J.H.G.M. van Geffen and P.A. Davies

Geophys. & Astrophys. Fluid Dynam.90, 1--41 (1999)

In the course of
its motion through the domain, the monopole encounters a topographic
ridge with a cosine shape (Fig. 1), oriented in different ways and with the
possibility of having different heights and widths. From the results
presented in this paper it is clear that the width and height of the ridge
are important for what happens to a vortex that encounters the ridge. Even
more important, however, is the orientation of the ridge, *i.e.* the degree to
which the topography deforms the contours of potential vorticity from the
*beta*-effect, which lie equidistant from each other and parallel to the
*x*-axis.
The nature of the interactions occuring as the controlling parameters are
varried is extremely complex, requiring a parametric case study approach to
understand the associated dynamics.

A ridge along the *x*-axis causes only a minimal deformation of these
contours and hence only a small disturbance in the trajectory of the
monopole that encounters the ridge. It seems that the vortex can only
cross the top of the ridge once it has positive potential vorticity
sufficiently close to its (north)west side, and the vortex
achieves this by moving westward along the ascending side of the
ridge (Fig. 10).
If the ridge is along the line *y=x* the contours from the *beta*-effect
are deformed more and the monopole's trajectory differs more from the
ridge-less case than for a ridge along the *x*-axis.
In both cases, however, the vortex manages to cross the top of the ridge
after gathering sufficient positive potential vorticity, even for very
high and wide ridges ("very high" of course within the assumption of
two-dimensinal motions in the model).

A north-south oriented ridge causes considerable deformation of the contours
from the *beta*-effect: the local north coordinate induced by the ridge is
directed "uphill", *i.e.* perpendicular to the north coordinate of the
*beta*-effect.
Because of this, such a ridge can have a more significant effect on the fate
of the vortex, depending on the width and height, than the earlier ridges.
There does not seem to be a simple criterion to distinguish different regimes
of the monopole's evolution, but a few remarks can be made.
For low ridges, the disturbance is minimal and for these low ridges the width
is not of influence on the survival of the vortex; different widths lead,
however, to slightly different trajectories after the monopole has crossed
the ridge.
Somewhat higher ridges cause more disturbances in the monopole's trajectory
and deform its shape more, but the vortex can cross the ridge.

For a very high ridge, with a height of 0.4 times the fluid depth away from the ridge, the results can be summarised as follows. A very narrow ridge (smaller that the size of the monopole) seems to have little effect on the monopole itself, though its trajectory is shifted somewhat. Wider ridges can deform the vortex so much that it splits into two parts when it crosses the top of the ridge (where the secondary vortex may or may not also cross the ridge). And for even wider ridges, 3 to 4 times the size of the monopole, the deformation can lead to the destruction of the monopole if the monopole is not strong enough (anymore). The latter condition also depends on the strength of the viscous effects.

Higher ridges no doubt lead to more deformations and the possible
destruction of the monopole even for narrow ridges.
In principle it is even possible that the relative vorticity changes sign
when the monopole climbs the ridge. Let *omega _{0}* and

The computations described in this paper are performed with a monopole of
radius 0.5 in a *20 x 20* domain. If a larger domain is used, two
effects influence the evolution of the monopole. A detailed study of
this point falls outside the scope of the present paper, but a few remarks
are in order. As noted in Sect. 4.1, the
trajectory of the monopole will show a difference in the number and location
of the bends caused by the interaction with the Rossby-waves. Without any
topography, this has no significant effect on the average motion of the
monopole. If the monopole encounters a
ridge, however, there will be a difference in the evolution since the
monopole will reach the ridge at a slightly different location under a
somewhat different angle because of the bends in its trajectory.
Hence, the interaction of the monopole with the ridge-induced relative
vorticity --- clearly visible north and south of the monopole in, for
example, Fig. 13 --- will be a little different.
A larger domain also means that for a north-south ridge the generation of
relative vorticity at the ridge near the boundaries will be stronger, since
this depends on the north-south coordinate *y*, that is on the degree in
which the ridge deforms the contours of potential vorticity of the
*beta*-effect, which is more for larger |*y*|.
Since this stronger ridge-induced relative vorticity is at the same time
further away from the centre of the domain (where the monopole's evolution
takes place), the effect on the monopole's evolution is less than the effect
of the vorticity induced by the ridge near the monopole.
The combined effect of a slightly different approach of the north-south
ridge by the monopole and the stronger ridge-induced vorticity further away
(for larger |*y*|) is that there will be a small difference in the
monopole's evolution if a larger domain is used. But there is no indication
that a larger domain means better or more accurate results, since the
relative vorticity at the ridge is generated by the ridge, not by the
boundaries of the domain.
A *20 x 20* domain is thus sufficiently large and the results should be
considered with respect to this domain.

In all cases when the monopole actually crosses the top of the ridge,
it can only do so after it has gathered sufficient positive potential
vorticity on its (north)west side. For a north-south ridge, the vortex
achieves this by going north along the ascending (east) side of the ridge
(Fig. 12).
The computations discussed in this paper are made on a pure *beta*-plane,
with *f=f _{0}+beta y=beta y*, which means that
the domain has a part with negative (

Another point that will be addressed is the effect of a non-zero *f _{0}* in
the Coriolis parameter

**Note on notation here**

*
Greek characters and equations cannot (yet) be represented properly
on HTML-pages, hence they are given here in italics, however poor this
representation is.
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