Evolution of Coronal Mass Ejections and the Corresponding Forbush Decreases: Modeling vs. Multi-Spacecraft Observations

Abstract

One of the very common in situ signatures of interplanetary coronal mass ejections (ICMEs), as well as other interplanetary transients, are Forbush decreases (FDs), i.e. short-term reductions in the galactic cosmic ray (GCR) flux. A two-step FD is often regarded as a textbook example, which presumably owes its specific morphology to the fact that the measuring instrument passed through the ICME head on, encountering first the shock front (if developed), then the sheath, and finally the CME magnetic structure. The interaction of GCRs and the shock/sheath region, as well as the CME magnetic structure, occurs all the way from Sun to Earth, therefore, FDs are expected to reflect the evolutionary properties of CMEs and their sheaths. We apply modeling to different ICME regions in order to obtain a generic two-step FD profile, which qualitatively agrees with our current observation-based understanding of FDs. We next adapt the models for energy dependence to enable comparison with different GCR measurement instruments (as they measure in different particle energy ranges). We test these modeling efforts against a set of multi-spacecraft observations of the same event, using the Forbush decrease model for the expanding flux rope (ForbMod). We find a reasonable agreement of the ForbMod model for the GCR depression in the CME magnetic structure with multi-spacecraft measurements, indicating that modeled FDs reflect well the CME evolution.

Appendices

Appendix A: Diffusion Coefficient

In order to introduce energy dependence we allow the diffusion coefficient, $D$, to be a function of rigidity as well as of time, which can be expressed through an empirical formula as used in numerical models fitted to GCR measurements, as given by Potgieter (2013):

$$ D_{E}(P)=0.02\cdot 10^{22}\cdot k_{\mathrm{||},0}\cdot \beta \frac{P^{a}}{B}\Bigg[\frac{P^{c}+(P_{k})^{c}}{1+(P_{k})^{c}}\Bigg]^{( \frac{b-a}{c})}\,, $$

(10)

where $D_{E}$ is given in units $\mathrm{cm}^{2}\mathrm{s}^{-1}$, $P$ is rigidity in units GV, $B$ is the magnetic field in units nT, and $k_{\mathrm{||},0}$, $a$, $b$, $c$ and $P_{k}$ are parameters obtained empirically from the observation of the GCR spectrum using instruments such as Payload for Antimatter Matter Exploration and Light-nuclei Astrophysics (PAMELA: Adriani et al., 2011) onboard the Russian Resurs-DK1 satellite as by (Potgieter et al., 2014) or the Alpha Magnetic Spectrometer (AMS-02: Aguilar et al., 2013) experiment onboard the International Space Station as by, e.g., Corti et al. (2019). We note that the parameters and the dependence in Equation 10 is slightly different for these two studies involving PAMELA and AMS. It should be noted that Potgieter et al. (2014) studied the period of solar minimum 2006 – 2009, whereas Corti et al. (2019) studied the period around and after the solar maximum, 2011 – 2017. It is reasonable to assume that the perpendicular diffusion coefficient changes periodically with time (i.e. solar activity) not only due to the change of the interplanetary magnetic field (IMF) strength, but also the time-varying orientation and complexity of the IMF, closely related to the time-varying speeds and densities of the solar wind and reflected in the change of the parameters in Equation 10. In Figure 10a we combine the results for the perpendicular diffusion coefficient at Earth, $D_{E}$, calculated based on Potgieter et al. (2014) and Corti et al. (2019) for magnetic field strength $B=5$ nT and rigidity $P=1$ GV, where it can be seen that the resulting $D_{E}$ varies periodically in rough anti-correlation to the solar activity indicating that the two empirical formulas corresponding to these two different time-periods can be combined. These two studies therefore provide a calculating frame for the energy-dependent diffusion coefficients. In Figure 10b we show the rigidity dependence of the diffusion coefficient at Earth in 2014 calculated in this way, as well as the rigidity dependence of the initial diffusion coefficient, estimated at $R_{0}=15\mathrm{R_{SUN}}$ based on $D_{0}=D(R(t)/R_{0})^{-n_{B}}$ assuming a magnetic field expansion factor $n_{B}=1.8$.

Figure 10

a) Perpendicular diffusion coefficient at Earth calculated based on Potgieter et al. (2014) for 2006 – 2009 and Corti et al. (2019) for 2011 – 2016 (extrapolated to 2010) for magnetic field strength $B=5$ nT and rigidity $P=1$ GV. b) Rigidity dependence of the diffusion coefficient in 2014 at Earth and at $R_{0}=15~\mathrm{R}_{\odot }$.

Appendix B: The GCR Spectral Intensity

The ‘force–field’ approximation is used to describe the long-term GCR modulation and is typically valid for quiet-time periods. However, Usoskin et al. (2015) have shown that the same approximation can be used to describe the GCR spectrum during an FD. In this approximation all GCR modulation mechanisms are gathered into a single parameter called the modulation potential, $\Phi$, which influences the unmodulated local interstellar spectrum, $J_{\mathrm{LIS}}$, to yield the time-dependent differential energy spectrum of GCRs as observed near Earth:

$$ J(E,\Phi )=J_{\mathrm{LIS}}(E+\Phi ) \frac{(E)(E+2m_{0})}{(E+\Phi )(E+\Phi +2m_{0})}\,, $$

(11)

$$ J_{\mathrm{LIS}}(E)= \frac{1.9\times 10^{4}\cdot P(E)^{-2.78}}{1+0.4866P(E)^{-2.51}}\,, $$

(12)

where we assume all GCRs are protons, $E$ is their kinetic energy, $m_{0}$ their rest mass, $P(E)=\sqrt{E(E+2m_{0})}$ is the rigidity, and $\Phi$ is the modulation potential which is time-dependent and can be obtained empirically based on GCR measurements (Usoskin, Bazilevskaya, and Kovaltsov, 2011; Usoskin et al., 2017). However, it was shown by Gieseler, Heber, and Herbst (2017) that it is not sufficient to describe GCR intensities at Earth by only one rigidity-independent parameter $\Phi$, as it also depends on the energy range of interest and there are severe limitations at lower energies. Therefore, we use a modified force field approach by Gieseler, Heber, and Herbst (2017) in which the rigidity-dependent modulation parameter is given by

$$ \Phi (P) = \textstyle\begin{cases} \frac{\Phi _{\mathrm{USO11}}-\Phi _{\mathrm{PP}}}{P_{\mathrm{USO11}}-P_{\mathrm{PP}}} \cdot (P-P_{\mathrm{PP}})+\Phi _{\mathrm{PP}}, P< P_{\mathrm{USO11}} \\ \Phi _{\mathrm{USO11}}, P\ge P_{\mathrm{USO11}} \end{cases} $$

(13)

where $\Phi _{\mathrm{USO11}}$ is the solar modulation potential obtained for neutron monitors empirically by Usoskin, Bazilevskaya, and Kovaltsov (2011), $\Phi _{\mathrm{PP}}$ is the solar modulation potential derived from the 1.28 GV proton proxies IMP-8 helium and the ACE/Cosmic Ray Isotope Spectrometer (CRIS) carbon by Gieseler, Heber, and Herbst (2017), and $P_{\mathrm{USO11}}=13.83\pm 4.39$ GV and $P_{\mathrm{PP}}=1.28\pm 0.01$ GV are the corresponding mean rigidities. Equations 11–13 therefore provide a scheme to calculate GCR spectrum for a given event. We note that for the event presented in Section 4.3 the uncorrected and corrected solar modulation potentials are 0.681 and 0.97 GV, respectively (Gieseler, Heber, and Herbst, 2017).

Appendix C: The SOHO/EPHIN Response Function

In order to obtain an analytical form of the response function of the single detector F of the SOHO/EPHIN instrument, we use the GEometry ANd Tracking (GEANT 4) Monte Carlo simulation of the instrument performed by Kühl et al. (2015) for the omnidirectional isotropic flux of protons, given in Figure 1 of Kühl et al. (2015). The data points representing the GEANT 4 simulation are presented in Figure 11, where a fitting function is applied in order to derive an approximate analytical form of the functional dependency:

$$ R = 6.4\ln ^{3}(E)-0.42\ln ^{2}(E)+36.8\ln (E)+288.8. $$

(14)

Figure 11

GEANT 4 Monte Carlo simulation of the response function of the single detector F of the SOHO/EPHIN instrument (black dots) and a fitting function approximating its analytical form (dotted line).