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The description of linear force-free magnetic fields in terms of the Moses eigenfunctions of the curl operator begun previously is here completed by the derivation of a general expression for the field’s spherical curl transform. This enables the transform space representation of a given field to be determined and compared with that of other fields, assisting the analysis and classification of this type of magnetic field as well as providing a basis for generalization. The result obtained gives the spherical curl transform as a weighted projection of the vector Radon transform of the field on the appropriate curl eigenvector. The process is exemplified by the determination of the transforms of three fields: the simplest force-free magnetic field, and the Lundquist and classical spheromak fields. The latter two are both of interest as models of the magnetic fields of solar magnetic clouds, while the classical spheromak field is relevant to the design of nuclear fusion reactors as well. The use of the transform in generalizing the Lundquist field is briefly discussed. As before, all results apply equally well to the description of the Trkalian subset of Beltrami fields in fluid dynamics.

The Moses curl eigenfunctions are used to describe linear force-free magnetic fields, enabling a proof that such fields are defined entirely by the value of their curl transform on the unit hemisphere in transform space. This change of viewpoint suggests an orderly approach to the exploration and classification of the properties of such fields, which is briefly sketched. The simplest force-free fields defined on zero- , one- , and two-dimensional sets on the transform sphere are exhibited, and the possibility of fields with fractal support sets suggested. Connections with other mathematical descriptions are pointed out, as well as several promising directions for further exploration. All results apply equally well to the description of the Trkalian subset of Beltrami fields in fluid dynamics.

A one-dimensional time-dependent model of the ionosphere has been developed and applied to the study of a metallic ion sporadic-E layer observed in the Aladdin 1 experiment carried out at Eglin AFB, Florida, 20 November 1970. The model develops the molecular ion background ionosphere using a dynamic photochemical calculation from noon to a time near model sunset. A representative metallic ion altitude profile is then introduced, the divergence terms included in the continuity equations, and the integration carried forward. Introducing an ad hoc constant electric field of 2 mv m -• directed to the south, the model metallic ion sporadic-E layer forms at the proper altitude and reaches the measured peak density in about a half hour. Changing the initial metallic ion profile changes the time to reach peak density and the degree of asymmetry of the layer, but the layer altitude is determined asymptotically by the location of the convergent node of the vertical ion velocity profile. The background ionospheric density calculated with the model agrees within experimental error with the experimental profile. The calculations support the hypothesis that midlatitude sporadic-E layers are caused by neutral-wind-induced compression of metallic ions resulting from meteoric ablation in the lower E region. 3 of the model [Keneshea and MacLeo& 1970; MacLeod et al., 1973]. The terms in the second bracket which depend on the electric field have the same form as a neutral wind u'--E/O•, suggesting that their main effect on the motion will come from the upper E and F regions as p•--) 0. It will be seen below that these terms can also produce observable effects in the lower E region. The ion continuity equation. The general continuity equation for the ith ion species may be written allilar= Qi-Li + • ' (D•7rti-lli¾i) (6) where Q• and L• are the general source and loss terms for species i and depend on the n i number NUMERICAL MODELLING 373

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