  In this study we give a characterization of semi-geostrophic turbulence by
performing freely decaying simulations for the case of constant uniform
potential vorticity, a set of equations known as surface semi-geostrophic
approximation. The equations are formulated as conservation laws for potential
temperature and potential vorticity, with a nonlinear Monge-Amp\'{e}re type
inversion equation for the streamfunction, expressed in a transformed
coordinate system that follows the geostrophic flow. We perform model studies
of turbulent surface semi-geostrophic flows in a doubly-periodic domain in the
horizontal limited in the vertical by two rigid lids, allowing for variations
of potential temperature at one of the boundaries, and we compare them with the
corresponding surface quasi-geostrophic case. Results show that, while surface
quasi-geostrophic dynamics is dominated by a symmetric population of
cyclones-anticyclones, surface semi-geostrophic dynamics features a more
prominent role of fronts and filaments. The resulting distribution of potential
temperature is strongly skewed and peaked at non-zero values at and close to
the active boundary, while symmetry is restored in the interior of the domain,
where small-scale frontal structures do not penetrate. In surface
semi-geostrophic turbulence energy spectra are less steep than in the surface
quasi-geostrophic case, with more energy concentrated at small scales for
increasing Rossby number. Energy connected to frontal structures, lateral
strain rate and vertical velocities are largest close to the active boundary.
These results show that the semi-geostrophic model could be of interest for
studying the lateral mixing of properties in geophysical flows.
