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

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44 A Semi-Implicit, Three-Dimensional Model for Estuarine Circulation Table 2.1. Dimensionless numbers in 3-D equations. Rossby number Froude number Internal Froude number Name Symbol Definition Ratio between Estimate Ro Fr Fr i Unsteadiness number β Depth ratio — Horizontal Ekman number Vertical Ekman number Horizontal Péclet number Vertical Péclet number E H E V Pe H Pe V ur fL ---- ur ( gH) 1 --------------- ⁄ 2 ur gH ρ2 ρ – ⎛ 1 ----------------- ⎞ ⎝ ρ ⎠ 0 1 ---------------------------------- ⁄ 2 ω --- f D --- H AH fL2 ------- AV fH2 -------- urL ------- DH urH -------- DV fluid acceleration ------------------------------------------------- 2.5 10 Coriolis acceleration 2 – × fluid velocity --------------------------------------------- 2.5 10 surface wave speed 2 – × ---------------------------------------------fluid velocity 0.5 internal wave speed inertial period --------------------------------- 0.5 tidal period tidal amplitude ----------------------------------- 10 depth 1 – horizontal--------------------------------------------------------------------------momentum diffusion 10 Coriolis acceleration 4 – vertical momentum diffusion --------------------------------------------------------------------- 1 Coriolis acceleration mass advection ------------------------------------------------------------ 2.5 10 horizontal mass diffusion 2 × ----------------------------------------------------mass advection 2.5 10 vertical mass diffusion 2 ×
2. Governing Equations and Boundary Conditions 45 Because the northern reach of San Francisco Bay is influenced by freshwater inflow from the Sacramento−San Joaquin River Delta, significant density gradients can occur there. The forcing period (1/ω) was chosen to be approximately six hours, or one-quarter of the diurnal tidal cycle in San Francisco Bay, and corresponds with roughly the full range of local velocity change (from flood to ebb tide). For the above set of parameters, the corresponding magnitudes of the dimensionless numbers are shown in table 2.1. In the dimensionless equations, the chosen parameters scale each partial derivative term to approximately equal size and of order unity. The coefficients for each term then provide an order-of-magnitude estimate of the relative importance of the terms and groups of terms with respect to others. The relative magnitude of the terms in the momentum balance equation are shown in table 2.2 and in the salt balance equation in table 2.3. Owing to the strong tidal forcing in San Francisco Bay, the water surface slope terms are the largest in the momentum equation (table 2.2). The baroclinic, Coriolis, vertical momentum diffusion, and temporal acceleration terms also are very significant. The advective acceleration and horizontal momentum diffusion terms are the smallest terms in the estuarine momentum equation and generally are not of major significance. It is not recommended, however, that these terms be neglected. The need for some amount of horizontal momentum diffusion to reproduce horizontal circulation in side bays was discussed in section 2.3.1.1. The scaling analysis here also does not consider changes in the estuarine bathymetry or lateral geometry. Bathymetric and (or) geometric changes can induce significant horizontal velocity shear that may cause the advective acceleration and horizontal momentum diffusion terms to become locally significant. In San Francisco Bay, regions of high velocity gradients occur near the sharp bathymetric gradients at the edges of the deepwater navigation channels, near flow constrictions, and behind islands and headlands where eddies form. Table 2.2. Relative magnitude of the terms in the momentum equation. Term Coefficient Relative magnitude Water surface slope 4 Ro D Fr2 ------------ H Baroclinic 1 --------- Ro Fr 2 i Coriolis acceleration 1.0 1 Vertical momentum diffusion E V 1 Temporal acceleration β 0.5 Advective acceleration R o 0.025 Horizontal momentum diffusion E H 0.0001 Table 2.3. Relative magnitude of the terms in the salt equation. Term Coefficient Relative magnitude Advective transport 1.0 1 Vertical salt diffusion 1 ⁄ PeV 0.004 Horizontal salt diffusion 1 ⁄ PeH 0.004
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