R. Modzelewska

On the Relationship of the 27-day Variations of the Solar Wind Velocity and Galactic Cosmic Ray Intensity in Minimum Epoch of Solar Activity

We study the relationship of the 27-day variations of the galactic cosmic ray intensity with similar variations of the solar wind velocity and the interplanetary magnetic field based on observational data for the Bartels rotation period # 2379 of 23 November 2007 – 19 December 2007. We develop a three-dimensional (3-D) model of the 27-day variation of galactic cosmic ray intensity based on the heliolongitudinally dependent solar wind velocity. A consistent, divergence-free interplanetary magnetic field is derived by solving Maxwell’s equations with a heliolongitudinally dependent 27-day variation of the solar wind velocity reproducing in situ observations. We consider two types of 3-D models of the 27-day variation of galactic cosmic ray intensity, i) with a plane heliospheric neutral sheet, and ii) with the sector structure of the interplanetary magnetic field. The theoretical calculations show that the sector structure does not significantly influence the 27-day variation of galactic cosmic ray intensity, as had been shown before, based on observational data. Furthermore, good agreement is found between the time profiles of the theoretically expected and experimentally obtained first harmonic waves of the 27-day variation of the galactic cosmic ray intensity (with a correlation coefficient of 0.98±0.02). The expected 27-day variation of the galactic cosmic ray intensity is inversely correlated with the modulation parameter ζ (with a correlation coefficient of −0.91±0.05), which is proportional to the product of the solar wind velocity V and the strength of the interplanetary magnetic field B (ζ∼VB). The high anticorrelation between these quantities indicates that the predicted 27-day variation of the galactic cosmic ray intensity mainly is caused by this basic modulation effect.

The 27-Day Cosmic Ray Intensity Variations During Solar Minimum 23/24

We have studied the 27-day variations and their harmonics of the galactic cosmic ray (GCR) intensity, solar wind velocity, and interplanetary magnetic field (IMF) components in the recent prolonged solar minimum 23 24. The time evolution of the quasi-periodicity in these parameters connected with the Suns rotation reveals that their synodic period is stable and is aprox 26-27 days. This means that the changes in the solar wind speed and IMF are related to the Suns near equatorial regions in considering the differential rotation of the Sun. However, the solar wind parameters observed near the Earths orbit provide only the conditions in the limited local vicinity of the equatorial region in the heliosphere (within in latitude). We also demonstrate that the observed period of the GCR intensity connected with the Suns rotation increased up to aprox 33-36 days in 2009. This means that the process driving the 27-day variations of the GCR intensity takes place not only in the limited local surroundings of the equatorial region but in the global 3-D space of the heliosphere, covering also higher latitude regions. A relatively long period ( aprox 34 days) found for 2009 in the GCR intensity gives possible evidence of the onset of cycle 24 due to active regions at higher latitudes and rotating slowly because of the Suns differential rotation. We also discuss the effect of differential rotation on the theoretical model of the 27-day variations of the GCR intensity.

Dependence of the 27-day variation of cosmic rays on the global magnetic field of the Sun

We show that the higher range of the heliolongitudinal asymmetry of the solar wind speed in the positive polarity period (A > 0) than in the negative polarity period (A 0) than in 1985–1987 (A < 0). Subsequently, different ranges of the heliolongitudinal asymmetry of the solar wind speed jointly with equally important corresponding drift effect are general causes of the polarity dependence of the amplitudes of the 27-day variation of the GCR intensity. At the same time, we show that the polarity dependence is feeble for the last unusual minimum epoch of solar activity 2007–2009 (A < 0); the amplitude of the 27-day variation of the GCR intensity shows only a tendency of the polarity dependence. We present a three dimensional (3-D) model of the 27-day variation of GCR based on the Parker’s transport equation. In the 3-D model is implemented a longitudinal variation of the solar wind speed reproducing in situ measurements and corresponding divergence-free interplanetary magnetic field (IMF) derived from the Maxwell’s equations. We show that results of the proposed 3-D modeling of the 27-day variation of GCR intensity for different polarities of the solar magnetic cycle are in good agreement with the neutron monitors experimental data. To reach a compatibility of the theoretical modeling with observations for the last minimum epoch of solar activity 2007–2009 (A < 0) a parallel diffusion coefficient was increased by ∼40%.

Peculiarities of cosmic ray modulation in the solar minimum 23/24

We study changes of the galactic cosmic ray (GCR) intensity for the ending period of the solar cycle 23 and the beginning of the solar cycle 24 using neutron monitors experimental data. We show that an increase of the GCR intensity in 2009 is generally related with decrease of the solar wind velocity U, the strength B of the interplanetary magnetic field (IMF), and the drift in negative (Aneg) polarity epoch. We present that temporal changes of rigidity dependence of the GCR intensity variation before reaching maximum level in 2009 and after it, do not noticeably differ from each other. The rigidity spectrum of the GCR intensity variations calculated based on neutron monitors data (for rigidities greaten than 10 GV) is hard in the minimum and near minimum epoch. We do not recognize any non-ordinary changes in the physical mechanism of modulation of the GCR intensity in the rigidity range of GCR particles to which neutron monitors respond. We compose 2-D non stationary model of transport equation to describe variations of the GCR intensity for 1996-2012 including the Apos (1996-2001) and the Aneg (2002-2012) periods; diffusion coefficient of cosmic rays for rigidity 10-15 GV is increased by 30 percent in 2009 (Aneg) comparing with 1996 (Apos). We believe that the proposed model is relatively realistic and obtained results are satisfactorily compatible with neutron monitors data.

Stochastic approach to the numerical solution of the non-stationary Parker's transport equation

We present the newly developed stochastic model of the galactic cosmic ray (GCR) particles transport in the heliosphere. Mathematically Parker transport equation (PTE) describing non-stationary transport of charged particles in the turbulent medium is the Fokker-Planck type. It is the second order parabolic time-dependent 4-dimensional (3 spatial coordinates and particles energy/rigidity) partial differential equation. It is worth to mention that, if we assume the stationary case it remains as the 3-D parabolic type problem with respect to the particles rigidity R. If we fix the energy/rigidity it still remains as the 3-D parabolic type problem with respect to time. The proposed method of numerical solution is based on the solution of the system of stochastic differential equations (SDEs) being equivalent to the Parkers transport equation. We present the method of deriving from PTE the equivalent SDEs in the heliocentric spherical coordinate system for the backward approach. The advantages and disadvantages of the forward and the backward solution of the PTE are discussed. The obtained stochastic model of the Forbush decrease of the GCR intensity is in an agreement with the experimental data.

Cosmic ray heliospheric transport study with neutron monitor data

Determining transport coefficients for galactic cosmic ray (GCR) propagation in the turbulent interplanetary magnetic field (IMF) poses a fundamental challenge in modeling cosmic ray modulation processes. GCR scattering in the solar wind involves wave-particle interaction, the waves being Alfven waves which propagate along the ambient field (B). Empirical values at 1 AU are determined for the components of the diffusion tensor for GCR propagation in the heliosphere using neutron monitor (NM) data. At high rigidities particle density gradients and mean free paths at 1 AU in B can only be computed from the solar diurnal anisotropy (SDA) represented by a vector A (components Ar, Aϕ, Aθ) in a heliospherical polar coordinate system. Long-term changes in SDA components of NMs (with long track record and the median rigidity of response Rm ~ 20 GV) are used to compute yearly values of the transport coefficients for 1963–2013. We confirm the previously reported result that the product of the parallel (to B) mean free path (λ||) and radial density gradient (Gr) computed from NM data exhibits a weak Schwabe cycle (11y) but strong Hale magnetic cycle (22y) dependence. Its value is most depressed in solar activity minima for positive (p-) polarity intervals (solar magnetic field in the northern hemisphere points outward from the sun) when GCRs drift from the polar regions toward the helio-equatorial plane and out along the heliospheric current sheet (HCS), setting up a symmetric gradient Gθs pointing away from HCS. Gr drives all SDA components and λ|| Gr contributes to the diffusive component (Ad) of the ecliptic plane anisotropy (A). GCR transport is commonly discussed in terms of an isotropic hard sphere scattering (a.k.a billiard-ball scattering) in the solar wind plasma. We use it with a flat HCS model and the Ahluwalia-Dorman master equations to compute the coefficients α (= λ⊥/λ||) and ωτ (a measure of turbulence in the solar wind) and transport parameters λ||, λ⊥, Gr, Gθs, and an asymmetric gradient Gθa normal to the ecliptic plane. We study their dependence on rigidity (R), p-/n- intervals, sunspot numbers (SSNs), and solar wind parameters at 1 AU. λ|| exhibits a strong 22y dependence but Gr does not, explaining solar polarity dependence of λ|| Gr. The computed Gr values are an order of magnitude greater than those reported by colleagues making an ad hoc assumption that α is low (0.01). At high rigidities the drift contribution at 1 AU is small and unsteady. A new methodology is outlined to compute yearly GCR north–south anisotropy (Aθ) from the data for a single detector sorted for p-/n- intervals. We show that Gθa is the main contributor to Aθ in the steady state and Gθa is shown not correlated with the north–south excess SSNs.

27-day variation of the GCR intensity based on corrected and uncorrected for geomagnetic disturbances data of neutron monitors

We study 27-day variations of the galactic cosmic ray (GCR) intensity for 2005–2008 period of the solar cycle #23. We use neutron monitors (NMs) data corrected and uncorrected for geomagnetic disturbances. Besides the limited time intervals when the 27-day variations are clearly established, always exist some feeble 27-day variations in the GCR intensity related to the constantly present weak heliolongitudinal asymmetry in the heliosphere. We calculate the amplitudes of the 27-day variation of the GCR intensity based on the NMs data corrected and uncorrected for geomagnetic disturbances. We show that these amplitudes do not differ for NMs with cut-off rigidities smaller than 4-5 GV comparing with NMs of higher cut-off rigidities. Rigidity spectrum of the 27-day variation of the GCR intensity found in the uncorrected data is soft while it is hard in the case of the corrected data. For both cases exists definite tendency of softening the temporal changes of the 27-day variations rigidity spectrum in period of 2005 to 2008 approaching the minimum of solar activity. We believe that a study of the 27-day variation of the GCR intensity based on the data uncorrected for geomagnetic disturbances should be carried out by NMs with cut-off rigidities smaller than 4-5 GV.

THEORETICAL AND EXPERIMENTAL STUDIES OF THE 11-YEAR AND 27-DAY VARIATIONS OF THE GALACTIC COSMIC RAYS INTENSITY AND ANISOTROPY

The changes of the structure in the energy range of the interplanetary magnetic field (IMF) turbulence versus solar activity can be considered as one of the important reasons of the long period (11-year) modulation of galactic cosmic ray (GCR) intensity; the amplitude of the 27-day variation of GCR anisotropy is greater in the qA > 0 periods than in the qA < 0 periods of the solar magnetic cycles in a good correlation with the similar changes of the 27-day variation of GCR intensity.

Algorithms for Forward and Backward Solution of the Fokker-Planck Equation in the Heliospheric Transport of Cosmic Rays

Motion of charged particles in an inhomogeneous turbulent medium as magnetic field is described by partial differential equations of the Fokker-Planck-Kolmogorov type. We present an algorithm of numerical solution of the four-dimensional Fokker-Planck equation in three-dimensional spherical coordinates system. The algorithm is based on Monte Carlo simulations of the stochastic motion of quasi-particles guided by the set of stochastic differential equations corresponding to the Fokker-Planck equation by the Ito formalism. We present the parallel algorithm in Julia programming language. We simulate the transport of cosmic rays in the heliosphere considering the full three-dimensional diffusion tensor. We compare forward- and backward-in-time solutions of the transport equation and discuss its computational advantages and disadvantages.

Solution of the Stochastic Differential Equations Equivalent to the Non-stationary Parker Transport Equation by the Strong Order Numerical Methods

We present the newly developed stochastic model of the galactic cosmic ray (GCR) particles transport in the heliosphere. The model is based on the numerical solution of the Parker transport equation (PTE) describing the non-stationary transport of charged particles in the turbulent medium. We present the method of deriving from PTE the equivalent stochastic differential equations (SDEs) in the heliocentric spherical coordinate system for the backward approach. We present the formulas for the numerical solution of the obtained set of SDEs driven by a Wiener process in the case of the full three-dimensional diffusion tensor. We introduce the solution applying the strong order Euler-Maruyama, Milstein, and stochastic Runge-Kutta methods. We compare the convergence and stability of the solution for the listed methods. We also discuss the advantages and disadvantages of the presented numerical methods in the context of increasing the accuracy of the solution of the PTE. We present the comparison of the stochastic model of the Forbush decrease (Fd) of the GCR intensity with the experimental data.