Solar Wind Equations Research Articles

Radial evolution of the accuracy of ballistic solar wind backmapping

Context.Solar wind backmapping is a technique employed to connect in situ measurements of heliospheric plasma structures to their origin near the Sun. The most widely used method is ballistic mapping, which neglects the effects of solar wind acceleration and corotation and instead models the solar wind as a constant radial outflow whose speed is determined by measurements in the heliosphere. This results in plasma parcel streamlines that form an Archimedean spiral (the Parker spiral) when viewed in the solar corotating frame. This simplified approach assumes that the effects of solar wind acceleration and corotation compensate for each other in the deviation of the source longitude. Most backmapping techniques so far considered magnetic connectivity from a heliocentric distance of 1 au to the Sun.Aims.We quantify the angular deviation between different backmapping methods that depends on the location of the radial probe and on the variation in the solar wind speed with radial distance. We assess these differences depending on source longitude and solar wind propagation time.Methods.We estimated backmapping source longitudes and travel times using (1) the ballistic approximation (constant speed), (2) a physically justified method using the empirically constrained acceleration profile Iso-poly, derived from Parker solar wind equations and also a model of solar wind tangential flows that accounts for corotational effects. We compared the differences across mapped heliocentric distances and for different asymptotic solar wind speeds.Results.The ballistic method results in a Carrington longitude of the source with a maximum deviation of 4″ below 3 au compared to the physics-based mapping method taken as reference. However, the travel time especially for the slow solar wind could be underestimated by 1.5 days at 1 au compared to non-constant speed profile. This time latency may lead to an association of incorrect solar source regions with given in situ measurements. Neglecting corotational effects and accounting for acceleration alone causes a large systematic shift in the backmapped source longitude.Conclusions.Incorporating both acceleration and corotational effects leads to a more physics-based representation of the plasma trajectories through the heliosphere compared to the ballistic assumption. This approach accurately assesses the travel time and provides a more realistic estimate of the longitudinal separation between a plasma parcel measured in situ and its source region. Nonetheless, it requires knowledge of the radial density and Alfvén speed profiles to compute the tangential flow. Therefore, we propose a compromise for computing the source longitude that employs the commonly used ballistic approach and the travel times computed from the derived radial acceleration speed profile. Moreover, we conclude that this approach remains valid at all radial distances we studied and is not limited to data obtained at 1 au.

Possible Evidence for Shear-driven Kelvin–Helmholtz Instability along the Boundary of Fast and Slow Solar Wind in the Corona

This paper reports the first possible evidence for the development of the Kelvin–Helmholtz (KH) instability at the border of coronal holes separating the associated fast wind from the slower wind originating from adjacent streamer regions. Based on a statistical data set of spectroscopic measurements of the UV corona acquired with the UltraViolet Coronagraph Spectrometer on board the SOlar and Heliospheric Observatory during the minimum activity of solar cycle 22, high temperature–velocity correlations are found along the fast/slow solar wind interface region and interpreted as manifestations of KH vortices formed by the roll-up of the shear flow, whose dissipation could lead to higher heating and, because of that, higher velocities. These observational results are supported by solving coupled solar wind and turbulence transport equations including a KH-driven source of turbulence along the tangential velocity discontinuity between faster and slower coronal flows: numerical analysis indicates that the correlation between the solar wind speed and temperature is large in the presence of the shear source of turbulence. These findings suggest that the KH instability may play an important role both in the plasma dynamics and in the energy deposition at the boundaries of coronal holes and equatorial streamers.

Direct Detection of Solar Angular Momentum Loss with the Wind Spacecraft

Abstract The rate at which the solar wind extracts angular momentum (AM) from the Sun has been predicted by theoretical models for many decades, and yet we lack a conclusive measurement from in situ observations. In this Letter we present a new estimate of the time-varying AM flux in the equatorial solar wind, as observed by the Wind spacecraft from 1994 to 2019. We separate the AM flux into contributions from the protons, alpha particles, and magnetic stresses, showing that the mechanical flux in the protons is ∼3 times larger than the magnetic field stresses. We observe the tendency for the AM flux of fast wind streams to be oppositely signed to the slow wind streams, as noted by previous authors. From the average total flux, we estimate the global AM loss rate of the Sun to be 3.3 × 1030 erg, which lies within the range of various magnetohydrodynamic wind models in the literature. This AM loss rate is a factor of ∼2 weaker than required for a Skumanich-like rotation period evolution ( stellar age−1/2), which should be considered in studies of the rotation period evolution of Sun-like stars.

A Nearly Incompressible Turbulence-Driven Solar Wind Model

We develop a 1D steady-state turbulence driven-solar wind model by coupling recently developed nearly incompressible magnetohydrodynamic turbulence transport model equations and 1D hydrodynamic solar wind equations. The dissipation of majority component quasi-2D turbulence and minority slab turbulence generated by the emergence of the “magnetic carpet” from the photosphere is responsible for heating the coronal plasma. We solve the coupled quasi-2D and slab turbulence transport equations and the solar wind equations from the base of the solar corona until 20 solar radii. We find that i) the coronal temperature increases to ~ 2.5 × 106 K near the base of the solar corona; ii) the solar wind accelerates rapidly within a few solar radii; iii) turbulence energy densities decrease and correlation lengths increase with distance; iv) balanced quasi-2D turbulence at the coronal base becomes imbalanced, and imbalanced slab turbulence becomes more imbalanced with distance, and v) the normalized quasi-2D and slab residual energy becomes positive near the coronal base, and both energies become negative with increasing distance.

An equatorial solar wind model with angular momentum conservation and nonradial magnetic fields and flow velocities at an inner boundary

AbstractAn analytic, self‐consistent, theoretical model for the solar wind is developed that generalizes previous models to include all of the following: conservation of angular momentum, frozen‐in magnetic fields, both radial (r) and azimuthal (ϕ) components of the magnetic field (Br and Bϕ) and velocity (vr and vϕ) from the inner boundary rs to 1 AU, and the detailed tracing back of observations at 1 AU to the inner boundary and all intervening (r,ϕ). The new model applies near the solar equatorial plane, assumes constant radial wind speed at each heliolongitude, and enforces corotation at the inner boundary. It is shown that the new theoretical model can be reduced to the previous models in the appropriate limits. We apply the model to two solar rotations of Wind spacecraft data, one near solar minimum (1–27 August 2010) and one near solar maximum (1–27 July 2002). The model analytically predicts the Alfvénic critical radius ra from the radial Alfvénic Mach number observed at 1 AU. Typically, the values are less than 15 solar radii, in agreement with some recent observations, and vary with longitude. Values of vϕ(r,ϕ) are predicted from the model, being always in the sense of corotation but varying in magnitude with r and ϕ. Reasonable and self‐consistent results are found for Br(r,ϕ), Bϕ(r,ϕ), vϕ(r,ϕ), and n(r,ϕ) from rs to 1 AU. Both the azimuthal and radial magnetic fields at rs vary with time by more than an order of magnitude and usually |Br(rs,ϕs)|≥|Bϕ(rs,ϕs)|. Typically, though not always, magnetic contributions to the total angular momentum are small. Interestingly, however, the azimuthal flow velocities observed at 1 AU are not always in the corotation direction and usually have much larger magnitudes than predicted by the model. Conservation of angular momentum alone cannot explain these azimuthal velocities and the standard interpretation involving stream‐stream interactions and dynamical behavior seems reasonable. Issues regarding the model's applicability appear to be due to the assumptions of corotation and constant wind speed breaking down below the Alfvén critical radius.

THREE-FLUID, THREE-DIMENSIONAL MAGNETOHYDRODYNAMIC SOLAR WIND MODEL WITH EDDY VISCOSITY AND TURBULENT RESISTIVITY

We have developed a three-fluid, three-dimensional magnetohydrodynamic solar wind model that incorporates turbulence transport, eddy viscosity, turbulent resistivity, and turbulent heating. The solar wind plasma is described as a system of co-moving solar wind protons, electrons, and interstellar pickup protons, with separate energy equations for each species. Numerical steady-state solutions of Reynolds-averaged solar wind equations coupled with turbulence transport equations for turbulence energy, cross helicity, and correlation length are obtained by the time relaxation method in the corotating with the Sun frame of reference in the region from 0.3 to 100 AU (but still inside the termination shock). The model equations include the effects of electron heat conduction, Coulomb collisions, photoionization of interstellar hydrogen atoms and their charge exchange with the solar wind protons, turbulence energy generation by pickup protons, and turbulent heating of solar wind protons and electrons. The turbulence transport model is based on the Reynolds decomposition and turbulence phenomenologies that describe the conversion of fluctuation energy into heat due to a turbulent cascade. In addition to using separate energy equations for the solar wind protons and electrons, a significant improvement over our previous work is that the turbulence model now uses an eddy viscosity approximation for themore » Reynolds stress tensor and the mean turbulent electric field. The approximation allows the turbulence model to account for driving of turbulence by large-scale velocity gradients. Using either a dipole approximation for the solar magnetic field or synoptic solar magnetograms from the Wilcox Solar Observatory for assigning boundary conditions at the coronal base, we apply the model to study the global structure of the solar wind and its three-dimensional properties, including embedded turbulence, heating, and acceleration throughout the heliosphere. The model results are compared with plasma and magnetic field observations on WIND, Ulysses, and Voyager 2 spacecraft.« less

LINKING REMOTE IMAGERY OF A CORONAL MASS EJECTION TO ITS IN SITU SIGNATURES AT 1 AU

In a case study (June 6-7, 2008) we report on how the internal structure of a coronal mass ejection (CME) at 1 AU can be anticipated from remote observations of white-light images of the heliosphere. Favorable circumstances are the absence of fast equatorial solar wind streams and a low CME velocity which allow us to relate the imaging and in-situ data in a straightforward way. The STEREO-B spacecraft encountered typical signatures of a magnetic flux rope inside an interplanetary CME (ICME) whose axis was inclined at 45 degree to the solar equatorial plane. Various CME direction-finding techniques yield consistent results to within 15 degree. Further, remote images from STEREO-A show that (1) the CME is unambiguously connected to the ICME and can be tracked all the way to 1 AU, (2) the particular arc-like morphology of the CME points to an inclined axis, and (3) the three-part structure of the CME may be plausibly related to the in situ data. This is a first step in predicting both the direction of travel and the internal structure of CMEs from complete remote observations between the Sun and 1 AU, which is one of the main requirements for forecasting the geo-effectiveness of CMEs.

Comet Hyakutake (C/1996 B2): Spectacular disconnection event and the latitudinal structure of the solar wind

Images of comet Hyakutake (C/1996 B2) are analyzed in conjunction with solar wind data from spacecraft to determine the relationship between solar wind conditions and plasma tail morphology. The disconnection event (DE) on March 25, 1996 is analyzed with the aid of data from the IMP-8 and WIND Earth-orbiting spacecraft and the DE is found to be correlated with a crossing of the heliospheric current sheet. The comet was within 0.1 AU of Earth at the time of the DE and data from IMP-8 and WIND show no high-speed streams, significant density enhancements or shocks. The latitudinal variation in the appearance and orientation of the plasma tail are interpreted based on results from the Ulysses spacecraft. In the polar solar wind region, the comet has a relatively undisturbed appearance, no DEs were observed, and the orientation of the plasma tail was consistent with a higher solar wind speed. In the equatorial solar wind region, the comet's plasma tail had a disturbed appearance, a major DE was observed, and the orientation of the plasma tail was consistent with a lower solar wind speed. The boundary between the equatorial and polar regions crossed by comet Hyakutake in April 1996 was near 30°N (ecliptic) or 24°N (solar) latitude.

Resonant acceleration and heating of solar wind ions by dispersive ion cyclotron waves

We investigate the preferential acceleration and heating of solar wind alpha particles by the resonant cyclotron interaction with parallel‐propagating left‐hand‐polarized ion cyclotron waves. The Alfvén wave spectrum equation is generalized to multi‐ion plasmas and a Kolmogorov type of cascade effect is introduced to transfer energy from the low‐frequency Alfvén waves to the high‐frequency ion cyclotron waves, which are assumed to be entirely dissipated by the wave‐particle interaction. In order to distribute the dissipated wave energy among the alphas and protons, the quasi‐linear theory of the wave‐particle interaction is used along with the cold plasma dispersion relation, and a power law spectrum of the ion cyclotron waves is assumed, with the spectral index as a free parameter of the model. The set of three‐fluid solar wind equations and the Alfvén wave spectrum equation are then solved in order to find fast solar wind solutions. It is found that the effect of the alpha particles on the dispersion relation, omitted in most previous wave‐driven solar wind models, has a significant influence on the preferential acceleration and heating of the alphas, especially in the region close to the Sun. With this effect included, the alpha particles can be accelerated to a bulk flow speed faster than the protons by a few hundred kilometers per second and heated by the resonant cyclotron interaction to more than mass‐proportional temperature values at several solar radii. However, this mechanism does not yield a differential speed of the order of an Alfvén speed and a mass‐proportional temperature for the alphas beyond 0.3 AU, as observed, which confirms the same conclusion reached previously by Isenberg and Hollweg [1983] for nondispersive ion cyclotron waves.

Self-consistent and Time-dependent Solar Wind Models

We describe the first results from a self-consistent study of Alfven waves for the time-dependent, single-fluid magnetohydrodynamic (MHD) solar wind equations, using a modified version of the ZEUS MHD code. The wind models we examine are radially symmetrical and magnetized; the initial outflow is described by the standard Parker wind solution. Our study focuses on the effects of Alfven waves on the outflow and is based on solving the full set of the ideal nonlinear MHD equations. In contrast to previous studies, no assumptions regarding wave linearity, wave damping, and wave-flow interaction are made; thus, the models naturally account for the back-reaction of the wind on the waves, as well as for the nonlinear interaction between different types of MHD waves. Our results clearly demonstrate when momentum deposition by Alfven waves in the solar wind can be sufficient to explain the origin of fast streams in solar coronal holes; we discuss the range of wave amplitudes required to obtained such fast stream solutions.

Reply [to “Comments on ‘Alfvénic disturbances in the equatorial solar wind with a spiral magnetic field' by Yu‐Qing Lou”

Reply [to “Comments on ‘Alfvénic disturbances in the equatorial solar wind with a spiral magnetic field' by Yu‐Qing Lou”

Comments on “Alfvénic disturbances in the equatorial solar wind with a spiral magnetic field” by Yu‐Qing Lou

Comments on “Alfvénic disturbances in the equatorial solar wind with a spiral magnetic field” by Yu‐Qing Lou

On two‐component models of solar wind fluctuations

Matthaeus et al. first suggested a two‐component scenario for solar wind fluctuations which contains both parallel propagating Alfvén waves together with a population of quasi‐two‐dimensional turbulence. On the basis of these earlier ideas and further evidence, Tu and Marsch developed a two‐component model, in which small‐scale incompressible fluctuations are decomposed into Alfvén waves and convective structures, in an attempt to account for observed radial evolutions of energy density, normalized cross helicity σc, and Alfvén ratio rA of solar wind fluctuations. According to a recent analysis on two‐dimensional Alfvénic disturbances in the equatorial solar wind with a spiral magnetic field, we discuss the theoretical basis of this important two‐component concept and further indicate necessary ingredients that a more self‐consistent two‐component turbulence model should contain. From the perspective of Alfvénic disturbances in the solar wind, the wave and convective components coexist in a naturally intermixed manner. Our analysis specifically describes behaviors of Alfvénic disturbances of all scales, including the intermediate ones. In a fully developed spiral magnetic field at large radii, the criterion for separating radially propagating Alfvén waves and radially convective Alfvénic structures depends on whether |ƒ + mƒS| is greater or less than the characteristic frequency ƒc ≡ (FMU³/FB²)1/2, where ƒ is the perturbation frequency, ƒS is the solar rotation frequency, m is an integer for the azimuthal wavenumber, FM is the solar wind mass flux, FB is the radial magnetic flux, and U is the asymptotic wind speed. For typical solar wind parameters, the timescale corresponding to ƒc is ≳ 15 hours.

Supersonic Solutions of the Solar-Wind Equations

We reexamine the inviscid solar-wind equations with heat conduction from below and establish a fundamentally new approach for finding supersonic solutions. Although the problem is fourth order, only two independent integration constants can be arbitrarily assigned, since two boundary conditions that are required to specify well-behaved supersonic solutions determine the values of the other two constants. Valid numerical solutions can be integrated only by recognizing that the asymptotic values of temperature and kinetic energy are determined by the integration constants previously admitted. The Noble and Scarf (1963, Astrophys. J. 138, 1169-1191) solar-wind models are essentially accurate for practical purposes, but in a fundamental sense they are not self-consistent. The ratio of thermal energy to gravitational potential at the supersonic point, ( kTr/GMm) s, must lie between 0.4375 and about 0.405 to permit stable supersonic solutions for ionized hydrogen.

Alfvénic disturbances in the equatorial solar wind with a spiral magnetic field

We study a two‐dimensional propagation of Alfvénic perturbations within the solar equatorial plane (SEP) in the solar wind with a spiral magnetic field. Analytical solutions within the entire frequency range are derived at large radial distance r. Numerical solutions which smoothly pass through the Alfvén critical point rA are also obtained. Since the spiral magnetic field caused by the solar rotation scales as ∼r−1 at large r, the velocity and magnetic field perturbations, which are perpendicular to the SEP, remain small relative to the wind parameters in the entire radial range. The frequency criterion for the non‐WKB radial scalings of the perturbation variables to appear is (f + mf⊙)² < fc², where f is the perturbation frequency, f⊙ is the solar rotation frequency, m is the azimuthal wavenumber, and fc is the well‐known characteristic frequency determined by the wind parameters. We demonstrate that the process of continuous reflection in a spiral magnetic field and the effect of superposing Alfvénic perturbations with various f and m can lead to variations of the relative magnetic field fluctuation, the normalized cross helicity σc, the Alfvén ratio ra, the magnetic perturbation energy density Eb, and the kinetic perturbation energy density Eυ, which differ from the predictions of the conventional WKB theory. In particular, the non‐WKB radial scalings of the perturbation variables can appear at any timescales with appropriate values of m. The relevance of these results to data analyses on interplanetary fluctuations is discussed, and we emphasize the importance of obtaining fluctuation spectrum with respect to both f and m in the outer heliosphere.

Compressible fluctuations in the equatorial solar wind with a spiral magnetic field

We investigate basic properties of two‐dimensional propagation and advection of compressible magnetohydrodynamic (MHD) perturbations in an axisymmetric equatorial solar wind with a spiral magnetic field. Analytical perturbation solutions at large radii are derived. In principle, numerical solutions which smoothly pass through both MHD fast and slow critical points can be constructed. In addition to wide applications of these solutions to small‐scale small‐amplitude compressible MHD fluctuations within the equatorial solar wind, we view large‐scale MHD pressure waves and pressure‐balanced structures (PBSs) as they evolve into large heliocentric distances, as a kind of nonlinear manifestations of MHD fast‐mode and slow‐mode disturbances in the solar wind, respectively. We discuss radial evolution of MHD fast waves for large‐scale MHD pressure waves in their linear phase in the distant solar wind. We argue that it is natural to interpret PBSs in terms of MHD slow‐mode disturbances at large heliocentric distances. We propose a scenario to identify PBSs with MHD slow‐mode perturbations as well as their relation to MHD pressure waves based on the dissolution of a wave packet of compressible MHD disturbances in the interplanetary space. In principle, observational verification of this suggestion can be carried out by one or several coordinated spacecrafts. Furthermore, the analysis described here provides an important basis for two‐dimensional numerical simulations of nonlinear interactions of interplanetary disturbances. Finally, we indicate that the minimum variance direction of magnetic field fluctuations in the present formulation is along the radial direction. This result is consistent with the earlier WKB Alfvén wave normal analysis but is contrary to what is observed.

Shocks in the outer heliosphere: Voyager 2 observations from 18.9 AU to 30.2 AU (1986‐1989)

A search for shocks observed by Voyager 2 from 1986.0 to day 233, 1989, in the region between 18.9 AU and 30.2 AU revealed only five candidate fast forward shocks and no reverse shocks. No shock was observed during 1987 (near solar minimum). Only one candidate shock was observed during 1986, when the heliospheric current sheet (HCS) was still close to the equator. This, the weakest of the shocks, was associated with a weak corotating merged interaction region (CMIR). Only two shocks were observed during 1988, when solar activity and the inclination of the HCS were increasing. One of these shocks was associated with a CMIR, the other with a local merged interaction region (MIR) (a transient MIR localized in space). The shocks observed between 18.9 AU and 28.2 AU from 1986 through 1988 were relatively weak, the estimated magnetoacoustic Mach number being ≤ 1.5 for each of the three shocks. Since the shocks observed between 20 AU and 29 AU during the 3‐year period centered around solar minimum were weak and few in number, shocks were not important in determining the structure, dynamics, and thermodynamics of the equatorial solar wind. The strongest shock in this study was observed during 1989 at the front of a global merged interaction region (GMIR) (a MIR that extends around the Sun and is presumably formed by the coalescence of several transient ejecta and other interaction regions). A weak shock was also observed inside this GMIR.

A parameter study of the two-fluid solar wind

A two-fluid model of the solar wind was introduced by Sturrock and Hartle (1966) and Hartle and Sturrock (1968). In these studies the proton energy equation was integrated neglecting the heat conductive term. Later several authors solved the equations for the two-fluid solar wind model keeping the proton heat conductive term. Methods where the equations are integrated simultaneously outward and inward from the critical point were used. The equations were also integrated inward from a large heliocentric distance. These methods have been applied to cases with low coronal base electron densities and high base temperatures. In this paper we present a method of integrating the two-fluid solar wind equations using an iteration procedure where the equations are integrated separately and the proton flux is kept constant during the integrations. The technique is applicable for a wide range of coronal base densities and temperatures. The method is used to carry out a parameter study of the two-fluid solar wind.

Proton and alpha particle fluxes in the solar wind: Results of a three‐fluid model

A three‐fluid model for the solar wind, with continuity, momentum, and energy equations for the three species protons, alpha particles, and electrons, including electron heat conduction and a parameterized coronal heat source, is used to study the behavior of the resulting proton and alpha fluxes as a function of coronal energy input and the proton‐to‐alpha density ratio at the coronal base. While in model solutions without alpha particles the proton flux is a strongly increasing function of coronal temperature, quite opposite to the observed near constancy of the proton flux in the solar wind, solutions with alpha particle abundances of the order of a few percent at the coronal base have a reduced proton flux and much less variation of the proton flux. The mechanism by which the proton flux is reduced is a pileup of the alpha particle abundance in the region of the coronal temperature maximum and the increased frictional drag on the protons resulting therefrom. We find a range of model solutions in which the resulting proton flux is almost independent of coronal energy input or coronal temperature, and we conclude that proton‐alpha friction and dynamic accumulation of alphas in the corona can provide an explanation for the near constancy of the proton flux which is observed in the solar wind.

Conductive damping of coronal motions

Time‐dependent motions in the solar corona are subject to modification through conductive transport of thermal energy away from compression regions and into rarefaction regions. The evolution depends on how the energy is introduced—by an essentially thermal perturbation or by a density perturbation. The net effect of thermal conduction is to convert local kinetic energy in the transient into widely distributed thermal energy, which then may in turn be partially converted into kinetic energy of the bulk flow. Local motion in the transient is thereby damped. These effects are modeled here using a numerical solution of the time‐dependent solar wind equations for single‐fluid thermally conductive flow. The simulations of transient phenomena are made for ordinary collisional thermal conduction and compared with similar simulations done using polytropic flow. Between one and five solar radii, the gross character in the two cases is very similar. However, there are effects with thermal conduction that cannot be modeled with polytropic flow, illustrated here by temperature and velocity forerunners and a delayed and large‐amplitude trailing velocity low. If thermal conduction is artifically inhibited by simple reduction of the thermal conduction coefficient, relative velocity amplitudes remain larger in approximate proportion to the amount the coefficient is reduced.