Temperature structure coefficient

In analogy with the theory of the turbulent velocity field, Tatarskii[6] finds an inertial domain, between the outer scale of turbulence L and l where the temperature turbulent variations are defined by a statistical structure function

$$D_{T}(\Delta {\bf r}) = <(T({\bf r})-T({\bf r}+\Delta {\bf r}))^{2}>$$

has the form:

$$D_{T}(\Delta {\bf r}) = C_{\scriptscriptstyle T}^2 \Delta {\b... ...{3}} \;\;\;\;\;\;\;\; {\rm for} \;\; l \ll \Delta r \ll L \;$$ (2)

where $C_{\scriptscriptstyle T}^2$ is the temperature structure coefficient.
$C_{\scriptscriptstyle T}^2$ characterizes completely the local thermal turbulence at a give time, has SI units (K² m^-2/3) and is therefore formally defined as:

$$C_{\scriptscriptstyle T}^2 = \frac{<(T({\bf r})-T({\bf r}+\Delta {\bf r}))^{2}>} {\Delta {\bf r}^{2/3}}$$ (3)

for a separation $\Delta {\bf r}$ in the inertial domain.

is also related to the one-dimensional temperature spectrum which in the inertial domain has the form:

$$\Phi (\kappa_w) = 0.25 \ C_T^2 \kappa_w^{-5/3}$$ (4)

where $\kappa_w$ is the streamwise component of wavenumber.
Moreover, Tatarskii[6] notes that in the inertial domain $D_{T}(\Delta {\bf r})$ should be a function of only the dissipation rate of kinetic energy $\epsilon$, $\Delta {\bf r}$ and the temperature dissipation rate $\epsilon_\theta$. Dimensional reasoning leads then to

$$C_{\scriptscriptstyle T}^2 = a^2 \epsilon_\theta \epsilon^{-\frac{1}{3}}$$ (5)

where a² is a constant found to be equal to about 3.

Analog statistical properties may be applied to the index of refraction and one may define a structure coefficient of the index of refraction . From equation (1) and ignoring the very minor effect of humidity, $C_{\scriptscriptstyle N}^2$ is related to $C_{\scriptscriptstyle T}^2$ by:

$$C_{\scriptscriptstyle N}^2 = C_{\scriptscriptstyle T}^2 \... ....52\,10^{-3}\lambda^{-2} \right) \frac{P}{T^{2}}\right]^{2}$$ (6)

where $\lambda$ is the wavelength.