We develop a numerical model to study the time-dependent modulation of galactic cosmic rays (GCRs) in the inner heliosphere. In the model a time-delayed modified Parker heliospheric magnetic field (HMF) and a new diffusion coefficient model, NLGCE-F, from , are adopted. In addition, the latitudinal dependence of magnetic turbulence magnitude is assumed as ∼ (1 + sin 2 θ)/2 from the observations of Ulysses, and the radial dependence is assumed as ∼ r S , where we choose an expression of S as a function of the heliospheric current sheet (HCS) tilt angle. We show that the analytical expression used to describe the spatial variation of HMF turbulence magnitude agrees well with the Ulysses, Voyager 1, and Voyager 2 observations. By numerically calculating the modulation code we get the proton energy spectra as a function of time during the recent solar minimum, it is shown that the modulation results are consistent with the PAMELA measurements.

Galactic cosmic-ray (GCR) helium and heavier ions are important sources of space radiation, and their elemental spectra and composition can help us better understand the transport in both the galaxy and the heliosphere. A model based on the numerical solution of Parker’s transport equation is used to study the modulation of GCR helium and heavier ions in the inner heliosphere. The model incorporates a modified Parker heliospheric magnetic field, time-dependent diffusion and drift model, time-delayed heliosphere, and randomly determined solar magnetic polarity during solar maximum. We set the outer boundary of modulation at 85 au, and the reference unmodulated GCR energy spectra for GCR helium and heavier ions, which are assumed to have a general form, are determined by fitting the numerical results to the selected GCR measurements, e.g., BESS, ACE/CRIS, HEAO-3-C2, etc. In addition, we use the Sun’s polar magnetic field data from NSO/NISP to determine the possibility of A > 0 during the recent solar maximum, and it gives an improved numerical result during the period 2013–2015. Finally, the fitted unmodulated GCR energy spectra are used to study the long-term modulation of GCRs of helium and heavier ions, and the computed results show good agreement with various GCR measurements.

The 11-year and 22-year modulation of galactic cosmic rays (GCRs) in the inner heliosphere are studied using a numerical model developed by Qin and Shen in 2017. Based on the numerical solutions of Parker's transport equations, the model incorporates a modified Parker heliospheric magnetic field, a locally static time delayed heliosphere, and a time-dependent diffusion coefficients model in which an analytical expression of the variation of magnetic turbulence magnitude throughout the inner heliosphere is applied.Furthermore, during solar maximum, the solar magnetic polarity is determined randomly with the possibility of A > 0 decided by the percentage of the north solar polar magnetic field being outward and the south solar polar magnetic field being inward. The computed results are compared with several GCR observations, e.g., IMP 8, SOHO/EPHIN, Ulysses, Voyager 1 & 2, at various energies and show good agreement.It is shown that our model has successfully reproduced the 11-year and 22-year modulation cycles.