Cranmer et al. (2007) described a set of models in which the time-steady plasma properties along a one-dimensional magnetic flux tube are determined. These model flux tubes are rooted in the solar photosphere and are extended into interplanetary space. The numerical code developed in that work, called ZEPHYR, solves the one-fluid equations of mass, momentum, and energy conservation simultaneously with transport equations for Alfvénic and acoustic wave energy. ZEPHYR is the first code capable of producing self-consistent solutions for the photosphere, chromosphere, corona, and solar wind that combine: (1) shock heating driven by an empirically guided acoustic wave spectrum, (2) extended heating from Alfvén waves that have been partially reflected, then damped by anisotropic turbulent cascade, and (3) wind acceleration from gradients of gas pressure, acoustic wave pressure, and Alfvén wave pressure.
The only input “free parameters” to ZEPHYR are the photospheric lower boundary conditions for the waves and the radial dependence of the background magnetic field along the flux tube. The majority of heating in these models comes from the turbulent dissipation of partially reflected Alfvén waves (see also Matthaeus et al., 1999; Dmitruk et al., 2002; Verdini and Velli, 2007; Chandran et al., 2009a). Photospheric measurements of the horizontal motions of strong-field intergranular flux concentrations (i.e., G-band bright points) were used to constrain the Alfvén wave power spectrum at the lower boundary. This empirically determined power spectrum is dominated by wave periods of order 5 –10 minutes. It is important to note, however, that radio and in situ measurements find that most of the fluctuation power in the solar wind is at lower frequencies (i.e., periods of hours). We still do not yet know (1) if the shape of the power spectrum evolves significantly between the lower solar atmosphere and interplanetary space, or (2) if some low-frequency power is missed by the existing measurements of G-band bright point motions. In any case, as seen below, the resulting wave reflection and turbulent dissipation that comes from just the 5 – 10 minute periods appear to be sufficient to explain the observed levels of coronal heating and solar wind acceleration.
Non-WKB wave transport equations were solved to determine the degree of linear reflection at heights above the photospheric base (see Cranmer and van Ballegooijen, 2005). The resulting values of the Elsasser amplitudes , which denote the energy contained in upward () and downward () propagating waves, were then used to constrain the energy flux in the cascade. Cranmer et al. (2007) used a phenomenological form for the damping rate that has evolved from studies of Reduced MHD and comparisons with numerical simulations. The resulting heating rate (in units of erg s–1 cm–3) is given byet al., 1995; Zhou and Matthaeus, 1990; Breech et al., 2008; Podesta and Bhattacharjee, 2009; Beresnyak and Lazarian, 2009). Cranmer et al. (2007) used a standard assumption that scales with the cross-sectional width of the flux tube (Hollweg, 1986). The term in parentheses above is an efficiency factor that accounts for situations in which the cascade does not have time to develop before the waves or the wind carry away the energy (Dmitruk and Matthaeus, 2003). In open field regions, the cascade is “quenched” when the nonlinear eddy time scale becomes much longer than the macroscopic wave reflection time scale . In closed field regions, the correction factor may behave in an opposite sense as it does for open field regions (see, e.g., Gómez et al., 2000; Rappazzo et al., 2008). In most of the solar wind models, though, Cranmer et al. (2007) used in Equation (1) based on analytic and numerical results (Dobrowolny et al., 1980; Oughton et al., 2006), but they also tried to explore a stronger form of the quenching effect.
Figure 9 summarizes the results of varying the magnetic field properties while keeping the lower boundary conditions fixed. For a single choice for the photospheric wave properties, the models produced a realistic range of slow and fast solar wind conditions. A two-dimensional model of coronal holes and streamers at solar minimum reproduces the latitudinal bifurcation of slow and fast streams seen by Ulysses. An active-region-like enhancement of the magnetic field strength in the low corona generates a high mass flux and a slow wind speed, in agreement with observations of open field lines connected with active regions (see also Wang et al., 2009). As predicted by earlier studies, a larger coronal “expansion factor” naturally gives rise to a slower and denser wind, higher temperature at the coronal base, and lower-amplitude Alfvén waves at 1 AU.
In these models, the radial gradient of the Alfvén speed affects where the waves are reflected and damped, and thus whether energy is deposited below or above the Parker (1958a) critical point. Early studies of solar wind energetics (e.g., Leer and Holzer, 1980; Pneuman, 1980; Leer et al., 1982) showed that if there is substantial heating below the critical point, its primary impact is to “puff up” the hydrostatic scale height, drawing more particles into the accelerating wind and thus producing a slower and more massive wind. If most of the heating occurs at or above the critical point, the subsonic atmosphere is relatively unaffected, and the local increase in energy flux has nowhere else to go but into the kinetic energy of the wind (leading to a faster and less dense outflow). The ZEPHYR results shown in Figure 9 display this kind of dichotomy because the superradial expansion creates a much higher critical point over the equatorial regions than over the poles. Additional studies of how and where the mass flux and wind speed are determined include Withbroe (1988), Hansteen and Leer (1995), Hansteen et al. (1997), Janse et al. (2007), and Wang et al. (2009).
Perhaps more surprisingly, varying the coronal expansion factor in the models shown in Figure 9 also produces correlative trends that are in good agreement with in situ measurements of commonly measured ion charge state ratios (e.g., O7+/O6+) and FIP-sensitive abundance ratios (e.g., Fe/O). Cranmer et al. (2007) showed that the slowest solar wind streams – associated with active-region fields at the base – can produce a factor of 30 larger frozen-in ionization-state ratio of O7+/O6+ than high-speed streams from polar coronal holes, despite the fact that the temperature at 1 AU is lower in slow streams than in fast streams. Furthermore, when elemental fractionation is modeled using a theory based on preferential wave-pressure acceleration (Laming, 2004, 2009; Bryans et al., 2009), the slow wind streams exhibit a substantial relative buildup of elements with low FIP with respect to the high-speed streams. Although the WTD models utilize identical photospheric lower boundary conditions for all of the flux tubes, the self-consistent solutions for the upper chromosphere, transition region, and low corona are qualitatively different. Feedback from larger heights (i.e., from variations in the flux tube expansion rate and the resulting heating rate) extends downward to create these differences.
Another empirical “marker” of heliospheric stream structure is the proton specific entropy, or entropy per proton, which is often approximated as being proportional to , where is an empirical adiabatic index for solar wind protons (e.g., Burlaga et al., 1990; Pagel et al., 2004). When measured in regions of the (non-CME) heliosphere where corotating interaction regions have not yet formed shocks, this quantity is seen to clearly distinguish slow wind streams from fast wind streams. Figure 10 shows how the specific entropy is positively correlated with wind speed, both in measurements made by the Solar Wind Electron Proton Alpha Monitor (SWEPAM) instrument on ACE (McComas et al., 1998) and in the Cranmer et al. (2007) ZEPHYR models discussed above. Each model data point was computed independently of the others. The models had identical lower boundary conditions at the photosphere, and they differed from one another only by having a different radial dependence of the magnetic field. Because entropy should be conserved in the absence of significant small-scale dissipation, the quantity that is measured at 1 AU may be a long-distance proxy for the near-Sun locations of strong coronal heating. In other words, the comparison of measured and modeled solar wind entropy variations may be a key way to discriminate between competing explanations of solar wind acceleration.
Although Equation (1) describes the plasma heating rate in terms of the local properties of MHD turbulence, it is also possible to see that this expression gives a heating rate proportional to the mean magnetic flux density at the coronal base. As illustrated above in Figure 4, the mean field strength in the low corona is determined by both the photospheric field strength in the intergranular bright points and the total number of bright points that eventually merge their fields together in the low corona. The field strength at this merging height can thus be estimated as , where is the (nearly universal) photospheric bright-point field strength and is the area filling factor of bright points in the photosphere. The latter quantity appears to vary by more than an order of magnitude in different regions on the Sun, from about 0.002 (at low latitudes at solar minimum) to 0.1 (in active regions). If the regions below the merging height can be treated using approximations from “thin flux tube theory” (e.g., Spruit, 1981; Cranmer and van Ballegooijen, 2005), then it is possible to express each term in Equation (1) as a function of and the photospheric properties. For example, applies to thin flux tubes in pressure equilibrium, and thus at the merging height can be estimated as (where is the photospheric density). For Alfvén waves at low heights, , and so at the merging height scales like . Also, we assumed that . If the quenching factor in parentheses in Equation (1) is neglected, thenet al., 2003; Schwadron et al., 2006; Suzuki, 2006; Kojima et al., 2007; Pinto et al., 2009).
This work is licensed under a Creative Commons License.