DISCUSSION

There are two key model parameters that we will derive from the experimental data: the source vortcitiy ratioαiand the transition timescaleT_ibeyond which the time-dependent model is valid. We also investigate the initial vortex radiusR_i atT =T_i and find that the theoretical values provide poor estimates of the values seen experimentally. This is discussed in detail in section 8.2.1.3.

8.2.1.1 Source vorticity ratio

We begin our analysis with the source vorticity ratioα_i. In chapter 4 we presented PIV data that provided an estimate of the source vorticity for a subset of the experimental parameters.

To obtain estimates for the full parameter range we fitted the experimental data to the steady state current model. This gave some promising results, in particular demonstrating that a zero PV model is able to capture the key characteristics of the current in a steady state for Ro/Fr_c≳1/√

2. We look to follow the same method here, by fitting the data to the theory as closely as possible to provide estimated values forαi.

The fitted values ofα_iare derived by considering the qualitative fit to the experimental data across all features of the flow, that is: the vortex depth, vortex radius, current velocity and current width. The measurements of the vortex maximum depth are particularly important

to consider when deriving the value ofα_i, as they impose a lower bound according to the theoretical model as discussed previously in chapter 7. The lower bound is given by

α_i≥1− 1

H_max. (8.1)

Furthermore, the measurements of the vortex depth are made using the method of dye at-tenuation, which results in relatively small measurement error. The vortex radius, current velocity and current width are all measured using the method of front-tracking, which is more susceptible to measurement errors. Typical errors will be given when the data are presented in later sections. This is all taken into account when deriving the fitted values ofαifrom the data. We consider both of the vortex growth models presented in chapter 5 and in many cases we obtain different values ofαiwhen fitting the data to each of the shallow vortex and vortex current models. A subset of the experiments are chosen that allow for the best comparison with the theory. One particularly important criterion in selecting the experiments for comparison is the ratio of the transition timescaleT_iand the end time of the experiments.

To be able to compare the data with the theory we require that the transition timeT_iis much less than the time at which the experiment ends. Unfortunately, this means that for many of the experiments with low values ofI^∗a comparison with the theory is not possible. For low values ofI^∗the vortex is elongated along the boundary wall with a weak circulation and a shallow depth, meaning that it takes a longer time for the vortex to deepen and reach the source value ofH₀which defines the transition timeT_i. The current speed is also in general larger in this regime and the current rounds the perimeter of the tank in a shorter time forcing the experiment to be halted.

The fitted values ofα_ifor both of the shallow vortex and vortex current models are dis-played in figure 8.1 versus the ratio of the aspect ratiosI^∗. A total of seventeen experiments are used in the comparison with the experimental data. The plot shows that the fitted values for the vortex current model are in general higher than for the shallow vortex model. This is due to the faster rate of decay of the vorticity ratioαT for the shallow vortex model which results in more rapid changes to the flow properties, as seen in chapter 7. Theαivalues range from 0.59≤α_i≤0.99 and do not appear to show any trend relating to the value ofI^∗.

We next consider the values ofQ^∗=H^∗(1−α_i), which are plotted in figure 8.2 versusI^∗. The data show that the flow has a finite value of PV across the majority of the experimental parameter range for both models, with 0.01≤Q^∗≤0.74. The value of the PV does, however,

8.2 Model comparison 177

Figure 8.1The fittedαivalues that give the best agreement between the theory and the experimental data, versus the aspect ratio ratioI^∗.

Figure 8.2The fittedQ^∗values that give the best agreement between the theory and the experimental data, versus the aspect ratio ratioI^∗.

Figure 8.3The ratio of the fitted values ofαi derived from the time-dependent theoretical model αiT f it and the steady state current modelαiSS f it, versusI^∗.

remain quite small which explains the ability of the zero PV steady state current model of TL to be able to capture the key properties of the current forRo/Fr_c≳1/√

2 as seen in chapter 4. We also see thatQ^∗<1 for each experiment, which ensures that the theoretical model remains valid.

Finally, we compare the fitted values of the source vorticity ratioα_iderived in this chapter for the time-dependent theoretical model, with those derived in chapter 3 for the steady state current model. Figure 8.3 plots the ratio of the values versusI^∗. The data show that in general the fittedα_ivalues are slightly lower for the time-dependent shallow vortex model, whilst the values from the vortex current model agree very well with their steady-state counterparts.

The mean value of the ratio for the shallow vortex model is 0.89±0.14, while for the vortex current model the mean ratio value is 1.02±0.14. The largest discrepancy is seen at the lowest value ofI^∗, which is unsurprising as the flow in the lowI^∗regime is shallow and wide making it susceptible to the largest measurement errors. The level of agreement between the fitted values is promising as it suggests that fitting the data to the theory provides a valid and consistent method for estimating the source vorticity ratio when accurate measurements are unavailable.

Before we move onto a comparison of both the shallow vortex and vortex current models with the experimental data in the remainder of this chapter, it is worth reiterating that the fitted

8.2 Model comparison 179 values ofα_ifor both models have been derived qualitatively. That is, the theoretical curves for all of the flow features: the vortex depth, vortex radius, current velocity and current width are fitted to the data and the value ofα_ithat provides the best qualitative fit across all four features is used. For the preliminary comparison with the data in this thesis such a method is okay as we are concentrating on a proof-of-principle type comparison with the time-dependent theoretical model to assess its capability to capture the first order time-time-dependent physics of the flow. For future studies, a rigorous framework for model evaluation will be required, where the fitted values ofαiare calculated in the same way for each experiment.

One possible such method would be to use the value ofαithat minimises the least squares regression coefficient between the theoretical curves and the data across the four flow features.

8.2.1.2 Transition timescale

Another important parameter in the time-dependent model that must be derived from the experimental data is the transition timescaleT_i. We saw in chapter 7 that the vorticity ratio α_T decreases from its initial valueα_iforT >T_i, whereT_iis defined as the time at which the the vortex depth is equal to the depth of the sourceH₀. Our model is only valid forT >T_i.

We are able to obtain estimates for the timeT =T_ifor the vast majority of the experi-ments, with the exception of three runs where the vortex does not reach the source depth before the end of the experiment. These will be excluded from our analysis. Figure 8.4 plots the transition timescale as measured experimentally versus the depth ratioH^∗. The data show a transition in the behaviour forH^∗≲1. ForH^∗∼1 the maximum depth fixed by geostrophy, h₀, is approximately equal to the source depth. We expect geostrophic adjustment to occur after 1−2 rotation periods, corresponding toT_i∼4π−8π. The data show that forH^∗>1 the source depth is reached quickly as the outflow increases in depth towards the larger geostrophic value ofh₀. This is in contrast to the behaviour forH^∗∼1 where the transition timeT_ivaries on the order of the geostrophic adjustment timescale. The experiments with H^∗∼1 and smallT_i<10 have a larger volume fluxQthan those with largerT_i. This increases the source momentum-fluxQ²/Awhich acts to deepen the flow over a shorter timescale.

Figure 8.4The experimentally estimated value ofTifrom the vortex depth data versusH^∗=h₀/H₀.

8.2.1.3 Initial vortex radius

The transition timescale T_i is defined to be the time taken for the vortex depth to increase to the source depthH₀. During this period of transition the vortex radius will also increase, reaching a value ofR_iat timeT_i. The thoeretical model introduced in chapter 7 estimates the value ofR_ito be

R_i

R_d = 2√ 2

pα_i(2−α_i), (8.2)

which is the same for both the vortex current and shallow vortex models for the vortex growth.

In this section we compare the experimental measurements for the vortex radius with the theoretical predictions forR_i. The theoretical predictions according to (8.2) are calculated using the fitted values for αi discussed in section 8.2.1.1 and we use the experimentally measured values ofT_ifrom the vortex depth data in section 8.2.1.2.

Figure 8.5 plots the ratio of the experimentally estimatedR_ivalue and the theoretically predicted value, versusI^∗. The theoretical predictions for the shallow vortex model in both the parallel and perpendicular directions and those of the vortex current model are compared with the data. The figure shows that in general the value ofR_iachieved experimentally is less than that predicted by the theory. The best agreement is seen for the shallow vortex model for the radius in the perpendicular direction, but there is still a consistent overestimation of the intial radius by the theory. This is perhaps not too suprising as we have used a relationship

8.2 Model comparison 181

Figure 8.5The ratio of the experimentally estimatedRivalue and the theoretically predicted value versusI^∗.

between the vortex maximum depth and radius to derive the equation forR_igiven in (8.2), which we expect only to be valid forT ≥T_i. The data show that this relationship does not hold during the intial vortex formation forT <T_i.

The results seen here mean that we will use the experimental values forR_iwhen compar-ing the theory with the data in later sections. We only expect the time-dependent theoretical model of chapter 7 to be valid forT >T_iand therefore by using the experimental values for R_iandT_iwe will ensure a fair comparison with the data that is not skewed by the inability of the theory to predict the initial conditions of the vortex atT =T_i.