Three dimensional tensor of cosmic rays for various interplanetary magnetic field structure Michael V. Alania Siedlce University, Poland

energy change Parker, 1965

ANISOTROPIC DIFFUSION TENSOR 2D An important parameter for the modulation of GCR is a tensor of diffusion containing many uncertainties due to the structure of the IMF. The anisotropic diffusion tensor K ij of GCR for the two dimensional IMF ; i,j=1,2 in the spherical coordinate system B(r, ,  ) has the following form [Dorman, 1968] where  is the angle between the lines of the IMF and the radial direction from the Sun

Tensor 3D IMF [Alania 1978; 2002] Burger et al. (2008),

Generally an existence of the B(  ) component of the IMF is quite problematic. Rosenberg & Coleman ( 1969 ) and Polyavyn & Usmanov ( 1985)- apparently first recognized the possible deviation of the IMF’s lines from the equatorial plane; Nagashima et al. ( 1986) - theoretically mentioned on expected B(  ) component of the IMF; Kota & Jokipi, (1989 )- suggested that the large-scale field near the poles may be dominated by randomly-oriented transverse magnetic fields with magnitude much larger than the average spiral. The field direction is transverse to the radial direction most of the time instead of being nearly radial. Feynman, (1991]- mentioned that the latitudinal component B(  ) of the IMF is one of the most geo effective parameters. In general, B(  ) is used to estimate the disturbances of the Earth magnetosphere. Fisk (1996) and Fisk et al..,(2001,2005)-assumed that the polar coronal hole is symmetric with respect to the solar magnetic axis, and that the magnetic field expands nonradially to yield a uniform field farther away from the Sun. The footpoints of the magnetic field lines anchored in the photosphere experience differential rotation.

Then, if the magnetic axis of the Sun is assumed to rotate rigidly at theequatorial rate, differential rotation will cause a footpoint to movein heliomagnetic latitude and longitude, thus experiencing differentdegrees of nonradial expansion. The end result is a fieldline that moves in heliographic latitude, and the simple concept of ‘‘field lines on cones’’ of the Parker field (Parker 1958) breaks down. Obridko et al. ( 2004 )- recognized that the lines of the IMF are weakly diverged from the solar equatorial plane near the Earth’s orbit based on the analyses of the IMF data for the period of They explained this effect as a result of the super radial extension of solar wind related with the continuous solar activity. Burger et al.,(2004,2008)-a fisk-parker hybrid heliospheric magnetic field with a solar-cycle dependence Alania (1986) - the divergence of the IMF’s lines from the radial direction can be associated with the specific heliolatitudinal distribution of the solar wind velocity. Magnetic field lines are frozen in the moving high conductivity fluid (plasma) and they can not cross each other. For simplicity the magnetic field lines are considered as a radial for the spherical symmetric and constant solar wind velocity (fig.1).

DIFFERENCE BETWEEN THE PARKER AND FISK MODEL Fisk, 1996 Parker, 1963 The IMF lines originate at the same latitude on the Sun. In Parker’s model IMF lines stay inside cone – shaped bounds at the latitudes of origination In Fisk’s model IMF lines loop into other latitudes

We apply the Bernoulli principle- Pst + Pdyn = constant for any point of the motion fluid.

When the velocity increases (dynamic pressure-Pdyn), a decrease of the static pressure ( Pstat ) takes place, i.e. In the high solar wind velocity layer ( stream tube) the dynamic Pdyn pressure increases and the static (Pstat) pressure decreases comparing with the low solar wind velocity layer. So, the change of the solar wind velocity versus the heliolatitude causes the violence of the equilibrium between the neighboring located regions with different solar wind velocities. Generally stream tube of the heliosphere with the high solar wind velocity involves the solar wind plasma with the IMF lines from the lower velocity regions (fig.2).

P st1. P dyn2.> P dyn1 P st1+ P dyn1=const P st2 + P dyn2 = const P st2 < P st1 1 2 V1V1 V2V2 V 2 > V 1

Figure 2. Magnetic field lines in the heliosphere with asymmetry solar wind velocity vs heliolatitues. Dashed lines are undisturbed magnetic field lines ( directed radial) and the bending solid magnetic field lines are diverged according to our expectation based on the Bernoulli principle, Pstat + Pdyn = const In figure 3 is shown the case approximately corresponding to the distribution of the solar wind velocity versus the heliolatitude according to the ULYSSES data. In the figures 2 and 3 dashed lines are corresponding to the radial magnetic field lines, but – bending solid lines (in figure 3) show the divergence from the radial direction. The divergence of the IMF’s lines from the radial direction should be the motive of the appearance of the IMF’s latitudinal component B(  ). the solar wind velocity equqls ~400 km/s; for  1 ≤  ≤  2 the solar wind velocity is ~800 km/s ( according to the ULYSSES data).Dashed lines are undisturbed magnetic field lines ( directed radial) and the bending solid magnetic field lines are diverged according to the Bernoulli principle.

STRUCTURE OF THE HELIOSPHERE Associated with the deflection of the solar wind around a obstacle

11-year variation

The power type rigidity R spectrum of the GCR intensity variations (  D(R)/D(R)  R -  ) have been calculated according to Power of the rigidity spectrum Upper limiting rigidity beyond which the VARIATION of GCR intensity vanishes

The temporal changes of the semi annual average magnitudes of the GCR intensity variations by M neutron monitor data and the rigidity spectrum exponent Reference Point (RP)

In the Figure are presented the average values of  and of the PSD of the B y component of the IMF turbulence (in the frequency range ~ – Hz ) 1- ( ) 2- ( ) 3- ( ) 4- ( ) 5- ( )

Alania, Modzelewska, Wawrzynczak, 2010 Parker, 1963

Bartels Rotation # of 30 September 2007– 11 February 2008

Temporal changes of daily data (points) and an approximation of the first harmonic waves of the 27-day variations (dashed lines) of the solar wind velocity (a), and the GCR intensity (b) during the period of 23 November-19 December 2007 (BR# 2379).

Figure 3ab. Heliolongitudinal changes of the (a) and (b) components of the IMF measured at the Earth [OMNI] during the period of 23 November-19 December 2007 (BR# 2379) (points) and the first harmonic waves of the 27-day variations (dashed lines). Figure 4ab. Heliolongitudinal changes of the (a) and (b) components of the IMF near Earth orbit obtained as the solution of Maxwell’s equations for the solar wind velocity given by (1) taking into account sector structure of the IMF (points) and its first harmonic waves of the 27-day variations (dashed lines).

CONCLUSION 1. An existence of the heliolongitudinal dependence of the solar wind velocity can be promoted to the creation of the heliolatitudinal component B(  ) of the interplanetary magnetic field (IMF) according to the Bernoulli principle. 2. Values of B(  ) less than 1-2 nT does not noticeably influence on the results of solutions of the transport equation of GCR. 3. The rigidity spectrum exponent  of the long period variations of the GCR intensity variations should be considered as a new index to study the 11-year variations of GCR intensity.

Thank You

Parker, 1963 Nagashima, 1986

Jokipii, Kota, 1989 Fisk, 1996; Zurbruchen, Schwadron, Fisk, 1997 Burger and Hitge, 2004

Alania, Modzelewska, Wawrzynczak, 2010 Parker, 1963

Tensor 2D IMF

Tensor 3D IMF [Alania 1978; 2002] Burger et al. (2008),

We consider Maxwell’s equations (e.g., Parker 1963, Jackson, 1998): The current (Ohm’s Law ) takes the form (Jackson, 1998): THEORETICAL MODELING (1)(2) (3) Using (1), one can eliminate J from Eq. (3) and we get: (4) Taking curl of the Eq. (4) and the condition, we get (5) According to (2) we get magnetic induction equation (Jackson, 1998): For solar wind moving with the velocity V we can assume that, as far and. Hence we get an equation: Eq. (7) describes purely advection process. (7) (6)

Maxwell’s equations (e.g., Parker 1963, Jackson, 1998): the system of equations (8a)-(8b) for the stationary case in the heliocentric spherical coordinate system can be rewritten, as: THEORETICAL MODELING