Kilogauss-strength magnetic ﬁelds are often observed in intergranular lanes at the photosphere in the quiet Sun. Such ﬁelds are stronger than the equipartition ﬁeld B_e, corresponding to a magnetic energy density that matches the kinetic energy density of photospheric convection, and comparable with the ﬁeld B_p that exerts a magnetic pressure equal to the ambient gas pressure. We present an idealised numerical model of three-dimensional compressible magnetoconvection at the photosphere, for a range of values of the magnetic Reynolds number. In the absence of a magnetic ﬁeld, the convection is highly supercritical and is characterised by a pattern of vigorous, time-dependent, “granular” motions. When a weak magnetic ﬁeld is imposed upon the convection, magnetic ﬂux is swept into the convective downﬂows where it forms localised concentrations. Unless this process is significantly inhibited by magnetic diffusion, the resulting ﬁelds are often much greater than B_e, and the high magnetic pressure in these ﬂux elements leads to their being partially evacuated. Some of these ﬂux elements contain ultra-intense magnetic ﬁelds that are significantly greater than B_p. Such ﬁelds are contained by a combination of the thermal pressure of the gas and the dynamic pressure of the convective motion, and they are constantly evolving. These ultra-intense ﬁelds develop owing to nonlinear interactions between magnetic ﬁelds and convection; they cannot be explained in terms of “convective collapse” within a thin ﬂux tube that remains in overall pressure equilibrium with its surroundings.\u
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