A three-dimensional, fully time-dependent ab initio modulation model for cosmic rays is introduced in Chapter 5. Structures and properties of the heliosphere and the solar cycle relevant to time-dependent modulation of cosmic rays.
Introduction
Solar Activity
The Solar Wind
Figure 2.3 shows 101-day averages of the solar wind speed, density, temperature, and dynamic pressure observed by Voyager 2 (V2). The upper left panel also shows 101-day averages of the solar wind speed at 1 au.
The Heliospheric Magnetic Field: Parker Model
This theoretical description of the HMF is usually assumed to apply from this surface onwards. It is clear from Figure 2.6 that the Parker model provides a fairly accurate description of the observed magnitude of the HMF in the solar ecliptic plane.
Heliospheric Current Sheet and the Tilt Angle
Figure 2.7 shows the tilt angle observations using two different models for calculation from solar magnetic field maps [Hoeksema, 1992]. Now the falling phase of the pitch angle is over and the rising phase has begun.
Turbulence Models
The left panel of Figure 2.11, taken from Matthaeus et al. [2003] , illustrates the effects of pure slab turbulence on magnetic flux tubes. Various observational studies have also reported time dependences for different amounts of turbulence [see, e.g., Nel , 2015 ; Zhao et al., 2018].
Classification and Transport of Cosmic Rays
The coefficients in the field adjusted tensor can be related to those of the tensor in heliocentric spherical coordinates by [Burger et al., 2008]. The various elements of the diffusion tensor in heliocentric spherical coordinates can then be written as [Burger et al.,2008].
Drift and Diffusion
In the middle panel, a solar cycle-dependent background HMF magnitude is used to calculate the parallel λk and perpendicular λ⊥ MFPs. The effect of turbulence anisotropy on MFPs is shown in the bottom panel, which is similar to the middle panel, but now for a 60:40 2D and slab energy ratio. According to Engelbrecht et al.[2017] is the length scale corresponding to the turbulence-reduced drift coefficient given by
Minnie et al ., 2007b ;Tautz and Shalchi , 2012 ] for a wide range of turbulence conditions expected in the heliosphere.
Observations
Space Age Observations
The PAMELA mission was launched in 2006 and is particularly optimized to measure cosmic rays and their antiparticles, such as positrons and antiprotons [Casolino et al., 2008]. It is also suitable for studying particles of solar origin and particles trapped in the Earth's magnetosphere. One of the most important discoveries was the large positron excess with respect to electrons between 10 GeV and 100 GeV as well as the discovery of antiprotons trapped in the radiation belts around the Earth [Adriani et al., 2011].
Inferred Historic Observations
Summary
Porter, Galactic cosmic rays in the local interstellar medium: Voyager 1 observations and model results, Astrophysics. Shen, Modulation of Galactic Cosmic Rays in the Inner Heliosphere, Compared to PAMELA Measurements, Astrophys. Wang, The global nature of solar cycle variations of solar wind dynamic pressure, Geophys.
Wei, Modulation of Galactic Cosmic Rays by Helium in Nickel in the Inner Heliosphere, Astrophy.
Transport Model
Solutions to the Oughton et al.(2011) full TTM solved by Engelbrecht & Burger(2013a,2015b) for general solar minimum conditions are also shown. values result in smoother transitions. In the ecliptic plane (upper left panel of Figure 2 ), the radial dependences are somewhat different, but still yield results within the spread of the observations of Zank et al. (1996). The values used in equation (12) for the parametric fits in Oughton et al. (2011) TTM results concerning slab correlation scales in the ecliptic plane as.
Values used in equation(13) for parametric fits to the Oughton et al.(2011) TTM results regarding the 2D correlation scales in the ecliptic plane as.
Modulation Results and Discussion
Given the ab initio nature of the current model and its ability to reproduce observed intensity spectra during the previous three solar minima, the question arises what predictions can be made for the next solar minimum. Upper left panel: total magnetic variance as a function of radial distance, with observations thereof reported by Zank et al. (1996). The differences in the results presented here may be due to the fact that this study takes into account the effects of fundamental turbulence quantities on the diffusion and drift coefficients of CRs.
This is due to the fact that most of the modulation of galactic CRs occurs in the heliosheath (see, e.g., Stone et al. 2013 ), and as such, this result is only expected to change. if there were large changes in the threshold spectrum at 85au related to the solar cycle and magnetic polarity.
Summary and Conclusions
Over the past 50 years, studies of the transport and modulation of cosmic ray particles (CRs) through turbu-. Due to the complexity of the plasma geometry and the associated transport parameters, the relevant equation must be solved numerically. For the finer details of the SDE approach, the reader is referred to Yamada et al.
The latitude gradient, defined in terms of CR difference intensity by Zhang (1997) to be.
The propagation model
In particular, finite-difference-type schemes become less accurate and numerically unstable as the number of dimensions increases (e.g., Kopp et al., 2012). The transfer of CR from the local interstellar spectrum (LIS) to any position in the heliosphere (which constitutes the computational domain) is governed by Parker's (1965) cosmic ray transport equation (TPE), which is given in terms of the omnidirectional CR phase space density f0. related to the differential CR intensity with jT ¼p2f0). Since different pseudoparticles (i.e., different integrations of the same set of SDEs) are independent, we choose to integrate them for the same phase space position on different computing nodes using MPI.
The solutions from different computing cores are then combined using the MPI reduction routine.
Numerical implementation
The SDE solver
Random number generation
Galactic CR latitude gradients and relative amplitudes
Regarding the diffusion tensor (Eq.(9)), the expression for the parallel mean free path of the galactic proton used here is based on the results derived from the quasi-linear theory of Jokipii (1966) by Teufel and Schlickeiser (2003) as applied. of e.g. Burger et al. 2 shows Galactic CR proton differential intensities calculated at 1 AU and 0 azimuth in the ecliptic plane using the Engelbrecht and Burger (2015b) code, for positive and negative magnetic polarity cycles (left and right panels, respectively), choosing the number of pseudoparticles N¼10 000; N ¼100 000;N and N¼10 000 000. Galactic CR proton differential intensities calculated at 1 AU and 0azimuth in the ecliptic plane under A>0 (left panel) and A<0 (right panel5 and Burger) (201brecht) SDE solver.
The model is commonly used to study CR transport in the heliosphere, but aspects of it are applicable to any diffusion equation.
Discussion and conclusions
The independence of the different pseudoparticles makes this approach ideal for running on parallel computing architectures because we need almost no communication between different nodes or computing cores. Due to the longer processing times for scenarios with more pseudoparticles (larger N systems), the relative cost is reduced and the code scales better. Also Eric Mbele and Oscar Monama for technical assistance and CHPC for access to the new cluster.
Expressed opinions and conclusions are those of the authors and should not necessarily be attributed to NRF.
A brief history
An African first
The modulation model
The present study uses the diffusion tensor transformation proposed by Burger et al. (2008) to convert the diffusion tensor. The radial dependencies for both quantities have been modeled parametrically to follow the results of the turbulence transport model of Zank et al. (2018). This latter assumption does not perfectly reflect observations made by spacecraft (see, for example, Bieber et al. 1993; Matthaeus et al. 2007).
Drift coefficients, reduced in the presence of turbulence, are modeled according to the approach of Engelbrecht et al. 2017), so that the length scale corresponding to the drift coefficient is given by.
Results
The calculated results are again in good agreement with the data and can reproduce the clockwise rotation seen in the data. The bottom panel of Fig. 3 shows the threshold spectrum used in this study (Eq. (17), black line) as well as the 1au calculated differential intensity spectra compared to different spacecraft data taken at periods marked with different colors, which correspond to the indicated columns in the upper panel of the figure. This is particularly true for the period prior to 1989 and is reflected in the total overshoot of the data with calculated intensities of 1.28GV from 1989 to 1991 seen in the top panel of Figure 3.
Bottom panel: Boundary spectrum (equation(17), black line) and 1au model GCR proton differential intensity spectra calculated for comparison with spacecraft observations of the same taken at the time intervals indicated in the top panel of this figure, as a function of kinetic energy.
Long-term Modulation
Peaks in the HMF magnitude occur roughly when tilt angle values approach their local maxima, consistent with space-age observations. Furthermore, there is no indication of the peak/plateau signature of drift effects in the mean intensity profile. To test conclusions drawn regarding the role of drift in the historical modulation of GCRs by Caballero-Lopez et al. (2004a), model results were calculated using historical inclination angles reconstructed using sunspot observations along the lines suggested by Asvestari & Usoskin (2016).
In the future, this model will be used to revisit the study by Caballero-Lopez et al.
Summary and conclusions
The model only describes the cyclic behavior of the tilt angle based on the phase of the solar cycle as calculated using the sunspot record. This leads us to the conclusion that drift effects cannot be ignored when trying to estimate historical changes in the magnitude of the HMF by modeling cosmic ray fluxes in the past. This leads to the important conclusion that drift effects cannot be ignored when seeking to estimate historical changes in the magnitude of the HMF by modeling cosmic ray fluxes in the past.
Possible refinements and improvements to the model include the integration of more recent developments in diffusion coefficient models, an improved way of dealing with current plate drift, and consideration of the modulation of heavier elements in the model.
