The O’KEYPS Equation and 60 Years Beyond

Li, Dan

doi:10.1007/s10546-020-00585-y

Cited by 10 publications

(6 citation statements)

References 105 publications

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“…The streamwise and vertical turbulence integral length scales are obtained by fitting exponential functions to the autocorrelation as (Salesky et al., 2013):

\begin{align*}{R}_{uu}({\increment}x,0)=\mathrm{exp}\left(-\frac{\vert {\increment}x\vert }{{L}_{h}}\right),\end{align*}

\begin{align*}{R}_{ww}(0,{\increment}z)=\mathrm{exp}\left(-\frac{\vert {\increment}z\vert }{{L}_{v}}\right),\end{align*}

where R uu (∆ x ,0) and R ww (0,∆ z ) denote the streamwise and vertical autocorrelation which are calculated as

{R}_{\alpha \alpha }({\Delta }x,{\Delta }z)=\frac{\overline{{\alpha }^{\prime }(x+{\Delta }x,z+{\Delta }z){\alpha }^{\prime }(x,z)}}{\sqrt{\overline{{\alpha }^{\prime }{(x+{\Delta }x,z+{\Delta }z)}^{2}}}\sqrt{\overline{{\alpha }^{\prime }{(x,z)}^{2}}}}

with α denoting u and w , ∆ x , and ∆ z denote the streamwise and vertical spatial lags, and

{L}_{h}

and

{L}_{v}

denote the streamwise and vertical integral length scales (Li, 2020; Salesky et al., 2013). An example of deriving ...…”

Section: Methodsmentioning

confidence: 99%

“…with α denoting u and w, ∆x, and ∆z denote the streamwise and vertical spatial lags, and 𝐴𝐴 𝐴𝐴ℎ and 𝐴𝐴 𝐴𝐴𝑣𝑣 denote the streamwise and vertical integral length scales (Li, 2020;Salesky et al, 2013). An example of deriving 𝐴𝐴 𝐴𝐴𝑣𝑣 through fitting is given in Figure S1 in Supporting Information S1.…”

Section: Data Processingmentioning

confidence: 99%

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A Non‐Dimensional Index for Characterizing the Transition of Turbulence Regimes in Stable Atmospheric Boundary Layers

Shao,

Zhang,

et al. 2023

Geophysical Research Letters

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The transition from moderate to weak turbulence regimes remains a grand challenge for stable boundary layer parameterizations in weather and climate models. In this study, a critical horizontal Froude number (≈0.28) is proposed to characterize such a transition, which corresponds to the development of quasi two‐dimensional pancake vortices. Traditionally defined stability parameters corresponding to the critical horizontal Froude number are estimated and are consistent with values in the literature. The critical horizontal Froude number can recover previously used height‐ and site‐dependent mean wind speed thresholds. These findings offer a way to constrain the validity range of Monin‐Obukhov similarity theory in numerical models for weather and pollutants dispersion.

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“…The streamwise and vertical turbulence integral length scales are obtained by fitting exponential functions to the autocorrelation as (Salesky et al., 2013):

\begin{align*}{R}_{uu}({\increment}x,0)=\mathrm{exp}\left(-\frac{\vert {\increment}x\vert }{{L}_{h}}\right),\end{align*}

\begin{align*}{R}_{ww}(0,{\increment}z)=\mathrm{exp}\left(-\frac{\vert {\increment}z\vert }{{L}_{v}}\right),\end{align*}

where R uu (∆ x ,0) and R ww (0,∆ z ) denote the streamwise and vertical autocorrelation which are calculated as

{R}_{\alpha \alpha }({\Delta }x,{\Delta }z)=\frac{\overline{{\alpha }^{\prime }(x+{\Delta }x,z+{\Delta }z){\alpha }^{\prime }(x,z)}}{\sqrt{\overline{{\alpha }^{\prime }{(x+{\Delta }x,z+{\Delta }z)}^{2}}}\sqrt{\overline{{\alpha }^{\prime }{(x,z)}^{2}}}}

with α denoting u and w , ∆ x , and ∆ z denote the streamwise and vertical spatial lags, and

{L}_{h}

and

{L}_{v}

denote the streamwise and vertical integral length scales (Li, 2020; Salesky et al., 2013). An example of deriving ...…”

Section: Methodsmentioning

confidence: 99%

Section: Data Processingmentioning

confidence: 99%

A Non‐Dimensional Index for Characterizing the Transition of Turbulence Regimes in Stable Atmospheric Boundary Layers

Shao,

Zhang,

et al. 2023

Geophysical Research Letters

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“…Ψ m = z z0 [1 − Φ m (z ′ /L)]d ln z ′ is the M-O wind speed correction function; the analytic forms for Φ m and Ψ m differ in stable and unstable conditions, and have been determined empirically in decades past (Businger et al, 1971;Carl et al, 1973;Li, 2021). But Monin-Obukhov similarity theory and its assumptions (such as constant u * ), as well as established forms for Φ m , fail above the surface layer; 3 this motivates use of α in applications such as wind energy, as (1) does not directly rely on surface-layer assumptions.…”

Section: Shear Exponentmentioning

confidence: 99%

From shear to veer: theory, statistics, and practical application

Kelly

Laan

2023

Preprint

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Abstract. In the past several years, wind veer — sometimes called `directional shear' — has begun to attract attention due to its effects on wind turbines and their production, particularly as the length of manufactured turbine blades has increased. Meanwhile, applicable meteorological theory has not progressed significantly beyond idealized cases for decades, though veer's effect on the wind speed profile has been recently revisited. On the other hand the shear exponent (α) is commonly used in wind energy for vertical extrapolation of mean wind speeds, as well as being a key parameter for wind turbine loads calculations and design standards. In this work we connect the oft-used shear exponent with veer, both theoretically and for practical use. We derive relations for wind veer from the equations of motion, finding the veer to be composed of separate contributions from shear and vertical gradients of cross-wind stress. Following from the theoretical derivations, which are neither limited to the surface-layer nor constrained by assumptions about mixing length or turbulent diffusivities, we establish simplified relations between the wind veer and shear exponent for practical use in wind energy. We also elucidate the source of commonly-observed stress-shear misalignment and its contribution to veer, noting that our new forms allow for such misalignment. The connection between shear and veer is further explored through analysis of one-dimensional (single-column) Reynolds-averaged Navier-Stokes solutions, where we confirm our theoretical derivations as well as the dependence of mean shear and veer on surface roughness and atmospheric boundary layer depth in terms of respective Rossby numbers. Finally we investigate the observed behavior of shear and veer across different sites and flow regimes (including forested, offshore, and hilly terrain cases) over heights corresponding to multi-megawatt wind turbine rotors, also considering the effects of atmospheric stability. From this we find empirical forms for the probability distribution of veer during high-veer (stable) conditions, and for the variability of veer conditioned on wind speed. Analyzing observed joint probability distributions of α and veer, we compare the two simplified forms we derived earlier and adapt them to ultimately arrive at more universally applicable equations to predict the mean veer in terms of observed (i.e., conditioned on) shear exponent; lastly, the limitations, applicability, and behavior of these forms is discussed along with their use and further developments for both meteorology and wind energy.

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“…Turbulence kinetic energy cascades from larger to smaller eddies until it dissipates at the dissipative scales (Kolmogorov, 1941; Richardson, 1920). This concept of direct energy cascade forms the cornerstone of conventional atmospheric boundary layer theories, such as Monin‐Obukhov similarity theory (Katul et al., 2013; Li, 2020; Monin & Obukhov, 1954) and the scaling behavior of streamwise turbulent velocity variance (Banerjee et al., 2016) which enable parameterizations for momentum flux and turbulence kinetic energy in numerical models. Due to the exceptional performance of the concept of direct energy cascade in the conventional atmospheric boundary layers, these theories have been directly extended to describe typhoon boundary layers (Momen et al., 2021; Montgomery & Smith, 2017; Smith & Montgomery, 2010; Zhu et al., 2010, 2021).…”

Section: Introductionmentioning

confidence: 99%

A Physical Model for the Observed Inverse Energy Cascade in Typhoon Boundary Layers

Shao,

Zhang,

Tang

2023

Geophysical Research Letters

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The knowledge of the observed inverse energy cascade in typhoon boundary layers holds significant implications for understanding the formation, enhancement, and deformation of typhoons. This study reveals that the observed inverse energy cascade originates from the rapid rotation of typhoons. The transition from the direct energy cascade regime to the inverse energy cascade regime can be characterized by the Zeman length scale. The turbulence structures behave as two dimensional above the Zeman length scale. Through symmetry analysis, it is discovered that the ratio of the inverse energy cascade flux to the direct energy cascade flux is proportional to the turbulence Rossby number with a power of −2. These findings offer a framework for incorporating the inverse energy cascade into the turbulence parameterizations and improving typhoon modeling.

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The O’KEYPS Equation and 60 Years Beyond

Cited by 10 publications

References 105 publications

A Non‐Dimensional Index for Characterizing the Transition of Turbulence Regimes in Stable Atmospheric Boundary Layers

A Non‐Dimensional Index for Characterizing the Transition of Turbulence Regimes in Stable Atmospheric Boundary Layers

From shear to veer: theory, statistics, and practical application

A Physical Model for the Observed Inverse Energy Cascade in Typhoon Boundary Layers

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