Linear Predictability: A Sea Surface Height Case Study

A primitive equation ocean model has been applied to the study of mesoscale ocean dynamics and prediction using sea surface height information derived from a satellite‐borne altimeter. Results from a model of the Gulf of Mexico were directly compared with the altimeter data from GEOS 3 and Seasat reported by Marsh et al. (1984) and in situ hydrographic data analyzed by Maul and Herman (1985). In the eastern Gulf the amplitude and position of relative maxima of sea surface height variability were found to be similar for model, altimeter, and in situ data. In the central Gulf the model and the altimeter‐derived sea surface height variability maps were similar but differed considerably from the in situ results. The mean sea surface in the model and from the in situ data also agreed well in the eastern Gulf, but there was little agreement in the central Gulf. While there is no geoid adequate for determining an independent altimetric estimate of the mean dynamic height of the sea surface in the Gulf by differencing the geoid with the altimeter‐derived mean sea surface, the model dynamic mean sea surface can be subtracted from the mean surface of Marsh et al. (1984) to obtain a new estimate of the Gulf geoid, which may be the best geoid available in that region for several years. A benchmark experiment (“truth”) was integrated to statistical equilibrium and compared with results from four experiments in which the model was initialized with fields modified from archived benchmark data. The experiments differed only in the initialization fields. Each experiment was integrated for 100 days with inflow transport remaining constant throughout the integration. The idealized experiments were initialized geostrophically (1) with the exact sea surface and pycnocline height fields, (2) with only the exact sea surface heights and a pycnocline assumed to compensate it such that the deep pressure perturbations vanished, (3) just as in case 2 but with a geoid error component added on small spatial scales, and (4) just as in case 3 except the geoid error model included an additional contribution in strong geoid gradient regions. The sequence of four numerical experiments showed that (1) geostrophic initialization with exact sea surface and pycnocline height information provided accurate forecasts to 100 days, (2) even when only the sea surface height information was provided to the numerical model for geostrophic initialization and deep pressure perturbations assumed to vanish at the initial time, the forecasts were superior to persistence or climatology over the 100‐day forecast period, and (3) errors in the geoid on spatial scales comparable to the grid resolution of the model did not seriously degrade the forecast, even in dynamically active regions with large gradients in the geoid height where instability processes were observed to occur.

This study investigates the sea surface height (SSH) and current responses to synoptic winter wind in the Bohai, Yellow, and East China Seas (BYES) using a numerical model. The basin‐scale responses are primarily interpreted as two leading coastal trapped waves (CTWs) identified through the complex empirical orthogonal function analysis of SSH. One CTW is a Kelvin wave (KW) explaining 76% of the total SSH variance; the other is a wind‐forced lowest shelf wave (SW1) explaining 17% of the total SSH variance. The KW dominates the current in the Bohai and Yellow Seas (BYS), and the SW1 dominates the current in the western East China Sea (ECS). The KW is a free relaxation response to large wind‐driven SSH variations and propagates cyclonically in the BYES around one SSH node. The SW1 propagates cyclonically in the semienclosed BYS around two nodes and southward along the western ECS with two nodes on the shelf. The wavelength and e‐folding scale of amplitude vary spatially, producing regionally distinct features and mechanisms. In the semienclosed BYS, the associated geostrophic current is along the cophase line of SSH, and its magnitude is proportional to both wave amplitude and wavenumber. In the western ECS, the associated geostrophic current is perpendicular to the cophase line of SSH, and its magnitude is proportional to the wave amplitude and inversely proportional to the e‐folding scale of amplitude. This study reveals not only the basin‐scale SSH and current variations in the BYES but also new features of CTWs in a semienclosed embayment.

The Indonesian throughflow (ITF) from the Pacific to the Indian Ocean plays an important role in global ocean circulation and climate. Yet, continuous ITF measurement is difficult and expensive. The longest time series of in situ measurements of the ITF to date were taken in the Makassar Strait, the main pathway of the ITF. Here we have demonstrated a plausible approach to derive the ITF transport proxy using satellite altimetry sea surface height (SSH), gravimetry ocean bottom pressure (OBP) data, in situ measurements from the Makassar Strait from 1996 to 1998 and 2004 to 2011, and a theoretical formulation. We first identified the optimal locations of the correlation between the observed ITF transport through the Makassar Strait and the pressure gradients, represented by the SSH and OBP differences between the Pacific and Indian Oceans at a 1° × 1° horizontal resolution. The optimal locations were found centered at 162°E and 11°N in the Pacific Ocean and 80°E and 0° in the Indian Ocean, then were used in the theoretical formulation to estimate the throughflow. The proxy time series follow the observation time series quite well, with the 1993–2011 mean proxy transport of 11.6 ± 3.2 Sv southward, varying from 5.6 Sv during the strong 1997 El Niño to 16.9 Sv during the 2007 La Nina period, which are consistent with previous estimates. The observed Makassar mean transport is 13.0 ± 3.8 Sv southward over 2004–2011, while the SSH proxy (for the same period) gives an ITF mean transport of 13.9 ± 4.1 Sv and the SSH + OBP proxy (for 2004–2010) is 15.8 ± 3.2 Sv.

The sea surface height (SSH) variations observed by the TOPEX/POSEIDON satellite in the Yellow and East China Seas are examined over the 3 years from 1993 through 1995 assuming a linear response to wind stress across the region. The transfer function relating wind stress to SSH is modeled by a series of increasingly complex functions that gradually allow wind stress over successively broader regions and time periods to influence the SSH solution at a particular point. The SSH response to instantaneous wind stress at the same point in space implies an Ekman transport, but the response explains only a small fraction of the SSH variability. Relating SSH to wind stress averaged over two subregions and including both instantaneous and 12‐hour lagged wind stress indicates that a significant portion of SSH variability responds to winds that are remote in both space and time. An optimal estimation of the transfer function by minimizing the error variance leads to an extended empirical orthogonal function (EEOF) analysis of the wind stress field. The SSH response to each wind mode is determined. The analysis shows the principle variations in the wind stress to consist of northerly wind bursts during winter, which produce large SSH drops in the Bohai Bay and northern Yellow Sea region. The principal wind mode also is significantly related to variations along the shelf break near the Kuroshio Current. The third and fourth wind stress modes indicate the passage of typhoons.

Recent sea surface height (SSH) trends in the South Pacific are substantially greater than trends in the North Pacific. Here, we use the Estimating the Climate and Circulation of the Ocean Version 4 Release 4 ocean state estimate and the Ocean Reanalysis System 5 to identify the forcing and mechanisms underlying that meridional asymmetry during 2005–2015. Thermosteric contributions dominate the spatial structure in Pacific SSH trends, but contributions from local surface heat fluxes are small. Wind stress trends drive a spin‐up of the South Pacific subtropical gyre and a northward shift of the North Pacific subtropical gyre. A reduced gravity model forced with reanalysis winds qualitatively reproduces the meridional seesaw in sea level, suggesting that asymmetric trends in subtropical wind stress drive a cross‐equatorial heat transport. A reversal in forcing associated with this process could impact near‐term rates of coastal sea‐level change, particularly in Pacific Island communities.

Post-processing of complex altimetry-derived echo shapes is necessary over the Caspian Sea, in which the altimetry-inferred level fluctuations, derived from on-board ‘retrackers’, represent errors within a wide boundary close to the coastlines, i.e. up to ~10–30 km. Four different techniques were implemented to retrack 10 Hz radar waveforms of the TOPEX/Poseidon (T/P) mission and 20 Hz waveforms of its follow-on mission Jason-1. These include three previously introduced algorithms of the Off Center of Gravity, the standard NASA β-5, and Threshold Retracking, after some improvements. We also introduced a new method called ‘Extrema Retracking (ExtR)’, which works based on edge detection and extracting extremum points of the returned waveforms. Before implementing each technique, a two-step preprocessing was applied to exclude unwanted waveforms. Then, an unsupervised classification approach was applied to classify the remaining waveforms into multiple groups, according to their shape and geometrical properties. Finally, the waveforms of each group were post-processed using an appropriate retracking algorithm, and subsequently, the derived corrected ranges were converted to sea surface height (SSH) values. Our post-processed results were evaluated (1) by comparing the time series of SSHs with in situ tide gauge measurements as well as (2) by comparing the along-track retracked SSHs to their corresponding geoid profiles, computed from the EGM2008 model. For each evaluation, in situ measurements or geoid profiles were subtracted from the retracked SSH values to compute residuals. Thereafter, the mean and root mean squares of the residuals were calculated and interpreted, respectively, as a measure of bias and noise level of the processed SSHs. Our results indicated a reduction in noise and bias after the application of retrackers. Specifically, the bias and noise of the ExtR-derived SSH time series were decreased at locations close to the Anzali, Noshahr and Neka tide-gauge stations (along the southern coast). For example, the correlation between altimetry observations and tide gauge measurements close to the Anzali Port over the T/P era was increased from 0.31 (using original altimetry data) to 0.83 (using ExtR), which is equivalent to a noise reduction of 0.58 to 0.39 m. Comparisons of the along-track SSHs with geoid profiles indicated a significant mitigation of land contamination over the coastal areas.

Abstract. We investigate the spatial and temporal variability of sea surface height (SSH) in the North Atlantic basin using satellite altimeter data from October 1992–January 2004. Our primary aim is to provide a detailed description of such variability, including that associated with propagating signals. We also investigate possible correlations between SSH variability and atmospheric pressure changes as represented by climate indices. We first investigate interannual SSH variations by deriving the complex empirical orthogonal functions (CEOFs) of altimeter data lowpass-filtered at 18 months. We determine the spatial structure of the leading four modes (both in amplitude and phase) and also the associated principal component (PC) time series. Using wavelet analysis we derive the time-varying spectral density of the PCs, revealing when particular modes were strongest between 1992–2004. The spatial pattern of the leading CEOF, comprising 30% of the total variability, displays a 5-year periodicity in phase; signal propagation is particularly marked in the Labrador Sea. The second mode, with a dominant 3-year signal, has strong variability in the eastern basin. Secondly, we focus on the Azores subtropical frontal zone. The leading mode (35%) is strong in the south and east of this region with strong variations at 3- and 5-year periods. The second mode (21%) has a near-zonal band of low variance between 22°–27° N, sandwiched between two regions of high variance. Thirdly, we lowpass filter the altimeter data at a cutoff of 30 days, instead of 18 months, in order to retain signals associated with propagating baroclinic Rossby waves and/or eddies. The leading mode is the annual steric signal, around 46% of the SSH variability. The third and fourth CEOFs, 11% of the remaining variability, are associated with westward propagation which is particularly dominant in a "waveband" between 32°–36° N. For all three cases considered above, no significant cross-correlation is found between the North Atlantic Oscillation index and the amplitude of the leading four PCs of interannual SSH variability. The only exception is an anti-correlation found over the North Atlantic basin between the NAO and the 4th PC. In the subtropical front, the East Atlantic Pattern index is anti-correlated with the leading PC for SSH variations lowpass filtered at 30 days. Further investigation of forcing mechanisms is suggested using hindcasts from ocean general circulation models.

The gyre-scale, dynamic sea surface height (SSH) variability signifies the spatial redistribution of heat and freshwater in the ocean, influencing the ocean circulation, weather, climate, sea level, and ecosystems. It is known that the first empirical orthogonal function (EOF) mode of the interannual SSH variability in the North Atlantic exhibits a tripole gyre pattern, with the subtropical gyre varying out-of-phase with both the subpolar gyre and the tropics, influenced by the low-frequency North Atlantic Oscillation. Here, we show that the first EOF mode explains the majority (60–90 %) of the interannual SSH variance in the Labrador and Irminger Seas, whereas the second EOF mode is more influential in the northeastern part of the subpolar North Atlantic (SPNA), explaining up to 60–80 % of the regional interannual SSH variability. We find that the two leading modes do not represent physically independent phenomena. On the contrary, they evolve as a quadrature pair associated with a propagation of SSH anomalies from the eastern to the western SPNA. This is confirmed by the complex EOF analysis, which can detect propagating (as opposed to stationary) signals. The analysis shows that it takes about 2 years for sea level signals to propagate from the Iceland Basin to the Labrador Sea, and it takes 7–10 years for the entire cycle of the North Atlantic SSH tripole to complete. The observed westward propagation of SSH anomalies is linked to shifting wind stress curl patterns and to the cyclonic pattern of the mean ocean circulation in the SPNA. The analysis of regional surface buoyancy fluxes in combination with the upper-ocean temperature and salinity changes suggests a time-dependent dominance of either air-sea heat fluxes or advection in driving the observed SSH tendencies, while the contribution of surface freshwater fluxes (precipitation and evaporation) is negligible. We demonstrate that the most recent cooling and freshening observed in the SPNA since about 2010 was mostly driven by advection associated with the North Atlantic Current. The results of this study indicate that signal propagation is an important component of the North Atlantic SSH tripole, as it applies to the SPNA.

Altimeter data from May 1995 to February 2002, together with wind, heat, and fresh water fluxes and hydrographic data, are used to study the annual variability in sea surface height (SSH) in the Nordic Seas. The main contributions to the SSH are quantified with error estimates. The amplitude of the annual variation of the SSH ranges from 4 to 8 cm with largest variability in the deeper basins (Lofoten, Norwegian, and Greenland Basins) and along the Norwegian coast. The seasonal steric height can explain 1.5-4 cm of the SSH variation and arises from changes in the air-sea heat flux. However, the associated seasonal changes in the surface currents that arrive from the steric height and the heat flux are weak (a few mm/s). This is because their contributions to the SSH are nearly constant over large areas. From altimetry and hydrography, we calculated seasonal changes in bottom pressure and currents. In the Greenland and Norwegian Seas, the seasonal anomalous large-scale circulation is cyclonic during spring and anticyclonic during autumn. It is most apparent in the Norwegian Sea with a seasonal variation in speed of 1-2 cm/s. This variation can be explained by changes in the averaged wind-stress curl over the Norwegian Sea. The annual cycle of the Norwegian Atlantic slope current is also revealed. It is strongest in winter and weakest in summer with an annual amplitude in speed of 2-4 cm/s.

A large-scale view of oceanic variability from 2007 to 2015 in the global high resolution monitoring and forecasting system at Mercator Océan

An evaluation of high-resolution ocean reanalyses in the California current system

Numerical experiments were performed in order to investigate the effects of variations of the transport through the Korea/Tsushima Strait, an inlet of the Japan/East Sea, on the upper layer circulation in the JES based on a 10-month transport observation from May 1999 to March 2000 (Perkins et al., 2000). All external forcings to the model were annual mean fields, except the transport variation through the Korea Strait. In the experiments where the periodic variation of the transport repeated continuously sinusoidally by several periods, strong variability of sea surface height (SSH) was detected in the region extending from the Korea Strait to the Japanese coast due to the geostrophy of the buoyancy forcing at the Korea Strait. The region along the Korean coast is more sensitive to the long-term variations than the short-term (≤60-day period) ones. In two experiments forced by realistic and monthly mean transport, the difference of rms of sea surface height was largest at the Japanese coast and relatively large at the East Korean Warm Current separation region (128∼130°E, 39∼41°N) and to the east of Yamato Rise. The distribution of difference of eddy kinetic energy at 100 m depth between the two experiments was similar to that of the rms of SSH. In the distributions of mean SSH and mean kinetic energy at 100 m depth the realistic transport invokes eddy variability to interact with mean current resulting in the changes of the mean SSH and the mean kinetic energy at the East Korean Warm Current separation region, but it does not produce conspicuous changes in the mean fields of entire JES compared with the mean fields forced by the seasonal transport.

Abstract. We use output from a freely running NEMO model simulation for the equatorial Pacific to investigate the utility of linearly removing the local influence of vertical displacements of the thermocline from variations in sea surface height. We show that the resulting time series of residual sea surface height, denoted ηnlti, measures variations in near-surface heat content that are independent of the local vertical displacement of the thermocline and can arise from horizontal advection, surface heat flux, and diapycnal mixing processes. We find that the variance of ηnlti and its correlation with sea surface temperature are focused on the Niño4 region. Furthermore, ηnlti averaged over the Niño4 region is highly correlated with indices of central Pacific El Niño–Southern Oscillation (CP ENSO), and its variance in 21-year running windows shows a strong upward trend over the past 50 years, corresponding to the emergence of CP ENSO following the 1976/77 climate shift. We show that ηnlti can be estimated from observations, using satellite altimeter data and a linear multi-mode model. The time series of ηnlti, especially when estimated using the linear model, show pronounced westward propagation in the western equatorial Pacific, arguing for an important role for zonal advective feedback in the dynamics of CP ENSO, in particular for cold events. We also present evidence that the role of the thermocline displacement in influencing sea surface height increased strongly after 2000 in the eastern part of the Niño4 region, at a time when CP ENSO was particularly active. Finally, the diagnostic is easy to compute and can be easily applied to mooring data or coupled climate models.

The observed signature of mesoscale eddies in sea surface temperature and the associated heat transport

The increasing adoption of AI-based approaches in Earth system sciences has led to breakthroughs in modeling and forecasting, exemplified by state-of-the-art performance of neural weather forecasting systems [Bi et al., 2023, Lam et al., 2022]. In oceanography, Deep Learning techniques show significant promise for advancing ocean state modeling by combining both modeled and observational ocean datasets [Febvre et al., 2023, Martin et al., 2023, Wang et al., 2024]. However, in the context of ocean forecasting, the deployment of neural forecasting approaches faces challenges such as sparse observational data and uncertainties in existing datasets. Despite advances in ocean observation systems, the ocean remains under-sampled, complicating the training of robust forecasting models [Wang et al., 2024].This study presents the application of the 4DVarNet framework [Fablet et al., 2021, Fablet et al., 2023] in forecast mode, specifically for 7-day sea surface height (SSH) prediction. 4DVarNet employs an end-to-end Deep Learning strategy to forecast future SSH state from sparse satellite observational data. Using a variational data assimilation formulation, the framework combines a UNet with a convolutional LSTM to iteratively reconstruct future ocean state. The model was trained on synthetic altimetry observations sampled from the GLORYS12 operational reanalysis (2010–2019) and evaluated on independent Nadir altimetry tracks from 2023.The results demonstrate that 4DVarNet outperforms traditional state-of-the-art operational forecasting system GLO12 [Lellouche et al., 2013], achieving a normalized RMSE (nRMSE) score of 0.92 for lead time 0 compared to 0.86 for the baseline GLO12. The model shows superior accuracy across all forecast lead times, highlighting its potential advantage in operational oceanography. The framework improves the accuracy of the SSH forecast by effectively using gappy satellite altimetry data. This demonstrates a particular interest of applying the proposed method to other data sources, such as the SWOT altimetry mission, but also of implementing alternative learning strategies, including training on synthetic datasets and fine-tuning with real-world altimetry observations.In addition to the improved predictive performance compared to the state-of-the-art operational forecast system, this study establishes a standardized workflow for data processing, training, and evaluation, inspired by OceanBench framework [Johnson et al., 2023].This research highlights the potential of the 4DVarNet framework for short-term neural ocean forecasting. The proposed method efficiently handles sparse altimetry data and achieves significant performance in predicting 7-days gap-free SSH state at global scale, improving the accuracy by almost 65% in average compared to the baseline GLO12. Future studies should focus on improving the model by incorporating additional data sources, evaluating the impact of input and output resolutions and associated learning strategies (eg. patching), and exploring its applicability to additional variables describing ocean state other than the SSH.