# Eulerian and Lagrangian PV balances: Effect of the time resolution of the model output¶

## Conclusion¶

In this note, one can find tests of the Eulerian and Lagrangian PV balances for two experiments (exp4 that does not satisfy sufficiently the Eulerian and Lagragian PV balances, and exp6 that does) using model outputs with various temporal resolution: 1/4-day, 1-day and 2-day for exp4, 1/2-day, 1-day and 2-day for exp6.

In exp4, the error seems to converge and be irreducible while in exp6, the error keeps decreasing to zero with higher temporal resolution. To explain this difference, we need to describe the different components of the error. The error is composed of A) the actual deviation from the PV balance by the model equations and its discretization in time and space, B) the error in my calculation of the different terms in the PV balance from the model output. In all the figures, I show the error computed using either snapshots of u and v or averages of other model outputs. The difference between the two gave us an estimate for error B.

If the model does not satisfy well the PV balance, then the error will be dominated by error A. This is what, I think, is happening for exp4. Whatever the increase in model output resolution (and in the decrease of error B), the error A does not change. If, on the other hand, the model does satisfies well the PV balance, then the error will be dominated by error B. This is what, I think, is happening in exp6. In this case, error B keeps decreasing with model outputs with higher temporal resolution.

Thus, in exp6, the PV balance is satisfied as much as the accuracy of our test allows to tell. In exp4, our test is accurate enough to see that the model does not satisfy the PV balance.

What is still puzzling is why even in exp4, the net PV change is satisfied (all the errors go to zero at the end of the cycle), even if that net is actually smaller (by about an order of magnitude) than the magnitude of the PV change during one wave cycle? Maybe I should plot the error weighted by the PV to see if, as we should expect, the error in exp4 is larger than in exp6.

## exp4¶

Fig. 1 shows the trajectories of 4 parcels in exp4 using either the 1/4-day, 1-day or 2-day model outputs. Fig. 2 shows the error in the Eulerian and Lagrangian PV balance in each case. The trajectories appear to be the same at the zeroth order . For the Lagrangian PV balance, the error is close to be the same as well. The error is different for the Eulerian PV change but has the same order of magnitude so that with respect to the absolute balance, there is not much change.

Figure 1: Trajectories of 4 parcels in exp4 computing using either the 1/4-day (line), 1-day (dots) or 2-day (stars) model outputs.

Figure 2: Error in the (left) Eulerian PV balance at 4 fixed locations and (right) Lagrangian PV balance for 4 parcels in exp4. The parcels have started at the fixed locations at which the Eulerian PV balance is computed. Each color corresponds to a parcel. Lines: 1/4-day u and v snapshots; dots: 1/4-day averages; squares: 1-day u and v snapshots; circles: 1-day averages; stars: 2-day u and v snapshots; crosses: 2-day averages.

It is interesting that the differences are more dramatic in the Eulerian PV change. One explanation is that these differences come from mainly the meridional advective term which is the largest term in the balance. For the Lagrangian PV balance, because the differences in the trajectories using either the 2-day, 1-day and 1/4-day outputs are smaller, the differences in the error are also smaller.

## exp6¶

The same analysis is repeated for exp6 using either the 1/2-day, 1-day or 2-day model outputs. Again, the trajectories appear the same at the zeroth order for the three cases (Fig. 3). However, although in exp4 it seems that one has reached convergence so that the error computed with a model output of higher temporal resolution does is not reduced, in exp6, the error keeps decreasing to zero (Fig. 4). One can wonder if an irreducible error will be reached, as in exp4, or if the error does converge to zero.

Figure 3: Trajectories of 4 parcels in exp6 computing using either the 1/2-day (line), 1-day (dots) or 2-day (stars) model outputs.

Figure 4: Error in the (left) Eulerian PV balance at 4 fixed locations and (right) Lagrangian PV balance for 4 parcels in exp6. The parcels have started at the fixed locations at which the Eulerian PV balance is computed. Each color corresponds to a parcel. Lines: 1/2-day u and v snapshots; dots: 1/2-day averages; squares: 1-day u and v snapshots; circles: 1-day averages; stars: 2-day u and v snapshots; crosses: 2-day averages.