JournalofGeophysicalResearch: SpacePhysics

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1 JournalofGeophysicalResearch: SpacePhysics RESEARCH ARTICLE Key Points: Solar wind proton radial component temperature change slope is flatter than 4/3 Proton heating inconclusive for high normalized cross-helicity conditions A turbulent energy cascade is consistent with most energization of protons Correspondence to: B. J. Vasquez, bernie.vasquez@unh.edu Citation: Lamarche, L. J., B. J. Vasquez, and C. W. Smith (2014), Proton temperature change with heliocentric distance from 0.3 to 1 AU according to relative temperatures, J. Geophys. Res. Space Physics, 119, , doi:. Received 9 OCT 2013 Accepted 20 APR 2014 Accepted article online 25 APR 2014 Published online 16 MAY 2014 Proton temperature change with heliocentric distance from 0.3 to 1 AU according to relative temperatures Leslie J. Lamarche 1,2, Bernard J. Vasquez 1, and Charles W. Smith 1 1 Institute for the Study of Earth, Oceans, and Space and the Department of Physics, University of New Hampshire, Durham, New Hampshire, USA, 2 Department of Physics, University of Alaska, Fairbanks, Alaska, USA Abstract Helios spacecraft data excluding shocks, ejecta, and low-cadence plasma intervals are averaged into hour-long intervals and binned by heliocentric distance. In each distance bin, relative classes of fluctuation-normalized cross helicity and total energy are made with a further refinement of each of these classes according to the relative proton radial component of temperature. The relative classes of temperature itself are also examined. All temperatures in each class are fitted by a power law as a function of heliocentric distance to determine the power law index. The difference between this index and the adiabatic index for isotropic plasma can be the first-order indicator of heat addition to the plasma. Relative total energy has temperature indices and behaviors that can be consistent with heat addition from a turbulent energy cascade. Relative cross helicity also shows indices that can support heat addition, but the results are inconclusive on heat addition, especially at high cross helicity. A detailed knowledge of the thermal anisotropy, at least, is required in the case of high cross helicity. 1. Introduction Assuming a steady state, the trace of the gyrophase average pressure equation can be evaluated to find if there is a remainder that can be assigned to sources of thermal energy. These sources can include a turbulent energy cascade or exchange between differential streaming energy and thermal energy. This evaluation can be done independently for each species. The protons give the strongest indications of heat addition. Their relative concentration is highest among ions. Proton temperature correlates strongly with fluctuating magnetic field intensity [e.g., Grappin et al., 1990; Smith et al., 2006] and wind speed [e.g., Lopez and Freeman, 1986; Elliott et al., 2012], and this is consistent with the dissipation of a forward turbulent energy cascade that transports energy from larger to smaller scales. Third-moment studies find that the cascade rate is generally in agreement with the proton heating rate [e.g., Stawarz et al., 2009, 2011; Osman et al., 2011; Coburn et al., 2012]. Electrons have a similar relative concentration, but their pressure equation is harder to evaluate accurately to find heat addition because electrons have a large heat conduction flux. Electron temperatures do not vary even with wind speed [Newbury et al., 1998] and so do not show a clear dependence on turbulent intensities. Evaluations of the electron pressure equation find little indication for heat addition in slow wind and conflicting results in fast winds [e.g., Pilipp et al., 1990; Scime et al., 2001; Cranmer et al., 2009]. In evaluating the pressure equation, a key quantity for measurement is the total temperature T =(T + 2T ) 3, where T is the parallel component of temperature along the magnetic field and T is the perpendicular component. Yet the total temperature is often not publicly available since that involves considerable effort to determine T and T. Instead, the radial component of temperature T pr is most available and usually a very well determined spacecraft measurement. This temperature component is often used in heating studies, and its relation to total temperature can be assessed on a case by case basis. One typically starts at lowest order of the pressure equation where the thermal anisotropy and bulk flow acceleration are neglected. The total temperature can be replaced by the radial component at this order. In a time steady and spherically symmetric expanding wind, the temperature cools according to a 4 3 power law with increasing heliocentric distance in the adiabatic limit. Verma et al. [1995] derive an expression that combines in situ heating and adiabatic cooling in a time steady and spherical symmetric solar wind which we rewrite as dt pr dr T pr r = m p (3 2)V SW k B ε, (1) LAMARCHE ET AL American Geophysical Union. All Rights Reserved. 3267

2 Figure 1. Plot of the radial component temperature power law index ξ da in a solar wind with no heat addition as functions of spiral angle ψ 0 and the thermal anisotropy A 0 at 0.3 AU. Curves correspond to ψ 0 at 20 (solid line), 25 (dotted line), 30 (dashed line), and 40 (dash three dotted line). where ε is a heating rate per unit mass, r is the heliocentric distance, V SW is the solar wind speed, m p is the proton mass, and k B is the Boltzmann constant. Using ε the heating rate is expressed in a way that is most allied with a turbulent cascade rate. With T pr = T pr,0 (r 0 r) ξ, (1) can be solved for ε so that ( ) 4 3 ε = 3 ξ V SW k B T pr. (2) 2 r m p Vasquez et al. [2007, 2010] give considerable detail on how well this approximation works and what kind of corrections can be expected. Heating as a function of wind speed has been a major focus of research in order to characterize slow and fast wind conditions. To do this the temperature power law index was assessed in wind speed bins. Schwenn [1983] obtained these measurements using Helios spacecraft in near radial alignment. Wind speed was binned in constant increments of 100 km/s that were the same at all distances considered. The results had indices near 4 3 in slow winds with ξ = 1.3, and so possibly no heating within uncertainties, but far removed in fast winds with index near 0.7. Later, Arya and Freeman [1991] found that the slowest winds were significantly accelerated from 0.3 to 1 AU. This was shown using relative binning of the distribution of speed within each distance bin. The lowest relative speed class increased in speed toward 1 AU. Totten et al. [1995] applied this relative classification of speed to find temperature power law indices. They sought a determination of the amount of heating due to ambient turbulence and used a visual inspection of all data to exclude stream interaction regions. The results showed that the power indices are far removed from 4 3 even for slow winds and were generally in the range of 0.9 ± 0.1. The heating rate is then mainly proportional V SW T pr. Vasquez et al. [2007] showed that the power law index for total temperature was actually closer to 0.75 ± 0.1 as deduced from previous works which increases the heating rate by a factor of Hellinger et al. [2013] have recently completed a large statistical study of Helios spacecraft obtained plasma distribution function-derived moments. They provide power law fits as a function of distance for T and T in the forms T,0 r ξ and T,0 r ξ, respectively. An effective power index ξeff for total temperature can be obtained from their tabulated results for wind speed bins. Letting T r = ξ eff T r, then the value of ξ eff is given by ξ eff = ξ T,0 r ξ + 2ξ T,0 r ξ. (3) T,0 r ξ + 2T,0 r ξ Figure 2. Histogram of temperatures in various distance bins in units of Kelvin. There are 4633, 1625, 2401, and 5012 hour intervals sampled in the bins from left to right. LAMARCHE ET AL American Geophysical Union. All Rights Reserved. 3268

3 Table 1. Temperature Class Upper Limits in Units of Kelvin for Each Distance Bin a Distance 5% 15% 25% 35% 45% 55% 65% 75% 85% <0.4 69, , , , , , , , , ,760 77, , , , , , , , ,240 54,420 73,980 98, , , , , , ,240 43,300 63,140 87, , , , , , ,460 49,040 65,740 84, , , , , , ,760 37,140 50,920 67,400 88, , , , , ,780 36,740 47,600 62,240 77,720 99, , , ,920 a The classes in the upper row are identified by the midpoint of the percentile range. The 95% class has no preset upper limit. The 5% class has no lower limit but the practical one at zero. Using values found in Hellinger et al. [2013] for V SW bins of 300 km/s or greater, (3) has been evaluated between from 0.3 to 1 AU. The value of ξ eff is found within the range between 0.7 and The result deduced in Vasquez et al. [2007] for the power law index of total temperature is in agreement with this range. When the thermal anisotropies are known sufficiently well as they are for solar wind speed bins, then the radial component can be used to assess heat addition. This could also be extended to cases where the binned quantities are correlated with wind speed. There is a difficulty, however, if the anisotropy behavior is not known, and the possibility of a wind without heat addition or very little heat addition could be included in the bins examined. In such cases, the radial component temperature and its departure from 4 3 may not distinguish the role of heat addition. To illustrate this situation which does need to be considered here, we take a definite case with a constant wind speed and a Parker-like spiral magnetic field. In this case, the number density n is given by In the equatorial plane, the magnetic field intensity B varies in accord with n = n 0 r 2 0 r 2. (4) ( ) r 4 B = B 0 cos 2 0 ψ 0 r + r sin2 0 ψ 0, (5) r 2 Figure 3. Plot of average temperature in 10 relative bins and all bins as a function of heliocentric distance. Solid lines follow the average temperatures. Uncertainties of the mean provided by vertical bars on each temperature data point. Fits to average of all data and the 10 relative classes are shown with dashed lines. LAMARCHE ET AL American Geophysical Union. All Rights Reserved. 3269

4 Table 2. Power Law Index ξ and Estimated Uncertainty for T pr Itself Divided Into 10 Classes and for All Classes Combined TClass ξ 5% 0.85 ± % 0.94 ± % 1.00 ± % 1.01 ± % 0.96 ± % 0.89 ± % 0.83 ± % 0.78 ± % 0.75 ± % 0.60 ± 0.06 All 0.80 ± 0.01 where ψ 0 is the spiral angle at r 0. The spiral angle ψ as a function of r obeys tan ψ = tan ψ 0 (r r 0 ). If there is no heat addition, then the double adiabatic plasma would have temperature components with respect to the magnetic field according to and T = T,0 n 2 n 2 0 B 2 0 B 2 (6) B T = T,0. (7) B 0 The radial component of temperature in this case is given by T pr = T cos 2 ψ + T sin 2 ψ. (8) The effect on the power law index of the radial component of temperature ξ da can be estimated by taking the ratio of T pr at r and at r 0. Then ξ da is defined by ξ da = ln(t pr T pr,0 ). (9) ln(r r 0 ) Choosing r = 1AU and using (4) (8), the ratio T pr T pr,0, and so also ξ da, can be shown to be only a function of ψ 0 and the thermal anisotropy A 0 = T,0 T,0. Figure 1 plots ξ da in the range of the typical data to be examined in this paper and for a chosen range of thermal anisotropy. The reference distance is r 0 = 0.3, the range of ψ 0 from 20 to 40, and A 0 from 0.1 to 10. The value of ξ da can depart greatly from 4 3 even though no heat addition is occurring in this case. When A 0 is less than 1, ξ da can be larger than 4 3. When A 0 is greater than 1, ξ da is smaller than 4 3 and can attain values that fall into the observed range of indices. This latter case is the more concerning one. To reach A 0 1, the wind most likely needs additional external heating beyond the near Sun region. If this heating is halted near 0.3 AU for whatever reason, and then the wind receives no additional heating beyond, the radial temperature component power law index would be flatter than 4 3, but this would not correctly correspond to heat addition. In such a case, the behavior of T and T as a function of r would need to be known before a conclusion about heat addition could be made. In the past, relative binning proved important to understanding heat addition as a function of solar wind speed. In this paper we employ relative binning to new quantities. Third-moment studies have generally shown agreement between measured cascade rates and proton heating rates at 1 AU. The cascade concerns the net rate at which total energy per unit mass density E tot is transported from scale to scale. The value of E tot is given by E tot =(δv δv + δb δb) 2, where δv is the fluctuating velocity, and δb is the fluctuating magnetic field in Alfvén units based on the average density. Larger E tot corresponds to greater cascade rates when all else remains fixed. The value of the cascade rate is also affected by the cross-helicity σ c normalized by E tot where σ c is determined from σ c = 2 δv δb (δv) 2 +(δb) 2. (10) Figure 4. Plot of the distance power law index for temperature as a function of the relative classes. The notation is the sum of the enclosed quantity over the interval under consideration. We will only consider the magnitude of σ c in this paper, and so σ c will always range from 0 to 1. With higher σ c the cascade rate is expected to decrease and reach a zero value when σ c = 1 where the nonlinear terms vanish for incompressible MHD. LAMARCHE ET AL American Geophysical Union. All Rights Reserved. 3270

5 Table 3. Heating Rate ε for the AU Distance Bin as a Function of T pr Class and for All Classes Combined a TClass ε 5% 1307 ± 38 15% 2280 ± 60 25% 3270 ± 70 35% 4419 ± 95 45% 5757 ± % 7324 ± % 9815 ± % ± % ± % ± 989 All 8502 ± 105 a Rate is in units of J kg s. In this paper, the power law indices for T pr are found according to relative classes based on fluctuation-normalized cross helicity, fluctuation total energy, and temperature itself. These indices are for the most part significantly smaller than 4 3. Temperature and total energy can correlate with wind speed, so that thermal anisotropy in these cases may behave as it does with wind speed. In these cases, we would expect that the winds are heated. Relative binning for total energy and relative temperature do give a consistent result with a turbulent energy cascade in that the expected heating rate increases with increasing total energy. Trends with relative temperature alone show a power law index that varies nonmonotonically, and possible explanations are discussed. Relative cross-helicity bins have temperature indices that are also well removed from 4 3, even when the cross helicity is high. The thermal anisotropy in these cases is not known definitively. High cross-helicity states, which are of greatest interest, can involve a mixture of slow and fast winds. As such, the results are inconclusive regarding the role of heat addition in these states. The outline of this paper is as follows: Section 2 discusses criteria for data selection to obtain hourly data for analysis. The results of temperature fittings are presented and discussed in section 3. Section 4 summarizes and gives the conclusions. 2. Data Selection Helios 1 and 2 data with merged plasma and magnetic field data having a typical cadence of 40.5 s were obtained from ftp://nssdcftp.gsfc.nasa.gov/spacecraft_data/helios/helios1/merged/he1_40sec and ftp://nssdcftp.gsfc.nasa.gov/spacecraft_data/helios/helios2/merged/he2_40sec. All available data were considered. For Helios 1 the data range from year 1974 day 346 to year 1985 day 247 and Helios 2 year 1976 day 76 to year 1980 day 68. In contrast to the work of Totten et al. [1995], intervals containing stream interaction regions are not specifically removed. Third-moment determinations of the turbulent cascade rate most commonly use the formalism from the works of Politano and Pouquet [1998a, 1998b]. In these works, there can be no net velocity shear so that the application in the solar wind requires averaging over the velocity changes so that the net amount in the data is zero. There is some net heating due to stream interactions (about 5%), but it is not so large as to invalidate an assessment of the turbulent energy cascade contribution to solar wind heating [e.g., Burlaga and Ogilvie, 1973; Stawarz et al., 2011]. Of more importance is the elimination of intervals that contain shocks and solar ejecta structures. Leaving such intervals in the analysis can double the assessed cascade rates [e.g., Stawarz et al., 2009]. In this study, intervals of known shocks and magnetic clouds were excluded. Lists of shocks identified by Helios in the work of Lai et al. [2012] were obtained from the website Figure 5. Histogram of solar wind speed in various distance bins in units of km/s. LAMARCHE ET AL American Geophysical Union. All Rights Reserved. 3271

6 Figure 6. Histogram of proton number density in various distance bins in units of cm 3. SP11. Any times 6 h before and 36 h after the shocks were excluded to remove transients. A list of magnetic clouds was obtained from the dissertation work of Bothmer [1998]. Any times including the clouds were also excluded. Hourly data sets were composed with the restriction that at least 40 simultaneous magnetic field and proton velocity, density, and temperature measurements were available in the hour interval. The hour average values of the magnetic field, velocity, density, and temperature were obtained. About these averages, fluctuation magnetic field and velocity were then found so as to compute the average cross helicity and total energy of the fluctuations. The average heliocentric distance during the interval was also calculated. Publicly available hour data on temperatures and distances were compared to the one generated in this study. The two sets yield similar conclusions about the nature of relative temperature as a function of distance. We now turn to the analysis of the data. 3. Results and Discussion Analysis of Helios spacecraft hourly intervals is made and discussed here. Section 3.1 examines relative temperature alone. Section 3.2 considers relative σ c, and section 3.3 relative E tot Relative Temperature The data are first binned according to heliocentric distance. The innermost bin contains all data within 0.4 AU and covers a little more than 0.1 AU. Farther out, the bins have increments that are always 0.1 AU. The outermost bin ends at 1 AU. This yields seven distance bins. Figure 2 plots histograms of the distribution of T pr in several bins. Cool to hot winds are correlated with smaller to larger fluctuation amplitudes and wind speed. We would expect some discernible trends in the way cool winds evolve with distance as opposed to hotter winds. The approach that will be repeatedly employed here is to bin across the range of temperature into 10 classes that contain 10% of the distribution each. Table 1 lists the upper limits of the ranges as a function of distance. The common relative temperature classes can be joined across the distance bins to determine how their average temperature behaves. Table 4. Parameters for the 5% T pr Class a Distance T pr V SW n σ c E tot ψ a The distance in AU is the average of those in each distance bin. T pr in units of Kelvin, V SW in km/s, n in cm 3, E tot in km 2 s 2, and ψ in degrees. Figure 3 plots the average value of T pr as a function of distance. This figure provides an overview of the employed approach. The relative classes are distinguished by colors given in the legend to the right of the plot. The average of all data is plotted as well by the black solid line. Solid lines adjoin the actual data points where vertical bars correspond to the error of mean for that point. The error of the mean is approximately given by the range of temperature and divided by the square root of the number of LAMARCHE ET AL American Geophysical Union. All Rights Reserved. 3272

7 Table 5. Parameters for the 35% T pr Class Distance T pr V SW n σ c E tot ψ , , , , , , , , , , samples. Dashed lines are the fits by a least squares. In the fit, the average distance of the data in each bin is used and also the average temperature of each class under consideration. The values of ξ from the fit and uncertainty are given in Table 2. The focus of the analysis concerns the behavior of ξ. Figure 4 plots ξ as a function of relative T pr class. The trend is nonmonotonic with ξ maximizing in the intermediate classes. In each class ξ is considerably different from 4 3. If the thermal anisotropy here behaves as it does with solar wind speed, as is expected, then heat addition is occurring in each class. The amount is far more dependent upon the value of V SW T pr than ξ. Table 3 lists the value of ε for the distance bin between 0.9 and 1 AU as a function of T pr class and all combined. Equation (1) is evaluated with the average value of T pr, V SW and r and the value of ξ in Table 2 to calculate ε. The value of ε increases monotonically with increasing T pr class. Values in the other distance bins (not listed) follow the same trend. The behavior of ξ with T pr class is rather surprising. The simplest expectation for ξ would be for a monotonically decreasing function with increasing T pr class. Instead, the coolest winds in Figure 4 show a further departure from 4 3 than do neighboring intermediate classes. There must certainly be unavoidable mixing of plasma temperature states across the temperature boundaries which define the relative classes. For example, a measurement at 0.3 AU might correspond to a plasma state that does not sufficiently obey the fit function results and has either a small or large enough value of ξ with respect to the average value so as to cross into another class before reaching 1 AU. The potential effects of mixing have been studied using synthetic data with a known average law for the behavior of ξ. To examine the observed form closely, the average law was taken to be a piecewise linear interpolation of the results from Table 2. There is no statistical guidance concerning the actual and individual behavior of solar wind parcels. Thereby, the spread of individual ξ about the average has been treated as a Gaussian random variable with a standard deviation of 20%. The analysis of the synthetic data finds that mixing fractions increase with distance and so lessen the accuracy of classification by 1 AU. The fit functions, however, which are derived from all the distance bins are not influenced as greatly. When a nonmonotonic average law for ξ is adopted, it is recovered in the analysis of the synthetic data with a standard deviation of about For the observed ξ, the uncertainty of this quantity is added to the one found from the fit function uncertainty. The uncertainty for ε is based on propagation of error from ξ and from the error of mean for T pr. In regard to how a nonmonotonic profile could arise, Figure 5 plots the histogram for V SW in several distance bins, and Figure 6 for proton number density n. At the lowest speeds, wind speed does show an increase toward 1 AU which also affects the density. This makes the lowest T pr class potentially different than other neighboring classes. For spherical expansion and steady conditions nv SW r 2 is conserved. The effect of increasing V SW with r is then to have n fall faster than r 2. In the solar wind, acceleration for this lowest class could come from following faster winds that also compress the slower wind and contribute some heating due to this. With such interactions, the wind would not obey equations strictly based on spherical symmetry alone. Table 6. Parameters for the 95% T pr Class Distance T pr V SW n σ c E tot ψ , , , , , , , , , , , , , , Table 4 lists average r, T pr, V SW, n, σ c, E tot, and the spiral angle ψ of the background magnetic field with respect to the radial direction in the distance bins for the 5% T pr class. Tables 5 and 6 list these parameters for the 35% and 95% classes, respectively. The 5% class does show the largest amount of net acceleration from 0.3 to 1 AU. LAMARCHE ET AL American Geophysical Union. All Rights Reserved. 3273

8 Figure 7. Histogram of the absolute value of normalized cross helicity in various distance bins. A factor that is applicable to all classes concerns the angle ψ between the direction of the background magnetic field and radial direction of sampling. This angle indicates, on average, the relative contributions of T and T to the value of T pr. An exception exists when ψ = 54 [Vasquez et al., 2007]. At this angle, the total temperature matches the radial component regardless of the thermal anisotropy. For the 5% class, ψ is sufficiently close to 54 at all distances that the total temperature is near T pr, and the value of ξ applies well to both quantities. Higher-percentile classes have ψ significantly smaller than 54 especially near the Sun. In general, the observed T pr is weighted more toward T in the higher classes. Generally, T falls faster than T with distance. The value of ξ for T pr approaches 0.6 at the highest class, but this should be smaller than the value of ξ for total temperature. The total temperature index is likely closer to the wind speed average of Thereby, the rapid fall off of the index toward higher classes is likely due to the sampling of the radial component of temperature with respect to the background magnetic field. The value of ε in Table 3 for the highest T pr class could have a systematic error due to the smaller value of ξ used to evaluate (1) that amounts to as much as a 30% overestimate of the heating rate. From Tables 4 to 6 the average values of σ c and E tot increase with increasing T pr class. This is generally what is expected for the range of cool and slower winds to hotter and faster winds. We now turn to examine these factors Relative Normalized Cross Helicity Figure 7 shows histograms of σ c in several distance bins. Near 0.3 AU the value of σ c is concentrated near 0.8. With increasing distance the distribution of σ c spreads out, and the average value decreases. The large changes in σ c with distance motivate a relative binning approach. The distribution of σ c is divided, here, into five bins with 20% of the distribution in each. Table 7 lists the upper limits of the σ c ranges as a function of distance bin. A further refinement within each σ c is the relative binning of T pr into 10 bins as before. At 1 AU, the highest relative σ c bin has a lower boundary at σ c = 0.8. For each σ c class and each relative T pr class, least squares fits have been made as above. Figure 8 plots the value of ξ as a function of T pr class for the 10% (solid line), 30% (dotted line), 50% (dashed line), 70% (dash dotted line), and 90% (dash three dotted line) σ c classes. Table 8 gives the values of ξ for all Table 7. Normalized Cross-Helicity σ c Class Upper Limits for Each Distance Bin a classes. Mixing across the σ c bounding ranges is also expected. Synthetic data have not been tested in this regard. An uncertainty of 0.03 for relative σ c binning has been added onto the uncertainty of ξ to reflect the likelihood of increased uncertainty in this case. Distance 10% 30% 50% 70% 90% < a The classes in the upper row are identified by the midpoint of the percentile range. In the highest T pr class all curves in Figure 8 converge near ξ = Regardless of σ c class, the highest relative T pr class has ξ far from 4 3. At low relative T pr the curves are widely spread. The 10% σ c class has a fairly constant value of LAMARCHE ET AL American Geophysical Union. All Rights Reserved. 3274

9 Figure 8. Plot of the distance power law index for temperature for relative σ c classes as a function of the relative temperature. The σ c classes are 10% (solid line), 30% (dotted line), 50% (dashed line), 70% (dash dotted line), and 90% (dash three dotted line). ξ with T pr class. The other σ c classes tend to decrease with increasing T pr. The intermediate σ c classes are associated with the largest ξ which nears 1.2. The 90% σ c class does not, however, approach 4 3. Thermal anisotropy and its behavior with distance have not been evaluated as a function of relative cross helicity. It would be expected that there is some trend with wind speed. Lower and intermediate cross helicity states that contribute the bulk of the data probably have sufficient correspondence with anisotropies in wind speed bins to consider heat addition evaluated in accord with (1). Table 9 lists the value of ε for the distance bin between 0.9 and 1 AU as a function of T pr class and all combined in each σ c class. Equation (1) is evaluated with the average value of T pr, V SW, and r and the value of ξ in Table 8 to calculate ε. The uncertainty for ε is computed by the propagation of error from ξ and mean T pr. In each σ c class the value of ε increases monotonically with increasing T pr class. The same trend is seen in other distance bins (not listed). Under the assumptions used to evaluate ε, significant heat addition can be found even when the relative σ c is high. Within a given T pr class, the value of ε tends to increase with increasing σ c. This would not be expected of a turbulent energy cascade if all other conditions were equal. However, the average E tot is typically larger for a higher σ c class and so the results cannot be compared so directly in this manner. The highest cross-helicity class could be associated with the weakest cascade when all else compared is equal. Slow and fast winds can be associated with such states. A minority of samples such as this one class could be at variance with the bulk of measurements. If no heat addition or little heat addition is occurring, the true situation may not be correctly deduced from the value of ξ as discussed in section 1 with respect to Figure 1. In the present analysis, the T pr observations at high cross helicity are then inconclusive regarding heat addition. Measurements of T and T are needed to determine the heat addition. Third-moment measurements have raised the possibility that no heat addition occurs for high cross-helicity states near 1 AU. Smith et al. [2009] and Stawarz et al. [2010] evaluated cascade rates as a function of σ c using third moments determined from ACE spacecraft data near 1 AU. In their works, they found that the cascade was backward on average when σ c 0.8. If this characterized the true situation at 1 AU, the turbulent cascade could only contribute to plasma heating intermittently. Although the average cascade is backward and no turbulent associated heating occurs during sampled instances with a backward cascade, there will be other but less common sampled instances where a forward cascade operates and heating does occur. The average backward cascade would also have to be a localized effect at 1 AU since there is Table 8. Power Law Index ξ and Estimated Uncertainty for Relative σ c Classes Subdivided Into T pr Classes Tclass ξ:10% ξ:30% ξ:50% ξ:70% ξ:90% 5% 0.74 ± ± ± ± ± % 0.69 ± ± ± ± ± % 0.75 ± ± ± ± ± % 0.75 ± ± ± ± ± % 0.78 ± ± ± ± ± % 0.82 ± ± ± ± ± % 0.78 ± ± ± ± ± % 0.78 ± ± ± ± ± % 0.83 ± ± ± ± ± % 0.67 ± ± ± ± ± 0.09 All 0.76 ± ± ± ± ± 0.05 LAMARCHE ET AL American Geophysical Union. All Rights Reserved. 3275

10 Table 9. Heating Rate ε for the AU Distance Bin as a Function of Relative σ c Class and T pr Class a Tclass ε:10% ε:30% ε:50% ε:70% ε:90% 5% 1, 039 ± 53 1, 169 ± 60 1, 449 ± 60 1, 849 ± 93 2, 228 ± % 1, 559 ± 70 2, 063 ± 80 2, 463 ± 90 3, 084 ± 117 3, 794 ± % 2, 054 ± 89 2, 892 ± 110 3, 347 ± 124 4, 452 ± 177 5, 334 ± % 2, 612 ± 113 3, 588 ± 136 4, 529 ± 165 5, 828 ± 232 7, 473 ± % 3, 395 ± 146 4, 516 ± 169 5, 660 ± 212 6, 951 ± , 277 ± % 4, 539 ± 193 5, 827 ± 221 6, 940 ± 263 9, 166 ± , 756 ± % 6, 184 ± 268 7, 215 ± 281 8, 550 ± , 260 ± , 326 ± % 8, 789 ± 381 9, 662 ± , 489 ± , 205 ± , 508 ± % 12, 627 ± , 869 ± , 127 ± , 969 ± , 773 ± % 25, 536 ± , 283 ± 1, , 054 ± 1, , 270 ± 1, , 759 ± 1, 520 All 6, 123 ± 287 6, 948 ± 280 7, 823 ± 293 9, 708 ± , 523 ± 381 a Rate is in units of J kg s. no known small-scale energy source in the inner heliosphere to sustain the backward cascade. Possibly, the third-moment analysis under the high σ c conditions have not determined the true cascade rate [Podesta, 2010; Smith et al., 2010]. Hellinger et al. [2013] and Gogoberidze et al. [2013] have suggested that the effects associated with spherical expansion need to be taken into account when using third-moment equations at high values of σ c. Another concern for the analysis at high cross helicity is relative binning. The relative binning of cross helicity could produce special circumstances farther from the Sun. Individual plasma parcels with σ c can mix across the boundaries of the relative classes. The overall evolution is for the average σ c to decrease with increasing distance. This is theorized to be the overall effect of velocity shear in the solar wind [e.g., Roberts et al., 1987a, 1987b, 1992]. It is possible that the individual parcels experience an increase of σ c that is out of proportion to those that are decreasing. States that have low velocity shear at their location could undergo what is known as dynamical alignment [e.g., Matthaeus et al., 1983, 2008; Grappin et al., 1983; Ting et al., 1986; Stribling and Matthaeus, 1990, 1991; Servidio et al., 2008]. In these cases, σ c can increase toward 1. The fluctuating velocity and magnetic field tend to align with each other as the turbulence evolves and so give the name dynamical alignment. A condensation of such states may occur at the high σ c class that is not obeying Gaussian statistics and so not readily perceived by the methods employed here. The effect of this would be cumulative. Near the Sun, the average high cross-helicity state would be dominated by those associated with heat addition and decreasing σ c with distance. Farther from the Sun, this average behavior of the high σ c class would be overwhelmed with states that evolved from smaller σ c to larger. If these states were to dominate the high σ c class near 1 AU, then the question of their heat addition could not be determined in the present analysis even with a known thermal anisotropy. It may be resolved by considering temperature evolution beyond 1 AU. Figure 9. Histogram of the fluctuation total energy per unit mass density in various distance bins in units of km 2 s 2. LAMARCHE ET AL American Geophysical Union. All Rights Reserved. 3276

11 Table 10. The Value of E tot Class Upper Limits in Units of km 2 s 2 for Each Distance Bin a Distance 10% 30% 50% 70% < a The classes in the upper row are identified by the midpoint of the percentile range. The 90% class has no upper limit Relative Total Energy Figure 9 shows histograms of E tot in several distance bins. The average value of E tot decreases with increasing distance, as is expected for fluctuations in the expanding solar wind. Fluctuation amplitude correlates with temperature [e.g., Grappin et al., 1990] and so also with wind speed. The expectation here is that thermal anisotropy common with wind speed binning extends to this case of total energy binning. The distribution of E tot is divided into five bins of 20% each. Table 10 lists the upper limits of the E tot ranges as a function of distance bin. In each of these bins, relative T pr is subdivided into 10 bins as before. Fits are made for each class. Figure 10 plots the value of ξ as a function of T pr class for the 10% (solid line), 30% (dotted line), 50% (dashed line), 70% (dash dotted line), and 90% (dash three dotted line) E tot classes. Table 11 gives the values of ξ for all classes. As with σ c, an additional uncertainty of 0.03 is included to represent the increased uncertainties associated with binning for E tot. In the highest T pr class the curves in Figure 10 have end points that are well spread out. They are ordered, however, as expected in the turbulence paradigm because the 10% E tot class has the largest value of ξ, and each successive E tot class has a smaller ξ. The average T pr, V SW, and σ c differ here within a factor of 4 or so. Thereby, a significant change in ξ appears to be needed to get a larger rate of heat addition with increasing E tot. The ordering with E tot class breaks down toward the lower T pr classes. In the lowest T pr class, the curve end points are closer together with the exception of the lowest E tot class. Possibly, turbulence is not the only source contributing to heating at the lower T pr classes. Table 12 lists the value of ε for the distance bin between 0.9 and 1 AU as a function of T pr class and all combined in each σ c class. The value ε and its uncertainty are calculated according to the procedure used in Tables 3 and 9. In all E tot classes the value of ε increases with increasing T pr class. Within a T pr class the value of ε increases with increasing E tot class. These trends are seen in the other distance bins (not listed), and they are consistent with the expectations of a turbulent energy cascade providing the heat addition. Figure 10. Plot of the distance power law index for temperature for relative E tot classes as a function of the relative temperature. The E tot classes are 10% (solid line), 30% (dotted line), 50% (dashed line), 70% (dash-dotted line), and 90% (dash three dotted line). 4. Summary and Conclusions Helios 1 and 2 spacecraft plasma and magnetic field data at a typical cadence of 40.5 s have been averaged to determine an hourly data set that describes fluctuation and background parameters. Times near shocks and clouds were excluded as well as hour-long intervals with less than 40 data points. The generated hourly data were then analyzed in terms of relative binning to determine the behavior of the radial component of proton temperature with distance from the Sun. First, relative temperature was itself binned. Fits to the average temperature in each class as a function of distance determined power law indices that range between 0.6 and 1. These indices are far from 4 3. With the expectation that thermal anisotropies here are common to those found in solar wind speed bins, the proton wind is heated, as is expected in general. The indices did, however, vary nonmonotonically with LAMARCHE ET AL American Geophysical Union. All Rights Reserved. 3277

12 Table 11. Power Law Index ξ and Estimated Uncertainty for Relative E tot Classes Subdivided Into T pr Classes Tclass ξ:10% ξ:30% ξ:50% ξ:70% ξ:90% 5% 0.81 ± ± ± ± ± % 0.79 ± ± ± ± ± % 0.92 ± ± ± ± ± % 0.88 ± ± ± ± ± % 0.95 ± ± ± ± ± % 0.94 ± ± ± ± ± % 0.98 ± ± ± ± ± % 1.03 ± ± ± ± ± % 1.02 ± ± ± ± ± % 0.92 ± ± ± ± ± 0.10 All 0.97 ± ± ± ± ± 0.05 increasing temperature class. The index of 1 is reached at intermediate temperature classes. This behavior may be explained by the contribution of parallel temperature to the radial component of temperature toward higher temperature classes and the influence of solar wind acceleration for the lowest temperature class. Second, the absolute value of relative normalized cross helicity was binned and then further subdivided into temperature classes. Fits to temperature as a function of distance were made for each class. The results have indices that differ from 4 3. In particular, the highest cross-helicity class was far removed from 4 3. Heat addition may be occurring in the high cross-helicity states, but the results are inconclusive in that if heating of the wind halted near 0.3 AU at thermal anisotropies near or above unity, the resulting gradient of T pr would not be distinct enough. There are third-moment measurements at 1 AU which show that a backward cascade exists when the cross helicity is high. Results here have not ruled out this possibility. Finally, the relative total energy per unit mass density was binned and then further subdivided into temperature classes. Temperature fits give indices consistent with heat addition with the expectation of thermal anisotropy in accord with solar wind speed. At the lower temperature classes, the indices did not decrease and so depart from 4 3 as one would expect with increasing total energy. This could indicate that heating arises from other sources in addition to a turbulent energy cascade. Toward the higher temperature classes, indices did decrease with total energy and were consistent with a significant turbulent energy cascade. When the heating rate is calculated, its value increases with increasing T pr and E tot classes, and this is consistent with a turbulent energy cascade. Table 12. Heating Rate ε for the AU Distance Bin as a Function of Relative E tot Class and T pr Class a Tclass ε:10% ε:30% ε:50% ε:70% ε:90% 5% 841 ± 41 1, 341 ± 64 2, 033 ± 94 2, 893 ± 135 4, 391 ± % 1, 170 ± 49 2, 066 ± 82 3, 091 ± 121 4, 725 ± 191 7, 650 ± % 1, 579 ± 63 2, 615 ± 103 3, 910 ± 152 6, 114 ± , 432 ± % 1, 890 ± 77 3, 172 ± 122 4, 960 ± 192 7, 512 ± , 141 ± % 2, 275 ± 90 3, 742 ± 144 5, 935 ± 235 9, 465 ± , 005 ± % 2, 790 ± 111 4, 745 ± 187 7, 073 ± , 391 ± , 596 ± % 3, 443 ± 134 5, 766 ± 227 8, 367 ± , 789 ± , 728 ± % 4, 403 ± 169 7, 218 ± , 062 ± , 970 ± , 983 ± 1, % 5, 593 ± 217 9, 430 ± , 102 ± , 223 ± , 588 ± 1, % 8, 540 ± , 006 ± , 420 ± 1, , 675 ± 1, , 771 ± 2, 550 All 3, 128 ± 102 5, 185 ± 153 7, 589 ± , 979 ± , 140 ± 558 a Rate is in units of J kg s. LAMARCHE ET AL American Geophysical Union. All Rights Reserved. 3278

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