A Semi-Implicit, Three-Dimensional Model for Estuarine ... - USGS

A Semi-Implicit, Three-Dimensional Model for
Estuarine Circulation
By Peter E. Smith
Abstract
A semi-implicit, finite-difference method for the numerical solution of the three-dimensional equations for circulation in estu-
aries is presented and tested. The method uses a three-time-level, leapfrog-trapezoidal scheme that is essentially second-order
accurate in the spatial and temporal numerical approximations. The three-time-level scheme is shown to be preferred over a two-
time-level scheme, especially for problems with strong nonlinearities. The stability of the semi-implicit scheme is free from any
time-step limitation related to the terms describing vertical diffusion and the propagation of the surface gravity waves. The scheme
does not rely on any form of vertical/horizontal mode-splitting to treat the vertical diffusion implicitly. At each time step, the
numerical method uses a double-sweep method to transform a large number of small tridiagonal equation systems and then uses
the preconditioned conjugate-gradient method to solve a single, large, five-diagonal equation system for the water surface eleva-
tion. The governing equations for the multi-level scheme are prepared in a conservative form by integrating them over the height
of each horizontal layer. The layer-integrated volumetric transports replace velocities as the dependent variables so that the depth-
integrated continuity equation that is used in the solution for the water surface elevation is linear. Volumetric transports are com-
puted explicitly from the momentum equations. The resulting method is mass conservative, efficient, and numerically accurate.
1. Introduction
1.1 Background
Estuaries are a valuable natural resource. Each benefits the economy of the surrounding region, is important as habitat for
living creatures, and is cherished for scenic and recreational qualities. The exhaustible nature of estuarine resources requires that
they be afforded a high level of environmental protection. As centers of population growth, commerce, and industrial development,
estuaries frequently are subjected to change related to the human activities that take place within and around their shorelines. Activ-
ities such as altering freshwater inflow, dredging channels, reclaiming tidal wetlands, discharging municipal and industrial wastes,
and accidentally spilling toxic chemicals can degrade the quality of estuarine waters and adversely affect the biological populations
that are part of the estuarine ecosystem. An evaluation of the environmental effects of an estuarine change usually requires an
understanding of the physical processes of water circulation and mixing. The distribution and transport of estuarine salts, sedi-
ments, contaminants, and certain biological organisms (for example, plankton and larval fish) are governed by circulation and mix-
ing. Water replacement rates (flushing times) in estuaries are determined by the cumulative effects of long-term circulation patterns
and mixing processes.