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

3. Layer Averaging the Governing Equations
3.1 Introduction
3. Layer Averaging the Governing Equations 55
The present model is based on a z-coordinate system, as discussed in section 2.6.2. The grid system that is used in solving the
model equations is composed of layers, as is illustrated in figure 3.1. Because the interfaces between the layers are level planes,
this kind of model is sometimes called a multilevel model (Liu and Leendertse, 1978) to distinguish it from a multilayer model
which is more commonly used to identify a density-coordinate model. Because both multilevel and multilayer models are based
on the concept of layers, Cheng and others (1976) referred to both types of models as multilayer models but labeled them as Type I
and Type II models, respectively.
The hydrodynamic variables usually change considerably over the depth of an estuarine flow. In order to base the model computations
on mass and momentum fluxes in the different layers, it is necessary to prepare the governing equations for the finitedifference
method by integrating them over the height of each layer. By choosing the layer-integrated, volumetric transports as
dependent variables, this approach will insure that the model equations are in a conservative form. One advantage of this form is
that the depth-integrated continuity equation is linear. If primitive variables were used, terms involving products of the layer thicknesses
and the horizontal velocity components would appear in the discrete form of the depth-integrated continuity equation; when
the time-varying surface layer thickness and the surface velocity components are out of phase (as is typical of a standing wave),
the calculations for these terms are susceptible to phase errors and amplitude errors. Another benefit of integrating the governing
equations over layers is that the equations for a single-layer system reduce to a conservative form of the 2-D vertically averaged
equations; thus in an area of shallow water where only one layer is required, the 3-D model becomes automatically a 2-D model
without any modifications to the model coding.
The procedure for layer averaging the 3-D governing equations of Chapter 2 is defined in some detail in Chapter 3. Very few
approximations are made; the set of model equations for a layered system that is the result is fully three-dimensional. The form of
the layer-averaged equations is similar to that first presented by Leendertse and others (1973).
The starting point is the 3-D equations in approximately the form of equations 2.38 to 2.42 with the turbulent stresses represented
by τxx = – ρ0u′u′ , τxy = – ρ0u′v′ , τxz = – ρ0u′w′ , τyx = – ρ0v′u′ , τyy = – ρ0v′v′ , and τyz = – ρ0v′w′ ,
and the turbulent salt fluxes represented by Jx = – ρ0u′s′ , Jy = – ρ0v′s′ , and Jz = – ρ0w′s′ . The introduction of the eddy viscosity
and eddy diffusivity will come in a later step. The 3-D equations in the appropriate form are