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Eulerian and Lagrangian PV balances in various simulations

The Eulerian and Lagrangian PV balances have been computed for 4 fixed locations and 4 parcels, respectively, and for each of the simulations described in Tab. 1.

  -clinic dt -tropic dt SPLIT
exp4 1200 s 15 s YES
exp6 30 s 15 s NO
exp7 300 s 15 s YES
exp8 30 s 15 s YES

Table 1: Baroclinic and -tropic time step dt for each of the 4 simulations. The use or not of the SPLIT option is shown in the last column

Fig. 1 summarizes the results by showing the error in the respective PV balance. Figs. 2 to 9 show the actual PV balance. Even with a time step reduced by a factor 4 (from exp4 to exp7, the Lagrangian PV balance is not satisfactory (exp7; see also Fig. 7). Only with the time step reduced to 30 s, the Lagrangian PV balance is satisfactory, with or without the SPLIT condition (exp6 and exp8; see also Figs. 5 and 9). The simulation with the SPLIT condition (exp6) gives, however, the smallest error.


Figure 1: Error in the (left) Eulerian PV balance at 4 fixed locations and (right) Lagrangian PV balance for 4 parcels for the different simulations. The parcels have started at the fixed locations at which the Eulerian PV balance is computed. The trajectories of the parcels and the actual PV balances are shown in the next figures for each simulation. Dots lines are from using u and v outputs only and no vertical dissipation, plain lines are from using other model outputs: horizontal and vertical friction as well as advective terms (for the Eulerian PV balance only). See analysis_2_script.m in RESEARCH/MODELISATION/HIM/studies/PV_and_dissipation/forced_damped_wave/exp4/.

Because the PV balances seem to be sensitive to the time resolution of the model, one can ask if they are also sensitive to the time resolution of the output used to compute the trajectories. The answer is yes as Fig. 10 shows for exp6. One can wonder, thus, if exp4 might still satisfy the PV balance if the time resolution of the model output would have been larger.

Finally, does the advective scheme matter? exp9 is similar to exp6 except that the advective scheme of Arakawa and Hsu (energy and enstrophy conserving) is used, instead of Sadourny (energy conserving scheme). It appears that if you use model output u and v only 1) the advective scheme does matter and 2) Sadourny’s scheme might be better (Fig. 11). The reason why this affect only the calculation using model output u and v is not known.



Figure 2: Trajectories of the 4 parcels in exp4. The red square shows the initial position and the red bar the net displacement after one wave cycle. See analysis_2_script.m in RESEARCH/MODELISATION/HIM/studies/PV_and_dissipation/forced_damped_wave/exp4 to reproduce all the figures of this note.


Figure 3: (left) Eulerian and (right) Lagrangian PV balance in exp4. The Eulerian PV balances are performed at the initial position of the parcel and the Lagrangian one along their trajectories. Each row corresponds to one parcel, the trajectories of which are given in the previous figure at the same row number. Dotted and plain lines are obtained by using different modle outputs (see legend).



Figure 4: As in Fig. 2 but for exp6.


Figure 5: As in Fig. 3 but for exp6.



Figure 6: As in Fig. 2 but for exp7.


Figure 7: As in Fig. 3 but for exp7.



Figure 8: As in Fig. 2 but for exp8.


Figure 9: As in Fig. 3 but for exp8.

Sensitivity to the time resolution of the model output


Figure 10: As in Fig. 1 except that in all cases, we use exp6, the difference bring that the full 1-day resolution of the model outputs is used in the upper panels, while only every other day outputs are used in the lower panels. We can see that the error is as large as in exp4.

Sensitivity to the advective numerical scheme


Figure 11: As in Fig. 1 except for (upper) exp6 and (lower) exp9.