This is the first result concerning the relationship between the Lagrangian mean flow and the level of dissipation.

To obtain this relationship, the 3-layer model with the same configuration than in the previous simulations has been used (see for instance *the previous note*). Previously, I was struggling with instabilities arising even with relatively weak amplitude. Instead of keeping to work out this issue, I decided to increase the wind stress to get realistic (about 5-10 cm/s) wave amplitude but to add as well dissipation, starting with a high level of it. Although the wave amplitude is now realistic, I am pretty sure we are now in a moderately to strongly nonlinear regime –by looking, for instance, to the far-from-circle trajectories of parcels. The first experiment was run with a Laplacian friction with coefficient of K=1e4 cm2/s. Three more runs have been performed, each by decreasing the Laplacian coefficient by a factor of 2 or 5.

Each experiment has been run for 2000 days first. Contrary to previous experiments with no dissipation, the 100-day Eulerian mean flow is quasi-stationary after 200 days with no sign of developping instabilities (not shown). Because of the absence of instabilities, I decided not to use a sponge layer anymore, which did not change the result and further decreased the computing time. After day 2000, each experiment is run for another 100 days with a 1-day output in order to compute off-line the Lagrangian trajectories. The Lagrangian mean flow over the 100-day cycle is then deduced from the net displacement of parcels. A meridional line of parcels have been put to the west of and within the forcing area. Because there is a lot of information and I am struggling to understand if there is anything relevant, I only show in Fig. 1 the Lagrangian mean zonal flow to the west of the forcing area in the middle layer. I was, to be honest, excited to the possibility to see the *decrease* of the Lagrangian mean flow with the *decrease* of dissipation. Of course, it is the contrary that we observe –which would not surprise people not aware of the role of dissipation in allowing Lagrangian mean flow.

The problem with this result is that more work needs to be done before concluding anything. Several reasons can explain why the Lagrangian mean flow does not decrease with dissipation:

- With no dissipation and infinitively strong dissipation, the Lagrangian mean flow vanishes and can both increase and decrease with dissipation between these two limits. It is thus possible that the experiments are in the branch where the Lagrangian mean flow increases with weaker dissipation and further experiments with much weaker dissipation would be needed.
- The computation of the Lagrangian trajectories is wrong; one simple test would be to recalculate them with lower time and/or spatial resolution and see if the difference is significant or not. I am guessing that with 100 points per cycle, as we have now, time resolution should not be too much of a problem. I am not sure, however, with respect to the spatial resolution.
- The model does not respect the potential vorticity (PV) equation; a simple test would be to compare the PV obtained at each location along each trajectory with the PV obtained from the integration of the PV equation the same trajectory, following Holloway and Hua’s works.

The problem that I have if it was reason 1 is that I think the regime of the experiments is moderately to strongly nonlinear, which 1) might be unrealistic and 2) makes computation of Lagrangian trajectories less accurate. Again I should first obtain an experiment with satisfying amplitude, yet in the weakly to moderately nonlinear regime. I have some options for this purpose that I still need to explore.

Eric’s questions/comments and my answers:

*In your present run, is the primary disturbance unstable, so that there is an energy cascade?*

No, but the wave is nonlinear so its shape is significantly off that of a pure sinusoidal wave (Figs. 2 and 3).

*With your original configuration with no dissipation, you got very weak Lagrangian mean flow west of the forcing region. Did you try adding weak dissipation to that run?*

No.

*Dissipation may be necessary but not sufficient. In the case of the Y wave, one can see (sort of, with a lot of effort) how dissipation leads to a net PV tendency. Is it possible that this is not the case for the type of disturbances you are forcing?*

I am not sure I understand. We do see a mean Lagrangian flow, in accord with the present of dissipation, is not it?

*The effect of explicit dissipation on the mean flow may depend on the scales of the disturbances versus the scale of the mean, and on the scale-selectiveness of the form of dissipation used.*

This is a good point. Results may differ if the dissipation acts more or less efficiently on the scale of the Lagrangian mean flow than on the scale of the disturbance. But I imagine that the differences would appear in the strength of the Lagrangian mean flow and at the level of dissipation at which the Lagrangian mean is maximum, not at a fundamental level; in particular, I still imagine that we should see an increase and a decrease of the Lagrangian mean flow with decreasing dissipation. But I may be wrong.