events in a chosen spatial grid cell (i,j) as the total duration of the detected events ΔTstruct divided by the accumulated residence time of the spacecraft ΔTsc see also: 4P(i,j)=∑kΔTkstruct(i,j)ΔTsc(i,j), with k being the number of events found in each grid cell (i,j). In other words, P represents the percentage of observations containing a MM-like event at any given point in the magnetosheath. The accumulation of the events in the grid, including those crossing a cell boundary, is naturally taken into account using a bi-variate histogram accumulation.

An example of spacecraft residence time and MM-like detection probabilities is shown in Fig. a, b. Figure  is discussed further in Sect. . As mentioned above, because ΔTstruct is an underestimate of the total duration of MM-like structures in the magnetometer dataset, P is also underestimated: we thus refer to it as a “detection probability” of detecting MM-like structures.

To study how the distribution of MM-like structures varies during the time of the mission, we choose several controlling physical parameters, which are summarised in Fig. . We discriminate our results against the following.

Mars Year (MY): on average, one MY lasts 1.88 Earth years (687 Earth days). Because MAVEN started its scientific investigation in November 2014, we consider data from this date (after the autumn equinox of MY32) up to MY35 (ending on 7 February 2021). Precise start and end times for each MY were obtained from the equation of time of (their Eq. 14). For reference, MY32 = [31 January 2013–18 June 2015 12:34 UT], MY33 = [18 June 2015 12:34 UT–5 May 2017 11:45 UT], MY34 = [5 May 2017 11:45 UT–23 March 2019 11:32 UT] and MY35 = [23 March 2019 11:32 UT–7 February 2021 11:12 UT].

Figure  presents the PDF of the detected MM-like structures with respect to radial polar coordinate ρ=X′2+Y′2+Z′2 (panels a and d, in bins of 0.1 RM which is the resolution of our 2D distribution maps), EUV flux levels (panels b and e, in bins of 0.1 mW m-2) and Ls (panels c and f, in bins of 10∘). Each PDF is discriminated against high and low solar flux levels (panels a–c) and against Ls ranges (panels d–f, with Ls1–4 for NH spring, summer, autumn and winter) to illustrate the co-dependence of the studied parameters. Panels (b) and (f) represent the baseline statistics of IEUV and Ls parameters. They highlight the bins where the PDFs have larger values and hence the parameter has a good statistical coverage. For example, the sharp drop in PDF occurring for IEUV>3.4 mW m-2 in Fig. b is due to the smaller spacecraft orbital coverage above this threshold: this is also clearly seen in Fig. , where only MY32 and MY33 contribute to the statistics, with the threshold 〈I〉 just above the irradiance local peaks during MY34 and MY35. Successive Ls ranges are in contrast quite homogeneously distributed with relatively constant PDFs throughout (Fig. f), with the Ls2 range having the largest PDF overall. Note that in Fig. b and f, the finite width of EUV (single precision compared to the double precision IEUV threshold) and Ls bins results in expected overlaps at the borders between blue, red, orange and purple lines.

In Fig. a, d, we see a combined peak of the PDFs in the 1.2–1.3 RM bins, close to the position of the IMB Rss,imb=1.25–1.33 and Rtd,imb∼1.45; see. The PDF drops by almost half around 1.5 RM, roughly 0.2 RM ahead of the bow shock's variable subsolar position. Hence, a rather homogeneous ring of MM-like structures around the planet forms, centred on 1.2–1.3 RM, as already seen in Fig. . Overall, the distributions look very similar for all EUV and Ls ranges considered. However, we see two more prominent peaks of the PDF: one 0.1–0.2 RM inwards of the terminator standoff distance Rtd (true for all EUV conditions and Ls ranges) and the other 0.2 RM outwards of it (low EUV conditions, Ls2 contributing most, Ls4 the least due to reduced spatial coverage of MAVEN in these conditions). These two peaks correspond in Fig.  to the tail detections around 2.3RM from the centre of Mars and to the detection enhancement around the average shock position in the terminator plane around 3RM.

The co-dependence between EUV flux and Ls range is clearly seen in Fig. c and e. Obviously, aphelion and perihelion conditions mostly correspond to PDFs for low and high EUV solar fluxes, respectively. The main peak in the PDF appears in bin Ls=80–90∘ within the “Ls2” range (see also Fig. f), slightly after aphelion conditions. No peak in the PDF particularly stands out for Ls>180∘ and higher EUV fluxes, although a relatively lower PDF seems to occur around perihelion conditions. Complementarily, in Fig. f and e, range Ls2 (Ls=45–135∘, containing aphelion conditions) is dominated by low EUV fluxes, whereas Ls4 (Ls=225–315∘, containing perihelion conditions) is dominated by high EUV flux, with the first range strongly peaking at IEUV∼2.3 mW m-2 and the second range at IEUV∼3.3 mW m-2. Consequently, these two Ls ranges appear to be a good proxy of conditions driven by seasons only at an almost constant EUV flux, either low or high. The remaining two Ls ranges, Ls1 (Ls=315–45∘) and Ls3 (Ls=135–225∘), have similar distributions with respect to EUV flux and are thus more comparable to one another.

Overall, these preliminary results are consistent with past observations at Mars, be it with MGS or with MAVEN , who all noted that MM structures seemed to pile up against the IMB. This suggests that MMs, after being created upstream of their detection place, are convected down to it with the ambient plasma flow. Using the full MAVEN plasma complement and owing to the ambient plasma becoming less unstable to MMs the further away from the shock, the locus of generation of the MMs found by was inferred to be in the immediate wake of the Q-⟂ shock, a condition that seems to predominate in the 2014–2021 MAVEN dataset .

Figure  presents, for four MYs (panels a–d), the spacecraft residence time (left panels), the MM-like detection probability P (middle panels), and their relative difference (right panels) with respect to the total detection probability Ptot (taken from Fig. b), in the same format as in Fig. . Because MAVEN started observing late in Oct. 2014 (towards the end of MY32), the statistical coverage for MY32 is lower than for MY33–MY35, with a residence time in a grid cell on average about 3 times less than any other year. Similarly, because of the orbit being more compact around the planet during MY35 (panel d), the mean residence time in a given grid cell is significantly higher than for the other years. However, relatively high average residence times above 5 h ensure a good statistical coverage of the X′-Y′2+Z′2 spatial plane.

The detection probabilities are shown in percentages on the middle panels, when, as for Fig. , all grid cells for which ΔTsc<2 h are discarded to ensure an optimal statistics. For each MY, P reaches about 1 % at most. The higher probabilities appear close to the IMB in the magnetosheath at SZA angles ranging from the subsolar point (SZA≈0∘) to almost the terminator plane (SZA≈90∘), and in a lesser measure, around the bow shock's predicted position (in effect, in the wake of the true bow shock position). This is in line with our conclusions in Sect. . On the rightmost figure panels, negative (positive) percentages are represented by cold (hot) hues for which PMY<Ptot (PMY>Ptot).

As remarked before, interannual variability of P at Mars co-depends on EUV flux levels and, to a lesser degree, to exosphere variations (parameterised with Ls), although these latter effects are significantly damped over a full MY. For example, during MY32, the EUV flux levels were always high with MAVEN observing only at high Ls values Ls=[225–315∘]) as shown in Fig. . The predominance of red hues especially around the shock (rightmost plot of panel a) points to detections expanding outward under larger EUV fluxes, an effect following the well-known phenomenon of the shock's expansion into the solar wind in those conditions . However, due to the lower statistical orbital coverage during that year compared to MY33–MY35, it is difficult to draw further conclusions on the overall trend of this year's rate.

From MY33 to MY35, less and less detections are seen in and around the shock (less and less red cells on rightmost panels), with MM-like structures mostly confined to a narrow region lodged against the IMB (MY35, middle panels, even factoring the somewhat reduced spatial coverage due to MAVEN's altered orbit in 2019) and in the tail behind the terminator plane (MY33). The slight increase of EUV flux during the second part of MY35 does not seem to be enough to alter this general trend: in effect, global EUV levels for MY34 and MY35 are comparable. Looking more globally into the evolution of the number of detected MM-like events, the ratio of the number of detections to the time of residence of the spacecraft inside the shock steadily diminishes from MY33 to MY35 (from 0.143%, 0.128% to 0.113%; see Table ).

This behaviour mimics rather well the evolution of the average EUV flux during that time (Fig. ) and points to a modulating influence of the solar flux in the number of detections, and possibly their spatial distribution, throughout MY33–MY35. From Table , the fractional change between MY33 and MY34 and between MY34 and MY35 was 0.128%/0.143%∼0.113%/0.128%=0.89 (relative decrease in detection rates). During that time, the average EUV flux for MY33–MY35 was 2.99, 2.71 and 2.74mW m-2, leading to the rather similar fractional changes of 0.90 (MY33–MY34), whereas MY34–MY35 had a slight increase of 1.01. The influence of the EUV flux on MM detections is further investigated in Sect. .

As for the mission-wide results, the region containing most MM-like structures spans 2–3 grid cells on average, that is, 0.2–0.3 RM or ∼ 700–1000 km. At the subsolar point, this region fills a significant portion of the narrower magnetosheath. With the predicted position of the IMB from for the comparatively higher solar conditions encountered by MGS, MM-like structures seem to “leak” into the magnetosphere, although we cannot say for sure if these detections are in fact inward of the IMB or not. For example, an appreciable part of the detections seemingly present in the solar wind upstream of the average shock position are in fact just behind the actual shock, whose position continuously varies with the solar wind upstream conditions. A cursory examination of individual detections for a reduced dataset in December 2014 and September 2016 pleads in favour of this latter idea (see, for example, Fig. ).

At Mars, variations in EUV flux combine two main aspects: the solar cycle variations, on the one hand, and the variations of the EUV input due to the large eccentricity of Mars' orbit on the other (Fig. ). By modifying the global energy input to the Martian atmosphere–ionosphere–exosphere system, both aspects lead to variations in Mars' exospheric extent and ionisation levels and are among the key drivers of the bow shock and IMB positions .

More precisely, two combined effects are expected to take place with respect to the generation of MM structures. First, an increased EUV flux favours the expansion of the exosphere in the upstream solar wind, resulting in a swelling of all the plasma boundaries including that of the bow shock . Second, for a given static exosphere, an increased EUV flux also increases the local ionisation in the exosphere, thereby increasing the number of newly born ions and thus pickup ions created . In turn, these newly picked-up ions may contribute to heating in the perpendicular direction to the magnetic field, helping the plasma to become marginally unstable to the MM instability MMI <0; see. These two effects should lead to MMs becoming more frequent and extended in space for higher EUV flux levels, regardless of the nature of the shock.

To study how far this reasoning may hold, we now investigate in Fig.  how the EUV flux modifies the detection probability of MM-like structures. We consider two ranges: one for low EUV fluxes (IEUV<2.77 mW m-2, panel a) and one for high EUV fluxes (IEUV≥2.77 mW m-2, panel b), as previously defined in Fig. . To facilitate comparisons between the two conditions, we calculate in panel c the departure from the total detection probability, ΔPhi-lo/Ptot=Phi-Plo/Ptot, with Ptot being the total detection probability from Fig. b. A negative percentage (cold hues) implies in this way that Phi<Plo, whereas a positive value (hot hues) implies Phi>Plo.

On average, the spacecraft resides >20 h in a grid cell (left panels), with a very similar spatial coverage except in the solar wind upstream of the bow shock at the subsolar point, and in the deep magnetospheric tail in the antisolar direction. Thus we expect relatively more fluctuations in P in these regions. As previously adopted, a good statistics is ensured by discarding all grid cells for which the cumulated spacecraft residence time ΔTsc<2 h (about half a full orbit duration), which helps contain that effect. As the exosphere expands with increasing EUV flux, the obstacle to the solar wind flow grows in size, with the bow shock and IMB both swelling up. The swelling of the bow shock is illustrated in Fig.  by comparing the dashed shock curves of in panels a and b, representing “EUV low” and “EUV high” conditions, respectively, with the fixed curves of (“H19”) or those of (“E08”).

In Fig. c, we observe an outward displacement of the location of high detection probability P around the bow shock (red region for X′>0), corresponding to the outward displacement of the relevant boundaries with increasing EUV flux. This is consistent with the exosphere comparatively shrinking for low solar EUV flux, with pickup ion effects becoming less prominent around the shock , leading to less MM-unstable conditions there. Because Ls variations are averaged out during a full MY containing all four seasons, the EUV flux thus appears to be the main driver of this tendency. Moreover, the higher the solar activity and the EUV flux, the less events seem present in the deeper magnetosheath and upstream of the dayside IMB (slight dominance of colder hues there in panel c).

This is studied in more detail in Fig.  which presents the PDF of the relative difference between high and low EUV conditions (ΔPhi-lo/Ptot in Fig. c), first in the whole magnetosheath upstream of the average fitted IMB line ofpanel a and second in what we term the “deep magnetosheath” upstream of the average fitted IMB line and downstream of the nominal shock line ofpanel b, that is, the region between the two continuous black lines in Fig. c. The “deep magnetosheath” distribution is negatively skewed towards low EUV conditions (generalised extreme value [GEV] fit peaking at ΔPhi-lo/Ptot=-2%, skewness =-0.7, kurtosis =3.7), confirming our impression from Fig. c. In contrast, the “full magnetosheath” distribution is much more symmetric with a positive skewness (=0.15) and a larger kurtosis (=4.0) showing the presence of wider, more uniformly distributed tails. The positive skewness is due in part to the values above 60% corresponding to a geometric effect, that is, the outward displacement of the shock at large EUV conditions; when ignoring these large values, the corrected skewness becomes -0.62, close to the deep magnetosheath values.

On top of the solar activity-led expansion of the associated plasma boundaries, we thus note from Figs.  and that, on average (i) the detection probability in the magnetosheath is perceptibly higher for low EUV flux (〈P〉=0.121% on average per cell) than for high EUV flux (〈P〉=0.111%), especially in the near-subsolar magnetosheath and up to about SZA=45∘, and (ii) in contrast, detection probabilities appear higher in the tail for high EUV flux than for low EUV flux.

Following conclusion (i), we observe a perceptible decrease of the total detection probabilities with increasing EUV flux, when comparing them to the residence time NMM/ΔTinsc, from 0.133% to 0.120% (see Table , from low to high EUV flux). This is in part due to MY32, containing only high EUV fluxes and encompassing perihelion conditions, and for which we calculated noticeably smaller ratios than for other years. However, this conclusion is in direct disagreement with our first expectation that more pickup ions due to higher EUV fluxes would lead to the generation of more MM structures around Mars.

In that aspect, it is useful to compare these maps to similar ones found at Venus for MM-like detections and temperature anisotropy. At Venus, found higher detection levels in the magnetosheath for lower solar activity than at maximum activity. They also noted that the maximum of detection probability moved from just behind the shock at solar minimum down to the IMB at solar maximum. A similar evolution in the spatial distribution of the temperature anisotropy was recently presented by , with the maxima of anisotropy moving from SZA≈45 to SZA>60∘ between solar minimum and solar maximum conditions. In agreement with our findings here with conclusion (ii) at Mars, calculated comparatively higher anisotropies in the Venusian magnetospheric tail at solar maximum than at solar minimum. Finally, found also that the MM instability criterion was fulfilled significantly more often during solar minimum than during solar maximum conditions, in keeping with the higher detection levels at solar minimum. Our companion paper (Part 2) on the distribution of MM-like structures at Venus further discusses these aspects with the full Venus Express dataset .

Consequently, Mars and Venus qualitatively display a similar tendency towards higher MM detection probabilities in low solar activity conditions, a conclusion which appears in contradiction with the addition of pickup ions to the ambient plasma at higher EUV fluxes.

We suggest here two phenomena that likely play complementary roles in enforcing this apparently contradictory trend. First, have observed an enhanced turbulence due to the presence of so-called proton cyclotron waves (PCWs) becoming much more prominent the closer to perihelion conditions. This is in turn linked to the local plasma β, which plays a leading role in favouring the MM instability over the Alfvén Ion Cyclotron instability (typically for β≥1.5) as predicted by the theory of microinstabilities . In solar maximum conditions, the plasma β⟂ is on average lower than 1 in the solar wind and in the dayside magnetosheath, as shown by for Venus and for the solar wind. Consequently, the ion cyclotron instability may preferentially grow over MMs, resulting in significantly less MM detections in solar maximum conditions or for increased EUV fluxes. This would explain why large EUV fluxes (closer to perihelion conditions) have comparatively lower detection rates of MM-like structures, especially since PCWs have very different magnetic signatures (non-compressional, left-hand elliptically polarised and MVA direction at small angles to background B-field direction) and would not be captured by our detection criteria . Second, the distributions of pickup ions at Mars and comets are expected to be non-gyrotropic see e.g., which is known to modify the wave mode properties, the linear growth rate, the instability threshold, or even produce new wave modes that may consume the additional free energy seeand references therein. This may in turn favour other modes over MMs, although this specific question remains open. Further study of these two aspects is needed with the use of full plasma suite on board MAVEN; it is left for the future.

Dependence of MM-like structures on Martian season is presented in Fig. , with residence times on the left, detection probabilities in the middle and percentage departures from the total detection probability on the right, as before. In contrast to MY discrimination, the average residence times in a grid cell is quite similar for all four Ls ranges considered, with 〈ΔTsc〉∼12 h per grid cell, with the smallest residence time for Ls3= [135–225∘] (see also Fig. f and Table ). This is due to a combination of relatively large spatial extension for the orbit and missing that Ls range during MY32 (see Fig. ). Conversely, the highest residence time is for Ls2= [45–135∘] because of an orbital coverage less spatially extended in space and despite the Ls range being also missed during MY32.

The detection probabilities P (middle panels, ignoring cells for which ΔTsc<2 h) display a rather similar behaviour for all Ls ranges with two main distribution loci: one around the shock and the other immediately upstream of the IMB. The Ls range [Ls4=225–315∘] (Fig. d) includes perihelion condition (Ls=251∘, local maximum of EUV flux) as well as the peak of exospheric H density ∼Ls=270∘; see. It seems to contain, overall, smaller detection probabilities in the locus closest to the IMB than at aphelion (Ls=71∘, local minimum of EUV flux, included in Fig. b). This is in agreement with the findings of Sect. , where larger EUV flux levels create noticeably less MM-like structures in the subsolar magnetosheath than smaller EUV levels do, as a likely result of a combination of comparatively lower plasma-β leading to PCWs being the fastest growing mode in those conditions , and, possibly, non-gyrotropic effects.

From a statistical viewpoint, Fig. b for Ls2 (respectively, Fig. d for Ls4) resembles most Fig. a (Fig. b). This is in line with the conclusions of Sect.  and Fig. e, where we emphasised that the Ls2 (Ls4) range is representative of low (high) EUV flux conditions. Moreover, Ls1 and Ls3 ranges are mutually comparable as they have roughly similar PDFs over a rather large EUV flux range: these two ranges thus mostly display changes due to seasonal effects. In this view, the Ls1 range has a more evenly spread distribution of MM-like events around the shock, whereas the Ls3 range displays comparatively sharper and less spatially extended features. It is interesting to note that, in contrast to other Ls ranges, the Ls4 range displays its highest detection probability in the deep magnetospheric tail just behind the IMB (see Fig. d, middle plot and red region on the rightmost plot). This is a characteristic we have seen most often displayed at high EUV fluxes (Fig. b) and during MY33 (Fig. b), which is consistent with this Ls range encompassing perihelion conditions (see Fig. ).