2.1. Transfer entropy

Quantification of causal relationships between time-dependent variables can be achieved effectively using the framework of information theory (Shannon Reference Shannon1948). The core of this approach lies in Shannon entropy, which serves as the fundamental measure for assessing causal interactions. For a random variable $X$ with probability distribution function (p.d.f.) $p(x)$ , where $x$ is any measurement of $X$ , its Shannon entropy is defined as

(2.1) \begin{align} S(X) = -\sum _{x}p(x)\log p(x), \end{align}

which characterises the uncertainty inherent in $X$ . The joint entropy for a pair of variables $(X, Y)$ extends this measure to quantify their combined uncertainty, i.e.

(2.2) \begin{align} S(X,Y) = -\sum _{x} \sum _{y} p(x,y)\log p(x,y), \end{align}

where $Y$ represents another random variable and $p(x,y)$ denotes the joint p.d.f. of $X$ and $Y$ .

The conditional Shannon entropy is defined as

(2.3) \begin{align} S(X|Y) = S(X,Y)-S(Y), \end{align}

which quantifies the remaining uncertainty in $X$ given knowledge of $Y$ .

Building upon conditional entropy and leveraging the temporal asymmetry inherent in causal relationships, TE is formally defined as (Schreiber Reference Schreiber2000)

(2.4) \begin{align} T_{X \to Y}(\Delta t)=S(Y(t)|Y(t-\Delta t))-S(Y(t)|X(t-\Delta t),Y(t-\Delta t)), \end{align}

where $\Delta t$ denotes time lag. The term $S(Y(t)|Y(t-\Delta t))$ quantifies the uncertainty in $Y$ that is conditioned on the past state of $Y$ . Similarly, $S(Y(t)|X(t-\Delta t),Y(t-\Delta t))$ represents the residual uncertainty in $Y$ when conditioned on the historical states of $X$ and $Y$ simultaneously. Therefore, TE fundamentally measures the reduction in the uncertainty of $Y$ obtained by observing the historical evolution of $X$ . A non-zero positive TE indicates that there exists causality. This formulation naturally captures a crucial property of time series, i.e. inherent directionality (asymmetry) in the causal relationship between variables, which distinguishes it from other conventional analysis methods. Within this information-theoretic framework, TE emerges as a useful measure of causality that has found a number of applications in recent turbulence research (Materassi et al. Reference Materassi, Consolini, Smith and De Marco2014; Lozano-Durán et al. Reference Lozano-Durán, Bae and Encinar2020; Wang et al. Reference Wang, Chu, Lozano-Durán, Helmig and Weigand2021b , Reference Wang, Lozano-Durán, Helmig and Chu2022; Martínez- Sánchez et al. Reference Martínez-Sánchez, López, Le Clainche, Lozano-Durán, Srivastava and Vinuesa2023).

For a multivariate system, the TE from variable $V_j$ to $V_i$ can be computed as (Lizier Reference Lizier2014; Lozano-Durán et al. Reference Lozano-Durán, Bae and Encinar2020)

(2.5) \begin{align} T_{\!j \to i}(\Delta t) = S(V_i(t)|\boldsymbol{V}^{\not{j}}(t-\Delta t)) - S(V_i(t)|\boldsymbol{V}(t-\Delta t)), \end{align}

where $\boldsymbol{V}^{\not{j}}$ denotes the vector $\boldsymbol{V}$ excluding the $j$ th component.

2.2. Information flow

The framework of IF was derived from first principles (Liang & Kleeman Reference Liang and Kleeman2005; Liang Reference Liang2013), considering a minimal two-dimensional dynamical system capable of sustaining information transfer, as

(2.6) \begin{align} \frac {{\textrm {d}}x_1}{{{\textrm {d}}}t}=F_1(x_1,x_2,t), \quad \frac {{{\textrm {d}}}x_2}{{{\textrm {d}}}t}=F_2(x_1,x_2,t), \end{align}

where $F_1$ and $F_2$ represent general functions. This minimal dimensionality of the system is necessary and sufficient for demonstration. As the dynamical system evolves temporally, the joint p.d.f. $p(x_1,x_2)$ undergoes corresponding evolution governed by the Liouville equation (Lasota & Mackey Reference Lasota and Mackey2013) as

(2.7) \begin{align} \frac {\partial p(x_1,x_2)}{\partial t}+\frac {\partial }{\partial x_1}(F_1(x_1,x_2,t)p(x_1,x_2))+\frac {\partial }{\partial x_2}(F_2(x_1,x_2,t)p(x_1,x_2))=0. \end{align}

The temporal evolution rate of the Shannon entropy $\textrm {d}S_1/\textrm {d}t$ plays a fundamental role in quantifying the IF from $x_2$ to $x_1$ . To derive $\textrm {d}S_1/\textrm {d}t$ , the Liouville equation (2.7) is integrated over $x_2$ , i.e.

(2.8) \begin{align} \frac {\partial p(x_1)}{\partial t}+\frac {\partial }{\partial x_1}\int\!F_1(x_1,x_2,t)p(x_1,x_2) \textrm {d}x_2 = 0, \end{align}

in which the other terms vanish due to the compact support assumption for $p(x_1,x_2)$ . Multiplying (2.8) by $-(1 + \log p(x_1))$ and integrating it over $x_1$ gives

(2.9) \begin{align} \begin{aligned} \frac {\textrm {d}S_1}{\textrm {d}t} &=\iint \left [\log p(x_1)\frac {\partial (p(x_1,x_2) F_1(x_1,x_2,t))}{\partial x_1}\right ]\textrm {d}x_1\textrm {d}x_2\\ &=-E\left (\frac {F_1(x_1,x_2,t)}{p(x_1)}\frac {\partial p(x_1)}{\partial x_1}\right ), \end{aligned} \end{align}

where $E(\boldsymbol{\cdot })$ denotes the mathematical expectation.

Liang & Kleeman (Reference Liang and Kleeman2005) suggested that in evolving two-dimensional systems, the marginal entropy change of $x_1$ comes from two distinct mechanisms, i.e.

(2.10) \begin{align} \frac {\textrm {d} S_1}{\textrm {d}t}=\frac {\textrm {d}S_1^*}{\textrm {d}t}+L_{2 \to 1}, \end{align}

where $\textrm {d}S_1^*/\textrm {d}t$ represents the autonomous entropy production from $x_1$ itself and $L_{2 \to 1}$ quantifies the IF rate from $x_2$ to $x_1$ . The latter term $L_{2 \to 1}$ serves as the fundamental causality measure in IF theory. The total entropy rate $\textrm {d} S_1/\textrm {d}t$ can be calculated using (2.9), but determining the IF $L_{2 \to 1}$ requires knowledge of $\textrm {d}S_1^*/\textrm {d}t$ . By multiplying the Liouville equation (2.7) by $-(1 + \log p(x_1,x_2))$ and integrating over $x_1$ and $x_2$ (assuming compact support), the joint Shannon entropy evolution can be derived as

(2.11) \begin{align} \frac {\textrm {d} S}{\textrm {d}t} = E(\boldsymbol{\nabla }\boldsymbol{\cdot }\boldsymbol{F}), \end{align}

where $\boldsymbol{\nabla}$ denotes the divergence operator and $\boldsymbol{F}=[F_1(x_1,x_2,t),F_2(x_1,x_2,t)]^T$ represents the vector field of the coupled system. Building upon (2.11), Liang & Kleeman (Reference Liang and Kleeman2005) proposed that the autonomous contribution to the marginal entropy change of $x_1$ can be determined by

(2.12) \begin{align} \frac {\textrm {d} S_1^*}{\textrm {d}t} = E\left (\frac {\partial F_1(x_1,x_2,t)}{\partial x_1}\right )\!. \end{align}

Therefore, the IF from $x_2$ to $x_1$ is consequently obtained as

(2.13) \begin{align} L_{2 \to 1} = \frac {\textrm {d} S_1}{\textrm {d}t} - \frac {\textrm {d}S_1^*}{\textrm {d}t} = -E\left (\frac {F_1(x_1,x_2,t)}{p(x_1)}\frac {\partial p(x_1)}{\partial x_1}\right ) - E\left (\frac {\partial F_1(x_1,x_2,t)}{\partial x_1}\right )\!. \end{align}

Liang (Reference Liang2008) extended the formalism to stochastic dynamical systems. The theoretical foundation of the IF was further rigorously established by Liang (Reference Liang2016) for both deterministic and stochastic dynamical systems. For practical applications, Liang (Reference Liang2014) developed a maximum likelihood estimator that can express the IF purely in terms of covariances between two variables in the linear sense. For the directional IF from $x_2$ to $x_1$ , the estimator takes the form

(2.14) \begin{align} L_{2\to 1}=\frac {C_{11}C_{12}C_{2,{\textrm {d}}1}-C_{12}^{2}C_{1,{\textrm {d}}1}}{C_{11}^{2}C_{22}-C_{11}C_{12}^{2}}, \end{align}

where $C_{ij}=\textrm {cov}(V_i,V_j)$ denotes the covariance between system components $V_i$ and $V_j$ , $C_{i, { \textrm {d}}j}=\textrm {cov}(V_i,\dot {V}_j)$ represents the covariance between $V_i$ and the temporal derivative $\dot {V}_j$ .

The multivariate form of the IF was given by Liang (Reference Liang2021) similarly to the bivariate case, as

(2.15) \begin{align} L_{{\kern-1pt}j \to i}=\frac {\textrm {d}S_i}{\textrm {d}t}-\frac {\textrm {d}S_{i\not{j}}}{\textrm {d}t} \approx \frac {1}{\textrm {det} ({\unicode{x1D63E}})}\boldsymbol{\cdot }\sum \limits _{m=1}^d\!\Delta _{jm}C_{m,{\textrm {d}}i} \boldsymbol{\cdot }\frac {C_{ij}}{C_{ii}}, \end{align}

where $\unicode{x1D63E}$ represents the covariance matrix, $\textrm {det}(\boldsymbol{\cdot })$ denotes the determinant of a matrix, $\Delta _{ij}$ indicates the cofactors of $\unicode{x1D63E}$ and $d$ is the system dimensionality. Here $\textrm {d}S_i/\textrm {d}t$ is the temporal variation rate of $S(V_i)$ and $\textrm {d}S_{i\not{j}}/\textrm {d}t$ is the temporal variation rate of $S(\boldsymbol{V})$ with the effect of $V_j$ excluded, thus making $L_{{\kern-1pt}j \to i}$ a direct measure of the information transfer rate from $V_j$ to $V_i$ .

2.3. Synergistic–unique–redundant decomposition

A novel approach to causal analysis, termed SURD of causality, was recently proposed by Martínez-Sánchez et al. (Reference Martínez-Sánchez, Arranz and Lozano-Durán2024). Synergistic, unique and redundant decomposition provides a framework for causal quantification by decomposing information into redundant, unique and synergistic components, and can also account for unobserved confounders.

The SURD approach still operates within the information-theoretic framework, with mutual information serving as the fundamental measure for causality quantification. Within the framework of information theory, mutual information between two variables is formally defined as

(2.16) \begin{align} I(X;Y) = S(X)-S(X|Y)=S(Y)-S(Y|X)=S(X)+S(Y)-S(X,Y). \end{align}

Here, $S(X)$ and $S(X|Y)$ represent the Shannon entropy and the conditional Shannon entropy, as defined in (2.1) and (2.3), respectively. Mutual information quantifies the decrease in uncertainty associated with one random variable when another random variable is known. This reduction in uncertainty can be understood as the information jointly known by both variables, i.e. their shared information. Therefore, mutual information constitutes a symmetric metric, i.e. $I(X;Y)=I(Y;X)$ . It is noted that $X$ and $Y$ may denote a collection of variables.

Consider a system with $N$ observable variables at time $t$ , represented by the vector $\boldsymbol{\text{Q}}(t)=[Q_1(t),Q_2(t),\ldots ,Q_N(t)]$ , where each $Q_i(t)$ denotes a variable of interest. Information about the future state of the variable $Q_{{\kern-1pt}j}$ is quantified by $S(Q_{{\kern-1pt}j}(t+\Delta t))$ . According to the definition of mutual information, information about the future state of the variable $Q_{{\kern-1pt}j}$ can be decomposed into two components based on all observable variables, as

(2.17) \begin{align} S(Q_{{\kern-1pt}j}(t+\Delta t)) = I(Q_{{\kern-1pt}j}(t+\Delta t);\boldsymbol{\text{Q}}(t))+S(Q_{{\kern-1pt}j}(t+\Delta t)|\boldsymbol{\text{Q}}(t)), \end{align}

where $I(Q_{{\kern-1pt}j}(t+\Delta t);\boldsymbol{\text{Q}}(t))$ represents the reduction in uncertainty about the future state of $Q_{{\kern-1pt}j}$ given the current state of the variables $\boldsymbol{\text{Q}}$ , which can be interpreted as the information transferred from $\boldsymbol{\text{Q}}(t)$ to $Q_{{\kern-1pt}j}(t+\Delta t)$ . On the other hand, $S(Q_{{\kern-1pt}j}(t+\Delta t)|\boldsymbol{\text{Q}}(t))$ denotes the residual uncertainty in the future state of $Q_{{\kern-1pt}j}$ conditioned on the current state of the observable variables $\boldsymbol{\text{Q}}$ . This term represents information transferred to $Q_{{\kern-1pt}j}$ from unobserved sources beyond the system $\boldsymbol{\text{Q}}$ , which can be characterised as information leakage $\Delta I_{leak \to j}$ .

Based on the specific mutual information (DeWeese & Meister Reference DeWeese and Meister1999), the mutual information can be decomposed as

(2.18) \begin{align} I(Q_{{\kern-1pt}j}(t+\Delta t);\boldsymbol{\text{Q}}(t))=\sum _{i=1}^{N} \Delta I^U_{i \to j} + \sum _{\boldsymbol{\text{i}}} \Delta I^R_{\boldsymbol{\text{i}} \to j} + \sum _{\boldsymbol{\text{i}}} \Delta I^S_{\boldsymbol{\text{i}} \to j}, \end{align}

where $\Delta I^U_{i \to j}$ , $\Delta I^R_{\boldsymbol{\text{i}} \to j}$ and $\Delta I^S_{\boldsymbol{\text{i}} \to j}$ are unique, redundant and synergistic causalities, respectively. The symbol $\boldsymbol{\text{i}}$ denotes a higher-order combination of variables. The redundant causality represents the shared causal information common to all components of $\boldsymbol{\mathit{Q_i}}$ . This condition occurs when all variables in $\boldsymbol{\mathit{Q_i}}$ provide identical predictive information about $Q_{{\kern-1pt}j}(t+\Delta t)$ . The unique causality quantifies the distinct causal information that is exclusively attributable to $Q_i$ and cannot be derived from any other individual variable. This form of causality emerges when the observation of $Q_i$ provides greater informational gain outcomes of $Q_{{\kern-1pt}j}(t+\Delta t)$ compared with the observation of any other isolated variable in the system. The synergistic causality refers to the causal influence arising from the joint interaction of variables in $\boldsymbol{\mathit{Q_i}}$ . This form of causality is characterised by an informational gain in predicting $Q_{{\kern-1pt}j}(t+\Delta t)$ when the entire set $\boldsymbol{\mathit{Q_i}}$ is observed simultaneously, exceeding the information obtained from observing each variable in isolation. These components are derived through the computation of the expectations of specific mutual information terms.

Finally, the information on the future state of variable $Q_{{\kern-1pt}j}$ can be ultimately decomposed into four distinct contributions, i.e. unique, redundant, synergistic causalities and information leaks (Martínez-Sánchez et al. Reference Martínez-Sánchez, Arranz and Lozano-Durán2024):

(2.19) \begin{align} S(Q_{{\kern-1pt}j}(t+\Delta t)) = \sum _{i=1}^{N} \Delta I^U_{i \to j} + \sum _{\boldsymbol{\text{i}}} \Delta I^R_{\boldsymbol{\text{i}} \to j} + \sum _{\boldsymbol{\text{i}}} \Delta I^S_{\boldsymbol{\text{i}} \to j}+\Delta I_{leak \to j}. \end{align}

2.4. Implementation details

In this work we employ the histogram method to calculate the p.d.f. in TE. The number of bins in each dimension is usually set to 10 if the number of variables is less than 9. Otherwise, 5 bins are used per dimension due to the excessive storage requirement. This reduction in the number of bins does not affect the conclusion of causal inference, according to our tests in simple problems. Furthermore, we calculate the SURD using the multivariate compatible procedure introduced by Martínez-Sánchez et al. (Reference Martínez-Sánchez, Arranz and Lozano-Durán2024). This method involves discretising the variables into 50 equally spaced bins per dimension and computing the joint and marginal p.d.f. directly from the resulting histograms.

We normalise the TE and IF by the sum of the causal contributions from all source variables to the target variable. Similarly, we use total mutual information to normalise the redundant, synergistic and unique information components in SURD. This can yield the proportional contribution of each component to the overall information of the target variable. Consistent with Martínez-Sánchez et al. (Reference Martínez-Sánchez, Arranz and Lozano-Durán2024), we normalise the information leak associated with a variable by the total information encompassed in its future states. This procedure quantifies the relative contribution of unobserved variables to the information contained in the future states of the target variable.

In this study, our main interest lies in the causal interactions between variables. In the subsequent analysis, the self-information transfer (self-causality) is omitted by setting it to zero after normalisation. Moreover, following Martínez-Sánchez et al. (Reference Martínez-Sánchez, Arranz and Lozano-Durán2024), we restrict the presentation of our SURD results to third-order synergy. Higher-order synergistic effects are consequently subsumed within the information leak term. Finally, all components are sorted in descending order on the basis of their magnitudes in SURD.

3.1. The linear problem

We first examine causal relationships in a six-dimensional vector autoregressive process $\boldsymbol{x}=(x_1,\ldots ,x_6)^{ {T}}$ as

(3.1) \begin{align} {\boldsymbol{x}(n+1)=\boldsymbol{\alpha }+{\unicode{x1D63C}}\boldsymbol{x}(n)+{\unicode{x1D63D}}\boldsymbol{e}(n+1)}, \end{align}

where $\boldsymbol{\alpha }$ is a constant vector and $\unicode{x1D63C}$ denotes the coefficient matrix. The perturbation vector $\boldsymbol{e}=(e_1,\ldots ,e_6)^{ {T}}$ consists of independent standard normal random variables, and each $b_{ii}$ in the diagonal perturbation matrix $\unicode{x1D63D}$ quantifies the noise amplitude of the $i$ th component. In the present study of a strong perturbation, we set uniformly high noise levels of $b_{ii}=100$ $(i=1,\ldots ,6)$ . Following Liang (Reference Liang2021), the other parameters are specified as

(3.2) \begin{align} {\boldsymbol{\alpha }=(0.1,0.7,0.5,0.2,0.8,0.3)^{{T}}} \end{align}

and

(3.3) \begin{align} {{\unicode{x1D63C}}=\left [ \begin{array}{cccccc} 0 & 0 & -0.6 & 0 & 0 & 0 \\ -0.5 & 0 & 0 & 0 & 0 & 0.8 \\ 0 & 0.7 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0.7 & 0.4 & 0 \\ 0 & 0 & 0 & 0.2 & 0 & 0.7 \\ 0 & 0 & 0 & 0 & 0 & -0.5 \end{array} \right ]}. \end{align}

By numerically solving the system (3.1) with uniformly distributed random initial conditions, the numerical simulation produced a total of 500 realisations. Each realisation comprises six distinct time series, each spanning 500 000 time steps. To ensure that the system had reached its attractor, the first 100 000 time steps of each time series were discarded as transient. Consequently, the final dataset of each realisation consists of 400 000 time steps per series. In this study, the causal analysis is performed with a lag of one time step.

Figure 1.

Causal maps of the linear problem: (a) TE, (b) IF, (c) SURD. Red boxes: the ground-truth causal links in the system.

Figure 1. Long description

Panel A: A heatmap titled ’Transfer entropy’ shows the causal links between variables x1 to x6. The x-axis represents the cause, and the y-axis represents the effect. The color intensity indicates the strength of the causal link, with red boxes highlighting the ground-truth causal links. Panel B: A heatmap titled ’Information flow’ displays similar data with the x-axis as cause and y-axis as effect. The color intensity again indicates the strength of the causal link, with red boxes marking the ground-truth causal links. Panel C: Eight bar graphs show the normalized information leakage for different variables x1 to x6. Each subplot is labeled with the variable and the type of information leakage, with the y-axis representing the normalized value. The bars are color-coded to indicate different conditions or states.

Figure 1 presents the results of the causal analysis of the linear system. In this problem, if $x_j$ exists on the right-hand side of the equation of $x_i$ , $x_j \to x_i$ is regarded as a ground-truth causal link. All the ground-truth causal links in the system (3.1) are highlighted by the red bounding boxes in figure 1. It is seen that both TE and IF demonstrate consistent identifications of the causal links of the linear system. The detected non-zero causalities, i.e. $x_1 \to x_2$ , $x_2 \to x_3$ , $x_3 \to x_1$ , $x_4 \to x_5$ , $x_5 \to x_4$ , $x_6 \to x_2$ and $x_6 \to x_5$ , are in perfect agreement with the non-zero elements in the coefficient matrix (3.3).

The identification result by SURD is presented in figure 1(c), in which $\boldsymbol{\text{x}}$ denotes the set of all variables in the linear model. Higher than second-order redundant contributions are not explicitly shown in the figure due to limited space, and we mainly focus on low-order causal interactions, as the dynamics of the linear model dictate that any variable can be influenced by at most two other variables. It can be found that the inference result of SURD is in close agreement with the ground-truth causal links of the linear model. The link $x_3 \to x_1$ is expressed through unique information transfer $\Delta I_{3 \to 1}^U$ . The causalities of $x_1$ and $x_6$ to $x_2$ are manifested by a combination of unique ( $\Delta I_{6 \to 2}^U$ ), synergistic ( $\Delta I_{16 \to 2}^S$ ) and redundant ( $\Delta I_{16 \to 2}^R$ ) contributions, respectively, with the synergistic component being the most substantial. The link $x_2 \to x_3$ is identified as a predominant unique contribution $\Delta I_{2 \to 3}^U$ . The link $x_4 \to x_5$ involves a combination of synergistic ( $\Delta I_{45 \to 4}^S$ ) and redundant ( $\Delta I_{45 \to 4}^R$ ) contributions, while the causalities of $x_4$ and $x_6$ to $x_5$ are jointly governed by unique ( $\Delta I_{6 \to 5}^U$ ), synergistic ( $\Delta I_{46 \to 5}^S$ ) and redundant ( $\Delta I_{46 \to 5}^R$ ) mechanisms, in which the unique one is the most important.

From the above results, we can see that all three methods can successfully identify the causal links embedded in the linear system with strong noise. The SURD method can further distinguish contributions from unique, synergistic and redundant mechanisms, providing more detailed causal information.

3.2. The nonlinear problem

Next, we examine a nonlinearly coupled system to evaluate the performance of the causal inference methods in more complex dynamical systems.

Following Paluš et al. (Reference Paluš, Krakovská, Jakubík and Chvosteková2018), we consider the nonlinear system

(3.4) \begin{align} \begin{aligned} &\!\mathcal{X}=\left \{\!\! \begin{array}{ll} \textrm {d}x_1/\textrm {d}t = -\omega _1x_2(t) - x_3(t), \\ \textrm {d}x_2/\textrm {d}t = \omega _1x_1(t) + 0.15x_2(t), \\ \textrm {d}x_3/\textrm {d}t = 0.2 + x_3(t)\left [x_1(t)-10\right ]\!, \end{array} \right . \\ &\mathcal{Y}=\left \{\!\! \begin{array}{ll} \textrm {d}y_1/\textrm {d}t = -\omega _2y_2(t) - y_3(t) + \epsilon \left [x_1(t)-y_1(t)\right ]\!, \\ \textrm {d}y_2/\textrm {d}t = \omega _2y_1(t) + 0.15y_2(t), \\ \textrm {d}y_3/\textrm {d}t = 0.2 + y_3(t)\left [y_1(t)-10\right ]\!, \end{array} \right . \end{aligned} \end{align}

in which $\omega _1 = 1.015$ , $\omega _2 = 0.985$ and $\epsilon$ is the coupling strength parameter, which is set to $0.25$ in this study. Equation (3.4) is a unidirectionally coupled Rössler system, i.e. $\mathcal{X}$ is the driving or master system and $\mathcal{Y}$ is the driven system.

The nonlinear system is numerically solved using a variable-order numerical differentiation formula method (the ‘ode15s’ function in MATLAB) with a fixed time step of $\Delta t=0.001$ . The numerical solution generated a total of 500 realisations. Each realisation begins from a random initial condition and evolves for $N=500\,000$ time steps, with the first 100 000 steps discarded as transient to ensure that the solution reaches its attractor. Causal analysis is performed with a time lag of $2\Delta t$ to resolve the characteristic time scale of coupled dynamics (Paluš et al. Reference Paluš, Krakovská, Jakubík and Chvosteková2018).

Figure 2.

Causal maps of the nonlinear problem: (a) TE, (b) IF, (c) SURD. Red boxes: the causal links in the system.

Figure 2 presents the causal inference results of the nonlinear system. It is seen that both TE and IF correctly identify the primary driver–response causal relationships specified in (3.4), i.e. the link $x_1 \to y_1$ . However, when examining other causal links, some differences emerge between the two methods. It is seen that both methods can detect the causal links of $x_1 \leftrightarrow x_2$ and $y_1 \leftrightarrow y_2$ , but the IF can produce more pronounced values. In particular, the IF can clearly identify $x_1 \leftrightarrow x_3$ and $y_1 \leftrightarrow y_3$ , while the values in the TE result are quite small. In addition, much richer causalities are identified by TE than by IF. In figure 2(c) the variables are indexed as $(1,2,3) \to (x_1,x_2,x_3)$ and $(4,5,6) \to (y_1,y_2,y_3)$ to illustrate the SURD result. Similar causal links are found with the primary causalities identified by TE, e.g. the primary driver–response causal link $x_1 \to y_1$ manifested as the redundant causality $\Delta I_{14 \to 4}^R$ in SURD. The link $y_1 \to x_1$ identified by TE is manifested by the redundant causality $\Delta I_{14 \to 1}^R$ . The links $y_2 \leftrightarrow x_2$ are manifested by the redundant causalities $\Delta I_{25 \to 2}^R$ and $\Delta I_{25 \to 5}^R$ . The link $x_3 \to y_3$ is by the redundant one $\Delta I_{36 \to 6}^R$ .

3.3. The low-dimensional model of near-wall turbulence

Next, we analyse the causal relations in a low-dimensional model of the plane Couette flow with a sinusoidal body force (Moehlis et al. Reference Moehlis, Faisst and Eckhardt2004). It presents an improvement of the eight-mode model of Waleffe (Reference Waleffe1997). The self-sustaining process of near-wall turbulence, i.e. the streak-vortex cycle, can be well represented by the low-dimensional model. There are nine modes of the model, which are (i) the basic profile mode, (ii) the streak mode, (iii) the downstream vortex mode, (iv,v) the spanwise flow modes, (vi,vii) the normal vortex modes, (viii) a fully three-dimensional mode, and (ix) the mode of modification of the basic profile. The first eight modes are the same as those in Waleffe (Reference Waleffe1997). By applying the Galerkin projection to the Navier–Stokes equations without the pressure term, a set of nine ordinary differential equations is obtained for the temporal amplitude coefficients $a_i(t)$ ( $i=1$ to 9), featuring quadratic nonlinearities.

Figure 3.

Causal maps of the low-dimensional model of near-wall turbulence: (a) TE, (b) IF, (c) SURD. Red boxes: the causal links associated with the self-sustaining cycle of streaks and vortex.

Figure 3. Long description

The image contains three panels (a, b, and c) depicting different types of causal maps for the low-dimensional model of near-wall turbulence. Panel A shows a heatmap of transfer entropy (TE) with axes labeled ’Cause’ and ’Effect’. The color gradient ranges from light blue to dark blue, indicating varying levels of transfer entropy. Red boxes highlight specific causal links associated with the self-sustaining cycle of streaks and vortex. Panel B presents a heatmap of information flow (IF) with the same axes labels and color gradient as Panel A. Red boxes again mark significant causal links. Panel C consists of multiple bar graphs, each representing the ratio of leaked information to the information of a specific variable over time. Each bar graph is labeled with different variables (a1 to a9) and shows the contribution of various reactions (R numbers) to the leaked information. The y-axis represents the ratio, and the x-axis lists the reactions. Notable trends include specific reactions contributing significantly to certain variables, as indicated by taller bars.

Here we adopt the same parameters as Martínez-Sánchez et al. (Reference Martínez-Sánchez, López, Le Clainche, Lozano-Durán, Srivastava and Vinuesa2023). The system is numerically solved to generate 500 realisations, each spanning 4000 time units with a fixed time step of 0.01 (400 000 steps per realisation). The solution data are generated by applying a uniformly distributed random perturbation to initialise $a_4$ . The initial conditions are $(1, 0.07066, -0.07076, 0, 0, 0, 0, 0, 0)$ for the nine equations, which are solved using the variable-order differentiation method. In causal analysis, the time lag is set to one time step ( $\Delta t=0.01$ ).

Figure 3 shows the causal inference results of the low-dimensional model, which reveals distinct sensitivities of methodologies to capture the self-sustaining mechanism of near-wall turbulence. First, it can be seen that both TE and IF can successfully identify the core interactions, including the lift-up mechanism ( $a_1 \to a_2$ and $a_3 \to a_2$ ) and streak-to-vortex mechanism ( $a_2 \to a_3$ ). In figure 3(c), these two mechanisms are manifested through the redundant causalities of $\Delta I_{23 \to 2}^R$ and $\Delta I_{23 \to 3}^R$ . Second, Figure 3 also shows that TE identifies a wider spectrum of causalities, while IF detects a more compact one. It indicates that IF can detect primary energy transfers, while the TE shows a higher sensitivity to higher-order interactions, particularly spanwise flow effects ( $a_4 \to a_2$ , $a_5 \to a_3$ ) and three-dimensional mode couplings ( $a_6 \to a_2$ ). Third, the causalities identified by SURD exhibit a certain similarity with IF. The causal links $a_9 \leftrightarrow a_1$ identified by IF manifest themselves as redundant causalities $\Delta I_{19 \to 1}^R$ and $\Delta I_{19 \to 9}^R$ in the SURD result. The causal links $a_6 \leftrightarrow a_8$ correspond to the redundant causalities $\Delta I_{68 \to 6}^R$ and $\Delta I_{68 \to 8}^R$ . The causal links $a_7 \leftrightarrow a_8$ resemble the redundant causalities $\Delta I_{78 \to 7}^R$ and $\Delta I_{678 \to 8}^R$ .

3.4. Discussions

The results of the nonlinear problem and low-dimensional model of the near-wall turbulence problem imply that different methods may have different sensitivities to nonlinearity. It should be beneficial to identify the actual causal links and evaluate causal inference methods if the ground truth is known. However, it is often not possible for a nonlinear system, as nonlinear couplings can propagate causal influences across variables over finite time lags, making a ground truth solely based on dynamical equations infeasible.

Taking the low-dimensional model of near-wall turbulence as an example, the dynamical equations of the system are given in Appendix A. First, it is observed that $a_9$ does not appear on the right-hand side of $\textrm {d}a_1/\textrm {d}t$ , while all the three methods identify the causal link of $a_9 \to a_1$ . By subtracting (A9) from (A1), one can obtain a differential equation that contains only $a_1$ and $a_9$ (and their derivatives), i.e. (A11). This indicates that $a_1$ and $a_9$ are closely coupled with each other; hence, the identified causal link has a theoretical foundation. The second is the identified causal link of $a_2 \to a_3$ . It is seen that $a_2$ is absent in the dynamical equation of $a_3$ , i.e. (A3). By multiplying the equations of each mode by their respective coefficients and summing them, one can completely eliminate all crossing terms and obtain (A21), in which all coefficients are coupled. As a result, the reformulation of the dynamical equations reveals the possible existence of the causal link from $a_2$ to $a_3$ . The above two examples illustrate the complexity of nonlinear problems, in which the underlying causal relationships may not be explicit in dynamical systems.

Nevertheless, the key point is that all three methods successfully identify the primary driver–response relationship ( $x_1 \to y_1$ ) in the nonlinear system (§ 3.2) and the core streak–vortex self-sustaining cycle in the low-dimensional turbulence model (§ 3.3). These are the most physically important causal links in each system. The fact that the methods capture these essential mechanisms, despite their differences in secondary links, demonstrates their applicability in the analyses of complex nonlinear problems.

To further strengthen the reliability of the causal analysis of the turbulent channel flow, we adopt a conservative interpretation strategy in the following sections. The causal links that are consistently identified by at least two of the three methods are kept, which significantly reduces the uncertainty and risk of spurious detections. Moreover, sensitivity analyses with respect to the data length and histogram bin number for the verification cases are provided in the supplementary material is available at https://doi.org/10.1017/jfm.2026.12025, confirming the robustness of the methods.

4.1. An improved inner–outer decomposition method

The reason for the existence of the non-zero cross-RSS components lies in the fact that (4.2) does not take into account the non-zero contributions of the outer footprints from the cross-velocity component. For example, in the estimation of $u_L$ , only the contribution of $u$ itself is included in (4.2), while that of $v$ is neglected. Furthermore, so is $v_L$ . Therefore, we develop an improved inner–outer decomposition method that can include the effects of the cross-velocity components in calculating the outer footprint velocity $u_{i,L}$ . More specifically, we consider cross-contributions between $u$ and $v$ as their product constitutes the non-zero RSS components.

First, we apply the spectral linear stochastic estimation to calculate the correlated component of the streamwise and wall-normal velocity fluctuations at $y_o^+=100$ , as

(4.5) \begin{align} u^{c}(x,y_{o},z,t)= F_x^{-1} \left \{\hat {H}_{v}^{c}(\lambda _x)F_x \left [v(x,y_o,z,t) \right ] \right \}, \end{align}

(4.6) \begin{align} v^{c}(x,y_{o},z,t)= F_x^{-1} \left \{\hat {H}_{u}^{c}(\lambda _x)F_x \left [u(x,y_o,z,t) \right ] \right \}, \end{align}

where

(4.7) \begin{align} \hat {H}_{u}^c(\lambda _x)=\frac {\langle \hat {v}(\lambda _x,y_o,z,t) \overline {\hat {u}(\lambda _x,y_o,z,t)} \rangle _{z,t}}{\langle \hat {u}(\lambda _x,y_o,z,t) \overline {\hat {u}(\lambda _x,y_o,z,t)} \rangle _{z,t}}, \end{align}

(4.8) \begin{align} \hat {H}_{v}^c(\lambda _x)=\frac {\langle \hat {u}(\lambda _x,y_o,z,t) \overline {\hat {v}(\lambda _x,y_o,z,t)} \rangle _{z,t}}{\langle \hat {v}(\lambda _x,y_o,z,t) \overline {\hat {v}(\lambda _x,y_o,z,t)} \rangle _{z,t}}, \end{align}

are the scale-dependent kernel functions.

Next, the uncorrelated components of the streamwise and wall-normal velocity fluctuations at the reference height can be obtained by

(4.9) \begin{align} u^{uc}(x,y_{o},z,t)= u(x,y_o,z,t)-u^{c}(x,y_o,z,t), \end{align}

(4.10) \begin{align} v^{uc}(x,y_{o},z,t)= v(x,y_o,z,t)-v^{c}(x,y_o,z,t). \end{align}

In the following, we use $v^{uc}$ and $u^{uc}$ to calculate the outer footprint velocity components in $u$ and $v$ from the cross-velocity contributions, respectively, as the contributions from $v^c$ and $u^c$ have been included in $u$ and $v$ themselves. Consequently, the improved estimation for the outer footprint velocity components can be written as

(4.11) \begin{align} \begin{split} u_{L}(x,y,z,t) = & F_x^{-1} \left \{\hat {H}_{u,L}(\lambda _x,y)F_x \left [u(x,y_o,z,t) \right ] \right \} \\ &+F_x^{-1} \left \{\hat {H}_{v,L}^{uc}(\lambda _x,y)F_x \left [v^{uc}(x,y_o,z,t) \right ] \right \}\!, \end{split} \end{align}

(4.12) \begin{align} \begin{split} v_{L}(x,y,z,t) = & F_x^{-1} \left \{\hat {H}_{v,L}(\lambda _x,y)F_x \left [v(x,y_o,z,t) \right ] \right \} \\ &+F_x^{-1} \left \{\hat {H}_{u,L}^{uc}(\lambda _x,y)F_x \left [u^{uc}(x,y_o,z,t) \right ] \right \}\!. \end{split} \end{align}

In this formulation, the first term on the right-hand side remains identical to that in (4.2) and the second term is from the cross-velocity contribution. The terms $\hat {H}_{u,L}^{uc}$ and $\hat {H}_{v,L}^{uc}$ represent scale-dependent kernel functions for calculating the cross-velocity components, which are

(4.13) \begin{align} \hat {H}_{u,L}^{uc}(\lambda _x,y)=\frac {\langle \hat {v}(\lambda _x,y,z,t) \overline {\hat {u}^{uc}(\lambda _x,y_o,z,t)} \rangle _{z,t}}{\langle \hat {u}^{uc}(\lambda _x,y_o,z,t) \overline {\hat {u}^{uc}(\lambda _x,y_o,z,t)} \rangle _{z,t}}, \end{align}

(4.14) \begin{align} \hat {H}_{v,L}^{uc}(\lambda _x,y)=\frac {\langle \hat {u}(\lambda _x,y,z,t) \overline {\hat {v}^{uc}(\lambda _x,y_o,z,t)} \rangle _{z,t}}{\langle \hat {v}^{uc}(\lambda _x,y_o,z,t) \overline {\hat {v}^{uc}(\lambda _x,y_o,z,t)} \rangle _{z,t}}. \end{align}

Then the demodulated inner motions are obtained as

(4.15) \begin{align} u^*(x, y, z, t) = \frac {u_S(x, y, z, t)}{1 + \varGamma _{uu}(y) u_L(x, y, z, t)}, \end{align}

(4.16) \begin{align} v^*(x, y, z, t) = \frac {v_S(x, y, z, t)}{1 + \varGamma _{uv}(y) u_L(x, y, z, t)}, \end{align}

(4.17) \begin{align} w^*(x, y, z, t) = \frac {w_S(x, y, z, t)}{1 + \varGamma _{uw}(y) u_L(x, y, z, t)}, \end{align}

where $u^*$ , $v^*$ and $w^*$ represent the demodulated inner motions, while $\varGamma _{uu}$ , $\varGamma _{uv}$ and $\varGamma _{uw}$ denote the modulation coefficient. Both the demodulated inner motions and modulation coefficients are obtained iteratively. First, an initially prescribed $\varGamma _{uu}$ , $\varGamma _{uv}$ and $\varGamma _{uw}$ are used to compute $u^*$ , $v^*$ and $w^*$ via (4.15), (4.16) and (4.17).

Subsequently, the modulation coefficients are determined by

(4.18) \begin{align} M_{A}(u^{*}) = \frac { \langle E_{L}(u^{*}) u_{L} \rangle }{ \sqrt { \langle E_{L}(u^{*})^{2} \rangle } \sqrt { \langle u_{L}^{2} \rangle } }, \end{align}

(4.19) \begin{align} M_{A}(v^{*}) = \frac { \langle E_{L}(v^{*}) u_{L} \rangle }{ \sqrt { \langle E_{L}(v^{*})^{2} \rangle } \sqrt { \langle u_{L}^{2} \rangle } }, \end{align}

(4.20) \begin{align} M_{A}(w^{*}) = \frac { \langle E_{L}(w^{*}) u_{L} \rangle }{ \sqrt { \langle E_{L}(w^{*})^{2} \rangle } \sqrt { \langle u_{L}^{2} \rangle } }, \end{align}

where $E_{L}$ denotes the signal envelope obtained by the Hilbert transform. When the modulation coefficients meet $M_{A}(u^{*})=0$ , $M_{A}(v^{*})=0$ and $M_{A}(w^{*})=0$ , the correct solutions of $\varGamma _{uu}$ , $\varGamma _{uv}$ and $\varGamma _{uw}$ and $u^*$ , $v^*$ and $w^*$ are obtained. Otherwise, the $\varGamma _{uu}$ , $\varGamma _{uv}$ and $\varGamma _{uw}$ values must be adjusted iteratively until $M_{A}(u^{*})=0$ , $M_{A}(v^{*})=0$ and $M_{A}(w^{*})=0$ are satisfied.

Figure 5.

Velocity fluctuation intensities of the demodulated inner motions at friction Reynolds numbers $Re_\tau =1000,2000$ and $5200$ . (a) Streamwise velocity fluctuations, (b) wall-normal velocity fluctuations, (c) spanwise velocity fluctuations, (d) cross components of the decomposed RSS. The line colours correspond to the cases listed in table 1.

Figure 5(a–c) shows that the intensities of the three velocity fluctuations of the inner motions by the improved method remain Reynolds number independent across the three Reynolds numbers. Figure 5(d) illustrates the cross-components of the decomposed RSS, which are close to zero, indicating that there is no correlation between the cross-velocities of the inner and outer motions. Furthermore, we note that the kernel functions defined in (4.7), (4.8), (4.13) and (4.14) also exhibit Reynolds number independence, similar to those in Wang et al. (Reference Wang, Hu and Zheng2021a ).

For consistency, we maintain the same notation for inner and outer motions obtained through the newly improved decomposition method. All subsequent analyses exclusively use data derived from this new approach. It should be noted, however, that the decomposition of turbulent motions into inner and outer components may not be unique. Different decomposition strategies (e.g. Fourier filtering, proper orthogonal decomposition or wavelet-based scale separation) may yield different partitions of the velocity field. Consequently, the causal findings reported in the following sections may be specific to the present decomposition framework.

5.1. Dataset

The dataset used in the present analysis is from a time-resolved DNS of turbulent channel flow at $Re_\tau =1000$ . The numerical scheme combines fourth-order compact differences in the homogeneous directions and second-order central differences in the wall-normal direction for solving the incompressible Navier–Stokes equations on a staggered grid (Hu et al. Reference Hu, Wang, Wang, Wang and Zheng2018). The simulation was conducted in a computational domain of $L_x/\delta =4\pi$ , $L_y/\delta =2$ and $L_z/\delta =2\pi$ . Uniform grids were employed in the streamwise ( $x$ ) and spanwise ( $z$ ) directions with grid numbers of $N_x=N_z=1152$ , resulting in grid resolutions of $\Delta x^+=10.9$ and $\Delta z^+=5.5$ in wall units. In the wall-normal direction, a non-uniform grid with $N_y=384$ was used, resulting in a grid spacing distribution of $\Delta y_w^+=0.4$ at the wall and $\Delta y_c^+=13.2$ at the channel centre. The dataset contains a flow duration of 19.8 $\delta /u_\tau$ , with instantaneous flow fields stored in a time interval of $\Delta t^+=0.9$ , which yields 20 000 snapshots in total. The velocity and pressure fields at four wall-normal locations ( $y^+=15,\ 70,\ 100,\ 200$ ) are stored. As demonstrated in Appendix B, the present DNS dataset shows excellent agreement with the reference DNS data of Lee & Moser (Reference Lee and Moser2015) at the same Reynolds number, confirming the accuracy and reliability of the present data. The statistical convergence of causal inference is assessed in Appendix C.

5.2. One-point causal analysis

We first examine the causalities of the demodulated inner and outer motions at the same location in a wall-parallel plane, i.e. one-point causality. Time series for the variable vector $\boldsymbol{V}= (u^*,v^*,w^*,u_L,v_L,w_L,p )$ at 10 000 points uniformly distributed in a horizontal plane are adopted for the causal analysis.

Figure 6.

Causal maps for the demodulated inner and outer motions at $y^+=15$ : (a) TE, (b) IF, and (c) SURD. Yellow boxes: inter-scale causalities. Cyan boxes: intra-scale causalities. Red boxes: causalities identified by both TE and IF.

Figure 6. Long description

The image contains three panels of heatmaps and bar graphs. Panel A shows a heatmap titled ’Transfer entropy’ with axes labeled ’Cause’ and ’Effect’. The heatmap uses a color scale from 0 to 1 to represent the strength of causal interactions between different variables. Yellow boxes indicate inter-scale causalities, cyan boxes indicate intra-scale causalities, and red boxes indicate causalities identified by both transfer entropy and information flow. Panel B shows a similar heatmap titled ’Information flow’ with the same axes and color scale. Panel C contains multiple bar graphs, each representing different causal interactions. The bar graphs have different labels on the x-axis representing various causal interactions and the y-axis representing the strength of these interactions. The bar graphs show the relative strength of causal interactions for different variables, with some bars highlighted in different colors to indicate specific types of causalities.

Figure 7.

Causal maps for the demodulated inner and outer motions at $y^+=70$ : (a) TE, (b) IF, and (c) SURD. Yellow boxes: inter-scale causalities. Cyan boxes: intra-scale causalities. Red boxes: causalities identified by both TE and IF.

Figure 6 displays the causal maps at $y^+=15$ where the peak of streamwise turbulence intensity is located approximately. From figures 6(a) and 6(b), one can see that TE can identify more causal links than IF, which is similar to the results of the low-dimensional model of near-wall turbulence shown in figure 3. The prominent causal links identified by both TE and IF are highlighted via the red boxes in figure 6 and the following analysis. For example, one can find the intra-scale causal links of $u^* \leftrightarrow v^*$ , which represent the self-sustaining streak-vortex cycle of the inner motions. In contrast, we can hardly identify any evident inter-scale causal links between the inner motions and the outer footprints at $y^+=15$ . Second, it is found that pressure fluctuation is an active causal component. The notable causal links of $p \to u^*$ and $p \leftrightarrow v^*$ can be observed, which implies that pressure is actively involved in the self-sustaining near-wall turbulence cycle. This is understandable since the vortex centre is usually located in a low-pressure region and stagnation in a high-pressure region, both of which are typical flow topologies in turbulence.

In figure 6(c) we can see that the causal inference result by SURD shows similar characteristics to the primary causalities identified by IF, where the variables are indexed $(1,2,3,4,5,6,7) \to (u^*,v^*,w^*,u_L,v_L,w_L,p)$ . For example, the causal links of the inner streak-vortex generation cycle $u^* \leftrightarrow v^*$ are manifested as the redundant causality $\Delta I_{12 \to 1}^R$ and the unique causality $\Delta I_{1 \to 2}^U$ in the SURD result. The causal links of the outer footprint cycle $u_L \leftrightarrow v_L$ are identified by the redundant causality $\Delta I_{45 \to 4}^R$ and the unique causality $\Delta I_{4 \to 5}^U$ . The pressure-related causal links $p \leftrightarrow v^*$ are reflected by the redundant causalities $\Delta I_{127 \to 2}^R$ and $\Delta I_{12 \to 7}^R$ .

Figure 7 shows the causal maps at $y^+=70$ , which is close to the upper bound of the buffer layer and more susceptible to the influence of outer motions. In general, the causal features at this higher wall-normal location are quite similar to those at $y^+=15$ , demonstrating some universality. Specifically, intra-scale causal links $u^* \leftrightarrow v^*$ and $u_L \leftrightarrow v_L$ can be observed, indicating that the self-sustaining cycles of both the demodulated inner motions and outer footprints exist at this height. This evidence supports the self-sustaining mechanism of turbulent motions at all scales (Hwang & Cossu Reference Hwang and Cossu2010, Reference Hwang and Cossu2011; Hwang Reference Hwang2015; Rawat et al. Reference Rawat, Cossu, Hwang and Rincon2015; Hwang & Bengana Reference Hwang and Bengana2016; Cossu & Hwang Reference Cossu and Hwang2017; Bae et al. Reference Bae, Lozano-Durán and McKeon2021). There are also no evident inter-scale causal links between the inner motions and outer footprints at $y^+=70$ . Moreover, pressure fluctuations persist as an active component in the causal relationships at $y^+=70$ , where the causal links of $p \to u^*$ and $p \leftrightarrow v^*$ remain discernible. This confirms the critical role of pressure in the sustainment of near-wall turbulence. In figure 7(c) the causal links $u^* \leftrightarrow v^*$ correspond to the redundant causality $\Delta I_{127 \to 1}^R$ and the unique causality $\Delta I_{1 \to 2}^U$ in the SURD result. The causal links $u_L \leftrightarrow v_L$ are manifested as the redundant causalities $\Delta I_{45 \to 4}^R$ and the unique causality $\Delta I_{4 \to 5}^U$ . Furthermore, the causal links $p \to u^*$ and $p \to v^*$ are reflected by the synergistic causality $\Delta I_{17 \to 1}^S$ and the redundant causality $\Delta I_{127 \to 2}^R$ .

In summary, the one-point causal analysis has yielded the following insights. First, the demodulated inner motions and the outer footprints are demonstrated to be self-sustained and independent of each other, the latter of which is further demonstrated with other statistics in the supplementary material. This supports the idea that turbulent motions are self-sustained at all scales. In addition, the present causal inference results imply that the outer footprints are not passive with respect to their origin in the logarithmic layer but are active with local generation and evolution. Second, pressure plays a critical role in the self-sustaining cycles and may be regarded as a bridge connecting the inner and outer motions in the local sense.

5.3. Two-point causal analysis

Next, we investigate the causal relationship between velocities and pressure at two wall-normal locations, i.e. one in the near-wall region ( $y^+\lt 100$ ) and the other in the outer region ( $y^+\gt 100$ ). The flow data at 10 000 points uniformly distributed in the horizontal plane of each height are utilised for analysis, and the streamwise and spanwise coordinates of the two paired points are the same. The variables in the SURD result are indexed as $(1,2,3,4,5,6,7,8,9,10,11) \to (u_{1}^*,v_{1}^*,w_{1}^*,u_{L1},v_{L1},w_{L1},p_1,u_2,v_2,w_2,p_2)$ , in which the subscripts ‘1’ and ‘2’ indicate the inner and outer locations, respectively.

Figures 8(a) and 8(b) display the two-point causal maps of $y_1^+=15$ and $y_2^+=200$ by TE and IF. The result is found to be almost the same as the one-point result shown in figure 6. The intra-scale self-generation streak-vortex cycle of inner motions ( $u_{1}^* \leftrightarrow v_{1}^*$ ) is still evident. It suggests that the inclusion of the outer point in causal inference leaves the near-wall causal links essentially unaffected, confirming the stability of the autonomous regeneration mechanism of the inner motions. In addition, a similar causal cycle of the outer motions at $y_2^+=200$ is also observed, namely $u_2 \to v_2$ , which further supports the self-sustaining hypothesis of turbulent motions at all scales (Hwang & Cossu Reference Hwang and Cossu2010, Reference Hwang and Cossu2011; Hwang Reference Hwang2015; Rawat et al. Reference Rawat, Cossu, Hwang and Rincon2015; Hwang & Bengana Reference Hwang and Bengana2016; Cossu & Hwang Reference Cossu and Hwang2017; Bae et al. Reference Bae, Lozano-Durán and McKeon2021).

Figure 8.

Two-point causal maps for the inner and outer motions at $y^+=15$ and $y^+=200$ : (a) TE, (b) IF, and (c) SURD. Yellow boxes: causalities between the two heights. Green boxes: causalities at the same height.

Figure 8. Long description

Panel A: A heatmap displays transfer entropy with axes labeled ’Cause’ and ’Effect’. The color scale ranges from 0 to 1, indicating the strength of causal interactions. Yellow and green boxes highlight causalities between different heights and at the same height, respectively. Panel B: Another heatmap shows information flow with the same axes and color scale as Panel A. Yellow and green boxes again indicate causalities between different heights and at the same height. Panel C: A series of bar graphs illustrate specific uncertainty reduction for various interactions. Each subplot is labeled with a specific interaction (e.g., ΔI(→1)/I(u1(t + Δt); x(t))), and the bars represent different data points. The y-axis ranges from 0 to 1, showing the proportion of uncertainty reduction.

The causal relationships between the two wall-normal heights $y_1^+=15$ and $y_2^+=200$ are shown in the yellow boxes of figures 8(a) and 8(b). First, the IF can identify evident causal links of $u_{L1} \leftrightarrow u_{2}$ , $v_{L1} \leftrightarrow v_{2}$ and $w_{L1} \leftrightarrow w_{2}$ , demonstrating direct causal relationships between the outer motions at $y_2^+=200$ and their near-wall footprints at $y_1^+=15$ , which reflect the dynamical coherence of them in the forward-inclined LSMs (Brown & Thomas Reference Brown and Thomas1977; Chauhan et al. Reference Chauhan, Hutchins, Monty and Marusic2013; Liu et al. Reference Liu, Bo and Liang2017; Li et al. Reference Li, Hutchins, Zheng, Marusic and Baars2022). However, the TE can only detect ‘bottom-up’ causalities of $u_{L1} \to u_{2}$ , $v_{L1} \to v_{2}$ and $w_{L1} \to w_{2}$ . In addition, the causal link $u_{L1} \rightarrow v_{2}$ is also observed, indicating a statistical causality among cross-velocity components. It is important to note that these links may not imply a dynamic ‘bottom-up’ generation mechanism. Instead, these are signatures of the same large-scale coherent motions at different wall-normal locations. Therefore, these causal links quantitatively confirm the spatial convection and structural coherence of the LSMs between the outer layer and the near-wall region. Second, the absence of causal links of $(u_2, v_2, w_2) \leftrightarrow (u_{1}^*, v_{1}^*, w_{1}^*)$ implies that the outer motions at $y_2^+=200$ are causally uncorrelated with the near-wall demodulated inner motions at $y_1^+=15$ . This also supports the hypothesis of the self-sustaining mechanism of turbulent motions at different scales.

The pressure fluctuations at the two heights are also active in causal relations. The two-point causal inference results of $p_1$ at $y_1^+=15$ and $p_2$ at $y_2^+=200$ demonstrate that the pressure-related causal links remain consistent with the one-point result at the same height. Taking into account two-point causalities between velocity and pressure, only a weak link $v_{L1} \to p_{2}$ can be identified. Moreover, the causal link $p_{2} \leftrightarrow p_{1}$ reveals a two-way influence of inner and outer pressures, suggesting that pressure may act as a critical mediator or bridge to coupling the outer and inner motions.

From figure 8(c) we can see that the SURD result is consistent with the findings of TE and IF, providing additional insight into the causal relationships. The $u_{1}^*$ -targeted causal link $v_{1}^* \to u_{1}^*$ is manifested as the synergistic causality $\Delta I_{12 \to 1}^S$ . The $v_{1}^*$ -targeted causal link $u_{1}^* \to v_{1}^*$ corresponds to the unique causality $\Delta I_{1 \to 2}^U$ . The $u_{L1}$ -targeted causal links $v_{L1} \to u_{L1}$ and $u_{2} \to u_{L1}$ are reflected by the synergistic causality $\Delta I_{45 \to 4}^S$ and the redundant causality $\Delta I_{458 \to 4}^R$ . The $v_{L1}$ -targeted causal link $u_{L1} \to v_{L1}$ is related to the unique causality $\Delta I_{4 \to 5}^U$ . The $p_1$ -targeted causal links $u_{L1} \to p_{1}$ correspond to the unique causality $\Delta I_{4 \to 7}^U$ . The $u_2$ -targeted causal links $u_{L1} \to u_{2}$ and $v_{L1} \to u_{2}$ correspond to the unique causality $\Delta I_{4 \to 8}^U$ and the redundant causality $\Delta I_{458 \to 8}^R$ , respectively. The $v_2$ -targeted causal links $u_{L1} \to v_{2}$ , $v_{L1} \to v_{2}$ , $u_{2} \to v_{2}$ and $p_{2} \to v_{2}$ are manifested as the unique causality $\Delta I_{4 \to 9}^U$ , the redundant causalities $\Delta I_{458 \to 9}^R$ and the synergistic causality $\Delta I_{411 \to 9}^S$ , respectively. The $p_2$ -targeted causal links $u_{2} \to p_{2}$ correspond to the unique causality $\Delta I_{8 \to 11}^U$ . It is noteworthy that $u_{L1}$ is quite active under the optimal time delay, exhibiting many unique causalities.

Figures 9(a) and 9(b) show the two-point causal maps of $y_1^+=70$ and $y_2^+=200$ by TE and IF. It can be seen that the causal relationships at the same height are similar to figure 8, in which we can find the streak-vortex links of the inner and outer motions, i.e. $u_{1}^* \leftrightarrow v_{1}^*$ , $u_{L1} \leftrightarrow v_{L1}$ and $u_2 \leftrightarrow v_2$ . Compared with figure 8, a height dependence similar to the one-point causal inference is observed, i.e. the intra-scale causalities of outer footprints at $y_1^+$ are stronger with a higher wall-normal height. For the causalities between the two heights, the links $(u_{L1},v_{L1}) \leftrightarrow (u_2,v_2)$ demonstrate bidirectional causal relationships between the outer motions and their footprints at $y^+=70$ . These causal links also confirm the spatial convection and inclined structural coherence of the LSMs between the outer layer and the near-wall region.

Figure 9.

Two-point causal maps for the inner and outer motions at $y^+=70$ and $y^+=200$ : (a) TE, (b) IF, and (c) SURD. Yellow boxes: causalities between the two heights. Green boxes: causalities at the same height.

The pressure-related causal characteristics illustrated in figure 9 are basically similar to those in figure 8, where most of the links are of the same height. For two-point pressure–velocity causal links, $v_{L1} \to p_2$ and $u_{L1} \leftrightarrow p_2$ can be identified, demonstrating the non-local causal relationships between outer pressure and near-wall velocity footprints. The causal links $p_2 \leftrightarrow p_1$ indicate a two-way coupling between the inner and outer pressures. This result strengthens the role of pressure in bridging the inner and outer turbulent motions.

From figure 9(c) the $u_{1}^*$ -targeted causal links $u_{L1} \to u_{1}^*$ and $p_1 \to u_{1}^*$ are manifested as the redundant causality $\Delta I_{14 \to 1}^R$ and $\Delta I_{17 \to 1}^R$ , the synergistic causalities $\Delta I_{14 \to 1}^S$ and $\Delta I_{17 \to 1}^S$ , respectively. The $v_{1}^*$ -targeted causal link $p_{1} \to v_{1}^*$ corresponds to the redundant causality $\Delta I_{17 \to 2}^R$ . The causal link $u_{L1} \to v_{1}^*$ corresponds to the unique causality $\Delta I_{4 \to 2}^U$ . The $u_{L1}$ -targeted causal links $v_{L1} \to u_{L1}$ and $u_{2} \to u_{L1}$ are reflected by the synergistic causalities $\Delta I_{45 \to 4}^S$ and the redundant causalities $\Delta I_{48 \to 4}^R$ , $\Delta I_{458 \to 4}^R$ , respectively. The causal link $p_{1} \to u_{L1}$ is reflected by the synergistic causality $\Delta I_{47 \to 4}^S$ , which further illustrates the important role of pressure in the self-sustaining cycle. The $v_{L1}$ -targeted causal link $u_{L1} \to v_{L1}$ is related to the unique causality $\Delta I_{4 \to 5}^R$ . The causal links $p_{1} \to v_{L1}$ and $u_{2} \to v_{L1}$ are related to the redundant causality $\Delta I_{458 \to 5}^R$ and the synergistic causality $\Delta I_{47 \to 5}^S$ , respectively, which reveal the interactions between the outer motions and their footprints. The $p_1$ -targeted causal links $u_{L1} \to p_{1}$ and $u_{2} \to p_{1}$ correspond to the unique causality $\Delta I_{4 \to 7}^U$ and the redundant causality $\Delta I_{48 \to 7}^R$ , respectively. The $u_2$ -targeted causal links $u_{L1} \to u_{2}$ , $v_{2} \to u_{2}$ and $p_{2} \to u_{2}$ correspond to the redundant causalities $\Delta I_{48 \to 8}^R$ , $\Delta I_{489 \to 8}^R$ , the synergistic causality $\Delta I_{411 \to 8}^S$ and the unique causality $\Delta I_{4 \to 8}^U$ respectively. The $v_2$ -targeted causal links $u_{L1} \to v_{2}$ , $v_{L1} \to v_{2}$ , $u_{2} \to v_{2}$ and $p_{2} \to v_{2}$ are manifested as the redundant causalities $\Delta I_{48 \to 9}^R$ , $\Delta I_{458 \to 9}^R$ , the synergistic causality $\Delta I_{411 \to 9}^S$ and the unique causality $\Delta I_{4 \to 9}^U$ , respectively. The $p_2$ -targeted causal links $u_{L1} \to p_2$ and $u_{2} \to p_{2}$ correspond to the redundant causalities $\Delta I_{89 \to 11}^R$ , $\Delta I_{48 \to 11}^R$ , the synergistic causality $\Delta I_{48 \to 11}^S$ and the unique causalities $\Delta I_{8 \to 11}^U$ and $\Delta I_{4 \to 11}^U$ , respectively.

In summary, the two-point causal analysis has provided the following insights. First, the causally identified intra-scale streak-vortex self-sustaining cycles of the inner motions and outer footprints are the same as the one-point result. A similar causal cycle in the outer region is also found, which further supports the self-sustaining mechanism of turbulent motions at all scales. However, direct causal links between the inner and outer velocities (or footprints) are absent. Second, there exist two-way causal couplings between outer velocity components and their near-wall footprints as a result of the inclined LSMs. Moreover, there also exist two-way causal couplings between their cross-velocity components. We have also verified that the identified causal structures consistently persist at $Re_{\tau }=2000$ and with a different outer location ( $y_2^+=100$ ) as in the supplementary material, indicating the robustness of the present findings.

Figure 10.

Schematic diagram of the causal links between inner and outer motions. Black arrows: the streak-vortex self-sustaining cycle of outer motions; blue arrows: the streak-vortex self-sustaining cycle of outer footprints; red arrows: the streak-vortex self-sustaining cycle of inner motions; green arrows: the bidirectional causalities of the outer motions and their near-wall footprints; brown arrows: the inner-pressure-related causal links; magenta arrows: the outer-pressure-related causal links; cyan arrow: the causal links between the inner and outer pressure.

Figure 10. Long description

A schematic diagram of the causal links between inner and outer motions in wall-bounded turbulence. The diagram is divided into two regions: the inner region and the outer region, separated by a horizontal line labeled y+ = 100. The black arrows represent the streak-vortex self-sustaining cycle of outer motions. The blue arrows indicate the streak-vortex self-sustaining cycle of outer footprints. The red arrows show the streak-vortex self-sustaining cycle of inner motions. The green arrows illustrate the bidirectional causalities between the outer motions and their near-wall footprints. The brown arrows depict the inner-pressure-related causal links. The magenta arrows represent the outer-pressure-related causal links. The cyan arrow shows the causal links between the inner and outer pressure. The diagram includes labels for various motions and footprints, such as u2, v2, uL1, vL1, u1*, and v1*, indicating different components of the turbulent flow.