Lake Water Body in the Ice Season - Freezing of Lakes and the Evolution of their Ice Cover - page 211

The third equation presents an equilibrium condition for the entrainment rate: the left-hand
side has the energy consumption consisting of radiation by internal waves and entraining
of deep water, and the right-hand side has the energy production due to wind stress,
buoyancy and solar radiation. There ci i is the internal wave speed, u * is the friction velocity
of wind, B o is the buoyancy flux at the surface, and the factors s, m and n represent the

corresponding fractions of the kinetic energy of the mixed layer, wind stress and buoy-
ancy

fl

flux taking part in the entrainment (see Niiler and Kraus 1977 for more details).
A speci

fl

with
parameterized vertical temperature distribution has been widely used in recent years
(Mironov et al. 1991, 2004; Kourzeneva et al. 2008; Golosov and Kirillin 2010; Kirillin
2010). The model is based on similarity theory approach for the vertical structure of the
thermocline and variable mixed layer depth. Flake has been used in numerical weather
prediction and climate models to provide better estimate for the surface temperature and as
well it has been used in several other applications.

c mixed layer model — Flake, see http://www. fl ake.igb-berlin.de/ —

7.1.5 Turbulence Models

For a more detailed description of the vertical temperature velocity structure, advanced

turbulence models are needed. In Nordic lakes a k − e type 2nd order closure model has
been widely used (Svensson 1979), known better as the general equations solver

-

'
(Svensson 1998; Omstedt 2011). The quantities k and e are the turbulent kinetic energy and
the dissipation rate of turbulent kinetic energy, respectively. The system of equations for
the average (averaged over turbulent

' PROBE

fl fluctuations) temperature and velocity is

þ j Q s e j z

@ T

@ t ¼ @

K T @ T
@ z

ð 7 : 16a Þ

@ z

@ U

@ t ¼ f fk U þ @

K @ U
@ z

ð 7 : 16b Þ

@ z

cients are obtained from the turbulent kinetic energy and the dis-
sipation rate of turbulent kinetic energy as:

The diffusion coef

K ¼ C l k 2

K
K T ¼ r k e ;

ð 7 : 16c Þ

where is the Prandtl number. The 2nd order

closure provides the evolution of k and e through (Svensson 1979)

C l ¼ 0 : 09 is a model constant and r k ¼ 1 : 4

þ P s þ P b e

@ k

@ t ¼ @

r k @ k

ð 7 : 16d Þ

@ z @ z