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

30 A Semi-Implicit, Three-Dimensional Model for Estuarine Circulation
where D V0 is the neutral value of the eddy diffusivity without stratification. By using the Reynolds analogy, D V0 can be equated
to A V0 as defined by equation 2.31. The form of Φ s(Ri), as proposed by Munk and Anderson (1948), is
Φ
s
( Ri)
( 1 + β2Ri) α = 2 , (2.37)
where β 2 and α 2 are coefficients chosen by Munk and Anderson: β 2 = 3.33 and α 2 = −1.5. The numerical values for β 2 and α 2 are
not universal constants, and in all cases they should be checked against available data and adjusted as necessary. According to 2.29
and 2.37 (and using the coefficients chosen by Munk and Anderson [1948]), the vertical turbulent Schmidt number Sc V increases
with Ri and with the amount of stable stratification. Officer (1977, p. 17, table 1.2) published values from six estuaries for the
inverse of the vertical turbulent Schmidt number ( = D V/A V) which, when inverted into Schmidt numbers, lie in the range of 10 to
0.83. 14 For a Richardson number of 0.5, the Munk and Anderson formulas predict that Sc V = 1.8.
2.3.2 Treatment of the Pressure Term
The shallow-water flows considered herein are assumed to be essentially horizontal and, therefore, vertical accelerations and
velocities are negligible compared to gravity. As a result of this assumption, only the pressure and gravity terms are retained in the
x 3 -momentum equation, which then reduces to the hydrostatic pressure equation. 15 The Coriolis terms in the horizontal momentum
equations that involve the vertical velocity u 3 can also be neglected, as discussed in Appendix A. By incorporating these simplifi-
cations and representing the turbulence fluxes as shown previously, equations 2.14 to 2.16 can be written in coordinates
(x, y, z) = (x 1, x 2, x 3) as
∂u
∂v
∂w
----- + ---- + ------ = 0 , (2.38)
∂x
∂y
∂z
∂u
∂uu
∂uv
∂uw
1
----- + -------- + -------- + --------- – fv -----
∂t
∂x
∂y
∂z
ρ
0
p ∂ ∂
----- ----- ⎛ ∂u
A ----- ⎞ ∂
----- ⎛ ∂u
∂x
∂x⎝
H
A ----- ⎞ ∂ ∂u
= – + + ----
∂x⎠
∂y⎝
H
+ ⎛A----- ⎞
∂y⎠
∂z⎝
V
, (2.39)
∂z⎠
∂v
∂uv
∂vv
∂vw
1
---- + -------- + ------- + --------- + fu -----
∂t
∂x
∂y
∂z
ρ
0
p ∂ ∂
----- ----- ⎛ ∂v
A ----- ⎞ ∂
----- ⎛ ∂v
∂y
∂x⎝
H
A ----- ⎞ ∂ ∂v
= – + + ----
∂x⎠
∂y⎝
H
+ ⎛A-----⎞ ∂y⎠
∂z⎝
V , (2.40)
∂z⎠
0
-----
1
ρ0 p ∂ ρ
= – ----- – ----- g , and (2.41)
∂z
ρ0 ∂s
∂us
∂vs
∂ws
---- + ------- + ------- + -------- -----
∂ ⎛D----- ∂s⎞
-----
∂ ⎛
∂t
∂x
∂y
∂z
∂x⎝
H
D -----
∂s⎞
----
∂ ∂s
= +
∂x⎠
∂y⎝
H
+ ⎛D---- ⎞
∂y⎠
∂z⎝
V , (2.42)
∂z⎠
where u, v, and w are the velocities in the x, y, and z directions and f is the Coriolis parameter (see Appendix A). The hydrostatic
pressure equation is 2.41, which has been substituted for the z-momentum equation. The advective acceleration terms (∂uu/∂x,
∂uv/∂y, ∂uw/∂z, and so forth) in equations 2.39 and 2.40 are written in a conservative form (or divergence from) similar to that used
in equation 2.5. This form is obtained by adding the continuity equation (eq. 2.38), multiplied by the appropriate velocity
component, to the left side of the momentum equations (similar to eq. 2.6).
14 The values for AV and D V reported by Officer (1977) were for tidally averaged conditions.
15 Casulli and Stelling (1996) present a 3-D model in which the hydrostatic pressure assumption is not introduced and give examples showing when the
non-hydrostatic pressure component should not be neglected.