A. Adcroft, W. Anderson, V. Balaji, C. Blanton, M. Bushuk, C. O. Dufour, J. P. Dunne, S. M. Griffies, R. Hallberg, M. J. Harrison, I. M. Held, M. F. Jansen, J. G. John, J. P. Krasting, A. R. Langenhorst, S. Legg, Z. Liang, C. McHugh, A. Radhakrishnan, B. G. Reichl, T. Rosati, B. L. Samuels, A. Shao, R. Stouffer, M. Winton, A. T. Wittenberg, B. Xiang, N. Zadeh, and R. Zhang. The GFDL global ocean and sea ice model OM4.0: model description and simulation features. J. Adv. Mod. Earth Sys., 11(10):3167–3211, 2019. doi:10.1029/2019ms001726.


Alistair Adcroft and Robert Hallberg. On methods for solving the oceanic equations of motion in generalized vertical coordinates. Ocean Modelling, 11(1-2):224–233, January 2006. URL: (visited on 2019-01-02), doi:10.1016/j.ocemod.2004.12.007.


Alistair Adcroft, Robert Hallberg, and Matthew Harrison. A finite volume discretization of the pressure gradient force using analytic integration. Ocean Modelling, 22(3-4):106–113, January 2008. URL: (visited on 2019-01-02), doi:10.1016/j.ocemod.2008.02.001.


Akio Arakawa and Yueh-Jiuan G. Hsu. Energy conserving and potential-enstrophy dissipating schemes for the shallow water equations. Monthly Weather Review, 118:1960–1969, 1990. doi:10.1175/1520-0493(1990)118<1960:ECAPED>2.0.CO;2.


Akio Arakawa and Vivian R. Lamb. A Potential Enstrophy and Energy Conserving Scheme for the Shallow Water Equations. Monthly Weather Review, 109(1):18–36, January 1981. URL:, doi:10.1175/1520-0493(1981)109<0018:APEAEC>2.0.CO;2.


T. H. Bell. Lee waves in stratified flows with simple harmonic time dependence. J. Fluid Mech., 67(4):705–722, 1975. doi:10.1017/s0022112075000560.


Rainer Bleck. An oceanic general circulation model framed in hybrid isopycnic-Cartesian coordinates. Ocean Modelling, 4(1):55–88, 2002. URL: (visited on 2016-02-04), doi:10.1016/S1463-5003(01)00012-9.


Rainer Bleck and Linda T. Smith. A wind‐driven isopycnic coordinate model of the north and equatorial atlantic ocean: 1. model development and supporting experiments. JGR Oceans, 95:3273–3285, 1990. doi:10.1029/JC095iC03p03273.


K. Bryan and L. J. Lewis. A water mass model of the world ocean. J. Geophys. Res., 84(C5):2503, 1979. doi:10.1029/jc084ic05p02503.


Jr. Carpenter, Richard L., Kelvin K. Droegemeier, Paul R. Woodward, and Carl E. Hane. Application of the piecewise parabolic method (ppm) to meteorological modeling. Monthly Weather Review, 118:586––612, 1990. doi:<0586:AOTPPM>2.0.CO;2.


Phillip Colella and Paul R Woodward. The Piecewise Parabolic Method (PPM) for gas-dynamical simulations. Journal of Computational Physics, 54(1):174–201, April 1984. URL: (visited on 2017-02-09), doi:10.1016/0021-9991(84)90143-8.


G. Danabasoglu, S. C. Bates, B. P. Briegleb, S. R. Jayne, M. Jochum, W. G. Large, S. Peacock, and S. G. Yeager. The CCSM4 ocean component. J. Climate, 25(5):1361–1389, 2012. doi:10.1175/jcli-d-11-00091.1.


John K. Dukowicz and John R. Baumgardner. Incremental Remapping as a Transport/Advection Algorithm. Journal of Computational Physics, 160(1):318–335, May 2000. URL: (visited on 2019-03-27), doi:10.1006/jcph.2000.6465.


Dale R. Durran. Numerical Methods for Fluid Dynamics With Applications to Geophysics. Springer-Verlag New York, 2010. doi:10.1007/978-1-4419-6412-0.


Richard C. Easter. Two modified versions of bott's positive-definite numerical advection scheme. Monthly Weather Review, 121:297–304, 1993. doi:10.1175/1520-0493(1993)121<0297:TMVOBP>2.0.CO;2.


B. Fox-Kemper, G. Danabasoglu, R. Ferrari, S. M. Griffies, R. W. Hallberg, M. M. Holland, M. E. Maltrud, S. Peacock, and B. L. Samuels. Parameterization of mixed layer eddies. III: Implementation and impact in global ocean climate simulations. Ocean Modelling, 39(1):61–78, January 2011. URL:, doi:10.1016/j.ocemod.2010.09.002.


B. Fox-Kemper and R. Ferrari. Parameterization of mixed layer eddies. part ii: prognosis and impact. J. Phys. Oceangraphy, 38:1166–1179, 2008. doi:10.1175/2007JPO3788.1.


B. Fox-Kemper, R. Ferrari, and R. Hallberg. Parameterization of mixed layer eddies. part i: theory and diagnosis. J. Phys. Oceangraphy, 38:1145–1165, 2008. doi:10.1175/2007JPO3792.1.


Peter R. Gent and James C. Mcwilliams. Isopycnal mixing in ocean circulation models. J. Phys. Oceanogr., 20:150–155, 1990. doi:10.1175/1520-0485(1990)020<0150:IMIOCM>2.0.CO;2.


Peter R. Gent, Jurgen Willebrand, Trevor J. McDougall, and James C. McWilliams. Parameterizing Eddy-Induced Tracer Transports in Ocean Circulation Models. Journal of Physical Oceanography, 25(4):463–474, April 1995. URL: (visited on 2018-10-10), doi:10.1175/1520-0485(1995)025<0463:PEITTI>2.0.CO;2.


S. M. Griffies, M. Levy, A. J. Adcroft, G. Danabasoglu, R. W. Hallberg, D. Jacobsen, W. Large, and T. Ringler. Theory and numerics of the community ocean vertical mixing (cvmix) project. Technical Report, NOAA GFDL, 2015.


S.M. Griffies, A. Adcroft, and R.W. Hallberg. A primer on the vertical lagrangian-remap method in ocean models based on finite volume generalized vertical coordinates. Journal of Advances in Modeling Earth Systems, 2020. doi:10.1029/2019MS001954.


Stephen M. Griffies. Fundamentals of Ocean Climate Models. Princeton University Press, Princeton, USA, 2004. 518+xxxiv pages.


Stephen M. Griffies, Claus Böning, Frank O. Bryan, Eric P. Chassignet, Rüdiger Gerdes, Hiroyasu Hasumi, Anthony Hirst, Anne-Marie Treguier, and David Webb. Developments in ocean climate modelling. Ocean Modelling, 2(3):123–192, January 2000. URL: (visited on 2019-03-07), doi:10.1016/S1463-5003(00)00014-7.


Stephen M. Griffies and Robert W. Hallberg. Biharmonic Friction with a Smagorinsky-Like Viscosity for Use in Large-Scale Eddy-Permitting Ocean Models. Monthly Weather Review, 128(8):2935–2946, August 2000. URL: (visited on 2018-06-10), doi:10.1175/1520-0493(2000)128<2935:BFWASL>2.0.CO;2.


Robert Hallberg. Localized Coupling between Surface and Bottom-Intensified Flow over Topography. Journal of Physical Oceanography, 27(6):977–998, June 1997. URL: (visited on 2016-09-21), doi:10.1175/1520-0485(1997)027<0977:LCBSAB>2.0.CO;2.


Robert Hallberg. Stable split time stepping schemes for large-scale ocean modeling. Journal of Computational Physics, 135:54–65, 1997. doi:DOI:10.1006/jcph.1997.5734.


Robert Hallberg. Time integration of diapycnal diffusion and richardson number–dependent mixing in isopycnal coordinate ocean models. Monthly Weather Review, 128:1402–1419, 2000.


Robert Hallberg. A thermobaric instability of lagrangian vertical coordinate ocean models. Ocean Modelling, 2005. doi:10.1016/j.ocemod.2004.01.001.


Robert Hallberg and Alistair Adcroft. Reconciling estimates of the free surface height in Lagrangian vertical coordinate ocean models with mode-split time stepping. Ocean Modelling, 29(1):15–26, January 2009. URL: (visited on 2019-01-02), doi:10.1016/j.ocemod.2009.02.008.


M. J. Harrison and R. W. Hallberg. Pacific Subtropical Cell Response to Reduced Equatorial Dissipation. Journal of Physical Oceanography, 38(9):1894–1912, September 2008. URL: (visited on 2018-12-20), doi:10.1175/2008JPO3708.1.


Frank S. Henyey, Jon Wright, and Stanley M. Flatté. Energy and action flow through the internal wave field: An eikonal approach. Journal of Geophysical Research: Oceans, 91(C7):8487–8495, 1986. URL: (visited on 2018-12-20), doi:10.1029/JC091iC07p08487.


C. W. Hirt, A. A. Amsden, and J. L. Cook. An Arbitrary Lagrangian-Eulerian Computing Method for All Flow Speeds. Journal of Computational Physics, 135(2):203–216, August 1997. URL: (visited on 2016-10-05), doi:10.1006/jcph.1997.5702.


H. T. Huynh. Schemes and constraints for advection. In P. Kutler, J. Flores, and J.J. Chattot, editors, Fifteenth International Conference on Numerical Methods in Fluid Dynamics, volume 490. Springer, Berlin, Heidelberg, 1997. doi:10.1007/BFb0107151.


David R. Jackett and Trevor J. McDougall. Minimal adjustment of hydrographic profiles to achieve static stability. J. Atmos. Ocean. Tech., 12:381–389, 1995. doi:10.1175/1520-0426(1995)012<0381:MAOHPT>2.0.CO;2.


L. Jackson, R. Hallberg, and S. Legg. A Parameterization of Shear-Driven Turbulence for Ocean Climate Models. Journal of Physical Oceanography, 38(5):1033–1053, May 2008. URL: (visited on 2018-10-12), doi:10.1175/2007JPO3779.1.


Malte F. Jansen, Alistair J. Adcroft, Robert Hallberg, and Isaac M. Held. Parameterization of eddy fluxes based on a mesoscale energy budget. Ocean Modelling, 92:28–41, August 2015. URL: (visited on 2018-09-21), doi:10.1016/j.ocemod.2015.05.007.


P. D. Killworth and N. R. Edwards. A turbulent bottom boundary layer code for use in numerical ocean models. J. Phys. Oceanography, 29(6):1221–1238, 1999. doi:10.1175/1520-0485(1999)029<1221:atbblc>;2.


E. B. Kraus and J. S. Turner. A one-dimensional model of the seasonal thermocline II. the general theory and its consequences. Tellus, 19(1):98–106, 1967. doi:10.3402/tellusa.v19i1.9753.


W. G. Large, J. C. McWilliams, and S. C. Doney. Oceanic vertical mixing: A review and a model with a nonlocal boundary layer parameterization. Reviews of Geophysics, 32(4):363–403, 1994. URL: (visited on 2019-01-16), doi:10.1029/94RG01872.


Shian-Jiann Lin, Winston C. Chao, Y. C. Sud, and G. K. Walker. A class of the van leer-type transport schemes and its application to the moisture transport in a general circulation model. Mon. Wea. Rev., 122:1575–1593, 1994. doi:10.1175/1520-0493(1994)122<1575:ACOTVL>2.0.CO;2.


D. P. Marshall and A. J. Adcroft. Parameterization of ocean eddies: potential vorticity mixing, energetics and arnold first stability theorem. Ocean Modelling, 32:188–204, 2010. doi:10.1016/j.ocemod.2010.02.001.


T. J. McDougall, P.M. Barker, R.M. Holmes, R. Pawlowicz, S.M. Grif\/f\/ies, and P.J. Durack. The interpretation of temperature and salinity variables in numerical ocean model output, and the calculation of heat fluxes and heat content. Geoscientific Model Development, 14:6445–6466, 2021. doi:10.5194/gmd-14-6445-2021.


Trevor J. McDougall and Peter C. McIntosh. The Temporal-Residual-Mean Velocity. Part II: Isopycnal Interpretation and the Tracer and Momentum Equations. Journal of Physical Oceanography, 31(5):1222–1246, May 2001. URL: (visited on 2018-10-10), doi:10.1175/1520-0485(2001)031<1222:TTRMVP>2.0.CO;2.


Angelique Melet, Robert Hallberg, Sonya Legg, and Kurt Polzin. Sensitivity of the Ocean State to the Vertical Distribution of Internal-Tide-Driven Mixing. Journal of Physical Oceanography, 43(3):602–615, December 2012. URL: (visited on 2018-11-19), doi:10.1175/JPO-D-12-055.1.


F.J. Millero. Freezing point of seawater. In Eight report of the Joint Panel on Oceanographic Tables and Standards (JPOTS), number 28, 29–35. UNESCO technical papers in marine sciences, 1978. Annex 6. URL:


P. Muller. A\textquotesingle ha huliko\textquotesingle a workshop series. Technical Report, School of Ocean and Earth Science and Technology, 2003. doi:10.21236/ada618366.


M. Nikurashin and R. Ferrari. Radiation and dissipation of internal waves generated by geostrophic motions impinging on small-scale topography: application to the southern ocean. J. Phys. Oceanography, 40(9):2025–2042, 2010. doi:10.1175/2010jpo4315.1.


M. Nikurashin and R. Ferrari. Radiation and dissipation of internal waves generated by geostrophic motions impinging on small-scale topography: theory. J. Phys. Oceanography, 40(5):1055–1074, 2010. doi:10.1175/2009jpo4199.1.


R. C. Pacanowski and S. G. H. Philander. Parameterization of vertical mixing in numerical models of tropical oceans. J. Phys. Oceanography, 11(11):1443–1451, 1981. doi:10.1175/1520-0485(1981)011<1443:povmin>;2.


Kurt L. Polzin. Idealized solutions for the energy balance of the finescale internal wave field. J. Phys. Oceanogr., 34:231–246, 2004.


Kurt L. Polzin. An abyssal recipe. Ocean Modelling, 30(4):298–309, January 2009. URL: (visited on 2018-11-19), doi:10.1016/j.ocemod.2009.07.006.


Brandon G. Reichl and Robert Hallberg. A simplified energetics based planetary boundary layer (ePBL) approach for ocean climate simulations. Ocean Modelling, 132:112–129, December 2018. URL: (visited on 2018-11-16), doi:10.1016/j.ocemod.2018.10.004.


F. Roquet, G. Madec, T. J. McDougall, and P. M. Barker. Accurate polynomial expressions for the density and specific volume of seawater using the teos-10 standard. Ocean Modelling, 90:29–43, 2015.


Gary L. Russell and Jean A. Lerner. A new finite-differencing scheme for the tracer transport equation. Journal of Applied Meteorology, 20:1483–1498, 1981. doi:10.1175/1520-0450(1981)020<1483:ANFDSF>2.0.CO;2.


Robert Sadourny. The Dynamics of Finite-Difference Models of the Shallow-Water Equations. Journal of the Atmospheric Sciences, 32(4):680–689, April 1975. URL: (visited on 2019-07-28), doi:10.1175/1520-0469(1975)032<0680:TDOFDM>2.0.CO;2.


A. Shao, A.J. Adcroft, R.W. Hallberg, and S.M. Griffies. A general-coordinate, nonlocal neutral diffusion operator. Journal of Advances in Modeling Earth Systems, 2020. doi:10.1029/2019MS001992.


Andrew Shao, Alistair Adcroft, Robert Hallberg, and Stephen Griffies. A new, general-coordinate, non-local neutral diffusion operator. 2019.


A. F. Shchepetkin and J. C. McWilliams. The regional ocean modeling system (roms): a split-explicit, free-surface, topography-following coordinates oceanic model. Ocean Modeling, 9:347–404, 2005.


H. L. Simmons, S. R. Jayne, L. C. St. Laurent, and A. J. Weaver. Tidally driven mixing in a numerical model of the ocean general circulation. Ocean Modell., 6:245–263, 2004. doi:10.1016/S1463-5003(03)00011-8.


L. C. St Laurent, H. L. Simmons, and S. R. Jayne. Estimating tidally driven mixing in the deep ocean. Geophysical Research Letters, 29(23):21–1–21–4, December 2002. URL: (visited on 2018-11-19), doi:10.1029/2002GL015633.


Shan Sun, Rainer Bleck, Claes Rooth, John Dukowicz, Eric Chassignet, and Peter Killworth. Inclusion of thermobaricity in isopycnic-coordinate ocean models. Journal of Physical Oceanography, 29:2719–2729, 1999.


J. S. Turner. Turbulent entrainment: the development of the entrainment assumption, and its application to geophysical flows. J. Fluid Mech., 173:431–471, 1986. doi:10.1017/s0022112086001222.


L. Umlauf and H. Burchard. Second-order turbulence closure models for geophysical boundary layers. a review of recent work. Continental Shelf Res., 25(7-8):795–827, 2005. doi:10.1016/j.csr.2004.08.004.


Geoffrey K. Vallis. Atmospheric and Oceanic Fluid Dynamics: Fundamentals and Large-scale Circulation. Cambridge University Press, Cambridge, 2nd edition, 2017. 946 + xxv pp.


Martin Visbeck, John Marshall, Tom Haine, and Mike Spall. Specification of eddy transfer coefficients in coarse-resolution ocean circulation models. J. Phys. Oceanogr., 27:381–402, 1997. doi:10.1175/1520-0485(1997)027<0381:SOETCI>2.0.CO;2.


D. Wang. Entrainment laws and a bulk mixed layer model of rotating convection derived from large-eddy simulations. Geophys. Res. Lett., 2003. doi:10.1029/2003gl017869.


Laurent White and Alistair Adcroft. A high-order finite volume remapping scheme for nonuniform grids: The piecewise quartic method (PQM). Journal of Computational Physics, 227(15):7394–7422, July 2008. URL: (visited on 2019-01-02), doi:10.1016/


Laurent White, Alistair Adcroft, and Robert Hallberg. High-order regridding-remapping schemes for continuous isopycnal and generalized coordinates in ocean models. Journal of Computational Physics, 228(23):8665–8692, December 2009. URL: (visited on 2018-12-15), doi:10.1016/


Daniel G. Wright. An Equation of State for Use in Ocean Models: Eckart's Formula Revisited. Journal of Atmospheric and Oceanic Technology, 14(3):735–740, June 1997. URL:, doi:10.1175/1520-0426(1997)014<0735:AEOSFU>2.0.CO;2.


W. R. Young. Dynamic enthalpy, Conservative Temperature, and the seawater Boussinesq approximation. Journal of Physical Oceanography, 40:394–400, 2010. doi:10.1175/2009JPO4294.1.


S. Zilitinkevich and D. V. Mironov. A multi-limit formulation for the equilibrium depth of a stably stratified boundary layer. Boundary-Layer Meteorology, 81(3-4):325–351, 1996. doi:10.1007/bf02430334.


IOC, SCOR, and IAPSO. The international thermodynamic equation of seawater-2010: calculation and use of thermodynamic properties. Intergovernmental Oceanographic Commission, Manuals and Guides No. 56, UNESCO, edition, 2010. 196pp.