[Paper Review] Effects of the tilted and wavy current sheet on the solar modulation of galactic cosmic rays

Shoko Miyake, S. Yanagita|arXiv (Cornell University)|Oct 26, 2006

Solar and Space Plasma Dynamics3 citations

TL;DR

This study develops a 3D, time-dependent stochastic differential equation (SDE) model to simulate galactic cosmic ray (GCR) modulation in the heliosphere, incorporating a dynamically evolving, tilted, and wavy heliospheric current sheet (HCS). The model successfully reproduces the 22-year solar modulation cycle and qualitatively captures the asymmetric intensity modulation for positively and negatively charged particles, though it fails to reproduce the observed spectrum for protons with qA > 0, likely due to oversimplified HCS structure and missing transverse magnetic field components in polar regions.

ABSTRACT

Transport equation of the galactic cosmic ray (GCR) is numerically solved for qA>0 and qA<0 based on the stochastic differential equation (SDE) method. We have developed a fully time-dependent and three-dimensional code adapted for the wavy heliospheric current sheet (HCS). Results anticipated by the drift pattern are obtained for sample trajectories and distributions of arrival points at the heliospheric boundary for GCR protons. Our simulation reproduced a 22-year cycle of solar modulation which is qualitatively consistent with observations. Energy spectra of protons at 1 AU are calculated and compared with the observation by BESS.

Motivation & Objective

• To investigate how the tilt and waviness of the heliospheric current sheet (HCS) affect the solar modulation of galactic cosmic rays (GCRs).
• To develop a fully time-dependent, three-dimensional numerical model based on stochastic differential equations (SDEs) to simulate GCR transport in a realistic HCS geometry.
• To reproduce the 22-year modulation cycle observed in GCR intensity and compare simulated proton spectra with BESS observations.
• To identify limitations in current models, particularly the failure to reproduce the observed spectrum for positively charged particles (qA > 0).

Proposed method

• The study uses a stochastic differential equation (SDE) formulation equivalent to Parker’s transport equation, incorporating solar wind flow, diffusion tensor, and gradient-curvature drift velocity.
• The heliospheric magnetic field (HMF) is modeled as a Parker spiral, and the HCS is represented by a time- and radially-dependent tilt function that evolves over the 11-year solar cycle.
• The HCS structure is defined by a dynamic equation that includes solar rotation and time-varying tilt angles, allowing signal propagation from the Sun to 80 AU at solar wind speed (~400 km/s).
• Particles are traced backward from 1 AU to the heliospheric boundary (80 AU), with initial conditions set at 1 AU on the equatorial plane, and arrival time, energy loss, and trajectory distributions recorded.
• The drift velocity field method is used to compute particle drifts across the HCS, with polarity-dependent behavior for qA > 0 and qA < 0.
• Simulated proton energy spectra are normalized to BESS observations from 1998 and 2000 and compared to assess model accuracy.

Experimental results

Research questions

• RQ1 How does the time-dependent, radially propagating tilt and waviness of the HCS influence the modulation of galactic cosmic rays?
• RQ2 Can a 3D, time-dependent SDE model reproduce the observed 22-year modulation cycle in GCR intensity?
• RQ3 Why does the model fail to reproduce the observed energy spectrum for protons with qA > 0, despite capturing the general modulation trend?
• RQ4 How do particle trajectories and arrival distributions differ between qA > 0 and qA < 0 polarities due to HCS drift effects?

Key findings

• The model successfully reproduces the 22-year modulation cycle in galactic cosmic ray intensity, with a time lag in intensity changes due to finite signal propagation from the Sun to the heliospheric boundary.
• For qA < 0, the simulated proton energy spectrum agrees well with BESS observations from 2000, particularly during high tilt (70°) and negative solar polarity phases.
• For qA > 0, the model underestimates the observed proton intensity, suggesting insufficient modulation, likely due to the absence of transverse magnetic field components in polar regions that suppress drifts.
• Particles with qA > 0 arrive at 1 AU in only 4.5 days on average, indicating weak modulation, while those with qA < 0 take 80 days, consistent with stronger confinement and drift along the HCS.
• The latitude distribution of arrival points at 80 AU reflects the HCS structure: qA < 0 particles cluster near the HCS, while qA > 0 particles distribute near the poles, depending on the observer’s location relative to the HCS.
• The model's failure to reproduce the qA > 0 spectrum highlights the need to include more realistic HCS dynamics, such as random transverse fields and non-steady-state evolution, in future models.