Here (Fig. 1), we show the result of a calculation where Lagrangian means have been computed over 90-day long cycles and these means have been then further averaged out over 10 of these cycles.

Good: The time rate of change has not blown out and the other terms are similar to what we get when we calculate the Lagrangian means over the actual wave cycles (100 days). Good: Also, the nonlinear term is nearly equal to the terms computed from the ensemble-averaged Lagrangian means. Bad (but not too bad): The time rate of change has not canceled (even if it does when the cycles are 100-day long).

Fig. 2 shows that, actually, the time rate of change does get weaker and weaker, the more cycles are included in the average. Given the decay, we hope that it should get small relative to the other terms after including about 20 cycles. Actually, if we would have included 9, instead of 10 cycles, we would have already get the picture we want, with negligible time rate of change (Fig. 3).

computed with `theory_test_several_cycles_script.m` in `RESEARCH/MODELISATION/HIM/studies/PV_and_dissipation/forced_damped_wave/exp20` on the main `ipu1` disk. The Matlab file is `diag_VC_ten_cycles_90day_long_exp20_2.mat` in that same directory.