2.46.2. Convection-diffusion modulation for small energy galactic CR particles

According to Eq. 2.46.7 and Table 2.46.1, for small energy galactic CR particles it is necessary to take into account the additional time-lag caused by the particle diffusion in the interplanetary space. In a first approximation we use the quasi-stationary model of convection-diffusion modulation described by Dorman (2003a), and here developed by taking into account the diffusion time-lag:

ln (n (R, robs, t )) = A (Xo ,0, t„ t2 )-B (Xo ,0, tu t2 )x F ft, Xo ,0,W (t - X* // 1 (2-46.8)

where

X * = X + C (R, t )x(X - Xobs))xo - X - Xobs) . (2.46.10)

Different Approaches can be considered for small energy convection-diffusion modulation:

1-st Approach: C(R,t) = 0 - no diffusion time-lag. This Approach was used by Dorman (2003a,b) for NM energies; for small energies this Approach will be used for comparison.

2-nd Approach: C(r, t) = C(r)av ~ (Cmax + Cmin V2 , where Cmax and Cmin are listed in Table 1, and C (R )av = 0.087 for 3 GV, 0.15 for 1 GV, and 0.48 for 0.3 GV are obtained.

3-rd Approach: C(R, t) is determined by Eq. (2.46.4). In Fig. 2.46.1 the dependences of C from tilt angle a, calculated on the basis of results obtained in Burger and Potgieter, 1999; see also Section 2.37), are shown.

The dependences shown in Fig. 2.46.1 can be approximated for R = 3 GV by

with correlation coefficient 0.978. On the basis of Eq. 2.45.6 (in Section 2.45.3), the Eq. 2.46.11 can be presented through sunspot number W as

For 1 GV we obtain

C(1 GV, a) = 0.00116 a+ 0.190, C(1 GV,W) = 0.000407 W + 0.206, (2.46.12) with correlation coefficient 0.997. For 0.3 GV it will be

C(0.3 GV, a) = 0.00156 a+ 0.394, C(0.3 GV,W) = 0.000545 W + 0.415, (2.46.13)

with correlation coefficient 0.980. Eq. 2.46.11a - 2.46.11b and Eq. 2.46.12 - 2.46.13 can be combined approximately as (here particles rigidity R is in GV)

C(R, a) = (-3.94R+1.63)10-3 a - 0.142ln(R) + 0.213, (2.46.14)

C(R,W) = (-1.38R+5.68)10-4W - 0.148ln(R) + 0.227. (2.46.15)

2.46.3. Small energy CR long-term variation caused by drifts

According to the main idea of the drift mechanism (see Jokipii and Davila, 1981; Jokipii and Thomas, 1981; Lee and Fisk, 1981; Kota and Jokipii, 1999; Burger and Potgieter, 1999; Ferreira et al., 1999), we assume that the drifts depend on the value of tilt angle a and change sign during periods of the SMF polarity reversal (see drift approach 3 according to Dorman, 2003a). We used data of tilt-angles for the period May 1976-September 1993. On the basis of these data we determined the correlation between a and W for 11 month-smoothed data as determined by Eq. 2.45.6 (in Section 2.45.3) with correlation coefficient 0.955±0.013. We assume that the drift effect is proportional to the theoretical value in dependence on tilt-angle a (or in dependence on the sunspot number W through Eq. 2.45.6) with negative sign for general solar magnetic field A > 0 and positive sign for A < 0, and in the period of reversal we suppose linear transition through 0 from one polarity cycle to another. The theoretical expected values of convection-diffusion modulation Acd and drift modulation A^r have been determined from Fig. 2.37.1 - 2.37.4 in Section 2.37 (from Burger and Potgieter, 1999); we assume that in these figures the average of curves for A > 0 and A < 0 characterizes the convection-diffusion modulation (which does not depend on the sign of general solar magnetic field), and the difference between these curves represents the double drift modulation (which depends on the sign of general solar magnetic field). Fig. 2.46.2 shows the drift modulation A^r (relative to the intensity out of Heliosphere) for R = 3, 1, and 0.3 GV derived from theoretical results of Burger and Potgieter (1999) by taking into account Eq. 2.45.6 (in Section 2.45.3).

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