The classical problem of irrotational long waves on the surface of a shallow layer of an ideal fluid moving under the influence of gravity as well as surface tension is considered. A systematic procedure for deriving an equation for surface elevation for a prescribed relation between the orders of the two expansion parameters, the amplitude parameter α and the long wavelength (or shallowness) parameter β, is developed. Unlike the heuristic approaches found in the literature, when modifications are made in the equation for surface elevation itself, the procedure starts from the consistently truncated asymptotic expansions for unidirectional waves, a counterpart of the Boussinesq system of equations for the surface elevation and the bottom velocity, from which the leading order and higher order equations for the surface elevation can be obtained by iterations. The relations between the orders of the two small parameters are taken in the form β = O(α n ) and α = O(β m ) with n and m specified to some important particular cases. The analysis shows, in particular, that some evolution equations, proposed before as model equations in other physical contexts (like the Gardner equation, the modified KdV equation, and the so-called 5th-order KdV equation), can emerge as the leading order equations in the asymptotic expansion for the unidirectional water waves, on equal footing with the KdV equation. The results related to the higher orders of approximation provide a set of consistent higher order model equations for unidirectional water waves which replace the KdV equation with higher-order corrections in the case of non-standard ordering when the parameters α and β are not of the same order of magnitude. The shortcomings of certain models used in the literature become apparent as a result of the subsequent analysis. It is also shown that various model equations obtained by assuming a prescribed relation β = O(α n ) between the orders of the two small parameters can be equivalently treated as obtained by applying transformations of variables which scale out the parameter β in favor of α. It allows us to consider the nonlinearity-dispersion balance, epitomized by the soliton equations, as existing for any β, provided that α → 0, but leads to a prescription, in asymptotic terms, of the region of time and space where the equations are valid and so the corresponding dynamics is expected to occur.
An efficient method for precipitation recycling calculations based on the general bulk recycling model is developed and applied to the Amazon basin region. The results are compared with the estimates provided by simplified recycling models, and with the estimations obtained in other studies. Then the numerical procedure is adjusted to the modified recycling model incorporating some effects of incomplete vertical mixing of water in the tropospheric column as described in Part I of this paper. The modified numerical model is applied to the Amazon basin region and the results are compared with those from the unmodified model. It is seen from the results that the effects related to an incomplete vertical mixing produce significant changes in the level and distribution of precipitation recycling over the Amazon basin. The general changes in the recycling patterns occurring as a result of modifications include a significant increase in the contribution of recycled water to precipitation in the eastern part of the region and weakening continental recycling gradients. The regional recycling ratio values for the Amazon basin estimated by the modified model are significantly higher than the values provided by the unmodified model. The results of application of the models to several other regions on the globe are described in short. In particular, a comparison of the results with the estimates by the modified Budyko model is made for different regions, and it is found that the modified Budyko's model estimates are, in general, closer to the modified model numerical results than it was for the unmodified models. The results for precipitation recycling over the central United States are presented in more detail. For this region, the information on the degree of incomplete vertical mixing available from the GCM tracer transport modeling results has been used in the modified model calculations. It has been found that, as distinct from the Amazon basin, the effects related to an incomplete vertical mixing do not produce significant changes in the level and distribution of precipitation recycling over the central United States.
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