Trends in the M$$_2$$ ocean tide observed by satellite altimetry in the presence of systematic errors

Abstract Trends in the deep-ocean M $$_2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mmultiscripts> <mml:mrow/> <mml:mn>2</mml:mn> <mml:mrow/> </mml:mmultiscripts> </mml:math> barotropic tide, deduced from nearly three decades of satellite altimetry and recently presented by Opel et al. (Commun Earth Environ 5:261, https://doi.org/10.1038/s43247-024-01432-5 , 2024), are here updated with a slightly longer time series and with a focus on potential systematic errors. Tidal changes are very small, of order 0.2 mm/year or less, with a tendency for decreasing amplitudes, which is evidently a response to the ocean’s increasing stratification and an increasing energy loss to baroclinic motion. A variety of systematic errors in the satellite altimeter system potentially corrupt these small trend estimates. The Dynamic Atmosphere Correction (DAC), derived from an ocean model and used for de-aliasing, introduces a spurious trend (exceeding 0.1 mm/year in places) caused by changes in ECMWF atmospheric tides. Both operational and reanalysis atmospheric tides have spurious trends over the altimeter era. Tidally coherent errors in satellite orbits, including from use of inconsistent tidal geocenter models, are more difficult to bound, although differences between two sets of satellite ephemerides are found to reach 0.1 mm/year for M $$_2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mmultiscripts> <mml:mrow/> <mml:mn>2</mml:mn> <mml:mrow/> </mml:mmultiscripts> </mml:math> . Orbit errors are more deleterious for some other constituents, including the annual cycle. Tidal leakage in the “mesoscale correction,” needed here to suppress non-tidal ocean variability, is a known potential problem, and if the leakage changes over time, it impacts ocean-tide trend estimation. Tests show the error is likely small in the open ocean ( $$&lt;0.04$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>&lt;</mml:mo> <mml:mn>0.04</mml:mn> </mml:mrow> </mml:math> mm/year) but large in some marginal seas ( $$&gt;0.2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>&gt;</mml:mo> <mml:mn>0.2</mml:mn> </mml:mrow> </mml:math> mm/year). Potential contamination from other altimeter corrections (e.g., ionospheric path delay) is likely negligible for M $$_2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mmultiscripts> <mml:mrow/> <mml:mn>2</mml:mn> <mml:mrow/> </mml:mmultiscripts> </mml:math> but can be difficult to bound.