  As was demonstrated in earlier studies, turbulence can result in a negative
contribution to the effective mean magnetic pressure, which, in turn, can cause
a large-scale instability. In this study, hydromagnetic mean-field modelling is
performed for an isothermally stratified layer in the presence of a horizontal
magnetic field. The negative effective magnetic pressure instability (NEMPI) is
comprehensively investigated. It is shown that, if the effect of turbulence on
the mean magnetic tension force vanishes, which is consistent with results from
direct numerical simulations of forced turbulence, the fastest growing
eigenmodes of NEMPI are two-dimensional. The growth rate is found to depend on
a parameter beta_* characterizing the turbulent contribution of the effective
mean magnetic pressure for moderately strong mean magnetic fields. A fit
formula is proposed that gives the growth rate as a function of turbulent
kinematic viscosity, turbulent magnetic diffusivity, the density scale height,
and the parameter beta_*. The strength of the imposed magnetic field does not
explicitly enter provided the location of the vertical boundaries are chosen
such that the maximum of the eigenmode of NEMPI fits into the domain. The
formation of sunspots and solar active regions is discussed as possible
applications of NEMPI.
