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 omega0 and omegah be the relative vorticity of the monopole away from the ridge and at height h=1-H, respectively. Conservation of potential vorticity (neglecting viscous effects for a moment) then implies that omegah=omega0(1-h)-hf. For a positive monopole a change of sign thus takes place if h>omega0/(omega0+f). For a ridge with h=1/3 this occurs if omega0<f/2=beta y/2. This means either a very weak monopole, or a strong beta-effect or a very large y-value -- all of which seems not realistic within the model. The higher the ridge, the more plausible such a change of sign. But since a height of 0.4 seems to be about the maximum allowed height within the assumption of two-dimensional motions, this cannot be investigated with the model.
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=f0+beta y=beta y, which means that the domain has a part with negative (y<0) and a part with positive (y>0) potential vorticity when the monopole approaches the ridge (see for instance the top-left panel of Fig. 12). In other words, the ridge is centred on the equator (y=0), where f changes sign. It therefore seems likely that the fate of the monopole also depends on the initial y-position of the vortex. This point is currently under study and results will be presented elsewhere.
Another point that will be addressed is the effect of a non-zero f0 in the Coriolis parameter f=f0+beta y. For motions in the absence of topography the value of f0 does not matter since it is a constant that drops out of the equations: it only appears in derivatives. If the fluid depth H does depend on position, however, this is no longer the case since the potential vorticity omegap=(omega+f)/H appears in the derivatives. The value of f0 will therefore matter for the evolution of the monopole when it is on the ridge and hence f0 may determine the actual fate of the vortex.
Note on notation here
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