Modulation in steady-state has been well studied by previous works (Potgieter 2013; Potgieter et al. 2014; Zhao et al. 2014). Potgieter et al. (2014) studied the modulation of proton spectra with the PAMELA data from 2006 to 2009 July, and they concluded that the recent solar minimum was “diffusion dominated.” In the work of Potgieter et al. (2014), parameters in diffusion coefficients and drift coefficients were adapted to observations (see also Potgieter et al. 2015; Raath et al. 2016). Zhao et al. (2014) studied the modulation of the GCR energy spectra during the past three solar minima using an empirical diffusion coefficient model according to Zhang (1999). They found that decreased perpendicular diffusion in polar direction, which is in contrast to the assumption of enhanced diffusion in polar regions that was used to explain the observed Ulysses CR gradients (see, e.g., Potgieter 2000), and increased parallel diffusion might be the reason for the record high-level of GCR intensity measured at Earth. Since the diffusion coefficients describe the scattering of GCRs by random fluctuations in the IMF, turbulence quantities are needed in diffusion theory. In the solar wind, the evaluation of turbulence is well described by magnetohydrodynamic (MHD) theory (Marsch & Tu 1989; Zhou & Matthaeus 1990), and the turbulence transport throughout the heliosphere has been studied over the years (e.g., Zank & Matthaeus 1993; Zank et al. 1996, 2012, 2017; Matthaeus et al. 1999; Smith et al. 2001; Breech et al. 2008; Hunana & Zank 2010; Oughton et al. 2011; Wiengarten et al. 2016). With the turbulence transport models (TTMs), the diffusion tensor can be calculated (e.g., Zank et al. 1998; Pei et al. 2010a; Engelbrecht & Burger 2013; Zhao et al. 2017). Theoretical and numerical works have shown that drift coefficients can be reduced in the presence of turbulence (e.g., Jokipii 1993; Fisk & Schwadron 1995; Giacalone & Jokipii 1999; Candia & Roulet 2004; Stawicki 2005; Minnie et al. 2007; Tautz & Shalchi 2012); see also the first-order approach in Engelbrecht et al. (2017). The reduced drift coefficients, which are obtained from fitting the simulation results (Burger & Visser 2010; Tautz & Shalchi 2012), are also used together with the turbulence transport theory to study the modulation of GCRs (e.g., Engelbrecht & Burger 2013). Using the nearly incompressible (NI) MHD TTM developed by Zank et al. (2017), Zhao et al. (2017) also showed the effect of both weak and moderately strong turbulence on drift coefficients. Since there exists close coupling between turbulence, solar wind, and energetic particles, some work combined large-scale solar wind flow with small-scale fluctuations in a self-consistent way (see, e.g., Usmanov et al. 2011, 2014, 2016; Wiengarten et al. 2015; Shiota et al. 2017) to study the spatial variations of the diffusion coefficients (see, e.g., Chhiber et al. 2017). Furthermore, using a diffusion coefficient model according to Giacalone & Jokipii (1999), Guo & Florinski (2016) studied the modulation of GCRs by CIRs at 1 au. They combined the small-scale turbulence transport with the MHD background for the simulation of cosmic-ray transport to show short-term modulation effects.
Citation: Zank & Matthaeus 1993
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