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

3. Layer Averaging the Governing Equations 67
which is very similar in form to the momentum acceleration terms expressed by 3.28 and 3.29. Equation 3.53 is valid for all layers
subject to the boundary conditions of ( ws)
1⁄ = ( ws)
2 km + 1⁄ = 0 , which cause the kinematic conditions at the free surface and the
2
bottom of the water column to be satisfied; the kinematic conditions prevent the advective flux of salt through the boundaries.
Integration of the turbulent diffusion terms in equation 3.5 results in
1 ∂(
hJx) ∂(
hJ-----
k y)
⎛ k
----------------- + ----------------- + ( J
ρ0 ∂x
∂y
z)
– ( J
k – 1⁄ z)
⎞ , (3.54)
⎝ 2 k + 1⁄ 2⎠
which is subject to the boundary conditions of no turbulent salt flux at the water column free surface and bottom; this condition is
enforced by setting27 Jz = J
⁄ z = 0 . Substituting the concept of a horizontal eddy diffusivity in the form
+ ⁄
into 3.54 results in
( ) 1 2
( ) km 1 2
( Jx) k
ρ ⎛ ∂s
0 D ---- ⎞
∂s
=
⎝ H ( J
∂x⎠
y)
= ρ ⎛
k 0 D ---- ⎞
H
(3.55)
k
⎝ ∂y⎠k
∂
---- D
∂x
Hh s ∂ ⎛ ---- ⎞ ∂
---- D
⎝ ∂x⎠k∂y
Hh s ∂ ( J
⎛ ---- ⎞ z)
k 1 ( J
– ⁄ 2 z)
k + 1
2
+ + ------------------- – ------------------- ⁄
(3.56)
⎝ ∂y⎠kρ0
ρ0 which is the final form of the layer-averaged turbulent salt flux terms. The form of 3.56 is close to that of the layer-averaged
turbulent stress terms in the momentum equations expressed by 3.51 and 3.52.
The two integral terms in equation 3.53 are salt dispersion terms representing advective fluxes of salt in the x- and y-directions
caused by vertical non-uniformities of the velocity and salinity over a layer. If the 3-D velocity and salt concentration do not vary
with z over a layer, these terms will vanish. In general, if layer heights are small enough to avoid a large change in velocity and
salinity across the layers, these terms will be small. Here these terms will be dropped. If for a certain application these terms are
significant, the magnitude of D H can be adjusted.
27 The conditions of no turbulent flux of salt at the water column free surface and bottom are formally represented by
and
∂z
1⁄ ∂z
1
2
0 ( Jx) ---------
⁄ 2
= – – ( J 1⁄ 2 ∂x
y)
--------- + J 1⁄ 2 ∂y
z
( ) 1⁄ 2
∂zkm
+ 1⁄ ∂z
2
km + 1⁄ 2
0 = – ( Jx) ------------------- – ( J
km + 1⁄ 2 ∂x
y)
------------------- + ( J
km + 1⁄ 2 ∂y
z)
.
km + 1⁄ 2
Because the terms involving the boundary slopes are already eliminated in 3.54, the conditions are satisfied simply by requiring that
( Jz) = ( J 1⁄ z)
= 0 in 3.56.
2 km + 1⁄ 2