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Test of the PV testΒΆ

I have been concerned with the potential vorticity (PV) test: compute the PV along the trajectories of parcels from local velocity and thickness field and compare it to the PV computed from integration of the PV equation. I did that for one of the experiment with Laplacian dissipation and realistic amplitude of the wave field and the test failed suggesting that the model does not satisfy the PV equation.

Before concluding this, however, I wanted to test my test. To do so, I performed a numerical simulation with (very) weak forcing: This should make any advective term negligible and the problem should be close to linear. Unfortunately or fortunately, the test of the test failed.

The amplitude of the flow is in the order of 1e-4 m/s and the flow averaged over the 100-day cycle is of the order of 1e-7 m/s. The trajectories of six parcels are shown in Figs. 1 and 2: The parcels barely move around and they all form a closed trajectory except for the parcel at 10E and 30N –why? I do not know.

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Figure 1: 100-day averaged zonal velocity in the middle layer. Supersimposed are the trajectories (black) of 6 parcels during one 100-day cycle.

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Figure 2: Trajectories of the six parcels shown in Fig. 1.

The PV computed from the local velocity and from the integration of the PV equation are shown in Fig. 3. In this figure, the latter has been estimated in two ways: one by computing the dissipative term from the u and v field and one by computing it directly from one model output, diffu and diffv. The two estimates are nearly the same suggesting that I know what I am plotting: The term is KH*L(zeta)/h, where KH is the diffusion coefficient (500 m2/s in this case), L is the Laplacian operator, zeta is the relative vorticity and h the thickness. The PV is defined as (f+zeta)/h where f is the Coriolis parameter.

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Figure 3: PV during one cycle and along the trajectories of the six parcels shown in Figs. 1 and 2: from the local velocity field (blue), from the integration of the dissipative term calculated from u and v (red dots), from the integration of the dissipative term calculated from diffu and diffv, outputs of the model (red dashes).

The final values and the overall evolution of the PV are recovered but not the exact evolution; it is not too far off but it is off.

The failure of the test suggests that I am doing a mistake in my test; I cannot believe that the model would not respect the PV equation in this nearly linear case. Except maybe the resolution is still too coarse? In this case, the same test applied to a simulation with a finer resolution should show a decrease in the difference between the two PV estimates. Any other idea?