1. Introduction
A fundamental result underpinning much of our understanding of the multi-scale behaviour of turbulence from a statistical perspective is due to Kolmogorov (Reference Kolmogorov1941a,Reference Kolmogorovb), who derived scaling laws for the moments of the velocity differences as a function of their spatial separation, i.e. the structure functions. However, moving beyond a simple statistical description has proven difficult due to the suite of vortices at different scales with varying topologies that generates a local physics that may depart significantly from the average picture, as well as the non-local action of the pressure terms on the vortical dynamics (Ohkitani & Kishiba Reference Ohkitani and Kishiba1995; Buaria & Pumir Reference Buaria and Pumir2023). Classification of vortical components of the flow into sheets and tubes (Horiuti Reference Horiuti2001) is based on the eigenvalues of terms derived from the rotation rate and strain rate tensors, which are in turn obtained from the velocity gradient tensor (VGT). The statistics of the VGT are typically examined in the space of the second and third invariants (with the first invariant zero for incompressible flow). See Johnson & Wilczek (Reference Johnson and Wilczek2024) for a review of recent work on the multi-scale behaviour of the VGT as well as Chevillard et al. (Reference Chevillard, Meneveau, Biferale and Toschi2008), Wilczek & Meneveau (Reference Wilczek and Meneveau2014) and Johnson & Meneveau (Reference Johnson and Meneveau2016) for efforts to model the conditional statistical behaviour of the VGT dynamics for homogeneous isotropic turbulence (HIT). In these statistical approaches, topological considerations are often left implicit, but connections may be readily made as, for example, the second invariant for the VGT, known as 
$Q$, has been commonly adopted as a coherent flow structure identification criterion (Hunt, Wray & Moin Reference Hunt, Wray and Moin1988; Dubief & Delcayre Reference Dubief and Delcayre2000).
When we attempt to translate these concepts from HIT to boundary layers and channel flows, the key difference is that the presence of the wall provides a means to generate and sustain turbulent motions (Lumley Reference Lumley1964). The dynamics of these near-wall motions is closely associated with the attached eddy hypothesis (Townsend Reference Townsend1976), and a great deal of recent work has demonstrated the general correctness of this formalism. For example, Jiménez & Hoyas (Reference Jiménez and Hoyas2008) were able to show that the squared velocity intensities scaled logarithmically away from the wall, as predicted in the original theory, and Hwang (Reference Hwang2015) determined a self-similarity in the spanwise length of self-sustaining motions, which could be linked explicitly to the structures postulated in the attached eddy hypothesis.
In addition, evidence has also emerged that the larger scales in a channel flow can maintain themselves rather than being the consequence of an amalgamations of smaller-scale streaks or hairpin vortices (Hwang & Cossu Reference Hwang and Cossu2010). This self-sustaining process is similar to that previously identified by Hamilton, Kim & Waleffe (Reference Hamilton, Kim and Waleffe1995) and Schoppa & Hussain (Reference Schoppa and Hussain2002) in the near-wall region for the smallest attached eddies. Hamilton et al. (Reference Hamilton, Kim and Waleffe1995) initiated a Couette flow numerically with just the vortices present and any cross-flow fixed (to prevent vortex decay) and then looked for the development of streaks. It was found that the streamwise vortices acting on the mean flow produced the streaks by momentum redistribution. However, breakdown was a consequence of the instability of the streaks rather than the mean flow or the streamwise vortices, with vortex regeneration a consequence of nonlinear interactions. A critical threshold for vortex circulation needed to be maintained for this cycle to be self-sustaining, with the circulation peaking during breakdown and then decaying, with streaks forming during the periods of vortex decay.
In terms of the scale-by-scale organization of flow structures and the development of an energy spectrum, Goto (Reference Goto2008) proposed that HIT was organized such that smaller-scale vortex tubes form in the regions of high strain between larger, anti-parallel tubular structures and wrap around them in a perpendicular orientation. Extending this work to wall-bounded flows, Motoori & Goto (Reference Motoori and Goto2019) found that vortices sufficiently small relative to the distance from the wall were formed by stretching due to the action of vortices twice as large. This is similar to the observation by Cardesa, Vela-Martín & Jiménez (Reference Cardesa, Vela-Martín and Jiménez2017) that, in the turbulence cascade, packets of energetic motions of a given scale first appear within packets with double this scale, and dissipate within those that are half as large. In contrast, and related to the attached eddy hypothesis, Motoori & Goto (Reference Motoori and Goto2019) found that the vortices scaling on the distance from the wall developed as a consequence of the mean gradients in the flow field rather than this non-local stretching mechanism. The anti-parallel arrangement of vortices as a function of the scale of the cascade has also been detected in vortex ring collision experiments (McKeown et al. Reference McKeown, Ostilla-Mónico, Pumir, Brenner and Rubinstein2020). Those authors found that, at a given scale, pairs of anti-parallel vortices interact according to the Biot–Savart law to generate a subsequent flow structure via the elliptical instability (Kerswell Reference Kerswell2002) and proposed that this mechanism was analogous to that identified by Goto (Reference Goto2008).
The manner by which the vortices develop and the flow dissipates energy may be considered from the perspective of enstrophy production and strain production, the terms contributing to the third invariant of the VGT (Carbone & Wilczek Reference Carbone and Wilczek2022). Taylor (Reference Taylor1937) observed that enstrophy production is positive in the mean and hypothesized that vortex stretching is the principal means by which turbulent flows dissipate energy. However, further work by Betchov (Reference Betchov1956) showed that, in fact, vortex compression is critical. Betchov used an expression due to Townsend (Reference Townsend1951) for the relation between average strain production and average enstrophy production in the frame of the principal axes of the strain rate field to illustrate this
\begin{equation} {-}\langle e_{1}e_{2}e_{3}\rangle = \langle e_{1}\varOmega_{1}^{2} + e_{2}\varOmega_{2}^{2} + e_{3}\varOmega_{3}^{2}\rangle. \end{equation}
Here, the 
$e_{i}$ values are the eigenvalues of the strain rate tensor (ordered in descending magnitude in this instance) and the 
$\varOmega _{i}$ values are the rotations about an axis parallel to the principal axes of the strain rate field. Incompressibility means that 
$e_{2}$ and 
$e_{3}$ must have the same sign and this is opposite to 
$e_{1}$. Vortex stretching corresponds to 
$e_{1} > 0$, but Betchov noted that for both sides of (1.1) to be positive requires 
$e_{1} < 0$ given that 
$\varOmega _{1}$ is attenuated but 
$\varOmega _{2}$ and 
$\varOmega _{3}$ are enhanced in such a compressive ‘jet collision’ regime.
Greater understanding of this result stems from the work by Jiménez et al. (Reference Jiménez, Wray, Saffman and Rogallo1993), who showed that enstrophy production is correlated to both the enstrophy and the total strain, but more strongly to the latter, and from the work of Tsinober (Reference Tsinober2001), who showed that strain production is also associated with the total strain, but exhibits very little relation to the enstrophy. The reason for this complexity is that the non-local nature of a turbulent flow field as a consequence of the action of the pressure terms (Ohkitani & Kishiba Reference Ohkitani and Kishiba1995) means that any given point may have high values for strain production and small values for enstrophy production despite the fact that, on average, they are in balance.
Consequently, the enstrophy production plays a key role in vortical dynamics and exhibits a complex relation with other important quantities. It is given by 
$\omega _{i}^{A}S_{ij}^{A}\omega _{j}^{A}$, where 
$\boldsymbol {A}$ is the VGT, 
$A_{ij} = \partial u_{i} / \partial x_{j}$, 
$S_{ij}$ is the strain rate and 
$\omega _{i}$ is the vorticity vector, 
$\omega _{i} = -\epsilon _{ijk}\varOmega _{jk}$, where 
$\epsilon _{ijk}$ is the Levi-Civita symbol and
\begin{gather} \boldsymbol{S}_{A} = \tfrac{1}{2}(\boldsymbol{A} + \boldsymbol{A}^{*}), \end{gather}
\begin{gather}\boldsymbol{\varOmega}_{A} = \tfrac{1}{2}(\boldsymbol{A} - \boldsymbol{A}^{*}), \end{gather}
are the strain and rotation tensors, respectively. The enstrophy, total strain and their respective production terms are the constituents of the aforementioned second, 
$Q$, and third, 
$R$, invariants of the VGT
\begin{gather} Q = \tfrac{1}{2}(\Vert\boldsymbol{\varOmega}_{A}\Vert^{2} - \Vert\boldsymbol{S}_{A}\Vert^{2}), \end{gather}
\begin{gather}R ={-}\mbox{det}(\boldsymbol{S}_{A}) - \mbox{tr}(\boldsymbol{\varOmega}_{A}^{2}\boldsymbol{S}_{A}), \end{gather}
where 
$\mbox {tr}(\cdots )$ and 
$\mbox {det}(\cdots )$ are the trace and determinant operators, respectively. Thus, positive 
$Q$ (Hunt et al. Reference Hunt, Wray and Moin1988) identifies locations where the enstrophy exceeds the total strain, while although (1.1) shows that on average there is a balance between the two terms on the right-hand side of (1.5), the distinctly non-Gaussian joint distribution for 
$Q$ and 
$R$ is biased towards 
$R < 0$ for the enstrophy-dominated regions with 
$Q > 0$ and towards 
$R > 0$ in the strain-dominated regions (
$Q < 0$), leading to the development of the well-known Vieillefosse tail to this joint distribution (Vieillefosse Reference Vieillefosse1984; Meneveau Reference Meneveau2011).
In this study we use a Schur decomposition of the VGT to gain greater understanding of the interplay between the parts of the enstrophy production that are associated with the eigenvalues of the VGT and those that would traditionally be kinematic quantities associated with the alignment structure between the vorticity vector and the strain rate eigenvectors. Taking the spatial derivative of the Navier–Stokes equations gives an evolution equation for the dynamics of 
$\boldsymbol {A}$, for an incompressible fluid where 
$\mbox {tr}(\boldsymbol {A}) = 0$
\begin{equation} \frac{\partial \boldsymbol{A}}{\partial t} + (\boldsymbol{u} \boldsymbol{\cdot} \boldsymbol{\nabla})\boldsymbol{A} ={-}\left(\boldsymbol{A}^2-\frac{\mbox{tr}(\boldsymbol{A}^2)}{3}\boldsymbol{I}\right) - \boldsymbol{H} + \nu \nabla^2 \boldsymbol{A}, \end{equation}
where 
$\boldsymbol {H}$ is the anisotropic/deviatoric part of the pressure Hessian 
$\nu$ is the kinematic viscosity, and 
$\boldsymbol {H}\equiv \boldsymbol {\nabla }\boldsymbol {\nabla } p -\nabla ^2p\boldsymbol {I}/3$, where p is the pressure, and 
$\boldsymbol {I}$ is the identity matrix. Cantwell (Reference Cantwell1992) simplified (1.6) by neglecting the viscous term and the deviatoric part of the pressure Hessian to give two coupled ordinary differential equations for the Lagrangian evolution of the VGT
\begin{equation} \left.\begin{gathered} \frac{{\rm d} Q}{{\rm d} t} ={-}3 {R} \\ \frac{{\rm d} R}{{\rm d} t} = \frac{2}{3} {Q}^{2}. \end{gathered}\right\} \end{equation}
We note that 
$Q$ and 
$R$ are normal terms that may be written in terms of the eigenvalues of the VGT, 
$\lambda _{i}$
\begin{gather} Q = (1 - \delta_{ij}) \sum \lambda_{i}\lambda_{j} \end{gather}
\begin{gather}R = \prod_{i} \lambda_{i}. \end{gather}The semi-autonomous set of ordinary differential equations in (1.7) is therefore only related to the eigenvalues of the VGT, meaning that the deviatoric pressure Hessian must contribute to the non-normal flow properties, which in turn must be related to the non-local effect of the deviatoric part of the pressure Hessian (Ohkitani & Kishiba Reference Ohkitani and Kishiba1995).
One way to isolate the non-normal contributions to the flow dynamics has been to decompose 
$\boldsymbol {A}$ using a Schur decomposition (Keylock Reference Keylock2018, Reference Keylock2019). This framework has been applied to study turbulence close to interfaces (Boukharfane et al. Reference Boukharfane, Er-raiy, Parsani and Chakraborty2021), in the near-wake region of spatially developing flows (Beaumard, Buxton & Keylock Reference Beaumard, Buxton and Keylock2019) and near the wall (Keylock Reference Keylock2022). It provides a means to represent non-normal processes explicitly rather than in terms of alignment structure. Hence, for a tensor, 
$\boldsymbol {A}$, we have that 
$\boldsymbol {A} = \boldsymbol {U} \boldsymbol {T} \boldsymbol {U}^{*}$ for the Schur decomposition and 
$\boldsymbol {A} = \boldsymbol {E} \boldsymbol {\varLambda } \boldsymbol {E}^{-1}$ for the eigenvalue decomposition, where the asterisk is the conjugate transpose. If a tensor is normal, i.e. 
$\boldsymbol {A}\boldsymbol {A}^{*} = \boldsymbol {A}^{*}\boldsymbol {A}$, then these two decompositions are equivalent. The primary advantage of analysing a tensor with the Schur decomposition is that the rotation vectors in the matrix, 
$\boldsymbol {U}$, are always unitary, while the primary advantage of the eigen decomposition is that the eigenvalue matrix, 
$\boldsymbol {\varLambda }$, is diagonal.
Work on using the Schur decomposition to distinguish the normal and non-normal dynamics explicitly has begun to feed into engineering model development. For example, Yu, Zhao & Lu (Reference Yu, Zhao and Lu2021) used an index, 
$\kappa _{B,C}$ of the relative magnitude of the normal and non-normal parts of the VGT from Keylock (Reference Keylock2018) to show that conditioning the subgrid-scale energy and enstrophy dissipation on the sign of this index is a useful means to build a subgrid closure for large-eddy simulations of compressible flow reflecting the varying nature of these terms near the wall. The Schur decomposition approach is also beginning to have an impact in non-Newtonian fluid mechanics (Yao & Zatloukal Reference Yao and Zatloukal2024).
The analysis by Yu et al. (Reference Yu, Zhao and Lu2021) has highlighted the relevance of treating the normal and non-normal dynamics separately near the wall for compressible flow. However, while 
$\kappa _{B,C}$ provides information on the relative magnitude of the norms for the normal and non-normal dynamics as a function of space and time, it does not connect explicitly to the dynamics of enstrophy and strain production at the wall. Such understanding is important both for control applications (determining the extent to which a sensor is recording local or non-local information) and refining wall closures, such as that proposed by Yu et al. (Reference Yu, Zhao and Lu2021). In this study we focus upon the enstrophy production near the wall. In the next section we outline our Schur decomposition approach before deriving new expressions for the enstrophy production in terms of the mean and fluctuating quantities (Motoori & Goto Reference Motoori and Goto2019) as well as the normal and non-normal contributions. We then use a direct numerical simulation of a turbulent channel flow with a shear velocity Reynolds number of 
${\sim }1000$ (Li et al. Reference Li, Perlman, Wan, Yang, Burns, Meneveau, Chen, Szalay and Eyink2008; Graham et al. Reference Graham2016) to establish which terms drive the dynamics of a near-wall channel flow.
2. Normal and non-normal contributions to the velocity gradient tensor dynamics
Analysis of the VGT based on the Schur decomposition (Schur Reference Schur1909) was used by Li, Zhang & He (Reference Li, Zhang and He2014) to undertake a triple decomposition of the flow field into a rigid body rotation term, a shear term and a stretching/compressive term (see also, Kolář et al. Reference Kolář, Šístek, Cirak and Moses2013; Nagata et al. Reference Nagata, Watanabe, Nagata and da Silva2020; Liu, Yu & Gao Reference Liu, Yu and Gao2022). The rigid body rotation component then underpins the Liutex definition of a coherent structure (Xu, Cai & Liu Reference Xu, Cai and Liu2019), while Zhu (Reference Zhu2021) has defined a real Schur flow and used this notion to explore, in particular, the dynamics of rotational and compressible flows. Our motivation for the use of the Schur decomposition comes from the earlier discussion of the restricted Euler model for the dynamics of the VGT (Cantwell Reference Cantwell1992) and the need to find a means to incorporate the non-normal dynamics into models that extend this formalism.
We follow Keylock (Reference Keylock2018) and consider the VGT to be formed additively by two constituents
\begin{equation} \boldsymbol{A} = \boldsymbol{B} + \boldsymbol{C}, \end{equation}
where the eigenvalue-related normal contributions are in 
$\boldsymbol {B}$ and 
$\boldsymbol {C}$ contains the non-normal contribution. The complex Schur decomposition of 
$\boldsymbol {A}$ is
\begin{equation} \boldsymbol{A} = \boldsymbol{U} \boldsymbol{T} \boldsymbol{U}^{*}, \end{equation}
where 
$\boldsymbol {U}$ is unitary i.e. 
$\boldsymbol {U}\boldsymbol {U}^{*} = \boldsymbol {I}$, and the asterisk indicates the conjugate transpose. The upper triangular tensor 
$\boldsymbol {T} = \boldsymbol {\varLambda } + \boldsymbol {N}$ consists of a diagonal matrix of eigenvalues, 
$\boldsymbol {\varLambda }$, where 
$\varLambda _{i,i} = \lambda _{i}$, and the non-normal contribution, 
$\boldsymbol {N}$. Decomposing 
$\boldsymbol {T}$ in this way permits a reconstruction
\begin{equation} \left.\begin{gathered} \boldsymbol{B} = \boldsymbol{U} \boldsymbol{\varLambda} \boldsymbol{U}^{*}\\ \boldsymbol{C} = \boldsymbol{U} \boldsymbol{N} \boldsymbol{U}^{*}. \end{gathered}\right\} \end{equation}
Numerical algorithms for calculating the eigenvalues and eigenvectors of non-symmetric matrices make use of the QR algorithm (Francis Reference Francis1961), and this approach is also central to obtaining the Schur decomposition (Golub & van Loan Reference Golub and van Loan2013). Givens’ rotations or Householder reflections are used to reduce a matrix to an upper Hessenberg form, which is in turn reduced to 
$\boldsymbol {T}$ using iterations of the QR decomposition. The Schur rotation vectors may then be obtained by post-multiplication. The Schur and eigendecompositions are equal for the special case that 
$\boldsymbol {A}$ is normal, which is equivalent to stating that 
$\boldsymbol {A} = \boldsymbol {B}$. For a particular ordering of the eigenvalues along the diagonal of 
$\boldsymbol {\varLambda }$, the values of 
$\boldsymbol {T}$ and 
$\boldsymbol {U}$ are fixed, but change if this order varies, or if the eigenvalues are complex and a real form for the Schur transform is adopted (resulting in a block-diagonal form for the normal part of the dynamics). For a recent detailed treatment of the linear algebra of the Schur transform and a proof of its connection to the triple decomposition of the VGT see Kronborg & Hoffman (Reference Kronborg and Hoffman2023). These authors argue that the real Schur form should be ordered with the real eigenvalue in the top-left location to align the coordinate system with the rotation axis. This may be contrasted with Zhu (Reference Zhu2018) and the analysis of the real Schur form as a two component, two-dimensional flow coupled to a one component, three-dimensional flow. For work using the Schur form for coherent flow structure identification see Gao & Liu (Reference Gao and Liu2018) and Dong, Gao & Liu (Reference Dong, Gao and Liu2019).
Returning to (1.4) and (1.5), non-normality contributes equally to each of the terms on the right-hand side of these two equations, meaning it cancels out when their difference is found. Explicitly, we have that
\begin{gather} \Vert\boldsymbol{\varOmega}_{A}\Vert^{2} = \Vert\boldsymbol{\varOmega}_{B}\Vert^{2} + \Vert\boldsymbol{\varOmega}_{C}\Vert^{2}, \end{gather}
\begin{gather}\Vert\boldsymbol{S}_{A}\Vert^{2} = \Vert\boldsymbol{S}_{B}\Vert^{2} + \Vert\boldsymbol{\varOmega}_{C}\Vert^{2}, \end{gather}
where 
$\Vert \boldsymbol {\varOmega }_{C}\Vert ^{2} = \Vert \boldsymbol {S}_{C}\Vert ^{2}$ is used to eliminate the latter term.
Similarly, (1.5) may be written as
\begin{gather} -\mbox{det}(\boldsymbol{S}_{A}) ={-}\mbox{det}(\boldsymbol{S}_{B}) -\mbox{det}(\boldsymbol{S}_{C}) + \mbox{tr}(\boldsymbol{\varOmega}_{C}^{2}\boldsymbol{S}_{B}) \end{gather}
\begin{gather}\mbox{tr}(\boldsymbol{\varOmega}_{A}^{2}\boldsymbol{S}_{A}) = \mbox{tr}(\boldsymbol{\varOmega}_{B}^{2}\boldsymbol{S}_{B}) -\mbox{det}(\boldsymbol{S}_{C}) +\mbox{tr}(\boldsymbol{\varOmega}_{C}^{2}\boldsymbol{S}_{B}), \end{gather}
where the three terms on the right-hand side of (2.6) are referred to as the normal strain production, non-normal production and interaction production, respectively, and the first term on the right-hand side of (2.7) is the normal enstrophy production. The two new terms contain the non-normal effects and, at least for HIT, the distribution function for the non-normal production is approximately symmetric while the interaction production is preferentially positive (Keylock Reference Keylock2018). Hence, it is the latter that drives the general positivity of 
$-\mbox {det}(\boldsymbol {S}_{A})$ and 
$\mbox {tr}(\boldsymbol {\varOmega }_{A}^{2}\boldsymbol {S}_{A})$ because the sign of 
$-\mbox {det}(\boldsymbol {S}_{B})$ is dictated by the sign of the product of the strain rate eigenvalues, which is positive for 
$R > 0$ and negative for 
$R < 0$, while 
$\Vert \boldsymbol {\varOmega }_{B}\Vert ^{2} = 0$ and 
$\mbox {tr}(\boldsymbol {\varOmega }_{B}^{2}\boldsymbol {S}_{B}) = 0$ are zero if the strain rate tensor eigenvalues are all real or, otherwise, the sign for 
$\mbox {tr}(\boldsymbol {\varOmega }_{B}^{2}\boldsymbol {S}_{B})$ is opposite to that for 
$-\mbox {det}(\boldsymbol {S}_{B})$.
An alternative perspective on these dynamics, studied by Galanti, Gibbon & Heritage (Reference Galanti, Gibbon and Heritage1997) and Gibbon (Reference Gibbon2002), is via terms defined by the dot and cross-products of the vorticity and the vortex stretching vector, 
$\boldsymbol {\sigma } = \boldsymbol {S}_{A}\boldsymbol {\omega }_{A}$
\begin{gather} \alpha(\boldsymbol{x},t) = \frac{\boldsymbol{\omega}_{A}\boldsymbol{\cdot}\boldsymbol{\sigma}}{\boldsymbol{\omega}_{A}\boldsymbol{\cdot}\boldsymbol{\omega}_{A}} \end{gather}
\begin{gather}\boldsymbol{\chi}(\boldsymbol{x},t) = \frac{\boldsymbol{\omega}_{A}\times\boldsymbol{\sigma}}{\boldsymbol{\omega}_{A}\boldsymbol{\cdot}\boldsymbol{\omega}_{A}}, \end{gather}
where the scalar 
$\alpha$ is in the form of a Rayleigh coefficient. Thus, 
$\boldsymbol {\chi }$ measures the degree of perpendicularity of the vorticity and the vortex stretching. This angular treatment is in the spirit of work examining the alignment between the vorticity and the strain rate (Ashurst et al. Reference Ashurst, Kerstein, Kerr and Gibson1987). In contrast, by adopting unitary rotation matrices in (2.2), information that is typically held in the alignment kinematics is moved into 
$\boldsymbol {N}$ in our approach. Thus, the above statement about the general positivity of 
$\mbox {tr}(\boldsymbol {\varOmega }_{C}^{2}\boldsymbol {S}_{B})$ implies an alignment between the normal strain rate and non-normal vorticity. Keylock (Reference Keylock2018) found this to be the most important way to generate the preferential alignment between the vorticity vector and the eigenvector for the second strain eigenvalue, 
$e_{2}$ (Kerr Reference Kerr1985; Ashurst et al. Reference Ashurst, Kerstein, Kerr and Gibson1987), where 
$R > 0$ and 
$Q < 0$ (i.e. close to the Vieillefosse tail). In this region, with local vorticity weak, the orientation of 
$\omega _{A}$ was dictated by its non-normal component and this was coupled to a strong mutual alignment between the eigenvectors for the intermediate strain eigenvalues for both normal and non-normal strain tensors. Where the normal vorticity was stronger (
$Q > 0$, 
$R< 0$) the non-normal vorticity played a necessarily smaller role in dictating the alignment for 
$\omega _{A}$. However, in these regions the strain rate is dominated by the non-normal component and while 
$\omega _{B}$, and thus 
$\omega _{A}$, was aligned with the eigenvector of the most extensive eigenvalue for 
$\boldsymbol {S}_{B}$, this in turn was aligned with the intermediate eigenvector for 
$\boldsymbol {S}_{C}$, which drove the nature of 
$\boldsymbol {S}_{A}$.
The Reynolds decomposition analysis of near-wall enstrophy production by Motoori & Goto (Reference Motoori and Goto2019) used the term on the left-hand side of (2.7). The intention in this manuscript is to apply the Schur decomposition to this term to explain the complex results those authors obtained, such as the changes in sign of the contributing terms with distance from the wall. The next section briefly reviews the simulation used in this study, which is in the public domain, aiding reproducibility of this work. We then study the enstrophy production in detail, with sampling of vertical profiles that are separated in time and space to obtain pseudo-independent samples that are then averaged to obtain converged results. This approach is complemented by the visualization of the various components of the enstrophy production using the same snapshot of the flow field throughout the paper.
3. The numerical simulation
In this study we make use of the Johns Hopkins Turbulence database channel flow simulation (Li et al. Reference Li, Perlman, Wan, Yang, Burns, Meneveau, Chen, Szalay and Eyink2008; Wan et al. Reference Wan, Chen, Eyink, Meneveau, Perlman, Burns, Li, Szalay and Hamilton2016), which is a direct numerical simulation of a wall-bounded channel flow with periodic boundary conditions in the longitudinal and transverse directions (Graham et al. Reference Graham2016). No-slip conditions are imposed at the top and bottom walls and the flow was initially driven to maintain a bulk velocity of 
$U = 1$ until stationarity was achieved. A mean pressure gradient was then applied to give the same shear stress as the previous step. The equations were solved with the PoongBack code (Lee, Malaya & Moser Reference Lee, Malaya and Moser2013), with data stored at 
$2048 \times 512 \times 1536$ spatial positions for a domain size of 
$L_{x} = 8{\rm \pi} h$, 
$L_{y} = 2h$ and 
$L_{z} = 3{\rm \pi} h$, with 
$h$ the channel half-height. Thus, the computational cells had a spacing of 
$x^+ = 12.26$ and 
$z^+ = 6.13$ wall units in the longitudinal and transverse directions, respectively, while the wall-normal resolution varied in the range 
$0.02 < y^+ < 6.16$ from the first node from the wall to the centreline. The shear velocity Reynolds number was 
$\mbox {Re}_{\tau } \equiv u_{\tau } h / \nu = 1000$, with a bulk velocity Reynolds number of 
$\mbox {Re} \equiv U h / \nu = 20\,000$ and a kinematic viscosity of 
$\nu = 5 \times 10^{-5}$ (dimensionless). The best-fit mean velocity profile attained the logarithmic shape conforming to the law-of-the-wall for 
$30 \lesssim y^+ \lesssim 250$. The simulation was run for just under 26 dimensionless time steps, with results stored every 
$\delta t = 0.0065$ steps. Further details of the simulation may be found in Wan et al. (Reference Wan, Chen, Eyink, Meneveau, Perlman, Burns, Li, Szalay and Hamilton2016).
Figure 1(a) shows a selected region in terms of the threshold for 
$Q$, with 
$Q > 4\overline {|Q|}$ shown in blue and 
$Q < -4\overline {|Q|}$ in red. The other three panels show the 
$|Q| > 4\overline {|Q|}$ regions in grey and the locations where the normal enstrophy, normal strain rate and non-normality exceed 
$6\overline {|Q|}$ in blue, red and green, respectively. Features that can be seen here include, in (a), longitudinally oriented, larger diameter strain-dominated structures at the wall, with positive 
$Q$ structures wrapped around them. Panels (b,c) show a spatial association between regions with strong normal enstrophy and normal strain and these have a predominant transverse orientation. The non-normality shown in green in (d) exhibits a particularly strong concentration near the wall.
Figure 1. The visualized flow structures for a 
$x^+ = 256$ by 
$y^+ = 170$ by 
$z^+ = 256$ domain are shown in (a) with positive 
$Q$ in blue and negative in red. The remaining panels show all large 
$|Q|$ regions in grey and then highlight the individual terms in (2.4) and (2.5), with 
$\Vert \boldsymbol {\varOmega }_{B}\Vert ^{2}$ in (b), 
$\Vert \boldsymbol {S}_{B}\Vert ^{2}$ in (c) and 
$\Vert \boldsymbol {\varOmega }_{C}\Vert ^{2}$ in (d). Values are made dimensionless using 
$(\frac {1}{2}\langle \Vert \boldsymbol {\varOmega }_{A}\Vert ^{2} \rangle )^{{3}/{2}}$.
4. Normal and non-normal components of the enstrophy production
Moving to index notation, we adopt a Reynolds decomposition where, for example, 
$\boldsymbol {\omega }^{A}=\bar {\boldsymbol {\omega }}^{A} + \boldsymbol {\omega }'^{\,A}$, with 
$\overline {\boldsymbol {\omega }}^{A}$ the mean vorticity vector at a constant distance from the wall, 
$y$, and 
$\boldsymbol {\omega }^{A'}$, the fluctuating part at the same height. Hence, given the spatial homogeneity in 
$x$ and 
$z$, averaging is undertaken in time, and in the longitudinal and transverse directions. It may then be shown (Motoori & Goto Reference Motoori and Goto2019) that the symmetries in a turbulent boundary-layer or channel flow lead to
\begin{equation} \left.\begin{gathered} \bar{\omega}^{A}_{1} = \bar{\omega}^{A}_{2} = 0\\ \bar{S}_{13}^{A} = \bar{S}_{31}^{A} = \bar{S}_{23}^{A} = \bar{S}_{32}^{A} = \bar{S}_{33}^{A} = 0, \end{gathered}\right\} \end{equation}which results in an average enstrophy production given by
\begin{equation} \overline{\omega_{i}^{A}S_{ij}^{A}\omega_{j}^{A}} = 2 \overline{\bar{\omega}_{i}^{A}S_{ij}'^{\,A}\omega_{j}'^{\,A}} + \overline{\omega_{i}'^{\,A}\bar{S}_{ij}^{A}\omega_{j}'^{\,A}} + \overline{\omega_{i}'^{\,A}S_{ij}'^{\,A}\omega_{j}'^{\,A}}. \end{equation}
We refer to the last term on the right-hand side of (4.2) as the ‘fluctuating enstrophy production’. The simulation by Motoori & Goto (Reference Motoori and Goto2019) employed a Reynolds number based on the shear velocity and momentum thickness of 3170, and the patterns seen in their results (figure 5 of their paper) are replicated in our results as given by the black lines in figure 2. In our figure, results are non-dimensionalized using the mean enstrophy from the wall to 
$y^+ = 70$, the top of the domain considered in this study, i.e. 
$\frac {1}{2}\langle ||\varOmega _{A}||^{2} \rangle ^{{3}/{2}}$, where the angled braces are the average over all spatial directions including 
$y$. This value corresponded approximately to that observed on average at 
$y^+ = 16$.
Figure 2. The most important term contributing to the budgets for 
$2 \overline {\bar {\omega }_{i}^{A}S_{ij}^{A'}\omega _{j}^{A'}}$ (a), 
$\overline {\omega _{i}^{A'}\bar {S}_{ij}^{A}\omega _{j}^{A'}}$ (b) and 
$\overline {\omega _{i}^{A'}S_{ij}^{A'}\omega _{j}^{A'}}$ (c) for 
$0 \le y^+ \le 70$. For notational compactness, the 
$i$ and 
$j$ subscripts have been removed and the 
$B$ and 
$C$ superscripts changed to subscripts in the legends. The values for the terms formulated by Motoori & Goto (Reference Motoori and Goto2019) are given in black, the quantities only involving non-normal terms are shown as solid grey lines. Quantities that involve a mixture of normal and non-normal terms are shown as dotted or dash-dotted grey lines. All terms are normalized by 
$(\frac {1}{2}\langle ||\varOmega _{A}||^{2} \rangle )^{{3}/{2}}$.
Motoori & Goto (Reference Motoori and Goto2019) showed that the first term on the right-hand side of (4.2) dominates the budget for 
$y^+ \le 10$ and peaks at 
$y^+ = 3.5$, with a small magnitude negative peak at 
$y^+ = 14$. The second term on the right-hand side of (4.2) is typically the smallest magnitude term and is initially negative in sign until 
$y^+ = 5$, reaching a negative peak at 
$y^+ = 2.5$. It attains a positive peak at 
$y^+ = 10$, which is approximately 50 % larger in magnitude than the negative peak. The fluctuating enstrophy production in (c) exhibits a positive peak at 
$y^+ = 2.8$, and a secondary maximum at 
$y^+ = 10$. This term then decays more slowly than the others and, consequently, dominates the budget for 
$y^+ > 30$. Motoori & Goto (Reference Motoori and Goto2019) showed that this is the case as a far out as at least 
$y^+ = 1000$. However, in this study we focus on the more complex interplay for the various terms near the wall seen in the figure and thus focus attention on 
$y^+ \le 70$.
Applying the Schur decomposition to (4.2) results in the following sets of terms:
\begin{align} \overline{\bar{\omega}_{i}^{A}S_{ij}'^{\,A}\omega_{j}'^{\,A}} &= \overline{\bar{\omega}_{i}^{B}S_{ij}'^{\,B}\omega_{j}'^{\,B}} + \overline{\bar{\omega}_{i}^{B}S_{ij}'^{\,B}\omega_{j}'^{\,C}} + \overline{\bar{\omega}_{i}^{B}S_{ij}'^{\,C}\omega_{j}'^{\,B}} + \overline{\bar{\omega}_{i}^{B}S_{ij}'^{\,C}\omega_{j}'^{\,C}} \nonumber\\ &\quad + \overline{\bar{\omega}_{i}^{C}S_{ij}'^{\,B}\omega_{j}'^{\,B}} + \overline{\bar{\omega}_{i}^{C}S_{ij}'^{\,B}\omega_{j}'^{\,C}} + \overline{\bar{\omega}_{i}^{C}S_{ij}'^{\,C}\omega_{j}'^{\,B}} + \overline{\bar{\omega}_{i}^{C}S_{ij}'^{\,C}\omega_{j}'^{\,C}} , \end{align}
\begin{align} \overline{\omega_{i}'^{\,A}\bar{S}_{ij}^{A}\omega_{j}'^{\,A}} &= \overline{\omega_{i}'^{\,B}\bar{S}_{ij}^{B}\omega_{j}'^{\,B}} + 2\overline{\omega_{i}'^{\,B}\bar{S}_{ij}^{B}\omega_{j}'^{\,C}} + \overline{\omega_{i}'^{\,B}\bar{S}_{ij}^{C}\omega_{j}'^{\,B}} \nonumber\\ &\quad + 2\overline{\omega_{i}'^{\,C}\bar{S}_{ij}^{C}\omega_{j}'^{\,B}} + \overline{\omega_{i}'^{\,C}\bar{S}_{ij}^{B}\omega_{j}'^{\,C}} + \overline{\omega_{i}'^{\,C}\bar{S}_{ij}^{C}\omega_{j}'^{\,C}} , \end{align}
\begin{align} \overline{\omega_{i}'^{\,A}S_{ij}'^{\,A}\omega_{j}'^{\,A}} &= \overline{\omega_{i}'^{\,B}S_{ij}'^{\,B}\omega_{j}'^{\,B}} + 2\overline{\omega_{i}'^{\,B}S_{ij}'^{\,B}\omega_{j}'^{\,C}} + \overline{\omega_{i}'^{\,B}S_{ij}'^{\,C}\omega_{j}'^{\,B}} \nonumber\\ &\quad + 2\overline{\omega_{i}'^{\,C}S_{ij}'^{\,C}\omega_{j}'^{\,B}} + \overline{\omega_{i}'^{\,C}S_{ij}'^{\,B}\omega_{j}'^{\,C}} + \overline{\omega_{i}'^{\,C}S_{ij}'^{\,C}\omega_{j}'^{\,C}} . \end{align} It was found that the great majority of the terms in these expansions do not have a significant impact on the enstrophy budget near the wall. Thus, in figure 2 we show each term from the left-hand sides of (4.3)–(4.5) and the two or three terms from the right-hand sides of these equations that contribute in practice to the near-wall dynamics. Inspection of figure 2(a) shows that three terms of the eight on the right-hand side of (4.3) are important: the purely non-normal term (grey, solid line), which mimics the left-hand side most closely, 
$2\overline {\bar {\omega }_{i}^{C}S_{ij}'^{\,B}\omega _{j}'^{\,C}}$ (grey, dot-dashed), which is positive throughout, and a negative contribution from 
$2\overline {\bar {\omega }_{i}^{C}S_{ij}'^{\,C}\omega _{j}'^{\,B}}$ (grey, dashed). The primary peak at 
$y^+ = 4$ is due to a similar magnitude contribution from the two terms involving 
$\bar {\omega }_{i}^{C}$ and 
$\omega _{j}'^{\,C}$, while the negative peak at 
$y^+ = 13$ is a consequence of 
$2\overline {\bar {\omega }_{i}^{C}S_{ij}'^{\,C}\omega _{j}'^{\,B}}$ and the change in sign of 
$2\overline {\bar {\omega }_{i}^{C}S_{ij}'^{\,C}\omega _{j}'^{\,C}}$ at 
$y^+ = 8.5$.
Figure 2(b) shows that just two of the six terms in (4.4) are sufficient to explain the rather complex behaviour of this term. Apart from being slightly negative very near the wall, 
$2\overline {\omega _{i}'^{\,C}\bar {S}_{ij}^{\,C}\omega _{j}'^{\,B}}$ (grey, dashed line) is generally positive and drives the positive peak, while 
$2\overline {\omega _{i}'^{\,C}\bar {S}_{ij}^{\,C}\omega _{j}'^{\,C}}$ changes sign at 
$y^+ = 8.5$ and drives the near-wall negative peak. Overall, the budget for (4.5) near the wall reduces to contributions from 
$\overline {\omega _{i}'^{\,C}S_{ij}'^{\,C}\omega _{j}'^{\,C}}$ (grey solid line) and 
$\overline {\omega _{i}'^{\,C}S_{ij}'^{\,B}\omega _{j}'^{\,C}}$ (grey dot-dashed line), with both of these terms positive throughout. The secondary maximum at 
$y^+ = 12$ for 
$\overline {\omega _{i}'^{\,A}S_{ij}'^{\,A}\omega _{j}'^{\,A}}$ also seen in the results of Motoori & Goto (Reference Motoori and Goto2019) is much more pronounced for the purely non-normal term. The larger magnitude 
$\overline {\omega _{i}'^{\,C}S_{ij}'^{\,B}\omega _{j}'^{\,C}}$ term also exhibits two maxima, but these are both further from the wall, with the vertical offset between the maxima for the two terms acting to smooth the upper maximum for the overall fluctuating enstrophy production. It is also noticeable that the persistent positivity of 
$\overline {\omega _{i}'^{\,A}S_{ij}'^{\,A}\omega _{j}'^{\,A}}$ away from the wall is driven more by 
$\overline {\omega _{i}'^{\,C}S_{ij}'^{\,B}\omega _{j}'^{\,C}}$ than the purely non-normal term, which has decayed towards zero by 
$y^+ = 70$. Thus, the driver for enstrophy production away from the wall is the normal fluctuating straining of the non-normal contribution to the fluctuating vorticity. This may also be seen in the last column of table 1, where this term accounts for over 50 % of the fluctuating enstrophy production at all heights, while the purely non-normal term decreases from 44 % near the wall to 11 % at 
$y^+ = 70$. Consistent with this, further from the wall, the purely normal term (second column) increases dramatically in importance as one moves away from the wall, from a 0.6 % contribution at 
$y^+ = 3$ to 18.6 % at 
$y^+ = 70$.
Table 1. Values for the constituents of the fluctuating enstrophy production, 
$\overline {\omega _{i}'^{\,A}S_{ij}'^{\,A}\omega _{j}'^{\,A}}$, shown in figure 2(c) and expressed as a percentage of the value for 
$\overline {\omega _{i}'^{\,A}S_{ij}'^{\,A}\omega _{j}'^{\,A}}$ at selected values of 
$y^+$.
An instantaneous snapshot of the two key terms in the fluctuating enstrophy production budget is shown in figure 3 superimposed onto regions of high 
$|R|$, with locations with high 
$|Q|$ shown in grey for reference. The general characteristics of the mean profiles may be discerned in the fields shown. For example, the general positivity of both terms (blue), the greater persistence of this positivity with height in the case of 
$\overline {\omega _{i}'^{\,C}S_{ij}'^{\,B}\omega _{j}'^{\,C}}$ in (a) and the greater predominance of negative production regions in (b), helping to explain the less positive mean profile for 
$\overline {\omega _{i}'^{\,C}S_{ij}'^{\,C}\omega _{j}'^{\,C}}$ above the viscous layer in figure 2(c).
Figure 3. Snapshots of the same flow field shown in figure 1. The grey areas show locations with large values for 
$|Q|$. Otherwise, the colours reflect the values for the final two terms in (4.5) that are given as grey dash-dotted and grey lines, respectively, in figure 2(c). These fields are draped onto regions of high 
$|R|$ and all terms are normalized by 
$(\frac {1}{2}\langle ||\varOmega _{A}||^{2} \rangle )^{{3}/{2}}$.
That figure 2(c) and table 1 highlight the importance of 
$\overline {\omega _{i}'^{\,C}S_{ij}'^{\,B}\omega _{j}'^{\,C}}$ is interesting in terms of the analysis of HIT by Keylock (Reference Keylock2018). The positivity of this term has a direct analogy to the result from that study where the ‘interaction production’ (the normal straining of the non-normality) was largely responsible for driving the simultaneous positivity of the strain production and enstrophy production budgets in HIT. However, in a boundary layer, given the small magnitude of normal strain near the wall, it is not obvious that this term should dominate over the purely non-normal 
$\overline {\omega _{i}'^{\,C}S_{ij}'^{\,C}\omega _{j}'^{\,C}}$, particularly for small, 
$y^+$. In addition, the two maxima at 
$y^+ = 3$ and 
$y^+ = 10$ for 
$\overline {\omega _{i}'^{\,A}S_{ij}'^{\,A}\omega _{j}'^{\,A}}$ in figure 2(c) are reflected in both the constituent terms shown in that panel, and the reasons for this are likely to be due to contributions from different parts of the vorticity and strain rate at different heights. Hence, in the next section we decompose the two most important terms in figure 2(c) to determine which components of the strain rate and rotation rate tensors drive these terms.
5. The terms driving the fluctuating enstrophy production
5.1. Individual components of the fluctuating enstrophy production
Figure 4(a,b) shows the profile of 
$\overline {\omega _{i}^{'\,C}S_{ij}^{'\,B}\omega _{j}^{'\,C}}$ from figure 2(c) as a black line (CBC), with the six constituents of this term shown as grey lines, with three of the six terms in each of these two panels. Similarly, (c,d) show 
$\overline {\omega _{i}^{'\,C}S_{ij}^{'\,C}\omega _{j}^{'\,C}}$ from figure 2(c) as a black line (CCC), with its six constituents. The sum of the three constituent terms shown in grey in each panel is then shown as a black dashed (a,c) or dash-dotted line (b,d). The components making up each line are given in the legend, where the commas delimit the vorticity from the strain rate and then from the vorticity again. If the strain term is an off-diagonal component, it is left implicit in the legend that the value has been multiplied by two due to the symmetry of the strain rate tensor. It is clear from (a,b) that the largest contribution to 
$\overline {\omega _{i}^{'\,C}S_{ij}^{'\,B}\omega _{j}^{'\,C}}$, particularly at the wall, comes from 
$\overline {\omega _{3}^{'\,C}S_{33}^{'\,B}\omega _{3}^{'\,C}}$. The implication of this is that close to the wall the only really significant component of the fluctuating normal strain tensor is the transverse term. Panels (a,c) show the three terms that involve 
$\omega _{3}^{'\,C}$ and it is clear from (a) that their combined effect accounts for much of the relevant behaviour from figure 2(c), although with a deficit for 
$5 < y^+ < 50$. Panel (b) shows that, above the viscous sub-layer, the other three components all act to counter this deficit. In particular, the vertical term, 
$\overline {\omega _{2}^{'\,C}S_{22}^{'\,B}\omega _{2}^{'\,C}}$, makes the principal contribution and its increase away from the wall explains the more diffuse peak to the second maximum for 
$\overline {\omega _{i}^{'\,C}S_{ij}^{'\,B}\omega _{j}^{'\,C}}$ compared with 
$\overline {\omega _{i}^{'\,C}S_{ij}^{'\,C}\omega _{j}^{'\,C}}$ in figure 2(c).
Figure 4. Vertical profiles for the constituent elements of the two dominant terms for the fluctuating enstrophy production. The components of 
$\overline {\omega _{i}^{'\,C}S_{ij}^{'\,B}\omega _{j}^{'\,C}}$ are given in (a,b) and 
$\overline {\omega _{i}^{'\,C}S_{ij}^{'\,C}\omega _{j}^{'\,C}}$ are shown in (c,d). In each case, the solid black line is equivalent to the appropriate case from figure 2(c) and the dashed black line is the sum of the terms shown in grey in that panel. The legend indicates the nature of the component terms, which are shown in grey and, for example, ‘1,13,3’ in (a) indicates 
$2\overline {\omega _{1}^{'\,C}S_{13}^{'\,B}\omega _{3}^{'\,C}}$. All terms are non-dimensionalized by 
$(\frac {1}{2}\langle ||\boldsymbol {\varOmega }_{A}||^{2} \rangle )^{3/2}$.
The primary difference between (c) and (a) is that the sum of the three terms shown exceeds the values for the term itself, implying negative contributions on average from the remaining three terms, which are shown in (d). Hence, the unexpected dominance of 
$\overline {\omega _{i}^{'\,C}S_{ij}^{'\,B}\omega _{j}^{'\,C}}$ given how small the normal parts of the tensor are near the wall is because of negative contributions to the budget for the term that dominates (d), 
$\overline {\omega _{1}^{'\,C}S_{12}^{'\,C}\omega _{2}^{'\,C}}$. Thus, it is the non-normal contribution to the fluctuating shear in the dominant longitudinal–vertical plane that acts to reduce the purely non-normal contribution to fluctuating enstrophy production. The three terms in (c) are of similar magnitude, in contrast to (a). Of these, 
$\overline {\omega _{1}^{'\,C}S_{13}^{'\,C}\omega _{3}^{'\,C}}$ is the largest very close to the wall, with 
$\overline {\omega _{2}^{'\,C}S_{23}^{'\,C}\omega _{3}^{'\,C}}$ greater for 
$y^+ > 5$. Thus, the important positive and negative contributions to 
$\overline {\omega _{i}^{'\,C}S_{ij}^{'\,C}\omega _{j}^{'\,C}}$ all involve off-diagonal components of the non-normal strain rate, in marked contrast to the normal strain rate tensor in (a,b).
The instantaneous behaviour associated with the profiles in figure 4 is given in figure 5 using the same snapshot as in earlier figures but with the 
$z$-axis extent reduced by a factor of two and the 
$y$-axis restricted to 
$y^+ \le 70$ to aid visualization of the near-wall characteristics. Panel (a) indicates the combined contribution from 
${\omega _{i}^{'\,C}S_{ij}^{'\,C}\omega _{j}^{'\,C}}$ and 
${\omega _{i}^{'\,C}S_{ij}^{'\,B}\omega _{j}^{'\,C}}$, while other panels show specific terms. The median values for each quantity in this snapshot are stated above each panel to the right, with the standard deviation in brackets. The spatial median values here have a qualitative similarity to the profiles in figure 4 such that, for example, 
$\overline {\omega _{3}^{'\,C}S_{33}^{'\,B}\omega _{3}^{'\,C}}$ makes the largest magnitude contribution and 
$2\overline {\omega _{1}^{'\,C}S_{12}^{'\,C}\omega _{2}^{'\,C}}$ is negative on average. The smallest magnitude contribution on average comes from 
$\overline {\omega _{3}^{'\,C}S_{33}^{'\,C}\omega _{3}^{'\,C}}$, but the large spatial standard deviation indicates a great deal of local variability, as seen by the relatively extensive area of strongly positive and negative regions for this term near the wall compared with the other panels in the bottom row.
Figure 5. Visualization of selected components of the fluctuating enstrophy production using the same snapshots as in figure 1 but with the transverse dimension reduced by a factor of two and the vertical to 
$y^+ \le 70$ to highlight the near-wall behaviour. Panel (a) gives the sum of the terms given by black lines in figure 4(a,c) while five selected individual components making a major contribution to the budget are shown in the subsequent panels. For each panel, the quantity shown is stated above and to the left and the numbers above and to the right are the spatial median (standard deviation in brackets) of the values for this snapshot.
Enstrophy production can be conceived as the product of the vortex stretching and the vorticity. Thus, the terms driving the fluctuating enstrophy production in figure 4 are unpacked in figure 6 into the correlations between strain tensor components and the non-normal vorticity, with the normal strain in (a) and the non-normal in (b). Hence, when one unpacks 
$2\overline {\omega _{1}^{'\,C}S_{13}^{'\,B}\omega _{3}^{'\,C}}$, one finds that the negative behaviour for 
$y^+ < 5$ in figure 4(a) results from the correlation between the strain term and the vorticity oriented with a longitudinal axis (dark grey, dot-dashed line in figure 6a) but that the overall low magnitude of this component of the enstrophy production is a consequence of the approximate zero correlation between 
$S_{13}^{'\,B}$ and 
$\omega _{3}^{'\,C}$ for all 
$y^+$ apart from very near the wall, where the small positive value is sufficient for negative enstrophy production to be seen in the profile in figure 4(a). The correlation between 
$S_{33}^{'\,B}$ and 
$\omega _{3}^{'\,C}$ remains significantly positive throughout the profile, while that between 
$S_{23}^{'\,B}$ and 
$\omega _{2}^{'\,C}$ declines to zero by the start of the inertial region. In contrast, in (b), we see that the correlation between 
$S_{33}^{'\,C}$ and 
$\omega _{3}^{'\,C}$, while significantly non-zero, is less than that seen for 
$S_{33}^{'\,B}$ and 
$\omega _{3}^{'\,C}$ for 
$y^+ \gtrsim 10$, explaining our primary result that the interaction between normal strain and non-normal vorticity is key to explaining enstrophy production at the wall.
Figure 6. The correlation between fluctuating strain and vorticity components. (a) Shows selected relations between 
$S^{B}_{ij}$ and 
$\omega ^{C}_{i}$, while those between 
$S^{C}_{ij}$ and 
$\omega ^{C}_{i}$ are in (b). Dark grey, black and grey lines are for 
$\omega _{1}$, 
$\omega _{2}$ and 
$\omega _{3}$, respectively. Dashed, dot-dashed and then dotted lines are for 
$S_{12}$, 
$S_{13}$ and 
$S_{23}$, respectively, while normal components are given by a thicker solid line. The legend indicates the components of the strain tensor and vorticity separated by a comma.
5.2. Large fluctuating shear events
Motoori & Goto (Reference Motoori and Goto2019) showed that the larger flow structures near the wall were coupled to the mean gradient, with the orientation of the smaller structures dictated more by the presence of the surrounding larger structures. Hence, given the dominant contribution to the mean dynamics from the direction of mean shear, 
$\bar {S}_{12}^{A}$, we considered large fluctuating events to be those where 
$\vert S_{12}^{'\,C}\vert > \bar {S}_{12}^{A}$. In practice, exceedances for 
$\vert S_{12}^{'\,B}\vert$ were of very similar probability and distributed on a very similar set of points. These probabilities are given in figure 7(a) and very few negative occurrences (grey line) arise until 
$y^+ > 8$. Hence, the results in (b,c) for the negative enstrophy production are truncated at this height. The probability of the negative occurrences then increases rapidly such that, at the bottom of the inertial regime at 
$y^+ \sim 30$, they are as likely as the positive occurrences. That the bottom of the inertial regime marks this transition in the relative probability of positive and negative occurrences is something that will be investigated in future work.
Figure 7. Vertical profiles of the key components of the dominant contributions to the fluctuating enstrophy production where 
$\vert S_{12}^{'\,C}\vert > \bar {S}_{12}^{A}$. (a) Shows the probability of exceeding this threshold while (b) and (c) show the dominant fluctuating enstrophy production terms involving the normal and non-normal strain rate, respectively. Negative occurrences for the threshold exceedance are in grey and positive are in black. Results are not shown in (b,c) for the negative cases for 
$y^+ < 8$ as there were too few instances occurring. Values are non-dimensionalized by 
$\frac {1}{2}\langle \Vert \boldsymbol {\varOmega }_{A}\Vert ^{2} \rangle ^{3/2}$.
The results shown in (b,c) are conditioned on the 
$\vert S_{12}^{'\,C}\vert > \bar {S}_{12}^{A}$ cases. Hence, that the values for 
$\overline {\omega _{3}^{'\,C}S_{33}^{'\,B}\omega _{3}^{'\,C}}$ in (b) are greater than those for 
$\overline {\omega _{3}^{'\,C}S_{33}^{'\,C}\omega _{3}^{'\,C}}$ in (c) further demonstrates the crucial role played by the interaction between the normal and non-normal parts of the tensor, in this case for rare events with a magnitude some thirty times greater than the average picture in figure 4. Despite the increase in probability of threshold exceedance with height, strong fluctuating enstrophy production from the two terms shown is concentrated in the near-wall region (particularly, for the purely non-normal component) as a consequence of the decline in the magnitude of 
$\omega _{3}^{'\,C}$ away from the wall. However, an additional effect is that, for 
$y^+ > 8$, where the number of negative exceedances becomes statistically significant, the difference between the enstrophy production components for the negative component and the positive component is much greater in (b) than (c). Indeed, in figure 7(c) at 
$y^+ = 8$, the two are within 10 % of each other, while at the same height in figure 7(b) there is a factor of ten difference. This lack of symmetry in the results involving the normal strain indicates that enstrophy production by 
$S_{33}^{'\,B}$ is strongly dependent on positive shear fluctuations in the longitudinal–vertical plane. Hence, the analysis of these extreme instances indicates that the driving mechanism for near-wall enstrophy production is asymmetric in nature, tied to positive shear fluctuation in particular.
6. Discussion and conclusion
Following a Reynolds decomposition of the enstrophy production equation, Motoori & Goto (Reference Motoori and Goto2019) found that the fluctuating enstrophy production term was the most important to the budget for enstrophy production in the near-wall region, but the profile they obtained (see figure 2c) has a complex shape, with a double peak, implying different terms are contributing to the fluctuating enstrophy production at different heights. To investigate this phenomenon in greater detail, we have undertaken a Schur decomposition of the velocity gradient tensor to isolate the normal (denoted by a ‘B’) and non-normal (denoted by a ‘C’) contributions to the strain rate and rotation rate tensors (Keylock Reference Keylock2018; Xu et al. Reference Xu, Cai and Liu2019). Using this approach we have found that four particular terms dominate: 
$\overline {\omega _{3}^{'\,C}S_{33}^{'\,B}\omega _{3}^{'\,C}}$, 
$\overline {\omega _{1}^{'\,C}S_{13}^{'\,C}\omega _{3}^{'\,C}}$, 
$\overline {\omega _{2}^{'\,C}S_{23}^{'\,C}\omega _{3}^{'\,C}}$ and 
$\overline {\omega _{3}^{'\,C}S_{33}^{'\,C}\omega _{3}^{'\,C}}$, with the first the most important. That this term is greater than the final term is surprising given that the strong mean and fluctuating shear near the wall induces a non-local response on individual flow structures that is captured by the non-normal terms. However, 
$\omega _{3}^{'\,C}$ and 
$S_{33}^{'\,B}$ are strongly correlated (figure 6a). In addition, when assessing the budgets more generally, one component of the purely non-normal contribution to the fluctuating enstrophy production, 
$\overline {\omega _{1}^{'\,C}S_{12}^{'\,C}\omega _{2}^{'\,C}}$, is negative on average, as seen in figure 4(d). This acts to reduce the overall positive contribution from 
$\overline {\omega _{i}^{'\,C}S_{ij}^{'\,C}\omega _{j}^{'\,C}}$ relative to 
$\overline {\omega _{i}^{'\,C}S_{ij}^{'\,B}\omega _{j}^{'\,C}}$.
The initial decomposition of the fluctuating enstrophy production into normal and non-normal components in (4.5) showed that the double peaks in the vertical profile for the purely non-normal term were more marked than that for 
$\overline {\omega _{i}^{'\,C}S_{ij}^{'\,B}\omega _{j}^{'\,C}}$. This is also unexpected as the non-local contribution of the pressure field to the strain would be expected to have a smoothing influence. The explanation comes from the changing sign of the correlations between the non-normal strain rate and non-normal vorticity with height shown in figure 6(b), which combine to yield a sharp double-peaked profile in figure 2(c). In contrast, while different correlations peak at different heights for the association between normal strain and non-normal vorticity, these remain positive throughout the near-wall region, smoothing the profile.
The interaction between the normal part of the strain rate and the non-normal part of the vorticity captured in 
$\overline {\omega _{3}^{'\,C}S_{33}^{'\,B}\omega _{3}^{'\,C}}$ implicitly captures an important part of the model of Motoori & Goto (Reference Motoori and Goto2019) outlined in the introduction, where small vortices are oriented at ninety degrees to larger structures (see also Moffatt, Kida & Ohkitani Reference Moffatt, Kida and Ohkitani1994). Motoori & Goto stated (p.1103) that
‘
$\ldots$large-scale vortices are stretched and created by the mean flow and they tend to align with the mean shear
$\ldots$, whereas small-scale vortices apart from the wall are mainly stretched by the strain-rate field induced by large-scale vortices’.
Their figure 14 provides an example of such a phenomenon arising in the near-wall region. Fluctuations in the near-wall flow field in the plane of mean shear as a consequence of the action of larger vortices aligned with the mean shear will be captured in our formulation in the term 
$S_{12}^{'\,C}$. Naturally, this then introduces a non-normal component to the associated vorticity, 
$\omega _{3}^{'\,C}$, even though at this stage there may be no normal vorticity (the eigenvalues of the VGT are all real). What our results have shown is the strong coupling between this term and the transverse normal strain rate 
$S_{33}^{'\,B}$, with a correlation coefficient that remains above 0.2 for 
$y^+ < 60$ (figure 6a). In contrast, the correlation to 
$S_{33}^{'\,C}$ is less than 0.1 by 
$y^+ \sim 10$. The generation of a normal/local component to the strain rate means that its stretching action will then further increase the vorticity until a normal vortical component develops. It then follows that significant enstrophy production can begin to arise from this normal term. Table 1 shows that, while 
$\overline {\omega _{i}^{'\,C}S_{ij}^{'\,B}\omega _{j}^{'\,C}}$ makes up over 50 % of the enstrophy production for the whole near-wall region examined in this study (
$\kern 1.5pt y^+ < 70$), the normal fluctuating term, 
$\overline {\omega _{i}^{'\,B}S_{ij}^{'\,B}\omega _{j}^{'\,B}}$ increases from 2.3 % at 
$y^+ = 5$ to 18.6 % at 
$y^+ < 70$. It is this latter term that is the one that features in the definition of the third invariant of the VGT, as shown in (2.7). Hence, analysis of the third invariant neglects the near-wall mechanism that is the focus on this study.
In their recent review of the field, Johnson & Wilczek (Reference Johnson and Wilczek2024) describe the work of Tom, Carbone & Bragg (Reference Tom, Carbone and Bragg2021) as a case where an attempt has been made to provide a detailed analysis of the terms associated with the decomposition of the VGT into the restricted Euler, viscous and deviatoric pressure Hessian contributions. In that work, the terms were projected into the strain rate (i.e. 
$\boldsymbol {S}_{A}$) eigenframe and were considered as a function of a filter length, 
$L_{f}$, a scale-dependent time frame and dissipation rate for scales ranging from seven Kolmogorov scales, 
$\eta$, to a few hundred 
$\eta$, with the integral scale at 812
$\eta$. With the exception of the viscous terms, the variance of which decayed by four orders of magnitude, the variance of most other terms decayed by an order of magnitude over this scale range, while the mean values typically decayed for 
$7 < L_{f}/\eta < 30$ and then remained relatively constant. The enstrophy production term associated with the intermediate strain eigenvector was the term that decayed the most, implying a reduction in the alignment between the vorticity vector and the intermediate strain eigenvector with increasing scale (Danish & Meneveau Reference Danish and Meneveau2018).
At larger scales than those considered by Tom et al. (Reference Tom, Carbone and Bragg2021) the non-Gaussian nature of the joint probability distribution function of 
$Q$ and 
$R$ is suppressed (Naso & Pumir Reference Naso and Pumir2005) and this has implications for this work as the redistribution of the mass of the joint distribution function for 
$Q$ and 
$R$ towards the left Vieillefosse tail at larger scales (i.e. 
$R < 0$, 
$Q^{3} + \frac {27}{4}R^{2} \sim 0$) increases the probability of vorticity gaining a normal component. In a boundary-layer context, where a larger scale implies greater distance from the wall, this effect implies the increasing importance of fluctuating normal enstrophy production with greater distance from the wall, as seen in the first column of table 1. Given the need to condition any similar analysis to that of Tom et al. (Reference Tom, Carbone and Bragg2021) on distance from the wall, little work has so far taken a similar approach to the multi-scale analysis of the VGT in a boundary layer. In addition, while orientation into the strain eigenframe standardizes the analysis for all scales, it makes analysing the relative orientation of flow features at one scale conditioned on those at smaller scales rather complicated.
A further complexity with a boundary-layer analysis compared with HIT is the difference between behaviour that is affected directly by the thickness of the viscous/transitional sub-layers, the intermediate-scale behaviour that scales with the neighbouring vortices of double the size (Motoori & Goto Reference Motoori and Goto2019) and larger-scale behaviour that is coupled to the mean gradient. We have focused on establishing the small-scale mechanisms for the first of these in this study. However, we note that the largest observed fluctuations in the strain rate field were those aligned with the direction of mean shear, and the mean shear would induce a large-scale vorticity of a similar sense to that found here, but in the normal part, i.e. 
$\omega _{3}^{'\,B}$ rather than 
$\omega _{3}^{'\,C}$. Hence, we would again expect the normal vorticity to play a greater role away from the wall, as seen in table 1.
Our results have implications for near-wall flow control and drag reduction because, if one wishes to re-laminarize a flow by damping vortex formation, it would seem that a key process to disrupt is the coupling between 
$S_{33}^{'\,B}$ and 
$\omega _{3}^{'\,C}$, which we show in this paper is key to enstrophy production at the wall. That these are transverse components to the strain rate and vorticity provides support, from a very different starting point, for the work proposing that near-wall drag reduction is best achieved by an actuator that acts on the spanwise flow component (e.g. Quadrio Reference Quadrio2011; Ricco, Skote & Leschziner Reference Ricco, Skote and Leschziner2021). An experimental study of this mechanism by Marusic et al. (Reference Marusic, Chandran, Rouhi, Fu, Wine, Holloway, Chung and Smits2021) found that it operated by disrupting the near-wall streaks and functioned at all Reynolds numbers. The amplitude spectrum of the wall shear stress at 
$\mbox {Re}_{\tau } \in \{10^{3}, 10^{4}, 10^{5}\}$ had a peak at 
$T^+ \equiv u_{\tau }/(f\nu ) \sim 80$ for the unperturbed flow, where 
$f$ is the frequency. Actuation of these scales at 
$T_{osc}^+ = 140$ reduced the energy peak by 
$\sim 60\,\%$ and moved the energy content to smaller time scales. Given that the coupling between 
$S_{33}^{'\,B}$ and 
$\omega _{3}^{'\,C}$ drives the small-scale enstrophy production, it would seem that disrupting this coupling is relevant to small-scale drag reduction. However, as the large scales gain energy at higher Reynolds numbers, their footprint begins to be felt more acutely in terms of the modulation of the smaller scales (Marusic, Mathis & Hutchins Reference Marusic, Mathis and Hutchins2010; Keylock et al. Reference Keylock, Ganapathisubramani, Monty, Hutchins and Marusic2016). This non-local effect might be hypothesized to increase the importance of the contribution to enstrophy production from the purely non-normal term. In which case, the form of decomposition adopted in this study would map into the physical differences between small- and large-scale activation found by Marusic et al. (Reference Marusic, Chandran, Rouhi, Fu, Wine, Holloway, Chung and Smits2021). This may be a fruitful avenue for future research to explore as higher Reynolds numbers become accessible to simulation.
Acknowledgements
The author is extremely grateful to Professor S. Goto for hosting him in Osaka and to Professor Goto and Dr Y. Motoori for discussions about this work.
Funding
The author was funded by a Japan Society for the Promotion of Science Bridge Fellowship and Leverhulme International Fellowship 2023-014.
Declaration of interests
The author reports no conflict of interest.
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