A magnetorotational instability in the solar near-surface shear layer.

The Sun rotates differentially in both latitude and depth. There is a region of near-surface shear, the outer 5\% by radius, where the rotation rate decreases with radius. This shear is sufficient to trigger the classical magneto-rotational instability. The linear instability is assessed in spherical shell geometry with a dipolar magnetic field and two shear profiles, while ignoring the dynamical influence of convection and buoyancy. The spherical eigenproblem and local Cartesian eigenproblem are shown to have some similarities, but geometry impacts the eigenfunctions and hence the growth rates. The influence of density stratification is also investigated, showing that it raises the effective magnetic energy of the system and can quench the instability if the stratification is large enough. For solar-like values of a density described by an adiabatic polytrope, it can enhance the growth depending on the assumed strength of the magnetic field at the upper boundary. The nonlinear problem is then considered in the sphere shell, using the eigenfunctions as an initial condition.