  We propose a Discontinuous Galerkin (DG) scheme for the numerical solution of
the Hydrostatic Stokes equations in Oceanography. This new scheme is based on
the introduction of the symmetric interior penalty (SIP) technique for the
Hydrostatic Stokes mixed variational formulation. Recent research showed that
stability of the mixed formulation of Primitive Equations requires LBB
(Ladyzhenskaya--Babu\v{s}ka--Brezzi) inf-sup condition and an extra hydrostatic
inf-sup restriction relating the pressure and the vertical velocity. This
hydrostatic inf-sup condition invalidates usual Stokes continuous finite
elements like Taylor-Hood $P_2/P_1$ or bubble $P_{1,b}/P_1$. Here we consider
$P_k/P_k$ discontinuous finite elements and, using adequate LBB-like and
hydrostatic discrete inf-sup conditions we can demonstrate stability of the SIP
DG scheme in the natural energy norm for this problem. Finally, according
numerical tests are provided.
