The focused transport equation

3.12 Summary

4.1.4 The focused transport equation

From the particle speeds (figure 4.2) and the length of the magnetic field lines (figure 4.1), the minimum travelling time from the inner boundary at0.05AU outward along the Parker HMF is calculated for different solar wind speeds and shown in figure 4.3. The85keV electrons are indicated by a vertical dashed green line to guide the eye. The minimum amount of time for a particle to travel between the Sun and the Earth at the speed of light is∼ 8minutes (or7.2 AU/hr), shown as a horizontal red dashed line and serves as an asymptote for higher energy particles.

56 4.1. INTRODUCTION

λ_|| = λrr

cos²ψ. (4.23)

Dµµcan then in turn be inferred from equating the right side of equation 4.23 to

λ||(r, φ) = 3v 8

Z +1

−1

(1−µ²)²

Dµµ(r, µ, φ)dµ, (4.24)

as defined byHasselmann and Wibberenz [1970]. Combining equation 4.22 with equation 4.24 gives

λ||(r, φ) = 3v 8

Z +1

−1

(1−µ²)

Dµµ(r, φ)(|µ|^q−1+H)dµ

= 3v

8Dµµ(r, φ) · K,

(4.25)

whereKhas been substituted for the integral. Accordingly, by rearranging the terms of equa- tion 4.25, the pitch-angle diffusion coefficient is determined by

D_µµ(r, φ) = 3v

8λ_||K. (4.26)

Incorporating equation 4.23 again, gives

D_µµ(r, φ) = 3vK 8

cos²ψ

λ_rr (4.27)

whereψis the winding angle from equation 4.4. Thus, by equation 4.27, the pitch-angle diffu- sion coefficient can be calculated by defining a constant radial mean free path. More compli- cated expressions forD_µµ exist and examples hereof are found inStrauss et al.[2017a] (equa- tions18 and20in that paper) andStrauss and le Roux [2019] (equation24in that paper). The perpendicular diffusion coefficientD⊥is specified in the local HMF aligned coordinate system and takes on the form

D⊥ = D_⊥^(rr) D_⊥^(rφ) D_⊥^(φr) D^(φφ)_⊥

!

= D⊥sin²ψ D⊥sinψcosψ D⊥sinψcosψ D⊥cos²ψ

!

. (4.28)

See Strauss and Fichtner [2015] for a discussion concerning the importance of including the pitch-angle dependence ofD⊥when modeling SEP transport. It is assumed that perpendicular diffusion is efficient close to the Sun and therefore, as argued byDr¨oge et al.[2010], the assump- tion that the ratioλ⊥/λ_||is radially constant would not hold since the result would be a spatial distribution of particles at1AU much wider than is observed for SEP events.Dr¨oge et al.[2010]

then suggest that the perpendicular mean free path rather scales with the gyroradius of the particle, that is, with the magnetic field strength and with the pitch-angle of the particle such that

Λ⊥(r, ψ, µ) =α·λ_||(r)·(r/1AU)²·cosψ(r)·p

1−µ² (4.29)

whereαrepresents the free parameter to simulate the relative contributions of the perpendicu- lar and parallel diffusion.

All terms in the transport equation must be cast into spherical spatial coordinates to be solved in the transport model. Following the standard definition of the divergence operator in spher- ical coordinates, termAfrom equation 4.21 is transformed to

∇ ·(µvˆbf) = 1 r²

∂

∂r µvcosψr²f

| {z }

1

+ ∂

∂φ

−µvsinψf r

| {z }

2

. (4.30)

Note that since the model is solved in the equatorial plane, allsinθterms are equal to1and that since the Parker HMF has no meridional component equation 4.30 also has noθdependency.

Terms1and2are the streaming terms inrand inφ, respectively. TermsBandCfrom equation 4.21 has only an explicit pitch-angle dependence and do not need to be transformed to spherical coordinates and therefore retain their current form. TermDis specified as

(D^x_⊥· ∇f) =

"

D^rr_⊥ D_⊥^rφ D_⊥^φr D_⊥^φφ

#

·

" _∂f

∂r 1 rsinθ

∂f

∂φ

#

=

"

D^rr_⊥^∂f_∂r +D_⊥^rφ_rsin¹ _θ^∂f_∂φ D_⊥^φr∂f_∂r +D_⊥^φφ_rsinθ^{1 ∂f}_∂φ

#

=U ,~ (4.31)

where the standard definition of the gradient of a function in spherical coordinates is used.

Evaluating the divergence of each side of equation 4.31 gives

∇ ·(D^x_⊥· ∇f) =∇ ·U~

= 1 r²

∂

∂r r²U_r

| {z }

a

+ 1

rsinθ

∂

∂φ(U_φ)

| {z }

b

. (4.32)

Where the termsaandbexpand to

a= 1 r²

∂

∂r r²D^rr_⊥∂f

∂r +D_⊥^rr∂²f

∂r² + 1 r²

∂

∂r

rD^rφ_⊥∂f

∂φ +1 r

∂

∂r ∂f

∂φ

D_⊥^rφ, (4.33) and

b= D^φr_⊥ r

∂²f

∂φ∂r +D_⊥^φφ r²

∂²f

∂φ². (4.34)

Finally, combining the results of the transformations of equation 4.21 and the results of equa- tions 4.33 and 4.34 with equation 4.32, give the final form of the focused transport equation in spherical coordinates

58 4.1. INTRODUCTION

∂f

∂t +

streaming in r

z }| {

1 r²

∂

∂r

µvcosψ r²f +

streaming inφ

z }| {

∂

∂φ

−µvsinψ

r f

+

focusing

z }| {

∂

∂µ

1−µ² 2L vf

=

diffusion in r

z }| {

1 r²

∂

∂r r²D^rr_⊥∂f

∂r +D^rφ_⊥ r

∂²f

∂r∂φ+D_⊥^rr∂²f

∂r²

+

diffusion inµ

z }| {

∂

∂µ

D_µµ∂f

∂µ

+

diffusion inφ

z }| {

1 r²

∂

∂r

rD^φr_⊥∂f

∂φ +D^φr_⊥ r

∂²f

∂r∂φ+D^φφ_⊥ r²

∂²f

∂φ²,

(4.35)

while assuming thatD_⊥^rφ has no azimuthal dependence. When equation 4.35 is solved forf, the omni-directional particle intensity is calculated using

F(r, φ, t) = 1 2

Z +1

−1

f(r, φ, µ, t)dµ, (4.36)

and the first-order anisotropy as

A(r, φ, t) = 3 R+1

−1 µf dµ R+1

−1 f dµ , (4.37)

wheref is the pitch-angle dependent intensity measured in a given viewing direction andµis the average pitch-angle cosine for that direction.

Equation 4.35 is solved numerically and follows the same principles as discussed byStrauss and Fichtner[2015]. The transport equation is solved using the so-called operator splitting tech- nique also used byMarchuk[1990],Hatzky[1999], andLampa and Kallenrode[2009]. The focused transport equation is split along spatial and pitch-angle coordinates and along first order and second order terms which give six one-dimensional differential equations.Strauss and Fichtner [2015] provide an example of these sets of equations for theµdimension (see their equations A9 and A10) andHeita[2019] also provides an example when this operator splitting technique is applied to the Roelof equation [Roelof, 1969]. The last three terms of equation 4.35 (diffusion inr,µ, andφ) are the diffusion terms and are solved by a simple explicit time-forward central difference scheme with an accuracy of∆f ∼ O(∆t) +O(∆x)². The streaming terms (both in r andφ) as well as the focussing term of equation 4.35 give rise collectively to the set of ad- vection equations and are solved with a combination of an upwind scheme and a Van Leer flux limiter [van Leer, 1974] which give an accuracy of ∆f ∼ O(∆t²) +O(∆x²). Accordingly, the entire numerical scheme has a numerical accuracy of∆f ∼ O(∆t) +O(∆x²). Next, the boundary conditions are set up. An injection function (which is discussed in more detail in the next section) is specified atr₀ = 0.05AU and the model is only solved up to 3AU. Periodic boundary conditions are used for φ. As shown by Strauss and Fichtner[2015], the boundary conditions applied toµshould be carefully considered to ensure particle conservation is main-

Figure 4.4: The numerical scheme solving theµdiffusion and advection equations. The figure is from the appendix ofStrauss and Fichtner[2015].

tained. Figure 4.4 shows an example of theµgrid used by this transport model close toµ= 1.

fi specifies the cell centers whilsti=N ±1/2specifies the cell faces. The blue arrow indicates fluxes entering the cell and the green arrow indicates fluxes exiting the computational cell. The red arrows show that since the pitch-angle diffusion and the focusing terms are zero at±1, the fluxes through the cell face ati = N + 1/2are equal to zero, therefore the time step of fi is determined by

f_i=N^t+∆t=f_i=N^t + ∆t

∆µF_i=N−1/2^t (4.38)

withF_i=N−1/2^t either the diffusion or advective flux entering or leaving the last computational cell. Consequently, the advective flux term becomes

F_i=N−1/2^advective = v(1−µi)² 2L f_i^t

i=N−1/2

, (4.39)

and the diffusive flux becomes

F_i=N−1/2^diffusive =−D˜µµ

∂f

∂µ

i=N−1/2

, (4.40)

with

D˜_µµ ≈ 1

2(D_µµ,i=N+D_µµ,i=N−1), (4.41)

and

∂f

∂µ

_i=N−1/2

≈ 1

∆µ f_i=N^t −f_i=N^t ₋₁

. (4.42)

60 4.1. INTRODUCTION

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Time (hr)

10

³

10

²

10

¹

10

⁰

Normalised Injection

^acc

= 0.05 hr

acc

= 0.10 hr

acc

= 0.15 hr

acc

= 0.20 hr

acc

= 0.25 hr

Figure 4.5: The normalised isotropic Reid-Axford injection function for different acceleration times. The escape time is held constant atτesc = 1hour.