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Results with 3 layers, oscillating forcing, with or without sponges


Two simulations are forced by a 100-day wind forcing. They differ with one having a sponge layer along the western boundary. Although, overall, the periodic and Eulerian mean flow appears similar in both runs, the presence of the sponge layer prevents the development of instabilities that propagate into the interior from the western boundary. Although the sponge layer is not ideal, we focus on the simulation with the sponge layer to understand the dynamics.

A weak (0.1 cm/s) Eulerian mean flow is obtained in the forcing area in the middle layer but no net displacement of salinity anomaly is observed even after 3 cycles (300 days), although the displacement is significant during a cycle. This suggests that the Lagrangian mean flow is zero and the weak Eulerian mean flow is cancelled by the Stokes drift. The presence of Stokes drift in the forcing area, but not to the west of the area, is also consistent with what we expect from mid-latitude non-divergent plane Rossby waves. A measure of nonlinearity shows that although the amplitude seems weak in absolute, the situation is nonetheless weakly nonlinear.

Thus although we are not dealing with an ideal frictional layer along the western boundary, the results of the simulations seems to be consistent with theory.

Some possible next steps are also discussed.

Run overview

The two simulations are identical to exp2_i and exp2_j (3 layers, one with a sponge along the western boundary), except that the wind forcing is now oscillating with a period of 100 days as in Haidvogel and Rhines (1983). The simulation with the sponge is exp2_l.


The oscillating flow

The flow is dominated by a 100-day periodic motion with amplitude of about 0.5–2 cm/s in the upper layer and 0.25–1 cm/s in the middle layer, and negligible flow in the middle layer once the spin-up is achieved (Figs. 1 and 2). Fig. 2 shows also that after day 900, a nearly stationary state has been reached.


Figure 1: Snapshot of U in exp2_l on day 1496 (with sponge): (a) upper, (b) middle and (c) lower layers.


Figure 2: U at x = 555 km and y = 1230 km in exp2_l (with sponge).

The Eulerian mean-flow

The 100-day Eulerian mean flow in the upper layer is composed of an eastward flow surrounded by two westward flows on each side, all at the latitudes of the forcing area (Fig. 3). The amplitude is in the order of 0.5 cm/s. In the middle layer, a weak circulation of the order of 0.1 cm/s is found in the forcing area. In the run without sponge, a similarly weak circulation is found along the southern boundary, suggesting that it is linked with Kelvin or Rossby waves that radiate from the western boundary. In the run with sponge, there is a stronger circulation near the western boundary located near the latitudes of the forcing, likely to be an artifice of the sponge layer. The Eulerian mean flow in the bottom layer is comparatively negligible.

exp2_k (without sponge) exp2_l (with sponge)
Fig3a Fig3b
Figure 3: Time-mean of zonal velocity (U) between days 1400 and 1500 for exp2_k (left) and exp2_l (right): (a) upper, (b) middle and (c) lower layers.

Notice that with a sponge, much less small-scale instabilities are found near the western boundary in the upper layer. On the other hand, as mentioned above, an artififical circulation is obtained near the western boundary in the middle layer.

The time evolution of the 100-day Eulerian mean flow in the two runs is shown in Figs. 4 and 5. The presence of the sponge prevents instabilities to accumulate and propagate from the western boundary. In both cases, however, the strong circulation in the middle of the layer at the beginning of the simulation is gone after about 500 days and a permanent circulation located in the forcing area (a westward flow at y = 1110 km) appears after 900 days.


Figure 4: 100-day averaged U at y = 1110 km in exp2_k (without sponge).


Figure 5: 100-day averaged U at y = 1110 km in exp2_l (with sponge).

Figs. 6 and 7 shows the time evolution of the 100-day Eulerian mean flow at two specific locations both at y = 1110 km: west of the forcing area and in the forcing area. If we focus on the run with the sponge (Fig. 7), we see that to the west of the forcing area, the mean flow has reached a quasi-steady state in the upper layer, and is nearly zero in the middle layer. In the forcing area, the mean flow in the upper and middle layer have about the same amplitude and have reached a quasi-steady state after day 900 besides some 100-day fluctuations.


Figure 6: U at y = 1110 km in exp2_k (without sponge): (a) west of the forcing area (x = 555 km) and (b) within the forcing area (x = 1110 km). Color code is as in Fig. 2.


Figure 7: U at y = 1110 km in exp2_l (with sponge): (a) west of the forcing area (x = 555 km) and (b) within the forcing area (x = 1110 km). Color code is as in Fig. 2.

The Lagrangian mean flow

With no dissipation in the interior in the middle layer, the 100-day Lagrangian mean flow should be zero. Is the weak but non-zero Eulerian mean flow at the location of the forcing cancelled by the Stokes drift or does it violate the theory? The Eulerian mean flow is in the order of 0.1 cm/s so after 3 cycle (300 days), we should see a net displacement of water parcels of about 0.23 °. To have an idea of the Lagrangian mean flow, two meridional bars to the west and within the forcing area of salinity anomaly have been initialized on day 1500 and Fig. 8 shows the distribution of the anomaly after 3 cycles for the upper (Fig. 8a) and middle layer (Fig. 8b). Fig. 8c also shows that the Lagrangian motion during a cycle is not negligible, suggesting that the absence of net displacement should correspond to a real cancellation of the Lagrangian mean flow.


Figure 8: Salinity anomaly initialized on day 1496 as two meridional bars centered at x = 550 and 1110 km: (a) upper layer at the end of the third cycle (day 1796), (b) middle layer at the end of the third cycle (day 1796) and (c) middle layer at half the third cycle (day 1746).

The possibility that the Stokes drift cancels everywhere in the middle layer the Eulerian mean flow is consistent with the prediction of where we should find non-zero Stokes drift. To the west of the forcing area, waves are a priori plane mid-latitude non-divergent Rossby waves that have zero Stokes drift consistent with zero Eulerian mean flow there. Within the forcing area, the Rossby waves are being formed and are not plane and it is likely that there is a Stokes drift consistent with the non-zero Eulerian mean flow there.

How much nonlinear is the flow?

Although the amplitude of the motion in these simulations seem weak (taken conservatively as 0.25 cm/s from Fig. 2), the phase speed of the Rossby waves involved are also weak and if the ratio of the amplitude to the phase speed is a measure of nonlinearity, we obtain a ratio of 0.04 for the first-baroclinic-mode long Rossby waves and 0.15 for the second-baroclinic long Rossby waves. Thus, we could say that we are not in the linear regime but in a weakly nonlinear one.

Next steps

Next steps would be to:

  • Increase the amplitude of the forcing and change the ocean configuration, so that the waves have a more realistic amplitude without being more nonlinear. One possibility would be to make the second and third layers much thicker?
  • Increase the nonlinearity of the problem and see if the results stay consistent with the theory.
  • Replace the sponge layer with Rayleigh friction and see if there is any difference.
  • Introduce weak dissipation in the interior and see if a non-zero Lagrangian mean flow appears.
  • Study the sensitivity of the non-zero Lagrangian mean flow to 1) type of dissipation and 2) resolution.
  • Perform the diagnostic to check if PV is conserved in the middle layer.



Regarding next steps, I think I would start with the second and the 4th in your list. For the latter, I would start with Laplacian horizontal friction.

Other configuration changes require some more consideration, and I am not sure what makes the most sense. Starting with the two above, both of which are extremely simple to do, may provide some guidance.