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Victor Réville

Publications and source records attributed to Victor Réville.

At least 19 recordsLinked to original sources
arXiv2026

Bifurcations in the Heliosphere: Global-scale Coronal Pseudostreamer Evolution and the Creation of Secondary Heliospheric Current Sheets

The heliospheric current sheet (HCS) is the boundary between open magnetic fields of opposite polarity in the solar wind that connect back to their respective open flux regions in the corona. The HCS is often described as a disc-like sheet warped by the combined effects of the inclination of the magnetic equator on the Sun, solar rotation, and solar wind expansion. The HCS is an extension of the helmet streamer belt-the closed-flux region resulting from the global dipole component of the observed photospheric magnetic field. In the absence of a strong dipole, a dominant quadrupole can result in two separate HCSs. Here we report on a set of idealized, axisymmetric solar wind simulations designed to study the transition of the single-HCS corona and inner heliosphere into a multiple-HCS configuration, run with the WindPredict-AW PLUTO MHD code. We systematically vary the relative strengths of the dipole/quadrupole contributions to the global field and show this results in the formation of a large-scale, closed-flux pseudostreamer that interacts with the ambient solar wind. We find that when the dipole and quadrupole magnitudes are approximately equal, our model corona can oscillate between a one- and two-HCS configuration on the timescale of days-to-weeks. We characterize the plasma dynamics of the transition of a global-scale pseudostreamer into a localized, secondary helmet-streamer belt with its own HCS surrounding a newly-opened, opposite-polarity coronal hole. We discuss the implications of our results for solar cycle evolution and the dynamic coupling of the highly structured corona and inner heliosphere.

astro-ph.SR↗
arXiv2026

Statistical Energy Budget of the Solar Wind with Parker Solar Probe and Solar Orbiter : 1 - Unbalanced Total Energy Radial Evolution

The magneto-hydrodynamic (MHD) description of the solar wind has long been regarded as one of the most successful frameworks for investigating solar wind heating and acceleration. Under a set of simplifying assumptions, it provides a reduced large-scale description that is widely used to study the energetic role of electromagnetic fluctuations in the solar wind energy budget. In practice, applying simplification to MHD can result in the loss of some physics compared to an exact set of equations. However, an observational energy budget can help us to verify whether any energetic information has been lost in the process of simplification. To do so, we directly test the validity of the large-scale non-linear ideal MHD energy budget with solar wind observations from Parker Solar Probe (PSP) and Solar Orbiter (SO). After verifying the reliability of the measurements through comparison with previous studies of similar solar wind properties, we test the conservation of the total energy predicted by the MHD theory. We find that the average total energy increases with radial distance by $56\%$ ($\pm9\%$) between 14 and 203 solar radii. The increase in kinetic energy is not sufficiently compensated by the decrease of the thermal and electromagnetic contributions. We investigate the major sources of observational uncertainty and find that they cannot account for the observed energy increase. These results suggest that the large-scale theoretical description may neglect an energetically significant contribution to solar wind acceleration. Assuming that such a contribution exists, we predict that this extra energy would scale with radial distance as $r^{-0.54\pm0.38}$.

astro-ph.SR↗
arXiv2026

Evolution and Impact of Switchbacks Throughout the Heliosphere

Magnetic switchbacks are large-amplitude fluctuations in the interplanetary magnetic field, and appear frequently in the near-Sun solar wind explored recently by Parker Solar Probe: these new observations have prompted many new studies into their properties and origins. Here, we first review what is known about how switchbacks evolve as they travel away from the Sun: both in terms of their expansion-driven growth and their decay due to various processes like turbulence, reconnection, dispersion, parametric instability, and interaction with interplanetary shocks. We then review the current state of knowledge on how switchbacks impact the physics of the solar wind as a whole: in terms of the turbulent cascade, acceleration and heating of the wind, modification of the open solar flux and scattering of energetic particles. Finally, we suggest future studies to further our understanding of switchback evolution and impacts on the heliosphere.

physics.space-ph↗
arXiv2026

On the Radial Evolution of the Solar Wind : The Source Alignment Method Applied to Parker Solar Probe and Solar Orbiter Observations

The properties of the solar wind, as measured in situ throughout the heliosphere, depend both on the characteristics of its coronal source and on the intrinsic processes governing its interplanetary evolution. Recently, radial and Parker spiral alignment techniques have been applied to Parker Solar Probe (PSP) and Solar Orbiter (SO) observations to investigate the radial evolution of the same solar wind parcel. These studies have shown that the solar wind can undergo significant acceleration even beyond its primary acceleration region (i.e., above 15 solar radii). However, such radial and Parker spiral alignments are rare in practice, which limits the statistical significance and general applicability of the results. We introduce a new source alignment technique designed to overcome these limitations. Using magnetic backmapping, we associate similar solar wind streams observed by the two spacecraft based on the proximity of their photospheric footpoints, combined with additional in-situ stream similarity criteria. Applying the source alignment method to PSP and SO observations, we identify a total of 548 alignment intervals, each lasting 30 minutes. By constructing statistics over all alignments, we find that the solar wind speed increases by an average of 45% per radial decade (approximately 147 km/s) between the two probes. This result demonstrates that solar wind acceleration in the inner heliosphere remains significant compared to that occurring below 15 solar radii. Among the different studied plasma parameters, the radial evolution of the electron temperature and plasma density, show the strongest anti-correlation with the increase in bulk velocity.

astro-ph.SR↗
arXiv2026

Magnetic switchback formation: a review of proposed mechanisms

Magnetic switchbacks are large amplitude deflections of the magnetic field within the solar wind. They are Alfvénic in character and so are associated with a spike in velocity and a generally small variation in local plasma density. Early orbits of Parker Solar Probe revealed that the solar wind near the Sun is dominated by these structures, and therefore, they may be playing an important role in the energy budget and acceleration of the young solar wind. In this review, we present an overview of different mechanisms that have been proposed for how switchbacks could be formed. We group the mechanisms by whether they predominantly act in the low solar atmosphere or within the solar wind (in situ). We focus on mechanisms that can create reversals of the ambient magnetic field direction and, thus, account for the most extreme perturbations. The general consensus is that mechanisms in the lower solar atmosphere do not form such reversals on their own but provide the seed perturbations, flows, or particle beams necessary for in situ mechanisms to create switchbacks within the solar wind. Switchback observations thus likely contain an imprint of the coronal source of the seed perturbation or flow, which is evolved further locally by one of several plausible in situ mechanisms. We discuss the strengths and weaknesses of each mechanism and outline future observational and theoretical tests that could help differentiate between them.

astro-ph.SR↗
arXiv2025

On the Propagation and Damping of Alfvenic Fluctuations in the Outer Solar Corona and Solar Wind

We analyze \textit{Parker Solar Probe} and \textit{Solar Orbiter} observations to investigate the propagation and dissipation of Alfvénic fluctuations from the outer corona to 1~AU. Conservation of wave-action flux provides the theoretical baseline for how fluctuation amplitudes scale with the Alfvén Mach number $M_a$, once solar-wind acceleration is accounted for. Departures from this scaling quantify the net balance between energy injection and dissipation. Fluctuation amplitudes follow wave-action conservation for $M_a < M_a^{b}$ but steepen beyond this break point, which typically lies near the Alfvén surface ($M_a \approx 1$) yet varies systematically with normalized cross helicity $σ_c$ and fluctuation scale. In slow, quasi-balanced streams, the transition occurs at $M_a \lesssim 1$; in fast, imbalanced wind, WKB-like scaling persists to $M_a \gtrsim 1$. Outer-scale fluctuations maintain wave-action conservation to larger $M_a$ than inertial-range modes. The turbulent heating rate $Q$ is largest below $M_a^{b}$, indicating a preferential heating zone shaped by the degree of imbalance. Despite this, the Alfvénic energy flux $F_a$ remains elevated, and the corresponding damping length $Λ_d = F_a/Q$ remains sufficiently large to permit long-range propagation before appreciable damping occurs. Normalized damping lengths $Λ_d/H_A$, where $H_A$ is the inverse Alfvén-speed scale height, are near unity for $M_a \lesssim M_a^{b}$ but decline with increasing $M_a$ and decreasing $U$, implying that incompressible reflection-driven turbulence alone cannot account for the observed dissipation. Additional damping mechanisms -- such as compressible effects -- are likely required to account for the observed heating rates across much of the parameter space.

astro-ph.SR↗
arXiv2025

A parametric study of solar wind properties and composition using fluid and kinetic solar wind models

The physical processes in the solar corona that shape the solar wind remain an active research topic. Modeling efforts have shown that energy and plasma exchanges near the transition region plays a crucial role in modulating solar wind properties. Although these regions cannot be measured in situ, plasma parameters can be inferred from coronal spectroscopy and ionization states of heavy ions, which remain unchanged as they escape the corona. We introduce a new solar wind model extending from the chromosphere to the inner heliosphere, capturing thermodynamic coupling across atmospheric layers. By including neutral and charged particle interactions, we model the transport and ionisation processes of the gas through the transition region, the corona and into the solar wind. Instead of explicitly modeling coronal heating, we link its spatial distribution to large-scale magnetic field properties. Our results confirm that energy deposition strongly affects wind properties through key mechanisms involving chromospheric evaporation, thermal expansion, and magnetic flux expansion. For sources near active regions, the model predicts significant solar wind acceleration, with plasma outflows comparable to those inferred from coronal spectroscopy. For winds from large coronal holes, the model reproduces the observed anticorrelation between charge state and wind speed. However, the predicted charge state ratios are overall lower than observed. Inclusion of a population of energetic electrons enhances both heavy ion charge states and solar wind acceleration, improving agreement with observations.

astro-ph.SR↗
arXiv2025

Magnetized winds of M-type stars and star-planet magnetic interactions: uncertainties and modeling strategy

M-type stars are the most common stars in the universe. They are ideal hosts for the search of exoplanets in the habitable zone (HZ), as their small size and low temperature make the HZ much closer in than their solar twins. Harboring very deep convective layers, they also usually exhibit very intense magnetic fields. Understanding their environment, in particular their coronal and wind properties, is thus very important, as they might be very different from what is observed in the solar system. The mass loss rate of M-type stars is poorly known observationally, and recent attempts to estimate it for some of them (TRAPPIST-1, Proxima Cen) can vary by an order of magnitude. In this work, we revisit the stellar wind properties of M-dwarfs in the light of the latest estimates of $\dot{M}$ through Lyman-$α$ absorption at the astropause and slingshot prominences. We outline a modeling strategy to estimate the mass loss rate, radiative loss and wind speed, with uncertainties, based on an Alfvén wave driven stellar wind model. We find that it is very likely that several TRAPPIST-1 planets lie within the Alfvén surface, which imply that these planets experience star-planet magnetic interactions (SPMI). We also find that SPMI between Proxima Cen b and its host star could be the reason of recently observed radio emissions.

astro-ph.SR↗
arXiv2025

Evolution of MHD turbulence in the expanding solar wind: residual energy and intermittency

We conduct 3D magnetohydrodynamic (MHD) simulations of decaying turbulence in the solar wind context. To account for the spherical expansion of the solar wind, we implement the expanding box model. The initial turbulence comprises uncorrelated counter-propagating Alfvén waves and exhibits an isotropic power spectrum. Our findings reveal the consistent generation of negative residual energy whenever nonlinear interactions are present, independent of the normalized cross helicity $σ_c$ and compressibility. The spherical expansion facilitates this process. The resulting residual energy is primarily distributed in the perpendicular direction, with $[S_2(\mathbf{b})-S_2(\mathbf{u})] \propto l_\perp$ or equivalently $-E_r \propto k_\perp^{-2}$. Here $S_2(\mathbf{b})$ and $S_2(\mathbf{u})$ are second-order structure functions of magnetic field and velocity respectively. In most runs, $S_2(\mathbf{b})$ develops a scaling relation $S_2(\mathbf{b}) \propto l_\perp^{1/2}$ ($E_b \propto k_\perp^{-3/2}$). In contrast, $S_2(\mathbf{u})$ is consistently shallower than $S_2(\mathbf{b})$, which aligns with in-situ observations of the solar wind. We observe that the higher-order statistics of the turbulence, which act as a proxy for intermittency, depend on the initial $σ_c$ and are strongly affected by the expansion effect. Generally, the intermittency is more pronounced when the expansion effect is present. Finally, we find that in our simulations although the negative residual energy and intermittency grow simultaneously as the turbulence evolves, the causal relation between them seems to be weak, possibly because they are generated on different scales.

astro-ph.SR↗
arXiv2025

On stellar hotspots due to star-planet magnetic interactions: How much power can actually be transmitted to the chromosphere?

Star-Planet Magnetic Interactions (SPMI) have been proposed as a mechanism for generating stellar hot-spots with energy outputs on the order of $10^{19-21}$ watts. This interaction is primarily believed to be mediated by Alfvén waves propagating towards the star. The stellar atmosphere dictates where and how much of this incoming energy can actually be deposited as heat. The stellar transition region separating the chromosphere from the corona of cool stars gives rise to a significant variation of the Alfvén speed over a short distance, and therefore a reflection of the Alfvén waves at the transition region is naturally expected. We aim to characterize the efficiency of energy transfer due to SPMI by quantifying a frequency dependent reflection of the wave energy at the stellar transition region and its transmission to the stellar chromosphere. Magnetohydrodynamic simulations are employed to model the frequency-dependent propagation of Alfvén waves through a realistic background stellar wind profile. The transmission efficiency as a function of the wave frequency is quantified, and further analysis is conducted to characterize the overall energy transfer efficiency of SPMI in several candidate systems where chromospheric hotspots have been tentatively detected. Low-frequency waves experience greater reflection compared to high-frequency waves, resulting in reduced energy transfer efficiency for lower frequencies. Conversely, the parametric decay instability of Alfvén waves substantially diminishes the energy transfer efficiency at higher frequencies. As a result, there exists a frequency range where energy transfer is most efficient. A significant fraction of the Alfvén wave energy is reflected at the stellar transition region, and in most realistic scenarios, the transmission efficiency to the chromosphere is found to be approximately 10 percent.

astro-ph.SR↗
arXiv2024

Dominance of 2-Minute Oscillations near the Alfvén Surface

Alfvén waves, considered one of the primary candidates for heating and accelerating the fast solar wind, are ubiquitous in spacecraft observations, yet their origin remains elusive. In this study, we analyze data from the first 19 encounters of the Parker Solar Probe (PSP) and report dominance of 2-minute oscillations near the Alfvén surface. The frequency-rectified trace magnetic power spectral density (PSD) of these oscillations indicates that the fluctuation energy is concentrated around 2 minutes for the ``youngest'' solar wind. Further analysis using wavelet spectrograms reveals that these oscillations primarily consist of outward-propagating, spherically polarized Alfvén wave bursts. Through Doppler analysis, we show that the wave frequency observed in the spacecraft frame can be mapped directly to the launch frequency at the base of the corona, where previous studies have identified a distinct peak around 2 minutes ($\sim 8$ mHz) in the spectrum of swaying motions of coronal structures observed by SDO AIA. These findings strongly suggest that the Alfvén waves originate from the solar atmosphere. Furthermore, statistical analysis of the PSD deformation beyond the Alfvén surface supports the idea of dynamic formation of the otherwise absent $1/f$ range in the solar wind turbulence spectrum.

astro-ph.SR↗
arXiv2024

Impact of far-side structures observed by Solar Orbiter on coronal and heliospheric wind simulations

Solar Orbiter provides unique capabilities to understand the heliosphere. In particular, it has made observations of the far-side of the Sun and provides unique information to improve space weather monitoring. We aim to quantify how far-side data will affect simulations of the corona and the interplanetary medium, especially in the context of space weather forecasting. We focused on a time period with a single sunspot emerging on the far-side in February 2021. We used two different input magnetic maps: one with the far-side active region and one without. We used three different coronal models: a semi-empirical model (potential field source surface or PFSS) and two different magnetohydrodynamic models (Wind Predict and Wind Predict-AW). We compared all the models with both remote sensing and in situ observations. We find that the inclusion of the far-side active region in the various models has a small local impact due to the limited amount of flux of the sunspot (at most 8% of the total map flux), which leads to coronal hole changes of around 7% for all models. Interestingly, there is a more global impact on the magnetic structure seen in the current sheet, with clear changes in the coronal hole boundaries visible in extreme ultra-violet (EUV) on the western limb. For the Wind Predict-AW model, we demonstrate that the inclusion of the far-side data improves both the structure of the streamers and the connectivity to the spacecraft. In conclusion, the inclusion of a single far-side active region may have a small local effect with respect to the total magnetic flux, but it has global effects on the magnetic structure, and thus it must be taken into account to accurately describe the Sun-Earth connection. The flattening of the heliospheric current sheet for all models reveals an increase of the source surface height, which affects the open and closed magnetic field line distributions.

astro-ph.SR↗
arXiv2023

Modeling the formation and evolution of solar wind microstreams: from coronal plumes to propagating Alfvénic velocity spikes

We investigate the origin of mesoscale structures in the solar wind called microstreams defined as enhancements in solar wind speed and temperature that last several hours. They were first clearly detected in Helios and Ulysses solar wind data and are now omnipresent in the "young" solar wind measured by Parker Solar Probe and Solar Orbiter. These recent data reveal that microstreams transport a profusion of Alfvénic perturbations in the form of velocity spikes and magnetic switchbacks. In this study we use a very high-resolution 2.5 MHD model of the corona and the solar wind to simulate the emergence of magnetic bipoles interacting with the pre-existing ambient corona and the creation of jets that become microstreams propagating in the solar wind. Our high-resolution simulations reach sufficiently high Lundquist numbers to capture the tearing mode instability that develops in the reconnection region and produces plasmoïds released with the jet into the solar wind. Our domain runs from the lower corona to 20 Rs, this allows us to track the formation process of plasmoïds and their evolution into Alfvénic velocity spikes. We obtain perturbed solar wind flows lasting several hours with velocity spikes occurring at characteristic periodicities of about 19 minutes. We retrieve several properties of microstreams measured in the pristine solar wind by Parker Solar Probe, namely an increase in wind velocity of about 100 km/s during the streams passage together with superposed velocity spikes of also about 100 km/s released into the solar wind.

astro-ph.SR↗
arXiv2023

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

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. We pursued the previous modelling works of Reville (2020b, 2022) 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. 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 3-7Rs. 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 ~90-180min, ~7-10hr and ~25-50hr timescales, which are compatible with WISPR and past observations. This work shows strong evidence for a tearing-induced magnetic reconnection contributing to the long-observed high variability of the slow solar wind.

astro-ph.SR↗
arXiv2023

HelioCast: heliospheric forecasting based on white-light observations of the solar corona. I. Solar minimum conditions

We present a new 3D MHD heliospheric model for space-weather forecasting driven by boundary conditions defined from white-light observations of the solar corona. The model is based on the MHD code PLUTO, constrained by an empirical derivation of the solar wind background properties at 0.1au. This empirical method uses white-light observations to estimate the position of the heliospheric current sheet. The boundary conditions necessary to run HelioCast are then defined from pre-defined relations between the necessary MHD properties (speed, density and temperature) and the distance to the current sheet. We assess the accuracy of the model over six Carrington rotations during the first semester of 2018. Using point-by-point metrics and event based analysis, we evaluate the performances of our model varying the angular width of the slow solar wind layer surrounding the heliospheric current sheet. We also compare our empirical technique with two well tested models of the corona: Multi-VP and WindPredict-AW. We find that our method is well suited to reproduce high speed streams, and does -- for well chosen parameters -- better than full MHD models. The model shows, nonetheless, limitations that could worsen for rising and maximum solar activity.

astro-ph.SR↗
arXiv2023

Proton and electron temperatures in the solar wind and their correlations with the solar wind speed

The heating and acceleration of the solar wind remains one of the fundamental unsolved problems in heliophysics. It is usually observed that the proton temperature $T_i$ is highly correlated with the solar wind speed $V_{SW}$, while the electron temperature $T_e$ shows anti-correlation or no clear correlation with the solar wind speed. Here we inspect both Parker Solar Probe (PSP) and WIND data and compare the observations with simulation results. PSP observations below 30 solar radii clearly show a positive correlation between proton temperature and wind speed and a negative correlation between electron temperature and wind speed. One year (2019) of WIND data confirm that proton temperature is positively correlated with solar wind speed, but the electron temperature increases with the solar wind speed for slow wind while it decreases with the solar wind speed for fast wind. Using a one-dimensional Alfvén-wave-driven solar wind model with different proton and electron temperatures, we for the first time find that if most of the dissipated Alfvén wave energy heats the ions instead of electrons, a positive $T_i-V_{SW}$ correlation and a negative $T_e-V_{SW}$ correlation arise naturally. If the electrons gain a small but finite portion of the dissipated wave energy, the $T_e-V_{SW}$ correlation evolves with radial distance to the Sun such that the negative correlation gradually turns positive. The model results show that Alfvén waves are one of the possible explanations of the observed evolution of proton and electron temperatures in the solar wind.

astro-ph.SR↗
arXiv2022

Acceleration of polytropic solar wind: Parker Solar Probe observation and one-dimensional model

The acceleration of the solar coronal plasma to supersonic speeds is one of the most fundamental yet unresolved problem in heliophysics. Despite the success of Parker's pioneering theory on an isothermal solar corona, the realistic solar wind is observed to be non-isothermal, and the decay of its temperature with radial distance usually can be fitted to a polytropic model. In this work, we use Parker Solar Probe data from the first nine encounters to estimate the polytropic index of solar wind protons. The estimated polytropic index varies roughly between 1.25 and 1.5 and depends strongly on solar wind speed, faster solar wind on average displaying a smaller polytropic index. We comprehensively analyze the 1D spherically symmetric solar wind model with polytropic index $γ\in [1,5/3]$. We derive a closed algebraic equation set for transonic stellar flows, i.e. flows that pass the sound point smoothly. We show that an accelerating wind solution only exists in the parameter space bounded by $C_0/C_g < 1$ and $(C_0/C_g)^2 > 2(γ-1)$ where $C_0$ and $C_g$ are the surface sound speed and one half of the escape velocity of the star, and no stellar wind exists for $γ> 3/2$. With realistic solar coronal temperatures, the observed solar wind with $γ\gtrsim 1.25$ cannot be explained by the simple polytropic model. We show that mechanisms such as strong heating in the lower corona that leads to a thick isothermal layer around the Sun and large-amplitude Alfvén wave pressure are necessary to remove the constraint in $γ$ and accelerate the solar wind to high speeds.

astro-ph.SR↗
arXiv2022

The preferential orientation of magnetic switchbacks, implications for solar magnetic flux transport

We investigate the orientation of the magnetic field deflections in switchbacks (SB) to determine if they are characterised by a possible preferential orientation. We compute the deflection angles of the magnetic field relative to the Parker spiral direction for encounters 1 to 9 of the PSP mission. We first characterize the distribution of these deflection angles for calm solar wind intervals, and assess the precision of the Parker model as a function of distance to the Sun. We then assume that the solar wind is composed of two populations, the background calm solar wind and the population of SB, characterized by larger fluctuations. We model the total distribution of deflection angles we observe in the solar wind as a weighed sum of two distinct normal distributions, each corresponding to one of the populations. We fit the observed data with our model using a MCMC algorithm and retrieve the most probable mean vector and covariance matrix coefficients of the two Gaussian functions, as well as the population proportion. We first observe that the accuracy of the spiral direction in the ecliptic is a function of radial distance, in a manner that is consistent with PSP being near the solar wind acceleration region. We then find that the fitted switchback population presents a systematic bias in its deflections compared to the calm solar wind population. This result holds for all encounters but E6, and regardless of the magnetic field main polarity. This implies a marked preferential orientation of SB in the clockwise direction in the ecliptic plane, and we discuss this result and its implications in the context of the existing switchback formation theories. Finally, we report the observation of a 12-hour patch of SB that systematically deflect in the same direction, so that the magnetic field vector tip within the patch deflects and returns to the Parker spiral within a given plane.

astro-ph.SR↗