  We derive a semi-geostrophic variational balance model for the
three-dimensional Euler--Boussinesq equations on the non-traditional $f$-plane
under the rigid lid approximation. The model is obtained by a small Rossby
number expansion in the Hamilton principle, with no other approximations made.
We allow for a fully non-hydrostatic flow and do not neglect the horizontal
components of the Coriolis parameter, i.e., we do not make the so-called
"traditional approximation". The resulting balance models have the same
structure as the "$L_1$ balance model" for the primitive equations: a kinematic
balance relation, the prognostic equation for the three-dimensional tracer
field, and an additional prognostic equation for a scalar field over the
two-dimensional horizontal domain which is linked to the undetermined constant
of integration in the thermal wind relation. The balance relation is elliptic
under the assumption of stable stratification and sufficiently small
fluctuations in all prognostic fields.
