Extremes, intermittency, and time directionality of atmospheric turbulence at the crossover from production to inertial scales

The effects of mechanical generation of turbulent kinetic energy and buoyancy forces on the statistics of air temperature and velocity increments are experimentally investigated at the crossover from production to inertial range scales. The ratio of an approximated mechanical to buoyant production (or destruction) of turbulent kinetic energy can be used to form a dimensionless stability parameter $\zeta$ that classifies the state of the atmosphere as common in many atmospheric surface layer studies. We assess how $\zeta$ affects the scalewise evolution of the probability of extreme air temperature excursions, their asymmetry, and time directionality. The analysis makes use of high-frequency turbulent velocity and air temperature time-series measurements collected at $z=5\mathrm{m}$ above a grass surface at very large frictional Reynolds numbers ${\text{Re}}_{*}=u_{*}z/\nu >1\times {10}^5$ ($u_{*}$ is the friction velocity and $\nu$ is the kinematic viscosity of air). A multitime measure of the imbalance between forward and backward phase-space trajectories is employed to investigate the time-directional properties of the scalar (temperature) field. Using conventional higher-order structure functions, we find that temperature exhibits larger intermittency and wider multifractality when compared to the longitudinal velocity component, consistent with laboratory studies and simulations conducted at lower ${\text{Re}}_{*}$. We find that the magnitude of $\zeta$, rather than the sign of the heat flux, impacts the distribution of scalar increments at separation scales well within the inertial subrange. Conversely, the direction of the heat flux fingerprints the observed time-directionality properties of the scalar field in the first two decades of inertial subrange scales. These combined findings demonstrate that external boundary conditions, and in particular the magnitude and sign of the sensible heat flux, have a significant impact on temperature advection-diffusion dynamics within the inertial range.

Figure 1

Normalized second-order structure functions for (a) vertical velocity and (b) temperature are shown for runs that are weakly unstable (blue dashed lines), strongly unstable (red solid lines), and stable (black dash-dotted lines). Black lines indicate the value 1 and the 2/3 power law for reference; vertical dashed lines correspond to the dimensionless scales $\tau =0.05$ (smallest scale not impacted by instrument path length), $\tau =1$ (integral scale of the flow), and $\tau =5$ (typical scale larger than $I_w$, while small enough not to be impacted by statistical convergence issues in structure functions calculations). (c) Integral scales of the flow for $s=T$ (circles) and $s=w$ (crosses) as a function of the stability parameter $\left|\zeta \right|$ and (d) their ratio $I_T$ to $I_w$ again as a function of $\left|\zeta \right|$, where stable runs ($\zeta >0$) are indicated by black squares.

Figure 2

Sequences of velocity and temperature fluctuations extracted from a strongly unstable run [run 8, $\zeta =-0.52$, $I_w=1.74s$, column (a)] and a stable or near-neutral one [run 34, $\zeta =0.07$, $I_w=2.18s$, column (b)]. The presence of ramps and inverted-ramplike structures, respectively, is marked by dashed vertical lines. Column (c) illustrates a phase-randomized sequence obtained from run 34 (top), a series of synthetic ramps with durations exponentially distributed with mean $2I_w$ (middle), and the surrogate time series obtained merging the above sawtooth pattern with the phase-randomized time series (bottom), where the relative weight of the ramps $\alpha$ was set equal to 0.5.

Figure 3

(a) Average values of the scaling exponents for longitudinal velocity $u$ (triangles), vertical velocity $w$ (squares), and temperature $T$ (circles). Black solid lines and dashed lines show, respectively, the Kolmogorov and the She-Leveque predictions for the longitudinal velocity structure functions. Exponents are computed from scales ranging between $\tau =0.05$ and $\tau =0.2$. (b) Scaling exponents for temperature only; Mean and standard deviation values are computed over all the runs and are indicated by circles and vertical bars, respectively. Data from Mydlarsky and Warhaft [61] (squares), Antonia et al. [55] (triangles), Meneveau et al. [56] (asterisks), and Ruiz-Chavarria et al. (diamonds) [57] are shown for comparison. The KOC scaling (black line) and results from the Kraichnan model [58] (dashed line) as reported in [4] are also presented for reference.

Figure 4

Normalized probability density functions observed for increments of (a) longitudinal velocity, (b) vertical velocity, and (c) air temperature at large scales ($\tau =2$, top panels) and small scales ($\tau =0.05$, bottom panels). The figure includes data from runs in the strongly unstable class ($\zeta <-0.5$, shown in red) and near-neutral class ($\left|\zeta \right|<0.072$, blue). Black lines show the standard Gaussian distribution for reference.

Figure 5

Functions $q_0\left(\mathrm{\Delta }T\right)$ and $r_0\left(\mathrm{\Delta }T\right)$ estimated from the conditional derivatives of the original temperature time series, for the two classes of strongly unstable (red solid lines) and near-neutral runs (blue dashed lines). The same quantities are reported for phase-randomized surrogate time series for comparison (gray circles). Results are shown for the central body of the PDF (within $3\sigma$ from the mean) for illustration purposes. Results are computed for (a) and (b) a lag equal to the integral timescale of the flow $\tau =1$ and (c) and (d) the smaller time lag $\tau =0.1$. Black lines $q_0=1$ and $r_0=-\mathrm{\Delta }T$ correspond to the standard Gaussian distribution.

Figure 6

Evolution across scales $\tau$ of (a) the $q$-Gaussian tail parameter $q$, and of the stretched exponential shape parameter $\eta$ obtained from a separate fit to the (b) left and (c) right tails of the distribution of temperature increments. Data from two stability classes are included: strongly unstable ($\zeta <-0.5$, red circles) and near-neutral conditions ($\left|\zeta \right|<0.072$, blue triangles). Black lines and shaded areas indicate average values and standard deviations, respectively, computed over the entire data set.

Figure 7

Tail parameters of the PDF of temperature increments across stability conditions $\zeta$. Results include the $q$-Gaussian tail parameter $q$ [column (a)] and the stretched exponential shape parameter $\eta$, obtained from fitting the left [column (b)] and right [column (c)] tails of the distribution $p\left(\mathrm{\Delta }T\right)$. Values of $q$ and $\eta$ are reported for large scales ($\tau =5$, top panels) and small scales ($\tau =0.05$, bottom panels). Triangles denote strongly unstable runs ($\zeta <-0.5$), squares denote stable runs ($\zeta >0$), and circles refer to slightly unstable runs ($-0.5<\zeta <0$).

Figure 8

Normalized third-order structure functions $S\left(\tau \right)$ and $F\left(\tau \right)$ at the crossover from inertial to production scales. Vertical dashed lines indicate the integral timescales and horizontal solid lines show the constant values (a) 0.25 and (b) 0.4. Results are shown for near-neutral runs ($\left|\zeta \right|<0.072$, blue dashed lines), strongly unstable runs ($\left|\zeta \right|>0.5$, red solid lines), and runs with intermediate values of $\left|\zeta \right|$ (black dash-dotted lines).

Figure 9

Measures of (a) asymmetry $S_T^3$ and (b) time irreversibility $R_{\tau }$ computed for temperature increments for scales varying from $\tau =0.05$ to $\tau =5$. The plots include stable runs (black dashed lines), weakly unstable runs (blue dash-dotted lines), and strongly unstable runs (red solid lines). For reference, the same quantities are computed for phase-randomized time series (cyan) and synthetic time series with sawtooth positive (blue) and inverted (black) ramps. Shaded regions correspond to the $1\sigma$-confidence intervals over 34 realizations of the surrogate time series. The relative weight and mean duration of the synthetic ramps were set to $\alpha =0.4$ and $2I_w$, respectively.

Figure 10

(a) Mean and standard deviation over 34 time series of $\langle Z_{\tau }\rangle$ computed for scales varying from $\tau =0.05$ to $\tau =20$. Values of $\langle Z_{\tau }\rangle$ are shown for original temperature records (red) and surrogate time series obtained by phase randomization (green). For comparison, the same analysis is reported for fractional Brownian motion with Hurst exponent $H=1/3$ (blue). (b) Comparison of $\langle Z_{\tau }\rangle$ for temperature (red), longitudinal velocity (yellow), and vertical velocity (green). Also shown are the (c) Kolmogorov-Smirnov test average rejection rate and (d) average $P$ value computed for all the temperature time series (cyan for mean value and $1\sigma$-confidence interval) and for different stability classes: strongly unstable runs ($\zeta <-0.5$, red), near-neutral runs ($\left|\zeta \right|<0.072$, blue), and intermediate values ($0.072<\left|\zeta \right|<0.5$, black). The KS test was performed at the 0.05 significance level, corresponding to the horizontal line in (d). The vertical dashed line marks the integral timescale $I_w$.

Figure 11

(a) Quantity $I_s$ and its expected value 0.4 (black horizontal line) for the four stable runs in the data set. (b) Normalized Ozmidov length for the same runs.