Philip A. Isenberg

The effect of curvature on flux-rope models of coronal mass ejections

The large-scale curvature of a flux rope can help propel it outward from the Sun. Here we extend previous two-dimensional flux-rope models of coronal mass ejections to include the curvature force. To obtain analytical results, we assume axial symmetry and model the flux rope as a torus that encircles the Sun. Initially, the flux rope is suspended in the corona by a balance between magnetic tension, compression, and curvature forces, but this balance is lost if the photospheric sources of the coronal field slowly decay with time. The evolution of the system shows catastrophic behavior as occurred in previous models, but, unlike the previous models, flux ropes with large radii are more likely to erupt than ones with small radii. The maximum total magnetic energy that can be stored before equilibrium is lost is 1.53 times the energy of the potential field, and this value is less than the limiting value of 1.662 for the fully opened field. As a consequence, the loss of ideal MHD equilibrium that occurs in the model cannot completely open the magnetic field. However, the loss of equilibrium does lead to the sudden formation of a current sheet, and if rapid reconnection occurs in this sheet, then the flux rope can escape from the Sun. We also find that the held can gradually become opened without suffering any loss of equilibrium if the photospheric field strength falls below a critical value. This behavior is analogous to the opening of a spherically symmetric arcade in response to a finite amount of shear.

A hemispherical model of anisotropic interstellar pickup ions

We present an analytical model for the distribution function of interstellar pickup ions in a radially directed heliospheric magnetic field which naturally results in anisotropic particle distributions. The model includes the effects of convection and spatial transport in the solar wind, adiabatic deceleration in the radial flow, adiabatic focusing in the radial field, and pitch angle scattering toward isotropy in the frame of the solar wind. The pitch angle scattering is approximated by the hemispherical assumption: we take the scattering to be very efficient within each pitch angle range μ ≷ 0 (where μ is the cosine of the pitch angle) but inhibited between the hemispheres separated by μ = 0. The analytical solution is obtained for the case where the scattering rate across μ = 0 scales as the particle speed divided by the radial position of the fluid parcel. The model distribution functions can be used to interpret recent observations of anisotropic pickup ions.

Catastrophic evolution of a force-free flux rope : a model for eruptive flares

We present a self-consistent, two-dimensional, magnetohydrodynamic model of an eruptive flare based on an ideal-MHD coronal magnetic field configuration which is line-tied at the photosphere and contains a force-free flux rope. If the flux rope is not too large, the gradual disappearance of the photospheric field causes the flux rope to lose equilibrium catastrophically and jump to a higher altitude, releasing magnetic energy in the process. During the jump, an extended current sheet forms below the flux rope, and subsequent reconnection of this current sheet allows the flux rope to escape into the outer corona. A critical flux-rope radius, which depends on the form of the photospheric field, divides configurations which undergo a catastrophic loss of equilibrium from those which do not

Turbulent Heating of the Solar Wind by Newborn Interstellar Pickup Protons

Spacecraft missions to the outer heliosphere have shown that the thermal protons that make up the solar wind are hotter than simple adiabatic expansion would predict. We examine a theory that describes this heating by using the time-varying 1 AU measurements as input and comparing the predictions with observations in the outer heliosphere. Inside 20 AU wind shear and shocks provide the dominant energy source to drive the turbulence. Outside 20 AU little remains to inject energy into the fluctuations except newborn interstellar pickup protons. The theory is built on a combination of two-dimensional magnetohydrodynamic turbulence concepts and the latest kinetic theory describing the scattering of newborn interstellar pickup protons. We find that application of the theory to the observations produces encouraging agreement at the same time that it illuminates latitudinal effects associated with solar minimum conditions. A remaining challenge is to close the gap of a factor of 2 between observed and predicted proton temperatures beyond 40 AU. For this, we suggest that further structural development of the theory is needed, rather than ad hoc adjustment of the model parameters, which are reasonably well constrained by theory, simulation, and observations.

TURBULENT HEATING OF THE DISTANT SOLAR WIND BY INTERSTELLAR PICKUP PROTONS

The solar wind heating observed in the outer heliosphere can be attributed to the turbulent dissipation of wave energy generated by the isotropization of interstellar pickup protons. We calculate the fluctuation energy provided by the pickup protons, assuming that the ambient fluctuations are dominated by turbulent processes. This assumption is in contrast to previous estimates of this energy, which obtained a bispherical result in the absence of turbulent interactions. We find that when turbulent processes dominate the wave generation, a more isotropic distribution of pickup protons is produced and less energy is injected into the solar wind fluctuations. When this form of the pickup proton energy source is used in a phenomenological model of turbulent evolution in the solar wind, we find good agreement with the solar wind temperature measurements by the Voyager 2 spacecraft available at present out to almost 70 AU. The efficient spectral smoothing by the turbulent transport may also explain the absence of observable pickup proton–generated waves in the outer heliosphere. Subject headings: solar wind — turbulence Neutral atoms that flow slowly into the heliosphere from the surrounding interstellar medium may become ionized by solar ultraviolet radiation or by charge exchange with ions in the outflowing solar wind. At that moment, the new ions stream toward the Sun with respect to the reference frame of the plasma with a speed close to the solar wind speed Vsw. These suddenly energetic ions, termed ‘‘ pickup ions,’’ then enter into an involved interaction with the solar wind particles and fields that is still not completely determined. In broad outline this interaction consists of several steps, which take place on distinct timescales when the solar wind

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.

The kinetic shell model of coronal heating and acceleration by ion cyclotron waves: 2. Inward and outward propagating waves

We extend the kinetic shell model of the cyclotron resonant interaction between coronal hole protons and outward propagating ion cyclotron waves presented in the first paper of this series [Isenberg et al., 2001]. That work showed that the resonant dissipation of outward propagating waves produced proton distributions that were unstable to the generation of inward propagating waves. Here we include the kinetic shell interaction with the inward waves, assuming that both the wave generation and wave dissipation proceed much faster than all other processes in the collisionless coronal hole. In this case, the entire proton distribution will be resonant with one set of waves or the other and is thus composed of constant-energy shells in both the sunward and antisunward regions of velocity space. The evolution of the distribution as the plasma flows away from the Sun is then described by following the motion of these shells under the action of the nonresonant forces in the coronal hole. We find that the distribution consists of a core population which circulates through velocity space and a halo which continually expands to higher energy. The halo population is shown to be essential to obtain acceleration of the bulk proton plasma through its response to the mirror force. We present an illustrative calculation of this system which assumes the waves to be dispersionless. We find for this case that the halo particles soon reach extremely high energies, leading to a continuous, rather than declining, acceleration of the plasma. We suggest that these properties are due to the dispersionless assumption and that an improved model incorporating wave dispersion may give more reasonable quantitative results.

Prominence eruptions and coronal mass ejections triggered by newly emerging flux

Using a simple model for the onset of solar eruptions, we investigate how an existing magnetic configuration containing a flux rope evolves in response to new emerging flux. Our results show that the emergence of new flux can cause a loss of ideal MHD equilibrium under certain circumstances, but the circumstances which lead to eruption are much richer and more complicated than one might expect given the simplicity of the model. The model results suggest that the actual circumstances leading to an eruption are sensitive not only to the polarity of the emerging region, but also to several other parameters, such as the strength, distance, and area of the emerging region. It has been suggested by various researchers that the emergence of new flux with an orientation which allows reconnection with the preexisting flux (a process sometimes referred to as tether cutting) will generally lead to destabilization of the coronal or prominence magnetic field. Although our results can replicate such behavior for certain restricted classes of boundary conditions, we find that, in general, there is no simple, universal relation between the orientation of the emerging flux and the likelihood of an eruption.

Turbulence-driven Solar Wind Heating and Energization of Pickup Protons in the Outer Heliosphere

We improve and extend the dominant turbulence model of Isenberg et al., which describes the turbulent heating of the distant solar wind by the dissipation of wave energy generated by the isotropization of interstellar pickup protons. We first modify some of the detailed expressions in Isenberg et al. to fully incorporate the effects of wave dispersion in the pitch-angle scattering of the newly ionized pickup protons. The corrected pickup proton distributions have a more involved shape, but the energy given up to the waves is only slightly smaller than that found in Isenberg et al. We then treat the effects of second-order Fermi acceleration of the pickup protons, which occurs through the same resonant cyclotron interaction that controls the pitch-angle scattering. In principle, the pickup proton energization is an energy sink that was not included in Isenberg et al. and so could modify the model results. In addition, since this turbulent-evolution model makes specific predictions of the intensity of resonant waves as a function of heliocentric radius, our quantitative results for the pickup proton evolution are on a firmer footing than possible in previous models. We find that neither of these modifications produces significant changes in the solar wind heating predicted by Isenberg et al. Thus, the results of the dominant-turbulence model appear to be robust in that they are not sensitive to the incorporation of second-order effects.

A dispersive analysis of bispherical pickup ion distributions

We investigate the properties of a bispherical shell distribution of pickup ions when dispersion of the resonant waves is taken into account. We also extend the method of Johnstone et al. [1991] and Huddleston and Johnstone [1992] for determining the spectra of the quasi-linear self-generated waves to include dispersion. Our specific results assume that all waves propagate along the average magnetic field, and that the waves satisfy the cold electron-proton dispersion relation. The major effect of including dispersion in the analysis is that the particles in a bispherical distribution retain more of their initial energy than in the nondispersive treatment, so the resulting wave spectra are less intense than predicted when dispersion is neglected. The dispersive analysis also encounters the well-known gap in the cyclotron resonance with fast-mode waves which is absent in the nondispersive picture. We present detailed results for pickup protons in the two extreme configurations of magnetic field perpendicular to, and parallel to, the solar wind flow for a range of values of VA/VSW. We find that when pickup protons are scattered to a bispherical distribution with VA/VSW = 0.1, typical of conditions in the solar wind, the discrepancy in the total energy density of the self-generated waves between the dispersive and nondispersive results amounts to 18% in the perpendicular configuration. This discrepancy falls between 9% and 73% in the parallel configuration, depending on the extent of the particle transport through the resonance gap. These discrepancies are substantially larger for larger values of VA/VSW. These results indicate that dispersive effects can be important in the correct modeling of the quasi-linear resonant wave-particle interaction between pickup ions and self-generated waves in the solar wind.