We employ a turbulence transport theory to the radial evolution of the solar wind at both low and high latitudes. The theory includes cross helicity, magnetohydrodynamic (MHD) turbulence, and driving by shear and pickup ions. The radial decrease of cross helicity, observed in both low and high latitudes, can be accounted for by including sufficient shear driving to overcome the tendency of MHD turbulence to produce Alfvénic states. The shear driving is weaker at high latitudes leading to a slower evolution. Model results are compared with observations from Ulysses and Voyager

Bavassano, B.

Bieber, J.W.

Breech, Ben

Matthaeus, William H.

Minnie, J.

Oughton, Sean

Parhi, S.

English

Name not available

Radial evolution of cross helicity at low and high latitudes in the solar wind

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RADIAL EVOLUTION OF CROSS HELICITY AT LOW AND HIGH LATITUDES IN THE SOLAR WINDB. Breech1, W. H. Matthaeus1, J. Minnie2, S. Oughton3, S. Parhi1, J. W. Bieber1, and B. Bavassano41Bartol Research Institute, University of Delaware, Newark, Delaware 19716, USA2School of Physics, North-West University, Potchefstroom, South Africa3Department of Mathematics, University of Waikato, Hamilton, New Zealand4Istituto di Fisica dello Spazio Interplanetario, Consiglio Nazionale delle Ricerche, Rome, ItalyABSTRACTWe employ a turbulence transport theory to the radialevolution of the solar wind at both low and high lati-tudes. The theory includes cross helicity, magnetohy-drodynamic (MHD) turbulence, and driving by shear andpickup ions. The radial decrease of cross helicity, ob-served in both low and high latitudes, can be accountedfor by including sufficient shear driving to overcome thetendency of MHD turbulence to produce Alfve´nic states.The shear driving is weaker at high latitudes leading toa slower evolution. Model results are compared with ob-servations from Ulysses and Voyager.1. INTRODUCTIONThe evolution of solar wind turbulence is a challengingspace plasma physics problem, and one that is central inunderstanding various features of the heliosphere includ-ing radial temperature, solar energetic particles and mod-ulation of galactic cosmic rays. Relatively complete for-malisms for turbulence transport in a weakly inhomoge-neous medium have been developed using several com-plementary approaches [1, 2, 3]. Frequently further ap-proximations are imposed to achieve a physically trans-parent model. One of those simplifications is to the caseof zero cross helicity or, equivalently, equal admixtures ofinward and outward Alfve´nic fluctuations. This is proba-bly well satisfied beyond a heliocentric distance of a fewAstronomical Units (AU), but it is marginal from 1–3 AU,and is definitely not a reasonable simplification at dis-tances less than 1 AU from the sun [4, 5, 6, 7, 8].Recent work [9, 10] has focused on developing a turbu-lence transport model for the solar wind which allows formixed cross helicities. In this paper, we apply the modelto the solar wind at both low and high latitudes. The ap-plication of the model differs in the two regimes by rea-sonable changes to characteristic parameters.2. MODEL AND PARAMETERSSeveral features of MHD turbulence are relevant to un-derstanding the evolution of solar wind fluctuations.First, the general scenario of cascade and decay of ho-mogeneous MHD turbulence is found to proceed in muchthe same way as in hydrodynamics [11]. Second, MHDcascades can be strongly anisotropic. Spectral transferis stronger in the directions perpendicular to the large-scale mean magnetic field, and is weaker in the paralleldirection, leading to relatively stronger cross-field gradi-ents of magnetic, velocity, and small-scale density fluc-tuations [12]. Third, for the weakly inhomogeneous so-lar wind flow, a generalization of WKB theory [2, 1, 13]describes the transport of “locally homogeneous” MHDturbulence. Fourth, for low plasma-frame Mach num-ber the local turbulence can be described approximatelyby a nearly incompressible (NI) MHD theory [14, 15] inwhich the leading-order nonlinear effects are anisotropicand incompressible. These factors may be assembled intoa quantitative description of turbulence transport and de-cay that, with some simplifications, can be applied to thesolar wind in relatively tractable form [16, 17, 18, 19].In the absence of strong large-scale velocity shear,dynamic alignment [20, 21] tends to increase theAlfve´nicity, in the sense that the cross helicity Hc =12〈v · b〉 of the velocity v and magnetic field b fluctua-tions decreases more slowly than does the incompressibleenergy density (per unit mass) E = 12〈|v|2 + |b|2〉. (An-gle brackets denote an appropriate averaging procedure.)Quantitatively, dynamic alignment is signified by growthof the normalized cross helicity σc = 2Hc/E = (Z2+ −Z2−)/(Z2+ + Z2−), with Z2± = 〈|v ± b|2〉. Shear driv-ing generally supplies equal energy in “forward” (Z+,say) and “backward” (Z−) type fluctuations, which inlinear theory are eigenmodes associated with unidirec-tional propagation. Hence shear driving opposes dynamicalignment, which on its own would act to increase theAlfve´nicity.Here we will compare the predictions of a four-equationturbulence model with observations of the radial decreaseof normalized cross helicity. The turbulence model in-cludes equations for turbulence energy Z2 = (Z2+ +Z2−)/2, similarity or correlation lengthscale λ, protontemperature T , and normalized cross helicity σc. Theequations are steady-state, and employ a uniform large-scale solar wind speed U and specified radial profiles forthe Alfve´n speed and density. Turbulence is considered tobe locally homogeneous and incompressible, followingan MHD adaptation of a Ka´rma´n–Taylor phenomenology[16, 22]. The equations are,dZ2dr= −Z2r+Csh −MσDrZ2+E˙PIU−αλUf+(σc)Z3,(1)dλdr=MσD − Cˆshrλ−βαλE˙PIUZ2+βUf+(σc)Z, (2)dσcdr= αf ′(σc)ZUλ−[Csh −MσDr+E˙PIUZ2]σc, (3)dTdr= −43Tr+13mpkBαUf+(σc)Z3λ. (4)Three functions of the cross helicity were introduced tosimplify the notation. They are,f±(σc) =(1− σ2c )1/22[(1 + σc)1/2 ± (1− σc)1/2],(5)andf ′(σc) =[σcf+(σc)− f−(σc)]≈σc − σ3c2. (6)The model depends upon various parameters, includingKa´rma´n–Taylor constants, chosen here as α = 2β = 0.8,a mixing constant M = 1/2, σD = 〈|v|2 − |b|2〉/2E =−1/3 (assumed constant). See, e.g., [22].Turbulence is driven by two effects. First, large-scaleshear instability supplies turbulence energy. This is rep-resented by constants Csh and Cˆsh that control the shearstrength. (Here we take Cˆsh = 0 appropriate to driv-ing at roughly the correlation scale.) Second, scatteringof ionized interstellar neutrals (pickup ions) supplies en-ergy through wave-particle interactions, represented hereby the term E˙PI . The form of pickup ion driving weemploy is the same as in [18] (for an updated form, see[19]). Pickup ion driving of turbulence is important be-yond 10 AU and is not the focus of the present study.The main focus of the present paper is the radial evolu-tion of normalized cross helicity σc. Using the approxi-mation given in Eq. (6), we examine threshold estimatesof the interplanetary conditions needed to account for thetypical observation that σc decreases with increasing he-liocentric distance [23, 7]. Rearranging Eq. (3) we finddσcdr≈[α2(1− σ2c )ZUλ−Csh −MσDr−E˙PIUZ2]σc.(7)Thus, σc decays towards zero provided the square-bracketed term is negative. Consequently the observa-tions require, approximately, thatCsh −MσD +rE˙PIUZ2> α(1− σ2c )2rZUλ. (8)Note that usually MσD ≈ −1/6 << Csh [3]. In themiddle heliosphere, say 0.3 AU < r < 10 AU, pickupion effects are not important, and Eq. (8) simplifies toCsh > α(1− σ2c )2rZUλ. (9)The fraction rZ/λU , interpretable as the ratio of the ex-pansion and nonlinear timescales, is often of order unityin the solar wind. We see that there exists a threshold inshear strength above which σc decreases with radius.The shear constant is estimated using ∆U/∆r = CshU/r[2, 16]. The strength of shear driving of turbulence isexpected to differ in low latitude and high latitude solarwind. In particular, a lower value of Csh is expected in thehigh latitude fast wind, due to the absence of large streaminterfaces. We find [10] that Csh ∼ 1.5 for the low lati-tudes and Csh ≈ 1/2 for the high latitudes works well toaccount for the solar wind turbulence evolution. Note thatthese values of Csh are high enough that the conditiongiven in equation (9) should be satisfied for high valuesof σc thought to be apparent in the innermost heliosphere(say, r = 0.3 AU). We can thus expect a decrease in thecross helicity in the innermost part of the heliosphere.3. LOW LATITUDE RESULTSWe used the model to compute solutions at both low andhigh latitudes and compare the results to observations.Figure 1 shows a sample solution in the ecliptic plane,where the cross helicity is observed to decrease with he-liocentric distance [5, 6, 7, 8].Note that we have used parameters (see caption) in thetransport model that are comparable to values employedby [18] The model results are compared to observationalpoints adapted from [7], as well as two 1 AU data pointsfrom the OMNI data set. Evidently the transport theoryaccounts reasonably well for the observed decrease of σcvs. r, providing further support for the suggestion [24]that Alfve´nicity is reduced by turbulence driven by shearin solar wind stream structure. Note, however, that theadapted data points are not properly sorted, e.g., by windspeed. Consequently a more careful comparision withobservations is needed. The present results also motivatefurther examination of latitude effects on solar wind crosshelicity [25, 23, 26].4. HIGH LATITUDE RESULTSFigure 2 shows solutions computed by the model at highlatitude. The solutions are compared with hourly σc val-Figure 1. Radial evolution of normalized cross helicityσc at low latitudes, near the ecliptic plane. Observa-tional values extracted from Helios and Voyager data,are suggested by the symbols, which are adapted from[7] (courtesy of D. A. Roberts). Additionally, two datapoints from the OMNI data set are also included. The pa-rameters used for the model solution are appropriate tolow latitudes; shear strength Csh = 1.5, mixing parame-ter MσD = −1/6, a standard form of the pickup drivingterm (see Smith et al.2001), and constants α = 2β = 0.8,with boundary data at 0.3 AU specified as Z2 = 2200km2/sec2, λ = 0.01 AU and σc = 0.7ues from Bavassano et al. (2000a,b), computed fromUlysses data. There is no attempt to examine system-atic variation with latitude—we plot values of σc for tworanges of latitude (moderately high 25–55◦ and very high55–80◦). All points have θ > 25◦.As there is considerable spread in the data, we have plot-ted three solutions for σc(r) from the theory. The threesolutions differ in the imposed inner boundary values forthe turbulence amplitude Z2 and the correlation scale λ.Relative to low latitude solutions (figure 1), these solu-tions use lower Csh, higher solar wind speed, and lowerdensity [27]. The three solutions show that a moderate(and realistic) variation of inner boundary values pro-duces theoretical curves that vary in a way that is com-parable to the scatter in the Ulysses data. A fourth solu-tion is also shown, with identical parameters, except thatσc = 1.0 (maximal value) at the inner boundary, and Cshwas reduced. This solution represents a pure Alfve´nicstream, with higher σc, as might be inferred from somestudies (e.g., Goldstein et al., 1995) ).We recall the suggestion [Bavassano et al., 2000a,b] thatthe cross helicity saturates, i.e., approaches a terminalnonzero value in the high latitude wind. This tendency isapparent in Figure 2: Between 1.5 AU and 3 AU there isa downward trend. However there is no clear downwardtrend for data beyond 3 AU. Note the substantial spreadin observed values of σc(r) near any r.1 10 100-0.20.00.20.40.60.81.0σc(r)25-55o55-80oFigure 2. Radial evolution of normalized cross helicityσc at high latitudes. Observational points from Ulyssesdata in two latitude bands (see legend). Three solu-tions for σc(r) from the transport equations are shown,for a fixed latitude of 75◦ and varying values of turbu-lence level and correlation scale at the boundary: Dottedcurve: Z2 = 10000km2 s−2 and λ = 0.02AU; Solidcurve: Z2 = 6000 km2 s−2 and λ = 0.04AU; Dashedcurve Z2 = 4500 km2 s−2 and λ = 0.07AU. All otherparameters are fixed. Appropriate to high latitudes [27],we take U = 774 km/s and an Alfve´n speed at 1 AU of51 km/s. The driving parameters are Csh = 0.5 andMσD = −1/6, with a standard pickup driving term[18].Note that lower Z2 and higher λ at the inner bound-ary produces more rapid radial decrease of σc.5. CONCLUSIONWe have presented a turbulence transport model that hasbeen extended to include non-zero cross helicities [9, 10].The model accounts reasonably well for observations inboth the high and low latitude solar wind. The parame-ters characterizing the model differ in the two regimes inreasonable ways, including lower shear in the high lati-tude wind. It is beyond the scope of the current paper tofully explore the implications of the model solutions atdifferent latitudes. Such a study will be presented in thefuture.REFERENCES1. Marsch, E. and Tu, C.-Y. Dynamics of correlationfunctions with Elsa¨sser variables for inhomogeneousMHD turbulence. J. Plasma Phys., 41:479, 1989.2. Zhou, Y. and Matthaeus, W. H. Transport and tur-bulence modeling of solar wind fluctuations. J. Geo-phys. Res., 95:10 291, 1990.3. Matthaeus, W. H., Oughton, S., Pontius, D., andZhou, Y. Evolution of energy containing turbulenteddies in the solar wind. J. Geophys. Res., 99:19 267–19 287, 1994.4. Belcher, J. W. and Davis Jr., L. Large-amplitudeAlfve´n waves in the interplanetary medium, 2. J.Geophys. Res., 76:3534–3563, 1971.5. Bavassano, B., Dobrowolny, M., Mariani, F., andNess, N. F. Radial evolution of power spectra of in-terplanetary Alfve´nic turbulence. J. Geophys. Res.,87:3617, 1982.6. Bavassano, B., Dobrowolny, M., Fanfoni, G., Mari-ani, F., and Ness, N. 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Alfv´ enic turbulence in the polar wind: A statistical study on cross helicity and residual energyvariations.

Anisotropy in MHD turbulence due to a mean magnetic ﬁeld.

Dynamics of correlation functions with Els¨ asser variables for inhomogeneous MHD turbulence.

Evolution of energy containing turbulent eddiesin thesolar wind.

Evolution of turbulent magnetic ﬂuctuation power with heliocentric distance.

Fully developed anisotropic hydromagnetic turbulence in interplanetary space.

Growth of correlations in magnetohydrodynamic turbulence.

Heating of the low-latitudesolar wind by dissipation of turbulent magnetic ﬂuctuations.

Nearly incompressible ﬂuids. II: Magnetohydrodynamics, turbulence, and waves.

Numerical simulations of large-scale solar wind ﬂuctuations observed by Ulysses at high heliographic latitudes.

On the evolutionofoutwardandinwardAlfv´ enicﬂuctuationsin the polarwind. J.Geophys.Res., 105:15959– 15964,

Originandevolutionofﬂuctuations in the solar wind: Helios observations and HeliosVoyager comparisons.

Phenomenology for the decay of energy-containingeddies in homogeneous MHD turbulence.

Phenomenology of hydromagnetic turbulence in a uniformly expandingmedium.

Properties of magnetohydrodynamic turbulence in the solar wind as observed by Ulysses at high heliographic latitudes.

Radial evolution of cross helicity in high-latitude solar wind.

Radial evolution of power spectra of interplanetary Alfv´ enic turbulence.

Solar wind observations over Ulysses’ ﬁrst full polar orbit.

spatial transport, and heating of the solar wind.

Spectral evolution and cascade of solar wind Alfv´ enic turbulence.

Statistical properties of MHD ﬂuctuations associated with high-speed streams from Helios-2 observations. Solar Phys.,

Transport and turbulence modeling of solar wind ﬂuctuations.

Transport of cross helicity and the radial evolution of Alfv´ enicity in the solar wind.

Transport theory and the WKB approximation for interplanetary MHD ﬂuctuations.

Turbulent heating of the distant solar wind by interstellar pickup protons.

Velocitysheargenerationof solarwind turbulence.

Waves and turbulence in the solar wind.

Radial evolution of cross helicity at low and high latitudes in the solar wind

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