2.1. Adjoint representation of vorticity

We start from the Helmholtz equation that governs the forward evolution of vorticity,

(2.1) \begin{eqnarray} \frac {\partial \boldsymbol {\omega }}{\partial t} + \boldsymbol {u}\boldsymbol \cdot \boldsymbol {\nabla }\boldsymbol {\omega }-\boldsymbol {\omega }\boldsymbol\cdot \boldsymbol {\nabla } \boldsymbol {u}-\nu {\nabla}^2 \boldsymbol {\omega }= \boldsymbol {0}, \end{eqnarray}

where $\boldsymbol {\omega } = \boldsymbol {\nabla } \times \boldsymbol {u}$ is the vorticity and $\boldsymbol {u}$ is the velocity, and $\nu$ is the fluid kinematic viscosity. This parabolic partial differential equation is not time-reversible due to the viscous term. As such, it cannot be solved backward in time to obtain vorticity at an earlier time from its final value at time $T$ . In addition, a standard approach to derive an adjoint to the forward system would start by introducing an additional evolution equation for the velocity. Rather than adopt this standard approach, we introduce a key assumption and derive an adjoint equation that will be shown to be equivalent to the stochastic Cauchy invariant. Specifically, we will freeze the forward trajectory of the velocity $\boldsymbol {u}(\boldsymbol {x},t)$ , or assume its knowledge. This step is different from the standard adjoint derivation where variations are considered for all state variables (Giles & Pierce Reference Giles and Pierce1997; Luchini & Bottaro Reference Luchini and Bottaro2014, Reference Luchini and Bottaro2024). Here, we are not concerned with variations in the vorticity due to change in the flow trajectory in state space. Instead, we are concerned with the ‘origin of vorticity’ backward in time, along a frozen trajectory in state space. This assumption was inspired by, and is fully consistent with, the stochastic formulation as will be shown in the next section.

With the above assumption, the standard procedure for deriving forward-adjoint duality yields the relation

(2.2) \begin{eqnarray} \begin{aligned} &\left (\boldsymbol {\Omega }, \frac {\partial \boldsymbol {\omega }}{\partial t}+\boldsymbol {u} \boldsymbol \cdot \boldsymbol {\nabla } \boldsymbol {\omega }-\boldsymbol {\omega } \boldsymbol \cdot \boldsymbol {\nabla } \boldsymbol {u}-\nu {\nabla}^2 \boldsymbol {\omega }\right ) =\left (\boldsymbol {\omega }, \frac {\partial \boldsymbol {\Omega }}{\partial \tau }-\boldsymbol {u} \boldsymbol \cdot \boldsymbol {\nabla } \boldsymbol {\Omega }-\boldsymbol {\nabla } \boldsymbol {u} \boldsymbol \cdot \boldsymbol {\Omega }-\nu {\nabla}^2 \boldsymbol {\Omega }\right ) \\ &\quad+\int _D \boldsymbol {\Omega }\left (\boldsymbol {x},T\right ) \boldsymbol \cdot \boldsymbol {\omega }\left (\boldsymbol {x},T\right ) {\textrm{d}}V -\int _D \boldsymbol {\Omega }\left (\boldsymbol {x},s\right ) \boldsymbol \cdot \boldsymbol {\omega }\left (\boldsymbol {x},s\right ) {\textrm{d}}V \\&\quad + \int _s^T \oint _{\partial D} \left [\left (\boldsymbol {\Omega } \boldsymbol \cdot \boldsymbol {\omega }\right ) \boldsymbol {u}\boldsymbol \cdot \boldsymbol {n} - \nu \left ( \left ( \boldsymbol {n}\boldsymbol \cdot \boldsymbol {\nabla } \boldsymbol {\omega } \right ) \boldsymbol \cdot \boldsymbol {\Omega } - \left ( \boldsymbol {n}\boldsymbol \cdot \boldsymbol {\nabla } \boldsymbol {\Omega }\right ) \boldsymbol \cdot \boldsymbol {\omega } \right )\right ] \, {\textrm{d}}S \, {\textrm{d}}t. \end{aligned} \end{eqnarray}

In the above expression, the adjoint variable is the vector $\boldsymbol {\Omega }$ , the space–time inner product is $ ( \boldsymbol {f}, \boldsymbol {g} ) = \int _s^T \int _D \boldsymbol {f}\boldsymbol \cdot \boldsymbol {g} \, {\textrm{d}}V \, {\textrm{d}}t$ , where ${\textrm{d}}V$ is the volume element in the flow domain $D,$ ${\textrm{d}}S$ is the area element on the boundary surface $\partial D,$ $\boldsymbol {n}$ is the outward pointing surface normal and $\tau = T-t$ is the reverse time. The derivation can be considered as a variational calculation, in which $\boldsymbol {\Omega }$ represents a field of Lagrange multipliers to enforce the Helmholtz equation. Variation over $\boldsymbol {\omega }$ then yields the evolution equation for the adjoint vorticity,

(2.3) \begin{equation} \frac {\partial \boldsymbol {\Omega }}{\partial \tau } - \boldsymbol {u}\boldsymbol \cdot \boldsymbol {\nabla }\boldsymbol {\Omega } - \boldsymbol {\nabla } \boldsymbol {u} \boldsymbol \cdot \boldsymbol {\Omega } -\nu {\nabla}^2 \boldsymbol {\Omega } = \boldsymbol {0}. \end{equation}

The remaining terms relate the vorticity field at the final time $T$ to the initial vorticity field at time $s$ and the vorticity boundary conditions over the interval $ [s,T ]$ :

(2.4) \begin{eqnarray} \begin{aligned} \int _D \boldsymbol {\Omega }\left (\boldsymbol {x},T\right ) \boldsymbol \cdot \boldsymbol {\omega }\left (\boldsymbol {x},T\right ) {\textrm{d}}V = & \int _D \boldsymbol {\Omega }\left (\boldsymbol {x},s\right ) \boldsymbol \cdot \boldsymbol {\omega }\left (\boldsymbol {x},s\right ) {\textrm{d}}V \,\,\,\\ & + \int _s^T \oint _{\partial D} \left [ \nu \left ( \left ( \boldsymbol {n}\boldsymbol \cdot \boldsymbol {\nabla } \boldsymbol {\omega } \right ) \boldsymbol \cdot \boldsymbol {\Omega } - \left ( \boldsymbol {n}\boldsymbol \cdot \boldsymbol {\nabla } \boldsymbol {\Omega }\right ) \boldsymbol \cdot \boldsymbol {\omega } \right )\right ]\,{\textrm{d}}S\,{\textrm{d}}t, \end{aligned} \end{eqnarray}

where we have discarded the term $\int _s^T \oint _{\partial D} (\boldsymbol {\Omega } \boldsymbol \cdot \boldsymbol {\omega } ) \boldsymbol {u}\boldsymbol \cdot \boldsymbol {n}\,{\textrm{d}}S {\textrm{d}}t$ because it vanishes at periodic boundaries, at no-penetration walls and at the far field if the velocity decays. To obtain a useful representation of vorticity from (2.4), we must impose additional initial and boundary conditions for $\boldsymbol {\Omega }$ .

We now denote as $\boldsymbol {\Omega }^k$ the particular adjoint field which enables us to trace back from the $k{\textrm {th}}$ component of the vorticity vector at a point in space $\boldsymbol {x}_f$ and time $T$ to its origin. In other words, $\boldsymbol {\Omega }^k$ will capture how earlier vorticity $\boldsymbol {\omega } (\boldsymbol {x},s )$ was tilted, stretched and diffused to generate $\omega _{k} (\boldsymbol {x}_{f},T )$ . This adjoint field is defined at $t=T$ according to

(2.5) \begin{equation} \boldsymbol {\Omega }^k\left (\boldsymbol {x},T\right ) = \boldsymbol {e}_k \delta \left (\boldsymbol {x}-\boldsymbol {x}_f\right ), \end{equation}

which, substituted in (2.4), yields the relation between the terminal and initial vorticity,

(2.6) \begin{eqnarray} \begin{aligned} \omega _k\left (\boldsymbol {x}_f,T \right ) =& \int _D \boldsymbol {\Omega }^k\left (\boldsymbol {x},s\right ) \boldsymbol \cdot \boldsymbol {\omega }\left (\boldsymbol {x},s\right ) {\textrm{d}}V \,\,\, \\ & + \int _s^T \oint _{\partial D} \nu \left ( \left ( \boldsymbol {n}\boldsymbol \cdot \boldsymbol {\nabla } \boldsymbol {\omega } \right ) \boldsymbol \cdot \boldsymbol {\Omega }^k - \left ( \boldsymbol {n}\boldsymbol \cdot \boldsymbol {\nabla } \boldsymbol {\Omega }^k\right ) \boldsymbol \cdot \boldsymbol {\omega } \right ) \,{\textrm{d}}S {\textrm{d}}t. \end{aligned} \end{eqnarray}

Hence, $\boldsymbol {\Omega }^k,$ $k=1,2,3$ are Eulerian vector fields that map the corresponding $k{\textrm {th}}$ component of vorticity to its origin in a viscous flow.

In the presence of solid boundaries, a choice of homogeneous Dirichlet $\boldsymbol {\Omega }^k=\boldsymbol {0}$ or homogeneous Neumann $\boldsymbol {n}\boldsymbol \cdot \boldsymbol {\nabla } \boldsymbol {\Omega }^k=\boldsymbol {0}$ boundary conditions can be adopted for the adjoint vorticity. The Dirichlet condition leaves a surface integral $- \int _s^T \oint _{\partial D} \nu ( \boldsymbol {n}\boldsymbol \cdot \boldsymbol {\nabla } \boldsymbol {\Omega }^k ) \boldsymbol \cdot \boldsymbol {\omega }\,{\textrm{d}}S\,{\textrm{d}}t$ , where the wall-normal flux of the adjoint samples the wall vorticity over time. Equation (2.6) then yields the solution at time $t=T$ of the Helmholtz equation (2.1), with initial data at time $t=s$ and Dirichlet boundary conditions for vorticity over the interval $[s,T].$ On the other hand, the zero Neumann condition leaves a surface integral $ \int _s^T \oint _{\partial D} \nu ( \boldsymbol {n}\boldsymbol \cdot \boldsymbol {\nabla } \boldsymbol {\omega } ) \boldsymbol \cdot \boldsymbol {\Omega }^k \,{\textrm{d}}S\,{\textrm{d}}t$ , where the adjoint field samples the diffusive influx of vorticity at the wall $\nu \boldsymbol {n}\boldsymbol \cdot \boldsymbol {\nabla } \boldsymbol {\omega }$ over time. In that case, (2.6) then yields the solution at time $t=T$ of the Helmholtz equation (2.1), with initial data at time $t=s$ and Neumann boundary conditions for vorticity over the interval $[s,T].$ Note that either boundary condition for the adjoint variable can be adopted for tracing back the origin of the vorticity within the bulk. However, only the Neumann version is compatible with the initial condition $\boldsymbol {\Omega }^k (\boldsymbol {x},T ) = \boldsymbol {e}_k \delta (\boldsymbol {x}-\boldsymbol {x}_{f} )$ for a point on the wall, and hence only the Neumann boundary condition can be adopted for tracing back the origin of the wall vorticity, or equivalently of the wall stress.

The representation (2.6) is not unique. Other duality relations can be derived starting from different forms of the vorticity equation, for example, by replacing the viscous term by $\nu \boldsymbol {\nabla }\times \boldsymbol {\nabla }\times \boldsymbol {\omega }$ and repeating the derivation. As we show in § 2.2, however, the representation (2.6) is distinguished by the fact that it is equivalent to the stochastic Lagrangian representation of Constantin & Iyer (Reference Constantin and Iyer2008). To facilitate the comparison, we gather the three adjoint fields for $k=1,2,3,$ interpreted as column vectors, to form the rows of a matrix $\unicode{x1D76E}= [{\boldsymbol \Omega }^1,{\boldsymbol \Omega }^2,{\boldsymbol \Omega }^3]^\top$ , which satisfies

(2.7) \begin{eqnarray} \frac {\partial \unicode{x1D76E}}{\partial \tau } - \boldsymbol {u}\boldsymbol \cdot \boldsymbol {\nabla }\unicode{x1D76E} - \unicode{x1D76E}(\boldsymbol {\nabla } \boldsymbol {u})^\top - \nu {\nabla}^2\unicode{x1D76E} = 0. \end{eqnarray}

Rewriting the duality relation (2.6), we see that $\unicode{x1D76E}$ acts on the initial vorticity vector and boundary sources to give the vorticity vector at the target position and time by the formula

(2.8) \begin{equation} \boldsymbol {\omega }\left (\boldsymbol {x}_f,T \right ) = \int _D {\unicode{x1D76E} }\left (\boldsymbol {x},s\right ) \boldsymbol {\omega }\left (\boldsymbol {x},s\right ) {\textrm{d}}V + \int _s^T \oint _{\partial D} \nu \left ( {\unicode{x1D76E} } \left ( \boldsymbol {n}\boldsymbol \cdot \boldsymbol {\nabla } \boldsymbol {\omega } \right ) - \left ( \boldsymbol {n}\boldsymbol \cdot \boldsymbol {\nabla } {\unicode{x1D76E} }\right ) \boldsymbol {\omega } \right )\,{\textrm{d}}S\,{\textrm{d}}t. \end{equation}

The physical interpretation of the Eulerian matrix $\unicode{x1D76E}$ is most important. The field $\unicode{x1D76E}(\boldsymbol {x},t)$ is the volume density of mean deformation experienced by vorticity from space–time point $(\boldsymbol {x},t)$ to the target point $(\boldsymbol {x}_f,T)$ , as will emerge from the proof of equivalence in the following section. In particular, for a smooth Euler solution obtained in the inviscid limit, the matrix $\unicode{x1D76E}$ coincides with the deformation tensor $\unicode{x1D63F}$ of standard continuum mechanics. Another instructive connection in the inviscid limit is to the evolution of an infinitesimal fluid element with volume $\delta V = \delta \boldsymbol {l} \boldsymbol \cdot \delta \boldsymbol {A}$ , where $\delta \boldsymbol {l}$ is the line element and $\delta \boldsymbol {A}$ is the area vector element. The forward vorticity vector satisfies the same evolution equation as $\delta \boldsymbol {l}$ , which is the foundation for our intuition regarding vorticity tilting and stretching. Each row of $\unicode{x1D76E}$ is an adjoint-vorticity vector $\boldsymbol {\Omega }$ , which satisfies the same evolution equation as $\delta \boldsymbol {A}$ (see (3.1.5) of Batchelor (Reference Batchelor2000), which is identical to (2.3) with $\tau =-t$ and $\nu =0$ ). Therefore, terms such as $ (-\boldsymbol {\nabla } \boldsymbol {u} \boldsymbol \cdot \boldsymbol {\Omega } )$ have an intuitive physical interpretation as stretching and tilting of the area vector element, i.e. enlarging and rotating the area.

In summary, $\unicode{x1D76E}$ represents the action of advection, stretching, tilting and also viscous diffusion on the initial vorticity and boundary source, which transforms them to produce the target vorticity vector. As we will see by aid of an example from channel flow in § 3, this interpretation permits one to understand intuitively the adjoint field and to connect its behaviour with the Lagrangian dynamics of the flow vorticity.

In § 2.2, we very succinctly review the stochastic Lagrangian representation and explain its connection with the adjoint-vorticity equations (2.7). The proof requires basic knowledge of stochastic calculus, and can be skipped without loss of continuity. The most important consequence is a more precise statement of the above physical interpretation, in terms of Lagrangian particle trajectories.

2.2. Equivalence with stochastic Lagrangian approach

In the forward Lagrangian description (Constantin & Iyer Reference Constantin and Iyer2008), a particle position $\widetilde {{\boldsymbol X}}_s^t({\boldsymbol a})$ depends on time $t$ and the particle label $\boldsymbol a$ , where the label can be defined in terms of the initial position $\widetilde {{\boldsymbol X}}_s^s({\boldsymbol a}) = {\boldsymbol a}$ at the initial time $s$ . In the backward time approach (Constantin et al. Reference Constantin2011), which is our current interest, one uses instead the back-to-label map $\widetilde {{\boldsymbol A}}_T^t({\boldsymbol x}_f)$ , which starts from the terminal particle position ${\boldsymbol x}_f$ at the terminal time $T$ and traces back to its original label $\boldsymbol a$ at the initial time $s$ . The forward and inverse satisfy $\widetilde {{\boldsymbol X}}_t^T\circ \widetilde {{\boldsymbol A}}_T^t=\textrm { Id}$ , where Id is the identity map. In presence of viscosity, this back-to-label map can be computed by solving the backward It $\bar {\textrm o}$ stochastic differential equation (SDE) for stochastic Lagrangian particle trajectories,

(2.9) \begin{equation} \hat {d}\widetilde {{\boldsymbol A}}_T^t({\boldsymbol x}_f)={\boldsymbol u}(\widetilde {{\boldsymbol A}}_T^t({\boldsymbol x}_f),t){\textrm{d}}t +\sqrt {2\nu }\,\hat {\textrm{d}}\widetilde {{\boldsymbol W}}(t), \quad s\lt t\lt T; \qquad \widetilde {{\boldsymbol A}}_T^T({\boldsymbol x}_f) ={\boldsymbol x}_f, \end{equation}

with ${\textrm{d}}t\lt 0$ and where $\widetilde {\boldsymbol W}$ is Brownian motion. A few such trajectories are plotted in figure 1. Along these trajectories, a material element undergoes tilting and stretching which can be encoded in the deformation matrix,

(2.10) \begin{align}\widetilde{\kern-2pt \unicode{x1D63F}}_T^{\,t}({\boldsymbol x}_f):=\left . ({\boldsymbol \nabla _a}\widetilde {{\boldsymbol X}}^T_t({\boldsymbol a}))^\top \right |_{{\boldsymbol a}=\widetilde {{\boldsymbol A}}_T^t({\boldsymbol x}_f)} =({\boldsymbol \nabla _{x_f}}\widetilde {{\boldsymbol A}}_T^t({\boldsymbol x}_f))^{-\top }, \end{align}

and is computed by solving the ordinary differential equation (ODE),

(2.11) \begin{equation} d\,\widetilde{\kern-2pt \unicode{x1D63F}}_T^{\,t}({\boldsymbol x}_f)=-\,\widetilde{\kern-2pt \unicode{x1D63F}}_T^{\,t}({\boldsymbol x}_f) \boldsymbol \cdot (\boldsymbol {\nabla u})^\top (\widetilde {{\boldsymbol A}}_T^t({\boldsymbol x}_f),t){\textrm{d}}t, \quad s\lt t\lt T \end{equation}

backward in time. We now have all the ingredients to relate vorticity at the terminal position and time $({\boldsymbol x}_f,T)$ to its back-in-time origin,

(2.12) \begin{equation} {\boldsymbol \omega }({\boldsymbol x}_f,T) ={\mathbb E}\left [\widetilde{\kern-2pt \unicode{x1D63F}}_T^{\,s}({\boldsymbol x}_f) {\boldsymbol \omega }(\widetilde {{\boldsymbol A}}_T^s({\boldsymbol x}_f),s)\right ]. \end{equation}

The above expression describes the terminal vorticity as the expectation over back-in-time stochastic Lagrangian particles that all emanate from ${\boldsymbol x}_f$ at time $T$ . It is interesting to note that the above expectation is independent of the initial time $s\lt T$ . In fact, the random variable in the expectation has been shown to be a backward martingale, or a statistically conserved quantity backward in time. It has therefore been dubbed the stochastic Cauchy invariant (Eyink et al. Reference Eyink, Gupta and Zaki2020a ), since it generalises the invariant of Cauchy (Reference Cauchy1815) for Euler solutions to viscous Navier–Stokes solutions.

The stochastic Lagrangian representation using backward evolution in time and the adjoint representation of vorticity can be shown to be exactly equivalent. We start by rewriting the above expectation as

(2.13) \begin{equation} {\boldsymbol \omega }({\boldsymbol x}_f,T) =\int {\mathbb E}\left [\widetilde{\kern-2pt \unicode{x1D63F}}_T^{\,s}({\boldsymbol x}_f) \delta ^3\left (\boldsymbol {x}-\widetilde {{\boldsymbol A}}_T^s({\boldsymbol x}_f)\right ) {\boldsymbol \omega }(\boldsymbol {x},s)\right ] \mathrm{d}^3\boldsymbol {x}. \end{equation}

Since the vorticity is no longer a stochastic variable in this representation, we can move it outside the expectation,

(2.14) \begin{equation} {\boldsymbol \omega }({\boldsymbol x}_f,T) =\int {\mathbb E}\left [\widetilde{\kern-2pt \unicode{x1D63F}}_T^{\,s}({\boldsymbol x}_f) \delta ^3\left (\boldsymbol {x}-\widetilde {{\boldsymbol A}}_T^s({\boldsymbol x}_f)\right )\right ] {\boldsymbol \omega }(\boldsymbol {x},s) \mathrm{d}^3\boldsymbol {x}. \end{equation}

Comparing the above form to the duality relation in the adjoint approach, we define the adjoint field using the backward stochastic flow as

(2.15) \begin{eqnarray} {\unicode{x1D76E} }({\boldsymbol x},t) := {\mathbb E}\left [\widetilde{\kern-2pt \unicode{x1D63F}}_T^{\,t}({\boldsymbol x}_f) \delta ^3({\boldsymbol x}-\widetilde {{\boldsymbol A}}_T^t({\boldsymbol x}_f))\right ], \end{eqnarray}

so that the stochastic representation (2.14) is rewritten as

(2.16) \begin{equation} {\boldsymbol \omega }({\boldsymbol x}_f,T)= \int _D {\unicode{x1D76E} }({\boldsymbol x},s) {\boldsymbol \omega }({\boldsymbol x},s)\, \mathrm{d}^3x, \quad s\lt T. \end{equation}

This expression coincides with the adjoint representation (2.8) derived in the previous section, in the absence of solid boundaries.

To derive the evolution equation for $\unicode{x1D76E}$ , we evaluate the time derivative of (2.15),

(2.17) \begin{eqnarray} \partial _t {\unicode{x1D76E} }({\boldsymbol x},t){\textrm{d}}t = {\mathbb E}\left [ \partial _t\,\widetilde{\kern-2pt \unicode{x1D63F}}_T^{\,t} \delta ^3({\boldsymbol x}-\widetilde {{\boldsymbol A}}_T^t)\right ]{\textrm{d}}t + {\mathbb E}\left [\widetilde{\kern-2pt \unicode{x1D63F}}_T^{\,t} \hat {d}_t \delta ^3({\boldsymbol x}-\widetilde {{\boldsymbol A}}_T^t)\right ]. \end{eqnarray}

The first time-derivative $\partial _t\,\widetilde{\kern-2pt \unicode{x1D63F}}_T^{\,t}$ on the right-hand side is given by (2.11), and the backward It $\bar {\textrm o}$ rule in used for the second term,

(2.18) \begin{eqnarray} \hat {d}_t \delta ^3({\boldsymbol x}-\widetilde {{\boldsymbol A}}_T^t) = -(\hat {d}_t\widetilde {{\boldsymbol A}}_T^t{\boldsymbol \cdot }{\boldsymbol \nabla }_{\boldsymbol {x}}) \delta ^3({\boldsymbol x}-\widetilde {{\boldsymbol A}}_T^t) -\nu {\nabla}^2_{\boldsymbol {x}} \delta ^3({\boldsymbol x}-\widetilde {{\boldsymbol A}}_T^t) {\textrm{d}}t. \end{eqnarray}

Substituting in (2.17), we obtain using (2.9) and (2.11) that

(2.19) \begin{equation} \partial _t{\unicode{x1D76E} } = - \unicode{x1D76E}(\boldsymbol {\nabla } \boldsymbol {u})^\top -\boldsymbol {u}\boldsymbol \cdot \boldsymbol {\nabla }\unicode{x1D76E} - \nu {\nabla}^2 \unicode{x1D76E}, \end{equation}

which coincides with the adjoint vorticity equation (2.7) after switching to reversed time $\tau =T-t.$

Equation (2.15) is a fundamental result of this section, which provides the direct link between the adjoint and stochastic Lagrangian representations, and gives the precise physical intepretation of the adjoint vorticity field $\unicode{x1D76E}(\boldsymbol {x},t)$ , as noted earlier, as the volume density of mean Lagrangian deformation experienced by vorticity from space–time point $(\boldsymbol {x},t)$ to the target point $(\boldsymbol {x}_f,T)$ .

The equivalence between the adjoint formulation of § 2.1 and the stochastic Lagrangian representation applies also in wall-bounded flows. The Dirichlet and Neumann conditions on the adjoint, introduced in the previous section, are equivalent to stochastic counterparts. In the Dirichlet case, the stochastic Lagrangian process stops at the wall (Constantin et al. Reference Constantin2011; Eyink et al. Reference Eyink, Gupta and Zaki2020a ), while in the Neumann case, the stochastic Lagrangian process reflects at the wall (Wang et al. Reference Wang, Eyink and Zaki2022a ). The details are provided in Appendix A. This equivalence enhances the importance of both methods. The adjoint vorticity field now gains an intuitive Lagrangian interpretation, whereas the stochastic representation gains a partial differential equation (PDE) implementation which is much more computationally efficient than direct Monte Carlo averaging over Lagrangian trajectories.

3.1. Comparison of Dirichlet and Neumann conditions

For our first demonstration of the back-in-time tracking of vorticity using the adjoint, we will examine a single stress maxima and then proceed to report results from the ensemble of 109 cases. To compare the choice of Dirichlet versus Neumann boundary conditions on the adjoint variable, we must consider a point within the bulk of the fluid since only the Neumann condition is compatible with the initial condition $\boldsymbol {\Omega }^k (\boldsymbol {x},T ) = \boldsymbol {e}_k \delta (\boldsymbol {x}-\boldsymbol {x}_{f} )$ sampling a point on the wall. We therefore initialise the adjoint variable above the location of the stress maximum, at a height $y^+_f=5$ . The particle stress maximum that we consider here is marked by the red circle in figure 2( $b$ ).

Figure 3. Back-in-time contributions to the target spanwise vorticity $\omega _z^* = \omega _z(\boldsymbol {x}_f, T)$ for $(a)$ Dirichlet and $(b)$ Neumann boundary condition on the adjoint. The starting location is at $y_f^+=5$ , above the wall-stress maximum shown in figure 2 $(b)$ marked by a red circle. ( ) The total vorticity is evaluated from (2.8) and compared with ( ) the reference value $\omega _z^*$ . Internal and boundary contributions: $\mathcal {I}_x^z$ ; $\mathcal {I}_y^z$ ; $\mathcal {I}_z^z$ ; $\mathcal {I}^z$ ; $\,.\,.\,.\,.\,\mathcal {B}^z$ .

In figure 3, we report the contributions to the dominant spanwise-vorticity at the target point, by evaluating the terms in the integral (2.6) with $k=z$ ,

(3.1) \begin{eqnarray} \begin{aligned} \omega _z\left (\boldsymbol {x}_f,T \right ) =& \int _D \boldsymbol {\Omega }^z\left (\boldsymbol {x},s\right ) \boldsymbol \cdot \boldsymbol {\omega }\left (\boldsymbol {x},s\right ) {\textrm{d}}V \,\,\, \\ & + \int _s^T \oint _{\partial D} \nu \left ( \left ( \boldsymbol {n}\boldsymbol \cdot \boldsymbol {\nabla } \boldsymbol {\omega } \right ) \boldsymbol \cdot \boldsymbol {\Omega }^z - \left ( \boldsymbol {n}\boldsymbol \cdot \boldsymbol {\nabla } \boldsymbol {\Omega }^z\right ) \boldsymbol \cdot \boldsymbol {\omega } \right ) \,{\textrm{d}}S {\textrm{d}}t. \end{aligned} \end{eqnarray}

The time horizon is chosen to be $T=30$ (or $T^+=347$ ), which is equivalent to 1.92 large-eddy turnover times $h/u_{\tau }$ . The two panels correspond to the Dirichlet and Neumann boundary conditions. The horizontal grey line marks the value of the target spanwise vorticity at $\omega _z(\boldsymbol {x}_f,T) = -13.5$ . The black solid line shows the sum of the right-hand side terms, plotted as a function of the reverse time $\tau =T-t$ preceding the event. The constancy of this value is a consequence of adjoint duality and the observed horizontal line demonstrates the accuracy of our numerical method.

In addition, however, the reconstruction by the above formula gives precise, detailed information about the origin of the vorticity. In figure 3, we have divided the target spanwise vorticity into three interior contributions:

(3.2) \begin{equation} \mathcal {I}^z_i(s) = \int _D \Omega _i^z(\boldsymbol {x},s) \, \omega _i(\boldsymbol {x},s)\, {\textrm{d}}V, \qquad i=x,y,z \end{equation}

that result from twisting/tilting of vorticity from the $i=x,y$ directions and stretching of vorticity in the $i=z$ direction, at time $s\lt T$ . We also report in the figures the boundary contribution for Dirichlet ( $D$ ) or Neumann ( $N$ ) conditions:

(3.3) \begin{align} \mathcal {B}_D^z(s) &= -\int _s^T\oint _{\partial D} \nu \frac {\partial \boldsymbol {\Omega }^z}{\partial n}\boldsymbol \cdot \boldsymbol {\omega }\,{\textrm{d}}S\,{\textrm{d}}t = \int _s^T\oint _{\partial D} B_D^z(s) \,{\textrm{d}}S\,{\textrm{d}}t, \end{align}

(3.4) \begin{align} \mathcal {B}_N^z(s) &=\int _s^T\oint _{\partial D} \nu \frac {\partial \boldsymbol {\omega }}{\partial n} \boldsymbol \cdot \boldsymbol {\Omega }^z \,{\textrm{d}}S\,{\textrm{d}}t, =\int _s^T\oint _{\partial D} B_{N}^z(s) \,{\textrm{d}}S\,{\textrm{d}}t, \end{align}

arising from either the wall vorticity or its flux, respectively, over the considered time interval $[s,T]$ . To indicate partial integrals of any quantity $\mathcal {C}$ over variables $A\in \{x,z,t\}$ , we will adopt the notation $\overline {\mathcal {C}}^{A}$ , etc.

The various contributions for the Dirichlet boundary condition are plotted in figure 3 $(a)$ . There is a very substantial increase of the wall contribution over approximately twenty viscous time units. This back-in-time abrupt return to the wall corresponds in the forward evolution to ‘abrupt lifting’ (Sheng et al. Reference Sheng, Malkiel and Katz2009). However, this process contributes only approximately 50 % of the target vorticity. The rest arises from the wall only after a much longer interval of several hundred viscous time units, corresponding to a slow diffusion process. The interior contribution at intermediate times arises almost entirely from $\Omega _z^z,$ indicated by the red dashed line, which corresponds to the spanwise stretching of pre-existing spanwise vorticity. By contrast, tilting and stretching make negligible contributions. These results are consistent with the high-stress event analysed by Eyink et al. (Reference Eyink, Gupta and Zaki2020b ) numerically using the stochastic Lagrangian approach. However, with the Monte Carlo algorithm of that earlier study, conservation within a few percent was possible for only approximately 100 viscous time units, even averaging over $N=10^7$ sample paths. In addition, the number of samples required for accurate reconstruction grew exponentially in backward time, so that integrating further was prohibitively expensive. Here, we obtain higher accuracy further back in time for much lower computational cost.

The various contributions in the case of Neumann boundary conditions are plotted in figure 3 $(b)$ . The contribution from the wall is now much less significant, and grows very slowly and non-monotonically. Even after $300$ viscous time units, the wall contributes only 26 % of the total. Most importantly, the dominant contribution arises from interior vorticity. Just as for the Dirichlet case, this interior contribution arises very predominantly from $\Omega _z^z,$ which corresponds to the spanwise stretching of pre-existing spanwise vorticity. A key difference, however, is that spanwise stretching now accounts for the majority of the total vorticity rather than the wall term. It is perhaps important to underscore that this result is the first of its kind in two ways. First, earlier analysis of the Neumann condition using the stochastic Lagrangian approach has only been attempted in a transitional boundary layer, to determine the origin of skin-friction increase at the onset of turbulence spots (Wang et al. Reference Wang, Eyink and Zaki2022a ), and has never been applied in the fully turbulent regime. Second, the importance of the spanwise stretching on internal vorticity is precisely the mechanism suggested by Lighthill (Reference Lighthill1963) for concentration and magnification of spanwise vorticity in near-wall turbulence. Whether this behaviour is statistically typical, beyond this single analysed event, is a question that we will address later in this section.

Figure 4. Iso-surfaces of $\Omega _{z}^z/max |\Omega _{z}^{z}|$ as a function of backward time at $\tau =\{0.4, 1.4, 3.4\}$ for $(a)$ Dirichelet and $(b)$ Neumann boundary conditions. The line contours are the wall values for $(a)$ $\nu (\partial \Omega _{z}^z / \partial y)$ and $(b)$ $\Omega _{z}^z$ . The vertical axes are stretched for clarity of the visualisation.

For a detailed view of the space–time origin of vorticity, we plot in figure 4 iso-surfaces of the adjoint field $\Omega _z^z$ and iso-contours of the wall terms ( $\nu \partial \Omega _z^z/\partial y$ for Dirichlet, $\Omega _z^z$ for Neumann). The figure shows three time instances together, $\tau =\{0.4, 1.4, 3.4\}$ , which are selected to be within the interval of fast changes in the contributions in the Dirichlet case (see figure 3). The expanding iso-surfaces in backward time represent the spreading of the adjoint field, which accounts for the accumulated stretching rate of earlier vorticity as it is transported by advection and viscous diffusion. The level sets of the wall terms clearly lag behind those of the interior. This observation is easily understood by the faster streamwise fluid velocities at greater distances from the wall. The most prominent difference between the two figures is related to the interior contribution. While the iso-surfaces appear similar away from the wall, they are qualitatively different in the near-wall region. Specifically, the iso-surfaces of the Neumann adjoint have a large lobe near the wall, which is entirely missing for the Dirichlet case. This difference is intuitive since the Neumann condition is akin to an adiabatic condition while the Dirichlet counterpart annihilates the adjoint at the wall. In the stochastic formulation, the former condition reflects the particles when they hit the wall, whereas the latter absorbs the particles and hence depletes the adjoint field.

The total contribution to the target vorticity arises, however, not only from the adjoint field, but also from its products with the interior vorticity and with the wall vorticity or its flux. These initial vorticities and wall conditions are advected and diffused, stretched and rotated to the target point, where they are fused by viscosity to yield the resultant vorticity. These fluid dynamical processes are all precisely represented by the adjoint vorticity, or density of the deformation field. Figures 5–9 illustrate how these processes operate in the specific high-wall-stress event discussed thus far.

The Dirichlet case is shown first in figures 5 and 6, for the two backward times $\tau =1.4$ and $\tau =3.4,$ respectively. The target point is at $y^+=5$ and its horizontal coordinates are taken as the origin in the wall $xz$ -plane. Panels $(a)$ of both figures focus on the interior contribution to the target vorticity. The three panels show a side view at the target location: (a-i)colour contours of the spanwise vorticity $\omega _z$ and lines for $\Omega _z^z$ ; (a-ii)a zoomed-in view of $\Omega _z^z$ and (a-iii) their instantaneous product $\Omega _z^z\omega _z$ . The last term, once integrated over volume, is the largest interior contribution to the target vorticity even if it is decaying while the wall value is increasing. To interpret these results, it is helpful to recall that $\Omega _z^z$ is the volume density of mean spanwise stretching. The colour contours in panels (ii) for the two times show clearly that the adjoint backward in time samples regions further upstream and vertically higher. The vorticity $\omega _z,$ however, is strongly concentrated near the wall, with narrow tongues ejected into the interior. Thus, the product $\Omega _z^z\omega _z$ arises also mainly from the near-wall region, although the contours are attenuated because of the Dirichlet condition on $\Omega _z^z$ .

Panels $(b)$ of figures 5 and 6 relate to the contribution from the wall vorticity. The three panels show a top view in the wall plane: (b-i)colour contours of the spanwise vorticity $\omega _z$ and lines for $\nu \partial \Omega _z^z/\partial y$ ; (b-ii)a zoomed-in view of $\nu \partial \Omega _z^z/\partial y$ and (b-iii) the time integral of their product over the interval $[T-\tau ,T]$ , which represents the cumulative contribution to the target vorticity per area. Recall that the quantity in panels (b-ii) represents the density per area and time of mean deformation of wall vorticity, which ultimately reaches the target location. The figures clearly illustrate that the adjoint samples the wall farther upstream backward in time. However, the product field which is the boundary contribution shown in panels (b-iii) is retarded relative to the adjoint field and to the interior contribution in panels (a-iii), because it is cumulative and is dominated by early times when the adjoint field first reaches the wall.

Figure 5. Contributions to the terminal vorticity at $\tau =1.4$ , for Dirichlet boundary conditions. (a-i) Side view showing colour contours of the spanwise vorticity and line contours of the adjoint $\Omega _{z}^{z}$ . (a-ii) Zoomed-in view with colours showing $\Omega _{z}^{z}$ . (a-iii) Contribution to the terminal vorticity by spanwise stretching $\Omega _{z}^{z}\omega _z$ . These side-view plots are all vertically stretched. (b-i) Top view of the wall, showing colour contours of the spanwise vorticity and lines contours of $\nu \partial \Omega _{z}^{z} / \partial y$ . (b-ii) Zoomed-in view with colours showing $\nu \partial \Omega _{z}^{z} / \partial y$ . (b-iii) Boundary contribution $\overline {B_D^z}^t$ .

Figure 6. Same as figure 5, except showing the contributions to terminal vorticity at $\tau =3.4$ , and the contours levels are adjusted as marked.

Figure 7. Contributions to the terminal vorticity at $\tau =1.4$ , for Neumann boundary conditions. (a-i) Side view showing colour contours of the spanwise vorticity and lines contours of the adjoint $\Omega _{z}^{z}$ . (a-ii) Zoomed-in view with colours showing $\Omega _{z}^{z}$ . (a-iii) Contribution to the terminal vorticity by spanwise stretching $\Omega _{z}^{z}\omega _z$ . These side-view plots are all vertically stretched. (b-i) Top view of the wall, showing colour contours of $-\nu \partial \omega _{z} / \partial y$ and line contours of $\Omega _{z}^{z}$ . (b-ii) Zoomed-in view with colours showing $\Omega _{z}^{z}$ . (b-iii) Boundary contribution $\overline {B_N^z}^t$ .

Figure 8. Same as figure 7, except showing the contributions to the terminal vorticity at $\tau =3.4$ , and the contours levels are adjusted as marked.

The analogous plots in figures 7 and 8 for the Neumann case show the same event and the same times, for direct comparison. Panels $(a)$ in the two new figures show the analogous interior quantities. The interior contributions may appear similar to those for Dirichlet boundary conditions except, as already noted, there is a much larger near-wall region due to the adiabatic condition. As a result, the interior contribution for the Neumman case is much larger and is in fact the overall dominant term as reported in the overall duality balance (see figure 3). The boundary contribution in the Neumann case is particularly interesting since it is related to the vorticity flux at the wall. The relevant quantities are reported in panels (b) of figures 7 and 8: (b-i) colour contours of the Lighthill vorticity source $(-\nu \partial \omega _z/\partial y)$ and lines for the adjoint $\Omega _z^z$ ; (b-ii) a zoomed-in view of $\Omega _z^z$ and (b-iii) the time integral of their product over the interval $[T-\tau ,T]$ , which again represents the cumulative contribution to the target vorticity per area. The Lighthill source in panels (b-i) is notable in having values of both positive and negative sign, distinctly different from the spanwise vorticity itself which is almost always the same sign as the mean vorticity. Since at the wall, $(-\nu \partial \omega _z/\partial y)=\partial p/\partial x$ , this quantity has a negative mean value in the pressure-driven channel flow. However, this mean is more than an order of magnitude smaller than the fluctuations and its relative magnitude decreases proportional to $ 1/Re_\tau$ . This very small mean pressure gradient was already argued by Lighthill (Reference Lighthill1963) to be unable to account for the strong near-wall concentration of spanwise vorticity, writing that ‘even in an accelerating flow, the rate of production… is too small’ (p. 98). The argument by Lighthill (Reference Lighthill1963) is not entirely compelling, however, because it does not take account of the very large fluctuating values of the pressure gradient. Nevertheless, panels (b-i,b-ii) show that the adjoint samples vorticity sources of both signs and the cumulative contribution may suffer extensive cancellation, and hence the small total boundary contribution in figure 3 at early times. Notice, in fact, that for this particular example, the boundary contributions at both times, $\tau =\{1.4, 3.4\}$ , have net positive sign which is opposite to the target vorticity.

Figure 9. Comparison of the internal contributions to the terminal vorticity at the large backward time $\tau = 10$ , using $(a)$ Dirichlet and $(b)$ Neumann boundary conditions. (i) Side view showing contours of spanwise vorticity, overlaid by lines of the adjoint $\Omega _{z}^{z}$ . Solid and dashed line contours represent positive and negative values of $\Omega _z^z$ , respectively. (ii) Contours of the contribution of vorticity stretching $\Omega _{z}^{z} \omega _{z}$ . The plane is located at the spanwise location where the target vorticity is sampled. These side-view plots are all vertically stretched.

In figure 9, we directly compare the interior contributions using Dirichlet and Neumann boundary conditions for the same event, at an earlier time $\tau =10$ . At this time, the target vorticity in the Dirichlet case is almost entirely from the wall, while for the Neumann case, the majority contribution remains due to the interior. In panels (i) of these figures, the spanwise vorticities $\omega _z$ are the same, but the level sets of $\Omega _z^z$ , while very similar far from the wall, show that the Neumann adjoint field samples the near-wall region much more densely. Because this region is also the site of the strongest magnitudes of $\omega _z$ , the volumetric contributions $\Omega _z^z\omega _z$ plotted in panels (ii) show much larger values for the Neumann case and is heavily localised near the wall.

Returning to figure 9(i), we may note that, at these early times, some of the contours of $\Omega _z^z$ are negative, which can be explained by adjoint chaos in the buffer layer (Wang et al. Reference Wang, Wang and Zaki2022b ): the adjoint vorticity is advected, tilted and stretched by the turbulent velocity field. Furthermore, to make our observations regarding the adjoint fields near the wall more quantitative, we plot in figure 10 the $L^1$ -norms $\int _{S(y)}|\Omega _z^z|\,{\textrm{d}}x\,{\textrm{d}}z$ as functions of $y^+$ and backward time $\tau =T-t$ . These show a vertical spreading in $y$ , for both the Dirichlet and Neumann boundary conditions, as back-in-time advection extends the adjoint field upward. There is also a strengthening of magnitudes with $\tau$ , associated with the accumulated stretching. Most importantly, while the Dirichlet condition depletes the near-wall adjoint field, the Neumann condition preserves the near-wall magnitudes, which corresponds to a high density of stretching in this region. For this reason, the near-wall stretching of internal vorticity remains the dominant term in the overall makeup of the terminal vorticity (see figure 3).

Thus far, we have considered a particular high-stress event, and examined it in detail using the adjoint fields with Dirichlet and Neumann boundary conditions. Whether the reported trends are statistically robust is examined in the next section.

Figure 10. Comparison of horizontally integrated $|\Omega _{z}^{z}|$ as a function of backward time, for $(a)$ Dirichlet and $(b)$ Neumann boundary conditions.

3.2. Ensemble analysis of spanwise vorticity above stress maxima

In the remainder of this section, we analyse the entire ensemble of 109 local stress-maxima that were identified. To continue the comparison of the Dirichlet and Neumann adjoint boundary conditions, we must examine a point in the interior of the fluid domain. We therefore continue the analysis of the spanwise vorticities at $y^+=5$ , above these local wall maxima. Our aim here is to identify generic features in the development of such strong near-wall vorticities. We will then exploit the Neumann condition, which can be applied for the analysis of vorticity points that are directly on the wall, to study the origin of the extrema in the spanwise wall shear stress.

Figure 11. Ensemble of 109 target vorticity events, at $y^+=5$ above a wall-stress maximum, tracked back in time using the Dirichlet condition. Grey lines are the $(a)$ interior vorticity stretching ${\mathcal I}^z_z=\Omega _{z}^{z} \omega _{z}$ and $(b)$ boundary contributions to the target vorticity, normalised by the target values $\omega _z^* = \omega _z(\boldsymbol {x}_f, T)$ . () Ensemble-averaged value; () $\pm$ the standard deviation.

Figure 12. Same as figure 11, except that the target vorticity is tracked back in time using the Neumann condition.

We begin by plotting in figure 11, as a function of reverse time $\tau$ , the two largest fractional contributions to the target vorticity when the Dirichlet boundary condition is adopted. In panel $(a)$ , we report the spanwise stretching of interior spanwise vorticity, $\mathcal {I}_z^z$ as defined in (3.2) for $i=z$ , and in panel $(b)$ , we plot the wall-vorticity contribution, $\mathcal {B}_{D}^z$ as defined in (3.3). The light curves are the 109 ensemble members, and the red lines mark the mean plus-and-minus a standard deviation. We see that many of the features observed for the results of the single realisation plotted in figure 3(a) hold quite generally. The entire contribution arises at very small $\tau$ (immediately before the target time) from $\mathcal {I}_z^z$ and the decline in this contribution in backward time is compensated by a growth in the contribution from $\mathcal {B}_{D}^z.$ There is remarkable similarity in the time histories for all realisations, although with clear fluctuations and occasional large excursions. In all cases, the contribution from spanwise stretching of interior vorticity declines to near 8 % within a few hundred viscous times and the contribution from wall vorticity rises to near 93 % in the same time. The development of large $\omega _z^*$ at $y^+=5$ appears to occur in a very regular fashion when traced back to wall vorticity, which is demonstrated by the small standard deviation for vorticity stretching and boundary contributions in figure 11.

The analogous data for the adjoint with Neumann boundary conditions are shown in figure 12. We again plot the fractional contributions to $\omega _z^*$ from the interior $\mathcal {I}_z^z$ in panel $(a)$ and from the boundary term $\mathcal {B}_{N}^z$ in panel $(b)$ . Note that the latter samples the wall vorticity flux, as defined in (3.4). We see again that many of the features observed for the results of the single realisation plotted in figure 3 $(b)$ hold quite generally. The stretching of interior spanwise vorticity, $\mathcal {I}_z^z$ , accounts for an appreciable fraction of the total, whether for the individual cases or for the average. Different from the specific realisation plotted in figure 3, some of the curves of $\mathcal {I}_z^z$ persist in exceeding the target value. In addition, while the boundary contribution $\mathcal {B}_{N}^z$ generally grows backward in time, some of the events have the opposite sign to the target vorticity. In other words, the flux of vorticity from the wall reduces the observed value and must be compensated by other terms. Of course, this is quite understandable given the space plots of the vorticity source $\sigma _z$ over the wall in figures 7–8(b-i), which reveal that the instantaneous flux takes on large values of both signs. We know that the boundary flux contribution must grow to 100 % extremely far back in time since all the vorticity originates ultimately at the walls. However, even at the final time $\tau _2=30$ in figure 12, the mean contribution over the ensemble is only approximately 31 %. We see much larger fluctuations over the ensemble for Neuman boundary conditions than we did in the Dirichlet case, which means that the origin of high- $\omega _z$ events at $y^+=5$ appears much more idiosyncratic in terms of wall vorticity flux than it does in terms of wall vorticity. This state of affairs is reflected in the appreciable increase of the standard deviation of the interior and wall contributions in figure 12. At early time $\tau \approx 1$ , the figure shows sharp distributions with $\mathcal {I}_z^z/\omega _z^*\simeq 1$ and $\mathcal {B}_{N}^z/\omega _z^*\simeq 0$ , whereas at $\tau =30$ , both distributions have become quite broad, with $\mathcal {I}_z^z$ contributing 75 % of $\omega _z^*$ on average and $\mathcal {B}_{N}^z$ contributing 31 % on average, but with standard deviations of order 54 % for $\mathcal {I}_z^z$ and 70 % for $\mathcal {B}_{N}^z.$ Additional statistical characterisation of the results in figures 11 and 12 are presented in Appendix B.

Figure 13. Ensemble of 109 target vorticity events, at $y^+=5$ above a wall-stress maximum, tracked back in time showing the sum $(\mathcal {I}_z^z+\mathcal {B}^z_{D/N})/\omega _z^*$ of interior stretching and boundary contributions for $(a)$ Dirichlet $(D)$ and $(b)$ Neumann $(N)$ boundary conditions.

In figure 13, we report the sum of the two dominant terms, namely the stretching of interior spanwise vorticity $\mathcal {I}_z^z$ and the boundary contribution $\mathcal {B}^z_{D/N}$ . Figure 13 $(a)$ is for the Dirichlet case and figure 13 $(b)$ is the Neumann counterpart. We see that the sum of these two terms is close to $\omega _z^*$ for all reverse times, so that we can infer that the other interior terms, $\mathcal {I}_x^z$ and $\mathcal {I}_y^z$ , are relatively small, certainly on average. In addition, for the Neumann case, the variance in the sum is reduced relative to the individual terms (compare figures 13 $b$ and 12).

In summary, the key contribution to the vorticity at $y^+=5$ , above a high-wall-stress event, is due to the combined effects of stretching on interior vorticity and the boundary. When the adjoint satisfies the homegeneous Dirichlet condition, it samples the wall vorticity, which progressively becomes the more important of the two contributions in this case ( $50\, \%$ by $\tau ^+\approx 20$ and exceeds $70\, \%$ by $\tau ^+ \gtrsim 50$ ). However, this choice does not account for the origin of that wall vorticity itself. In the Neumann case, the dominant contribution within the entire interval of reverse time that we considered ( $\tau ^+ \le 347$ ) was primarily due to the spanwise stretching of interior vorticity. The Neumann condition also has the advantage of being applicable directly at the wall, starting from the high-stress events themselves – the focus of the subsequent analyses.

3.3. Ensemble analysis of wall-stress maxima

Starting from each of the 109 high-stress events on the wall, we have tracked back their origin using the adjoint approach with Neuman boundary conditions. The general trends are similar to the results from $y^+=5$ , so we will here report new quantities that were not previously discussed to expand the scope of the analysis. In figure 14( $a$ ), we report the horizontally integrated interior contribution, normalised by the target value and ensemble averaged over the 109 events. The contours are plotted as a function of $y^+$ and reverse time. The results show that the interior term is concentrated in the near-wall region, $y^+ \lesssim 15$ . This trend is the combined effect of the adjoint vorticity being large near the wall, as we have seen in figure 10, for example, and the vorticity being largest at the wall. In figure 14 $(b)$ , we report the distribution of the interior terms $\mathcal {I}_{(x,y,z)}^z$ and their sum, at $\tau =30$ . The streamwise and wall-normal contributions $\mathcal {I}_{(x,y)}^z$ are, as anticipated, distributed near zero. The spanwise stretching term is by far the largest, accounting for approximately $75\,\%$ of the target value. In addition, this term accounts for the majority of the total interior contribution.

Figure 14. $(a)$ Horizontally integrated interior contribution to the wall vorticity at stress maxima, as a function of backward time. The integral is normalised by the target vorticity and ensemble averaged. $(b)$ Distribution of interior contributions at $\tau =30$ , all normalised by the target value $\omega _z^*$ . ( ) The total interior contribution (dashed line), and its spanwise $\mathcal {I}_{z}^z$ (red), wall-normal $\mathcal {I}_{y}^z$ (green) and streamwise $\mathcal {I}_{x}^z$ (blue) components.

The boundary term is examined in figure 15. The distribution is of the fractional boundary contributions from the ensemble of events, at $\tau =30$ or $\tau ^+=347$ . Interestingly, this distribution has a small positive mean of approximately $0.25$ . To test if the small mean value we observe arises entirely from the mean streamwise pressure gradient, we consider the contributions from separate components of the vorticity source,

(3.5) \begin{equation} \mathcal {B}_{Ni}^z(s) =\int _s^T\oint _{\partial D} \Omega ^z_i\sigma _i\,{\textrm{d}}S\,{\textrm{d}}t, \qquad i=x,y,z, \end{equation}

where $\sigma _i=-\nu \partial \omega _i/\partial y$ , and furthermore decompose the spanwise term into two parts,

(3.6) \begin{equation} \mathcal {B}_{Nz}^z = \mathcal {B}_{N\langle z \rangle }^z + \mathcal {B}_{Nz^\prime }^z, \end{equation}

where $\mathcal {B}_{N\langle z \rangle }^z$ arises from the mean source $\overline {\sigma }_z=\partial \overline {p}/\partial x$ and $\mathcal {B}_{Nz^\prime }^z$ is from the fluctuations $\sigma^{\prime}_z=\partial p^{\prime}/\partial x$ . The contribution from the mean pressure gradient does have a small positive net value of approximately $0.12$ , approximately half of the distribution mean but with a very small variance (which arises solely from the variance of $\Omega _z^z$ averaged over target points). As such, this term does not account for the observed variance of $\mathcal {B}_{N}^z$ and only half of its mean. We have found instead that $\mathcal {B}_{Ny}^z$ , due to the tilting of wall-normal vorticity injected at the wall, accounts for the distribution of the boundary term, as shown by the results plotted in figure 15( $a$ ).

To understand the last observation, we note from (3.5) that a persistent correlation between the two fields, $\Omega _i^z$ and $\sigma _i$ , is required to produce a substantial contribution to the space–time integral for $\mathcal {B}_{Ni}^z$ . As observed already by Lighthill (Reference Lighthill1963),

(3.7) \begin{equation} \sigma _z=\partial p/\partial x, \qquad \sigma _x = -\partial p/\partial z, \end{equation}

whereas $\sigma _y=-\nu \partial \omega _y/\partial y$ is not directly related to pressure. Furthermore, it was discovered by Kim & Hussain (Reference Kim and Hussain1993) from numerical simulations that pressure propagates in the buffer and viscous layers with a phase speed of approximately 13 $\,u_\tau$ , which is higher than the phase speed of the velocity perturbations in this region, which is only 9.5 $\,u_\tau$ . Thus, if the adjoint fields $\Omega _i^z$ for $i=x,y,z$ all propagate with the phase speed of the velocity perturbations, then they will become rapidly uncorrelated with $\sigma^{\prime}_z$ and $\sigma^{\prime}_x$ , but may remain correlated with $\sigma^{\prime}_y$ over long times. Because the fluctuating source terms $\sigma^{\prime}_i$ have zero means with equal contributions of both signs, the decorrelation will lead to strongly depleted contributions to $\mathcal {B}_{Nz^{\prime}}^z$ and $\mathcal {B}_{Nx}^z$ compared with $\mathcal {B}_{Ny}^z$ at large backward times. To test this hypothesis, we measured the phase speeds of all of the wall fields, vorticity source $\sigma _i(\boldsymbol {x},t)$ and wall adjoint vorticity $\Omega _{Ni}^z(\boldsymbol {x},t)$ for $i=x,y,z$ . For the source, since the field is homogeneous in the streamwise direction and statistically stationary in time, we used a standard method of space–time correlations. The results are plotted in figure 16 and show that, as expected, $\sigma^{\prime}_x$ and $\sigma^{\prime}_z$ have phase speeds close to $13\,u_\tau$ , whereas $\sigma^{\prime}_y$ has a phase speed of approximately $9\,u_\tau$ . Since the fields of $\Omega _i^z$ are not streamwise homogeneous or statistically stationary, we used a different method. We evaluated the horizontal shifts that maximise the correlation between fields at successive reverse times, and thus evaluated the upstream displacement of the adjoint field as a function of $\tau$ and then averaged this quantity over the ensemble of 109 independent samples of local stress maxima. The results are plotted in figure 17 and show that the phase speeds of the adjoint are approximately $ 9.5\,u_\tau$ , similar to those of the near-wall velocity. Figures 16 and 17, together, support our proposed explanation why $\mathcal {B}_{Ny}^z$ is the dominant part of the wall vorticity source contribution to the target vorticity.

Figure 15. Distribution of boundary contributions to the wall vorticity at the stress maxima, at $\tau =30$ , all normalised by the target value $\omega _z^*$ . $(a)$ Total boundary contribution $\mathcal {B}_{N}^z$ (dotted line); contribution $\mathcal {B}_{N\langle z \rangle }^z$ due to the mean streamwise pressure gradient (dark red) and contribution $\mathcal {B}_{Ny}^z$ due to tilting of wall-normal vorticity injected at the boundary (green). $(b)$ Contributions from $\mathcal {B}_{Nz^\prime }^z$ due to fluctuating streamwise pressure-gradient (red) and from $\mathcal {B}_{Nx}^z$ due to twisting of streamwise vorticity produced at the wall (blue).

Figure 16. Two-point space–time correlation of the wall-normal flux of $(a)$ streamwise, $(b)$ wall-normal and $(c)$ spanwise vorticity. The three marked lines correspond to speeds $c=\{0.8, 0.59, 0.84\}$ , or equivalently $c/u_\tau = \{12.5, 9.2, 13.1\}$ .

Figure 17. Phase speed for wall adjoint vorticity in its $(a)$ streamwise, $(b)$ wall-normal and $(c)$ spanwise components. Here, $x_{N}=\sum _{n=1}^{N-1}\Delta x_{n}$ , where $\Delta x_{N}, \Delta z_{N}=\mathrm {argmax}_{\Delta x_{N}, \Delta z_{N}} \mathrm {Cov}[q(x,z,\tau _N ),q(x+\Delta x,z+\Delta z,\tau _N +\Delta \tau )]$ . The three marked velocities are $c=\{0.59, 0.58, 0.54\}$ , or equivalently $c/u_\tau = \{9.2, 9.0, 8.4\}$ .

The present analysis demonstrates that the stress extrema in channel flow are primarily due to stretching of interior near-wall vorticity for reverse times of the order of $\tau ^+ \simeq 347$ . The contribution of the vorticity flux at the wall, or the Lighthill source term, is relatively small and does not necessarily have the correct sign for individual events. On average, this term is small, with half of its mean arising from the externally applied streamwise pressure gradient. More interestingly, the distribution of the boundary term is essentially set by the flux of wall-normal vorticity that is subsequently tilted to generate a spanwise stress. The analysis highlights the incisive utility of the adjoint vorticity equation (2.8) and its value in tracing back the origin of vorticity in viscous incompressible flows in a manner as insightful as the original inviscid formulation, but with a rigorous account for the influence of viscosity.