J.H.G.M. van Geffen and P.A. Davies
Dynam. Atmosph. & Oceans 32, 1--26 (2000)
The monopole, with initial radius a=0.5 (it grows somewhat due to viscous effects as time goes on) has positive vorticity -- i.e. it is cyclonic (anti-cyclonic) in sense in the northern (southern) hemisphere. This vortex moves to the north-west as a result of the beta-effect, where it encounters an isolated, cosine-shaped, axisymmetric topographic feature, representing a seamount, with a maximum height of hmax=0.4 (relative to the default fluid depth H=1) and a radius R varying between R=a=0.5 and R=4a=2.0. This seamount is located somewhere between far south (f0=-5) and far north (f0=+5) of the equator, the equator being located at yeq=-f0/beta; for this study beta=0.3 and f0 is varied. If the monopole is to model a real-life vortex in the Earth's ocean with a diameter of, say, 105 m then the scaling is such that f0=+5 represents roughly a latitude of 15o north with yeq approx -17 (Section 3.1).
The results presented in the present paper show that there is a distinction in the possible outcome of the encounter depending on whether the seamount has a horizontal scale comparable with that of the vortex (R<=1.0) or larger (R=>1.5). And there is also a difference between encounters taking place in the northern (f0>0) and southern (f0<0) hemisphere, especially if the distance to the equator is large.
Relatively small seamounts (R<=1.0) in the southern hemisphere or at the equator (Fig. 5) affect the monopole's trajectory slightly, and the position of the vortex some time after the encounter is similar to the position the monopole would have had if no topography had been present; the difference in these positions increases for larger |f0|, i.e. further south. For the larger seamouns (R=>1.5; Fig. 5) there is some difference in the monopole's trajectory if f0 is not too strong, in which case the monopole's eventual position is further to the north than if there is no topography. If the latter is located far to the south, however, the monopole can be rebounded along its direction of approach and the seamount acts as a complete barrier to the monopole. In all the encounters taking place in the southern hemisphere or at the equator the monopole retains its integrity, though there is some deformation of the monopole as it moves. And if the monopole passes or crosses the topography, it does so east of the top of the seamount.
For seamounts in the northern hemisphere (f0>0) comparable in size with that of the vortex (R<=1.0), the monopole crosses the seamount to the west of the top without significant deformation (Fig. 6); the "end point" of the vortex' motion is roughly the same as without a seamount present. For larger seamounts (R>1.5; Figs. 8, 9, and 11) north of the equator the encounter leads to complicated trajectories and strong deformations. At low latitudes the monopole is deformed severely but survives the crossing and then follows a trajectory with several small loops and bends to the north-west. At higher latitudes the vortex can be trapped by the topography, with a trajectory consisting of orbits around the top, with small loops superimpossed on its motion. In some cases the vortex is able subsequently to escape from the topographic trapping, while in other cases it is shown to be destroyed by the interaction with the topography-induced vorticity. For a seamount located even further to the north, the topography-induced vorticity is so strong that the monopole is deflected along the south side of the mountain to the west, with considerable deformations and a complicated trajectory. It has been shown that the monopole is also able to turn to the south-east and again encounter the topography, with subsequent disintegration of the vortex, if the topographic feature is very large (Fig. 11).
The simulations discussed in the present paper have been carried out with a positive monopole, i.e. one that has cyclonic (anti-cyclonic) vorticity in the northern (southern) hemisphere, and which travels to the north-west. For a negative monopole, which travels to the south-west, cyclonic/north and anti-cyclonic/south have to be interchanged. Hence, a negative monopole can be rebounded in the northern hemisphere and trapped on the southern hemisphere by a seamount, and there is no basic difference between positive and negative vortices. At least that is the situation in numerical simulations as those presented here; in the laboratory it is difficult to have stable negative vortices (anti-cyclones on the northern hemisphere; see Kloosterziel and Van Heijst, 1991).
The monopole used in the present study has a single signed vorticity distribution, i.e. it has a non-vanishing circulation. A so-called isolated monopole, which has a total circulation equal to zero, with a positive (negative) core also travels to the north-west (south-west) due to the beta-effect, but along a trajectory more to the north (south) than the non-isolated monopole (Carnevale et al., 1991; Van Heijst, 1994). This difference is caused by the presence of an oppositely signed ring of vorticity around the core for the vortex, part of which is shed in the form of small vortices as the monopole moves, hence the interaction with the ambient vorticity field will be different, though the basic interaction mechanism is the same (see Carnevale et al., 1991); this matter has not been investigated any further in this study.
The precise structure of the topographic feature has some influence on the outcome too, of course. The seamount used here, given by Eq. (7), has the profile of half a cosine: hmaxcos(r pi/2R). A topography with a less steep slope near its foot -- such as a full-cosine function hmax[cos(r pi/R)-1]/2 -- allows the monopole to climb a little further towards the top. Near the top, however, such a topographic profile is steeper again than the half-cosine counterpart and the "effective" size of the full-cosine mountain is therefore somewhat smaller than that of the half-cosine of Eq. (7). Thus, the range of possible outcomes of the encounter shifts a little to larger seamount radii, but the different regimes are similar.
For a negative hmax the topography is obviously a pit. If the (positive) monopole encounters the pit from the south-east, the effect of the pit on the monopole's evolution will be similar to the effect of the mountain discussed in Section 3, but with north and south of the equator interchanged: a pit in the northern hemisphere can rebound the vortex, and the monopole can be trapped by a pit in the southern hemisphere. The details of the evolution of the monopole will no doubt be somewhat different though. Figure 12 shows as an example a pit with the half-cosine profile given by Eq. (7) with radius R=1.0 and hmax=-0.4 at three latitudes. In the equatorial case the monopole crosses the pit and ends up more to the west than in the case of a mountain (cf. Fig. 5). In the northern hemisphere, with f0=+5, the pit deflects the monopole in a way similar to a mountain on the southern hemisphere with f0=-5 (Fig. 5). And if the pit is in the southern hemisphere (f0=-5) it traps the monopole, unlike for the mountain of the same radius in the northern hemisphere (Fig. 6), where the monopole crosses the mountain. The reason for the latter difference is that as the monopole descends into the pit it become narrower and stronger, whereas it becomes wider and weaker as it climbs the mountain. A monopole crossing a pit is thus less susceptible to deformations than a monopole crossing a mountain under otherwise-idential conditions. The example shows that the pit topography case deserves a study of the full parametric range of values for f0 and R, but that falls outside the scope of the present paper.
In summary, what happens to a monopolar vortex when it encounters the topographic feature depends mostly on whether the encounter takes place in the northern or southern hemisphere, and on the lateral size of the topography relative to the size of the vortex: the bigger the seamount and/or the further away from the equator, the more dramatic the influence can be on the monopole's evolution, leading to the possible destruction of the vortex.
Note on notation here
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