Articles | Volume 8, issue 10
https://doi.org/10.5194/gmd-8-3071-2015
© Author(s) 2015. This work is distributed under
the Creative Commons Attribution 3.0 License.
the Creative Commons Attribution 3.0 License.
https://doi.org/10.5194/gmd-8-3071-2015
© Author(s) 2015. This work is distributed under
the Creative Commons Attribution 3.0 License.
the Creative Commons Attribution 3.0 License.
ECCO version 4: an integrated framework for non-linear inverse modeling and global ocean state estimation
G. Forget
CORRESPONDING AUTHOR
Dept. of Earth, Atmospheric and Planetary Sciences, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
J.-M. Campin
Dept. of Earth, Atmospheric and Planetary Sciences, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
P. Heimbach
Dept. of Earth, Atmospheric and Planetary Sciences, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
Institute for Computational Engineering and Sciences, The University of Texas at Austin, Austin, TX 78712, USA
Jackson School of Geosciences, The University of Texas at Austin, Austin, TX 78712, USA
C. N. Hill
Dept. of Earth, Atmospheric and Planetary Sciences, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
R. M. Ponte
Atmospheric and Environmental Research, Inc., Lexington, MA 02421, USA
C. Wunsch
Dept. of Earth and Planetary Sciences, Harvard University, Cambridge, MA 02139, USA
Related authors
Susanna Winkelbauer, Gaël Forget, Michael Mayer, Romain Bourdalle-Badie, Andrea Cipollone, Leopold Haimberger, Keith Haines, Satoshi Osafune, Yuanyuan Song, Andrea Storto, Chunxue Yang, and Hao Zuo
EGUsphere, https://doi.org/10.5194/egusphere-2026-5435, https://doi.org/10.5194/egusphere-2026-5435, 2026
This preprint is open for discussion and under review for Ocean Science (OS).
Short summary
Short summary
Ocean currents carry vast amounts of heat through the Atlantic and strongly influence climate. We compared estimates from 17 global ocean reanalyses with observations to assess how well they reproduce this heat transport and its changes over time. Most reanalyses capture the main patterns and variations, but important differences remain. The reanalyses also suggest a decrease in northward heat transport since the early 1990s, although its magnitude remains uncertain.
Gaël Forget
EGUsphere, https://doi.org/10.5194/egusphere-2026-4643, https://doi.org/10.5194/egusphere-2026-4643, 2026
This preprint is open for discussion and under review for Ocean Science (OS).
Short summary
Short summary
The sunlit ocean layer, from the surface to 200 meters, plays a crucial role in the Earth's climate. Our research introduces the concept of its energy imbalance, driven by ocean heat uptake and transport. We analyzed this imbalance over four decades and found it varies substantially, with ocean heat transport convergence driving regional warming patterns. This research provides a framework for understanding global trends and regional patterns, with implications for climate change impacts.
Bror F. Jönsson, Christopher L. Follett, Jacob Bien, Stephanie Dutkiewicz, Sangwon Hyun, Gemma Kulk, Gael L. Forget, Christian Müller, Marie-Fanny Racault, Christopher N. Hill, Thomas Jackson, and Shubha Sathyendranath
Geosci. Model Dev., 16, 4639–4657, https://doi.org/10.5194/gmd-16-4639-2023, https://doi.org/10.5194/gmd-16-4639-2023, 2023
Short summary
Short summary
While biogeochemical models and satellite-derived ocean color data provide unprecedented information, it is problematic to compare them. Here, we present a new approach based on comparing probability density distributions of model and satellite properties to assess model skills. We also introduce Earth mover's distances as a novel and powerful metric to quantify the misfit between models and observations. We find that how 3D chlorophyll fields are aggregated can be a significant source of error.
Rachael N. C. Sanders, Daniel C. Jones, Simon A. Josey, Bablu Sinha, and Gael Forget
Ocean Sci., 18, 953–978, https://doi.org/10.5194/os-18-953-2022, https://doi.org/10.5194/os-18-953-2022, 2022
Short summary
Short summary
In 2015, record low temperatures were observed in the North Atlantic. Using an ocean model, we show that surface heat loss in December 2013 caused 75 % of the initial cooling before this "cold blob" was trapped below the surface. The following summer, the cold blob re-emerged due to a strong temperature difference between the surface ocean and below, driving vertical diffusion of heat. Lower than average surface warming then led to the coldest temperature anomalies in August 2015.
Ehud Strobach, Andrea Molod, Donifan Barahona, Atanas Trayanov, Dimitris Menemenlis, and Gael Forget
Geosci. Model Dev., 15, 2309–2324, https://doi.org/10.5194/gmd-15-2309-2022, https://doi.org/10.5194/gmd-15-2309-2022, 2022
Short summary
Short summary
The Green's functions methodology offers a systematic, easy-to-implement, computationally cheap, scalable, and extendable method to tune uncertain parameters in models accounting for the dependent response of the model to a change in various parameters. Herein, we successfully show for the first time that long-term errors in earth system models can be considerably reduced using Green's functions methodology. The method can be easily applied to any model containing uncertain parameters.
Susanna Winkelbauer, Gaël Forget, Michael Mayer, Romain Bourdalle-Badie, Andrea Cipollone, Leopold Haimberger, Keith Haines, Satoshi Osafune, Yuanyuan Song, Andrea Storto, Chunxue Yang, and Hao Zuo
EGUsphere, https://doi.org/10.5194/egusphere-2026-5435, https://doi.org/10.5194/egusphere-2026-5435, 2026
This preprint is open for discussion and under review for Ocean Science (OS).
Short summary
Short summary
Ocean currents carry vast amounts of heat through the Atlantic and strongly influence climate. We compared estimates from 17 global ocean reanalyses with observations to assess how well they reproduce this heat transport and its changes over time. Most reanalyses capture the main patterns and variations, but important differences remain. The reanalyses also suggest a decrease in northward heat transport since the early 1990s, although its magnitude remains uncertain.
Gaël Forget
EGUsphere, https://doi.org/10.5194/egusphere-2026-4643, https://doi.org/10.5194/egusphere-2026-4643, 2026
This preprint is open for discussion and under review for Ocean Science (OS).
Short summary
Short summary
The sunlit ocean layer, from the surface to 200 meters, plays a crucial role in the Earth's climate. Our research introduces the concept of its energy imbalance, driven by ocean heat uptake and transport. We analyzed this imbalance over four decades and found it varies substantially, with ocean heat transport convergence driving regional warming patterns. This research provides a framework for understanding global trends and regional patterns, with implications for climate change impacts.
Carl Wunsch, Sarah Williamson, and Patrick Heimbach
Ocean Sci., 19, 1253–1275, https://doi.org/10.5194/os-19-1253-2023, https://doi.org/10.5194/os-19-1253-2023, 2023
Short summary
Short summary
Data assimilation methods that couple observations with dynamical models are essential for understanding climate change. Here,
climateincludes all sub-elements (ocean, atmosphere, ice, etc.). A common form of combination arises from sequential estimation theory, a methodology susceptible to a variety of errors that can accumulate through time for long records. Using two simple analogs, examples of these errors are identified and discussed, along with suggestions for accommodating them.
Bror F. Jönsson, Christopher L. Follett, Jacob Bien, Stephanie Dutkiewicz, Sangwon Hyun, Gemma Kulk, Gael L. Forget, Christian Müller, Marie-Fanny Racault, Christopher N. Hill, Thomas Jackson, and Shubha Sathyendranath
Geosci. Model Dev., 16, 4639–4657, https://doi.org/10.5194/gmd-16-4639-2023, https://doi.org/10.5194/gmd-16-4639-2023, 2023
Short summary
Short summary
While biogeochemical models and satellite-derived ocean color data provide unprecedented information, it is problematic to compare them. Here, we present a new approach based on comparing probability density distributions of model and satellite properties to assess model skills. We also introduce Earth mover's distances as a novel and powerful metric to quantify the misfit between models and observations. We find that how 3D chlorophyll fields are aggregated can be a significant source of error.
Rachael N. C. Sanders, Daniel C. Jones, Simon A. Josey, Bablu Sinha, and Gael Forget
Ocean Sci., 18, 953–978, https://doi.org/10.5194/os-18-953-2022, https://doi.org/10.5194/os-18-953-2022, 2022
Short summary
Short summary
In 2015, record low temperatures were observed in the North Atlantic. Using an ocean model, we show that surface heat loss in December 2013 caused 75 % of the initial cooling before this "cold blob" was trapped below the surface. The following summer, the cold blob re-emerged due to a strong temperature difference between the surface ocean and below, driving vertical diffusion of heat. Lower than average surface warming then led to the coldest temperature anomalies in August 2015.
Ehud Strobach, Andrea Molod, Donifan Barahona, Atanas Trayanov, Dimitris Menemenlis, and Gael Forget
Geosci. Model Dev., 15, 2309–2324, https://doi.org/10.5194/gmd-15-2309-2022, https://doi.org/10.5194/gmd-15-2309-2022, 2022
Short summary
Short summary
The Green's functions methodology offers a systematic, easy-to-implement, computationally cheap, scalable, and extendable method to tune uncertain parameters in models accounting for the dependent response of the model to a change in various parameters. Herein, we successfully show for the first time that long-term errors in earth system models can be considerably reduced using Green's functions methodology. The method can be easily applied to any model containing uncertain parameters.
Carl Wunsch
Ocean Sci. Discuss., https://doi.org/10.5194/os-2021-113, https://doi.org/10.5194/os-2021-113, 2021
Preprint withdrawn
Short summary
Short summary
Combinations of observations with dynamical, chemical, etc, models are essential tools for understanding of climate change. By "climate" is meant all of the sub-elements including ocean, atmosphere, ice, et al. A common form of combination arises from sequential estimation theory, a methodology susceptible to a variety of errors that can accumulate through time for long records. Using two simple analogues, many of these errors are identified here, with suggestions for accommodating them.
Cited articles
Adcroft, A. and Campin, J.: Rescaled height coordinates for accurate representation of free-surface flows in ocean circulation models, Ocean Model., 7, 269–284, 2004.
Adcroft, A., Hill, C., and Marshall, J.: A new treatment of the Coriolis terms in C-grid models at both high and low resolutions, Mon. Weather Rev., 127, 1928–1936, 1999.
Adcroft, A., Campin, J.-M., Hill, C., and Marshall, J.: Implementation of an atmosphere-ocean general circulation model on the expanded spherical cube, Mon. Weather Rev., 132, 2845–2863, https://doi.org/10.1175/MWR2823.1, 2004a.
Adcroft, A., Hill, C., Campin, J.-M., Marshall, J., and Heimbach, P.: Overview of the formulation and numerics of the MITGCM, in: Proceedings of the ECMWF Seminar Series on Numerical Methods, Recent Developments in Numerical Methods for Atmosphere and Ocean Modelling, 139–149, ECMWF, available at: http://mitgcm.org/pdfs/ECMWF2004-Adcroft.pdf (last access: 29 April 2015), 2004b.
Andersen, O. B. and Knudsen, P.: DNSC08 mean sea surface and mean dynamic topography models, J. Geophys. Res.-Oceans, 114, C11001, https://doi.org/10.1029/2008JC005179, 2009.
Balmaseda, M. A., Hernandez, F., Storto, A., Palmer, M., Shi, L., Smith, G., Toyoda, T., Valdivieso, M., Alves, O., Barnier, B., Boyer, T., Chang, Y.-S., Chepurin, G. A., Ferry, N., Forget, G., Fujii, Y., Good, S., Guinehut, S., Haines, K., Ishikawa, Y., Keeley, S., Köhl, A., Lee, T., Martin, M., Masina, S., Masuda, S., Meyssignac, B., Mogensen, K., Parent, L., Peterson, D., Yin, Y., Vernieres, G., Wang, X., Waters, J., Wedd, R., Wang, O., Xue, Y., Chevallier, M., Lemieux, J.-F., Dupont, F., Kuragano, T., Kamachi, M., Awaji, T., Cantalbiano, A., Wilmer-Becker, K., and Gaillard, F.: The Ocean Reanalyses Intercomparison Project (ORA-IP), Proceedings of the Institute of Marine Engineering, Science, and Technology, J. Operational Oceanogr., 8, 80–97, https://doi.org/10.1080/1755876X.2015.1022329, 2015.
Barnier, B., Madec, G., Penduff, T., Molines, J.-M., Treguier, A.-M., Le Sommer, J., Beckmann, A., Biastoch, A., Böning, C., Dengg, J., Derval, C., Durand, E., Gulev, S., Remy, E., Talandier, C., Theetten, S., Maltrud, M., McClean, J., and De Cuevas, B.: Impact of partial steps and momentum advection schemes in a global ocean circulation model at eddy-permitting resolution, Ocean Dynam., 56, 543–567, 2006.
Barth, A., Beckers, J.-M., Troupin, C., Alvera-Azcárate, A., and Vandenbulcke, L.: divand-1.0: n-dimensional variational data analysis for ocean observations, Geosci. Model Dev., 7, 225–241, https://doi.org/10.5194/gmd-7-225-2014, 2014.
Blessing, S., Kaminski, T., Lunkeit, F., Matei, I., Giering, R., Köhl, A., Scholze, M., Herrmann, P., Fraedrich, K., and Stammer, D.: Testing variational estimation of process parameters and initial conditions of an earth system model, Tellus A, 66, 22606, https://doi.org/10.3402/tellusa.v66.22606, 2014.
Buckley, M. W., Ponte, R. M., Forget, G., and Heimbach, P.: Low-frequency SST and upper-ocean heat content variability in the North Atlantic, J. Climate, 27, 4996–5018, 2014.
Buckley, M. W., Ponte, R. M., Forget, G., and Heimbach, P.: Determining the origins of advective heat transport convergence variability in the North Atlantic, J. Climate, 28, 3943–3956, 2015.
Campin, J., Adcroft, A., Hill, C., and Marshall, J.: Conservation of properties in a free surface model, Ocean Model., 6, 221–244, 2004.
Campin, J.-M., Marshall, J., and Ferreira, D.: Sea ice–ocean coupling using a rescaled vertical coordinate, Ocean Model., 24, 1–14, 2008.
Chaudhuri, A. H., Ponte, R. M., Forget, G., and Heimbach, P.: A comparison of atmospheric reanalysis surface products over the ocean and implications for uncertainties in air–sea boundary forcing, J. Climate, 26, 153–170, 2013.
Comiso, J.: Bootstrap sea ice concentrations for NIMBUS-7 SMMR and DMSP SSM/I, Digital Media, National Snow and Ice Data Center, https://doi.org/10.5067/J6JQLS9EJ5HU, 1999.
Dail, H. and Wunsch, C.: Dynamical reconstruction of upper-ocean conditions in the Last Glacial Maximum Atlantic, J. Climate, 27, 807–823, https://doi.org/10.1175/JCLI-D-13-00211.1, 2014.
Danabasoglu, G., Yeager, S. G., Bailey, D., Behrens, E., Bentsen, M., Bi, D., Biastoch, A., Böning, C., Bozec, A., Canuto, V. M., Cassou, C., Chassignet, E., Coward, A. C., Danilov, S., Diansky, N., Drange, H., Farneti, R., Fernandez, E., Fogli, P. G., Forget, G., Fujii, Y., Griffies, S. M., Gusev, A., Heimbach, P., Howard, A., Jung, T., Kelley, M., Large, W. G., Leboissetier, A., Lu, J., Madec, G., Marsland, S. J., Masina, S., Navarra, A., Nurser, A. G., Pirani, A., Salas y Mélia, D., Samuels, B. L., Scheinert, M., Sidorenko, D., Treguier, A.-M., Tsujino, H., Uotila, P., Valcke, S., Voldoire, A., and Wang, Q.: North Atlantic simulations in Coordinated Ocean-ice Reference Experiments phase {II} (CORE-II). Part I: Mean states, Ocean Model., 73, 76–107, https://doi.org/10.1016/j.ocemod.2013.10.005, 2014.
Dee, D. P., Uppala, S. M., Simmons, A. J., Berrisford, P., Poli, P., Kobayashi, S., Andrae, U., Balmaseda, M. A., Balsamo, G., Bauer, P., Bechtold, P., Beljaars, A. C. M., van de Berg, L., Bidlot, J., Bormann, N., Delsol, C., Dragani, R., Fuentes, M., Geer, A. J., Haimberger, L., Healy, S. B., Hersbach, H., Hólm, E. V., Isaksen, L., Kållberg, P., Köhler, M., Matricardi, M., McNally, A. P., Monge-Sanz, B. M., Morcrette, J.-J., Park, B.-K., Peubey, C., de Rosnay, P., Tavolato, C., Thépaut, J.-N., and Vitart, F.: The ERA-Interim reanalysis: configuration and performance of the data assimilation system, Q. J. Roy. Meteor. Soc., 137, 553–597, 2011.
Denham, C. R.: Seagrid orthogonal grid maker for matlab, US Geological Survey, 384, available at: http://woodshole.er.usgs.gov/staffpages/cdenham/public_html/seagrid/seagrid.html (last access: 29 April 2015) 2000.
Duffy, P., Eby, M., and Weaver, A.: Effects of sinking of salt rejected during formation of sea ice on results of an ocean–atmosphere-sea ice climate model, Geophys. Res. Lett., 26, 1739–1742, 1999.
Dutkiewicz, S., Sokolov, A. P., Scott, J., and Stone, P. H.: A Three-Dimensional Ocean-Seaice-Carbon Cycle Model and its Coupling to a Two-Dimensional Atmospheric Model: Uses in Climate Change Studies, MIT Joint Program on the Science and Policy of Global Change, available at: http://hdl.handle.net/1721.1/18091 (last access: 29 April 2015), 2005.
Ekman, V. W.: On the influence of the earth's rotation on ocean currents, Ark. Mat. Astron. Fys., 2, 1–53, 1905.
Fekete, B. M., Vörösmarty, C. J., and Grabs, W.: High-resolution fields of global runoff combining observed river discharge and simulated water balances, Global Biogeochem. Cy., 16, 15.1–15.10, https://doi.org/10.1029/1999GB001254, 2002.
Fenty, I. and Heimbach, P.: Coupled sea ice–ocean-state estimation in the Labrador Sea and Baffin Bay, J. Phys. Oceanogr., 43, 884–904, 2013.
Ferreira, D., Marshall, J., and Heimbach, P.: Estimating eddy stresses by fitting dynamics to observations using a residual-mean ocean circulation model and its adjoint, J. Phys. Oceanogr., 35, 1891–1910, https://doi.org/10.1175/JPO2785.1, 2005.
Ferreira, D., Marshall, J., and Rose, B.: Climate determinism revisited: multiple equilibria in a complex climate model, J. Climate, 24, 992–1012, 2011.
Ferron, B. and Marotzke, J.: Impact of 4D-variational assimilation of WOCE hydrography on the meridional circulation of the Indian Ocean, Deep-Sea Res. Pt. II, 50, 2005–2021, 2003.
Follows, M. J., Dutkiewicz, S., Grant, S., and Chisholm, S. W.: Emergent biogeography of microbial communities in a model ocean, Science, 315, 1843–1846, 2007.
Forget, G.: Mapping ocean observations in a dynamical framework: a 2004–06 ocean atlas, J. Phys. Oceanogr., 40, 1201–1221, 2010.
Forget, G.: The observed abyssal variability puzzle, Geophys. Res. Lett., in preparation, 2015.
Forget, G. and Ponte, R.: The partition of regional sea level variability, Prog. Oceanogr., 137, 173–195, 2015.
Forget, G. and Wunsch, C.: Estimated global hydrographic variability, J. Phys. Oceanogr., 37, 1997–2008, 2007.
Forget, G., Ferron, B., and Mercier, H.: Combining Argo profiles with a general circulation model in the North Atlantic. Part 1: Estimation of hydrographic and circulation anomalies from synthetic profiles, over a year, Ocean Model., 20, 1–16, 2008a.
Forget, G., Mercier, H., and Ferron, B.: Combining Argo profiles with a general circulation model in the North Atlantic. Part 2: Realistic transports and improved hydrography, between spring 2002 and spring 2003, Ocean Model., 20, 17–34, 2008b.
Forget, G., Maze, G., Buckley, M., and Marshall, J.: Estimated seasonal cycle of North Atlantic eighteen degree water volume, J. Phys. Oceanogr., 41, 269–286, 2011.
Forget, G., Ferreira, D., and Liang, X.: On the observability of turbulent transport rates by Argo: supporting evidence from an inversion experiment, Ocean Sci. Discuss., 12, 1107–1143, https://doi.org/10.5194/osd-12-1107-2015, 2015.
Fukumori, I.: A partitioned Kalman filter and smoother, Mon. Weather Rev., 130, 1370–1383, 2002.
Fukumori, I., Menemenlis, D., and Lee, T.: A near-uniform basin-wide sea level fluctuation of the Mediterranean Sea, J. Phys. Oceanogr., 37, 338–358, 2007.
Fukumori, I., Wang, O., Llovel, W., Fenty, I., and Forget, G.: A near-uniform fluctuation of ocean bottom pressure and sea level across the deep ocean basins of the Arctic Ocean and the Nordic Seas, Prog. Oceanogr., 134, 152–172, 2015.
Gaspar, P., Grégoris, Y., and Lefevre, J.-M.: A simple eddy kinetic energy model for simulations of the oceanic vertical mixing: tests at station papa and long-term upper ocean study site, J. Geophys. Res., 95, 16179–16193, 1990.
Gent, P. and Mcwilliams, J.: Isopycnal mixing in ocean circulation models, J. Phys. Oceanogr., 20, 150–155, 1990.
Giering, R. and Kaminski, T.: Recipes for adjoint code construction, ACM T. Math. Software, 24, 437–474, 1998.
Giering, R., Kaminski, T., and Slawig, T.: Generating efficient derivative code with TAF: adjoint and tangent linear Euler flow around an airfoil, Future Gener. Comp. Sy., 21, 1345–1355, 2005.
Goldberg, D. N. and Heimbach, P.: Parameter and state estimation with a time-dependent adjoint marine ice sheet model, The Cryosphere, 7, 1659–1678, https://doi.org/10.5194/tc-7-1659-2013, 2013.
Griewank, A.: Achieving logarithmic growth of temporal and spatial complexity in reverse automatic differentiation, Optim. Method. Softw., 1, 35–54, 1992.
Griewank, A. and Walther, A.: Evaluating Derivatives: Principles and Techniques of Algorithmic Differentiation, Society for Industrial and Applied Mathematics, Philadelphia, PA, https://doi.org/10.1137/1.9780898717761, 2008.
Griffies, S. and Greatbatch, R.: Physical processes that impact the evolution of global mean sea level in ocean climate models, J. Marine Syst., 51, 37–72, 2012.
Hansen, P. C.: Analysis of discrete ill-posed problems by means of the L-curve, SIAM Rev., 34, 561–580, 1992.
Hascoët, L. and Pascual, V.: The Tapenade automatic differentiation tool: p}rinciples, model, and specification, {ACM T. Math. Software, 39, 20.1–20.43, https://doi.org/10.1145/2450153.2450158, 2013.
Heimbach, P., Hill, C., and Giering, R.: Automatic generation of efficient adjoint code for a parallel Navier–Stokes solver, in: Computational Science – ICCS 2002, Springer, 1019–1028, 2002.
Heimbach, P., Hill, C., and Giering, R.: An efficient exact adjoint of the parallel MIT general circulation model, generated via automatic differentiation, Future Gener. Comp. Sy., 21, 1356–1371, 2005.
Heimbach, P., Forget, G., Ponte, R., Wunsch, C., Balmaseda, M., Awaji, T., Baehr, J., Behringer, D., Carton, J., Ferry, N., Fischer, A., Fukumori, I., Giese, B., Haines, K., Harrison, E., Hernandez, F., Kamachi, M., Keppenne, C., Köhl, A., Lee, T., Menemenlis, D., Oke, P., Remy, E., Rienecker, M., Rosati, A., Smith, D., Speer, K., Stammer, D., and Weaver, A.: Observational requirements for global-scale ocean climate analysis: lessons from ocean state estimation, in: Proceedings of the OceanObs09 Conference: Sustained Ocean Observations and Information for Society, Venice, Italy, Vol. 2, https://doi.org/10.5270/OceanObs09.cwp.42, 2009.
Heimbach, P., Menemenlis, D., Losch, M., Campin, J.-M., and Hill, C.: On the formulation of sea-ice models. Part 2: Lessons from multi-year adjoint sea-ice export sensitivities through the Canadian Arctic Archipelago, Ocean Model., 33, 145–158, 2010.
Heimbach, P., Wunsch, C., Ponte, R. M., Forget, G., Hill, C., and Utke, J.: Timescales and regions of the sensitivity of Atlantic meridional volume and heat transport: toward observing system design, Deep-Sea Res. Pt. II, 58, 1858–1879, 2011.
Holland, W. R. and Malanotte-Rizzoli, P.: Assimilation of altimeter data into an ocean circulation model: space versus time resolution studies, J. Phys. Oceanogr., 19, 1507–1534, 1989.
Hoppe, C. M., Elbern, H., and Schwinger, J.: A variational data assimilation system for soil–atmosphere flux estimates for the Community Land Model (CLM3.5), Geosci. Model Dev., 7, 1025–1036, https://doi.org/10.5194/gmd-7-1025-2014, 2014.
Hoteit, I., Cornuelle, B., Köhl, A., and Stammer, D.: Treating strong adjoint sensitivities in tropical eddy-permitting variational data assimilation, Q. J. Roy. Meteor. Soc., 131, 3659–3682, 2005.
Hoteit, I., Cornuelle, B., Kim, S., Forget, G., Köhl, A., and Terrill, E.: Assessing 4D-VAR for dynamical mapping of coastal high-frequency radar in San Diego, Dynam. Atmos. Oceans, 48, 175–197, 2009.
Hoteit, I., Hoar, T., Gopalakrishnan, G., Collins, N., Anderson, J., Cornuelle, B., Köhl, A., and Heimbach, P.: A MITgcm/DART ensemble analysis and prediction system with application to the Gulf of Mexico, Dynam. Atmos. Oceans, 63, 1–23, 2013.
Ives, D. C. and Zacharias, R. M.: Conformal mapping and orthogonal grid generation, J. Propul. Power, 5, 327–333, 1989.
Jackett, D. R. and McDougall, T. J.: Minimal adjustment of hydrographic profiles to achieve static stability, J. Atmos. Ocean. Tech., 12, 381–389, 1995.
Jiang, Z., Hui, W., and Kamachi, M.: The improvement made by a modified TLM in 4DVAR with a geophysical boundary layer model, Adv. Atmos. Sci., 19, 563–582, 2002.
Kalmikov, A. G. and Heimbach, P.: A Hessian-based method for uncertainty quantification in global ocean state estimation, SIAM J. Sci. Comput., 36, S267–S295, 2014.
Köhl, A.: Evaluation of the GECCO2 ocean synthesis: transports of volume, heat and freshwater in the Atlantic, Q. J. Roy. Meteor. Soc., 141, 166–181, https://doi.org/10.1002/qj.2347, 2014.
Köhl, A. and Stammer, D.: Optimal observations for variational data assimilation, J. Phys. Oceanogr., 34, 529–542, 2004.
Köhl, A. and Stammer, D.: Decadal sea level changes in the 50-Year GECCO Ocean Synthesis, J. Climate, 21, 1876–1890, https://doi.org/10.1175/2007JCLI2081.1, 2008.
Köhl, A., Stammer, D., and Cornuelle, B.: Interannual to decadal changes in the ECCO global synthesis., J. Phys. Oceanogr., 37, 529–542, https://doi.org/10.1175/2513.1, 2007.
Köhl, A., Siegismund, F., and Stammer, D.: Impact of assimilating bottom pressure anomalies from GRACE on ocean circulation estimates, J. Geophys. Res., 117, C04032, https://doi.org/10.1029/2011JC007623, 2012.
Large, W. and Yeager, S.: Diurnal to decadal global forcing for ocean and sea-ice models: the data sets and flux climatologies, Technical Report TN-460+STR, NCAR, 2004.
Large, W., McWilliams, J., and Doney, S.: Oceanic vertical mixing: a review and a model with a nonlocal boundary layer parameterization, Rev. Geophys., 32, 363–403, 1994.
Liang, X., Wunsch, C., Heimbach, P., and Forget, G.: Vertical redistribution of oceanic heat content, J. Climate, 28, 3821–3833, https://doi.org/10.1175/JCLI-D-14-00550.1, 2015.
Liu, C., Köhl, A., and Stammer, D.: Adjoint-based estimation of eddy-induced tracer mixing parameters in the global ocean, J. Phys. Oceanogr., 42, 1186–1206, 2012.
Llovel, W., Willis, J., Landerer, F., and Fukumori, I.: Deep-ocean contribution to sea level and energy budget not detectable over the past decade, Nature Climate Change, 4, 1031–1035, 2014.
Locarnini, R., Mishonov, A., Antonov, J., Boyer, T., Garcia, H., and Levitus, S.: World Ocean Atlas 2005 Volume 1: Temperature, NOAA Atlas NESDIS, S. Levitus, Ed. NOAA Atlas NESDIS 61, US Gov. Printing Office, Wash., D.C., 182 pp., 2006.
Losch, M., Menemenlis, D., Campin, J.-M., Heimbach, P., and Hill, C.: On the formulation of sea-ice models. Part 1: Effects of different solver implementations and parameterizations, Ocean Model., 33, 129–144, 2010.
Losch, M., Strass, V., Cisewski, B., Klaas, C., and Bellerby, R. G.: Ocean state estimation from hydrography and velocity observations during EIFEX with a regional biogeochemical ocean circulation model, J. Marine Syst., 129, 437–451, 2014.
Lyman, J. and Johnson, G.: Estimating global ocean heat content changes in the upper 1800 m since 1950 and the influence of climatology choice, J. Climate, 27, 1945–1957, 2014.
Marotzke, J., Giering, R., Zhang, K. Q., Stammer, D., Hill, C., and Lee, T.: Construction of the adjoint MIT ocean general circulation model and application to Atlantic heat transport sensitivity, J. Geophys. Res.-Oceans, 104, 29529–29547, 1999.
Marshall, J., Adcroft, A., Hill, C., Perelman, L., and Heisey, C.: A finite-volume, incompressible Navier Stokes model for studies of the ocean on parallel computers, J. Geophys. Res., 102, 5753–5766, 1997.
Marshall, J., Ferreira, D., Campin, J., and Enderton, D.: Mean climate and variability of the atmosphere and ocean on an aquaplanet, J. Atmos. Sci., 64, 4270–4286, 2007a.
Marshall, J., Ferreira, D., Campin, J.-M., and Enderton, D.: Mean climate and variability of the atmosphere and ocean on an aqua-planet, J. Atmos. Sci., 64, 4270–4286, 2007b.
Marshall, J., Anderson, A., Dewar, W., Doney, S., Edson, J., Ferrari, R., Forget, G., Fratantoni, D., Gregg, M., Joyce, T., Kelly, K., Lozier, S., Lumpkin, R., Maze, G., Paster, J., Samelson, R., Silverthorne, K., Skyllingstad, E., Straneo, F., Talley, L., Thomas, L., Toole, J., and Weller, R.: Observing the cycle of convection and restratification over the Gulf Stream and the subtropical gyre of the North Atlantic Ocean: preliminary results from the CLIMODE field campaign, B. Am. Meteorol. Soc., 90, 1337–1350, 2009.
Marshall, J., Armour, K. C., Scott, J. R., Kostov, Y., Hausmann, U., Ferreira, D., Shepherd, T. G., and Bitz, C. M.: The ocean's role in polar climate change: asymmetric Arctic and Antarctic responses to greenhouse gas and ozone forcing, Philos. T. R. Soc. A, 372, 20130040, https://doi.org/10.1098/rsta.2013.0040, 2014.
Marzocchi, W. and Jordan, T. H.: Testing for ontological errors in probabilistic forecasting models of natural systems, P. Natl. Acad. Sci. USA, 111, 11973–11978, 2014.
Maze, G., Forget, G., Buckley, M., Marshall, J., and Cerovecki, I.: Using transformation and formation maps to study the role of air–sea heat fluxes in North Atlantic Eighteen Degree Water formation, J. Phys. Oceanogr., 39, 1818–1835, 2009.
Mazloff, M. R., Heimbach, P., and Wunsch, C.: An eddy-permitting Southern Ocean state estimate, J. Phys. Oceanogr., 40, 880–899, 2010.
McCaffrey, K., Fox-Kemper, B., and Forget, G.: Estimates of ocean macro-turbulence: structure function and spectral slope from Argo profiling floats, J. Phys. Oceanogr., 45, 1773–1793, https://doi.org/10.1175/JPO-D-14-0023.1, 2015.
Menemenlis, D., Hill, C., Adcroft, A., Campin, J., Cheng, B., Ciotti, B., Fukumori, I., Koehl, A., Heimbach, P., Henze, C., Lee, T., Stammer, D., Taft, J., and Zhang, J.: NASA supercomputer improves prospects for ocean climate research, EOS T. Am. Geophys. Un., 86–96, https://doi.org/10.1029/2005EO090002, 2005.
Mercier, H.: Determining the general circulation of the ocean: a nonlinear inverse problem, J. Geophys. Res.-Oceans, 91, 5103–5109, 1986.
MITgcm Group: MITgcm Release 1 Manual, Technical report, available at: http://mitgcm.org/public/sealion/ (last access: 29 April 2015), Massachusetts Institute of Technology, Cambridge, MA 02139, USA, 346 pp., 2002.
Moore, A. M., Arango, H. G., Broquet, G., Powell, B. S., Weaver, A. T., and Zavala-Garay, J.: The Regional Ocean Modeling System (ROMS) 4-dimensional variational data assimilation systems: Part I – System overview and formulation, Prog. Oceanogr., 91, 34–49, https://doi.org/10.1016/j.pocean.2011.05.004, 2011.
Munk, W. H.: Abyssal recipes, in: Deep Sea Research and Oceanographic, Vol. 13, Elsevier, 707–730, https://doi.org/10.1016/0011-7471(66)90602-4, 1966.
Myers, G. J., Sandler, C., and Badgett, T.: The art of software testing, John Wiley & Sons, 2011.
Nguyen, A., Menemenlis, D., and Kwok, R.: Improved modeling of the Arctic halocline with a subgrid-scale brine rejection parameterization, J. Geophys. Res.-Oceans, 114, C11014, https://doi.org/10.1029/2008JC005121, 2009.
Olbers, D., Wenzel, M., and Willebrand, J.: The inference of North Atlantic circulation patterns from climatological hydrographic data, Rev. Geophys., 23, 313–356, 1985.
Pavlis, N. K., Holmes, S. A., Kenyon, S. C., and Factor, J. K.: The development and evaluation of the Earth Gravitational Model 2008 (EGM2008), J. Geophys. Res.-Sol. Ea., 117, B04406, https://doi.org/10.1029/2011JB008916, 2012.
Piecuch, C., Ponte, R., Heimbach, H., and Forget, G.: Sensitivity of Ocean-Model Estimates of Contemporary Global and Regional Sea-Level Changes to Geothermal Heat Flow, Ocean Model., under review, 2015.
Ponte, R. M.: Oceanic response to surface loading effects neglected in volume-conserving models, J. Phys. Oceanogr., 36, 426–434, https://doi.org/10.1175/JPO2843.1, 2006.
Ponte, R. M., Wunsch, C., and Stammer, D.: Spatial mapping of time-variable errors in Jason-1 and TOPEX/Poseidon sea surface height measurements, J. Atmos. Ocean. Tech., 24, 1078–1085, 2007.
Prinn, R. G., Heimbach, P., Rigby, M., Dutkiewicz, S., Melillo, J. M., Reilly, J. M., Kicklighter, D. W., and Waugh, C.: A Strategy for a Global Observing System for Verification of National Greenhouse Gas Emissions, Tech. rep., MIT Joint Program on the Science and Policy of Global Change, 2011.
Purkey, S. G. and Johnson, G. C.: Warming of global abyssal and deep Southern Ocean waters between the 1990s and 2000s: contributions to global heat and sea level rise budgets*, J. Climate, 23, 6336–6351, 2010.
Purser, R. and Rančić, M.: Smooth quasi-homogeneous gridding of the sphere, Q. J. Roy. Meteor. Soc., 124, 637–647, 1998.
Quinn, K. J. and Ponte, R. M.: Estimating weights for the use of time-dependent gravity recovery and climate experiment data in constraining ocean models, J. Geophys. Res., 113, C12013, https://doi.org/10.1029/2008JC004903, 2008.
Quinn, K. J. and Ponte, R.: Uncertainty in ocean mass trends from GRACE, Geophys. J. Int., 181, 762–768, https://doi.org/10.1111/j.1365-246X.2010.04508.x, 2010.
Rančić, M., Purser, R., and Mesinger, F.: A global shallow-water model using an expanded spherical cube: gnomonic versus conformal coordinates, Q. J. Roy. Meteor. Soc., 122, 959–982, 1996.
Redi, M. H.: Oceanic isopycnal mixing by coordinate rotation, J. Phys. Oceanogr., 12, 1154–1158, 1982.
Reynolds, R. W., Rayner, N. A., Smith, T. M., Stokes, D. C., and Wang, W.: An improved in situ and satellite SST analysis for climate, J. Climate, 15, 1609–1625, 2002.
Risien, C. and Chelton, D.: A global climatology of surface wind and wind stress fields from eight years of QuikSCAT scatterometer data, J. Phys. Oceanogr., 38, 2379–2413, 2008.
Roquet, F., Charrassin, J.-B., Marchand, S., Boehme, L., Fedak, M., Reverdin, G., and Guinet, C.: Delayed-mode calibration of hydrographic data obtained from animal-borne satellite relay data loggers, J. Atmos. Ocean. Tech., 28, 787–801, 2011.
Ryskin, G. and Leal, L.: Orthogonal mapping, J. Comput. Phys., 50, 71–100, 1983.
Scharroo, R., Lillibridge, J., Smith, W., and Schrama, E.: Cross-calibration and long-term monitoring of the microwave radiometers of ERS, TOPEX, GFO, Jason, and Envisat, Mar. Geod., 27, 279–297, https://doi.org/10.1080/01490410490465265, 2004.
Schott, F. and Zantopp, R.: On the effect of vertical mixing on the determination of absolute currents by the beta spiral method, Deep-Sea Res., 27, 173–180, 1980.
Shchepetkin, A. F. and McWilliams, J. C.: The regional oceanic modeling system (ROMS): a split-explicit, free-surface topography-following-coordinate ocean model, Ocean Model., 9, 347–404, 2005.
Speer, K. and Forget, G.: Chapter 9 – Global distribution and formation of mode waters, in: Ocean Circulation and Climate: a 21st Century Perspective, 211–226, https://doi.org/10.1016/B978-0-12-391851-2.00009-X, 2013.
Stammer, D.: Adjusting internal model errors through ocean state estimation, J. Phys. Oceanogr., 35, 1143–1153, https://doi.org/10.1175/JPO2733.1, 2005.
Stammer, D., Wunsch, C., Giering, R., Eckert, C., Heimbach, P., Marotzke, J., Adcroft, A., Hill, C., and Marshall, J.: The Global ocean circulation during 1992–1997, estimated from ocean observations and a general circulation model, J. Geophys. Res.-Oceans, 107, 1.1–1.27, https://doi.org/10.1029/2001JC000888, 2002.
Stammer, D., Ueyoshi, K., Köhl, A., Large, W., Josey, S., and Wunsch, C.: Estimating air–sea fluxes of heat, freshwater, and momentum through global ocean data assimilation, J. Geophys. Res, 109, 8691–8702, 2004.
Steele, M., Morley, R., and Ermold, W.: PHC: a global ocean hydrography with a high-quality Arctic Ocean, J. Climate, 14, 2079–2087, 2001.
Stommel, H.: The delicate interplay between wind-stress and buoyancy input in ocean circulation: the Goldsbrough variations*, Tellus A, 36, 111–119, 1984.
Sverdrup, H. U.: Wind-driven currents in a baroclinic ocean; with application to the equatorial currents of the eastern Pacific, P. Natl. Acad. Sci. USA, 33, 318–326, 1947.
Thacker, W. C. and Long, R. B.: Fitting dynamics to data, J. Geophys. Res.-Oceans, 93, 1227–1240, 1988.
Toole, J., Krishfield, R., Timmermans, M.-L., and Proshutinsky, A.: The ice-tethered profiler: Argo of the Arctic, Oceanography, 24, 126–135, 2011.
Trefethen, L.: SCPACK user's guide, Tech. rep., Technical Report 89-2, MIT Numerical Analysis Report, 1989.
Tziperman, E. and Thacker, W. C.: An optimal-control/adjoint-equations approach to studying the oceanic general circulation, J. Phys. Oceanogr., 19, 1471–1485, 1989.
Tziperman, E., Thacker, W. C., Long, R. B., and Hwang, S.-M.: Oceanic data analysis using a general circulation model. Part I: Simulations, J. Phys. Oceanogr., 22, 1434–1457, 1992a.
Tziperman, E., Thacker, W. C., Long, R. B., Hwang, S.-M., and Rintoul, S. R.: Oceanic data analysis using a general circulation model. Part II: A North Atlantic model, J. Phys. Oceanogr., 22, 1458–1485, 1992b.
Utke, J., Naumann, U., Fagan, M., Tallent, N., Strout, M., Heimbach, P., Hill, C., and Wunsch, C.: OpenAD/F: a modular open-source tool for automatic differentiation of Fortran codes, ACM T. Math. Software, 34, 18, https://doi.org/10.1145/1377596.1377598, 2008.
Verdy, A., Mazloff, M. R., Cornuelle, B. D., and Kim, S. Y.: Wind-driven sea level variability on the California coast: an adjoint sensitivity analysis, J. Phys. Oceanogr., 44, 297–318, 2014.
Weaver, A. and Courtier, P.: Correlation modelling on the sphere using a generalized diffusion equation, Q. J. Roy. Meteor. Soc., 127, 1815–1846, 2001.
Wells, D.: Prime Numbers: the Most Mysterious Figures in Math, John Wiley & Sons, Hoboken, NJ, 2011.
Wilkin, J. and Hedström, K.: User's manual for an orthogonal curvilinear grid-generation package, Institute of Marine and Coastal Sciences, Rutgers University, available at: http://www.marine.rutgers.edu/po/tools/gridpak/grid_manual.ps.gz (last access: 29 April 2015), 1998.
Wilson, C., Chipperfield, M. P., Gloor, M., and Chevallier, F.: Development of a variational flux inversion system (INVICAT v1.0) using the TOMCAT chemical transport model, Geosci. Model Dev., 7, 2485–2500, https://doi.org/10.5194/gmd-7-2485-2014, 2014.
Wunsch, C.: Determining the general circulation of the oceans: a preliminary discussion, Science, 196, 871–875, 1977.
Wunsch, C.: Acoustic tomography and other answers, in: It's the water that makes you drunk. A celebration in Geophysics and Oceanography – 1982. In honor of Walter Munk on his 65th birthday, Scripps Institution of Oceanography Reference Series 84-5, Scripps Institution of Oceanography of California, San Diego, La Jolla, CA, 47–62, 1984.
Wunsch, C.: Discrete Inverse and State Estimation Problems: with Geophysical Fluid Applications, Cambridge University Press, 2006.
Wunsch, C. and Heimbach, P.: Practical global oceanic state estimation, Physica D, 230, 197–208, 2007.
Wunsch, C. and Heimbach, P.: The global zonally integrated ocean circulation, 1992–2006: seasonal and decadal variability, J. Phys. Oceanogr., 39, 351–368, https://doi.org/10.1175/2008JPO4012.1, 2009.
Wunsch, C. and Heimbach, P.: Dynamically and kinematically consistent global ocean circulation and ice state estimates, in: Ocean Circulation and Climate: a 21st Century Perspective, 103, 553–579, https://doi.org/10.1016/B978-0-12-391851-2.00021-0, 2013a.
Wunsch, C. and Heimbach, P.: Two decades of the Atlantic meridional overturning circulation: anatomy, variations, extremes, prediction, and overcoming its limitations, J. Climate, 26, 7167–7186, 2013b.
Wunsch, C. and Heimbach, P.: Bidecadal thermal changes in the Abyssal Ocean, J. Phys. Oceanogr., 44, 2013–2030, 2014.
Wunsch, C. and Minster, J.-F.: Methods for box models and ocean circulation tracers: mathematical programing and nonlinear inverse theory, J. Geophys. Res.-Oceans, 87, 5647–5662, 1982.
Wunsch, C., Ponte, R., and Heimbach, P.: Decadal trends in sea level patterns: 1993–2004, J. Climate, 20, 5889–5911, 2007.
Yu, L. and Weller, R. A.: Objectively analyzed air–sea heat fluxes for the global ice-free oceans (1981–2005), B. Am. Meteorol. Soc., 88, 527–539, 2007.
Zanna, L., Heimbach, P., Moore, A. M., and Tziperman, E.: Optimal excitation of interannual Atlantic meridional overturning circulation variability, J. Climate, 24, 413–427, 2011.
Short summary
The ECCO v4 non-linear inverse modeling framework and its reference solution are made publicly available. The inverse estimate of ocean physics and atmospheric forcing yields a dynamically consistent and global state estimate without unidentified sources of heat and salt that closely fits in situ and satellite data. Any user can reproduce it accurately. Parametric and external model uncertainties are of comparable magnitudes and generally exceed structural model uncertainties.
The ECCO v4 non-linear inverse modeling framework and its reference solution are made publicly...