This study extends our previous work (McCloughan & Suslov, J. Fluid Mech. , vol. 887, 2020, A23), where the existence of a saddle-node bifurcation of steady axisymmetric electrolyte flows driven by the Lorentz force in a shallow annular domain was first reported. Here we perform further weakly nonlinear analysis over a wider range of the governing parameters to demonstrate that the previously reported saddle-node bifurcation is a local feature of a global fold catastrophe, which, in turn, is a part of cusp catastrophe occurring as the thickness of the fluid layer increases. The amplitude equation characterising multiple flow solutions in the finite vicinity of catastrophe points is derived. The sensitivity of its coefficients and solutions to the distance from the catastrophe points is assessed demonstrating the robustness of the used analytical procedure. The asymptotic flow solution past the catastrophe point is subsequently obtained and its topology is explored confirming the existence of the secondary circulation in the bulk of flow (two-tori background flow structure). The latter is argued to lead to the appearance of experimentally observable vortices on the fluid surface. The rigorous justification of this conjecture is to be given in Part 2 of the study.

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A circumferential flow of a conducting fluid in an annular channel can be created by the action of a Lorentz force arising as a result of the interaction between an applied vertical magnetic field and a radial electric current flowing through the electrolyte. Quite unexpectedly, experiments revealed that a robust vortex system appears near the outer cylindrical wall in such flows. McCloughan and Suslov (J. Fluid Mech. 887:A23, 2020) (McCS) reported comprehensive linear stability results of such a flow for variable Lorentz forcing. Here we complement that study by investigating the flow structure as a function of the channel aspect ratio. Remarkably, despite the completely different physical nature of parametric dependences, dynamic in McCS and purely geometric here, we show that in both scenarios vortices appear on a background of a steady axisymmetric flow at the boundary between two counter-rotating toroidal structures and have a similar energy distributions. The two studies demonstrate the robustness of the mechanism responsible for the vortex formation: Rayleigh's inviscid centrifugal instability aided by radial shear in the boundary layer near the outer cylindrical wall. References P. A. Davidson. An introduction to magnetohydrodynamics. Cambridge University Press, 2nd edition, 2017. doi:10.1017/CBO9780511626333. J. McCloughan and S. A. Suslov. Linear stability and saddle–node bifurcation of electromagnetically driven electrolyte flow in an annular layer. J. Fluid Mech., 887:A23.1–30, 2020. doi:10.1017/jfm.2020.29. J. Perez-Barrera, J. E. Perez-Espinoza, A. Ortiz, E. Ramos, and S. Cuevas. Instability of electrolyte flow driven by an azimuthal Lorentz force. Magnetohydrodynamics, 51(2):203–213, 2015. http://mhd.sal.lv/contents/2015/2/MG.51.2.4.R.html. S. A. Suslov, J. Perez-Barrera, and S. Cuevas. Electromagnetically driven flow of electrolyte in a thin annular layer: Axisymmetric solutions. J. Fluid Mech., 828: 573–600, 2017. doi:10.1017/jfm.2017.551.

Comprehensive linear stability study of flow in an annular layer of electrolyte driven by the action of the Lorentz force is conducted following the analysis of steady axisymmetric solutions of Suslov et al. (J. Fluid Mech., vol. 828, 2017, pp. 573-600). It is shown that an experimentally observed instability in the form of anticyclonic moving vortices reported in Perez-Barrera et al. (Magnetohydrodynamics, vol. 51 (2), 2015, pp. 203-213) develops on a background of the basic flow consisting of two tori with the opposite azimuthal vorticity components. It is found that, while the background flow is driven electromagnetically, the appearance of vortices is purely due to hydrodynamic effects: shear of the flow and centrifugal inertial forcing. The current study has also revealed that the unstable two-torus basic flow has a stable single-torus counterpart, both emanating from a saddle-node bifurcation of steady states when the Lorentz force is sufficiently strong. The transition from a one-torus to two-torus flow at weaker forcing is abrupt and leads to the appearance of vortices as soon as it occurs. The ranges of layer depths and Reynolds numbers for which vortices develop on a steady background are determined. Subsequently, weakly nonlinear amplitude expansion is used to find an approximate unsteady solution beyond the saddle-node bifurcation.

We describe the application of the synoptic transport equation to simulate the temporal evolution of the magnetic flux over the solar surface. This provides a means of predicting each day both the synoptic maps for the Carrington rotation starting the next day and the instantaneous map of the solar flux over the whole solar surface for the next day. The reliability of the predicted synoptic maps is tested by comparing the locations of the zero-flux contour with those of the observed maps produced by the National Solar Observatory, Kitt Peak and with the locations of Hα filaments measured on filtergrams obtained by the Big Bear Solar Observatory. We conclude that the best match at high latitudes is obtained by long-term simulations (over 20 rotations) with flux updates each rotation between latitudes ± 60°. We illustrate the use of the simulations to describe the evolution of the polar fields at the time of the polarity reversals in Cycle 23. The reconstruction of the instantaneous maps is tested by comparison with full-disk magnetograms. The method provides a simple means of estimating the large-scale flux distribution over the whole surface. It does not take account of flux emerging after the central meridian passage each rotation so it is only approximate in the activity belts but provides a reliable map beyond those latitudes.

The evolution of magnetic flux at the solar surface is widely modeled by the flux transport equation. This describes the distribution of flux from instant to instant over the whole surface but does not describe how the synoptic map for one Carrington rotation evolves into the synoptic map for the next rotation. We derive the correct synoptic evolution equation, show that a simple version yields extremely accurate predictions of synoptic maps and discuss the implications for previous studies of the evolution of surface magnetic structures. We also note that the procedure yields a method of reconstructing an approximate map of the flux over the whole surface at any instant.