In this paper, written as a general historical and technical introduction to the various contributions of the collection “Solar and Stellar Dynamo: A New Era”, we review the evolution and current state of dynamo theory and modelling, with emphasis on the solar dynamo. Starting with a historical survey, we then focus on a set of “tension points” that are still left unresolved despite the remarkable progress of the past century. In our discussion of these tension points we touch upon the physical well-posedness of mean-field electrodynamics; constraints imposed by magnetic helicity conservation; the troublesome role of differential rotation; meridional flows and flux transpost dynamos; competing inductive mechanisms and Babcock–Leighton dynamos; the ambiguous precursor properties of the solar dipole; cycle amplitude regulation and fluctuation through nonlinear backreaction and stochastic forcing, including Grand Minima; and the promises and puzzles offered by global magnetohydrodynamical numerical simulations of convection and dynamo action. We close by considering the potential bridges to be constructed between solar dynamo theory and modelling, and observations of magnetic activity in late-type stars.

We compare spectra of the zonal harmonics of the large-scale magnetic field of the Sun using observation results and solar dynamo models. The main solar activity cycle as recorded in these tracers is a much more complicated phenomenon than the eigen solution of solar dynamo equations with the growth saturated by a back reaction of the dynamo-driven magnetic field on solar hydrodynamics. The nominal 11(22)-yr cycle as recorded in each mode has a specific phase shift varying from cycle to cycle; the actual length of the cycle varies from one cycle to another and from tracer to tracer. Both the observation and the dynamo model show an exceptional role of the axisymmetric ℓ₅ mode. Its origin seems to be readily connected with the formation and evolution of sunspots on the solar surface. The results of observations and dynamo models show a good agreement for the low ℓ₁ and ℓ₃ modes. The results for these modes do not differ significantly for the axisymmetric and non-axisymmetric models. Our findings support the idea that the sources of the solar dynamo arise as a result of both the distributed dynamo processes in the bulk of the convection zone and the surface magnetic activity.

Observations of proxies of the magnetic helicity in the Sun over the past two solar cycles revealed reversals of the helicity hemispheric sign rule (negative in the North and positive in the South hemispheres). We apply the mean-field solar dynamo model to study the reversals of the magnetic helicity sign for the dynamo operating in the bulk of the solar convection zone. The evolution of the magnetic helicity is governed by the conservation law. We found that the reversal of the sign of the small-scale magnetic helicity follows the dynamo wave propagating inside the convection zone. Therefore, the spatial patterns of the magnetic helicity reversals reflect the processes which contribute to generation and evolution of the large-scale magnetic fields. At the surface, the patterns of the helicity sign reversals are determined by the magnetic helicity boundary conditions at the top of the convection zone. We demonstrate the impact of fluctuations in the dynamo parameters and variability in dynamo cycle amplitude on the reversals of the magnetic helicity sign rule. The obtained results suggest that the magnetic helicity of the large-scale axisymmetric field can be treated as an additional observational tracer for the solar dynamo and it probably can be used for the solar activity forecast as well.

European species of Nuphar are amongthe most accessible members of the basal angiosperm grade, but detailed studies using scanning electron microscopy are lacking. We provide such data and discuss them in the evolutionary context. Dorsiventral monopodial rhizomes of Nuphar bear foliage leaves and non-axillary reproductive units (RUs) arranged in a Fibonacci spiral. The direction of the phyllotaxis spiral is established in seedlings apparently environmentally and maintained through all rhizome branching events. The RUs can be located on dorsal, ventral or lateral side of the rhizome. There is no seasonality in timing of their initiation. The RUs usually form pairs in positions N and N + 2 along the ontogenetic spiral. New rhizomes appear on lateral sides of the mother rhizome. A lateral rhizome is subtended by a foliage leaf (N) and is accompanied by a RU in the position N + 2. We hypothesize a two-step process of regulation of RU/branch initiation, with the second step possibly involving environmental factors such as gravitropism. Each RU has a short stalk, 1-2 scale-like phyllomes and a long-pedicellate flower. We support a theory that the flower is lateral to the RU axis. The five sepals initiate successively and form two whorls as 3 + 2. The sepal arrangement is not ‘intermediate’ between whorled and spiral. Mechanisms of phyllotaxis establishment differ between flowers and lateral rhizomes. Petal, stamen and carpel numbers are not precisely fixed. Petals are smaller than sepals and form a whorl. They appear first in the sectors of the outer whorl sepals. The stamen arrangement is whorled to chaotic. The merism of the androecium tends to be the same as in the corolla. Flowers with odd numbers of stamen orthostichies are found. These are interpreted as having a non-integer merism of the androecium (e.g., 14.5). Carpels form a whorl in N. lutea and normally alternate with inner whorl stamens. Sterile second whorl carpel(s) are found in some flowers of N. pumila.