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A simplified ab initio cosmic-ray modulation model: Construction and predictive capabilities Katlego Moloto Eugene Engelbrecht Adri Burger July 17, 2017 Centre for Space Research North-West University Table of contents 1. Problem Statement

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SLIDE 1

A simplified ab initio cosmic-ray modulation model:

Construction and predictive capabilities

Katlego Moloto Eugene Engelbrecht Adri Burger July 17, 2017

Centre for Space Research North-West University

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SLIDE 2

Table of contents

  • 1. Problem Statement
  • 2. Modulation Model
  • 3. Sample Solutions

2

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SLIDE 3

Problem Statement

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SLIDE 4

Problem Statement

Kinetic energy (GeV) 0.1 1 Differential intensity (part.m-2.s-1.sr-1.MeV-1) 0.1 1

1977: Evenson et al. (1983) 1977: von Rosenvinge et al. (1979) 1998: Sanuki et al. (2000) 1965: Fan et al. (1966) 1965: Ormes and Webber (1968) 1965: Balasubrahmanyan et al. (1965) 1996: McDonald et al. (1998) 1987: McDonald et al. (1998) December 2009 PAMELA

Previous solar minima spectra Blue open symbols: A > 0 cycles Red filled symbols: A < 0 cycles

Potgieter et al. (2013, ICRC PROC)

  • Explain observed

modulation self-consistently

3

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Problem Statement

Kinetic energy (GeV) 0.1 1 Differential intensity (part.m-2.s-1.sr-1.MeV-1) 0.1 1

1977: Evenson et al. (1983) 1977: von Rosenvinge et al. (1979) 1998: Sanuki et al. (2000) 1965: Fan et al. (1966) 1965: Ormes and Webber (1968) 1965: Balasubrahmanyan et al. (1965) 1996: McDonald et al. (1998) 1987: McDonald et al. (1998) December 2009 PAMELA

Previous solar minima spectra Blue open symbols: A > 0 cycles Red filled symbols: A < 0 cycles

Potgieter et al. (2013, ICRC PROC)

  • Explain observed

modulation self-consistently

  • Input realistic

solar minimum conditions

3

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SLIDE 6

Problem Statement

Kinetic energy (GeV) 0.1 1 Differential intensity (part.m-2.s-1.sr-1.MeV-1) 0.1 1

1977: Evenson et al. (1983) 1977: von Rosenvinge et al. (1979) 1998: Sanuki et al. (2000) 1965: Fan et al. (1966) 1965: Ormes and Webber (1968) 1965: Balasubrahmanyan et al. (1965) 1996: McDonald et al. (1998) 1987: McDonald et al. (1998) December 2009 PAMELA

Previous solar minima spectra Blue open symbols: A > 0 cycles Red filled symbols: A < 0 cycles

Potgieter et al. (2013, ICRC PROC)

  • Explain observed

modulation self-consistently

  • Input realistic

solar minimum conditions

  • Should give

predictive capacity to modulation code

3

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SLIDE 7

Problem Statement

Kinetic energy (GeV) 0.1 1 Differential intensity (part.m-2.s-1.sr-1.MeV-1) 0.1 1

1977: Evenson et al. (1983) 1977: von Rosenvinge et al. (1979) 1998: Sanuki et al. (2000) 1965: Fan et al. (1966) 1965: Ormes and Webber (1968) 1965: Balasubrahmanyan et al. (1965) 1996: McDonald et al. (1998) 1987: McDonald et al. (1998) December 2009 PAMELA

Previous solar minima spectra Blue open symbols: A > 0 cycles Red filled symbols: A < 0 cycles

Potgieter et al. (2013, ICRC PROC)

  • Explain observed

modulation self-consistently

  • Input realistic

solar minimum conditions

  • Should give

predictive capacity to modulation code

  • This can be done

using an ab initio approach

3

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SLIDE 8

Introduction

  • Ab initio approach to modulation (e.g. Engelbrecht and Burger

2013a and b, APJ) requires turbulence spectra as input for diffusion tensor and not just B or alpha

4

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Introduction

  • Ab initio approach to modulation (e.g. Engelbrecht and Burger

2013a and b, APJ) requires turbulence spectra as input for diffusion tensor and not just B or alpha

  • Simplified version of Engelbrecht and Burger (2015, APJ) model is

used in this study; results are preliminary and qualitative rather than quantitative

4

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Introduction

  • Ab initio approach to modulation (e.g. Engelbrecht and Burger

2013a and b, APJ) requires turbulence spectra as input for diffusion tensor and not just B or alpha

  • Simplified version of Engelbrecht and Burger (2015, APJ) model is

used in this study; results are preliminary and qualitative rather than quantitative

  • Part of a project to study long-term modulation and solar-cycle

dependence of turbulence parameters, thus time dependence

4

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Introduction

  • Ab initio approach to modulation (e.g. Engelbrecht and Burger

2013a and b, APJ) requires turbulence spectra as input for diffusion tensor and not just B or alpha

  • Simplified version of Engelbrecht and Burger (2015, APJ) model is

used in this study; results are preliminary and qualitative rather than quantitative

  • Part of a project to study long-term modulation and solar-cycle

dependence of turbulence parameters, thus time dependence

  • The code used in this project is a 3D Steady-state SDE code.

4

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Modulation Model

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Model

∂f ∂t = − (Vsw + vd) · ∇f + ∇ · (KS · ∇f ) + 1 3(∇ · Vsw) ∂f ∂ ln P

  • The term (Vsw + vd) · ∇f describes the outward convection of

cosmic rays by the solar wind and cosmic ray drift.

5

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Model

∂f ∂t = − (Vsw + vd) · ∇f + ∇ · (KS · ∇f ) + 1 3(∇ · Vsw) ∂f ∂ ln P

  • The term (Vsw + vd) · ∇f describes the outward convection of

cosmic rays by the solar wind and cosmic ray drift.

  • The term 1

3(∇ · Vsw) ∂f ∂ ln P describes adiabatic energy changes

5

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Model

∂f ∂t = − (Vsw + vd) · ∇f + ∇ · (KS · ∇f ) + 1 3(∇ · Vsw) ∂f ∂ ln P

  • The term (Vsw + vd) · ∇f describes the outward convection of

cosmic rays by the solar wind and cosmic ray drift.

  • The term 1

3(∇ · Vsw) ∂f ∂ ln P describes adiabatic energy changes

  • and the remaining term ∇ · (KS · ∇f ) describes diffusion.

5

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Model

  • Strauss. (2010, MSC)
  • A colatitude

dependent solar wind speed is used.

  • Parker Heliospheric

magnetic field is used.

  • neutral sheet drift by

Burger (2012, APJ)

6

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Model

100 50 50 100 100 50 50 100 50 100

Engelbrecht and Burger (2010, ASR)

  • A colatitude

dependent solar wind speed is used.

  • Parker Heliospheric

magnetic field is used.

  • neutral sheet drift by

Burger (2012, APJ)

7

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Model

  • A colatitude

dependent solar wind speed is used

  • Parker Heliospheric

magnetic field is used.

  • neutral sheet drift by

Burger (2012, APJ)

8

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Steady-state 3D Model

  • Steady-state 3D model using “effective” values of 1 AU observations

as input, taking into account outward convection by the solar wind

9

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Effective Values

10 30 50 70 1980 1985 1990 1995 2000 2005 2010 2015 Tilt Angle [degrees] Time [Years] Classic 22 23 24 A > 0 A < 0 A > 0 A < 0 A < 0 21 Spot value 12Months 24Months

10

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Solar Minimum Values

Year Magnetic Field Variance Tilt Angle 1987 5.8 8.3 4.1 1997 5.7 9.4 4.5 2009 4.4 5.7 11.2

  • Nel. (2015, MSC)

11

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Variance

10-3 10-2 10-1 100 101 102 103 100 101 102 δB2 [nT2] R [AU] Ecliptic Zank et al. 1996 Smith et al. 2001 100 101 102 R [AU] Polar 1987 1997 2009

Based on Oughton et al. (2011, JGR) Based on Bavassano et al. (2000, JGR)

12

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Correlation Scales

10-3 10-2 10-1 100 100 101 102 λc [AU] R [AU] Ecliptic Smith et al. 2001, e-folding Smith et al. 2001, integration 100 101 102 R [AU] Polar 2D Slab Weygand et al. 2011 Weygand et al. 2011

Based on Oughton et al. (2011, JGR) Based on Wicks et al. (2010, PRL)

13

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Diffusion Tensor

  • Parallel diffusion based on QLT (Teufel & Schlickeiser 2003, A&A)

λ = 3s √π(s − 1) R2 bkmin

  • Bo

δBslab,x 2 b 4√π + 2 √π(2 − s)(4 − s) b Rs

  • 14
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Diffusion Tensor

  • Parallel diffusion based on QLT (Teufel & Schlickeiser 2003, A&A)

λ = 3s √π(s − 1) R2 bkmin

  • Bo

δBslab,x 2 b 4√π + 2 √π(2 − s)(4 − s) b Rs

  • Perpendicular diffusion based on NLGC (Matthaeus et al. 2003,

APJL; Shalchi et al. 2004) λ⊥ ≈ 2ν − 1 4ν a2F2(ν)l2D √ 3δB2

2D

B2

  • 2/3

λ1/3

  • 14
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SLIDE 26

Diffusion Tensor

  • Parallel diffusion based on QLT (Teufel & Schlickeiser 2003, A&A)

λ = 3s √π(s − 1) R2 bkmin

  • Bo

δBslab,x 2 b 4√π + 2 √π(2 − s)(4 − s) b Rs

  • Perpendicular diffusion based on NLGC (Matthaeus et al. 2003,

APJL; Shalchi et al. 2004) λ⊥ ≈ 2ν − 1 4ν a2F2(ν)l2D √ 3δB2

2D

B2

  • 2/3

λ1/3

  • Drift coefficient derived by Engelbrecht et al (2017, APJ)

κA = v 3 RL

  • 1 +

λ ⊥ RL 2 δBT

2

B0

2

−1

14

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Sample Solutions

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MFP Rigidity

10-4 10-3 10-2 10-1 100 101 0.1 1 10 λ [AU] Rigidity [GV] λ|| λ⊥ λΑ 1987 2009 1997

15

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MFP Radial

10-4 10-3 10-2 10-1 100 101 102 1 10 100 λ [AU] R [AU] λ|| λ⊥ λΑ 1987 2009 1997

16

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Spectra Comparison

10-2 10-1 100 101 102 10-2 10-1 100 101 jT [MeV.s.m2.sr]-1 Kinetic Energy [GeV] LIS 1987 1997 2009 IMP8, A < 0 Pamela, A < 0 IMP8, A > 0

17

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Spectra Comparison

10-2 10-1 100 101 102 10-2 10-1 100 101 jT [MeV.s.m2.sr]-1 Kinetic Energy [GeV] LIS 1987 1997 2009 2019 IMP8, A < 0 Pamela, A < 0 IMP8, A > 0

18

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Summary and Conclusions

  • Simplified ab initio approach with observational inputs for large

scale structures and turbulence quantities relevant to cosmic-ray modulation yields results in reasonable with all three solar minimum data sets.

  • Preliminary prediction for cycle 25 solar minimum is for intensity

even greater than in 2009

  • Model is still in the process of extension and refinement to full

time dependence

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