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REVIEW 3 major objections 4 minor 58 references

Internal tides in the Mediterranean Sea

T0 review · 3 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read Internal tides are widespread in the Mediterranean Sea, with primary sources at the Gibraltar Strait, the Sicily Strait/Malta Bank, and the Hellenic Arc, and semidiurnal waves travelling hundreds of kilometres.

desk verdict First basin-wide internal tide mapping for the Mediterranean, with a genuinely new generation site at the Hellenic Arc; worth a serious referee despite energy totals that need a fix. read the letter →

arxiv 2411.19790 v1 pith:EBVNK5CW submitted 2024-11-29 physics.ao-ph

Bethany McDonagh , Jin-Song von Storch , Emanuela Clementi , Nadia Pinardi This is my paper · ORCID
classification physics.ao-ph
keywords internaltidesMediterraneanSeabarotropic-to-baroclinicenergyconversionNEMOICON-OsemidiurnaldiurnalHellenicArc
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper gives the first basin-wide, model-based census of internal tides in the Mediterranean Sea, a nearly enclosed sea whose weak surface tides have made internal waves easy to overlook. Using two independent high-resolution ocean models, it argues that internal tides are widespread, with the strongest generation at the Gibraltar Strait, the Sicily Strait/Malta Bank, and the Hellenic Arc. From those sites, twice-daily (semidiurnal) internal tides travel hundreds of kilometres into the Algerian, Tyrrhenian, and Ionian seas, while daily (diurnal) internal tides stay trapped against the seabed. A sympathetic reader would care because this map changes where ocean models should put tidal mixing: instead of being confined to a few straits, internal-tide energy is carried into deep basins, and the Hellenic Arc appears as a previously unknown source for the eastern Mediterranean.

What carries the argument

The load-bearing tool is the barotropic-to-baroclinic energy conversion rate $C = \int_{-D}^{\eta} g\rho' W \, dz$, where $\rho'$ is the tidal density perturbation and $W$ the vertical tidal velocity; positive $C$ marks places where the surface tide feeds internal waves, and maps of $C$ define the generation sites. Supporting diagnostics are vertical-mean baroclinic kinetic energy maps at M2 and K1 frequencies, wavenumber spectra along propagation paths, and first-mode wavelengths obtained by solving the Sturm-Liouville eigenvalue problem for vertical modes, which computes theoretical wavelengths from the stratification and separates true internal-tide peaks from Mediterranean seiches at nearby frequencies. Two models with different grids, tidal forcing, bathymetry, and mixing closures are run for the same month, so features common to both are treated as the robust signal.

What would settle it

Deploy moored current-and-density arrays for at least a lunar month across the Camarinal Sill, the Malta Bank, and the Hellenic Arc, and compute the observed conversion $C = \int_{-D}^{\eta} g\rho' W \, dz$; if the basin-wide total comes out far below the 1.36–2.89 GW range, or no coherent semidiurnal beams radiate from the Hellenic Arc into the Ionian Sea, the paper's central map is refuted.

Watch

Extended reading notes

Core claim

On the models' own terms, the discovery is a coherent basin-wide geography of internal tides in a sea often treated as nearly tideless. By computing the conversion of barotropic tidal energy into baroclinic (internal) tidal energy, $C = \int_{-D}^{\eta} g\rho' W \, dz$, from harmonic analysis of hourly model currents and densities, the paper locates the strongest generation at the Camarinal Sill in the Gibraltar Strait, the Sicily Strait and Malta Bank, and along the Hellenic Arc. The M2 (semidiurnal) internal tide is resolved as beams that leave these sites and cross hundreds of kilometres into the Algerian Sea, the Tyrrhenian Sea, and the Ionian Sea, with first-mode wavelengths of roughly 41–72 km depending on model and region; the K1 (diurnal) internal tide remains bottom-trapped near topography, with generation more widespread in the eastern basin in the NEMO model. Basin-integrated energy conversion totals 2.89 GW in NEMO and 1.36 GW in ICON-O, and the paper attributes the difference chiefly to diurnal-tide generation and to differences in bathymetry, stratification, and barotropic tides between the two configurations.

Load-bearing premise

The load-bearing premise is that the two hydrostatic models, validated only for surface tides against a global tide atlas and not against observed internal wave fields, still reproduce where and how strongly internal tides are generated and how far they travel; the paper's own conclusion asks for satellite and cruise data to test this.

Editorial extensions

If this is right

  • Basin-scale tidal mixing in the Mediterranean should not be treated as local to straits: semidiurnal internal tides carry energy hundreds of kilometres into the Algerian, Tyrrhenian, and Ionian seas before dissipating.
  • The Hellenic Arc, if confirmed by observations, is a new source region for the eastern Mediterranean's internal wave field and helps explain previously reported semidiurnal energy in the Ionian Sea and Cretan Passage.
  • The Mediterranean's total internal-tide generation, 1.36–2.89 GW, is small next to the roughly 1.7 TW global conversion, but it falls in a basin where tidal mixing is often set to near zero in regional models, so omitting internal tides would under-mix deep water.
  • The resolved first-mode M2 wavelengths of roughly 41–72 km are consistent with the models' horizontal resolution, meaning the main beams are captured while higher modes, which dissipate locally, are not.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The paper does not analyse seasons other than March 2022; a natural extension is that internal-tide generation at the Hellenic Arc and propagation into the Ionian Sea should strengthen or weaken with the seasonal pycnocline, so an annual model run would show whether the mapped geography is permanent.
  • The gap between the two models' basin totals (1.36 GW vs 2.89 GW) means the true conversion value is bracketed but not pinned down; direct estimates from moored arrays could narrow this range and test whether the Mediterranean's internal-tide energy budget is closer to the lower or upper estimate.
  • Because neither model resolves sub-kilometre topography such as the Messina Strait, the paper's own list of unresolved sites implies the basin totals may be lower bounds if such narrow straits also generate internal tides.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The manuscript presents a basin-wide numerical study of internal tides in the Mediterranean Sea using two independently developed high-resolution ocean models (NEMO v3.6 at 1/24° and ICON-O at R2B9), each providing hourly output for March 2022. The authors compute the barotropic-to-baroclinic energy conversion term C, map baroclinic kinetic energy and tidal beams for the M2 and K1 constituents, and extract internal tide wavelengths from wavenumber spectra that are compared with Sturm-Liouville solutions. The main claims are that internal tides are widespread; that the Gibraltar Strait, Sicily Strait/Malta Bank, and the Hellenic Arc are the primary generation sites; that semidiurnal internal tides propagate hundreds of kilometres; and that the basin-integrated internal tide generation is 2.89 GW in NEMO and 1.36 GW in ICON.

Significance. If the results hold, this is the first basin-wide map of internal tide generation and propagation in the Mediterranean and it identifies the Hellenic Arc as a previously undocumented strong generation region. The paper's use of two independent models with qualitatively consistent spatial patterns is a strength, as is the direct computation of C from diagnosed model fields with no parameter inversion. The comparison of spectral peaks to Sturm-Liouville wavelengths is a useful self-consistency check, and the barotropic-tide assessment against TPXO9 in Appendix B is good practice. However, the quantitative energy totals are not currently robust as reported: the positive-only summation of C and the single-month sample mean the headline numbers should be treated as preliminary rather than as a validated energy budget.

major comments (3)
  1. [Section 3.1, Table 3] The statement that 'only points with positive values are included in the calculation' means that the values 2.89 GW and 1.36 GW in Table 3 are positive-part sums of the signed conversion term C, not the total barotropic-to-baroclinic conversion. Because negative C represents genuine local conversion from baroclinic to barotropic energy through interference, discarding it systematically inflates the totals and could easily change the NEMO/ICON ratio of about 2.1. The authors should report the net (signed) integral, the separate positive and negative contributions (or at least the omitted negative sum), and a sensitivity test of the filter.
  2. [Section 3.1, Eq. (2)] As printed, Eq. (2) is not correct: the right-hand side, U·∇D + (D+z)·∇U, equals +∇·[(D+z)U], while the left-hand side, −∇(D+z)U, appears to denote the negative of that divergence; moreover, (D+z)·∇U is dimensionally inconsistent as a scalar. If the intended formula is the standard barotropic vertical velocity W = −∇·[(D+z)U], the RHS must be −U·∇D − (D+z)∇·U. Since the sign of C determines which points are counted as 'generation' in the positive-only sum, this diagnostic needs to be corrected and verified.
  3. [Section 2.2 and Section 5] All results are drawn from a single month (March 2022), yet the abstract and Table 3 present the generation totals as general Mediterranean values. Given the known seasonal variability of internal tides, the authors should either qualify the totals as March 2022 estimates or provide evidence that this month is representative. In addition, Section 5 explicitly acknowledges the absence of validation against internal-tide observations; given the central role of the quantitative conversion rates, a comparison with available in-situ or altimetric internal-tide data in at least one generation region (e.g., Gibraltar or Sicily Strait) would materially strengthen the claims.
minor comments (4)
  1. [Appendix A] Equations (A.1)-(A.5) are garbled in places (e.g., 'q = T ∗ 2', 'j = T ∗ − 1 2 , ..., T ∗ 2'); they should be carefully typeset and checked for mathematical correctness.
  2. [Section 3.3 and Table 4] The Sturm-Liouville wavelengths are computed from each model's own N2 and mean depth, so the agreement with the spectral peaks is a self-consistency check rather than an independent theoretical validation; the text should state this more explicitly.
  3. [Table 3] The boundaries of the regions used for the area-weighted means in Table 3 are not defined quantitatively; providing the masks or coordinates would improve reproducibility.
  4. [General] There are numerous typographical issues, including 'These works suggest' in Section 1, inconsistent spacing around C, and broken author names in the reference list (e.g., 'M¨ uller'); a careful proofread is needed.

Circularity Check

0 steps flagged · score 1.0 of 10

No circular derivation: the central claims are model diagnostics and the flagged positive-only C sum is a bias, not a circular step.

full rationale

The central internal-tide maps, propagation patterns, and energy totals are diagnostics computed directly from model-simulated fields (C from ρ′ and W; kinetic energy from harmonic analysis of currents), not outputs of an inversion or fit. The Sturm-Liouville wavelength comparison (Section 3.3, Appendix D) uses the same model's N² field to compute theoretical mode wavelengths and compares them with the model's own wavenumber spectra; this is an internal consistency check rather than an independent validation, but it does not force the spectral peaks and is not a circular reduction. The positive-only summation of C in Section 3.1 is a genuine methodological limitation that likely inflates the reported 2.89 GW and 1.36 GW totals by discarding negative baroclinic-to-barotropic conversion, but it is a bias in the estimator, not a case of a prediction being equivalent to its input by construction. Self-citations (McDonagh 2024; McDonagh et al. 2024) concern model tuning choices and interpretation of near-inertial interactions; they are not load-bearing for the basin-wide generation and propagation claims. External control exists for barotropic tides via TPXO9 in Appendix B, and the paper explicitly flags the absence of direct internal-tide validation in Section 5. That caveat concerns correctness and uncertainty, not circularity.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The paper's central results are diagnostics from two ocean model configurations. The free parameters are inherited from the models (mixing closures and bathymetry modifications) and are not fitted to the internal tide results in this study. The axioms are the modeling choices and mathematical assumptions on which the analysis rests.

free parameters (2)
  • NEMO TKE closure tuning parameters = not specified; tuned following McDonagh (2024)
    Vertical mixing coefficients are tuned in the NEMO configuration, affecting stratification and hence internal tide generation and propagation. The values are not provided in this paper (Section 2.1).
  • Gibraltar Strait bathymetry modifications = specific grid points altered
    Several points in the Gibraltar Strait are manually modified to improve mass transport and reduce spurious mixing (Section 2.1, after McDonagh 2024). This directly influences the model's internal tide generation at the primary site.
assumptions (4)
  • domain assumption Hydrostatic primitive equations with Boussinesq approximation are adequate for resolving first-mode internal tides in the Mediterranean Sea.
    Both NEMO and ICON-O are hydrostatic models (Section 2). The paper analyzes low-mode internal tides, for which hydrostatic theory is standard, but the assumption is not tested against non-hydrostatic solutions or observations.
  • ad hoc to paper One month (March 2022) of hourly output is representative of internal tide generation and propagation in the Mediterranean.
    Internal tides depend on stratification, which varies seasonally. The paper uses a single month without discussing representativeness or seasonal sensitivity (Section 2.2).
  • domain assumption Flat-bottom assumption in the Sturm-Liouville eigenvalue calculation is acceptable for computing first-mode internal tide wavelengths.
    Appendix D states 'primarily that a flat bottom is assumed' and calls it 'a caveat to this analysis.' The assumption is violated over the steep topography of the analyzed regions.
  • standard math Harmonic analysis over 744 hours cleanly separates the eight tidal constituents.
    The Foreman et al. (2009) method is standard, but a one-month record limits separation of constituents with close frequencies (e.g., S2 and K2).

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Pith. "Pith review of Internal tides in the Mediterranean Sea." pith.science (2026). https://pith.science/paper/EBVNK5CW

@misc{pith2026241119790,
  author       = {Pith},
  title        = {Pith review of: Internal tides in the Mediterranean Sea},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/EBVNK5CW}},
  note         = {Machine review of arXiv:2411.19790}
}
read the original abstract

The generation and propagation sites of internal tides in the Mediterranean Sea are mapped through a comprehensive high-resolution numerical study. Two ocean general circulation models were used for this: NEMO v3.6, and ICON-O, both hydrostatic ocean models based on primitive equations with Boussinesq approximation, where NEMO is a regional Mediterranean Sea model with an Atlantic box, and ICON a global model. Internal tides are widespread in the Mediterranean Sea. The primary generation sites: the Gibraltar Strait, Sicily Strait/Malta Bank, and Hellenic Arc, are mapped through analysis of the tidal barotropic to baroclinic energy conversion. Semidiurnal internal tides can propagate for hundreds of kilometres from these generation sites into the Algerian Sea, Tyrrhenian Sea, and Ionian Sea respectively. Diurnal internal tides remain trapped along the bathymetry, and are generated in the central Mediterranean Sea and southeastern coasts of the basin. The total energy used for internal tide generation in the Mediterranean Sea is 2.89 GW in NEMO and 1.36 GW in ICON. Wavelengths of the first baroclinic modes of the M2 tide are calculated in various regions of the Mediterranean Sea where internal tides are propagating, comparing model outputs to a theory-based calculation. The models are also intercompared to investigate the differences between them in their representation of internal tides.

Figures

Figures reproduced from arXiv: 2411.19790 by the authors.

Figure 1
Figure 1. Map of NEMO model domain with model bathymetry in coloured contours. Key [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Maps of the barotropic to baroclinic energy conversion term ( [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. Maps of the barotropic to baroclinic energy conversion term ( [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
Figures from the paper (10 more)
Figure 4
Figure 4. Figure 4: Model bathymetry in key regions for internal tide generation: Gibraltar [PITH_FULL_IMAGE:figures/full_fig_p012_4.png]
Figure 5
Figure 5. Figure 5: Maps of the barotropic to baroclinic energy conversion term ( [PITH_FULL_IMAGE:figures/full_fig_p013_5.png]
Figure 6
Figure 6. Figure 6: Vertical mean baroclinic kinetic energy in the Mediterranean Sea during March [PITH_FULL_IMAGE:figures/full_fig_p015_6.png]
Figure 7
Figure 7. Figure 7: Maps of baroclinic M2 current amplitude at the closest vertical level to 150m for [PITH_FULL_IMAGE:figures/full_fig_p016_7.png]
Figure 8
Figure 8. Figure 8: Maps of baroclinic M2 current amplitude at the closest vertical level to 300m for [PITH_FULL_IMAGE:figures/full_fig_p017_8.png]
Figure 9
Figure 9. Figure 9: Maps of baroclinic M2 current amplitude at the closest vertical level to 1000m [PITH_FULL_IMAGE:figures/full_fig_p017_9.png]
Figure 10
Figure 10. Figure 10: Spectra of zonal (u) and meridional (v) currents at three points: the Gibraltar [PITH_FULL_IMAGE:figures/full_fig_p018_10.png]
Figure 11
Figure 11. Figure 11: Wavenumber energy density spectra of M2 baroclinic currents for four regions [PITH_FULL_IMAGE:figures/full_fig_p020_11.png]
Figure 12
Figure 12. Figure 12: Vertical profiles of the Brunt V¨ais¨al¨a frequency for the NEMO simulation (blue) [PITH_FULL_IMAGE:figures/full_fig_p023_12.png]
Figure 13
Figure 13. Figure 13: Model bathymetry for a. NEMO, b. ICON, and c. Difference (ICON - NEMO). [PITH_FULL_IMAGE:figures/full_fig_p024_13.png]

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