Deformations Compatibility Equations in the General Shell Theory into the Relative Coordinate System with Projected Deformations

This paper presents a new analytical method for obtaining new deformations compatibility equations or, new Saint-Venant’s identities, into the relative coordinate system with projected deformations by applying the hypothesis of the lineal shell theory in general flexion state. The method proposed generalizes the compatibility conditions established by A.L. Goldenveizer for the shell theory. On the other hand, the new equations include the deformations compatibility equations by other authors: Flügge, Saint-Venant, Love-Kirchhoff, Timoshenko, Goldenveizer, and Reissner-Mindlin. The results showed an increase of knowledge in general shell theory, and provide inverse and semi-inverse solutions, whose systems solution correspond to the hyper-statics degrees of the physical model, and not to their degrees of freedom.

Global Tropopause Altitudes in Radiosondes and Reanalyses

Accurate depictions of the tropopause and its changes are important for studies such as stratosphere-troposphere exchange and climate change.Here, the fidelity of primary lapse-rate tropopause altitudes and double tropopause frequencies in four modern reanalyses (ERA-Interim, JRA-55, MERRA-2, and CFSR) is examined using global radiosonde observations. In addition, long-term trends (1981–2015) in these tropopause properties are diagnosed in both the reanalyses and radiosondes. It is found that ERA-Interim, JRA-55, and CFSR reproduce observed tropopause altitudes with little bias and error comparable to the model vertical resolution, while MERRA-2 tropopause altitudes are biased 500–600 m high. All reanalyses underestimate the double tropopause frequency (up to 30 % lower than observed), with the largest biases found in JRA-55 and the smallest in CFSR. The underestimates in double tropopause frequency are primarily attributed to the coarse vertical resolution of the reanalyses. Significant increasing trends in both tropopause altitude (40–120 m per decade) and double tropopause frequency (≥ 3 % per decade) were found in both the radiosonde observations and reanalyses over the 35-year analysis period. ERA-Interim, JRA-55, and MERRA-2 broadly reproduce the patterns and signs of observed significant trends, while CFSR is inconsistent with the remaining datasets. These trends were diagnosed in both the native Eulerian coordinate system of the reanalyses and in a relative latitude coordinate system where the tropopause break (the discontinuity in tropopause altitude between the tropics and extratropics) was used as the reference latitude in each hemisphere. The tropopause break-relative coordinate facilitates the evaluation of tropopause behavior within the tropical and extratropical reservoirs and revealed significant differences in trend estimates compared to the traditional Eulerian analysis. Notably, increasing tropopause altitude trends were found to be of greater magnitude in tropopause break-relative coordinates and increasing double tropopause frequency trends were found to occur primarily poleward of the tropopause break in each hemisphere.