Geoscience Reference
In-Depth Information
y
!
y
y
y
u
M
u
M
u
z
u
A
u
z
u
A
F
v
ı
y
u
n
1
C
ı
x
v
n
1
D
ı
x
y
u
x
u
!
x
y
v
M
v
z
v
A
ı
y
v
n
1
C
ı
y
2
(15.21)
y
v
where the mixing coefficient is modeled according the Smagorinsky formula
"
1
x
ı
x
u
n
2
1
2x
ı
2x
v
n
y
2
C
1
2
1
2y
ı
2y
u
n
x
C
A
M
D
C
Smag
xy
y
ı
y
v
n
2
#
2
1
C
(15.22)
with the magnitude of the eddy coefficient being scaled by the constant
C
smag
.The
vertical mixing coefficients are computed using the turbulence closure by Mellor
and Yamada in either 2 or 2.5 version.
The computation for the free-surface mode is governed by the equations:
u
u
gı
x
˛
1
n
C
1
C
˛
2
n
C
˛
3
n
1
C
D
x
y
u
u
u
u
ı
2t
.D
u
/
D
y
D
G
u
(15.23)
2t
gı
y
˛
1
n
C
1
C
˛
2
n
C
˛
3
n
1
C
D
v
v
x
y
v
v
v
v
ı
2t
.D
/
D
x
D
G
v
v
(15.24)
2t
y
u
u
u
n
1
u
u
n
C
1
u
u
n
xy
2t
ı
2t
D
ı
x
ˇ
1
D
C
ˇ
2
D
C
ˇ
3
D
x
v
ˇ
1
v
n
1
v
n
C
1
v
n
v
v
v
ı
y
D
C
ˇ
2
D
C
ˇ
3
D
C
xyDQ;
(15.25)
wh
ere
ˇ
1
,
ˇ
2
a
nd
ˇ
3
are positive constants define by the user with
ˇ
1
C
ˇ
2
C
ˇ
3
D
1
,
u
v
D
G
v
are the vertical integrals of all the terms in the right hand side
of (
15.14
)and(
15.15
) respectively, with the exception of the surface elevation
gradient terms and the vertical mixing, and
G
u
and
D
D
D
x
and
D
D
y
. The free-
u
v
D
D
/
n
C
1
from
the time discretized (
15.23
)and(
15.24
)into(
15.25
), resulting in an elliptic equation
that is solved for the surface elevation at time level n C
1
/
n
C
1
and
u
v
surface mode (
15.25
) is solved by first substituting
.D
u
.D
v
, which is then substituted
back in (
15.23
)and(
15.24
) for computing the barotropic transports
u
v
D
u
and
D
v
from which the barotropic velocities are obtained.
The vertical discretization uses a combination of sigma layers and z-levels in a
three-tiered distribution with (1) free sigma layers near the surface that expand and
contract with the free surface elevation, (2) fixed sigma layers that do not vary with
the free surface, and (3) fixed z levels that allow for partial bottom cells for a better
match of the bottom topography.
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