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

Horizontal momentum equations,
and
Salt transport equation,
2. Governing Equations and Boundary Conditions 43
β u* ∂
--------
∂u* u* ∂u*v*
∂u*w*
Ro
+ Ro --------------- + -------------- + ---------------- – v*
∂t*
∂x*
∂y*
∂z*
Fr2 D
– -------- ---
H
ζ* ∂
= --------
(2.69)
∂x*
D
---ζ*
H
∫ --------
Ro --------
∂ρ*
–
dz*
Fr2 ∂x*
i
z*
+ EH --------
∂ ⎛∂ ---------u*
⎞ --------
∂ ∂u*
+ ⎛--------- ⎞
∂x*
⎝ ∂x*
⎠ ∂y*
⎝∂y* ⎠
∂
E -------
∂u*
+ ⎛---------- ⎞
V∂z*
⎝ ∂z*
⎠
β v* ∂
--------
∂ u*v*
∂v*v*
∂v*w*
Ro
+ Ro -------------- + -------------- + --------------- + u*
∂t*
∂x*
∂y*
∂z*
Fr2 D
– -------- ---
H
ζ* ∂
= --------
(2.70)
∂y*
Ro
D
---ζ*
H
∫
∂ρ*
– --------
-------- dz*
Fr2 ∂y*
i
z*
EH ∂
-------- ⎛∂v* --------- ⎞ ∂ ∂v*
+ -------- ⎛--------- ⎞
∂x*
⎝∂x* ⎠ ∂y*
⎝∂y* ⎠
∂
E -------
V∂z*
w* ∂
+ + ⎛----------- ⎞ ;
⎝ ∂z*
⎠
β
------
Ro
s* ∂
------- +
∂t*
∂u* s* ∂v*s*
∂w*s*
-------------- + -------------- + ---------------
∂x*
∂y*
∂z*
=
1
---------
PeH ∂
-------- ⎛∂s* -------- ⎞ ∂ ∂s*
+ -------- ⎛--------⎞ ∂x*
⎝∂x* ⎠ ∂y*
⎝∂y* ⎠
1
---------
PeV ∂ ∂s*
+ ------- ⎛------- ⎞
∂z*
⎝∂z* ⎠.
(2.71)
The coefficients of most terms now form recognizable dimensionless numbers that are identified in table 2.1. In arriving at the two
horizontal momentum equations (2.69 and 2.70), the eddy coefficients were assumed to be constant. Also, the coefficients of the
Coriolis terms were arranged to be unity in these two equations. The salt transport equation (2.71) was arranged so that the
coefficient of the advection terms was unity.
Estimates of the scaling parameters and coefficients that are typical of the northern reach of San Francisco Bay are
L = 100 km H = 10 m
D = 1 m u r = 0.25 m/sec
f = 10 −4 sec −1 ω = 5 × 10 −5 sec −1
ρ 1 = 10 3 kg/m 3
s 1 = 1 s 2 = 34
ρ 2 = 1. 025 × 10 3 kg/m 3
ρ 0 = 10 3 kg/m g = 10 m/sec 2
A H = 10 2 m 2 /sec A V = 10 –2 m 2 /sec
D H = 10 2 m 2 /sec D V = 10 –2 m 2 /sec