Solar Wind Equations Research Articles

Multiple transonic solutions with a new class of shock transitions in steady isothermal solar and stellar winds

A new class of shock transitions are shown to arise in the transonic solutions of the steady isothermal solar wind equations when, for example, momentum deposition gives rise to multiple critical points in the flow. These shock transitions between critical solutions occur for a certain range of the parameters characterizing the momentum deposition function. In the presence of such shock transitions, the isothermal wind equations admit multiple transonic solutions, namely a continuous solution passing through an inner critical point and solutions involving a shock transition between critical solutions. These multiple transonic solutions have the same flow speed at the base but different supersonic flow speeds at infinity. An additional interesting feature of the isothermal equations is the equivalence of nonradial flow tube divergence and momentum addition in giving rise to multiple critical points and hence to multiple transonic solutions with shock transitions. The physical relevance of these properties for astrophysical systems such as the inner solar wind, flows in extragalactic jets and accretion discs are discussed.

Unsteady, thermally conductive coronal flow

A numerical algorithm using implicit time-differencing is applied to the solar wind equations allowing, for the first time, solutions including thermal conduction to be found for time-dependent flow traversing the subsonic to supersonic velocity transition region. Sample solutions are shown that demonstate the distinctive differences introduced by including thermal conduction, in comparison to the commonly available solutions assuming polytropic flow. Also, it is found that steady solutions are produced at least as quickly using a time-dependent relaxation to the steady state as when solving the steady-state equations.

Solutions of the two-fluid solar wind equations: Adiabatic and conduction dominated solutions

Solutions of the two-fluid solar wind equations: Adiabatic and conduction dominated solutions

Solar wind models with Alfven waves

Solutions to the one-fluid solar-wind equations that include pressure and heating due to outwardly propagating Alfven waves are presented. The solutions are given for two forms of the heat flux: the classical Spitzer (1962) formula and a 'collisional' heat flux. Both high-speed and low-speed solar-wind streams are considered. A comparison of solutions with and without waves shows that the solutions without waves cannot come close to reproducing observations of total mass and energy flux at 1 AU, while those with waves are capable of bringing nearly all flow variables into reasonable agreement with the observations. The required wave flux is found to be about 100,000 erg/sq cm per sec for the high-speed streams and an order of magnitude less for the low-speed streams. It is concluded that the pressure and heating due to outwardly propagating Alfven waves are important elements of solar-wind dynamics that cannot be neglected and that wave heating is especially important for the protons in high-speed streams.

The spiral field inhibition of thermal conduction in two‐fluid solar wind models

The two‐fluid solar wind equations, including inhibition of heat conduction by the spiral magnetic field, have been solved for steady radial flow, and the results are compared with those of our previous study of two‐fluid models with straight interplanetary field lines. The main effects of the spiral field conduction cutoff are to bottle up electron heat inside 1 AU and to produce adiabatic electron (and proton) temperature profiles at large heliocentric distances. Otherwise, the spiral field models are nearly identical with straight field models with the same temperatures and velocity at 1 AU, except for models associated with very low coronal base densities (n0 ∼ 106 cm−3 at 1 Rs). Low base density spiral models give a nearly isothermal electron temperature profile over 50–100 AU together with high velocities and temperatures at 1 AU. In general, high‐velocity models do not agree well with observed high‐velocity streams; lower‐velocity states can be represented reasonably well at 1 AU, but only for very high proton temperatures (Tp ∼ 2Te) at the coronal base. For spherically symmetric base conditions the straight field and spiral field models can be regarded, in lowest order, as approximations to the polar and equatorial three‐dimensional flows, respectively. This viewpoint suggests a pole to equator electron temperature gradient in the region 1‐10 AU, which would be associated with a meridional velocity of ∼0.5–1.0 km/s, diverging away from the equatorial plane. The formalism developed in this paper shows rather stringent limits to the mass loss rate for conductively driven winds and, in particular, illustrates that putative T Tauri outflows could not be conductively driven.

A reexamination of two-fluid solar wind models

The two-fluid solar-wind equations have been solved by a method which is approximately 50 times faster than any previously developed, through the use of asymptotic expansions which are self-consistently iterated upon to find a solution that passes through the critical point. The energy assumptions in two-fluid solar-wind models are reexamined, and the conclusions are as follows: (1) proton thermal conduction may not be neglected, (2) the Coulomb logarithm must be calculated as a function of radius, and (3) the electron and proton temperatures at the base need not be equal, even when the time scale for energy exchange between the species is an order of magnitude smaller than the expansion time at the base. It is possible to reproduce reasonable quiet-time solar-wind parameters at 1 AU, but only if the proton temperature is approximately twice the electron temperature at 1 solar radius. This may indicate that extended proton heating is important in the outer solar corona. Winds with velocities at 1 AU of 450 km/s are generated without nonthermal energy deposition but require high proton temperatures as well as very low densities at the base. Higher-velocity solutions are not possible in a spherically symmetric geometry for reasonable particle fluxes at 1 AU, and it is suggested that these higher-velocity states probably require additional heating, acceleration mechanisms, or nonradial flow.

Effects of rapidly diverging flow, heat addition, and momentum addition in the solar wind and stellar winds

The roles of rapid flow tube divergence and heat and momentum addition in one-fluid models of the solar wind and stellar winds and the role of heat addition in two-fluid models of the solar wind ae explored. It is found that under certain circumstances heat addition, momentum addition, or the rapid divergence of a flow tube can produce more than one critical point in the solution topologies of the solar and stellar wind equations. For the solar wind, additional critical points associated with rapid flow tube divergence (e.g., in coronal holes) and/or with momentum addition can lead to high expansion speeds near the coronal base, and these high speeds, in conjuction with a rapid flow tube divergence, may in certain cases produce an increased conductive energy supply to the solar wind. If such an increased conductive energy supply is produced, the need for energy addition above the coronal base is correspondingly reduced. In addition, the high flow speeds at low altitudes make it possible for any required energy addition to occur relatively near the coronal base, so that the need for extended heating of the solar wind may be substantially less than has been suggested in the past. For radiation-drivenmore » stellar winds the additional critical points associated with rapid flow tube divergence can lead to supersonic flow much deeper in the stellar atmosphere than is predicted by radial, spherically symmetric flow models, and this might resolve the apparent conflict between radiation-driven wind models and certain observations of Of stars. In two-fluid solar wind models including proton heat addition it is shown that the radial electron temperature profile has a sufficiently important dynamical influence that the accurate treatment of electron energy transport is a primary prerequisite to obtaining quantitatively significant results. (AIP)« less

An asymptotic analysis of the viscous two-fluid solar wind equations

The method of matched asymptotic expansions is used to analyze the asymptotic solutions of the two-fluid solar wind equations far from the Sun in a small viscosity limit. In effect, small viscous terms are superposed on the Lucas inviscid supersonic solutions characterized, for various solution pairs (..mu.., ..nu..), by the electron and ion temperatures varying like T/sub e/, T/sub i/proportionalr/sup -//sup mu/, r/sup -//sup nu/ as the heliocentric radius r..-->..infinity. In the r..-->..infinity, viscosity ..-->..0 limit, the (4/3,4/3) solution is found to be uniformly valid; the (2/5,4/5), (2/5,2/5), (14/25,2/5) solutions merge into viscous (2/3,2/3) -type supersonic solutions; the (2/7,2/5), (2/7,2/7), (2/5,2/7) solutions ultimately become subsonic through a diffuse shock, with an associated finite gas pressure at infinity; and the (2/7,6/7), (18/35,2/7) solutions merge into unphysical solutions, the ion and electron temperatures increasing without bound in the respective cases. It is suggested that further numerical work is required to determine which of the viscous supersonic solutions best fit solar wind observations. (AIP)

Viscous solutions of the two-fluid solar wind equations

The two-fluid equations describing the ideal, steady, viscous solar wind are examined, and supersonic solutions are sought in which the electron and ion temperatures vary as inverse powers of the radial distance from the Sun. Just two solutions are found, and these are analogous to those found by Whang et al. (1966) and Dahlberg (1970) in one-fluid theory.

On the effects of viscosity in one-fluid solar wind theory

In order to examine the influence of small viscous effects on the in viscid solutions of Parker, Whang & Chang, and Durney the ideal, steady, one-fluid solar wind equations are examined in the limit: Reynolds number → ∞, Prandtl number → 0 (i. e. viscosity → 0, thermal conduction fixed). The method of matched asymptotic expansions is employed, and it is found that the inclusion of viscosity has considerable effect on the solutions far from the Sun. Small viscous terms cause the velocity to drop below the supersonic value of the Parker solution to subsonic values through a diffuse shock, the temperature still following the variation ∝ r -2/7 at large dis­tances r . The result of small viscous effects on the Whang & Chang solution is that the temperature changes from a variation ∝ r -2/5 to follow the r -2/3 law of Whang, Liu & Chang, the velocity remaining supersonic, as r → ∞. The Durney solution is found to be the leading order part of a solution that is uniformly valid as r → ∞ in the limit considered.

Confirmation of known numerical solutions for the quiet solar wind equations

Confirmation of known numerical solutions for the quiet solar wind equations

On the effect of latitude dependent base conditions on the structure of the solar wind

The isothermal solar wind equations are solved for the case where the coronal conditions vary with latitude. It is found that the solutions are not uniquely determined by the base density but require knowledge of the injection angle of the fluid. Even for the case of spherically symmetric density at the corona, the solutions are not unique and form a one parameter set, but the latitude variation decreases rapidly with increasing heliocentric distance.

A note on the solution of the saturation flux limited solar wind equations

The solution curves of the differential equations determining the behavior of the solar wind are calculated for the case where the heat flux has its maximum value 3/2 nkTvth. All the supersonic solutions are asymptotically adiabatic, T ∼r-4/3.

The asymptotic behavior of the supersonic solutions of the two-fluid solar wind equations

Three different asymptotic branches of the two-fluid equations are found with τeαλm, τpαλn, where, λ is the inverse distance from the Sun, with (m, n) = (2/7, 2/7), (2/7, 6/7), (4/3, 4/3); other special solutions are also found but they correspond to special choices of density and temperature at the corona. In all the (4/3, 4/3) solutions, the electron and proton temperatures tend to equality at large distances.

On the Domains of Existence of the Three Types of Supersonic Solutions of the Inviscid Solar-Wind Equations

view Abstract Citations (6) References (7) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS On the Domains of Existence of the Three Types of Supersonic Solutions of the Inviscid Solar-Wind Equations Durney, B. R. ; Werner, N. Abstract The approximate energy equation for radial distances larger than the critical point is solved with "critical point" boundary conditions. It is shown that if E = (41/24) (12/5)5/3 (35/2A)2/3 = e "' (where E is the residual energy per particle at infinity, A = 5.8 X 106/C, and C is the mass flow) then the equation has the solution T . If 6 < , the asymptotic behavior of the temperature is whereas if e > 6 "' then T for large values of r. This clarifies the domains of existence of the Parker, Whang and Chang, and supersonic solutions of the solar-wind equations. Publication: The Astrophysical Journal Pub Date: February 1972 DOI: 10.1086/151315 Bibcode: 1972ApJ...171..609D full text sources ADS |

Asymptotic solutions of the viscous solar wind equations

There is one and only one solution of the viscous solar wind equations which is asymptotically represented for large distances from the sun by a formal solution of Y. C. Whang, C. K. Liu, and C. C. Chang as a power series in the inverse one third power of the distance.

Convergent solutions of the inviscid solar wind equations

Formal solutions of the inviscid solar wind equations as power series in the inverse one seventh power and in the inverse one fifth power of the distance from the sun are shown to be convergent.