Correlation of speed and temperature in the solar wind
W. H. Matthaeus,1 H. A. Elliott,2 and D. J. McComas2
Received 27 January 2006; revised 16 May 2006; accepted 5 July 2006; published 11 October 2006.
[1] We show that the well-known correlation between solar wind speed and temperature
can be understood as a consequence of an elementary symmetry of the solar wind transport equations. Under a reasonably wide range of circumstances, even when including nonadiabatic turbulence effects, the solutions of the transport equations may depend only upon the ratior/Uof distance to solar wind speed. Applied to the temperature equation, the familiar correlation emerges immediately. For a turbulence model of heating, this property is obtained in regions where the Alfve´n speed is much smaller than the flow speed, where pickup ions are negligible, and where the flow is locally a spherical, constant-speed expansion. These fundamental properties, illustrated here using analysis of ACE data, clarify why the correlation is reduced in ensembles that include highly nonspherical effects such as CMEs.
Citation: Matthaeus, W. H., H. A. Elliott, and D. J. McComas (2006), Correlation of speed and temperature in the solar wind, J. Geophys. Res.,111, A10103, doi:10.1029/2006JA011636.
1. Introduction and Background
[2] The existence of a statistical correlation between
large-scale solar wind flow speedUand proton temperature T has been noted by many authors, e.g., Burlaga and Ogilvie [1970], Lopez and Freeman [1986], and others. However, we are unaware that any physical explanation for it has been offered in the literature.
[3] Although this familiar ‘‘TU’’ correlation is neither
ubiquitous nor well understood, it does occur frequently enough and under a wide-enough range of conditions that it can be useful in understanding other solar wind phenomena. Recently, for example, Richardson and Wang [2003, see alsoRichardson and Smith, 2003] used an empirical relation
DT=ADUto relate deviationDTof temperatureTand the deviation DUof solar wind speed Ufrom their respective nearby mean values. In this form, no physical import is given the quantity A that provides the constant of propor-tionality. This relation was used to compute an ad hoc, but effective, improvement to the predicted temperature as function of radiusT(r) obtained from a turbulence transport theory [Smith et al., 2001]. The Richardson relationship can be viewed as a version of theTUrelation that is, in some sense, a local version of a global formulation in which a functional dependenceT(U) emerges. To the extent that this might be assumed, and the mapping is suitably smooth, the differential immediately yieldsdT = (dT(U)/dU)dUas an approximation for small changes d U in U and other parameters fixed. This of course begs the issue of how the functional relation arises in the first place. It is the purpose
of the present note to show how this global functional form arises under a wide variety of circumstances in the solar wind. Not only do the useful differential forms emerge, with physically based proportionality, but the model also points to the key role of boundary conditions and nonuniform expansion in affecting departures from theTUscaling. In particular such departures explain why ICMEs (Interplane-tary Coronal Mass Ejections), well known to depart greatly from spherical expansion rates [Skoug et al., 2000], do not show a typical degree ofT Ucorrelation.
2. Plausible Symmetry of Solar Wind
Temperature and Transport Equations
[4] To understand theTUcorrelation, begin with the
steady state equation governing the proton temperature, in a uniform spherical expansion in the radial coordinate r (standard spherical coordinates), at constant flow speedU. Assuming an ideal gas equation of state, the equation of interest is dT drþ 4 3 T r ¼ 1 3 mp kB Q rð Þ U ; ð1Þ
whereQ(r) is a function that describes addition of internal energy from all sources. (Proton mass mp, Boltzmann constant kB; see Verma et al. [1995] and Williams et al. [1995]). Note that in this simple treatment we ignore the possible differences in the temperatures of plasma compo-nents (ions, electrons, etc.; seeMarsch [1991]). When this ‘‘heating function’’ (energy per unit time per unit volume) vanishes, the expansion is adiabatic and the temperature falls off asT(r)/r4/3. It is, however, well known that with some exceptions, the solar wind expansion is nonadiabatic, and therefore internal energy is deposited in the plasma, and the heating function is nonzero [Burlaga and Ogilvie, 1970; Lopez and Freeman, 1986]. Although a particular
formula-Here for Full Article
1Bartol Research Institute and the Department of Physics and
Astronomy, University of Delaware, Newark, Delaware, USA.
2Space Sciences Division, Southwest Research Institute, San Antonio,
Texas, USA.
Copyright 2006 by the American Geophysical Union. 0148-0227/06/2006JA011636$09.00
tion of the heat function will be discussed below, the mail results discussed below are, remarkably, obtained equally well whether Q = 0, or if Q 6¼ 0, it possesses a certain plausible symmetry, as we now explain.
[5] Multiplying equation (1) byU, one sees easily that the
terms on the left-hand side depend only upon z = r/U. Consequently whenQ= 0 the solution is a function only of r/U, and moreover, this property is preserved ifQ= Q(z). This symmetry implies, for a specified boundary condition at r = 0, that the solution for the temperature is also a function of z, i.e., T = T(r/U). This has immediate con-sequences for solutions that differ in their value ofU. (Note that we are examining the properties of a family of constant speed flows, differing in speedUand possibly in boundary conditions; see section 5. We are not examining cases with U=U(r).)
[6] Suppose a solutionT(r;U) is known. For the moment
let us assume that the boundary condition is fixed and specified atr= 0. (We revisit this issue later in section 5.) The associated solution with speed aU, for any positive number a, is related to the original solution by
T rð;aUÞ ¼T rð =a;UÞ: ð2Þ If we assume differentiability with respect to both the coordinaterand the parameterU, a differential form of this relation is @T @U¼ r U @T @r: ð3Þ
[7] One can see from equation (2) that a positive
corre-lation is expected between changes in speed and changes in temperature, whenever the standard profile T(r; U) is a decreasing function of r for fixed U. Relative to this reference state, suppose that the speed is increased by a factoratoaU. Then the new temperature at positionris the value that had been at positionr/aat the original speedU. We may wish to relate small changes in temperatureDTthat are associated with small changes in speedDUat fixedr. We can approximate as DT/DU @T(r)/@U, which becomes exact asDU!0. From equation (3) we conclude that
DT ¼ r U
@T rð Þ
@r DU: ð4Þ
[8] Other relations related to these may also be useful.
For example, if the standard temperature solution is a power lawT(r) =T(r0)(r0/r)
a
[seeMarsch, 1991], again assuming the r U symmetry, the variations of T are related to variations ofUas
DT¼aT
UDU: ð5Þ
It is interesting to note that a global (not differential) application of the above reasoning, whenTraleads to the expectation thatT/Ua, which of course will reduce to equation (5) for smallDU. Relations such as equations (4) and (5) apply immediately when the expansion is adiabatic,
i.e.,Q= 0 in equation (1). We will explore below, in section 5, an example of a nonzero heat function that preserves the required symmetry.
3. Spherical Expansion, Observations, and the
T U Correlation
[9] We have shown that a T U correlation is readily
obtained when the temperature and heat function depend only uponr/U. It is likely that this correlation is routinely encountered in interplanetary data because this symmetry is not difficult to obtain approximately, in a constant speed, spherically expanding solar wind. There are, how-ever, obvious exceptions to this approximation, such as in the inner heliosphere, whereUis not uniform andU(r) has strong radial variation, or near interplanetary shocks or stream interaction regions. However, for a pure spherical expansion, various terms fall into place, such asU r ! U@/@r and r U ! 2U/r, and the required symmetry emerges. It is possible that other symmetries may also give rise to a relationship between T and U, but for now we adopt the working hypothesis that spherical expansion is the main reason a T U correlation has so often been reported.
[10] Observations support the viewpoint that the solar
wind expansion is, on average, very nearly spherical. In particular, measurement of proton density from the inner to the outer heliosphere indicate the proton density has ap-proximately an r2 dependence, which is a well-known consequence of a constant speed spherical expansion. He-lios 1 and 2 measurements of the inner heHe-liosphere indicate the proton density has an r2.1 dependence [Schwenn, 1990], and Ulysses observations from 1 – 5 AU indicate the proton density is well represented by anr2dependence [Phillips et al., 1995;Goldstein et al., 1996]. Outer helio-sphere Voyager 2 observations indicate the density has an r1.93dependence [Richardson et al., 1996].
[11] Exploration of the relationship betweenTandUhas
continued to be of interest in empirical studies [Elliott et al., 2005]. To illustrate here the relationship between the proton temperature and the solar wind speed in the bulk solar wind and in ICMEs, we analyze hourly average Solar Wind Electron Proton Alpha Monitor (SWEPAM) [McComas et al., 2000] level 2 data from day 023 of 1998 to day 138 of 2005 using the same techniques asElliott et al.[2005]. See Figure 1. The data are first sorted into three categories: ‘‘likely’’ ICMEs, ‘‘possible’’ ICMEs, and non-ICME. The ‘‘likely’’ ICME data satisfy three criteria: high alpha to proton density ratio (na/np> 0.08), high O7+to O6+density ratio (n(O7+)/n(O6+) > 1), and low proton beta (b < 0.1). Possible ICMEs satisfies any one of those criteria or are taken within a day of satisfying a criterion. The non-ICME data are the data remaining after removing all ‘‘possible’’ ICMEs. The proton speed and temperature are positively correlated in the non-ICME bulk solar wind (Figure 1, left); however, this correlation does not exist in the likely ICME data, and the higher-speed likely ICME data is much cooler than expected.
[12] In order to look for the effects of dynamic evolution,
we subdivided the non-ICME data into three categories: compression, rarefactions, and ‘‘other.’’ To do this, we examine the slope of the 2-day running average proton
speed time profile. Positive slopes are labeled as compres-sions and negative slopes are labeled as rarefactions. How-ever, if the magnitude of the slope is <2.2104km s2, it is classified as ‘‘other.’’ The compressions are systemati-cally hotter than the rarefactions since such positive speed-time slopes occur when fast wind runs into slow wind. See Figure 1 (right). Therefore compressions and rarefactions have different speed-temperature relationships with the compressions having a larger temperature-speed slope (685 K km1 s) than the rarefactions (483 K km1 s). The more dramatic departures from the T U correla-tion observed for ICMEs may reflect strong expansion close to the Sun, reducing the temperatures. To the extent that ICME evolution departs greatly from the overall symmetric spherical expansion, the conditions outlined above that give rise to the T U correlation would be absent.
[13] As another illustration, let us examine the relation
equation (5) in the context of published analyses of Voyager plasma data in the range 1 AU to 10 AU [e.g., Richardson et al., 1995]. Suppose we use an average solar wind speed of 440 km/s and consider that DU is the deviation from this value. LetDT be the departure from a reference value of temperature, which we take to be T = 35,000 K, typical of conditions around 5 AU. Further-more, let us assume that the temperature in the region of interest is varying radially as T1/ra, with a= 1, which is less steep than the ideal gas adiabatic law, but not as shallow as the a 1/2 found by Richardson when data beyond 10 AU, affected presumably by pickup ion heat-ing [Williams et al., 1995], are included. Assembling these values into equation (5) we find the numerical relation
DT= 79.5 [K s km1] * DUexpected to be valid near 5 AU. This is almost precisely the same as the relation
DT = (V 440 km/s) * 80[K s km1] proposed by Richardson and Wang[2003] as an empirical correction to solar wind transport solutions [Matthaeus et al., 1999; Smith et al., 2001]. This correction was found to provide a remarkably detailed agreement with Voyager radial
profiles. The present development provides a plausible explanation of the efficacy of this procedure.
4. Turbulence Formulation of the Heat Function
[14] So far we have assumed that the heat function Q
satisfies the property that Q = Q(r/U). A special case is adiabatic expansion, for whichQ= 0. For the nonadiabatic case Q6¼0, it remains to establish that Qmight maintain the required symmetry for a suitably wide range of conditions. For this it is necessary to adopt a particular model for in situ solar wind heating. One possibility is turbulent dissipation [Coleman, 1968; Matthaeus et al., 1999], or in a related view, the damping of Alfve´n waves. A similar analysis can be carried out with other heating models with regards to their impact on the T U correlation.
[15] In the case of a strong low-frequency MHD cascade,
a simple approximate expression for deposition of heat due to turbulence is
Q¼Afþð Þsc Z3
l; ð6Þ
where A is a constant, f+(sc) is a dimensionless function of the normalized cross helicitysc, to be defined below,Z is the turbulence amplitude (in units of speed) and lis a similarity (correlation) length. (For details, see Matthaeus et al. [1999, 2004]). A very similar heat function is found for turbulent heating by shocks [MacLow, 1999] so that the above formulation may be applicable beyond weakly compressible turbulence. In the equation (6), there is no explicit appearance of either the radial coordinate ror the wind speed U in this expression. Consequently, the heat function Q will be a function of r/U if and only in the underlying dynamical variables, Z, l, and sc, are functions of this ratio and not of r or U in any other combination.
Figure 1. (left) Temperature-speed scatterplot of all ACE level 2 non-ICME data (black) and the ‘‘likely’’ ICME data from day 023 of 1998 to day 138 of 2005. (right) The compressions (orange) and rarefactions (blue) in the non-ICME data. Here the data have been divided into 25 km s1speed bins and then fit with a line. The vertical bars indicate the standard deviation.
[16] Therefore one examines the structure of the steady
state model equations for turbulence energy,Z2
UdZ 2 dr ¼ A U rZ 2þE_ PI afþ l Z 3; ð7Þ similarity scale,l, U@l @r ¼B U r l _ lPIþbfþð ÞZsc ; ð8Þ
and cross helicity,sc,
Udsc dr ¼C U r sc _ EPI Z2scþaf 0 s c ð ÞZ l: ð9Þ
The quantities A, B, and C are constants related to shear, expansion, and turbulence geometry,aandb are constants related to decay of turbulence, and f0(sc) and f+(sc) are functions of the dimensionless cross helicity sc. The influence of pickup ions is through the the terms E_PI and
_
lPI. Again, for details, see [Zank et al., 1996;Matthaeus et al., 1999;Smith et al., 2001;Matthaeus et al., 2004]. Together, these three equations determine the heating rate through equation (6), which then acts as a source of internal energy on the right-hand side of the temperature equation (1).
[17] Examination quickly reveals that none of the terms
included in the above model, with the possible exception of the pickup-ion-related terms, will violate therUsymmetry required to maintain theTUcorrelation. Therefore where pickup ion effects are negligible, that is, within about 10 AU of the Sun [Williams et al., 1995], the solutions forZ2,l, and
scare approximately functions only ofr/U. The above form of the model holds only when the Alfve´n speed is much smaller than the flow speed U [Zank et al., 1996], so the conclusions, as stated, apply only to the approximate range of distances 0.5 AU <r< 10 AU. In these circumstances,Qalso will be a function only ofr/U, and thetUcorrelation will be maintained in the associated nonadiabatic expansion.
[18] For the radial variation of the pickup ion terms, see
Smith et al. [2001] (especially Figure 5 and associated discussion) andIsenberg et al.[2003]. The situation is less clear in the outer heliosphere, as approximate constancy inr of the pickup-ion energy supply, outside of 10 AU, (which would restore the r/U symmetry), can be suggested from theory but has not been demonstrated in general.
[19] In conclusion, up to the level of approximation
represented by the turbulence transport model in equations (7) – (9), one expects that the heat function will permit maintenance of aT Ucorrelation in the large-scale solar wind, as described in sections 2 and 3.
5. High-Speed and Low-Speed Wind and the Role
of Boundary Conditions
[20] Fast and slow solar wind emerge from different
regions of the Sun, and may be accelerated at different altitudes, and heated by different processes. This departs
greatly from the idealized circumstances assumed above. It is, however, useful to contrast the fast and slow wind in the very simplified framework proposed above by examining boundary conditions and transport effects that influence the TUcorrelation.
[21] So far we have assumed thatT(r= 0) is specified and
fixed for all solutions with varying U. This maintains the symmetryT(r,U) =T(ar,aU) for any constanta. In more general circumstances we might specify the boundary conditionT(r0,U) at some positionr0> 0. This can affect
theTUcorrelation.
[22] One possibility is that boundary conditions
them-selves obey the same symmetry as the transport equations. Then, for example, for solutions with speedsUandU0=aU, the boundary conditions would be chosen as T(r0, U) =
T(ar0,aU) =T(r00,U0). In this case, the boundary conditions
are applied at different positionsr0andr00=ar0. Clearly, if
the boundary conditions breathe radially with varying speed in the same way that the transport equations do, then the conclusions arrived at above will be maintained. This will not however be the most general case, and one would expect that arbitrary variations of the boundary temperature at points other than the origin (r0 = 0) will violate the
symmetry that leads toT(r,U) = T^(r/U).
[23] To make this point explicitly, let us continue to treat
the problem as spherically symmetric, with solar wind speed a parameter, but now include boundary effects. If temper-atures T(r, U2) and T(r, U1) are measured at different
wind speeds, but at the same heliocentric radial position, we might wish to account for their difference DT. In general,
T rð;UÞ ¼T rð 0;UÞ þ
Z r r0
dr0@T rð 0;UÞ=@r0; ð10Þ
so we can separate the contributions from transport effects from variations due to the boundary conditionsT(r0,U) by
writingDT= DTtransport+ DTboundary where DTboundary = T(r0, U2) T(r0, U1). An estimate for DTtransport, the
change in the integral in equation (10) due to the speed change, is available, for example, from equations (4) or (5) under the assumption that therU symmetry is present. Clearly, the conclusions relating to the T Ucorrelation emerge only when either DTboundary is negligible or it admits the same symmetries as DTtransport. On the face of it, this seems unlikely, and the point of the present section is to emphasize this potential weakness in the global application of the the arguments given in the previous sections.
[24] In spite of this discouraging note, let us briefly recall
the temperature and speed differences between high-speed high-latitude wind, as observed for example by Ulysses and the low-speed low-latitude wind. At 1 AU the temperature of high-latitude wind is generally nearTf= 2 – 3105K and the speedUf750 km/s. The low-latitude slow wind,Us 350 km/s is cooler on average at 1 AU withTs 7 – 8 104K. Suppose we assume that both types of wind are described by the same class of solutions, with T(r, U) = T(ar,aU), and in addition, thatT(r,U)1/ra, the boundary conditions assumed for the moment to permit needed symmetry. We can then map the expanded position ar of
the faster (a> 1) wind back tor and deduce immediately thatT(r,U) =aaT(r,aU). Ifa= 1 and noting thata=Uf/ Us, then T rð ;UsÞ Us ¼T r;Uf Uf : ð11Þ
These assumptions imply that the temperature ratio equals the speed ratio, Tf/Ts = Uf/Us and that formally the temperature is proportional to the speed. This is a pure, or extreme, form of T= U correlation, and it seems unlikely that the solar wind would precisely satisfy all of the approximations leading to equation (11). In fact, the numbers also do not fully support this conclusion: The speed ratio is Uf/Us = 750/350 = 2.14 while in the most favorable interpretation,Tf/Ts= 2105/8104= 2.5. The ratios are, however, rather close to one another. Alterna-tively, and perhaps more plausibly, we can use the above reasoning to determine the effective radial index a. Then, sinceTs=aaTfanda=Uf/Us, we deduce that (Uf/Us)a= Tf/Ts, and therefore the implied average radial power law index is a= log(Tf/Ts)/log(Uf/Us) log(2.5)/log(2.14) 1.2. This corresponds to an average polytropic index g = (a + 2)/2 = 1.6, corresponding to a nonadiabatic expansion. These are curiously not far from typical values quoted for speeds around 300 – 400 km/s [Freeman, 1988; Marsch, 1991] although faster wind usually has a shallower index when the fit includes mapping back to the coronal base.
[25] There are many open issues regarding the base
temper-atures and effective source surfaces of the fast and slow winds. Any treatment like the present one must be a vast oversimplification. However, the fact that the temperatures lie in approximate proportion to the wind speeds suggests, among other things, that the observed temperatures vary inversely with the convection time from the source region, provided that the source radius is much less than 1 AU. If the boundary conditions are not consistent with the assumed symmetry, then it is coincidental that this ratio is nearly restored by transport to the observation point at Ulysses.
6. Discussion
[26] We have discussed a simple symmetry of the
spherically expanding steady solar wind that can give rise to a wind speed-temperature (T U) correlation. This has been frequently, but not ubiquitously, observed, but as far as we know, not explained. Whenever spherical symmetry is approximately valid, the solutions for the heat function and the temperature can have the symmetry that only depend upon the ratio r/U, which is a transit time to the position of interest in the heliosphere from its effective origin in the corona. Evidently these conditions can be widely met, for example, both by adiabatic expansion as well as, in suitable approximations, by a driven dissipative turbulently heated (nonadiabatic) solar wind. Under these conditions the T U correlation can emerge either globally or locally, as a consequence of the breathing of a solar wind solution radially outward with increasing speed. For a simple solution of this type,
plasma parcels with equal convective age will have the same temperature. Boundary conditions complicate the global interpretation, but the observed average differences in temperature in high-speed and low-speed solar wind point to a substantial difference in the temperature expected in these two types of wind at a radially fixed source surface. Conversely, the implied differences of temperature at such a boundary are in approximate proportion to the speed. We conclude that using basic symmetry properties of the solar wind expansion, the empirical basis of the T U correlation might be understood in fairly general terms.
[27] Acknowledgments. This work was supported by NASA grants
from the RSSW1AU program (NASA NNG04GA54G) and the Cluster GI program (NNG05GG83G).
[28] Amitava Bhattacharjee thanks the reviewers for their assistance in evaluating this paper.
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H. A. Elliott and D. J. McComas, Southwest Research Institute, 6220 Culebra Road, P. O. Drawer 28510, San Antonio, TX 78228-0510, USA. ([email protected]; [email protected])
W. H. Matthaeus, Bartol Research Institute and Department of Physics and Astronomy, University of Delaware, Newark, DE 19716, USA. ([email protected])