A model for the periodic water wave problem and its long wave amplitude equations

We are interested in the validity of the KdV and of the long wave NLS approximation for the water wave problem over a periodic bottom. Approximation estimates are non-trivial, since solutions of order O(ε^2 ), resp. O(ε), have to be controlled on an O(1/ε^3 ), resp. O(1/ε^2 ), time scale. In contrast to the spatially homogeneous case, in the periodic case new quadratic resonances occur and make a more involved analysis necessary. For a phenomenological model we present some results and explain the underlying ideas. The focus is on results which are robust in the sense that they hold under very weak non-resonance conditions without a detailed discussion of the resonances. This robustness is achieved by working in spaces of analytic functions. We explain that, if analyticity is dropped, the KdV and the long wave NLS approximation make wrong predictions in case of unstable resonances and suitably chosen periodic boundary conditions. Finally we outline, how, we think, the presented ideas can be transferred to the water wave problem

Mediterranean winter rainfall in phase with African monsoons during the past 1.36 million years

Mediterranean climates are characterized by strong seasonal contrasts between dry summers and wet winters. Changes in winter rainfall are critical for regional socioeconomic development, but are difficult to simulate accurately1 and reconstruct on Quaternary timescales. This is partly because regional hydroclimate records that cover multiple glacial–interglacial cycles2,3 with different orbital geometries, global ice volume and atmospheric greenhouse gas concentrations are scarce. Moreover, the underlying mechanisms of change and their persistence remain unexplored. Here we show that, over the past 1.36 million years, wet winters in the northcentral Mediterranean tend to occur with high contrasts in local, seasonal insolation and a vigorous African summer monsoon. Our proxy time series from Lake Ohrid on the Balkan Peninsula, together with a 784,000-year transient climate model hindcast, suggest that increased sea surface temperatures amplify local cyclone development and refuel North Atlantic low-pressure systems that enter the Mediterranean during phases of low continental ice volume and high concentrations of atmospheric greenhouse gases. A comparison with modern reanalysis data shows that current drivers of the amount of rainfall in the Mediterranean share some similarities to those that drive the reconstructed increases in precipitation. Our data cover multiple insolation maxima and are therefore an important benchmark for testing climate model performance

Winter Precipitation Forecast in the European and Mediterranean Regions Using Cluster Analysis

The European climate is changing under global warming, and especially the Mediterranean region has been identified as a hot spot for climate change with climate models projecting a reduction in winter rainfall and a very pronounced increase in summertime heat waves. These trends are already detectable over the historic period. Hence, it is beneficial to forecast seasonal droughts well in advance so that water managers and stakeholders can prepare to mitigate deleterious impacts. We developed a new cluster-based empirical forecast method to predict precipitation anomalies in winter. This algorithm considers not only the strength but also the pattern of the precursors. We compare our algorithm with dynamic forecast models and a canonical correlation analysis-based prediction method demonstrating that our prediction method performs better in terms of time and pattern correlation in the Mediterranean and European regions

Quadratic Life Span of Periodic Gravity-capillary Water Waves

We consider the gravity-capillary water waves equations for a bi-dimensional fluid with a periodic one-dimensional free surface. We prove a rigorous reduction of this system to Birkhoff normal form up to cubic degree. Due to the possible presence of three-wave resonances for general values of gravity, surface tension, and depth, such normal form may be not trivial and exhibit a chaotic dynamics (Wilton ripples). Nevertheless, we prove that for all the values of gravity, surface tension, and depth, initial data that are of size \u3b5 in a sufficiently smooth Sobolev space leads to a solution that remains in an \u3b5-ball of the same Sobolev space up times of order \u3b5 122. We exploit that the three-wave resonances are finitely many, and the Hamiltonian nature of the Birkhoff normal form