andP, as discussed in Section 3.2, must be considered. At energies respectively much higher and much lower than the rest-mass energy, it follows thatP ∝EandP ∝E1/2, so that Eq. 3.53 becomes
j ∝Eγ(s)withγ(s)∈
s+ 2
1−s, s+ 2 2−2s
. (3.54)
To demonstrate,γ(s)is shown in Figure 3.12 as function of kinetic energy for CR electrons and ions. As a result of the smaller rest-mass energy of electrons, the relationship of their kinetic en- ergy with rigidity becomes linear at lower energies, and hence the transition of their accelerated spectra to the smaller high-energy spectral index also occurs at lower energies than for their heavier CR counterparts. The form of Eq. 3.54, where the spectral index of shock-accelerated spectra arising from a monoenergetic source depends only ons, is encountered repeatedly in further chapters, although the nature of the dependence onsmay differ depending on other involved modulation parameters, e.g. those contained inKt. The idealised solution of Eq. 3.46 given by Eq. 3.54 for an infinite plane shock hence provides a guideline for the forms one can expect numerically computed shock-accelerated spectra to have. A notable deviation (in a 2-D geometry) occurs when the diffusion length scale,κrr/Vsw, of a particle becomes comparable to the shock radius, that is, when
κrr Vsw
−
+ κrr
Vsw
+
&rT S. (3.55)
In such an event particles detect the curvature of the shock and escape, which results in the termination of the accelerated power-law distribution. The establishment of the accelerated spectrum is also time-dependent, since the power law of e.g. Eq. 3.54 can only be achieved up to such an energy as there is time for particles to be accelerated to. Such an acceleration time may be expressed in terms of momentum,p, as
τa= 3 Vsw− −Vsw+
Z p p0
"
κrr
Vsw
−
+ κrr
Vsw
+# dp0
p0 , (3.56)
wherep0 (≡ Pmin(Ze)/c) denotes the initial momentum. More on these quantities follow in Chapter 4. See also the related discussions byJones and Ellison[1991].
1996, 1999] being notable examples. The progenitor of the 2-D model applied in the present study was initially developed by le Roux et al.[1996]. CR drifts were subsequently included [Haasbroek, 1997], and variants of this model have been widely applied in CR modulation stud- ies [e.g.Ferreira, 2002;Langner, 2004;Ngobeni, 2006, 2015]. Recently, this model was also applied byStrauss[2010] in the study of ACRs, for which the effects of stochastic acceleration (momen- tum diffusion) had to be included.
The TPE (Eq. 3.42), valid ∀θ, ∀P, r ∈ [r, rHP], r 6= rT S, and the matching condition (Eq.
3.50), valid∀θ,∀P,r=rT S, provide the complete set of equations governing CR transport and acceleration. The TPE is solved using the Locally One-Dimensional (LOD) method [Steenkamp, 1995;Langner, 2004], which relies on a technique called fractional splitting to write the TPE as separate equations, each dependent only on a single computational dimension. Consider the TPE as written in Eq. 3.42 in the following form:
∂f
∂t =a0
∂2f
∂r2 +b0
∂2f
∂θ2 +c0
∂f
∂r +d0
∂f
∂θ +e0
∂f
∂lnP , (3.57)
which is written in terms of rigidity. Note that since no sources are specified using the source term,Qs, it is omitted from this discussion. The coefficients are specified as
a0 = κrr
b0 = κθθ r2 c0 = 1
r2
∂
∂r r2κrr + 1
rsinθ
∂
∂θ(κθrsinθ)−Vsw d0 = 1
r2
∂
∂r(rκrθ) + 1 r2sinθ
∂
∂θ(κθθsinθ) e0 = 1
3r2
∂
∂r r2Vsw
. (3.58)
The LOD method applied to Eq. 3.57 yields the following set of equations radial equation: 1
3
∂f
∂t = a0∂2f
∂r2 +c0∂f
∂r (3.59)
polar equation: 1 3
∂f
∂t = b0∂2f
∂θ2 +d0∂f
∂θ (3.60)
energy equation: 1 3
∂f
∂t = e0
∂f
∂lnP. (3.61)
Note that each of these equations are only valid on a third of a time step, that is, for t0 <
t < t0 + ∆t/3, t0+ ∆t/3 < t < t0 + 2∆t/3, andt0 + 2∆t/3 < t < t0+ ∆t. Note that the ra- dial and polar equations are second-order parabolic partial differential equations (PDEs) that can be solved using the Crank-Nicholson method. The energy equation is also a second-order parabolic PDE if the momentum diffusion term (Eq. 3.43) is present. Its solution in this form is discussed in greater detail byStrauss[2010]. Without the momentum diffusion term, however, it is a first-order hyperbolic PDE, and can either be solved using the method of characteristics [Langner, 2004] or the explicit upwind method [Haasbroek, 1997]. To be solved simultaneous to the above equations is the matching condition of Eq. 3.50, which is also a first-order hyperbolic PDE in the variableslnP andθ. It is solved using the Wendroff’s implicit method in the direc-
0 50 100 150 200 250 300 0
20 40 60 80 100 120 140
Radial grid point number, i
Radial distance (AU)
Figure 3.13:Radial distances versus their corresponding grid points. The vertical and horizontal lines denote the TS position atrT S = 94 AU and i = 200, at which the grid spacing is much finer than elsewhere in the heliosphere. The HP at 122 AU is ati=300.
tion of particle drifts [Steenkamp, 1995]. The matching condition and Eqs. 3.59, 3.60 and 3.61 are furthermore expanded using finite difference schemes, which is discussed thoroughly by Steenkamp[1995] andLangner[2004] and thus not repeated here.
Note finally the boundary conditions and numerical grid sizes applied in the model: The outer- most iteration of the numerical scheme is a simple linear time grid, which continues until such a time at which sufficient convergence of the numerical solution is attained; this is usually after some 40000 iterations when heating due to DSA is included, and on average around 23000 if DSA is suppressed. The radial grid(ri, i= 1 → nr, nr = 300) is transformed to create an un- even spacing of grid points, which are condensed near the TS, but are further apart away from it, as illustrated and discussed byLangner[2004]; see also Figure 3.13. This is for better numer- ical accommodation of the discontinuous SW speed decrease at the TS. The inner modulation boundary is located atr1 ≈0.8AU, and is a reflective boundary, so that
∂f
∂r
r=r1
= 0, (3.62)
which implies that no particles enter or leave this boundary. The outer boundary,rnr(≡rHP), is a free escape boundary, with
f(rnr, θ, P, t) = 0, (3.63)
for ACRs (Chapter 4), but
f(rnr, θ, P, t) =jHPS(rnr, θ, P, t)/P2, (3.64) when GCRs are considered. Here, jHPS represents the differential intensity value associated with the input spectrum, which is specified at the HP in this study, is the same at all polar angles, and does not change with time. These are specified in Chapter 5 for GCR electrons. On the other hand, the polar grid (θj, j = 1 → nj, nj = 37)is equally spaced, with∆θ = 2.5o. The TPE is solved in this study only for the regionθ1 = 0o → θnj = 90o for a spherically
symmetrical heliosphere. Hence, it is assumed that ∂f
∂θ
θ=0,π/2
= 0. (3.65)
Lastly, the rigidity grid(Pk, k= 1→nk, nk= 175)is logarithmic, withPminvaried as required;
the default values are 30 MV for ACRs and 1 MV for GCR electrons. The maximum rigidity is determined byPmax = Pminexp(nk dP), withdP varied between 0.040 and 0.062 as required.
Zero modulation is assumed to occur atPmax, which for GCRs implies that intensities equal those of the input spectrum at all radial distances and latitudes at this rigidity. SeeSteenkamp [1995] for a discussion on numerical stability.