Fig. 1 first shows the PDf in time of the zonal velocity U west of the forcing area and around the central latitude in the middle layer. We see that an instability occurred with a low-frequency wave of period about 600 days has been generated. This wave interacts with the initial wave to generate multiple peaks around the frequency peak of the initial wave. A similar feature appears around each harmonic of the initial wave frequency.

Fig. 2 shows the LPV analysis using 12 cycles separated by 50 days each (over a total of 600 days). We can compare this with the analysis made over two cycles only (Fig. 4 in *this note*). As expected, the time rate of change is much weaker and we almost have a the steady balance between the advective term and the dissipative term.

Fig. 3 shows how the rms of each term varies with a greater number of cycles included. We see that the rms of the time rate of change keeps decreasing while the rms of the advective and dissipative terms stabilize and become equal. We may need, however, 4 or 6 more cycles to maybe get the rms as low as 5% of the rms of the advective and dissipative term.

The only thing I am a bit disappointed is the structure of the Lagrangian-mean flow. Unfortunately, I did a mistake in writing the output of `theory_test_light_several_cycles_script.m` (now corrected) so I do not have the Lagrangian-mean flow averaged over all these cycles. But the advective and dissipative terms shown in Fig. 2 should now be close to the meridional component of the Lagrangian-mean flow. If that is true then the structure of Lagrangian-mean flow becomes very complicated.

Computed with `theory_test_light_several_cycles_script.m` in `RESEARCH/MODELISATION/HIM/studies/PV_and_dissipation/forced_damped_wave/exp25` on the main disk on `ipu1`. The Matlab file is `diag_VC_twenty_cycles_100day_long_day2015_to_day2300_exp25_light.mat` in that same directory. Fig. 3 was produced using `LPV_steady_balance_test_conv.m` in `RESEARCH/MODELISATION/HIM/studies/PV_and_dissipation/forced_damped_wave/exp25`.