Thermodynamics of Seasonal Lake Ice - Freezing of Lakes and the Evolution of their Ice Cover - page 97

4.1.4 Turbulent Heat Exchange with the Atmosphere

The atmospheric boundary layer is in turbulent state, and the exchange of heat between
lakes and atmosphere can be modelled using the turbulent boundary layer theory. The

ows and can be
expressed as bulk exchange laws. Heat is transferred as sensible and latent heat, and
together with the momentum transfer the bulk formulae read (see Sect. 2.2.4 ) :

first-order models are based on the analogy of laminar and turbulent fl

s a ¼ q a C az U az U az ¼ q a u 2 ð 4 : 9a Þ

Q c ¼ q a c p C Hz T az T 0 ð Þ U az ð 4 : 9b Þ

Q e ¼ q a L ES C ez q az q 0 ð Þ U az ð 4 : 9c Þ

where u * is the friction velocity, the subscript ES refers to evaporation or sublimation
depending on the surface, and the lower index z has been added to indicate where the
quantities depend on the altitude. The transfer coef

cients C az , C Hz , C ez in general depend

on the surface roughness and strati cation of the atmospheric surface layer.

fluxes onto the surface are based on the cross-covariances
of the given quantity with the vertical velocity. The direct formulae for the momentum,
sensible heat and latent heat

In exact terms, the turbulent fl

fl fluxes are, respectively (e.g., Tennekes and Lumley 1972):

s a ¼ q a u 0 w 0

h i ð 4 : 10a Þ

Q c ¼ q a c p h 0 w 0

h i ð 4 : 10b Þ

Q e ¼ q a L ES q 0 w 0 h i ð 4 : 10c Þ

where the quantities marked with ' are fl fluctuations and the brackets < · > stand for time

averaging, e.g. h ¼ hhiþh 0 ; hh 0 i ¼0 ; hh 0 ðÞh 0 ð s Þi ¼ 0 ð t 6 ¼ s Þ

, and the x-axis has been

aligned with the direction of the mean wind. These fluxes can be measured directly with

eddy covariance measurement techniques or they can be estimated from the mean
boundary-layer characteristics with methods of different degrees of complexity.
In the limit U a →

fl

0 this system gives zero fl fluxes. In such conditions, however,

turbulent fluxes can arise from buoyant (free) convection. In the free convection limit, the

sensible heat

fl

fl flux is (Louis 1979):

r

gz 0
h 0

ðh h 0 Þ 3 = 2 w 0 h 0 h i 0 ¼ 0 : 26 ð 4 : 11 Þ

Turbulent exchange of a property Q is proportional to its vertical gradient times the
eddy diffusivity K, F Q *

∂ ∂

K Q/ z (see Tennekes and Lumley 1970); here the vertical