Simulating Heliospheric and Solar Particle Diffusion using the Parker Spiral Geometry
aa r X i v : . [ a s t r o - ph . H E ] N ov JOURNAL OF GEOPHYSICAL RESEARCH, VOL. ???, XXXX, DOI:10.1029/,
Simulating Heliospheric and Solar Particle Diffusionusing the Parker Spiral Geometry
R. C. Tautz, A. Shalchi, and A. Dosch Abstract.
Cosmic Ray transport in curved background magnetic fields is investigatedusing numerical Monte-Carlo simulation techniques. Special emphasis is laid on the So-lar system, where the curvature of the magnetic field can be described in terms of theParker spiral. Using such geometries, parallel and perpendicular diffusion coefficients haveto be re-defined using the arc length of the field lines as the parallel displacement andthe distance between field lines as the perpendicular displacement. Furthermore, the tur-bulent magnetic field is incorporated using a WKB approach for the field strength. Us-ing a test-particle simulation, the diffusion coefficients are then calculated by averagingover a large number of particles starting at the same radial distance from the Sun andover a large number of turbulence realizations, thus enabling one to infer the effects dueto the curvature of the magnetic fields and associated drift motions.
1. Introduction
It is accepted that, in general, the Solar magnetic field canbe described through an Archimedean spiral as originallysuggested by
Parker [1958]. Today, the Parker model isthe starting point for countless theoretical—both analyticaland numerical—investigations. Observations show generalagreement with the model [
Forsyth et al. , 1996], especiallyin the ecliptic plane and regarding the most probable fielddirection, although the magnetic field lines are less tightlywound as predicted by the model. However, there are overalldeviations from the Parker structure if the field is measuredfar away from the ecliptic plane, as has been done by theUlysses spacecraft [
Smith et al. , 2001]. It has been shownthat such can be explained by the motion executed by thefoot points of magnetic field lines across coronal loop holeboundaries [
Schwadron and McComas , 2005], thus givingrise to a “sub-Parker” structure. Using numerical simula-tions,
Riley and Gosling [2007] confirmed the connection ofa more radial magnetic field and coronal holes. Further-more, at high Solar latitudes there are asymmetries in theazimuthal field component, and the field lines are generallymore inclined towards the equator due to the interactionwith Solar wind plasma [
Forsyth et al. , 1996].The linear radial decrease of the (azimuthal) field com-ponent that was predicted by the Parker model was con-firmed from the investigation of Pioneer 10 data [
Parker andJokipii , 1976], although some deviations have been knowndepending on the phase of the Solar cycle; for example,the radial component is independent on Solar latitude dur-ing Solar minima and maxima, and the overall field mag-nitude is smaller than expected [
Smith , 2004]. Further-more, near the Sun the magnetic field has to be describedthrough a superposition of dipole, quadrupole, and currentsheet structures, thus giving rise to extensive modifications Zentrum f¨ur Physik und Astronomie, TechnischeUniversit¨at Berlin, Hardenbergstraße 36, D-10623 Berlin,Germany. Department of Physics and Astronomy, University ofManitoba, Winnipeg, Manitoba R3T 2N2, Canada. CSPAR, University of Alabama in Huntsville, HuntsvilleAL 35805, USA.Copyright 2018 by the American Geophysical Union.0148-0227/18/$9.00 of the Parker structure especially during Solar minima [
Ba-naszkiewicz et al. , 1998]. In defense of Parker, however,it should be noted that his model describes the field onlyoutside of the zone where field lines execute a rigid-bodyrotation.It has been argued that, due to some difficulties with van-ishing magnetic field divergence, the standard Parker spiralhas to be superposed by other field components, preferablyhomogeneous [
Goldstein Jr. , 1998] with a strength of a frac-tion of nT. Such could be interpreted as the local galacticmagnetic field, for which estimates are available throughcarefully distinguishing the excellently measured Solar fieldand the large-scale field as inferred from pulsar rotation rates[
Rand and Kulkarni , 1989]. However, it should be notedthat the form of the magnetic field can be inferred fromthe study of suprathermal electrons (with energies largerthan 70 eV), which are aligned with the heliospheric mag-netic field (and have therefore been named “strahl”), be-cause such a particle distributions broadens with distance[
Owens et al. , 2008]. Independently from other models, suchleads to a well-formed Parker geometry.A modified model [
Fisk , 1996] includes the reversion ofpolarity and thus deviates from the Parker model becausethe magnetic field components now depend on the fieldpolarity. From the solution of the transport equation, itwas shown that the energy spectra of protons arriving atEarth show significant differences depending on the under-lying magnetic field model. Similarly, the transport equationhas been solved for solar energetic particle events (SEP) us-ing both finite difference [
Kallenrode , 1993] and stochasticmethods [
Zhang et al. , 2009]. Such particles have sufficientlyhigh energy to pose a threat to both astronauts and spaceinfrastructure such as satellites. Based on a magnetic field inthe form of a Parker spiral, the results underline again thatperpendicular diffusion is important if one wants to under-stand the development of the SEP during the time it needs toarrive on Earth. Therefore, the knowledge of the magneticfield geometry is important if one attempts to understandthe propagation of particles, which is also important for aprediction of “space weather” [
Pommois et al. , 2002].The smooth, large-scale field described by the Parker—and similar, extended—models is, however, only half thetruth; in reality, the heliospheric magnetic field is highlyvariable on all scales, which can be described in termsof waves, fluctuations, or more general turbulence models[
Smith , 1989]. Thus, the spiral structure becomes evidentonly after (time and space) averaging of the magnetic field,
TAUTZ ET AL.: PARKER SPIRAL GEOMETRY showing that the turbulence strength is of the same orderof magnitude as the background field. Moreover, as knownfrom observations, the small- and middle-scale structures,can in turn influence the underlying spiral structure [
Dalinet al. , 2002] by changing the inclination angle of the mag-netic field lines. It is generally assumed that the domi-nant constituents are magnetohydrodynamic Alfv´en waves[
Smith , 1989;
Tautz , 2010a], which propagate in oppositedirections along the background magnetic field lines [
Cableand Lin , 1998], and convective, two-dimensional structures[
Schmidt and Marsch , 1995]. However, other waves suchas magnetosonic waves have also been used, thereby givingrise to anisotropies in the power spectra used to characterizethe turbulence [
Chashei , 2000]. Furthermore, magnetosonicwaves dominate the stochastic acceleration of charged par-ticles [
Schlickeiser and Miller , 1998;
Tautz , 2010a].The motion of charged particles in such systems canbe described by a diffusion process [see
Schlickeiser , 2002;
Shalchi , 2009, for detailed introductions]. Although theproblem cannot be considered to be completely solved, sig-nificant progress has been made in describing the motionof energetic particles such as Cosmic Rays in a turbulentmedium immersed in a homogeneous background magneticfield. For example, simulation results in different turbu-lence geometries [
Giacalone and Jokipii , 1999;
Qin et al. ,2002a, b;
Tautz , 2010b] could be reproduced using non-linear extensions [e. g.,
Matthaeus et al. , 2003;
Shalchi et al. ,2004;
Shalchi , 2006;
Qin , 2007;
Tautz et al. , 2008a;
Doschet al. , 2009;
Shalchi , 2010] of the quasi-linear transport the-ory [
Jokipii , 1966].As soon as it comes to curved background fields, however,matters are even less understood. The concept of adiabaticfocusing has attracted attention recently [e. g.,
Kunstmann ,1979;
Spangler and Basart , 1981;
Bieber et al. , 2002;
Schlick-eiser and Shalchi , 2008], which describes the modification ofdiffusion and transport equations due to a spatial gradient inthe magnetic field. Other models simply prescribe a curvedspatial geometry by accepting a background magnetic fieldwith non-zero divergence [
Minnie et al. , 2007]. But a satisfy-ing solution has not been achieved yet for real magnetic con-figurations that are not uniformly converging or diverging—such as the Parker spiral. In principle, the real field lines,which are distorted by the presence of turbulence, can beinferred using a field line wandering (or field line randomwalk) approach [
Jokipii and Parker , 1969;
Matthaeus et al. ,1995;
Shalchi and Kourakis , 2007]. By specifying the turbu-lence power spectrum, a direct calculation of the stochasticfield lines is then possible, showing excellent agreement withHelios 2 data. Furthermore,
Kobylinski [2001] showed thatthe diffusion tensor can be calculated analytically by apply-ing two rotations to the parallel and perpendicular diffusioncoefficients, although, in general, such a transformation willalways be ambiguous.Furthermore, the question of “diffusivity”, i. e., whetherthe diffusion coefficients attain finite values, is still notsolved satisfactorily. Whereas, in perpendicular scatter-ing in magnetostatic slab turbulence (in Alfv´enic slab tur-bulence, diffusion is recovered
Shalchi et al.
Tautz
Tautz et al. , 2008b]. However, for anisotropic non-slab turbulence, such is not clear [
Zimbardo et al. , 2006].There are several cases where sub- or even super-diffusionhas been found in numerical simulations [
Tautz and Shalchi ,2010] and in observations of electrons accelerated at inter-planetary shocks [
Perri and Zimbardo , 2007]. It has evenbeen thought that the underlying process might not be aclassic diffusion process but a L´evy random walk [
Zimbardo ,2005]. However, such analyses would first require a stabledescription of the turbulence, because the results depend sensitively on the details of the turbulence model and itsparameters.In this article, the transport of test-particles is investi-gated using a numerical Monte Carlo technique [
Giacaloneand Jokipii , 1999;
Micha lek , 2001;
Tautz , 2010b]. The mag-netic field is composed of a large-scale Parker-type field anda turbulent component, which is assumed to be isotropic. Itis known from observations [
Matthaeus et al. , 1990;
Bieberet al. , 1994;
Chashei , 2000] that the turbulent magnetic fieldcomponent is most likely not isotropic. There are even stud-ies that incorporate such effects in sophisticated turbulencemodels [
Sridhar and Goldreich , 1994;
Goldreich and Srid-har , 1995]. The problem, however, is that the basic trans-port theories are not sufficiently understood yet to allow forthe isolation of small effects. Here, therefore, an isotropicturbulence spectrum will be used [
Tautz et al. , 2006, 2008a]that uses a constant energy range. Numerically, the well-known method will be applied of superposing a homogeneous(in our case: the Parker spiral) magnetic field with a tur-bulent component calculated by the summation over planewaves with random directions of propagation and randomphase angles [
Tautz , 2010b]. Thereby, the diffusion coef-ficients are calculated from the mean square deviation. Itwill be shown how “parallel” and “perpendicular” particledisplacements must be transformed using concepts from dif-ferential geometry. The numerical ansatz will be relativisti-cally correct, although such might not be necessary for theenergies considered. In a second paper [
Tautz and Vocks ,2010], the effects of Whistler wave turbulence will be stud-ied, which is important for the formation of the “strahl” elec-trons that have been mentioned above. Here, in contrast,magnetostatic turbulence will be employed. Although, ashas already been mentioned, plasma waves may have a se-vere impact on the diffusion coefficients especially if theirelectric fields are of significant magnitude, such a configura-tion represents the most basic case that has to be understoodfirst before implementing more advanced features. The ap-proach is, to some extent, comparable to a method used by
Pei et al. [2006], although in their work focus was laid onthe calculation of SEP onset times. It was found that, inthe presence of turbulent field component, the onset timeswere generally reduced in comparison to a smooth Parkermagnetic field.The present paper is organized as follows: In Sec. 2,the geometry of the Parker spiral is introduced and it isshown how appropriate modifications can be made to the(numerical) implementation of the simulation code that cal-culates the diffusion coefficients. In Sec. 3, the setup ofthe test-particle simulations and the turbulence generationare briefly explained. In Sec. 4, the simulation results arepresented and compared both to previous calculations andsimulations in homogeneous background magnetic fields. Fi-nally, Sec. 5 provides a summary and conclusions regardingfuture work.
2. Parker Spiral Geometry
According to
Parker [1958], the large-scale spiral patternof the Solar magnetic field in interplanetary space (withouttaking into account the turbulent component) is determinedthrough an equation relating the radial distance from theSun, r , and the azimuth angle in the ecliptic plane, φ , as rb − − ln (cid:16) rb (cid:17) = v SW bω ⊙ ( φ − ψ ) , (1)where the parameter ψ is the azimuth angle of the field lineat the co-rotation radius. Furthermore, v SW ≈
400 km/s isthe outward velocity of the solar wind, ω ⊙ is the angular AUTZ ET AL.: PARKER SPIRAL GEOMETRY
X - 3 velocity of the sun, and b = 46 × − AU is a distance be-yond which any direct influence of the sun may be neglected[cf.
Parker , 1958]. Physically, b marks the end of the region,where the field lines execute a “rigid rotation” and, there-fore, point radially outwards; i. .e., for r < b the field linespoint radially outward while they corotate with the Sun’ssurface and, thus, Eq. (1) is not valid. Here, however, onlythe case r > b will be considered.In what follows, quantities and relations will be referredto Figs. 1 and 2, which illustrate the explanations givenhere. Solving Eq. (1) for the azimuth angle φ as a functionof the radius, r , yields φ ( r ) = ψ + bζ h rb − − ln (cid:16) rb (cid:17)i , (2)with ζ = v SW /ω ⊙ ≈ Zank et al. , 2004].Let us now consider an arbitrary particle in the eclipticplane (black dot) at the cartesian coordinates ( x s , y s ) denot-ing the starting point on a magnetic field line (black dashdotted line). The starting point may be expressed throughthe new coordinates ( r s , φ s ) using the transformations r s = p x + y (3a) φ s = arctan (cid:18) y s x s (cid:19) , (3b)where, on evaluating the arctan function, attention has tobe paid to choosing the right quadrant. The inner field lineazimuth angle, ψ , can be determined by back-tracing thefield line to the radius r = b . Since the starting point lieson that field line, ψ can simply be calculated by ψ = φ s − bζ h r s b − − ln (cid:16) r s b (cid:17)i , (4)which is visualized in Fig. 1.Usually the end point of the particle trajectory, ( x e , y e ),will be located on a different (but nearby) field line (seebrown dot on the brown dashed line in Fig. 2). Particu-larly, the particle has drifted from one field line to anotherdue to diffusion processes. According to the definition of theparallel and perpendicular (running) diffusion coefficients, κ k , ⊥ ( t ) = (cid:10) ( ∆s k , ⊥ ) (cid:11) t , (5)one has to calculate the parallel and perpendicular meansquare displacements, respectively, which are denoted by h ∆ i . The perpendicular displacement is associated with theshortest distance—perpendicular to the field lines—betweenthe ending point and the initial field line, leading to the tar-get point (red dashed line and red dot). In a uniform mag-netic field such is simply a straight line perpendicular tothe background magnetic field lines. Likewise, the paralleldisplacement is the shortest distance between the startingpoint and the target point; in a uniform magnetic field, itwould be a straight line, too. But how does that have to betranslated for a curved background magnetic field such asthat described by the Parker spiral?The natural answer is motivated by differential geometry.The perpendicular displacement is characterized by a curveperpendicular to all field lines, which runs through the endpoint and the target point. As a first approach it is as-sumed that such a curve can be approximated by a straightline as in Euclidean geometry (red dashed line), based on thefact that the turbulence in interplanetary space is moderate[ Bieber et al. , 1994] and, therefore, perpendicular diffusionis weak, too. Furthermore, it was shown by
Shalchi andDosch [2009] that, even in strong isotropic turbulence, onegenerally has κ ⊥ ≪ κ k .This also motivates the assumption that the particle tra-jectory ends on nearby field lines and states that the parallel displacement is given through the arc length of the initial(black dash dotted line) field line to a (red dot) target point,where the shortest distance to the (brown dot) particle endpoint is perpendicular on all field lines. Hence, instead of astraight line the “shortest distance” is now a geodesic fol-lowing the geometry of the Parker spiral. As an approxi-mation, however—owed to the fact that the turbulence ininterplanetary space is weak and that, therefore, perpen-dicular diffusion is weak, too—the calculation of the targetpoint proceeds as follows: One simply calculates the pointfrom which the distance—this time taken as a straight lineas in Euclidean geometry—to the particle trajectory’s endpoint is minimal (red dashed line). Then, the perpendiculardisplacement, denoted by ∆s ⊥ , is simply given through thedistance between the target point, r t , and the end point, r e ,as ∆s ⊥ = min r t ∈ R + n(cid:2) r t cos (cid:0) φ ( r t ) (cid:1) − x e (cid:3) + (cid:2) r t sin (cid:0) φ ( r t ) (cid:1) − y e (cid:3) o , (6)which function is visualized in Fig. 3.The only intricacy is to find the correct minimum, sincethe function in Eq. (6) has multiple local minima for φ +2 πk with k approximately an integer number. Hence, a methodin three steps will be used. First step: to identify a minimumof any function f , three points a < b < c are needed with f ( b ) < f ( a ) and f ( b ) < f ( c ) are necessary—which is called“bracketing” of the minimum [ Press et al. , 2007, p. 490].Assuming φ t ≈ φ e and assuming that the particle will notcomplete one or more full orbits around the Sun, one cantherefore search for a minimum starting from r s . By itera-tively solving for r ( φ e ≈ φ t ) starting with r = r s one has,therefore, r i +1 = b h (cid:16) r i b (cid:17)i + ζ ( φ e − ψ ) , (7)which will be stopped after a few steps. For very small dis-placements ( r e , φ e ) ≈ ( r s , φ s ), i. e., after short times, how-ever, the iteration can yield wrong results. Second step: Inthat case, a fail-safe method will determine an approximatebracketing condition from calculating r ( φ s − π ) ≤ r s ≤ r ( φ s + 2 π ), where again use is made of the assumption thata given particle will not complete a full orbit around theSun. Third step: Based on the bracketing of the minimum,it is then a simple task to calculate the exact minimum us-ing, e. g., Brent’s method [ Press et al. , 2007, pp. 496–502]without derivative information [see also
Forsythe et al. § Brent ∆s k , is calculated through the arc length ofthe field line between the target point and the starting point,yielding ∆s k = L ( r t ) − L ( r s ) , (8)where the arc length function, L , can be calculated analyt-ically and is given through L ( r ) = Z r d s s s (cid:18) d φ d s (cid:19) (9a)= r − b p ( r − b ) + ζ ζ + ζ (cid:18) r − bζ (cid:19) . (9b)For instance, the trajectory parameters from Fig. 2 (with ζ = 1)have been used to obtain ∆s k /b = 20 . ∆s ⊥ /b = 2 . x - y plane—has been consid-ered. To incorporate drift motion, the vertical displacement, - 4 TAUTZ ET AL.: PARKER SPIRAL GEOMETRY ∆z = z e − z s , is calculated so that, from the comparison of ∆s ⊥ and ∆ z , the drift effect can be inferred.Furthermore, the (background) magnetic field compo-nents, which are designed to cover mainly the eclipticplane, are described using spherical coordinates through [see Parker , 1958;
Burger and Hitge , 2004] B r ( r, θ, φ ) = B (cid:18) br (cid:19) (10a) B θ ( r, θ, φ ) = 0 (10b) B φ ( r, θ, φ ) = B (cid:18) br (cid:19) r − bζ sin θ, (10c)with B = 1830 nT [ Zank et al. , 2004]. Note that thestrength of the magnetic field is maximal in the eclipticplane, and that no care has been taken of the current sheetand the polarity reversal [cf.
Fisk , 1996;
Burger and Hitge ,2004]. For small r , the field is purely radial and scales with r − , whereas, according to Eqs. (10), for large r the fieldbecomes more and more azimuthal and scales with r − , inagreement with Pioneer 10 data [ Parker and Jokipii , 1976];see also Fig. 2 and Fig. 6 of
Parker [1958].When the diffusion coefficient is calculated in the waydescribed here, the curved space defined by the large-scalemagnetic field is mapped to a Euclidean space. The diffu-sion coefficient can include the additional terms coming outof the divergence in the curved space. Thus, the diffusioncoefficient carries a meaning different from local diffusioncoefficient defined in the curved space. Note also that, untilhere, only the smooth spiral field lines have been considered.Both in the real world and in the simulation code, the back-ground field will be superposed by a turbulent component.However, diffusion coefficients will be calculated in referenceto the mean field component.
3. Monte-Carlo Simulations
For the simulation of cosmic ray scattering processes,a modified version of the recently developed
Padian code[
Tautz , 2010b] has been used, which traces the trajectoriesof a large number of test particles (typically 10 ) for a suf-ficiently long time (typically 10 Larmor orbits). In doingso, it can be decided whether the transport is diffusive, un-perturbed, or subdiffusive, and the diffusion coefficients canbe determined. For the integration of the equation of mo-tion, an integration algorithm with adaptive step sizes suchas the Bulirsch-Stoer method [see
Stoer and Bulirsch , 2002;
Press et al. , 2007, pp. 921–928] is used, which limits the rel-ative deviation in the particle rigidity, ∆R/R , to be smallerthan 10 − %. Here, the particle is expressed as momentumper unit charge times speed of light, i. e., R = p c/e . Notethe difference to several previous articles [e. g., Tautz et al. ,2008a;
Tautz , 2010b], where a dimensionless “rigidity”-likequantity was defined through R = γm v / ( Ωℓ ), where Ω denotes the gyro-frequency.In comparison to the original formulation [ Tautz ,2010b], the equation of motion—i. e., the Newton-Lorentzequation—has to be slightly modified due to the non-constant background magnetic field. Furthermore, SI unitsare used so that the rigidity and the magnetic field strengthscan be given in megaVolt (MV) and nanoTesla (nT), respec-tively. The trajectory x ( t ) (normalized to a characteristiclength scale ℓ ; see below) is therefore calculated as a func-tion of the dimensionless time τ = vt/ℓ usingdd τ x ℓ = 1 R R (11a)dd τ R = aR R × [ B ( r ) ˆ e B + δB ( r ) ˆ e δB ] , (11b) where Eq. (11b) corresponds to ˙ p = q v × B (SI units). Thescaling factor is given by a = ℓ c/ ≈ .
45 m / s fortypical parameters (see below). Note that Eqs. (11) arevalid for all particle species and that the rest frame of theSun has been used.The unit vectors ˆ e B and ˆ e δB point in the directions ofthe Parker field and the turbulent magnetic field, respec-tively. In the ecliptic plane, the total background mag-netic field strength is, approximately, determined throughEq. (10) as B ( r ) = 1830 nT (cid:18) br (cid:19) s (cid:18) rζ (cid:19) . (12)Following Zank et al. [1996, Sec. 3.1], a WKB approxima-tion [see also
Matthaeus et al. , 1994] leads to an equationfor the turbulent magnetic field energy per mass, E δB = δB / (8 πρ ), as d E δB d r + E δB r = 0 , (13)yielding E δB ∝ r − . Equation (13) was motivated by theWal´en relation [ Matsuoka et al. , 2002], i. e., ∆ B ∝ ρ∆ v ,where ∆ B and ∆ v are changes in the magnetic field andthe plasma velocity, respectively. Based on the Wal´en re-lation, the ratio of kinetic and magnetic energy (both permass) are constant. Therefore, with the additional assump-tion that, in the ecliptic plane, the mass density behaves as ρ ∝ r − , one has (cid:18) δBδB ref (cid:19) = (cid:16) r ref r (cid:17) , (14)where δB ref = 4 nT is the turbulent field strength at r ref =1 AU. Accordingly, the turbulent magnetic field strength isgiven through [ Zank et al. , 2004;
Shalchi et al. , 2010, Ap-pendix B] δB ( r ) = 4 nT ( r [AU]) − / , (15)where r is the radial distance from the Sun in astronomi-cal units (AU). Based on Eq. (12), the relative strength ofturbulent and background magnetic field depends on r as δBB ≈ s r [AU]1 + ( r [AU]) . (16)The turbulence model, which, to some extent, is basedon a model used by Giacalone and Jokipii [1999], gener-ates random magnetic fluctuations by the superposition of alarge number N of plane waves (typically 512) with randomdirections of propagation, random phases, and amplitudesas prescribed by the turbulence spectrum of Tautz et al. [2006, 2008a] that has the form G ( k ) = (cid:2) ℓ k ) (cid:3) − ν ≈ ( ℓ k ) − / , (17)where the parameter ν = 5 / k − / [cf. Podesta et al. , 2007]and where ℓ is the isotropic turbulence bend-over scale.Note that, in the Solar system, the bend-over scale, ℓ , maydepend on the position [ Bruno and Carbone , 2005]. Thespectrum from Eq. (17) has a constant energy range (i. e.,the range k < ℓ − ). For a more general spectrum, a formulasimilar to that of Eq. (17) can be found, e. g., in Shalchiand Weinhorst [2009].The spatially fluctuating but time-independent (i. e.,magnetostatic) magnetic field is then calculated as δ B ( x, y, z ) = Re N X n =1 e ′⊥ A ( k n ) e i ( k n z ′ + β n ) , (18) AUTZ ET AL.: PARKER SPIRAL GEOMETRY
X - 5 where the sum extends over N logarithmically spacedwavenumber values k n . The amplitude function A ( k n ) ∝ p G ( k n ) is related to the turbulence spectrum and e ′⊥ isa unit vector in the direction perpendicular to z ′ . Theprimed coordinates are obtained from a rotation matrix,whose angles are randomly generated for each summand n .The parameter β is the random phase for each wave mode.Thus, Eq. (18) generates an isotropic, time-independent(i. e., magnetostatic) turbulent magnetic field. Accordingly,the orientation of ( x, y, z ) is irrelevant; however, for conve-nience the z direction is kept perpendicular to the eclipticplane. The divergence of δ B is kept zero due to the fact that e ′⊥ ⊥ k n for all summands, which corresponds to ∇· δ B = 0in Fourier space.Care must be taken about the minimum and maximumwave numbers. Whereas the analytical form of the spec-trum from Eq. (17) extends over all wavenumbers, such isnot possible in computer simulations. There are two majorconditions that have to be fulfilled, which are: (i) the reso-nance condition stating that there must exist a wavenumberso that kµR L = 1 is fulfilled; and (ii) the time scale condi-tion stating that Ωt max k min R L <
1. The second conditionrequires a maximum turbulence scale (defined through k − )to be larger than the distance traveled by the particle to en-sure the particle cannot move out of the system.From simulations in a homogeneous background magneticfield [e. g., Tautz , 2009], one knows that both conditions (i)and (ii) do not depend on the turbulence bend-over scale ℓ ; instead, they depend on the maximum and minimumscale of the system (given by the minimum and maximumwavenumber, respectively). Condition (ii) therefore statesthat the particles must not travel farther than the systemscale L max = k − . Although, in the simulation code, theturbulence is generated wherever the particle position is,particles start to free-stream once the condition is violated.In that case, one finds κ ⊥ ∝ t − , which is equivalent to aconstant perpendicular mean square displacement and indi-cates a purely parallel motion.In Fig. 4, the magnetic field strength is shown as a func-tion of the distance from the Sun. In the Padian simula-tion code, the field strength is usually normalized to unity;here, however, the field strength decreases with increasingdistance. Furthermore, the (average) magnitude of the tur-bulent field is shown in comparison to the background fieldstrength, indicating a maximum at 1 AU.By integrating the particle trajectory as given byEq. (11), the mean square displacement is then determinedas described in Sec. 2. Finally, the diffusion coefficientsand the mean free paths are calculated by averaging over allparticles and all different turbulence realizations.
4. Simulation Results and Comparison
For the simulation runs, the particle energies were cho-sen in the range 10 MV ≤ R ≤ MV. All runs werecarried out using a total number of 1000 particles in 25different turbulence realizations. It is interesting that, incontrast to simulations involving homogeneous backgroundmagnetic fields, here small particle energies require consid-erably longer computation times than high energies.Furthermore, all particles are assumed to start in theecliptic plane with an equal initial radial distance to thesun of r s = 10 ℓ . Since the bend-over scale is usually takento be ℓ = 0 .
03 AU, a heliocentric distance of 1 AU corre-sponds to 33 ℓ . While, in the absence of a turbulent mag-netic field component, the particles follow the magnetic fieldlines, there is extensive scattering if the magnetic field is tur-bulent (see Fig. 5). Especially at low particle energies, thereeven seems to be confinement-like mechanisms (not shown in the figure), where particles repeatedly move back and forthalong the same magnetic field line without covering a largenet distance.In Fig. 6, the resulting mean free paths in the directionsparallel, perpendicular, and vertical to the Sun’s magneticfield are shown as a function of the normalized dimension-less time, vt/ℓ . (Note that, here, the term “perpendicular”is used to describe a direction in the ecliptic plane, whereas“vertical” is oriented normal to the ecliptic plane.)The comparison of the perpendicular and vertical meanfree paths shows no qualitative difference, which leads to theconclusion that a drift effect can be neglected. The resultthat turbulent magnetic fields suppress particle drift mo-tions is supported by the work of Minnie et al. [2007], whofound that, if scattering processes are important, all driftmotions are superseded by diffusion.Furthermore, note that all three mean free paths in Fig. 6seem to approach a constant, thus indicating a diffusivebehavior. In contrast to a configuration with a homoge-neous background magnetic field as investigated in
Tautzand Shalchi [2010], there is no indication here of a (evenslightly) subdiffusive behavior—at least not for sufficientlyhigh rigidities
R >
10 MV.In Fig. 7, the parallel mean free path is shown as afunction of the particle rigidity. Furthermore, a compari-son is shown to the Palmer consensus range (shaded box)consisting of mean free path values from various observa-tional studies [see
Palmer , 1982;
Bieber et al. , 1994, andreferences therein].Note that all simulation results were obtained for an ini-tial radial distance r s = 10 ℓ . However, the difference toadditional simulation runs with r s = 30 ℓ turned out to benegligible. Therefore, the radial dependence of the magneticfield strength seems to have only a small effect on the meanfree paths—in contrast to the turbulent magnetic field. Ac-cording to quasi-linear theory [e. g., Jokipii , 1966], the tur-bulence strength relates to the parallel mean free path as B / | δB | ∝ p λ k , which behavior has also been observedin simulations [ Tautz , 2010a] for intermediate turbulencestrengths; for strong turbulence, such may be different [cf.
Shalchi et al. , 2009].In Fig. 8, the ratio of the perpendicular/vertical and par-allel mean free path is shown as a function of the particlerigidity. In agreement with previous work [e. g.,
Giacaloneand Jokipii , 1999;
Shalchi et al. , 2004;
Zank et al. , 2004;
Tautz et al. , 2006], λ ⊥ /λ k tends to be slightly reduced as theparticle rigidity increases. The overall values are somewhathigher than normally observed, because, using the estimateof Bieber et al. [2004], i. e., λ ⊥ /λ k ≈ ( δB/B ) /
6, one has, ata radial distance of 10 and 20 AU yields values for λ/ ⊥ /λ k ofapproximately 0.05 and 0.07, respectively, whereas the sim-ulation results are all well above λ ⊥ /λ k > . Minnie et al. [2007].
5. Summary and Conclusion
In this article, particle scattering and diffusion param-eters have been calculated for the Solar system. The in-vestigation was based on the Parker spiral model for thebackground magnetic field together with a WKB model toaccount for the turbulent magnetic field component, whichvaries with the radial distance. For the first time, the Parkergeometry of the Sun’s magnetic field has been implementedin a test-particle simulation code.The prediction of negligible drift motions could be con-firmed (Fig. 9). It is obvious from Fig. 7, the results for theparallel mean free path shown here are too small, except for - 6
TAUTZ ET AL.: PARKER SPIRAL GEOMETRY high rigidities R ≥ MV. There are three possible expla-nations:First, in the present version an isotropic turbulence modelhas been used. Usually, a slab/2D model consisting of twoturbulence components with wavevectors parallel and per-pendicular to the background magnetic field are used forapplications in the Solar system [
Bieber et al. , 1994, 1996],which increases the parallel mean free path [e. g.,
Tautzand Shalchi , 2010]. On the other hand, it has been shownby
Weinhorst and Shalchi [2010] that a more realistic tur-bulence model, which overcomes the singularities inherentin the slab/2D formulation, can reproduce data calculatedfrom spacecraft measurements better. However, the turbu-lence model that was designed in the style of the so-calledMaltese cross [
Matthaeus et al. , 1990] has not yet been im-plemented into simulation codes.Second, the WKB model that has been used for thestrength of the turbulent magnetic field could overestimatethe true turbulence level. To allow for better adjustment ofthe turbulence parameters, more measurements of the tur-bulence level at different radial positions would be necessary.Third, for other choices of the correlation scales, espe-cially for different scales in the directions parallel and per-pendicular to the background magnetic field, the resultingtransport parameters would be different [see also
Pei et al. ,2010]. Especially, a different choice of parallel and perpen-dicular turbulence bend-over scales could modify the result-ing mean free paths [
Matthaeus et al. , 2003;
Shalchi , 2009,Sec. 5.4.4].Future work should, therefore, try to implement a Mal-tese cross based turbulence model to account for the variousanisotropies between the parallel and the perpendicular tur-bulent magnetic field components [e. g.,
Narita and Glass-meier , 2010;
Chen et al. , 2010]. The effects due to the Solarwind would also be worth studying. Furthermore, a reli-able description is necessary of the turbulent magnetic fieldstrength on the radial distance from the Sun.
Acknowledgments.
The work of R. C. Tautz was partiallysupported by the German Academy of Natural Scientists Leopold-ina Fellowship Programme through grant LDPS 2009-14 fundedby the Federal Ministry of Education and Research (BMBF).
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Angle Φ H r LH x,y L Angle Ψ H r L r - - x (cid:144) b - y (cid:144) b Figure 1.
Illustration of the angles used to describe theParker spiral. The parameter ψ corresponds to the an-gle at which the large-scale field line (solid line) starts,whereas the variable φ ( r ) determines the course of thefield line in dependence on the radius, r . StartEndTarget Trajectory Field lines - - - x (cid:144) b - - y (cid:144) b Figure 2. (Color online) The meaning of “parallel”and “perpendicular” in Parker spiral geometry. Start-ing (black dot) on the black dash dotted field line, theparticle trajectory (blue solid line) ends (brown dot) onthe brown dashed field line. As an approximation, theperpendicular distance between the two field lines (redshort-dashed line) is taken to be a straight line, wherethe target point (red dot) has to be calculated throughminimization of the perpendicular distance. r t D ¦ r (cid:144) b D ¦ H r L(cid:144) b Figure 3.
The function from Eq. (6), the minimumof which (black dot) has to be found in order to calcu-late the radius of the target point, r t , and the associatedperpendicular and parallel displacements, ∆s ⊥ and ∆s k ,respectively. The correct minimum is that next to thestarting point, for which r s /b = 4 .
25 30 35 40 45 50 r (cid:144) l B @ nT D
30 40 50 r (cid:144) l ∆ B (cid:144) B Figure 4.
The decrease of the background magneticfield strength with radial distance from the Sun as pre-scribed by the Parker spiral through Eqs. (10). At r = r s = 33 ℓ , one has B = 1 .
68 nT. The inset showsthe relative strength of the turbulence in terms of thebackground magnetic field.
AUTZ ET AL.: PARKER SPIRAL GEOMETRY
X - 9 −40 −20 0 20 40 60−20020406080 x/l y / l −80 −60 −40 −20 02030405060708090 x/l y / l Figure 5.
Particle trajectories without (upper panel,dashed lines) and with (lower panel, solid lines) magneticturbulence. As a guide to the eye, the magnetic field linesaccording to the Parker spiral (with, in the lower panel,superposed by the turbulent component) are shown asdotted lines. Furthermore, the dash dotted line roughlymarks the Earth orbit around the Sun. −1 −4 −3 −2 −1 R = 10 MV λ [ AU ] vt/l Figure 6.
The mean free paths normalized to the astro-nomical unit, κ/ ( vℓ ), as a function of the dimensionlesssimulation time, vt/ℓ . The solid line shows the paral-lel mean free path (i. e., the coefficient of the diffusionalong the curved magnetic field lines of the Parker spi-ral), whereas the dashed line show the mean free pathperpendicular but in the ecliptic plane. The dottedline shows the vertical mean free path normal to theecliptic plane, which is defined in the classic way, i. e., λ z = 3 h ( ∆z ) i / (2 vt ). −1 −3 −2 −1 R [MV] λ || [ AU ] Figure 7.
The mean free path as a function of the par-ticle rigidity. The error bars indicate the simulation re-sults, and the shaded area illustrates the
Palmer [1982,see also
Bieber et al. −1 λ ⊥ / λ || −1 R [MV] λ z / λ || Figure 8.
The ratio of the perpendicular and the parallelmean free path (upper panel) and the ratio of the verticaland the parallel mean free path (lower panel). The errorbars are calculated from the individual estimated meanerrors of the individual mean free path components. - 10