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

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20 A Semi-Implicit, Three-Dimensional Model for Estuarine Circulation It is common to simplify the two terms on the left side of equation 2.5 by first expanding, using the product rule for differentiation, into the form ρ˜ ∂ũ i ∂ρ˜ ∂ρ˜ ( ũ j ) ∂ũi ------- + ũ ----- ∂t i + ũ --------------- ∂t i + ρ˜ u ------- ∂x j . (2.6) j ∂x j The second and third terms of this expression sum to zero, because they represent the continuity equation (eq. 2.1) multiplied by the velocity. Using this simplification, the equation of motion becomes ρ˜ ∂ũi ∂ũi ------- ρ˜ ũ ------- ∂p˜ + ∂t j – ------ – 2ε Ω ∂x j ∂x ijk j ρ˜ ũk ρ˜ gi ρ˜ ν i ∂2ũi = + + -------------- . (2.7) ∂x j ∂x j On the left side of equation 2.7 are the inertia or acceleration terms that represent the rate of change of linear momentum of a unit volume of the fluid. The first term is a temporal acceleration, and the second is the advective acceleration. The right side of equation 2.7 lists the forces causing the acceleration. The first term is the pressure force term, which in an estuary can result from horizontal gradients in the water surface, density, and atmospheric pressure. The second term is the Coriolis body-force term, which is derived in detail in Appendix A. The Coriolis force is actually a pseudoforce that arises solely as a result of the rotating reference frame. (In an estuary in the northern hemisphere, the Coriolis force acts perpendicular to, and to the right of, the velocity vector; it is proportional to the magnitude of the velocity.) The third term on the right side of equation 2.7 is the gravity body-force term, which in the chosen reference system is defined by ρ˜ g i = ( 00 , , – ρ˜ g) . The fourth term is the viscous-shear term and represents internal forces in the fluid resulting from the fluid viscosity. A widely used approximation to equation 2.7, first introduced by Boussinesq (1903), is known as the Boussinesq approximation. It consists essentially of neglecting variations in density insofar as they affect the mass of the fluid but retaining them where they affect the weight. To clarify the Boussinesq approximation, the relations ρ˜ = ρ 0 + ρ˜ ′ and p˜ = p 0 + p˜ ′ can be substituted into equation 2.7, where ρ0 and p0 are the reference values of density and pressure that satisfy hydrostatic equilibrium such that ∂p 0 ⁄ ∂xi = – ρ 0 gi , and ρ ˜ ′ and p˜ ′ are small deviations from the reference values caused by the fluid motion and stratification in combination. The result, after cancelling terms and dividing by ρ 0 , is (similar to Turner [1973], p. 9) ⎛ ρ˜ ′ 1 + ----- ⎞ ⎝ ρ ⎠ 0 ∂ũ ⎛ i ∂ũi⎞ 1 ⎜------- + ũ ------- ⎝ ∂t j ⎟ ----- ∂x j⎠ ρ 0 ∂p˜ ′ ρ – ------- 2ε Ω ∂x ijk j 1 i ˜ ⎛ ′ + ----- ⎞ ρ˜ ′ – ũk -----g ρ ⎝ ρ ⎠ ρ i 1 0 0 ˜ ′ ∂ ⎛ + ----- ⎞ν ⎝ ρ ⎠ 0 2ũi = + + -------------- . (2.8) ∂x j ∂x j Here the density ratio ρ ˜ ′ ⁄ ρ 0 appears as only a small correction to the inertia, Coriolis, and viscous-shear terms, but it is of great importance to the gravity term where it appears alone. The Boussinesq approximation neglects the effect of density variations in each term except for the gravity term. This simplification allows an average value for the density of the flow to be used in the horizontal momentum equations (x 1- and x 2-directions) where the gravity term is zero; the actual density must be considered only in the x 3-equation. Equation 2.7 with the Boussinesq approximation becomes ∂ũi ∂ũi ρ ------- 0 ρ ∂t 0 ũ ------- ∂p˜ + j – ------ – 2ε Ω ∂x j ∂x ijk j ρ 0 ũ k ρ i ˜ g i ρ 0 ν ∂2ũi = + + -------------- . (2.9) ∂x j ∂x j
2. Governing Equations and Boundary Conditions 21 In a stratified estuarine flow, the lighter (lower density) water near the surface will actually tend to accelerate faster under an applied pressure gradient than the heavier (higher density) water near the bottom. This influence of the vertical density variation on the fluid inertia is neglected in the Boussinesq approximation. For the normal range of stratification in estuaries this is equivalent to less than a 2 percent correction. It therefore appears reasonable to neglect this factor in calculating the inertial response of the flow. A recent evaluation of the Boussinesq approximation in ocean models is given by Mellor and Ezer (1995). 2.2.3 Dissolved Species Conservation The principle of conservation of salt in an estuary requires the time rate of change in salt mass within a control volume to be balanced by the net flux of salt from the control volume. For a control volume at rest relative to the estuary, the salt flux is com- posed of both advection (transport by the motion of the fluid) and molecular diffusion (transport by molecular motions). The dif- ferential equation expressing the conservation principle for the instantaneous salt concentration s˜ is known as the advection-diffusion equation and may be written as (Fischer and others, 1979) ∂s˜ ---- ∂t ∂( ũis˜ ) + -------------- λ ∂xi ∂2s˜ = -------------- . (2.10) ∂xi∂x i The coefficient λ is called the molecular diffusivity (assumed constant) and has units of (length) 2 /time. The molecular-diffusion term involving λ is based on Fick’s law of diffusion (Fischer and others, 1979, p. 30−34), which relates the mass flux to the concentration gradient of the solute. Molecular diffusion is a result of the random scattering of solute particles by the constant motion and collision of molecules in the fluid; in equation 2.10, the rate of molecular diffusion in a moving fluid is assumed to equal the rate for a fluid at rest. In turbulent estuarine flows, the contribution of molecular diffusion to the salt flux is almost always insignificant, and the advection by the instantaneous velocity is the dominant mechanism. In section 2.3.1, the concept of turbulent diffusion is introduced; it is many times more efficient in promoting the mixing of salt than is molecular diffusion. 2.2.4 Equation of State To close the set of governing equations 2.4, 2.9, and 2.10, an equation of state in the form ρ˜ = f( s˜ , Θ˜ ) (2.11) is needed to relate the density to salinity and temperature Θ ˜ ( )The density represented by ρ˜ is normally chosen to be the potential density (Pond and Pickard, 1986, p. 8), which is computed as a function of salinity and potential temperature at atmospheric pressure rather than actual temperature. Potential density is used to eliminate any effect of the compressibility of water when comparing water densities at significantly different depths. 9 The equation of state used in the model is given in Appendix B. 9 The distinction of using potential density instead of actual density is not important in estuaries unless the water depths exceed about 100 m. For a change in pressure of 100 m of water, the density of sea water (s ~ 35) changes by approximately 0.5 kg/m 3 , which is roughly equivalent to the effect caused by a change in salinity at constant pressure of 0.6. Because a vertical density distribution caused by the compressibility of water does not affect the water-column stability, the potential density is the correct indicator of stability.
Page 1: In cooperation with the Interagency
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