2. Components of the pressure Poisson equation

Following Mansour et al. (Reference Mansour, Kim and Moin1988) and Kim (Reference Kim1989), and using the Einstein summation convention where roman indices are just component indices in that same Cartesian frame, the fluctuating pressure satisfies

(2.1) \begin{equation} \partial _i\partial _i p = \underbrace {-\,2\,\frac {\textrm {d}U}{\textrm {d}y}\,\partial _x v}_{\textit{linear }(\mathcal{L})} + \underbrace {-\,\partial _{\!j}\partial _k \left (u_{\!j} u_k\right )}_{\textit{nonlinear }(\mathcal{Q})} +\overline {\,\partial _{\!j}\partial _k \left (u_{\!j} u_k\right )}\!\text{ ,} \end{equation}

where $\tilde {u}_i = U_i + u_i$ . In the resolvent analysis framework (McKeon & Sharma Reference McKeon and Sharma2010), $\mathcal Q$ is the dilatational part of the nonlinear forcing to the linearised Navier–Stokes operator, namely the gradient of the Reynolds stress tensor (Rosenberg & McKeon Reference Rosenberg and McKeon2019; Blanco et al. Reference Blanco, Hanifi, Henningson and Cavalieri2024). Rewritten in vorticity/strain form by considering the symmetric and antisymmetric parts of the Reynolds stress gradient tensor (Bradshaw & Koh Reference Bradshaw and Koh1981), the equation reads

(2.2) \begin{equation} \partial _i\partial _i p = \underbrace {2\,\varOmega _z\,\partial _x v}_{\mathcal{L}} + \underbrace {\Big (\overbrace {\frac 12\,\omega _i\omega _i}^{\mathcal{T}} \;-\; \overbrace {S_{ij}S_{ij}}^{\mathcal{S}}\Big ) \;-\;\overline {\Big (\frac 12\,\omega _i\omega _i - S_{ij}S_{ij}\Big )}}_{\mathcal{Q}}\!\text{ ,} \end{equation}

where

(2.3) \begin{align} S_{ij} &= \frac 12(\partial _i u_{\!j} + \partial _{\!j} u_i),\qquad \omega _i = \varepsilon _{ijk}\,\partial _{\!j} u_k,\qquad \varOmega _z = -\,\frac {\textrm {d}U}{\textrm {d}y}\text{ ,} \end{align}

with $\mathcal Q = \mathcal T - \mathcal S$ .

We define the pressure components by

(2.4a) \begin{align} \partial _i\partial _i p_{\mathcal{L}} &= \mathcal{L}, &\qquad (\partial _{y} p_{\mathcal{L}})\big |_{y=\pm 1} &= 0, \\[-12pt]\nonumber \end{align}

(2.4b) \begin{align} \partial _i\partial _i p_{\mathcal{Q}} &= \mathcal{Q}, &\qquad (\partial _{y} p_{\mathcal{Q}})\big |_{y=\pm 1} &= 0, \\[-12pt]\nonumber \end{align}

(2.4c) \begin{align} \partial _i\partial _i p_{\nu } &= 0, &\qquad (\partial _{y} p_{\nu })\big |_{y=\pm 1} &= Re^{-1}\,\partial _{\!j}\partial _{\!j} v \text{ ,} \end{align}

with $Re \equiv \delta U_e / \nu$ , and $U_e$ is the centreline/edge velocity in the channel/BL. It follows that

(2.5) \begin{equation} p = p_{\mathcal{L}} + p_{\mathcal{Q}} + p_{\nu }\text{ ,} \end{equation}

where $p_{\mathcal{L}}$ , $p_{\mathcal{Q}}$ and $p_{\nu }$ are the linear, nonlinear and viscous components of the pressure, respectively.

Because the analysis below is restricted to $k=\sqrt {k_x^2+k_z^2}\gt 0$ , the wall-parallel mean $\overline {\mathcal Q}(y)$ contributes only to the $(k_x,k_z)=(0,0)$ mode and therefore does not appear explicitly in the Fourier-space expressions below.

Let $\widehat {\boldsymbol{\cdot }}$ denote the $(k_x,k_z)$ Fourier transform and $k=\sqrt {k_x^2+k_z^2}\gt 0$ . In a channel $y\in [-1,1]$ with homogeneous Neumann boundary conditions, the solution of (2.1) for the linear and nonlinear sources $\mathcal{M}\in \{\mathcal{L},\mathcal{Q}\}$ admits the integral representation

(2.6a) \begin{align} \widehat {p}_{\mathcal{M}}(k,y;\delta ^+) = \int _{-1}^{1} g(k;\,y,y')\,\widehat {\mathcal{M}}(y',k;\delta ^+)\,\textrm {d}y'\text{ ,} \\[-12pt]\nonumber \end{align}

(2.6b) \begin{align} g(k;\,y,y') = -\frac {\cosh \!\big (k(1-y_\gt )\big )\,\cosh \!\big (k(1+y_\lt )\big )}{k\,\sinh (2k)} \text{ ,} \end{align}

with $y_\lt = \min (y,y')$ and $y_\gt = \max (y,y')$ , where $y'\in [-1,1]$ is the source-point integration variable. The viscous component $p_{\nu }$ is obtained through enforcement of no slip via the Neumann boundary condition.

Evaluating (2.6) at either wall yields a convolution between the sources and a single-wall kernel. For the bottom wall ( $y=-1$ ),

(2.7a) \begin{align} \widehat {p}_{\mathcal{M},w^-}(k;\delta ^+) &= \int _{-1}^{1} G^{-}(k,y')\,\widehat {\mathcal{M}}(y',k;\delta ^+)\,\textrm {d}y', \;\; G^{-}(k,y') = -\,\frac {1}{k}\,\frac {\cosh \!\big (k(1-y')\big )}{\sinh (2k)}, \end{align}

and top wall ( $y=+1$ ),

(2.7b) \begin{align} \widehat {p}_{\mathcal{M},w^+}(k;\delta ^+) &= \int _{-1}^{1} G^{+}(k,y')\,\widehat {\mathcal{M}}(y',k;\delta ^+)\,\textrm {d}y', \;\; G^{+}(k,y') = -\,\frac {1}{k}\,\frac {\cosh \!\big (k(1+y')\big )}{\sinh (2k)}. \end{align}

For later convenience, we define the inner-scaled distances from the bottom and top walls as

(2.8) \begin{equation} y_-^+ \equiv \delta ^+(1+y), \qquad y_+^+ \equiv \delta ^+(1-y), \end{equation}

or, equivalently, $y_\pm ^+ \equiv \delta ^+(1\mp y)$ .

Writing the hyperbolic functions in exponential form makes the wall-normal attenuation explicit,

(2.9) \begin{equation} G^{-}(k,y') = -\frac {1}{k} \frac {\textrm {e}^{-k(1+y')}+\textrm {e}^{-k(3-y')}}{1-\textrm {e}^{-4k}} \;\;\xrightarrow [k\gg 1]{}\;\; -\frac {1}{k}\textrm {e}^{-k(1+y')} , \end{equation}

so that each Fourier mode decays exponentially with the wall-normal distance from the source to the wall. Equivalently, the full-field kernel in (2.6) has the leading-order form

(2.10) \begin{equation} g(k;y,y') \; \sim \; -\frac {1}{2k}\textrm {e}^{-k|y-y'|}, \end{equation}

demonstrating the expected elliptic attenuation: contributions at wavenumber $k$ are exponentially suppressed away from their source with characteristic scale $1/k$ . Below, we focus on the wall pressure and its decomposition,

(2.11) \begin{equation} p_w = (p_{\mathcal{L}})_w + (p_{\mathcal{Q}})_w + (p_{\nu })_w, \end{equation}

with each component obtained by substituting $\widehat {\mathcal{M}}$ into (2.7), and we drop the $\boldsymbol{\cdot }_w$ subscript for brevity. In the bottom-wall attribution maps below, $y$ denotes source distance from that wall, so $G(k,y)$ denotes the corresponding single-wall kernel; for $k\gg 1$ , $G(k,y)\sim -k^{-1}\textrm {e}^{-ky}$ .

Figure 2.

Instantaneous and mean r.m.s. profiles of the pressure across a half-channel width at $\delta ^+\approx 550$ and a comparison to the r.m.s. reported by Lee & Moser (Reference Lee and Moser2015). The linear, nonlinear and viscous contributions to the variance as well as the cross terms are shown. Light curves indicate instantaneous profiles, while the darker/dashed curves indicate the corresponding mean profiles.

The dataset used derives from a previously published channel-flow DNS at $\delta ^+\approx 550$ (Huang et al. Reference Huang, Toedtli, Chini and McKeon2026) that employs the method of Flores & Jiménez (Reference Flores and Jiménez2006). Figure 2 shows a comparison with results from Lee & Moser (Reference Lee and Moser2015) (as reported in Panton et al. (Reference Panton, Lee and Moser2017)) and confirms that we match $\langle p^2\rangle ^+$ at comparable $\delta ^+$ . Figure 2 shows that this case lies close to the DNS of Lee & Moser (Reference Lee and Moser2015) and that the viscous contribution is already small compared with the linear and nonlinear terms, as noted by Kim (Reference Kim1989) and Jimenez & Hoyas (Reference Jimenez and Hoyas2008). We use $\delta ^+\approx 550$ as an illustrative dataset: it is not sufficient to settle asymptotic scaling, but it is sufficient to expose the source localisation and wall-mapping mechanisms on which the present interpretation rests. The nonlinear pressure variance is comparable with or larger than the linear contribution everywhere except very near the wall, with the mean-square nonlinear source peaking near $y^+\approx 20$ , in line with previous work (Kim Reference Kim1989; Panton et al. Reference Panton, Lee and Moser2017; Yang & Yang Reference Yang and Yang2022). The cross terms contributing to the variance are also shown. Although Yang & Yang (Reference Yang and Yang2022) show that such cross terms can contribute significantly, in the present case they remain smaller than the dominant linear and nonlinear self-contributions. We therefore focus below on the mechanisms associated with $\mathcal{L}$ and $\mathcal Q$ .

3. Source-term structure

Figure 3.

Ring-averaged $y^+$ – $k$ spectra of (a) linear $\mathcal{L}$ and (b) nonlinear $\mathcal{Q}$ sources, as well as the corresponding wall-pressure attribution maps to the bottom wall for (c) linear pressure and (d) nonlinear pressure. (c, d) The black dashed line indicates $k\,y=1$ , and the two grey lines indicate $\xi =0.3$ and $\xi =3$ . The kink in the lines is due to the change from log to linear scaling at $y^+=10$ toward the channel wall.

The structure of the source terms is illustrated in figure 3. Figure 3(a) shows that the $\mathcal{L}$ energy is concentrated near $y^+\approx 10$ and at relatively high $k$ , whereas figure 3(b) shows that $\mathcal{Q}$ peaks farther out ( $y^+\approx 20$ ) and extends to the highest resolved $k$ . As (2.2) shows, $\mathcal{Q}$ is the difference of two large, correlated quadratic forms. Strong cancellations occur when it is mapped onto the wall, which makes it susceptible to sign changes and partial cancellation. When mapped to the wall, this yields the positive and negative attribution regions seen in figure 3(d).

3.1. Variance attribution

Figures 3(c) and 3(d) highlight how the source terms are mapped to the wall-pressure fluctuations by showing the instantaneous wall-pressure attribution maps for $\mathcal{L}$ and $\mathcal{Q}$ . Here, consistent with the bottom-wall maps, $y$ denotes source distance from the bottom wall. The attribution is defined as the signed fractional contribution of sources at distance $y$ to the wall-pressure mode at wavenumber $k$ . We define the attribution function $f(y,k)$  as

(3.1) \begin{equation} f(y,k) = \frac { \text{Re} \{ G(k,y)\,\widehat {\mathcal{M}}(y,k)\,\widehat {p}^*(k)\}}{|\widehat {p}(k)|^2} \text{ ,} \end{equation}

where $\text{Re}$ denotes the real part. For each $k$ , $\int _{0}^{2} f(y,k)\,\textrm {d}y = 1$ ; positive values indicate contributions that are in phase with the wall-pressure mode, whereas negative values indicate cancellation. Figure 3(c) shows that the attribution associated with $\mathcal{L}$ is positive everywhere. For $k \lesssim 10$ it is distributed over a range of wall-normal distances, whereas with increasing $k$ it becomes concentrated over a narrower band. The attribution associated with $\mathcal{Q}$ in figure 3(d) is similar for $k\gtrsim 10$ . For $k\lesssim 10$ , however, the nonlinear map develops alternating positive and negative regions, indicating that the wall-pressure mode is assembled through partial cancellation among source layers, including appreciable contributions from both sides of the channel centreline.

3.2. Elliptic attenuation

Heuristically, the elliptic attenuation reflects the balance between the algebraic spectral decay of the source and the exponential decay of the Green’s kernel with wall-normal distance. For a wall-parallel wavenumber magnitude $k$ at height $y$ , the wall-weighted amplitude scales as

(3.2) \begin{equation} |G(k,y)\,\widehat {\mathcal{M}}(y,k)| \sim \frac {\textrm {e}^{-k y}}{k}\times \big [\text{algebraic source decay in }k\big ], \end{equation}

where the factor $\textrm {e}^{-ky}$ comes from the Green’s kernel for $k\gg 1$ (2.9), so that contributions from a layer at $y$ are rapidly attenuated once $k\,y\gg 1$ . Accordingly, we use this simplifying assumption to define the effective envelope of wall-source communication by a fixed iso-attenuation criterion on the kernel; for instance,

(3.3) \begin{equation} \textrm {e}^{-k y} = \textrm {e}^{-1}\quad \Longleftrightarrow \quad \xi \equiv k\,y = 1, \end{equation}

where $\xi =\mathcal{O}(1)$ denotes one $\textrm {e}$ -fold of attenuation. For $\xi \ll 1$ , the kernel hardly filters (and the communication to the wall is then controlled by the source spectrum), whereas for $\xi \gg 1$ , the exponential damping dominates and suppresses the contribution. The line $\xi =1$ therefore serves as a practical elliptic attenuation boundary in $(k,y)$ -space.

Figures 3(c) and 3(d) plot the $\xi = 1$ line as an approximate envelope: high- $k$ contributions from a given layer lie predominantly on or slightly above this curve, while the low- $k$ contributions ( $k \lesssim 10$ ) extend more broadly, with $\xi =1$ providing an upper bound on their influence. This low- $k$ behaviour is influenced by channel-specific effects, in particular the cancellation of contributions across the centreline and the relative (to zero pressure gradient BL flows) suppression of the largest scales. To keep the theoretical picture consistent with single-wall geometries, we therefore focus, for the present envelope argument, on the region $k\gt 10$ , and reiterate that the $e^{-ky}$ decay used in (2.9) is the large- $k$ ( $k\gg 1$ ) asymptotic form of the Green’s kernel. Figures 3(c) and 3(d) also plot lines at $\xi =0.3$ and $\xi =3$ to give a sense of the $O(1)$ range of the elliptic attenuation.

3.3. Positive and negative $\mathcal{Q}$

It is useful to distinguish the local enstrophy–strain imbalance,

(3.4) \begin{equation} \mathcal{B}\equiv \mathcal{T}-\mathcal{S}, \end{equation}

from the nonlinear Poisson source,

(3.5) \begin{equation} \mathcal{Q}=\mathcal{B}-\overline {\mathcal{B}}. \end{equation}

Regions with $\mathcal{B}\gt 0$ are locally enstrophy-dominated, whereas regions with $\mathcal{B}\lt 0$ are locally strain-dominated. By contrast, the sign of $\mathcal{Q}$ indicates whether the local enstrophy–strain imbalance lies above or below its mean, not simply whether $\mathcal{T}$ is larger or smaller than $\mathcal{S}$ . Thus, $\mathcal{Q}\gt 0$ and $\mathcal{Q}\lt 0$ identify positive and negative nonlinear source events, respectively.

To isolate these two classes of events, we define

(3.6) \begin{equation} \mathcal{Q}^\oplus =\max (\mathcal{Q},0), \qquad \mathcal{Q}^\ominus =\max (-\mathcal{Q},0), \end{equation}

and examine their r.m.s. profiles separately.

3.4. Root mean square of the source components

Figure 4.

Profiles of the r.m.s. of the Poisson source components versus $y^+$ . (a) The individual components, $\mathcal{L}$ , $\mathcal{T}$ , $-\mathcal{S}$ and $\mathcal{Q}$ , and the sign-split parts, $\mathcal{Q}^\oplus$ and $\mathcal{Q}^\ominus$ ((2.1), (2.2) and (3.6)). (b) Log–log view of selected curves.

Profiles of the r.m.s. of the Poisson-source terms versus $y^+$ quantify the available source amplitude prior to cancellation by sign and by Green-function weighting. Figure 4(a) shows that $\mathcal{T}$ and $\mathcal{S}$ are individually much larger than $\mathcal{L}$ and $\mathcal{Q}$ , so that the nonlinear source arises from a strong cancellation between large enstrophy and strain contributions. Figure 4(b) shows the selected r.m.s. profiles on logarithmic axes. The linear source $\mathcal{L}$ peaks at $y^+\approx 10$ , whereas $\mathcal{Q}$ peaks farther from the wall, at $y^+\approx 15$ – $20$ , and remains larger than $\mathcal{L}$ over most of the domain. Over the inertial layer, both $\mathcal{L}$ and $\mathcal{Q}$ decay approximately as $y^{+-1}$ .

The $\mathcal{Q}^\oplus /\mathcal{Q}^\ominus$ split isolates positive and negative nonlinear source events, rather than enstrophy- and strain-dominated events. Very near the wall, $\mathcal{Q}^\ominus _{\mathit{rms}} \gt \mathcal{Q}^\oplus _{\mathit{rms}}$ . Beyond $y^+\approx 7$ , $\mathcal{Q}^\oplus _{\mathit{rms}}$ rises rapidly and becomes comparable to, and near the peak slightly larger than, $\mathcal{Q}^\ominus _{\mathit{rms}}$ ; farther from the wall the two remain of the same order.

4. Direct source density of the linear contribution

Figure 5.

Linear-source variance-density decomposition at $\delta ^+\approx 550$ . (a) Direct signed wall-normal source density of the linear pressure variance, $\mathcal S_{\mathcal L}^+(y^+)$ , defined by (4.2); the solid curve shows the positive part and the dotted curve shows the magnitude of the negative part. (b) Inner-scaled magnitudes of the two local factors entering $\mathcal L$ : $|2\varOmega _z^+|$ , $\textrm {r.m.s.}(\partial _{x^+}v^+)$ , and their pointwise product. Comparing (a) and (b) distinguishes local source amplitude from the Green-function-weighted, phase-aware contribution to wall-pressure variance.

The $\delta ^+$ -invariant inner function of the wall-pressure spectrum modelled by Massey et al. (Reference Massey, Smits and McKeon2026b) is defined only once $\delta ^+$ is sufficiently large. In practice, the proposed $B_{\mathcal L}$ offset in (1.1) is the asymptotic value obtained when that inner contribution is integrated to $\langle p^{+2}\rangle$ . The question is therefore whether the linear self-contribution can generate a logarithmic dependence as the wall-normal domain expands with $\delta ^+$ .

The r.m.s. profiles discussed above quantify the amplitude available before wall mapping. Figure 5(b) shows that the local product $|2\varOmega _z^+|\,\textrm {r.m.s.}(\partial _{x^+}v^+)$ is weighted toward the near-wall region. This is suggestive, but not sufficient: it does not include the Green-function weighting, wall-normal correlations orcancellations between source layers.

A direct test is obtained from the wall-normal source density of the variance. For a source component $\mathcal M\in \{\mathcal L,\mathcal Q\}$ , multiplying (2.7) by $\widehat p_{\mathcal M}^{\,*}$ , summing over wall-parallel Fourier modes, and averaging gives

(4.1) \begin{equation} \langle p_{\mathcal M}^{2}\rangle = \int _{-1}^{1}\mathcal S_{\mathcal M}(y)\,\text{d}y, \end{equation}

with the signed wall-normal source density

(4.2) \begin{equation} \mathcal S_{\mathcal M}(y) = \sum _{\boldsymbol{k}} \text{Re} \left \{ G_w(k,y) \left \langle \widehat {\mathcal M}(\boldsymbol{k},y)\, \widehat p_{\mathcal M}^{\,*}(\boldsymbol{k}) \right \rangle \right \}\!. \end{equation}

Equivalently,

(4.3) \begin{equation} \mathcal S_{\mathcal M}(y) = \int _{-1}^{1} \sum _{\boldsymbol{k}} G_w(k,y)G_w(k,y') \text{Re} \left \{ \left \langle \widehat {\mathcal M}(\boldsymbol{k},y) \widehat {\mathcal M}^{\,*}(\boldsymbol{k},y') \right \rangle \right \} \text{d}y'. \end{equation}

These expressions retain the Green-function weighting, wall-normal correlations and phase cancellations omitted by $\textrm {r.m.s.}(\mathcal M)$ . The plus-normalised density is defined so that $\mathcal S_{\mathcal M}^+(y^+)\,\text{d}y^+$ contributes to $\langle p_{\mathcal M}^{+2}\rangle$ .

Figure 5(a) shows the resulting $\mathcal S_{\mathcal L}^+(y^+)$ . The density rises through the viscous and buffer layers, peaks at near-wall distances and then decays through the logarithmic and outer regions more rapidly than the $(y^+)^{-1}$ guide. The negative part of the signed density is small and does not form a compensating outer contribution.

This observation is most naturally interpreted as a statement about the possible asymptotic contribution. If, over an inertial range $y_0^+\ll y^+\ll \delta ^+$ , the density were Reynolds-number invariant in inner units, then a non-cancelling $\mathcal S_{\mathcal L}^+\sim C/y^+$ tail would produce a contribution $C\ln (\delta ^+/y_0^+)$ as $\delta ^+$ increases. Any steeper decay leads instead to a contribution that approaches a constant as the upper limit extends. The measured $\mathcal S_{\mathcal L}^+$ in figure 5(a) is consistent with the latter behaviour.

The key point is therefore that the variance density, not the local amplitude, determines the Reynolds-number dependence. The difference between panels (a) and (b) highlights this: the local product $|2\varOmega _z^+|\,\textrm {r.m.s.}(\partial _{x^+}v^+)$ does not account for Green-function weighting or cancellation and does not predict the decay of $\mathcal S_{\mathcal L}^+$ .

The conclusion from this section is necessarily limited. The $\delta ^+\approx 550$ dataset does not establish the asymptotic Reynolds-number decomposition of $\langle p^{+2}\rangle$ . It does show that, in the present flow, the exact self-source density of $\langle p_{\mathcal L}^{+2}\rangle$ is near-wall dominated and decays more rapidly than the form required to produce a logarithmic contribution. This is consistent with interpreting $B_{\mathcal L}$ as a predominantly inner-set contribution, while any $\ln \delta ^+$ growth of the total variance arises from components whose source densities do not decay with the expanding of the inertial layer. Subsequent work explores this conclusion more concretely (Massey et al. Reference Massey, Zimmerman, Klewicki and McKeon2026a ). The next section considers this possibility for $\mathcal Q$ .

5. Fissures as compact carriers of the source terms

Figure 6.

Schematic of the physical process that yields the $\ln \delta ^+$ growth of $\langle p^{+2}\rangle$ . Vortex stretching/tilting produces thin, sheet-like fissures that carry the Poisson sources compactly across the inertial layer and imprint at the wall through an elliptic Green’s mapping that favours low wall-parallel wavenumbers.

Moving away from the mid- $\delta ^+$ dataset, we aim to describe the mechanistic link between the pressure-Poisson sources and the vortical-fissure picture of the inertial layer. The theoretical arguments presented rest on the compact spatial support of the sources within the fissures. We define compact support as regions where the source terms are significantly non-zero and spatially localised. Figure 6 summarises the formation and alignment of these fissures and their variation with $\delta ^+$ . We consider a single fissure neighbourhood and introduce local coordinates

(5.1) \begin{equation} (s,n,t)=(x,y,z)+O(\theta ),\qquad \theta \ll 1, \end{equation}

with $s$ (approximately streamwise), $n$ (approximately wall-normal) and $t$ (approximately spanwise) (Cui & Jacobi Reference Cui and Jacobi2025). In the fissure frame, the velocity is decomposed into a fissure-aligned (skeletal) probabilistic mean (Bautista et al. Reference Bautista, Ebadi, White, Chini and Klewicki2019) and residuals,

(5.2a) \begin{align} \tilde {u}^{(f)}(s,n,t) &= \mathcal{U}^{(f)}(n) + u^{(f)}(s,n,t) \approx \mathcal{U}^{(f)}(y) + u^{(f)}(x,y,z), \\[-12pt]\nonumber \end{align}

(5.2b) \begin{align} \tilde {v}^{(f)}(s,n,t) &= v^{(f)}(s,n,t) \approx v^{(f)}(x,y,z), \\[-12pt]\nonumber \end{align}

(5.2c) \begin{align} \tilde {w}^{(f)}(s,n,t) &= w^{(f)}(s,n,t) \approx w^{(f)}(x,y,z), \end{align}

where $\mathcal{U}^{(f)}$ is the fissure-aligned most probable (UMZ-like) streamwise profile and $u^{(f)}$ , $v^{(f)}$ and $w^{(f)}$ are zero mean under a fissure-aligned conditional average, $\langle (u_i)^{(f)}\rangle _{\!f}=0$ . The global mean remains $U_i=(U(y),0,0)$ . Consistent with the UMZ–vortical-fissure picture, the skeletal profile is a near step with jump $\Delta \mathcal{U}^{(f)}=O(u_\tau )$ across thickness $\delta ^{(f)}$ , such that

(5.3) \begin{equation} \partial _{y} \mathcal{U}^{(f)} \sim \frac {\Delta \mathcal{U}^{(f)}}{\delta ^{(f)}} \approx \frac {u_\tau }{\delta ^{(f)}} \quad \text{with }\ \delta ^{(f)}\ll \ell _{s,t}. \end{equation}

This yields the local velocity gradient

(5.4) \begin{equation} \partial _i \tilde {u}_{\!j}^{(f)} \approx \underbrace {\begin{pmatrix} 0 & 0 & 0\\[5pt] \partial _{y} \mathcal{U}^{(f)} & 0 & 0\\[5pt] 0 & 0 & 0 \end{pmatrix}}_{\textit{mean fissure shear}} +\partial _i u_{\!j}^{(f)} \, . \end{equation}

Before turning to the spectral content of the source decomposition that follows, we emphasise that $k$ throughout this section refers to the wall-parallel wavenumber magnitude entering the Green’s function in (2.7). The wall-normal thinness of a fissure (its small $\delta ^{(f)}$ ) does not enter this Fourier decomposition; what matters for wall transmission is the source’s footprint in the wall-parallel $(x,z)$ plane. A fissure is thin in $y$ but coherent over along-fissure scales $\ell _{s,t}\gg \delta ^{(f)}$ , so its wall-parallel spectrum is concentrated at relatively low $k$ even though the source amplitude within the fissure is large and locally sheet-like.

To expose compact support of the nonlinear source $\mathcal{Q}=-\partial _i\partial _{\!j}(u_i u_{\!j})$ in local coordinates, we write the fluctuations as the sum of a skeletal contribution (support confined to the fissure) and residuals that are zero mean in the fissure frame,

(5.5) \begin{equation} u_i \approx \underbrace {\big (\mathcal{U}^{(f)} - U\big )}_{U^{(sk)}(y)} + \underbrace {(u^{(f)},v^{(f)},w^{(f)})}_{u^\star (s,n,t)}. \end{equation}

Substituting (5.5) yields

(5.6) \begin{align} \mathcal{Q} &\,=\; -\partial _i\partial _{\!j}(U^{(sk)}_i U^{(sk)}_{\!j})\; -\,2\,\partial _i\partial _{\!j}\big (U^{(sk)}_i u^\star _{\!j}\big )\; -\,\partial _i\partial _{\!j}\big (u^\star _i u^\star _{\!j}\big ). \end{align}

Because $U^{(sk)}$ has only an $s$ component depending on $n$ , $\partial _i\partial _{\!j}(U^{(sk)}_i U^{(sk)}_{\!j})=0$ . The residual product $u^\star _i u^\star _{\!j}$ in the last term is volumetric and turbulent throughout the UMZ interior, with no preferred along-plane coherence length, so its wall-parallel spectrum is broadband with substantial content at high $(k_x,k_z)$ ; under the elliptic Green’s mapping, this high- $k$ content is exponentially attenuated before reaching the wall. The cross-term simplifies to

(5.7) \begin{equation} -\,2\,\partial _x\partial _{y}\big (U^{(sk)} v^{(f)}\big ) = -\,2\,\partial _x\big[(\partial _{y} U^{(sk)})\,v^{(f)} + U^{(sk)}\,\partial _{y} v^{(f)}\big]. \end{equation}

In the fissure neighbourhood, $\delta ^{(f)}\ll \ell _{s,t}$ , so $\partial _n\gg \partial _{s,t}$ and the first product, proportional to $\partial _{y} U^{(sk)}\sim u_\tau /\delta ^{(f)}$ , dominates. Thus,

(5.8) \begin{equation} {\;\mathcal{Q}^{(f)} \approx -\,2\,\partial _x\partial _{y}\big (U^{(sk)}\,v^{(f)}\big )\;} \end{equation}

is compactly supported on the fissure. Integrating across the $O(\delta ^{(f)})$ thickness yields the fissure-source density

(5.9) \begin{equation} \int \mathcal{Q}^{(f)}\,\text{d}y \approx -2\partial _x[\![ U^{(sk)}v^{(f)} ]\!], \qquad [\![ \boldsymbol{\cdot }]\!] \equiv \text{jump across the fissure.} \end{equation}

The spectral content of $\mathcal{Q}^{(f)}$ is therefore set by the along-fissure variations of the flux $U^{(sk)} v^{(f)}$ , whose scales are $O(\ell _{s,t})\gg \delta ^{(f)}$ , so the resulting wall-parallel content sits at low $k$ .

For the linear source, $\partial _n U_s\neq 0$ only within the fissure. Thus,

(5.10) \begin{equation} {\;\mathcal{L}^{(f)} = -\,2\,(\partial _n U)\,\partial _s v^{(f)} \approx -\,2\,(\partial _{y} U)\,\partial _x v^{(f)}\;} \end{equation}

is likewise localised by $\partial _n U$ , with its spectrum set by the along-fissure gradients $\partial _s v^{(f)}$ (low $k$ ). In short, both $\mathcal{Q}^{(f)}$ and $\mathcal{L}^{(f)}$ are fissure supported and streamwise biased in wavenumber.

Because the skeletal shear $\partial _{y} U^{(sk)}$ is not directly accessible from instantaneous fields, we adopt the instantaneous wall-normal shear $\partial _{y} u$ as a direct operational proxy, and define the fissure support via a soft threshold $\chi _{\!f}$ applied to $|\partial _{y} u|$ . Substituting $(\partial _{y} u)_{\!f}=\chi _{\!f}\,\partial _{y} u$ for $\partial _n U$ in (5.8)–(5.10) yields the source proxies shown in figure 7. Across this snapshot, $69\,\%$ of $|\mathcal{Q}^{(f)}|$ and $91\,\%$ of $|\mathcal{L}^{(f)}|$ lie within the identified fissure support, confirming that the retained sources are localised on the high-shear skeleton rather than distributed through the surrounding uniform-momentum interiors.

Figure 7.

Instantaneous $x$ – $y$ snapshot of the fissure-supported source proxies, with $\partial _{y} u$ used as a direct proxy for the skeletal shear $\partial _{y} U^{(sk)}$ (see text). (a) Thresholded shear indicator $|(\partial _{y} u)_{\!f}|/q_{99}$ , with $(\partial _{y} u)_{\!f}=\chi _{\!f}\,\partial _{y} u$ , $\chi _{\!f}$ a soft (tanh) threshold defining the fissure support, and $q_{99}$ the $99$ th percentile of $|\partial _{y} u|$ over the analysed wall-normal band (robust colour-bar scale). (b) Nonlinear proxy $\mathcal{Q}^{(f)}=-2\partial _x[(\partial _{y} u)_{\!f}\,v]$ and (c) linear proxy $\mathcal{L}^{(f)}=-2(\partial _{y} u)_{\!f}\,\partial _x v$ , each normalised by its r.m.s. Contours in (b, c) repeat the thresholded fissure support.

Finally, the wall imprint of these compact sources is dictated by the elliptic Green’s function. Because the wall-parallel Green’s weight decays approximately as $e^{-k\,y}$ , the broadband wall-parallel content of UMZ-interior fluctuations is strongly filtered, whereas the fissure source – although thin in $y$ – projects onto low wall-parallel wavenumbers because of its large along-fissure extent ( $\ell _{s,t}\gg \delta ^{(f)}$ ) and therefore transmits efficiently to the wall. This picture is consistent with the UMZ–vortical-fissure structure in which $\Delta U=O(u_\tau )$ across fissures of thickness $\delta ^{(f)+}\sim (\delta ^+)^{1/2}$ (Klewicki Reference Klewicki2013; Bautista et al. Reference Bautista, Ebadi, White, Chini and Klewicki2019).

6. Estimate of $\boldsymbol{A}_{\mathcal{Q}}$

We now use the fissure-supported nonlinear source and the elliptic mapping to obtain an order-of-magnitude estimate for the prefactor $A_{\mathcal Q}$ in (1.1). From (5.9), a thin fissure at height $y$ supplies a sheet-integrated nonlinear source. Across the inertial region the streamwise velocity jump satisfies $\Delta U = O(u_\tau )$ , and we write $\Delta U^+ \equiv \Delta U/u_\tau$ . We denote the inertial-layer r.m.s. of $v^+$ by $v_{\textit {rms}}^+$ . Approximating $\partial _x v^{(f)}\sim v_{\textit {rms}}/\ell _s$ , with $\ell _s$ an along-fissure correlation length, the sheet source density scales as

(6.1) \begin{equation} \int \mathcal{Q}^{(f)} \,\textrm {d}y \sim \frac {2 \Delta U\, v_{\textit {rms}}}{\ell _s} \text{ .} \end{equation}

In the log region, attached-eddy arguments (Townsend Reference Townsend1951) and the UMZ–vortical-fissure picture suggest that the relevant wall-parallel scales grow linearly with $y$ . Following Wei et al. (Reference Wei, Fife, Klewicki and McMURTRY2005) and Klewicki (Reference Klewicki2021), we write the typical layer width as $W(y)=y/\phi _c$ , with the Fife similarity parameter $\phi _c\approx 1.62$ . Then taking

(6.2) \begin{equation} \ell _s (y) = \alpha W(y) = \alpha \frac {y}{\phi _c} \text{ ,} \end{equation}

where $\alpha =O(1)$ measures the ratio of along-fissure correlation length to UMZ width. The segment-length statistics of large-scale UMZ interfaces in Cui & Jacobi (Reference Cui and Jacobi2025) show that the wavy segments that compose strongly linear edges have a characteristic length of order $0.1h$ over a wide range of streamwise extents, while the reanalysis of UMZ thicknesses in Heisel et al. (Reference Heisel, de Silva, Hutchins, Marusic and Guala2020) and Cui & Jacobi (Reference Cui and Jacobi2025) confirms that $H(y)\sim y$ throughout the logarithmic region. Together, these results support an $O(1)$ ratio $\ell _s/W(y)$ and hence $\alpha =O(1)$ . In inner units this gives $y^+/\ell _s^+ = \phi _c/\alpha$ , independent of $y^+$ over the inertial layer.

The wall kernel behaves as $G\sim -k^{-1}\textrm {e}^{-k y}$ for $k\gg 1$ (see (2.9)). The attribution maps in figure 3 and the envelope in (3.3) indicate that the dominant contribution to the wall pressure from a layer at $y$ comes from wall-parallel wavenumbers satisfying $k y \approx 1$ . Evaluating the kernel on this line gives a factor $y\,\textrm {e}^{-1}$ . Combining this with the sheet source scaling yields a single-fissure contribution in inner units

(6.3) \begin{equation} p^{(1)+} \sim C_{\mathcal Q}^{1/2}\big (2\Delta U^+v_{\textit {rms}}^+\big ) \textrm {e}^{-1}\frac {y^+}{\ell _s^+} \text{ ,} \end{equation}

where $C_{\mathcal Q}=O(1)$ collects kernel prefactors, the modest anisotropy of the fissure-aligned source and weak bandwidth effects associated with the $k y\simeq 1$ envelope. Using $y^+/\ell _s^+=\phi _c/\alpha$ then gives a single-layer variance

(6.4) \begin{equation} \big \langle (p^{(1)+})^2\big \rangle = C_{\mathcal Q}\big (2\Delta U^+v_{\textit {rms}}^+\big )^{2} \textrm {e}^{-2}\left (\frac {\phi _c}{\alpha }\right )^{2} \text{ .} \end{equation}

In the UMZ–fissure hierarchy, the characteristic layer positions satisfy $y_{i+1}=\phi _c y_i$ , so the number of inertial-layer levels from $y^+\approx \sqrt {\delta ^+}$ up to $y^+\approx \delta ^+/2$ is

(6.5) \begin{equation} N(\delta ^+) \approx \frac {1}{2\ln \phi _c}\,\ln \delta ^+ \text{ .} \end{equation}

Assuming that correlations between distinct layers are weak, the nonlinear contribution to the wall-pressure variance may be written as

(6.6) \begin{equation} \langle p^{+2}\rangle _{\mathcal Q} \approx B_{\mathcal Q} + N(\delta ^+)\big \langle (p^{(1)+})^2\big \rangle \text{ ,} \end{equation}

so that

(6.7) \begin{equation} A_{\mathcal Q} = \frac {C_{\mathcal Q}}{2\ln \phi _c} \left (2\Delta U^+v_{\textit {rms}}^+\frac {\phi _c}{\alpha }\textrm {e}^{-1}\right )^{2} \text{ .} \end{equation}

It is convenient to factor out the purely geometrical numerical prefactor associated with the inertial-layer hierarchy. Writing (6.7) as

(6.8) \begin{equation} A_{\mathcal Q} = C_{\mathcal Q}\,\mathcal M(\phi _c) \left (\frac {\Delta U^+v_{\textit {rms}}^+}{\alpha }\right )^{2} \text{ ,} \end{equation}

we define

(6.9) \begin{equation} \mathcal M(\phi _c) = \frac {2\phi _c^{2}}{\ln \phi _c}\,\textrm {e}^{-2} \text{ .} \end{equation}

For the Fife similarity parameter $\phi _c \approx 1.62$ , this gives $\mathcal M(\phi _c)\approx 1.47$ , so that the remaining sensitivity of $A_{\mathcal Q}$ is contained entirely in the factors $\Delta U^+$ , $v_{\textit {rms}}^+$ , $\alpha$ and $C_{\mathcal Q}$ .

Measurements of the velocity jump across UMZ interfaces in high- $Re$ BL particle image velocimetry and DNS show $\Delta U^+=O(1)$ and typically slightly greater than unity. The UMZ/VF model of Bautista et al. (Reference Bautista, Ebadi, White, Chini and Klewicki2019) and the mean-vorticity analysis of Klewicki (Reference Klewicki2013) both indicate that $\Delta U$ scales on $u_\tau$ with a range of $\Delta U^+\in [1, 1.6]$ and $\langle \Delta U^+\rangle \gtrsim 1$ (Bautista et al. Reference Bautista, Ebadi, White, Chini and Klewicki2019), while the interface study of De Silva et al. (Reference De, Charitha, Philip, Hutchins and Marusic2017) finds velocity jumps concentrated in the range $\Delta U^+ \in [1, 2]$ . DNS channel flow (Hoyas & Jiménez Reference Hoyas and Jiménez2006) up to $\delta ^+ \approx 2,000$ likewise show that the inertial-layer wall-normal r.m.s. velocity satisfies $v_{\textit {rms}}^+ = O(1)$ , with representative values $v_{\textit {rms}}^+\sim [0.9, 1.1]$ . Taken together, these results motivate using a central estimate $\Delta U^+\approx 1.1$ and $v_{\textit {rms}}^+\approx 1$ for order-of-magnitude purposes.

The segment-length and spacing statistics of Cui & Jacobi (Reference Cui and Jacobi2025) further support an $O(1)$ ratio of along-edge coherence length to UMZ thickness, so for the estimate below we take $\alpha \approx 1$ . Finally, $C_{\mathcal Q}$ encapsulates the modest spread of Green’s function kernel weights, bandwidth and anisotropy about the idealised $k y\simeq 1$ line; in the absence of any strong bias we likewise treat it as an $O(1)$ factor.

Using the central values $\Delta U^+\approx 1.1$ , $v_{\textit {rms}}^+\approx 1$ , $\alpha \approx 1$ and $C_{\mathcal Q}\approx 1$ in (6.7) yields

(6.10) \begin{equation} A_{\mathcal Q}\approx 2 \text{ .} \end{equation}

If instead we let $\Delta U^+$ and $v_{\textit {rms}}^+$ range over the empirical values quoted above, (6.7) gives a broader, but still order-one, prediction,

(6.11) \begin{equation} 1.2 \lesssim A_{\mathcal Q} \lesssim 4.6 \text{ .} \end{equation}

Thus, the fissure-based picture combined with wall distance scaling robustly produces an $A_{\mathcal Q}$ of order unity, consistent with the logarithmic growth of $\langle p^{+2}\rangle$ inferred from figure 1(b).