Toroidal Flux Ropes Research Articles

On the Influence of the Solar Wind on the Propagation of Earth-impacting Coronal Mass Ejections

Coronal mass ejections (CMEs) are subject to changes in their direction of propagation, tilt, and other properties as they interact with the variable solar wind. We investigated the heliospheric propagation of 15 Earth-impacting CMEs observed during 2010 April to 2018 August in the field of view (FOV) of the Heliospheric Imager (HI) on board the STEREO. About half of the 15 events followed self-similar expansion up to 40 R ⊙. The remaining events showed deflection either in latitude, longitude, or a tilt change. Only 2 events showed significant rotation in the HI1 FOV. We also use toroidal and cylindrical flux rope fitting on the in situ observations of interplanetary magnetic field and solar wind parameters to estimate the tilt at L1 for these 2 events. Although the sample size is small, this study suggests that CME rotation is not very common in the heliosphere. We attributed the observed deflections and rotations of CMEs to a combination of factors, including their interaction with the ambient solar wind and the influence of the ambient magnetic field. These findings contribute to our understanding of the complex dynamics involved in CME propagation and highlight the need for comprehensive modeling and observational studies to improve space weather prediction. In particular, HI observations help us to connect observations near the Sun and near the Earth, improving our understanding of how CMEs move through the heliosphere.

Rotation of a Stealth CME on 2012 October 5 Observed in the Inner Heliosphere

Coronal mass ejections (CMEs) are subject to changes in their direction of propagation, tilt, and other properties. This is because CMEs interact with the ambient solar wind and other large-scale magnetic field structures. In this work, we report on the observations of the 2012 October 5 stealth CME using coronagraphic and heliospheric images. We find clear evidence of a continuous rotation of the CME, i.e., an increase in the tilt angle, estimated using the graduated cylindrical shell (GCS) reconstruction at different heliocentric distances, up to 58 R ⊙. We find a further increase in the tilt at L1 estimated from the toroidal and cylindrical flux rope fitting on the in situ observations of interplanetary magnetic field (IMF) and solar wind parameters. This study highlights the importance of observations of Heliospheric Imager (HI), on board the Solar Terrestrial Relations Observatory. In particular, the GCS reconstruction of CMEs in the HI field of view promises to bridge the gap between the near-Sun and in situ observations at the L1. The changes in the CME tilt have significant implications for the space weather impact of stealth CMEs.

Modeling a Coronal Mass Ejection from an Extended Filament Channel. II. Interplanetary Propagation to 1 au

We present observations and modeling results of the propagation and impact at Earth of a high-latitude, extended filament channel eruption that commenced on 2015 July 9. The coronal mass ejection (CME) that resulted from the filament eruption was associated with a moderate disturbance at Earth. This event could be classified as a so-called “problem storm” because it lacked the usual solar signatures that are characteristic of large, energetic, Earth-directed CMEs that often result in significant geoeffective impacts. We use solar observations to constrain the initial parameters and therefore to model the propagation of the 2015 July 9 eruption from the solar corona up to Earth using 3D magnetohydrodynamic heliospheric simulations with three different configurations of the modeled CME. We find the best match between observed and modeled arrival at Earth for the simulation run that features a toroidal flux rope structure of the CME ejecta, but caution that different approaches may be more or less useful depending on the CME–observer geometry when evaluating the space weather impact of eruptions that are extreme in terms of their large size and high degree of asymmetry. We discuss our results in the context of both advancing our understanding of the physics of CME evolution and future improvements to space weather forecasting.

The Emergence of Toroidal Flux Ropes with Different Twist Rising at the Same Speed

The role of twist in the emergence of magnetic flux ropes into the solar atmosphere has remained unclear for some time. Although many studies have investigated how the photospheric properties of active regions resulting from the simulated emergence of magnetic flux ropes from the convection zone with different twists compare to the observed properties of active regions, these simulations have a wide range of magnetic flux rope radii, depths, and initial configurations, making it challenging to form a complete picture of the role of any one variable in the emergence process. Twist, in particular, has been difficult to analyze because isothermally buoyant magnetic flux ropes with different twists also experience different accelerations. In this paper, we develop an analytical model of a toroidal magnetic flux rope in approximate vertical force balance in the convection zone. We numerically implement this model in a stratified atmosphere, and then subtract off a twist-independent density to make magnetic flux ropes buoyant in a twist-independent way, ensuring that the initial acceleration of each magnetic flux rope is approximately the same. We perform numerical simulations to obtain a parameter study of toroidal magnetic flux ropes with different twist rising at the same speed. We analyze the photospheric and coronal properties of the active regions resulting from the emergence of these magnetic flux ropes, and argue that the Parker instability is responsible for many of the features observed in the simulations.

Identification of kink instability in 3D helical flux ropes at VEST

Local helicity injection (LHI) is a non-inductive startup and current drive method via Taylor relaxation for the spherical torus. In achieving Taylor relaxation, it has been suggested that kink instability in 3D helical flux ropes plays an important role. However, the role and occurrence of kink instability during LHI have yet to be validated. Experimentally, determining the kink mode in a flux rope relies on measuring internal information using a probe. However, for LHI, the 3D geometry complicates this measurement process. Here, we propose a new approach for determining the kink modes of 3D helical flux ropes without any internal probe measurements. It is confirmed by this approach that flux ropes exhibit two different kink modes. With increasing plasma current in the flux ropes, a transition from the coherent internal kink mode to the external kink mode is observed. Kink mode properties such as rotating frequency calculated from the kink theory agree well with the magnetic signature driven by the kink mode. During the LHI experiment in the versatile experiment spherical torus, three distinguishable phases are confirmed by the approach, consistent with NIMROD simulation. Before driving the toroidal plasma current, the external kink mode is observed for 3D helical flux ropes. As the toroidal plasma current increases, the external kink mode disappears while generating broadband internal modes instead of coherent internal kink of flux ropes. Decoupling between the toroidal plasma and flux rope results in both decay of toroidal plasma current and re-appearance of the external kink mode in the flux ropes.

Interplanetary Magnetic Flux Ropes as Agents Connecting Solar Eruptions and Geomagnetic Activities

We investigate the solar wind structure for 11 cases that were selected for the campaign study promoted by the International Study of Earth-affecting Solar Transients (ISEST) MiniMax24 Working Group 4. We can identify clear flux rope signatures in nine cases. The geometries of the nine interplanetary magnetic flux ropes (IFRs) are examined with a model-fitting analysis with cylindrical and toroidal force-free flux rope models. For seven cases in which magnetic fields in the solar source regions were observed, we compare the IFR geometries with magnetic structures in their solar source regions. As a result, we can confirm the coincidence between the IFR orientation and the orientation of the magnetic polarity inversion line (PIL) for six cases, as well as the so-called helicity rule as regards the handedness of the magnetic chirality of the IFR, depending on which hemisphere of the Sun the IFR originated from, the northern or southern hemisphere; namely, the IFR has right-handed (left-handed) magnetic chirality when it is formed in the southern (northern) hemisphere of the Sun. The relationship between the orientation of IFRs and PILs can be taken as evidence that the flux rope structure created in the corona is in most cases carried through interplanetary space with its orientation maintained. In order to predict magnetic field variations on Earth from observations of solar eruptions, further studies are needed about the propagation of IFRs because magnetic fields observed at Earth significantly change depending on which part of the IFR hits the Earth.

Magnetic cloud fit by uniform-twist toroidal flux ropes

Context. Detailed studies of magnetic cloud observations in the solar wind in recent years indicate that magnetic clouds are interplanetary flux ropes with a low twist. Commonly, their magnetic fields are fit by the axially symmetric linear force-free field in a cylinder (Lundquist field), which in contrast has a strong and increasing twist toward the boundary of the flux rope. Therefore another field, the axially symmetric uniform-twist force-free field in a cylinder (Gold-Hoyle field) has become employed to analyze magnetic clouds.Aims. Magnetic clouds are bent, and for some observations, a toroidal rather than a cylindrical flux rope is needed for a local approximation of the cloud fields. We therefore try to derive an axially symmetric uniform-twist force-free field in a toroid, either exactly, or approximately, and to compare it with observations.Methods. Equations following from the conditions of solenoidality and force-freeness in toroidally curved cylindrical coordinates were solved analytically. The magnetic field and velocity observations of a magnetic cloud were compared with solutions obtained using a nonlinear least-squares method.Results. Three solutions of (nearly) uniform-twist magnetic fields in a toroid were obtained. All are exactly solenoidal, and in the limit of high aspect ratios, they tend to the Gold-Hoyle field. The first solution has an exactly uniform twist, the other two solutions have a nearly uniform twist and approximate force-free fields. The analysis of a magnetic cloud observation showed that these fields may fit the observed field equally well as the already known approximately linear force-free (Miller-Turner) field, but it also revealed that the geometric parameters of the toroid might not be reliably determined from fits, when (nearly) uniform-twist model fields are used. Sets of parameters largely differing in the size of the toroid and its aspect ratio yield fits of a comparable quality.

Toroidal Flux Ropes with Elliptical Cross Sections and Their Magnetic Helicity

Axially symmetric constant-alpha force-free magnetic fields in toroidal flux ropes with elliptical cross sections are constructed in order to investigate how their alphas and magnetic helicities depend on parameters of the flux ropes. Magnetic configurations are found numerically using a general solution of a constant-alpha force-free field with an axial symmetry in cylindrical coordinates for a wide range of oblatenesses and aspect ratios. Resulting alphas and magnetic helicities are approximated by polynomial expansions in parameters related to oblateness and aspect ratio. These approximations hold for toroidal as well as cylindrical flux ropes with an accuracy better than or of about 1%. Using these formulae, we calculate relative helicities per unit length of two (probably very oblate) magnetic clouds and show that they are very sensitive to the assumed magnetic cloud shapes (circular versus elliptical cross sections).

The 17 March 2015 storm: the associated magnetic flux rope structure and the storm development

The objective of this study is (1) to determine the magnetic cloud (MC) structure associated with the 17 March 2015 storm and (2) to gain an insight into how the storm developed responding to the solar wind conditions. First, we search MC geometries which can explain the observed solar wind magnetic fields by fitting to both cylindrical and toroidal flux rope models. Then, we examine how the resultant MC geometries can be connected to the solar source region to find out the most plausible model for the observed MC. We conclude that the observations are most consistently explained by a toroidal flux rope with the torus plane nearly parallel to the ecliptic plane. It is emphasized that the observations are characterized by the peculiar spacecraft crossing through the MC, in that the magnetic fields to be observed are southward throughout the passage. For understanding of the storm development, we first estimate the injection rate of the storm ring current from the observed Dst variation. Then, we derive an expression to calculate the estimated injection rate from the observed solar wind variations. The point of the method is to evaluate the injection rate by the convolution of the dawn-to-dusk electric field in the solar wind and a response function. By using the optimum response function thus determined, we obtain a modeled Dst variation from the solar wind data, which is in good agreement with the observed Dst variation. The agreement supports the validity of our method to derive an expression for the ring current injection rate as a function of the solar wind variation.

Toroidal linear force-free magnetic fields with axial symmetry

Aims. Interplanetary magnetic flux ropes are often described as linear force-free fields. To account for their curvature, toroidal configurations must be used. The aim is to find an analytic description of a linear force-free magnetic field of the toroidal geometry in which the cross section of flux ropes can be controlled. Methods. The solution is found as a superposition of fields given by linear force-free cylinders tangential to a generating toroid. The cylindrical field is expressed in a series of terms that are not all cylindrically symmetric. Results. We found the general form of a toroidal linear force-free magnetic field. The field is azimuthally symmetric with respect to the torus axis. It depends on a set of coefficients that enables controlling the flux rope shape (cross section) to some extent. By varying the coefficients, flux ropes with circular and elliptic cross sections were constructed. Numerical comparison suggests that the simple analytic formula for calculating the helicity in toroidal flux ropes of the circular cross section can be used for flux ropes with elliptic cross sections if the minor radius in the formula is set to the geometric mean of the semi-axes of the elliptic cross section.

CATASTROPHE VERSUS INSTABILITY FOR THE ERUPTION OF A TOROIDAL SOLAR MAGNETIC FLUX ROPE

The onset of a solar eruption is formulated here as either a magnetic catastrophe or as an instability. Both start with the same equation of force balance governing the underlying equilibria. Using a toroidal flux rope in an external bipolar or quadrupolar field as a model for the current-carrying flux, we demonstrate the occurrence of a fold catastrophe by loss of equilibrium for several representative evolutionary sequences in the stable domain of parameter space. We verify that this catastrophe and the torus instability occur at the same point; they are thus equivalent descriptions for the onset condition of solar eruptions.

How do fits of simulated magnetic clouds correspond to their real shapes in 3-D?

Abstract. Magnetic clouds are important objects for space weather forecasters due to their impact on the Earth's magnetosphere and their consequences during geomagnetic storms. Being considered as cylindrical or toroidal flux ropes, their size, velocity, magnetic field strength, and axis orientation determine its impact on Earth. Above mentioned parameters are usually extracted from model fits using measurements from one-spacecraft crossings of these structures. In order to relate solar events with these spacecraft observations, the parameters are then compared to situation at the Sun around a most probable source region with a goal to correlate them with near-Sun observed quantities for prediction purposes. In the past we performed three-dimensional simulations of magnetic cloud propagation in the inner heliosphere. Simulated spacecraft measurements are fitted by models of magnetic clouds and resulting parameters are compared with real shapes of magnetic clouds which can be directly obtained from our simulations. The comparison shows that cloud parameters are determined quite reliably for spacecraft crossings near the cloud axis.

THREE-DIMENSIONAL NONLINEAR EVOLUTION OF A MAGNETIC FLUX TUBE IN A SPHERICAL SHELL: INFLUENCE OF TURBULENT CONVECTION AND ASSOCIATED MEAN FLOWS

We present the first 3D MHD study in spherical geometry of the non-linear dynamical evolution of magnetic flux tubes in a turbulent rotating convection zone. We study numerically the rise of magnetic toroidal flux ropes from the base of a modelled convection zone up to the top of our computational domain where bipolar patches are formed. We compare the dynamical behaviour of flux tubes in a fully convective shell possessing self-consistently generated mean flows such as meridional circulation and differential rotation, with reference calculations done in a quiet isentropic zone. We find that two parameters influence the tubes during their rise through the convection zone: the initial field strength and amount of twist, thus confirming previous findings in Cartesian geometry. Further, when the tube is sufficiently strong with respect to the equipartition field, it rises almost radially independently of the initial latitude (either low or high). By contrast, weaker field cases indicate that downflows and upflows control the rising velocity of particular regions of the rope and could in principle favour the emergence of flux through Omega-loop structures. For these latter cases, we focus on the orientation of bipolar patches and find that sufficiently arched structures are able to create bipolar regions with a predominantly East-West orientation. Meridional flow seems to determine the trajectory of the magnetic rope when the field strength has been significantly reduced near the top of the domain. Finally differential rotation makes it more difficult for tubes introduced at low latitudes to reach the top of the domain.

Linear force-free field of a toroidal symmetry

Context. Interplanetary flux ropes are often described by linear force-free fields. To account for their curvature, toroidal configurations valid for large aspect ratios are used. However, in some cases, flux ropes need to be approximated by a toroid with a low aspect ratio. Aims. The aim is to find an analytical description of a linear force-free magnetic field of toroidal geometry, not restricted to large aspect ratios. Methods. The solution is found as a superposition of fields given by linear force-free cylinders tangential to a generating toroid. Results. The obtained solution describes toroidal flux ropes with reasonable shapes and magnetic field values. The field is exactly linear force-free for arbitrary aspect ratios. The new solution can be applied for solar and interplanetary flux ropes, astrophysical objects, and laboratory plasma.

A Three‐dimensional Line‐tied Magnetic Field Model for Solar Eruptions

We introduce a three-dimensional analytical model of a coronal flux rope with its ends embedded in the solar surface. The model allows the flux rope to move in the corona while maintaining line-tied conditions at the solar surface. These conditions ensure that the normal component of the coronal magnetic field at the surface remains fixed during an eruption and that no magnetic energy enters the corona through the surface to drive the eruption. The model is based on the magnetic configuration of Titov & Demoulin, where a toroidal flux rope is held in equilibrium by an overlying magnetic arcade. We investigate the stability of this configuration to specific perturbations and show that it is subject to the torus instability when the flux rope length exceeds a critical value. A force analysis of the configuration shows that flux ropes are most likely to erupt in a localized region near the apex, while the regions near the surface remain relatively undisturbed. Thus, the flux rope will tend to form an aneurysm-like structure once it erupts. Our analysis also suggests how the flux rope rotation seen in some eruptions and simulations may be related to the observed orientation of the overlying arcade field. This model exhibits the potential for catastrophic loss of equilibrium as a possible trigger for eruptions, but further study is required to prove this property.

Damped Oscillations of Coronal Loops

A mechanism of damped oscillations of a coronal loop is investigated. The loop is treated as a thin toroidal flux rope with two stationary photospheric footpoints, carrying both toroidal and poloidal currents. The forces and the flux-rope dynamics are described within the framework of ideal magnetohydrodynamics (MHD). The main features of the theory are the following: i) Oscillatory motions are determined by the Lorentz force that acts on curved current-carrying plasma structures and ii) damping is caused by drag that provides the momentum coupling between the flux rope and the ambient coronal plasma. The oscillation is restricted to the vertical plane of the flux rope. The initial equilibrium flux rope is set into oscillation by a pulse of upflow of the ambient plasma. The theory is applied to two events of oscillating loops observed by the Transition Region and Coronal Explorer (TRACE). It is shown that the Lorentz force and drag with a reasonable value of the coupling coefficient (cd) and without anomalous dissipation are able to accurately account for the observed damped oscillations. The analysis shows that the variations in the observed intensity can be explained by the minor radial expansion and contraction. For the two events, the values of the drag coefficient consistent with the observed damping times are in the range cd≈2 – 5, with specific values being dependent on parameters such as the loop density, ambient magnetic field, and the loop geometry. This range is consistent with a previous MHD simulation study and with values used to reproduce the observed trajectories of coronal mass ejections (CMEs).

Model for the Coupled Evolution of Subsurface and Coronal Magnetic Fields in Solar Active Regions

According to Babcock's theory of the solar dynamo, bipolar active regions are Ω-shaped loops emerging from a toroidal field located near the base of the convection zone. In this paper, a mean field model for the evolution of a twisted Ω-loop is developed. The model describes the coupled evolution of the magnetic field in the convection zone and the corona after the loop has fully emerged into the solar atmosphere. Such a coupled evolution is required to fully understand what happens to the coronal and subsurface fields as magnetic flux cancels at polarity inversion lines on the photosphere. The jump conditions for the magnetic field at the photosphere are derived from the magnetic stress balance between the convection zone and corona. The model reproduces the observed spreading of active region magnetic flux over the solar surface. At polarity inversion lines, magnetic flux submerges below the photosphere, but the component of magnetic field along the inversion line cannot submerge, because the field in the upper convection zone is nearly radial. Therefore, magnetic shear builds up in the corona above the inversion line, which eventually leads to a loss of equilibrium of the coronal fields and the lift-off of a coronal flux rope. Fields that submerge are transported back to the base of the convection zone, leading to the repair of the toroidal flux rope. Following Martens and Zwaan, interactions between bipoles are also considered.

Buoyant magnetic flux ropes in a magnetized stellar envelope

Context.The context of this paper is buoyant toroidal magnetic flux ropes, which is a part of flux tube dynamo theory and the framework of solar-like magnetic activity.

On the Nature of the X-Ray Bright Core in a Stable Filament Channel

In a search for the cause of the intense heating revealed by X-ray emission in filament channels, we have simulated the evolution of a twisted toroidal flux rope emerging quasi-statically into the corona. Initially, the simulated flux rope remains confined in equilibrium as the stored magnetic energy increases. With enough twist buildup, there is a sudden catastrophic loss of equilibrium and total expulsion of the flux rope. We focused on the quasi-static phase in which a current sheet forms within the flux rope cavity, along the so-called bald-patch separatrix surface (BPSS). This comprises an envelope of field lines that graze the anchoring lower boundary, enclosing the detached helical field that supports the prominence. Significant magnetic energy dissipation and heating are expected to center around such current sheets. The heating that should result provides a plausible explanation for the hot X-ray sources, although they appear to be colocated with cool material. If our physical picture is correct, then the development of X-ray bright cores or sigmoids in a filament channel suggests the presence of a BPSS separating the helical field of a twisted flux rope in stable confinement from the surrounding untwisted fields.

Potential magnetic fields around flux ropes

A method of calculation of potential magnetic fields around cylindrical and toroidal flux ropes is presented. The flux rope has an arbitrary orientation with respect to ambient field. The problem is solved analytically. Expressions for the magnetic field components, diamagnetic force, and its moment are found. Results obtained can be used for estimations of magnetic forces in systems where magnetic flux ropes and potential fields coexist, e.g., in the solar corona.