2.1. Constructing the ARMA model in boundary-layer turbulence

The ARMA model applies to a time series, which links to a spatial second-order structure function under Taylor's frozen hypothesis (Taylor Reference Taylor1938). For a measured streamwise turbulent velocity time series with a time step $\Delta t_0$, mean streamwise velocity $U$ and wall-normal position $z$, we extract a coarse series that does not resolve the small-scale inertial range. So the time step $\Delta t$ of the extracted series satisfies

(2.2)\begin{equation} U\Delta t \approx\eta_1 z, \end{equation}

where $\eta _1$ corresponds to the scale that separates the inertial and dynamic ranges. The new series can be obtained by extracting one point for every $\Delta t/\Delta t_0$ points from the original series.

The larger scale $\eta _2$ bounds the memory depth of the model, i.e.

(2.3)\begin{equation} U\Delta t \max\{p, q\} \approx\eta_1 z\max\{p, q\}\approx\eta_2 z, \end{equation}

where $p$ and $q$ are undetermined, but at least one of them should equal $\eta _2/\eta _1$. Then in the large-scale range, the autocorrelation (cf. (A13)) takes the form

(2.4)\begin{equation} R_{uu}\left(r\right) =\displaystyle\sum_{i=1}^{p}c_i \exp\left(\frac{r}{\Delta x}\ln \lambda_i\right), \quad r\geq\eta_2 z, \end{equation}

with the corresponding second-order structure function

(2.5)\begin{equation} \langle \Delta u^2 \rangle =2\langle u^2 \rangle -2\displaystyle\sum_{i=1}^{p}c_i \exp\left(\frac{r}{\Delta x}\ln \lambda_i\right), \quad r\geq\eta_2 z, \end{equation}

where a set of exponential functions approximate the large-scale behaviour of $\langle \Delta u^2 \rangle$, and $c_i$ and $\lambda _i$ with $i=1,\ldots,p$ are constants related to the model parameters (cf. (A13) and (A14)). In (2.5), a higher order $p$ means the more likely it is to better approximate $\langle \Delta u^2 \rangle$. However, the parameters $c_i$ need to be determined by the autocorrelation of the data, and the roots $x_i$ need to be determined by the model parameters through the fitting procedure. We take a first-order approximation to obtain a more intuitive expression, i.e. $p=1$, then (2.3) leads to $q=\eta _2/\eta _1$. Additionally, the model equation (cf. (2.1)) is reduced to

(2.6)\begin{equation} u_t=\alpha u_{t-\Delta t}+\epsilon_t+\sum_{j=1}^{q}\beta_j\epsilon_{t-j\Delta t}. \end{equation}

Here, the velocity $u$ at the time $t$ depends on the value of $u$ at the previous time $t-\Delta t$ and is subject to random effects with memory depth $q$. The properties of autocorrelation of the ARMA model are presented in Appendix A.

When the two-point displacement is beyond the dynamic range, i.e. $k>q$, the autocorrelation becomes (cf. (A13))

(2.7)\begin{equation} R_{uu}(k\Delta x)=\alpha R_{uu}\left((k-1)\Delta x\right)=\alpha^{k-q}R_{uu}(q\Delta x)=\frac{R_{uu}(q\Delta x)}{\alpha^q}\exp\left(k\ln\alpha\right). \end{equation}

Replacing $k\Delta x$ with $r$, (2.7) becomes

(2.8)\begin{equation} R_{uu}\left(r\right) =\frac{R_{uu}(\eta_2 z)}{\alpha^q}\exp\left(\frac{\ln\alpha}{\Delta x}r\right), \quad r>\eta_2z. \end{equation}

Thus, the autocorrelation implies an exponential decay in the large-scale range since $\alpha <1$, which is similar to the approximation for correlations in homogeneous turbulence (Taylor Reference Taylor1921; Tennekes Reference Tennekes1979, Reference Tennekes1982).

The AR part of the ARMA model is regarded as the memory effect of flow and mainly controls large-scale motions, and the MA part does not contribute to the large-scale autocorrelation directly. If there is no MA part, the autocorrelation of the ARMA model decays exponentially as the displacement increases. In the dynamic range, the MA part contributes to the autocorrelation (cf. (A9)) and with a further constraint that the MA coefficients follows a power expression with exponent $-2$, the autocorrelation has a logarithmic dependence on the displacement, which is discussed in Appendix B.

2.2. Linking between ARMA and AEM

Based on the analytical expression of the ARMA model, similar to the AEM, the linear additive form of the MA part in the ARMA model leads to the logarithmic expression. AEM interprets the logarithmic expression of the dynamic range of second-order structure function based on the streamwise velocity representation (Yang et al. Reference Yang, Baidya, Johnson, Marusic and Meneveau2017)

(2.9)\begin{equation} u=\sum_{i=1}^{N_z}a_i, \end{equation}

where $a_i$ is the random velocity increment induced by the attached eddies of size $\delta /2^i$ and $N_z$ is the number of attached eddies contributing to $u$. Namely, the fluctuating velocity $u$ is generated by random eddies with vertical sizes ranging from $z$ to $\delta$, and other potential effects are neglected. Noteworthily, the ARMA model equation (2.1) also contains random variables and the velocity memory. Similar to AEM, we interpret the MA part of the ARMA model as the random effect arising from turbulent eddies. The MA coefficients can be seen as the eddy population density and the random increment $\epsilon$ has the same strength. The above analysis is consistent with the interpretation of the ARMA model by Faranda et al. (Reference Faranda, Pons, Dubrulle, Daviaud, Saint-Michel, Herbert and Cortet2014b) that the AR part is linked to the contribution of the large scales and the MA part corresponds to the effects of eddies and correlation structure. This physical picture also guides us to characterize the dynamic range, from which the model orders are determined.

As mentioned before, the model orders are determined according to the scale information of boundary layers. By Taylor's frozen hypothesis, we can obtain $\Delta x=U\Delta t$ (cf. (2.2)) and in the ARMA(1, $q$) model, $\Delta x$ is considered as the minimum attached eddy's size, which is of $O(z)$. The memory depth of the model corresponds to the maximum attached eddy's size, $q\Delta x\approx \eta _2 z$, which is of $O(\delta )$ and represents the largest scale of the dynamic range. These scales are also important in AEM. Therefore, the ARMA model can be regarded as a stochastic form of AEM: a set of random white noise, whose scale range is consistent with the sizes of attached eddies, drives the fluctuating velocity. Additionally, in the ARMA model, the MA coefficients $\beta _i$ with $i$ from 1 to $q$ quantify the distribution of attached eddies’ effects (see Appendix B). As shown in figure 8(b) (§ 3.2), the $-2$ exponent in the power function of MA coefficients is consistent with the eddy population density in some AEM studies (de Silva, Hutchins & Marusic Reference de Silva, Hutchins and Marusic2016; Hu, Dong & Vinuesa Reference Hu, Dong and Vinuesa2023). In addition, since the model's stationarity requires $\alpha <1$, the lag term in (2.6) damps this linear dynamical system energized by random noise. Unlike the linear additive process model (cf. (2.9)) proposed for AEM, the ARMA(1, $q$) model hypothetically presents a simple stochastic model of AEM, from which the long-term behaviour of the autocorrelation can be explored. Though Davidson & Krogstad (Reference Davidson and Krogstad2009) provided a general expression for autocorrelation resulting from the attached eddies’ contribution, eddy shape needs to be assumed, and they did not discuss the expression's large-scale behaviour. One advantage of the ARMA model is that the large-scale autocorrelation is obtained without assuming the eddy shape due to the capturing of the memory effect.

2.3. Asymptotic expression of second-order structure function

Even though the ARMA model provides an analytical expression for the second-order structure function in the dynamic and large-scale ranges, this complicated expression is not practically convenient. So in this section, we obtain a global expression covering the inertial, dynamic and large-scale ranges of the second-order structure function by asymptotic matching. As aforementioned, the second-order structure function follows a logarithmic expression in the dynamic range (cf. (1.4)). More generally, in shear flows where the ratio of energy production rate to dissipation rate $P/\varepsilon$ varies with wall-normal location $z$, which is not rare in ASL turbulence, the two ranges also match at an intermediate scale influenced by $P/\varepsilon$ (Zhang et al. Reference Zhang, Xie and Zheng2022). For the general case, (1.2) becomes

(2.10)\begin{equation} \langle \Delta u^{+2} \rangle = \frac{C_2}{\left(\kappa P/\varepsilon \right)^{2/3} } \left(\frac{r}{z} \right) ^{\xi_2} = M_2 \left(\frac{r}{z} \right) ^{\xi_2} ,\quad l_\nu\ll r< z, \end{equation}

where $M_2$ may be a height-dependent constant due to the presence of $P/\varepsilon$. Assuming that the dimensionless lower and upper ends of the dynamic range are $r/z=\eta _1$ and $r/z=\eta _2$, respectively, we can perform the Taylor expansion on (2.10) and (1.4), and obtain the general form of (1.4) by matching at $r/z=\eta _1$:

(2.11)\begin{equation} \langle \Delta u^{+2} \rangle = A_2+B_2\ln\left( \frac{r}{z\eta_1} \right) ,\quad \eta_1< \frac{r}{z}< \eta_2, \end{equation}

where

(2.12a,b)\begin{equation} A_2=M_2\eta_1^{\xi_2},\quad B_2=M_2\eta_1^{\xi_2}\xi_2, \end{equation}

and they are obtained by matching (1.3) and (2.11) at $\eta _1$. The matching process links the dynamic-range coefficients $A_2$ and $B_2$ with the inertial-range coefficients $M_2$ and $\xi _2$, reducing two free parameters. The empirical value of $\eta _1$ is $O(1)$ (Davidson, Krogstad & Nickels Reference Davidson, Krogstad and Nickels2006a; Davidson & Krogstad Reference Davidson and Krogstad2009; de Silva et al. Reference de Silva, Marusic, Woodcock and Meneveau2015; Xie et al. Reference Xie, de Silva, Baidya, Yang and Hu2021), corresponding to a length scale of $O(z)$. For the canonical case, by setting $C_2=2$, $\kappa =0.4$, $\xi _2=2/3$ and $\eta _1=1$, we calculate from (2.12a,b) to obtain $A_2\approx 3.68$ and $B_2\approx 2.46$, whose values are close to previous research (Meneveau & Marusic Reference Meneveau and Marusic2013; de Silva et al. Reference de Silva, Marusic, Woodcock and Meneveau2015; Xie et al. Reference Xie, de Silva, Baidya, Yang and Hu2021).

In § 2.2, the ARMA model is analogous to a stochastic AEM and the attached eddies are modelled as a set of random effects. The logarithmic dependence of $\langle \Delta u^{+2} \rangle$ can also be obtained from the ARMA model under the assumption of self-similarity with $\beta _{i}\sim i^{-2}$. The details can be found in Appendix B.

In the large-scale range with scales larger than $\eta _2 z$, conventional AEM does not provide information on the structure functions. The advantage of the ARMA(1, $q$) model is that it naturally provides an exponential autocorrelation for large scales (cf. (2.8)), i.e.

(2.13)\begin{equation} \langle \Delta u^{+2} \rangle = 2\langle u^{+2} \rangle -D_2 \exp \left(-E_2 \frac{r}{z} \right) ,\quad \frac{r}{z}>\eta_2, \end{equation}

where $D_2$ and $E_2$ are positive constants. Assuming that there is no distinguished range between the dynamic and the large-scale ranges, we match the expression in these ranges to determine three unknown parameters $\eta _2$, $D_2$ and $E_2$. By defining $r' = r/z$, we expand (2.11) and (2.13) to

(2.14a)$$\begin{gather} \langle \Delta u^{+2} \rangle =A_2+B_2\ln\left(\frac{r'_0}{\eta_1}\right) +\frac{B_2}{r'_0}\,\mathrm{d} r' -\frac{B_2}{2r'^2_0}\left(\mathrm{d} r'\right)^2+\cdots,\end{gather}$$

(2.14b)$$\begin{gather}\langle \Delta u^{+2} \rangle =2\langle u^{+2} \rangle -D_2 \exp\left(-E_2 r'_0 \right) +D_2E_2\exp\left(-E_2 r'_0 \right)\,\mathrm{d} r'\nonumber\\ -\frac{D_2E_2^2}{2}\exp\left(-E_2 r'_0 \right)\left(\mathrm{d} r'\right)^2+\cdots. \end{gather}$$

Matching the leading, the first- and second-order terms at $r'_0=\eta _2$, we obtain

(2.15a)$$\begin{gather} D_2=B_2 \mathrm{e}, \end{gather}$$

(2.15b)$$\begin{gather}\eta_2=\frac{1}{E_2}= \eta_1\exp\left( \frac{2\langle u^{+2} \rangle -B_2-A_2}{B_2}\right). \end{gather}$$

Thus, (2.13) becomes

(2.16)\begin{equation} \langle \Delta u^{+2} \rangle =2\langle u^{+2} \rangle -B_2\exp \left(1 - \frac{r}{z\eta_2} \right) ,\quad \frac{r}{z}>\eta_2 . \end{equation}

At the intermediate scale within distinct ranges, the dominant terms exhibit comparable magnitudes, while higher-order terms might also hold significance (Tong & Ding Reference Tong and Ding2019). Here, to obtain a concise expression, we neglect these higher-order terms for simplicity and find that direct matching of low-order terms works well. Equations (2.10), (2.11) and (2.16) constitute the global expression for the second-order structure function, where the coefficients are not all independent. With known turbulent intensity $\langle u^{+2} \rangle$ and three small-scale parameters $M_2$, $\xi _2$ and $\eta _1$, the global expression for the second-order structure function is determined. For a fixed height, the global expression applied to three horizontal scale ranges is determined by four parameters: $\langle u^{+2} \rangle$, $M_2$, $\xi _2$ and $\eta _1$. Additionally, the height dependence is captured by the logarithmic dependence of $\langle u^{+2} \rangle$, $\langle u^{+2} \rangle =A_1-B_1 \log (z/\delta )$, thus, we have in total five parameters: $A_1$, $B_1$, $M_2$, $\xi _2$ and $\eta _1$.

Assuming that the boundary layer thickness captures the outer scale for large-scale motions, we set the large-scale range's characteristic scale $\eta _2z$ of $O(\delta )$. In the logarithmic region, the streamwise turbulence intensity follows (Marusic et al. Reference Marusic, Monty, Hultmark and Smits2013; Meneveau & Marusic Reference Meneveau and Marusic2013)

(2.17)\begin{equation} \langle u^{+2} \rangle =A_1-B_1\ln \left( \frac{z}{\delta} \right), \end{equation}

where $B_1$ is the Townsend–Perry constant and $A_1$ is a flow-dependent constant. The size of the attached eddies that contribute to the velocity ranges from $z$ to $\delta$, which is consistent with $\delta$ as the characteristic length scale of the large-scale range. Substituting (2.17) into (2.15b), we get

(2.18)\begin{equation} \eta_2=\eta_1 C_\eta \left( \frac{\delta}{z}\right) ^{2B_1/B_2}, \end{equation}

where $C_\eta =\exp [ ( 2A_1-B_2-A_2 )/B_2]$ and $\eta _1$ is of $O(1)$. Considering a canonical situation with $B_2=2B_1$ (Davidson et al. Reference Davidson, Nickels and Krogstad2006b; Davidson & Krogstad Reference Davidson and Krogstad2014), for example, $B_1=1.25$ (Meneveau & Marusic Reference Meneveau and Marusic2013) and $B_2=2.5$ (Xie et al. Reference Xie, de Silva, Baidya, Yang and Hu2021), we obtain $\eta _2\sim \delta /z$, which is consistent with our analysis and the assumption that $\eta _2z\sim \delta$. Also, rearranging (2.15b), we obtain

(2.19)\begin{equation} \langle u^{+2} \rangle =\frac{A_2+B_2}{2}+\frac{B_2}{2}\ln \left(\frac{\eta_2}{\eta_1}\right). \end{equation}

Taking the approximation $\eta _2/\eta _1\sim \delta /z$, (2.19) recovers the expression for streamwise turbulent intensity (2.17) and $B_2=2B_1$. Even though $B_2$ may not equal twice of $B_1$ in real ASL measurement data, which we show in figure 10(b) (§ 3.3), the assumed exponential expression in the large-scale range links the dynamic range and $r\to \infty$, and is consistent with the matching procedure by Davidson & Krogstad (Reference Davidson and Krogstad2014). The above results show that the proposed exponential function for $\langle \Delta u^{+2} \rangle$ based on the ARMA model is consistent with and bridges previous theories.

2.4. Implication of the global expression

2.4.1. Characteristic length scales calculated from the global expression

In this section, we derive the expression of characteristic length scale from the previously obtained expression of $\langle \Delta u^{+2} \rangle$. The characteristic length scale $L_x$ is defined by introducing an artificial threshold $T$ of the autocorrelation:

(2.20)\begin{equation} R_{uu}(L_x/2)=T \langle u^2 \rangle . \end{equation}

Though studies show that the threshold value does not change the trend of obtained length scales (Zhou et al. Reference Zhou, Adrian, Balachandar and Kendall1999), we believe the reason is that the threshold values are usually small. The length-scale behaviour could change when different thresholds locate in different ranges of the second-order structure function. For example, the thresholds corresponding to $\eta _1$ and $\eta _2$ are

(2.21a)$$\begin{gather} T_{\eta_1}=1-\frac{M_2 \eta_1^{\xi_2}}{2\langle u^{+2} \rangle }, \end{gather}$$

(2.21b)$$\begin{gather}T_{\eta_2}=\frac{B_2}{2\langle u^{+2} \rangle }. \end{gather}$$

The dependences of $T_{\eta _1}$ and $T_{\eta _2}$ on $z/\delta$ are shown in figure 3, showing that the detected characteristic length scale depends on the choice of the threshold value, e.g. if $T=0.05$, the threshold locates in the large-scale range, and if $T=0.4$, the threshold locates in the dynamic range.

Figure 3. Variation of $T_{\eta _1}$ and $T_{\eta _2}$ with $z/\delta$ according to (2.21a) and (2.21b), where $M_2=3.62$, $B_1=1.25$, $A_1=1.73$ and $B_2=2.5$.

Taking $\langle \Delta u^{+2} \rangle =2(1-T )\langle u^{+2} \rangle$ in different ranges of $\langle \Delta u^{+2} \rangle$, we can obtain the analytical expression for the length scale $L_x/\delta$ predicted for different thresholds:

(2.22)\begin{align} \frac{L_x}{\delta} =\left\{ \begin{array}{@{}ll} 2\eta_1 C_\eta \left(\dfrac{z}{\delta}\right)^{1-2B_1/B_2}\left\{ 1-\ln\left[ \dfrac{2T}{B_2}\left( A_1-B_1 \ln \left( \dfrac{z}{\delta}\right) \right) \right] \right\} , & T< T_{\eta_2},\\ 2\eta_1\exp \left[ \dfrac{2(1-T)A_1 -A_2}{B_2} \right] \left( \dfrac{z}{\delta} \right)^{1-2B_1(1-T)/B_2}, & T_{\eta_2}< T< T_{\eta_1},\\ 2\dfrac{z}{\delta}\left[ \dfrac{2(1-T)}{M_2}\left(A_1-B_1\ln \left( \dfrac{z}{\delta} \right) \right) \right]^{1/\xi_2} , & T>T_{\eta_1}. \end{array} \right. \end{align}

Since $\langle u^{+2} \rangle$ is expressed in terms of $A_1$ and $B_1$, there is an extra parameter in the expression for $L_x$ compared with the global expression for the second-order structure function. The main independent parameters that control the behaviour of $L_x$ are $A_1/B_2$ and $B_1/B_2$, and other parameters are close to their canonical value. It is interesting to show that when $T< T_{\eta _2}$, the characteristic length has a double-log dependence on the distance to the wall, which is one of our main results and is checked below by experimental data. Additionally, for a threshold value corresponding to the dynamic and inertial ranges, the behaviours of $L_x$ can also be described by (2.22). If one wants to observe the large characteristic scales corresponding to the flow structure with the dynamics, the threshold value should not be chosen to be greater than the minimum value of $T_{\eta _2}$, as shown in figure 3.

2.4.2. Lower bound for the streamwise turbulent intensity in the logarithmic region

Assuming that the streamwise autocorrelation function decays with increasing scale, which is commonly found in experiments and numerical simulations of boundary-layer turbulence, the value of $\langle \Delta u^{+2} \rangle$ at $r/z=\eta _2$ should be larger than that at $r/z=\eta _1$ in the dynamic range. Therefore, (2.11) and (2.16) imply

(2.23)\begin{equation} 2\langle u^{+2} \rangle -B_2\geq A_2. \end{equation}

With $A_2=B_2/\xi _2$ and $B_2=C_2\kappa ^{-2/3}\eta _1^{\xi _2}\xi _2$, we obtain

(2.24)\begin{equation} \langle u^{+2} \rangle \geq\frac{1}{2}\left(1+\frac{1}{\xi_2}\right)C_2\kappa^{-2/3}\eta_1^{\xi_2}\xi_2. \end{equation}

Taking $C_2=2$, $\kappa =0.4$, $\xi _2=2/3$ and $\eta _1=1$, we find a lower bound for the turbulence intensity in the logarithmic region $\langle u^{+2} \rangle \geq 3.07$, which is justified by previous results $\langle u^{+2} \rangle$ (Hutchins et al. Reference Hutchins, Chauhan, Marusic, Monty and Klewicki2012; Wang & Zheng Reference Wang and Zheng2016). In addition, this lower bound depends on the imbalance between energy production and dissipation because $\eta _1$ increases as dissipation increases (Zhang et al. Reference Zhang, Xie and Zheng2022). More numerical and experimental results are needed to further check the validity of this lower bound.

2.4.3. Estimation of the integral length scale

Knowing the global expression for second-order structure function, the integral length scale $\mathcal {L}$ can be calculated as

(2.25)\begin{equation} \mathcal{L}=\int_0^{\infty} \frac{\langle u_1u_2 \rangle }{\langle u^2 \rangle }\,\mathrm{d} r =\int_0^{\infty} \frac{\langle u^2 \rangle -\langle \Delta u^2 \rangle /2}{\langle u^2 \rangle }\,\mathrm{d} r =\int_0^{\infty} \left(1-\frac{\langle \Delta u^{+2} \rangle }{2\langle u^{+2} \rangle }\right)\,\mathrm{d} r. \end{equation}

Denoting $r/z$ as $\eta$, and using (2.10), (2.11) and (2.16), we get

(2.26)\begin{align} \frac{\mathcal{L}}{z}&=\int_0^{\infty} \left(1-\frac{\langle \Delta u^{+2} \rangle }{2\langle u^{+2} \rangle }\right)\,\mathrm{d} \eta\nonumber\\ &\approx \int_0^{\eta_1} \left( 1-\frac{M_2 \eta^{\xi_2}}{2\langle u^{+2} \rangle } \right)\,\mathrm{d} \eta +\int_{\eta_1}^{\eta_2} \left[ 1-\frac{A_2+B_2 \ln (\eta/\eta_1)}{2\langle u^{+2} \rangle } \right]\,\mathrm{d} \eta \nonumber\\ &\quad +\int_{\eta_2}^{\infty} \left[ 1-\frac{2\langle u^{+2} \rangle -B_2\mathrm{e}^{1-\eta/\eta_2}} {2\langle u^{+2} \rangle }\right]\,\mathrm{d} \eta \nonumber\\ &= I_1+I_2+I_3. \end{align}

Referring to (2.12a,b) and (2.15b), we can express $M_2$, $A_2$ and $\eta _2$ as functions of $B_2$, $\eta _1$ and $\xi _2$. Denoting $B_2/(2\langle u^{+2} \rangle )$ as $\mu$, the integrals in (2.26) can be calculated as

(2.27a)$$\begin{gather} I_1=\int_0^{\eta_1} \left( 1-\frac{\mu \eta^{\xi_2}}{\eta_1^{\xi_2} \xi_2} \right)\,\mathrm{d} \eta = \eta_1-\frac{\mu}{\xi_2 (\xi_2+1)}\eta_1, \end{gather}$$

(2.27b)$$\begin{gather}I_2=\int_{\eta_1}^{\eta_2} \left[ 1- \frac{\mu}{\xi_2}-\mu \ln \left(\frac{\eta}{\eta_1}\right) \right]\,\mathrm{d} \eta = \left(\frac{\mu}{\xi_2}-1-\mu\right)\eta_1+2\mu \eta_2, \end{gather}$$

(2.27c)$$\begin{gather}I_3=\int_{\eta_2}^{\infty} \left[ \mu\mathrm{e}^{1-\eta/\eta_2} \right]\,\mathrm{d} \eta = \mu \eta_2, \end{gather}$$

where the second integral uses $\ln (\eta _2/\eta _1)=1/\mu -1-1/\xi _2$ (cf. (2.15b)). Then we obtain

(2.28)\begin{equation} \frac{\mathcal{L}}{z}=-\frac{\xi_2 \mu}{\xi_2+1}\eta_1+3\mu \eta_2. \end{equation}

Again using (2.15b), (2.28) can be further expressed as

(2.29a)\begin{equation} \frac{\mathcal{L}}{z}=\left[ 3\mu\exp \left( \frac{1}{\mu}-1-\frac{1}{\xi_2}\right) -\frac{\xi_2 \mu}{\xi_2+1} \right]\eta_1=f_1(\mu,\xi_2)\eta_1 \end{equation}

or

(2.29b)\begin{equation} \frac{\mathcal{L}}{z}=\left[ 3\mu-\frac{\xi_2 \mu}{(\xi_2+1)\exp \left( 1/\mu-1-1/\xi_2\right)} \right]\eta_2=f_2(\mu,\xi_2)\eta_2. \end{equation}

Here, the expressions are reformed to compare with two characteristic scales corresponding to $\eta _1$ and $\eta _2$. The values of $f_1$ and $f_2$ with $\xi _2=2/3$ are shown in figure 4. The parameter $\mu$ that guarantees $f_1>1$ and $f_2<1$ is often achieved in boundary-layer turbulence, e.g. with $B_2 = 2.36$ and the lower bound of streamwise turbulent intensity $\langle u^{+2} \rangle \geq 3.07$ estimated in § 2.4.2. Therefore, in the logarithmic region, the integral length scale $\mathcal {L}$ locates in the dynamic range, which is consistent with $\mathcal {L}$ estimated in the neutral ASL and experiments (Hutchins & Marusic Reference Hutchins and Marusic2007; Gustenyov et al. Reference Gustenyov, Egerer, Hultmark, Smits and Bailey2023).

Figure 4. Variations of (a) $f_1$ and (b) $f_2$ with $\mu =B_2/(2\langle u^{+2} \rangle )$ for $\xi _2=2/3$.

Throughout the remainder of the article, we apply the ARMA(1, $q$) model on the streamwise velocity of ASL turbulence and test our theoretical expressions (2.16), (2.22) and (2.29) at $Re_\tau \sim O(10^6)$.

3.1. Experimental facility and data pretreatment

The ASL data used in this work come from QLOA, which is built on the flat dry lakebed of Qingtu Lake in western China (E: $103^\circ 40^\prime 03^{\prime \prime }$, N: $39^\circ 12^\prime 27^{\prime \prime }$) and provides the highest order of magnitude friction Reynolds number data to date. Also, QLOA is the unique observation site where multi-point measurements can be performed simultaneously, including three-dimensional turbulent velocities, temperature, humidity, PM10 concentration and electric field, and so forth. The high-quality data of clear-air and sand-laden ASL flows proved suitable for turbulent boundary layer studies (Wang & Zheng Reference Wang and Zheng2016; Liu, Wang & Zheng Reference Liu, Wang and Zheng2019a; Wang, Gu & Zheng Reference Wang, Gu and Zheng2020; Liu, He & Zheng Reference Liu, He and Zheng2021). The main tower is 32 m high and has measurement positions at 0.9, 1.71, 2.5, 3.49, 5, 7.15, 8.5, 10.24, 14.65, 20.96 and 30 m, approximately logarithmically aligned. Eight low towers of 5 m high are arranged at an equal distance of 30 m in the prevailing wind direction, and twelve low towers of 5 m high are arranged at an equal distance of 5 m in the spanwise direction. The data used in this study were obtained from the main tower, whose height lies roughly in the logarithmic region of the atmospheric boundary layer. The three-component sonic anemometers (Campbell scientific, CSAT-3B) installed on the tower measure the velocities and temperature synchronously, with a sampling frequency of 50 Hz, a velocity measurement range of 0–45 m s$^{-1}$, a minimum velocity resolution of 0.001 m s$^{-1}$, a temperature measurement range of $-$40–60 $^\circ$C, a minimum temperature resolution of 1 $^\circ$C and a wind direction recording range of 0–359$^\circ$. The sonic anemometers are all connected to data acquisition instruments that are time-synchronized with the global positioning system to ensure data synchronization.

Due to the uncertainty and uncontrollability of field measurement, the measured data need to be selected and pretreated. The same pretreatment procedures as previous ASL turbulence studies using QLOA data are implemented here. The raw data are partitioned into hourly time series, and then the pretreatments including wind direction adjustment, detrending manipulation, stratification stability judgment and stationary wind selection is carried out. Although the streamwise direction of QLOA is designed to be consistent with the prevailing wind direction, the wind direction changes during the field measurement. So the measured data need to be adjusted as

(3.1a,b)\begin{equation} u = u_{m} \cos\omega + v_{m} \sin\omega, \quad v = v_{m} \cos\omega - u_{m} \sin\omega, \end{equation}

where $u_{m}$ and $v_{m}$ are streamwise and spanwise velocities measured by the anemometers, and $\omega$ is the angle between the actual wind direction and the streamwise direction of QLOA averaged over all heights. After adjusting the wind direction, the long-term synoptic signal also needs to be detrended by a low-pass filter with a cut-off wavelength of 20$\delta$ to extract the turbulence fluctuating signal.

In addition, thermal convection often occurs in ASL turbulence, affecting the budget of turbulent kinetic energy (Wyngaard & Coté Reference Wyngaard and Coté1971; Wyngaard, Coté & Izumi Reference Wyngaard, Coté and Izumi1971). The Monin–Obukhov stratification parameter (Monin & Obukhov Reference Monin and Obukhov1954), which characterizes the ratio of the buoyancy and shear effects, is often used to evaluate the stratification conditions and is defined as

(3.2)\begin{equation} \frac{z}{L} =- \frac{\kappa z g \langle w \theta \rangle }{\langle \theta \rangle u _ {\tau}^ {3 }}, \end{equation}

where $L$ is the Obukhov length (Obukhov Reference Obukhov1946), $g$ is the acceleration of gravity, and $\langle w \theta \rangle$ is the mean vertical heat flux defined as the covariance of the vertical wind velocity $w$ and the measured potential temperature $\theta$. If $|z/L|\ll 1$, the ASL flow is shear-dominant and the density stratification is negligible. In our analysis, the Monin–Obukhov stability parameter is calculated at $z=1.71$ m, and we select the near-neutral data with $|z/L|<0.06$. Another effective indicator of the stratification stability is the $k_x^{-5/3}$ scaling of spectra at low wavenumber (Tong & Nguyen Reference Tong and Nguyen2015; Tong & Ding Reference Tong and Ding2019). If the $k_x^{-1}$ scaling is observed instead of $k_x^{-5/3}$, the ASL flow can be considered to be shear dominant and therefore near-neutral. The streamwise spectra of the ASL data used in this work exhibit no significant $k_x^{-5/3}$ range at low wavenumber, as shown, for example, in figure 9(a) (§ 3.2).

Since we focus on fully developed stationary turbulence, selecting the statistically stationary ASL data is necessary. A non-stationary index $\gamma$ is used to judge the stationarity, which is defined as

(3.3)\begin{equation} \gamma = | ( \sigma _ {M} - \sigma_ {I} ) / \sigma _ {I} | \times 100\,\%, \end{equation}

where $\sigma _{M}=\sum _{i=1}^{12}\sigma _i/12$, $\sigma _1, \sigma _2, \ldots, \sigma _{12}$ are the streamwise velocity variances of one-twelfth part of the entire time interval and $\sigma _{I}$ is the variance of the overall time interval. Stationarity requires ASL data over the interval to meet $\gamma <30\,\%$ and we pick the half-hour with the smallest $\gamma$ in an hour as our time interval.

Other quantities used here are defined as follows, consistent with previous studies at the QLOA site under the near-neutral stratification conditions. The friction velocity $u_\tau$ is approximated by the peak value of $(-\langle uw \rangle )^{1/2}$. The air kinematic viscosity $\nu$ is calculated from the measured mean temperature under standard atmospheric pressure. Additionally, the ASL thickness $\delta$ is estimated as 150 m to evaluate the friction Reynolds number $Re_\tau$. Some basic flow information of the selected intervals, including the mean velocity profile, the streamwise turbulent intensity and the shear Reynolds stress, are shown in figure 5. Comparison with ASL data obtained by Hutchins et al. (Reference Hutchins, Chauhan, Marusic, Monty and Klewicki2012) and theoretical results of boundary-layer turbulence proves that our data are consistent with the logarithmic region of the boundary layers.

Figure 5. Profiles of statistics for selected intervals at ${Re}_\tau = O(10^6)$. (a) Log-linear plot of the mean streamwise velocity. The open symbols are current QLOA data and the dashed line corresponds to the log-law with $\kappa = 0.41$ (Pope Reference Pope2000). (b) Logarithmic decay of streamwise turbulent intensity. The blue symbols is the ASL result of Hutchins et al. (Reference Hutchins, Chauhan, Marusic, Monty and Klewicki2012) at ${Re}_\tau = O(10^6)$. The dashed and dotted lines correspond to (2.17) with $B_1=0.55$ and $0.80$, respectively. (c) Dimensionless shear Reynolds stress. The blue symbols are ASL data from Hutchins et al. (Reference Hutchins, Chauhan, Marusic, Monty and Klewicki2012) and the dashed line is the similarity formulation from Chauhan (Reference Chauhan2007).

3.2. Justifying the ARMA model

In this section, we justify that the ARMA model can well capture the second-order statistics of the ASL data in the dynamic and large-scale ranges. The procedural workflow for applying the ARMA model to boundary-layer turbulence is visually depicted in figure 6. (I) In the first step, ASL data selection and pretreatment are performed. These details are described in § 3.1. (II) Once the streamwise velocity series is obtained, we need to determine two characteristic scales $\eta _1$ and $\eta _2$. Based on AEM, we have $\eta _1=O(1)$, so we choose the interval for the extracted velocity series as $\Delta t/\Delta t_0=z/(U\Delta t_0)$. This resampling process filters out the inertial range. (III) In § 2.1, we specify the model orders as $p=1$ and $q=\eta _2/\eta _1$. The largest scale $\eta _2$ of the dynamic range can be estimated by (2.15b). Alternatively, a more accurate approach to determine $\eta _1$ and $\eta _2$ is by obtaining $\eta _1$ from the inertial and dynamic ranges of the ASL data according to (2.10)–(2.11), and then calculating $\eta _2$ with the help of (2.15b). Specifically, the power fit for scale range less than $\eta _1$ yields $M_2$ and $\xi _2$, then $A_2$, $B_2$ and $\eta _2$ can be obtained by (2.12a,b) and (2.15b), and then it is convenient to optimize $\eta _1$ that minimizes the total absolute error of the global expression by the optimization method, such as the ‘fmincon’ function in MATLAB. We adopt the latter approach in this work. (IV) Knowing the value of $\eta _1$ and $\eta _2$, we can obtain $q$ and use the ARMA(1, $q$) model to fit the extracted series. The fourth step uses the same approach as the statistical practice, which is available in many function packages, and here we use the ‘arima’ function in R. (V) After the model fitting procedure, an essential fifth step involves evaluating the model's goodness of fit. A common approach is to scrutinize the residuals resulting from the model fitting process; if these residuals exhibit characteristics akin to white noise, we validate the model. Conventional practices for this purpose encompass well-known statistical tests such as the Ljung–Box and Breusch–Godfrey tests (Liu, Erdem & Shi Reference Liu, Erdem and Shi2011). In the present study, the model residuals successfully pass the Ljung–Box test with a p-value larger than 0.05, indicating that the ARMA model captures the correlation of boundary-layer turbulence well.

Figure 6. Diagrammatic flow of applying the ARMA model to boundary-layer turbulence.

There are two significant distinctions from traditional statistical practice in our procedure. First, the precondition of stationarity is required before employing the ARMA model. The traditional statistical modelling methods require only the removal of mean value by applying a moving average or a differential operator and the removal of periodic signals. Considering the physics of the ASL, we also need to ensure neutrality and filter large-scale synoptic signals. Second, the typical approach for ascertaining the model orders involves statistical criteria such as the Akaike or Bayesian information criterion (Choi Reference Choi2012). In contrast, within this study, we establish the model order based on the characteristic scale of the boundary layer, which again is restricted to ASL physics. This approach enhances the model's interpretability and endows the model with heightened physical significance.

Figure 7 shows an example of fitting the ARMA model. The half-hour raw fluctuating velocity series is presented in figure 7(a). Additionally, the reproduced coarse series is shown in figure 7(b). Using the ARMA model, we can regenerate the velocity series with almost the same velocity variance of raw data. Moreover, compared with the autocorrelation calculated from raw data, the autocorrelation generated by the ARMA(1, $q$) model fits well, which is shown in figure 7(c). Figure 7(d) shows the corresponding second-order structure function, and the consistency between the result of the ARMA model and the proposed global expression indicates the validity of the approximate exponential function in the large-scale range. These results validate the applicability of using the ARMA(1, $q$) model to describe the boundary-layer streamwise velocity series.

Figure 7. (a) Raw fluctuating velocity $u$; (b) reproduced fluctuating velocity $\tilde u$; (c) comparison of autocorrelations of ASL data and ARMA model in normal-normal coordinates; and (d) comparison of the global expression, second-order structure functions of ASL data and ARMA model in log-normal coordinates, where $u_\tau =0.37$ m s$^{-1}$, ${Re}_\tau =3.78\times 10^6$ and $z^+=2.27\times 10^4$.

As discussed in § 2.1, we simplify the attached eddies with continuously varying sizes to a series of eddies whose sizes are approximated as a multiple of $z$ up to $\delta$. The model parameters $\beta _i$ quantify these attached eddies’ effects on the current position's acceleration. Figure 8(a) shows that at all heights, as the distance $i\Delta x$ increases, the effect of corresponding attached eddies ($\beta ^2_i$) decreases, indicating that the attached eddies with larger size contribute less to the fluctuating velocity. In addition, the model's memory depth is restricted to the dynamic range, where autocorrelation shows a logarithmic behaviour. Figure 8(b) shows that the self-similar random effect with a $-2$ power scaling of $\beta _i$ leads to a logarithmic autocorrelation, which is discussed in detail in Appendix B.

Figure 8. Fitted model parameters $\beta _{i}$, $i=1,2,\ldots,q$, with ${Re}_\tau =3.78\times 10^6$.

Here, the ARMA(1, $q$) model is used to model the dynamic range and to obtain the large-scale range's behaviour as a natural extension. To justify the large-scale behaviour, in figure 9, we show the premultiplied spectra, whose peak is used to distinguish large- and very-large-scale motions (Wang & Zheng Reference Wang and Zheng2016). Even though small scales are not resolved by the ARMA(1, $q$) model, its premultiplied spectra well captures that of the ASL data. Note that since the one-dimensional spectra at low wavenumber contain an aliasing effect, one needs to be careful when examining large-scale information from one-dimensional spectra. Improved approaches include examining two-dimensional spectra (Tong & Nguyen Reference Tong and Nguyen2015) and employing Fourier cutoff techniques (Ding et al. Reference Ding, Nguyen, Liu, Otte and Tong2018).

Figure 9. (a) Power spectra $\varPhi _{uu}$ and (b) premultiplied spectra $k_x\varPhi _{uu}$ obtained from ASL data and ARMA(1, $q$) model, respectively. Here ${Re}_\tau =3.78\times 10^6$ and $z^+=7.55\times 10^5$.

3.3. Asymptotic global expression of the second-order structure function

Because the exact expression of the second-order structure function calculated from the ARMA(1, $q$) model is complicated, for analytic simplicity, we derive a global asymptotic expression for the second-order structure function $\langle \Delta u^{+2} \rangle$ in § 2.3 (cf. (2.10), (2.11) and (2.16)), which is justified in figure 10(a). The uncertainty ranges of parameters $M_2$, $\xi _2$ and $\eta _1$ are discussed in Appendix C. Additionally, $\eta _2$ is calculated from (2.15b) with the optimized $\eta _1$. The parameters of inertial and dynamic ranges are close to the results of de Silva et al. (Reference de Silva, Marusic, Woodcock and Meneveau2015).

Figure 10. (a) Comparison of $\langle \Delta u^{+2} \rangle$ obtained from ASL data and global asymptotic expression at ${Re}_\tau =3.78\times 10^6$. The intermediate scales of theoretical curves are as follows: Curve 1 – $\eta _1 = 1.06$, $\eta _2 = 27.71$; Curve 2 – $\eta _1 = 0.87$, $\eta _2 = 23.12$; Curve 3 – $\eta _1 = 0.73$, $\eta _2 = 15.42$. The black curves correspond to expressions for inertial and dynamic ranges presented by de Silva et al. (Reference de Silva, Marusic, Woodcock and Meneveau2015). (b) Comparison of $\eta _2$ obtained from ASL data and the expression (2.18) with $A_1/B_2=1.15$, $A_2/B_2=1/\xi _2=1.50$, $B_1/B_2=0.41$ and $B_1=0.80$.

According to (2.18), $\eta _2/\eta _1$ is a power function of $z/\delta$, which we check in figure 10(b). This exponent is conventionally regarded to be $1$, corresponding to the empirical values of $B_1=1.25$ and $B_2=2.5$. However, in ASL data, $B_1$ often deviates from 1.25, and some researchers argue that $B_1$ may decrease with increasing $Re_\tau$ (Monkewitz Reference Monkewitz2022). In addition, $B_2$ depends on $\eta _1$ (cf. (2.11)), which is influenced by the ratio between the energy production and dissipation rate. So instead of restricting the values of $B_1$ and $B_2$ individually, we take the realistic values obtained from ASL data to test our expressions.

Furthermore, we undertake validation of the global expression through direct numerical simulation (DNS) and experimental results. Illustrated in figure 11(a), the triangle symbols correspond to the DNS data at ${Re}_\tau \approx 5200$ obtained from Johns Hopkins turbulence database (Lee & Moser Reference Lee and Moser2015). Although the logarithmic slope $B_2$ of DNS data ranges from 1.46 to 1.78, smaller than the previously reported range of 2.16 to 2.5, the global expression matches the data well. The circle and square symbols in figure 11(a) correspond to the experiment results at ${Re}_\tau = 19\,000$, as detailed by Hutchins et al. (Reference Hutchins, Nickels, Marusic and Chong2009) and portrayed in figure 3(a) of de Silva et al. (Reference de Silva, Marusic, Woodcock and Meneveau2015). Combining these results from experiment, DNS and ASL data, we conclude that the proposed global expression for the second-order structure function is applicable across friction Reynolds numbers from $O(10^4)$ to $O(10^6)$. Since $B_2$ varies significantly with $z^+$ in DNS data, it is more appropriate to check (2.15b) rather than (2.18). The relationship between $\eta _2$ and $\eta _1$ obtained by asymptotic matching conforms well to the data, as shown in figure 11(b).

Figure 11. (a) Comparison of $\langle \Delta u^{+2} \rangle$ obtained from DNS and experiment data and global asymptotic expression. Here, $z^+$ locates in the logarithmic region. The triangle symbols correspond to the DNS data at ${Re}_\tau \approx 5200$ (Lee & Moser Reference Lee and Moser2015), and the circle and square symbols correspond to the experiment data ${Re}_\tau = 19\,000$ (Hutchins et al. Reference Hutchins, Nickels, Marusic and Chong2009; de Silva et al. Reference de Silva, Marusic, Woodcock and Meneveau2015). (b) Comparison of $\eta _2$ obtained from DNS data and the expression (2.15b) with $A_1=1.72$ and $B_1=1.50$.

Introducing the two matching scales $\eta _1$ and $\eta _2$ makes a more lucid examination of behaviours across distinct scale ranges feasible. Within the transport equation for the second-order structure function, the shear production term $\langle \Delta u\Delta w \rangle \,\mathrm {d} U/\mathrm {d} z$ injects energy into the shear boundary layers. So we explore the scaling behaviour of $\langle u_1w_2+u_2w_1 \rangle$. Figure 12(a) shows that the energy flux remains relatively constant within the inertial range (scales less than $\eta _1 z$). Additionally, in the dynamic range, $\langle u_1w_2+u_2w_1 \rangle$ follows a $-1$ power law that leads to the logarithmic expression for the structure functions (Xie et al. Reference Xie, de Silva, Baidya, Yang and Hu2021). Conversely, scales exceeding $\eta _2 z$ exhibit negligible production. So $\eta _2 z$ is an approximate indicator of the energy injection scale in boundary layers. With negligible energy flux, the autocorrelation in the active range decays exponentially.

Figure 12. Behaviour of covariance of streamwise velocity $u$ and vertical velocity $w$ in the (a) dynamic range and (b) large-scale range.

One main result of the present study is obtaining the expression of the second-order structure function in the large-scale range, which is beyond previous works on the inertial and dynamic ranges (Davidson et al. Reference Davidson, Nickels and Krogstad2006b; Davidson & Krogstad Reference Davidson and Krogstad2014; de Silva et al. Reference de Silva, Marusic, Woodcock and Meneveau2015). In our extended expression, the upper bound of the dynamic range scales as $\eta _2$. Thus, with scale normalized by $\eta _2 z$, collapse of data of different ${Re}_\tau$ in figure 13(a) justifies this characteristic scale. In the large-scale range, the exponential expression of the second-order structure functions obtained from the ARMA(1, $q$) model is justified in figure 13(b).

Figure 13. (a) Dynamic and large-scale ranges of second-order structure function with new scale $\eta _2 z$ at different high friction Reynolds numbers ${Re}_\tau$. (b) Exponential scaling of the second-order structure function for scales larger than $\eta _2 z$.

3.4. The characteristic length scale

The asymptotic global expression for the second-order structure function provides a theoretical expression for characteristic length scales $L_x$ identified by a threshold of autocorrelation. The well-fitting result of the global expression implies the validity of (2.22). There are three independent parameters in (2.22): $\eta _1$, $A_1/B_2$ and $B_1/B_2$. The empirical scale $\eta _1$ and already defined constants $A_1/B_2$ and $B_1/B_2$ characterize the variation of $L_x$. As shown in figure 14(a), the previous results of $L_x$ are summarized by Liu et al. (Reference Liu, Wang and Zheng2017) with an approximate logarithmic expression. However, other studies found anomalous behaviour of $L_x$ (Önder & Meyers Reference Önder and Meyers2018), suggesting that $L_x$ may be variant under different flow conditions. Our double-log expression (2.22) with turning parameters well captures all data, implying it as a robust relation.

Figure 14. (a) Characteristic length scales summarized by Liu et al. (Reference Liu, Wang and Zheng2017) and the data comes from Lee & Sung (Reference Lee and Sung2011), Volino, Schultz & Flack (Reference Volino, Schultz and Flack2007), Tutkun et al. (Reference Tutkun, George, Delville, Stanislas, Johansson, Foucaut and Coudert2009), Hutchins & Marusic (Reference Hutchins and Marusic2007), Hutchins et al. (Reference Hutchins, Chauhan, Marusic, Monty and Klewicki2012), Liu et al. (Reference Liu, Wang and Zheng2017) and Önder & Meyers (Reference Önder and Meyers2018). Equation (2.22) provides the prediction of parameters $A_1/B_2$ and $B_1/B_2$. (b) Some ASL data that deviate from the $L_x$ summarized by Liu et al. (Reference Liu, Wang and Zheng2017) can also be characterized by (2.22). The height-averaged ratio of energy production and dissipation rate $P/\varepsilon$ for the time intervals corresponding to the circle, triangle and square symbols are 1.84, 1.47 and 1.37, respectively.

To further justify (2.22), in figure 14(b), we select three sets of ASL data with abnormal $L_x$. Equation (2.22) also describe these behaviours with parameters $A_1/B_2$ and $B_1/B_2$ obtained from the streamwise fluctuating velocity variance and second-order structure function of the corresponding data sets. Note that in (2.22), in addition to $A_1/B_2$ and $B_1/B_2$, there is another parameter $\eta _1$ that needs to be determined, which is of order unity. In figure 14(a), $\eta _1$ is assumed to be 1 while it is taken as its average value at different heights in figure 14(b). The well-fitting results show that (2.22) is a good candidate to explain the variation of $L_x$.

Here our primary parameter of interest is $B_1/B_2$, which although has an accepted value of $0.5$, deviates in different flows. As can be observed from the expression for $L_x$ in the large-scale range (cf. (2.22)), the extent to which $B_1/B_2$ deviates from 0.5 leads mainly to the growth rate of $L_x/\delta$ with $z/\delta$. Figure 14(b) shows that, usually, a smaller value of $B_1/B_2$ corresponds to a larger set of $L_x/\delta$. In uncontrollable ASL measurements, flow parameters that deviate from the standard zero-pressure-gradient boundary layer are not rare, e.g. production and dissipation are not locally balanced, and the height dependence of turbulent kinetic energy can vary (cf. (2.17)). The unbalanced energy production and dissipation affects the intermediate scale $\eta _1$ between inertial and dynamic ranges (Zhang et al. Reference Zhang, Xie and Zheng2022), thus impacting the value of $L_x$.

With the global expression, we can estimate the integral length scale $\mathcal {L}$ approximately. For ASL data, because errors at large scales have a significant impact on the calculation of $\mathcal {L}$, we integrate the dimensionless autocorrelation up to the first 0.01 crossing scale, similar to the approach adopted by O'Neill et al. (Reference O'Neill, Nicolaides, Honnery and Soria2004) and Tritton (Reference Tritton2012). As shown in figure 15, (2.29a) characterizes the variation of $\mathcal {L}$ well. Since $f_1$ is sensitive to the value of $\mu$, see figure 4, we determine the value of $\mu$ from ASL data rather than using empirical values and the small-scale power exponent is set as $\xi _2=2/3$. Additionally, as discussed in § 2.4.3, (2.29) indicates that $\mathcal {L}$ locates in the dynamic range, suggesting that $\mathcal {L}$ is a characteristic scale of the attached eddies in the logarithmic region.

Figure 15. Comparison of integral length scale $\mathcal {L}$ and approximate expression (2.29a). The autocorrelation is integrated up to the first 0.01 crossing. The parameter $\mu$ in $f_1$ is obtained from ASL data and $\xi _2=2/3$.