The Quasi-Geostrophic Model

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

where is the streamfunction. Equation 4.24 corresponds to a barotropic and geostrophic ocean with a rigid lid approximation. The wind forcing is altered to include the influence of the water depth in order to better mimic the shallow water equations. The discretization of (4.24) is done using second order center differencings. The streamfunction formulation ( ) leads to a linear pentagonal system of equations to solve at each time step. We used the simple leapfrog time integration and the viscous term is discretized by the conventional five-point Laplacian. We are interested in testing different formulations of the Jacobian in (4.24), as the formulation of this term may have consequences for the vorticity budget for the same reasons mentioned previously for the C-grid model.

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

where notation is found in Section 2.2.2 and is the ensemble of indices for points whose location lies in the interior domain. By defining

(4.26) | |||

(4.27) | |||

(4.28) | |||

(4.29) |

we recast the vorticity budget in the following form

(4.30) |

One main characteristic of QG vorticity budgets is the explicit contribution of the beta term, . This contribution is hidden in for the SW models. Therefore, we define here to be , where represents the integration of the Jacobian term over . We focus our study on the behavior of both and . As for the C-grid model, a minimum requirement is that goes to zero at infinite resolution. This also applies to and separately. We propose to test three different numerical formulations of the Jacobian,

Representation of the Jacobian in (4.25) has been extensively
considered by Arakawa arakawa66 and AL77. From the latter, we
borrow the notation ** J_{i}**, where

(4.31) |

AL77 proposed the ** J_{3}** form of the Jacobian which
conserves energy in doubly periodic domains

(4.32) |

The

It is interesting to note that the ** J_{3}** 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

Using ** J_{1}**, 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 ** J_{3}** and

In terms of vorticity budget, we are interested in the behavior of the
advective contribution,
,
with increasing resolution for the ** J_{3}**and the

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 ** J_{3}** and

(4.34) |

since , by definition of no-permeability. The west-east asymmetry due to the beta effect imposes that with

One last point we would like to make is related to similarities
mentioned above, between the ** J_{3}**-QG and the B combination of the SW
model. Figure 4.13 shows
,
and
with
increasing resolution for

To conclude, except for the ** J_{1}** 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

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