SOLAR WIND SCALING LAW N. A. Schwadron and D. J. McComas

The Astrophysical Journal, 599:1395 103, 2003 December 20 # 2003. The American Astronomical Society. All rights reserved. Printed in U.S.A. SOLA WIND SCALING LAW N. A. Schwadron and D. J. McComas Southwest esearch Institute, P.O. Drawer 28510, San Antonio, TX 78228; nschwadron@swri.edu eceived 2003 June 5; accepted 2003 August 29 ABSTACT We derive a simple, robust scaling law, which explains naturally how the well-known anticorrelation between final solar wind speed and freezing-in temperature results from the loss of radiated energy in the corona. Furthermore, if the Sun injects roughly fixed electromagnetic energy per particle, this law provides a unified theory for the source of solar wind: fast tenuous solar wind from dark coronal holes; slow dense wind from hotter, brighter regions; and bound but unstable plasma in extremely hot regions, which may be related to solar transients. The scaling law is not an extension or a new variant of previous solar wind models, but rather a requirement on all solar wind models, which should apply quite generally to magnetically driven winds. Thus, this paper reveals an underlying connection between the solar wind and its coronal source. Subject headings: solar wind Sun: corona 1. INTODUCTION 1395 The solar wind s high degree of organization by speed is among its best-known features. Near solar minimum, fast (700 800 km s 1 ) solar wind is emitted from high-latitude coronal holes (Krieger, Timothy, & oelof 1973; Nolte et al. 1977; von Steiger et al. 1997; McComas et al. 1998). At lower latitudes, within 10 30 of the current sheet, is a slower, more variable solar wind (300 500 km s 1 ) (McComas et al. 1998). The final solar wind speed is strongly anticorrelated with its coronal freezing-in temperature (Geiss et al. 1995), which is determined from the charge-state distributions of heavy elements. The freezingin temperature is set where the solar wind draws ions out faster than they can equilibrate (through ionization and recombination) to the local electron temperature (Geiss et al. 1995). Models show that typical freeze-in heights for C, O, and Si occur below 350 Mm from the solar surface, with C freezing in at the lowest altitudes, nearest the transition region (Bürgi & Geiss 1986). There is also an anticorrelation between solar wind speed and the abundance of elements such as Fe, Si, and Mg (Geiss et al. 1995; von Steiger et al. 2000); these elements have low first-ionization potentials (FIPs) and are therefore ionized lower in the photosphere than other common solar wind elements. The anticorrelations between solar wind speed and the freezing-in temperature and abundance of low-fip elements shows that the fundamental differences between sources of fast and slow wind extend all the way down to the photosphere. There is extensive literature on solar wind acceleration models (e.g., Parker 1958; Isenberg 1991 and references therein; Marsch 199 and references therein; Hansteen & Leer 1995; Axford & McKenzie 1997; McKenzie, Axford, & Banaszkiewicz 1997). These models typically assume a spatial heating profile, which thereby prescribes the heating and subsequent acceleration of the solar wind. Others (Wang & Sheeley 1991, 199, 2003; Sandbaek, Leer, & Hansteen 199) have tried to understand the anticorrelation between final solar wind speed and the flux tube expansion rate near the Sun inferred from potential field models (Levine, Altschuler, & Harvey 1977; Wang & Sheeley 1990; Wang 1995). Finally, several recent studies (Fisk 2003; Gloeckler, Zurbuchen, & Geiss 2003) have tried to understand the observed anticorrelation between final solar wind speed and freezing-in temperature; however, they missed the critical role of radiated energy loss. This paper is not an extension or a new variant of any of these previous solar wind models. ather, we show that the solar wind speed (or equivalently, energy) is determined by a simple and robust scaling law, where the final wind speed is a strong function of the imum temperature near the Sun (near or below a scale height). The scaling law is a requirement on all solar wind models, explains large-scale features of the solar wind, and should apply quite generally to magnetically driven winds. 2. THE SCALING LAW The solar wind is the result of the heating of the Sun s outer atmosphere to millions of degrees. The heating is exceptionally rapid, causing the fast solar wind to become supersonic at less than 3 solar radii (Withbroe 1988). The source of energy for this heating is undoubtedly electromagnetic, e.g., turbulence or magnetohydrodynamic waves. Although there are many variants for injection of electromagnetic energy (e.g., Parker 1988, 1991; Fisk, Schwadron, & Zurbuchen 1999), the basic concept is simple and quite general. Consider a small volume of plasma with transverse field strength B 0 and density n 0 that is injected into the base of an open flux tube. The magnetic and plasma energy (per particle) of this small volume, mv 2 A0 ¼ B 2 0 =ðn 0 Þ, is dissipated to heat the particles. Only the transverse field components transported with the particles raise the field energy of the flux tube and can be dissipated into heat. In this respect, waves, turbulence, current sheets, and kinked field lines are all possible magnetic energy sources, whereas a static field line with ionized particles flowing along it would inject only particle energy. Since the quantity mv 2 A0 is the injected electromagnetic energy per particle, the energy flux (or Poynting flux) into the corona is f 0 mv 2 A0, where f 0 is the particle flux near the base of a flux tube. After injection of electromagnetic energy, the only significant loss of energy occurs radiatively. The plasma is heated very rapidly, from thousands to millions of degrees in a thin

1396 SCHWADON & McCOMAS Vol. 599 layer referred to as the transition region. Across the transition region, the plasma pressure is essentially constant, causing the density to drop rapidly with increasing temperature and height. There are two competing processes that determine how much energy is lost through radiation: electrons in the plasma are highly mobile and efficiently transmit heat from the temperature imum (below or near 1 scale height) down into the transition region where it is radiated away; the outward flux of particles acts to advect thermal energy out of the radiative region and to reduce the density, thereby suppressing radiative energy loss. Thus, the total radiated energy per particle is the energy conducted by electrons minus the advected thermal energy. Quantitatively, the electron heat flux is q e ¼ 0 T 5=2 rt where 0 10 6 K ergs cm 1 s 1 (Spitzer 1962). Thus, the energy (per particle) conducted down into the transition region is the heat flux divided by the particle flux, C 0 0 T=ðf 0 LÞ, where L is the length from the photospheric base at which the imum temperature T near or below 1 scale height is achieved, f 0 is the particle flux, and C 0 is a dimensionless constant. Including the competing effect of advected thermal energy, kt, we find that the total radiated energy per particle is C 0 0 T=ðf 0 LÞ C 1 kt where C 1 is another dimensionless constant. A detailed derivation of these terms and dimensionless constants, C 0 ¼ 0:91 and C 1 ¼ 5, is presented in Appendices A and B. The form for the radiated energy is remarkably robust, although the dimensionless constants may vary slightly for different sets of assumptions (for example, a nonuniform heating rate). For temperature ima above 1 scale height s p ¼ 2kT =ðmg Þ, the total radiated energy is characterized by considering an effective temperature imum near 1 scale height, since the density and radiative loss rate (which scales with the square of the electron density) decrease sharply above s p. Observationally, this effective temperature imum (referred to hereafter as the imum source temperature) is best approximated by the C freezing-in temperature, since it is determined at the lowest altitudes, nearest the transition region (Geiss et al. 1995). The final essential element of the scaling law is gravitation. The gravitational potential energy, GM m=, reduces the final solar wind energy, since particles must climb out of the Sun s gravitational well (M ¼ 2 10 33 gis the solar mass, and ¼ 7 10 10 cm is the solar radius). Particles near the solar surface also have rotational energy due to solar rotation; but this is a small effect (less than 1% of the gravitational energy) and thus neglected here. Combining the three energy terms, injected electromagnetic energy minus radiated energy and gravitational potential energy, we arrive at the final solar wind energy scaling law:! m 2 u2 f ¼ mv 2 0 T A0 C 0 f 0 L C 1kT GM m ; ð1þ where u f is the final solar wind speed. We have approximated the final solar wind energy as all bulk flow energy, which is observed to be 98% of the solar wind s energy at 1 5 AU. 1 Figure 1 schematically illustrates the elements of 1 Typical sound speeds and Alfvén speeds in the solar wind are less than 10% of the solar wind speed beyond 1 AU (McComas, Gosling, & Skoug 2000a), which implies that the bulk flow energy constitutes more than 98% of the total energy. the scaling law superimposed on an image of the Sun with an illustration of two open flux tubes emanating from the solar surface; one that produces fast solar wind from cool coronal hole material, and the other that provides slow solar wind from the hotter, more radiative material outside the coronal hole. The scaling law predicts a strong anticorrelation between solar wind speed and imum source temperature due to the strong temperature dependence of the radiative term. There are two parameters in the scaling law: one is the effective Alfvén speed v A0, which characterizes the injected electromagnetic energy; and the other is f 0 L, the base flux times length (or height) to the source temperature imum. Given the parameters v A0 and f 0 L, the scaling shows a imum speed of u 2 ¼ 2v 2 A0 2GM = at low temperatures (consistent with eq. [17] of Fisk et al. 1999, where the critical role of radiation was missed). In addition, there is a cutoff in temperature T co above which the plasma flow is subsonic and therefore bound: where T co T co þ T co ; T co ¼ 0:8 mu 2 2=7 f 0 L= 0 and T co ¼ T co =½0:53mu 2 =ðk T co Þ 1Š : 3. A UNIFIED THEOY FO THE SOLA WIND The solar wind scaling law applies very generally. As a simplified but reasonable approximation, we consider the implications of the solar wind scaling law when the injected electromagnetic energy per particle (/mv 2 A0 ) and the average base flux times length, f 0 L, are roughly constant. The latter assumption requires explanation. Since both particle flux times area and magnetic flux times area are constants on the flux tube, their ratio must also be a constant. Thus, the average base flux can be expressed as f 0 ¼ B 0 ð f 1 =B 1r Þ, where f 1 3 10 8 cm 2 s 1 is the particle flux and B 1r 30 lg is the radial magnetic field strength at 1 AU. Since, on average, both the particle and magnetic flux are roughly constant at 1 AU (McComas et al. 2000b), the average base flux is proportional to the average base field strength, f 0 / B 0. Furthermore, since the coronal magnetic field strength falls off with height, we expect that B 0 L, and therefore f 0 L, should be roughly constant. This behavior is exemplified by a recent study (Mandrini, Démoulin, & Klimchuk 2000) in which the coronal field strengths in active region loops of 50 300 Mm were seen to fall off roughly as B 0 / L 0:88, supporting the near constancy of B 0 L (and therefore f 0 L) for scale sizes larger than 50 Mm. The injected Poynting flux is f 0 mv 2 A0. Thus, a fixed value of v A0 leads to a Poynting flux f 0 mv 2 A0 / B 0, since f 0 / B 0. But the flux-tube area scales inversely with field strength, as required by conservation of magnetic flux x B ¼ 0. Hence, referencing the bottom of flux tubes to a nearly fixed field strength, the injected power (Poynting flux times flux-tube area) and the injected particle rate (particle flux times flux-tube area) are nearly constant. Figure 2 (top) shows the solar wind energy and scaling law terms for parameters v A0 ¼ 750 km s 1 and B 0 L ¼ 368 G Mm, consistent, for example, with field strength B 0 8 D

No. 2, 2003 SOLA WIND SCALING LAW 1397 Fig. 1. Illustration of solar wind sources showing elements of the energy scaling law. Flux tube illustrations are superimposed on a remote observation of the solar atmosphere taken by SOHO EIT (shown is the line ratio Fe xii 195 Å/Fe ix 171 Å, which is an effective thermometer of the solar atmosphere with dark to bright emissions ranging from 0.9 to 1.5 MK, the same range of observed C freezing-in temperatures). G, comparable to the open field strength in coronal holes, 2 and length L 6 Mm, which is slightly larger than the size scale of supergranules (30 Mm). Also shown in Figure 2 2 The open field strength in coronal holes is given by B 1r ðr 1 = Þ 2 D exp, where r 1 ¼ 1 AU and D exp is a factor representing the solid area expansion from the coronal hole. Given typical values of D exp ¼ 5:8 (Munro & Jackson 1977) and B 1r ¼ 30 lg (McComas et al. 2000b), we find a coronal hole open field strength 8 G. (bottom) is the solar wind speed vs. imum source temperature. The results based on equation (1) are plotted with published data for the C freezing-in temperature (von Steiger et al. 2000) from the SWICS instrument on Ulysses (Gloeckler et al. 1992). Above the temperature cutoff shown in Figure 2 only subsonic bound solutions are available. This regime is inherently time-dependent and unstable: plasma and electromagnetic energy are continually injected, leading to a growth in total density, but the energy is radiated away too

1398 SCHWADON & McCOMAS Vol. 599 Fig. 2. Top: Energy partition of terms in the scaling law, eq. (1). Bottom: Solar wind speed vs. imum source temperature. Points in blue are published data (von Steiger et al. 2000) from the SWICS instrument on Ulysses (Gloeckler et al. 1992) for the C freezing-in temperature with solar wind speeds and standard deviations from SWOOPS. The observed solar wind has bimodal properties: a fast wind (upper solid curve), a slow wind (lower solid curve), and a transition between these states (dashed curve) (McComas, Elliott, & von Steiger 2002). There is an upper limit for nontransient speeds in which radiative losses are insignificant, and a lower speed limit for large temperatures where there is so much radiated energy that the solar wind is subsonic and therefore bound. quickly to form a steady wind. This unstable bound regime may be related to a variety of highly time-dependent, unstable phenomena (e.g., in the form of coronal mass ejections, transient streams, and solar flares) exhibited by the Sun s corona, although detailed discussion of the instability is beyond the scope of this paper. The agreement with observations shown in Figure 2 suggests a source for all solar wind in which there is a nearly fixed injection of electromagnetic energy (per particle). Thus, this theory unifies what are apparently distinct sources for fast and slow solar wind. While the ranges shown in Figure 2 may be typical of the solar wind, much faster transient flows and higher C freeze-in temperatures are infrequently observed. Variations in the parameters of the scaling law may cause variations from typical cutoffs and speeds. In addition, the scaling law applies only for steady-state solar wind, and transient events are likely to exhibit markedly different behavior. The nearly fixed electromagnetic energy (per particle) agrees qualitatively with recent remote observations (Schrijver et al. 1998) showing the reconfiguration of the supergranule networks in about a day, as is consistent with a large injection of Poynting flux and particle flux onto open field lines due to the implied frequent reconnection between open field lines and small loops with scale sizes less than the supergranule. This reconfiguration of the network is observed to be global and homogeneous, suggesting an equally homogeneous injection of electromagnetic energy and mass. Our observationally supported assumption of a fixed injection of electromagnetic energy (per particle) is in contrast to Fisk (2003), where radiated energy was neglected and the injected electromagnetic energy (per particle) had to be varied in an ad hoc way using a set of free parameters related to loops taken to be the sources of solar wind. Gloeckler et al. (2003) then showed that the observed anticorrelation between freeze-in temperature and solar wind

No. 2, 2003 SOLA WIND SCALING LAW 1399 speed could be made consistent for a specially chosen set of these parameters. However, here we show that this anticorrelation is just the natural result of radiated energy loss. While not required, our two assumptions are also consistent with a possible loop source for the solar wind, with fast/cool solar wind from the small loops that reconnect with open flux tubes or near continuously open flux tubes in coronal holes; and slow/hot solar wind from larger and hotter loops that reconnect with open flux tubes surrounding coronal holes (Fisk & Schwadron 2001). For a fixed injection of electromagnetic energy (per particle), loops act as a conduit for the solar wind. Over time, loops grow either through reconnection with other loops or through heating and expansion. In both cases, loops act as particle and magnetic field energy reservoirs. When the loops finally reconnect with open field lines, their stored particles and field energy are released to form the solar wind. Larger loops are generally observed to be hotter, longer lived, and brighter (Bray et al. 1991; Feldman & Widing 1993; Feldman, Widing, & Warren 1999); therefore they must radiate larger amounts of energy than small loops or flux tubes that are almost continually open. Thus, the solar wind scaling law shows that solar wind from dim, cool, small loops or from cool continuously open field regions should be more energetic and tenuous than solar wind from large, hot, bright loops that are more rapidly heated (the volumetric heating rate is _E H ¼ C 0 0 T=L 2, and therefore scales very strongly with imum temperature, T,as shown in Appendix B). This increased heating of larger loops is interesting in the context of observed elemental abundances that show strong enhancements relative to photospheric abundances of low- FIP elements (such as Fe, Si, and Mg; von Steiger et al. 2000) in slow solar wind. Presumably, increased heating of larger loops would cause increased heating rates just below the transition region, leading to enhancements of low-fip elements due to increased scale heights of the ionized, predominantly low-fip material (Schwadron, Fisk, & Zurbuchen 1999). In the unified theory, B 0 L is a constant. Thus, if both fast and slow solar wind arise from loops, then the larger hot loops must have weaker field strengths than the small cool loops. Wang (1995), however, found no correlation between final solar wind speed and source field strength, which was inferred from potential field models. Similarly, Mandrini et al. (2000) found that for scales smaller than 50 Mm, loops near active regions had roughly constant field strengths. Taken together, these studies suggest that on small length scales (e.g., less than 50 Mm), field strengths may not fall off with height, and B 0 L may not be a constant. Even with more complicated forms of B 0 L, however, the strong temperature dependence of the radiative energy loss ensures that for a fixed injection of electromagnetic energy (per particle), stronger heating at less than a scale height leads to a slower and more dense solar wind. Thus, the simple assumption that B 0 L is constant applies for large scales (e.g., larger than 50 Mm), and the detailed behavior of B 0 as a function of L on smaller scales may lead to small deviations from the model results in Figure 2. The scaling law can also be used to understand flux-tube expansion effects. On average, the base field strength B 0 is weaker where the expansion is most rapid. For fixed values of L and imum temperature T, equation (1) shows that the radiative loss, C 0 0 T=ðf 0 LÞ C 1 kt, is largest where the field B 0 (and therefore flux f 0 / B 0 ) is weakest. Thus, with a fixed injected electromagnetic energy per particle, the final speed is anticorrelated with the base expansion rate, consistent with inferences from potential field modeling (Levine et al. 1977; Wang & Sheeley 1990; Wang 1995). The slowdown in solar wind speed is not the result of a change in the volumetric heating rate, _E H ¼ C 0 0 T=L 2. Instead, in a rapidly expanding flux tube, plasma parcels require more time to transit the radiative region and therefore lose more energy. The difficulty with invoking expansion effects to explain slow wind is that charge-state measurements (e.g., von Steiger et al. 2000) show much larger source temperatures and therefore more strongly heated sources of slow wind. Unlike expansion models, however, the unified theory requires only more rapid heating at the base of slow wind; it is independent of whether or not the solar wind arises on loops, or undergoes significant expansion.. CONCLUSIONS In this paper we have derived a scaling law for the solar wind in which its final energy (speed) is the injected electromagnetic energy minus the radiated and gravitational potential energy. This scaling law is not an extension or a new variant of previous solar wind models, but rather a requirement on all solar wind models, which should apply quite generally to magnetically driven winds. The term in the scaling law representing radiated energy is strongly temperature dependent. Accordingly, the scaling law shows why fast solar wind is emitted from cool dark coronal holes, whereas slow solar wind is emitted from the hotter and brighter regions. The scaling law naturally provides for a unified theory for the source of solar wind in which all solar wind originates with a roughly fixed injected electromagnetic energy (per particle), but slow solar wind is emitted from hotter regions that are more highly radiative, and bound but unstable plasma exists in extremely hot regions, potentially related to solar transients. Thus, we have revealed an underlying connection between the solar wind and its coronal source. This work was supported by Southwest esearch Institute and is a part of NASA s Ulysses/SWOOPS and ACE/SWEPAM programs. N. A. S. was also supported by the NASA Solar and Heliospheric Physics Program, grant NA-01-01-SHP-10. The authors would like to thank Vadim Soloviev and Heather Elliott. We also really appreciate the thoughtful and detailed reviews provided by an anonymous referee.

100 SCHWADON & McCOMAS Vol. 599 APPENDIX A THE SCALING LAW: DEIVATION The final state of a magnetically driven stellar wind is calculated utilizing a magnetohydrodynamic (MHD) equation expressing conservation of total energy density (e.g., Powell et al. 1999): where the total energy density is the energy flux is the hydrodynamic enthalpy is @E tot @t þ D x F E ¼ _E rad ; E tot ¼ u2 2 þ p 1 þ B2 8 þ F ; F E ¼ uh hd þ c E µ B þ q e þ u F ; H hd ¼ u2 2 þ 1 p ; ðaþ and _E rad is the volumetric energy-loss rate due to radiation. In these expressions, the mass density is, the solar wind velocity is u, the particle pressure is p, the magnetic field vector is B, the Poynting flux is ce µ B=, the electron heat flux is q e, the ratio of specific heats is, and we take the external forces to be a gradient of a potential, F ext ¼ m F. Note that a heating term on the right-hand side of equation (A1) is absent. It does not appear because the Poynting flux is included in the energy flux; therefore, all heating due to dissipation of electromagnetic waves and turbulence appears as a divergence of the Poynting flux. We integrate the ensemble-averaged steady-state limit of equation (A1) over the volume of a flux tube that extends from an inner radial distance r 0 to an outer radial distance r 1. Using Gauss s law with an energy flux directed along the flux tube, we find Z Z Z r1 ds 1 x hf E i 1 ¼ ds 0 x hf E i 0 dvh _E rad i : ða5þ r 0 Here ds i x hf E i i indicates surface integration of the flux-tube boundary at r i (i ¼ 0 or 1). The angle brackets indicate ensemble averaging, with h...i i ensemble averaging at r i. The volume integral of the radiation loss over the entire flux tube is r1 r 0 dvh _E rad i. We evaluate the final energy flux at r 1 ¼ 1 AU, where the solar wind is supersonic, and the total energy flux is strongly dominated by the ram energy, u 2 f hf E i 1 ðuþ f 2 ^e r : ða6þ Here the subscript f indicates average final values of variables near 1 AU, well beyond the sonic point. Based on solar wind observations, we have taken the average solar wind flow to be directed outward radially (^e r is the unit vector in the radial direction). Although there are additional contributions to the energy flux at 1 AU due to MHD waves, electron heat flux, thermal enthalpy, and correlated fluctuations in the wind speed and density, these contributions are small (d2% of the ram energy). At the inner radial position, r 0, which we take slightly below the transition region where temperatures are low (5000 K), the bulk flow is slow, and the electron heat conduction nearly vanishes since the temperature gradient is weak. In this region, the dominant terms for the energy flux are the Poynting flux and gravitational potential loss: hf E i 0 c he µ Bi 0 hui GM 0 : ða7þ We recover an expression for the final energy of the wind by substituting equations (A6) and (A7) into equation (A5), and dividing by the particle transfer rate _N ¼ ds i x hnui i (for i ¼ 0 and 1): D ða1þ ða2þ ða3þ mu 2 f 2 ¼ c r1 ds0 x he µ Bi 0 r 0 _N dvh _E rad i _N GM m : ða8þ The electromagnetic energy that drives the solar wind is the first term on the right-hand side of equation (A8), the net source rate of magnetic energy (Poynting flux) divided by the particle transfer rate. It has been argued that the energy and mass flux needed to drive the fast solar wind is derived from the emergence of small loops within supergranules, which when they reconnect with open field lines release Poynting flux and mass flux that propagate into the base of the corona as kinked field

No. 2, 2003 SOLA WIND SCALING LAW 101 lines (Fisk et al. 1999). This scenario is supported by observations showing that the network magnetic fields of supergranules are reconfigured very quickly, on the order of a day (Schrijver et al. 1997, 1998). In this case, the Poynting flux is given by Z c B ds 0 x he µ Bi 0 ¼ 2 0? u 0A 0 ; ða9þ where B 0? is the ensemble-averaged strength of emerging magnetic loops (in the plane of the inner flux tube surface S 0 with area A 0 ). Note that we have defined the inner boundary of the flux tube such that the average flow u 0 is parallel to the central axis of the flux tube. The particle transfer rate into the flux tube in this case is Z _N ¼ ds 0 x hnui 0 ¼ f 0 A 0 ; ða10þ where f 0 is the particle flux carried by emerging loops. Accordingly, the first term on the right-hand side of equation (A8) is c ds0 x he µ Bi 0 B ¼ m 2 0? ¼ mv _N 2 A0 ; ða11þ 2 0 where 0 is the ensemble-averaged mass density of emerging loops, and v A0 is the ensemble-averaged Alfvén speed. Typical Alfvén speeds are quite large in the low corona (700 km s 1 ). For clarity, we have discussed the electromagnetic energy source in the context of emerging magnetic loops. In the most general terms, however, the quantity mv 2 A0 is the injected electromagnetic energy per particle, or equivalently, the area integrated Poynting flux divided by the particle transfer rate, ðc=þ ds 0 x he µ Bi 0 = _N. Equation (A8) applies for any form of injected electromagnetic energy, so long as the energy flux is directed along the flux tube. The radiative loss rate scales with the square of the electron density (osner, Tucker, & Vaiana 1978) multiplied by a strongly temperature dependent term that is largest for T > 500; 000 K. Thus, it is significant low in the corona, where densities and temperatures are large. Taking a uniform heating rate that generates a imum temperature T at length L below or near 1 scale height, we find (as described in the next section) that the net radiative loss is L 0 dvh _E rad i 0 T C 0 _N f 0 L C 1kT ; ða12þ where C 0 ¼ 0:91 and C 1 ¼ 5. The first term of the right-hand side of equation (A12) is caused by conduction of heat from the temperature imum down to the base of the flux tube, where radiative losses are most significant. The second term on the right-hand side of (A12) arises because of advective loss of thermal energy, which depresses the pressure and thereby limits radiative loss. Folding these terms (A12) along with the expression for the Poynting flux (A11) into the energy equation (A8), we find the solar wind energy scaling law:! mu 2 f 2 0 T mv2 A0 C 0 f 0 L C 1kT GM m : ða13þ APPENDIX B TOTAL ADIATIVE ENEGY LOSS IN A FLOWING PLASMA We derive below the integrated radiative energy loss, L 0 dvh _E rad i= _N, equation (A12). The derivation extends scaling laws for static loops (osner et al. 1978; Serio et al. 1981) to magnetic flux tubes with flowing plasma. We solve the steady-state limit of the energy equation (A1), @ @z 1 up þ @q e @z ¼ _E H _E ; ðb1þ where the conductive heat flux is q e ¼ 0 T 5=2 @T=@z 10 6 T 5=2 @T=@z in cgs units (Spitzer 1962), the coordinate along the magnetic field is given by z, the flow velocity along the field is given by u, and we take for simplicity a uniform field strength (and therefore flux tube cross section). Note that we have considered only the flux of thermal energy (/up) in the hydrodynamic enthalpy, since the equation is solved for low in the corona, where the thermal pressure dominates the ram pressure. We have also neglected the effect of gravity because the gravitational potential changes very little close to the Sun through the transition region where radiative losses are significant. We have also now explicitly included on the right-hand side a heating term, _E H, which is equivalent to the negative divergence of the Poynting flux, _E H ¼ c x he µ Bi=ðÞ. The left-hand side of the energy equation can be rewritten in terms of the particle flux f 0 and the heat conduction: 1 f 0kq e þ dq2 e dt ¼ 2 0T 5=2 ð _E H _E Þ : ðb2þ This result is obtained by transforming the enthalpy gradient using the property that in steady state the particle flux (times D

102 SCHWADON & McCOMAS Vol. 599 area) is constant, ð 1Þ 1 ½@ðupÞ=@zŠ ¼ 2½ 0 ð 1ÞŠ 1 f 0 kq e T 5=2. We have also transformed the heat flux gradient, @q e =@z ¼ð@T=@zÞð@q e =@TÞ ¼ ð2 0 Þ 1 T 5=2 @q 2 e=@t. Equation (B2) is then integrated from T 0 to T to solve for q e ðtþ, q e ðtþ 2 þ 1 f 0kQ e ðtþ ¼F ðtþ F H ðtþ ; ðb3þ T where F ;H ðtþ ¼2 0 T 0 dt 0 ðt 0 Þ 5=2 _E ;H ; the integrated heat flux is Q e ðtþ ¼ T T 0 dt 0 q e ðt 0 Þ, and we have taken q e ðt 0 Þ¼0. The rate of radiative energy loss takes the form _E ¼½p 2 0 =ðk2 T 2 ÞŠðTÞ, where the function ðtþ is detailed by osner et al. (1978) (note that the precise form for the radiative energy loss has been discussed in numerous studies; see Aschwanden & Schrijver 2002). For temperatures T > 0:13 MK, F ðtþ ½ 0 p 2 0 =ð2k2 ÞŠ 1 T, where 1 ¼ 10 18:8 K 1=2 cm 3 ergs s 1.Wehave also taken a uniform pressure p 0 (this restriction will be relaxed further on). Assuming uniform heating, we find F H ðtþ ð 0 _E H =7ÞT. At the temperature imum, T, the heat conduction vanishes since there is no temperature gradient. Thus, we find the following requirement on the heating rate Substituting this result into (B3) we obtain q e ðtþ 2 ¼ F ðt Þ _E H ¼ 7 1p 2 0 8k 2 T 5=2! T T T T 7f 0kQ e ðt Þ ð 1Þ 0 T 1 f 0kQ e ðt Þ : ðbþ " # Q e ðtþ Q e ðt Þ T T : ðb5þ Note that the fraction Q e ðtþ=q e ðt Þ must vary from 0 to 1 as the temperature varies from T 0 to T. As a first-order approximation, we take Q e ðtþ Q e ðt Þ T ; ðb6þ T which allows Q e ðt Þ to be solved for explicitly, 2sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 1 0 p 2 0 Q e ðt Þ¼ 1 T T 2k 2 þ 2 1f 0 kt 2 þ 2 1f 0 kt 5 ; ðb7þ 1 1 where 1 ¼ 1 0 dxðx xþ 1=2 ¼ 0:50. Since 0 T 5=2 q 1 e is proportional to the temperature gradient of length, we can integrate it to solve for the position of the temperature imum, 2sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 L ¼ 2 0 T 1 0 p 2 0 T 2k 2 þ 2 1 1f 0 kt 2 þ 2 1f 0 kt 5 ; ðb8þ 1 1 where 2 ¼ 1 0 dx x5=2 ðx x Þ 1=2 ¼ 0:72. In more familiar terms, (B8) is a scaling law somewhat modified relative to the static scaling law of osner et al. (1978): p 0 L ¼ T 3 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 C B f 0 LT 5=2 ; ðb9þ C A where C A ¼ 1: 10 3 Kcm 1=3 dyne 1=3 and C B ¼ 1:3 10 9 cm s K 5=2. Note that with zero flux, we recover precisely the previously derived static scaling law (osner et al. 1978). We can now solve for the net radiated energy using equations (B1), (B), (B7) and (B8): L 0 dv h L _E rad i 0 ¼ dz _E ¼ L _E H 0 T C 1 kt ¼ C 0 _N f 0 f 0 f 0 L C 1kT ; ðb10þ where C 0 ¼ 72 2= ¼ 0:91 and C 1 ¼ 2=ð 1Þ ¼5 for a standard ratio of specific heats, ¼ 5=3. The result is remarkably simple, but relies on one important approximation, equation (B6). We find, however, that these scaling relations are extremely accurate, as demonstrated in Figure 3, which compares the approximated scaling laws given in equations (B9) and (B10) with results from the full numerically integrated solution to equation (B3). Finally, we revisit the assumption of constant pressure. When this assumption is relaxed, it has been shown that on a static loop the required heating rate is reduced, L _E H =f 0 ¼ð7 2 =Þ 0 T =ðf 0LÞ expð L=s p Þ ; where s p 2kT =ðmg s Þ¼ðT =MKÞð50 MmÞ is the pressure scale height and ¼ 16=25 (Serio et al. 1981). The exponential falloff is due to a sharply decreasing density and radiative loss for heights greater than a scale height. Note, however, that remote observations indicate that the heating rate peaks well below a scale height (Aschwanden, Schrijver, & Alexander 2001) (heating scale heights in loops are seen to be 12 5 Mm, covering only about 20% of the loop length), and

No. 2, 2003 SOLA WIND SCALING LAW 103 Fig. 3. adiated energy L 0 dvh _E rad i= _N and length L of the temperature imum vs. particle flux f 0 for the dynamic scaling law derived here. We compare analytic approximations of eqs. (B10) and (B8) with numerically integrated solutions of eq. (B3). Values for p 0 and T are indicated in the figure. Also indicated by the upper x-axis are the field strengths B 0 and 3 times the velocity u 0 that would produce these particle fluxes at the base of the flux tube given a particle flux at 1 AU of f 1 ¼ 3 10 8 cm 2 s 1 and a radial field strength B 1r ¼ 3 10 5 G [where f 0 ¼ f 1 ðb 0 =B 1r Þ¼10 13 B 0 ]. consistently, in situ observations indicate a weak gradient in freezing-in temperatures, which are fixed above the transition region (e.g., von Steiger et al. 2000 finds for fast solar wind a freezing-in temperature for C, T C 0:95 MK, and for O, T O 1:08). Thus, remote and in situ observations consistently suggest that large temperatures are achieved below a scale height, with a weak positive gradient of temperature extending above a scale height to the true temperature imum. Accordingly, there is a small conductive heat flux above a scale height, and the derivation above applies for a imum temperature T below or near 1 scale height. Aschwanden, M. J., & Schrijver, C. J. 2002, ApJS, 12, 269 Aschwanden, M. J., Schrijver, C. J., & Alexander, D. 2001, ApJ, 550, 1036 Axford, W. I., & McKenzie, J. F. 1997, in Cosmic Winds and the Heliosphere, ed. J.. Jokipii, C. P. Sonett, & M. S. Giampapa (Tucson: Univ. Arizona Press), 31 Bray,. J., Cram, L. E., Durrant, C. J., & Loughhead,. E. 1991, Plasma Loops in the Solar Corona (Cambridge: Cambridge Univ. Press) Bürgi, A., & Geiss, J. 1986, Sol. Phys., 103, 37 Feldman, U., & Widing, K. G. 1993, ApJ, 1, 381 Feldman, U., Widing, K. G., & Warren, H. P. 1999, ApJ, 522, 1133 Fisk, L. A. 2003, J. Geophys. es., 108, 1157 Fisk, L. A., & Schwadron, N. A. 2001, ApJ, 560, 25 Fisk, L. A., Schwadron, N. A., & Zurbuchen, T. H. 1999, J. Geophys. es., 10, 19765 Geiss, J., et al. 1995, Science, 268, 1033 Gloeckler, G., Zurbuchen, T. H., & Geiss, J. 2003, J. Geophys. es., 108, 1158 Gloeckler, G., et al. 1992, A&AS, 92, 267 Hansteen, V. H., & Leer, E. 1995, J. Geophys. es., 100, 21577 Isenberg, P. A. 1991, in Geomagnetism, Vol., ed. J. A. Jacobs (San Diego: Academic Press), 1 Krieger, A. S., Timothy, A. F., & oelof, E. C. 1973, Sol. Phys., 29, 505 Levine,. H., Altschuler, M. D., & Harvey, J. W. 1977, J. Geophys. es., 82, 1061 Mandrini, C. H., Démoulin, P., & Klimchuk, J. A. 2000, ApJ, 530, 999 Marsch, E. 199, Adv. Space es., 1(), 103 McComas, D. J., Elliott, H. A., & von Steiger,. 2002, Geophys. es. Lett., 29, 28 McComas, D. J., Gosling, J. T., & Skoug,. M. 2000a, Geophys. es. Lett., 27, 237 McComas, D. J., et al. 1998, Geophys. es. Lett., 25, 1. 2000b, J. Geophys. es., 105, 1019 EFEENCES McKenzie, J. F., Axford, W. I., & Banaszkiewicz, M. 1997, Geophys. es. Lett., 2, 2877 Munro,. H., & Jackson, B. V. 1977, ApJ, 213, 87 Nolte, J. T., Krieger, A. S., oelof, E. C., & Gold,. E. 1977, Sol. Phys., 51, 59 Parker, E. N. 1958, ApJ, 128, 66. 1988, ApJ, 330, 7. 1991, ApJ, 372, 719 Powell, K., oe, P., Linde, T., Gombosi, T., & Zeeuw, D. L. D. 1999, J. Comput. Phys., 15, 28 osner,., Tucker, W. H., & Vaiana, G. S. 1978, ApJ, 220, 63 Sandbaek, O., Leer, E., & Hansteen, V. H. 199, ApJ, 36, 390 Schrijver, C. J., Title, A. M., van Ballegooijen, A. A., Hagenaar, H. J., & Shine,. A. 1997, ApJ, 87, 2 Schrijver, C. J., et al. 1998, Nature, 39, 152 Schwadron, N. A., Fisk, L. A., & Zurbuchen, T. H. 1999, ApJ, 521, 859 Serio, S., Peres, G., Vaiana, G. S., Golub, L., & osner,. 1981, ApJ, 23, 288 Spitzer, L. 1962, Physics of Fully Ionized Gases (2d ed; New York: Interscience) von Steiger,., Geiss, J., & Gloeckler, G. 1997, in Cosmic Winds and the Heliosphere, ed. J.. Jokipii, C. P. Sonett, & M. S. Giampapa (Tucson: Univ. Arizona Press), 581 von Steiger,., Schwadron, N. A., Hefti, S., Fisk, L. A., Geiss, J., Gloeckler, G., Wilken, B., Wimmer-Schweingruber,. F., & Zurbuchen, T. H. 2000, J. Geophys. es., 105, 27217 Wang, Y.-M. 1995, ApJ, 9, L157 Wang, Y.-M., & Sheeley, N.., Jr. 1990, ApJ, 355, 726. 1991, ApJ, 372, L5. 199, J. Geophys. es., 99, 6597. 2003, ApJ, 587, 818 Withbroe, G. L. 1988, ApJ, 325, 2