Geoscience Reference
In-Depth Information
Denoting
K
=
κ
/(
ρ
0
c
0
),
K
being then the thermal diffusivity, we have:
þ
1
e
k
n
z
h
i
Q
si
q
0
c
0
@
T
@
t
þ
u
r
T ¼
r
K
H
r
T
Þþ
@
@
z
K
V
@
T
@
z
ð
Þ
ð
ð
7
:
4
Þ
where the underlined quantities refer to the horizontal plane. The left-hand side gives the
material rate of change of temperature (local change plus advection), and the right-hand
side terms are horizontal diffusion, vertical diffusion and penetration of solar radiation into
the lake. This equation can be solved with prescribed velocity
field or simultaneously with
k
−
1
, solar radiation does not reach the lake
bottom and the bottom heat storage remains passive. The surface boundary conditions of
Eq. (
7.4
) read:
ʾ −
≫
a 3-dimensional circulation model. If
b
z¼
n
¼
Q
0
@
T
@
z
ð
7
:
5
Þ
q
0
c
0
The vertical temperature gradient at the surface closes the surface heat balance. The
bottom boundary condition can be based on the temperature or the heat
fl
ux. In
fl
ow and
out
ow introduce also open boundary conditions.
Since cooling is primarily a vertical process, one-dimensional (vertical) models can tell
much of its physics. These vertical models are classi
fl
ed into analytical models, mixed-
layer models, and turbulence models.
Integration of Eq. (
7.4
) from the level d to the surface gives
dt
¼K
V
@
T
h
i
Q
T
q
0
c
0
z¼
n
Þ
d
T
þ
1
e
k
n
d
ð
Þ
ð
n
d
@
z
z¼d
ð
7
:
6
Þ
Z
n
w
@
T
þ
@
z
u
r
T
þr
K
H
r
T
½
ð
Þ
dz
d
where T
is the mean temperature across the integrated layer. The horizontal advection and
diffusion terms have been moved together into the brackets [
] in the integral, and they
need to be parameterized in vertical models. Vertical advection cannot be integrated in
general form and it is usually neglected as a small term. The depth of lake is H =
·
d.
Assuming that d = constant and water level variations to be small compared to the
depth,
ʾ −
constant. Then, also assuming that the lake depth is much
larger than the optical thickness of lake water, H
ʾ ≪
d, we can take H
≈
≫ ʺ
−
1
, Eq. (
7.6
) can be written as
dT
dt
¼
Q
0
þ
Q
T
þ
Q
b
þ C
H
ð
7
:
7
Þ
q
0
c
0
H
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