This paper presents a differential geometry based model for the analysis and computation of the equilibrium property of solvation. Differential geometry theory of surfaces is utilized to define and construct smooth interfaces with good stability and differentiability for use in characterizing the solvent-solute boundaries and in generating continuous dielectric functions across the computational domain. A total free energy functional is constructed to couple polar and nonpolar contributions to the salvation process. Geometric measure theory is employed to rigorously convert a Lagrangian formulation of the surface energy into an Eulerian formulation so as to bring all energy terms into an equal footing. By minimizing the total free energy functional, we derive coupled generalized Poisson-Boltzmann equation (GPBE) and generalized geometric flow equation (GGFE) for the electrostatic potential and the construction of realistic solvent-solute boundaries, respectively. By solving the coupled GPBE and GGFE, we obtain the electrostatic potential, the solvent-solute boundary profile, and the smooth dielectric function, and thereby improve the accuracy and stability of implicit solvation calculations. We also design efficient second order numerical schemes for the solution of the GPBE and GGFE. Matrix resulted from the discretization of the GPBE is accelerated with appropriate preconditioners. An alternative direct implicit (ADI) scheme is designed to improve the stability of solving the GGFE. Two iterative approaches are designed to solve the coupled system of nonlinear partial differential equations. Extensive numerical experiments are designed to validate the present theoretical model, test computational methods, and optimize numerical algorithms. Example solvation analysis of both small compounds and proteins are carried out to further demonstrate the accuracy, stability, efficiency and robustness of the present new model and numerical approaches. Comparison is given to both experimental and theoretical results in the literature. KeywordsGeneralized Poisson-Boltzmann equation; Biomolecular surface formation and evolution; Potential driving geometric flows; Solvation free energy; Multiscale models I IntroductionAmong the various components of molecular interactions, electrostatic interactions are of special importance 6,8,41,44,55,59,71, 142 -144 , 169 ,170 because of their long range and influence on polar or charged molecules -including water, aqueous ions, and amino or nucleic acids. Electrostatic interactions are ubiquitous for any system of charged or polar molecules, such as * Please address correspondence to Guowei Wei. wei@math.msu.edu. (proteins, nucleic acids, lipid bilayers, sugars, etc.) in their aqueous environment. Electrostatic solute-solvent interactions, therefore, are of central importance in analyzing molecular structure and modeling the intramolecular and intermolecular interactions of macromolecules in simulations. There are two types of solvation models: 131, 142 , 143 , 169 explicit ...

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Systematic measurements of the surface drift current, the wind profile over the water surface and the wave spectra have been made for (i) pure wind-waves, (ii) a coexisting system of wind-waves and swell propagating against the wind, and (iii) a coexisting system of wind-waves and swell propagating in the direction of the wind. The surface drift current is gradually intensified by the swell propagating against the wind when the swell steepness increases. The maximum increase of the surface drift velocity caused by the opposing swell is about 46 % of the surface drift velocity for pure wind-waves at the same wind speed. Such a phenomenon was not observed when the swell was propagating in the direction of the wind.

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