We present a refinement of the Fisk-Parker hybrid field of Burger and Hitge which now includes a region bordering the solar rotational equator where magnetic field footpoint motion occurs only through diffusive reconnection. The hybrid field, therefore, only occurs above a certain latitude in a given hemisphere, and in the equatorial region the field is a pure Parker field. We also propose a simple qualitative model for the solar cycle dependence of the hybrid field, taking into account changes in the tilt angle of the heliospheric current sheet and the latitudinal extend of the polar coronal hole on the photosphere and on the source surface over the course of a solar activity cycle. We find that the amplitude of magnetic field fluctuations for assumed solar minimum parameters would not be observable above the background noise (see Roberts and coworkers). We also show that for these parameters, periodicities associated with differential footpoint motion would be barely distinguishable from rigid rotation at the solar equatorial rate. We point out that the question of periodicities in magnetic field data is perhaps more complicated than previously thought. We confirm the result of Burger and Hitge that a Fisk-type heliospheric magnetic field provides a natural explanation for the observed linear relationship between the amplitude of the recurrent cosmic-ray variations and the global latitude gradient (see Zhang). We show that this relationship holds for helium, protons, and electrons. Moreover, we show that the constant of proportionality is larger when qA > 0 than when qA < 0, as inferred from observations by Richardson and coworkers.

The Fisk model of the heliospheric magnetic field (HMF) has proved to be quite difficult to implement in three-dimensional numerical modulation codes. We have developed a divergence-free Fisk-Parker hybrid HMF that is easier to implement and that should be a reasonable approximation for the case when field lines open into the heliosphere both in the polar coronal holes and at low latitudes. To study the effects of this hybrid field on galactic cosmic rays, we solve the three-dimensional steady state Parker transport equation numerically, covering the entire model heliosphere. We find that latitudinal cosmic-ray gradients are reduced as expected, but only during epochs when particles are drifting from the polar regions of the heliosphere toward the ecliptic plane. We investigate 26 day recurrent variations of both protons and electrons and find that the amplitude of these variations are directly proportional to the latitudinal intensity gradient, independent of species, in agreement with observational studies. In the absence of drifts in the code, electrons and protons obey different but still linear relationships.

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Mathematics forms an integral part in the training of scientists and engineers. In recent history the South African school system has experienced several changes in school curricula. In 1994 the traditional knowledge-based curricula were replaced by an outcomes-based curriculum. Owing to implementation problems which resulted in resistance from teachers and the general public, revisions followed of which the National Curriculum Statement (NCS) and Curriculum Assessment Policy Statements had the most direct effect in terms of preparation for tertiary mathematics. We report here on an investigation of the basic mathematical knowledge of three student cohorts representing three curricula, namely the last cohort that received the traditional knowledge-based curriculum, and the first cohorts that received the two outcomes-based curricula. The results indicate that changes in the mathematical content of the curricula did not impact negatively on the basic mathematical knowledge of students enrolled for tertiary mainstream mathematics. The only exception is Euclidean geometry, for which certain topics were transferred to an optional paper in the NCS curriculum. Significance: • The introduction of outcomes-based curricula in South Africa initiated a discourse on the preparedness of first-year students for programmes with mainstream mathematics. • The availability of a homogeneous set of samples and a uniform test provided a unique opportunity to compare the basic mathematical knowledge of first-year natural science and engineering students entering university from three different exit-level school curricula.