Geoscience Reference
In-Depth Information
The conservation of mass leads to the continuity equation
∂
∂
u
x
+
∂
∂
u
x
+
∂
∂
u
x
1
2
3
=
0
(3.21)
1
2
3
The two horizontal components of the Reynolds equations read
2
2
+
2
∂
∂
u
t
∂
∂
u
x
∂
∂
u
x
∂
∂
u
x
∂
∂
u
x
+
∂
∂
u
x
∂
∂
u
x
−=−
∂
∂
1
ρ
p
x
H
V
1
+
u
1
+
u
1
+
u
1
fu
+
v
1
1
v
1
(3.22a)
1
2
3
2
M
2
2
M
2
1
2
3
1
1
2
3
and
2
2
+
2
∂
∂
u
t
∂
∂
u
x
∂
∂
u
x
∂
∂
u
x
∂
∂
u
x
+
∂
∂
u
x
∂
∂
u
x
+=−
∂
∂
1
ρ
p
x
H
V
2
+
u
2
+
u
2
+
u
2
fu
+
v
2
2
v
2
(3.22b)
1
2
3
1
M
2
2
M
2
1
2
3
2
1
2
3
where the total time derivative has been split into the inertial term that describes the
local acceleration and a second term that describes the advective acceleration.
For the vertical component the hydrostatic approximation is used
∂
∂
p
x
=−
ρ
g
(3.23)
3
The conservation equations for heat and salt read
+
2
2
2
∂
∂
Q
x
∂
∂
T
t
∂
∂
T
x
∂
∂
T
x
∂
∂
T
x
∂
∂
T
x
+
∂
∂
T
∂
∂
T
x
1
ρ
H
V
+
u
+
u
+
u
=
v
v
+
s
(3.24a)
1
2
3
T
2
2
T
2
x
c
1
2
3
1
2
3
p
3
and
+
2
2
2
∂
∂
S
t
∂
∂
S
x
∂
∂
S
x
∂
∂
S
x
∂
∂
S
x
+
∂
∂
S
∂
∂
S
H
V
+
u
+
u
+
u
=
v
v
(3.24b)
1
2
3
S
2
2
S
2
x
x
1
2
3
1
2
3
with
V
T
and
V
S
the turbulent diffusivities for temperature and salinity, respectively.
Finally, the equation of state reads
ρ =
fTSp
(,,)
This completes the discussion of the hydrodynamic equations.
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