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

170 A Semi-Implicit, Three-Dimensional Model for Estuarine Circulation
Momentum equations,
Salt transport equation,
∂Hû
∂Hûû
∂Hûvˆ
--------- ------------- ------------ H
∂tˆ
∂xˆ
∂yˆ
ûω ∂
+ + + --------- – fHvˆ gH
∂σ
ζ ∂
= – -----
∂xˆ
gH
------ H
ρ
0
∂
---- ρ
∂xˆ
ˆ 0
∫
∂H
dσ′ ------ σ
∂xˆ
σ
ρˆ
0
∫
∂
–
+ ----- dσ′
∂σ
σ
∂
---- ⎛ ∂û
HA ----- ⎞ ∂
---- ⎛ ∂û
∂xˆ
⎝ H∂xˆ
⎠
HA ----- ⎞ ∂
∂yˆ
⎝ H∂yˆ
⎠
H.O.T. -----
∂σ
A V
------
H
û ∂
+ + + + ⎛ ----- ⎞
⎝ ∂σ⎠
and
∂
---------
Hvˆ
------------
∂Hûvˆ
------------
∂Hvˆvˆ
H
∂tˆ
∂xˆ
∂yˆ
vˆω ∂
+ + + --------- + fHû gH
∂σ
ζ ∂
= – -----
∂yˆ
–
gH
------ H
ρ0 ∂
---- ρ
∂yˆ
ˆ 0
∫ dσ′
σ
∂Hsˆ
∂Hûsˆ
∂Hvˆsˆ
--------- ------------ ------------ H
∂tˆ
∂xˆ
∂yˆ
ωsˆ ∂
+ + + -------- =
∂σ
∂H
------ σ
∂yˆ
ρˆ
0
∫
∂
+ ----- dσ′
∂σ
σ
----
∂ ⎛ ∂
HA ---vˆ
⎞ ----
∂ ⎛ ∂
∂xˆ
⎝ H HA ---vˆ
⎞
∂xˆ
⎠ ∂yˆ
⎝ H H.O.T. -----
∂
∂yˆ
⎠
∂σ
AV ------
H
vˆ ∂
+ + + + ⎛ ----- ⎞ ;
⎝ ∂σ⎠
(C.14)
(C.15)
∂
---- ⎛HD ----
∂sˆ
⎞ ----
∂ ∂sˆ
∂xˆ
⎝ H + ⎛HD ---- ⎞
∂xˆ
⎠ ∂yˆ
⎝ H + H.O.T. -----
∂
∂yˆ
⎠
∂σ
DV ------
H
sˆ ∂
+ ⎛ ----- ⎞ .
(C.16)
⎝ ∂σ⎠
The higher order terms (H.O.T.) in equations C.14, C.15, and C.16 are those due to all but the first term on the right sides of
operators C.10 and C.11. The H.O.T. contain horizontal gradients of the water depth and water surface slope and often are neglected
in practice (Mellor and Blumberg, 1985). Neglecting these terms is convenient because their inclusion can add considerably to the
cost of running a model and can lead to a less stable and accurate numerical formulation. In regions of steep bottom slopes,
however, the omission of the H.O.T. in solving the salt transport equation can cause spurious vertical diffusion of salt, especially
in cases where significant salt stratification is present. In these cases the complete transformation must be included or a special
numerical solution strategy, such as the finite-volume method applied to a σ grid by Stelling and van Kester (1994), should be used.