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