Over the last couple of days, I performed a set of experiments that spans the two-dimensional phase space of forcing magnitude and Laplacian viscosity (Tab. 1). Two different values of forcing magnitude and five of Laplacian viscosity are used. Some experiments are also run with a sponge layer along the western boundary. In none of the two sets of experiment with constant forcing and decreasing Laplacian viscosity is observed a decrease of the Lagrangian-mean flow, the main quest of this work. This is disappointing and either we are forgetting something in the theory or I am not in the correct regime. We could already conclude that this regime is not prevalent but I would differ: other known and important processes, like the formation of regular patterns in Rayleigh-Benard convection, are rarely observed except in very special regime *via* careful and controlled experiments.

Besides this arguable point, the time series of the 100-day Eulerian-mean flow at 2 specific locations within and to the west of the forcing area in the upper and middle layers are plotted in Figs. 1 and 2. We see that once the dissipation is strong enough (cyan, magenta and red curves), the flow has reached a steady state. For experiments with weak or not dissipation (blue), the flow can still vary strongly even after 4000 days. For these experiments, the calculation of the Lagrangian-mean flow between days 2000 and 2100 (Figs. 3 and 4) are thus not representative. The cause of this low-frequency instability is not known (numerical or real?).

KH = 0 m2/s | KH = 10 m2/s | KH = 50 m2/s | KH = 100 m2/s | KH = 250 m2/s | |
---|---|---|---|---|---|

MAG = 100 | exp2_l |
exp2_l2 |
exp2_l5 |
||

MAG = 500 | exp2_w6
exp2_w6_sponge |
exp2_w5 | exp2_w4 | exp2_w3
exp2_l5bis |

**Table 1:** Values of the forcing magnitude MAG and Laplacian viscosity KH for the set of experiments. Names in bold correspond to experiments with a sponge layer along the western boundary. MAG is the coefficient used in `surface_forcing.c` but it does **not** correspond to a wind magnitude of 100 or 500 dyn/m2.

The Eulerian-mean and Lagrangian-mean zonal flows for all experiments are plotted in Figs. 3 and 4. In every case, the Lagrangian-mean flow is simply decreasing with increasing Laplacian viscosity. As concluded in previous notes, it seems that if the regime where the Lagrangian-mean flow increases with increasing viscosity exists, then this regime might be for experiments where the flow does not have a steady or quasi-steady state.

What are the alternatives? 1) Are the instabilities baroclinic instabilities that could be avoided by changing the bottom friction in the lower layer? 2) Is the use of **Laplacian** viscosity the “problem” and should we go back rather to a two-layer model and use only the bottom linear friction (our initial decision to shift to three layers was our expectation that with no friction in the middle layer, no Lagrangian-mean flow should be found; unfortunately, this regime is hard to reach without all these instabilities)? Other alternatives?