An update of Leighton's solar dynamo model

Abstract: In 1969 Leighton developed a quasi-1D mathematical model of the solar dynamo, building upon the phenomenological scenario of Babcock(1961). Here we present a modification and extension of Leighton's model. Using the axisymmetric component of the magnetic field, we consider the radial field component at the solar surface and the radially integrated toroidal magnetic flux in the convection zone, both as functions of latitude. No assumptions are made with regard to the radial location of the toroidal flux. The model includes the effects of turbulent diffusion at the surface and in the convection zone, poleward meridional flow at the surface and an equatorward return flow affecting the toroidal flux, latitudinal differential rotation and the near-surface layer of radial rotational shear, downward convective pumping of magnetic flux in the shear layer, and flux emergence in the form of tilted bipolar magnetic regions. While the parameters relevant for the transport of the surface field are taken from observations, the model condenses the unknown properties of magnetic field and flow in the convection zone into a few free parameters (turbulent diffusivity, effective return flow, amplitude of the source term, and a parameter describing the effective radial shear). Comparison with the results of two-dimensional flux transport dynamo codes shows that the model captures the essential features of these simulations. We carry out a parameter study over the four-dimensional parameter space and identify the parameter ranges that provide solar-like solutions. Dipole parity is always preferred and solutions with periods around 22 years and a correct phase difference between flux emergence in low latitudes and the strength of the polar fields are found for a return flow speed around 2~m/s, turbulent diffusivity <80~km^2/s, and dynamo excitation not too far above the threshold (linear growth rate <0.1/yr).