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Summary of results from a large set of experiments

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.

../../../../../_images/mU_west_within_upper_layer_MAG100.png

Figure 1: 100-day mean zonal Eulerian flow in the upper (panels a and b) and middle layer (panels c and d) west of the forcing area (5°E; panels a and c) and within the forcing area (10°E; panels b and d) for the set of experiments with a forcing magnitude MAG=100 and varying Laplacian viscosity. Curves with squares correspond to experiments with a sponge layer along the western boundary.

../../../../../_images/mU_west_within_upper_layer_MAG500.png

Figure 2: As in Fig.1 but for the set of experiments with a forcing magnitude MAG=500.

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.

../../../../../_images/UE_UL_MAG100.png

Figure 3: Eulerian-mean (left) and Lagrangian-mean (right) zonal flow between days 2000 and 2100 in the set of experiments with a forcing magnitude MAG=100 and varying Laplacian viscosity. All experiments were performed with a sponge layer along the western boundary.

../../../../../_images/UE_UL_MAG500.png

Figure 4: As in Fig.3 but for the set of experiments with a forcing magnitude MAG=500. The ‘S’ in title indicates experiments with a sponge layer along the western boundary.

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?