3. Extracting the resonant fundamental harmonic response (FHR)

As we are actuating the eigenmode (TS wave) of a system with quadratic nonlinearities, a resonant response with harmonics must be expected. To extract those harmonic coherent structures, we apply SPOD to the symmetric ( ${\mathbf{q}}^S$ ) and anti-symmetric ( ${\mathbf{q}}^A$ ) components individually, yielding energy-optimal modes at each frequency in each symmetry class. Figure 2(a,b) contrasts the ${\mathbf{q}}^S$ and ${\mathbf{q}}^A$ spectra. The ${\mathbf{q}}^S$ spectrum is dominated by low-rank peaks at the TS frequency ( $f_k/f_1=1$ ) and its harmonics, characteristic of dominant coherent structures, while the noise-like ${\mathbf{q}}^A$ spectrum is broadband with slow rank decay, showing no energetically preferred modes. Thus, the FHR is symmetric, time-periodic and represents the expected deterministic result from symmetric, periodic actuation of the Navier–Stokes equations, as predicted by classical frameworks like harmonic balance (Rigas et al. Reference Rigas, Sipp and Colonius2021). The FHR can be assembled by a linear combination of the first symmetric SPOD modes $\boldsymbol{\unicode{x03C8}}^{(1)}_{k}$ of the harmonic peaks $f_k=f_n$ ( $n \in \mathbb{Z}$ ) in figure 2(a):

(3.1) \begin{equation} \tilde {\mathbf{q}}({{\mathbf{x}}},t) \equiv \sum _{n} \boldsymbol{\unicode{x03C8}}^{(1)}_{n} a^{(1)}_{n} e^{(\mathrm{i}2\pi f_n t)} + \bar {\mathbf{q}}({{\mathbf{x}}}) = \tilde {\mathbf{q}}_{\setminus 0}({{\mathbf{x}}}, t) + \bar {\mathbf{q}}({{\mathbf{x}}}), \end{equation}

with the time-averaged mean flow $\bar {\mathbf{q}}({{\mathbf{x}}})$ , and the block-wise estimated dominant SPOD expansion coefficients $a^{(1)}_{n}$ . We compute the SPOD using $N_{\textit{FFT}} = 512$ , eight periods per block and 50 % block overlap. Most notably, figure 2(c) shows that $\tilde {\mathbf{q}}({{\mathbf{x}}},t)$ exhibits compact spatial support when fully assembled and, while the resulting vortical structures closely resemble hairpin ‘packets’ or ‘forests’ in their late stages (often characterised as ‘chaotic’ in the turbulence literature (Adrian Reference Adrian2007; Wu & Moin Reference Wu and Moin2009)), we show that such seemingly disordered structures in fact arise deterministically here as the system’s resonant FHR to periodic actuation of its eigenmode.

As illustrated in figure 1 and § 2.1, subtracting the deterministic FHR from the full data yields a turbulent, cyclo-stationary fluctuation ${\mathbf{q}}^{\prime\prime} \equiv {\mathbf{q}} - \tilde {\mathbf{q}}$ . Further splitting this fluctuation into symmetry classes, ${\mathbf{q}}'' = {\mathbf{q}}^s + {\mathbf{q}}^a$ , isolates the data components of temporal and spatial symmetry breaking by construction. As $\tilde {\mathbf{q}}$ is symmetric and periodic, ${\mathbf{q}}^s$ must contain all non-periodic symmetric fluctuations that break time symmetry, while any indication of ${\mathbf{q}}^a$ captures the onset of spanwise asymmetry. This results in a pivotal multiscale separation. We later show in figure 4(c) that the rapid surge of ${\mathbf{q}}^s$ and ${\mathbf{q}}^a$ occurs with a simultaneous decline in the $\tilde {\mathbf{q}}$ amplitude, allowing us to locate exactly where symmetry breaking arises.

Figure 2.

SPOD energy spectra for (a) ${\mathbf{q}}^S$ symmetric and (b) ${\mathbf{q}}^A$ anti-symmetric components; compare with Sayadi et al. (Reference Sayadi, Hamman and Moin2013). (c) Superposition of $N_n$ = {1, 2, 4, 8, 16, 32} dominant SPOD modes $\boldsymbol{\unicode{x03C8}}^{(1)}_{n}$ at harmonic peaks $f_n$ (3.1); $Q$ -criterion ( $Q=10^3$ ) coloured by $u$ .

Figure 3.

Local power spectral densities of different quantities as a streamwise function of ${\textit{Re}}_x$ , coloured by the ratio of the dominant local SPOD energy over the full local energy $\lambda^{(1)}(x)/\sum _i \lambda^{(i)}(x)$ . (a) ${\mathbf{q}}^S$ full symmetric data. (b) ${\mathbf{q}}^s$ symmetric fluctuation. (c) Zoom on amplified region, progressive ablation of STPOD modes from ${\mathbf{q}}^s$ : first two modes (i.e. ${\mathbf{q}}^{\bullet }$ ; resulting in ${\mathbf{q}}^{s}-{\mathbf{q}}^{\bullet }={\mathbf{q}}^{\circ }$ symmetric residual) versus eight modes subtracted.

We next adopt a local perspective to gain a better overview of the streamwise evolution of these components. To this end, figure 3 plots local integral- $y^*$ – $z^*$ -plane power spectral densities for each streamwise station, coloured by the local SPOD energy ratio $\lambda^{(1)}(x)/\sum _i \lambda^{(i)}(x)$ to quantify the mode dominance. High values (blue) denote energy focused in one leading mode, while low values (red) reflect energy spread across many modes. This spatially resolved analysis reveals the statistics and dynamics that dominate at each location and frequency, guiding a subsequent search for coherent structures. The fundamental spectral evolution starts with the TS wave, marked in figure 3(a), whose energy level we use as reference for other components (above TS, significant; below TS, weak). Its FHR $\tilde {\mathbf{q}}$ manifests as dominant peaks that grow with increasing ${\textit{Re}}_x$ , while non-harmonic components remain negligible in the weak background spectrum. The initial flow is thus deterministic. As the flow gains energy towards $C_f^{{\textit{max}}}$ (at ${\textit{Re}}_x \approx 4 \times 10^5$ ), the non-harmonic spectrum gains amplitude with decreasing mode dominance, indicating increasing stochasticity, while the harmonic peaks lose both amplitude and their initial mode dominance. Since the flow is cyclo-stationary and all its energy is initially confined to $\tilde {\mathbf{q}}$ (see figure 4 c), the emergence of broadband content reflects energy transfer from the harmonics to ${\mathbf{q}}^s$ , marking the transition from periodic to time-symmetry-broken states. Beyond $C_f^{{\textit{max}}}$ , the spectrum converges towards a fully broadband, low modal dominance state, indicating statistical stationarity and turbulence.

Remarkably, figure 3(b) shows that even after removing the deterministic FHR $\tilde {\mathbf{q}}$ , the ${\mathbf{q}}^s$ fluctuation spectrum still exhibits prominent harmonic peaks with high modal dominance. This crucial finding indicates that fluctuations remain largely phase-locked to the harmonics initially and can also be represented by few dominant modes at these frequencies, thereby motivating a targeted search for coherent structures that dominate symmetry breaking.

Figure 4.

(a) ${\mathbf{q}}^s$ symmetric and (b) ${\mathbf{q}}^a$ anti-symmetric STPOD modal energy spectra; dominant modes with periodic dynamics coloured magenta. (c) Streamwise amplitude development of select STPOD modes $\boldsymbol{\unicode{x03D5}}^{s/a}_{\,m}({\boldsymbol{x}},\tau ) a^{s/a}_{m,j}$ , enveloped by the total ${\mathbf{q}}^S$ , ${\mathbf{q}}^s$ and ${\mathbf{q}}^{a}$ components. The TS wave input amplitude level serves as significance threshold.

4. Dynamics and statistics of temporal and spatial symmetry breaking

We apply STPOD to the ${\mathbf{q}}^s$ and ${\mathbf{q}}^a$ fluctuations using the method outlined in § 2.3 to identify dominant symmetry breaking modes with minimal assumptions. As we centre the data around the FHR $\tilde {\mathbf{q}}$ (which would otherwise be obtained as the first mode if not subtracted), $\tilde {\mathbf{q}}$ can be interpreted as the fundamental space–time mode, leading to the full decomposition

(4.1) \begin{equation} {\mathbf{q}}({{\mathbf{x}}},t) = {\mathbf{q}}^s({{\mathbf{x}}},t) + {\mathbf{q}}^a({{\mathbf{x}}},t) + \tilde {\mathbf{q}}({{\mathbf{x}}}, t) = \sum _{m} \left ( \boldsymbol{\unicode{x03D5}}_m^{s}({{\mathbf{x}}},\tau ) a_{m,j}^{s} + \boldsymbol{\unicode{x03D5}}_m^{a}({{\mathbf{x}}},\tau ) a_{m,j}^{a} \right ) + \tilde {\mathbf{q}}({{\mathbf{x}}}, t), \end{equation}

where $a_{m,j}^{s/a}$ with $j\in [1,N_{T_1}]$ are realisation-dependent expansion coefficients describing amplitude modulations of mode $\boldsymbol{\unicode{x03D5}}_m^{s/a}$ across different flow realisations (periods) $j$ . To further resolve the time dynamics of these space–time modes, we propose a modal multi-time scale approach via a nested space-only POD, separating spatial and temporal information:

(4.2) \begin{equation} \begin{aligned} \boldsymbol{\unicode{x03D5}}_m({{\mathbf{x}}},\tau) & = {{\mathbf{U}}}_m({\boldsymbol{x}}) {\mathbf{a}}_m^{\mathrm{POD}}(\tau), \\{\mathbf{q}}({{\mathbf{x}}},t) &= \sum _{m} {{\mathbf{U}}}_m({\mathbf{x}}) {\mathbf{a}}_m^{\mathrm{POD}}(\tau) a_{m,j} + {{\mathbf{U}}}_0({{\mathbf{x}}}) {\mathbf{a}}_0^{\mathrm{POD}}(t), \end{aligned} \end{equation}

with space-only POD modes ${\mathbf{U}}_m({\boldsymbol{x}})$ and local POD expansion coefficients ${\mathbf{a}}_m^{\mathrm{POD}}(\tau)$ . We choose POD for its $L_2$ -optimality and objectivity about time dynamics, unlike dynamic mode decomposition or Fourier methods, which assume linear or periodic dynamics. This decomposition now allows for a clear separation of time scales: the fast local dynamics ${\mathbf{a}}_m^{\mathrm{POD}}(\tau)$ within a realisation ( $\tau \in [0,T_1]$ ) and slow global modulation $a_{m,j}$ across realisations, where the instantaneous time becomes $t = \tau+(j-1)T_1$ . Furthermore, this enables a dynamical systems analysis by resolving mode trajectory shapes in phase space with ${\mathbf{a}}_m^{\mathrm{POD}}(\tau)$ (Cvitanović & Gibson Reference Cvitanović and Gibson2010). For visualisation, plotting the dominant three coefficients of the ${\mathbf{a}}_m^{\mathrm{POD}} (\tau)$ vector typically suffices, as higher coefficients generally remain periodic even when dominant ones are not.

Figure 5.

Symmetry breaking modes from STPOD. Deterministic mode $\tilde {\mathbf{q}}$ (FHR), first three symmetric modes ( $\boldsymbol{\unicode{x03D5}}^s_{1-3}$ ) and first two anti-symmetric modes ( $\boldsymbol{\unicode{x03D5}}^a_{1-2}$ ). (a) ${\mathbf{a}}_m^{\mathrm{POD}}(\tau)$ trajectory phase-spaces and (b) instantaneous isosurfaces at same phase. Symmetric modes shown by $Q$ -criterion ( $Q=10^3$ ) coloured by $u$ -velocity; anti-symmetric modes shown by $w$ -velocity ( $w=\pm 0.002$ ). (c) Symmetric and (d) anti-symmetric mode relative $\text{L}_2$ -norm distances between consecutive phases. Red markers denote distance from last to first phase (periodicity mismatch), error bars show the existing min $/$ max consecutive phase distances within the period, normalised by the average phase distance in a given mode. See supplementary movie 2.

4.1. Onset of quasi-periodicity and aperiodicity

Figure 4 plots the energy spectra and streamwise amplitude development of the STPOD modes. Up to ${\textit{Re}}_x \lesssim 3.5\times 10^5$ , the flow is governed entirely by the deterministic, spatially compact mode (FHR) of the early transition. Figure 4(c) shows that, as $\tilde {\mathbf{q}}$ falls below the total amplitude, new dominant symmetric STPOD modes $\boldsymbol{\unicode{x03D5}}_1^s$ and $\boldsymbol{\unicode{x03D5}}_2^s$ surge above the TS wave level at ${\textit{Re}}_x \approx 3.5\times 10^5$ . They too are vortical and compact, seen in figure 5(b), and mark the onset of variance from the purely periodic $\tilde {\mathbf{q}}$ . The phase space analysis in figure 5(a,c) reveals that these modes, remarkably, also exhibit $T_1$ -periodic dynamics:

(4.3) \begin{equation} \boldsymbol{\unicode{x03D5}}_m^s \left ( \boldsymbol{x}, \tau \rightarrow T_1 \right ) \begin{cases} = \boldsymbol{\unicode{x03D5}}_m^s \left ( {\mathbf{x}}, \tau \rightarrow 0 \right ), & m=\{1,2\}, \\ \neq \boldsymbol{\unicode{x03D5}}_m^s \left ( {\mathbf{x}}, \tau \rightarrow 0 \right ), & m \geqslant 3. \end{cases} \end{equation}

The critical distinction lies in their non-constant expansion coefficients $a_{1,j}^s$ and $a_{2,j}^s$ that vary over long time scales. This creates a region of geometrically quasi-periodic dynamics where the state is modulated by different periodic trajectories, i.e. for $3.4\times 10^5 \lesssim Re_x \lesssim 3.7\times 10^5$ ,

(4.4) \begin{equation} {\mathbf{q}}({{\mathbf{x}}},t) \approx \tilde {\mathbf{q}}({{\mathbf{x}}},t) + {\mathbf{q}}^{\bullet }({{\mathbf{x}}},t) = \tilde {\mathbf{q}}({{\mathbf{x}}}, t) + a_{1,j}^s\boldsymbol{\unicode{x03D5}}_1^s({{\mathbf{x}}},\tau) + a_{2,j}^s \boldsymbol{\unicode{x03D5}}_2^s({{\mathbf{x}}},\tau). \end{equation}

These trajectories are statistically periodic, but instantaneously depart from the deterministic state $\tilde {\mathbf{q}}$ in shapes and directions given by $\boldsymbol{\unicode{x03D5}}_1^s$ and $\boldsymbol{\unicode{x03D5}}_2^s$ , with amplitudes and timings governed explicitly by expansion coefficients $a_{1,j}^s$ and $a_{2,j}^s$ that vary slowly over the periods $j$ . This quasi-periodic dynamic is a categorical departure from the previous deterministic behaviour, as the system now explores multiple periodic trajectories in a structured manner.

The onset of chaos begins with the emergence of $\boldsymbol{\unicode{x03D5}}_3^s$ around ${\textit{Re}}_x \gtrsim 3.7\times 10^5$ . This mode breaks the pattern of local periodicity, exhibiting a clear jump between its first and last phase (figure 5 a,c). All higher modes ( $m \geqslant 3$ ) are also aperiodic and located increasingly further downstream. This creates the spectral redistribution from harmonic to broadband frequencies, shown in figure 3(c). A zoom into the region of critical fluctuation growth reveals the spectral contents of the modes. Removing the periodic modes $\boldsymbol{\unicode{x03D5}}_1^s$ and $\boldsymbol{\unicode{x03D5}}_2^s$ eliminates precisely the mode-dominant harmonic peaks, while removing higher modes flattens the spectrum across all frequencies towards fully broadband characteristics. Since modes that are not $T_1$ -periodic can, by definition, not be captured solely within the harmonic frequencies, they must fill the broadband spectrum between peaks, and are thus indicative of chaos and turbulence.

The progression from periodic to chaotic dynamics seen here echoes the dynamical systems view of wall turbulence (Cvitanović & Gibson Reference Cvitanović and Gibson2010; Kawahara & Kida Reference Kawahara and Kida2001; Viswanath Reference Viswanath2007), where trajectories recurrently visit the available unstable periodic orbits that ‘scaffold’ the state space. Our hierarchical breakdown – from a single periodic base state ( $\tilde {\mathbf{q}}$ ), through quasi-periodic modulations ( $\boldsymbol{\unicode{x03D5}}_1^s$ , $\boldsymbol{\unicode{x03D5}}_2^s$ ), to chaotic excursions via non-periodic modes ( $\boldsymbol{\unicode{x03D5}}_3^s$ and higher) – offers a data-driven perspective on this mechanism.

4.2. Onset of spanwise asymmetry

Spatial symmetry breaking follows a remarkably similar route. The anti-symmetric component ${\mathbf{q}}^A = {\mathbf{q}}^a$ remains energetically negligible until ${\textit{Re}}_x\lesssim 3.8\times 10^5$ (figure 4 c), from where it rapidly amplifies towards the amplitude of ${\mathbf{q}}^S$ . To isolate the modes that initiate asymmetry onset, we compute the STPOD for ${\mathbf{q}}^a$ with a spatial weight focused on the amplified region of interest ( ${\textit{Re}}_x \leqslant 4.0\times 10^5$ ) prior to the onset of saturated downstream turbulence. Most notably, the first anti-symmetric mode $\boldsymbol{\unicode{x03D5}}_1^a$ also exhibits local periodicity (figure 5 a,d)

(4.5) \begin{equation} \boldsymbol{\unicode{x03D5}}_m^a \left ( {\mathbf{x}}, \tau \rightarrow T_1 \right ) \begin{cases} = \boldsymbol{\unicode{x03D5}}_m^a \left ( {\mathbf{x}}, \tau \rightarrow 0 \right ), & m = 1, \\ \neq \boldsymbol{\unicode{x03D5}}_m^a \left ( {\mathbf{x}}, \tau \rightarrow 0 \right ), & m \geqslant 2, \end{cases} \end{equation}

despite a total absence of anti-symmetric forcing or mean flow. Similar to (4.4), $\boldsymbol{\unicode{x03D5}}_1^a$ forms quasi-periodic asymmetric variations with $\tilde {\mathbf{q}}$ , while spectral analysis analogous to figure 3(b,c) (omitted for brevity) confirms that higher modes again become broadband. Thus, a central finding here is that both spatial and temporal symmetry breaking emerge through coherent space–time structures in distinct dynamical hierarchies, not as random fluctuations.

5. Inter-modal $/$ inter-symmetry energy transfer

We introduce the symmetry-decomposed Navier–Stokes equations (NSE). By inserting the decomposition (2.2) into the incompressible NSE, followed by reflection about the symmetry plane and subsequent addition or subtraction of the reflection, we obtain respectively

(5.1) \begin{align} \hspace {-14pt}\text{(Symmetric NSE)}\;\;\hspace {5pt} \partial _t u_i^{S} {\kern-1pt} +{\kern-.5pt} u_j^S\,\partial _{x\!_j} u_i^{S}{\kern-1pt} +{\kern-.5pt} u_j^{A}\,\partial _{x\!_j} u_i^{A} &= -\partial \!_{x_i} p^{S} {\kern-1pt} +{\kern-.5pt} \nu \,\partial _{{x\!_j}{x\!_j}} u_i^{S}, \hspace {8pt} \partial \!_{x_i} u_i^{S}=0, \end{align}

(5.2) \begin{align} \hspace {-5pt}\text{(Anti-sym. NSE)}\;\;\hspace {5pt} \partial _t u_i^{A} {\kern-1pt} +{\kern-.5pt} u_j^{S}\,\partial _{x\!_j} u_i^{A} {\kern-1pt} +{\kern-.5pt} u_j^{A}\,\partial _{x\!_j} u_i^{S} &= -\partial \!_{x_i} p^{A} {\kern-1pt} +{\kern-.5pt} \nu \,\partial _{x\!_j x\!_j} u_i^{A}, \hspace {6pt} \partial \!_{x_i} u_i^{A}=0. \end{align}

As the cyclo-stationary anti-symmetric component is purely fluctuating ( $u_i^{A}=u_i^{a}$ ), its energy budget follows from dotting (5.2) with $u_i^{a}$ and phase-averaging $\widetilde {(\boldsymbol{\cdot })}$ , yielding the equation

(5.3)

in which the production pathway is shear extraction from the symmetric field consisting of FHR ( $\mathcal{P}^{\star \to a}$ ) and turbulent ( $\mathcal{P}^{s\to a}$ ) production for a phase-averaged TKE . For the symmetric NSE, we insert the cyclo-stationary decomposition $u_i^{S} = \widetilde {u}_i + u_i^{s}$ , and take dot products with $\widetilde {u}_i$ and $u_i^{s}$ each, followed by phase-averaging, to derive TKE budgets $k^{\star }\equiv \widetilde {u}_i \widetilde {u}_i/2$ and for the FHR and the symmetric fluctuations, respectively,

(5.4)

(5.5)

The three budgets are linked by pairs of equal-and-opposite production terms, which create directed energy transfers via harmonic ( $\mathcal{P}^{\star \to s}$ , $\mathcal{P}^{\star \to a}$ ) or inter-symmetry ( $\mathcal{P}^{s\to a}$ ) pathways. Similarly, we derive an inter-modal energy transfer between the leading modes of each symmetry class and their residual turbulent fluctuations. The symmetric fluctuation is split into $u_i^{s}=u_i^{\bullet }+u_i^{\circ }$ . With the respective dot products and phase-averaging, the TKE budget of the leading quasi-periodic symmetric modes (i.e. ${\mathbf{q}}^{\bullet } \!=\! a_{1}^s\boldsymbol{\unicode{x03D5}}_{1}^s \!+ \! a_{2}^s\boldsymbol{\unicode{x03D5}}_{2}^s$ ) is derived as

(5.6)

and the TKE budget for the symmetric residual fluctuation is given by a conjugate equation

(5.7)

We note that the sum of (5.6) and (5.7) exactly recovers the total symmetric TKE budget (5.5). Both feature conjugate source terms, $\mathcal{S}^{\circ \to \bullet }$ and $\mathcal{S}^{\bullet \to \circ }$ , that couple the energy flow between the leading modes and residual fluctuations, as well as standard production terms by the FHR ( $\mathcal{P}^{\star \to \bullet /\circ }$ ) and, notably, interaction terms with the anti-symmetric fluctuation $(\mathcal{T}^{\bullet /\circ \leftarrow a}\!-\mathcal{P}^{\bullet /\circ \rightarrow a})$ . Thus, we apply the split $u_i^{a}=u_i^{{\blacktriangle }}+u_i^{{\vartriangle }}$ to likewise derive the TKE budget of the leading quasi-periodic anti-symmetric mode $m=1$ (i.e. ${\mathbf{q}}^{{\blacktriangle }}\!=a_{1}^a\boldsymbol{\unicode{x03D5}}_{1}^a$ ) as

(5.8)

and the respective conjugate TKE budget for the anti-symmetric residual fluctuation as

(5.9)

Again, both (5.8) and (5.9) feature conjugate source terms ( $\mathcal{S}^{{\vartriangle }\to {\blacktriangle }}$ , $\mathcal{S}^{{\blacktriangle }\to {\vartriangle }}$ ), production terms with FHR and symmetric fluctuations, and recombine to form the total budget (5.3).

Figure 6.

Inter-modal/inter-symmetry energy transfer and production terms. (a) Symmetric and (b) anti-symmetric terms in integral- $y^*$ , time-averaged view. (c,d) Integral- $y^*$ – $z^*$ , time-averaged view, normalised by their total TKE (and small $\epsilon$ to avoid division by 0), yielding a TKE-specific efficacy of each pathway relative to the local energy. See supplementary movie 3. Legend: ( $\bullet /\circ$ ) quasi-periodic $/$ residual symmetric; ( $\blacktriangle /\triangle$ ) quasi-periodic $/$ residual anti-symmetric; ( $\star$ ) FHR.

Figure 6 reveals that in both symmetry classes, the production from the FHR to the leading modes ( $\mathcal{P}^{\star \rightarrow \bullet /{\blacktriangle }}$ ) arises and peaks upstream of their residual production ( $\mathcal{P}^{\star \rightarrow \circ /{\vartriangle }}$ ). This establishes a distinct streamwise succession: dominant modes are energised first by the FHR, serving as precursor for the subsequent growth of residual fluctuations. The mechanism underlying this transition is the inter-modal transfer, which exhibits an equal-and-opposite source-sink structure, i.e. $\mathcal{S}^{\bullet \rightarrow \circ }\!\approx \! -\mathcal{S}^{\circ \rightarrow \bullet }$ and $\mathcal{S}^{{\blacktriangle }\rightarrow {\vartriangle }}\!\approx \! -\mathcal{S}^{{\vartriangle }\rightarrow {\blacktriangle }}$ , visible in both their ${\textit{Re}}_x$ – $z^*$ footprints and line plots. Accordingly, we report only the transfer $/$ production terms essential for energetic coupling of the TKE budgets (transport terms $\mathcal{T}$ are numerically negligible). Most notably, the residual production arises only in lockstep with the inter-modal transfer, and not before or independently, implying a handoff mechanism wherein the residual field does not grow until energised by the transfer from the leading modes. This sequenced, directed energy transfer suggests that the leading modes act as energetic gateway, enabling the flow of energy from the deterministic FHR to the broadband turbulent residuals.

In the anti-symmetric case, the residual production $\mathcal{P}^{\star \rightarrow {\vartriangle }}$ , once energised via $\mathcal{S}^{{\blacktriangle }\rightarrow {\vartriangle }}$ , rapidly exceeds the efficacy of the mode production as it peaks in tandem with the nonlinear inter-symmetry production $\mathcal{P}^{s\rightarrow {\vartriangle }}$ . This term acts as a potent source for the anti-symmetric residual and as sink for the symmetric budget (green dashed lines in figure 6 c,d), confirming negligible back-coupling (i.e. $(\mathcal{T}^{s\leftarrow {\vartriangle }}\!\!-\!\mathcal{P}^{s\rightarrow {\vartriangle }}) \!\approx \! -\mathcal{P}^{s\rightarrow {\vartriangle }}$ ). Its rise occurs downstream of the initial mode production $\mathcal{P}^{\star \rightarrow {\blacktriangle }}$ and in tandem with the surging residual production, suggesting a sequenced breakdown: anti-symmetry initially arises via the dominant mode, but is subsequently amplified by nonlinear inter-symmetry transfer to produce the anti-symmetric broadband residual. Across all energy pathways, the quantities are spatially coherent in the ${\textit{Re}}_x$ – $z^*$ plane (figure 6 a,b), demonstrating that symmetry-broken turbulence is energetically driven by space–time coherent modes rather than random noise.