# Small-Scale Anisotropy in Stably Stratified Turbulence; Inferences Based on Katabatic Flows

^{1}

^{2}

^{3}

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## Abstract

**:**

_{λ}(i.e., at Taylor Reynolds numbers) and, therefore, did not provide convincing evidence of anisotropy penetration into viscous sublayers. Nocturnal katabatic flows having configurations of stratified parallel shear flows and developing on mountain slopes provide high Reynolds number data for testing the notion of anisotropy at viscous scales, but obtaining appropriate time series of the data representing stratified shear flows devoid of unwarranted atmospheric factors is a challenge. This study employed the “in situ” calibration of multiple hot-film-sensors collocated with a sonic anemometer that enabled obtaining a 90 min continuous time series of a “clean” katabatic flow. A detailed analysis of the structure functions was conducted in the inertial and viscous subranges at an Re

_{λ}around 1250. The results of DNS simulations by Kimura and Herring were employed for the interpretation of data.

## 1. Introduction

_{λ}is larger, thus allowing for some comparisons. Additional references are given in the discussion chapter.

## 2. Methodology

_{devd}(22:50–23:10 MDT). This subinterval was considered as fully developed turbulence, since flow variations were modest, and appearance of small-scale bursting events [12] was very limited (less than 2% of the time).

## 3. Stratified Turbulence in Field Experiments and DNS

#### 3.1. Puzzling Observations in the MATERHORN Campaign

_{λ}was 1250, whereas in the DNS computations the Re

_{λ}was 264 and 383. While in the inertial subrange (r/η = 20 ÷ 200), both the qualitative and quantitative agreements were very good, in the viscous subrange (r/η ≤ 10) the results were conspicuously different, and the difference increased with the decrease in separation. It is worth noting that at small normalized separations r/η < 10, the scaling exponents were even of different signs for the field and DNS data. Since, in the viscous subrange, one can expect linear dependence between velocity and separation, e.g., [15], the scaling exponent for the conventional third-order structure function should be about zero to satisfy Kolmogorov’s Self-Similar Hypothesis (KSSH) [16].

_{3}divided by the second-order structure function L

_{2}to the power 1.5, e.g., ${H}_{3}(r)={L}_{3}(r)/{({L}_{2}(r))}^{3/2}\text{}$, which also represents the skewness of the structure function. In Figure 2, we separately consider the third- and second-order structure functions, both normalized according to KSSH [16].

^{1}. However, the second-order structure function in Figure 2b corresponds to a different type of dependence, namely, $\Delta u(x,r)$~r

^{5/6}. This explains the scaling exponent observed for the canonical third-order structure function in Figure 1.

^{0.87}for third-order and $\Delta u(x,r)$~r

^{0.91}, for second-order structure functions. It is worth noting that the expectation of identical behavior by odd and even structure functions is generally unjustified, despite it being widely quoted and often used, in particular, in the Extended Self-Similarity (ESS) approach. In previous research [4], we found that all even (second, fourth, and sixth in this study) structure functions yield $\Delta u(x,r)$~r

^{5/6}in the viscous subrange. Then, it was found that all the odd structure functions (first, third, and fifth in this study) constructed using the modulus of velocity increment $\left|\Delta u(x,r)\right|$ yield a similar relation, namely, $\Delta u(x,r)$~r

^{5/6}in the viscous subrange; here, $\Delta u(x,r)$ is the characteristic velocity increment at selected separation r. The consistent and significant difference between the power exponents of the third- and second-order structure functions is responsible for the substantial power exponent 0.5 at the small scales of the conventional third-order structure function/skewness. In isotropic DNS, this power exponent is about −0.1, which is much closer to zero, though of a different sign. This result is somewhat perplexing, given the expectation that, due to the local isotropy at small scales, the power exponent tends to be zero. Even more puzzling is that, in the inertial subrange, the behavior of the canonical third-order structure function in the DNS and field experiments shows full (qualitative and quantitative) agreement.

_{λ}~O(100) [18] or at a higher Re

_{λ}~O(1000) [19], which was again ascribed to shear penetrating to smaller scales. To quote [19], “The results show that PLI is untenable, both at the dissipation and inertial scales, at least to R

_{λ}~1000, and suggest it is unlikely to be so even at higher Reynolds numbers.” See more considerations in the Discussion section.

_{λ}~O(1000) or the DNS computations of stably stratified turbulence.

_{λ}(<100), and, thus, high-quality field experiments with continuous measurements at high sampling rates are called for. In fact, during the entire MATERHORN campaign, only a few records provided “clean” data sets for the nocturnal stably stratified turbulence strongly affected by thermal stratification. It should be stressed that these measurements were possible due to our novel calibration approach that enabled in situ calibration of a multi-hot-film probe based on the simultaneous measurements of low-frequency 3D-velocity data of a collocated sonic anemometer. Employing machine learning (neural network training) enabled the calibration of the hot-film probe in situ, thus avoiding problems with the hot-film’s potential deterioration in hostile field environments. The efficacy of this calibration method [21] was tested in a series of papers [22,23,24] and proved to be very efficient, even in the presence of noise to some degree.

_{λ}but also to the elegant methodology of velocity data analysis based on Craya–Herring decomposition. The latter approach allows for the separation of the entire oscillating flow into 2 types of modes: horizontal and vortical; vertical and wavy [2,23]. In Section 3.2, the puzzling results (the penetration of the anisotropy caused by stratification into small scales) of our study are discussed in light of [2].

#### 3.2. DNS of Stably Stratified Turbulence by Kimura and Herring 2012 [2]

_{2}(d, u, x) for the longitudinal velocity component u at separation d (r in our notations) in the x-direction can be presented as the superposition of two terms following Craya–Herring decomposition (expression 4.4 in [2]).

_{⊥}and k

_{z}components of vector number

**k**are in the horizontal plane and the vertical direction, respectively; and J

_{0}and J

_{1}denote the Bessel function of the corresponding order. While ${\Phi}_{1}$ contributes only to velocities in the horizontal plane and is determined by vertical vorticity component ω

_{z}, ${\Phi}_{2}$ is contributing to both horizontal and vertical velocity components and can be determined using only the vertical velocity component w, as presented in expressions 2.9 and 2.10, respectively [2]. Since, in a stratified flow, the vertical direction coincides with gravitational acceleration g, the second term of the decomposition may account for the internal waves. Indeed, the behavior of the second-order structure function due to the first term only, evaluated in [2], is practically the same as that of the isotropic turbulence (see Figure 14 in [2]), while adding the second term accounting for buoyancy leads to behavior of the second-order structure function that substantially differs from that of the isotropic turbulence.

#### 3.3. A Simplified Model

_{1}(x), representing pure HIT (isotropic and asymmetrical), and u

_{2}(x), the strongly stratified turbulence (anisotropic and symmetric, with respect to the vertical), with a low correlation between the two. The PDF of u

_{1}is essentially asymmetrical for the longitudinal velocity derivative, while the PDF of u

_{2}may be assumed to be symmetrical. It follows that the third-order structure function for $\Delta u(x,r)$ is determined by u

_{1}only.

_{3}(r), following above assumptions,

_{1}and u

_{2}, and the fourth term is zero due to the symmetry of the probability density function of u

_{2}, leaving the first term as the nonzero term. The situation is different for the higher-order odd longitudinal structure functions.

^{5/6}dependence (i.e., scaling exponent p*5/6; Figure 9 in [4]) as all the even structure functions. It is obvious that all the structure functions for the absolute velocity increments include both contributions (u

_{1}and u

_{2}). Therefore, the scaling exponent, in general, can be different from that of the homogeneous turbulence. Relatively weak anomalies only start to appear at p = 6. However, presently, we are unable to offer a sound explanation for the distinct shape (5/6 scaling exponent) in the above dependence.

## 4. Discussion

_{b}, the buoyancy Reynolds number, only. The authors also performed direct numerical simulations to investigate the anisotropy of stratified turbulence and the transition to isotropy at small length scales. Turbulence was generated by forcing large-scale vortical modes, an approach that is broadly consistent with geophysical stratified turbulence. The authors’ results suggest that Re

_{b}≥ 500 is required to obtained the same degree of small-scale isotropy seen in the unstratified turbulence at a similar Re.

**randomly forced at small scales**. Their interests are mainly related to the inverse cascade. The same remarks are applicable to the comprehensive study of Delache et al. [35], which dealt with anisotropy in freely decaying rotating turbulence.

_{λ}and, therefore, did not provide convincing evidence of anisotropy penetration into the viscous subrange. Measurements in the atmosphere during the MATERHORN project could provide such evidence as discussed in this paper, where the viscous subrange was accessed via a specialized (combo) probe, which is an assembly of a high (space-time)-resolution, multi-sensor hot-film probe array collocated with a sonic that measures the full velocity vector at a low-frequency resolution.

_{λ}around 1250, which can be considered substantial. The seminal paper of Kimura and Herring [2] on the DNS of stably stratified turbulence was employed for data interpretation.

_{λ}~1250. This result contradicts the hypothesis of the postulate of local isotropy (PLI), which posits enhanced isotropy, whence the separation (r/η) is diminishing. In fact, Cambon et al. [17] predicted kindred behavior when external anisotropic forces, such as rotation (Coriolis) and stable stratification (buoyancy), do not produce turbulent energy.

_{z}and the internal waves determined by vertical velocity component u

_{z.}While the former term is essentially isotropic on the horizontal plane, the latter term is responsible for anisotropy, which we used to qualitatively interpret the observations.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Conflicts of Interest

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**Figure 2.**Normalized according to KSSH third-order (

**a**) and second-order (

**b**) structure functions obtained from the field data, computed using longitudinal velocity increment $\Delta u(x,r)$.

**Figure 3.**Second-order structure functions for squared normalized buoyancy frequency N

^{2}= 1, 10, 50, and 100 as a function of separation d. The assessed power exponent in viscous subrange is about 5/3. (Reproduced with permission from Kimura and Herring [2], J. Fluid Mech. 698, 19. Copyright 2012, Cambridge University Press.)

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**MDPI and ACS Style**

Kit, E.; Fernando, H.J.S.
Small-Scale Anisotropy in Stably Stratified Turbulence; Inferences Based on Katabatic Flows. *Atmosphere* **2023**, *14*, 918.
https://doi.org/10.3390/atmos14060918

**AMA Style**

Kit E, Fernando HJS.
Small-Scale Anisotropy in Stably Stratified Turbulence; Inferences Based on Katabatic Flows. *Atmosphere*. 2023; 14(6):918.
https://doi.org/10.3390/atmos14060918

**Chicago/Turabian Style**

Kit, Eliezer, and Harindra J. S. Fernando.
2023. "Small-Scale Anisotropy in Stably Stratified Turbulence; Inferences Based on Katabatic Flows" *Atmosphere* 14, no. 6: 918.
https://doi.org/10.3390/atmos14060918