SUMMARYAn implicit method is developed for solving the complete three-dimensional (3D) Navier-Stokes equations. The algorithm is based upon a staggered ÿnite di erence Crank-Nicholson scheme on a Cartesian grid. A new top-layer pressure treatment and a partial cell bottom treatment are introduced so that the 3D model is fully non-hydrostatic and is free of any hydrostatic assumption. A domain decomposition method is used to segregate the resulting 3D matrix system into a series of two-dimensional vertical plane problems, for each of which a block tri-diagonal system can be directly solved for the unknown horizontal velocity. Numerical tests including linear standing waves, nonlinear sloshing motions, and progressive wave interactions with uneven bottoms are performed. It is found that the model is capable to simulate accurately a range of free-surface ow problems using a very small number of vertical layers (e.g. two-four layers). The developed model is second-order accuracy in time and space and is unconditionally stable; and it can be e ectively used to model 3D surface wave motions.

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SUMMARYAn implicit ÿnite di erence model in the co-ordinate system is developed for non-hydrostatic, twodimensional vertical plane free-surface ows. To accurately simulate interaction of free-surface ows with uneven bottoms, the unsteady Navier-Stokes equations and the free-surface boundary condition are solved simultaneously in a regular transformed domain using a fully implicit method in two steps. First, the vertical velocity and pressure are expressed as functions of horizontal velocity. Second, substituting these relationship into the horizontal momentum equation provides a block tri-diagonal matrix system with the unknown of horizontal velocity, which can be solved by a direct matrix solver without iteration. A new treatment of non-hydrostatic pressure condition at the top-layer cell is developed and found to be important for resolving the phase of wave propagation. Additional terms introduced by the co-ordinate transformation are discretized appropriately in order to obtain accurate and stable numerical results. The developed model has been validated by several tests involving free-surface ows with strong vertical accelerations and non-linear waves interacting with uneven bottoms. Comparisons among numerical results, analytical solutions and experimental data show the capability of the model to simulate free-surface ow problems.

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Entrainment of phosphorus across the thermocline can be an important nutrient source for phytoplankton in stratified lakes. In eutrophic stratified lakes, seasonal entrainment can be responsible for delayed recovery following a decrease in external phosphorus load. We compared seasonal estimates of entrainment derived from single-and multilocation thermocline migration approaches. Entrainment estimates from these methods were similar. A sampling approach based on a single centralized location produced whole-lake entrainment estimates in general agreement with the mean value from the multilocation approach (the single-location approach reasonably reproduced patterns at a weekly and seasonal time scale). In this study, it was not essential to account for spatial variation in estimating annual rates of entrainment. We also estimated entrainment based on a threedimensional (3-D) hydrodynamic model over a short period of significant thermocline migration. The 3-D model was most useful in exposing spatial and temporal variation in temperature and phosphorus profiles that are otherwise difficult to observe. Spatial variation in phosphorus profiles was associated with upwelling of metalimnetic water represented by the 3-D model. These transient dynamics, though a relatively small portion of an annual phosphorus budget, may supply nutrients to epilimnetic phytoplankton during periods of nutrient limitation. There is a need for further research that combines 3-D hydrodynamic modeling with field collection of biological and chemical data.