The structured solar wind

The propagation of SEPs between the Sun and the Earth (and beyond) is heavily dependent on the state of the solar wind which can be extremely variable in both space and time [Kocharov et al., 2009]. The Parker HMF geometry is most commonly assumed when simulating the quiet- time transport of SEPs, and including the usual definition of turbulence, the magnetic field is written as

B~ =B~₀+~b (6.1)

whereB~0is the Parker HMF and~bis the fluctuating turbulent component. When~bis averaged overenoughtime, the fluctuating component disappears, such that

h~bi= 0. (6.2)

The question remains what constitutes enough time? If the HMF is averaged over one solar rotation of ∼ 27 days, small-scale turbulent structures will be smeared out. However, if av- eraging over30 second intervals, it will lead to a large fluctuating component and deviation

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94 6.2. THE STRUCTURED SOLAR WIND

Figure 6.1: Left: Illustration of the mean magnetic field, B~₀, conventionally assumed when simulating SEP transport. In this caseh~bi= 0afterappropriateaveraging and the perpendicular diffusion coefficients are all directed parallel to each other and, per definition, perpendicular to the mean field line. Right: Illustration of the mean magnetic field with a perturbed background Parker HMF whereh~bi= 0afterappropriateaveraging, butδ ~B 6= 0. In this case the perpendicu- lar diffusion coefficients are transverse to the field line, but not parallel to each other, and also not perpendicular to the Parker HMF.

from the standard Parker field geometry even during solar-quiet periods. Strauss et al.[2020]

explains that the particles under consideration, e.g. SEPs, determine what part of the magnetic field is considered as turbulence or deterministic structures. SEPs interact with one instance of the magnetic field structure during a short amount of time, and therefore, the averaging time is finite. If the averaging time isshort enough, such thathBi 6=~ B0, then the field structure could rather take the form

B~ =B~0+δ ~B+~b (6.3)

where δ ~B represents the deviations from the standard Parker HMF geometry and the SEPs register this as coherent, deterministic structures, whereas small-scale fluctuations,~b, will be experienced as turbulence withh~bi = 0. This has significant consequences for SEP transport, specifically in the calculation ofD⊥. Figure 6.1 shows the two scenarios discussed in equations 6.1 and 6.3. The left panel represents the scenario of equation 6.1 where the deviation from the standard Parker HMF is added and afterenoughaveragingh~bi ∼0, such that the perpendicular diffusion, D⊥, is directed as indicated in the figure. The right panel of figure 6.1 shows the scenario applicable to equation 6.3. Here, it is assumed thatδ ~B 6= 0, and this leads to a per- turbed Parker background HMF where the perpendicular diffusion is indeed transverse to the field, but not parallel to each other due to the added perturbation. This magnetic field structure could in principle explain the wide-spread events where very efficient perpendicular diffusion is observed.

In this study, equation 6.3 represents a so-called evolving, structured solar wind that devi- ates, on short time scales during solar-quiet periods, from the long-term averaged Parker HMF.

However, care should be taken when averaging over different time scales. Figure 6.2 shows a typical 2D turbulence power spectrum with the perpendicular length scale,L⊥, on the hori- zontal axis. It is assumed thatL⊥ ∼k⁻¹_⊥ , wherek⁻¹_⊥ is the perpendicular wave number. This figure illustrates how the level of turbulence a particle will experience will change depending on the averaging time.

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2D turbulence power spectrum 1 hour averaged B-field data 6 hour averaged B-field data

Figure 6.2: A typical 2D turbulence power spectrum. The length scales for1and6hour mag- netic field data is indicated by the red dashed and dash-dot vertical lines, respectively. The grey shaded region represents the turbulence not observed in the data due to averaging, while the blue shaded region represents the observed coherent structures.

One hour averaged magnetic field data has a length scale of 0.014AU (determined by the Taylor hypothesis [Taylor, 1938]) and is indicated in figure 6.2 by the vertical red dashed line.

The red shaded region around the dashed line represents the range of solar wind speeds from 400to 800kms⁻¹ with the dashed line at 600kms⁻¹. The grey shaded region represents the small-scale turbulence that will not be observed in the resultant1hour averaged magnetic field data. These small-scale features will be averaged out during the averaging process. The same is shown for6hour averaged magnetic field data where the length scale is0.086AU and the range of solar wind speeds is also indicated by the red shaded region around the dash-dot vertical line. The blue shaded region represents the coherent structures the particle will interact with and is defined by theδ ~B term in equation 6.3. If, for example, the averaging takes place over one solar rotation of27 days, the length scale will be6.22 AU which results in ignoring most of the turbulence shown in figure 6.2 and confirms then the magnetic field structure shown in the left panel of figure 6.1. It should be noted that, since one SEP will sample several different structures during its transport between the Sun and the Earth, it would be near impossible to simulate exactly what structures each SEP experiences during its journey. Hence, a statistical description of fluctuations presented in e.g. figure 6.2 is used.

96 6.3. MAPPING MAGNETIC FIELD LINES