The geomagnetic field is a vector sum of a magnetic dipolar field (the dominant part), a nondipole field related to the heterogeneous structure of the deep Earth's interior, magnetic properties of the crustal rocks, and magnetic field of external sources.

The resultant vector at the planetary surface differs significantly from the dipole magnetic field and is accompanied by a nonuniform magnetic gradient, particularly in the cross-longitudinal direction.

This means that when advancing towards Earth's surface, particles start experiencing magnetic irregularities, especially in the lower part of their spiral motion along the geomagnetic field lines.

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The additional cross-longitudinal magnetic gradient affects the speed of trapped and quasitrapped particles’ drift (simultaneously perpendicular to the geomagnetic field lines, the radius of their curvature, and the gradient vector), described by Eq. (5.10):

(5.10)vdrift=mq⋅B2v⊥2⋅B×∇B2B+vII2⋅ρ×Bρ2

where B is the magnetic vector, ρ is the radius of curvature of the geomagnetic field lines, vII and v⊥ are projections of particles velocities parallel and perpendicular to geomagnetic field line, respectively, and q and m are particles’ charge and mass, respectively.

The first term in the brackets corresponds to the magnetic gradient perpendicular to the field lines, while the second term relates to their curvature.

Close to the point of the magnetic mirror, the field aligned particles’ velocity is approaching zero, so the drift velocity is determined mainly by the cross-latitudinal and cross-longitudinal magnetic gradients (the first term in Eq. 5.9).

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Under the influence of the bi-directional magnetic gradient (in the x–y plane, with x directed to the east, while y is to the north), the protons, entering Earth's atmosphere from the west (at the lowest part of their circular trajectories around geomagnetic field lines), are shifted south-westwards when entering regions with a positive cross-longitudinal gradient, and south-eastwards in regions with a negative gradient (see Eq. 5.9).

Consequently, the overall westward drift (forced by the magnetic curvature and cross-latitudinal gradient) is reduced by the eastward component, exerted by the cross-longitudinal magnetic gradient in regions with negative longitudinal magnetic gradient such as the East American–Atlantic region, Eastern Asia–Western Pacific, and the South Pacific Ocean.

 

Furthermore, the drift-aligned electric field—expelling the confined particles outside the magnetic trap (due to the (E × B)/B2 electric drift; see Section 5.2)—is reduced significantly in these regions. As a result, only a few particles have a ‘chance’ to be lost in the atmosphere in these regions, and the ground-based neutron monitors should measure low counting rates.

Oppositely, in regions with positive cross-longitudinal gradients (i.e. North-Western America, Eastern Europe-Western Asia, the central part of the South Atlantic Ocean, and the South Indian Ocean) the southward component, induced by the cross-longitudinal magnetic gradient, changes slightly the direction but not the amplitude of the westward drift, impelled by the magnetic curvature and latitudinal gradient.

Consequently, in these regions the drift-induced charge separation, and related electric field, will intensively expel the charged particles outside the magnetic trap.

 

Furthermore, these particles interact with the atmospheric molecules, creating secondary electrons, ions, and nuclear products, giving rise to the ionization of the lower atmosphere, and to the radiation measured by the ground-based neutron monitors.

Moreover, the effect of the cross-longitudinal magnetic heterogeneity should be stronger in regions with a steeply decreasing negative gradient and in the rising part of the positive cross-longitudinal gradient, i.e. in regions of geomagnetic field strengthening.

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The validity of these theoretical considerations is presented in Fig. 5.5B, which compares the maps of the cross-longitudinal geomagnetic gradient (contours) with annual mean values of NMs counting rates (differently sized stars).

In order to eliminate the altitude dependence of received radiation, shown are only NMs with altitudes less than 500 m. Although the map of the near-surface particles’ intensity is quite rough (due to the relatively small number of NMs and their irregular distribution globally), Fig. 5.5B illustrates fairly well the fact that the lowest counting rates are detected in regions with longitudinaly decreasing geomagnetic field.

This effect could be a reasonable explanation for the higher particle intensity encountered in Moscow compared to Inuvik, Tule, Nain, and other neighbouring stations (Kiel, Oulu, Apatite).

Similarly, the dose measured in Norilsk and Irkutsk (situated in a region with a positive longitudinal geomagnetic field gradient) is higher than that detected in Tixie Bay, Yakutsk, and Magadan, which are placed in a region with a decreasing (along the path of the arrival protons) geomagnetic field.