ON THE RELATIVE CONSTANCY OF THE SOLAR WIND MASS FLUX AT 1 AU

The solar wind proton flux density at 1 AU typically varies between ∼2 × 10⁸ cm⁻² s⁻¹ in high-speed streams and ∼4 × 10⁸ cm⁻² s⁻¹ in slow wind (see, e.g., Feldman et al. 1978; Goldstein et al. 1996). As noted by Leer et al. (1982) and Withbroe (1989), this constancy is rather surprising, given that the mass flux from a corona dominated by gravity and thermal pressure forces is expected to be highly sensitive to the assumed temperature or rate of energy deposition in the subsonic region.

Sandbæk et al. (1994) argued that the heating rate in the lower corona must scale linearly with the magnetic field in order to keep the mass flux at Earth constant. Using a radiative energy-balance model similar to that developed by Hammer (1982) and Withbroe (1988), Wang (1994) found that very large energy fluxes at the Sun, of order 10⁶ erg cm⁻² s⁻¹, were required to produce the mass fluxes observed from small coronal holes at sunspot maximum; moreover, photospheric magnetograms showed their footpoint fields to be much stronger than those in the polar holes that dominate near sunspot minimum. Because the magnetic field falls off rapidly with height in these active-region holes, most of the energy is deposited near the coronal base and goes into increasing the enthalpy flux rather than accelerating the wind, which thus ends up with low speeds.

In this Letter, we revisit the question of the constancy of the solar wind mass flux, taking a systematic approach that distinguishes among different types of coronal hole wind. As in Wang (1995), we use mass and energy conservation to relate the in situ properties of the solar wind to magnetograph measurements at the Sun, but extend our statistical analysis from the ecliptic to encompass the wind at all latitudes throughout the solar cycle.

In the following, the subscripts "0" and "1" are used to indicate quantities evaluated at the coronal base and at 1 AU, respectively. The coronal base is here defined as the height where the electron temperature equals 0.5 MK and corresponds roughly to the top of the transition region; beyond this point, which lies above the chromospheric canopy (thus allowing the use of low-resolution magnetograph data), radiative losses can normally be neglected.

Let v denote the proton bulk speed, ρ the proton mass density, m_p the proton mass, n = ρ/m_p the proton number density, p the thermal pressure, B the radial field strength, r heliocentric distance, L heliographic latitude, and ϕ Carrington longitude. The conservation of mass along a magnetic flux tube yields

$$\begin{equation} \rho _1v_1 = \left(\frac{B_1}{B_0}\right)\rho _0v_0. \end{equation} \tag{ 1 }$$

The total energy flux density of the solar wind may be written as

$$\begin{equation} F_{\rm tot} = F_{\rm w} - \rho v\left(\frac{GM_\odot }{r}\right), \end{equation} \tag{ 2 }$$

where we have separated out the nongravitational contribution

$$\begin{equation} F_{\rm w} = F_{\rm m} + F_{\rm A} + F_{\rm c} + \case{5}{2}pv + \case{1}{2}\rho v^3. \end{equation} \tag{ 3 }$$

The terms on the right-hand side of Equation (3) represent the heating rate due to the dissipation of "mechanical" energy in the corona, the "wave" energy flux, the conductive heat flux, the enthalpy flux, and the bulk kinetic energy flux, respectively. Given that the last term is observed to provide the main contribution at 1 AU, the conservation of the total energy along a flux tube yields

$$\begin{equation} F_{\rm w0}\simeq \rho _0v_0\left(\frac{GM_\odot }{R_\odot }\right)\, +\, \left(\frac{B_0}{B_1}\right)\frac{1}{2}\rho _1v_1^3 = \rho _0v_0\left(\frac{GM_\odot }{R_\odot } + \frac{1}{2}v_1^2\right)\!. \end{equation} \tag{ 4 }$$

The dominant contributions to F_w0 are expected to be the mechanical and wave energy fluxes, with the former being dissipated at heliocentric distances r ≲ 2 R_☉ and the latter (possibly in the form of Alfvén waves) acting to provide additional acceleration in the outer corona. However, our results do not depend on any specific assumptions about the heating and acceleration mechanisms of the solar wind. For convenience, we refer to F_w0 as the "solar wind energy flux density" at the coronal base, keeping in mind that it does not include the gravitational potential energy.

Since v₁, ρ₁, and B₁ are known from in situ observations and B₀ can be determined from magnetograph measurements, Equations (1) and (4) may be used to derive the mass and energy flux densities at the coronal base. In carrying out this procedure for the near-Earth solar wind, we employ 8 hr averages of the hourly proton bulk velocity, proton density, and radial component of the interplanetary magnetic field (IMF) recorded between 1998 February and 2009 August by the Advanced Composition Explorer (ACE).¹ In addition, to characterize the global solar wind, we employ daily averages of the same quantities measured by Ulysses between 1990 October and 2009 June.² We have not attempted to remove the contribution of coronal mass ejections (CMEs), interplanetary shocks, and streamer blobs from the in situ data; because of the large number of data points used in this study, their effect on the overall statistics is expected to be small. According to Webb & Howard (1994), ∼15% of the near-ecliptic mass flux at solar maximum comes from CMEs.

The photospheric field data are in the form of 27.3 day synoptic maps from the Mount Wilson Observatory (MWO) and the Wilcox Solar Observatory (WSO), which have a resolution of order 5° in longitude; as in Wang et al. (2009b), we average together the two data sets (which optimizes the agreement between the observed and predicted coronal streamer structure during solar cycle 23). The line-of-sight magnetograph measurements are corrected for the saturation of the Fe i 525.0 nm line profile by multiplying by the latitude-dependent factor (4.5 − 2.5sin²L), and are deprojected by dividing by cos L (see Wang & Sheeley 1995; Ulrich et al. 2009). A potential-field source-surface extrapolation is then applied to map the field out to r = R_ss = 2.5 R_☉. Field lines that cross the source surface are considered to be "open"; their footpoint areas have been shown to coincide approximately with observed coronal holes throughout the solar cycle (see, e.g., Wang et al. 1996; Neugebauer et al. 1998; Liewer et al. 2004). Following the prescription of Schatten (1971), we introduce sheet currents in the region r > R_ss by matching a potential field with l = 0 monopole component to |B_r(R_ss, L, ϕ)| and then restoring the polarity of the field lines. (As demonstrated in Wang & Sheeley (1995), this procedure reproduces the long-term variation of the IMF sector structure both in and out of the ecliptic plane.) The footpoint (R_☉, L₀, ϕ₀) of the open field line that intersects the spacecraft is found by tracing inward from 1 AU, allowing for the solar rotation and resulting longitude shift during the wind propagation time (∝v⁻¹₁).

The scatter plots in Figure 1 illustrate some of the statistical relationships derived for the in-ecliptic wind by tracing back from the ACE spacecraft to the coronal base. (In this section, subscript "E" is used to denote quantities evaluated at ACE, located near Earth at the L1 point.) A data point is plotted for every 5° of longitude during Carrington rotations (CRs) 1933–2086 (1998 February 18 through 2009 August 20).

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Figure 1(a) shows log₁₀(F_w0) plotted against log₁₀(B₀). The solar wind energy flux at the coronal base is seen to increase almost linearly with the footpoint field strength, over a range of more than 2 orders of magnitude in both quantities (the dashed line indicates a slope of 1 in the log–log plot). Note that we have made no assumptions about the nature of the energy flux at the Sun in obtaining this empirical result, which therefore provides support for the hypothesis that the magnetic field is the main source of the heating and acceleration of the solar wind.

Figure 1(b) shows a nearly perfect correlation between log₁₀(F_w0) and log₁₀(n₀v₀). This result, which follows directly from Equation (4), expresses the fact that a major portion of the energy flux at the coronal base goes into lifting the mass out of the Sun's gravitational well.

As indicated by Figure 1(c), the statistical relationships in Figures 1(a) and (b) together imply that the mass flux density at the coronal base increases almost linearly with the footpoint field strength. (Here and in Figure 1(a), some of the scatter toward high values of n₀v₀ and F_w0 may be associated with CMEs and corotating interaction regions.)

Figure 1(d) confirms that the mass flux density at Earth tends to decrease slightly with increasing wind speed, while Figure 1(e) shows that it increases slightly with the mass flux density at the coronal base. The relative constancy of n_Ev_E = B_E(n₀v₀/B₀) follows immediately from the relative constancy of B_E and the tendency for n₀v₀ to increase linearly with B₀ (Figure 1(c)). The radial IMF strength undergoes much less variation than the footpoint fields, both because it depends only on the lowest-order multipoles of the photospheric field (principally the dipole and quadrupole components) and because the open flux is distributed isotropically far from the Sun, as deduced from Ulysses magnetometer measurements (Balogh et al. 1995; Smith et al. 2001; Smith & Balogh 2008).

As expected from the continuity equation, n_Ev_E shows a slight tendency to increase with B_E (Figure 1(f)). This dependence should be regarded as a geometrical effect rather than as a consequence of coronal heating: for a given B₀ (which determines n₀v₀ through some heating process), the net expansion undergone by the flux tube between the Sun and 1 AU is smaller (and n_Ev_E is larger), the greater the radial IMF strength. Here, it should be remarked that the parameters B₀ and B_E are to a large extent independent of each other (their logarithms having a correlation coefficient of 0.17), since B₀ represents the field strength at the footpoint of a single flux tube, whereas B_E measures the total open flux of the Sun.

We may obtain further information about the sources of the in-ecliptic wind by plotting v_E, log₁₀(B₀), log₁₀(n₀v₀), and log₁₀(n_Ev_E) against sin |L₀|, where L₀ denotes the footpoint latitude (Figure 2). We note the presence of two distinct sources of slow solar wind, emanating from low and very high latitudes, respectively. The low-latitude component, associated with small coronal holes located near active regions, is characterized by very strong footpoint fields and high mass and energy flux densities at the coronal base, and dominates the near-Earth wind around sunspot maximum. The high-latitude component, characterized by weak footpoint fields and low mass and energy flux densities at the coronal base, comes from just inside the polar hole boundaries and makes its appearance toward sunspot minimum. As indicated by Figure 2(d), however, the mass flux densities of both components of the slow wind end up almost the same at Earth, because the net Sun–Earth expansion factor is much greater for the active-region holes than for the polar hole boundaries.

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We now perform a similar analysis for the solar wind at Ulysses, whose latitudinal trajectory ranged between L = −80° and L = +80°. Subscript "U" is used to denote quantities measured at the spacecraft.

As illustrated in Figure 3, the statistical relationships between F_w0, B₀, n₀v₀, n_Uv_U, and B_U are very similar to those found using the in-ecliptic ACE data (compare Figure 1). (To correct for the varying heliocentric distance of the spacecraft, B_U and n_Uv_U have been scaled to the values they would have at 1 AU by multiplying by the factor (r_U/r_E)².) The new feature is the dense cluster of points representing the polar hole wind during the last two cycle minima. This clustering reflects the fact that the axisymmetric polar holes near sunspot minimum have a relatively narrow range of footpoint field strengths (B₀ ∼ 5–15 G) and a roughly constant angular size (measured from the pole) of ∼30°. Despite its much higher asymptotic speeds (v_U ∼ 750 km s⁻¹), the wind from the polar hole interiors is characterized by substantially lower energy fluxes at the coronal base (F_w0 ∼ 4 × 10⁵ erg cm⁻² s⁻¹) than the wind from small active-region holes (F_w0 ≳ 10⁶ erg cm⁻² s⁻¹). As noted by Leer & Holzer (1980), it is the location of the heating relative to the coronal base, not the magnitude of the heating itself, that determines the final wind speed; the location of the heating is in turn determined by the rate at which the magnetic field falls off in the corona, or equivalently, by the rate of flux-tube expansion inside the source surface (Wang et al. 2009a).

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From their comparison of Ulysses plasma data taken at high latitudes during the last two sunspot minima, McComas et al. (2008) found that the wind speed was ∼3% slower and the mass flux as much as ∼20% lower during the most recent minimum. At the same time, both the polar fields and the IMF were 35%–40% weaker at the end of solar cycle 23 than at the beginning, while the polar holes were ∼20% smaller (see, e.g., Smith & Balogh 2008; Kirk et al. 2009; Wang et al. 2009b). Since the polar holes provide most of the Sun's open flux near solar minimum (see Figure 9(b) in Wang et al. 2009b), the decrease in the IMF strength mainly reflects the weakening of the polar fields, and to a lesser extent the shrinking of the polar hole areas. From Figure 3, we see that the weaker footpoint fields imply smaller energy and mass flux densities at the coronal base; the mass flux densities at Ulysses are also lower, both because of the decrease in the coronal heating rate (leading to smaller values of n₀v₀) and because of the increase in the net expansion factor due to the shrinkage of the polar holes.

The somewhat slower polar wind observed during the recent sunspot minimum is consistent with the inverse correlation between wind speed and the rate of flux-tube divergence inside the source surface, as measured by the quantity

$$\begin{equation} f_{\rm ss} = \left(\frac{R_\odot }{R_{\rm ss}}\right)^2\frac{B_0}{B_{\rm ss}}, \end{equation} \tag{ 5 }$$

where B_ss denotes the field strength at the point where the flux tube intersects r = R_ss = 2.5 R_☉ (see Levine et al. 1977; Wang & Sheeley 1990; Arge & Pizzo 2000; Abramenko et al. 2009; Luhmann et al. 2009). The observed decrease in the polar hole area implies larger values of f_ss and hence lower asymptotic wind speeds. In Figure 4, we show scatter plots of v_U versus f_ss, n_Uv_U versus B₀, and F_w0 versus B₀, constructed separately for the 1993–1996 and 2006–2008 polar passes, when Ulysses flew above heliographic latitude 40°. The median values of these quantities during the earlier sunspot minimum are f_ss = 4.2, B₀ = 10.6 G, v_U = 763 km s⁻¹, n_Uv_U = 1.9 × 10⁸ cm⁻² s⁻¹, and F_w0 = 5.2 × 10⁵ erg cm⁻² s⁻¹; for the 2006–2008 period, the corresponding values are f_ss = 5.1, B₀ = 7.3 G, v_U = 740 km s⁻¹, n_Uv_U = 1.5 × 10⁸ cm⁻² s⁻¹, and F_w0 = 3.8 × 10⁵ erg cm⁻² s⁻¹. The lower mass fluxes at Ulysses during the later polar pass are mainly a consequence of the weaker coronal heating, with F_w0 being ∼26% smaller in 2006–2008 than in 1993–1996; because B₀ and B_U both underwent a substantial decrease, the contribution of the geometrical expansion effect is relatively minor, with the net Sun–Ulysses expansion factor f_U ∝ B₀/B_U increasing by only ∼4%. The lower polar-hole speeds during the recent activity minimum are due to the energy being deposited at lower heights when the field diverges more rapidly above the coronal base (see Cranmer et al. 2007; Wang et al. 2009a).

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By using conservation equations to relate in situ observations of the solar wind to photospheric field measurements, we have arrived at a better understanding of why the solar wind mass flux is relatively constant at 1 AU. The steps in our argument may be summarized as follows.

• 1.  
  We have verified empirically that the energy flux density at the coronal base, F_w0, increases roughly linearly with the footpoint field strength, B₀, with the slope being close to unity in a log–log plot.
• 2.  
  Since most of the energy flux at the coronal base goes into lifting the mass out of the Sun's gravitational well, F_w0 also increases linearly with the proton flux density at the coronal base, n₀v₀.
• 3.  
  From (1) and (2), it follows that n₀v₀ increases roughly linearly with B₀.
• 4.  
  The radial IMF strength, B₁, depends only on the slowly varying, lowest-order multipoles of the photospheric field; moreover, it is independent of latitude and longitude at 1 AU. As a result, B₁ varies far less than B₀.
• 5.  
  From (3) and (4), it follows that the proton flux density at 1 AU, n₁v₁ = B₁(n₀v₀/B₀), undergoes much less variation than either n₀v₀ or B₀.

Because B₀ may vary by up to 2 orders of magnitude, depending on how close the footpoint of the flux tube lies to an active region, the solar wind energy and mass flux densities are very far from constant at the Sun. The mass flux density ends up relatively constant at 1 AU because the large values of n₀v₀ associated with active-region holes are offset by the geometrical effect of the net flux-tube expansion, which is also proportional to B₀.

The underlying empirical result of this study, that F_w0 ∝ B₀, provides an important constraint on heating mechanisms in coronal holes and the solar wind. This scaling law, first suggested by Sandbæk et al. (1994), has now been established from observations taken both in and out of the ecliptic and throughout the solar cycle. A direct dependence of the local heating rate on the field strength as a function of height may also provide the physical basis for the inverse correlation between the asymptotic wind speed and coronal flux-tube divergence, if it is assumed that the heating is concentrated at lower heights when the magnetic field falls off rapidly above the coronal base. It should be noted that this tendency for v₁ to decrease with increasing f_ss does not require the energy flux density at the coronal base to be constant, as implied by Sandbæk et al. (1994) and Withbroe (1989); because v₁ ∝ (F_tot0/n₀v₀)^1/2, all that is required is that n₀v₀ increase more rapidly with f_ss than F_tot0, as is indeed found empirically to be the case (compare Figures 4 and 5 in Wang 1995). The latter result can be understood physically from the fact that, for a given energy flux density at the coronal base, the mass flux density is larger, the closer to the coronal base that the energy is deposited.