The thermal turbulence which causes the effect of seeing can
quantified by the local temperature structure coefficient 
,
which is defined in section 
.
We have also seen that the temperature structure coefficient is
related to the dissipation rates of kinetic energy and temperature:
where 
is a constant equal to about 3.
Introducing the eddy coefficients 
for momentum and 
for
temperature, the respective dissipation rates can be expressed
in tensor notation as:
The last term at the right side accounts for buoyancy effects.
If the mean characteristics generally depend only on one geometrical
coordinate, as it is the case in a stationary atmospheric boundary
layer with the
height z above the ground, the above expressions become:
Inserting in () one gets:
[Wyngaard] has analyzed in detailed, on the basis of
experimental data, the parameterisation of 
in terms of the
temperature and velocity fields in the atmospheric surface layer using
the so-called similarity theory.
Similarity theory is a method by which statistical mean and turbulent values in a flow/temperature field, when properly adimensionalized, are assumed to be universal constant or functions of a stability parameter. The adimensionalizing quantities, called scaling variables, and the stability parameter can be chosen in different ways by obeying to some simple rules (see for instance [Hull], pp. 347-361).
Here the scaling variables taken are the height z and the temperature
gradient 
, while the Richardson number was used for
the stability parameter:
Noting that similarity theory predicts that 
and
, hence
, when adimensionalized are universal function of Ri,
[Wyngaard] derived the expression
The function f(Ri), obtained from experimental measurements
is plotted in fig. and
is a good illustration of the fundamental asymmetry of thermal
turbulence, hence seeing, with respect to the sign of the temperature
gradient.
As a numerical exercise we have computed 
by means of
expression () as a function of 
for three different speed rms 
values at 15 meters
height above the ground (fig. ).
One will note that
the effect of even small variations of 
on
the local 
is very significant. The achievement of low seeing
implies very small temperature gradients, particularly in unstable
conditions.
An exception is given by the case of a stable gradient with low
mechanical turbulence. This is possibly the plainest demonstration
that quiet inversion layers have very favorable seeing
characteristics.
The variations of mechanical turbulence have opposite effects on
depending if the thermal conditions are unstable or stable.
For unstable conditions and a same 
,
decreases with increasing turbulence.
For stable conditions 
increases dramatically with increasing turbulence. This means for instance that the artificial inversion obtained by chilling the dome floor in some observatories (CFHT, ESO 2.2-m) does achieve a low seeing only as long as no wind turbulence enters the dome.
Figure: The function f(Ri)
in equation (5.7) - from [Wyngaard]
Figure: Computation of 
versus 
in the
atmospheric surface layer, 15 m above the ground
By choosing other scaling variables, namely the friction velocity
and the surface heat flux q (in K m s
), and as the
stability
parameter the ratio 
, where L is the Monin-Obukhov length
[Wyngaard] obtains another expression for 
:
where 
is an empirical function evaluated from
experimental data as:
In presence of
a strong turbulent flow, L is large and therefore
close to the surface
is constant 
and equation () becomes:
We note that this expression may be derived also directly from
the general expression ().
When friction effects predominate over buoyancy
the second term of equation () may be neglected.
Putting 
we obtain:
With this approximation and using a common parameterisation for the K factors:
where k is the Von Karman constant (
0.4),
is the friction velocity, 
the velocity rms
and q the vertical heat flux,
expression () can be elaborated as
Near the surface the heat flux q is practically equal to the surface
flux 
, which in a turbulent surface layer is
proportional to 
.
Therefore in a turbulent near-neutral surface layer
is proportional to 
hence
to 
which is the square of turbulence intensity 
.
One then finds that 
is directly related to both the squares of
turbulence intensity and temperature difference:
can also be put in relation with the outer scale of turbulence
. Following [Tatarskii], the outer scale of turbulence is related
to 
as
inserting this expression into () we obtain
The free convection case
When the flow is strongly unstable, that is when, approaching the free
convection condition, buoyancy predominates over
friction effects such that 
,
expression () becomes
inserting in equation ()and using the definition of L,
one obtains an expression in which 
disappears:
This relationship between surface flux, height and 
is
graphically illustrated in fig. below.
Another expression for the free convection case can be obtained
quite simply from equation (), noting that the function
becomes about 3.6 for 
(cf. fig. ):
which has the same form as equation () and
where the distance z may be interpreted as a length scale parameter
which characterizes flow mixing in the free convection circulation
process.
Figure: Relationship between 
,
height and surface
heat flux in free convection over a horizontal surface
