  Strongly stratified hydromagnetic turbulence has recently been identified as
a candidate for explaining the spontaneous formation of magnetic flux
concentrations by the negative effective magnetic pressure instability (NEMPI).
Much of this work has been done for isothermal layers for which the density
scale height is constant throughout. We now study the validity of earlier
conclusions about the size and growth rate of magnetic structures in the case
of polytropic layers, which scale height decreases sharply towards the surface.
To allow for a continuous transition from isothermal to polytropic layers, we
employ a generalization of the exponential function known as the q-exponential.
Now, the top of the polytropic layer shifts with the polytropic index such that
the scale height at some reference height is always the same. We use both
mean-field and direct numerical simulations of forced stratified turbulence to
determine the resulting flux concentrations. Magnetic structures begin to form
at a depth where magnetic field strength is about 3-4% the local equipartition
field strength with respect to the turbulent kinetic energy. Unlike the
isothermal case where stronger fields can give rise to magnetic flux
concentrations at larger depths, in the polytropic one the growth rates
decreases for structures deeper down. For vertical fields, magnetic structures
of super-equipartition strengths are formed because such fields survive
downward advection, unlike NEMPI under horizontal magnetic fields. The
horizontal cross-section of such structures is approximately circular. Results
based on isothermal models can be applied locally to polytropic layers. For
vertical fields, magnetic flux concentrations of super-equipartition strengths
form, which supports suggestions that sunspot formation might be a shallow
phenomenon.
