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Notes on “On particle tracking in Eulerian ocean models” by Haidvogel (Ocean Modelling, 1982)

Haidvogel studies these two questions:

  1. Can fixed-grid ocean models be used to produce individually accurate particle trajectories?
  2. Can fixed-grid ocean models be used to reproduce accurate paticle statistics?

He demonstrates that the answer to the both questions is likely to be “no” [..] in the cases where nonlinear processes play an important dynamical role.

To demonstrate this, he performs the integration over time of the same barotropic non-divergent vorticity equation used in Haidvogel and Rhines (1983). Five clusters of 16 floats each are released and the float trajectories are computed using a high-order Runge-Kutta integration technique. The interpolated velocity field is obtained using the Fourier series representation used to resolve the numerical model. “Although inefficient, the resulting interpolation is highly accurate”.

To estimate if these trajectories obtained numerically are accurate, the criterion that each particle needs to satisfy the potential vorticity (PV) equation is applied. Hua (1994, JPO) used basically the same criterion. The PV is estimated by integrating the PV equation; this estimate is considered as the true PV, given the accuracy of the particle trajectory. The PV is then estimated independently by computing it directly from the Eulerian field. With infinite resolution, the two should collapse (is this true? Eric seemed to say that this is not necessarily true; maybe yes in this case because the model simulates directly the PV equation). Comparison of the two estimates of PV from the simulation shows that PV is not conserved from the Eulerian point of view.

“This non-conservation of vorticity following the floats must ultimately be related, of course, to some sort of discretization/truncation error” (because, again, here the model simulates directly the PV equation; in the case of primitive-equation model, the numerics of the model could violate the PV equation in some cases). The author points to the advective term as the source of error; in this case, it is the lack of nonlinear interactions at scale lower than the scale allowed by the Fourier representation –in other words, as usual, a subgrid-scale process (NOTA: It is not clear to me why the estimate of the Lagrangian trajectory would not also suffer of the same problem).

The truncation error in the Eulerian model can be seen as an additional source/sink term in the PV equation. This additional (aliasing) term is computed (how?) and when the PV computed from the modified PV equation is estimated, the Eulerian and Lagrangian PV are now nearly equal. In the presented simulation, this aliasing term is the largest of the source/sink terms, even though the scale of the main dynamical features (the eddies) are sufficiently resolved a prori. This modified Lagrangian PV is the PV conserved in the model, although it is physically wrong.

It is not clear if the effect of the aliasing terms is important or not on the trajectories themselves. The text and Fig. 11 are not clear.

The author notices correctly that we are in general interested in the ensemble average of particles, not individual particles themselves; Could the statistics be insensitive to the resolution? Lagrangian statistics are calculated and compared for two simulations differing in resolution (the high-resolution model being considered more accurate with less aliasing error). In the low-resolution model, particles move apart more rapidly than in the high-resolution model. Estimates of particle diffusivity would then be quite different between the two simulations.

Does all of this matter? Yes, if this is the cause of the absence of deep equatorial circulation, or other deep flows such as the various re-circulation gyres below the eastward extension of western boundary currents, in state-of-the-art numerical models.