The meandering of alluvial rivers may be forced by normal faulting due to tectonically altered topographic gradients of the river valley and channel at and near the fault zone. Normal faulting can affect river meandering by either instantaneous (e.g. surface‐rupturing earthquakes) or gradual displacement. To enhance our understanding of river channel response to tectonic faulting at the fault zone scale we used the physics‐based, two‐dimensional morphodynamic model Nays2D to simulate the responses of a laboratory‐scale alluvial river with vegetated floodplain to various faulting and offset scenarios. The results of a model with normal fault downstepping in the downstream direction show that channel sinuosity and bend radius increase up to a maximum as a result of the faulting‐enhanced valley gradient. Hereafter, a chute cutoff reduces channel sinuosity to a new dynamic equilibrium value that is generally higher than the pre‐faulting sinuosity. A scenario where a normal fault downsteps in the upstream direction leads to reduced morphological change upstream of the fault due to a backwater effect induced by the faulting. The position within a meander bend at which faulting occurs has a profound influence on the evolution of sinuosity; fault locations that enhance flow velocities over the point bar during floods result in a faster sinuosity increase and subsequent chute cutoff than locations that enhance flow velocity directed towards the floodplain. This upward causation from the bend scale to the reach and floodplain scale arises from the complex interactions between meandering and floodplain and the nonlinearities of the sediment transport and chute cutoff processes. Our model results provide a guideline to include process‐based reasoning in the interpretation of geomorphological and sedimentological observations of fluvial response to faulting. The combination of these approaches leads to better predictions of possible effects of faulting on alluvial river meandering. Fault displacement leads to a transient response of the longitudinal bed profile and channel sinuosity. Channel sinuosity and the pacing of morphological responses are highly sensitive to the relative position within a meander bend at which faulting occurs. Upward causation from the bend scale to the reach and floodplain scale arises from the complex interactions between meandering and floodplain and the nonlinearities of the sediment transport and chute cutoff processes and has ramifications for reconstructions. Meandering rivers form by interactions between flow, sediment, bed topology and channel curvature and migrate across their floodplains where they form a distinct morphology through meander initiation, expansion and cutoff  shift in channel pattern (e.g. from scroll bar‐dominated to chute‐dominated meandering) to an intra‐pattern adjustment such as a change in sinuosity of a meandering river channel  or gradual displacement. Both have a different effect on river morphodynamics, based on the ratio of vertical displacement and sediment transport capacity of the river. The reoccurrence interval of large surface‐rupturing earthquakes, that are able to produce co‐seismic vertical displacements of the earth's surface, is often relatively large. This hampers the possibility of collecting observational data of river response to co‐seismic vertical displacements. Slow tectonic deformation such as creep forces river morphodynamics on a timescale of thousands of years, obstructing the collection of observational data as well.Over the last decades numerical modelling has provided the opportunity to study the effects of both intrinsic and extrinsic forcing factors on meandering river and floodplain dynamics Crosato, . NumericHence, our understanding of the morphodynamic and morphological responses of meandering rivers to various faulting scenarios on the scale of multiple meander bends around a fault zone is still limited. Therefore, the objective of this study is to determine the effects of both instantaneous and gradual faulting scenarios on meandering rivers at the scale of multiple meander bends along a normal fault.To achieve this objective, we subject a river and its floodplain in a physics‐based, two‐dimensional morphodynamic model Nays2D , the Otofuke River (Japan), the upstream part of the Meuse River in the Netherlands Figure  and on pThe model consists of a fixed rectilinear computational grid with 0.1 m square grid cells, representing a domain of 4.5 m wide and 60 m long. The initial straight channel was 0.5 m wide and 0.04 m deep. An initial lateral offset of +0.4 m was used for the 0.5 m fixed‐width inlet to steer morphological change. The slopes of the banks were 45°. The model setup applies bedload transport only  morphodynamic change of the meandering channel reached a dynamic equilibrium.For all model scenarios we applied a minimum of 100 hydrological cycles of a 50 min hydrograph (83.3 h). The hydrograph varied between 1 and 6.4 L/s. The initial topography was run for 2.5 h to re‐establish hydrodynamics and vegetation cover on the loaded initial planform before a faulting perturbation was applied to the model.The bathymetry of the model scenarios was used for both qualitative and quantitative map comparison and was detrended with the initial valley gradient before the comparisons. The main channel is identified as the maximum value in each cross‐section of the detrended bed elevation times flow velocity to the power of three  rift system , the abrupt scenarios show aggradation at this position. For the first outer bend downstream of the fault zone (i.e. at about 34 m), all subsidence scenarios show an increase in bed level and reduced channel depth over time compared to those immediately after the faulting perturbation  shows an 11% increase in meander amplitude during the model run Figures , Table 3The results of the large instant offset simulations show a different pattern. In these results, the meander amplitude increases with a maximum of 87% of its original size, and the location shifts downstream till 62.5 h. Hereafter, a chute cutoff occurs and the main channel is straightened just downstream of the fault zone, which reduces the meander amplitude ~20% and doubles the wavelength of the meander Figures . The sinEnhanced bank erosion occurs in response to the erosion and scouring at the pools in the outer part of the meander bends, which increases the outer bank slope and leads to mass failure when the angle of bed slope exceeds a critical angle Figure . The enhOver time, erosion and sedimentation around the fault zone reduce the gradient over the fault. In turn, this reduces the lateral displacement rate and change in channel sinuosity Figure . MoreoveAs the sinuosity of the river channel increases due to outer bank erosion and lateral migration of the channel, sedimentation occurs in the channel Figure . In addiThe abrupt subsidence scenario is used to test the effect of different amounts of fault offset as this scenario results in the most obvious sinuosity change Figure . The amoThe rate of sinuosity increase with increasing fault offsets can be explained by the fact that a larger offset results in a higher gradient and, therefore, a larger flow velocity. In turn, this leads to increased bank erosion, lateral displacement and thus a sinuosity increase of the channel.The exception of the 0.030 m model run is probably the result of the large amount of sediment that is deposited downstream, at the foot of the fault Figures  and 11. The effects of the uplift scenarios on the bed topography are inferred from longitudinal and perpendicular cross‐sections Figures  and 13. The consistency in bed topography on the hanging wall can be attributed to the development of a backwater in front of the fault in these scenarios, which is caused by the obstruction to flow posed by the uplifted downstream footwall. The backwater effect reduces flow velocities in the channel on the hanging wall and, during floods, over the point bar and floodplain Figure . The decThe scenarios with relative uplift show that the main channel remains at a fixed position around the fault Figure . An exceThe sinuosity of the channel remains fairly constant over time  vegetation is crucial in the morphodynamics of meandering rivers. Hence, different model settings may have affected the outcome of our model runs significantly, which should be taken into account when comparing our model results to natural river systems. This does not affect, however, conclusions derived from comparisons between the model runs.Our model results, compared with the controls, show that a fault displacement of an alluvial meandering river with downstream migrating meander bends results in multiple, time‐dependent, morphological adjustments of the river channel such as erosion and sedimentation along the longitudinal river profile, sinuosity increase and (chute) cutoff of the channel Figures , and 16.The above indicates that without a detailed reconstruction of the response of a meandering channel, our abductive inferences from the morphological or sedimentological records might lead to erroneous conceptual models of the relation between faulting and alluvial river response. Our model results provide guideline to include process‐based reasoning in the interpretation of geomorphological and sedimentological observations of fluvial response to faulting. The combination of these approaches will help to understand the natural processes involved in more detail and might therefore lead to better predictions of possible effects of faulting on river morphodynamics.The numerical modelling of the morphodynamic response of an alluvial meandering river to various faulting scenarios in this study showed longitudinal and lateral adaptions of the channel. After an initial longitudinal riverbed adaption, sinuosity of the river channel increases as a result of the faulting‐enhanced valley gradient at the fault, which causes enhanced bank erosion and lateral displacement. This upward causation from the bend scale to the reach and floodplain scale arises from the complex interactions between meandering and floodplain and the nonlinearities of the sediment transport and chute cutoff processes.Both the longitudinal and lateral adaptions are restricted to a zone of approximately 2 meander wavelengths around the fault for the applied offsets and diminish both upstream and downstream of the fault. The effects of fault‐bounded vertical deformation on meandering rivers are, hence, relatively local around the fault zone.The transient response to faulting should be considered in the interpretation of morphological and sedimentological observations as it is crucial to understand if an observed morphological response is the final outcome of the morphodynamic adaption or part of an ongoing transient response.Channel sinuosity and the pacing of morphological responses are highly sensitive to the relative position within a meander bend at which faulting occurs. Fault locations that enhance flow velocities over the point bar result in a faster sinuosity increase and subsequent chute cutoff than locations that enhance flow velocity towards the floodplain. As a result, in a meandering river system with downstream migrating bends, the timing of faulting is decisive for the morphodynamic response of the river to faulting.


Magnetically controlled growing rods (MCGR’s) have revolutionized the treatment of early-onset scoliosis (EOS) because painless lengthenings can be done in the outpatient clinic without anesthesia. Untreated EOS leads to respiratory insufficiency and reduced life expectancy. However, MCGR’s have inherent complications like non-functioning of the lengthening mechanism. We quantify an important failure mechanism and give advice on how to avoid this complication. The magnetic field strength was measured on new/explanted rods at different distances between the external remote controller and the MCGR and likewise in patients before/after distractions. The magnetic field strength of the internal actuator decayed fast with increasing distances and plateaued at 25–30 mm approximating zero. Two new and 12 explanted MCGRs was used for the lab measurements of the elicited force using a forcemeter. At a distance of 25 mm, the force was reduced to approximately 40% (ca. 100 N) compared to zero distance (ca. 250 N), most so for explanted rods. This is used to point out the importance of minimizing the implantation depth to ensure proper functionality of the rod lengthening in clinical use for EOS patients. A distance of 25 mm from skin to MCGR should be considered a relative contraindication to clinical use in EOS patients. Early-onset scoliosis (EOS), defined as scoliosis  that presents itself before the age of 102, is a rare, but often severe condition. It is well-known that left untreated, EOS may lead to increased disability and potentially life-threatening conditions such as respiratory insufficiency and pulmonary hypertension and a reduced life expectancy3. The etiologies of EOS are idiopathic, neuromuscular, congenital, and syndromic with a wide spectrum of underlying diagnoses2. The treatment goals are three-fold 1) preservation of thoracic growth and lung growth and pulmonary function, 2) preservation of quality of life and 3) correction of deformity4.The treatment of early onset scoliosis (EOS) by magnetically controlled growing rods (MCGR) has become the standard of care since the first article by Cheung et al.5, and the repeated spine surgeries in children have also been shown to have adverse psychological effects6. A large multicentre study comparing MCGR with TGR reported improvements in quality-of-life measures and concluded that the reduced number of surgeries lessens the psychosocial burden7. However, an increasing number of MCGR failure mechanisms have been observed11, and approximately 10% of patients undergo unplanned revisions12 most often because of anchor failure, failure of the distraction mechanism14, fracture of the internal magnet15 or a non-functioning MCGR16. Titanium wear has raised concerns17. The forces elicited by the MCGR depends on the distance between the MCGR and the ERC (implant depth) and has not so far been quantified. The amount of lengthening dependent on implant depth has been described by Seidel et al.18 even though he offers no direct explanation for his findings and the R2 values for the linear regression model are quite low (0.25–0.36) which means that only a fourth to approximately a third of the variance is explained by the linear regression model presented by Seidel et al.18. In fact, surprisingly little is known of the interaction between the internal magnet/actuator and the ERC considering that this interaction is essential for the desired effect—rod lengthening. Recently, the initial optimism described above has been dampened by reports of low survival rates of the MCGR lengthening mechanism at 2 years19. On the other hand, one could object that this is dependent on the right indications for surgery.The main advantage of MCGRs compared to traditional growing rods (TGR) is pain-free lengthening performed in the outpatient clinic every few months after the initial surgery with the help of an external remote controller (ERC). This compares favourably with the TGR, where manual distraction and open surgery in general anaesthesia must be performed every 6–9 months. This increases the risk of wound infectionWe aim to specify one correct indication for MCGR use by measuring the lengthening forces at varying distances between the ERC and the MCGR to quantify the role of implantation depth. Furthermore, the magnetic field of the magnet in the MCGR as a function of the distance to the magnet was measured in the laboratory and in vivo to provide insight into the magnetic coupling mechanism that drives the distraction.14, it is possible to outline the construction of the MCGR, the lengthening mechanism, and the interaction between ERC and MCGR: An internal dipolar magnet mounted on a screw mechanism is brought into rotation, which leads to an extension of the rod that drives the distraction. The rotation is initiated by an interaction with an external rotating magnetic field generated by the ERC. More specifically, when the magnetic dipole moment By combining the limited information provided by the manufacturer (Nuvasive Inc.) with the literatureThe a priori hypothesis was that the magnetic field strength of the magnets in MCGRs would decrease with increasing distances between the ERC and the MCGR and likewise that the force elicited by the MCGR would decrease with increasing distances between the ERC and the MCGR. All rods investigated were from the MAGEC® system  with diameters of 4.5 or 5.5 mm (explanted rods) and 4.5 mm (new rods). The diameter at the magnet position was 9.8 mm for all rods. 12 explanted MCGRs from six patients and two new rods were tested using the laboratory setup shown in Fig. The demographics of the patients and the MCGRs are shown in Table In four cases, magnetic field measurements were performed on patients before and after lengthening for both the right and left rods. The distance between the skin and magnetometer probe was increased by stacking 5 mm plastic spacers to control the distance (up to 25 mm) from the body surface. The additional distance due to the implant depth was measured using an ultrasonic probe . The magnetometer probe was hand-held tangentially to the body surface immediately above the demarcation of the magnet on the skin and the maximum field value was recorded.Finally, the coupling between the ERC and MCGR was investigated as a function of the distance between ERC and MCGR by measuring the force elicited by the MCGR during lengthening in a laboratory setup, as shown in Fig. .20, allowing the rod to lengthen during interaction with the ERC like in vivo. Force measurements were carried out on 2 new unused rods and 12 explanted rods. Measurements were repeated 6 times for each new rod and 3 times for each used rod.The MCGR was fixed at both ends and plastic guides along the rod were used to prevent rod-bending during ERC activation. The ERC was placed directly above the magnet, and the distance between the ERC and MCGR was precision controlled using acrylic spacers. The ERC was activated, and the maximum obtainable force was measured with a force meter (Sauter FH500) at increasing distances between the ERC and MCGR (zero distance is the surface of the MCGR). The distraction method used was the distraction-to-stall method—when the maximal distraction force (up to approximately 250 N) is obtained the internal actuator automatically stops and a clunk is felt. Two differently termed rods were used (“standard” and “offset”). In addition to a fixed mount, we also added a spring at one end of the rod Fig. f, inspirThe influence of the initial rotational magnet position was also investigated for two magnet positions (field pointing in the direction to the ERC and at 90° to the ERC) of the MCGR.All measurements involving patients were registered in the electronic health records . The study was approved by the local ethics board of the Head and Ortho Center of the University Hospital. Informed consent from a parent or legal guardian was obtained in all instances. Besides, all experiments were performed in accordance with the 1964 Helsinki declaration and its later amendments.The magnetic field strength measurements for the 12 explanted rods and one new rod are shown in Fig. In Fig. Examples of field measurements for randomly selected patients are shown in Fig. In Fig. The measured forces produced by interaction with the ERC as a function of the distance between the ERC and the MCGR are shown in Fig. All force measurements presented so far are peak values measured during actuation with the ERC. We observed that the force relaxed to a lower value after the actuation was stopped, and the ERC was removed from the rod. Three examples are shown in Fig. The measurements of the magnetic field strength as a function of the distance to the internal actuator magnet provided detailed insight into the distance dependence of the field Fig. a. There 21.The measured magnetic field from the explanted MCGR’s were in accordance with the in vivo measurements from the lengthening sessions Fig. c. Our me13. The reason for this was not further investigated because general failure mechanisms have already received much attention11.A clear failure mechanism we observed was that some of the explanted rods did not produce a force Table , as obseOn the new rods we noted that there were considerable variances in the measured forces. The average forces of the two new rods shown in Fig. It was furthermore investigated if there was a correlation between the measured force for the explanted functioning MCGRs and the time of use in the patients or the total expansion length, but no correlations were observed .The current study has shown that the angular position of the magnet is left randomly oriented after each distraction Fig. b. It cou.18 have shown that the distance between the ERC and MCGR is one important predictor of obtainable lengthening. Our measurements of force vs. distance clearly demonstrate that the force decreases significantly with distance, and that the distance dependence is quite severe, also at distances typically used in vivo. Thus, the force produced by the MCGR is more than halved at distances around 20–25 mm, where the force produced by the MCGR is approximately 85–120 N. This spells difficulties with increasing number of distractions, as it has been shown in vivo that increasing forces are necessary proportional to the number of lengthenings22. However, the interval between lengthenings differs between MCGR’s and TGR’s. Nonetheless, several other studies25 has corroborated the findings of Noordeen22. Mean forces of 200–400 N are required to elongate TGRs 2 mm24. In several of our patients, we measured distances of 20–30 mm from the skin surface to the MCGR  our advice is to carefully consider whether MCGRs is the best choice of treatment.Supplementary Figures.


We present a family of radial solutions (given in Eulerian coordinates) to the three-dimensional Euler equations in a fluid domain with a free surface and having finite depth. The solutions that we find exhibit vertical structure and a non-constant vorticity vector. Moreover, the flows described by these solutions display a density that depends on the depth. While the velocity field and the pressure function corresponding to these solutions are given explicitly through (relatively) simple formulas, the free surface defining function is specified  implicitly by a functional equation which is analysed by functional analytic methods. The elaborate nature of the latter functional equation becomes simpler when the density function has a particular form leading to an explicit formula of the free surface. We subject these solutions to a stability analysis by means of a Wentzel–Kramers–Brillouin (WKB) ansatz. The momentum conservation equations together with the equation of mass conservation (proposed by Euler in the middle of the eighteenth century) are widely used today when tackling fluid flow problems. In spite of tremendous progress made over the past two and a half centuries many important questions concerning fluids still remain unanswered. Along the previous lines we note, for instance, the scarcity of explicit solutions. Motivated by this circumstance we set out for a study of the three-dimensional Euler equations with a free surface from the perspective of explicit solutions. While less significant for engineering applications, explicit solutions are interesting for a variety of reasons. Indeed, such solutions may play an appreciable role in the validation of the governing equations. On the other hand, in an oceanographical context, they provide the basis for physically more realistic flows by way of asymptotic, or perturbative methods ; the latIt is to note that the only known explicit solution to the nonlinear  gravity water wave problem for an inviscid, incompressible and homogeneous flow of infinite depth was pointed out by Gerstner in 1809, cf. , rediscoThe Gerstner wave solution was extended by Constantin , 12 and The unsatisfactory situation regarding the lack of explicit solutions is mitigated by the availability of rigorous studies on the existence of exact solutions to the nonlinear water wave problem , 52, 54 Certainly, the Lagrangian perspective offers important insights into the flow evolution by following the path of a particular particle, but, on the other hand, it is highly significant to know the velocity field, the pressure function and the shape of the free surface at any given time instant and physical location, setting known as the Eulerian picture. Complying with the previous ideas we present in this paper some explicit solutions (in Eulerian coordinates) to the nonlinear inviscid and incompressible three-dimensional water wave problem. While the Gerstner wave solution supplies explicit formulas for the velocity field and for the pressure function, (some) of the solutions presented here are explicit also in terms of the free surface. A further by-product of our solutions is that they also accommodate a variable pressure on the free surface which appears to be of relevance in modeling the event of storms in the ocean . In the Another remarkable feature of the solution is that the flow it prescribes accommodates a density stratification that varies  with depth. A noteworthy aspect is that the density is instrumental in conferring the velocity field a non-trivial vertical structure.Density stratification has an essential role in geophysical fluid dynamics, particularly in the equatorial region where large scale ocean flows display marked density variations leading to a layering of the flow: fluid layers of different densities organize themselves so that the higher densities are found below lower densities, cf. , 16, 20.After a brief presentation of the water wave problem in Sect. x, y, z, we assume the water flow to be bounded below by the bed t denotes the time variable.We formulate here the governing equations for free surface water flows. Working in a Cartesian coordinate system of coordinates u, v, w) represents the velocity field. Under the assumption that the water is inviscid, the equations of momentum conservation areP is the pressure function and g denotes the gravitational constant.The guiding principles for water flow propagation refer to mass conservation and Newton’s second law of motion. The equation of mass conservation relates the rate of change of density to the field of motion. More precisely, denoting with While both , 2.2) a a2.2) arIn the next section we introduce a family of solutions to the problem –2.5)..2.5).We first lay out the ansatz giving the explicit formulas of the velocity field and then, availing of the Euler’s equations , we deriP we have to solve the systemP in the above system we obtainc.A computation shows thatClearly,ty field .We set out to determine the free surface nditions –2.5). U. U\docump, which represent a small deviation from the pressure We will resort now to the implicit function theorem to prove the existence of tisfying . To thistisfying . The surved from  by buildondition  is satisWhile, in general, is not possible to establish an explicit formula for the shape of the free surface p is chosen such thatLet us assume that ave from \documentTaking To reinforce the relevance of the explicit solutions to the full nonlinear governing equations –2.5) we we2.5) wu, v, w) from  from  satisfy u, v, w) given in . To be more precise, we use for F is a scalar function  and We consider perturbations given in  and of tgiven in , respectB satisfiesF is solution to the equationFrom the requirement that the perturbed quantities momentum  and (2.2B ansatz  and idenF for which the right hand side in system (x(t), y(t), z(t)) the streamlines of the basic flow  given in (t. Proceeding to solve (c such thatt. It follows thatx(t), y(t), z(t)) of the basic flow  , y(t), z(t) in (F of the perturbation such thatt. We will be concerned first with the homogeneous systemThe purpose of the first part of this section is to show that there are choices for the phase function n system  vanishesgiven in . This mee system  we obtaieader to  for detaL). Then all solutions y of the Hill’s equationt. Assume tt, provided the functions f and Let  formula  such thaf and t holdst. From  and y(t) we obtain the claim ,,\documenformulas  we infers system  are bounLet sic flow  given bysic flow  where the system , and thue system  are bounf(c) and We will seek to find a particular solution e system  that is e system  is bounde system  and the atisfies . Routineatisfies  can be wation of  becomes4~2p from  if and ot verify  of the tWe state now the main result on the stability of perturbations along the streamlines of the basic flow .f and F of the perturbation is constant or satisfies (Let us consider the flowthe flow  along thatisfies .f from the definition of u, v is equal to 1. In this case it is possible to derive explicit formulas for the trajectories of the amplitudes of the trajectories of the basic flow. More precisely, we easily see that the amplitudes We address now the stability issue of the basic flow  for the 32. From  and appe32. From  we find F of the perturbation (The flowrbations  along thurbation  is consturbation .


The Martian interaction with the solar wind leads to the formation of a bow shock upstream of the planet. The shock dynamics appear complex, due to the combined influence of external and internal drivers. The extreme ultraviolet fluxes and magnetosonic Mach number are known major drivers of the shock location, while the influence of other possible drivers is less constrained or unknown such as crustal magnetic fields, solar wind dynamic pressure, or the Interplanetary Magnetic Field (IMF) intensity, and orientation. In this study, we compare the influence of the main drivers of the Martian shock location, based on several methods and published datasets from Mars Express (MEX) and Mars Atmosphere Volatile EvolutioN (MAVEN) missions. We include here the influence of the crustal fields, extreme ultraviolet fluxes, solar wind dynamic pressure, as well as  magnetosonic Mach number and Interplanetary Magnetic Field parameters (intensity and orientation angles). The bias due to the cross‐correlations among the possible drivers is investigated with a partial correlations analysis. Several model selection methods (Akaike Information Criterion and Least Absolute Shrinkage Selection Operator regression) are also used to rank the relative importance of the physical parameters. We conclude that the major drivers of the shock location are extreme ultraviolet fluxes and magnetosonic Mach number, while crustal fields and solar wind dynamic pressure are secondary drivers at a similar level. The IMF orientation also plays a significant role, with larger distances for perpendicular shocks rather than parallel shocks. The relative importance of the drivers of the Martian shock location is studied according to the AIC and LASSO model selection methodsThe Mach number and EUV fluxes are the primary drivers of the shock location, crustal fields, and solar wind pressure are secondary driversThe angle between the shock normal and the Interplanetary Magnetic Field appears as a significant driver of the shock location The Martian interaction with the solar wind leads to the presence of a bow shock upstream of the planet, as well as several other plasma boundaries, that is, the Induced Magnetospheric Boundary or the PhotoElectron Boundary. The boundaries are dynamic and depend on internal and external drivers  location, based on the data obtained by the Mars Global Surveyor (MGS), Mars Express (MEX), and Mars Atmosphere and Volatile EvolutioN (MAVEN) missions. The first driver analyzed was the solar wind (SW) dynamic pressure Crider, . Edberg Mms, and SW dynamic pressure), the influence of other potential drivers is less understood. In particular, the influence of the crustal field is poorly constrained by previous studies. Most of the works suggested a possible influence of the crustal fields, based on hemispheric differences between the north and south locations of BS . Pole versus equator, dawn/dusk, or north/south asymmetries were observed depending on the authors and often interpreted as the result of the strong mass loading of pickup ions, or of anisotropies of the magnetosonic wave velocity .In this study, we investigate the possible drivers of the BS location based on both MAVEN and MEX BS crossings datasets. Instead of focusing on only few drivers (and independently from each other) as most authors, we compare and rank the relative importance of the drivers of the Martian BS based on several methods, including a partial correlation approach to take into account possible biases due to mutual correlation, and the Akaike Information Criterion (AIC) and Lasso regularization model selection methods to compare the relative importance of the drivers. This allows providing a comprehensive view of the Martian BS dynamics with respect to internal and external drivers. Figure The paper is organized as follows: in Section 22.1We use in this study the same datasets as G22, which we refer to for a more detailed description of the lists of MAVEN and MEX crossings considered. Here we provide a brief description.The MEX BS crossings dataset was derived by BH16 from the MEX ASPERA‐3 ELS data Barabash et al. , includiA large number of BS crossings for MEX and MAVEN ‐ due to their respective orbital periods of 6.7 and 4.5 hr ‐ allows for a large spatial and temporal coverage that enables statistical analyses. Significant differences between both datasets need to be considered. First, MEX covers a large period through a whole solar cycle, while the MAVEN dataset corresponds to a shorter period where the mean EUV level (given by the solar 10.7 cm radio flux) was larger by ∼14% compared with the MEX dataset. We thus focus on the dynamics rather than on absolute BS location due to the different solar conditions encountered. Second, the MEX spacecraft is not equipped with a magnetic field instrument, which prevents us from using MEX data to investigate the influence of IMF related parameters .RTD, is calculated by:RM is the Martian radius (3390 km), L and e are the semi‐latus rectum and eccentricity, and the focus of the conic is located at . We used for the MEX crossings the  values by Hall, Lester, Sánchez‐Cano, et al. , which would finally lead to very different RTD values according to the above equation.Note the one dimensional approach allows the investigation of the presence of any large scale and permanent influence on the BS location, including asymmetries of the BS shape or a permanent influence of IMF intensity and orientation angles, as studied by numerous previous authors : zero‐order linear Pearson correlation coefficients, unpaired t‐test evaluates the test statistics associated with the correlation, and compares it with the statistics of the null hypothesis H0: if the test statistics t is larger than a threshold  defined for a given risk, or equivalently if the p‐value is smaller than the risk considered (5% by default), then the correlation is considered significant, otherwise the correlation factor is considered not significantly different from 0. We will in the rest of the paper mention that p‐values are negligible when they are smaller than 10−5 and tag them with ”(n)”. Unpaired t‐tests can also be used to determine if there is a significant difference between two groups of data. They assess whether the two groups show different mean values, with p‐values providing the significance of this difference. Throughout the paper, correlations factors, as well as significance tests statistics or p‐values by default, corresponding to direct linear correlations.First, simple linear Pearson correlation coefficients are used to investigate linear relationships between two variables. Hypothesis tests are used to calculate the significance of the correlation. A y and x0, after controlling for the influence of other variables xi. G22 successfully used this technique to disentangle the influence of the crustal fields on the Martian BS location despite a significant cross correlation with EUV in the MAVEN data. It was also used by several authors for the solar wind Earth interaction  which becomes critical in the case of the Martian BS that may be influenced by a number of parameters.Second, we use a partial correlation approach in Section y = a + ∑ixi ⋅ bi with a constant and bi slopes), but can alternatively correspond to power laws y=a∏ixibi then linearized by a logarithm. Note that the linear assumption is weak and our method does not need true linear relationships to remain valid since at first order most of the regular relationships can be considered as linear (or power law like). The significance of the partial correlation factors is given by a t‐test to compare the statistics with the null assumption H0, and provides p‐values that need to be smaller than the risk chosen (5% by default) to consider the partial correlation as significant. The 5% risk level corresponds to a 2 standard deviations tolerance interval for a gaussian probability distribution.The correlations are considered linear  and the relative influence of each of these drivers. Ranking the relative importance of the drivers can be achieved by model selection approaches such as the Akaike information criterion or the LASSO method.The Akaike information criterion , but to provide a better understanding of the drivers of the BS location, and compare their relative importance.We use in Section 33.1http://amda.cdpp.eu/) for 10–120 nm wavelengths, and for MEX from the solar 10.7 cm radio flux index extrapolated to Mars assuming a 1/rMarsSun2 law; using solar radio flux values for MAVEN leads to almost identical results since both parameters are strongly correlated (with a correlation factor of 0.98 (n)); however, we choose the FISM model for MAVEN since it is available and slightly more precise than the radio flux proxy, and the absolute values of EUV are of no interest in our study where we focus on methods which are unit independent1. Solar Extreme UltraViolet fluxes (EUV) for MAVEN and MEX: the EUV values were determined for MAVEN from the FISM model  provided by respectively MEX/IMA and MAVEN/SWIA data; a restriction on the quality factor values for MEX/IMA data is applied, which reduces the number of MEX crossings available with this driver (this explains why the median Mms (no magnetic field measurements were available for MEX) is calculated by Mms=vsw/cs2+vA2 with vsw SW speed, cs sound speed, vA Alfvén speed, with the electron temperature assumed equal to the proton temperature and a polytropic index γ = 5/33. SW magnetosonic mach number (Mms) for MAVEN: the MAVEN km altitude, directly based on the results by G22:4. crustal magnetic fields for MAVEN and MEX: several parameters for the crustal field were used and derived in particular from the Morschhauser et al.  model at4.1.B2/(2μ0)) at the sub‐spacecraft longitude/latitude of the crossingcrustal field pressure 4.3.σ = 60°, based on G22), to increase the local influence of crustal fields in the averaging procedureaveraged crustal field pressure convolved with a Gaussian‐like weighting function  to focus on the influence of the strongest crustal source region4.5.ϕMSO) of this strongest crustal source region to account for the larger impact of crustal fields when this crustal fields region is on the daysideMSO longitude  or perpendicular g et al.  for MAVE5.4.θvn angle between the normal of the BS and the SW velocity vector5.5.θbv angle between the SW velocity vector and the IMF vector, also called cone angle5.6.the MSE pole versus equator asymmetry, determined by the absolute cosine of the angle between the IMF and the location of the crossing projected in the terminator plane ; this angle was called ”clock angle” of the crossing by several authors for studies of Venus , SW dynamic pressure, and the crustal fields through the angular distance from the strongest crustal source region introduced by G22. As shown by previous authors, the BS is located at higher altitudes in response to stronger EUV fluxes or lower Mms values, as well as under weaker SW dynamic pressures or spacecraft locations closer to the strongest crustal source region in the southern hemisphere. The MAVEN panels (upper panels) clearly show that larger Mms values lead to smaller RTD values. Enhanced EUV leads to higher BS altitudes for both MAVEN and MEX datasets. The influence of SW dynamic pressure and angular distance to the strongest crustal source region is also visible in the observations by both missions: the mean profile (black dashed line) shows that despite very strong data scattering  the MAVEN and MEX BS RTD values decrease with increasing SW dynamic pressure by hundreds of km; the panels b and f show the same results as G22, with decreasing RTD values  from the region around the strongest crustal fields ‐ with an angular extent of 40/50° corresponding to the size of the strongest crustal source region ‐ to regions away from the strongest crustal source region, with slight peaks around angular distances of 120–150° that may correspond to the influence of localized crustal sources indeed present at such angular distances. Note that the ranges of SW dynamic pressures are different between both missions: this may be due to either different time periods, or to the different types of instruments and methods used to extract the dynamic pressure; however, even if a real difference was existing, this would not impact our conclusions that are based on the variability of the drivers, with analysis for both missions that independent from each other.Figure RTD values (see details below) for both MAVEN/MEX missions.In order to show the influence of the possible drivers of the BS altitude comprehensively and comparably, Figure The figure provides two different ways of comparison between the various drivers. The lower panel provides the linear Pearson correlation coefficients between the terminator altitudes of the BS and the parameters. The upper panel uses the following approach: each set of parameters  was separated into low  and high  subsets; the median terminator altitudes of the ”low” (blue) and ”high” (red) subsets are then determined for each parameter to show how low/high regimes of each driver influences the terminator altitude of the BS.cm radio flux: 49 ⋅ 10−22 W/m2/Hz) was thus larger by ∼14% compared with the MEX dataset (43 ⋅ 10−22 W/m2/Hz). Given the major influence of EUV on the BS location (see below), the larger EUV conditions associated with the MAVEN dataset certainly contribute to the slightly larger BS distances observed (by ∼5%) for MAVEN than for MEX. The compared magnetosonic Mach number conditions, which also have a major influence on the BS location, may also have induced this absolute difference in the BS altitude between both missions, however, these conditions are not known precisely for the pre‐MAVEN period.The main differences between both missions are the smaller changes observed  for MEX compared to MAVEN, and lower terminator altitudes for MEX. Regarding the first difference, the much larger dataset for MEX  induces a stronger mixing of different influences and a greater difficulty to distinguish one driver from another. Systematic differences may also exist between MAVEN and MEX due to different instruments and methods for determining the BS locations. The second difference most probably originates from the EUV levels encountered at the MAVEN and MEX crossings. The large MEX dataset allows for a wide range of EUV conditions, with a full solar cycle (including the lower and extended ever recorded solar minimum), while the MAVEN dataset corresponds to a period with an active Sun in 2014 and 2015 (where most of our MAVEN crossings occurred) below a declining activity in 2016–2017. The mean EUV level of the MAVEN crossings (given by the solar 10.7 θbn) or stronger IMF intensity, (d) decreasing SW dynamic pressure, (e) large cosine of clock and cone angles, and (f) other angular parameters of the IMF (θvn or the MSE pole vs. equator asymmetry). The apparent relative influence of the (non‐IMF related) drivers is consistent between the two spacecraft, with smaller variabilities and lower absolute altitudes for MEX than for MAVEN due to a larger amount of data and the different EUV conditions. A small difference can however be noticed in the relative importance of crustal fields versus SW dynamic pressure or EUV from MAVEN to MEX, with crustal fields appearing slightly more influent in the case of MAVEN: this is due to mutual correlation between parameters, that is, between EUV and crustal fields as discussed in details by G22, or between EUV and SW dynamic pressure as detailed in the next section that discusses the cross correlations between parameters. The apparent strong influence of the IMF intensity will also be discussed.Overall, based on the correlation coefficients or absolute variations related to individual parameters, the BS terminator altitude seems to increase with (in a descending order of importance and for both MAVEN/MEX) (a) and (b) increasing EUV fluxes or decreasing magnetosonic Mach number  for the MEX dataset was reached on 22 October 2014 and was induced by a combination of large crustal fields, a relatively low SW dynamic pressure, and most importantly one of the largest EUV conditions encountered. These extreme EUV conditions were related to the event consisting of M and X type solar flares originating from the same active region 12,192 from 19th to 22 October 2014, surprisingly not accompanied with coronal mass ejections  for the MEX dataset was reached on 25 June 2010, at a period with very low EUV , relatively strong SW dynamic pressure, and over the strongest crustal source region. No extreme solar CME‐type event happened closely before this crossing to our knowledge, except for the slow CME mentioned by Manchester et al.  nor the extreme high  BS terminator distances correspond to extreme solar events as listed by Lee et al. , Mms (MAVEN dataset), and crustal fields .The schematics show, beyond the statistically significant correlation  between a number of drivers and the BS location (with in particular the major influence of EUV and Mms), cross correlations between most of the drivers themselves. One notes that the direct correlations between the major drivers and the BS RTD. In the case of the MAVEN dataset, the cross correlations between the crustal field parameters considered and the BS location is for example, smaller than the cross correlations between the crustal field parameters and the EUV. At least a part of the influence of the crustal fields in the MAVEN dataset shows up through the cross influence of EUV which is a major driver of the BS location. G22 investigated this issue and used the partial correlation approach to investigate whether the crustal fields had a significant influence not only in the MEX dataset (where no such bias exists) but also in the MAVEN dataset. They concluded that indeed a part of the apparent crustal fields' influence was due to the cross correlation with the EUV conditions, but controlling for the EUV influence in the analysis still leads to very significant partial correlations of the angular distance to the strongest crustal source region or of ϕMSO the MSO longitude of the strongest crustal source region center with the BS RTD . Controlling over the EUV then leads to similar correlations factors and similar relative influence of the EUV and crustal fields parameters between MEX and MAVEN based on the slopes of the multivariate regression performed with the partial correlation analysis.The cross correlations among the drivers must be considered to avoid biased interpretations, particularly when they are comparable or even greater than the direct zero‐order correlation with x1 and x2 that influence the parameter y (in our case RTD) and that can be cross correlated to each other, three correlation factors can be considered: r12, ry1 and ry2. When the product of the signs of the three correlation factors is positive , this means the influence of variables x1 and x2 on y is to some extent overestimated when considering only the direct correlation factors ry1 and ry2, while their influence is underestimated when the product of the signs of the correlation factors is negative. Depending on the relative signs of the correlations, a part of the increase or decrease of y due to a correlation with one of the variables  may be indeed partially due to or hidden by the increase of the second variable  that is not only correlated with y but also with x1. This can be illustrated by the example of the influence of the SW dynamic pressure, as detailed below.Even when cross correlations are not strong enough to interfere with the influence of a driver, they can lead to an overestimation or an underestimation of the influence of the drivers. If one considers two variables nPa  with a linear assumption for MAVEN/MEX) compared to the correlations of RTD with EUV fluxes (strong positive correlation) or magnetosonic Mach number (strong negative correlation), and also smaller than correlations with crustal field pressure or angular distance. However, these direct correlations of the SW dynamic pressure previously mentioned in Section nPa during the low EUV periods, and of ∼1 nPa during the high EUV periods. Enhanced EUV fluxes push the BS to higher altitudes, so that enhanced EUV fluxes occurring at the same time as slightly enhanced solar wind dynamic pressure may partially hide the compression of the BS due to small solar wind dynamic pressure pulses. With a partial correlation analysis assuming a linear regression (or power law with similar results) and control over the EUV fluxes and ‐ for MAVEN only ‐ magnetosonic Mach number, the partial correlations between the solar wind dynamic pressure and the BS RTD altitude become larger at −0.24(n) and −0.14(n), respectively, for MAVEN and MEX. After controlling for cross‐correlations with the major drivers, the solar wind dynamic pressure thus appears as a driver with a similar influence (regarding correlation factors) as the crustal field pressure or angular distance.The SW dynamic pressure influence on the BS location is clearly visible when simple scatter plots with the BS location are performed, with in particular fewer high altitudes crossings at pressures above 1 r et al. ). HoweveRTD terminator altitude of the MAVEN and MEX BS crossings versus candidate drivers, after removing the linear dependence on the major drivers . The resulting p‐values of the partial linear correlation coefficients are also shown. After controlling for the major drivers of the BS location, most of the possible drivers still keep a significant correlation with the BS location. This includes the crustal fields , SW dynamic pressure, as well as the IMF intensity and several IMF orientation angles  and the MSE pole versus equator asymmetry, with correlations that are more or less obvious by eye . However, if these residuals confirm that a number of physical parameters can be possible secondary drivers of the Martian BS location, the schematics in Figure Mms and EUV, which is discussed in the next section focusing on the influence of the IMF characteristics.We show in Figure 3.3θbn angle is discussed in Section We focus on the MAVEN dataset in this section to analyze the influence of the IMF (intensity and orientation) on the Martian BS location. We discuss in more detail the impact of the IMF, whose characteristics include complex cross correlations beyond those with the major drivers. We also compare with results obtained in the literature at planets where the IMF influence was more widely investigated. The specific influence of the RTD altitudes (Pearson correlation factor of 0.26 (n)), in contradiction with the simple effect of an increased external magnetic pressure. A positive correlation could be due to the fact that an enhanced IMF induces a stronger draping around the planet, which could consequently push the BS further. However, Figure Mms ). Including also the SW dynamic pressure in the partial correlation in addition to EUV and Mms shows a much‐reduced correlation between the IMF and RTD with a Pearson linear correlation coefficient of −0.07 . The real influence (if any) of the total IMF intensity on the BS location thus is difficult to be distinguished due to complex mutual correlations with other significant drivers.The IMF intensity is apparently significantly positively correlated with the Mms (and of SW dynamic pressure according to the authors), they suggested that the IMF components have differential effects, with the BS expanding along the MSO Y/Z axis for respectively increasing BZ/BY IMF components and a stronger influence of BX in the tail. However, the analysis of our dataset does not show such differential influence on the BS RTD altitudes: the IMF components have little influence compared to the total IMF on RTD, with for example, a partial correlation significance of BZ with RTD below the null H0 assumption when focusing on BS crossings around the MSO Y axis sector. An influence of the IMF BX component in the tail is however difficult to identify, due to the limited number of MAVEN crossings in the tail (only 24 crossings occurred at X < −1 RM).Wang, Xie, et al.  recentlyθvn has an apparent small negative but still statistically significant direct correlation with the BS RTD values, but controlling for the EUV fluxes (or for the other main drivers) with the partial correlation approach reduces the significance of the correlation well below the H0 null assumption . Similarly, the significance of the already small correlation between the cosine of the cone angle θbv and the BS distance becomes non‐significant  after controlling for the θbn influence. No significant cone angle dependence can thus be seen as was observed at Venus by Alexander et al.  the clock angle distributions are highly inhomogeneous with ∼70% of the crossings with an IMF lying within 30° around the MSO Y axis and (b) cross‐correlations modify the apparent influence of the clock angle of the IMF. The residuals of the RTD dependance after controlling for both EUV fluxes and Mms already reduce the clock angle influence  and the BS location close to the significance limit . An influence of the clock angle thus appears still possible but small, which agrees with the findings by XF17.The clock angle of the IMF appeared as a significant driver in the previous analysis based on direct correlations, with BS located farther in response to low clock angles, that is, for an IMF along the MSO e Figure , but thee Figure  is even Y − Z plane, equivalent to the MSE frame representation. At Venus, Alexander et al. , as a function of the angle from the IMF vector in a radial way as performed by Russell et al.  (thus similar to a polar vs. equator asymmetry but shifted by ∼30–45°), an asymmetry also suggested when focusing around close to terminator BS crossings only. The lower panel also suggests this asymmetry appears stronger at large cone angles, in agreement with an increased V→XB→ influence. Unpaired Student's t‐tests were performed and infirm the presence of a North versus South asymmetry suggested by Edberg et al. (−4/5.6 ⋅ 10−5 for cone angles below/larger than 60°) before considering possible cross correlations. The partial correlation between the absolute cosine of the angle from the IMF and the BS location after controlling over EUV and Mms keeps significant  as soon as we remove the EUV and Mms influence by considering residuals. A close to equator versus pole asymmetry is thus suggested by the MAVEN direct data analysis. Chai et al.  or of ai et al.  mentione3.4θbn the angle between the BS normal and the IMF direction, available for the MAVEN dataset. This angle is a distinguishing indicator between quasi‐parallel  and quasi‐perpendicular  BS. Our correlation analysis revealed a rather strong correlation (0.19 (n)) between the MAVEN BS RTD altitude and the sine of θbn (or 0.22 (n) with θbn itself). Figure θbn (θbn is considered positive). This profile suggests indeed that perpendicular BS happens on average at higher altitudes than parallel BS, which is also observed in the residuals of Figure θbn influence, with a large t/tH0 ratio above 8  when controlling over the magnetosonic Mach number, EUV and any of the other variables considered in this study . Below, we investigate the physical reasons that could induce an influence of the θbn angle with perpendicular BS at higher terminator altitudes than parallel BS.Another parameter suggested as significant in Section νms defined by:cs sound speed and vA Alfvén speed. Based on this equation, it appears that perpendicular BS (θbn close to 90°) leads to larger velocity values of the magnetosonic wave (the wave velocity being thus anisotropic), which could induce a further location of the BS.Several authors mentioned a similar influence at planetary BS, with further perpendicular BS than parallel BS, usually for the reason detailed below. The (Martian) BS is indeed a fast mode type of BS , with an influence mostly expected in the tail and for low Mach values (where the anisotropy has more time to develop). Dmitriev et al.  in particular since this asymmetry is reduced at low SZAs where the wave anisotropy is expected to lead to less influence than at higher SZAs where the BS is further and the travel time of the wave is larger. Wang et al.  but could not explain it. Overall, a number of Earth and Venus studies, as well as few Mars studies, thus mentioned a possible influence of the anisotropic wave velocity on the BS, essentially based on geometrical arguments rather than based on quantitative arguments.At Mars, a few early works also mentioned the influence of g et al.  first usg et al.  mentionecs2+VA2, with 90% of the MAVEN BS crossings showing a ratio afastcs2+VA2 above 0.9. Second, one would expect, if an anisotropic wave speed was the reason for the θbn influence, that the BS RTD altitude would be more strongly correlated with νms rather than with θbn or its sine. The direct correlations of νms and the sine of θbn with RTD are actually similar (respectively 0.20 (n) and 0.19 (n)), but much of the influence of the wave speed on the BS location is due to the Mms influence through the Alfvén velocity that is closely related to the wave speed νms (with a very strong correlation of −0.71 (n)): the partial correlation of νms with RTD becomes small and even slightly negative (−0.06) after controlling over the influence (assumed linear at first order) of EUV and (mostly) Mms. Similarly, the partial correlation between the sine of θbn and RTD keeps strong and almost unchanged (correlation factor 0.16 (n)) when controlling for the fast magnetosonic wave speed value, thus suggesting an independent influence of the θbn. The apparent influence of the wave speed is thus due to the magnetosonic Mach number influence rather than the IMF orientation. Moreover, the analysis of the influence of θbn shows no specifically stronger influence at low SZAs or low Mach number regimes that would be expected with an anisotropic wave speed influence  after controlling over Mms and EUV, while the θbn influence does not reduce after controlling for a linear influence of Mms or other variables.However, in our case calculating the wave speed with Equation i et al. ). HoweveBX component) that parallel/perpendicular BS correspond to respectively dawn/dusk sectors . Moreover, the MEX dataset shows a reversed situation, with dawn BS occurring at further altitudes for intermediate SZAs until the terminator where the situation is reversed with dusk shocks at further altitudes than dawn shocks.Apart from the anisotropy of the wave speed explanation, several authors linked the perpendicular versus parallel BS differences to dawn versus dusk asymmetries observed at planetary BS, in particular since the Parker spiral average configuration implies . If we keep the same dataset but use other conic parameters by Edberg et al.  =  instead of  for XF17 ‐ again the correlation factor is changed by only 0.01 with negligible p‐value. The choice of the conic parameters thus does not affect our observation of the θbn influence.A last possible uncertainty on the influence of the k et al. , to keepg et al.  ‐ that iθbn angle. This is however not observed in our Martian dataset where the cone angle has no significant influence, with even slightly smaller BS RTD values for radial IMF orientations.Finally, alternative explanations were provided by several authors to explain the presence of BS expansions. Thomas and Winske  developeθbn angle of the IMF, suggested by a number of authors in the Earth and Venusian BS literature based essentially on geometric arguments, does not appear convincing in our quantitative analysis of the MAVEN data.The analysis of the MAVEN dataset thus shows that quasi‐perpendicular BS occurs at higher altitudes than quasi‐parallel ones, independently from the influence of other possible drivers. However, the exact origin of this influence is unclear. The common explanation of the fast magnetosonic wave speed dependence on the 3.5Understanding the dynamics of the plasma boundaries includes understanding both the relationship with individual drivers  and the relative importance of each driver. The results discussed in Sections 3.5.1The Akaike information criterion , or a power law dependence RTD=a∏iXibi. Table We performed AIC calculations with a dedicated R code, assuming either a linear dependence of the Martian BS terminator distance θbn angle of the IMF on a similar level as well, before crustal fields  and solar wind dynamic pressure at a comparable level. Regarding these two, the ranking differs between MAVEN and MEX, with angular distance ranked above solar wind dynamic pressure for MEX but after it in the MAVEN case . ϕMSO appears less than or as influent as the angular distance depending on the mission. The IMF intensity and clock angle are also parameters of influence whose removal lead to information loss, but less than other parameters. This is also the case for the equator to pole asymmetry in the MSE frame discussed in Section The AIC approach applied to linear regression models ‐ see below for power law models ‐ for MAVEN and MEX thus confirms the main conclusions brought in the previous sections, with EUV fluxes and magnetosonic mach number (for MAVEN) being the major drivers whose removal imply very strong loss of information, with the θbv and θvn parameters removal do not imply information loss (compared to a simple constant) and are thus considered not significant by AIC analysis.Moreover, the θvn and θbv). One can note that linear or power law dependence laws are widely considered for EUV, magnetosonic Mach number, or crustal field pressure, but it is difficult to anticipate functional forms for the angular characteristics of the IMF, however, the robustness of the AIC results for both linear and power law models and their coherence with more direct previous analysis suggests the ranking proposed is consistent. Moreover, in first order, most of the relationships can be considered linear (or power laws) which makes the linear model a weak assumption.The use of power law functional forms instead of linear relationships leads only to slight ranking changes for the MAVEN least influent parameters , the weight of the sum of the coefficients in the optimization process becomes eventually strong, which leads the algorithm to eventually set to zeros some of the coefficients, thus selecting the most significant variables only. For large λ values, more coefficients of the regression are set to zero, thus removing the associated predictors from the model selected.As a complementary approach to the AIC results, we performed an analysis with the Least Absolute Shrinkage Selection Operator method  to remove the specific dynamics of each predictor  and thus allow to make the regression coefficients comparable to each other with the same mean (0) and standard deviation (1) values for each driver distribution. The definition of the training set to learn from and of the test set and the determination of the best regularization λ value is obtained from a cross‐validation procedure with 10‐folds to avoid overfitting.We used the glmnet R package to compute the LASSO procedure. The variables introduced in the linear regression model are the same as for the AIC procedure, and were at first standardized  but decreasing for larger Mms (negative coefficient).Figure λ values the largest coefficient among the four variables included . The SW dynamic pressure and angular distance coefficients are close with no penalty (slightly larger for angular distance), while the SW dynamic pressure coefficient drops more rapidly than the one for angular distance with increasing penalties, reaching zero at large penalties. The coefficient for ϕMSO the longitude of the strongest crustal source region is the fourth coefficient in absolute value and reaches 0 at a large penalty.The MEX results show that EUV displays for all θbn, and then of solar wind dynamic pressure, which does not reach zero values at high penalty. The angular distance coefficient is also significant at a low penalty, before the cosine of ϕMSO. The lowest coefficients are then those of the clock angle and MSE pole versus equator parameter. The lowest coefficients, that are non‐zero with no penalty but are set to zero rapidly by the penalty term are the cosines of the IMF orientation angles θbv (cone angle) and θvn, and to a lesser extent the IMF intensity.The MAVEN curves, which include the IMF intensity and orientation parameters, show first that EUV and magnetosonic Mach number are always the largest coefficients. Then the largest coefficients are those of the sine of RTD predictors. The best model corresponds to small penalty terms (respectively 1.2 and 2.4 for MEX and MAVEN), confirming the significance of most of the variables included in the model according to LASSO. However, if the cross validation procedure leads to non zero coefficients for the least significant variables , several of the folds lead to zero values of these coefficients , making their significance weak or inexistent according to LASSO. One can note that using an initial power law functional form (linearized with a logarithm) for the LASSO regression instead of an initial linear form leads to similar results: this leads to different coefficients but the relative values are very close to the linear form conclusions, as well as the evolution of the coefficients as a function of the penalty term.Table θbn angle), and then solar wind dynamic pressure or crustal fields (depending on the dataset). The IMF intensity and orientation angles (mostly the clock angle and MSE pole vs. equator) are predictors of nonnull importance (except probably θbv and θvn) but are weak.Overall, the LASSO results are consistent with the previous results based on partial correlations and the AIC approach. The EUV and magnetosonic Mach are the major drivers of the BS location, before parallel versus perpendicular BS .4when the magnetosonic Mach number decreases (first order importance): the BS is a fast magnetosonic standing wave induced by the supersonic flow encountering the ionized Mars obstacle, which propagates with a velocity determined by the magnetosonic Mach number; the magnetosonic Mach drives do not only the jump conditions at the shock through Rankine‐Hugoniot relations but also the shape of the BS, in particular, the flaring of the BS with a shock cone inversely proportional to the Mach number according to MHD modeling, which modifies the cross section of the BS and its terminator distancefor enhanced extreme ultraviolet fluxes (first order importance): the EUV fluxes increase the ionization rate of both the Martian collisional atmosphere and its exosphere, which adds mass to the solar wind flow through the pickup of ions and thus slows down the solar wind, then creating a larger apparent obstacle that pushes the BS further; the EUV fluxes also heat the atmosphere, thus increasing the ionospheric scale height and expanding the exosphere; this influence of EUV happens at various timescales, from intense brutal solar flares to seasonal variations induced by the eccentricity of Mars' orbit and depending on the hemisphereθbn angle between the IMF and the normal to the BS is a significant driver of the Martian BS, with several possible reasons but none being conclusive; the most probable physical process is the expected dependence of the anisotropic wave velocity of fast mode magnetosonic waves on the θbn angle that is often mentioned in the Earth's and Venusian BS literature based on MHD modeling ; however, the anisotropy of the wave speed is expected to be significant mostly for low Mach numbers and in the tail, that is, when and where the BS cross section is maximum, as the travel time of the wave; moreover, our data analysis suggests that the observed influence of θbn on the BS is not associated with a significant anisotropy of the wave speedin the case of perpendicular BS: our statistical analysis of the MAVEN data shows the with crustal magnetic fields (second order importance) through two ways: though an influence at the spacecraft location , and through an amplification of this influence all the more than the strongest crustal source region center is located close to noon; the crustal fields may impact through several processes on the BS: they first add internal magnetic pressure, increasing the apparent size of the obstacle, then they modify and induce currents that propagate through the induced magnetosphere, interacting with the magnetosheath plasma and the draping of the IMF around the planet, which in the end pushes further the BS; moreover, when the strongest crustal source region is located close to noon (compared to midnight), the travel time for the magnetosonic wave is reduced allowing the BS standing wave to form further upstream the solar wind flow; finally, G22 revealed the existence of a strong coupling between the crustal field influence on the BS and the ionosphere, through a strong correlation with the Total Electronic Content that is a tracer of the ionospheric variability , but also for the thermosphere‐ionosphere coupling ; a partwith reduced solar wind dynamic pressure (second order importance): the solar wind dynamic pressure is widely known as the major driver of Earth's bow shock, and also plays a significant role at Mars; an increased steady incident pressure indeed compresses the whole induced magnetosphere, and thus pushes the BS closer to the planet, while a variable pressure can also induce more complex phenomena such as transients with Hot Flow Anomalies as observed at Earth , or significantly reduced after controlling for the first order importance drivers and for the θbn influence with using partial correlations.other variables possibly influence the MAVEN BS location, such as the IMF intensity, clock angle, or other IMF orientation angles that could generate asymmetries of the BS surface; large cone angles may for example, in principle be related to increased Based on the previous analyses ‐ direct analysis of the drivers, as well as partial correlations or model selection methods such as AIC or LASSO methods ‐, the statistical study of the Martian BS crossings datasets by MAVEN and MEX shows that the terminator distance of the BS increases:RTD dataset with the following procedure: we first perform a regression of the RTD dataset with respect to the various possible drivers and identify the slope value associated with a specific driver we want to test ; then we modified the specific influence of this specific driver, by removing (little influence dataset) from the real RTD dataset, adding 10 times (strong influence dataset), or adding twice with also noise (noisy moderate influence dataset) the slope associated with this specific driver. When applying the partial correlations, AIC and LASSO methods to these three artificial RTD datasets , one gets results that are coherent with the expectations: all methods show an increased and coherent ranking of the specific driver for an increased influence of this driver , while the other drivers influence keeps unchanged. This simple test thus confirms these methods are able to identify any significant driver that impacts the RTD dataset, that is, any driver that shows a recurrent pattern, whatever its nature and its reference coordinate system . These methods are more powerful than direct analysis regarding the detection of minor drivers whose influence may be hidden/underestimated/overestimated based on simple direct analysis due to cross correlations with major drivers.We showed in the previous sections how to direct simple analysis can lead to an underestimation or an overestimation of the influence of minor drivers when these are cross correlated with major drivers. The use of specific methods such as partial correlations, AIC or LASSO, allows for an efficient disentangling of the various drivers at play. To check the efficiency of these methods, we performed a simple test: we kept for MAVEN the real driver's datasets (thus keeping the existing cross correlations), but artificially modified the real Parametrizing in detail the BS location as a function of its physical drivers is beyond the scope of this paper since this needs a detailed theoretical understanding of the response of the BS to each driver and to a combination of these drivers that may be inter‐related as demonstrated above. However, previous authors tried to infer functional forms of the influence of the main drivers on the plasma boundaries locations, either from empirical relations or theoretical developments, and it may thus be interesting to compare some of these results with our analysis.bLASSOi with direct slopes of classic regressions (bi in 2.2), one simply needs to divide the LASSO coefficient of a specific variable by the standard deviation σi of the variable (or of its logarithm if a power law function is considered) since the variables are standardized in the LASSO approach: bi=bLASSOi/σi.One can in particular compare the LASSO regression results Table  that proPSW−1/6 SW dynamic pressure dependence for the nose distance of the Earth's magnetopause and BS, later used at different planets including Mars  based on a proxy of the IMF at Mars in the absence of direct measurement by MEX. The MAVEN LASSO coefficients for a linear model (RTD = a + ∑iXi ⋅ bi) lead to a slope for Mms of −0.06 that is of the same order as the −0.1 slope by Edberg et al.  while it is the contrary for MAVEN as discussed in the previous sections. If we consider a purely local influence of the crustal fields by taking the local crustal field pressure instead of the angular distance , the power law index associated is reduced by a factor of 4 and thus becomes smaller  than the c et al.  results θbn), and may be strongly cross‐correlated as shown above. Each driver also possesses its own timescales, for example, EUV impacting in general at larger timescales than the solar wind dynamic pressure or magnetosonic Mach number. Besides, our limited time resolution of the characteristics of the highly dynamic SW could lead to a slight underestimation of their influence on the BS location.Our results, when compared with a similar approach, are thus consistent with previous Martian studies that mostly focused on individual or few main parameters of influence at a time and for a specific mission. However, it is clear no simple empirical law can account for the whole dynamics of the BS boundary, due to the complex interplay of the physical processes at work that can eventually not be represented by combined linear or power law functions , with dIt would also be interesting to further compare with a consistent method the dynamics of the BS with the dynamics of the inner plasma boundaries  since the dynamics of a boundary may impact another boundary, such as at Earth between the magnetopause and the BS  and external/internal drivers. These techniques are indeed mature and proved efficient in space physics to detect plasma phenomena  or to i5The recent studies of the Martian environment, thanks in particular to the Mars Express and Mars Atmosphere Volatile EvolutioN missions, underline the complexity of the Martian interaction with the incident solar wind, which shapes the plasma boundaries. The bow shock of the planet is known to depend on extreme ultraviolet fluxes and magnetosonic Mach number, while the influence of other possible drivers is less constrained or unknown such as crustal magnetic fields or the Interplanetary Magnetic Field intensity and orientation.In this study, we analyzed the influence of a number of possible internal and external drivers of the shock location, based the first time on two missions (MAVEN and MEX). We used consistent methods that allow us to compare both mission's datasets, as well as to take into account the cross correlations between the drivers that can modify the interpretation of the data.We first analyzed the direct influence of the possible drivers of the shock based on linear correlation coefficients or absolute variations of the shock location. Then, we showed that many of the possible drivers are correlated to each other due to their nature and/or the trajectory of spacecraft. We used a partial correlation approach to investigate in detail the impact of these cross‐correlations on the interpretation of the MAVEN and MEX shock crossings datasets. This approach appears crucial to investigate the influence of second order drivers on the shock that may be linked to each other or to stronger drivers. Finally, we used two known different model selection methods, called Akaike Information Criterion  that could generate asymmetries of the shock surface, for example, a possible close to equator versus pole asymmetry in the Mars Sun Electric field frame as identified at Venus and probably related to mass loading effects  and solar wind dynamic pressure are significant but play a secondary role of similar importance. The results for the magnetosonic Mach number and for the IMF‐related parameters are based only on the MAVEN dataset, due to the absence of a magnetic field for MEX. The analysis of the MAVEN data also shows perpendicular shocks are located at significantly higher distances than parallel shocks, but no conclusive explanation was reached despite a number of possible reasons studied, including an anisotropy fast magnetosonic wave velocity depending on the Providing an analytical proxy of the shock location that accounts precisely for the combined influence of its drivers appears unrealistic given the complex interplay of the internal and external physical processes at work, each of them possessing its own timescales. Better understanding of the plasma boundaries dynamics and in particular how internal stimuli propagate through the Martian induced magnetosphere until the bow shock will need a modeling approach including the complex crustal field topology and temporal dynamics, as well as a comparison of the dynamics of the various boundaries.In future work, we plan to investigate the use of machine learning and artificial intelligence techniques that are efficient to provide automatic catalogs of events  and eventually identify complex nonlinear relationships between the boundary's location and external/internal drivers. These techniques are indeed mature and proved efficient in space physics to detect plasma phenomena or to identify parameters of influence.