Lundquist Solution Research Articles

On the Quasi-Three Dimensional Configuration of Magnetic Clouds.

We develop an optimization approach to model the magnetic field configuration of magnetic clouds, based on a linear force‐free formulation in three dimensions. Such a solution, dubbed the Freidberg solution, is kin to the axisymmetric Lundquist solution, but with more general “helical symmetry.” The merit of our approach is demonstrated via its application to two case studies of in situ measured magnetic clouds. Both yield results of reduced χ 2 ≈ 1. Case 1 shows a winding flux rope configuration with one major polarity. Case 2 exhibits a double‐helix configuration with two flux bundles winding around each other and rooted on regions of mixed polarities. This study demonstrates the three‐dimensional complexity of the magnetic cloud structures.

The Magnetic Field Geometry of Small Solar Wind Flux Ropes Inferred from Their Twist Distribution

This work extends recent efforts on the force-free modeling of large flux rope-type structures (magnetic clouds, MCs) to much smaller spatial scales. We first select small flux ropes (SFRs) by eye whose duration is unambiguous and which were observed by the Solar Terrestrial Relations Observatory (STEREO) or Wind spacecraft during solar maximum years. We inquire into which analytical technique is physically most appropriate. We consider three models: (i) linear force-free field ($\bigtriangledown\times$ B = $\alpha (r) $ B) with a specific, prescribed constant $\alpha$ (Lundquist solution), and (ii) with $\alpha$ as a free constant parameter (Lundquist-alpha solution), (iii) uniform twist field (Gold-Hoyle solution). We retain only those cases where the impact parameter is less than one-half the FR radius, $R$, so the results should be robust (29 cases). The SFR radii lie in the range [$\sim$ 0.003, 0.059] AU. Comparing results, we find that the Lundquist-alpha and uniform twist solutions yielded comparable and small normalized $\chi^2$ values in most cases. We then use Grad-Shafranov (GS) reconstruction to analyze these events further. We then considered the twist per unit length, $\tau$, both its profile through the FR and its absolute value. We find $\tau$ to lie in the range [5.6, 34] turns/AU. The GH model-derived $\tau$ values are comparable to those obtained from GS reconstruction. We find that twist unit length ($L$) is inversely proportional to $R$, as $\tau \sim 0.17/R$. We combine MC and SFR results on $\tau (R)$ and give a relation which is approximately valid for both sets. The axial and azimuthal fluxes, $F_z$ and $F_\phi$, vary as $\approx 2.1 B_0 R^2 \times10^{21}$ Mx and $F_{\phi}/L \approx 0.36 B_0 R \times10^{21}$Mx/AU. The relative helicity per unit length, $H/L \approx 0.75 B_0^2 R^3$$\times 10^{42}$ Mx$^2$/AU.

Trkalian fields and Radon transformation

We write the spherical curl transformation for Trkalian fields using differential forms. Then we consider Radon transform of these fields. The Radon transform of a Trkalian field satisfies a corresponding eigenvalue equation on a sphere in transform space. The field can be reconstructed using knowledge of the Radon transform on a canonical hemisphere. We consider relation of the Radon transformation with Biot–Savart integral operator and discuss its transform introducing Radon–Biot–Savart operator. The Radon transform of a Trkalian field is an eigenvector of this operator. We also present an Ampere-law type relation for these fields. We apply these to Lundquist solution. We present a Chandrasekhar–Kendall-type solution of the corresponding equation in the transform space. Lastly, we focus on the Euclidean topologically massive Abelian gauge theory. The Radon transform of an anti-self-dual field is related by antipodal map on this sphere to the transform of the self-dual field obtained by inverting space coordinates. The Lundquist solution provides an example of quantization of topological mass in this context.

Analysis of large scale MHD quantities in expanding magnetic clouds

Magnetic clouds (MCs) transport the magnetic flux and helicity released by the Sun. They are generally modeled as a static flux rope traveling in the solar wind, though they can present signatures of expansion. We analyze three expanding MCs using a self-similar free radial expansion model with a cylindrical linear force-free field (i.e., Lundquist solution) as the initial condition. We derive expressions for the magnetic fluxes, the magnetic helicity and the magnetic energy per unit length along the flux tube. We find that these quantities do not differ more than 25 % when using the static or expansion model.

Asymmetric magnetic field inside a cylindrical flux rope

AbstractIn order to consider an asymmetric field distribution inside a cylindrical tube with a circular cross section, with the field magni-tude maximum reached at a point different from the geometrical centre of the tube, we propose a new solution obtained in bi-cylin-drical coordinates. If the parameter a of the system is very large, the solution approaches the well known symmetric Lundquistsolution. The new solution models a field magnitude shift due to relative motion of the cylinder and the ambient solar wind. Ifthe cylinder is moving much faster than the ambient medium, its field maximum is shifted forward, otherwise it is shifted in theopposite direction. 2004 COSPAR. Published by Elsevier Ltd. All rights reserved. Keywords: Magnetic clouds; Modelling; Interplanetary magnetic field 1. IntroductionForce-free interpretation of interplanetary magneticclouds showed a good agreement with observation fora number of events, starting from analysis by Burlaga(1988), where an axially symmetric linear solution byLundquist (1950) was used. On the other hand not allclouds can be treated as axially symmetric. In Vandasand Romashets (2003) a linear force-free solution forclouds with an elliptic cross section was derived. Sincethe ratio b of plasma pressure over magnetic pressureis sometimes high enough, some investigators developedand applied non-force-free models for data interpreta-tion (e.g., Mulligan and Russell, 2001; Hidalgo et al.,2002a,b; Ruzmaikin et al., 2003; Russell and Mulligan,2003). Such solutions contain additional parameters,which surely lead to better agreement between modelsand observations. Here we propose two new solutions,one force-free and one non-force-free. Depending onthe averaged b in a magnetic cloud, one can use a formeror a latter one. The both solutions can be regarded asgeneralizations of the Lundquist solution, because theiradditional parameters characterize deviations from it.So there is a natural link between fits obtained earlierby the Lundquist solution and fits by our new models.2. ModelIn order to consider an asymmetric field distributioninside a cylindrical tube, with the field magnitude maxi-mum reached at a point different from the geometricalcentre of the tube, it is convenient to use bi-cylindricalcoordinates l, g, and Z, which are related to the Carte-sian coordinates by the following relationships (see, e.g.,Moon and Spencer, 1961):x ¼asingcoshl cosg; ð1Þy ¼asinhlcoshl cosg; ð2Þ

New Force-Free Models of Magnetic Clouds

AbstractMagnetic clouds are thought to be large flux ropes propagating through the heliosphere. Their twisted magnetic fields are mostly modeled by a constant-alpha force-free field in a circular cylindrical flux rope (the Lundquist solution). However, the interplanetary flux ropes are three dimensional objects. In reality they possibly have a curved shape and an oblate cross section. Recently we have found two force-free models of flux ropes which takes into account the mentioned features. These are (i) a constant-alpha force-free configuration in an elliptic flux rope (Vandas & Romashets 2003, A&A, 398, 801), and (ii) a non-constant-alpha force-free field in a toroid with arbitrary aspect ratio (Romashets & Vandas 2003, AIP Conf Ser. 679, 180). Two magnetic cloud observations were analyzed. The magnetic cloud of October 18-19, 1995 has been fitted by Lepping et al. (1997, JGR, 102, 14049) with use of the Lundquist solution. The cloud has a very flat magnetic field magnitude profile. We fitted it by the elliptic solution (i). The magnetic cloud of November 17-18, 1975 has been fitted by Marubashi (1997) with use of a toroidally adjusted Lundquist solution. The cloud has a large magnetic field vector rotation and a large magnetic field magnitude increase over the background level. We fitted it by the toroidal solution (ii). The both fits match the rotation of the magnetic field vector in a comparable quality to the former fits, but the description of the magnetic field magnitude profiles is remarkable better. It is possible to incorporate temporal effects (expansion) of magnetic clouds into the new solutions through a time-dependent alpha parameter as in Shimazu & Vandas (2002, EP&S, 54, 783).

Comparison of force-free flux rope models with observations of magnetic clouds

Recently, we have found a force-free solution inside an elliptic cylinder for magnetic cloud models. For a diminishing oblateness, the solution tends to the widely used Lundquist constant-alpha force-free solution inside a circular cylinder. The solution may include effects of a flux rope expansion. A comparison of this new solution and the Lundquist solution with magnetic cloud observations is done for magnetic clouds with flat magnetic field magnitude profiles. In such cases, oblateness improves fits of magnetic field magnitude profiles in clouds significantly.

Magnetic clouds of oblate shapes

There is a growing evidence that cross-sections of magnetic clouds (interplanetary magnetic flux ropes) have oblate shapes. This follows from theoretical results as well as from interpretation of in situ measurements. MHD simulations show that a leading part of a cloud, an apex, has an oblate cross-section, and this may be very extreme in some cases. Interpretations of magnetic cloud observations by non-force-free models, multispacecraft observations, and analyses of the cloud bow shock stand-off distance also indicate that a cross-section of some magnetic clouds is oblate. Recently we have found a force-free solution with constant alpha in an elliptic cylinder. It is a direct generalization of the widely used Lundquist constant-alpha force-free solution inside a circular cylinder. Comparisons of this solution and the Lundquist solution with observations are shown.

Observable Properties of the Breakout Model for Coronal Mass Ejections

We compare the model for coronal mass ejections (CMEs) with observed general properties of CMEs by analyzing in detail recent high-resolution MHD simulations of a complete breakout CME. The model produces an eruption with a three-part plasma density structure that shows a bright circular rim outlining a dark central cavity in synthetic coronagraphic images of total brightness. The model also yields height-time profiles similar to most three-part CMEs, but the eruption speed by 2.5 R☉ is of order the Alfven speed, indicative of a fast CME. We show that the evolution of the posteruptive flare loop and chromospheric ribbons determined from the model are in agreement with observations of long-duration flares, and we propose an explanation for the long-standing observation that flares have an impulsive and gradual phase. A helical magnetic flux rope is generated during eruption and is consistent with a large class of interplanetary CME observations. The magnetic fields in this flux rope are well approximated by the Lundquist solution when the ejecta are at 15 R☉ and beyond. Furthermore, the interior density structure of the magnetic flux rope appears to have some of the basic features of an average magnetic cloud profile at 1 AU. Future simulation improvements and more stringent observational tests are discussed.

A Direct Method to Estimate Magnetic Helicity in Magnetic Clouds

Magnetic clouds are extended and magnetized plasma structures that travel from the Sun toward the outer heliosphere, carrying an important amount of magnetic helicity. The magnetic helicity quantifies several aspects of a given magnetic structure, such as the twist, kink, and the number of knots between magnetic field lines, the linking between magnetic flux tubes, etc. Since the helicity is practically conserved in the solar atmosphere and the heliosphere, it is a useful quantity to compare the physical properties of magnetic clouds to those of their solar source regions. In this work we describe a method that, assuming a cylindrical geometry for the magnetic cloud structures, allows us to calculate their helicity (per unit length) content directly from the observed magnetic field values. We apply the method to a set of 20 magnetic clouds observed by the WIND spacecraft. To test its reliability we compare our results with the helicity computed using a linear force-free field model under cylindrical geometry (i.e. Lundquist's solution).To search for other articles by the author(s) go to: http://adsabs.harvard.edu/abstract_service.html

Geometric considerations of the evolution of magnetic flux ropes.

We use flux conservation and magnetohydrodynamics (MHD) theory to discuss essential differences in the nature of the evolution of two analytical solutions describing magnetic flux tubes evolving in time. The first of these maintains the elongation of the tube, while the second maintains a constant angular extension with respect to a possible pointlike source. In the first case, free expansion of the plasma (density N) occurs only in a direction perpendicular to the flux-tube x axis. In the second case, isotropic evolution is considered. In both cases it is assumed that at initial time t(0) the flux-tube B field is the force-free magnetostatic Lundquist solution, which energetically corresponds to the most stable state for any flux-tube structure. We show that for each case conservation of magnetic flux is enough to establish the scaling with time of the B field. While both expansions may correspond to the evolution of observed flux tubes in the heliosphere, the isotropic expansion appears to capture consistently essential features associated with the actual observations of expanding coronal mass ejections within 30 solar radii. For isotropic expansion of the plasma the force-free nature of the B field is preserved for all time. As an example the MHD solutions are applied to an interplanetary magnetic cloud observed with the spacecraft Wind, which passed Earth's vicinity on June 2, 1998.

A force-free field with constant alpha in an oblate cylinder: A generalization of the Lundquist solution

A force-free magnetic field with constant alpha for a circular cylindrical flux rope (Lundquist solution) is widely used to describe the magnetic field configuration in interplanetary flux ropes. Observations as well as MHD simulations indicate that interplanetary flux ropes are not circular but have an oblate shape. Here we present an analytical solution for a force-free magnetic field with constant alpha in an elliptic flux rope which may be regarded as a direct generalization of the Lundquist solution. An alternative simpler solution for a force-free magnetic field with constant alpha in an oblate flux rope is discussed.