We also investigated the influence of coastline discretization in
quasi-geostrophic (QG) models, although our main interest in this thesis
is focused on the shallow water (SW) models.
QG models solve the vorticity
equation directly. It seems therefore a reasonable assumption that these
models should yield more accurate vorticity budgets than do SW and
primitive equation models. The vorticity equation used is
As for the SW C-grid model, the vorticity budget for the QG
model is defined only on an interior sub-domain, half a grid point inside
the model basin. This follows from the fact that the vorticity equation
is only solved at interior points (see Figure 4.1).
The discretized vorticity budget is
Representation of the Jacobian in (4.25) has been extensively
considered by Arakawa arakawa66 and AL77. From the latter, we
borrow the notation Ji, where J is the discretized Jacobian and itakes values between 1 to 7, depending on the discretized formulation. The
simplest representation is the J1 Jacobian, where
AL77 proposed the J3 form of the Jacobian which
conserves energy in doubly periodic domains
It is interesting to note that the J3 formulation is similar in structure to the advective terms in the SW vorticity equation when the B combination, discussed above, is employed. For example, if we take and , then J3 can be recast as . The advective term for the B combination in the vorticity equation takes the form of . Hence the two formulations use a divergence form of the advection. Moreover, the viscous term in the SW vorticity equation derived using the -
Using J1, the solutions are very different for positive and negative values of the rotation angle of the basin. Positive angles are characterized by larger kinetic energy and stronger oscillations of a Rossby basin mode (curve b of Figure 4.9), which appears to be unstable at low resolution. However, with increasing resolution (curves d-f of Figure 4.9), the kinetic energy for both positive and negative angles seems to converge to the value of kinetic energy for the non-rotated basin cases (curves a,d). Nonetheless, we prefer to discard this formulation of the Jacobian for the rest of the discussion, due to its low level of accuracy at moderate resolutions.
On the other hand, solutions using J3 and J7 appear stable and converge reasonably well with increasing resolution to the same value of kinetic energy, for both rotated and non-rotated basins (Fig. 4.10). Therefore, this results contrasts with those of the SW model for which the convergence was only obtained for the B combination. The QG model appears to be less sensitive to grid rotations and advective formulations.
In terms of vorticity budget, we are interested in the behavior of the advective contribution, , with increasing resolution for the J3and the J7 Jacobians. Specifically, we are interested in how the convergence order for differs in the QG model compared to the SW model. As mentioned, is made of two independent contributions, and . depends directly on the Jacobian formulation but does not. Figure 4.11 shows the convergence of in rotated and non-rotated basins for the two considered Jacobians. is close to second order in non-rotated basins for both Jacobians. At 30o rotation, however, the convergence order is closer to unity for J3 but second order for J7.
We now analyze the convergence order for
the second contribution to
Figure 4.12 shows the convergence for
in rotated and
non-rotated basins under J3 and J7.
The results appear independent of the Jacobian formulation, as
expected. The convergence order is however unity, in
contrast with results for
This result comes readily from the
traditional treatment of the
term. The proof is given in a square
One last point we would like to make is related to similarities mentioned above, between the J3-QG and the B combination of the SW model. Figure 4.13 shows , and with increasing resolution for J3 and under -30o rotation angle. is negative, goes through an minimum and, then increases toward zero, whereas is positive and decreasing to zero. Hence, appears to go through a pool of negative values, just as the B results showed. This contrasts with results using J7 for which takes positive values for both negative and positive rotation angle (not shown).
To conclude, except for the J1 Jacobian, the QG model is less sensitive to the basin rotation, in contrast with results for the SW model. Convergence orders for the advective flux of vorticity, , on the other hand, are order 1 or less--comparable to what was found for the SW simulations. In the QG case, this low order of convergence is related to the beta contribution, . Using J7, varies between 5% (high resolution) and 20% (coarse resolution) of the wind input depending on the rotation angle. These results are somewhat better than those obtained in the SW simulations.
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