A&A 456, 359-365 (2006)
DOI: 10.1051/0004-6361:20054624
B. Li - X. Li
Institute of Mathematical and Physical Sciences, University of Wales Aberystwyth, SY23 3BZ, UK
Received 1 December 2005 / Accepted 31 May 2006
Abstract
Aims. The classic Weber-Davis model of the solar wind is reconsidered by incorporating alpha particles and by allowing the solar wind to flow out of the equatorial plane in an axisymmetrical configuration.
Methods. In the ion momentum equations of the solar wind, the ion gyro-frequency is many orders of magnitude higher than any other frequency. This requires that the difference between proton and alpha velocity vectors be aligned with the background magnetic field. With the aid of this alignment condition, the governing equations of the multi-fluid solar wind are derived from the standard transport equations. The governing equations are numerically solved along a prescribed meridional magnetic field line located at colatitude
at 1 AU and a steady state fast solar wind solution is found.
Results. A general analysis concludes, in agreement with the Weber-Davis model, that the magnetic field helps the coronal plasma to achieve an effective corotation out to the Alfvénic radius, where the poloidal Alfvénic Mach number M_{T} equals unity (M_{T} is defined by Eq. (20)). The model computations show that, magnetic stresses predominate the angular momentum loss of the Sun. For the fast wind considered, the proton contribution to the angular momentum loss, which can be larger than the magnetic one, is almost completely canceled by the alpha particles that develop an azimuthal speed in the direction opposite to the solar rotation. The Poynting flux associated with the azimuthal components is negligible in the energy budget. However, the solar rotation can play some role in reducing the relative speed between alpha particles and protons for low latitude fast solar wind streams in interplanetary space.
Key words: Sun: solar wind - Sun: magnetic fields - stars: winds, outflows
The solar angular momentum loss rate consists of the particle contribution and that contained in magnetic stresses . The comparison of measurements of these quantities with models, the Weber-Davis analysis (1967) in particular, has yielded divergent results. Missions before Helios measured a total angular momentum flux consistent with the Weber-Davis model (about 10^{30} dyne cm sr^{-1}), but the measured azimuthal angle of the bulk flow was generally greater than at 1 AU (or equivalently 7 km s^{-1} for an average slow wind of 400 km s^{-1}) (see Pizzo et al. 1983, and references therein). Such a large azimuthal flow speed implies that particles play a far more important role than magnetic stresses in reducing the angular momentum of the Sun. However, in the Weber-Davis model, 3/4 of the angular momentum flux at 1 AU is due to magnetic stresses. The Helios data show that is dyne cm sr^{-1}, in which is about dyne cm sr^{-1} (Pizzo et al. 1983). Although the measured magnitude of is smaller than that computed in the Weber-Davis model, the distribution of angular momentum flux between particles and magnetic stresses is largely compatible with their prediction. An equally important finding concerns further distribution of between two major ion species in the solar wind, namely, protons and alpha particles. Alpha particles are found to carry an angular momentum flux in the direction of counter-rotation with the Sun. This flux is substantial enough to offset the proton contribution which could be comparable to the magnetic one. This finding cannot be addressed by the Weber-Davis model where the solar wind was treated as a bulk flow and only protons were considered.
Apart from being essential in the problem of solar angular momentum loss, the azimuthal ion motions may also provide a possible means to resolve a long standing observational puzzle, namely that alpha particles are observed to flow faster than protons in the fast solar wind. The differential streaming in the fast wind could be as pronounced as 150 km s^{-1} at 0.3 AU before decreasing to about 40 km s^{-1} at 1 AU (Marsch et al. 1982). Such a behavior has yet to be understood. One possible mechanism is the coupling between the azimuthal and meridional motions, facilitated by the strong magnetic field (McKenzie et al. 1979; Hollweg & Isenberg 1981). Although the Poynting flux may still be negligible (cf. Acuna & Whang 1976; Alexander & de La Torre 1995; Hu et al. 2003), the proposed coupling is expected to limit, at least to a non-trivial extent, the ion differential streaming. As pointed out by Hollweg & Isenberg (1983), one shortcoming of the formulation of McKenzie et al. (1979) is that protons are privileged over alphas: the azimuthal magnetic field is assumed to be determined entirely by the protons whose azimuthal flow is neglected. Hence the formulation cannot properly account for the azimuthal dynamics of protons or alphas. In addition, the formulae are applicable only to the equatorial plane where the slow solar wind prevails at solar minimum.
The goal of this paper is to extend the Weber-Davis model by including alpha particles self-consistently. This approach allows us to assess the individual contributions of ion flows and magnetic stresses to the angular momentum loss of the Sun. The effect of the coupling between azimuthal and meridional motions in limiting the proton-alpha differential streaming will also be explored quantitatively. Given that the differential streaming is more prominent in the fast wind, which in general flows out of the equatorial plane, it is necessary to formulate the model such that it treats both protons and alphas on an equal footing, and allows the solar wind to flow outside the equatorial plane. In this sense, this paper also extends the model of McKenzie et al. (1979).
The paper is organized as follows. The derivation of the governing equations is given in the appendix. Section 2 details the physical model and also describes the assumptions on the background poloidal magnetic field and the ion heating mechanism. A general analysis is then given in Sect. 3. Section 4 presents the numerical results and the effect of the solar rotation. In Sect. 5, the main results are summarized.
The solar wind is assumed to consist of electrons (e), protons (p) and alpha particles (). Since the role of alpha particles is not necessarily minor, their contribution has to be self-consistently taken into account (Li et al. 1997). This is done by rewriting the momentum equations (Schunk 1977) in the flux tube frame, instead of the standard spherical coordinate system ( ). Central to the derivation is that the ion-cyclotron frequency is many orders of magnitude higher than any other frequency present in the ion momentum equations (McKenzie et al. 1979). Here m_{k} is the mass of ion species k ( ), Z_{k} is the charge of species k in units of the electron charge e, B_{l} denotes the meridional magnetic field strength, and c is the speed of light. The derivation is provided in the appendix, the resulting governing equations are quoted here to save space.
Given these assumptions, the governing equations take the form
Figure 1: a) The poloidal magnetic field configuration given as contours of the magnetic flux function. The equator points upward. The line of force on which the model equations are solved is displayed by the thick contour. This field line is located at at 1 AU and originates from on the Sun. b) Radial distribution of the poloidal magnetic field strength B_{l} along the designated field line. At 1 AU, B_{l} is . | |
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The energy and momentum exchange rates
and
are due to
Coulomb collisions of species
with all the remaining
ones (Schunk 1977),
To avoid complications associated with the cross-field force balance, we choose to prescribe the background poloidal magnetic field by adopting an analytical model given in Banaszkiewicz et al. (1998). In the present implementation, the model magnetic field consists of dipole and current-sheet components only. A set of parameters M=3.6222, Q=0, K=1.0534 and a_{1}=2.5 are chosen such that the last open magnetic field line is anchored at on the Sun, and the poloidal magnetic field strength is 3.3 at at 1 AU, compatible with Ulysses measurements (Smith & Balogh 1995).
Figure 1a shows the magnetic field configuration in the meridional plane. The thick solid line represents the field line along which we will find solar wind solutions. This field line is rooted at colatitude on the Sun, and reaches at 1 AU, which corresponds to the edge of the fast stream observed by Ulysses (McComas et al. 2000). Plotted in Fig. 1b is the radial profile of the poloidal magnetic field strength B_{l} along the designated field line.
To produce fast solar wind solutions,
an empirical energy flux, launched from the Sun and in the direction of ,
is assumed to heat ions only.
This energy flux is assumed to dissipate
at a rate Q with a characteristic length ,
i.e.,
In the computations, the following parameters
Before solving Eqs. (1) to (7) to find solar wind solutions, one can conduct an analysis to reach some general conclusions.
The conservation law for the angular momentum, Eq. (14), is valid for an arbitrary flux tube in an azimuthally symmetric solar wind. The single-fluid version (or equivalently the two-fluid one) of this conservation law has already been obtained by, e.g., Low & Tsinganos (1986) and Hu et al. (2003).
For the solar wind,
is valid near 1 ,
but
holds at 1 AU.
Hence, there must exist a point between 1
and 1 AU where M_{T}=1.
At this location, which will be termed the Alfvénic point,
is singular unless the numerator in Eq. (19) vanishes,
(22) |
Hence the conclusion of Weber & Davis (1967) still holds: the magnetic field helps the coronal plasma to achieve an effective corotation to the Alfvénic point, as long as the poloidal Alfvénic Mach number M_{T} is defined by Eq. (20).
Figure 2: Results derived from a 1.5D 3-fluid solar wind model which incorporates the azimuthal components self-consistently. The radial distribution of a) the densities of protons and alpha particles (solid lines), as well as electrons (dashed line), b) the poloidal flow speeds of protons ( ) and alphas ( ), and c) the temperatures of electrons (), protons () and alpha particles (). The error bars in a) are the upper and lower limits for the electron density derived by Fisher & Guhathakurta (1995). The error bars in c) represent the uncertainties of UVCS measurements for the effective proton temperature reported by Kohl et al. (1998). Please note that both measurements are made for polar coronal holes. | |
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Combining the governing equations in the steady state, one can
derive an energy conservation law,
Equations (1) to (7)
are solved by using a fully implicit numerical scheme
(Hu et al. 1997).
From an arbitrary initial guess, the equations are advanced
in time until a steady state is achieved.
The computational domain extends from 1
to 1.2 AU.
At 1 ,
ion densities as well as species temperatures are fixed,
Figure 2 displays the radial distribution of
(a) the species densities ,
and ,
(b) poloidal flow speeds
and
,
and (c) species temperatures ,
and .
The model yields the following parameters at 1 AU,
To examine the differential streaming further, is plotted in Fig. 3a. The poloidal flow speeds of protons ( ) and alpha particles ( ) are replotted in Fig. 3b (a different scale is used, see Fig. 2b). In addition, model results from the corresponding computation that neglects the solar rotation (i.e., ) are plotted as dotted lines for comparison. For the ease of description, we shall call the model with (without) azimuthal components the 1.5D (1D) model. It is found that the effect of the azimuthal components on the poloidal dynamics can be adequately represented by the flow speed profiles. Below the local maximum of 78.6 km s^{-1} at 7.28 , Fig. 3a shows no difference in the profile between 1D and 1.5D models. The differential streaming, , for both models plummets from nearly zero at the coronal base to about -44.6 km s^{-1} at 1.44 , and rises thereafter to the local maximum. Interestingly, in the 1D model, beyond the local maximum undergoes only a modest decrease to 66.3 km s^{-1} at 1 AU, while in the 1.5D model is 48.7 km s^{-1} at 1 AU. This further reduction in the differential streaming is achieved through a slight rise in the profile accompanied by a modest deceleration of alpha particles (Fig. 3b).
This behavior is not surprising since
in the poloidal momentum equation (Eq. (2)),
(25) |
Figure 4 displays the radial profiles of
(a)
,
(b) the azimuthal speeds of protons (
), alpha particles (
)
and electrons (
),
and (c) the specific contribution of protons (
),
alpha particles (
)
and the magnetic stresses (
)
to the angular momentum flux
(cf. Eq. (15)).
In addition, the sum
,
which gives the overall particle
contribution
,
is also plotted.
Given in dotted line is ,
the ratio of the Poynting flux to the total energy flux
(cf. Eq. (23)).
In Fig. 4c, the dashed line is used to plot negative values.
The asterisks in Fig. 4b denote the
Alfvénic point, which is located at
.
Figure 3: Radial distributions of a) the differential streaming, , and b) the poloidal flow speeds of protons ( ) and alpha particles ( ). Solid lines are used to plot the 1.5D model, whereas dotted lines are used for the corresponding 1D model which neglects the solar rotation. | |
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Figure 4: Radial distributions of a) where is the magnetic azimuthal angle, b) the azimuthal speeds of protons , alpha particles as well as electrons , c) the relative importance of the proton fluid , the alpha fluid , the sum of the two , and the magnetic stresses in the total angular momentum loss of the Sun (please see Eq. (16) in text). In addition, the ratio of the Poynting flux to the total energy flux, , is plotted as dotted line. The dashed line represents negative values. In panel b), the asterisks denote the Alfvénic point, where the poloidal Alfvénic Mach number (defined by Eq. (20)) equals unity. | |
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From Fig. 4a, it is obvious that only beyond, say 10 , does a spiral angle develop. This can be explained in view of Eqs. (12), (13): within 10 the left hand side is much smaller than the poloidal flow speed on the right hand side for any species. On the other hand, in interplanetary space, the species azimuthal speed is much smaller than , the Parker theory for the spiral magnetic field is recovered, i.e., , where v_{l} can be taken as the poloidal speed of any species.
In the inner corona, both protons and alpha particles tend to corotate with the Sun: and are positive (Fig. 4b). The azimuthal speed of the alpha particles is slightly larger than that of the protons below 2 , and from there on, the alpha particles are gradually turned opposite to the solar rotation. becomes negative beyond 5.71 , eventually reaches -24.7 km s^{-1} at 1 AU. On the other hand, the proton azimuthal speed increases from a local minimum of 0.78 km s^{-1} at 12.5 monotonically to 4.8 km s^{-1} at 1 AU.
The behavior of the azimuthal flow speeds can be explained
by Eqs. (17) and (18).
Near the coronal base, both
and
are far from unity.
It then follows from Eqs. (17) and (18) that
(27) |
Now let us move on to Fig. 4c. It can be seen that, from the coronal base to 1 AU, magnetic stresses play a dominant role in the total angular momentum budget, the particle contribution is no more than . However, the individual angular momentum flux carried by protons or alpha particles is not necessarily small in magnitude. As a matter of fact, protons contribute more to the total angular momentum flux than magnetic stresses do beyond 101 . However the proton contribution is virtually canceled by the alpha particles that counter-rotate with the Sun. This can be understood in light of Eqs. (17) and (18). As has been described, far away from the Alfvénic point, , both and are mainly determined by the terms associated with the differential streaming. From the identity , one can see that for , and tend to have the same magnitude but opposite sign. At this point, we can also see from the dotted line in Fig. 4c that although the solar rotation introduces appreciable difference in the meridional dynamics, the Poynting flux never exceeds of the total energy budget. Needless to say, its contribution to the solar wind acceleration is in fact determined by its difference between 1 and 1 AU.
At 1 AU, the model yields a total angular momentum loss of dyne cm sr^{-1}, in which the magnetic part is dyne cm sr^{-1}, consistent with measurements (Pizzo et al. 1983; Marsch & Richter 1984). However, the absolute azimuthal speed or is larger than the measured values (although these quantities can only be determined with a modest precision). Moreover, never turns negative, in this sense at variance with the measurements: Pizzo et al. (1983) and Marsch & Richter (1984) showed that particles in the fast wind tend to carry a negative angular momentum flux. Pizzo et al. (1983) suggested that the discrepancies between the model and measurements may be removed by including the stream interaction in the super-Alfvénic region. This is however beyond the scope of this paper.
The main aim of this paper is to extend the Weber-Davis analysis (Weber & Davis 1967) on the transport of the angular momentum from the Sun by including alpha particles and by allowing the solar wind to flow out of the equatorial plane in an axisymmetrical configuration. Following McKenzie et al. (1979), we exploit the fact that the gyro-frequency of ions is many orders of magnitude higher than any other frequency in ion momentum equations. From this it follows that the difference between proton and alpha velocities must be in the direction of the magnetic field. Using this alignment condition, the governing equations are then derived from the standard five-moment transport equations.
The model equations also enable us to examine quantitatively the effect of azimuthal components in limiting the proton-alpha differential streaming in the fast wind. For simplicity, we choose to solve the governing equations on a prescribed poloidal magnetic field line located at a colatitude of at 1 AU, corresponding to the edge of the fast stream observed by Ulysses at solar minimum conditions (McComas et al. 2000). The effects of the azimuthal components on the meridional dynamics, if any, are optimal in this regard. These effects are directly shown by a comparison of two models with and without azimuthal components.
The main results can be summarized as follows:
The model also suffers from the inconsistency that the force balance in the direction perpendicular to the poloidal magnetic field is replaced by prescribing a background magnetic field. In a more rigorous treatment, the poloidal magnetic field should be derived self-consistently. In principle, such a task can be accomplished by adopting an iterative approach: the parallel and perpendicular force balance are solved alternately until a convergence is met (Pneuman & Kopp 1971; Sakurai 1985). By doing so, the angular momentum loss from the Sun can be obtained self-consistently for all poloidal flux tubes. An accurate estimate of the duration over which the angular momentum of the Sun is completely removed is then possible (see Hu et al. 2003).
The present paper is aimed at presenting a rather general analysis of the angular momentum loss from a magnetized rotating object for flows assuming axial symmetry and incorporating two major ion species. Although for the present Sun, the centrifugal and magnetic forces are so weak that they have little impact on the meridional dynamics (especially below the Alfvénic point), a similar study as presented in the text can be carried out for stars that rotate at a faster rate or have a stronger magnetic field than the Sun.
Acknowledgements
This research is supported by a PPARC rolling grant to the University of Wales Aberystwyth. We thank Shadia Rifai Habbal for her input. We thank the anonymous referee for his/her comments which helped to improve this paper.
In this appendix, it is shown how the 5-moment transport equations are reduced to the governing equations in Sect. 2.1. The approach adopted here closely follows that by McKenzie et al. (1979; see also Hollweg & Isenberg 1981). The original derivation of McKenzie et al. (1979) is restricted to the equatorial flow, and ions other than protons are treated as test particles. Employing the same spirit, we extend their derivation to general flows assuming axial symmetry. In addition, all ion species are treated on an equal footing, which is particularly important for the solar wind since alpha particles can not be seen as test particles. The central point is that, due to the presence of a strong magnetic field (in the sense that the ion gyro-frequency is many orders of magnitude higher than any other frequency in the momentum equations), the difference vector between proton and alpha velocities must be aligned with the magnetic field.
First of all, let us examine the momentum equation for species
(Schunk 1977),
The plasma in question consists of two ion species, protons (p) and
an additional one (i).
(Subscript i is used here to indicate that in principle the equations
to be developed are also applicable if other ion species than alpha particles
is considered.)
As the frequency in question is well below the electron plasma frequency,
the expression for
follows from quasi-neutrality,
(A.4) |
Substitution of Eq. (A.2) into Eq. (A.1) for ion species k ()
then leads to
Now it becomes necessary to separate explicitly the poloidal and azimuthal
components of the magnetic field and species velocities, namely,
It proves useful to work in the flux tube frame, whose base vectors are
where
The ion gyro-frequency is many orders of magnitude higher than
any other frequency in the momentum equation.
This has two consequences.
First, from Eq. (A.12),
v_{j2}-v_{k2} is
far smaller than v_{k3} from an order-of-magnitude estimate.
Since
(see Eq. (A.9)),
both
and
should be very small and
can be safely neglected unless they appear alongside the ion-cyclotron frequency.
Second, Eq. (A.11) leads to
(A.13) |
Solving Eq. (A.12) for v_{j2}-v_{k2} and then substituting it into Eq. (A.10), one arrives at the poloidal momentum equation (Eq. (2)). It is interesting to note that the magnetic field does not appear explicitly in this equation (except for the term ), although it plays an essential role in coupling the azimuthal and meridional motions. Combining Eq. (A.12) for p and i, one can obtain Eq. (5). The partial differentiation with respect to time t in the equation is merely for numerical purpose.
In closing, we note that the p and i versions
of Eq. (A.11) can be combined to
yield a force balance condition in the
direction,