A&A 494, 361-371 (2009)
DOI: 10.1051/0004-6361:200810992
B. Li - X. Li
Institute of Mathematics and Physics, Aberystwyth University, SY23 3BZ, UK
Received 19 September 2008 / Accepted 27 November 2008
Abstract
Context. The Helios measurements of the angular momentum flux L of the fast solar wind lead to a tendency for the fluxes associated with individual ion angular momenta of protons and alpha particles,
and ,
to be negative (i.e., in the sense of counter-rotation with the Sun). However, the opposite holds for the slow wind, and the overall particle contribution
tends to exceed the magnetic contribution .
These two aspects are at variance with previous models.
Aims. We examine whether introducing realistic ion temperature anisotropies can resolve this discrepancy.
Methods. From a general set of multifluid transport equations with gyrotropic species pressure tensors, we derive the equations governing both the meridional and azimuthal dynamics of outflows from magnetized, rotating stars. The equations are not restricted to radial flows in the equatorial plane but valid for general axisymmetric winds that include two major ion species. The azimuthal dynamics are examined in detail, using the empirical meridional flow profiles for the solar wind, constructed mainly according to measurements made in situ.
Results. The angular momentum flux L is determined by the requirement that the solution to the total angular momentum conservation law is unique and smooth in the vicinity of the Alfvén point, defined as where the combined Alfvénic Mach number
.
has to consider the contributions from both protons and alpha particles. Introducing realistic ion temperature anisotropies may introduce a change of up to
in L and up to 1.8
in azimuthal speeds of individual ions between 0.3 and 1 AU, compared with the isotropic case. The latter has strong consequences on the relative importance of L_{P} and
in the angular momentum budget.
Conclusions. However, introducing ion temperature anisotropies cannot resolve the discrepancy between in situ measurements and model computations. For the fast-wind solutions, while in extreme cases L_{P} may become negative,
never does. On the other hand, for the slow solar wind solutions examined, L_{P} never exceeds ,
even though
may be less than the individual ion contribution, since
and
always have opposite signs for the slow and fast wind alike.
Key words: Sun: rotation - Sun: magnetic fields - solar wind - stars: rotation - stars: winds, outflows
The angular momentum loss of a rotating star due to its outflow influences the rotational evolution of the star considerably, and is therefore of astrophysical significance in general (see e.g., Weber & Davis 1967; Mestel & Spruit 1987; Belcher & MacGregor 1976; Bouvier et al. 1997). However, direct tests of in situ measurements against theories such as those presented by Weber & Davis (1967) are only possible for the present Sun. A substantial number of studies have been conducted and were compiled in the comprehensive paper by Pizzo et al. (1983), who themselves paid special attention to the Helios measurements of specific angular momentum fluxes. The measurements, further analyzed by Marsch & Richter (1984), are unique in that they allow the individual ion contribution from protons and alpha particles to the solar angular-momentum loss rate per steradian L to be examined. For instance, despite the significant scatter, the data exhibit a distinct trend for to be positive (negative) for solar winds with proton speeds below (above) 400 . A similar trend for is also found on average. The magnetic contribution , on the other hand, is remarkably constant. A mean value of 10^{29} can be quoted for the solar winds of all flow speeds and throughout the region from 0.3 to 1 AU. For comparison, the mean values of angular momentum fluxes carried by ion flows in the slow solar wind are and 10^{29} (see Table II of Pizzo et al. 1983). The overall particle contribution to L is then 10^{29} , which tends to be larger than . It is noteworthy that a more recent study by Scherer et al. (2001) showed how examining the long-term variation of the non-radial components of the solar wind velocity and the corresponding angular momentum fluxes can help us understand the heliospheric magnetic field better.
Alpha particles should be placed on the same footing as protons from the perspective of solar wind modeling, given their non-negligible abundance and the fact that there tends to exist a substantial differential speed . As shown by the Helios measurements, a amounting to up to of the local proton speed may occur in both the fast and slow solar winds (Marsch et al. 1982a,b), with the latter being exemplified by an event that took place on day 117 of 1978, when a positive was found at 0.3 AU (Marsch et al. 1981). That on the average in the slow wind simply reflects that the events with positive and negative occur with nearly equal frequency (Marsch et al. 1982a). As for the alpha abundance relative to protons, a value of ( ) is well-established for the fast (slow) solar wind (e.g., McComas et al. 2000). Therefore alpha particles can play an important role as far as the energy and linear momentum balance of the solar wind are concerned. When it comes to the problem of angular momentum transport, it was shown that in interplanetary space not only the angular momentum flux carried by the alpha particles but also that convected by the protons are determined by the terms associated with (Li & Li 2006). This essentially derives from the requirement that the proton-alpha velocity difference vector be aligned with the instantaneous magnetic field. As a consequence, these terms have no contribution to the overall angular momentum flux convected by the ion flow L_{P}, which turns out to be smaller than in all the models examined in the parameter study by Li et al. (2007). This, together with the fact that is always positive (i.e., in the sense of corotation with the Sun), is at variance with the Helios measurements.
A possible means to reconcile the measurements and the model computation is to incorporate the species temperature anisotropies. This is because the total pressure tensor summed over all species s participates in the problem of angular momentum transport via the component where and are relative to the magnetic field (see e.g., Weber 1970, hereafter referred to as W70). While the overall loss rate per steradian L may not be significantly altered, the azimuthal speed of the solar wind and therefore the particle part of L may be when compared with the isotropic case. Note that in the treatment of W70 the solar wind was seen as a bulk flow and the ion species are not distinguished. On the other hand, the formulation by Li & Li (2006) did not take into account the pressure anisotropy, which is a salient feature of the velocity distribution functions for both protons and alpha particles as revealed by the Helios measurements (Marsch et al. 1982a,b). It therefore remains to be seen how introducing the pressure anisotropy influences individual ion azimuthal speeds. Moreover, the simple, prescribed functional form for assumed in W70 needs to be updated in light of the more recent particle measurements.
The aim of the present paper is to extend the W70 study in three ways. First, we shall follow a multicomponent approach and examine the angular momentum transport in a solar wind comprising protons, alpha particles and electrons where a substantial proton-alpha particle velocity difference exists. Second, although following W70 we use a prescribed form of for simplicity, this prescription is based on the Helios measurements, and also takes into account other in situ and remote sensing measurements. Third, unlike W70 where the model equations are restricted to the equatorial plane, the equation set we shall derive is appropriate for a rather general axisymmetrical, time-independent, multicomponent, thermally anisotropic flow emanating from a magnetized rotating star. We note that a similar set of equations, which was also restricted to radial flows, was derived by Isenberg (1984) who worked in the corotating frame of reference and neglected the azimuthal dynamics altogether. The functional dependence on the radial distance and flow speed of the magnetic spiral angle was prescribed instead. His approach is certainly justifiable for the present Sun, but a self-consistent treatment of the azimuthal dynamics is required when flows from other stars are examined. This is because many stars either have a stronger magnetic field or rotate substantially faster than the Sun.
The paper is organized as follows. We start with Sect. 2 where a description is given for the general multifluid, gyrotropic transport equations, based on which the azimuthal dynamics of the multicomponent solar wind is examined. Then Sect. 3 describes the adopted meridional magnetic field and flow profiles. The numerical solutions to the angular momentum conservation law are given in Sect. 4. In Sect. 5, we shall discuss how examining the angular momentum transport in a multicomponent solar wind can also shed some light on the spectra of ion velocity fluctuations induced by Alfvénic activities. Finally, Sect. 6 summarizes the results. The equations of and a discussion on the poloidal dynamics are presented in the Appendix.
Presented in this section is the mathematical development of the equations that govern the angular momentum transport in a time-independent solar wind which consists of electrons (e), protons (p) and alpha particles (). Each species s ( ) is characterized by its mass m_{s}, electric charge e_{s}, number density n_{s}, mass density , velocity , and partial pressure tensor . If measured in units of the electron charge e, e_{s} may be expressed by e_{s} = Z_{s} e with by definition.
To simplify the mathematical treatment, a number of assumptions have been made and are collected as follows:
The equations appropriate for a multi-component solar wind plasma with gyrotropic species pressure tensors may be found by neglecting the electron inertia ( ) in the equations given by Barakat & Schunk (1982). Following the same procedure as given in the Appendix A.1 in Li & Li (2006), one may find
In Eq. (2), the subscript j stands for the ion species other than k, namely, for and vice versa. As can be seen, in addition to the term , the Lorentz force possesses a new term in the form of the cross product of the ion velocity difference and magnetic field. Physically, this new term represents the mutual gyration of one ion species about the other, the axis of gyration being in the direction of the instantaneous magnetic field. Furthermore, Eq. (5) is the time-independent version of the magnetic induction law, which states that the magnetic field is frozen in the electron fluid. It may be readily shown that the effects of the electron pressure gradient, the Hall term, and the momentum exchange rates as contained in the generalized Ohm's law can be safely neglected given the large spatial scale in question (a formal evaluation of the different terms can be found in Sect. 2.1 of Li et al. 2006).
To proceed, we choose a flux tube coordinate system, in which the base vectors are
,
where
The fact that v_{s N} ( ) is negligible means that the system of vector equations may be decomposed into a force balance condition across the poloidal magnetic field and a set of transport equations along it. In the present paper, however, we simply replace the force balance condition by prescribing an analytical meridional magnetic field configuration. Moreover, we examine in detail only the azimuthal dynamics, leaving a brief discussion on the poloidal one in the Appendix.
The
component of the magnetic induction law (5) gives
(9) |
Figure 1: Adopted meridional magnetic field configuration in the inner corona. Here only a quadrant is shown in which the magnetic axis points upward, and the thick contours labeled F and S delineate the lines of force along which the fast and slow solar wind solutions are examined, respectively. Also shown is how to define the geometrical factor R, and the base vectors and of the flux tube coordinate system (see Sect. 2). | |
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Another equation that enters into the azimuthal dynamics is the component of the total momentum. In the present case, it reads
(11) |
(12) |
For a time-independent flow
.
It then follows that
(14) |
Substituting Eqs. (10) into (13), one may find
As discussed in detail by Li & Li (2006), when species temperature anisotropy is absent (
and therefore
), for Eq. (17) to possess a solution that passes smoothly through
the two constants A_{L} and
have to be related by
(21) |
In principle, one needs to solve Eqs. (A.1) to (A.4) together with Eq. (17) simultaneously to gain a a quantitative insight. In the present paper, we refrain from doing so because from previous experience it proves difficult to yield the flow profiles that satisfactorily reproduce in situ measurements such as made by Helios. Take the proton-alpha speed difference in the fast solar wind for example. It is observationally established that closely tracks the local Alfvén speed in the heliocentric range r> 0.3 AU (Marsch et al. 1982a). So far this fact still poses a theoretical challenge: adjusting the ad hoc heating parameters, or fine-tuning the cyclotron resonance mechanism is unable to produce such a behavior (see, e.g. Hu & Habbal 1999). We therefore adopt an alternative approach by prescribing the background meridional flow profiles that mimic the observations and then examining what consequences the species anisotropies have on the azimuthal dynamics.
For the meridional magnetic field, we adopt an analytical model given by Banaszkiewicz et al. (1998). In the present implementation, the model magnetic field consists of the dipole and current-sheet components only. A set of parameters M=2.2265, Q=0, K=0.9343 and a_{1}=1.5 are chosen such that the last open magnetic field line is anchored at heliocentric colatitude on the Sun, while at the Earth orbit, the meridional magnetic field strength B_{l} is 3 and independent of colatitude , consistent with Ulysses measurements (Smith & Balogh 1995).
The background magnetic field configuration is depicted in Fig. 1, where the thick contours labeled F and S represent the lines of force along which we examine the fast and slow solar wind solutions, respectively. Tube F (S), which intersects the Earth orbit at 70 (89) colatitude, originates from ( ) at the Sun where the meridional magnetic field strength B_{l} is 3.93 (3.49) G.
The background meridional flow parameters are found by adopting a three-step approach described as follows:
Figure 2: Radial distribution between 1 and 1 AU of the adopted meridional flow parameters for the fast ( left column) and slow ( right) solar wind. a) and c), the meridional flow speeds of protons ( ) and alpha particles ( ). b) and d), the ion temperatures (dotted lines), (dashed lines), and (solid lines) where . The construction of is described in Sect. 3.2. The error bars in b) and d) represent the uncertainties of the UVCS measurements of the effective proton temperature as reported by Kohl et al. (1998) for a coronal hole, and by Frazin et al. (2003) for a streamer, respectively. Note that both measurements are typical of solar minimum conditions. Moreover, the asterisks in a) and c) denote the Alfvén point, where the meridional Alfvénic Mach number (defined by Eq. (18)) equals unity. | |
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In this study and will serve as free parameters. The Helios measurements indicate that 0.3 and 0.6 for the fast solar wind with , while 0.7 and 0.6 for the slow solar wind with (Marsch et al. 1982a,b). Theoretically, one may expect that the pair may not occupy the whole rectangle bounded by the given values in the space, since too strong an anisotropy can drive the system unstable with respect to a number of instabilities when the plasma is comparable to unity. Given that the lower limit of or is only slightly lower than 1, the ion-cyclotron instability can be shown to be unlikely to occur (see, e.g., Eq. (3) in Gary et al. 1994). However, the firehose instability may be relevant since it happens when is sufficiently larger than and . Note that the alpha particles with a non-negligible abundance drifting relative to protons may complicate the situation considerably given that in addition to the firehose, electromagnetic ion/ion instabilities may also be relevant and the occurrence of such instabilities is not restricted to the cases where the parallel is large (Hellinger & Trávnícek 2006). Nevertheless, we only compare the modeled with the non-resonant firehose criterion such as found via the dispersion relation of Alfvén waves (see Eq. (23) in Isenberg 1984). Specializing to an electron-proton-alpha plasma, the dispersion relation dictates that instability occurs when where ( ) with being the Alfvén speed determined by the bulk mass density . Using this criterion it is found that the modeled flow profiles are all stable with the only exception being for the segment in the fast wind with the largest values of and .
Figure 2 gives the radial distributions between 1 and 1 AU of the flow parameters for the fast and slow solar wind in the left and right panels, respectively. Figures 2a and c depict the meridional ion speeds and , while the ion temperatures (the dotted curves), (dashed) and T_{k} (solid) are given in Figs. 2b and d ( ). The values for the temperature anisotropy adopted for the construction are and for the fast wind, and and for the slow wind. In Fig. 2b, the error bars represent the uncertainties of the UVCS measurements for the proton effective temperature, made for a polar coronal hole as reported by Kohl et al. (1998). Similar measurements by Frazin et al. (2003) along the edges of an equatorial streamer are given in Fig. 2d. Moreover, the asterisks in Figs. 2a and c mark the location of the Alfvén point as defined by Eq. (18).
For the fast (slow) solar wind it is found that at 1 AU the meridional proton speed is 607 (304) , the proton flux is 2.8 (3.84) in units of 10^{8} , the alpha abundance is 4.56% (3.6%), and the meridional component of the proton-alpha velocity difference is 23 (5) . These values are consistent with in situ measurements such as made by Ulysses (McComas et al. 2000). Moreover, the fast (slow) solar wind reaches the Alfvén point at 10.7 (13.3) , beyond which increases only slightly with increasing r. On the other hand, for AU the meridional alpha speed decreases rather than increases with r as a consequence of the prescribed profile. If examining the ratio of to the meridional Alfvén speed , one may find that for the fast solar wind this ratio decreases only slightly from 0.98 at 0.3 AU to 0.82 at 1 AU, while for the slow wind it shows a substantial variation from 0.88 at 0.3 AU to 0.29 at 1 AU. The modeled can be seen to agree with the Helios measurements as given by Fig. 11 of Marsch et al. (1982a). Note that a value of at 0.3 AU is not unrealistic for slow solar winds, even larger values have been found by Helios 2 when approaching perihelion (Marsch et al. 1981). Moving on to the temperature profiles, one may see that the profiles inside 5 are in reasonable agreement with the UVCS line-width measurements for both the fast and slow solar wind.
Having described the meridional magnetic field and flow profiles, we may now address the following questions: to what extent is the total angular momentum loss of the Sun affected by the ion temperature anisotropies? and how is the angular momentum budget distributed among particle momenta, the magnetic torque, and the torque due to ion temperature anisotropies? To this end, let us first examine the fast and then the slow solar wind solutions. In the computations, we take 10^{-6} rad s^{-1}, which corresponds to a sidereal rotation period of 25.38 days.
Figure 3 presents the radial profiles of (a) the proton azimuthal speed ; (b) the alpha one ; and (c) the ion angular momentum fluxes L_{k} ( ), their sum L_{P}, the flux due to the magnetic torque , and that due to temperature anisotropies (see Eq. (16)). Note that the dash-dotted curves in Fig. 3c plot negative values. In Figs. 3a and b, the ion azimuthal speeds for the isotropic model with identical meridional flow parameters are given by the dashed lines for comparison. The fast wind profile corresponds to and .
For the chosen and , it is found that . Consequently, the total angular momentum loss rate per steradian L is 1.8 (here and hereafter in units of 10^{29} ) in the anisotropic case, and is only modestly enhanced compared with the isotropic case, for which L=1.71. Furthermore, Figs. 3a and b indicate that the radial dependence of the ion azimuthal speed or in the anisotropic model is similar to that in the isotropic one. For instance, both models yield that with increasing distance the alpha particles develop an azimuthal speed in the direction of counterrotation with the Sun: becomes negative beyond 7.95 (8.35) in the anisotropic (isotropic) model. The difference between the isotropic and anisotropic cases becomes more prominent at large distances where becomes increasingly significant, as would be expected from Eq. (17). Take the values of and at 1 AU. The isotropic (anisotropic) model yields that (3.46) and that (-11.7) at 1 AU. Note that the changes introduced to the ion azimuthal speeds by pressure anisotropies (0.9 for both protons and alpha particles) play an important role in the distribution of the angular momentum budget L among different contributions, as shown by Fig. 3c. The proton contribution exceeds for and attains 5.07 at 1 AU, significantly larger than the magnetic part at the same location. In fact, the overall particle contribution L_{P}, which increases with distance, overtakes the magnetic contribution from 170.5 onwards, despite the fact that the alpha contribution tends to offset the proton one. The dominance of L_{P} over happens in conjunction with the increasing importance of , the flux due to total pressure anisotropy which is in the direction of counterrotation with the Sun. In contrast, without pressure anisotropies, at 1 AU it turns out that even though a value of 3.73 is found for , it is almost cancelled by an of -3.49. The resulting L_{P} is thus 0.23, substantially smaller than , which is nearly identical to the value found in the anisotropic model. This contrast between anisotropic and isotropic cases is understandable since it follows from Eq. (13) that, given that the constant A_{L} does not vary much from the isotropic to anisotropic model, the change of L_{P} should be largely offset by that of .
Figure 4 expands the obtained results by displaying the dependence on and of (a) the factor , (b) the proton azimuthal speed and (c) the alpha one at two different distances plotted by the different linestyles indicated in (b), as well as (d) the constituents comprising the angular momentum flux at 1 AU. In addition to the individual ion contributions and , and the magnetic one , the overall particle contribution is also given. Note that instead of is plotted in Fig. 4d. Moreover, the horizontal bars on the left of Figs. 4b and c represent the azimuthal ion speeds derived in the isotropic case at the corresponding locations for comparison. The open circles correspond to the cases where . It turns out that at any given each parameter varies monotonically from the value with , represented by the end of the arrow, to the value with given by the arrow head. In Fig. 4b the arrows have been slightly shifted from one another to avoid overlapping.
Table 1: Profiles for some solar wind parameters in the region r> 0.3 AU.
Figure 3: Radial distributions of a) the proton azimuthal speed , b) the alpha one , and c) various contributions to the angular momentum budget in an solar wind with ion temperature anisotropies. In a) and b), the profiles derived for a solar wind with identical flow parameters where ion temperature anisotropies are neglected are given by dashed lines for comparison. Panel c) depicts the individual ion angular momentum fluxes and , their sum L_{P}, and the fluxes associated with the magnetic stresses , and with the temperature anisotropies (see Eq. (16)). The dash-dotted lines represent negative values. | |
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Figure 4: Values of several parameters as a function of , the proton temperature anisotropy at 1 AU. a) The factor , the deviation of which from unity represents the correction to the total angular momentum loss due to the introduction of ion pressure anisotropies; b) and c) the proton and alpha azimuthal speeds and at two different heliocentric distances given by different line styles as indicated in b); d) various components in the angular momentum flux at 1 AU including individual ion contribution and , the overall particle contribution , as well as the contribution from magnetic stresses . Note that instead of is given in d). The short horizontal bars in panels b) and c) represent the azimuthal ion speeds derived in the isotropic model for comparison. Furthermore, in panel b) the curve corresponding to 1 AU is slightly shifted relative to that for 0.3 AU to avoid the two overlapping each other. The open circles correspond to the cases where is fixed at 1.3, where is the alpha temperature anisotropy at 1 AU. At a given each parameter varies monotonically from the value with , represented by the end of the arrow, to the value with given by the arrow head. The ranges in which and vary are determined from the Helios measurements (see text for details). | |
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Figure 5: Similar to Fig. 4 but for the slow solar wind. Here the open circles correspond to the cases where is fixed at 1.4, and the arrow represents how the specific parameter varies at a given when increases from 0.8 to 2.0. | |
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From Fig. 4a one can see that decreases with increasing or , ranging from 1.101 at the upper left to 1.058 at the lower right corner. The deviation of from unity, albeit modest, indicates that the changes introduced in the total angular momentum loss due to the ion pressure anisotropies are not negligible. From Figs. 4b and c one can see that between 0.3 to 1 AU, the magnitude of the azimuthal speeds of both species decreases with increasing distance. Furthermore, at either 0.3 or 1 AU, both and increase when or increases. Take the values at 1 AU for instance. One can see that ranges from 2.36 to 3.46 , while varies between -12.8 and -11.7 . For the majority of the solutions both and tend to be larger in the algebraic sense than the corresponding values in the isotropic model, which yield 2.54 and -12.6 for protons and alpha particles, respectively. However, at 0.3 AU or tends to be smaller in the anisotropic than in the isotropic case. Now and vary in the intervals [2.37, 3.77] and [-15.8, -14.2] , respectively. For comparison, the isotropic model yields a ( ) of 3.33 (-14.7) . Now let us examine the specific angular momentum fluxes , and at 1 AU. Figure 4d indicates that has the weakest parameter dependence, which is easily understandable given that to a good approximation where k may be taken to be p or (see Eq. (10)). Besides, the parameter dependence of is rather modest, varying by 10% from -3.55 to -3.23 when or changes. On the other hand, changes substantially, ranging between 3.46 and 5.07. Hence the overall particle contribution L_{P} also shows a significant parameter dependence. In particular, L_{P} may exceed when . For the solutions examined, L_{P} can be found to be positive and attain its maximum of L_{P}=1.84 when . Only for the lowest values of and can one find a negative L_{P} of -0.084. Moreover, the protons always show a partial corotation, i.e., . From this we conclude that the ion temperature anisotropies are unlikely the cause of the tendency for or L_{P} to be negative for the fast solar wind as indicated by the Helios measurements (Marsch & Richter 1984; Pizzo et al. 1983).
Figure 5 presents, in the same fashion as Fig. 4, the dependence on and of various quantities derived for the slow solar wind. A comparison with Fig. 4 indicates that nearly all the features in Fig. 5 are reminiscent of those obtained for fast solar wind solutions. However, some quantitative differences exist nonetheless. For instance, when is held fixed, all the examined parameters for the slow wind vary little even though changes considerably from 0.8 to 2. In contrast, the parameters for the fast wind show an obvious dependence. This difference can be largely attributed to the fact that in the slow wind the ions are substantially cooler than in the fast wind. Figure 5a shows that ranges from 0.94 to 1.016. In other words, relative to the isotropic case, the solar angular momentum loss rate per steradian in the anisotropic models may be enhanced or reduced by up to . If examining Figs. 5b and c, one may find that at both 0.3 and 1 AU, the azimuthal speeds of both ion species, and , are larger algebraically in the anisotropic models than in the isotropic one. The difference between the two is more prominent at 0.3 AU, where the isotropic model yields that , whereas the anisotropic models yield that with increasing , increases from 3.76 to 5.04 , and varies between -17.8 to -16.3 . As for the ion azimuthal speeds at 1 AU, one can see that varying leads to a varying between 1.18 and 1.72 , and a ranging from -5.85 to -5.31 . The corresponding changes in the specific ion angular momentum fluxes are shown by Fig. 5d, which indicates that the proton one increases with increasing from 2.52 to 3.67, and likewise, the alpha one increases from -1.83 to -1.66. On the other hand, the flux associated with magnetic stresses hardly varies, and a value of 3.36 can be quoted for all the models examined. Therefore in the parameter space explored, may be smaller than , which is however offset by the alpha contribution that is always in the direction of counter-rotation to the Sun. In fact, the alpha contribution is so significant that the overall particle contribution L_{P} never exceeds . In other words, incorporating ion temperature anisotropy cannot resolve the outstanding discrepancy between previous models and observations concerning the relative importance of particle and magnetic contributions in the angular momentum budget of the solar wind.
As demonstrated by Li & Li (2008), the discussion on the angular momentum transport also allows us to say a few words on the frequency spectra S_{k} (f) ( ) of the ion velocity fluctuations during Alfvénic activities in the fast solar wind in the super-Alfvénic portion where . This is due to the well-known change of the properties of Alfvénic fluctuations around some , where is the speed of center of mass evaluated at the Alfvén point (see e.g., Li & Li 2008; Heinemann & Olbert 1980). For typical fast wind parameters, 10^{-5} . While the fluctuations with frequencies are genuinely wave-like and may be described by the WKB limit given the slow spatial variation of flow parameters in the region in question, those with behave in a quasi-static manner and may be described by the solutions to the angular momentum conservation law which also governs the zero-frequency fluctuations. As shown by Li & Li (2008) who neglected the species temperature anisotropy, in the region AU which will be explored by the Solar Orbiter and Solar Probe, the ratio of the alpha to proton velocity fluctuation amplitude can be an order-of-magnitude larger for than for . Hence one may expect that, if the proton velocity fluctuation spectrum is somehow smooth around , then the alpha one will show an apparent spectral break. Now let us revisit this problem in light of the discussion presented in this paper and see what changes the pressure anisotropies may introduce.
Figure 6: Radial dependence of the ratio of the alpha to the proton velocity fluctuation amplitudes induced by Alfvénic activities in super-Alfvénic portions of the fast solar wind. The dashed curves correspond to the isotropic model, while the hatched areas give the possible range may occupy when the parameters and vary in the ranges given in text. Both the zero-frequency (upper portion) and WKB ( lower) estimates are given. | |
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Restrict ourselves to either the high-latitude region or the region inside say 100
such that the magnetic field may be seen as radial. Furthermore, suppose that the waves are propagating parallel to the magnetic field in the empirical fast wind profiles detailed in Sect. 3. Figure 6 presents the radial dependence of
in the region between 40 and 100
for both the zero-frequency (upper part) and WKB (lower part) solutions. For comparison, the dashed curves represent the corresponding results in the isotropic model. To construct Fig. 6, all the possible values of
and
have been examined. As a result, at any radial location the ratio
varies from model to model, and the range in which this ratio may occupy is given by the hatched area. The zero-frequency solutions are obtained by solving Eq. (20), while for hydromagnetic WKB Alfvén waves it is well known that
,
where
is the wave phase speed and given by (e.g., Barnes & Suffolk 1971; Isenberg 1984)
From Fig. 6 one can see that the zero-frequency and WKB solutions are well separated from each other, in the isotropic and anisotropic cases alike. For the isotropic model, in the zero-frequency case increases monotonically from 3.79 at 40 to 4.68 at 100 . On the other hand, in the WKB case it decreases first from 0.22 at 40 and attains its minimum of 0.083 at 60.3 and then increases to 0.15 at 100 . The difference in between the zero-frequency and WKB solutions may be slightly smaller in the anisotropic than in the isotropic case for some combinations of , but the difference is still quite significant. From this we can conclude that, with realistic ion temperature anisotropies included, the alpha velocity fluctuation spectrum during Alfvénic activities will also show an apparent break near , if the proton one is smooth there. This break is entirely a linear property, and has nothing to do with the nonlinearities that may also shape the fluctuation spectra.
This study has been motivated by the apparent lack of an analysis on the angular momentum transport in a multicomponent solar or stellar wind with differentially flowing ions and species temperature anisotropy. Moreover, there has been an outstanding discrepancy between available measurements and models concerning the relative importance of the particle L_{P} and magnetic contribution to the solar angular momentum loss rate per steradian L. The Helios measurements indicate that for fast (slow) solar wind with (400) , L_{P} tends to be negative (positive), with the positive sign denoting the direction of corotation with the Sun. Furthermore, L_{P} tends to be larger than in the slow wind. The behavior of L_{P} derives from that of individual ion angular momentum fluxes, and , thereby calling for a multifluid approach.
Starting with a general set of multifluid transport equations with gyrotropic species pressure tensors, we have derived the equations for both the angular momentum conservation (Eqs. (10) and (20) in Sect. 2), and the energy and linear momentum balance (Eqs. (A.1) to (A.4) in the Appendix). These equations are not restricted to radial outflows in the equatorial plane, instead they are valid for arbitrary axisymmetrical winds that include two major ion species, and therefore are expected to find applications in general outflows from late-type stars. To focus on the problem of angular momentum transport, we refrained from solving the full set of equations governing the meridional dynamics. Rather, we constructed, largely based on the available in situ measurements, the empirical profiles for the meridional magnetic field and flow parameters. Only the ion temperature anisotropies are considered, i.e., the electron temperature is seen as isotropic. For both the fast and slow solar wind profiles, we solved the angular momentum conservation law (Eqs. (10) and (20)) to examine how the azimuthal speeds of protons and alpha particles , as well as the individual components in the solar angular momentum budget are influenced by the ion temperature anisotropies. To this end, solutions to the isotropic version are obtained for comparison.
Our main conclusions are:
Acknowledgements
We thank the referee (Dr. Horst Fichtner) for his very helpful comments. This research is supported by an STFC rolling grant to Aberystwyth University.
In Sect. 2, we have demonstrated that the vector equations governing a time-independent multicomponent solar wind with species temperature anisotropy are allowed to be decomposed into a force balance condition across the poloidal magnetic field and a set of transport equations along it. The azimuthal dynamics has been discussed in the text, whereas this Appendix provides some discussion on the poloidal dynamics. In particular, we shall derive the equations governing the poloidal motion v_{k l} of ion species ( ), and the species temperatures ( ) in rather general situations.
Due to the presence of v_{kN} in the l component of the ion momentum Eq. (2), one may expect that the N-component of Eq. (2) has to be solved. In fact, there is no need to do so because v_{k N} appears only in the difference v_{j N}-v_{k N}, which may be found from the component of Eq. (2). Substituting v_{j N}-v_{k N} into the l component of Eq. (2) will then eliminate the cumbersome and v_{k N}. Note that this technique, first devised by McKenzie et al. (1979), ensures the conservation of not only total momentum but also total energy (see Li & Li 2006). Specifically, the resulting equations for the poloidal dynamics are
Introducing azimuthal components may influence the ion flow speeds v_{k l} both directly and indirectly. The direct consequence is that azimuthal components may introduce into the reduced meridional momentum Eq. (A.2) an effective force (see the first three terms). Note that in a corotating frame the magnitude of the ion velocity becomes , from relation (A.5) one may see that in such a frame all particles move in the same centrifugal potential . Therefore in effect the introduced force tends to reduce the magnitude of the ion speed difference with increasing distance as tends to increase. This effect has been explored in detail in Li & Li (2006) and Li et al. (2007), where it is shown that the influence may play an important part in the force balance for the solar wind. In fact, introducing solar rotation alone is able to reproduce the profile measured by Ulysses beyond 2 AU if a proper value of is imposed there. On the other hand, v_{k l} may be altered indirectly by the modified pressure gradient force due to changes in the temperatures, which in turn are caused by the changes in the heat fluxes (the third term in Eqs. (A.3) and (A.4)) and through the adiabatic cooling (the second term). A detailed discussion on the former requires a specific form for the heat flux, which is beyond the scope of the present paper. As a consequence, we shall focus on the latter instead.
Neglecting the terms in the second pair of square parentheses, Eqs. (A.3)
and (A.4) give
(A.6) |
For completeness, we note that the force balance condition across the N direction comes from the N component of the total momentum, which reads
(A.7) |