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

82 A Semi-Implicit, Three-Dimensional Model for Estuarine Circulation
4.3.1.1 Momentum equations
The solution of the momentum equations is separated into two stages in which first the explicit terms and then the implicit
terms are evaluated. The separation is not a form of time-splitting but simply a grouping of terms for convenience in making the
computer program modular and facilitating the presentation of equations. The finite-difference equation for the x-direction
1
momentum equation is written so it is centered within layer k at the horizontal point ( i + ⁄ 2)Δx
, jΔy ; the explicit stage is repre-
sented by
Ûi + 1⁄ 2,
jk , = Ui 1⁄ 2
n – 1
+ , j, k 2Δt ( ADVx) n ( CORx) n
[ – +
( BCLINICx) n
+ – + ( HDIFFx) n 1
– ] i 1⁄ 2
+ , j, k,
(4.23)
where the terms in brackets refer to the discretized form of the corresponding groupings of terms in equation 3.59. The bracketed
terms are expanded fully using the leapfrog scheme and displayed in Appendix D (eqs. D.1–D.4). The symbol ˆ
( )in equation 4.23
denotes a solution for the layer volumetric transport which includes only the contribution from the explicit terms. The
corresponding finite-difference equation for the explicit stage of the y-direction momentum equation is centered within layer k at
1
the point iΔx, ( j + ⁄ 2)Δy
and is represented by
n – 1
Ûi, j+
1⁄ 2,
k Uij , + 1⁄ 2,
k 2Δt ( ADVy) n ( CORy) n
( BCLINICy) n
– ( HDIFFy) n 1 – = + [ – –
+
] i, j+
1⁄ 2,
k . (4.24)
Here the bracketed terms refer to groupings of terms in equation 3.60, and they are expanded fully in Appendix D (eqs. D.5–D.8).
All terms in equations 4.23 and 4.24, except horizontal diffusion (HDIFF), are centered in time at time level n to achieve secondorder
numerical accuracy. The horizontal diffusion is written backward-in-time at time level n − 1 because the centering of that
term can result in a weak instability; although formally this uncentered treatment of the horizontal diffusion introduces first-order
truncation error, the actual size of the error should be miniscule. Presently in the model, the horizontal momentum exchange (or
eddy viscosity) coefficient AH is defined at the center of each cell with whole integers of the indices i, j, k.
The shear stress terms in equations 3.59 and 3.60 can be replaced by introducing the concept of the vertical momentum
exchange (or eddy viscosity) coefficient: 34
τxz ∂u
------ = A ----ρ
V ,
∂z
τyz ∂v
------ = A ---ρ
V . (4.25)
∂z
Then the finite-difference equation for the implicit stage of the x-momentum equation is written as
n + 1
+ , j, k=
Ui 1⁄ 2
+
⎛
⎝
n
g
n + 1
n + 1 n – 1
n – 1
⎛( U⁄ h)
i + 1⁄ 2,
j, k– 1–
( U⁄ h)
i + 1⁄ 2,
j, k ui + 1
2
--------------------------------------------------------------------------- ⁄ , j, k– 1–
ui + 1⁄ 2,
j, k⎞
⎜ + ------------------------------------------------------- ⎟
⎝ n + 1
n – 1 ⎠
⁄ , , – ⁄
⁄ , , – ⁄
Δt
– ----- hn i + 1⁄ 2,
j, k
Δx
ρn ⎛ i + 1⁄ 2,
j, 1⎞
⎜--------------------- ⎟ ⋅ ( ζn + 1
⎝ρn i + 1, j – ζn + 1
i, j + ζn – 1
i + 1, j – ζn – 1
i, j )
⎠
i + 1⁄ 2,
j, k
Ûi + 1⁄ 2,
j, k
Δt⎜A Vi + 1⁄ 2 jk 1 ⋅
, , – ⁄ 2
hi + 1
2 j k 1
2
hi + 1
2 j k 1
2
n + 1
n + 1
n – 1 n – 1
n ⎛( U⁄ h)
i + 1
2
A ⁄ , jk , – ( U⁄ h)
i + 1⁄ 2,
j, k+ 1 ui + 1
Vi
---------------------------------------------------------------------------- ⁄ 2,
jk , – ui + 1⁄ 2,
jk , + 1⎞⎞
– ⋅
+ 1⁄ 2,
jk , + 1 ⎜ + ------------------------------------------------------ ⎟⎟,
(4.26)
⁄ 2 ⎝ n + 1
n – 1 ⎠⎠
⁄ , , + ⁄
⁄ , , + ⁄
hi + 1
2 jk 1
2
hi + 1
2 jk 1
2
34 The z-derivatives of u and v in equations 4.25 are mathematically undefined at the layer interfaces because the layer-averaged velocities are
discontinuous there. A z-derivative at an interface is therefore understood to imply a finite-difference quotient involving the layer variables above and below the
interface.