{"id":"01d35e5b-d57d-4423-850d-48d856833add","arxiv_id":"1908.11796","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"In high-Reynolds Taylor-Couette flow, most momentum transport occurs at Taylor vortex in/outflow regions via plume bursts, and the vortices host azimuthally traveling waves similar to wavy Taylor vortex flow but with much larger wavenumber.","lead":"Experiments on turbulent Taylor-Couette flow show that angular momentum is carried mostly by plume bursts in the inflow and outflow regions of large Taylor rolls, and that the rolls themselves host azimuthally traveling waves even at very high Reynolds numbers. The results strengthen the analogy between Taylor-Couette flow and Rayleigh-Bénard convection and give a new organizing picture of turbulent rotating shear flows.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Traveling-wave claim hinges on interpreting CPOD mode 1 as a single coherent wave; phase and wavenumber from Eqs. (7.5)-(7.6) are only meaningful if the mode is quasi-monochromatic, which is not demonstrated.","rationale":"The reader's verdict is CONDITIONAL and its weakest-assumption identification matches the core of my concern: the CPOD mode is assumed to be a single coherent traveling wave without verifying that the phase functions are linear or that the mode is not a superposition of multiple waves or a standing component. This is the central load-bearing point because the paper's most novel claim is the detection of azimuthally traveling waves in the ultimate regime, and the quantitative wave parameters (frequency, wavenumber, phase speed) are derived directly from those phase functions. If the assumption fails, the wave numbers in Table 2 and the comparison to wavy Taylor vortex flow are not meaningful. The paper does provide independent evidence for the first part of the central claim (localized angular momentum transport in vortex in/outflow regions) through PDFs, joint velocity distributions, and spectra, so that part is less critically affected. The proposed concrete test, a joint spatiotemporal spectrum, directly checks whether the traveling wave exists in the raw data rather than only in the CPOD reconstruction, and would settle the concern. Therefore, the CONDITIONAL verdict remains appropriate: the central claim is plausible but would be materially strengthened by this check.","tokens_in":27090,"tokens_out":7908,"duration_ms":75360,"concrete_test":"Compute the two-dimensional Fourier transform (over azimuthal angle ϕ and time t) of the radial velocity fluctuations at the vortex-center height for cases C1 and C4. If a distinct spectral ridge appears at k ≈ 9 and f ≈ 1.80 Hz (C1) or 3.67 Hz (C4), consistent with the reported phase speeds, the traveling-wave interpretation is supported. If the spectral energy is broadly distributed, or if the ridge is not statistically significant compared to surrogate data with randomized Fourier phases, then the CPOD interpretation in Section 7 is not reliable.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The most load-bearing assumption is that the first complex POD mode, capturing only about 17% (classical) and 12% (ultimate) of the fluctuation energy, represents a single coherent azimuthally traveling wave whose phase speed and wavenumber can be read from the temporal and spatial phase functions in Eqs. (7.5)-(7.6). This requires the temporal coefficient a1(t) to be quasi-monochromatic, with an essentially linear phase advance, and the spatial mode to contain a single azimuthal wavenumber, with a linear azimuthal phase. The paper does not report the time series of φ1(t) or the azimuthal profile of Φ1(r=0.5,ϕ); it reports only the resulting averages fw = 1.80 Hz and 3.67 Hz and k ≈ 9. If a1(t) is a superposition of several frequencies, or if the mode contains multiple azimuthal m-components, then the instantaneous frequency dφ1/dt and the local wavenumber ∂Φ1/∂ϕ are not well-defined, and the reported wave parameters may be artifacts of applying the Hilbert transform to broadband turbulence. The limited azimuthal field of view, which the paper itself notes in §6.1 as restricting the resolvable range of azimuthal scales, further complicates the interpretation: an oscillation within a sector does not by itself establish a global azimuthal mode around the entire circumference. The central claim that turbulent Taylor vortices support azimuthally traveling waves, especially in the ultimate regime, therefore rests on an unverified single-mode, quasi-monochromatic assumption.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This experimental paper reports planar PIV measurements in horizontal planes of a Taylor-Couette facility with radius ratio η=0.714 and aspect ratio Γ=18.3, covering shear Reynolds numbers up to ReS=3.5×10^5 and rotation ratios µ=0 and µ=µmax. The authors validate their velocity statistics and torque results against literature data and DNS, then use the data to characterize the local angular momentum transport, the statistics of the net convective Nusselt number, azimuthal energy co-spectra, and large-scale flow organization. The two headline claims are that the local angular momentum transport is concentrated in the vortex inflow and outflow regions, and that a complex proper orthogonal decomposition reveals azimuthally traveling waves superimposed on turbulent Taylor vortices in both the classical and ultimate regimes, with wavenumber approximately 9 and phase speeds 0.35Δω and 0.11Δω, respectively. The paper also draws an analogy to Rayleigh-Bénard convection based on plume-like small-scale structures and exponential tails in the transport PDFs.","tokens_in":27433,"tokens_out":11191,"duration_ms":109396,"significance":"The transport-location claim is well supported: the meridional maps of the net convective Nusselt number, the joint PDFs of the radial and azimuthal velocity fluctuations, and the inflow/outflow contribution profiles form a coherent picture, and the global Nusselt numbers agree with independent torque measurements and DNS. The comparison of the PDFs to the product-of-Gaussians prediction is a reference normalization rather than a fitted model, so the statistics section is not circular. If the traveling-wave claim holds, the paper would provide experimental evidence of wavy-Taylor-vortex-like azimuthal waves in highly turbulent Taylor-Couette flow, a result of considerable interest to the rotating-turbulence and convection communities, and it would extend the TC-RB analogy. The main weakness is that the traveling-wave conclusion currently rests on the interpretation of a single CPOD mode that captures only 12–17% of the fluctuation energy, and the paper does not provide the mode diagnostics needed to exclude alternative interpretations.","major_comments":[{"comment":"The traveling-wave claim, which is a headline result, is not yet fully established because the phase-based extraction of fw and k assumes that CPOD1 is a single quasi-monochromatic traveling wave. The paper reports only the averaged values fw=1.80 Hz, fw=3.67 Hz, and k≈9, but does not show that the temporal phase φ1(t) advances linearly in time, that the azimuthal phase profile Φ1(r=0.5,ϕ) is linear, or that the mode's azimuthal Fourier spectrum contains a single dominant wavenumber. This matters because CPOD1 captures only 17% (classical) and 12% (ultimate) of the fluctuation energy, and because §6.1 states that the limited azimuthal field of view prevents resolving the large-scale peak. I request that the authors add (i) the time trace of φ1(t) and its time derivative, (ii) the azimuthal profile of Φ1 at r≈0.5 and its derivative, (iii) the azimuthal wavenumber spectrum of CPOD1, and (iv) a statement of the angular extent of the PIV field of view. Without these diagnostics, the global wavenumber and phase speed are not well defined, and the result should be presented as a local traveling wave within the measured sector.","section":"§7.2–7.3, Eqs. (7.5)–(7.6)"},{"comment":"The quantitative statement that the vortex outflow contributes up to approximately 60% of the net convective transport in the inner gap region depends on the operational definition of the in- and outflow regions. The manuscript describes locating the in- and outflow by the extrema of the mean radial velocity profile and trimming the axial interval to one vortex pair by excluding the gray data points in Fig. 3e, but it does not specify how the axial bands assigned to the in- and outflow are chosen or how sensitive the resulting fractions are to that choice and to the number of heights included. Since this 60% figure is presented as a striking result, I ask the authors to provide a sensitivity test with respect to the band width and the axial averaging interval, or to soften the quantitative claim accordingly.","section":"§4, Fig. 11"},{"comment":"The claim that azimuthally traveling waves are detected in the ultimate regime is currently based on a single flow state, case C4 at ReS=6.68×10^4 and µ=−0.36. The higher-Reynolds-number cases at µmax, namely C5 and C7, are not analyzed with the CPOD procedure, even though these are the cases for which the inflow/outflow transport asymmetry is strongest. To support the general statement 'in the ultimate regime,' the authors should either extend the CPOD analysis to the highest-Re µmax case or explicitly limit the claim to the specific Reynolds number studied in §7.3 and note this limitation in the abstract.","section":"§7.3"}],"minor_comments":[{"comment":"The sentence 'the columns are assigned to the spatial grid points (1:M) and rows to the time signals (1:N)' contradicts the displayed matrix, where rows correspond to spatial points; please correct the dimension description.","section":"§7.1, Eq. (7.3)"},{"comment":"There are typographical errors in the references: 'Donelly' should be 'Donnelly' and 'Suanto et al.' appears to be a misspelling of 'Sutanto et al.'; also, the LaTeX control word 'greaterorequalslant' appears in the introduction and should be fixed.","section":"References and §1"},{"comment":"The Gaussian-product prediction of Eq. (5.2) is evaluated using ρP from case C8 (ReS=3.51×10^5, µ=0); please state whether the shape comparison is robust to using another pure-inner-rotation case, since the predicted tail slope depends on ρP.","section":"§5, Fig. 12"},{"comment":"The phrase 'the limited range of the azimuthal coordinate' is never quantified; please state the angular extent of the PIV field of view, because this directly affects the interpretation of the large-scale spectral peak and the CPOD results.","section":"§6.1"},{"comment":"The CPOD mode plots are shown without colorbars or quantitative scales; adding them would help the reader judge the mode amplitude and the relative strength of the radial and azimuthal components.","section":"Figs. 18 and 21"},{"comment":"The text states that the temporal power spectrum for the ultimate-regime case is 'not shown'; please include the spectrum or at least report the peak frequency and its width quantitatively in the text.","section":"§7.3"},{"comment":"The classical-regime phase speed is given as 0.35ω1 in the text and 0.35Δω in Table 2; the two expressions coincide for µ=0, but the notation should be unified to avoid confusion.","section":"Table 2"},{"comment":"The generalization from Taylor-Couette flow to pipe and channel flow is a speculation; please mark it as a hypothesis rather than a conclusion, or remove it from the summary.","section":"§8"}],"recommendation":"major_revision","confidential_remarks":"The traveling-wave claim is the most prominent and potentially most cited result of the paper, but the current evidence is insufficient to rule out a superposition of modes or a sector-localized oscillation. I would ask the editor to insist on the additional mode diagnostics proposed in the major comments. The transport and plume parts of the paper are solid and could stand on their own; the authors should either strengthen the wave claim or explicitly reframe it as a more limited observation."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things to know. First, the paper gives the clearest experimental evidence I've seen that in turbulent Taylor-Couette flow at the torque maximum, nearly all net angular momentum transport happens in narrow axial bands at the vortex in/outflow, carried by intermittent plume-like bursts. That claim is well supported. Second, the authors use complex POD to report azimuthally traveling waves on the turbulent Taylor rolls, with wavenumber about 9, in both the classical and ultimate regime. That claim is plausible but not fully nailed down.\n\nThe transport result is solid. The measurements are validated against literature torque and velocity data. The joint PDFs and meridional flux maps are convincing. The PDF tails showing exponential behavior and the RB analogy are a nice addition. The spectral analysis identifying plumes with k_phi d in [10,20] is consistent with the transport picture.\n\nThe wave claim is the soft spot. The first CPOD mode captures only 12–17% of the fluctuation energy. The paper treats it as a single coherent traveling wave and reads off a frequency and wavenumber from the phase of the temporal coefficient and the spatial mode. It doesn't show the time series of the phase derivative or the azimuthal profile of the spatial phase, so we can't tell whether the mode is quasi-monochromatic. If a1(t) contains multiple frequencies, the Hilbert transform gives a time-averaged number that may not correspond to a real wave. Also, the limited azimuthal field of view, which the paper notes in §6.1, means the apparent modulation could be a local oscillation rather than a global mode. That said, the space-time diagram does show coherent diagonal bands, and the spectral peak in a1(t) matches the full-field radial velocity, so I don't think the claim is baseless. But the wave parameters need uncertainty estimates, and the single-mode assumption should be tested.\n\nThe PDF tails are based on 1500 snapshots per height; that's a minor limitation for extreme event statistics, but the qualitative features are clear enough.\n\nOverall, the central transport claim holds up. The wave detection is an interesting observation that deserves a serious referee, but the paper would benefit from either more diagnostics or a more cautious interpretation.\n\nWho's this for: anyone working on TC or RB convection, and the POD/turbulence community. I'd bring it to my reading group, and I'd probably cite the transport localization if I write about TC.\n\nRecommendation: send it to peer review. Ask the authors to add robustness checks on the CPOD mode (e.g., instantaneous frequency over time, azimuthal phase profile, maybe a two-mode fit) and to quantify uncertainties on the wave parameters. That's a reasonable revision, not a deal-breaker.","headline":"Solid transport localisation, plausible but under-verified wave detection; worth refereeing with requests for CPOD diagnostics.","tokens_in":27951,"tokens_out":2822,"would_cite":true,"duration_ms":26124,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"Turbulent Taylor-Couette flow, even in the ultimate regime at shear Reynolds numbers up to 350,000, retains large-scale azimuthally traveling waves atop its Taylor vortices, and most angular momentum transport is concentrated in the…","keywords":["Taylor-Couette flow","turbulent Taylor vortices","angular momentum transport","Nusselt number","plumes","azimuthal traveling waves","complex proper orthogonal decomposition","wavy Taylor vortex flow"],"falsifier":"A two-dimensional Fourier transform of the first-mode space-time field, with azimuth on one axis and time on the other, would settle the matter: one sharp ridge with frequency over wavenumber equal to the quoted phase speed confirms a single traveling wave, while multiple peaks or a broad smeared ridge would show that the complex-POD phase extraction averaged over several wave components.","tokens_in":2021,"feed_emoji":"🌀","tokens_out":8232,"duration_ms":123708,"temperature":0.7,"pith_summary":"This paper uses planar velocity measurements at 23 heights in a Taylor-Couette apparatus with radius ratio 0.714 to ask how large-scale Taylor rolls shape small-scale turbulence and angular momentum transport up to shear Reynolds numbers of 350,000. It argues that when rolls are prominent, the net angular-momentum flux is not spread through the gap but concentrated in the vortex inflow and outflow, where radial and azimuthal velocity fluctuations are strongly correlated; the outflow near the inner cylinder alone can carry about 60 percent of the transport. Those bursts of flux show up as exponential tails in local Nusselt-number probability distributions and as small-scale plumes with azimuthal extent $k_\\phi d \\in [10,20]$. By applying a complex proper orthogonal decomposition at the vortex center, the paper also detects azimuthally traveling waves superimposed on the turbulent Taylor vortices in both the classical and ultimate regimes, with azimuthal wavenumber about 9 and phase speeds of the same order as earlier wavy Taylor-vortex measurements. These results link small-scale plume statistics to the large-scale roll structure and suggest that the wavy-vortex phenomenon survives into a regime where the boundary layers are fully turbulent.","feed_headline":"Waves survive in ultimate Taylor-Couette turbulence","feed_subtitle":"Experiments to Reynolds number 350,000 find waves on turbulent vortices; most momentum flux sits in narrow bands.","key_machinery":"Two tools carry the argument. The first is the net convective angular-momentum flux, expressed as a quasi-Nusselt number $r^2\\langle u'_r u'_\\phi \\rangle / J_{\\rm lam}$, evaluated on cylindrical surfaces and at specific vortex positions; this localizes the transport. The second is complex proper orthogonal decomposition (CPOD), in which each velocity-fluctuation field is made complex with a Hilbert transform and then decomposed by singular value decomposition. In CPOD, a mode whose imaginary spatial pattern is azimuthally shifted from its real part represents a wave traveling around the cylinder; the time derivative of the mode's phase gives the wave frequency, and the azimuthal derivative of its spatial phase gives the wavenumber. The first CPOD mode is the one that shows a clean traveling-wave pattern, and the same mode's structure is compared with the known wavy Taylor vortex pattern. Azimuthal energy co-spectra serve as a third piece, identifying the plume scale and showing that the large-scale spectral peak vanishes when the rolls are absent.","core_discovery":"The central claim is that the organization of fully turbulent Taylor-Couette flow is set by large-scale Taylor rolls whose inflow and outflow bands are the main highways for angular momentum, and that these rolls are not axisymmetric: they carry azimuthally traveling waves. This is established for radius ratio 0.714 with pure inner-cylinder rotation in the classical regime and with counter-rotation at the torque maximum in the ultimate regime. The net convective Nusselt number, built from the correlated radial and azimuthal velocity fluctuations, peaks in the inflow and outflow; its local probability distribution has its most probable value at zero and exponential positive tails that grow with Reynolds number; and pre-multiplied azimuthal co-spectra place the underlying plumes at wavenumbers $k_\\phi d \\approx 10$ to $20$, strongest where the rolls pull fluid off the walls. The first complex-POD mode captures about 17 percent of the fluctuation energy in the classical case and about 12 percent in the ultimate case, shows real and imaginary parts azimuthally shifted by a quarter wavelength, and yields a wave frequency of 1.80 Hz in the classical regime and 3.67 Hz in the ultimate regime, an azimuthal wavenumber of about 9, and phase speeds of roughly $0.35\\Delta\\omega$ and $0.11\\Delta\\omega$. The similarity between these Nusselt-number distributions and those of heat flux in Rayleigh-Benard convection is presented as evidence that the same plume mechanism transports angular momentum in both flows.","pith_inferences":["If confirmed, the detected wave provides a candidate mechanism for setting the spanwise scale of large-scale superstructures in Taylor-Couette and related turbulent flows; checking whether the same $k_\\phi \\approx 9$ signature appears in direct numerical simulations at comparable parameters would test that idea.","The wave was characterized only at the vortex center; a natural extension is to track the phase across several heights to test whether the wave is a rigidly traveling azimuthal modulation or a more complex three-dimensional pattern.","Because the ultimate-regime phase speed is slower relative to the imposed shear, the wave may be a passive remnant of the roll structure rather than an active participant in the momentum transport; varying the rotation ratio at fixed shear Reynolds number while monitoring wave amplitude would separate roll strength from Reynolds-number effects."],"forward_implications":["At torque-maximum rotation, global torque measurements alone understate the spatial concentration of transport: up to about 60 percent of the net convective flux occurs in the outflow band near the inner cylinder, so models of ultimate Taylor-Couette transport should treat the inflow and outflow bands as the controlling regions.","Local Nusselt-number probability distributions peak at zero and develop exponential tails, implying that the time-averaged flux is carried by rare, strong plume events; statistical descriptions of the transport must reproduce this intermittency.","Azimuthal traveling waves with wavenumber about 9 appear whenever Taylor rolls are present, in both the classical and ultimate regimes, and not when the rolls are absent, so the large-scale roll state is intrinsically three-dimensional rather than axisymmetric.","Because the detected wave speeds are of the same order as the classical wavy Taylor vortex speeds, the wavy-vortex instability is a plausible continuation into the turbulent regime rather than an unrelated large-scale mode."],"supporting_citations":[{"why":"Defines the angular momentum current Jω and the quasi-Nusselt number Nuω that the transport analysis builds on.","marker":"Eckhardt et al. (2007a)"},{"why":"Provides the phase diagram and transition criterion used to classify classical versus ultimate regimes and to set the torque-maximum rotation ratio.","marker":"Ostilla-Monico et al. (2014c)"},{"why":"Supplies the comparative Nusselt-number probability distribution analysis between Taylor-Couette and Rayleigh-Benard flow that the exponential-tail interpretation extends.","marker":"Brauckmann et al. (2016b)"},{"why":"Reports plume velocity measurements in Taylor-Couette flow that support identifying the detected small-scale structures as plumes.","marker":"van der Veen et al. (2016b)"},{"why":"Supplies the wavy Taylor-vortex wave speeds against which the detected wave speeds are compared.","marker":"King et al. (1984)"},{"why":"Documents the reappearance of azimuthal waves in turbulent Taylor-Couette flow and motivates searching for them at higher Reynolds numbers.","marker":"Wang et al. (2005)"},{"why":"Provides the Hilbert-transform-plus-singular-value-decomposition complex POD procedure used to extract traveling modes.","marker":"Harlander et al. (2011)"},{"why":"Establishes that the time derivative of the complex POD phase equals the angular frequency, the step that converts phase into wave frequency.","marker":"Suanto et al. (1998)"}],"fun_headline_variants":["Traveling waves ride Taylor vortices in ultimate turbulence","Momentum rides plumes in Taylor-Couette rolls","Wavy vortices persist to extreme Taylor-Couette flow","Plumes and waves organize ultimate Taylor-Couette flow","Inflow-outflow bands carry momentum in Taylor-Couette"],"cache_read_input_tokens":30080,"weakest_assumption_plain":"The traveling-wave result rests on the assumption that the strongest decomposition mode is a single coherent disturbance circling the cylinder; if that mode instead mixes several disturbances or contains a standing part, the quoted frequency, wavenumber, and speed are not well defined.","fun_headline_variants_meta":{"raw":{"variants":["Traveling waves ride Taylor vortices in ultimate turbulence","Momentum rides plumes in Taylor-Couette rolls","Wavy vortices persist to extreme Taylor-Couette flow","Plumes and waves organize ultimate Taylor-Couette flow","Inflow-outflow bands carry momentum in Taylor-Couette"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000236,"raw_usage":{"total_tokens":1606,"prompt_tokens":1151,"completion_tokens":455,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":767,"completion_tokens_details":{"reasoning_tokens":372}},"tokens_in":767,"tokens_out":455,"duration_ms":4119,"temperature":1.0,"reasoning_tokens":372,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T10:06:00.990864+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A two-dimensional Fourier transform of the first-mode space-time field, with azimuth on one axis and time on the other, would settle the matter: one sharp ridge with frequency over wavenumber equal to the quoted phase speed confirms a single traveling wave, while multiple peaks or a broad smeared ridge would show that the complex-POD phase extraction averaged over several wave components.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the wavy Taylor-vortex wave speeds against which the detected wave speeds are compared."},{"cited_title":", Olsen, M","cited_arxiv_id":null,"evidence_quote":"Documents the reappearance of azimuthal waves in turbulent Taylor-Couette flow and motivates searching for them at higher Reynolds numbers."},{"cited_title":", von Larcher, T","cited_arxiv_id":null,"evidence_quote":"Provides the Hilbert-transform-plus-singular-value-decomposition complex POD procedure used to extract traveling modes."},{"cited_title":", Zheng, Q","cited_arxiv_id":null,"evidence_quote":"Establishes that the time derivative of the complex POD phase equals the angular frequency, the step that converts phase into wave frequency."}],"review_version":1}