Geoscience Reference
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Table 7.2
Scaling the equation of motion of the water body in ice-covered lakes
Inertia
Advection
acceleration
Coriolis
Pressure gradient
(residual)
Viscous
friction
UT
−
1
U
2
L
−
1
UH
−
2
fU
g
ʔ
H/L
10
−
8
ms
−
2
10
−
9
ms
−
2
10
−
7
ms
−
2
10
−
7
ms
−
2
10
−
11
ms
−
2
To balance the equation, the pressure gradient must be of the same magnitude as the
Coriolis acceleration and is thus obtained as the residual. It is clear by the continuity
equation that vertical advection
L that horizontal
viscous friction is even smaller than vertical. The leading balance is between the Coriolis
acceleration and the pressure gradient, i.e. thermohaline circulation in ice-covered lakes
follows the geostrophic
horizontal advection, and by H
≫
*
fl
flow balance. This is also re
fl
ected in the small Rossby number
10
−
2
. There is a large difference in the circulation characteristics between
ice-free and ice-covered lakes, since in ice-free conditions the
Ro = U/(Lf)
*
fl
flow is turbulent, and
friction due to the wind forcing overcomes the Coriolis term.
Increasing the
1cms
−
1
, the importance of the advective
acceleration increases but only at U
*
10 cm s
−
1
it becomes comparable to the Coriolis
acceleration. Taking the length scale as 100 m does not bring advection to the level of the
geostrophic balance. For the time-scale of 10
4
fl
flow velocity to U
*
s, local acceleration becomes equal to
Coriolis acceleration, whatever is the
flow, is
quite stable so that the local acceleration becomes a very small term. Since Rossby
number is the ratio of advection to Coriolis acceleration, the geostrophic balance holds in
frictionless stationary
fl
flow velocity. But beneath an ice cover the
fl
1.
The role of friction depends on the quality of the
fl
flow as long as Ro
≪
flow, which is described by the
Reynolds number Re = UH/v. Taking Re <10
3
and Re >10
4
as the criteria for laminar and
turbulent
fl
1mms
-1
, laminar state follows for
H < 1 m and turbulent state for H > 10 m. Thus laminar
fl
flow, respectively, it is seen that for U
*
turbulent transition conditions
would be typical, while in extreme cases laminar or turbulent
-
fl
flow could be reached. The
vertical kinematic friction can in general be expressed as
F
V
¼
@
@
z
K
V
@
u
@
z
ð
7
:
22
Þ
10
−
6
m
2
s
−
1
(molecular viscosity) for laminar
where K
v
is kinematic viscosity, K
v
=v=1.8
×
10
−
2
1m
2
s
−
1
for turbulent
fl
ow, K
v
*
fl
flow, and for the laminar
turbulent transition regime
-
-
10
−
4
m
2
s
−
1
(Petrov et al. 2007). In the transition regime, the scale of the friction is not
well known but anyway it is much lower than the other terms in the momentum equation; the
in
K
v
*
fl
uence is limited to the thin surface and bottom boundary layers.
The sediment heat
fl
flux creates thermohaline circulation just after freeze-up. Then this
heat
flux is at highest and horizontal density gradients form between shallow regions and
pelagic zones. The circulation pattern consists of the downslope
fl
fl
flow of the warm dense
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