On the transition from 3D to 2D transport equations for a study of long-term cosmic-ray intensity variations in the heliosphere

The exact two-dimensional (2D) transport equation (TPE) for galactic cosmic ray (GCR) intensity in the heliosphere averaged over longitude, derived by averaging the full three-dimensional (3D) steady-state TPE over longitude is considered. This exact 2D TPE is equal to that with the averaged 3D TPE coefficients but with the “source-term” 𝑄 due to 3D modulation effects. In particular, 𝑄 is equal to the longitude convolution of the longitudinal variances of the coefficients and the intensity as used in the 3D TPE and as applicable to the modulation of GCR intensity. In previous work we suggested an expression for 𝑄 without solving the 3D TPE for the simplest case when the only characteristic heliospheric feature depending on helio-longitude is the polarity of the heliospheric magnetic field. The motivation for the estimating and studying the structure of the source-term of the exact 2D TPE is two-fold: 1) solving 2D instead of 3D TPE for simulating the long-term variations of GCR intensity looks more easy and logical and 2) studying the structure of the source-term can improve our understanding of the simulated GCR modulation. This study is focused on calculating and analizing the same longitude convolution as 𝑄 numerically, solving the steady-state 3D TPE for the above mentioned simplest case. Beside studying the structure of this calculated source-term, 𝑄1, it is also compared with that estimated without solving the 3D TPE, 𝑄2