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174 A Semi-Implicit, Three-Dimensional Model for Estuarine Circulation n ( HDIFFs) ---- ∂ D i, j, k ∂x Hh s ∂ ⎛ ---- ⎞ ---- ∂ D ⎝ ∂x⎠k∂y Hh s ∂ n – 1 = ⎛ + ⎛ ---- ⎞ ⎞ = ⎝ ⎝ ∂y⎠k⎠ i, j 1 4( Δx) 2 n – 1 n – 1 n – 1 n – 1 + ---------------- ( ( hi + 1, jk , + hi, j, k) ⋅ ( DH + D i + 1, jk , H ) i, j, k n – 1 n – 1 × ( si + 1, j, k– sijk , , ) n – 1 n – 1 n – 1 n – 1 – ( hi, j, k+ hi – 1, j, k) ⋅ ( DH + D i, j, k H ) i – 1, j, k n – 1 n – 1 × ( sijk , , – si – 1, j, k) ) 1 4( Δy) 2 n – 1 n – 1 n – 1 n – 1 + ---------------- ( ( hi, j+ 1, k + hi, j, k) ⋅ ( DH + D i, j+ 1, k H ) i, j, k n – 1 n – 1 × ( sij , + 1, k – si, j, k) n – 1 n – 1 n – 1 n – 1 – ( hi, j, k+ hij , – 1, k) ⋅ ( DH + D ijk , , H ) i, j– 1, k n – 1 n – 1 × ( si, j, k– si, j– 1, k) ) . (D.10)
Appendix E - Computation of the Vertical Eddy Coefficients Appendix E - Computation of the Vertical Eddy Coefficients 175 This appendix defines the discretized equations used in the model to compute the vertical distribution of the eddy viscosity (A V) for diffusion of momentum and the eddy diffusivity (D V) for diffusion of salt mass. The eddy viscosity under neutral (unstratified) conditions is computed using the Prandtl mixing length formula (eq. 2.31) in the discretized form ( AV0) n Λ i, j, k – 1⁄ 2 2 ⎛un ( ) n i + 1⁄ 2 , j, k– 1 – un i + 1⁄ 2 , jk , + un i – 1⁄ 2 , j, k– 1 – un i – 1⁄ 2 , j, k⎞ ijk , , – 1⁄ ⎜------------------------------------------------------------------------------------------------------------------------ ⎟ 2 hn i, j, k– 1 + hn ⎝ i, j, k ⎠ 2 ⎛ = ⋅ ⎜ ⎝ ⎛vn i, j + 1⁄ 2 , k – 1 – vn ij , + 1⁄ 2 , k + vn i, j– 1⁄ 2 , k – 1 – vn i, j– 1⁄ 2 , k⎞ ⎜----------------------------------------------------------------------------------------------------------------------- ⎟ hn ijk , , – 1 + hn ⎝ i, j, k ⎠ 2 ⎞ ⎟ ⎠ 1⁄ 2 + . (E.1) The mixing length Λ is estimated at the layer interfaces for location i, j using equation 2.32. Equation E.1 is evaluated for A V0 at each layer interface except at the free surface and bottom where the wind and frictional stresses are applied. 56 To introduce the effect of stratification on the eddy viscosity, a stability function (eq. 2.27) is used: n AV ijk , , – 1⁄ 2 ( AV0 ) n n ( 1 + β i, j, k– 1⁄ 1Rii, j, k– 1⁄ ) 2 2 α1 = . (E.2) The coefficients β 1 = 10.0 and α 1 = –0.5 were used by Munk and Anderson (1948); these coefficients are variables in the model and can be adjusted for best fit of the available data. The Richardson number R i is discretized from equation 2.28 as Rii, j, k– 1⁄ 2 – g ( ρn ijk , , – 1 – ρn i, j, k) = --------------------------------------------------- ⋅ -------------------------------------------------- ( ρn i, j, k– 1 + ρn i, j, k) ⁄ 2 ( hn i, j, k– 1 + hn ijk , , ) ⁄ 2 ⎛un i + 1⁄ 2 , jk , – 1 – un i + 1⁄ 2 , j, k+ un i – 1⁄ 2 , j, k– 1 – un i – 1⁄ 2 , jk , ⎞ ⎜------------------------------------------------------------------------------------------------------------------------ ⎟ hn i, j, k – 1 + hn ⎝ ijk , , ⎠ 2 ⎛ × ⎜ ⎝ ⎛vn i, j+ 1⁄ 2 , k – 1 – vn i, j+ 1⁄ 2 , k + vn i, j– 1⁄ 2 , k – 1 – vn i, j– 1⁄ 2 , k⎞ ⎜----------------------------------------------------------------------------------------------------------------------- ⎟ hn i, j, k– 1 + hn ⎝ i, j, k ⎠ 2 ⎞ ⎟ ⎠ 1 – + . (E.3) 56 For conditions where a wind stress generates significant wave action on the free surface, the magnitude of the eddy viscosity may be larger than estimated by equation E.1 because of the turbulence associated with oscillatory water motions. In those cases a wind-wave component, as mentioned in section 2.3.1.1, should be included in the model.
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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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2 A Semi-Implicit, Three-Dimensiona
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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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