  In order to treat the multiple time scales of ocean dynamics in an efficient
manner, the baroclinic-barotropic splitting technique has been widely used for
solving the primitive equations for ocean modeling. Based on the framework of
strong stability-preserving Runge-Kutta approach, we propose two high-order
multirate explicit time-stepping schemes (SSPRK2-SE and SSPRK3-SE) for the
resulting split system in this paper. The proposed schemes allow for a large
time step to be used for the three-dimensional baroclinic (slow) mode and a
small time step for the two-dimensional barotropic (fast) mode, in which each
of the two mode solves just need to satisfy their respective CFL conditions for
numerical stability. Specifically, at each time step, the baroclinic velocity
is first computed by advancing the baroclinic mode and fluid thickness of the
system with the large time-step \textcolor{black}{and the assistance of some
intermediate approximations of the baroctropic mode obtained by substepping
with the small-time step}; then the barotropic velocity is corrected by using
the small time step to re-advance the barotropic mode under an improved
barotropic forcing produced by interpolation of the forcing terms from the
preceding baroclinic mode solves; lastly, the fluid thickness is updated by
coupling the baroclinic and barotropic velocities. Additionally, numerical
inconsistencies on the discretized sea surface height caused by the mode
splitting are relieved via a reconciliation process with carefully calculated
flux deficits. Two benchmark tests from the "MPAS-Ocean" platform are carried
out to numerically demonstrate the performance and parallel scalability of the
proposed SSPRK-SE schemes.
