\begin{thebibliography}{10}

\bibitem{adcroft2006}
Alistair Adcroft and Robert Hallberg.
\newblock On methods for solving the oceanic equations of motion in generalized
  vertical coordinates.
\newblock {\em Ocean Modelling}, 11(1-2):224--233, January 2006.

\bibitem{adcroft2008}
Alistair Adcroft, Robert Hallberg, and Matthew Harrison.
\newblock A finite volume discretization of the pressure gradient force using
  analytic integration.
\newblock {\em Ocean Modelling}, 22(3-4):106--113, January 2008.

\bibitem{arakawa1990}
Akio Arakawa and Yueh-Jiuan~G. Hsu.
\newblock Energy conserving and potential-enstrophy dissipating schemes for the
  shallow water equations.
\newblock {\em Monthly Weather Review}, 118:1960--1969, 1990.

\bibitem{arakawa1981}
Akio Arakawa and Vivian~R. Lamb.
\newblock A {Potential} {Enstrophy} and {Energy} {Conserving} {Scheme} for the
  {Shallow} {Water} {Equations}.
\newblock {\em Monthly Weather Review}, 109(1):18--36, January 1981.

\bibitem{bleck2002}
Rainer Bleck.
\newblock An oceanic general circulation model framed in hybrid
  isopycnic-{Cartesian} coordinates.
\newblock {\em Ocean Modelling}, 4(1):55--88, 2002.

\bibitem{bleck1990}
Rainer Bleck and Linda~T. Smith.
\newblock A wind‐driven isopycnic coordinate model of the north and
  equatorial atlantic ocean: 1. model development and supporting experiments.
\newblock {\em JGR Oceans}, 95:3273--3285, 1990.

\bibitem{carpenter1990}
Jr. Carpenter, Richard~L., Kelvin~K. Droegemeier, Paul~R. Woodward, and Carl~E.
  Hane.
\newblock Application of the piecewise parabolic method (ppm) to meteorological
  modeling.
\newblock {\em Monthly Weather Review}, 118:586--–612, 1990.

\bibitem{colella1984}
Phillip Colella and Paul~R Woodward.
\newblock The {Piecewise} {Parabolic} {Method} ({PPM}) for gas-dynamical
  simulations.
\newblock {\em Journal of Computational Physics}, 54(1):174--201, April 1984.

\bibitem{dukowicz2000}
John~K. Dukowicz and John~R. Baumgardner.
\newblock Incremental {Remapping} as a {Transport}/{Advection} {Algorithm}.
\newblock {\em Journal of Computational Physics}, 160(1):318--335, May 2000.

\bibitem{durran2010}
Dale~R. Durran.
\newblock {\em Numerical Methods for Fluid Dynamics With Applications to
  Geophysics}.
\newblock Springer-Verlag New York, 2010.

\bibitem{easter1993}
Richard~C. Easter.
\newblock Two modified versions of bott's positive-definite numerical advection
  scheme.
\newblock {\em Monthly Weather Review}, 121:297--304, 1993.

\bibitem{fox-kemper2011}
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.
\newblock Parameterization of mixed layer eddies. {III}: {Implementation} and
  impact in global ocean climate simulations.
\newblock {\em Ocean Modelling}, 39(1):61--78, January 2011.

\bibitem{gent1995}
Peter~R. Gent, Jurgen Willebrand, Trevor~J. McDougall, and James~C. McWilliams.
\newblock Parameterizing {Eddy}-{Induced} {Tracer} {Transports} in {Ocean}
  {Circulation} {Models}.
\newblock {\em Journal of Physical Oceanography}, 25(4):463--474, April 1995.

\bibitem{SMGbook}
Stephen~M. Griffies.
\newblock {\em Fundamentals of Ocean Climate Models}.
\newblock Princeton University Press, Princeton, USA, 2004.
\newblock 518+xxxiv pages.

\bibitem{griffies2000}
Stephen~M. Griffies and Robert~W. Hallberg.
\newblock Biharmonic {Friction} with a {Smagorinsky}-{Like} {Viscosity} for
  {Use} in {Large}-{Scale} {Eddy}-{Permitting} {Ocean} {Models}.
\newblock {\em Monthly Weather Review}, 128(8):2935--2946, August 2000.

\bibitem{hallberg1997}
Robert Hallberg.
\newblock Localized {Coupling} between {Surface} and {Bottom}-{Intensified}
  {Flow} over {Topography}.
\newblock {\em Journal of Physical Oceanography}, 27(6):977--998, June 1997.

\bibitem{hallberg1997a}
Robert Hallberg.
\newblock Stable split time stepping schemes for large-scale ocean modeling.
\newblock {\em Journal of Computational Physics}, 135:54--65, 1997.

\bibitem{hallberg2005}
Robert Hallberg.
\newblock A thermobaric instability of lagrangian vertical coordinate ocean
  models.
\newblock {\em Ocean Modelling}, 8, 2005.

\bibitem{hallberg2009}
Robert Hallberg and Alistair Adcroft.
\newblock Reconciling estimates of the free surface height in {Lagrangian}
  vertical coordinate ocean models with mode-split time stepping.
\newblock {\em Ocean Modelling}, 29(1):15--26, January 2009.

\bibitem{hirt1997}
C.~W. Hirt, A.~A. Amsden, and J.~L. Cook.
\newblock An {Arbitrary} {Lagrangian}-{Eulerian} {Computing} {Method} for {All}
  {Flow} {Speeds}.
\newblock {\em Journal of Computational Physics}, 135(2):203--216, August 1997.

\bibitem{huynh1997}
H.~T. Huynh.
\newblock Schemes and constraints for advection.
\newblock In P.~Kutler, J.~Flores, and J.J. Chattot, editors, {\em Fifteenth
  International Conference on Numerical Methods in Fluid Dynamics}, volume 490.
  Springer, Berlin, Heidelberg, 1997.

\bibitem{TEOS2010}
{I}{OC}, {SCOR}, and {IAPSO}.
\newblock {\em The international thermodynamic equation of seawater-2010:
  calculation and use of thermodynamic properties}.
\newblock Intergovernmental Oceanographic Commission, Manuals and Guides No.\
  56, UNESCO, 2010.
\newblock 196pp.

\bibitem{jackett1995}
David~R. Jackett and Trevor~J. McDougall.
\newblock Minimal adjustment of hydrographic profiles to achieve static
  stability.
\newblock {\em J. Atmos. Ocean. Tech.}, 12:381--389, 1995.

\bibitem{large1994}
W.~G. Large, J.~C. McWilliams, and S.~C. Doney.
\newblock Oceanic vertical mixing: {A} review and a model with a nonlocal
  boundary layer parameterization.
\newblock {\em Reviews of Geophysics}, 32(4):363--403, 1994.

\bibitem{lin1994}
Shian-Jiann Lin, Winston~C. Chao, Y.~C. Sud, and G.~K. Walker.
\newblock A class of the van leer-type transport schemes and its application to
  the moisture transport in a general circulation model.
\newblock {\em Mon. Wea. Rev.}, 122:1575--1593, 1994.

\bibitem{mcdougall2001}
Trevor~J. McDougall and Peter~C. McIntosh.
\newblock The {Temporal}-{Residual}-{Mean} {Velocity}. {Part} {II}: {Isopycnal}
  {Interpretation} and the {Tracer} and {Momentum} {Equations}.
\newblock {\em Journal of Physical Oceanography}, 31(5):1222--1246, May 2001.

\bibitem{millero1978}
F.J. Millero.
\newblock Freezing point of seawater.
\newblock In {\em Eight report of the Joint Panel on Oceanographic Tables and
  Standards (JPOTS)}, number~28, pages 29--35. UNESCO technical papers in
  marine sciences, 1978.
\newblock Annex 6.

\bibitem{roquet2015}
F.~Roquet, G.~Madec, T.~J. McDougall, and P.~M. Barker.
\newblock Accurate polynomial expressions for the density and specific volume
  of seawater using the teos-10 standard.
\newblock {\em Ocean Modelling}, 90:29--43, 2015.

\bibitem{russell1981}
Gary~L. Russell and Jean~A. Lerner.
\newblock A new finite-differencing scheme for the tracer transport equation.
\newblock {\em Journal of Applied Meteorology}, 20:1483--1498, 1981.

\bibitem{sadourny1975}
Robert Sadourny.
\newblock The {Dynamics} of {Finite}-{Difference} {Models} of the
  {Shallow}-{Water} {Equations}.
\newblock {\em Journal of the Atmospheric Sciences}, 32(4):680--689, April
  1975.

\bibitem{shao2019-in-review}
Andrew Shao, Alistair Adcroft, Robert Hallberg, and Stephen Griffies.
\newblock A new, general-coordinate, non-local neutral diffusion operator,
  2019.

\bibitem{shchepetkin2005}
A.~F. Shchepetkin and J.~C. McWilliams.
\newblock The regional ocean modeling system (roms): A split-explicit,
  free-surface, topography-following coordinates oceanic model.
\newblock {\em Ocean Modeling}, 9:347--404, 2005.

\bibitem{sun1999}
Shan Sun, Rainer Bleck, Claes Rooth, John Dukowicz, Eric Chassignet, and Peter
  Killworth.
\newblock Inclusion of thermobaricity in isopycnic-coordinate ocean models.
\newblock {\em Journal of Physical Oceanography}, 29:2719--2729, 1999.

\bibitem{GVbook}
Geoffrey~K.\ Vallis.
\newblock {\em Atmospheric and Oceanic Fluid Dynamics: Fundamentals and
  Large-scale Circulation}.
\newblock Cambridge University Press, Cambridge, 2nd edition, 2017.
\newblock 946 + xxv pp.

\bibitem{white2008}
Laurent White and Alistair Adcroft.
\newblock A high-order finite volume remapping scheme for nonuniform grids:
  {The} piecewise quartic method ({PQM}).
\newblock {\em Journal of Computational Physics}, 227(15):7394--7422, July
  2008.

\bibitem{white2009}
Laurent White, Alistair Adcroft, and Robert Hallberg.
\newblock High-order regridding-remapping schemes for continuous isopycnal and
  generalized coordinates in ocean models.
\newblock {\em Journal of Computational Physics}, 228(23):8665--8692, December
  2009.

\bibitem{wright1997}
Daniel~G. Wright.
\newblock An {Equation} of {State} for {Use} in {Ocean} {Models}: {Eckart}'s
  {Formula} {Revisited}.
\newblock {\em Journal of Atmospheric and Oceanic Technology}, 14(3):735--740,
  June 1997.

\end{thebibliography}