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General Results for all Geometries

In Figure 5.5, we show the different statistical mean circulations obtained in the five attempted geometries. An ``X'' marks when a statistical mean could not be reached. Such is the case when the solution jumps to the high energy branch. When this occurs, the sea level tilt implied through geostrophic balance by the unreasonably strong currents quickly leads to zero layer thickness, at which point the integration is halted. This happens for the intermediate and high Re in the Geometry I and II and for high Re in Geometries III and IV. We achieve a reasonable statistical mean for all considered Re in Geometry V. As the Reynolds number is increased, the recirculation tends to move eastward and northward and strengthens. In Geometry V, the recirculation is nearly round, whereas, it is more elongated for the other geometries. The other interesting point to note in Geometry V is related to the position of the recirculation relative to the bumps. Between Re=0.5 and Re=1.2, the recirculation strengthens, but is somewhat trapped between Bumps 2-3. However, at Re=3.5, it jumps to the next indentation (Bumps 1-2; see labeling in Figure 5.2). The general result is therefore that the presence of bumps along the coastline inhibits and retards the jump to the high energy branch for the Munk problem with free-slip boundary condition. However, the radius of curvature of the coastline has to be fairly small (i.e., smaller than the radius of deformation) in order to achieve reasonable circulations under high Reynolds numbers.

Figure 5.5: Mean elevation fields for the five geometries using the SE model. When no steady state could be reached because the solution jumps to the high energetic branch, an 'X' is drawn instead.
$\nu =700$m$^2\cdot$s-2 $\nu=300$m$^2\cdot$s-2 $\nu =100$m$^2\cdot$s-2
Re = 0.5 Re = 1.2 Re = 3.5
\includegraphics[scale=0.25]{images/cer2_h.ps} X X
\includegraphics[scale=0.25]{images/cerc12_h.ps} X X
\includegraphics[scale=0.25]{images/cerc10_h.ps} \includegraphics[scale=0.25]{images/cerc11_h.ps} X
\includegraphics[scale=0.25]{images/cerc2_h.ps} \includegraphics[scale=0.25]{images/cerc8_h.ps} X
\includegraphics[scale=0.25]{images/cerc4_h.ps} \includegraphics[scale=0.25]{images/cerc6_h.ps} \includegraphics[scale=0.25]{images/cerc7_h.ps}

What is of interest is the vorticity structure for all these geometries. Figure 5.6 shows the relative vorticity field for Geometry IV and V and for different Reynolds numbers. One general characteristic is that these fields are less smooth than those for the stream function or the elevation field. This relates to the fact that the vorticity corresponds to the second order derivatives of the stream function. The vorticity field is therefore noisier and more difficult to resolve. Nonetheless, the results from the SE model are very encouraging when compared to those obtained from finite element models for which the vorticity fields are generally much noisier. One basic feature is that the vorticity approximates the form of a positive Dirac delta function close to the tip of the bumps. Therefore, dynamical processes close to the tip are rather complex, irregular and difficult to resolve using a high order formulation. However, the use of a discontinuous SE formulation seems to be of some help in resolving these irregularities by not propagating them to neighboring elements. The largest peaks are observed where the velocity is the largest. The magnitude of these peaks ranges between 10-5 and 10-4s-1. Where the magnitude of these peaks goes beyond 10-5, a tail of positive vorticity forms downstream of the peaks. Hence, the excess of positive vorticity is advectively transported downstream. Of course, these peaks increase the local gradient of vorticity pointing outward. They are therefore directly related to the mechanism which balances the vorticity budget and limits the size of the recirculation. Furthermore, we note that a positive vorticity wall surrounds the recirculation zone. This wall is consistent with the presence of a region of low velocity outside the recirculation zone (i.e., a region of strong shear). We note also that, for Geometry V and Re=3.5, a thin filament of large negative vorticity is located near the western boundary. An important remark concerns the Rossby number, Ro in the presence of bumps. By measuring the ratio $\zeta /f_0$, we note that Ro is above 0.1 for Geometry III and reaches about unity for Geometry V. As predicted, Rocan be fairly large in the presence of bumps which invalidates the QG approximation.

Figure 5.6: Mean vorticity field for the Geometries IV and V using the SE model. When no steady state could be reached because the solution jumps to the high energetic branch, an 'X' is draw instead. The influence of the bumps is clearly seen by the abrupt jump in the vorticity field.
$\nu =700$m$^2\cdot$s-2 $\nu=300$m$^2\cdot$s-2 $\nu =100$m$^2\cdot$s-2
Re = 0.5 Re = 1.2 Re = 3.5
\includegraphics[scale=0.25]{images/cerc2_vor.eps} \includegraphics[scale=0.25]{images/cerc8_vor.eps} X
\includegraphics[scale=0.25]{images/cerc4_vor.eps} \includegraphics[scale=0.25]{images/cerc6_vor.eps} \includegraphics[scale=0.25]{images/cerc7_vor.eps}

We also show the power input (the rate of energy put in by the wind), P, in Figure 5.7. P describes how the circulation adjusts to the wind pattern in order to minimize its energy. In general, it shows that the increase in P is much less than that in Re. This means that the circulation adjusts in such a way that reducing $\nu$ by a factor of two does not lead to a doubling of P. It would be interesting to verify if some simple scaling arguments reproduce this result. However, it is difficult to derive a scaling for P since it cannot be estimated on boundary layer considerations alone but requires also the knowledge of the interior circulation. The figure shows that the rate of increase is larger for the regular geometry than for the irregular geometries. In fact, the rate of increase is rather similar for the two irregular geometries, although there is a general shift toward lower values of P as the bumps grow in size. In contrast with results in double gyre experiments [Scott and Straub, 1998] where P tends to decrease with increasing Re, the single gyre circulations tend to have difficulties in minimizing P. This stems for the single-gyre circulation being to stable. Under the double-gyre wind forcing, the recirculation is usually highly unstable and counter-gyres develop above a certain Re (the four gyres structure observed by Greatbatch and Nadiga, 2000; also visible in Scott and Straub, 1998).

Figure 5.7: Power input by the wind using the mean fields with respect to the boundary Reynolds number.

Role of the Transients for Geometry V at High Reynolds Number

We now focus on the results of the high Reynolds number, Re=3.5, and Geometry V. In particular, we are interested in the role of the transients in achieving a steady state. One way to investigate the role of the transients is to plot maps of the standard deviation for the elevation. Figure 5.8 reveals that a belt of strong anomalies exist south of the recirculation. This belt extends northward to Bump 16 and 15, and westward close to Bump 5 where it reaches a maximum. The western part of the recirculation is also a local maximum of the deviation. It is along this belt that we observe strong eddies going around the recirculation and moving westward. We can further refine this kind of analysis by generating the same kind of maps but for selected frequencies.

Figure 5.9 reveals the activity of the eddies of period over 200 days. This figure is very similar to Figure 5.8. It reveals that the main contribution to the standard deviation comes from the slow modes. The maximum is located in the eddy-belt as previously introduced, south-east of the recirculation with another but slightly weaker maximum close to the western boundaries. That the eddies tends to intensify in proximity of the recirculation and not at the boundary probably means that they strongly interact with the recirculation.

Figure 5.8: Mean standard deviation of the elevation.

Figure 5.9: Mean standard deviation of the elevation for frequencies with period above 200 days.

Figure 5.10: Mean standard deviation of the elevation for frequencies with period between 17 and 200 days.

Figure 5.11: Mean standard deviation of the elevation for frequencies with period between .6 and 17 days.

Figure 5.10 shows standard deviations for periods between 17 days and 200 days. By isolating these periods, we hope to emphasize the influence of small eddies. A strong signal is visible south-west of the recirculation near Bump 5. It may be due to larger eddies and Rossby waves interacting with the western boundary and bouncing back at shorter wavelengths. The other noticeable point is that the western part of the recirculation is mainly active in this band of frequencies. Consistent with these findings, we noted that weak eddies of scale above the radius of deformation are produced on the southern flank of eastern bumps. The trajectory of these eddies instead of being simply westward is actually more to the south-west in the absence of strong currents. The eddies seem to originate from large shift of the elevation in interaction with the bumps. The strongest eddies originate from this mechanism but at higher latitudes. There, they interact with the recirculation and intensify.

Finally, Figure 5.11 shows the standard deviation for periods between 0.6 and 17 days. This figure mainly shows the inertial gravity and Kelvin waves. The maximum standard deviation for this figure is ten times smaller than the mean standard deviation of Fig. 5.8. Of interest is to note the spatial patterns of the Kelvin waves along the boundaries. The across-stream length scale tends to decrease near the bump tips, and decrease between bumps. This is evidence that the Kelvin waves are distorted by the presence of the bumps. The ``packing'' itself varies along the boundaries of the basin. The packing is loose in the eastern part of the basin and very severe in the western part, especially at Bump 1 and 2. The packing is then less and less severe as the Kelvin waves move anti-clockwise away from the recirculation. These variations in the packing of the Kelvin waves is related to the strength of the boundary currents. These currents are very strong near the recirculation, weaker away and absent in the eastern part of the basin. Figure 5.11 shows also two other interesting regions. One is the edge of the recirculation in the interior of the basin, where the inertial currents separate from the boundaries. There, the standard deviation peaks close to Bump 1 and sheds a tail along the edge of the recirculation. Presumably, because of the strong inertial currents, the Rossby number is large in this region and the inertial currents are slightly geostrophically unbalanced and produce inertial-gravity waves. A second region of interest is between Bump 1 and 2. There, the pattern due to the Kelvin waves is distorted because of the separation from the west flank of Bump 1. A reasonable explanation is that the Kelvin waves are disrupted by the encounter with the strong inertial currents of the recirculation and generate other gravity waves at Bump 1.

Transients may be essential in assuring lower energy levels by transferring the energy down-scale. This down-scale transfer can happen in two ways: either the eddies transfer the energy to inertial-gravity waves by interactions along the western walls, or in the recirculation zone through geostrophic imbalances. These small-scale inertial-gravity waves then dissipate the energy if their scale is close to the dissipative range. The recirculation location appears to be the most important from a plot of the divergence field (not shown). An intense dipole is present right at Bump 2, in front of the recirculation zone. The divergence may be related to a strong forcing of the Kelvin waves observed in Figure 5.11. The Kelvin waves are characterized by a mode two wave (two crests, two troughs around the basin) with period 8.3 days (Fig 5.12). The fact that these Kelvin waves correspond to a free mode of oscillation and are very regular both spatially and temporally suggests a resonant interaction5.2. Some irregularities are visible, though. These arise from interactions with the tip of the bumps along the western boundary. In order to emphasize the nonlinear energetic transfer to the Kelvin waves and possible viscous dissipative effects, we analyze the amplitude of the Kelvin waves as they propagate along the boundaries.

Since the amplitude of the Kelvin waves is dependent on the Coriolis parameter, f, we need to first separate the Coriolis effect from production or dissipative effects in order to make a clear diagnostic on these waves. To this aim, we can use the following rule (see also Gill, 1982, p. 379-380): For a Kelvin wave propagating along a southern boundary, we have

\eta = h_0 e^{-y/L_{Ro}} F(x-ct) \\
u = \fra...
...0}{c} e^{-y/L_{Ro}} F(x-ct) \\
v = 0 \mbox{~,}
\end{displaymath} (5.11)

where the southern boundary is located at y=0 km, $c=\sqrt{gh}$ and LRo is the radius of deformation given by LRo = c/f. The linearized total energy of this Kelvin wave is, after simplification:

te(x,y,t) = gh2 /2 + H u2 /2 = g h20 F2(x-ct) e-2y/LRo (5.12)

After integration over space, the energy becomes

\begin{displaymath}TE = \int\int te(x,y,t) \; dx dy= g h^2_0 \frac{L_{Ro}}{2} \int F^2(x-ct) dx
\end{displaymath} (5.13)

Now, we assume that the same Kelvin wave moves along a meridional wall conserving TE, with no change in structure ( $\int F^2 dx$ is now a constant independent of the orientation of the wall) but a change in amplitude and in the radius of deformation, LRo. As LRo changes with latitude, h0changes inversely as the square root of LRo for the total energy to be conserved. And as LRo is inversely proportional to f, h0 is therefore proportional to the square root of f. Thus, we can correct the amplitude of the Kelvin waves for the beta effect by using the relation:

\begin{displaymath}h_0' = \frac{h_0}{\sqrt{f_0+\beta y}} \mbox{~.}
\end{displaymath} (5.14)

The elevations along the boundaries were first corrected with respect to change in the envelope (passage of an eddy or global shift of the circulation strength) using a 17 day smoother. From this time series, h0 was computed using the difference of maximum and the minimum elevation observed at one location during a 17 day time window.

Figure 5.13 shows both h0 and h0' along the boundaries as the averaged value over the last 6 years of simulation. Along the western boundary, as the Kelvin waves pass the tip of the bumps, they encounter counter currents. The strength of these currents is not strong enough to stop the Kelvin waves, but does slow them and induces the peaks of Figure 5.13 and the packing in Figure 5.11. It is also apparent that there is a continuous decline in the amplitude of the Kelvin waves as they leave the western region of production and move anti-clockwise. This decline is probably due to viscous effects which are large for the scale of the width of the Kelvin wave. There is apparent but weak modification of the waves as they passed the tip of the eastern bumps where we measured radius of curvature of 18 km which are consistent with Figure 5.11. Therefore, the Kelvin waves tend to follow the coastline even when the radius of curvature is below the radius of deformation. The Kelvin waves cannot reflect on the eastern wall as Rossby waves because their frequency is too high for Rossby waves to exist. There is, however, the possibility that Kelvin waves generate inertial-gravity waves along the eastern boundary, as they go around the bumps and diffract some energy. For the eddy viscosity used and taking a velocity of 3 cm/s along the eastern boundaries, features below 3 km lie in the dissipative range. Therefore, these Kelvin waves must be largely dissipative themselves, directly or by further cascade to inertial gravity waves. Thus, the Kelvin waves provide one mechanism for the dissipation of the energy at this particular Reynolds number (not necessarily true at higher Re).

Of interest is to note that the amplitude of the Kelvin waves is not constant during the simulation (Fig. 5.14). In fact, we note that the amplitude is anti-correlated to the total energy (Fig. 5.15), the amplitude being highest when the total energy is the lowest. One explanation may be that, as the amplitude of the Kelvin modes grows, more energy can be dissipated via these waves. If the amplitude of the Kelvin waves grows, the reason lies in stronger interactions with the recirculation. These interactions may be related to the strong instabilities of the recirculation. It is difficult to explain why there should be a 180o phase lag between the energy in the Kelvin waves and the total energy, which represents mostly the geostrophic modes. A 180o phase lag would appear if all the energy lost in the geostrophic modes went into the Kelvin waves with very weak dissipation. However, Figure 5.13 implies nearly a 70% drop in amplitude for a Kelvin wave going along the perimeter of the basin (in 20 days). This suggests a very strong dissipation, inconsistent with the long period variations of Figures 5.14 and 5.15 (about 500 to 1000 days).

Figure 5.12: Hovmöller diagram of the filtered elevation with respect to time and location along the boundary. The elevation is given in meters. Note the strong regularity of the Kelvin waves. They are characterized by a mode 2 wave with period 8.3 days.

Figure 5.13: Amplitude of the Kelvin wave in meters along the boundary averaged over 6 years. The solid line represents the original amplitude, h0, and the dashed line represents the corrected amplitude, h0', with respect to the Coriolis parameter.

Figure 5.14: Time series of the amplitude of the oscillations at (x=500 km, y=0 km). The time series is for instance plotted in Figure 5.12 all along the boundary. From this series at the specified location, the maximum and minimum were taken in a 52-point running window (approximated 17 days). The difference of these two quantities divided by two yields the amplitude at a particular time.

Figure 5.15: Total energy for the last 6 years of simulation. Note that the amplitude of the Kelvin wave of Figure 5.14 tend to be anti-correlated with the total energy.

The maps of the standard deviation of the elevation field reveals that the recirculation zone is very active. Transient geostrophic eddies tend to amplify in the proximity of the recirculation. Energy leaks from the recirculation to these eddies and to the Kelvin waves. In order to emphasize the instabilities in the circulation zone, Figure 5.16 shows a sequence of snapshots taken of the relative vorticity every 20 days between day 5705 and day 5985. This particular sequence was chosen because it shows rapid change of the recirculation zone itself. For instance, on days 5705, 5785 and 5885, the recirculation minimum has shifted to the west and is weaker, whereas the recirculation is the strongest for days 5745, 5825 and 5925 after the minimum has shifted back to the east, close to the position of the edge of strong positive vorticity. Consistent with the eastward shift and the intensification of the recirculation, a tail of positive vorticity is shed along its edge. Rapid changes in the recirculation patterns mean that particles are not trapped indefinitely inside but escape regularly. This mechanism may prevent the formation of a Fofonoff gyre.

Figure 5.16: Instantaneous vorticity field in the vicinity of the recirculation. We focus of the period between 5705 and and 5925 days in a region limited in the south by y=-250 km, in the west by x=-200 km and in the east by x=200 km. Bumps 1 and 16 are visible along the northern boundary.
\includegraphics[scale=0.4]{images/vort1.eps} \includegraphics[scale=0.4]{images/vort2.eps} \includegraphics[scale=0.4]{images/vort3.eps} \includegraphics[scale=0.4]{images/vort4.eps} \includegraphics[scale=0.4]{images/vort5.eps} \includegraphics[scale=0.4]{images/vort6.eps}
\includegraphics[scale=0.4]{images/vort7.eps} \includegraphics[scale=0.4]{images/vort8.eps} \includegraphics[scale=0.4]{images/vort9.eps} \includegraphics[scale=0.4]{images/vort10.eps} \includegraphics[scale=0.4]{images/vort11.eps} \includegraphics[scale=0.4]{images/vort12.eps}

From a vorticity balance point of view, the transients transport the excess of vorticity produced in the interior to the walls. However, to be effective, such a transport needs to act across the streamlines. In a steady state, the vorticity balance across a streamline is

\oint \frac{\overline{\mbox{\boldmath$\tau$ }}}{\overline{h...
...erline{\zeta' \mbox{\bf u}'} \cdot \mbox{\bf n}\; dl \mbox{~.}
\end{displaymath} (5.15)

The transport of the mean vorticity by the mean currents does not contribute to this balance( $\overline{\mbox{\bf u}} \cdot \mbox{\bf n}=0$), therefore (5.15) simplifies to

\oint \frac{\overline{\mbox{\boldmath$\tau$ }}}{\overline{h...
...erline{\zeta' \mbox{\bf u}'} \cdot \mbox{\bf n}\; dl \mbox{~,}
\end{displaymath} (5.16)

where vorticity transport by eddies and viscous flux balance the wind input.

In order to illustrate the eddy transport, Figure 5.17 shows three sub-figures. The first is the curl of the wind input over the domain ( $\mbox{\bf k}\cdot \mbox{\boldmath$\nabla$ }\times \frac{\overline{\mbox{\boldmath$\tau$ }}}{\overline{h}}$), the second is the divergence of the eddy transport of vorticity ( $\mbox{\boldmath$\nabla$ }\cdot
\overline{\zeta' \mbox{\bf u}'}$) and the third is a vector-plot of the eddy transport normal to the mean streamlines as to emphasize the across-streamline component. The first two figures emphasize the local sources and sinks to the vorticity budget. The darker regions in the first sub-figure are stronger sinks of vorticity and the dark (light) regions in the second sub-figures are sources (sinks) of vorticity. In the first sub-figure, it is apparent that most of the domain is a sink of vorticity, consistent with the idea of a single-gyre forcing. However, due to the strong gradient present in the elevation field, the southern part of the recirculation is a very weak source of vorticity whereas the northern part and more specifically the regions surrounding Bump 1 and 2 are strong sinks of vorticity. The divergence of the eddy transport of vorticity (the second sub-figure) shows much finer scales and more noise. The basic features of this sub-figure are the presence of two arcs along the eastern and southern edges of the recirculations of opposite signs. The interior arc is a region of convergence of the eddy transport (source of vorticity) whereas the exterior arc is a region of divergence (sink). Regions close to Bump 3, 4 and 5 are mostly sources of vorticity whereas the regions between bumps tend to be weak sinks.

The maximum magnitude of the eddy transport is comparable to the value of the wind input. Three active regions are evident on the third sub-figure. The first is the recirculation zone, the second is directly southwest of this and the last region is southeast of the recirculation. The transport in the recirculation zone is outward-oriented along the western edge and inward along the southern and parts of the eastern edges. On this sub-figure, the two arcs of convergence (source of vorticity) and divergence (sink) are recognizable. The net forcing over the recirculation region appears to be weakly positive. This strong activity along the edge of the recirculation is another evidence that the transients are important in preventing the recirculation from growing and filling the entire domain, as it does when the solution jumps to the high energetic branch.

Southwest of the recirculation (the second region), the transport is mainly westward and southward oriented. Southeast of the recirculation (the third region), it is mainly eastward and southward oriented with an additional northward component closer to the eastern wall. The two other regions emphasize the eddy activity in the eddy belt, as defined above. Since westward propagating eddies with negative (positive) relative vorticity tend to migrate north (south), it follows that the eddy vorticity transport should be southward. This is consistent with the sub-figure which shows a main southward orientation. As the belt tends to surround the recirculation, the eddies propagate first to the southwest and then to the west. The eddy vorticity transport seems to adjust to this and tend to be oriented first to the southeast and then to the south, following a main leftward orientation with respect to the eddies.

From this analysis, it appears that the role of the mean vorticity transport (not shown) is not to be underestimated since the eddy transport seems to mainly remove the excess of vorticity from inner streamlines to outer streamlines. Close to the western boundary, the eddy transport shows no particular eastward orientation (which would be the signature of transport into the interior of positive vorticity produced at the wall). The mean vorticity transport is therefore still necessary to bring the excess of vorticity to the walls. This is done through several streamlines lying in the viscous sublayer, in particular around Bump-2, where the relative vorticity is at its maximum.

Figure 5.17: (a) Local wind input to the vorticity in Geometry V. (b) Local divergence of the eddy transport of vorticity in Geometry V computed using the last 6 years of simulation. (c) Vector plot of the eddy transport of vorticity normal to the streamlines in Geometry V computed using the last 6 years of simulation. The elevation field is plotted as an analog of the streamlines on each sub-figures.
\includegraphics[scale=0.5]{images/ekman.eps} \includegraphics[scale=0.5]{images/div_uzeta.eps}
(a) (b)

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Frederic Dupont