Small-scale Magnetic Flux Ropes Research Articles

A Tempered Fractional Kinetic Transport Theory for Energetic Particle Interaction with Quasi-two-dimensional Turbulence in the Large-scale Solar Wind

Observational evidence is accumulating that turbulence in the solar wind is intermittent (non-Gaussian) because of the strong presence of a quasi-two-dimensional (quasi-2D), low-frequency turbulence component containing nonpropagating, closed, small-scale magnetic flux ropes with open meandering field lines in between. le Roux & Zank showed how one can derive fractional focused and Parker-type transport equations that model large-scale anomalous transport in the solar wind as the outcome of energetic particle interaction with quasi-2D turbulence. In this follow-up paper this theory is developed further to address certain limitations. (i) The second moment of the Lévy probability distribution function (PDF) specified in the theory for the particle step size is infinite, indicating unphysical transport. (ii) The expected transition of energetic particle transport from anomalous to normal diffusion beyond a certain critical transport distance was not included. (iii) The competition between anomalous diffusion and advection is not properly sustained at late times. Shortcomings (i) and (ii) are addressed by introducing an exponentially truncated Lévy PDF for the energetic particle step size in the theory, resulting in revised tempered fractional focused and Parker-type transport equations featuring tempered fractional derivatives that enable modeling of tempered Lévy flights. Furthermore, these equations are cast in a tempered fractional telegrapher form to investigate whether the fractional wave equation part of the equation can restore causality in unscattered particle transport during early times and in Lévy flights during intermediate times (Lévy walks). They are also transformed into a tempered fractional Fokker–Planck form to overcome limitation (iii).

Axial Flux Evolution of Small-scale Magnetic Flux Ropes from 0.06 to 10 au

Small-scale magnetic flux ropes (SMFRs) fill much of the solar wind, but their origin and evolution are debated. We apply our recently developed, improved Grad–Shafranov algorithm for the detection and reconstruction of SMFRs to data from Parker Solar Probe, Solar Orbiter, Wind, and Voyager 1 and 2 to detect events from 0.06 to 10 au. We observe that the axial flux density is the same for SMFRs of all sizes at a fixed heliocentric distance but decreases with distance owing to solar wind expansion. Additionally, using the difference in speed between SMFRs, we find that the vast majority of SMFRs will make contact with others at least once during the 100 hr transit to 1 au. Such contact would allow SMFRs to undergo magnetic reconnection, allowing for processes such as merging via the coalescence instability. Furthermore, we observe that the number of SMFRs with higher axial flux increases significantly with distance from the Sun. Axial flux is conserved under solar wind expansion, but the observation can be explained by a model in which SMFRs undergo turbulent evolution by stochastically merging to produce larger SMFRs. This is supported by the observed log-normal axial flux distribution. Lastly, we derive the global number of SMFRs above 1015 Mx near the Sun to investigate whether SMFRs begin their journey as small-scale solar ejections or are continuously generated within the outer corona and solar wind.

A Closer Look at Small-scale Magnetic Flux Ropes in the Solar Wind at 1 au: Results from Improved Automated Detection

Small-scale interplanetary magnetic flux ropes (SMFRs) are similar to ICMEs in magnetic structure, but are smaller and do not exhibit coronal mass ejection plasma signatures. We present a computationally efficient and GPU-powered version of the single-spacecraft automated SMFR detection algorithm based on the Grad–Shafranov (GS) technique. Our algorithm can process higher resolution data, eliminates selection bias caused by a fixed 〈B〉 threshold, has improved detection criteria demonstrated to have better results on an MHD simulation, and recovers full 2.5D cross sections using GS reconstruction. We used it to detect 512,152 SMFRs from 27 yr (1996–2022) of 3 s cadence Wind measurements. Our novel findings are the following: (1) the SMFR filling factor (∼ 35%) is independent of solar activity, distance to the heliospheric current sheet, and solar wind plasma type, although the minority of SMFRs with diameters greater than ∼0.01 au have a strong solar activity dependence; (2) SMFR diameters follow a log-normal distribution that peaks below the resolved range (≳104 km), although the filling factor is dominated by SMFRs between 105 and 106 km; (3) most SMFRs at 1 au have strong field-aligned flows like those from Parker Solar Probe measurements; (4) the radial density (generally ∼1 detected per 106 km) and axial magnetic flux density of SMFRs are higher in faster solar wind types, suggesting that they are more compressed. Implications for the origin of SMFRs and switchbacks are briefly discussed. The new algorithm and SMFR dataset are made freely available.

Calcium Bright Knots and the Formation of Chromospheric Anemone Jets on the Sun

Space-based observations show that the solar atmosphere from the solar chromosphere to the solar corona is filled with small-scale jets and is linked with small-scale explosions. These jets may be produced by mechanisms similar to those of large-scale flares and such jets may be related to the heating of the corona and chromosphere as well as the acceleration of solar wind. The chromospheric anemone jets on the Sun remain puzzling because their footpoints (or bright knots) have not been well resolved and the formation process of such enigmatic small-scale jets remains unclear. We propose a new model for chromospheric jets using the 3D magnetohydrodynamic simulations, which show that the continuous, upward rising of small-scale twisted magnetic flux ropes in a magnetized solar chromosphere drives small-scale magnetic reconnection and the launching of several small-scale jets during the evolution of the chromospheric anemone jets. Our new, self-consistent, 3D computer modeling of small-scale, but ever-changing flux rope emergence in the magnetized solar atmosphere is fully consistent with observations and provides a universal mechanism for nanoflare and jet formation.

Series of Small-scale Low Plasma β Magnetic Flux Ropes Originating from the Same Longitudinal Region: Parker Solar Probe Observations

In this study, we report on small-scale magnetic flux ropes (SMFRs) observed as a compact series in a narrow Carrington longitudinal range during three Parker Solar Probe (PSP) encounters. First, during ∼1.5 days of PSP's inbound part of Encounter 4, we identified a series of 11 SMFRs within 1.°4 in longitude over the radial distance of ∼8.4 R ☉ (from ∼44 to 35 R ☉). The identified SMFRs lasted from ∼0.5 to 1.8 hr, and adjacent events were separated mostly by a few hours and up to ∼6.5 hr at the longest, but some events were very closely spaced with intervals of a few ∼tens of minutes or less apart. Most of the identified SMFRs are successfully fitted to the force-free model. The SMFRs are clearly distinguished from the surroundings by a notable reduction in plasma β, which itself was comparably low (less than unity) in the background plasma. Furthermore, the magnetic field and plasma flow within the SMFRs fluctuated significantly less than the more turbulent surroundings. The fluctuations in the surrounding medium exhibited occasional Br polarity reversal (possibly switchbacks) and were Alfvénic to a large extent with far weaker compressional components. The majority of these key features with some differences have also been found in the series of SMFRs and their surroundings identified within 1.°3 or less in longitude during Encounters 1 and 5. We speculate that these SMFRs were repetitively generated by successive reconnection within a very narrow angular zone located close to the Sun but not necessarily at the same radial position.

A Machine Learning Approach to Understanding the Physical Properties of Magnetic Flux Ropes in the Solar Wind at 1 au

Interplanetary magnetic flux ropes (MFRs) are commonly observed structures in the solar wind, categorized as magnetic clouds (MCs) and small-scale MFRs (SMFRs) depending on whether they are associated with coronal mass ejections. We apply machine learning to systematically compare SMFRs, MCs, and ambient solar wind plasma properties. We construct a data set of 3-minute averaged sequential data points of the solar wind’s instantaneous bulk fluid plasma properties using about 20 years of measurements from Wind. We label samples by the presence and type of MFRs containing them using a catalog based on Grad–Shafranov (GS) automated detection for SMFRs and NASA's catalog for MCs (with samples in neither labeled non-MFRs). We apply the random forest machine learning algorithm to find which categories can be more easily distinguished and by what features. MCs were distinguished from non-MFRs with an area under the receiver-operator curve (AUC) of 94% and SMFRs with an AUC of 89%, and had distinctive plasma properties. In contrast, while SMFRs were distinguished from non-MFRs with an AUC of 86%, this appears to rely solely on the 〈B〉 > 5 nT threshold applied by the GS catalog. The results indicate that SMFRs have virtually the same plasma properties as the ambient solar wind, unlike the distinct plasma regimes of MCs. We interpret our findings as additional evidence that most SMFRs at 1 au are generated within the solar wind. We also suggest that they should be considered a salient feature of the solar wind’s magnetic structure rather than transient events.

Nature of Turbulence inside Small-scale Magnetic Flux Ropes near the Sun: Parker Solar Probe Observations

In this study, we probe the turbulence characteristic within the small-scale magnetic flux ropes (SSMFRs) close to the Sun using Parker Solar Probe (PSP) magnetic field data. The study includes 50 SSMFRs observed by PSP during Encounter 1, 2, and 3 between 35.74 R ☉ and 142.29 R ☉ distance from the Sun. We observed that the average spectral index for all the selected SSMFR events is –1.49 ± 0.21. In line with expectations, we also saw average ∣σ m ∣ values close to zero throughout the inertial range. We also observed that the size of the eddy at the highest frequency is much smaller than the size of the SSMFRs, indicating anisotropy within it. Thus, our finding supports anisotropic models that feature the Iroshnikov–Kraichnan index. Our findings agree with the turbulence properties of the solar wind near the Sun. We also observe low ∼0.1 compressibility, indicating SSMFRs are dominant by Alfvénic fluctuations. In light of this, we believe such an incompressible MHD spectrum results from nonlinear interactions between Alfvénic fluctuations. As a result, our research contributes to understanding the energy cascade process and its transport in solar plasma within the inner heliosphere.

Variability of the slow solar wind: New insights from modelling and PSP-WISPR observations

Aims. We analyse the signature and origin of transient structures embedded in the slow solar wind, and observed by the Wide-Field Imager for Parker Solar Probe (WISPR) during its first ten passages close to the Sun. WISPR provides a new in-depth vision on these structures, which have long been speculated to be a remnant of the pinch-off magnetic reconnection occurring at the tip of helmet streamers. Methods. We pursued the previous modelling works of Réville et al. (2020, ApJ, 895, L20; 2022, A&A, 659, A110) that simulate the dynamic release of quasi-periodic density structures into the slow wind through a tearing-induced magnetic reconnection at the tip of helmet streamers. Synthetic WISPR white-light (WL) images are produced using a newly developed advanced forward modelling algorithm that includes an adaptive grid refinement to resolve the smallest transient structures in the simulations. We analysed the aspect and properties of the simulated WL signatures in several case studies that are typical of solar minimum and near-maximum configurations. Results. Quasi-periodic density structures associated with small-scale magnetic flux ropes are formed by tearing-induced magnetic reconnection at the heliospheric current sheet and within 3 − 7 R⊙. Their appearance in WL images is greatly affected by the shape of the streamer belt and the presence of pseudo-streamers. The simulations show periodicities on ≃90 − 180 min, ≃7 − 10 h, and ≃25 − 50 h timescales, which are compatible with WISPR and past observations. Conclusions. This work shows strong evidence for a tearing-induced magnetic reconnection contributing to the long-observed high variability of the slow solar wind.

Anomalous Transport of Energetic Particles Interacting with Dynamic Small-Scale Magnetic Flux Rope Structures

le Roux and Zank [25] showed previously how one can derive from first principles a pitch-angle dependent fractional diffusion-advection kinetic equation to model the anomalous diffusion of energetic particles interacting with small-scale magnetic flux ropes (SMFRs) in the inner heliosphere on the basis of the standard focused transport equation. This equation has the following limitations: (1) The asymptotic power law of a Lévy distribution was specified to model the non-Gaussian statistics of the disturbed energetic particle trajectories generated during energetic particle interaction with numerous SMFRs. The second moment (variance) and higher moments of the Lévy distribution are infinite, indicating over-efficient non-local transport that is scale-free. (2) The theory does not naturally allow for a transition of anomalous transport to normal diffusion, or to a different anomalous diffusion state. An outline of a derivation is presented in which an exponentially truncated Lévy distribution was specified instead, resulting in a tempered fractional diffusion-advection kinetic equation that addresses these two concerns.

Statistical Study of Ejections in Coronal Hole Regions As Possible Sources of Solar Wind Switchbacks and Small-scale Magnetic Flux Ropes

The omnipresence of transient fluctuations in the solar wind, such as switchbacks (SBs) and small-scale magnetic flux ropes (SMFRs), have been well observed by the in situ observation of Parker Solar Probe (PSP), yet their sources are not clear. Possible candidates fall into two categories: solar origin and in situ generation in the solar wind. Among the solar-origin scenarios, the small-scale activities (such as ejections and eruptions) in coronal hole (CH) regions, where solar wind originates, are suggested as candidates. Using full-disk extreme ultraviolet images from Atmospheric Imaging Assembly on board the Solar Dynamic Observatory, we identify small-scale ejections in CH regions during PSP Encounters 5, 7, and 8, and study their statistical properties. These ejections belong to two categories: standard jets and blowout jets. With 27,832 ejections identified in 24 days (about 2/3 of them are blowout jets), we updated the expected frequency for PSP to detect their counterparts in the heliospace. The ejections we identified are comparable to the frequency of PSP-detected SMFRs, but they are insufficient to serve as the only producer of SBs or SB patches. Certain smaller events missed by this study, such as jetlets, may fill the gap.

Parallel and Momentum Superdiffusion of Energetic Particles Interacting with Small-scale Magnetic Flux Ropes in the Large-scale Solar Wind

A recently developed time-dependent fractional Parker transport equation is solved to investigate the parallel and momentum superdiffusion of energetic charged particles in an inner heliospheric region containing dynamic small-scale flux ropes (SMFRs). Both types of superdiffusive transport are investigated with fractional transport terms containing a fractional time integral combined with normal spatial or momentum derivatives. Just as for normal diffusion, accelerated particles form spatial peaks with a maximum amplification factor that increases with particle energy. Instead of growth of the spatial peaks until a steady state is reached as for normal diffusion, parallel superdiffusion causes the peaks to dissipate into plateaus followed by a rollover at late times. The peaks dissipate at a faster rate when parallel transport is more superdiffusive. Furthermore, the accelerated particle spectral distribution function inevitably becomes an f 0 ∝ p −3 spectrum at late times in the test particle limit near the particle source despite the potential for spectral steepening from other transport terms. All this is a product of the growing domination of parallel spatial and especially momentum superdiffusion over other transport terms with time. Such extreme late time effects can be avoided by a transition to a normal diffusive state. Finally, fitting spatial peaks observed during SMFR acceleration events with the solution of the fractional Parker transport equation can potentially be used as a diagnostic for estimating the level of spatial and momentum superdiffusion in these events and how the levels of superdiffusion vary with distance from the Sun.

Small-scale Magnetic Flux Ropes in Stream Interaction Regions from Parker Solar Probe and Wind Spacecraft Observations

Using in situ measurements from the Parker Solar Probe and Wind spacecraft, we investigate the small-scale magnetic flux ropes (SFRs) and their properties inside stream interaction regions (SIRs). Within SIRs from ∼0.15 to 1 au, SFRs are found to exist in a wide range of solar wind speeds with more frequent occurrences after the stream interface, and the Alfvénicity of these structures decreases significantly with increasing heliocentric distances. Furthermore, we examine the variation of five corresponding SIRs from the same solar sources. The enhancements of suprathermal electrons within these SIRs persist at 1 au and are observed multiple times. An SFR appears to occur repeatedly with the recurring SIRs and is traversed by the Wind spacecraft at least twice. This set of SFRs has similarities in variations of the magnetic field components, plasma bulk properties, density ratio of solar wind alpha and proton particles, and unidirectional suprathermal electrons. We also show, through the detailed time-series plots and Grad–Shafranov reconstruction results, that they possess the same chirality and carry comparable amounts of magnetic flux. Lastly, we discuss the possibility for these recurring SFRs to be formed via interchange reconnection, maintain the connection with the Sun, and survive up to 1 au.

Investigating Particle Acceleration by Dynamic Small-scale Flux Ropes behind Interplanetary Shocks in the Inner Heliosphere

We recently extended our Parker-type transport equation for energetic particle interaction with numerous dynamic small-scale magnetic flux ropes (SMFRs) to include perpendicular diffusion in addition to parallel diffusion. We present a new analytical solution to this equation assuming heliocentric spherical geometry with spherical symmetry for all SMFR acceleration mechanisms present in the transport theory. With the goal of identifying the dominant mechanism(s) through which particles are accelerated by SMFRs, a search was launched to identify events behind interplanetary shocks that could be explained by our new solution and not classical diffusive shock acceleration. Two new SMFR acceleration events were identified in situ for the first time within heliocentric distances of 1 astronomical unit (au) in Helios A data. A Metropolis–Hastings algorithm is employed to fit the new solution to the energetic proton fluxes so that the relative strength of the transport coefficients associated with each SMFR acceleration mechanism can be determined. We conclude that the second-order Fermi mechanism for particle acceleration by SMFRs is more important than first-order Fermi acceleration due to the mean compression of the SMFRs regions during these new events. Furthermore, with the aid of SMFR parameters determined via the Grad–Shafranov reconstruction method, we find that second-order Fermi SMFR acceleration is dominated by the turbulent motional electric field parallel to the guide/background field. Finally, successful reproduction of energetic proton flux data during these SMFR acceleration events also required efficient particle escape from the SMFR acceleration regions.

Near-orthogonal Orientation of Small-scale Magnetic Flux Ropes Relative to the Background Interplanetary Magnetic Field

Small-scale magnetic flux ropes (SMFRs) have been identified at a large range of heliospheric distances from the Sun. Their features are somewhat similar to those of larger-scale flux rope structures such as magnetic clouds (MCs), while their occurrence rate is far higher. In this work, we examined the orientations of a large number of SMFRs that were identified at 1 au by fitting to the force-free model. We find that, while most of the SMFRs lie mostly close to the ecliptic plane, as previously known, their azimuthal orientations relative to the Sun–Earth line are found largely at two specific angles (slightly less than 45° and 225°). This latter feature in turn leads to a strong statistical trend in which the axis of SMFRs lies at a large tilt angle relative to (most often nearly orthogonal to) the corresponding background interplanetary magnetic field directions in the ecliptic plane. This feature is different from previous reports on SMFRs—and in stark contrast to the cases of MCs. This is an important observational constraint that should be considered for understanding SMFR generation and propagation.

Investigating Superdiffusive Shock Acceleration at a Parallel Shock with a Fractional Parker Equation for Energetic-particle Interaction with Small-scale Magnetic Flux Ropes

It has been suggested before that small-scale magnetic flux rope (SMFR) structures in the solar wind can temporarily trap energetic charged particles. We present the derivation of a new fractional Parker equation for energetic-particle interaction with SMFRs from our pitch-angle-dependent fractional diffusion-advection equation that can account for such trapping effects. The latter was derived previously in le Roux & Zank from the first principles starting with the standard focused transport equation. The new equation features anomalous advection and diffusion terms. It suggests that energetic-particle parallel transport occurs with a decaying efficiency of advection effects as parallel superdiffusion becomes more dominant at late times. Parallel superdiffusion can be linked back to underlying anomalous pitch-angle transport, which might be subdiffusive during interaction with quasi-helical coherent SMFRs. We apply the new equation to time-dependent superdiffusive shock acceleration at a parallel shock. The results show that the superdiffusive-shock-acceleration timescale is fractional, the net fractional differential particle flux is conserved across the shock ignoring particle injection at the shock, and the accelerated particle spectrum at the shock converges to the familiar power-law spectrum predicted by standard steady-state diffusive-shock-acceleration theory at late times. Upstream, as parallel superdiffusion progressively dominates the advection of energetic particles, their spatial distributions decay on spatial scales that grow with time. Furthermore, superdiffusive parallel shock acceleration is found to be less efficient if parallel anomalous diffusion is more superdiffusive, while perpendicular particle escape from the shock, thought to be subdiffusive during SMFR interaction, is reduced when increasingly subdiffusive.

Effects of Pressure Anisotropy on the Geometry of Magnetic Flux Rope

This paper aims to examine the effects of pressure anisotropy on the geometry of magnetic flux rope using the newly developed two-dimensional magnetohydrostatic reconstruction associated with pressure anisotropy. A small-scale magnetic flux rope observed by the Magnetospheric Multiscale mission, in the magnetosheath reconnection outflow during an outbound magnetopause crossing, is demonstrated. At the center of the flux rope, the magnetic field strength was enhanced with decreasing plasma pressure. The entire flux rope was mostly occupied by the pressure anisotropy of p ⊥ > p ∥, where the subscripts ∥ and ⊥ denote the components parallel and perpendicular to the local magnetic field, respectively. The estimated aspect ratio of the width to the length of the flux rope from reconstruction was ∼0.326 for p ⊥ > p ∥ and ∼0.389 for isotropic pressure. By comparing the magnetic field map from the isotropic Grad–Shafranov reconstruction, the results show for p ⊥ > p ∥ that (1) the width of the flux rope is reduced, leading to a small aspect ratio of the flux rope, and (2) the circular field line of the flux rope is contracted. Moreover, an experiment is conducted for p ∥ > p ⊥ by exchanging p ⊥ and p ∥ of the flux rope, for which the isotropic pressure is less affected. The experimental results indicate that the effects of p ∥ > p ⊥ on the geometry of the flux rope are opposite to that of p ⊥ > p ∥. The overall finding may provide new insight into charged particle acceleration within magnetic flux ropes/islands in anisotropic plasmas.

Small-scale Magnetic Flux Ropes and Their Properties Based on In Situ Measurements from the Parker Solar Probe

Abstract We report small-scale magnetic flux ropes via the in situ measurements from the Parker Solar Probe during the first six encounters, and present additional analyses to supplement our prior work in Chen et al. These flux ropes are detected by the Grad–Shafranov-based algorithm, with their durations and scale sizes ranging from 10 s to ≲1 hr and from a few hundred kilometers to 10−3 au, respectively. They include both static structures and those with significant field-aligned plasma flows. Most structures tend to possess large cross helicity, while the residual energy is distributed over wide ranges. We find that these dynamic flux ropes mostly propagate in the antisunward direction relative to the background solar wind, with no preferential signs of magnetic helicity. The magnetic flux function follows a power law and is proportional to scale size. We also present case studies showing reconstructed two-dimensional (2D) configurations, which confirm that both the static and dynamic flux ropes have a common configuration of spiral magnetic field lines (also streamlines). Moreover, the existence of such events hints at interchange reconnection as a possible mechanism for generating flux rope-like structures near the Sun. Lastly, we summarize the major findings, and discuss the possible correlation between these flux rope-like structures and turbulence due to the process of local Alfvénic alignment.

Statistical Study of Small-Scale Interplanetary Magnetic Flux Ropes in the Vicinity of the Heliospheric Current Sheet

The small-scale interplanetary magnetic flux ropes (SIMFRs) are common magnetic structures in the interplanetary space, yet their origination is still an open question. In this article, we surveyed 63 SIMFRs found within 6-day window around the heliospheric current sheet (HCS) and investigated their axial direction, as well as the local normal direction of the HCS. Results showed that the majority (48/63) of the SIMFRs were quasi-parallel to the associated HCS (i.e., the axial direction of SIMFRs was quasi-perpendicular to the normal direction of the associated HCS). They also showed that the SIMFRs quasi-parallel to the associated HCS statistically had shorter duration than the cases quasi-perpendicular. The results indicate that most of these SIMFRs may be generated in the nearby HCSs.

Characteristics of Suprathermal Electrons in Small-Scale Magnetic Flux Ropes and Their Implications on the Magnetic Connection to the Sun

Small-scale magnetic flux ropes (SMFRs) are observed more frequently than larger-scale magnetic flux ropes (e.g., magnetic clouds) in interplanetary space. We selected 235 SMFRs by applying cylindrical linear force-free fitting to 20-year observations of the Wind satellite, which meets the criteria of low beta, low temperature, an enhanced magnetic field, and a rotation feature. By examining the pitch angle distribution of suprathermal electrons for these events, we found that approximately 45.1% of the SMFRs were accompanied by unidirectional beams (strahl). A much smaller percentage of SMFRs (∼10.7%) were associated with bidirectional beams. We also found a small percentage (∼7.2%) of (sunward) conic distributions during SMFR events. Last, the remaining ∼37.0% of SMFRs were associated with complex electron distributions. The unidirectional beams and most of the conics (together corresponding to ∼50% of the total 235 SMFRs) imply open-field SMFRs with only one end connected to the Sun. For ∼37.7% of the unidirectional beam SMFRs, the local IMF field polarity was orthogonal or inverted (possibly due to interchange reconnection). Based on the solar wind conditions around the bidirectional beams, we suggest that more than half of the bidirectional beams were not necessarily closed-field-line SMFRs.

Multifluid MHD Studies of the Ionospheric Magnetic Flux Ropes at Mars

Abstract We use a multifluid MHD model to study the formation of the small-scale magnetic flux rope (MFR) in the ionosphere of Mars. It is found that a straight flux tube can turn into a twisted MFR during its sinking into the lower ionosphere, caused by the differential compression at different parts of the tube. Further studies show that the MFR is not a force-free structure, in which there is always a pressure gradient force to balance the magnetic force. Moreover, we find that the ionospheric ions inside the MFR can be squeezed out from the two ends of the MFR, with an outflow flux of about 3.5 × 107 cm−2 s−1, which can play an important role for the ion escape from Mars. In addition, we find that both the sinking speed and the ratio of the ion gyroradius to the tube radius can affect the characteristics of MFR. An MFR can be more quickly formed for a larger sinking speed, but it can be also more quickly filled by the ambient ions. Oppositely, the MFR is more difficult to be filled when the ratio of the ion gyroradius to the tube radius is smaller. Once the MFR is filled, the compression is prevented by an outward pressure gradient force, and both the field strength and the rope helicity begin to decrease. Our results can explain the different altitude dependences of MFRs observed at Mars and Venus.