We consider the problem of two-dimensional projectile motion in which the resistance acting on an object moving in air is proportional to the square of the velocity of the object (quadratic resistance law). It is well known that the quadratic resistance law is valid in the range of the Reynolds number: 1 × 103 ∼ 2 × 105 (for instance, a sphere) for practical situations, such as throwing a ball. It has been considered that the equations of motion of this case are unsolvable for a general projectile angle, although some solutions have been obtained for a small projectile angle using perturbation techniques. To obtain a general analytic solution, we apply Liao's homotopy analysis method to this problem. The homotopy analysis method, which is different from a perturbation technique, can be applied to a problem which does not include small parameters. We apply the homotopy analysis method for not only governing differential equations, but also an algebraic equation of a velocity vector to extend the radius of convergence. Ultimately, we obtain the analytic solution to this problem and investigate the validation of the solution.

We have used an AGCM (atmospheric general circulation model)-based Chemistry Transport Model (ACTM) for the simulation of methane (CH 4 ) in the height range of earth's surface to about 90 km. The model simulations are compared with measurements at hourly, daily, monthly and interannual time scales by filtering or averaging all the timeseries appropriately. From this model-observation comparison, we conclude that the recent (1990)(1991)(1992)(1993)(1994)(1995)(1996)(1997)(1998)(1999)(2000)(2001)(2002)(2003)(2004)(2005)(2006) trends in growth rate and seasonal cycle at most measurement sites can be fairly successfully modeled by using existing knowledge of CH 4 flux trends and seasonality. A large part of the interannual variability (IAV) in CH 4 growth rate is apparently controlled by IAV in atmospheric dynamics at the tropical sites and forest fires in the high latitude sites. The flux amplitudes are optimized with respect to the available hydroxyl radical (OH) distribution and model transport for successful reproduction of latitudinal and longitudinal distribution of observed CH 4 mixing ratio at the earth's surface. Estimated atmospheric CH 4 lifetime in this setup is 8.6 years. We found a small impact (less than 0.5 ppb integrated over 1 year) of OH diurnal variation, due to temperature dependence of reaction rate coe‰cient, on CH 4 simulation compared to the transport related variability (order of G15 ppb at interannual timescales). Model-observation comparisons of seasonal cycles, synoptic variations and diurnal cycles are shown to be useful for validating regional flux distribution patterns and strengths. Our results, based on two emission scenarios, suggest reduced emissions from temperate and tropical Asia region (by 13, 5, 3 Tg-CH 4 for India, China and Indonesia, respectively), and compensating increase (by 9, 9, 3 Tg-CH 4 for Russia, United States and Canada, respectively) in the boreal Northern Hemisphere (NH) are required for improved model-observation agreement.

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[1] Temporal variations of carbon monoxide (CO) were observed simultaneously at seven surface stations located in east Asia/western North Pacific from 24°N to 43°N during the East Asian Regional Experiment (EAREX) 2005 campaign in March 2005. Three major pollution events with enhanced CO levels were recorded around the same time at four stations over the East China Sea and at two northern stations of Japan. These pollution events were also observed 3-4 d later at Minamitorishima, located far from the Asian continent. A synoptic weather analysis showed that all of the major CO enhancements were brought about by the passages of cold fronts associated with the eastward migrating cyclonic development. The CO distribution simulated by a three-dimensional transport model showed that the polluted air masses exported from the continent were trapped behind the cold fronts and then merged into elongated belts of enriched CO before spreading over the western North Pacific. Transport of regionally tagged CO tracer simulated by the model indicated that the Chinese and Korean emissions were the major contributors to the pollution over the East China Sea, while the Japanese emissions had impacts at relatively higher latitude regions during the campaign. The simulation results also showed that the CO enhancements detected at Minamitorishima were caused by a long-range transport of pollution emissions from various regions in east Asia. The CO-enriched plumes from Southeast Asia and south Asia emissions were found above the boundary layer in the frontal zone but not at the surface.