Rigorous bounds on the nonlinear saturation of instabilities to parallel shear flows

A novel method is presented for obtaining rigorous upper bounds on the finite-amplitude growth of instabilities to parallel shear flows on the beta-plane. The method relies on the existence of finite-amplitude Liapunov (normed) stability theorems, due to Arnol'd, which are nonlinear generalizations of the classical stability theorems of Rayleigh and Fjørtoft. Briefly, the idea is to use the finite-amplitude stability theorems to constrain the evolution of unstable flows in terms of their proximity to a stable flow. Two classes of general bounds are derived, and various examples are considered. It is also shown that, for a certain kind of forced-dissipative problem with dissipation proportional to vorticity, the finite-amplitude stability theorems (which were originally derived for inviscid, unforced flow) remain valid (though they are no longer strictly Liapunov); the saturation bounds therefore continue to hold under these conditions.

Nonlinear stability and the saturation of instabilities to axisymmetric vortices

Nonlinear stability theorems are presented for axisymmetric vortices under the restriction that the disturbance is independent of either the azimuthal or the axial coordinate. These stability theorems are then used, in both cases, to derive rigorous upper bounds on the saturation amplitudes of instabilities. Explicit examples of such bounds are worked out for some canonical profiles. The results establish a minimum order for the dependence of saturation amplitude on supercriticality, and are thereby suggestive as to the nature of the bifurcation at the stability threshold.

An evaluation of the correlation between open solar flux and total solar irradiance

The correlation between the coronal source flux F_{S} and the total solar irradiance I_{TS} is re-evaluated in the light of an additional 5 years' data from the rising phase of solar cycle 23 and also by using cosmic ray fluxes detected at Earth. Tests on monthly averages show that the correlation with F_{S} deduced from the interplanetary magnetic field (correlation coefficient, r = 0.62) is highly significant (99.999%), but that there is insufficient data for the higher correlation with annual means (r = 0.80) to be considered significant. Anti-correlations between I_{TS} and cosmic ray fluxes are found in monthly data for all stations and geomagnetic rigidity cut-offs (r ranging from −0.63 to −0.74) and these have significance levels between 85% and 98%. In all cases, the t is poorest for the earliest data (i.e., prior to 1982). Excluding these data improves the anticorrelation with cosmic rays to r = −0:93 for one-year running means. Both the interplanetary magnetic field data and the cosmic ray fluxes indicate that the total solar irradiance lags behind the open solar flux with a delay that is estimated to have an optimum value of 2.8 months (and is within the uncertainty range 0.8-8.0 months at the 90% level).

Long-term drift of the coronal source magnetic flux and the total solar irradiance

We test the method of Lockwood et al. [1999] for deriving the coronal source flux from the geomagnetic aa index and show it to be accurate to within 12% for annual means and 4.5% for averages over a sunspot cycle. Using data from four solar constant monitors during 1981-1995, we find a linear relationship between this magnetic flux and the total solar irradiance. From this correlation, we show that the 131% rise in the mean coronal source field over the interval 1901-1995 corresponds to a rise in the average total solar irradiance of {\Delta}I = 1.65 +/- 0.23 Wm^{-2}.

Investigation of the accuracy of historical irradiance products and interannual variability of solar irradiance using Met Office ground data

Met Office station data from 1980 to 2012 has been used to characterise the interannual variability of incident solar irradiance across the UK. The same data are used to evaluate four popular historical irradiance products to determine which are most suitable for use by the UK PV industry for site selection and system design. The study confirmed previous findings that interannual variability is typically 3–6% and weighted average probability of a particular percentage deviation from the mean at an average site in the UK was calculated. This weighted average showed that fewer than 2% of site-years could be expected to fall below 90% of the long-term site mean. The historical irradiance products were compared against Met Office station data from the input years of each product. This investigation has found that all products perform well. No products have a strong spatial trend. Meteonorm 7 is most conservative (MBE = −2.5%), CMSAF is most optimistic (MBE = +3.4%) and an average of all four products performs better than any one individual product (MBE = 0.3%)

The only way is up: retail format saturation and the demise of the American five and dime store, 1914-1941

We examine a classic ‘wheel of retailing’ episode – the abandonment of the five and dime pricing formula by American variety chains. These switched from a conventional product lifecycle, focusing on cost reduction through standardisation, to a reverse path up the ‘service cost - unit value’ continuum. We show that, rather than reflecting deteriorating managerial acumen, this was a response to the continued imperative for growth following retail format saturation. Firm-specific (rather than format-specific) competitive advantages were too weak for any chain to be confident it could win a within-format price war, making inter-format competition through raising price points more attractive.