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

60 A Semi-Implicit, Three-Dimensional Model for Estuarine Circulation
1 where the subscripts ⁄ 2 and km 1⁄ 2
∂zkm
+ 1⁄ ∂z
2
km + 1⁄ 2
ukm + 1 -------------------
⁄ v
2
km + 1 -------------------
⁄ – w
2
km + 1⁄ 2
∂x
+ = 0 , (3.18)
∂y
+ refer to the free surface and the bottom, respectively. Applying the boundary conditions to
equation 3.15 results in the following forms of the continuity equations for surface, middle, and bottom layers:
Surface layer,
Middle layers,
Bottom layer,
wk – 1⁄ – w
2 k + 1⁄ 2
∂ζ
–
∂t
----- w3⁄ 2
∂U1
∂V1
+ --------- + -------- = 0 ; (3.19)
∂x
∂y
∂Uk
∂Vk
+ -------- + -------- = 0 k = 2, 3 , ... , km – 1 ; (3.20)
∂x
∂y
wkm – 1⁄ 2
∂Ukm
∂Vkm
+ ------------ + ------------ = 0 . (3.21)
∂x
∂y
In equation 3.19, the substitution ζ = z1 ⁄ 2 has been made to use conventional notation. By combining equations 3.20 and 3.21,
it is possible to write an expression for the vertical velocity component at an interface k – 1⁄ 2 as
wk – 1⁄ 2
km
∂Uk
∂V
= – ⎧ k ⎫
⎨-------- + -------- ⎬
. (3.22)
∑
⎩ ∂x
∂y
⎭
k = 2
This expression can be evaluated for w3⁄ and substituted into equation 3.19 to yield this expression for the water surface elevation:
2
km
∂ζ
∂U
-----
k ∂V
+ ⎧ k ⎫
∂t
⎨-------- + -------- ⎬
= 0 . (3.23)
∑
⎩ ∂x
∂y
⎭
k = 1
A variation of equation 3.23 is obtained by integrating the continuity equation over the entire depth of flow. This is written
⎛ km ⎞ ⎛ km ⎞
∂ζ
∂
----- ---- ⎜ U ⎟ ∂
+
∂t
∂x⎜∑k+
---- ⎜ V ⎟
⎟ ∂y
∑ k = 0 , (3.24)
⎜ ⎟
⎝ ⎠ ⎝ ⎠
k = 1
k = 1
km
km
where the quantities ∑ Uk and ∑ Vk represent the vertically-integrated components of the volume-transport velocity.
k = 1 k = 1
Equation 3.24 is the actual form of the continuity equation that is used in the model in determining the water surface elevation.