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Synchronous Rolling Study on an MR Tanker

2026-05-22 · weather-routing

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Abstract

On 27 January 2026, around 07:00 local (06:00 UTC), a laden MR tanker northbound through the Bay of Biscay in a tender condition (GM 1.6 m) experienced a peak roll of 22°. We hindcast the event from the ship's ECDIS track (ship's clock UTC+2), the ERA5 wave spectrum (Google ARCO) sampled along that track, and the vessel's hydrostatics. The conditions were a textbook synchronous-roll resonance: a persistent west-north-west Atlantic swell (~14 s mean, ~16 s dominant period) on the port quarter, whose encounter period held near the ship's 18.3 s natural roll period for essentially the whole transit (frequency ratio r ≈ 0.9–1.2). A single-degree-of-freedom roll model reproduces the resonance geometry; matching the observed 22° at resonance implies an effective damping ζ ≈ 0.045 — far below the model's 0.10 default (which gives ~10°, a 2.2× under-prediction). The model also carries a hard RAO cap (denominator floor → RAO ≤ 10) that tops the roll at ~20° here, structurally below the observed 22°. Both are model-fidelity gaps the calibration exposes.

1. The model

Roll is treated as a forced single-degree-of-freedom oscillator. For a regular wave of height Hs, length λ = gT²/2π and encounter angle μ (0° = following, 180° = head) on a hull of beam B:

  • Excitation (effective wave-slope, after Bhattacharyya): θ_static = π·Hs/λ·|sin μ|. Longer waves are gentler-sloped, so a long swell excites less roll per metre of height than a short steep sea.
  • Encounter frequency: ω_e = |ω − ω²·V·cos μ / g|, with ω = 2π/T the wave frequency and V the ship's speed. Following and quartering seas lengthen the encounter period above the wave period; head seas shorten it.
  • Dynamic amplification (RAO): 1/√[(1 − r²)² + (2ζr)²], with r = ω_e/ω_roll the frequency ratio and ζ the fraction of critical roll damping. The amplification peaks sharply at r = 1 (synchronism), capped there at 1/(2ζ).
  • Roll amplitude: φ = θ_static · RAO.

The natural roll period is taken from the IMO Intact Stability Code weather criterion, T_roll = 2·C·B/√GM with C = 0.373 + 0.023(B/d) − 0.043(L/100). A tender ship (low GM) has a long roll period, which both lowers the resonant encounter frequency and reduces the wave-radiation damping available at that frequency.

Resonance proximity is classified against IMO MSC.1/Circ.1228: synchronous rolling when T_e ≈ T_roll, parametric when T_e ≈ ½T_roll, and the following/quartering high-run zone when the sea is within ~45° of astern.

2. The reconstruction

  • Track, heading, speed: the ship's ECDIS voyage log (108 fixes through Biscay on 27 Jan), giving position, course, gyro heading and SOG (~11 kt) and the ship-measured wind.
  • Loading: master-reported GM 1.6 m; drafts 10.6 m forward / 11.7 m aft (mean 11.15 m, 1.1 m trim by the stern); 71 RPM; displacement ≈ 49,700 t from the vessel's hydrostatic table. → T_roll = 18.3 s.
  • Sea state: ERA5 reanalysis from the Google ARCO archive (0.25°, hourly) — combined sea plus the swell and wind-sea partitions (significant height, mean period, mean direction) and 10 m wind — sampled at every fix.
  • Outcome: observed peak roll 22° (bridge log).

The roll is driven by the full sea, not the swell alone: the response variances of the two near-resonant wave systems add, so the operative significant wave height is the combined Hs ≈ 7 m.

3. Results & interpretation

The storm wind peaked at 45.9 kn (ECDIS) around 00:00 UTC, building the sea to Hs ≈ 7 m. The heavy roll came later — at the incident around 06:00 UTC (07:00 local) — on the residual long swell as the wind eased. The hazard was swell-driven and decoupled from the instantaneous wind: a moderate-looking long swell on the quarter, not a wall of wind, was the danger. The encounter period held near the 18.3 s roll period (r ≈ 0.9–1.2) and the IMO synchronous + quartering zones were satisfied for essentially the whole transit; at the incident r = 0.92 — just below 1, on the long-encounter side.

The model reproduces this geometry. At its default damping (ζ = 0.10) it predicts a ~10° resonant peak — under-predicting the observed 22° by ~2.2×. The roll amplitude is therefore governed not by the (well-determined) resonance geometry but by the (poorly-determined) damping: calibrating to 22° at resonance yields an equivalent-linear ζ ≈ 0.045. Separately, the deployed kernel floors the RAO denominator at 0.1 (RAO ≤ 10), so it cannot exceed ~20° here regardless of damping — a hard structural cap below the observed 22°.

The speed trap (confirmed by the bridge). In following/quartering seas the encounter period lengthens with speed (ω_e = ω − ω²V cos μ/g). The master reported that reducing speed increased the roll, and increasing it reduced the roll but was capped by the engine load limiter. That speed-response is the decisive constraint: it places the ship just above the resonant speed, on the long-encounter-period flank. Reproducing it requires a dominant swell period near 16 s — longer than ERA5's energy-weighted mean swell period (13.9 s), i.e. the peak rather than the mean of the swell band, and consistent with ERA5 under-resolving the longest swell. With Tp ≈ 16 s on the port quarter the resonant speed is ≈ 8 kn: at the ship's ~11–12 kn, slowing toward 8 kn drove it into resonance (roll rising 14°→21°), while the proper escape — speeding up to push the encounter period clear of the 18.3 s roll period — was blocked by the engine load limiter (the engine was already at its torque/load ceiling in the seaway). Caught between can't-slow and can't-speed-up, the only remaining lever was a course alteration off the quarter — exactly the IMO MSC.1/Circ.1228 prescription. At a tender GM the synchronous band is wide and easily re-entered, so the alteration must be decisive.

4. Why the effective damping is so low

Ship roll damping (Ikeda) sums wave-radiation, skin-friction, eddy, bilge-keel, lift and appendage contributions. For this ship and condition every term points low:

  • Wave-radiation damping is small at 18.3 s. It is strongly frequency-dependent and falls toward zero at low frequency; a tender ship rolling at ω_roll ≈ 0.34 rad/s radiates very little. This is the dominant reason and a direct consequence of the low GM.
  • Full-form hull, rounded bilges → little eddy/form damping.
  • Bilge keels are modest relative to a ~50,000 t roll inertia, and their damping is amplitude-dependent and saturates at large angles.
  • Quartering, not beam, seas → gentler cross-flow at the bilges, so even the bilge keels do less.
  • Low speed (11 kt) → little lift damping.

A large tanker at synchronous roll genuinely lives near ζ ≈ 0.03–0.06, so the calibrated 0.045 is physically reasonable. As an equivalent-linear, steady-state value it also absorbs effects the single-DOF model omits — transient wave-group build-up (a run of resonant waves overshoots the steady-state amplitude), irregular-sea spectral spreading, and any under-estimation of the largest exciting waves by the 0.25° reanalysis — so the underlying hydrodynamic damping may sit slightly lower. Either way it is far below the model's 0.10 default.

5. Implications for the model

  1. Damping must be physics-based, not a constant. Replace the fixed ζ = 0.10 with an Ikeda-style component model (hull geometry + bilge keels + speed + amplitude), so damping follows the loading and sea instead of a single fudge factor.
  2. Roll period from the actual GM per voyage, not a fixed default — a tender condition can lengthen it from 14 s to 18 s and move the entire resonance picture.
  3. A transient / irregular-sea (spectral) roll solver to capture wave-group build-up, which the steady-state RAO cannot.
  4. Surface the encounter, swell and roll periods per leg so an operator (or the router) can see synchronous/parametric proximity directly — and route to keep T_e clear of T_roll.

Coordinates & reproducibility

  • Times: ECDIS timestamps are the ship's clock (UTC+2); UTC = clock − 2 h, local = clock − 1 h.
  • Incident (peak roll): 2026-01-27 06:00 UTC (07:00 local), 45.59°N 008.18°W, heading 056°T, SOG 12.1 kn.
  • Storm wind peak: 00:00 UTC, 44.58°N 008.93°W, 45.9 kn (ECDIS).
  • Track: 22:00 UTC (26 Jan) 44.24°N 009.19°W → 21:08 UTC (27 Jan) 48.10°N 006.14°W (NNE through Biscay). The full 108-fix track with per-fix heading, SOG, swell Hs/Tp/direction, combined Hs and reconstructed roll is in track_coordinates.csv (and incident_dataset.json).
  • ERA5 source (anonymous, free): Google ARCO store gs://gcp-public-data-arco-era5/ar/full_37-1h-0p25deg-chunk-1.zarr-v3 (xarray + gcsfs token="anon"). Variables: significant_height_of_combined_wind_waves_and_swell, peak_wave_period, significant_height_of_total_swell, mean_period_of_total_swell, mean_direction_of_total_swell, significant_height_of_wind_waves, mean_period_of_wind_waves, mean_direction_of_wind_waves, 10m_u_component_of_wind, 10m_v_component_of_wind. Window: 2026-01-27 00–23 UTC, hourly; bbox lat 43.25–48.75°N, lon 350–355°E (= 010–005°W), sampled to the track positions/times.
  • Ship: GM 1.6 m; drafts 10.6 m F / 11.7 m A; displacement ≈ 49,700 t; B 32.0 m, Lbp 179.6 m; hydrostatics from the vessel stability fixture.
  • Pipeline: studies-lib/{s3b_era5.py (ERA5 sampling), s3c_reconstruct.py (seakeeping + calibration + speed sweep), s3d_report.py (PDF)}.

References

  • IMO MSC.1/Circ.1228, Revised guidance to the master for avoiding dangerous situations in adverse weather and sea conditions (2007).
  • IMO 2008 Intact Stability (IS) Code, weather-criterion roll period.
  • Y. Ikeda, Prediction methods of roll damping of ships (component method).
  • R. Bhattacharyya, Dynamics of Marine Vehicles (1978) — wave-slope roll excitation.