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

58 A Semi-Implicit, Three-Dimensional Model for Estuarine Circulation
Leibnitz’ rule relates the derivative of a 3-D variable to its layer average. For a generic variable F(x, y, t), the Leibnitz rule,
applied to an integral over a layer, produces
or, after rearranging and inserting F k,
DEPTH
y
-z
z
k – 1⁄ 2
∫z
k + 1⁄ 2
∂
∂F
∂zk
– 1⁄ ∂z
2
k + 1⁄ 2
---- F dz
= ----- dz
+
∂x
∂x
k – 1 ---------------
⁄ – F 1 ---------------
2 ∂x
k + ⁄
(3.8)
2 ∂x
z
k – 1⁄ 2
∫∂
F
----- dz
∂x
z
k + 1⁄ 2
u 6
( z)
“
z
k – 1⁄ 2
∫ F
z
k + 1⁄ 2
∂
∂zk
– 1⁄ ∂z
1
2
k + ⁄ 2
= ---- ( h
∂x
kFk) – Fk – 1 ---------------
⁄ + F 1 ---------------
2 ∂x
k + ⁄ . (3.9)
2 ∂x
The integral over a layer for a product of two variables, F and G, is related to the product of the two layer-averaged variables as
follows:
= < ( Fk + Fk ″ ) ( Gk + Gk ″ )>
=
+ + +
= hkFk Gk + .
(3.10)
u ( z)
=
u ( z )
u 6
u k + uk z
u VELOCITY
k = 1
k = 2
k = 3
k = 4
k = 5
k = 6
Figure 3.2. Diagram of a 3-dimensional horizontal velocity profile u(z) approximated by layer-averaged
values u k (eq. 3.6). The deviation of u k from u is u k ″ k ″ .
“
( )
u 5
u 4
u 3
u 2
u 1
x