{"id":"2d47e613-ca9f-47b0-ae28-86c83c533a7f","arxiv_id":"2505.24075","paper_version":2,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":2.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"A perspective review of methods for extracting perturbation dynamics about time-varying base flows, identifying unresolved challenges in defining the base flow and extending operator-based analyses to aperiodic flows.","lead":"This paper is a perspective review of methods for studying flow structures that form on top of unsteady, time-changing base flows. It surveys data-driven, operator-based, and causality methods, and argues that canonical test cases and new techniques are needed.","discovery_kind":"review","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified: the paper is a perspective/review whose central claims are adequately scoped and internally consistent.","rationale":"The reader and I land on the same verdict (ACCEPT, high confidence). The weakest assumption identified by the reader—that a meaningful base-flow/perturbation decomposition exists for aperiodic flows—is real but the paper itself handles it responsibly. Section 2.2 explicitly discusses exact analytical solutions, numerical solutions, attracting trajectories, filtering/averaging procedures, triple decomposition, and exact coherent states as candidate definitions, and it warns that non-exact base flows are models rather than strict dynamical-systems descriptions. The paper is a perspective piece, so it is not required to resolve the base-flow definition problem; it is required to state it accurately and survey the existing tools, which it does. I considered whether the review's somewhat heterogeneous collection of methods (causality analysis, network-based methods, HHT, wavelet analysis) stretches the 'extracting dominant dynamics about unsteady base flows' frame, but the paper consistently ties each method back to the time-varying-base-flow setting and to the stated goal of understanding perturbation dynamics. I also considered whether the claimed low number of aperiodic-flow studies could be an artifact of self-selection, but the paper cites a representative set of such studies across multiple groups and explicitly frames the low number as a motivation for future work rather than an absolute fact requiring exhaustive citation. No mathematical steps in the paper are used as load-bearing evidence for a novel result, so correctness risk is appropriately low. The deep-learning omission is acknowledged in the concluding remarks, removing any concern of silent bias. Overall, I find no significant objection that would justify changing the verdict.","tokens_in":28081,"tokens_out":1592,"duration_ms":15931,"concrete_test":"As a verification step worth running: independently audit the review's claim in §3.2 that only a low number of studies treats transient growth about aperiodic base flows by performing a targeted literature search (e.g., Web of Science/Google Scholar with terms like 'transient growth' + 'time-dependent base flow' + 'non-periodic') and checking whether any substantive aperiodic-flow transient-growth studies are missing; if a major body of such work exists, the call-to-action strength would need slight softening, though the paper's core advice would remain intact.","verdict_should_be":"UNCHANGED","load_bearing_attack":"I read this as a perspective/review paper, not a methods paper, so the central claim is precisely the call to action in §3.2: the field should develop libraries of canonical transient flows and study transient growth of perturbations about aperiodic base flows. For that claim to hold, three conditions must be met: (i) the reviewed methods genuinely exist and work for unsteady/aperiodic base flows, (ii) the obstacles to their application are real and correctly identified, and (iii) the review does not materially misrepresent the state of the art. All three conditions are met. The methods surveyed (nonmodal stability, OTD modes, harmonic/wavelet/space-time resolvent variants, causality analysis, and others) are published in peer-reviewed venues and are presented with appropriate caveats about inputs and computational cost in Table 1 and §3. The obstacles—especially the base-flow definition problem in §2.2—are explicitly flagged rather than glossed over, including the statement that 'unless the unsteady base flow is an exact solution to the NSE, care must be taken' and that filtered or averaged base flows turn the analysis into 'a model beyond strict adherence to the dynamical systems theory.' The acknowledged omission of deep learning methods is also self-identified in the concluding remarks. No internally inconsistent claims or unsupported load-bearing steps emerged from my reading.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript is a perspective/review article that surveys methods for extracting dominant flow structures and their dynamics when the base flow is unsteady and aperiodic, rather than steady or periodic. It begins by formulating the perturbation equations about a time-varying base flow, highlighting the role of the forcing term \\check f(t) and the subtlety that a filtered or averaged base flow is not an exact solution of the Navier–Stokes equations. The paper then reviews data-driven modal decompositions (POD, SPOD, DMD and their time-localized variants), operator-based methods (nonmodal stability analysis, optimally time-dependent (OTD) modes, and harmonic, wavelet-based, and space-time resolvent analyses), causality analysis (Granger and information-theoretic), and other approaches (wavelets, network-based techniques, Hilbert-Huang transform). A summary table (Table 1) lists the typical inputs and computational considerations of each method. The central thesis, stated in Section 3.2, is that aperiodic flows are understudied and that the community should develop libraries of canonical transient flow systems and investigate transient growth in these flows. The paper concludes by acknowledging the omission of deep learning methods and calling for expanded use and further development of the reviewed techniques.","tokens_in":28279,"tokens_out":9174,"duration_ms":87013,"significance":"If the paper's assessment is correct, it serves an important community function by consolidating a disparate literature and identifying a concrete research direction: the construction of benchmark problems for aperiodic and transient base flows. The strengths of the paper include its honest treatment of the base-flow definition problem (Section 2.2), where the authors clearly state that non-exact base flows turn the analysis into a modeling effort beyond strict dynamical systems theory; the recognition of computational costs and trade-offs between matrix-based and matrix-free approaches; and the explicit self-identification of missing topics such as deep learning. The paper does not present new mathematical results or machine-checked code, but it is a well-scoped perspective that makes falsifiable predictions only in the weak sense of calling for future benchmark development. The methods surveyed are all published in peer-reviewed venues and are described with appropriate caveats, and the review appears to materially represent the state of the art. The main value is in the synthesis and the call to action, which is likely to be useful to researchers entering this area.","major_comments":[],"minor_comments":[{"comment":"In the text following Eq. (13), the statement that the transient growth is simply the leading singular value G(t) = σ1 of A(t) is inconsistent with the definition of G(t) as a ratio of squared norms; the maximum of that ratio is σ1^2, not σ1. Please correct the sentence to G(t) = σ1^2.","section":"Section 3.2, after Eq. (13)"},{"comment":"The block matrix shown in Eq. (33) does not display the off-diagonal blocks in the lower-left corner, making it difficult to verify that the matrix has the claimed Toeplitz structure. Please add the missing entries or use a more explicit notation, such as defining the entries T_{k,n} = ikωδ_{k,n} - \\hat L_{k-n}.","section":"Section 3.4.1, Eq. (33)"},{"comment":"The definition of the initial perturbation q'_{01}(t) is unclear: \\hat Y_r(t0)^T \\hat y_1(t) appears to have incompatible dimensions, and the normalization of q'_{01} is not specified. Please clarify the formula so that the ratio g1(t) is well-defined and consistent with the preceding text.","section":"Section 3.3, Eq. (27)"},{"comment":"The subsets Ω_b and Ω_f are defined using m ∈ Z and r ∈ Z, but a set indexed by all integers would be infinite. Please use nonnegative integers or an explicit finite range, e.g., m ∈ {0,1,...,M}.","section":"Section 3.4.1, after Eq. (31)"},{"comment":"There is a parenthesis mismatch in the notation \\tilde H_{t(η,ξ))}; it should be \\tilde H_{t(η,ξ)}. Please fix the typographical error.","section":"Section 3.4.2, Eq. (38)"},{"comment":"The phrase 'the eigenvector with the largest real eigenvalue' should be 'the eigenvector whose eigenvalue has the largest real part,' since eigenvalues are generally complex. Please revise the wording.","section":"Section 2.1, after Eq. (9)"},{"comment":"The product formula for A(nΔt) is exact only when L is piecewise constant over each interval. The text does call it a numerical approximation, but a brief clarifying comment would help avoid ambiguity for readers who might mistake it for the exact fundamental solution of a continuously time-varying system.","section":"Section 3.2, Eq. (12)"}],"recommendation":"minor_revision","confidential_remarks":"This is a perspective paper that fits the journal's scope well and makes no overreaching technical claims. The refereed issues are limited to typos and notation; I recommend forwarding with minor revision. Given the authors' transparency about limitations, the paper is likely to be influential in motivating benchmark development for transient flows."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Zoe,\n\nQuick take: this is a perspective/review that does exactly what it claims. It maps the current toolkit for analyzing perturbations about unsteady base flows, restates the relevant equations correctly, and is honest about the central difficulty—defining the base flow itself. There are no new results here and that is fine, because the contribution is organizational: a comparative table, a structured survey, and a call to build libraries of canonical transient flows.\n\nWhat actually earns credit: the paper gets the math right in the places I checked. The perturbation equations (4)-(5), the OTD formulation, and the resolvent variants are all standard and correctly presented. The treatment of the base-flow problem in §2.2 is more candid than most reviews: filtered or averaged base flows are explicitly labeled as 'a model beyond strict adherence to the dynamical systems theory.' The omission of deep learning methods is acknowledged in the conclusions, which is the right way to handle scope.\n\nThe soft spots are proportionate. First, the paper's central framework rests on a base/perturbation split that may not be well-defined for aperiodic flows. The authors flag this but do not offer practical guidance on how to choose the base flow in a given problem. That is understandable for a perspective, but it limits the actionable value. Second, the coverage is uneven: operator-based methods get deep treatment, while data-driven methods (POD/DMD variants) are covered more thinly. The deep learning omission is acknowledged, but it is a large hole for a paper about 'extracting dominant dynamics' in 2025. Third, there are minor typos in equations (15) and (33); nothing load-bearing.\n\nThe citation pattern looks fine. Self-citations are present but relevant, and the references are appropriate for a survey.\n\nWho this is for: a graduate student or researcher wanting a quick map of nonmodal, OTD, and time-varying resolvent methods, plus a clear statement of open problems. It deserves a serious referee. With a few typo fixes and maybe a short paragraph on practical base-flow selection, it would be a solid perspective piece.\n\nRecommendation: send it to peer review. It is a legitimate synthesis even if it doesn't move the needle on any single method.","headline":"A perspective/review that correctly maps the methods for unsteady base flows and honestly flags the base-flow definition problem; no new results, but a useful synthesis that deserves referee time.","tokens_in":28797,"tokens_out":2633,"would_cite":false,"duration_ms":27301,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"For flows whose base state itself changes in time, the standard modal and stability toolkit falls short, and the paper maps the alternatives that still work.","keywords":["unsteady base flows","aperiodic flows","transient growth","nonmodal stability analysis","optimally time-dependent modes","resolvent analysis","causality analysis","fluid flow modal analysis"],"falsifier":"A concrete test is to take an aperiodic flow with a known exact time-varying solution, such as an accelerating or decelerating channel, and compute the optimal transient growth twice: once about the exact solution and once about a temporally filtered approximation of the same field. If the growth envelopes or dominant perturbation shapes differ qualitatively, the base-flow choice controls the conclusions; if they nearly coincide, the modeling caveat is benign in that regime.","tokens_in":27896,"feed_emoji":"🌊","tokens_out":9402,"duration_ms":84595,"temperature":0.7,"pith_summary":"The paper argues that the standard methods for extracting dominant flow dynamics—proper orthogonal decomposition, dynamic mode decomposition, linear stability analysis, and resolvent analysis—were built around steady or time-periodic base flows and do not automatically work when the base flow itself evolves aperiodically in time. It reviews the emerging alternatives: data-driven decompositions adapted to transient windows, nonmodal stability analysis with optimal perturbations, optimally time-dependent (OTD) modes, time-varying resolvent variants, causality analysis, and other localized spectral tools. The authors' central call is that the field should build libraries of canonical transient flow systems and study transient growth of perturbations in these flows as systematically as it has been done for steady flows. This matters because pipes with sudden changes in flow rate, stalling airfoils, and forming vortices are exactly the aperiodic cases where analysis guidance is thinnest.","feed_headline":"Unsteady flows need time-aware stability tools","feed_subtitle":"A review maps nonmodal, OTD, and resolvent methods for aperiodic base states.","key_machinery":"The central object is the time-dependent linear operator $L[\\bar{q}(t)] = \\nabla N|_{\\bar{q}(t)}$ obtained by linearizing the nonlinear dynamics around the unsteady base flow, together with the fundamental solution operator $A(t) = \\prod_{j=1}^{n} e^{L[\\bar{q}(j\\Delta t)]\\Delta t}$ that propagates perturbations forward. The leading singular values of $A(t)$ give the transient growth envelope and the optimal initial perturbation; the evolving orthonormal basis that tracks the dominant directions of growth is the OTD subspace, whose evolution follows $dU_r/dt = L U_r - U_r(U_r^T L U_r - \\Phi)$; and time-localized or frequency-domain inverses of the operator define the resolvent variants for unsteady flows. This operator perspective is what ties the reviewed methods together: each approach is a different way of asking where and when perturbations can amplify about a moving base state.","core_discovery":"The paper's central claim is that perturbation dynamics about an unsteady base flow can still be extracted, provided the unsteady base state $\\bar{q}(t)$ is chosen deliberately and one of several time-aware frameworks replaces the frozen-time eigendecomposition. The organizing equation is the linearized perturbation equation $dq'(t)/dt = L[\\bar{q}(t)] q'(t) + \\check{f}(t)$, and the reviewed methods differ mainly in how they treat the forcing $\\check{f}(t)$ and how they represent time. Nonmodal stability computes the growth envelope through the fundamental solution operator $A(t)$ and its singular values; OTD mode analysis evolves an orthonormal basis that tracks the most amplified perturbation directions; harmonic, wavelet-based, and space-time resolvent analyses generalize the input-output operator to periodic and aperiodic base flows; and causality methods infer directional interactions with no restriction on the base flow's temporal evolution. The paper also contends that these methods are underused, that the choice of base flow is load-bearing, and that using a time-averaged or filtered base flow is a modeling effort rather than strict adherence to dynamical systems theory.","pith_inferences":["A testable extension is to assemble a benchmark suite of aperiodic flows with known exact time-varying base solutions—such as channel flows with time-varying wall motion—and run nonmodal, OTD, and time-varying resolvent analyses side by side to see whether their growth predictions agree.","Because the same base-split issue arises whenever a background state drifts, the paper's framework could transfer to cardiovascular pulsatile flow, atmospheric blocking events, or any climate signal treated as a slow base plus fast perturbations.","Comparing the nonmodal growth envelope with OTD-based amplification on the same unsteady flow would test the paper's conditional equivalence claim: the optimal perturbation must lie in the initial OTD subspace for the two answers to coincide.","The dimensionality of time-resolved resolvent operators grows with the number of temporal collocation points, so a practical next step is to establish error bounds for randomized and timestepping approximations in aperiodic settings before these methods become routine."],"forward_implications":["If the field adopts canonical transient base-flow test cases, future methods can be compared on common ground, just as pipe and Couette flows served as standard test beds for steady-state stability studies.","Nonmodal and OTD analyses can identify transient amplification in flows where frozen-time eigenanalysis predicts only decay, including accelerating and decelerating channels, vortex-airfoil interactions, and pitching airfoils.","Wavelet-based and space-time resolvent analyses can deliver time-localized forcing-response pairs for aperiodic flows, at a computational cost that will motivate the randomized and timestepping accelerations the paper mentions.","Causality analyses can build maps of directional interactions among flow components without assuming a stationary base flow, complementing the operator-based methods.","Results computed about a time-averaged or filtered base flow should be interpreted as outcomes of a model, so the paper's framing implies that such analyses need explicit checks of convergence, noise, and averaging effects."],"supporting_citations":[{"why":"Provides the standard modal-analysis toolbox of POD, DMD, stability, and resolvent methods whose steady-flow assumptions the review identifies as limiting.","marker":"[5]"},{"why":"Establishes hydrodynamic stability without eigenvalues, the theoretical basis for transient growth analysis about nonnormal and time-varying operators.","marker":"[14]"},{"why":"Sets up the resolvent/input-output framework for statistically stationary flows that the harmonic, wavelet-based, and space-time variants extend.","marker":"[16]"},{"why":"Supplies exact time-varying base-flow solutions for accelerating and decelerating channel flows, used as a demonstration case for optimal perturbations.","marker":"[44]"},{"why":"Provides the adjoint method that lets optimal perturbations be computed without explicitly forming the fundamental solution operator.","marker":"[64]"},{"why":"Defines nonmodal stability theory, the framework the paper argues should be applied more widely to aperiodic unsteady base flows.","marker":"[78]"},{"why":"Derives the OTD mode evolution equations from a minimization principle, the mathematical core of the OTD subsection.","marker":"[103]"},{"why":"Introduces wavelet-based resolvent analysis, one of the main methods the review presents for time-varying, aperiodic base flows.","marker":"[137, 138]"},{"why":"Presents space-time resolvent analysis with sparsity promotion, the other main method for aperiodic time-varying base flows.","marker":"[142, 129]"}],"fun_headline_variants":["Unsteady base flows demand time-resolved stability analysis","Nonmodal and OTD methods track aperiodic base flow dynamics","Review maps time-aware instability tools for unsteady flows","Extracting perturbation dynamics about time-varying base states","Aperiodic flows benefit from moving stability frameworks"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that every flow of interest can be split into an unsteady base state $\\bar{q}(t)$ and perturbations $q'(t)$ in a principled way, because the paper itself notes that a filtered or averaged base flow is a modeling choice rather than an exact solution to the Navier-Stokes equations.","fun_headline_variants_meta":{"raw":{"variants":["Unsteady base flows demand time-resolved stability analysis","Nonmodal and OTD methods track aperiodic base flow dynamics","Review maps time-aware instability tools for unsteady flows","Extracting perturbation dynamics about time-varying base states","Aperiodic flows benefit from moving stability frameworks"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000206,"raw_usage":{"total_tokens":1378,"prompt_tokens":910,"completion_tokens":468,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":526,"completion_tokens_details":{"reasoning_tokens":391}},"tokens_in":526,"tokens_out":468,"duration_ms":4929,"temperature":1.0,"reasoning_tokens":391,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T12:36:05.619434+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A concrete test is to take an aperiodic flow with a known exact time-varying solution, such as an accelerating or decelerating channel, and compute the optimal transient growth twice: once about the exact solution and once about a temporally filtered approximation of the same field. If the growth envelopes or dominant perturbation shapes differ qualitatively, the base-flow choice controls the conclusions; if they nearly coincide, the modeling caveat is benign in that regime.","supporting_citations":[],"review_version":1}