1 Introduction

Turbulent motion, the predominant form of atmospheric motion within the atmospheric boundary layer (ABL), plays a crucial role in the exchange between the earth surface and the atmosphere (Caughey et al. 1979; Stull 1988; Garratt 1994; Bo 2021). Theories of atmospheric turbulence serve as the fundamental basis for research in the atmospheric boundary layer and the atmospheric environment. Among these theories, the method of nondimensionalization stands out as a practical and significant approach. The primary concept of nondimensionalization involves normalizing specific variables in fluid equations. This process simplifies the mathematical model of physical processes and uncovers new characteristic scales (Blake et al. 1990; Alhama et al. 2012). Unlike the method of dimensional analysis, which is used to derive similarity relationships (Catalano et al. 2012; Fan et al. 2021; Vulfson and Nikolaev 2022) and involves manually selecting and combining controlling variables (Batchelor 1954), nondimensionalization introduces characteristic scales naturally through dimensional balance. This approach endows these scales with greater physical significance. In recent years, the method of nondimensionalization has gained recognition for its efficacy in addressing practical problems, including the derivation of characteristic scales and governing equations (Sánchez et al. 2017; Alhama et al. 2018). This underscores its potential applicability in the field of atmospheric turbulence.

Typical achievements of researches concerning atmospheric turbulence include the K41 theory on the turbulence energy spectrum and structure functions (Kolmogorov 1941), the Monin–Obukhov (M–O) similarity theory and its variants on turbulence characteristic quantities with stability parameter (Monin and Obukhov 1954; Obukhov 1971; Businger et al. 1971; Yaglom 1977; Businger 1988; Jacovides et al. 1992; Zilitinkevich and Calanca 2000; Zilitinkevich and Esau 2007, etc.), and there are also numerous works on the turbulence simulations and the use of similarity theory to develop parameterizations used in climate models (Yamada and Mellor 1975; Stull 1984; Foken 2008; Akon and Kopp 2018; Breedt et al. 2018; Xu et al. 2019, etc.).Among those works, M–O similarity has the merit of providing relationships between turbulence statistics and stratification, and thus is widely used in turbulence research. However, the application of M–O similarity theory tends to be limited, as a result of its strict assumptions, such as steady motions and flat underlying surfaces (Stull 1988; Garratt 1994). In real-world scenarios, these assumptions are typically not met. Instead, they are approximated using detrending algorithms and double coordination rotations (McMillen 1988; Rannik and Vesala 1999; Moncrieff et al. 2005). However, these methods also present certain drawbacks (Kaimal and Finnigan 1994; Aubinet et al. 2012). Moreover, under stable stratifications, where turbulence and the land surface become decoupled (Derbyshire 1990, 1995a, 1995b; Mahrt and Vickers 2002; Ohya and Uchida 2008), the characteristic quantities of turbulence can no longer be universally described by a specific function of the stability parameter. The problem has also been put forward in the appliance of M–O similarity theory in non-hydrostatic models, where the existence of viscous sublayer makes a difference, and the land–atmosphere inconsistency can profoundly affect the processes within boundary layer (Janjić, 1994; Chen et al. 1997; Janjic 2019).

Several reasons concerning the invalidation of M–O similarity theory have been put forward, such as the precision and reliability of observation, and the practicality of the theory. Advanced observational techniques, such as the instruments across multiple layers (Lüers and Bareiss 2011; Satyanarayana et al. 2014; Huo et al. 2015; Muhsin et al. 2016), the sophisticated observation schemes (Varshney and Poddar 2011; Ren et al. 2019), and diverse underlying surfaces (Zhang et al. 2014; Wei et al. 2021; Bo et al. 2022), enriches the turbulence statistics and depicts its fine structures. Given the absence of general analytical solutions for Navier–Stokes equations and the prohibitive cost of numerical simulations (Samaali et al. 2007; Snaiki and Wu 2019), characteristic scales in M–O similarity theory, such as the Obukhov length, were usually introduced through dimensional analysis, which integrates the Prandtl number to reflect wind shear and buoyancy effects (Yaglom 1977; Foken 2006). Discussions on incorporating virtual and potential temperatures (Businger 1988), and alternative characteristic lengths—rotational scales (Zilitinkevich 1972; Zilitinkevich and Mironov 1996), local flux-based lengths (Nieuwstadt 1984a; 1984b), and gradient-based scales (Sorbjan 2006, 2010, 2016), highlight the ongoing evolution of the theory. However, not all the variables concerning the atmospheric boundary layer are included in a single length scale, and it usually needs a series of strict assumptions to apply empirical relationships, which makes it uncertain whether the information from Obukhov length alone is enough in similarity theory to describe all the characteristics of turbulence.

In this context, the method of nondimensionalization holds significant potential for advancing the understandings on atmospheric turbulence. The utilization of this method within the realms of atmospheric and oceanic science can be traced back to the inception of the Rossby deformation radius. This involved the decomposition of characteristic scales and physical quantities for a specific variable, which leads to the acquisition of a final expression achieved through a balance of dimensions in the governing equations (Pedlosky 1987). There are also trials with similar methodology in finding new characteristic scales and similarity relationships (George et al. 2000; Johansson et al. 2003; Yano and Bonazzola 2009; van der Laan et al. 2020). In the domain of atmospheric boundary layer, Yano and Wacławczyk (2022) analyzed controlling equations of turbulence, decomposed the variables, combined the characteristic scales under certain rules, and finally got concise dimensionless equations and several newly defined characteristic scales. From the aforementioned studies, it can be inferred that one of the key benefits of nondimensionalization is the objective selection of characteristic scales and dimensionless parameters. This approach can complement physical intuition and serve as a viable alternative. However, current research primarily focuses on theoretical analysis, and some assumptions lack a solid physical foundation. Consequently, there is a pressing need for comprehensive quantitative analyses and in-depth discussions.

In this study, we examine the Reynolds-decomposed turbulence equations, which bear resemblance to the work of Yano and Wacławczyk (2022). However, we approach the nondimensionalization method from a distinct perspective. In Section 2, we nondimensionalize the turbulence equations, defining dimensionless parameters such as tensors \({\varvec B}\), \({\varvec{\Gamma}}\), and \({\varvec T}\), and their physical significance is thoroughly discussed. Additionally, we demonstrate that essential dimensionless parameters, including the aspect ratio \(\boldsymbol{\alpha }\) and the Reynolds number \(Re\), play analogous roles to those described by existing theories. The characteristic length scale, \({z}^{s}\), is also introduced from the variants of parameter \({\varvec B}\). Section 3 introduces the data used to perform quantitative analysis. The observational data are selected from the station in Horqin Sandy Land (for its homogeneous and flat underlying surface) from June to August, 2019, and the characteristic scales are pre-defined from standard deviation and amplitudes of observed variables. In Section 4, the similarity relationships between turbulence statistics and \({z}^{s}\)-based stability parameters are investigated to explore the prospect of \({z}^{s}\) in similarity theory. Through the evaluation on the time series of newly introduced dimensionless parameters, some novel properties of atmospheric turbulence are revealed and discussed, and new similarity relationships that complements the M–O theory are established, which offers a practical approach for estimating turbulent fluxes. Section 5 provides the conclusions and prospects of the method of nondimensionalization.

2 Nondimensionalization

2.1 Transformation of turbulence equations

As outlined in Section 1, the primary objective of nondimensionalizing the ABL system is to identify novel characteristic scales that can replace traditional ones, such as the Obukhov length in similarity theories. Consequently, a reanalysis of turbulence equations is necessitated. Turbulence equations are typically characterized as Reynolds-decomposed atmospheric equations, from which averaged equations are subtracted. Under the assumption that air motion is an incompressible flow of an ideal gas, and with the application of the Boussinesq approximation, the governing equations of atmospheric turbulence can be expressed through a series of fluctuating quantities:

$$\frac{\partial {{u}_{i}}^{\prime}}{\partial t}+{\overline{u} }_{j}\frac{\partial {{u}_{i}}^{\prime}}{{\partial x}_{j}}+{{u}_{j}}{\prime}\frac{\partial {\overline{u} }_{i}}{{\partial x}_{j}}+\frac{\partial }{{\partial x}_{j}}\left({{u}_{i}}^{\prime}{{u}_{j}}^{\prime}-\overline{{{u }_{i}}^{\prime}{{u}_{j}}^{\prime}}\right)=-\frac{1}{\overline{\rho }}\frac{\partial {p}^{\prime}}{{\partial x}_{i}}+{\delta }_{i3}g\frac{{\theta }^{\prime}}{{\theta }_{0}}-2{\varepsilon }_{ijk}{\omega }_{j}{{u}_{k}}^{\prime}+\nu \frac{{\partial }^{\prime}{{u}_{i}}^{\prime}}{{\partial {x}_{j}}^{\prime}},$$
(1a)
$$\frac{\partial {\theta }^{\prime}}{\partial t}+{\overline{u} }_{j}\frac{\partial {\theta }^{\prime}}{{\partial x}_{j}}+{{u}_{j}}^{\prime}\frac{\partial \overline{\theta }}{{\partial x}_{j}}+\frac{\partial }{{\partial x}_{j}}\left({{u}_{j}}^{\prime}{\theta }^{\prime}-\overline{{{u }_{j}}^{\prime}{\theta }^{\prime}}\right)={\nu }_{\kappa }\frac{{\partial }^{2}{\theta }^{\prime}}{{\partial {x}_{j}}^{2}},$$
(1b)
$$\frac{\partial {{u}_{i}}^{\prime}}{\partial {x}_{i}}=0,$$
(1c)
$$\frac{{\rho }^{\prime}}{\overline{\rho }}=-\frac{{\theta }^{\prime}}{\overline{\theta }}.$$
(1d)

where \({x}_{i}=x, y, z\) are the directions of prevailing wind, cross wind, and vertical wind, \({u}_{i}=u, v, w\) are the corresponding wind speeds, \(t\) is the time, \(\theta\) is the potential temperature, \({\theta }_{0}\) is the reference value of \(\theta\), \(\rho\) is the density of air, and \(p\) is the air pressure. The parameter \(g\) is the numerical value of gravitational acceleration, \({\omega }_{j}\) is the component of Earth angular velocity vector, \(\nu\) is the molecular viscosity, and \({\nu }_{\kappa }\) is the thermal diffusivity. The \({\varepsilon }_{ijk}\) is Levi–Civita symbol, the overline denotes the mean value, and the prime denotes the fluctuated value.

In order to nondimensionalize these equations, another principle is required to further decompose them, which can be called “scale decomposition” (Pedlosky 1987; Yano and Bonazzola 2009; Yano and Wacławczyk 2022):

$${V}^{\prime}={V}^{s}\cdot {V}^{\dag},$$
(2a)
$$\overline{V }={\overline{V} }^{s}\cdot {\overline{V} }^{\dag},$$
(2b)
$$\frac{\partial }{\partial t}=\frac{1}{{t}^{s}}\cdot \frac{\partial }{{\partial t}^{\dag}},$$
(2c)
$$\frac{\partial }{{\partial x}_{j}}=\frac{1}{{{x}_{j}}^{s}}\cdot \frac{\partial }{{{\partial x}_{j}}^{\dag}},$$
(2d)
$$\frac{\partial \overline{V}}{{\partial x }_{j}}={\left(\frac{\partial \overline{V}}{{\partial x }_{j}}\right)}^{s}\cdot {\left(\frac{\partial \overline{V}}{{\partial x }_{j}}\right)}^{\dag}$$
(2e)

where \(V\) stands for basic meteorological variables (e.g., wind speed, temperature, pressure, etc.). The superscript “\(s\)” designates positive-definite characteristic scales of the variables, which have the same dimensions as the original variables, and “\(\dag\)” designates a nondimensionalized physical variable, with only the fluctuation pattern remained. One can draw an analogy to this process: a simple harmonic motion can be decomposed to an amplitude (characteristic scale) and a unit trigonometric function (nondimensionalized variable). As there are only partial derivatives left for nondimensionalized physical variables denoted by “\(\dag\)”, therefore, there would not be extra derivatives on scales denoted by “\(s\)”, unless the characteristic scales themselves are defined by derivatives.

Note that for averaged basic variables, the decomposition of spatial gradients of takes different form as Eq. (2e). This is because the gradients of averaged wind speed and potential temperature are dominated by processes larger than atmospheric turbulence (for example, \(\frac{\partial \overline{u}}{\partial z }\) depends more on the synoptic condition, and \(\frac{\partial \overline{\theta }}{\partial z}\) is determined by the solar radiation). The spatial gradients of other variables (such as covariance), however, take the same form as fluctuations as Eq. (2d) as they are closely linked to the characteristics of atmospheric turbulence. Applying Eqs. (2) to (1), the turbulence equations can be expanded as:

$$\frac{{{u}_{i}}^{s}}{{t}^{s}}\frac{\partial {{u}_{i}}^{\dag}}{\partial {t}^{\dag}}+{{\overline{u} }_{j}}^{s}\frac{{{u}_{i}}^{s}}{{{x}_{j}}^{s}}\cdot {{\overline{u} }_{j}}^{\dag}\frac{\partial {{u}_{i}}^{\dag}}{\partial {{x}_{j}}^{\dag}}+{{u}_{j}}^{s}{\left(\frac{\partial {\overline{u} }_{i}}{{\partial x}_{j}}\right)}^{s}\cdot {{u}_{j}}^{\dag}{\left(\frac{\partial {\overline{u} }_{i}}{{\partial x}_{j}}\right)}^{\dag}+\frac{{\left({{u}_{i}}^{\prime}{{u}_{j}}^{\prime}\right)}^{s}}{{{x}_{j}}^{s}}\cdot \frac{\partial {\left({{u}_{i}}^{\prime}{{u}_{j}}^{\prime}\right)}^{\dag}}{\partial {{x}_{j}}^{\dag}}-\frac{{\left(\overline{{{u }_{i}}^{\prime}{{u}_{j}}^{\prime}}\right)}^{s}}{{{x}_{j}}^{s}}\cdot \frac{\partial {\left(\overline{{{u }_{i}}^{\prime}{{u}_{j}}^{\prime}}\right)}^{\dag}}{\partial {{x}_{j}}^{\dag}}=-\frac{1}{\overline{\rho }}\frac{{p}^{s}}{{{x}_{i}}^{s}}\cdot \frac{\partial {p}^{\dag}}{\partial {{x}_{i}}^{\dag}}+{\delta }_{i3}g\frac{{\theta }^{s}}{{\theta }_{0}}\cdot {\theta }^{\dag}-2{\varepsilon }_{ijk}{\omega }_{j}{{u}_{k}}^{s}\cdot {{u}_{k}}^{\dag}+\nu \frac{{{u}_{i}}^{s}}{{{{x}_{j}}^{s}}^{2}}\cdot \frac{{\partial }^{2}{{u}_{i}}^{\dag}}{\partial {{{x}_{j}}^{\dag}}^{2}},$$
(3a)
$$\frac{{\theta }^{s}}{{t}^{s}}\cdot \frac{\partial {\theta }^{\dag}}{\partial {t}^{\dag}}+{{\overline{u} }_{j}}^{s}\frac{{\theta }^{s}}{{{x}_{j}}^{s}}\cdot {{\overline{u} }_{j}}^{\dag}\frac{\partial {\theta }^{\dag}}{\partial {{x}_{j}}^{\dag}}+{{u}_{j}}^{s}{\left(\frac{\partial \overline{\theta }}{{\partial x}_{j}}\right)}^{s}\cdot {{u}_{j}}^{\dag}{\left(\frac{\partial \overline{\theta }}{{\partial x}_{j}}\right)}^{\dag}+\frac{{\left({{u}_{j}}^{\prime}{\theta }^{\prime}\right)}^{s}}{{{x}_{j}}^{s}}\cdot \frac{\partial {\left({{u}_{j}}^{\prime}{\theta }^{\prime}\right)}^{\dag}}{\partial {{x}_{j}}^{\dag}}-\frac{{\left(\overline{{{u }_{j}}^{\prime}{\theta }^{\prime}}\right)}^{s}}{{{x}_{j}}^{s}}\cdot \frac{\partial {\left(\overline{{{u }_{j}}^{\prime}{\theta }^{\prime}}\right)}^{\dag}}{\partial {{x}_{j}}^{\dag}}={\nu }_{\kappa }\frac{{\theta }^{s}}{{{{x}_{j}}^{s}}^{2}}\cdot \frac{{\partial }^{2}{\theta }^{\dag}}{\partial {{{x}_{j}}^{\dag}}^{2}},$$
(3b)
$$\frac{{{u}_{i}}^{s}}{{{x}_{i}}^{s}}\cdot \frac{\partial {{u}_{i}}^{\dag}}{\partial {{x}_{i}}^{\dag}}=0,$$
(3c)
$$\frac{{\rho }^{s}}{\overline{\rho }}{\rho }^{\dag}+\frac{{\theta }^{s}}{\overline{\theta }}{\theta }^{\dag}=0.$$
(3d)

According to the full expansion of the turbulence equations, all the elements in these equations are divided into dimensionless physical variables multiplied by a combination of characteristics scales, and the latter is to be further defined (see Section 2.3). However, an important premise of this decomposition has not been determined, namely the definition of all the characteristic scales.

2.2 Pre-definition of characteristic scales

The criteria of determining characteristic scales from the statistics of observational data turns out to be diverse, as is discussed in Section 1. In this study, the characteristic scales for average quantities can be defined straightforwardly by their absolute values:

$$\begin{array}{c}{{\overline{u} }_{i}}^{s}\equiv \left|{\overline{u} }_{i}\right|,\\ {\left(\frac{\partial {\overline{u} }_{i}}{{\partial x}_{j}}\right)}^{s}\equiv \left|\frac{\partial {\overline{u} }_{i}}{{\partial x}_{j}}\right|.\end{array}$$
(4a)

For fluctuated quantities, the scales should be able to be directly extracted, or indirectly combined from observational data. The characteristic scales of wind speed fluctuation \({{u}_{i}}^{s}\) can also be defined as

$${{u}_{i}}^{s}\equiv {\sigma }_{{u}_{i}},$$
(4b)

where \(\sigma\) is the observational standard deviation which can be directly extracted from observation, and is also positive definite as the scales in Eq. (4a). As a measure of the amount of variation of dispersion, standard deviation is able to reflect the characteristic of turbulence. There is also another kind of definition of characteristic scales based on the amplitude (or intensity) of turbulent motions extracted from the method of Hilbert-Huang transform (a method devoted to non-linear and non-stationary signals, which is adaptive without priori assumptions, introduced in Huang et al. 1996; Huang et al. 1998; Huang et al. 2008; the introduction and steps are given in Appendix. A), by which other potential features of turbulent motions might be unearthed. It is reasonable to use one definition at a time, rather than mixing them together in a single time of calculation.

The characteristic scales of potential temperature fluctuation \({\theta }^{s}\) and pressure fluctuation \({p}^{s}\) can also be accordingly defined as

$${\theta }^{s}\equiv {\sigma }_{\theta },$$
(4c)
$${p}^{s}\equiv {\sigma }_{p}.$$
(4d)

For the characteristic scales of covariance \({\left({{u}_{i}}^{\prime}{{u}_{j}}^{\prime}\right)}^{s}\) and \({\left(\overline{{{u }_{i}}^{\prime}{{u}_{j}}^{\prime}}\right)}^{s}\), one might take it for granted mistakenly that \({\left({{u}_{i}}^{\prime}{{u}_{j}}^{\prime}\right)}^{s}={\left(\overline{{{u }_{i}}^{\prime}{{u}_{j}}^{\prime}}\right)}^{s}={{u}_{i}}^{s}\cdot {{u}_{j}}^{s}\). However, in order to leave turbulent transport information in \({\left({{u}_{i}}^{\prime}{{u}_{j}}^{\prime}\right)}^{\dag}\) and \({\left(\overline{{{u }_{i}}^{\prime}{{u}_{j}}^{\prime}}\right)}^{\dag}\), the scales should have the same definition as

$${\left({{u}_{i}}^{\prime}{{u}_{j}}^{\prime}\right)}^{s}\equiv {\sigma }_{{{u}_{i}}^{\prime}{{u}_{j}}^{\prime}},$$
(5a)
$${\left(\overline{{{u }_{i}}^{\prime}{{u}_{j}}^{\prime}}\right)}^{s}\equiv \left|\overline{{{u }_{i}}^{\prime}{{u}_{j}}^{\prime}}\right|.$$
(5b)

Here \({\sigma }_{{{u}_{i}}^{\prime}{{u}_{j}}^{\prime}}\) refers to the standard deviation of \({{u}_{i}}^{\prime}{{u}_{j}}^{\prime}\) within a data sample (0.5 or 1 h in practice), and \(\overline{{{u }_{i}}^{\prime}{{u}_{j}}^{\prime}}\) is the averaged covariance in the data sample. The definition of characteristic scale of density fluctuation \({\rho }^{s}\) is easily determined by Eq. (3d). However, not all the scales can be directly defined. For example, \({t}^{s}\) cannot be defined according to standard deviation as time is independent variable of observations. The length scales \({{x}_{i}}^{s}\) are also difficult to define, as the scale of atmospheric turbulence is not directly measured. Under this circumstance, the combination of other scales can potentially provide reference on the definition of these scales.

2.3 Combination of characteristic scales

As is listed above, the fully expanded turbulence equations feature a series of combination of scales in front of dimensionless physical variables, the requirement of simplifying the equations promotes us to recombine these scales. To make it clear, defining the relationships among dimensionless physical variables can help eliminate the combination terms, and defining new dimensionless parameters can help simplify these terms.

Making Eqs. (3c) and (3d) satisfy this requirement is quite simple, because one can set:

$${\alpha }_{i}\equiv \frac{{{u}_{i}}^{s}}{{w}^{s}}, {\beta }_{i}\equiv \frac{{{u}_{i}}^{s}}{{{\overline{u} }_{i}}^{s}}, {\alpha }_{\theta \rho }\equiv \frac{{\rho }^{s}}{\overline{\rho }}\cdot \frac{\overline{\theta }}{{\theta }^{s}},$$
(6)

where \({\alpha }_{i}\) can be intuitively defined as aspect ratios revealing the comparative scale between horizontal and vertical fluctuations, as their relationship with the traditional aspect ratio \({\alpha }_{0}\) is \({{\alpha }_{x}}^{2}+{{\alpha }_{y}}^{2}={{\alpha }_{0}}^{2}\); \({\beta }_{i}\) is the turbulence intensity of different directions of motion; \({\alpha }_{\theta \rho }\) is a state parameter. The time scale can also be defined as:

$${t}^{s}\equiv \frac{{x}^{s}}{{\overline{u} }^{s}}=\frac{{y}^{s}}{{\overline{v} }^{s}}=\frac{{z}^{s}}{{\overline{w} }^{s}} ,$$
(7)

which takes the form similar to Taylor's hypothesis of frozen turbulence. However, this doesn’t require the assumption of \({\beta }_{i}\ll 1\), as the length scales are to be defined. In this way, the scale combination “\(\frac{{{u}_{i}}^{s}}{{{x}_{i}}^{s}}\)” in Eq. (3c) can be written as \(\frac{{u}^{s}}{{z}^{s}}\cdot \frac{{\overline{w} }^{s}}{{\overline{u} }^{s}}\), \(\frac{{v}^{s}}{{z}^{s}}\cdot \frac{{\overline{w} }^{s}}{{\overline{v} }^{s}}\), and \(\frac{{w}^{s}}{{z}^{s}}\cdot \frac{{\overline{w} }^{s}}{{\overline{w} }^{s}}\) and therefore the term “\(\frac{{\overline{w} }^{s}}{{z}^{s}}\)” can be eliminated with the remaining:

$${\beta }_{i}\cdot \frac{\partial {{u}_{i}}^{\dag}}{\partial {{x}_{i}}^{\dag}}=0,$$
(8a)

from which the equation is simplified with only one dimensionless parameter remained. Equation (3d) can also be rewritten in a simple form:

$${\alpha }_{\theta \rho }\cdot {\rho }^{\dag}+{\theta }^{\dag}=0.$$
(8b)

It’s also reasonable to combine \({\alpha }_{x}\), \({\alpha }_{y}\), and \({\alpha }_{z}\) in a unit vector \({{\varvec{E}}}_{\boldsymbol{\alpha }}\), but it involves the re-scaling of \({\varvec{u}}\), and is thus set aside. To deal with Eq. (3a), extracting a factor of “\(\frac{{{u}_{i}}^{s}}{{t}^{s}}\)” with the help of Eq. (7), and it can be transformed into a dimensionless equation, with a series of dimensionless parameters in front of dimensionless physical variables:

$$\frac{\partial {{u}_{i}}^{\dag}}{\partial {t}^{\dag}}+{{\overline{u} }_{j}}^{\dag}\frac{\partial {{u}_{i}}^{\dag}}{\partial {{x}_{j}}^{\dag}}+{\left(\frac{\partial {\overline{u} }_{i}}{{\partial x}_{j}}\right)}^{s}\cdot \frac{{{x}_{j}}^{s}}{{{u}_{i}}^{s}}\cdot \frac{{{u}_{j}}^{s}}{{{\overline{u} }_{j}}^{s}}\cdot {{u}_{j}}^{\dag}{\left(\frac{\partial {\overline{u} }_{i}}{{\partial x}_{j}}\right)}^{\dag}+\frac{{\left({{u}_{i}}^{\prime}{{u}_{j}}^{\prime}\right)}^{s}}{{{u}_{i}}^{s}{{\overline{u} }_{j}}^{s}}\cdot \frac{\partial {\left({{u}_{i}}^{\prime}{{u}_{j}}^{\prime}\right)}^{\dag}}{\partial {{x}_{j}}^{\dag}}-\frac{{\left(\overline{{{u }_{i}}^{\prime}{{u}_{j}}^{\prime}}\right)}^{s}}{{{u}_{i}}^{s}{{\overline{u} }_{j}}^{s}}\cdot \frac{\partial {\left(\overline{{{u }_{i}}^{\prime}{{u}_{j}}^{\prime}}\right)}^{\dag}}{\partial {{x}_{j}}^{\dag}}=-\frac{{p}^{s}}{\overline{\rho }{{u}_{i}}^{s}{{\overline{u} }_{i}}^{s}}\cdot \frac{\partial {p}^{\dag}}{\partial {{x}_{i}}^{\dag}}+{\delta }_{i3}g\frac{{\theta }^{s}}{{\theta }_{0}}\frac{{{x}_{i}}^{s}}{{{u}_{i}}^{s}{{\overline{u} }_{i}}^{s}}\cdot {\theta }^{\dag}-2{\varepsilon }_{ijk}{\omega }_{j}\frac{{{u}_{k}}^{s}}{{{u}_{i}}^{s}}\frac{{{x}_{i}}^{s}}{{{\overline{u} }_{i}}^{s}}\cdot {{u}_{k}}^{\dag}+\frac{\nu }{{{\overline{u} }_{i}}^{s}}\frac{{{x}_{i}}^{s}}{{{{x}_{j}}^{s}}^{2}}\cdot \frac{{\partial }^{2}{{u}_{i}}^{\dag}}{\partial {{{x}_{j}}^{\dag}}^{2}}.$$
(9)

For every term in Eq. (9), the combination of characteristic scales with superscript “s” can be defined as a new parameter to mathematically transform it into a concise and dimension-balanced form. For example, the combination in the second term can be defined as with only a one parameters in front of the dimensionless variable. Following this rule, the newly introduced parameters can be defined as:

$${\text{B}}_{\text{ij}}\equiv {\left(\frac{\partial {\overline{u} }_{i}}{{\partial x}_{j}}\right)}^{s}\cdot \frac{{{x}_{j}}^{s}}{{{u}_{i}}^{s}}\cdot \frac{{{u}_{j}}^{s}}{{{\overline{u} }_{j}}^{s}}, {\Gamma }_{\text{ij}}\equiv \frac{{\left({{u}_{i}}{\prime}{{u}_{j}}{\prime}\right)}^{s}}{{{u}_{i}}^{s}{{\overline{u} }_{j}}^{s}}, {\text{T}}_{\text{ij}}\equiv \frac{{\left(\overline{{{u }_{i}}{\prime}{{u}_{j}}{\prime}}\right)}^{s}}{{{u}_{i}}^{s}{{\overline{u} }_{j}}^{s}},$$
(10a)
$${F}_{j}\equiv {\omega }_{j}\frac{}{}\frac{{{x}_{i}}^{s}}{{{u}_{i}}^{s}{\overline{u}_i}^{s}},Re\equiv \frac{{\overline{w} }^{s}{z}^{s}}{\nu },$$
(10b)
$${\alpha }_{p,i}\equiv -\frac{1}{\overline{\rho }}\frac{{p}^{s}}{{{u}_{i}}^{s}{{\overline{u} }_{i}}^{s}}, {\alpha }_{\nu ,j}\equiv {\left(\frac{{\overline{{u }_{j}}}^{s}}{\overline{w} }\right)}^{-2}, {\alpha }_{\theta }\equiv g\frac{{\theta }^{s}}{{\theta }_{0}}\frac{{z}^{s}}{{w}^{s}{\overline{w} }^{s}},$$
(10c)

where tensor \({\varvec B}\), \({\varvec{\Gamma}}\), and \({\varvec T}\) are considered as key parameters in nondimensionalization, and will be analyzed in detail in the following sections; vector \(\varvec F\) can be called geostrophic parameter linked with Coriolis effect, which is ignored for its small order of magnitude; \(Re\) is a new expression of the Reynolds number; \({\alpha }_{\nu ,j}\) is the reverse square of the aspect ratio of average wind; vector \({\boldsymbol{\alpha}_p}\) can be called pressure parameter, as it is defined by a proportion of static and dynamic pressure; \({\alpha }_{\theta }\) can be called buoyancy parameter, as it describes the effects of temperature disturbance. Equation (3a) can thereby be derived into a vector equation:

$$\frac{\partial {\varvec u}^\dagger}{\partial t^\dagger}+\overline {\varvec u}^\dagger\cdot\left(\nabla^\dagger {\varvec u}^\dagger\right)+{\varvec u}^\dagger\cdot\left[\mathrm {\varvec B}\circ\left(\nabla^\dagger\overline {\varvec u}^\dagger\right)^T\right]+\nabla^\dagger\cdot\left[{\varvec \Gamma}\circ\left({\varvec u}^\dagger {\varvec u}^\dagger\right)\right]-\nabla^\dagger\cdot\left[\mathrm {\varvec T}\circ\left(\overline{{\varvec u}^\dagger {\varvec u}^\dagger}\right)\right]={\boldsymbol\alpha}_p\cdot\left(\nabla^\dagger p^\dagger\right)+\frac1{Re}\nabla^\dagger\cdot\left[\nabla^\dagger\left({\boldsymbol\alpha}_v\circ\overline {\varvec u}^\dagger\right)\right]+\alpha_\theta\theta^\dagger k-2 {{\varvec F}}\times {\varvec u}^\dagger,$$
(11)

where the operator “\(\circ\)” denotes Hadamard product of 2 matrices (Styan 1973), the superscript “\(T\)” denotes the transpose matrix, the bold styled variables are vectors, and a pair of neighboring 2 vectors denotes a dyadic tensor (including neighboring ∇ and vector, Jeffreys and Jeffreys 1999; Tai 1997). Note that \({\nabla }^{\dag}\) only acts on variables with superscript “\(\dag\)”, for the 4th and 5th term at the left side and the 2nd term at the right side of Eq. (11). In contrast, the original vector equation of the atmosphere is:

$$\frac{\partial {\varvec{u}}}{\partial t}+{\varvec{u}}\cdot \left(\nabla {\varvec{u}}\right)=-\frac{1}{\rho }\nabla p+\nu {\nabla }^{2}{\varvec{u}}-g{\varvec{k}}-2{\varvec{\omega}}\times {\varvec{u}}.$$
(12)

Compare Eq. (11) to (12), the difference exists only in the combination of \({{\varvec{u}}}^{\dag}\), \({\overline{{\varvec{u}}} }^{\dag}\), \({\nabla }^{\dag}\), and 3 tensors \({\varvec B}\), \({\varvec{\Gamma}}\), and \({\varvec T}\) controlling them. Repeat this work to Eq. (3b), by defining:

$${\beta }_{\theta ,j}\equiv {\left(\frac{\partial \overline{\theta }}{{\partial x}_{j}}\right)}^{s}\cdot \frac{{{x}_{j}}^{s}}{{\theta }^{s}}\cdot \frac{{{u}_{j}}^{s}}{{{\overline{u} }_{j}}^{s}}, {\gamma }_{\theta ,j}\equiv \frac{{\left({{u}_{j}}{\prime}{\theta }{\prime}\right)}^{s}}{{{\overline{u} }_{j}}^{s}{\theta }^{s}}, {\tau }_{\theta ,j}\equiv \frac{{\left(\overline{{{u }_{j}}{\prime}{\theta }{\prime}}\right)}^{s}}{{{\overline{u} }_{j}}^{s}{\theta }^{s}},$$
(13a)
$$Pr\equiv \frac{\nu }{{\nu }_{\kappa }},$$
(13b)

where \(Pr\) is the new expression of Prandtl number, and \({\beta }_{\theta ,j}\), \({\gamma }_{\theta ,j}\), and \({\tau }_{\theta ,j}\) are parameters concerning the thermodynamics of the turbulence which will be discuss in the following sections. Based on these definitions, Eq. (3b) can also be written in vector form:

$$\frac{\partial\theta^\dagger}{\partial t^\dagger}+\overline u^\dagger\cdot\left(\nabla^\dagger\theta^\dagger\right)+u^\dagger\cdot\left[\beta_\theta\circ\left(\nabla^\dagger\overline\theta^\dagger\right)\right]+\nabla^\dagger\cdot\left[\gamma_\theta\circ\left(u^\dagger\theta^\dagger\right)\right]-\nabla^\dagger\cdot\left[\tau_\theta\circ\left(\overline{u^\dagger\theta^\dagger}\right)\right]=\frac1{Pr\cdot Re}\nabla^\dagger\cdot\left[{\boldsymbol\alpha}_v\circ\left(\nabla^\dagger\theta^\dagger\right)\right],$$
(14)

from which 3 elements controlled by vectors \({{\varvec{\beta}}}_{\theta}\), \({{\varvec{\gamma}}}_{\theta}\), and \({{\varvec{\tau}}}_{\theta}\) can also be extracted from the original thermodynamic equation:

$$\frac{\partial \theta }{\partial t}+{\varvec{u}}\cdot \left(\nabla \theta \right)={\nu }_{\kappa }{\nabla }^{2}\theta ,$$
(15)

2.4 Estimating the length scale

The most important parameter in M–O similarity theory is the characteristic length, namely the Obukhov length. As there are still 3 length scales, namely \({x}^{s}\), \({y}^{s}\), and \({z}^{s}\), left to be determined during the process of nondimensionalization, whether they can be used to substitute Obukhov length in M–O theory is under investigation. In Section 2.3, there are various newly defined parameters, such as \({F}\) describing the effects of geostrophic force, which is usually neglected in observational practice; \(Re\) is Reynolds number, which is useful in telling the condition of flow and comparing different theories, rather than in direct use. Here, to estimate the characteristic length, the information in \({\varvec B}\) and \({{\varvec{\beta}}}_{\theta}\) is considered to be worth investigation.

The tensor \({{\varvec B}}\) is defined through nondimensionalizing the additional advection term of momentum, and it describes the intensity of wind shear. In practical observations, the underlying surface is flat, and it is inevitably assumed that there is only \(\frac{\partial }{\partial z}\) left in tensor \({\varvec B}\), making it degenerate to a vector:

$${\varvec B}={\varvec{\beta}}={\left(\frac{\partial \overline{u}}{\partial z }\right)}^{s}\cdot \frac{{z}^{s}}{{u}^{s}}\cdot \frac{{w}^{s}}{{\overline{w} }^{s}}{\varvec{i}}+{\left(\frac{\partial \overline{v}}{\partial z }\right)}^{s}\cdot \frac{{z}^{s}}{{v}^{s}}\cdot \frac{{w}^{s}}{{\overline{w} }^{s}}{\varvec{j}}+{\left(\frac{\partial \overline{w}}{\partial z }\right)}^{s}\cdot \frac{{z}^{s}}{{\overline{w} }^{s}}{\varvec{k}}.$$
(16)

As one of the targets of nondimensionalization is to simplify the controlling equations, one can extract \({z}^{s}\cdot \frac{{w}^{s}}{{\overline{w} }^{s}}\) from each term, and the remaining coefficients can be written as \({\left(\frac{\partial {\overline{u} }_{i}}{\partial z}\right)}^{s}/{{u}_{i}}^{s}\). To balance the equations, the vector \({\varvec{\beta}}\) needs to be a unit vector, and the connection among the characteristic scales can thereby be given as follows:

$${z}^{s}\cdot \sqrt{\sum {\left[{\left(\frac{\partial {\overline{u} }_{i}}{\partial z}\right)}^{s}/{{u}_{i}}^{s}\right]}^{2}}\cdot \frac{{w}^{s}}{{\overline{w} }^{s}}=1.$$
(17a)

In this way, the original momentum equation is further simplified. Notice that \({z}^{s}\) is explicitly expressed in Eq. (17a), it can be used to define the characteristic length scale:

$${z}^{s}=\frac{{\overline{w} }^{s}}{{w}^{s}}\cdot {\left\{\sum {\left[{\left(\frac{\partial {\overline{u} }_{i}}{\partial z}\right)}^{s}/{{u}_{i}}^{s}\right]}^{2}\right\}}^{-1/2}.$$
(17a')

This definition describes the contribution of wind shear to the atmospheric turbulence. The same process can also be applied to vector \({{\varvec{\beta}}}_{\theta}\) from the additional advection term of potential temperature, which describes the intensity of thermal convection. As the horizontal gradient is also omitted for potential temperature, there is only \({\beta }_{\theta ,w}\) left, and the connection among the characteristic scales in \({\beta }_{\theta ,w}\) can be defined as:

$$\frac{{z}^{s}}{{\overline{w} }^{s}}\cdot {\left(\frac{\partial \overline{\theta }}{\partial z}\right)}^{s}\cdot \frac{{w}^{s}}{{\theta }^{s}}=1,$$
(17b)

in which the \({z}^{s}\) is also explicitly expressed, making Eq. (17b) another definition of characteristic length scale:

$${z}^{s}={\theta }^{s}\cdot \frac{{\overline{w} }^{s}}{{w}^{s}}/{\left(\frac{\partial \overline{\theta }}{\partial z}\right)}^{s}.$$
(17b')

After this normalization process, the parameter \({{\varvec{\beta}}}_{\theta}\) is eliminated from the controlling equations, and this kind of definition quantifies the length scale when thermal convection (or buoyancy) is dominant.

However, Eqs. (17a') and (17b') are 2 different kinds of definitions, and they may not coexist in a certain condition, but according to the physical significance of tensor \({\varvec B}\) and vector \({{\varvec{\beta}}}_{\theta}\), they can be distinguished by their ratio, which yields the characteristic scale of Richardson number, \(Rs\):

$$Rs\equiv \frac{1}{{\theta }^{s}}{\left(\frac{\partial \overline{\theta }}{\partial z}\right)}^{s}/\sqrt{\sum {\left[{\left(\frac{\partial {\overline{u} }_{i}}{\partial z}\right)}^{s}/{{u}_{i}}^{s}\right]}^{2}},$$
(18)

and the \({z}^{s}\) can be written as a piecewise function to make a completed definition of length scale including the contribution of both dynamic and thermodynamic factors:

$${z}^{s}=\left\{\begin{array}{c}\frac{{\overline{w} }^{s}}{{w}^{s}}\cdot {\left\{\sum {\left[{\left(\frac{\partial {\overline{u} }_{i}}{\partial z}\right)}^{s}/{{u}_{i}}^{s}\right]}^{2}\right\}}^{-1/2} Rs<1, \\ {\theta }^{s}\cdot \frac{{\overline{w} }^{s}}{{w}^{s}}/{\left(\frac{\partial \overline{\theta }}{\partial z}\right)}^{s} Rs\ge 1.\end{array}\right.$$
(19)

When \(Rs<1\), the wind shear takes charge, and \(Rs\ge 1\) corresponds to the condition of thermal convection being dominant. However, when there is \(Rs\sim 1\), the two definitions are nearly equal and can yield close value of \({z}^{s}\). In comparison, the definition of the traditional length scale, Obukhov Length \(L\), is:

$$L=-\frac{{{u}_{*}}^{3}}{\kappa \frac{g}{\overline{\theta }}\overline{{w }^{\prime}{\theta }^{\prime}}},$$
(20)

where \({u}_{*}=\sqrt[4]{{\left(-\overline{{u^{\prime}w^{\prime}}}\right)}^{2}+{\left(-\overline{{v^{\prime}w^{\prime}}}\right)}^{2}}\) is the friction velocity, and \(\kappa\) is von-Karman constant. Note that \({R}_{s}\) takes the similar form to gradient Richardson number:

$$Ri=\frac{\frac{g}{{\theta }_{0}}\left(\frac{\partial \overline{\theta }}{\partial z}\right)}{\sum {\left(\frac{\partial {\overline{u} }_{i}}{\partial z}\right)}^{2}},$$
(21)

and their physical significance can also be considered the same, namely the comparative intensity of thermal convection and wind shear.

Now the new length scale \({z}^{s}\) is directly derived from the newly defined parameters \({\varvec B}\) and \({{\varvec{\beta}}}_{\theta}\), without presupposing any other relationships among different scales. It is noteworthy that the definition of \({z}^{s}\) does not constrain other parameters such as \({\boldsymbol{\alpha }}_{p}\) and \({\alpha }_{\theta }\), and there are no other preassigned combinations of characteristic scales. This is the most important difference from other length scales. As \({z}^{s}\) is determined by 3 physical criteria: wind shear, thermal convection, and stratification state of the atmosphere, if the laws of turbulence (such as the exponential pattern of normalized standard deviation, cf. Figures 1 and 2 in Zhang et al. 2014 and Fig. 2 in Wei et al. 2021) can be reproduced by a factor of \(z/{z}^{s}\), for both stable and unstable atmosphere, then the effect of \({z}^{s}\) can be confirmed (likewise for \({x}^{s}\) and \({y}^{s}\), as they are linked by aspect ratio), and it can be used in the existing or new similarity theories.

Fig. 1
figure 1

The time series of some pre-defined characteristic scales (in SI units) in July, 2019

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Fig. 2
figure 2

The time series of newly introduced aspect ratio. The upper panel for the 1-month aspect ratio time series (blue points denote \({\alpha }_{x}\), and red ones \({\alpha }_{y}\)); the lower panel for the magnified time series within a week. The stable and unstable stratifications are distinguished by Obukhov length \(L\), and are denoted by solid and hollow points, respectively

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2.5 Exploration of new similarity relationships

In this section, we concentrate on the properties of the newly defined parameters \({\varvec{\Gamma}}\), \({\varvec T}\), \({{\varvec{\gamma}}}_{\theta}\), and \({{\varvec{\tau}}}_{\theta}\). As per the definitions provided in Eqs. (5) and (10), \({\varvec{\Gamma}}\) is characterized by the standard deviation of wind speed covariance, normalized by the product of a standard deviation and an average. This particular definition is subtly distinguishing for a second-order tensor, given that \({\varvec{\Gamma}}\) remains asymmetric even under homogeneous conditions, and so it is for \({\varvec T}\). Notably, its form bears resemblance to the cross-correlation coefficient, and can be represented as follows:

$${\Gamma }_{\text{ij}}=\frac{{\left({{u}_{i}}^{\prime}{{u}_{j}}^{\prime}\right)}^{s}}{{{u}_{i}}^{s}{{\overline{u} }_{j}}^{s}}=\frac{{\sigma }_{{{u}_{i}}^{\prime}{{u}_{j}}{\prime}}}{{\sigma }_{{{u}_{i}}^{\prime}}{\sigma }_{{{u}_{j}}^{\prime}}}\cdot \frac{{\sigma }_{{{u}_{j}}^{\prime}}}{{\overline{u} }_{j}}=\frac{{\sigma }_{{{u}_{i}}^{\prime}{{u}_{j}}^{\prime}}}{{\sigma }_{{{u}_{i}}^{\prime}}{\sigma }_{{{u}_{j}}^{\prime}}}\cdot {\beta }_{i}.$$
(22a)

In this way, \({\varvec{\Gamma}}\) can be expressed as the normalized variation of covariance, weighted by the turbulence intensity. This provides a physical interpretation of \({\varvec{\Gamma}}\): it represents the response, or the rate of change, of the coupling of wind speeds to their fluctuating conditions. Consequently, it can be aptly termed as the sensitivity of dynamic coupling. As for \({\varvec T}\), the similar definition makes it the same role as \({\varvec{\Gamma}}\):

$${\text{T}}_{\text{ij}}=\frac{{\left(\overline{{{u }_{i}}^{\prime}{{u}_{j}}^{\prime}}\right)}^{s}}{{{u}_{i}}^{s}{{\overline{u} }_{j}}^{s}}=\frac{\overline{{{u }_{i}}^{\prime}{{u}_{j}}^{\prime}}}{{\sigma }_{{{u}_{i}}^{\prime}}{\sigma }_{{{u}_{j}}^{\prime}}}\cdot \frac{{\sigma }_{{{u}_{j}}{\prime}}}{{\overline{u} }_{j}}=\frac{\overline{{{u }_{i}}^{\prime}{{u}_{j}}^{\prime}}}{{\sigma }_{{{u}_{i}}^{\prime}}{\sigma }_{{{u}_{j}}^{\prime}}}\cdot {\beta }_{i},$$
(22b)

but the numerators in the definition are the averaged covariances in data samples, which allows for the designation of normalized turbulent momentum flux. And \({{\varvec{\tau}}}_\theta\) can be named as the normalized turbulent sensible heat flux from its thermodynamics nature. The parameters \({{\varvec{\gamma}}}_\theta\) and \({{\varvec{\tau}}}_\theta\) bear similarity to \({\varvec{\Gamma}}\) and \({\varvec T}\), as they share the same format of definition. Their only difference lies in the fact that \({{\varvec{\gamma}}}_\theta\) and \({{\varvec{\tau}}}_\theta\) are both vectors rather than tensors, because there is only a single component equation to describe the thermodynamics of turbulence.

It is necessary to explain the difference in the processing method concerning \({\varvec B}\) and \({\varvec{\Gamma}}\), \({\varvec T}\) (\({{\varvec{\beta}}}_\theta\) and \({{\varvec{\gamma}}}_\theta\), \({{\varvec{\tau}}}_\theta\)). In the definition of \({\varvec B}\) (\({{\varvec{\beta}}}_\theta\)), there was a kind of important characteristic scale, namely \({z}^{s}\), needed to be determined, whereas no such scale was required for \({\varvec{\Gamma}}\) and \({\varvec T}\) (\({{\varvec{\gamma}}}_\theta\) and \({{\varvec{\tau}}}_\theta\)). In this situation, \({\varvec B}\) (\({{\varvec{\beta}}}_\theta\)) was employed to define the new scale during the process of nondimensionalization, while \({\varvec{\Gamma}}\) and \({\varvec T}\) (\({{\varvec{\gamma}}}_\theta\) and \({{\varvec{\tau}}}_\theta\)) were utilized to characterize turbulence properties. Despite the different functions of these dimensionless parameters, they are all derived from the nondimensionalized turbulence equations, making it meaningful to analyze and compare them. Consequently, a new type of similarity relationship can be naturally inferred. In other words, the relationship between \({\varvec{\Gamma}}\), \({\varvec T}\) to the stability parameter by \({z}^{s}\) is worth exploring. If the observational data can reveal a universal relationship, for instance, \({\Gamma }_{\text{uw}}={\Gamma }_{\text{uw}}(z/{z}^{s})\), the validity and productivity of nondimensionalization can be affirmed.

For a better understanding of the types and physical significance of characteristic scales and dimensionless parameters, the corresponding elements are listed in Tables. 1 and 2. The physical significance of dimensionless parameters will be analyzed in detail in Section 4.

Table 1 The types of definition of characteristic scales and their physical significance
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Table 2 The types of definition of dimensionless parameters and their physical significance
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3 Data processing and analysis

3.1 Observational site

To ensure the effectiveness of quantitative analysis, it is necessary to have a nearly ideal homogeneous and flat underlying surface for data analysis. In this study, the observational station is selected at the Horqin Sandy Land ( \(42^\circ\)  \(56^\prime\) N,  \(120^\circ\)  \(42^\prime\) E), which locates in the Naiman County of Inner Mongolia, China. This location is chosen primarily due to its typical semi-arid continental monsoon climate, and the surrounding vegetation predominantly comprises low and open shrubs (Zhao et al. 2007; Park and Park 2014; Li and Zhang 2015), thereby ensuring near-ideal observational conditions. Li and Zhang (2015) provides more information about the details of climate, underlying surface and the observational instruments at the Horqin station.

The utilized data are acquired from a 20-m boundary layer meteorological observational tower, situated approximately 10 km north of downtown Naiman, at an altitude of 363 m. To enhance the analysis of nondimensionalization effectiveness, it is essential that the span of observational data extends for a minimum of one month. The data spanning from June 1 to August 31, 2019 are selected for the data analysis.

3.2 Data requirement and instruments

To perform verifications on nondimensionalization and similarity relationships, a series of average fields observation, gradient observation, and turbulence observation are needed, as Eqs. (4)-(5) show.

For average fields observation, there are instruments of wind speed (010C, Met One Instruments Inc., USA) at 2, 4, 16, and 20 m and air pressure (PTB110, Vaisala Co., Finland) at 2 m, which are continuously and automatically recorded at 10 min intervals and processed into 30 min averages. The gradient of wind speed and temperature are acquired by the difference of average fields.

For turbulence observation, there are eddy covariance (EC) system at 8 and 16 m, the 8 m one of which contains a three-axis sonic anemometer (CSAT3, Campbell Scientific Co., USA) for determining the 3-dimensional wind speeds and virtual temperature fluctuations, and the 16 m one contains a three-axis sonic anemometer (CSAT3B, Campbell Scientific Co., USA). The EC sensors are oriented west (270 \(^\circ\)) and their sampling frequency is 10 Hz. Besides, there are also self-developed fast-response air pressure sensor at 4, 8, and 16 m, whose frequency response was approximately 1 Hz and its accuracy exceeded 0.05 hPa. For detailed information on this pressure sensor, please refer to Wei et al. (2021) and Wei et al. (2022).

In this study, quantified verifications of the nondimensionalization and similarity relationships are conducted. To avoid misunderstandings, the data used in the figures below are listed:

  • Time span: Only the data from July 1st to July 31st are plotted, to show the most typical patterns (the pattern in June and August have been tested similar with July);

  • Turbulence data: the data collected by the EC system at the height of 8 m;

  • Gradient data: the differentiation of data collected by the multiple-layer instruments from 4 m, 8 m, and 16 m.

There are 2 kinds of definitions for characteristic scales, yet only the definition by standard deviation is used in the detailed analysis. The definition by amplitudes is also evaluated, and can draw the same conclusion, please refer to the figures in Appendix. B.

3.3 Data processing

The raw turbulence observational data were pre-processed using the EddyPro software (Advanced 7.0.9, LI-COR Biosciences, Inc., USA), including error flags, despiking (Vickers and Mahrt 1997), coordinate rotation, and detrending. The averaging time scale of the pre-processing is 30 min. The process of coordinate rotation is the standard double rotation for the calculation of EC system (Wilczak et al. 2001), while for the analysis of nondimensionalization, there is only a single coordinate rotation. This rotation sets the u-wind as the average wind direction of multiple heights (4 m, 8 m, and 16 m). The target of usually performed double rotation is to make averaged v and w-wind to be zero, but during a limited period of time on non-flat terrain, the average value of w-wind is not guaranteed to be zero. In this study, the underlying surface is chosen to be nearly homogeneous and flat, and rigorous tests have been performed to make sure the long-time averaged w-wind is zero, and in data samples the w-wind is irrelevant of u-, v-wind. For the selected wind speed of no more than 5 m/s, the influence of instrument’s swinging can also be neglected. By one rotation, the average value and gradient of u, v, and non-zero w-wind can be correctly extracted from the raw data, and the direction of wind is also consistent from low level to high level of observations, which is exactly of concern in this study.

According to Eq. (4), the time series of some pre-defined characteristic scales calculated from the data in July, 2019 are shown in Fig. 1. It is shown that there are diurnal variations for most characteristic scales, based on which high-frequency variabilities can be found for \(\overline{u }\), \(\overline{v }\), and \(\overline{w }\), on which the most attention will be cast in later analysis.

To remove inaccuracies caused by the field observations, the data were selected in a less set of criteria, compared to the standard EC system, the reject criteria are:

  1. (1)

    the angle between the direction of the sonic anemometer sensing probe and wind direction was larger than ± \(150^\circ\) ;

  1. (2)

    the mean wind speeds were smaller than 0.1 m/s or larger than 5 m/s;

  2. (3)

     the friction velocity \({u}_{*}\) was smaller than 0.05 m/s;

  1. (4)

    the absolute value of sensible heat flux was smaller than 5 \(\text{W}/{\text{m}}^{2}\) ;

  1. (5)

    the data acquired during the sunrise and sunset hour.

4 Results

4.1 Basic parameters of turbulent motions

The method of nondimensionalization can generate several parameters through the combination of characteristic scales. Some of these parameters bear resemblance to, or can be correlated with, certain traditional basic parameters in atmospheric science. In this section, we will analyze the properties and physical significances of these basic parameters. The objective is to ascertain whether this method is in accordance with existing turbulence theories, specifically, whether these parameters can fulfill roles that are similar to or identical with those of traditional parameters.

Figure 2 presents the newly introduced aspect ratio of turbulence, \({\alpha }_{i}\), computed from the ratio of characteristic scales of wind speed. This aspect ratio appears to maintain a consistent value around 2, exhibiting a stable order of magnitude during fluctuations. This suggests that turbulence tends to develop more vigorous in horizontal direction, particularly in areas without constraints, such as the top of the boundary layer. The higher outliers can be attributed to the minimal value of \({w}^{s}\) during periods of calm weather at night. During nights when vertical momentum is still present (such as 16th ~ 21st, July), the aspect ratios consistently hover around 2. Furthermore, the aspect ratio of averaged wind, \({\alpha }_{\nu ,i}\), is similar in pattern to \({\alpha }_{i}\) with a higher order of magnitude, and is thus omitted.

Aside from the state parameter \({\alpha }_{\theta \rho }\) which is unable to analyze due to the inability of observing \(\rho^{\prime}\), two additional parameters are introduced in Eq. (10c), namely the pressure parameter \({\boldsymbol{\alpha }}_p\) and the buoyancy parameter\({\alpha }_{\theta }\). As depicted in Fig. 3, both \({\boldsymbol{\alpha }}_p\) and \({\alpha }_{\theta }\) span multiple orders of magnitude, exhibiting highly similar patterns of change. This is attributable to the fact that the components of these two parameters (the average wind speed, and the fluctuation of air pressure and temperature) can be influenced by synoptic processes. For instance, a cold front passage occurred from July 15th to 21st, during which both \({\boldsymbol{\alpha }}_p\) and \({\alpha }_{\theta }\) experienced a sudden decrease followed by a gradual increase. Furthermore, the background weather conditions can significantly impact the proportion of static and dynamic pressure, as reflected by the wide range of \({\boldsymbol{\alpha }}_p\) values. Conversely, the variation of potential temperature is confined within a short time span, resulting in \({\alpha }_{\theta }\) being less dispersed. Here a special condition of \({\alpha }_{\theta }\equiv 1\) is specially marked, on which \({z}^{s}\) can be acquired as the function of other characteristic scales, suggested by Eq. (10c). In fact, on adding assumptions that only \(\frac{\partial \overline{v}}{\partial z }=\frac{\partial \overline{w}}{\partial z }=0\) and\({{\alpha }_{p, x}=\alpha }_{\theta }\equiv 1\), the scale \({z}^{s}\) defined in Eq. (16) can be expressed in the same form like:

$$z^\ast=\left\{\begin{array}{lc}\overline u^\ast/\left(\frac{\mathrm d\overline u}{\mathrm dz}\right)^\ast&Ri<1,\\\overline u^\ast/\left(\frac{\partial\overline b}{\partial z}\right)^{\ast1/2}&Ri>1\end{array}\right.$$
(23)

introduced in Yano and Wacławczyk (2022), where \(b=g\frac{{\theta }^{*}}{{\theta }_{0}}\) is the buoyancy term and ∗ denotes the characteristic scale. However, Fig. 8 implies that the assumption of \({\alpha }_{\theta }\equiv 1\) tends not to be satisfied, and introducing of \({z}^{s}\) is thus necessary.

Fig. 3
figure 3

The time series of other newly introduced parameters \({\boldsymbol{\alpha }}_p\) and \({\alpha }_{\theta }\). The stable and unstable stratifications are distinguished by Obukhov length \(L\), and are denoted by solid and hollow points, respectively. The red dashed line denotes a limiting condition for \({\alpha }_{\theta }\equiv 1\)

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4.2 Key parameters in nondimensionalization

In the process of nondimensionalizing the turbulence equations, a number of additional dimensionless parameters can be derived, surpassing those obtained from the original equations. In this section, a comprehensive analysis of these key parameters will be conducted to examine their properties and evolving patterns, uncover new aspects of turbulence information, and substantiate the efficacy of nondimensionalization.

Figure 4 presents the time series of certain components of the key parameters introduced in the momentum and thermal equations. The parameters \({\Gamma }_{\text{u},\text{w}}\) and \({\gamma }_{\theta ,w}\) predominantly cluster around \({10}^{1}\), exhibiting no discernible diurnal pattern, and they share an extremely high similarity. This can be rationalized from two distinct perspectives. From a physical standpoint, the standard deviation indicates the intensity of covariance fluctuations, which can be similar for two variables despite differences in their average values. The diurnal pattern of flux is counterbalanced by the diurnal patterns of \({u}^{s}\) and \({\theta }^{s}\), and the relatively stable \({\overline{w} }^{s}\) also contributes to their similar pattern. From a mathematical perspective, their definitions hold the same factors of turbulent intensity revealed by Eqs. (22), and therefore share the similar pattern from the diurnal variation of \({\beta }_{i}\). For parameters \({\text{T}}_{\text{u},\text{w}}\) and \({\tau }_{\theta ,w}\), the changing patterns are different from \({\Gamma }_{\text{u},\text{w}}\) and \({\gamma }_{\theta ,w}\), respectively, in the range of data and the response to synoptic process. For example, when there is a cold front passage from July 15th to 21st, the local minima keep around 0.5 for \({\Gamma }_{\text{u},\text{w}}\) and \({\gamma }_{\theta ,w}\), but the local minima of both \({\text{T}}_{\text{u},\text{w}}\) and \({\tau }_{\theta ,w}\) increase from \({10}^{-1}\) to \({10}^{1}\) and decreased within that week. Moreover, the comparison of \({\text{T}}_{\text{u},\text{w}}\) and \({\tau }_{\theta ,w}\) reveals a less similarity, compared with that of \({\Gamma }_{\text{u},\text{w}}\) and \({\gamma }_{\theta ,w}\). This can be attributed to the discrepancy of the turbulent transport between momentum and sensible heat, as suggested by the form of \({\phi }_{m}\) and \({\phi }_{h}\) in flux-profile relationships.

Fig. 4
figure 4

The time series of key dimensionless parameters introduced in nondimensionalization. From top to bottom: the sensitivity of dynamic coupling of u and w-wind, \({\Gamma }_{\text{u},\text{w}}\); the normalized turbulent momentum flux, \({\text{T}}_{\text{u},\text{w}}\); the sensitivity of vertical thermodynamic coupling, \({\gamma }_{\theta ,w}\); the normalized turbulent sensible heat flux, \({\tau }_{\theta ,w}\). The stable and unstable stratifications are distinguished by Obukhov length \(L\), and are denoted by solid and hollow points, respectively

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Over the course of an entire month, the prevailing wind direction and speed can exhibit significant variations in response to synoptic conditions. However, the magnitude of these key parameters remains relatively stable, irrespective of whether they exhibit a stable distribution or a diurnal pattern. Consequently, it can be inferred that there may exist novel types of similarity properties inherent in the dynamic and thermodynamic characteristics of atmospheric turbulent motions.

4.3 Length scale \({{\varvec{z}}}^{{\varvec{s}}}\) and its difference from Obukhov length

Figure 5 illustrates the numerical value of \({z}^{s}\) as determined by Eq. (16), along with the relationship between \({z}^{s}\) and Obukhov length \(L\), calculated from 30-min data samples. The comparison between \({z}^{s}\) and \(L\) can be revealed in 2 ways: from the perspective of range, the \({z}^{s}\) are positive-definite from \({10}^{-2}\) to \({10}^{1}\), while \(L\) can vary from \(-{10}^{3}\) to \({10}^{3}\); in terms of changing patterns, the complicated relationship and the correlation coefficient of -0.053 (unstable) and 0.209 (stable) indicate a relatively low level of correlation between these two types of length scales. These differences underscore that, despite similar definitions (incorporating both dynamics and thermodynamics), \({z}^{s}\) and \(L\) are completely different in essence. Therefore, the similarity relationships based on the stability parameter \(z/{z}^{s}\) can reflect physically different information compared to the conventional usage of \(z/L\).

Fig. 5
figure 5

The relationship between newly defined \({z}^{s}\) and Obukhov length \(L=-\frac{{{u}_{*}}^{3}}{\kappa \frac{g}{\overline{\theta }}\overline{{w }^{\prime}{\theta }^{\prime}}}\). The upper panel for unstable stratifications distinguished by \(L<0\), and the lower panel for stable stratifications distinguished by \(L\ge 0\); the difference in traditional \(Ri\) and newly introduced \(Rs\) is marked out: hollow points for \(|Ri|\ge 1\), solid points for \(|Ri|<1\), red points for \(Rs\ge 1\), and blue points for \(Rs<1\)

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The different ranges and functions of \(Ri\) and \(Rs\) are also revealed in Fig. 5 by different colors and styles of points, as they quantify the comparative intensity of thermal convection and wind shear. The results of \(|Ri|\)  indicate that thermal convection predominates in 30% of cases in unstable stratifications and only in a handful of cases in stable stratifications. Conversely, the results for \(Rs\) suggest that thermal convection is the dominant factor in the majority of cases in unstable stratifications, while wind shear prevails in most stable conditions. The discrepancy between \(|Ri|\) and \(Rs\) can be attributed to the one coordinate rotation, which deviates from the conventional pre-processing for \(Ri\). Based on the conditions of the observational station and simple tests of traditional methods, it is evident that thermal convection contributes significantly to the development of atmospheric turbulence when the stratification is unstable. Consequently, the use of \(Rs\), rather than \(Ri\), in subsequent processes is deemed appropriate.

Typically, considerable attention is devoted to two types of limiting cases, where \(|Ri|(Rs)\to 0\) and \(|Ri|(Rs)\to \infty\). Given that there is no phase transition between these two cases, and considering that the two definitions of \({z}^{s}\) yield similar values near \(Rs=1\), the simple boundary of \(Rs=1\) can be justified for its convenience in differentiating stratification states. It should be noted that since \({z}^{s}\) and \(Rs\) are positive definite, their signs can no longer serve as criteria for stratification state. Therefore, when the stratification state is required, additional parameters should be provided. Consequently, the similarity relationship based on \({z}^{s}\) can be taken as a supplement to the M–O similarity theory. However, under certain conditions, the sensible heat flux \(\overline{{w }^{\prime}{\theta }^{\prime}}\), the gradient of potential temperature \(\frac{\partial \overline{\theta }}{\partial z}\), or the boundary of \(Rs=1\) (although it is not a strict criterion), can also be employed to distinguish the stratification. As a result, the analysis of the similarity relationship based on \({z}^{s}\) will also be categorized into two categories of stable and unstable stratifications.

4.4 The similarity relationships based on \({{\varvec{z}}}^{{\varvec{s}}}\)

In Section 2.5, the potential similarity relationships among new length scale \({z}^{s}\) and newly defined parameters \({\varvec{\Gamma}}\), \({\varvec T}\), \({{\varvec{\gamma}}}_\theta\), and \({{\varvec{\tau}}}_\theta\) have been presented. In this section, the quantified verification of these similarity relationships will be conducted. Figure 6 illustrates the relationship between the components of \({\varvec{\Gamma}}\) and the stability parameter defined by \({\zeta }^{s}=z/{z}^{s}\). From these similarity relationships, a major difference from M–O similarity can be observed, that both the values of \({\varvec{\Gamma}}\) and \({\zeta }^{s}\) span several orders of magnitude. The values of \({\Gamma }_{\text{iu}}\) and \({\Gamma }_{\text{iv}}\) reveal a slight decrease with the increase of \({\zeta }^{s}\), which can be fitted into exponential functions, with both stable exponents for \({\Gamma }_{\text{iu}}\) and \({\Gamma }_{\text{iv}}\). However, the results of the correlation coefficients indicate that these components do not exhibit a direct correlation with the stratification conditions. For the components \({\Gamma }_{\text{iw}}\), on the contrary, the value significantly (correlation coefficients \(r>0.9\) and confidence level \(p<1\times {10}^{-5}\)) increases with \({\zeta }^{s}\). There is a subtle difference between two types of stratification states, which is similar to the conditions of M–O similarity. Here, the calculation of the Obukhov length is not necessary; temperature gradients can be used instead. Therefore, two similar exponential relationships can be fitted to the data points, with the same overall fitting parameters and the exponents of 4/3.

Fig. 6
figure 6

The similarity relationship of dimensionless parameter \({\varvec{\Gamma}}\) and stability factor \({\zeta }^{s}=z/{z}^{s}\) in double logarithmic coordinate. The unstable and stable stratifications are distinguished by \(\frac{\partial \overline{\theta }}{\partial z}\), and are denoted by solid and hollow points. For \({\Gamma }_{\text{iu}}\) and \({\Gamma }_{\text{iv}}\), the data points of 2 stratification states are fitted to red lines, and for \({\Gamma }_{\text{iw}}\), the data points in unstable and stable stratifications are fitted to red lines and blue dashed lines, respectively

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The low values of \({\Gamma }_{\text{iu}}\) and \({\Gamma }_{\text{iv}}\) can be rationalized from two perspectives. On one hand, the characteristics of turbulent motions in different directions provide a fundamental value. For \({\Gamma }_{\text{uv}}\) and \({\Gamma }_{\text{vu}}\), the horizontal turbulent transport naturally assumes a low value due to the homogeneous and flat underlying surface, accompanied by a weak diurnal variation. On the other hand, the turbulence intensity term, \(\frac{{\sigma }_{{{u}_{j}}{\prime}}}{{\overline{u} }_{j}}\), makes a major contribution to the value of \({\varvec{\Gamma}}\). As the turbulent motions in different directions are of similar scales while the averaged vertical motion is considerably constrained, the sensitivity term \({\Gamma }_{\text{iw}}\) can be substantially amplified to reveal the properties of vertical motions. This explains why \({\Gamma }_{\text{uu}}/{\Gamma }_{\text{wu}}\) and \({\Gamma }_{\text{vv}}/{\Gamma }_{\text{wv}}\) can also reflect the degree of wind speeds coupling, providing insight into the condition of turbulent kinetic energy (TKE)/vertical turbulent flux, but their values are still limited with large dispersion and unclear patterns, making them less effective in application than components of \({\Gamma }_{\text{iw}}\).

The increasing of \({\Gamma }_{\text{iw}}\) with \({\zeta }^{s}\) is associated with the physical information consist in \({z}^{s}\). A lower \({\zeta }^{s}\) corresponds to a higher \({z}^{s}\), and from its definition, it can be inferred that there is a weaker vertical fluctuation and smaller vertical gradients. Under these conditions, the interaction between the u- and w-wind speeds is limited, resulting in a low level of turbulent momentum transport. As the stability parameter \({\zeta }^{s}\) increases, the wind speeds begin to interact, driven by larger vertical gradients, and a turbulent motion of greater intensity can contribute to a higher level of turbulent transport. From this relationship, an empirical exponent of 4/3 can be summarized to encapsulate new aspects of information in similarity theory. As the empirical relationship is fitted mathematically, it is essential to conduct investigations to ensure that the positive correlation originates from actual physical processes rather than from the self-correlation of common factors, which is often a significant issue for the M–O similarity relationships (Klipp and Mahrt 2004), too. The evaluations on the self-correlation prove that the new similarity relationship is rooted in physical phenomena, rather than being a data illusion, and detailed discussions can be found in Appendix. C.

Figure 7 shows the relationship between the components of \({{\varvec{\gamma}}}_\theta\) and the stability parameter defined by \({\zeta }^{s}=z/{z}^{s}\). Similar to \({\varvec{\Gamma}}\), there is also a decreasing trend for \({\gamma }_{\theta ,u}\) and \({\gamma }_{\theta ,v}\) with low correlation coefficients, and a 4/3 exponentially increasing trend for \({\gamma }_{\theta ,w}\) with high correlation coefficients, which verifies the effectiveness of the \({z}^{s}\)-based similarity relationship concerning thermodynamics of atmospheric turbulence.

Fig. 7
figure 7

The similarity relationship of dimensionless parameter \({{\varvec{\gamma}}}_\theta\) and stability factor \({\zeta }^{s}=z/{z}^{s}\) in double logarithmic coordinate. The unstable and stable stratifications are denoted by solid and hollow points. For \({\gamma }_{\theta ,u}\) and \({\gamma }_{\theta ,v}\), the data points of 2 stratification states are fitted to red lines, and for \({\gamma }_{\theta ,w}\), the data points in unstable and stable stratifications are fitted to red lines and blue dashed lines, respectively

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The analysis on \({\varvec{\Gamma}}\) and \({{\varvec{\gamma}}}_\theta\) elucidates the degree of coupling between wind speeds and potential temperature, which reflects the characteristics of turbulent transports, while \({T}\) and \({{\varvec{\tau}}}_\theta\) directly demonstrate the relationships between the values of turbulent fluxes and stratification conditions. Figure 8 depicts the relationship between \({T}\), \({{\varvec{\tau}}}_\theta\), and the stability parameter defined by \({\zeta }^{s}=z/{z}^{s}\). Here only components concerning vertical turbulent transport are analyzed, as they are the most important ones in the land–atmosphere exchange. Despite that the relationships of \({\text{T}}_{\text{uw}}\) and \({\text{T}}_{\text{ww}}\) are similar to the 4/3 exponential relationships of \({\varvec{\Gamma}}\) and \({{\varvec{\gamma}}}_\theta\), the majority of points are concentrated around \({\zeta }^{s}\sim {10}^{0}\) to \({10}^{3}\), exhibiting a strong linearity. Consequently, the empirical relationships for \({T}\) and \({{\varvec{\tau}}}_\theta\) are established as simple linear ones for ease of application. It can be inferred that the normalized fluxes (and TKE) are linearly correlated with the stability parameter, exhibiting high correlation coefficients, which can be conveniently employed to quantify turbulent transport:

Fig. 8
figure 8

The similarity relationship of \({T}\), \({{\varvec{\tau}}}_\theta\) and stability factor \({\zeta }^{s}=z/{z}^{s}\) in double logarithmic coordinate. The unstable and stable stratifications are denoted by solid and hollow points and are fitted into red and blue dashed lines, respectively

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$${\text{T}}_{\text{ij}}=\frac{\overline{{{u }_{i}}^{\prime}{{u}_{j}}^{\prime}}}{{\sigma }_{{u}_{i}}{\overline{u} }_{j}}={f}_{\text{T},\text{ij}}\left({\zeta }^{s}\right),$$
(24a)
$${\tau }_{\theta ,i}=\frac{\overline{{{u }_{i}}^{\prime}{\theta }^{\prime}}}{{\overline{u} }_{i}{\sigma }_{\theta }}={f}_{\tau ,i}\left({\zeta }^{s}\right),$$
(24b)
$$\overline{{{u }_{i}}^{\prime}{{u}_{j}}^{\prime}}={\sigma }_{{u}_{i}}{\overline{u} }_{j}\cdot {f}_{\text{T},\text{ij}},$$
(24a')
$$\overline{{{u }_{i}}^{\prime}{\theta }^{\prime}}={\sigma }_{\theta }{\overline{u} }_{i}\cdot {f}_{\tau ,i}.$$
(24b')

And the turbulent fluxes can be calculated as:

$$\tau =-\rho \sqrt{{\left(-\overline{{u }^{\prime}{w}^{\prime}}\right)}^{2}+{\left(-\overline{{v }^{\prime}{w}^{\prime}}\right)}^{2}},$$
(25a)
$$H=\rho {C}_{p}\overline{{w }^{\prime}{\theta }^{\prime}}.$$
(25b)

In this way, the quantification of turbulent transport can be achieved through empirical functions, \({f}_{\text{T},\text{ij}}\) and \({f}_{\tau ,i}\), which can be defined by experimental data. And in application, the turbulent fluxes can be estimated without resorting to the complicated computation of covariance, thereby eliminating the need for stringent assumptions such as a flat underlying surface and steady motion. Rather, the standard deviations and gradients of average fields suffice for the estimation of turbulent flux. The fluxes, as computed based on Eqs. (24) can also be compared with the flux-profile relationship in the M–O similarity theory (Businger et al. 1971; Businger 1988).

Figure 9 presents a comparative analysis of the fluxes derived from the EC system, the M–O similarity theory, and the newly proposed similarity relationships. It is discernible that the fluxes computed using the M–O theory can exhibit significant deviations from those obtained via the EC system, under both unstable and stable stratification conditions. This discrepancy can be attributed to the fact that the data selection was not strictly confined to near-ideal conditions. In contrast, the new similarity relationship demonstrates a higher degree of comparability with the EC fluxes, particularly for smaller flux values. Nevertheless, an issue of underestimation persists for two types of fluxes in the unstable boundary layer and for the sensible heat flux under stable conditions. The improved accuracy in estimating turbulent fluxes offered by this method suggests promising prospects for its future applications. If the new relationships are found to be universally applicable across diverse underlying surfaces, they could serve as an effective supplement to existing similarity theories. Furthermore, it would be reasonable to extend the method of nondimensionalization to other observations of atmospheric turbulence. This approach could also serve as a reference for the simulation of turbulence and the development of climate models with parameterization schemes focused on atmospheric turbulence.

Fig. 9
figure 9

The comparison among the fluxes calculated from the EC system, the M–O similarity theory (hollow points), and the new similarity relationship from nondimensionalization (solid points). The left 2 columns correspond to unstable stratifications and the right ones for stable stratifications

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5 Conclusions

In the present study, the principle of nondimensionalization is applied to turbulence equations through the process of scale decomposition. This process led to the definition of several characteristic scales. In order to achieve the dimensional balance and unify the combination of these characteristic scales, a variety of parameters are introduced. These include established parameters such as the Reynolds number Re and aspect ratio \(\alpha\), and other newly defined parameters. From the dimensionless parameter tensor \({\varvec B}\) and dimensionless parameter vector \({{\varvec{\beta}}}_\theta\), the new length scale \({z}^{s}\) can be defined as composite of wind shear and thermal convection terms. Other dimensionless parameters such the \({\varvec{\Gamma}}\), \({\varvec T}\), \({{\varvec{\gamma}}}_\theta\), and \({{\varvec{\tau}}}_\theta\) are evaluated for their potential applicability in the development of new similarity relationships.

Following the theoretical examination of nondimensionalization, observational data from the Horqin Sandy Land in Naiman County of Inner Mongolia are employed to quantitatively validate the effects of nondimensionalization and the introduced parameters. An investigation into the time series of these parameters demonstrate that some fundamental parameters (e.g., \(\boldsymbol{\alpha }\) and \(Re\)) are in accordance with existing turbulence theories. Moreover, the newly introduced parameters \({\varvec{\Gamma}}\), \({\varvec T}\), \({{\varvec{\gamma}}}_\theta\), and \({{\varvec{\tau}}}_\theta\) have shown potential in elucidating the properties of atmospheric turbulence. A new length scale \({z}^{s}\) is defined, which takes the form of a combination of both wind shear and thermal convection. Although it partly bears some resemblance to the Obukhov length, the new scale \({z}^{s}\) exhibits distinct characteristics. The relationships between the new dimensionless parameters \({\varvec{\Gamma}}\), \({{\varvec{\gamma}}}_\theta\) and the stability parameter based on \({z}^{s}\) exhibit a 4/3 exponentially increasing pattern in vertical directions. The relationships between the new parameters \({\varvec T}\), \({{\varvec{\tau}}}_\theta\) and the stability parameter can be summarized in a linear pattern, facilitating the estimation of turbulent flux from basic observational statistics. These results indicate that the newly introduced variables are intimately associated with turbulent transport and are applicable in stable stratifications. However, certain limitations regarding the applicability of nondimensionalization is worth further exploration. For instance, in these new similarity relationships, the positive definite property of \({z}^{s}\)  implies that the stratification state cannot be reflected and should be assessed by traditional scales. Furthermore, the quantitative analysis was conducted using data from a single observational site with a near-ideal underlying surface. Therefore, the application of this method necessitates further evaluations and verifications across a broad range of observational sites with varying underlying surfaces and stratification states.

In conclusion, the application of nondimensionalization uncovers new characteristic scales of atmospheric turbulence and introduces new parameters pertinent to the stability conditions of the boundary layer. These scales and parameters, derived from turbulence equations, can be obtained from basic observations and have been demonstrated to satisfy new similarity relationships pertaining to key information of atmospheric turbulence, such as turbulent momentum and sensible heat flux. These results suggest that nondimensionalization is a practical approach for scaling and analyzing the properties of atmospheric turbulence. It holds potential to serve as a reference in a variety of turbulence and climate models for the parameterization schemes. Consequently, the method of nondimensionalization presents significant promise in the realm of atmospheric turbulence and is worth further exploration.