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

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168 A Semi-Implicit, Three-Dimensional Model for Estuarine Circulation The original vertical velocity w is related to ω by Expanding the operators D / Dt in the above expression leads to An expression for ∂w / ∂σ can be shown to be w Dσ ω = ------- . (C.2) Dt Dz ------ ----- D ( Hσ + ζ) Hω σ Dt Dt DH -------- Dt Dζ = = = + + ------ . (C.3) Dt w Hωu σ H ∂ ⎛ ------ ∂ζ + ------ ⎞ v σ ⎝ ∂x ∂x ⎠ H ∂ ⎛ ------ ∂ζ + ------ ⎞ σ ⎝ ∂y ∂y ⎠ H ∂ ------ ∂ζ = + + + ⎛ + ------ ⎞ . (C.4) ⎝ ∂t ∂t ⎠ ∂w ------ H ∂σ ω ∂ ------ ∂ζ ----- u ∂σ ∂t H ∂ ------ v ∂x H ∂ ⎛ + + -------- ⎞ ∂ σ ------ H ⎝ ∂y ⎠ ∂x u ∂ ----- ∂ ------ H ∂σ ∂y v ∂ ⎛ + ------ ⎞ ∂ ----ζ ⎝ ∂σ⎠ ∂x u ∂ ----- ∂ ----ζ ∂σ ∂y v ∂ = + + + ⎛ + ------ ⎞ . (C.5) ⎝ ∂σ⎠ By applying the familiar chain rule of calculus, the first-order derivatives in z coordinates are expressed in terms of σ coordinates by the following operators: z A y x z = 0 z = – h ( x, y ) ∂ ∂ ---- ---- ∂x ∂xˆ σ --- H H ∂ ------ ∂xˆ ∂ 1 ----- --- ∂σ H ζ ∂ ----- ∂xˆ ∂ = – – ----- , (C.6) ∂σ ∂ ∂ ---- ---- ∂y ∂yˆ σ --- H H ∂ ------ ∂yˆ ∂ 1 ----- --- ∂σ H ζ ∂ ----- ∂yˆ ∂ = – – ----- , (C.7) ∂σ ∂ 1 ---- --- ∂z H ∂ = ----- , and (C.8) ∂σ ∂ ∂ --- --- ∂t ∂tˆ σ --- H H ∂ ------ ∂tˆ ∂ 1 – ----- --- ∂σ H ζ ∂ ----- ∂tˆ ∂ = – ----- . (C.9) ∂σ σ B yˆ xˆ σ = 0 σ = – 1 Figure C.2. Schematic of a σ-grid shown in (A) z-coordinate space and (B) σ -coordinate space.
Appendix C - The σ Transformation 169 Differential operators for the second-order, spatial derivatives are also needed in transforming the diffusion terms. In complete form these are (Sheng, 1983, p. 224) ---- ∂ ∂x ∂ ⎛---- ⎞ --- 1 ⎝∂x⎠ H ∂ ---- H ∂xˆ ∂ ⎛ ---- ⎞ ----- ∂ σ ⎝ ∂xˆ ⎠ ∂σ H ∂ ------ ∂xˆ ∂ – ⎛ ---- ⎞ σ ⎝ ∂xˆ ⎠ ∂ ---- ∂xˆ H ∂ ------ ∂xˆ ∂ = – ⎛ ----- ⎞ ⎝ ∂σ⎠ 1 --- H H ∂ ⎛-------- ⎞ ⎝ ∂xˆ ⎠ 2 ∂ ----- ⎛σ2∂ ----- ⎞ ∂ ∂ζ ---- ----- ∂σ⎝ ∂σ⎠ ∂xˆ ∂xˆ ∂ ⎛ ----- ⎞ ∂ζ – ----- ⎝ ∂σ⎠ ∂xˆ ∂2 + – ----------- ∂σ∂xˆ ⎛∂ζ ------ ⎞ ⎝∂xˆ ⎠ 2 1 --- H ∂2 σ2 2 -------- --- ∂ H ζ ∂ ----- ∂xˆ H ∂ ------ ∂xˆ ∂ ----- σ ∂σ ∂ + + ⎛ ----- ⎞ , ⎝ ∂σ⎠ ---- ∂ ∂y ∂ ⎛---- ⎞ --- 1 ⎝∂y⎠ H ∂ ---- H ∂yˆ ∂ ⎛ ---- ⎞ ----- ∂ σ ⎝ ∂yˆ ⎠ ∂σ H ∂ ------ ∂yˆ ∂ – ⎛ ---- ⎞ σ ⎝ ∂yˆ ⎠ ∂ ---- ∂yˆ H ∂ ------ ∂yˆ ∂ = – ⎛ ----- ⎞ ⎝ ∂σ⎠ 1 --- H H ∂ ⎛-------- ⎞ ⎝ ∂yˆ ⎠ 2 ----- ∂ ⎛σ2----- ∂ ⎞ ∂ ∂ζ ---- ----- ∂σ⎝ ∂σ⎠ ∂yˆ ∂yˆ ∂ ⎛ ----- ⎞ ∂ – ----ζ ⎝ ∂σ⎠ ∂yˆ ∂2 + – ----------- ∂σ∂yˆ ⎛∂ζ ------ ⎞ ⎝∂yˆ⎠ 2 1 --- H ∂2 σ2 2 -------- --- ∂ H ζ ∂ ----- ∂yˆ H ∂ ------ ∂yˆ ∂ ----- σ ∂σ ∂ + + ⎛ ----- ⎞ , and ⎝ ∂σ⎠ ---- ∂ ∂z ∂ ⎛---- ⎞ --- 1 ⎝∂z⎠ H ∂ ----- ∂σ 1 --- H ∂ = ⎛ ----- ⎞ . ⎝ ∂σ⎠ In deriving operators C.6 to C.12, use was made of the identities ∂σ σ----- --- ∂x H H ∂ 1 ------ --- ∂x H ζ ∂ = – ⎛ + ------ ⎞ , ⎝ ∂x ⎠ ∂2σ x2 σ-------- --- ∂ H ∂2H x2 2σ --------- ∂ H2 ------ H ∂ ⎛-------- ⎞ ⎝ ∂x ⎠ 2 2 H2 ------ ζ ∂ ----- ∂x H ∂ 1 ------ --- ∂x H ∂2ζ x2 = – + + – -------- , and ∂ ⎛∂σ ------- ⎞ ⎝ ∂x ⎠ 2 σ2 H2 ------ H ∂ ⎛------- ⎞ ⎝ ∂x ⎠ 2 2σ H2 ------ ζ ∂ ----- ∂x H ∂ 1 ------ ∂x H2 ∂ζ ------ ⎛------ ⎞ ⎝∂x ⎠ 2 = + + , and similar identities for the gradient of σ with respect to y. Using these operators, the 3-D equations in z coordinates presented in Chapter 2 (equations 2.38, 2.49, 2.50, and 2.42) are transformed into the σ-coordinate system: Continuity equation, (C.10) (C.11) (C.12) ∂ζ ∂Hû ∂Hvˆ ----- --------- --------- H ∂t ∂xˆ ∂yˆ ω ∂ + + + ------ = 0 ; (C.13) ∂σ
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In cooperation with the Interagency
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U.S. Department of the Interior Dir
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vi 4.2 Semi-Implicit One-Dimensiona
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viii Figure 5.6. Graph showing solu
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x Tables Table 2.1. Dimensionless n
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Water surface Water surface Level p
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h 1 h 2 h 3 h 4 h 5 h 6 z 1 ⁄ 2 z
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A Semi-Implicit, Three-Dimensional Model for Estuarine ... - USGS

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