  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.
