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In this paper we apply second-order gauge-invariant perturbation theory toinvestigate the possibility that the non-linear coupling between gravitationalwaves (GW) and a large scale inhomogeneous magnetic field acts as anamplification mechanism in an `almost' Friedmann-Lemaitre-Robertson-Walker(FLRW) Universe. The spatial inhomogeneities in the magnetic field areconsistently implemented using the magnetohydrodynamic (MHD) approximation,which yields an additional source term due to the interaction of the magneticfield with velocity perturbations in the plasma. Comparing the solutions withthe corresponding results in our previous work indicates that, on super-horizonscales, the interaction with the spatially inhomogeneous field in the dustregime induces the same boost as the case of a homogeneous field, at least inthe ideal MHD approximation. This is attributed to the observation that the MHDinduced part of the generated field effectively only contributes on scaleswhere the coherence length of the initial field is less than the Hubble scale.At sub-horizon scales, the GW induced magnetic field is completely negligiblein relation to the MHD induced field. Moreover, there is no amplification foundin the long-wavelength limit.Comment: 14 pages revte

Inhomogeneous magnetic seed fields and gravitational waves within the magnetohydrodynamic limit