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

2. Governing Equations and Boundary Conditions
2.1 Introduction
2. Governing Equations and Boundary Conditions 17
This chapter develops the governing equations and boundary conditions used in the 3-D circulation model. The assumptions
and approximations inherent in the development of the equations are highlighted to ensure that the model is applied properly. In
section 2.2, the basic 3-D equations for the instantaneous flow that describe exactly the fluctuating turbulent variables are pre-
sented. Because the instantaneous-flow equations have long been recognized as intractable for applications to rivers and estuaries,
the time-averaged 3-D equations for the mean-flow (turbulent-averaged) variables are introduced in section 2.3. In this section, the
closure problem resulting from the equation-averaging process is discussed, and the concepts of an eddy viscosity and eddy diffu-
sivity are introduced. In section 2.4, the boundary conditions for an estuarine flow are specified. In section 2.5, a scaling analysis
is used to estimate the relative magnitude of the various terms in the differential equations for an estuary, using San Francisco Bay
as an example. Finally, section 2.6 discusses the horizontal and vertical coordinate transformations that are used in some existing
3-D models. These coordinate transformations are offered only as background because they are not used in the present model.
2.2 Instantaneous-Flow Equations
In estuaries, the flows are almost always turbulent. The exact equations describing the instantaneous turbulent motion origi-
nate from the conservation laws for mass, momentum, and salt. These equations are introduced in the following three sections.
The reference frame is a right-handed, Cartesian coordinate system with axes x i (i = 1, 2, 3) oriented such that the positive
x 1-axis is directed horizontally to the east, the positive x 2-axis is directed horizontally to the north, and the positive x 3-axis is
directed vertically upward along the line of action of gravity (fig. 2.1). In an estuary, the boundary is continuously in motion
because of the earth’s rotation. It is convenient to choose a reference frame at rest relative to the boundaries of the estuary rather
than one that is fixed in space (relative to the fixed stars). The reference frame is therefore chosen to be fixed on the earth’s surface
and rotating with the earth’s axis at an angular velocity Ω equal to 1 revolution per sidereal 8 day (7.29 x 10 − 5 radian/second).
The horizontal scale of many estuaries is sufficiently large that the earth’s rotation will introduce a significant term in the equa-
tion of motion. The area is not so large, however, that the curvature of the earth’s surface must be considered. The plane x 3 = 0 is
therefore considered to be a tangent plane to the geoid and coincides with mean sea level.
8 “One sidereal day = 23 hours 56 minutes 4 seconds = 86,164 seconds is the time required for the earth to rotate once about its axis, relative to the fixed
stars. Since the earth revolves about the sun it must turn a little further to point back to the sun and complete one solar day—hence the solar day is a little longer
than the sidereal day” (Pond and Pickard, 1986, p. 38).