SENSITIVITY OF COSMIC-RAY PROTON SPECTRA TO THE LOW-WAVENUMBER BEHAVIOR OF THE 2DTURBULENCE POWER SPECTRUM

N. E. Engelbrecht1,2

and R. A. Burger2

1 South African National Space Agency, Hermanus 7200, South Africa; n.eugene.engelbrecht@gmail.com2 Center for Space Research, North-West University, Potchefstroom, 2520, South Africa

Received 2015 October 2; accepted 2015 October 30; published 2015 December 1

ABSTRACT

In this study, a novel ab initio cosmicray (CR) modulation code that solves a set of stochastic transport equationsequivalent to the Parker transport equation, and that uses output from a turbulence transport code as input for thediffusion tensor, is introduced. This code is benchmarked with a previous approach to ab initio modulation. Thesensitivity of computed galactic CR proton spectra at Earth to assumptions made as to the low-wavenumberbehavior of the two-dimensional (2D) turbulence power spectrum is investigated using perpendicular mean freepath expressions derived from two different scattering theories. Constraints on the low-wavenumber behavior ofthe 2D power spectrum are inferred from the qualitative comparison of computed CR spectra with spacecraftobservations at Earth. Another key difference from previous studiesis that observed and inferred CR intensityspectra at 73 AU are used as boundary spectra instead of the usual local interstellar spectrum. Furthermore, theresults presented here provide a tentative explanation as to the reason behind the unusually high galactic protonintensity spectra observed in 2009 during the recent unusual solar minimum.

Key words: cosmic rays – diffusion – turbulence

1. INTRODUCTION

Although not new (see, e.g., Jokipii & Owens 1975; Jokipii& Levy 1977), the stochastic approach to solving the Parker(1965) transport equation (TPE) in cosmicray (CR) modula-tion studies has recently become quite popular (see, e.g., Zhang1999; Alanko-Huotari et al. 2007; Pei et al. 2010; Strauss et al.2011; Della Torre et al. 2012; Luo et al. 2013). This is due to anumber of factors. First, this approach allows one to solve thethree-dimensional (3D), energy-, and time-dependent ParkerTPE without the perceived disadvantages implied by the use ofa numerical grid, implicit to more traditional finite-differenceschemes such as the alternating direction implicit (ADI)scheme (see, e.g., Peaceman & Rachford 1955). Second, thisapproach readily lends itself to parallel computing techniques.Computer clusters are becoming less expensive and morepowerfeul, making computationally expensive techniques suchas this ever more feasible. Lastly, and perhaps mostimportantly, this approach is not plagued by the numericalstability issues that so often limit studies of CR modulationusing finite-difference techniques. With these considerations asmotivation, a stochastic technique is here applied to the studyof CR modulation in an ab initio approach. This entails treatingthe turbulence power spectra as the basic inputs for themodulation study, using diffusion coefficients derived fromthese spectra with recent, novel scattering theories, andmodeling the spatial dependence of the various basicturbulence quantities throughout the heliosphere with aturbulence transport model (TTM) that yields results inreasonable agreement with spacecraft observations of the same.

This paper is organized as follows. A detailed introduction tothe numerical technique followed in this approach, as well asother details pertaining to such a CR modulation code, is givenin Section 2. The code presented in this study incorporatesseveral refinements and recent advances in scattering theory nottaken into account in previous studies. It is benchmarkedagainst the results of Engelbrecht & Burger (2013a) inSection 2.1. A novel aspect of the current approach is the use

of CR boundary intensity spectra constructed using Voyagerobservations within the heliospheric termination shock as inputfor the stochastic modulation code, discussed in Section 2.2.This is motivated by the fact that previous ab initio codes, dueto assumptions implicit to the TTMs they used, could not takeinto account the effects of the heliosheath, a region where aconsiderable amount of CR modulation occurs. The new inputintensity spectra provide a way to take this modulation intoaccount without necessitating ad hoc assumptions as to particleand turbulence transport in the heliosheath. Their use alsoallows for the testing of assumptions as to the transportcoefficients employed on CR intensities computed from firstprinciples, when these intensities are compared to spacecraftobservations at Earth. Results from the Oughton et al. (2011)two-component TTM are used as inputs for an observationallyand theoretically motivated form of the two-dimensional (2D)turbulence power spectrum. New perpendicular mean free pathexpressions derived using this spectral form from the unifiednonlinear theory (UNLT) of Shalchi (2010) as well as thosederived by Engelbrecht & Burger (2013b) from the extendednonlinear guiding center (ENLGC) theory of Shalchi (2006) areemployed. This spectral form is also used to derive anexpression for the 2D ultrascale (see, e.g., Matthaeus et al.2007), which is a key input for the turbulence-reduced driftcoefficient of Burger & Visser (2010) used in this study. Thesetransport parameters are discussed in Section 3. The diffusionand drift coefficients have been shown by Engelbrecht &Burger (2014) to be extremely sensitive to assumptions maderegardingthe low-wavenumber behavior of the 2D turbulencepower spectrum, albeit using a different 2D spectral form anddifferent turbulence quantities in their calculations with a finite-difference ab initio modulation code. Taking this into account,various values and spatial scalings for the 2D outerscale, thelengthscale at which energy-containing range on the assumed2D turbulence power spectrum commences, are chosen basedon what little is currently known of this quantity. Thedifferences between the abovementioned ENLGC and UNLT

The Astrophysical Journal, 814:152 (13pp), 2015 December 1 doi:10.1088/0004-637X/814/2/152© 2015. The American Astronomical Society. All rights reserved.

1

perpendicular mean free path expressions as a function of theseassumed forms for the 2D outerscale are investigated at variousrigidities and as a function of heliocentric radial distance inSection 4. The Burger & Visser (2010) turbulence-reduced driftcoefficient is also sensitive to the behavior of the 2D outerscale,and is similarly studied. Finally, galactic proton CR intensitiesat 1 AU computed assuming these various forms of the 2Douterscale, for perpendicular mean free path expressionsderived from both scattering theories considered here, arepresented in Section 5, and compared with spacecraftobservations. Conclusions drawn as to the possible low-wavenumber behavior of the 2D turbulence power spectra atlarge radial distances are presented in Section 6.

2. THE MODULATION MODEL

The Parker CR transport equation in the absence of sinks andsources of energetic particles is given by

K V Vf

tf f

f

p

1

3 ln, 10

0 sw 0 sw0( ) ( )· · · · ( )

¶¶

= - + ¶

¶

with f0(r, p, t) the omnidirectional CR phase space density, as afunction of particle momentum p, and related to theobservationally reported CR differential intensity by jT=p2f0(see, e.g., Moraal 2013). Various processes modulating aninitialinterstellar CR differential intensity jLIS are containedwithin the above equation, including CR diffusion, drift due togradients in and curvatures of the heliospheric magnetic field(HMF), the effects of the outward convection of CRs by thesolar wind, and adiabatic cooling.

Following the approach outlined by Zhang (1999), Pei et al.(2010), and Strauss et al. (2011), Equation (1) can be written interms of a set of equivalent Itō stochastic differential equations(SDEs) given by

dx A x dt B x dW , 2i i ij

i j i i,( ) ( ) · ( )å= +

with i r E, , , ,[ ]q fÎ and xi(t) Ito processes (see, e.g., Zhang1999; Gardiner 2004). Furthermore, dWi satisfy a Weinerprocess, such that (Pei et al. 2010; Strauss et al. 2011)

dW t t dt , 3i ( ) ( ) ( )h=

where η(t) represents a pseudo-random,Gaussian-distributednumber between zero and one. Pseudo-random numbers aregenerated in this study following the same approach as that ofStrauss et al. (2011), by using the Mersenne Twister algorithmdeveloped by Matsumoto & Nishimura (1998).

In the present study, Equation (2) is solved in the time-backward manner using the Euler–Maruyama approximation(Maruyama 1955) employed by, e.g., Pei et al. (2010) andStrauss et al. (2011), so that the evolution of a large number Nof pseudo-particles with a particular energy at an initialspe-cified phase space point and time x t,i

o o( ) is iteratively followedto their exit points and times (and energies) x t,i

e e( ) at somespecified boundary, after which the average CR intensity at theinitial point is calculated with (Strauss et al. 2011)

j x tN

j x t,1

, , 4io o

k

N

i ke

ke

1LIS ,( )( ) ( )å=

=

using the assumed intensity at the boundary jLIS. Assuming ageneralized three-component HMF, Pei et al. (2010) show that

B

B

B

Br

Br

Br

B B B

22

,

2 ,

2,

2,

2,

2

sin,

0, 5

r r r

rr r rr

r r

r

1,1

2

2 2

2

1,2 2

2

1,3

2,2

2

2,3

3,3

2,1 3,1 3,2

( )

( )

⎛⎝⎜⎜

⎞⎠⎟⎟

⎛⎝⎜⎜

⎞⎠⎟⎟

k k k k k

k k k k k k k

k k k

k k k k

k k kk

k

k

kk

k k k

kk

k

q

=

-

+ + -

-

=-

--

=

=-

=

=

= = =

ff q f q qf

qf qq f qq ff

qf qq ff

f qf q ff

qf qq ffqq

qf

ff

f

ff

qq qf ff

qf

ff

ff

whereas the components of the vector A are given as

Ar r

rr

rV V

Ar r

rr

r

V

r

Ar r r

r

r

V

r

Ar r

r VE E

E EE

1 1

sin1

sinsin ,

1 1

sinsin

1

sin,

1

sin

1

sin

1

sin sin,

1

3

2. 6

r rr r

r d r

r

d

d

Eo

o

22

sw ,

22

2

2,

2 2 2 2

2 2

,

22

sw( ) ( ) ( )

kq f

k

q qk q

kq q

k q

q fk

q fk

qk

q qk

q

=¶¶

+¶¶

+¶¶

- -

=¶¶

+¶¶

+¶¶

-

=¶¶

+¶¶

+¶¶

-

=¶¶

+

+

f

q

q q qq

qfq

f ff ff

qff

The quantity Eo is the rest-mass energy of the species of CR inquestion, and κi,j denote elements of the diffusion tensor K inEquation (1). Note that the signs of the solar wind and driftspeeds Vsw and Vd, as well as of AE, are the opposite of thosepresented in, e.g., Pei et al. (2010). This isto make it explicitthat a time-backward approach is taken here in the solution ofEquation (2). The components of the drift velocity are, as inEngelbrecht & Burger (2013a), calculated following theapproach of Burger (2012). This allows for the calculation ofthe current sheet drift effects for large heliospheric tilt angles,even though the emphasis of the current study is on themodulation of galactic CR protons during solar minimumconditions.The elements of the diffusion tensor aretransformed in this

modelto a right-handed coordinate system aligned with a

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The Astrophysical Journal, 814:152 (13pp), 2015 December 1 Engelbrecht & Burger

general 3D HMF using (Burger et al. 2008)

cos sin cos sin

cos sin sin cos sin

sin cos cos cos sin

cos sin sin cos sin

cos sin sin cos

sin cos sin cos cos

sin cos cos cos sin

sin cos sin cos cos

sin cos

7

rr

r A

r A

r A

A

r A

A

2,3

2 2,2

2

2,3

2,2

,3

2,3

2,2

2,3

2 2,2

2

,3

,3

,3

2,3

2

( )( )( )( )( )( )( )( )

( )

k k y k y z k z

k k y k y k z z k y

k k k y y z k y z

k k y k y k z z k y

k k y k y z k z

k k k y y z k y z

k k k y y z k y z

k k k y y z k y z

k k y k y

= + +

= + - -

= - + -

= + - +

= + +

= - + +

= - + +

= - + -

= +

q

f

q

qq

qf

f

fq

ff

^ ^

^ ^

^

^ ^

^ ^

^

^

^

^

with the HMF winding angle denoted by ψ, and the angle ζ

defined as

B

B

B B

BB

B B

B

B B

sin ,

cos ,

sin ,

cos . 8

r

r

r

r

2 2

2 2

2 2( )

y

y

z

z

=-

=+

=+

=+

f

q

q

q

q

This means that the tangent of the HMF winding angle ψ

becomes

B

B Btan , 9

r2 2

( )y = -+

f

q

which reduces to the usual definition of Bf/Br in a Parker(1958) HMF without a meridional component. The parallel,perpendicular, and antisymmetric diffusion coefficients givenin Equation (7) will be discussed in more detail in Section 3.Note that in this study κ⊥,2=κ⊥,3, as axisymmetricperpendicular diffusion is assumed. These diffusion and driftcoefficients are extremely sensitive to assumptions made as tothe behavior of the turbulence power spectra throughout theheliosphere (see, e.g., Shalchi et al. 2010; Engelbrecht &Burger 2013b), and require as inputs values for turbulencequantities such as the magnetic variance. These turbulencequantities arecalculated hereusing the two-componentOughton et al. (2011) TTM. This model is solved as describedby Engelbrecht & Burger (2013b) for generic solar minimumconditions, utilizing the same boundary values, and modelingturbulence-related terms in exactly the same way. Theboundary values are chosento yield model results in reason-able agreement with spacecraft observations of turbulencequantities throughout the heliosphere, as described by Engel-brecht & Burger (2013a). In the solution of this TTM, thepickup ion contribution to the wavelike fluctuations is ignored,as is done by Engelbrecht & Burger (2013b). This is due to thefact that the ionization of neutral hydrogen generates fluctua-tions at or near the wavenumber corresponding to the proton

gyrofrequency (see, e.g., Isenberg et al. 2003; Isenberg 2005,and Cannon et al. 2014a, 2014b), close to the high-wavenumber dissipation range on the turbulence powerspectrum (see, e.g., Leamon et al. 2000). Therefore, itscontribution is not expected to significantly affect the level orwavenumber dependence of the fluctuation spectra at lowwavenumbers, and hence not to significantly affect thetransport for galactic cosmic ray protons with rigidities in therange (∼0.1–10 GV) of interest to this study. The Parker (1958)model for the HMF is employed, although by Equations (5) and(6) the modulation model presented here would also be able todeal with more complicated Fisk-type (Fisk 1996)HMFmodels, such as the Schwadron–Parker hybrid HMF proposedby Hitge & Burger (2010). The aim here is to model genericsolar minimum conditions, and to this end the latitudinal solarwind speed profile is modeled using a hyperbolic tangentfunction, so that Vsw=400 km s−1 in the ecliptic plane, and800 km s−1 over the poles, as observed by Ulysses (McComaset al. 2000). Drift along the heliospheric current sheet ismodeled following the approach of Burger (2012), assuming atilt angle of 10°. This approach yields results in goodagreement with the approach followed by Pei et al. (2010)and Strauss et al. (2012). These large-scale quantitiessimultaneouslyserveas inputs for the TTMand the SDEmodulation code.Use of the results of the Oughton et al. (2011) TTM as inputs

for expressions for turbulence power spectra leads to complexspatial dependences for diffusion coefficients. To illustrate theorigin of this dependence, contour plots of meridional slicestaken at 0° azimuth of the variances and correlation scales,calculated from the results of the Oughton et al. (2011) TTM,are shown in Figures 1 and 2, respectively. For any given radialspoke, the variances shown in Figure 1 decrease almostmonotonically with increasing radial distance due to theomission of pickup ion effects, a result consistent with thetheoretical findings of Hunana & Zank (2010). Both variancestend to be larger toward the poles in the inner heliosphere,reflecting the observations of higher levels of turbulence atthese latitudes reported by Forsyth et al. (1994), Bavassonoet al. (2000a, 2000b), and Erdös & Balogh (2005), with slabvariances becoming larger than 2D variances. Larger variancevalues at intermediate latitudes are due to increased stream-shear effects in the region where the slow solar wind transitionsinto the fast as a function of latitude.The corresponding correlation scales are shown in

Figure 2. Both scales broadly exhibit a latitudinal variation,with larger values toward the poles than in the ecliptic, inqualitative agreement with the observations reported byDasso et al. (2005) andWeygand et al. (2011), with largervalues for both quantities at latitudes corresponding toenhanced stream-shear effects. As a function of radialdistance, the 2D correlation scale increases monotonically,also in qualitative agreement with the Voyager observationsreported by Smith et al. (2001).

2.1. Benchmarking the SDE Model

As a benchmark of this newly developed model, compar-isons are made with the results of Engelbrecht & Burger(2013a) using identical input parameters. Figure 3 shows acomparison of results for positive and negative magnetic

3

The Astrophysical Journal, 814:152 (13pp), 2015 December 1 Engelbrecht & Burger

polarity cycles (red lines for the ADI code and blue lines withsquares for the stochastic code) as well as for the scenariowhere drift effects are neglected (thick black line for the ADIcode and thin black line with squares for the stochastic code).Selected spacecraft observations are indicatedby dots (McDo-nald et al. 1992) and triangles (Shikaze et al. 2007),respectively. The spacecraft data are shown simply to guidethe eye. Clearly, for both magnetic polarity cycles, as well asfor the no-drift scenario, the SDE approach yealds results ingood agreement with those computed using the fully ab initioADI modulation code of Engelbrecht & Burger (2013a). Themaximum difference in results computed with these codes isless than 10%. Note, however, that intensities computed withthe ADI code tend to decrease somewhat as magnitude of themomentum steps used is decreased. This would improveagreement between the approaches.

2.2. New Input Spectra

It has long been known that considerable CR modulationoccurs within the heliosheath (e.g., McDonald et al. 2000;Caballero-Lopez et al. 2010), and recent Voyager observationsreported by, e.g., Webber & McDonald (2013) andStone et al.(2013) confirm this fact. Figure 4 shows the Burger et al.(2008) input spectrum as a function of energy, as well asVoyager observations of proton intensities at 122 AU reported

by Stone et al. (2013; squares). This input spectrum, if used asa “local interstellar spectrum” at 100 AUas was done byEngelbrecht & Burger (2013a) at 100 AU, is clearly anoverestimate. Furthermore, the heliospheric structure is muchmore complicated than previously assumed (see, e.g., Opheret al. 2015), so the assumption of a spherically symmetricheliosphere without a termination shock as made by Engel-brecht & Burger (2013a) is most likely an oversimplification.These assumptions, however, are necessitated by the fact thatthe Oughton et al. (2011) TTM is derived using assumptionsthat are not valid in the heliosheath. To circumvent this issue,an observationally motivated input spectrum is proposed here,based on galactic CR proton intensities observed by Voyager 1and 2 before their termination shock crossings. Such aspectrum would already be modulated by whatever processesoccur in the heliosheath, and would provide a more realisticinput which can form the basis of comparisons of computedgalactic proton intensities with spacecraft observations of thesame in the inner heliosphere.Webber et al. (2008) report on galactic proton intensities

measured by both Voyager spacecraft prior to their respectivetermination shock encounters. These authors find a clearcharge-sign dependence, with intensities detected by Voyager 2during the negative polarity cycle (A< 0) being 1.4–1.7 timeslarger than those detected by Voyager 1 at 73 AU during thepositive polarity cycle. The Voyager 1 intensities, taken at

Figure 1. Slab (right panel) and 2D (left panel) variances calculated from the results of the Oughton et al. (2011) TTM, following the approach outlined byEngelbrecht & Burger (2013b).

4

The Astrophysical Journal, 814:152 (13pp), 2015 December 1 Engelbrecht & Burger

73 AU, are shown in Figure 4 (red triangles), along with thesame data, multiplied by a factor of 1.4 (blue triangles). Thelatter set of points will be taken to represent a lower estimate ofthe A<0 intensities at 73 AU. Observations of protonintensities of Earth shown in Figure 3 are shownonce moretoguide the eye. To take into account the charge-sign dependence

of the observations at 73 AU, two different input spectra werefit to Voyager observations. These fits (in units of particles perMeV per m2 per s per sr), based on the Burger et al. (2008)expression, are shown in Figure 4 (red and blue lines), and are

Figure 2. Correlation scales calculated from the results of the Oughton et al. (2011) TTM, following the approach outlined by Engelbrecht & Burger (2013b).

Figure 3. Galactic cosmic ray proton differential intensities calculated usingthe ADI ab initio modulation code of Engelbrecht & Burger (2013a; red lines)and the SDE approach employed here (blue lines and squares), for positive(solid lines) and negative (dashed lines) magnetic polarity cycles. Black linesand squares correspond to solutions where drift effects are switched off. Datashown are taken from McDonald et al. (1992) and Shikaze et al. (2007).

Figure 4. Galactic cosmic ray proton differential intensities observed byvarious spacecraft in different regions of the heliosphere, as well as themodulation code input spectra of Burger et al. (2008) and this study (black, red,and blue solid lines), as a function of kinetic energy. Squares denote Voyager 1observations at 122 AU reported by Stone et al. (2013), and triangles Voyager1 observations at 73 AU reported by Webber et al. (2008), while the black andwhite circles and the green stars represent 1 AU IMP8 and BESS97observations taken from McDonald et al. (1992) and Shikaze et al. (2007).

5

The Astrophysical Journal, 814:152 (13pp), 2015 December 1 Engelbrecht & Burger

given by

jP P

P P73 AU

11.4

0.85 4.0, 10T

A o

o

02.58

2.2

( )( )

( ) ( )=+

>-

-

where P denotes the particle rigidity in GV, Po = 1 GV, and

jP P

P P73 AU

14.4

0.85 4.0, 11T

A o

o

02.75

2.8

( )( )

( ) ( )=+

<-

-

respectively. Note that these input spectra do not take intoaccount anomalous protons, and therefore do not increase at thelowest energies (see, e.g., Fichtner 2001). The crossover of thenew input spectra, seen at ∼0.025 GeV, occurs at an energy toolow to affect the modulated intensity spectra to be presentedhere. In what follows, Equations (10) and (11) are used asboundary spectra in the ab initio SDE code at 73 AU, for therelevant magnetic polarity cycles.

3. THE DIFFUSION TENSOR

The diffusion coefficients parallel and perpendicular to theHMF κP and κ⊥ in Equation (7) are related to thecorresponding parallel and perpendicular mean free paths byκP,⊥=(v/3)λP,⊥, with v the particle speed. In this study, acomposite slab/2D model for turbulence is assumed (see, e.g.,Bieber et al. 1994). The galactic proton parallel mean free pathemployed in this study is that used by, e.g., Burger et al. (2008)and Engelbrecht & Burger (2013a), based on the results derivedfrom the quasilinear theory of Jokipii (1966) by Teufel &Schlickeiser (2003) for a slab turbulence power spectrum witha wavenumber-independent energy range, and an inertial rangewith a spectral index of −s. This expression is given by

s

s

R

k

B

B

R

s s

3

1

1

4

2

2 4, 12

o

s

2

min

2

sl2( )

·( )( )

( )

⎡⎣⎢

⎤⎦⎥

lp d

=-

+- -

-

where R=RLkmin is a function of the maximal Larmor radiusRL and the wavenumber kmin at which the inertial range on theslab turbulence power spectrum commences. Note that Bo andBsl

2d denote the uniform background HMF and the slabvariance, respectively.

The perpendicular mean free path used by Engelbrecht &Burger (2013b) is used in this study. This expression is derivedfrom the ENLGC theory of Shalchi (2006), which is based onthe NLGC theory proposed by Matthaeus et al. (2003), andyields results in good agreement with numerical test simula-tions of the perpendicular diffusion coefficient as well as withspacecraft observations of the same (see, e.g., Shalchi2006, 2009; Minnie et al. 2007a). From this theory, theperpendicular diffusion coefficient can be calculated, assumingmagnetostatic axisymmetric perpendicular fluctuations,

ka v

Bd k

S

v k3, 13ENLGC

2 2

02

32D

2

( ) ( )

òkl k

=+

^^ ^

where S2D denotes the 2D modal spectrum. Note that the modalspectrum is related to the omnidirectional spectrum byG k k S k2 .2D 2D( ) ( )p=^ ^ ^ Equation (13) is similar to the NLGCexpression given by Shalchi et al. (2004), but without the direct

contribution from the slab spectrum. This omission ismotivated by the fact that the numerical simulations of, e.g.,Qin et al. (2002) show that the slab contribution is subdiffusive(Shalchi 2006, 2010, 2015). Note that the quantity a is anumerical constant, assumed here to have a value of1 3 afterMatthaeus et al. (2003).A new perpendicular mean free path expression, derived

from the UNLT proposed by Shalchi (2010), is also employedin this study. In this theory, the perpendicular diffusioncoefficient for slab/2D composite turbulence can be calculatedusing

ka v

Bd k

S

v k3 4 3. 14UNLT

2 2

02

32D

2

( )( )

( )

òkl k

=+

^^ ^

Note that Equation (14) differs from the ENLGC result by onlya factor of 4/3 in the denominator of the integrand.Perpendicular mean free paths for these models arederivedherefor a 2D turbulence modal power spectrum that yields ak 1^- wavenumber dependence in the energy-containing range ofthe omnidirectional power spectrum, given by Engelbrecht &Burger (2013b) as

S kC B

k

k k

k k

k k

2

, ;

, ;

, ,

15

q

2D 1 2D 2D2

2D

out

1

out out1

2D1

out1

2D1

2D 2D1

( )

( )

( )( )

∣ ∣

∣ ∣

∣ ∣

( )

⎧

⎨⎪⎪

⎩⎪⎪

⎛⎝⎜

⎞⎠⎟

l dp

ll

l l

l l l

l l

=

´

<

<n

^-

^

-

^ ^-

^- -

^-

^-

^-

where the normalization constant C−1 is determined using

dk S k0

2D ( )ò¥

^ ^ = B2D2d (see, e.g., Matthaeus et al. 2007), with

B2D2d the 2D variance. The normalization constant for this

spectral form is given by Engelbrecht & Burger (2013b)

Cq

ln1

1

1

1. 161

out

2D

1

( )⎡⎣⎢⎢

⎛⎝⎜

⎛⎝⎜

⎞⎠⎟

⎞⎠⎟

⎤⎦⎥⎥

ll n

= ++

+-

-

-

The form of this spectrum is motivated by spacecraftobservations of turbulence at Earth (e.g., Bieber et al. 1993;Smith et al. 2006b). The quantity q represents the spectralindex of an “outer” range at small wavenumbers added to thisspectral form to ensure a finite 2D ultrascale (Matthaeus et al.2007), a quantity employed in the following, and is set to avalue of 3 based on physical considerations discussed byMatthaeus et al. (2007). The spectral index in the inertial rangeof this spectrum ν is set here at a Kolmogorov value of 5/3.Lastly, λout and λ2D are the length scales at which the energy-containing and inertial ranges commence. The perpendicularmean free path derived using this spectrum is given in a generalform by

a

B

C Bh h h

2,

17o

E U E U E U

2

21 2D 2D

2

,1, ,2, ,3,[ ]( )

ll dl

= + +^-

^^ ^ ^

where the subscript E/U denotes the theory (either ENLGC orUNLT) used in the derivation of the mean free path.

6

The Astrophysical Journal, 814:152 (13pp), 2015 December 1 Engelbrecht & Burger

When the UNLT of Shalchi (2010) is used to derive λ⊥

hq

Fq q

x

hx

y

h F y

11,

1

2,

3

2; ,

2log

1

1,

9

4 11,

1

2,

3

2; , 18

U U

UU

U

U U

,1,2D

2 12

,2,2D

2

2

,3,2D

2 12

( )

( )( )

⎜ ⎟

⎜ ⎟

⎛⎝

⎞⎠

⎡⎣⎢⎢

⎤⎦⎥⎥

⎛⎝

⎞⎠

l ll

l ll

ln

n n

=+

+ +-

=++

=+

+ +-

^^ -

^^

^

where

x3

2, 19U

out ( )

ll l

=^

y3

2. 20U

2D ( )

ll l

=^

Using the ENLGC theory of Shalchi (2006) for a k−1 energyrange, Engelbrecht & Burger (2013b) find that

hq

Fq q

x

hx

y

h F y

11,

1

2,

3

2; ,

2log

1

1,

3

11,

1

2,

3

2; , 21

E E

EE

E

E E

,1,2D

2 12

,2,2D

2

2

,3,2D

21

2

( )

( )( )

⎜ ⎟

⎜ ⎟

⎛⎝

⎞⎠

⎡⎣⎢⎢

⎤⎦⎥⎥

⎛⎝

⎞⎠

l ll

l ll

ln

n n

=+

+ +-

=++

=+

+ +-

^^ -

^^

^

where

x3

, 22Eout ( )

ll l

=^

y3

. 23E2D ( )

ll l

=^

The above expressions for the perpendicular mean free path areevaluated numerically. Following the approach of Engelbrecht& Burger (2013b), only the second terms of Equations (21) and(18) are evaluated, as the other terms contribute little to thefinal value of λ⊥ (see also Shalchi 2013, 2014).

Drift coefficients (κA) are modeled to take into account theeffects of turbulence (see, e.g., Jokipii 1993; Giacalone et al.1999; Minnie et al. 2007b), following variously the approachesof Bieber & Matthaeus (1997), Burger & Visser (2010), andTautz & Shalchi (2012). For the purposes of this study, onlythat proposed by Burger & Visser (2010) will be employed.This turbulence-reduced drift coefficient, which is a parame-trization of the results proposed by Bieber & Matthaeus(1997)to agree with the numerical simulation results of Minnieet al. (2007b), is given by

R

D

11

3, 24

L c s

c sg

,

,( ) ( )tl

lW =

^

where g R0.3 log 1.0L c s,( )l= + and λc,s is the slab correla-tion scale, is also sensitive to assumptions made as to the low-wavenumber behavior of the 2D turbulence power spectrum, asit is a function of the perpendicular field line random walkdiffusion coefficient D⊥. This quantity, given by (Matthaeus

et al. 1995), such that

D D D D1

24 , 25sl sl

22D2( ) ( )= + +^

where

DB

B

1

226s

oc ssl

2

2 , ( )dl=

and

DB

B

2, 27

ou2D

2D2

, 1 ( )d

l= -

which is a function of the 2D ultrascale, calculated byEngelbrecht & Burger (2013b) for the same spectral form asis assumed in this study as

Cq

1

1

1

1log . 28u, 1 1 2D

2 out

2D

12

( )⎡⎣⎢⎢

⎛⎝⎜

⎡⎣⎢

⎤⎦⎥

⎞⎠⎟

⎤⎦⎥⎥l l

nll

=-

++

+- -

In the following, the simplifying assumption is made that the2D turnover scale is equal to the 2D correlation scale.

4. THE SENSITIVITY OF DIFFUSION AND DRIFTCOEFFICIENTS TO CHANGES IN THE LOW-

WAVENUMBER BEHAVIOR OF THE 2D TURBULENCEPOWER SPECTRUM

In the present study, all turbulence quantities in Equation (15)except those pertaining to the “outer” range at the smallestwavenumbers are either modeled using the Oughton et al.(2011) TTM (as is the case for variances and correlation scales)or pre-specified based on spacecraft observations at Earth (as isthe case for the spectral indices in the inertial and energy-containing ranges). This leaves only the 2D outerscale λout andthe spectral index q of the outer range as candidates for aparameter study as to the effects of the low-wavenumberbehavior of the spectrum on CR modulation. Of thesequantities, the latter has been shown not to have a great effecton CR spectra computed with an ab initio modulation code(Engelbrecht 2013), while the former has been demonstrated tohave a significant effect (Engelbrecht & Burger 2014), albeitwith different diffusion coefficients and turbulence quantities.No spacecraft observations currently exist for either of thesequantities, thus a relatively ad hoc scaling for the 2D outerscalemust be chosen. This choice, however, can be guided to somedegree by inferences drawn from extant observations ofturbulence spectra at Earth, and from models of the transportof turbulent fluctuations throughout the heliosphere that canreproduce spacecraft observations, at least qualitatively, in theouter reaches of the heliosphere. The consensus from the latterappears to be that some form of driving still occurs at smallwavenumbers even at great radial distances in the heliosphere,at least within the termination shock (see, e.g., Zank et al. 1996;Breech et al. 2008; Oughton et al. 2011 and Zank et al. 2012).This motivates the assumption, implicit to the fixed formchosen for the 2D spectrum in Equation (15), of an energy-containing range of finite extent at all radial distancesconsidered, and therefore of a 2D outerscale consistentlygreater than the 2D turnover scale λ2D. Furthermore, observa-tions of turbulence spectra at 1 AU have long resolved theenergy-containing range, to the extent of one to two orders ofmagnitude in wavenumber, and yet do not find evidence of the

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The Astrophysical Journal, 814:152 (13pp), 2015 December 1 Engelbrecht & Burger

outer range. This difficulty is probably due to the fact that atsuch large scales non-turbulent effects due to, e.g., solarrotation, begin to impinge on observations (Goldstein &Roberts 1999).

A choice, then, of a form for the 2D outerscale would thenhave it be at least two orders of magnitude larger than the 2Dturnover scale at Earth. Motivated by these considerations, twosimple forms for this quantity are chosen in such a way so thatthe effects of changes in its magnitude can most readily bediscussed. The first approach assumes that λout would have thesame radial dependence as the 2D correlation scale, but remainlarger than it by a constant factor,

n r , 29cout ,2D ( ) ( )l l=

where λc,2D(r) is the 2D correlation scale yielded by theOughton et al. (2011) TTM, and n is a constant taken to assumevalues of 10, 100, and 1000. The case where n = 1000 leads toa value for λout at 73 AU of 69.7 AU is included as a potentialupper value for this quantity. To take into account the fact thatthe 2D outerscale needs to be at least two orders of magnitudelarger than the 2D turnover scale (at least at Earth) as well as toconsider an alternate radial dependence for this quantity, analternative scaling is also employed, so that

n 1 AU 30cout ,2D ( ) ( )l l=

with n chosen to be 100 and 1000, so that the outerscale in thiscase is constant. The value at Earth is that reported byWeygand et al. (2011), who find that λc,2D=0.0074±0.0007 AU. The results of the TTM employed here agree withthis Weygand et al. (2011) value at 1 AU. The various forms ofthe 2D outerscale discussed above, along with the slab and 2Dcorrelation scales yielded by the Oughton et al. (2011) TTM,are shown in the top panel of Figure 5 as a function ofheliocentric radial distance in the ecliptic up to 73 AU, theextent of the model heliosphere in the SDE modulation code.While the case where λout=10λc,2D (1 AU) is shown in thisfigure to illustrate thatbecause it becomes roughly equal toλc,2D, this case leads to a scenario where no energy-containingrange exists in the outer heliosphere, and is thus not consideredhere. All other 2D outerscales shown remain above λc,2D, andreach similar values at larger radial distances. Note that at 1 and73 AU the outerscales for both scaling choices are equal at theformer distance and very similar in value at the latter distancefor the cases where n = 100 or n = 1000 in Equation (29) andn = 10 or n = 100 in Equation (30), respectively.

The bottom panel of Figure 5 shows the modeled 2Domnidirectional spectra at 73 AU corresponding to the choicesof 2D outerscale described by Equation (29), calculated withEquation (15) using outputs from the Oughton et al. (2011)TTM. Spectra calculated for outerscales scaling as Equa-tion (30) are not shown, as these are indistinguishable fromthose in the figure due to the similarity in outerscales yieldedby both scalings at this radial distance. As all three spectra arecalculated assuming the same variance and turnover scalevalues, as well as using the same spectral indices for all threeranges, the net effect of modifying the 2D outerscale does notonly affect to the extent of the energy-containing range in termsof wavenumber, but also the level of the spectrum. Therefore,larger values of λout lead to lower levels of the spectra, at leastwhere their energy and inertial ranges overlap.

The parallel mean free path as well as the UNLT andENLGC perpendicular mean free paths are shown as a functionof rigidity in the ecliptic plane at 1 and 73 AU (top left andright panels) and as a function of heliocentric radial distance inthe solar ecliptic plane for a rigidity of 1.5 GV (bottom panels)in Figure 6, for the various choices of 2D outerscale discussedabove. Palmer (1982) consensus values are also shown atEarth. Only the choices of λout as defined by Equation (29) areshown in the top panels, as the scaling for this quantity definedby Equation (30) yields identical, or very similar, values forλout at the radial distances considered. This is not the case forthe bottom panels, where both scalings are shown in their ownpanelsto illustrate the effect of the different radial dependencesof λout. Figure 7 is in the same format, showing the length-scaleλA associated with κA as a function of rigidity and radialdistance for the various choices of λout considered here. Notethat the parallel mean free path remains unaffected by changesin λout, as this quantity has no bearing on the slab fluctuationspectrum used to derive λP. The parallel mean free pathremains well above the Palmer consensus range for therigidities shown here,due to the choice of boundary values

Figure 5. Top panel shows the assumed radial dependences for the 2Douterscales used in this study, along with the composite 2D and slab correlationscales yielded by the Oughton et al. (2011) TTM, also employed in this study.Bottom panel shows the modeled 2D omnidirectional power spectra G2D(k⊥)used in this study (see Equation (15) and the text below) for some of theassumed 2D outerscale values from the top panel, at 73 AU in the solar eclipticplane. The turbulence quantities used to plot these spectra are taken from theOughton et al. (2011) TTM. Note that only the 2D outerscale is varied fromspectrum to spectrum, and that in this figure, as in the rest of the study, q=3and ν=5/3.

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The Astrophysical Journal, 814:152 (13pp), 2015 December 1 Engelbrecht & Burger

for the TTM, to acquire solar minimum values for the variousturbulence quantities of Equation (12) is a function(see, e.g.,Engelbrecht & Burger 2013a). Considering both figuresbroadly, the results presented here differ considerably fromthose presented by Engelbrecht & Burger (2014), who alsoconsidered the effects of λout on the perpendicular mean freepath and drift scale. This is due to two reasons. First, the 2Dspectral form used here is different to that employed byEngelbrecht & Burger (2014), who assumed a flat energy-containing range. Second, Engelbrecht & Burger (2014) useoutputs from the Oughton et al. (2011) TTM which include thepickup ion contribution to the wavelike fluctuation energybeyond ∼10 AU. These effects are neglected here, resulting insignificant differences in the turbulence-related inputs betweenthe two studies, especially at larger radial distances. Theperpendicular mean free paths presented here, for instance, donot display the ∼P2 rigidity dependence reported by Engel-brecht & Burger (2014) below ∼1 GV at large radialdistancessimply because of the differences in the turbulencequantities employed. Qualitatively, however, from both studiesit is clear that a large value of λout leads to a larger valuefor λ⊥.

Turning now specifically to the results presented hereregarding the perpendicular mean free paths in Figure 6, it is

clear that the choice of λout has a significant effect on λ⊥ forboth scattering theories at all rigidities shown and at both radialdistances considered, with values for λ⊥ ranging, at Earth, fromvalues in excellent agreement with Palmer consensus values forthe perpendicular mean free path (ENLGC case withλout=10λ2D) to well within the Palmer consensus range forthe parallel mean free path (both theories, for λout=1000λ2D).At 73 AU, the range of perpendicular mean free path valuesspanning from the largest (UNLT case with λout=1000λ2D) tothe smallest (again the ENLGC case with λout=10λ2D) coversalmost three orders of magnitude. Results derived from theUNLT for a particular value of λout tend to be larger than thosederived using the ENLGC theory for a given value of the 2Douterscale. One exception to this behavior is the case whereλout=10λ2D, at least until ∼2 AU, as can be seen from thebottom panels of Figure 6. Beyond this distance, the UNLT λ⊥are consistently larger than the ENLGC λ⊥.Comparison of the drift scales as a function of rigidity at

Earth and 73 AU in the top panels of Figure 7 show that largerouterscales lead to smaller λA. This quantity is also verysensitive to the value used for λout, varying by almost twoorders of magnitude as a function of rigidity across the range ofvalues for λout considered in the top two panels of Figure 7.Note, however, that at the highest rigidities shown (above

Figure 6. Proton parallel and perpendicular mean free paths employed in this study, shown as a function of rigidity at 1 and 73 AU (top left and right panels) and as afunction of heliocentric radial distance in the solar ecliptic plane at a rigidity of 1.5 GV (bottom panels), for the various forms assumed in this study for the 2Douterscale λout. Palmer (1982) consensus values for the parallel and perpendicular MFPs are shown in the top left panelfor 1 AU. Note that the legend for the bottomright panel differs from those of the other three panels.

9

The Astrophysical Journal, 814:152 (13pp), 2015 December 1 Engelbrecht & Burger

∼2 GV at 1 AU), and at all rigidities at 73 AU, the drift scalesfor the cases where λout=10λ2D and λout=100λ2D reachvery similar values, with those for the former close displaying a∼P rigidity dependence like that expected of the weakscattering result, where λA=RL. The bottom panels of Figure 7show the radial dependences of the 1.5 GV drift scalescalculated using the forms for λout given in Equations (29)and (30). For the case where the outerscale is always 1000times larger than λ2D, the drift scale at ∼70 AU is ∼100 timeslarger than for the case where the outerscale is constant at 100times the value of λ2D at Earth. The difference between the twocases decreases significantly when the outerscale is decreasedby a factor of 10.

5. COMPUTED INTENSITIES

Using the above diffusion and drift coefficients, with resultsfrom the Oughton et al. (2011) TTM as inputs for basicturbulence quantities, the SDE code presented here is used tocompute galactic CR proton intensities at Earth for the variousforms of 2D outerscale described by Equations (29) and (30).This modulation code is run for generic solar minimum valuesfor both large-scale heliospheric quantities such as the solarwind speed profile and HMF magnitude, as well as for genericsolar minimum values for small-scale quantities pertaining toturbulence, as described in Section 2, using the observationallymotivated spectra discussed in the same section as input at

73 AU (Equations (10) and (11)). Note that the number ofpseudo-particles in the SDE code is set at N=5000 in thefollowing. Figure 8 shows these intensities, computed for bothpositive (red lines, A> 0) and negative (blue lines, A< 0)magnetic polarity cycles, as a function of kinetic energy usingperpendicular mean free path expressions derived from theUNLT (left panels) and ENLGC theories (right panels).Observed galactic proton spectra reported by McDonald et al.(1992) and Potgieter et al. (2013) are also shown. The legend inthe top right panel applies to all the panels except the bottomright one, which is provided with its own legend.In general, intensities computed using UNLT perpendicular

mean free paths are larger than their ENLGC counterparts. Thisreflects the general trend seen in Figure 6, where UNLTperpendicular mean free paths tend to be larger than ENLGCmean free paths for a given λout. The large effect of changes inλout on the perpendicular mean free paths and drift scalestranslates to large differences in the corresponding computedCR intensity spectra, but it is interesting to note that all thecomputed spectra remain quite close to the observations shownin Figure 8. Note that solutions for A>0 are larger than thosefor A<0, as shown by observations, only for the smallestvalue of the outerscale considered in each panel. However, thecrossover in intensities, for both the UNLT and ENLGCresults, occurs at energies below those at which thisphenomenon is observed to occur. For the cases where λout

Figure 7. Proton drift scales employed in this study, shown as afunction of rigidity at 1 and 73 AU (top left and right panels) and as a function of heliocentric radialdistance in the solar ecliptic plane at a rigidity of 1.5 GV (bottom panels), for the various forms assumed in this study for the 2D outerscale λout.

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The Astrophysical Journal, 814:152 (13pp), 2015 December 1 Engelbrecht & Burger

is assumed to be a constant multiple of the 2D correlation scaleyielded by the Oughton et al. (2011) TTM as this quantityvaries with radial distance, use of the smallest outerscales (withλout=10λ2D(r)) yields results quite close to the IMP8observations for both A>0 and A<0, with particularly goodagreement for the ENLGC intensities during A<0. For thischoice of λout, the A>0 and A<0 intensity spectra cross overat an energy slightly below ∼0.2 GeV, with A<0 intensitiesthen being larger at higher energies, in qualitative agreementwith neutron monitor observations (see, e.g., Reinecke &Potgieter 1994 and Potgieter et al. 2001). Similar behavioroccurs for the case where λout=10λ2D(1 AU), for bothscattering theories (panels (c) and (d) of Figure 8). Interest-ingly, spectra computed using ENLGC perpendicular meanfree paths with λout=100λ2D(r) (panel (b) of Figure 8) arelower than those computed using λout=10λ2D(r), a situationnot seen for the UNLT results in panel (a) of the same figure.This is probably due to the fact that at greater radial distances,the drift scale is larger relative to the ENLGC perpendicularmean free path than to the UNLT perpendicular mean free pathfor this particular choice of outerscale. For the casewhere λout=100λ2D(1 AU), the intensities computed usingboth UNLT and ENLGC mean free paths appear very nearlythe same as those computed assuming that λout=10λ2D(r): asshown in Figure 5, both these choices for λoutlead to similarvalues for this quantity, and hence to similar values for thetransport parameters it influences, at large radial distances.Lastly, intensities computed using both scattering theories

while assuming that λout=1000λ2D(r) are considerably higherthan those computed for smaller λout, with A>0 intensitiesagain falling below A<0 intensities. Intriguingly, the A<0intensities computed using UNLT perpendicular mean freepaths agree rather well with the unusually high A<0 protonintensities observed by PAMELA in 2009 December. Thisagreement extends to the lower energy PAMELA observations,which display an energy dependence somewhat different to thatshown by the IMP8 observations. This may be due to theslightly steeper rigidity dependence of the perpendicular meanfree paths calculated using λout=1000λ2D(r), as shown inFigure 6.

6. DISCUSSION AND CONCLUSIONS

The ab initio SDE CR modulation code presented herecombines such an approach for the first time with a TTM, andwas benchmarked against a modulation code that utilizes anADI numerical solver with identical transport parameters forthe Parker transport equation. From Section 5, it is clear thatcomputed galactic CR proton intensities are extremely sensitiveto the low-wavenumber behavior of the 2D turbulence powerspectrum, as well as to the choice of scattering theoryemployed in the derivation of the perpendicular diffusioncoefficient. This is, of course, due to the sensitivity of thisdiffusion coefficient, as well as that of the turbulence-reduceddrift coefficient considered here to assumptions made as to theenergy-containing range on the 2D power spectrum.

Figure 8. Galactic cosmic ray proton differential intensities at 1 AU computed using the SDE ab initio modulation code for different assumed 2D outerscales duringpositive (red lines) and negative (blue lines) magnetic polarity cycles. The left panels show results calculated using a perpendicular mean free path expression derivedfrom the UNLT, while the right panels show results computed using an ENLGC perpendicular mean free path expression. Data shown are taken from McDonald et al.(1992) and Potgieter et al. (2013).

11

The Astrophysical Journal, 814:152 (13pp), 2015 December 1 Engelbrecht & Burger

From Figure 8, it can be seen that only results calculatedassuming a 2D outerscale that is only moderately larger thanthe 2D turnover scale agree qualitatively with what is expectedof spacecraft observations at Earth during periods of “usual”solar activity, as represented by the IMP8 observations shownin the same figure. Also, the computed intensities appear to bemost sensitive to the values of λout at larger radial distances.This would imply that, given solar minimum conditions such asthose observed before the unusual solar minimum in 2009, theextent of the energy-containing range on the 2D spectrum atlarge radial distances would be less than an order of magnitudein wavenumber. This could possibly be tested by an analysis ofVoyager observations such as that presented by Fraternale et al.(2015). These authors, however, note that the Voyager 2 datadisplays many gaps, which could complicate such an analysis.An investigation of second-order structure functions (see, e.g.,Miranda et al. 2012; Ni & Xia 2013 and Wicks et al. 2013)calculated using these observations may, however, prove to beuseful in this regard. Galactic CR proton intensities calculatedwith larger values for λout appear to not qualitatively agree withthe IMP8 observations at all, especially with regards to the factthat intensities computed for A<0 magnetic polarity cyclesare larger than those computed for A>0 conditions. However,the A<0 solution calculated using UNLT perpendicular meanfree paths for the case where λout=1000λ2D(r) agree verywell with the unusually high intensities detected by PAMELAduring the (A< 0) solar minimum of 2009. Perpendicular meanfree paths calculated under the assumption thatλout=1000λ2D(r) also display a rigidity dependence (seeFigure 6) quite different to those shown by perpendicular meanfree paths calculated assuming smaller 2D outerscales. This isreminiscent of the findings of Potgieter et al. (2013), who fitvarious CR proton intensity spectra observed by PAMELA bychanging the rigidity dependence of the parametric parallelmean free path expression they employ, although these authorsfind that a flatter rigidity dependence is needed to fit the 2009intensity spectra. Given that the values for λout yielded by thischoice at large radial distances are probably unrealistic, thiscould be seen to be problematic. However, spacecraftobservations of turbulence quantities such as the magneticvariance have been reported to display a solar cycledependence (see, e.g., Bieber et al. 1993; Smith et al. 2006a;Burger et al. 2014). The TTM used in this study is set uptoyield turbulence quantities in agreement with observationstaken during generic solar minimum conditions. Burger et al.(2014) report thatduring the unusual 2009 solar minimum,turbulence quantities behaved differently than during othersolar minima, echoing the unusual behavior of the HMF duringthis period. From the bottom panel of Figure 5, it is clearthat,holding all other turbulence quantities constant, increasingthe value of the outerscale actsto lower the levels of theresulting 2D spectra relative to each other for wavenumbersabove ∼1.4 AU−1. Note that these wavenumbers correspondroughly to the Larmor radii of galactic CR protons withrigidities in the range of interest to this study. FromEquation (15) this effect could, however, be reproduced byholding λout at some constant value, and varying the value ofB .2D

2d The implication then is that it is the level of the 2Dturbulence power spectrum at large radial distances that ishaving such a marked effect on computed CR intensities. AsBurger et al. (2014) report an unusually low value for thevariance during 2009, it is then plausable that the lower spectral

level during this period may be the cause of the unusually highobserved CR intensities. Finally, as discussed in the previoussection, the smallest values of the outerscale used here lead toresults at least in qualitative agreement with spacecraftobservations, although the crossover between intensitiescomputed for A>0 and A<0 conditions occurs at too lowenergies for both scattering theories considered. Runs for evensmaller values of the outerscale considered here do not lead tobetter agreement with observations. This may be due to at leasttwo reasons. First, a more thorough understanding of the effectsof turbulence on the drift coefficient may resolve this issue, inthat the turbulence-reduced drift coefficient employed here maynot be realistic for turbulence conditions prevalent in the fastsolar wind (see Engelbrecht & Burger 2013b for more detail).Second, this study does not take into account any polarity cycledependence in the diffusion coefficients. Separate studies ofneutron monitor data by Chen & Bieber (1993) and Munakataet al. (2014) both find that the solar minimum parallel meanfree path during A<0 is larger than during A>0. As theperpendicular mean free paths presented here are functions ofλP, taking this into account would affect the computed protonintensities presented here. Third, whereas a steady state isassumed in the present study, a fully time-dependent approachis likely to change the results as well. In this case, intensitiesduring solar minimum are actually determined by conditions inthe period preceding solar minimum. This would requireknowledge of the solarcycle dependence of all of theturbulence quantities required for modulation models, not allof which have been studied in sufficient detail (Burgeret al. 2014).

N.E.E. thanks R. D. Strauss for many valuable discussions.This work is based on research supported in part by theNational Research Foundation of South Africa (Grant Numbers81834, 93592, and 96478). N.E.E. acknowledges support bythe South African National Space Agency.

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The Astrophysical Journal, 814:152 (13pp), 2015 December 1 Engelbrecht & Burger