# 02.13.13: LPV analysis of exp2_l over 2 and 13 cycles¶

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).

Figure 1: Time rate of change of QL (upper) and meridional component of the Lagrangian-mean flow (VL; lower) calculated over 2 cycles (left) and 13 cycles (right) around day 4200.

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.

Figure 2: Standard deviation of dQL/dt and VL east of 2.5°E (to exclude the sponge layer) depending on how many cycles have been used to compute the 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.