Well-posed boundary value problems for integrable evolution equations on a finite interval

We consider boundary value problems posed on an interval [0,L] for an arbitrary linear evolution equation in one space dimension with spatial derivatives of order n. We characterize a class of such problems that admit a unique solution and are well posed in this sense. Such well-posed boundary value problems are obtained by prescribing N conditions at x=0 and n–N conditions at x=L, where N depends on n and on the sign of the highest-degree coefficient n in the dispersion relation of the equation. For the problems in this class, we give a spectrally decomposed integral representation of the solution; moreover, we show that these are the only problems that admit such a representation. These results can be used to establish the well-posedness, at least locally in time, of some physically relevant nonlinear evolution equations in one space dimension.

Evaluation of the model representation of the evolution of convective systems using satellite observations of outgoing longwave radiation

We introduce a technique for assessing the diurnal development of convective storm systems based on outgoing longwave radiation fields. Using the size distribution of the storms measured from a series of images, we generate an array in the lengthscale-time domain based on the standard score statistic. It demonstrates succinctly the size evolution of storms as well as the dissipation kinematics. It also provides evidence related to the temperature evolution of the cloud tops. We apply this approach to a test case comparing observations made by the Geostationary Earth Radiation Budget instrument to output from the Met Office Unified Model run at two resolutions. The 12km resolution model produces peak convective activity on all lengthscales significantly earlier in the day than shown by the observations and no evidence for storms growing in size. The 4km resolution model shows realistic timing and growth evolution although the dissipation mechanism still differs from the observed data.