Comparison of two gravito-inertial modes along ϕ = 0 for the different formulations. The first mode, with $\widehat{\omega }=0.421N$, corresponds to the ₁g₂² mode impacted by the rotation, while the second one is closer to an inertial or gravito-inertial mode, $\widehat{\omega }=0.777N$. Velocities are given in units such that R_oω_d = 1.56 × 10¹⁰ cm s⁻¹. Left: Toroidal velocity u_ϕ at ϕ = 0 for $\widehat{\omega }=0.421N$ in the full formulation (Panel a), split formulation (Panel b) and with only effective Coriolis forcing (Panel c). Right: Toroidal velocity u_ϕ at ϕ = 0 for $\widehat{\omega }=0.777N$ in the full formulation (Panel a), split formulation (Panel b) and with only effective Coriolis forcing (Panel c).