ND (90) 50 10 50 ND{90) 50 10 50 ND(90) ND(90) 50 10 50 ND(90) 50 10 50 ND(90) Tilt angle (degrees) Tilt angle (degrees)

Fig. 2.37.4. The tilt angle and solar polarity-sign dependence of the ratios e7He++ and e-/p, normalized with respect to the minimum value for each ratio. "ND (90)" denotes a no-drift value at a hypothetical tilt angle of 90°. From Burger and Potgieter (1999).

From Fig 2.37.4 can be seen that the changes in the ratio becomes larger as the rigidity becomes smaller. In earlier studies (e.g., Potgieter and Burger 1990; Webber et al., 1990) the smooth transition from one polarity cycle to the next does not occur. The reason for this is that at large tilt angles, predecessors of the current two-dimensional model predicted a much flatter intensity-tilt response of positively charged particles during an A > 0 cycle, and therefore of negatively charged particles during an A < 0 cycle.

Apart from magnetic polarity, only the tilt angle is changed to obtain the present results. Since the tilt angle is a proxy for solar activity, we therefore employ drifts to construct a simplified solar-activity cycle. In Burger and Potgieter (1999) model, intensity-tilt profiles (Fig. 2.37.2) show three distinct regimes. During periods when the tilt is small, the well-known peaked profile for protons occurs when A < 0, and the "flat" profile (actually only flatter than the peaked profile) when A < 0. For larger tilt angles, a second regime occurs when the two profiles more-or-less track each other. Cane et al. (1999) find observational evidence for both regimes at neutron-monitor energies, but conclude that the second is not due to drift effects, in contrast to the results presented here. The third regime is when the A > 0 profile drops while the A < 0 profile flattens to converge to the no-drift intensity. Clearly, drifts are phased out as the tilt angle increases for both polarities.

The ratio of differently charged particle intensities throughout hypothetical solar activity cycle, where only the tilt angle changes, shows that during A > 0 cycles, the largest changes occur around solar minimum modulation, and the local maximum occurs at solar minimum. At other times, little change in the ratio occurs, but there is an sharp increase in the ratio going from an A > 0 to an A < 0 cycle, which decreases as the rigidity of the particles decreases. During an A < 0 cycle, changes in the ratio is typically larger than during the alternate cycle, especially at lower rigidities. At solar minimum modulation the ratio is at a local minimum. Burger and Potgieter (1999) note that the qualitative features of the ratio-tilt angle profiles for the electron/helium ratio agrees remarkably well with observations (Bieber et al., 1999a, b).

According to Burger and Potgieter (1999), before attempting a detailed comparison of the described results with observations, one should bear in mind the following:

(i) In a dynamical model, the symmetry with respect to solar minimum modulation is broken (le Roux and Potgieter, 1990).

(ii) Modulation caused by 'barriers' cannot be neglected during non solar minimum modulation periods (e.g., Potgieter and le Roux, 1992a,b).

(iii) The electron measurement may contain a sizable fraction of positrons (e.g., Evenson, 1998).

(iv) The state of the Heliosphere during the approach to solar maximum, is certainly different from that in the considered model (e.g., review by Jokipii and Wibberenz, 1998).

(v) The sign of the solar magnetic field does not change abruptly through solar maximum, and as a rule, this does not occur at a tilt angle of 90°.

2.38. CR drifts in a fluctuating magnetic fields

Giacalone et al. (1999) examine the drifts of CR particles in a fluctuating magnetic fields using direct numerical simulation of particle trajectories. They superimpose a randomly fluctuating magnetic field upon a background uniform field. Particle drifts in a magnetic field which has a mean which varies with position are a basic aspect of the motion of CR energetic particles. In general, the motion of CR is composed of the diffusive motion caused by the scattering of the particles due to the fluctuating part of magnetic field and the drift motions resulting from large-scale gradient and curvature of the average magnetic field. The nature of the diffusive transport, and the relation of the diffusion coefficients to the turbulent structure of the magnetic field has been extensively studied over the years. In particular, the diffusion parallel to the average magnetic field seems to be fairly well understood, whereas the perpendicular diffusion with coefficient k_|_ is less so (Fisk et al., M1998). In addition to the perpendicular and parallel diffusion, determined by the symmetric part of the diffusion tensor, a mean magnetic field produces in general an anti-symmetric part to the diffusion tensor, usually termed ka . In general the diffusion tensor may be write as

where j is the unit totally anti-symmetric tensor. The drift velocity \dr

(averaged over the nearly-isotropic distribution) may be shown to be precisely the divergence of the anti-symmetric part of the diffusion tensor (Jokipii et al., 1977). Depending on the situation, one may work in terms of either the drift velocity itself or the anti-symmetric diffusion tensor. In the following Giacalone et al. (1999) will use the term drift velocity or anti-symmetric diffusion tensor inter-changeably.

Giacalone et al. (1999) examine the nature of the gradient and curvature drifts in the presence of turbulent fluctuations. The standard expression for the drift velocity of a charged particle of mass m, charge q, momentum p, and speed v in a magnetic field B, in the limit that the scattering mean free path is much larger than the gyro-radius rg , is v dr = (pcv/3q -x(b/ B2), (2.38.2)

where c is the speed of light. The corresponding Ka = vrgj3 . This is the limit most-

frequently used, since Giacalone et al. (1999) expect that the mean free path is generally somewhat larger than the gyro-radius. A finite amount of scattering should reduce this somewhat. A simple analysis based on the venerable billiard ball scattering picture suggests that scattering by fluctuating magnetic field might reduce the drifts by a noticeable amount for CR in the Heliosphere (Burger and Moraal, 1990; Jokipii, 1993). Similarly some analyses of the modulation of galactic CR by the solar wind suggest that the drift motions in the Heliospheric magnetic field are significantly reduced from the classical value given above (e.g., Potgieter et al., 1989). In this special case the expressions for k_|_ and Ka become, in terms of the ratio n of the mean free path A to the gyro-radius rg ,

where K// is the parallel diffusion coefficient. Again vdr is the divergence of the anti-symmetric part of the diffusion tensor. Giacalone et al. (1999) utilize direct numerical simulations of particle motions in the turbulent magnetic field to analyze the effects of fluctuations on the drifts.

2.38.2. Analytical result and numerical simulations for CR particle drifts

Before proceeding to the results of the numerical simulations, Giacalone et al. (1999) first present an analytical result which enables to simplify and make more precise the numerical analysis. For simplicity in notation, Giacalone et al. (1999) assume without loss of generality that the average magnetic field, at least locally, is in the z direction, so that the perpendicular direc tions are x and y. In determining the transport coefficients from numerical simulations it is usual to work in terms of the Fokker-Planck transition moments ( Ax2) /At, etc (e.g., Giacalone and Jokipii,

1999). In this case the drift term appears in one or more of the first-order coefficients, for example (Ax^ZAt. However, this will only be non-zero when the magnetic field has spatial variation, and this is more complicated to compute numerically. Hence it is usually more convenient to work with the anti-symmetric diffusion coefficient, which is non-zero even if there are no gradients, and whose divergence is the drift velocity. But the obvious Fokker-Planck coefficient (AxAy)/At is obviously symmetric. The reason is that the divergence of the antisymmetric tensor is zero if the field does not vary, and in this case the antisymmetric coefficient does not appear in the diffusion equation. But it does appear in the equation for the streaming flux, or anisotropy. It must proceed differently.

It may be shown that, in general, the equation for the streaming flux in a simple system with no convection, may be written as

where the diffusion tensor Kj can be written as Kj = (v{Ax j) . It is easily seen that this also gives the anti-symmetric part of Kj. Furthermore, this form is much simpler to compute in a simulation.

According to Giacalone et al. (1999) the above result can be demonstrated as follows. At some time t, we may express the value of the distribution function f (xt, pt, t) in terms of the values of the pt = p\ and xt = x\ corresponding to the xt, pt at some other time t' (following the actual particle trajectories) by the exact relation f(, pt, t) = f (x\, p\, t), (2.38.5)

which is simply a restatement of Liouville's theorem. Now consider the situation where the time At = t'-t is many scattering times, but where the corresponding Axt = x\ -xi is much smaller than the scale of spatial variation of f (this is equivalent to the usual diffusion approximation). Then, since f is nearly isotropic and the momentum magnitude of a particle is constant, the spatial gradient df / dxj is approximately the same for all directions (all particles at a given p) and it may be write f (x'i,p'i, t) f (xi,pi, t) + Ax, (df (xi,pi, t)l'dxi). (2.38.6)

Therefore, since the p\ at x\ are scrambled relative to the pi, so that

the diffusive flux at xi, t,

Fi (p, xi, t ) = - J vfdQ = -J Vi AxjdQf = -V Axj) f = -Kyf, (2.38.8)

oxj^ ' dxj dxj which is the desired result Eq. 2.38.4. Here it was used of the fact that the diffusion coefficient k^ depends on the magnitude of p as well as xi and t, so the integral over Q sums over all the particles at a given p and the df /dxi properly weights the sum over particles. Below Giacalone et al. (1999) compute ka from the relationship Kj = (v,Aj .

2.38.3. Numerical simulations by integration of particle trajectories

Giacalone et al. (1999) integrate the trajectories of particles moving under the influence of a time-independent magnetic field of the form

The fluctuating component, 8B(r), is determined in a manner similar to that which was described previously in Giacalone and Jokipii, 1994, 1996, 1999). They are characterized by a discrete sum of individual stationary plane waves with random wave vectors, phases, and polarizations. The amplitudes are given by a

Kolmogorov-like power spectrum which is described mathematically in terms of

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