The sensitivity to variations in the inverse model can be investigated by running a number of different models, each varying from the reference state by one factor. The reference state was defined as the standard model and the preferred solution from this reference state was defined as the standard solution (section 5.6). The sensitivity of this reference state was examined by allowing each of its components to vary within a prescribed range. A number of models were used to illustrate the importance of various characteristics, and the effects of these variations on the net fluxes were inspected. Some models were deliberately ‘unreasonable’, producing a set of flux
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estimates significantly different to those obtained from the preferred solution from the standard model, so illustrating the importance of that particular model component. Other more ‘reasonable’ models were selected so as to provide a range of realistic solutions. The standard deviation of the solutions given by these latter models provided an estimate of the sensitivity of the inverse.
The model states and their ‘plausibility’ are categorised in Table 6.2 and the results are given in Table 6.3. If row and column weighting are not included (States 1 to 3), the model is implausible, since without the former, the property transport equations are affected by the magnitude of the property being transported, and without the latter, the solution is scaled by the cross-sectional area of its station pair. If the model includes layer-specific conservation equations, then diapycnal exchange must also be included else the model will be physically inconsistent (State 8). An unreasonable initial circulation (State 16) will produce an implausible solution since the inverse can only be expected to be close to correct if the first guess is realistic. A reasonable model will yield implausible solutions if too high or too low a rank is chosen (States 4,6,7). The choice of very low ranks will give solutions very close to the initial state, while choice of high ranks will yield solutions incorporating a high level of noise.
Model Specifications Plausibility
Model Solution State 1 standard model with no weighting ¥ ¥ State 2 standard model with column weighting only ¥ ¥ State 3 standardmodel with row weighting only ¥ ¥ State 4 standardmodel but rank 10 ÷ ¥ State 5 standard model but rank 30 ÷ ÷ State 6 standard model but rank 100 ÷ ¥ State 7 standard model but rank 140 ÷ ¥ State 8 standard model but no diapycnal velocities ¥ ¥ State 9 only full depth volume fluxes for each box, full rank ÷ ÷ State 10 layer specific volume fluxes ÷ ÷ State 11 layer specific salt fluxes ¥ ¥ State 12 layer specific salt and volume fluxes ÷ ÷ State 13 standard model but with bottom referenced geostrophy for initial velocity field ¥ ¥ State 14 standard model but with ekman fluxes at upper limit ÷ ÷ State 15 standard model but with ekman fluxes at lower limit ÷ ÷ State 16 standard model but no ekman fluxes ¥ ¥ State 17 standard model but with bottom triangles calculated by extrapolation of shear ÷ ÷ State 18 standard model but coarse realisation of denmark strait section ÷ ÷ State 19 standard model with subsampled fine resolution denmark strait section ÷ ÷ State 20 standard model with alternative crossover station ÷ ÷ State 21 standard model with alternative EGC ÷ ÷
Table 6.2: Specifications of alternative models to test inverse and oceanographic sensitivity. The ‘reasonable’ and ‘unreasonable’ model states and solutions are noted by a ÷ or a ¥, respectively.
6.4.1 Weighting scheme
If the system is solved at full rank then row weighting makes no difference to the solution. The standard solution, however, is not the full rank solution, but rather the solution for rank 43. Since residuals are allowed in the equations, the weighting of these residuals becomes important.
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Without column weighting the solutions for States 1 and 3 at rank 43 (Table 6.2) give similar results to the transport estimates from the initial velocity field (pre-inverse). The condition of requiring residuals to be zero within 3 standard deviations is only satisfied at ranks 148 and 145 respectively. With only column weighting (State 2) the same condition is satisfied at a rank of 13. Without the row weighting, however, the property transport equations are affected by the magnitude of the property being transported. In particular, the inflow of the NAW across the Greenland-Scotland Ridge is very low, which in turn reduces the total heat convergence within the Nordic Seas. This might be attributed to the anomalously high salinity on the Iceland-Scotland section (average salinity of >!35.25 in the upper waters) compared to all other sections: without row weighting the inverse reduces the northward transport across the section in order to satisfy the salinity constraints.
6.4.2 Choice of rank
The rank of the standard solution was chosen to be such that the residuals are required to be zero to within three standard deviations of the error estimate. To illustrate the effect the choice of rank has on the final result, the results are compared for arbitrary ranks of 10, 30, 100 and 140.
The higher rank solutions of 100 and 140 (States 6 and 7) have smaller residuals, but a less ‘reasonable’ circulation with reduced fluxes across the individual sections. In particular, the magnitudes of the Atlantic inflow across the Greenland-Scotland Ridge are reduced to 2.4!and!2.7!Sv. Also, the magnitudes of the East Greenland Current are roughly halved, as are the magnitudes of the dense overflows into the North Atlantic. The resultant heat fluxes across the Greenland-Scotland Ridge are reduced by almost a factor of 3, such that the total heat convergence in the Nordic Seas is reduced to an average of 5!TW (compared to the 123!TW of the standard solution).
The lower rank solution of 10 (State 4) gives generally higher fluxes across individual sections and greater heat fluxes. The lower rank solutions are close to the fluxes given by the initial velocity field (see section 5.5 and Tables 5.5 and 5.6), since the adjustments made by the inverse are small. The mid rank solution (30) is included as a plausible alternative to the standard solution.
6.4.3 Inclusion of diapycnal velocities
The solution given by State 8 illustrates the importance of the inclusion of diapycnal velocities when the inverse model is formulated to include layer-specific conservation. If, however, only top-to- bottom constraints are applied (see section 6.4.4 below), then the inclusion of diapycnal velocities is obviously irrelevant and has no effect on the solution. When fluxes between layers are not allowed although the total fluxes within the inverse boxes are constrained to be zero within the expected errors, the fluxes across individual sections are unrealistically low. Also, the solution for rank 43 without diapycnal velocities gives very large reference velocities with a standard deviation of
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3.4!cm!s-1 and peak value of 25.8!cm!s-1, compared to a standard deviation of 0.5!cm!s-1 and peak
value of 1.8!cm!s-1 for the standard solution.
6.4.4 Choice of constraints
A number of models are run with different constraints applied. For State 9 full depth volume conservation is required in each inverse box. Since there is only one constraint for each box, State 9 was solved to full rank. For States 10!-!12 layer-specific volume, layer-specific salinity and layer- specific volume and salinity constraints, respectively, are applied. These states are required to satisfy the same conditions as for the standard solution i.e. residuals are required to be zero within three standard deviations of the error estimate, and the solution is required to be within one standard
deviation of the a priori error estimate.
These models do go part way to providing a ‘reasonable’ representation of the circulation although since temperature flux conservation is not required, an accurate description of the heat fluxes, in particular, is lacking. State 11, where only layer-specific salinity conservation is required, does not satisfactorily reduce the flux residuals within each box since the salinity residuals are within the error estimate at very low ranks. This suggests that there is insufficient information in salinity alone to satisfactorily constrain the inverse solution.
6.4.5 Choice of initial velocity field
To illustrate the poor representation of the circulation that is obtained by using a level of no motion throughout the Nordic Seas, State 13 was required to be identical to the standard model with the exception that no direct velocities were applied. This allowed the impact of the direct velocity measurements in the solution to be evaluated. Since an initial level of no motion is prescribed at the deepest common level between stations, the initial velocity field is effectively set to bottom- referenced geostrophy. The solution for rank 43 is selected to allow a direct comparison to the standard solution (size of the residuals etc.). The solution is of poorer quality than that of the standard solution: there are larger residuals at the same rank, overflows and inflows are reduced to values inconsistent with literature and the bulk fluxes of the standard solution are not reproduced. In particular, the total inflow of AW across the Greenland-Scotland Ridge is reduced to 4.4!Sv (compared to 6.1!Sv in the standard solution). Although the 0.8!Sv inflow of Atlantic waters to the Barents Sea remains reasonable (1.0!Sv in the standard solution), this is at the expense of reducing the northward transport in the WSC through Fram Strait to zero. The net southward flow in the EGC is also reduced by about half, both in Fram Strait and futher south in the Greenland Sea
(~75°N). Although the overflows across the Iceland-Scotland Ridge are unchanged, the magnitude
of the Denmark Strait Overflow is reduced by almost half to 1.7!Sv.
This behaviour emphasises the importance of direct velocity observations (or transport estimates based on observations) in inverse box models to compensate for only being able to use a crude,
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diagonal representation of the a priori errors. Numerous recent inverse studies have found that incorporating direct velocity measurements makes a significant impact on the solution (Bacon, 1997; Bingham and Talley, 1991; Naveira Garabato et al., 2003; Joyce et al., 1986; Vanicek and Siedler, 2002). In particular, it is found here that the prevailing circulation of the Nordic Seas cannot be reasonably approximated with a level of no motion.