anp ( - ), where our best estimate for /* is between 1.5 and 1.7. The best fitting relation is Tp = (2.0 _+ 0.13) x 105 N '57.

Transcription

1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL 103, NO A7, PAGES 14,547-14,557, JULY 1, 1998 Ion energy equation for the high-speed solar wind: Ulysses observations W C Feldman, B L Barraclough, J T Gosling, D J McComas, and P Riley Space and Atmospheric Sciences Group, Los Alamos National Laboratory, Los Alamos, New Mexico B E Goldstein Jet Propulsion Laboratory, Pasadena, California A Balogh The Blackett Laboratory, Imperial College, London Abstract Ulysses data in the high-speed solar wind that cover a wide range of latitudes centered on the solar poles were studied to test whether a polytrope law can be used to close the ion energy equation Three approaches were taken We determined the correlation between proton temperature and density (1) in the free expansion of the highspeed solar wind between 15 and 48 AU, (2) in steepened microstreams at high latitudes, and (3) at the edges of the equatorial band of solar wind variability Strong correlations were,observed in all data subsets that are consistent with a single polytrope relation, T_ = anp ( - ), where our best estimate for /* is between 15 and 17 The best fitting relation is Tp = (20 _+ 013) x 105 N '57 1 Introduction A major problem for predicting the plasma state of stellar envelopes is developing a closure relation for the Boltzmann- Vlasov moment equations Although many suggestions have been made, a common expedient has been to replace the energy equation by a polytrop½ relation between pressure (P) and density (N), d/dt(p/n *) = 0 (1) [eg, Siscoe, 1983] If the flow is adiabatic, then,/* = / = Cp/C, which is 5/3 for an ideal, monatomic gas like the solar wind Here Cp and C are the specific heats at constant pressure and volume, respectively If the heat energy added to the gas is proportional to the temperature change, then dq = CdY, (2) * = (%- C)/(Cv- C) (3) [Chandrasekhar, 1939, p 86] An alternate way of expressing this relation is that the heat flux is proportional to the convected flux of internal energy [Parker, 1963; Feldman et al, 1978] Just such a relation has been postulated for the solar wind close to the Sun [Hollweg, 1974, 1976, 1978] Its physical underpinnings can be derived from the two-component nature of solar wind electron velocity distributions, corresponding to separate electron populations that are bound and unbound to the Sun by the solar ambipolar electric potential [Perkins, 1973; Feldman et al, 1975] Such a relation has been shown to hold in the ecliptic plane near 1 AU within compression regions on the leading edges of high-speed solar wind streams [Feldman et Copyright 1998 by the American Geophysical Union Paper number 98JA /98/98 JA $ ,547 al, 1978] The polytrope index for electrons in the high-speed portions of these compression regions is found to be /* _+ 005 This result stands in contrast to that obtained through measurement of the radial gradient of electron temperature between about 03 and 5 AU If the measured gradient is translated into a polytropic index, the solar wind expansion is seen to be nearly isothermal at high bulk speeds, with /* between about 1 and 12 [Feldman et al, 1979, 1996; Sittler and Scudder, 1980; Pilipp et al, 1990] Measurements of the radial gradient of proton temperatures in the relatively interaction-free high-speed solar wind have been made between 03 and 3 AU using the Helios and Ulysses plasma experiments [Marsch et al, 1982; Totten et al, 1995; Goldstein et al, 1996] Both data sets yield radial gradients characterized by power law indices that range between 07 and 11 (Note that reports of an index closer to 05 using Ulysses data [Feldman et al, 1996; Neugebauer et al, 1995] are incorrect because they used proton temperatures that were not properly corrected for the large angular sampling intervals in velocity space (see discussion of Goldstein et al [1996])) When translated to a proton polytrope index, the best estimate of the radial gradient of temperature yields 3/* - 15 [Goldstein et al, 1996] The self-consistency of this interpretation, however, has not been examined Indeed, an intercomparison of polytrope indices for electrons estimated from observations of simple compressions induced locally at stream-interaction regions at a given heliocentric distance [Feldman et al, 1978] with those estimated from measured radial gradients of electron temperature and density [Feldman et al, 1979, 1996; Sittler and Scudder, 1980; Pilipp et al, 1990] is inconsistent The purpose of the present work is to provide a critical evaluation of the use of a polytrope law as a substitute for the ion energy equation to describe the high-speed solar wind (HSSW), which is defined here to be flow at speeds greater than 600 km s- A similar study was conducted using Voyager data between 1 and 10 AU [Gazis and Lazurus, 1982; Gazis,

2 14,548 FELDMAN ET AL' ION ENERGY EQUATION FOR HIGH-SPEED SOLAR WIND N Pole Speed (km s -t) õ ø Nit : ; 45' '- ' Perihelion Equator [ '-: :[?';":"/ -- phellon Equator '- : ' 'j? [ ; -45 ø -45 ø SHSW 1ooo S Pole Figure 1 An overview of the various heliospheric flow regimes encountered by Ulysses summarized by the proton bulk speed, shown as a polar plot Subsets of Ulysses data that are used to define a polytrope relationship between Tp and Np are outlined by the circular sectors labeled I through V 1984] However, the results of these studies do not apply to relatively structure-free high-speed flows because the Voyager spacecraft observed very few intervals having speeds greater than 600 km s-, and the resultant data set was not filtered to exclude shocked gas or coronal mass ejections In contrast, Ulysses observations of the solar wind at high latitudes provide an ideal data set for this study because they provide clean examples of several distinctive flow types The flows above both solar poles over a broad range of latitudes are relatively uniform and free of the strong compressions that are common near the ecliptic plane They therefore can be used to test a polytrope relation because the solar wind is spherical, which causes both a smooth decrease in the density and a corresponding cooling of the ions Nevertheless, these polar flows often contain discrete microstreams that have compressive interaction regions at their leading edges [McComas et al, 1995; Neugebauer et al, 1995] that offer the opportunity to test a polytrope law at discrete heliocentric distances Ulysses data also include several examples of stronger, yet unshocked, compression and rarefaction regions at the edges of the equatorial band of solar wind variability, which can also be used to test the applicability of a polytrope law to the high-speed solar wind A description of the Ulysses data set assembled for our study is given in section 2 Average properties of the high-speed solar wind at polar latitudes are presented in section 3 Evaluation of measured radial gradients of all terms in the ion energy equation with regard to a polytrope law is given in section 4 This targeted to reorient the orbital plane of Ulysses to one that was nearly polar, passing first over the Sun's southern polar cap at a heliographic latitude of-802 ø at 229 AU on September 12, 1994, and over its northern polar cap at a latitude of ø at 202 AU on July 31, 1995 Our study of the internal energy state of ions in the high-speed solar wind uses data measured by the ion spectrometer of the Ulysses plasma experiment [Bame et al, 1992] between December 1, 1992, and February 24, 1997 We also use magnetic field data provided by the Ulysses magnetometer experiment Ulysses plasma data were separated into several subsets as shown in Figure 1 The first subset (identified as region I at the top of Figure 1) includes all high-speed solar wind data observed poleward of +50 ø latitude (the north high-speed solar wind (NHSSW) 0 > +50ø) This subset spanned times between May 12, 1995, and January 23, 1996, and radial distances between 15 and 32 AU This subset provides our primary source of radial-gradient information of the high-speed solar wind because it spans a time interval close to solar minimum when the streamer belt was nearly coplanar with the heliographic equator and no coronal mass ejections (CMEs) were detected [Gosling et al, 1997] The second subset (identified as region II at the bottom of Figure 1) includes all high-speed solar wind data observed poleward of -75 ø latitude (the south high-speed solar wind (SHSSW) 0 < -75ø) We purposely did not include data measured at more equatorial latitudes in this subset because at the time of Ulysses' south polar passage the Sun's presentation is followed by a study of discrete compressions streamer belt was tilted and warped, which produced occaand rarefactions observed at high latitudes in section 5 and of sional remnants of corotating interaction regions down to -60 ø the relatively strong (yet unshocked) compression and rarefaction heliographic latitude [Phillips et al, 1995] This set spanned regions observed at the edges of the equatorial solar wind in times between July 28 and October 20, 1994, and radial dissection 6 All results are summarized and discussed in section 7 tances between 20 and 26 AU 2 Ulysses Data Set Ulysses was launched in October 1990 into an orbit that intercepted Jupiter in February 1992 Passage by Jupiter was The next three subsets covered the southern and northern edges of the equatorial streamer belt The first occurred --10 months after passage by Jupiter as Ulysses headed southward (identified as region Ill at the lower right in Figure 1), between

3 FELDMAN ET AL: ION ENERGY EQUATION FOR HIGH-SPEED SOLAR WIND 14,549 December 1, 1992, at 51 AU and June 30, 1993, at 46 AU charge, ri) is the interplanetary electrostatic potential (normal- The next two subsets occurred as Ulysses was entering the ized to zero at infinity), V, and Vp are the alpha-particle and northern edge of the streamer belt near aphelion between proton bulk velocities, respectively, G is the gravitational con- January 23, 1996, at 32 AU and February 24, 1997, at 48 AU stant, M s is the mass of the Sun, and R is the heliocentric (regions IV and V at the upper right in Figure 1) Passages distance We note that Coulomb and wave-particle coupling through the northern and southern edges of the streamer belt between electrons and the ions have been ignored in the dernear perihelion (at the left in Figure 1) were not included in ivation of (2) and use has been made of the fact that 8V,p/Vp our study because the southern boundary crossing was complicated by detection of a coronal mass ejection (CME) [Gosling et al, 1995] and several shocks and the northern crossing was a sharp stream interface and not a compression [McComas et al, 1998] Care was taken to remove all data in each of the subsets that included CMEs or shocked gas CMEs were primarily identi- V = V + 8V + to (8) fied using the counterstreaming suprathermal electron criterion [eg, Gosling, 1990] However, this criterion was augmented to include regions of abnormally low electron and proton temperatures and high alpha abundances Although these criteria should catch most cases of CME gas, the resultant filtered data set was probably not completely uncontamiin (3) and integrating over all velocity phase space Here 8Vp and 8V are the proton and alpha-particle bulk velocities, respectively, relative to the center of mass velocity V and top and to are the respective proton and alpha-particle thermal velocities relative to Vp and V, respectively, where (top) = (to) = 0 After some algebraic calculations we find nated by CMEs Identification of forward and reverse shocks was made using the combined plasma and magnetic field data AK = F [05V 2 + Kt>(drift) + Ke(enthalpy) sets For those cases where pairs of forward and reverse shocks that bound corotating interaction regions (CIRs) could be identified, all gas between the shocks was edited out of the + Ko(heat flux) ] where the drift term Kz> is given by (9) accepted data set Otherwise, we exercised the conservative judgment to remove all data that appeared shocked before Kr : [SVp 2 + 2(6¾p ¾)2 + ( jyp o ¾)( JVp/V)2 detection of reverse shocks and after detection of forward shocks in the original Ulysses plasma data All data when the flow speed was less than 600 km s- were also removed so that (1-4N,/Np)/(4N,/Np)]/(8N,/Np), the enthalpy term KE is given by (10) the final data set should correspond only to the high-speed solar wind K r = 25kTp[1 - ¾p ¾/V 2 -t-(ra/rp)(na/np) 3 Ion Energy Flux Conservation Equation eri)[1 + 2(No/Np)] F r = AK + FM mp[ 1 -t- 4(No,/Np)] - F GMs/R, (5) which are both constants of the flow Here F3 and F E are the mass and energy flux, respectively, mp is the proton mass, Np and N, are the proton and alpha-particle number densities, respectively, A is the area of a stream tube, K is the ion kinetic energy flux given by K: I(OSmpVpVp2)l + I(0SmVvZ>l, (6) where the angle brackets denote an integral over the appropriate ion velocity distribution function, e is the electronic is small compared to unity, where 8Vap = Iv - Vpl The kinetic energy flux term in (2), K, can be evaluated by substituting Vp = V '--' Vp -t- OOp (7) (1-3- 8¾p' V/(V2(4No]Np)))]/(1 + 4No,/Np), (11) and the heat flux term Ke is given by Before exploring the validity of a polytrope closure relation K o = (Qp-t- Qo,)/Fw (12) for the ion Boltzmann-Vlasov equation in the high-speed solar wind, it is useful to determine the relative magnitude of the Here k is Boltzmann's constant, Tp and T, are proton and various terms in the ion energy equation This is best accom- alpha-particle temperatures, respectively, both assumed to be plished by determining their magnitudes averaged over that isotropic, and Qp and Q, are heat fluxes, calculated in the portion of the high-latitude wind that is poleward of 75 ø respective proton and alpha-particle center of mass frames heliographic latitude This region is chosen because it is far- These expressions can be simplified for application to the highthest away from the equatorial streamer belt and hence least speed solar wind at polar latitudes Here the average orientaperturbed by interaction with the low-speed solar wind and by tion of the magnetic field is nearly radial, so that averages of coronal mass ejections the vectors V, 8Vp, 8Va, Qp, and Q, are colinear and radial If The first- and third-moment integrals of the ion Boltzmann- we assume further that 8Vp/V << 1, then the drift term reduces to Vlasov equations can be cast in the form of mass and energy flux conservation equations that can be expressed as KD = 158V2 p(4n,/np)/(i + 4(N,/Np)) 2, (13) FM = mpnpv(1 + 4No,/Np)A (4) and the enthalpy term reduces to K r = 25kTp[1 + (ra/rp)(na/np)]/[1-3- 4(No]Np)], (14) The first term in (9) can be further broken into three components by writing the center of mass velocity as the sum of three components, V = V0 + 8V + 8Vr (15) where Vo is the average bulk center of mass speed, 8Va is the fluctuating component of V parallel to the interplanetary magnetic field (IMF) B (assumed to be parallel to Vo), and 8Vr is the fluctuating component transverse to B Whereas parallel fluctuations in V correspond to the microstreams identified in

4 14,550 FELDMAN ET AL: ION ENERGY EQUATION FOR HIGH-SPEED SOLAR WIND Table 1 Averages of Energy-Density Terms in the Ion Energy Equation for the High-Speed Solar Wind Term SHSSW, NHSSW, 0<-75 ø, 0> +75 ø, ergs ergs Bulk flow 4950 _ _+ 260 Ion enthalpy _+ 122 Electric potential energy _+ 41 Variance of bulk flow Alfv6n waves 893 _ _+ 28 Gravitational energy Alpha-proton relative flow 221 +_ _+ 119 Proton heat flux 022 +_ Electron heat flux Heliographic distance range, AU SHSSW, south high-speed solar wind; NHSSW, north high-speed solar wind the high-speed solar wind [McComas et al, 1995; Neugebauer et al, 1995], the transverse fluctuations in V correspond to Alfv6n waves [Smith et al, 1995] Under the assumption that both/5 V R and/svr are small compared to V0 and that (/SVR) = (/SVr) = 0, this expansion becomes FM(05V 2) = FM0(05V /5V ) + F,4 (16) where F u0 is equal to F u with V replaced by V0 and F4 is the flux of Alfv6n waves, given by r = (V + Vn)(B2/4,r)(aBr/B) 2 ( 7) In (17), ¾ is the Alfv6n speed, B is the magnitude of the interplanetary magnetic field, and /SBr is the fluctuation amplitude of the transverse component of the magnetic field Each term in the ion energy equation was averaged over all Ulysses hourly averaged data measured poleward of + 75 ø and collected in Table 1 Averages for the south high-speed solar wind (SHSSW) and north high-speed solar wind (NHSSW) are listed separately and ordered according to magnitude Although most of the terms in the left-hand column of Table 1 are self explanatory, some explanation of the inputs used to evaluate a few of these terms is helpful The magnitude of the electrostatic potential at heliocentric distance R relative to that at large distances was estimated using averages of the measured breakpoint energy between core and halo electron populations [eg, Feldman et al, 1975; McComas et al, 1992] The variance of 1-hour averaged center-of-mass bulk ion speed (/SV 2) measured poleward of _+ 75 ø listed fourth from the top is used to approximate the energy contained in the microstreams Radial variation of the Alfv6n wave flux F,was estimated using the fact that ((SBr/B) 2) 030 and is approximately constant in the high-latitude solar wind [Balogh et al, 1995; Smith et al, 1995] The energy contained in the relative streaming between alpha particles and protons was evaluated using 1-hour averages of the magnitude of the component of their vector velocity difference aligned along B An estimate of the maximum value of the proton heat flux was made by choosing the same fraction of the saturation heat flux (ktp/mp) ø' (15NkTp), observed by IMP 8 in the high- speed solar wind at Earth, = 024 [FeMman et al, 1977] Although it does not strictly belong in Table 1, the value of the electron heat flux from Scime et al [1995] is included for reference Inspection of Table i shows that the bulk convection term 05mpVo 2 dominates all the others by more than 2 orders of magnitude Next in magnitude comes the convected ion enthalpy term Although the next most important term appears to be the electrostatic potential, it represents a conservative potential energy (as does the gravitational potential included below) and therefore can exchange energy only with the ion bulk convection They both therefore have very little influence on the internal energy of the HSSW Indeed, they both have very little affect on the HSSW, in general, because their magnitudes are completely negligible compared to that of the bulk convection energy of the wind The next two terms in decreasing order of importance are the energy carried by microstreams and Alfv6n waves Although the microstreams carry slightly more internal energy than do the Alfv6n waves, their dissipation length is uncertain because of the possibility of transverse slippage parallel to the IMF Alfv6n waves may therefore be the more important source of ion heating in interplanetary space [see Tu and Marsch, 1995, and references therein] The relative fraction of internal energy carried by the proton heat flux is completely negligible (as is the heat flux carried by alpha particles, which was not included in Table 1 for that reason) Although the electron heat flux is about 5 times larger, its effect on ion heating is uncertain because the mechanism for its dissipation may lead only to a redistribution of electron internal energy [Gary et al, 1994] 4 Radial Gradient of Density and Temperature The subset of Ulysses data measured poleward of +50 ø latitude was used to determine the radial gradient of the most important terms in the ion energy equation The bulk speed variance term was not included in this study because, as will be shown in the section 5, very few of the bulk speed variations observed in the polar wind can be positively identified as steepened microstreams leading to compressive heating or expansive cooling Three terms remain: ion enthalpy, Alfv6n waves, and relative alpha-proton streaming in the center of mass reference frame Their radial variation in the NHSSW is shown in Figure 2 Also shown for reference is the radial variation of the proton number density Here averages for each of the internal energy terms and their sum were constructed over data binned in 01-AU increments between 15 AU and 32 AU (corresponding to latitudes poleward of +50ø) Inspection confirms the results summarized in Table 1 Enthalpy dominates all internal energy terms in the ion energy equation Alfv6n wave energy is next largest, followed by relative alpha-proton drift energy Radial variations of all terms can be parameterized quite well by power laws, all having correlation coefficients of r = 099 Whereas the ion enthalpy decreases with increasing R as R -q with q , the Alfv6n wave energy (based on hourly averaged magnetic variance data) decreases R- ' 6, and the relative alpha-proton drift energy decreases R- 192 The proton density also decreases a power law, R- 98, with a correlation coefficient of r = 099 The close fit of ion enthalpy to a power law in radial distance encourages the search for a polytrope relation Radial ion enthalpy and proton temperature are plotted versus proton density in Figure 3 As expected, a polytrope relation is confirmed, yielding an index of q for the ion enthalpy and 051 for the proton temperature, which corresponds to 3'* = 149 and 151, respectively

5 o FELDMAN ET AL' ION ENERGY EQUATION FOR HIGH-SPEED SOLAR WIND 14, Ion Enthalpy 850 8oo- F^/NV o dvap Term 10 '12 I 2 3 Heliocentric Distance (AU) Figure 2 Power law fits to the radial gradient of ion enthalpy, Alfv6n wave energy, the energy carried by the relative drift between alpha particles and protons, and the proton number density in the north high-speed solar wind poleward of +50 ø - ensity 01 ' 210 s E {:: 1 o 750- o 0 Q= 81 ' ' I ' ' ' ' ' ' I ' ' ' I ' ' I ' ' """ m= - I "_ '" , _{;',_*,_, Compressive Heating and Expansive Cooling in Microstreams Both SHSSW and NHSSW data subsets were scanned for signatures of compressive heating and expansive cooling near the leading and trailing edges of discrete speed enhancements, respectively When combined, these subsetspanned 372 days Time (Date, 1994) or slightly more than 1 year From this large data set, only six Figure 4 One of six discrete microstream compression and clean events were identified A representative example mea- rarefaction events observed by Ulysses in the high-latitude solar wind The traces from top to bottom correspond to proton bulk speed, proton density, radial proton temperature, and ' ø '--* the helium abundance,,,,,,, I,,, ' ' ' ' 002 Aug/20 Au 22 Aug/24 Aug/27 Aug/2S (D Q E o o I o s- Proton Temperature I 1 0 ' I Proton Density (cm'3) Figure 3 The correlation between ion enthalpy and radial proton temperature, and proton density in the north highspeed solar wind poleward of +50 ø latitude sured on August 27-28, 1994, is shown in Figure 4 From top to bottom are plots of the proton speed, density, radial temperature, and the helium abundance The discrete speedenhancement event is bracketed by solid vertical lines for purposes of identification Note the clean, correlated enhancements in density and temperature associated with an increase in the proton bulk speed A survey of all six events shows that although the density and temperature peaks clearly precede that of the bulk speed in four out of six events (as one might expect if they were caused by compressive heating), they occurred nearly simultaneously with that of the bulk speed in two of the events No variations of the helium abundance were observed during any of these events Scatterplots of temperature versus density for the six events are shown in Figure 5 Power laws fit to the data using the method of linear least squares are superimposed on each plot Correlation coefficients for the fits are reasonably good, 074 < r < 085 All parameters for the fits are collected in Table 2 We note that the average value for the power law index, q = 070 _+ 014, is approximately equal to that expected if the compressions were adiabatic, q = 3/ /3

6 14,552 FELDMAN ET AL' ION ENERGY EQUATION FOR HIGH-SPEED SOLAR WIND el e e e,, ()1'I PWI=I) ejn medwej uo ojd ()1'le!PeU) e JnlmedweL uolojd z i, i, i i,,,,, i i i i i ()1'le!peld) ejntmedwel uolojd (M'le!Pel:l) ejnlmedmej UOlOJd v s o I I I I Z e eee ee e ee ekee ' ()1'le!PeEI) ejnlmedtuel uolmd ( 1'le!PeEI) ejnlejedujel UOlOJ d

7 , FELDMAN ET AL: ION ENERGY EQUATION FOR HIGH-SPEED SOLAR WIND 14,553 Table 2 Selected Microstream Parameters Place Date a q r SHSSW August 27-29, x l0 s NHSSW May 11-13, x l0 s NHSSW August 27-29, x l0 s NHSSW January 10-13, x l0 s NHSSW January 17-18, x l0 s NHSSW January 18-19, x l0 s Average value 214 _ 037 x l0 s _ Tp = anp q and r = correlation coefficient However, careful inspection of all the data in the SHSSW and NHSSW subsets reveals several clear examples where the speed, density, and temperature do not vary in phase These examples are the rule not the exception This fact can be demonstrated by contrasting the correlation between proton temperature and density in the SHSSW (0 < -75 ø, where the radial range of Ulysses travel is sufficiently small that effects of the solar wind expansion presented in section 4 are not dominant) with that in the NHSSW (0 > +50 ø, where the radial gradient in solar wind density varies by more than a factor of 5) Both correlations are shown side by side in Figure 6 The solid lines in both plots were generated by minimizing the chi-square calculated using the perpendicular distance between each data point and the regression lines Whereas the correlation coefficient in the SHSSW is r = 027, that in the NHSSW is r = 083 When the NHSSW data subset is re- Figure 5 (opposite) Correlations between radial proton temperature and proton density for each of the six discrete microstream events observed in the high-latitude solar wind stricted to a radial range equal to that traversed by Ulysses in the SHSSW (0 < -75 ø) subset (not shown here), the correlation coefficient reduces to r = 041 We therefore conclude that uncorrelated variations in temperature and density dominate the microscale physics in the high-latitude solar wind but not the macroscale evolution of the wind This conclusion was also reached by McComas et al [1996] 6 Correlated Variations at the Edge of the Band of Solar Wind Variability Although Ulysses sampled the edge of the equatorial band of solar wind variability 4 times after passage by Jupiter in February 1992, only two of these samples are suitable for our present study The traversal of the equatorial belt during the fast latitude scan near perihelion was so fast that no clean examples of shock- and CME-free compressions and/or rarefactions were sampled We therefore concentrate on the two samples of belt edges observed near aphelion The southern edge was crossed between December 1, 1992, and June 30, 1993, and the northern edge was crossed between July 1, 1996, Log T (105K) = Log N (cm'a) 02 o, _ :- 0-2' z -" - : -: ;, ; - :,,-,,, :,2'½ ' ', : 4r :, -;:: i- ' ' Log T (105K) = Log N (cm'3) ''' I ''' I ''' I ''' I ''' I ', - ' -::';*'3, -01 ' , Log Density (cm-3) Log Density (cm-3) Figure 6 Correlations between radial proton temperature and proton density (right) in the northern high-speed solar wind (NHSSW) poleward of + 50 ø and (left) in the southern high-speed solar wind (SHSSW) poleward of - 75 ø

8 14,554 FELDMAN ET AL' ION ENERGY EQUATION FOR HIGH-SPEED SOLAR WIND pressions and expansions, q = 2/3 These values differ from that inferred from study of the free expansion of the highspeed solar wind at high northerly latitudes, q - 051, seen in Figures 3 and 6 but are close to that observed in microstream compressions and rarefactions in the HSSW, q _ 012, from Table 2 Although we do not know what causes the foregoing difference in polytrope indices, we speculate that it may reflect differences in the intensity of Alfv6n waves between the centers and edges of the polar coronal holes 7 Summary and Discussion 01 _U) The high-speed solar wind observed by Ulysses at latitudes poleward of the equatorial band of solar wind variability were (3 studied to explore whether a polytrope relation can be used to r OOl replace the ion energy equation Three different compression/ expansion processes were examined The first involved the free Nov/19 May/11 Jul/8 radial expansion of the high-latitude wind into the outer heliosphere The second involved the compressions and rarefac- Time (Date, )) tions associated with the evolution of microstreams in the high-latitude wind The last involved the compressions and rarefactions associated with corotating interaction regions that form at the edges of the low-latitude band of variability Study of the internal energy distribution of ions in the highlatitude solar wind shows that convected enthalpy flux dominates that carried in all other forms by at least a factor of 5, similar to results of previous studies We therefore expect the radial variation of proton temperature (which dominates the ion enthalpy term) to be close to adiabatic Indeed, the ion 400 enthalpy at high latitudes between 15 and 32 AU follows a power law with an index of q - 10 Since the density of this flow falls off as a power law with index q = 20, it is no surprise that a polytrope relation provides an excellent representation of the correlation between proton temperature and 01:3 density in this flow, with q (corresponding to 7* = 151) Although only six clean examples of microstream-driven compressions and rarefactions were identified at high southern 001,, (0 < -75 ø) and northern (0 > +50 ø) latitudes, they were Jul/12 S p/8 Nov/5 Jan/1 Feb/28 observed to support a polytrope law with index between 048 and 086 Averaging over six events, we find 7* = 170 _+ 012 Time (Date, ) A side product of this investigation is that the overwhelming majority of discrete speed variations in the high-latitude wind Figure 7 Proton speed and density variations at the (top) cannot be readily interpreted in terms of the steepening of southern and (bottom) northern edges of the equatorial streamer belt microstreams with increasing heliocentric distance (see, eg, discussion of McComas et al [1995] and Neugebauer et al [1995]) Instead, these speed variations are mostly associated with uncorrelated density and temperature variations Perhaps and February 24, 1997 Proton speed and density for both of these periods are shown in Figure 7 The data in these two intervals were filtered to exclude (1) CME events, (2) shocked gas within corotating interaction they are pressure-balanced structures [McComas e! al, 1995] or an ensemble of adjacent, discrete streamers that slip past one another without interacting along the interplanetary magnetic field, which trace the bulk flow streamlines regions, and (3) bulk speeds less than 600 km s - as described A similar study of compressions and rarefactions that occur in section 1 Correlations between radial proton temperature and density for both filtered sets are shown in Figure 8 Here as in Figure 6, the two regression lines were determined by at the high-speed edges of the equatorial band of solar wind variability yielded a similar result Although the scatter in the correlation between temperature and density was noticeably minimizing the chi-square calculated using the perpendicular larger at the southern edge than at the northern edge, both distance to each data point Inspection shows that although both sets show considerable scatter, the correlation between radial temperature and density was much cleaner for the northern edge belt encounter in than it was for the southern edge encounter in Power law indices for both data subsets are close to that expected for adiabatic comdata subsets could be well represented by a power law with indices close to q = 070 and 066, respectively It is interesting to note that both power law fits are also close to the adiabatic value q = 2/3 However, a careful inspection of both data sets shows that the southern edge set has more scatter than does the northern edge set We can only speculate that

9 1, FELDMAN ET AL: ION ENERGY EQUATION FOR HIGH-SPEED SOLAR WIND 14,555 Log T (10SK) = Log N (cm'3) 05 Log T (10SK) = Log N (crn'3) (!) {:), E I ; -1 Cl:l -06,:: 0 _1-08 :, -15 i, i,,, i i,, : : Log Density (cm'3), :,, I,,,, I,,, I Log Density (cm'3) Figure 8 Correlation between the radial proton temperature and proton density within the (left) southern and (right) northern edges of the equatorial streamer belt Both data subsetshown in Figure 7 were filtered to exclude coronal mass ejections and shocked gas, as well as bulk speeds less than 600 km s - the closer proximity of the observations by Ulysses at southern latitudes to solar maximum, when occurrence rates of CMEs are at their maximum, is the cause of this enhanced scatter Although we took care in eliminating data thought to be parts of CMEs, our criteria for identifying CME gas may not be sufficiently complete The foregoing results beg the question as to whether a single polytrope law, T - an q, can be used to represent all our subsets of high-speed solar wind data We approached this problem by choosing that value of 7* for which histograms of the entropy, "a" in each of the four subsets of high-speed solar wind, maximally overlapped These subsets are (1) the southern high-speed solar wind poleward of -75 ø (region II in Figure 1), (2) the northern high-speed solar wind poleward of +50 ø (region I in Figure 1), (3) the northern high-speed solar wind adjacent to the equatorial band of solar wind variability having latitudes between +36 ø and +50 ø (region IV in Figure 1), and (4) the high-speed solar wind within the northern and southern edges of the band of variability at aphelion (regions III and V in Figure 1) Values of the centroids and standard deviations of each of the four entropy histograms are given in Table 3 for 14 < 7* < 165 Standard deviations of the four entropy centroids for each value of 7', given at the bottom of Table 3, are shown in the left-hand plot of Figure 9 A para- bolic fit to the four highest gamma points shows a clearly defined minimum at 7* = 157 Data from all four high-speed data subsets were then summed to produce the combined histogram of entropy using 7* = 157 in the right-hand plot of Figure 9 It is seen to be a simple, narrow-peaked distribution having a centroid of 20 _+ 013 x l0 s K cm 3(ø'57) We therefore conclude that a single polytrope law can be used to replace the ion energy equation in the high-latitude solar wind between R - 15 AU and 48 AU, r = ( x ø-" However, the scatter in the data in the different subsets is sufficiently large that it is also possible that slightly different laws apply better to flow conditions at differing locations and different times in the high-latitude solar wind For example, all cases of short-term compressions and rarefactions induced by stream steepening (both at the leading edges of microstreams and at the leading and trailing edges of corotating interaction regions) appear to be fit best by 7* "' 3' = 5/3, the adiabatic value In contrast, the best fitting value of 7* in the radially expanding, high-latitude wind is about 151 As mentioned in section 1, such a value (which is less than the adiabatic value) requires heat input that is proportional to the temperature change (see (2)) A likely candidate for the carrier of the Table 3 Histograms of Entropy as a Function of Polytrope Index 7* Data Set 7 *= 140 7* = 145 7* * = 155 7* = *= 165 Region I, Figure _ _ _ _ _ _ 034 Region II, Figure _ _ _ _ _ 040 Regions III and V, Figure _ _ _ _ _ Region IV, Figure _ _ _ _ _+ 041 Average centroid 157 _ _ _ _ _ _+ 019 All entries are times l0 s

10 14,556 FELDMAN ET AL: ION ENERGY EQUATION FOR HIGH-SPEED SOLAR WIND 016 t I I I I I I t i 004/ I I I I I' I -200l 006 t I 10 s 2 10 s 3 10 s 4 10 s Gamma Entropy Figure 9 (left) Standard deviations of centroids of histograms of the proton entropy (assuming 140 < /* < 165) are shown for four different regions of the high-speed solar wind: (1) the southern high-speed solar wind poleward of -75 ø, (2) the northern high-speed solar wind measured between +36 ø and +50 ø heliographic latitude, (3) the northern high-speed solar wind poleward of +50 ø, and (4) the northern edge of the equatorial band of solar wind variability (right) Data from all four subsets are summed to form a single histogram of entropy (G = 157) required heat flux is Alfv6n waves because they are the next most important term in the ion energy equation (see Table 1 and also Tu and Marsch [1995]) If true, then inspection of Figure 2 shows that the waves that carry this energy flux must have periods longer than about a few thousand seconds (1- hour magnetic field variance data from Balogh et al [1995] and Smith et al [1995] were used in constructing Figure 2) This conclusion is consistent with previous analyses of the evolution of Alfv6n waves with increasing radial distance, which shows that the energy comes from the frequency point in the power density spectrum near the interface between an f- and f-5/3 dependence, where f is the observed wave frequency [see, eg, Tu and Marsch, 1995, and references therein] This point was shown to occur at periods that are longer than 1 hour in the Ulysses data in the high-latitude solar wind [Goldstein et al, 1995; Horbury et al, 1996] It is interesting to note that the best average value of ion entropy, (20 x 10s), between 15 and 48 AU derived from Ulysses data is, within uncertainties, equal to that derived from Helios I data, (213_+ 026 l0 s) [Totten et al, 1995] The fact that a single polytrope law appears to describe flow conditions on all streamlines in the high-speed solar wind between 03 and 49 AU and at a variety of latitudes implies that the plasma state at some distance in the outer corona within coronal holes is maximally randomized by some process or processes Otherwise, it would be difficult to understand why a single entropy parameter seems to characterize the flow state on all field lines that thread through the polar coronal holes Acknowledgments We wish to thank S P Gary for many useful conversations regarding wave-particle processes in the solar wind Work at Los Alamos was performed under the auspices of the US Department of Energy with financial support from NASA The Editor thanks Madhulika Guhathakurta and Shadia Habbal for their assistance in evaluating this paper References Balogh, A, E J Smith, B T Tsurutani, D J Southwood, R J Forsyth, and T S Horbury, The heliospheric magnetic field over the south polar region of the Sun, Science, 268, , 1995 Bame, S J, D J McComas, B L Barraclough, J L Phillips, K J Sofaly, J C Chavez, B E Goldstein, and R K Sakurai, The Ulysses solar wind plasma experiment, Astron Astrophys Suppl Set, 92, , 1992 Chandrasekhar, S, An Introduction to the Study of Stellar Structure, Dover, Mineola, NY, 1939 Feldman, W C, J R Asbridge, S J Bame, MD Montgomery, and S P Gary, Solar wind electrons, J Geophys Res, 80, , 1975 Feldman, W C, J R Asbridge, S J Bame, and J T Gosling, Plasma and magnetic fields from the Sun, in The Solar Output and its Variation, edited by O R White, pp , Colorado Assoc Univ Press, Boulder, Colo, 1977 Feldman, W C, J R Asbridge, S J Bame, J T Gosling, and D S Lemons, Electron heating within interaction zones of simple high- speed solar wind streams, J Geophys Res, 83, , 1978 Feldman, W C, J R Asbridge, S J Bame, J T Gosling, and D S Lemons, The core electron temperature profile between 05 and 10 AU in the steady state high-speed solar wind, J Geophys Res, 84, , 1979 Feldman, W C, J L Phillips, B L Barraclough, and C M Hammond, Ulysses observations of the solar wind out of the ecliptic plane, in Solar and Astrophysical Magnetohydrodynamic Flows, edited by K C Tsinganos, pp , Kluwer Acad, Norwell, Mass, 1996 Gary, S P, E E Scime, J L Phillips, and W C Feldman, The whistler heat flux instability: Threshold conditions in the solar wind, J Geophys Res, 99, 23,391-23,399, 1994 Gazis, P R, Observations of plasma bulk parameters and the energy balance of the solar wind between 1 and 10 AU, J Geophys Res, 89, , 1984 Gazis, P R, and A J Lazarus, Voyager observations of solar wind proton temperature: 1-10 AU, Geophys Res Lett, 9, , 1982 Goldstein, B E, E J Smith, A Balogh, T S Horbury, M L Goldstein, and D A Roberts, Properties of magnetohydrodynamic turbulence in the solar wind as observed by Ulysses at high heliographic latitudes, Geophys Res Lett, 22, , 1995

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