1. Mariette Hitge, Adri Burger
THE EFFECT OF AN AZIMUTHAL-DEPENDENT SOLAR WIND SPEED ON THE MODULATION OF GALACTIC COSMIC RAYS
Mariette Hitge, Adri Burger
Unit for Space Physics
Potchefstroom Campus of North-West University
South Africa
Today I’ll give a talk on the effect of azimuthal-dependent solar wind speed on the modulation of galactic CR. It is based on work done by myself and my co-worker Adri Burger.

2. Heliospheric magnetic fields The model used Results
OUTLINE
Introduction
Solar wind speed
Heliospheric magnetic fields
The model used
Results
Summary / Conclusions
I’ll firstly give an introduction on the talk, describing the purpose of the study.
Then I’ll describe the azimuthal solar wind speed used and give the expression for a Parker field with a azimuthal dependent solar wind speed.
After that I’ll describe the modulation model that was used to calculate the intensties of the CR.
I’ll then show the main results of the study for different solar wind speed scenario’s.
Lastly I’ll round the talk off giving the conclusions of the study.

3. INTRODUCTION
26-day recurrent variations in cosmic ray fluxes up to highest heliographic latitudes (Zhang 1997).
Linear relation between latitude gradient and amplitude of 26 day recurrent variations (Zhang 1997).
Studied with modulation model using Fisk-Parker hybrid field (Hitge & Burger 2010).
This study with modulation model using Parker field with solar wind speed dependent on azimuthal angle.
Zhang found recurrent variations in cosmic ray fluxes up to the highest heliographic latitudes for both solar polarity cycles covered by Ulysses, where there are no recurrent solar wind or magnetic field variations. The heliospheric neutral sheet and corotating interaction regions are confined below 40 degrees latitude.
Zhang also found a linear relation between the latitude gradient and the amplitude of 26-day recurrent variations in the fluxes of galactic cosmic rays for all species at all latitudes in the inner heliosphere covered by Ulysses and in the equatorial region covered by IMP8 for both solar polarity cycles. This means that particles with large latitude gradients are also strongly modulated by recurrent structures. Or, said in another way: particles whose flux undergo strong short-term variations, will also have a large pole-to-equator flux. These observations also implies that there is a dominant modulation mechanism that control both the global latitudinal distribution and the short-term temporal variation of fluxes.
This relation was studied with a modulation model using a Fisk-Parker hybrid field. Results similar to the Zhang-observations was found.
In this study the effect of a Parker field with an azimuthal dependent solar wind speed on recurrent variations in protons fluxes, was investigated.

4. SOLAR WIND SPEED km/s km/s
Here the expression for the solar wind speed dependent on azimuthal angle is shown. Theta is co-latitude and phi is the azimuthal angle. The solar wind speed in the polar regions and the average solar wind speed in the equatorial regions are taken to be the same for this study. This was done since the study focused on the effect of an azimuthal dependent solar wind speed only. Omega_A is the amplitude of the variation of the speed as percentage of the average speed. The period of the variation is half a solar rotation. The function fphi restricts the azimuthal dependence to the equatorial region. Larger values of delta corresponds to larger areas where the solar wind is dependent on azimuthal angle. The value for delta was taken as 30 degrees, unless otherwise stated. For a value of 30 degrees, the azimuthal dependence of the solar wind speed is confined to an area of 40 degrees latitude.

5. MODIFIED PARKER FIELD with
This is the expression for a Parker field with a solar wind speed dependent on azimuthal angle. A is the value of the radial component of the magnetic field at Earth, ro the heliocentric radial distance of Earth, rs the heliocentric radial distance of the source surface and Omega the equatorial rotation rate of the Sun. An extra term in the radial component has to be added to ensure that the field is locally and globally divergence free. The sign of this term can be positive or negative dependent on the sign of the derivative of the solar wind speed. The magnitude of the magnetic field change by a maximum of +/- 5% in the outer heliosphere in the equatorial region.
In the sketch a pure Parker field line (blue) and a modified Parker field line is drawn for a latitude or 20 degrees and a solar wind speed amplitude of 5%. The effect of the azimuthal dependent solar wind speed is seen clearly.

6. FISK-PARKER HYBRID FIELD (CONSTANT SOLAR WIND SPEED)
with
The assumptions that the footpoints of the field lines in the photosphere rotates rigidly with the sun and that the field lines are radially on the source surface, lead to a HMF with spiral field lines lying on cones of constant lat – the Parker field indicated by the blue line. The footpoints of the field lines are actually rotating differentially and the field lines are only coming from coronal holes. Coronal holes are areas on the surface of the sun that appears darker since it is cooler. From these areas, which are mainly in the polar regions of the sun, open field lines emerge and together with it, the fast component of the solar wind. The differential rotation of the footpoints in the photosphere, together with the non-radial expansion of the field lines originating from rigidly rotating polar coronal holes, result in excursions of the field lines in latitude as seen in the sketch – the red line.
In these expressions omega is the differential rotation rate of the footpoints (the difference between the rotation rates in the equatorial region and high latitudes), beta is the angle between the axis formed by the field line originating at the heliographic pole, and the rotation axis of the sun, phi* is the azimuthal angle in the coordinate system co-rotating with the sun and Fs is the transition function between the Parker field and the Fisk field. This magnetic field is a mixture of Fisk field and Parker field at high latitudes, a Fisk field at middle and a Parker field in the equatorial region. This field is divergence free for a constant solar wind speed only.

7. Steady state 3D numerical modulation model of Hattingh & Burger (1995)
THE MODULATION MODEL
Steady state 3D numerical modulation model of Hattingh & Burger (1995)
Spherical symmetric heliosphere
Outer boundary 50 AU
Outer boundary condition: LIS
Diffusion coefficients: QLT and NLGC
A steady state, 3D modulation model was used to solve the TPE for galactic protons in a spherical symmetric heliosphere.
The outer boundary was set at 50 AU due to computer restrictions and in order to keep the numerical model as simple as possible, since as realistic as possible diffusion coefficients were used. The TS and heliosheath were omitted since it does not have a significant effect on the modulation of protons in the inner heliosphere.
The LIS spectrum was set as boundary condition at the outer boundary.
The diffusion of CR due to the turbulence of the HMF and the curvature, gradient and NS drift of the CR due to the structure of the HMF, are described by a diffusion tensor including diffusion along and perpendicular to the HMF and drift. A parallel diffusion coefficient following from quasi linear theory, was used. For perpendicular diffusion a coefficient following from nonlinear guiding center theory was used. The drift scale is the Larmor radius of the particle in the weak scattering limit, inverse proportional to the magnitude of the magnetic field.

8. RESULTS: FISK-PARKER HYBRID FIELD (CONSTANT SOLAR WIND SPEED)
Effect of tilt angle of heliospheric neutral sheet
In these graphs and the graphs to follow, the relative amplitude of recurrent intensities of protons over one solar rotation is shown as function of latitudinal gradient for both solar polarity cycles. The amplitudes and gradients are calculated for energies ranging from 1 GeV to 10 MeV. The latitudinal gradients are calculated between 0 and 80 degrees latitude at 2 AU. The amplitudes are calculated at latitudes indicated on the graphs and at 2 AU.
Here the relative amplitudes and latitudinal gradients are shown for the Fisk-Parker hybrid field for neutral sheet tilt angles of 5 degrees and 10 degrees. These results are in accordance with observations. Recurrent variations in intensity are seen at higher latitudes, beyond latitudes where solar wind and magnetic field variations, the heliospheric neutral sheet and corotating interactions regions, occur. A linear relation is seen for both solar polarities and at all latitudes. At high latitudes the relative amplitudes for the A>0 polarity cycle during which protons drift from the polar regions to the equatorial region, is larger than for the A<0 solar cycle. In the equatorial region the amplitudes are similar for the two polarity cycles.

9. RESULTS: PARKER FIELD (CONSTANT SOLAR WIND SPEED
Effect of tilt angle of heliospheric neutral sheet
Here the effect of the tilt angle of the heliospheric neutral sheet for the Parker field with a constant solar wind speed, is shown. When compared with the results for the Fisk-Parker hybrid field, the linear relation is seen clearly for the A<0 solar cycle, but not so clear for the A>0 solar cycle. The relative amplitudes are also much smaller than for the Fisk-Parker hybrid field. The relative amplitudes decrease at higher latitudes for the A<0 solar cycle and almost diminishes for the A>0 solar cycle. It is clear that the Fisk-Parker hybrid field provides the mechanism for producing recurrent intensities at high latitudes as observed, while the Parker field can not.

10. RESULTS: MODIFIED PARKER FIELD
Effect of amplitude of solar wind speed
To produce these results, a Parker field with a azimuthal dependent solar wind speed, was used. Here the effect of the amplitude of azimuthal dependence of the solar wind speed, is shown. Near the Earth solar wind speed amplitudes of up to 30 % is seen. If amplitudes larger than 15% is used in this model, it produces negative latitudinal gradients and the linear relation is totally absent. Although a larger amplitude produces larger relative amplitudes, it can not produce recurrent variations in accordance with observations. The largest recurrent variations occur in the equatorial region, diminishing when going to higher latitudes.

11. RESULTS: MODIFIED PARKER FIELD
Effect of area of azimuthal dependence of solar wind speed
Here the effect of the area of azimuthal dependence of the solar wind speed, is shown. Although a larger area produces larger relative amplitudes, it also can not produce recurrent variations in accordance with observations. The largest recurrent variations occur in the equatorial region, diminishing when going to higher latitudes.

12. SUMMARY / CONCLUSIONS
The Fisk-Parker hybrid field produces results in accordance with observations: recurrent variations at high latitudes for both solar cycles and linear relation between recurrent variations and latitudinal gradients.
The Parker field, even with a azimuthal dependent solar wind speed, can not produce recurrent variations at high latitudes.
When studying phenomena averaged over solar rotations, the Fisk-Parker hybrid field and the Parker field give similar results.
When studying recurrent variations, a Fisk-like heliospheric magnetic field should be used.

13. Acknowledgements Adri for inspiring me to put this talk together.
Financial support by the South African National Research Foundation is acknowledged

14. SCHWADRON-PARKER HYBRID FIELD (LATITUDE-DEPENDENT SOLAR WIND SPEED)
with
The motion of the FPs between the fast and slow SWS regions at the sun create non-spiral configurations and large radial components of the field. The blue line is the PF and the red line the SF. It is clear that when the FP moves from an area of slow SWS to fast SWS, the radial component decreases and the polarity does reverse.
It increases when it moves from an area of fast SWS to slow SWS and the polarity doesnot reverse.