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

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14 A Semi-Implicit, Three-Dimensional Model for Estuarine Circulation which it is needed. Lynch and Werner (1987) and Walters (1992) chose an efficient spectral (harmonic) approach in the time domain for their 3-D finite-element models. These models separate the horizontal and vertical components of the solution much like mode-splitting models except that no time-stepping is employed. The model by Lynch and Werner (1987) is linear and allows either analytical or numerical (finite- element) solutions in the vertical dimension. The model by Walters (1992) includes nonlinear terms. Lynch and Werner (1991) use mode-splitting in a nonlinear, time-stepping, 3-D finite-element model in which the external mode is solved by way of a semi-implicit method with a wave-equation rearrangement of the governing equations. The wave-equation rearrangement has been prevalent in 2-D modeling (for example, Lynch and Gray, 1979; Kinnmark, 1986) and suppresses short wavelength solution oscillations that have long been the bane of finite-element modelers. In the external mode of the model by Lynch and Werner (1991), the friction and nonlinear part of the water surface slope term must be treated explicitly. A new approach for 3-D modeling has been demonstrated by Luettich and others (1994): velocity is replaced with shear stress as the dependent variable in the internal-mode computations of a mode-splitting, finite-element model. The rationale for changing variables is that, because shear stress normally varies more slowly near sheared boundaries, it can be resolved more readily than velocity. Luettich and others (1994) solve the external mode by using a harmonic model. Additional references on finite-element models can be found in the review papers by Westerink and Gray (1991) and Davies and others (1997). 1.5 The Present [1997] Model Numerous factors were considered in the design of the 3-D model described herein. Because the model will be used for cal- culations of the circulation in real estuaries under varying conditions, the equation formulation must be complete, incorporating a time-varying free surface, nonlinearities, and horizontal density-gradient forcing for prognostic purposes (a baroclinic model). The numerical scheme must be flexible enough to accept an advanced turbulence parameterization suitable for stratified flows later and be capable of providing accurate simulations under both periodic forcing from tides and aperiodic forcing by freshwater inflows and the wind. The shallowness of estuarine basins also requires using a quadratic (rather than linear) stress law for the bottom fric- tional-boundary condition. Owing to these very general requirements, the use of any of the various spectral approaches in time or space was not considered. Because the new 3-D finite-difference techniques (semi-implicit and ADI) obviate the need for mode- splitting to deal efficiently with gravity-wave propagation, the finite-difference approach was chosen for the numerical approximations in the new model. The objectives in the design of the numerical scheme for the model were that it be efficient for long-term (at least seasonal) simulations of estuaries and second-order accurate in the truncation errors to ensure that solutions are not overly damped or dif- fused by artificial viscosity. Implementing the numerical scheme using the conservation form of the governing equations for both the flow and salt transport also was considered desirable. Mass and momentum are more readily conserved by the conservation form of the basic equations, and the depth-integrated continuity equation is linear.
1. Introduction 15 Presently, the differences in the computational efficiencies of the new finite-difference, the ADI (Leendertse, 1989; Uitten- bogaard and others, 1992), and the semi-implicit (Casulli and Cheng, 1992) methods are difficult to assess and depend on the geometry and hydrodynamic conditions of the particular water body under study. As de Goede (1992) points out, semi-implicit methods require solving a pentadiagonal matrix system using a two-dimensional structure, whereas the ADI schemes solve only simple tridiagonal systems. The ADI schemes, however, may be limited because of geometry to a certain maximum time step for accuracy (the so-called ADI effect). The semi-implicit methods do not suffer from the ADI effect, but they may have a known time-step limitation related to the explicit treatment of other terms. The dependence on geometry and bathymetry by the ADI scheme is a disadvantage because errors due to the ADI effect can go undetected by an unaware user and cause misinterpretation of model results. Ultimately the decision was made to implement the semi-implicit approach in the new model. To simulate the hydrodynamics of a geometrically complex estuary, such as the Suisun Bay region of San Francisco Bay, the semi-implicit scheme may indeed be the most efficient approach. The semi-implicit scheme implemented herein closely follows the approach outlined by Casulli and Cheng (1992) for the implicit inclusion of vertical diffusion in a 3-D calculation without recourse to mode-splitting. The details of the overall scheme, however, differ from those in Casulli and Cheng (1992) in many significant ways. The time integration used here is a three-time- level, semi-implicit, leapfrog-trapezoidal method. Except for a small amount of first-order error introduced by the uncentered (in time) treatment of horizontal diffusion for stability considerations, the scheme is second-order accurate in the truncation errors of the finite-difference approximations in both space and time. All terms (other than horizontal diffusion), as well as nonlinear coef- ficients, are centered in time during the leapfrog and trapezoidal steps of the scheme to achieve second-order accuracy in time. The accurate evaluation of the nonlinearities in the equation of motion is important, because nonlinearities have a significant effect on the tidally averaged circulation in shallow estuaries. The governing equations for the multilevel scheme are prepared in a conser- vative form by integrating them over the height of each 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. The advection terms in the momentum and salt-transport equations are solved using explicit, leapfrog-trapezoidal integration rather than an ELM as used by Casulli and Cheng (1992). The leapfrog-trapezoidal approach does very well with the conservation of salt, which can be a problem with the ELM approach. The global matrix solution for water surface elevation in the new model is implemented using the conjugate-gradient method with a modified incomplete Cholesky preconditioner (Axelsson and Lindskog, 1986). This method was selected after testing a number of other preconditioners and acceleration methods.
Page 1: In cooperation with the Interagency
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