Internal wave viscosity reshapes global distribution of astronomical tidal energy

Every day, the gravitational tug-of-war between Earth, the Moon and the Sun drives vast quantities of water across the world’s oceans, and with that motion comes an enormous flux of energy. Where exactly that tidal energy is generated, how it travels, and where it ultimately dissipates are questions that have occupied physical oceanographers for decades, because the answers shape everything from the accuracy of tide predictions to the mixing that helps drive the global overturning circulation. Now, a new modeling study published in Ocean Dynamics offers a substantially refined global picture of the astronomical tidal energy budget, and in doing so delivers a first-of-its-kind estimate of a previously unquantified sink: approximately 0.24 terawatts of tidal energy dissipated through nonlinear advection in shallow seas.

The study, carried out by Jiaqi Guo and Xueen Chen of the Ocean University of China together with Pengyang Song of the Alfred Wegener Institute in Bremerhaven, centers on a seemingly technical but consequential change to a global tidal model: the introduction of an Internal Wave Viscosity, or IWV, parameterization. In ocean modeling, a parameterization is a compact mathematical scheme that stands in for physical processes too small or too complex to resolve directly on the model’s grid. The IWV scheme represents the drag exerted on the tides as energy leaks from surface-expressed barotropic tides into internal waves — slow, hidden undulations of the ocean’s density layers that radiate away from undersea ridges and other rough topography before breaking into turbulence.

Barotropic tides are the familiar kind: the entire water column rises and falls in unison, and their energy is concentrated near the surface where satellites can measure it. Baroclinic tides, by contrast, involve vertical shearing between layers of different density and carry energy into the ocean interior. The conversion between these two modes is the gateway through which mechanical tidal energy ultimately becomes deep-ocean turbulence, and turbulence, in turn, mixes heat, carbon and nutrients through the water column. Without a realistic representation of this conversion, global tidal models systematically overestimate how much barotropic tidal energy survives the deep ocean and crashes onto the continental shelves, where it drives shallow-water dynamics.

To quantify the effect, the team ran a global tidal model forced purely by the solar and lunar tidal potentials — a setup reminiscent of the classic benchmark calculations of the tidal equations performed in the late 1970s — and compared simulations with and without the IWV term. The reference standard was TPXO9, a widely used assimilative tidal solution derived from satellite altimetry by inverse modeling. The comparison focused on four principal tidal constituents: M2, the dominant semidiurnal lunar tide with a period of about 12.42 hours; S2, its solar counterpart at exactly 12 hours; and the diurnal constituents K1 and O1, which complete their cycles in roughly 23.93 and 25.82 hours respectively.

The improvements were striking. Adding IWV reduced the model’s errors in shallow-water regions by roughly 30 percent for M2, 18 percent for S2, 21 percent for K1 and 27 percent for O1. In the deep ocean, the parameterization contributed significantly to tidal dissipation, siphoning energy out of the barotropic tide at exactly the places — mid-ocean ridges, seamounts, and rough abyssal topography — where internal wave generation is known to occur. The authors conclude that IWV supplies a physically sufficient conversion of energy from barotropic to baroclinic tides, curtailing the artificially inflated tidal energy that unmodified models propagate toward the shelves, and yielding a more faithful simulation of how astronomical tides are generated, propagate and die.

Why should a viscosity term aimed at internal waves matter so much for the surface tide? The answer lies in the energetics. The nonlinear tidal energy equation used in the analysis shows that the global energy budget of astronomical tides depends closely on the distribution of water depth. Over deep basins, tidal flows interact with topographic roughness through stratification, spawning internal waves that act as an effective drag on the barotropic mode. If that drag is absent from a model, the residual energy must go somewhere — and in practice it spills into the shallow seas, producing tides that are too energetic and phases that drift away from observations. The IWV scheme closes this gap by routing the energy to its physical destination in the ocean interior.

Beyond correcting the deep-ocean energy ledger, the study produced a genuinely novel result. By evaluating the full nonlinear tidal energy equation globally, the researchers isolated, for the first time, a global dissipation of approximately 0.24 terawatts attributable to the nonlinear advection effect — the quadratic interactions of tidal currents with themselves and with spatially varying sea surface elevation in shallow water. Nonlinear advection matters because in shallow seas, where tidal currents are swift and water depths are small, different tidal constituents no longer behave independently. The M2 tide can interact with S2, K1 or O1, transferring energy among them and generating compound frequencies — the so-called shallow-water tides that appear as overtides and compound tides in coastal tidal records.

This discovery carries a double significance. Scientifically, it confirms that astronomical tidal constituents interact nonlinearly as they propagate through shallow regions, and it quantifies the energy fuelling the generation of shallow-water tides that classical linear theory cannot produce. Practically, it gives modelers a new budget term to respect: roughly a quarter of a terawatt — a nontrivial fraction of the approximately 3.7 terawatts that tidal dissipation estimates generally assign to the oceans — is consumed not by bottom friction or internal wave drag, but by the mathematics of nonlinearity itself. Any global model aspiring to accurate coastal tides, and any application relying on them, from storm surge forecasting to tidal energy siting, must account for this pathway.

The broader implications reach into some of the deepest questions in oceanography and Earth science. Since the landmark “Abyssal Recipes” work of Walter Munk and Carl Wunsch in 1998, oceanographers have known that maintaining the ocean’s stratified overturning circulation requires roughly two terawatts of mechanical mixing power, with tides supplying a substantial share. Satellite measurements of the M2 tide from the TOPEX/Poseidon mission, analyzed in detail in the early 2000s, confirmed that about a terawatt of that tide’s energy dissipates in the deep ocean rather than on the shelves. Simultaneously, lunar laser ranging and satellite tracking reveal that tidal friction is slowly braking Earth’s rotation, lengthening the day by milliseconds per century — a budget that must balance to within observational error. Every refinement of where tidal energy goes therefore tightens constraints on both the ocean’s mixing engine and the Earth-Moon system’s long-term evolution.

The methodology behind the new results combines several established tools in a fresh configuration. Bathymetry for the model was drawn from the ETOPO Global Relief Model maintained by NOAA, while the bottom buoyancy frequency — a measure of stratification that governs how readily internal waves can be generated — was computed from the World Ocean Atlas 2013 climatology. Harmonic analysis of the model output followed standard practice for tidal work, and validation against TPXO9 allowed the team to compute error reductions basin by basin and constituent by constituent. The use of a forward model forced only by the tidal potential, rather than one nudged toward observations, makes the energetics diagnostics particularly meaningful: the energy budget emerges from the dynamics rather than being imposed by data assimilation.

Support for the work came from the National Natural Science Foundation of China, under grants directed at the mechanisms of internal tide generation and evolution in the Luzon Strait and at interactions between internal waves and multi-scale ocean processes in the South China Sea — regions where some of the world’s most intense internal tides are found. The authors also acknowledged computing resources from the National Supercomputing Center in Jinan and the Marine Big Data Center at the Ocean University of China. That regional expertise feeding a global calculation is fitting: the physics validated here at planetary scale is the same physics that sculpts the turbulent underwater weather observed above the Luzon Strait ridges.

For the wider community, the message of the study is that the global tidal energy budget is not a fixed inheritance from twentieth-century calculations but an evolving account that sharpens as parameterizations improve. With IWV in place, the model describes a more reasonable cycle in which the astronomical tides born of celestial mechanics lose a proper share of their energy to the deep-ocean interior, arrive at the continental shelves with realistic amplitudes, and there surrender part of their remaining energy to the nonlinear interactions that breed shallow-water tides. The newly quantified 0.24-terawatt advection sink completes a picture in which generation, propagation and dissipation are, for the first time, balanced end to end on a global scale. As high-resolution global models increasingly simulate tides, eddies and the general circulation together, schemes like IWV — and budgets honest about nonlinearity — will be essential ingredients.