Lake Water Body in the Ice Season - Freezing of Lakes and the Evolution of their Ice Cover - page 220

7.3 Dynamics of Water Body Beneath Ice-Cover

7.3.1 Circulation

Thermohaline circulation 4 is the governing form of circulation in fully ice-covered lakes.
It results from the evolution of water temperature and salinity that builds horizontal
density gradients. The driving forces are the heat flux from the sediments and solar

radiation. Other forms of circulation result from atmospheric pressure variations and, in
lakes with signi

fl

cant through fl flow, from the boundary conditions. Except for the last case,

flow velocities are always very small, of the order of 1 mm s − 1 (e.g., Huttula et al.
2010). Due to the very low levels, in practice it is very dif

water fl

cult to measure the circulation

velocity field.

Circulation dynamics are based on the Navier-Stokes equation, equation of state, and
conservation laws for mass, temperature and salinity (Gill 1982; Cushmann-Roisin 1994):

@ u

@ t þ u r u þ f k u ¼ q r p þ v r 2 u

ð 7 : 21a Þ

q ¼ qð T ; S ; p Þ ð : Þ

7 21b

r u ¼0 ð 7 : 21c Þ

@fT; T ; S g
@ t

þ u rf T ; S g ¼v r 2 f T ; S gþf q ; 0 g ð 7 : 21d Þ

Thus we have seven unknowns and seven equations. In freshwater lakes, the salinity is
normally ignored, but beneath the ice cover it may have a signi

uence on density
and consequently on the thermohaline circulation. Wind-driven circulation is driven by
the wind stress, which steps into the surface boundary condition of the Navier-Stokes
equation, while thermohaline circulation is driven by the pressure gradient. In partially
ice-covered lakes, wind forcing through open water areas and drifting ice

cant in fl

fl oes can

generate a signi cant wind-driven circulation system (Fujisaki et al. 2012).

In fully ice-covered lakes, water velocities are small, and therefore advection and
friction are small. Scaling analysis shows the basic characteristics. Denote U — velocity

scale, T — T—time scale, H — vertical scale, and L — horizontal scale, and their typical values

1mms − 1 , T 10 5 s (1 day), H

are U 10 m, and L 1 km. Then we have for the

* * * *

terms in the Navier-Stokes equation (Table 7.2 ).

4 The term thermohaline circulation comes from oceanography;

' thermo ' refers to temperature and

' haline ' to salinity. This circulation is driven by horizontal density gradients.