  We prove short time existence and uniqueness of smooth solutions ( in
$C^{k+2,\alpha}$ with $k\geq 2$) to the 2-D semi-geostrophic system and
semi-geostrophic shallow water system with variable Coriolis parameter $f$ and
periodic boundary conditions, under the natural convexity condition on the
initial data. The dual space used in analysis of the semi-geostrophic system
with constant $f$ does not exist for the variable Coriolis parameter case, and
we develop a time-stepping procedure to overcome this difficulty.
