Properties and representativeness o f barotropic singular vectors

Three issues are discussed in this Section: the intra-seasonal variability of a barotropic instability index deduced from the singular vectors' amplification rates, the relation between the location of the fastest growing singular vectors and blocking, and the projection o f tendency fields onto the most unstable singular vectors.

4.4.1 Intra-seasonal variability o f a barotropic instability index

winter day as the largest singular value of the «-day propagator. This corresponds to the largest amplification factor of a singular vector evolving from day i to day i+n. Since we wish to relate this index to aspects o f the northern mid-latitude circulation, we have computed / ^ / , « j selecting for each

(i,n) the fastest-growing singular vectors which, at the optimization time, have their amplitude maxima between 45° and 70° N. (They coincide with the first singular vector in 80% of the cases, with the second singular vector in 10%). The instability index is plotted in Fig. 4.3b (dashed line) for an optimisation time interval of 4 days, together with its 5-day running mean (solid line). (A similar figure would have been obtained by considering the second fastest- growing singular vector instead of the first fastest-growing singular vector, while smaller fluctuations would have been seen if the instability index were defined by averaging the amplification factors of the first 5 fastest-growing singular vectors.) Four main peaks can be seen in the time-filtered instability index, with maxima respectively at days 10-12 (10-12 December 1990), 22-26 (22-26 December), 55-57 (24-26 January 1991) and 85-86 (23-24 February). The position o f these maxima compares reasonably well with the position of the local maxima in the planetary wave amplitude curve shown in Fig. 4.3a, although the quantitative correlation between the two indices is very weak (24%). Two alternative explanations can be given to this result: the first possibility is that the increased barotropic instability is purely a result of the amplification of planetary waves (and consequently o f the stronger horizontal vorticity gradients); the second one is that the high-latitude planetary wave index reflects the formation of blocking structures which are triggered by the stronger barotropic instability of the large-scale flow.

To decide which of the two explanations is more appropriate, it is useful to look in more detail at the relative timings of the wave-amplitude maxima and the instability maxima. If barotropic instability is simply driven by the wave amplitude, then the two maxima should be exactly in phase (remember that the index is plotted as a function of the initial time); if instead the planetary waves amplify because of the finite-time growth of barotropic singular vectors, then the wave amplitude maxima should lag the

instability maxima by approximately the optimization time (4-days in this case). Comparing again Figs. 4.3a and 4.3b, a lag of about 3-4 days can indeed be seen for the first and the last of the four peaks mentiqned above, but not for the strongest planetary wave maximum around day 56 (25 January).

Composites of the flow according to the barotropic instability index may reveal which of the two situations is prevailing. For each date we considered the most unstable singular vector growing o v e r the Pacific half­ hemisphere (between 90°E and 90°W) and over the Euro-Atlantic half­ hemisphere (defined as the complementary region), and thus we constructed two 'local' instability indices and Figures 4.4a-b show two composites o f 30 kPa stream function, obtained by averaging the observed fields in the initial day o f 4-day periods with larger-than-average values of (respectively) and (the average is taken over the 90-day period analyzed here). The difference between the 'Pacific unstable' and the 'Pacific stable' composite (characterized by lower-than-average is shown in Fig. 4.4c in terms o f stream function and in Fig. 4.4e in terms o f zonal wind component. Analogously, Fig. 4.4d shows the difference between the 'Euro- Atlantic unstable' and 'Euro-Atlantic stable' composites in terms of stream function, and Fig. 4.4f in terms of zonal wind component.

Considering the 'Pacific unstable' composite, we can see that over the eastern Pacific the difference map (Fig. 4.4c) has a blocking-type dipole structure with a high-latitude maximum equivalent to approximately 140 m of geopotential height, which reinforces the planetary wave amplitude in this region. Associated to this dipole structure one can see in Fig. 4.4e an increase of the zonal wind component in the western Pacific (implying a strengthening of the jet-stream), and a decrease in the eastern Pacific where the jet difluence is increased.

Although the similarity with Wallace and Gutzleds (1981) PNA pattern is not very strong, still the composite difference projects onto the negative phase of that teleconnection pattern, confirming Palmer's (1988) claims of stronger barotropic instability of negative PNA flows. An association between strong barotropic instability and enhanced planetary wave amplitude in the

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Figure 4.4. a); 'Pacific unstable' com posite (30 kPa stream function); b): as a) but for the 'Atlantic unstable' composite; c): difference between the 'Pacific unstable' and the 'Pacific stable' com posites (30 kPa stream function); d): as c) but for the difference between the Atlantic com posites; e) difference between the 'Pacific unstable' and the 'Pacific stable' com posites in terms o f zonal wind component; f): as e) but for the difference betw een the Atlantic com posites. Contour interval 15 10^ m 's'‘ for a-b), 5

PNA sector has also been reported by Betti and Navarra (1995).

The difference between the 'Euro-Atlantic unstable' and the 'Euro- Atlantic stable' composite (Fig. 4.4d) is characterized, over Europe, by a dipole similar to the Pacific one shown in Fig. 4.4c, but with a smaller meridional scale. The analysis of the difference in terms o f zonal wind component reveals an increase over the United States and the north-western Atlantic and a decrease over the eastern Atlantic and Northern Europe. As was the case for the Pacific, this wind anomaly reinforces the jet stream on the western side of the Atlantic, and increases the magnitude o f the longitudinal wind gradient across the ocean.

In summary, the composites suggest that, in both sectors, strong barotropic instability is preferentially associated with a rather mature stage of blocking development, when high-latitude anomalies have already increased the ridging on the eastern side of the oceans. Therefore, variations in barotropic instability appear to be driven by regional amplifications of planetary wave amplitude rather than being the cause o f them.

4.4.2 Location of. the fastest growing singular vectors

To investigate the relationship between finite-time barotropic instability and transition to blocking, we first ask whether the fastest growing singular vectors are indeed located in areas of blocking development. Figure 4.5 shows a Hovmoller diagram identical to that in Fig. 4.1, but with only positive contours (i.e. ridges) plotted. Superimposed on it, for a given day a square identifies the position (at final time) of the most unstable singular vector growing inside the latitudinal belt 45°-70° N. The square is located at the longitude corresponding to the stream function maximum or minimum, and its size is proportional to the singular vector amplification factor. The longitude corresponding to the maximum or minimum amplitude of the same singular vector at the initial time has also been identified, and is connected to the final position (i.e. the square) by a straight line. (In about 40% of the winter days, this simple visualization of the singular vector evolution did not provide meaningful results due to a rather delocalized structure of the most unstable

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