Structure functions in nocturnal atmospheric boundary layer turbulence

This paper analyzes odd and even higher-order moments for longitudinal velocity increment $\mathrm{\Delta }u(x,r)$, where $x$ is the longitudinal coordinate and $r$ is the separation distance, based on the canonical and a modified normalization for skewness of longitudinal velocity derivative $\partial u/\partial x$. Two types of data were used, stably stratified turbulence data from the nocturnal atmospheric boundary layer taken during the Mountain Terrain Atmospheric Modeling and Observations field campaign and from the direct numerical simulation of homogeneous and isotropic turbulence in a box at four Reynolds numbers and four different grid resolutions. Third moment data normalized by the same moment of third order for modulus $\left|\mathrm{\Delta }u(x,r)\right|$ representing modified skewness of the velocity increment showed a better collapse at all Reynolds numbers in the inertial and viscous subranges than canonical normalized skewness with normalization parameter ${\langle {\left(\mathrm{\Delta }u(x,r)\right)}^2\rangle }^{3/2}$, where $\langle ··\rangle$ represents the ensemble average. The analysis also considered odd $p\mathrm{th}$-order classical structure functions $\langle \mathrm{\Delta }u{(x,r)}^p\rangle$ with Kolmogorov-theory based normalization $\langle \mathrm{\Delta }u{(x,r)}^p\rangle /{\left(\varepsilon r\right)}^{p/3}$ for the inertial subrange, where ε is the rate of dissipation, and a modulus-based structure function $\langle {\left|\mathrm{\Delta }u(x,r)\right|}^p\rangle /{\left(\varepsilon r\right)}^{p/3}$. Both types of structure functions of order $p=1ndash;6$ were computed using different normalizations, and corresponding scaling exponents were assessed for the inertial and viscous subranges. Scaling for modulus-based structure functions in the viscous subrange was identified as $\langle {\left|\mathrm{\Delta }u(x,r)\right|}^p\rangle \propto r^{p·(5/6)}$. In the viscous subrange, the velocity increment varied linearly with $r$ for the classical third moment $\langle \mathrm{\Delta }u{(x,r)}^3\rangle \propto r^3$ based on the velocity increment while the classical fifth moment $\langle \mathrm{\Delta }u{(x,r)}^5\rangle$ did not provide any meaningful scaling exponent. A plausible qualitative explanation linking these effects to anisotropy of nocturnal stratified turbulence is proposed.

Figure 1

Canonical odd moments for all four datasets obtained in DNS: (a) third, (b) fifth, and (c) seventh moments.

Figure 2

Modified moments for all four DNS datasets. Color coding is the same as in Fig. 1.

Figure 3

Comparison of the third canonical (a) and modified (b) moments obtained in the field campaign [51, 52] with ${\mathrm{Re}}_{\lambda }=1200$ and DNS conducted for the two highest ${\mathrm{Re}}_{\lambda }=264$ and 383. The periods indicated in the legends are for time periods without bursts, determined according to the methodology of [52].

Figure 4

Comparison of fifth canonical moments (a) and modified moments (b) obtained in the field and DNS.

Figure 5

Normalized according to KSSH third-order (a) and second-order (b) structure functions in field experiments computed using longitudinal velocity increment $\mathrm{\Delta }u(x,r)$.

Figure 6

Normalized odd third-order (a) and first-order (b) structure functions in field experiments computed using the modulus of the velocity increment.

Figure 7

Normalized according to KSSH even fourth-order (a) and sixth-order (b) structure functions in field experiments computed using the longitudinal velocity increment.

Figure 8

Normalized fifth-order structure functions in field experiments computed using longitudinal velocity increment $\mathrm{\Delta }u(x,r)$ (a) and its modulus (b).

Figure 9

Scaling exponents of the structure functions in the inertial and viscous subranges for the velocity increment and its modulus evaluated for field data. Stars indicate that different scaling exponents of structure functions (exp-str-fun) were obtained with odd third and fifth structure functions for velocity increments and their modulus.