Gain of efficiency with a new time scheme in NEMO: Runge Kutta 3rd order

As Ocean General Circulation Models (OGCMs) become more complex, their computational efficiency remains a crucial challenge for ocean simulations. By using smart numerical choices, however, it is possible to increase the time step of an OGCM for a fixed spatial resolution, thereby reducing the computational cost and improving efficiency. To do this we are revisiting the NEMO time-stepping scheme. Modified Leapfrog (MLF) has been used so far. RK3 time stepping implemented in NEMO is a three-stage explicit RK scheme. It integrates the model from time step n to n+1. During the first stage ( from n to n+⅓ ) we build guesses of tracers and momentum using an euler time stepping : on the one hand tracers are estimated by integrating advection terms only, on the other hand momentum are estimated by integrating advection, pressure gradient and coriolis terms only. During the second stage ( from n to n+½ ) we use n+⅓ tracers and momentum fluxes from stage 1 to build tracers and momentum guesses at n+½ using the same equations. The last stage integrates full equations from n to n+1 with n+½ tracers and momentum flux guesses from stage 2. We integrate the barotropic mode from n to n+1 at the beginning of the time-step and at each RK stage the newly computed fields (velocity, transport) are corrected so that their vertical integral matches the barotropic fields linearly interpolated using times n and n+1. This method is referred to as "first-stage strategy" in Ducousso et al. (2024). The new RK3 implementation is now ready, it has been tested on all NEMO's components. Global simulations with sea ice and biogeochemistry at low resolution show that the model using RK3 is stable when doubling the time step. Nevertheless, some work remains before considering RK3 as MLF replacement into NEMO.