Again, Nope

The governing equations are timestep from

n (right-hand-side terms) to

n+1 (left-hand-side term: time rate of change). The vertically-integrated equations are integrated with smaller timestep to resolve fast dynamics due to gravity wave phenomena. For 3D total momentum coupling, the

ubar,

vbar, and

zeta need to be time-averaged at

n+1. So we need to timestep the vertically integrated equations beyond the time associated with

n+1 so the time-averaged quantities are

centered exactly at

n+1. This is achieved with the cosine-squared filter so we don't need to go all the way to

n+2 to achive such average. It is cleaver and efficient

Maybe you need read the literature of split-explicit timestepping or find someone at your intitution to explain it to you.