Theoretical expression for the countergradient vertical heat flux

A theoretical expression is derived from the heat-flux conservation equation for the counter potential-temperature gradient that can sustain an upward flux of sensible heat. This gradient is found to be γc = (g/θ) 〈θ′2〉/〈w′2〉, where 〈θ′2〉 is the potential temperature variance and 〈w′2〉 is the vertical velocity variance. The usual down-gradient eddy coefficient expression for the heat flux is obtained from the derivation only if γc is set to zero. Aircraft measurements of (g/θ) 〈θ′2〉/〈w′2〉 in the middle and upper portions of convective planetary boundary layers indicate that this expression for γc is of the same order of magnitude (near 0.7 × 10−5 °K cm−1) as the value deduced previously for γc from completely different considerations.