Table Of Contents

This Page

Results with stronger forcing

Summary

Two experiments are compared in this note. They are identical (oscillating forcing with a sponge layer along the western boundary) with the exception that the forcing in exp2_m is twice stronger than in exp2_l. The results are overall similar and they are consistent with theory: the Eulerian mean flow, arising from nonlinearity, is four times stronger in exp2_k and exp2_l in both the upper and middle layer and the Lagrangian mean flow in the middle layer (no dissipation) stays negligible in both cases to the mean flow in the upper layer.

Two caveats:
  1. it seems that there is an instability arising along the western boundary in exp2_m and not in exp2_l, despite the presence of the sponge layer,
  2. the Lagrangian mean flow in the middle layer is not shown to be exactly zero –not surprising given that we do not expect to obtain exactly zero for any practical calculation. It is only negligible compared to the mean flow in the upper layer. To check that the theory is verified, it would be nice to show that with decreasing dissipation, the Lagrangian mean flow goes to zero –or alternatively, the Lagrangian mean flow increases from the present value with increasing dissipation. This will constitute my next step.

Run overview

exp2_m is the same as exp2_l except that the surface forcing is twice stronger. Results from exp2_l has already been shown in this note.

Results

The oscillating flow

../../../../../_images/u_x555km_y1230km_exp2_m.png

Figure 1: U at x = 555 km and y = 1230 km in exp2_m.

As expected, the waves are twice stronger than in exp2_l (Fig. 1 to compare with Fig. 2 in this note) and the mean flow, that arises in the upper layer due to second-order nonlinearity, is about 4 times that in exp2_l.

The Eulerian mean flow

As in exp2_l, the Eulerian mean flow in the middle and lower layer is negligible compared to that in the first layer (see Fig. 3, right panels in this note). The residual Eulerian mean flow obtained in the middle layer is the same between exp2_l and exp2_m, with the amplitude of the latter being 4 times the amplitude of the former, consistent with the idea that this residual may be due to the Stokes drift of the waves (Fig. 2).

../../../../../_images/mu_mid_d1401_1496_exp2_l_exp2_m.png

Figure 2: Time-mean of zonal velocity (U)in the middle layer between days 1400 and 1500 for exp2_l (upper) and exp2_m (lower). The flow in the upper panel has been multiplied by 4.

The evolution of the Eulerian mean flow is also similar to exp2_l (Figs. 3 and 4 compared to Figs. 5 and 7 in this note), with the exception that an instability arises in the upper layer besides the presence of the sponge layer.

../../../../../_images/mu_coef_y1110km_exp2_m.png

Figure 3: 100-day averaged U at y = 1110 km in exp2_m. The amplitude has been divided by 4 for comparison with exp2_l.

../../../../../_images/mu_x550km_x1110km_y1110km_exp2_m.png

Figure 4: U at y = 1110 km in exp2_m: (a) west of the forcing area (x = 555 km) and (b) within the forcing area (x = 1110 km). Color code is as in Fig. 1. The amplitude has been divided by 4 for comparison with exp2_l.

The Lagrangian mean flow

With sponges, the use of salt anomalies to estimate the Lagrangian flow demands to define options and more layers that are not necessary for our problem. For these reasons, the Lagrangian mean flow is estimated from the computation of trajectories using 1-day output velocity field and a 4th-order Runge-Kutta algorithm. In both exp2_l and exp2_m, the Lagrangian mean flow in the middle layer is negligible compared to that in the upper layer. It is not exactly zero but the net mean flow is much smaller than the flow occurring during a cycle. This, however, does not still assure that the Lagrangian mean flow is exactly zero as expected from theory; notice, for instance, that the net mean flow with the stronger forcing is indeed stronger than the net mean flow with the weak forcing. An alternative way to check that the theory is respected would be to see a decrease of the net mean flow with decreasing dissipation, or alternatively, an increase of the net mean flow from the present value with increasing dissipation. This will be my next step.

../../../../../_images/mean_lagrangian_up_mid_d1501_1600_exp2_l_m.png

Figure 5: 100-day Lagrangian mean flow obtained in the upper (a) and middle (b) layers in exp2_l (black) and exp2_m (red).