  The ordered magnetic field observed via polarized synchrotron emission in
nearby disc galaxies can be explained by a mean-field dynamo operating in the
diffuse interstellar medium (ISM). Additionally, vertical-flux initial
conditions are potentially able to influence this dynamo via the occurrence of
the magneto-rotational instability (MRI). We aim to study the influence of
various initial field configurations on the saturated state of the mean-field
dynamo. This is motivated by the observation that different saturation behavior
was previously obtained for different supernova rates. We perform direct
numerical simulations (DNS) of three-dimensional local boxes of the vertically
stratified, turbulent interstellar medium, employing shearing-periodic boundary
conditions horizontally. Unlike in our previous work, we also impose a vertical
seed magnetic field. We run the simulations until the growth of the magnetic
energy becomes negligible. We furthermore perform simulations of equivalent 1D
dynamo models, with an algebraic quenching mechanism for the dynamo
coefficients. We compare the saturation of the magnetic field in the DNS with
the algebraic quenching of a mean-field dynamo. The final magnetic field
strength found in the direct simulation is in excellent agreement with a
quenched $\alpha\Omega$~dynamo. For supernova rates representative of the Milky
Way, field losses via a Galactic wind are likely responsible for saturation. We
conclude that the relative strength of the turbulent and regular magnetic
fields in spiral galaxies may depend on the galaxy's star formation rate. We
propose that a mean field approach with algebraic quenching may serve as a
simple sub-grid scale model for galaxy evolution simulations including a
prescribed feedback from magnetic fields.
