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

3. Layer Averaging the Governing Equations 65
The average pressure in any layer can be related to the average pressure in the layer above through the hydrostatic pressure
equation, which is approximated by
pk = pk 1 + ⁄ – ⁄ .
– gρk – 1 h
2 k 1
2
Here hk – 1⁄ is the average of the heights for layers k - 1 and k and represents the vertical distance between the centers of the two
2
layers. By differentiating this equation with respect to x and y, the pressure gradients for one layer can be determined from the layer
above. For the x-direction pressure gradient, the result is
∂pk ∂pk
– 1
= +
∂x
∂x
∂
-------------- g hk – 1⁄ ρ
2 k – 1⁄ 2
------------------------------
∂x
∂p
∂ρ
k – 1
k – 1⁄ 2
= -------------- + gh
∂x
k – 1⁄ +
2 ∂x
---------------- gρk – 1⁄ 2
∂hk
– 1⁄ 2 ----------------
∂x
∂p
gh
k – 1 k – 1
-------------- ---------------
∂x
2
ρ ∂ k – 1 ghk --------------- -------
∂x
2
ρ ∂ k gρk – 1
------- ---------------
∂x
2
h ∂ k – 1 gρk -------------- --------
∂x
2
h ∂ k
= + + + + -------
∂x
. (3.42)
In the surface layer, the x pressure gradient is approximated by
∂p1 ∂x
gρ1 --------
2
ζ ∂ gh1 ----- --------
∂x
2
ρ ∂ 1
= + -------- . (3.43)
∂x
Successively substituting equations 3.43 and 3.42 into equations 3.38 to 3.40, a general formula for the x pressure gradient term
can be seen by induction to be
1
---ρ0
z
k – 1⁄ 2
∫
∂p
hk ∂ζ
gh1 ----- dz
----- gρ ----- --------
∂x
ρ 1
0 ∂x
2
ρ ∂ 1 ghm – 1
-------- ----------------
∂x
2
ρ ∂ m – 1 ghm ---------------- ---------
∂x
2
ρ k
∂ m
= + + ⎛ + --------- ⎞
(3.44)
∑ ⎝ ∂x
⎠
m = 2
z
k + 1⁄ 2
where the summation is omitted for k = 1. The corresponding formula for the y pressure gradient term is
1
---ρ0
z
k – 1⁄ 2
∫
∂p
----- dz
∂y
hk ∂ζ
gh1 ----- gρ -----------ρ 1
0 ∂y
2
ρ ∂ 1 ghm – 1
-------- ----------------
∂y
2
ρ ∂ m – 1 ghm ---------------- ---------
∂y
2
ρ =
k
∂ m
+ + ⎛ + --------- ⎞
∑ ⎝ ∂y
⎠
m = 2
. (3.45)
z
k + 1⁄ 2
Now consider the integration of the stress terms in equations 3.2 and 3.3 over a layer. After integration and application of
equations 3.9 and 3.11 to the x-component terms, the result is
1
---ρ0
z
k – 1⁄ 2
∫
∂τxx
1 ∂τxy
1 ∂τxz
--------- dz
+ ----- --------- dz
+ ----- --------- dz
=
∂x
ρ0 ∂x
ρ0 ∂x
z
k + 1⁄ 2
z
k – 1⁄ 2
∫z
k + 1⁄ 2
z
k – 1⁄ 2
∫z
k + 1⁄ 2
1 ∂(
hτxx) ∂(
hτ-----
k xy)
∂z
1
⎛
k
k – ⁄ 2
-------------------- + -------------------- + – ( τ
ρ0 ∂x
∂y
xx)
--------------- 1 – τ
⎝
k – ⁄ 2 ∂x
xy
( ) 1 k – ⁄ 2
∂zk
+ 1⁄ 2
– – ( τxx) --------------- – τ
k + 1⁄ 2 ∂x
xy
( )
k + 1⁄ 2
∂z
1 k – ⁄ 2 --------------- + τ
∂y
xz
( ) 1 k – ⁄ 2
∂zk
+ 1⁄ 2 --------------- + τ
∂y
xz
( )
k + 1⁄ 2
⎞
⎠
. (3.46)