# Case of the dissipative nondivergent barotropic Rossy waveΒΆ

I calculated analytically the trajectory of parcels embedded in a plane nondivergent barotropic Rossby wave dissipated by Raileigh friction with a time scale TAU. Fig. 1 shows the trajectories and the evolution in time of its potential vorticity (PV) in the inviscid and viscid case. We see that as soon as there is dissipation, PV oscillates in time with a period equals to the wave period, as in exp6 (see, for instance, Fig. 6 of this note). The PV converges toward a mean value that is different from its initial value. As in exp6, the net PV change is periodic in x with a wavelength equals to that of the wave field (not shown). This confirms that the periodic pattern in the net PV change might be due to the change in amplitude of the wave field.

[NOTA: We need to be careful, however, as the case presented here is different from the case in exp6. Here, the dissipation is the cause of the change in wave amplitude while in exp6, the spin-up is the cause of such change, which is thus independent of the dissipation.]

The question is: is there a regime for which the amplitude of the PV oscillation is negligible compared to its final amplitude? In this case, no. This is illustrated in Fig.2 where a much weaker dissipation level is used to reduce the change of amplitude with time; the parallel consequence is that the net difference in PV between the initial and final position is also much smaller.

This might be different in the case where the amplitude is maintained, with the dissipation balanced by the pressure drag. We need for this a vertically propagating baroclinic Rossby wave, the amplitude of which decays with depth, not with time.

Figure 1: (a) Trajectory of the parcel that started at x=y=0. (b) Longitudinal and (c) latitudinal position of the parcel with time and (d) its potential vorticity (PV). Two cases are shown: the invisicid one (blue) and a viscid one with the time scale of dissipation (TAU) being equal to 800 days. See traj.m in RESEARCH/MODELISATION/HIM/studies/PV_and_dissipation/analytic_solutions/diss_baro_nondiv_plane_Rossby

Figure 2: PV of the parcels that started at x=y=0 in three cases: TAU=0, 800 and 8000 days.