Measuring how tidal patterns change over time

For more than a century, oceanographers have described the rise and fall of the sea with one of science’s most elegant and durable tools: harmonic analysis. By decomposing a water-level record into a handful of sinusoidal constituents whose frequencies are fixed by the motions of the Moon and Sun, researchers can predict tides at ports around the world with astonishing accuracy. Yet the method rests on a quiet assumption, that the tide is stationary, unchanging in amplitude and phase over time. Anyone who has watched a flood surge race up a river, or traced a burst of internal waves crossing a deep-ocean sill, knows that assumption frequently fails. What has been missing until now is a rigorous way to say just how badly it fails.

A team of Chinese oceanographers led by Haidong Pan of the First Institute of Oceanography in Qingdao has now closed that gap. In a study published in Ocean Dynamics, Pan and colleagues Xiaoqing Xu, Junchuan Sun, Fei Teng, Tengfei Xu and Zexun Wei introduce a comprehensive non-stationarity index, a single dimensionless number that measures how far a real tidal record departs from the idealized, perfectly periodic tide that classical harmonic analysis assumes. For the first time, river tides meandering through estuaries and internal tides oscillating in the deep ocean can be placed on the same quantitative footing, allowing direct comparison across tidal types, tidal ranges and tidal regimes that were previously studied in isolation.

The motivation for the index lies in the known limitations of the harmonic model itself. The classical approach, embodied in widely used software packages such as T_TIDE, fits constants, amplitudes and phases that are fixed over the analysis window. But in tidal rivers, friction, river discharge and changing channel depth cause tidal amplitudes to swell with the dry season and collapse during floods, while phases shift as the balance between river current and tidal current evolves. In the stratified open ocean, internal tides are generated episodically as barotropic flow interacts with rough topography, and their amplitudes and phases wander under the influence of mesoscale eddies, seasonal stratification and interference among multiple wave modes. When harmonic constants are fitted to such records, the mismatch between model and reality encodes precisely the non-stationary behavior the model cannot capture.

Pan’s team turned that mismatch into a measurement. The proposed index is built by comparing the observed water level or current time series against its stationary harmonic reconstruction and integrating several widely used statistical metrics into a unified framework. These include measures of the discrepancy between observed and predicted series, their correlation structure and their relative variability, concepts that echo the philosophy behind modern skill-metric frameworks such as the DISO (Decomposed Integrated Statistics-based Optimization) criteria, which were developed as a rethink of the familiar Taylor diagram. Rather than reporting variance explained, correlation and standard deviation separately, the non-stationarity index fuses such information into a single score: a value near zero signals a nearly stationary tide that harmonic analysis handles beautifully, while larger values flag progressively stronger time-dependence. Crucially, the index is defined so that it applies equally to water levels in a macro-tidal estuary, micro-tidal coastal records with diurnal, semi-diurnal or mixed regimes, and three-dimensional internal tidal currents, all within one consistent framework.

The researchers validated the index in two of the most intensively studied non-stationary tidal environments on Earth. The first is the Columbia River Estuary in the Pacific Northwest of the United States, the archetype of river tides. Decades of work there, notably by David Jay and collaborators, have documented how seasonal river discharge modulates tidal range, distorts tidal asymmetry and reshapes floodplain inundation, with consequences for salmonid habitat and sediment transport. Using water-level records from NOAA and USGS gauges distributed along the estuary, the team computed non-stationarity values spanning 0.078 to 0.592. The lower end of that range corresponds to the seaward reaches, where the ocean still dominates and the tide behaves nearly as the harmonic model predicts. Moving upriver, the index climbs steadily as river discharge, channel convergence and friction increasingly bend the tide out of shape, confirming quantitatively what fluvial tide models have long described qualitatively.

The second test case could hardly be more different: the Timor Passage, a deep, energetic gateway in the Indonesian seas through which the Indonesian Throughflow exchanges water between the Pacific and Indian Oceans. There the team analyzed current observations collected by the INSTANT (International Nusantara Stratification and Transport) mooring program, records that have previously revealed vigorous internal tides and strong mixing in the passage. The computed non-stationarity index ranged from 0.583 to 1.149, roughly an order of magnitude larger than the weakest river-tide signals in the Columbia system and clearly exceeding even its most non-stationary upstream reaches. The reason, the authors explain, lies in the internal tides’ multi-modal structure: several vertical wave modes coexist, each with its own amplitude and phase evolution, while baroclinic variability, background currents and episodic generation processes all superimpose additional time dependence. The comparison demonstrates that internal tides in the Timor Passage are substantially more non-stationary than any tide along the Columbia River channel, and it does so in a single common currency.

The practical implications reach well beyond estuarine and deep-sea dynamics. Tide tables and operational forecasts for sheltered harbors perform well precisely because their tides are nearly stationary; the new index offers a straightforward diagnostic for identifying locations where standard predictions will systematically fail. Coastal engineers assessing design water levels, flood-risk modelers projecting future extremes under sea-level rise, and satellite altimetry teams correcting for internal tide noise all depend on knowing how much a local tide deviates from constancy. The index also gives global ocean modelers a metric for evaluating how realistically their simulations capture time-varying tidal energetics, a growing concern as efforts proceed to embed tides and internal gravity waves in general circulation models that previously ignored them. And because the score is normalized, it can be mapped globally, potentially revealing hotspots of tidal non-stationarity from macrotidal rivers like the Yangtze, Qiantang and St. Lawrence to internal tide generators such as the Luzon Strait and the Indonesian archipelago.

The method arrives with working software. The researchers have incorporated a function that computes the non-stationarity index and its uncertainty directly into the S_TIDE MATLAB toolbox, the evolving non-stationary harmonic analysis package that Pan and colleagues have developed and applied previously to the Columbia River Estuary, the northern South China Sea and the deep Timor Passage. This means that any group already analyzing tidal records with modern tools can begin quantifying non-stationarity immediately, rather than adopting an entirely new processing chain.

The authors are careful to flag the method’s boundaries. The index, they note, can overestimate tidal non-stationarity when pronounced high-frequency non-tidal variations are present in the record, for example storm surges, wind-driven seiches or other aperiodic water-level fluctuations that a tidal analysis is never meant to capture. Because the score rewards closeness to a smooth harmonic reconstruction, any energetic non-tidal signal inflates the apparent non-stationarity even when the tide itself is stable. Users therefore need to apply the index judiciously, ideally after filtering or in settings where non-tidal variability is modest, and to interpret absolute values alongside physical knowledge of the site. The uncertainty estimate shipped with the toolbox is intended to help with exactly this kind of caveat-aware interpretation.

Even with those cautions, the study fills a conspicuous hole in tidal science. Researchers have developed an impressive arsenal of non-stationary techniques over the past two decades, from nonstationary harmonic analysis adapted for river tides, to empirical mode decomposition, to variational mode decomposition and continuously time-varying harmonic fits, each capable of tracking evolving amplitudes and phases. What none of these methods supplied was a standard yardstick, a way to say that the tide in one estuary is twice as non-stationary as the tide in another, or that a mooring in a deep passage records intrinsically wilder internal tides than a gauged harbor. By motivating the index directly from the failure modes of classical harmonic analysis and by demonstrating it across tidal types, tidal ranges and tidal regimes spanning the Columbia River Estuary and the Timor Passage, Pan and colleagues have given the community that yardstick. As tides themselves continue to shift under sea-level rise, dredging, damming and changing ocean stratification, being able to measure non-stationarity rather than merely describe it may prove essential for keeping humanity’s oldest prediction science honest.