  This paper describes a new 2D model for the photospheric evolution of the
magnetic carpet. It is the first in a series of papers working towards
constructing a realistic 3D non-potential model for the interaction of
small-scale solar magnetic fields. In the model, the basic evolution of the
magnetic elements is governed by a supergranular flow profile. In addition,
magnetic elements may evolve through the processes of emergence, cancellation,
coalescence and fragmentation. Model parameters for the emergence of bipoles
are based upon the results of observational studies. Using this model, several
simulations are considered, where the range of flux with which bipoles may
emerge is varied. In all cases the model quickly reaches a steady state where
the rates of emergence and cancellation balance. Analysis of the resulting
magnetic field shows that we reproduce observed quantities such as the flux
distribution, mean field, cancellation rates, photospheric recycle time and a
magnetic network. As expected, the simulation matches observations more closely
when a larger, and consequently more realistic, range of emerging flux values
is allowed (4e16 - 1e19 Mx). The model best reproduces the current observed
properties of the magnetic carpet when we take the minimum absolute flux for
emerging bipoles to be 4e16 Mx. In future, this 2D model will be used as an
evolving photospheric boundary condition for 3D non-potential modeling.
