The free convection case

When the flow is strongly unstable, that is when, approaching the free convection condition, buoyancy predominates over friction effects, equation (11) takes a form in which u_* disappears:

$$C_{T}^{2} = 2.68 \left( \frac{g}{T} \right)^{-\frac{2}{3}} \cdot \left( \frac{z}{q_s} \right)^{-\frac{4}{3}}$$ (15)

This relationship between surface flux, height and $C_{\scriptscriptstyle T}^2$ is graphically illustrated in fig. 2.

Another expression for the free convection case can be obtained quite simply from equation (10), noting that the function f(Ri) becomes about 3.6 for $Ri \ll 0$:

$$C_{T}^{2} \simeq 3.6 \left( \frac{dT}{dz} \right)^2 z^{\frac{4}{3}}$$ (16)

which has the same form as equation (14) and where the distance z may be interpreted as a length scale parameter which characterizes flow mixing in the free convection circulation process.