In *this note*, I show the LPV analysis of `exp2_l` (MAG=100, no dissipation, with a sponge along the western boundary) over 2 cycles. Because the time rate of change of QL was from being zero, I performed the LPV analysis over 13 cycles. Fig. 1 shows the time rate of change of QL and the meridional component of the Lagrangian-mean flow (equal to all purposes here to the component of the Lagrangian-mean flow across Lagrangian-mean PV contours) calculated over 2 cycles (left panels) and 13 cycles (right panels).

What we see is that the two quantities have decreased between using 2 cycles and using 13 cycles. This is a good result as we are trying to show that 1) the time rate of change should decrease the more cycles we use and 2) VL should be zero ultimately in the final balance without the time rate of change of QL.

But what I am realizing here is that we are effectively trying to show that each term should be zero at some point (because each terms in the final balance should be zero). This is of course not possible to show; the two terms shown in Fig. 1 will always be numerically non-zero.

So, the fact alone that both terms decrease is the best result we can get for this simulation. This also suggests that we should study instead a simulation with weak dissipation (for instance `exp25`, `exp2_w6` or `exp2_w6_sponge`) and find out how many cycles we need until the time rate of change becomes negligible compared to the component of the Lagrangian-mean flow across Lagrangian-mean PV contours (which can be taken as VL for all purposes here).

Fig. 2 shows the standard deviation of the two terms over the domain east of 2.5°E (to exclude the sponge layer). We see that, as expected, 1) VL is always canceling dQL/dt and 2) we might be able to use a simple test based on the calculation of QL and VL to know how many cycles we need to get the wanted balance.

Computed with `theory_test_several_cycles_script.m` in `RESEARCH/MODELISATION/HIM/studies/diss_train_of_eddies/exp2/exp2_l/analysis_1d` on the main disk on `ipu1`. The Matlab files is `diag_VC_thirteen_cycles_100day_long_day4215_to_day4500_exp2_l.mat` in that same directory.