Reproduction of Haidvogel and Rhines (1983)’s results - Run 1

Summary

Using the barotropic vorticity equation, a closed basin is forced by a 100-day wind stress, the curl of which is mostly a patch at the center of the basin. The curl is adjusted by a constant so that the domain-averaged curl is zero. For a certain value of the wind stress, a mean structure is obtained: eastward (prograde) flow at the latitudes of forcing and westward on each side. Strong nonlinear circulations are also obtained in the corners of the domain along the western boundary. These results are consistent with Haidvogel and Rhines (1983; hereafter HR83)’s ones.

An issue concerning the amplitude and thickness used in HR83 might, however, change the results.

Results

The model we use is the barotropic vorticity equation, the same as used in HR83. For the closed domain case, the differences between the two studies are:

• Only bottom friction –no horizonal friction– is used in the present run.
• The equation is resolved on a grid, not by using spatial modes (Chebyschev in x and sinusoids in y).
• The domain size is the same (about 20 deg. in longitude and latitude) but the resolution is different: 0.15 deg. in the present run everywhere, about 0.625 deg. (=20 deg./N where N=32 is the number of spatial modes used) constant in y but variable in x with higher resolution along the western boundary in HR83.
• The amplitude might be different. There is no mention of the constant density that should appear in the dimensionalization of the curl. We use a density of 1025 kg/m3 and a wind stress of 100 Pa (1000 dyn/cm2! I am obviously doing something wrong).
• Ekman number might be different: There is no mention of the thickness of the layer in HR83. We used a 500-m thick layer.

Besides those differences, the mean flow obtained is qualitatively similar between the two studies (Figs. 1 and 2) with eastward flow at the latitudes of forcing, westward flows on each side and strong nonlinear circulations in the corners along the western boundary. Notice small scale features in relative vorticity that might signal some numerical instability. These instabilities could disappear with the addition of lateral friction. The average was performed after day 500 in order to exclude the transients of the spin-up: A 100-day running mean shows that after day 500, the averaged features obtained are those of Fig. 1 (not shown).

Figure 1: 500-800-day averaged (a) streamfunction and (b) relative vorticity in Run 1. The dashed lines show the distance r0 from the domain center, where r0 is the exponential decay of the Gaussian (see 2.2 in HR83).

Figure 2: From HR83: Mean (a) streamfunction and (b) relative vorticity. Contours are every 400 m2/s in (a) and 3e-7 1/s in (b).

These mean features are due to nonlinearity: The same run without the advective term has mean features negligible compared to the above ones (not shown).