In boundary-layer turbulence, when the distance to the wall is within the logarithmic regime, the streamwise velocity structure function can be divided into three regimes by the horizontal scale in the order of distance to the wall, $z$, and the boundary-layer thickness, $\delta$. In the inertial range, when the streamwise two-point distance $r$ is less than $z$, the velocity structure function satisfies a power-function dependence on $r/z$. In the logarithmic range when $z<r<\delta$, the velocity structure function has a logarithmic dependence on $r/z$. In the outer range when $r>\delta$, the function approaches a constant value. This work shows that the above three regimes for the velocity structure function also apply to the mixed third-order structure function constituted by the velocity and potential temperature differences $\langle \delta u\delta {\theta }^2\rangle$. Here, $\delta u$ and $\delta \theta$ are the streamwise differences in streamwise velocity and potential temperature, respectively. We derive the logarithmic dependence of the mixed third-order structure function from the Kármán-Howarth-Monin equation for the potential energy with the help of the attached eddy hypothesis. The inertial range and logarithmic range are identified by analyzing data from the Qingtu Lake Observation Array. The transition scale between the two ranges decreases with an increase of $z/L$ in weak stable conditions where $L$ is the Obukhov length.

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• Atmospheric fluid dynamics
• Boundary layers
• Energy cascade
• Structure & turbulence of boundary layers
• Turbulence
• Navier-Stokes equation