A Primer on the Vertical LagrangianRemap Method in Ocean Models Based on Finite Volume Generalized Vertical Coordinates
Abstract
This paper provides a primer on the mathematical, physical, and numerical foundations of ocean models that are formulated using finite volume generalized vertical coordinate equations and that use the vertical Lagrangianremap method to evolve the ocean state. We consider the mathematical structure of the governing ocean equations in both their strong formulation (partial differential equations) and weak formulation (finite volume integral equations), thus enabling an understanding of their physical content and providing a physicalmathematical framework to develop numerical algorithms. A connection is made between the Lagrangianremap method and the ocean equations as written using finite volume generalized vertical budgets. Thought experiments are offered to exemplify the mechanics of the vertical Lagrangianremap method and to compare with other methods used for ocean model algorithms.
Key Points
 A pedagogical discussion is provided for ocean model dynamical cores that use the vertical Lagrangianremap method
 Mathematical physics is developed for finite volume ocean models that use generalized vertical coordinates
 A comparison is made between quasiEulerian, Arbitrary LagrangianEulerian, and vertical Lagrangian remapping methods
Plain Language Summary
Numerical ocean models are based on a variety of mathematical methods that provide the framework for developing computational algorithms suitable for solving the ocean equations on a computer. The vertical Lagrangianremap method is ideally suited for finite volume ocean models formulated using generalized vertical coordinates. It is gaining popularity among the ocean modeling community since it holds the potential to resolve longstanding limitations inherent with standard methods. However, there is some mystery surrounding the fundamentals of such ocean models. The goal of this paper is to dispel the mystery. We do so by providing a pedagogical treatment of the fluid mechanics that forms the basis for numerical ocean models formulated with finite volume generalized vertical coordinate equations and that use the vertical Lagrangianremap method.
1 Introduction
A finite volume generalized vertical coordinate (GVC) formulation of the ocean equations, when combined with the vertical Lagrangianremap method, enables a wide suite of vertical coordinate options. It also naturally allows for an exactly conserving realization of wetting and drying by following the numerical algorithms required for vanishing layers in isopycnal models. Since its inception for numerical ocean models by Bleck (2002), the vertical Lagrangianremap method has realized scientific and engineering maturity in the ocean modeling community. In particular, numerical algorithms based on the vertical Lagrangianremap method form the dynamical core for community ocean models such as HYCOM (Bleck, 2002) and MOM6 (Adcroft et al., 2019). These models have been configured in a variety of applications related to process physics, realistic operational prediction, and comprehensive earth system science. Use of the vertical Lagrangianremap method for ocean models is motivated by the numerical accuracy and flexibility it offers relative to alternatives. In response to this promise, we here offer a primer on ocean models that use the vertical Lagrangianremap method as framed within finite volume GVC equations.
1.1 An Outline of the LagrangianRemap Method
 Lagrangian step: fluid equations are solved in a Lagrangian reference frame whereby the grid mesh follows the fluid flow;
 regrid/regularization step: the grid mesh is moved to a specified target location; and
 remap step: the fluid state is conservatively remapped onto the target grid.
The regrid step generally specifies a movement of the grid relative to the fluid thus complementing the Lagrangian step whereby the grid moves with the fluid. The combined regridremap steps can be numerically identical to advection in an Eulerian method (Dukowicz & Baumgardner, 2000). When using GVCs, important differences from a fixed grid approach arise in practice due to the movement of the grid with the flow in the Lagrangian step, thus facilitating the reduction of crossgrid flow and the corresponding spurious calculation of crossgrid transport that occurs with traditional approaches (e.g., Griffies, Pacanowski, & Hallberg, 2000).
As realized in ocean models, the vertical Lagrangianremap method considers only the vertical direction for the Lagrangian step, with fixed horizontal positions for the mesh. In the ocean modeling community the vertical Lagrangianremap method is sometimes considered a particular realization of the Arbitrary LagrangianEulerian (ALE) method (Donea et al., 2004; Hirt et al., 1974). Indeed, ALE involves three steps very much aligned with those outlined above for Lagrangian remapping. However, ALE generally prescribes the grid motion, and that motion is not typically Lagrangian. Furthermore, Lagrangian methods date back to the origins of computational fluid mechanics (von Neumann & Richtmyer, 1950) whereas ALE evolved as a particular means to address limitations of early Lagrangian methods (Noh, 1964). We here follow terminology used in the computational physics community by referring to the method utilized by Bleck (2002) as vertical Lagrangian remapping rather than ALE.
1.2 Focus for This Paper and Presentation Style
Finite volumebased ocean equations written in GVCs provide the theoretical basis for many ocean models developed for applications ranging from the coast to the globe. These equations are key to a formulation of ocean model dynamical cores based on the vertical Lagrangianremap method. Our central goal for this paper is to accessibly present the physical, mathematical, and numerical basis for such models. That is, we aim to articulate concepts and methods forming the foundation of ocean models, thus allowing the methods to be understood sufficiently well that they can be scrutinized by a broad range of readers and in turn improved upon by future innovations. This paper thus supports a community understanding of modern numerical ocean models, and by doing so we hope to enhance the utility, transparency, integrity, and maturity of the science going into ocean models as well as the science emerging from their simulations.
To approach these goals we present the material in the form of a primer or tutorial. The reader is expected to have a firm understanding of ocean fluid mechanics yet not to have extensive experience with ocean model methods nor numerical algorithms. Correspondingly, we synthesize intellectual threads that underlie ocean model dynamical cores, with particular emphasis on finite volume formulations, GVCs, and vertical Lagrangian remapping. Although we focus on the dynamical core, we make some comment on subgrid scale (SGS) parameterizations where salient.
The debate over an optimal vertical coordinate has a long history in ocean modeling (e.g., Chassignet et al., 2006; Getzlaff et al., 2010; Griffies, Böning, et al., 2000; Hofmeister et al., 2010; Klingbeil et al., 2018; Leclair & Madec, 2011; Megann et al., 2010; White et al., 2009). However, this primer is not specifically concerned with the vertical coordinate choice, which will be mentioned only when pertinent, nor about the numerical algorithms for interpolation or remapping (e.g., Colella & Woodward, 1984; White & Adcroft, 2008). Instead, our focus concerns physical/mathematical methods that allow for any choice of vertical coordinate, with these methods independent of the vertical coordinate choice.
1.3 Mathematical Conventions
We make use of Cartesian tensors where convenient. Extensions to curvilinear horizontal coordinates are readily made through use of general covariance, with such extensions routinely realized in ocean codes simulating flow on the sphere. In essence, general covariance means that we can establish the physical validity of mathematical relations using any convenient coordinates (e.g., Cartesian) and then extend the relations to arbitrary coordinates by ensuring that tensor indices properly match, the Kronecker metric (i.e., identity tensor) is converted to a general metric, and partial derivatives are converted to covariant derivatives (see Chapter 21 of Griffies, 2004 for details).
General covariance applies to all three spatial directions, including the vertical. Nonetheless, we pay special attention to the needs of GVCs given that they offer the primary means to realize vertical Lagrangian methods in ocean models. Importantly, the use of GVCs requires us to respect their nonorthogonality properties. We recommend Starr (1945), Kasahara (1974), Chapter 6 of Griffies (2004), Young (2012), and Appendix C of the current paper for detailed mathematics of GVCs.
1.4 Content
We start the main portion of this paper in section 2 by developing the finite volume equations for an arbitrary moving region of fluid. This weak formulation of the ocean equations provides our starting point for the grid cell equations then derived in section 3. In section 4 we focus on the vertical Lagrangianremap method, aiming to expose the underlying conceptual picture forming the foundation for the method. Section 5 then provides a comparison of the vertical Lagrangianremap method to alternative approaches used to numerically time step the ocean equations. We close the main portion of the paper in section 6 by offering some concluding remarks and speculations.
Thereafter, we provide a suite of appendices that support our understanding of ocean model foundations. Appendix A summarizes the partial differential equations describing the motion of a seawater fluid element, with these equations comprising the strong formulation of the ocean equations. Appendix B documents the treatment of stresses (pressure and SGS friction) in ocean models. Appendix C provides the mathematical connection between the strong (differential) and weak (integral) formulations of the ocean equations. Appendix D offers a mathematical tutorial on GVCs. Appendix E tabulates the mathematical symbols used in this paper.
2 Finite Volume Equations for Moving Fluid Regions
One can formulate the equations of geophysical fluid mechanics by developing partial differential equations describing the motion of tiny (“infinitesimal”) elements of seawater, and we summarize the physical content of that description in Appendix A. However, differential equations do not provide our desired starting point for a finite volume formulation of discrete ocean model equations. We instead formulate integral budget equations for arbitrary closed moving fluid regions of finite extent, with our presentation inspired by similar treatments given by Donea et al. (2004) and Ringler (2011). The resulting finite volume budgets for scalars (e.g., seawater mass, tracer mass, and potential enthalpy) and vectors (e.g., linear momentum and vorticity) offer a weak formulation of the ocean fluid mechanical equations, whereas the corresponding partial differential equations provide a strong formulation. The terminology refers to the need to make more assumptions about smoothness of the flow state when working with the partial differential equations (i.e., “strong assumptions”), whereas the finite volume integral equations generally allow for nonsmooth flow features (i.e., “weak assumptions”). For many purposes of geophysical fluid mechanics, one can seamlessly move between the weak and strong formulations via the LeibnizReynolds transport theorem as detailed in Appendix C. We make use of the weak formulation as it provides a natural framework for developing the finite volume ocean equations.
2.1 Budget Equations for Scalar Fields
These equations constitute the weak form of the scalar budget equations for a finite region of fluid, with their connection to the strong form provided in Appendix C. The lefthand side of Equation 1 expresses the time tendency for regionintegrated mass of a scalar whose mass concentration (mass of scalar per mass of seawater) is C. Examples include salinity, S, Conservative Temperature times the constant heat capacity, c_{p} Θ, and biogeochemical trace species. Likewise, the lefthand side of Equation 2 is the time tendency for the total seawater mass in the region, with ρ the in situ density of seawater. The righthand side to these equations is the surface integral for the transport of these scalar properties through the region boundary. The first transport term arises from the difference between the fluid velocity, v, and the velocity of the boundary, v^{(b)}, with referred to as the diasurface transport of seawater mass (dimensions of mass per time) across a surface (see section D0.4 for more details), and the corresponding diasurface transport of tracer mass. The diasurface transport is projected onto the outward normal at the boundary so that the scalar budgets are unaffected by transport tangential to the boundary. The second transport term in the tracer Equation 1 arises from the SGS flux across the region boundary as parameterized by J. Notably, this flux generally incorporates surface and bottom boundary fluxes via Neumann boundary conditions placed on J.
2.2 Budget Equation for Linear Momentum
3 Finite Volume Ocean Model Grid Cell Budgets
We here specialize the finite volume scalar and linear momentum budgets from section 2 to develop budget equations for numerical ocean model grid cells such as those shown in Figure 2. The grid cells have top and bottom interfaces defined by surfaces of GVCs denoted by , with discrete values labeled s_{k ± 1/2} and with k an integer increasing downward. Along with interior transport of seawater between grid cells, we must also account for the transport of matter, enthalpy, tracers, and momentum across the surface and bottom boundaries of the ocean domain.
In these finite volume budgets, by virtue of the weak formulation, the grid is sufficiently described by the volume boundary positions alone, and there are no properties associated with a point at the center of a cell. Thus, in finite volume models, there is a general emphasis on reconstruction methods for calculating transports across surfaces rather than on finite difference methods for gradients between points.
3.1 Framing the Budgets
Lateral boundaries for interior grid cells are vertical so that the corresponding diasurface mass transport arises from the horizontal seawater velocity, . For cell boundaries defined by a constant ssurface, we make use of the GVC diasurface transport detailed in section D0.4. In that treatment, we define as the diasurface volume transport of seawater crossing a constant ssurface, per unit horizontal area. The diasurface transport serves as a “vertical velocity” in that multiplication by horizontal area element yields the instantaneous transport of seawater crossing the undulating ssurface. Following details in section D0.5, we introduce a corresponding diasurface SGS transport to account for unresolved processes such as diasurface diffusion that cause tracers and momentum to cross an ssurface.
At the surface ocean boundary, diasurface mass transport arises from precipitation, evaporation, and surface imposed river runoff (some models impose river runoff at depth as well). Diasurface transport of tracer and momentum across the ocean surface arises from airsea and icesea boundary fluxes, including turbulent and radiative heating (with shortwave radiation penetrating below the surface layer), turbulent stresses from winds, transport of enthalpy and tracers within the matter crossing the boundary (i.e., precipitation, evaporation, and runoff carry enthalpy, momentum, and tracer content), and other boundary tracer fluxes.
3.2 Scalar Budgets
3.3 Cell Averaged Fields
The finite volume budgets developed thus far consist of continuous fields integrated over a cell volume or cell face. A corresponding finite volume numerical model provides a discrete representation of the grid cell averages, with fluxes computed at the cell faces estimated through methods that utilize the cell averaged values. In this subsection we introduce grid cell averaged fields with a focus on issues related to the vertical direction, though note that similar issues arise for the horizontal.
The horizontal flux components are thickness weighted, with the thickness weighting reflecting the general case of a grid cell with a distinct thickness on each of its lateral cell faces (e.g., Figure 2). In a discrete model, the area averaged fluxes at cell faces are estimated via algorithms that make use of the cell averaged fields. There are many algorithms for this estimation with details outside of our scope.
3.4 Budget for Cell Averaged Scalars
3.5 Budget for Cell Averaged Linear Momentum
4 The Vertical LagrangianRemap Method
The finite volume GVC ocean equations developed in section 3 can be time stepped using a variety of numerical algorithms. We are interested here in three broad categories based on (i) those used in geopotential models (quasiEulerian); (ii) those inspired by isopycnal models (vertical Lagrangian) (Adcroft & Hallberg, 2006; Bleck, 2002); and (iii) those used in vertical ALE models as proposed by Leclair and Madec (2011) and Petersen et al. (2015). For example, the MITgcm of Adcroft and Campin (2004), the ROMS code of Shchepetkin and McWilliams (2005), the NEMO code of Madec (2008), and the MOM5 code of Griffies (2012) are formulated as GVC ocean models that use quasiEulerian numerical methods whereby the diasurface transport is diagnosed based on mass continuity (see section 5.2). In contrast, vertical Lagrangianremap methods as used by HYCOM (Bleck, 2002) and MOM6 (Adcroft et al., 2019) have direct control over the motion of coordinate surfaces, as does the vertical ALE method of Leclair and Madec (2011) and Petersen et al. (2015) though using distinct approaches. In this section we offer a conceptual description of the vertical Lagrangianremap method, and then in section 5 we compare numerical realizations of quasiEulerian, vertical ALE, and vertical Lagrangian remapping.
4.1 Essence of ALE and Vertical Lagrangian Remapping
The finite volume budgets from section 3 form the theoretical basis for the ALE method as reviewed by Donea et al. (2004), as well as Lagrangianremap methods. Both methods consider a grid mesh that moves as the fluid evolves (see Figure 1), with Lagrangian methods allowing the grid to follow the flow whereas ALE typically prescribes a distinction between fluid and grid motion. In essence, the methods are differentiated by specifications of the velocity, v^{(b)}, here measuring the velocity of the grid lines. A static grid ( ) represents a trivial limit of ALE in which the grid is Eulerian. A moving grid is Lagrangian by setting , whereas ALE grids are typically prescribed otherwise.
4.1.1 Concerning the Choice of Grids
A key constraint on the utility of grids is that they remain nonsingular in order to uniquely estimate the fluid state: The surfaces of constant GVC should never go vertical, should never cross over other coordinate surfaces, nor should they overturn. Furthermore, we generally wish to utilize grid surfaces in a “smart” manner that allows for accurate estimates of fluid properties (e.g., density, tracers, and velocity). These concerns are central to the problem of designing grids for realistic geophysical fluid motion (e.g., boundary currents, jets, fluid instabilities, and turbulent motions) with inhomogeneous properties (e.g., vast range of density structure between the tropics vs. high latitudes). One approach that follows the vertical ALE method was proposed by Leclair and Madec (2011) and Petersen et al. (2015). Their method (detailed in section 5) retains a suitable gridding by damping their moving grid back to a lowpass filtered state that remains nonsingular. For this approach the first arrow in Figure 1 is not a Lagrangian step and the second arrow (regridding) is absent.
4.1.2 Illustrating the Method
We illustrate the vertical Lagrangianremap method in Figure 3, which is a special case of the threedimensional Lagrangianremap method depicted in Figure 1. Here, the first step is Lagrangian just in the vertical. With a freely moving grid, the grid must be periodically reinitialized or regularized onto a target grid (this is the regrid step) to ensure it remains nonsingular and/or continues to offer an accurate representation of the fluid flow (e.g., by not losing too many grid layers as can occur in isopycnal models). A regrid step necessitates a corresponding remap step whereby the fluid state is estimated on the new grid. The remap updates the discrete representation of the ocean state by transforming (via interpolation) that representation onto the target grid. In principle it does not change the ocean state, whereas in practice there are changes due to interpolation errors. In section 4.3, we discuss the remaining steps to the algorithm illustrated in Figure 3.
4.1.3 Example Ocean and Sea Ice Realizations
Dukowicz and Baumgardner (2000) applied the notions of Lagrangian remapping to horizontal transport in sea ice models. They emphasized that remapping is operationally identical to advective transport. Furthermore, the method does not require regridding/remapping every time step. Instead, the regrid/remap step can occur after a few Lagrangian steps, particularly if the flow is laminar and/or the time steps are small.
Vertical Lagrangian remapping is a natural extension of methods used for isopycnal layered ocean models (see Bleck, 1998, for a review). Correspondingly, the numerical realization of the vertical Lagrangianremap method by Bleck (2002) facilitates a remapping of the ocean state over an arbitrary number of vertically displaced grid cells. This capability makes use of numerical technology developed to allow coordinate layers to vanish or inflate, which is an essential feature of isopycnal coordinate models used for realistic simulations. In turn, these algorithms lead to a natural realization of conservative wetting and drying with corresponding applications in studies of estuaries and moving iceshelf grounding lines (e.g., Goldberg et al., 2012a, 2012b).
Finally, we note that Leclair and Madec (2012b) and Petersen et al. (2015) present an implementation of vertical ALE within two ocean models. Their method is described in section 5.
4.2 The Hydrostatic Ocean Equations
In presenting the vertical Lagrangianremap method, we work with the finite volume grid cell budgets for a hydrostatic fluid using GVCs. We derived these budget equations in section 3, where the equations integrate the cell averaged fields and we summarize the equations in the form
The diasurface volume flux, (red term in the prognostic Equations 32a–32d), measures the flux of seawater that crosses the moving GVC surface (see section D0.4 for details). Hence, if the normal component to the ssurface velocity matches that of seawater fluid elements, then there is zero seawater that flows across the scoordinate surface. Time stepping the ocean equations with thus constitutes a vertical Lagrangian update to the ocean state since the scoordinate surfaces follow the flow when . In this manner we see how the GVC equations are ideally suited for a vertical Lagrangian time stepping of the ocean equations. We expand on this point next.
4.3 The Vertical Lagrangian Step
The first step of the vertical Lagrangianremap method (second panel in Figure 3) results from having the generalized vertical coordinate, s, follow the vertical motion of fluid elements so that . A surface maintaining means that no net seawater mass crosses the surface. If the fluid is homogeneous, then also means the surface is material; that is, no matter crosses it. However, we are concerned with real fluids containing multiple constituents and irreversible mixing. Mixing of fluid constituents acts to exchange matter between fluid elements while fixing the mass (or volume for an incompressible Boussinesq fluid) of each fluid element. Hence, the vertical Lagrangian step allows the ssurface to follow the motion of a seawater fluid element (the barycentric velocity of section A0.4) even as matter and momentum are irreversibly exchanged across the surface. Operationally, the Lagrangian step is realized by time stepping the fluid state with in the ocean Equations 32a–32f. To allow matter (tracers) and momentum to irreversibly cross the ssurfaces, we retain diffusion and friction in the tracer and momentum equations. This is also the step where boundary fluxes are incorporated to the fluid.
4.4 The Vertical Regrid Step
As emphasized above, the Lagrangian step allows for both reversible and irreversible changes to the ocean state, all while having the scoordinate surfaces vertically follow fluid elements. If continued indefinitely, the ssurfaces will generally drift far from their initial positions. If the grid were moving in three dimensions, it would become corrugated and filamentary, especially in the presence of turbulent motion. However, so long as the horizontal time stepping of transport is stable to thus ensure layer thicknesses remain positive, then the vertical Lagrangian method is more forgiving because all that can happen is the vertical resolution becomes heterogeneous. To ensure an accurate representation of the discrete model equations, it is necessary to include a second step that we call the vertical regrid step (also referred to as coordinate reinitialization or regularization), with this step depicted in the third panel of Figure 3.
The vertical regrid step occurs by laying down a new sgrid according to a prescribed regridding method associated with a target coordinate layout. There is great freedom in choosing this target grid. For example, we may choose a target grid according to surfaces of constant geopotential placed at prescribed vertical positions, as illustrated in the third panel of Figure 3. Any sensible alternative is available as well, such as prescribed potential density classes or a mixture of geopotentials, isopycnals, and terrainfollowing surfaces as in Bleck (2002). Indeed, as noted by Bleck (2002), there is no need to provide an analytical expression for the target grid; one only needs an algorithmic prescription. Furthermore, the vertical resolution of the target grid can differ between time steps. In this manner the vertical Lagrangianremap method can be considered “coordinate free.”
4.5 The Vertical Remap Step
The ocean state is known on the grid that moves with the flow under the Lagrangian step. Upon regridding, the ocean state must be estimated on the target grid via interpolation (or extrapolation if needed near boundaries). This estimation is known as the vertical remap step whereby we map the ocean state from its Lagrangiandisplaced grid onto the target grid. Every regrid step requires a remap step. The remap step is depicted in the fourth panel of Figure 3. Under a perfect realization of remapping, the ocean state does not evolve since no fluid elements move nor does any trace matter. The position of the sgrid changes during regridding, thus necessitating a new representation of the ocean state on the new grid. Furthermore, as seen in section 5.4, vertical remapping corresponds operationally to diasurface advection.
In practice, there is spurious evolution during the remapping step since a discrete remapping incurs discretization error. We see such errors in the final panel of Figure 3 whereby the fluid state has been fully remapped onto the new grid in preparation for the next time step. As they degrade the integrity of the simulated ocean state, such numerical errors also affect the mechanical energy by modifying the background potential energy, much like errors in traditional advection schemes (Ilicak et al., 2012). Reducing these numerical errors in vertical remapping, while maintaining scalar conservation (i.e., integrated scalar properties remain unchanged by the remapping algorithm), is a primary concern for vertical remapping methods such as those developed by White and Adcroft (2008). We provide more comments on this point in section 6.
4.6 Some Examples
We illustrate the vertical Lagrangianremap method by considering some examples for a Boussinesq fluid with a linear equation of state where buoyancy is proportional to Conservative Temperature. Although trivial oceanographically, these examples help to conceptually establish how the method works in practice.
4.6.1 Regridding/Remapping to Geopotentials
As already noted, Figure 3 illustrates the vertical Lagrangianremap method for target geopotential vertical coordinates where the scoordinate surfaces are initially horizontal. Furthermore, we assume that they are initially aligned with horizontal isopycnals. The Lagrangian step allows for the coordinate surfaces to evolve while maintaining fixed volume within the coordinate layers. It also allows for mixing to act on the tracer and velocity fields. The figure illustrates the Lagrangian step induced by a gravity wave in the absence of mixing. The wave causes the scoordinate surfaces to undulate along with the isopycnals. Since the ssurfaces are displaced by the wave, the regridding step reinitializes the scoordinate surfaces to their target geopotentials. The remapping step estimates the ocean state at the target positions thus completing the time stepping of the ocean state. Note that if the target grid was isopycnal, then there would be no regrid/remap step since the scoordinate surfaces remain fixed to their original isopycnal surfaces in the absence of mixing or boundary buoyancy forcing.
4.6.2 Regridding/Remapping to Isopycnals
Now consider the case of a wave in the presence of vertical diffusion and assume an isopycnal target grid. As before, we assume that the Lagrangian step is induced by a wave that causes the scoordinate surfaces to undulate just like the second panel of Figure 3. Additionally, vertical mixing causes the isopycnals to spread. Yet the coordinate surfaces do not feel diffusion since they evolve with . Hence, the isopycnals and scoordinate surfaces separate due to mixing in the fluid. The regrid step returns the scoordinate surfaces to their target isopycnals, and the remap step estimates the state within the new isopycnal positions (see section 6.1 for more on the challenges of finding robust positions for isopycnals).
4.6.3 Zero Fluid Particle Motion With Nonzero Mixing
To further emphasize the essence of the method, drop the wave motion yet retain vertical fluid mixing that acts on vertically stratified horizontal isopycnals. There is no motion of fluid elements in this case, and yet vertical mixing causes the isopycnals to move vertically. The Lagrangian step retains sisolines aligned with their initial geopotentials since there is no fluid motion. In the case of a geopotential target grid, there is no need to regrid or to remap since there is no fluid motion. All that happens is that density spreads relative to the static geopotentials. The complement holds for an isopycnal target grid whereby vertical mixing causes the scoordinates to lose their initial potential density, thus requiring a regrid step to return the sgrid to its target isopycnal grid. A regrid step in turn requires a remap step to estimate the ocean state on the target grid.
4.7 Summarizing Some Conceptual Points
When allowing the grid to move as in the vertical Lagrangianremap method, motion of fluid elements and thus the transport of fluid properties are measured relative to the moving grid. This transport is the diasurface transport detailed mathematically in section D0.4. If the grid lines are fixed, as in an Eulerian method, then advective transport arises from fluid moving across the fixed grid lines. For the vertical Lagrangianremap method the grid lines move vertically with the fluid during the Lagrangian step. Vertical advective transport arises only during the remap step as part of estimating the fluid state on the new grid. If there is no need to vertically remap, then there is no vertical advection (e.g., adiabatic flow with an isopycnal target grid). In short, an Eulerian method watches fluid move through fixed grids whereas the vertical Lagrangian method first follows the flow and then (if needed) vertically regrids relative to the fluid and remaps onto the target grid.
We emphasize that during the Lagrangian step the fluid experiences the full suite of mixing and boundary fluxes that lead to irreversible evolution of the ocean state. So although the vertical position of the grid lines moves with fluid particles during the Lagrangian step (as if there was no mixing), the fluid state evolves in the presence of mixing and boundary fluxes. Hence, the vertical Lagrangian step is not generally a reversible step.
5 Comparison of Numerical Algorithms
In this section we expose certain algorithm details required to realize the vertical Lagrangianremap method in a numerical model. In the process we compare this algorithm to a quasiEulerian algorithm and a vertical ALE algorithm. The quasiEulerian algorithm (see Chapter 12 of Griffies, 2004 for a summary) follows the general strategy of time stepping used in traditional geopotential coordinate models, with “quasi” used since the algorithm is applicable to some GVCs so that grid cells can have moving top and bottom interfaces. The vertical ALE algorithm we feature is based on that of Leclair and Madec (2011) and Petersen et al. (2015).
5.1 Focusing on the Thickness and Tracer Equations
Our aim is to sketch elements of the three distinct numerical algorithms and to compare them side by side. Our primary concern with questions of numerical stability is to note the need to resolve crossgrid vertical motions in time according to a vertical CFL condition with the quasiEulerian and vertical ALE methods, but not with the vertical Lagrangianremapping method. Additionally, we are not concerned with numerical accuracy, noting that while the chosen forward Euler scheme is only first order accurate in time overall and might be insufficient for certain applications, this limitation can be compensated within the spatial discretization.
To focus on the essential similarities and differences between the algorithms, we emphasize the distinct treatment by the algorithms of , the diasurface velocity, and the grid velocity, w^{grid}, with w^{grid} determining how the layer thicknesses evolve and hence how the coordinate interfaces move. For this purpose it is sufficient to consider just the prognostic equations for layer thickness and thicknessweighted tracer concentration in a hydrostatic Boussinesq ocean; to ignore all SGS terms and boundary fluxes; and to make use of a forward Euler time stepping scheme from time step (n) to time step (n + 1). Further terms such as those from the momentum equation add complexity important for the full ocean equations yet without providing new conceptual pieces to the algorithm discussion.
QuasiEulerian: Thickness tendency determined by an analytic vertical coordinate  
1  Layer motion via analytic coordinate  
2  Diagnose diasurface transport  
3  Horz advection thickness update  
4  Horz advection tracer update  
5  Vert advection thickness update  
6  Vert advection tracer update  
Vertical ALE: General thickness tendency with a general vertical coordinate  
1  General layer motion  
2  Diagnose diasurface transport  
3  Horz advection thickness update  
4  Horz advection tracer update  
5  Vert advection thickness update  
6  Vert advection tracer update  
Vertical Lagrangian remap: Coordinate free  
1  Layer motion via convergence of horz advection  
2  Horz advection thickness update  
3  Horz advection tracer update  
4  Regrid to the target grid  
5  Diagnose diasurface transport  
6  Remap tracer using diasurface transport 
 Note. For brevity we consider the Boussinesq equations for layer thickness and thicknessweighted tracer concentration. The quasiEulerian (QE) algorithm is a generalization of the traditional Eulerian algorithms originating with Bryan (1969), with the vertical grid defined analytically. The vertical ALE (VALE) algorithm allows for more general grid and does not require an analytic expression for the vertical grid. One example is the algorithm of Leclair and Madec (2011) and Petersen et al. (2015). The vertical Lagrangianremap (VLR) algorithm is based on the method detailed in section 4 and is generally coordinate free. Each algorithm has been written using operator splitting between vertical and horizontal to better compare.
Hence, we consider the thickness equation and thicknessweighted tracer equation in the continuous form
The thicknessweighted velocity is updated as per the momentum equation, but again, its consideration offers no new conceptual element to the following discussion of numerical algorithms. Table 1 provides a sidebyside comparison of the three numerical algorithms considered here. Details to this table are the focus for the remainder of this section.
5.2 QuasiEulerian (QE): Vertical Coordinate Determines ∂h/∂t
The quantity, w^{grid}, offers a convenient means to identify the distinct approaches used to determine motion of the grid. In the quasiEulerian algorithm detailed in this subsection, the grid motion is determined by motion of the vertical coordinate surfaces. This grid motion is more general for the vertical ALE and vertical Lagrangianremap methods. We emphasize that this grid motion is distinct from motion of seawater crossing the grid surfaces, which is measured by .
 It uses a diagnostic calculation for via the continuity Equation 44b.
 The time tendency of the layer thickness is known through the free surface Equation 40. More generally, it is known through an analytic specification of the vertical coordinate.
 All vertical motion, including gravity waves, must be resolved for advection. Hence, an explicit time realization of vertical advective transport requires the time step to respect the vertical advective CFL condition
(45)This condition concerns timeexplicit schemes and becomes rate limiting when cell thicknesses become small and/or when diasurface motion becomes large. Consequently, there have been a variety of algorithms developed to alleviate this time step constraint, such as the Courant numberdependent implicit scheme of Shchepetkin (2015); the fluxform semiLagrangian approach of Lin and Rood (1996); and the cell integrated semiLagrangian (CISL) scheme of Nair and Machenhauer (2002). As seen in sections 5.3 and 5.4, vertical ALE and vertical Lagrangian remapping address the time step constraint in a complementary manner by aiming to reduce the crosscoordinate flow.
5.3 Vertical ALE (VALE): Arbitrary ∂h/∂t
The quasiEulerian method can be generalized to allow the vertical grid to be unconstrained in its time tendency so that it is not necessarily tied to the free surface motion. In this manner, the grid cell thicknesses are no longer defined by an analytic vertical coordinate. Instead, they are prescribed by an algorithm as per a vertical ALE method.
To motivate our realization of vertical ALE, return to the z^{∗} coordinate of Equation 37. Observe that if the flow arises solely from free surface motion (i.e., barotropic flow), then there is zero diasurface transport so that, in this special case, z^{∗} is a Lagrangian coordinate. The coordinate of Leclair and Madec (2011) takes this idea one step further by absorbing relatively fast motion from internal layer undulations (e.g., isopycnal layers with gravity waves) into the layer motion. They do so by prescribing a method for evolution of the cell thicknesses that preferentially captures the faster motion within the grid and allows the slower motion to be handled much like a traditional z^{∗} algorithm.
 Any chosen fluid motion can be absorbed by the vertical grid. Correspondingly, the grid does not need to be defined analytically.
 An explicit time realization of vertical advective transport requires the time step to respect the vertical advective CFL condition . However, the grid motion is generally chosen to render smaller than with the quasiEulerian algorithm, thus allowing for a larger Δt and/or a smaller layer thicknesses. It is only that portion of the motion that is not captured by the grid that must be resolved for tracer and velocity advection.
 The vertical ALE algorithm allows for a Lagrangian grid by choosing
(51)so that , such as for an adiabatic isopycnal model.
 As shown by Leclair and Madec (2011) and Petersen et al. (2015), absorbing fast motion into the grid helps to reduce spurious crossgrid transport in the presence of linear gravity waves, in which case there is a reduced need to perform vertical advection. However, the method does not generally help reduce spurious transport associated with balanced motions such as geostrophic turbulence (e.g., Griffies, Pacanowski, & Hallberg, 2000; Ilicak et al., 2012).
5.4 Vertical Lagrangian Remapping (VLR)
Equations 52a52c constitute the Lagrangian portion of the algorithm, corresponding to the second panel (“Lagrangian”) in Figure 3. In particular, Equation 52b provides an intermediate or “predictor” value for the updated thickness, h^{†}, resulting from the vertical Lagrangian update. Similarly, Equation 52c performs a Lagrangian update of the thicknessweighted tracer to intermediate values, again operationally realized by dropping the contribution.
This step, and the remaining step (52f) for tracers, constitutes the remapping portion of the algorithm as depicted in the fourth panel of Figure 3. For example, if the prescribed coordinate surfaces are geopotentials, then and , in which case the remap step reduces to Cartesian vertical advection. Furthermore, notice the relative lateness of the diagnosis of for the vertical Lagrangianremapping algorithm as compared to the quasiEulerian and vertical ALE algorithms. The reason is that during the first half of vertical Lagrangian remapping, the equations are time stepped with . It is only for developing the remapping step that we need to diagnose according to Equation 53.
This realization of the vertical Lagrangianremap method allows for the use of numerical algorithms that remove the vertical CFL constraint. Conceptually, this freedom is afforded since vertical regrid step rearranges the grid without transmitting physical signals. If one designs the remapping algorithm to allow for remapping the ocean state over more than a single grid cell in the vertical, then there are no corresponding vertical CFL constraints for either tracer or velocity. This property is shared by other Lagrangian transport schemes (e.g., Lin, 2004). It is naturally facilitated by the vertical Lagrangianremap method, but it is not a necessary component to an algorithmic implementation, nor is it an exclusive feature of vertical Lagrangian remapping.
As for any generalized vertical coordinate layer model, it is essential to maintain positive definite layer thicknesses. Positive layer thicknesses can be ensured for timeexplicit methods if the horizontal (within layer) CFL constraint is maintained by the horizontal transport. By doing so, the layer thicknesses can properly respond to the horizontal divergence of mass within a layer. This horizontal CFL constraint is shared with the quasiEulerian and vertical ALE methods.
 If the numerical remapping algorithm allows for remapping the ocean state over more than a single grid cell in the vertical, as in MOM6 and HYCOM, then there are no corresponding vertical CFL constraints for either tracer or velocity. Positive layer thicknesses are ensured for timeexplicit methods if the horizontal (within layer) CFL constraint is maintained by the horizontal transport.

Equations 52a52f can be time stepped and subcycled in three groups, each with distinct time steps.

The Lagrangian dynamics in Equations 52a52c form the first group, and it typically requires the smallest time step sufficient to resolve gravity waves and inertial oscillations.
 The Lagrangian tracer update (52c) is the second, which is constrained by horizontal advection and diffusion.
 The regrid/remap equations 52d–52f form the third group. This update is constrained by drift of the coordinate surfaces and is thus a function of the vertical coordinates and flow regime.
The tracer updates are less constrained than the Lagrangian dynamics, thus allowing for a longer tracer time step than dynamics time step while still satisfying the CFL criteria. This feature is of particular use when incorporating many dozens of biogeochemical tracers found in comprehensive earth system and ecosystem models.

6 Challenges and Opportunities
Bleck (2002) pioneered the vertical Lagrangianremap method for ocean models and realized it algorithmically within the HYCOM code. It has taken the subsequent years for the broader modeling community to catch up to Bleck's insights and developments and to improve on certain of his original algorithmic details. Given the broad and growing community of ocean modelers making use of HYCOM and MOM6, we suspect that the vertical Lagrangianremap method will see more use and further algorithmic improvements. Furthermore, we hope that this primer will help to remove some mysteries surrounding the method among the community of model developers, practitioners, and interested spectators. We here close the main part of this paper by presenting some challenges and opportunities forming the focus of ongoing research.
6.1 Regridding, Remapping, and Spurious Mixing
The numerical integrity of a model using a vertical Lagrangianremapping dynamical core relies on two key properties of the simulation: (A) the amount of flow crossing surfaces of constant generalized vertical coordinate and (B) the accuracy of vertical remapping. When either are done poorly, the simulation degrades through spurious mixing (Griffies, Pacanowski, & Hallberg, 2000), with the degradation especially pertinent for climate simulations where errors can accumulate over time. As suggested by Griffies, Böning, et al. (2000), the use of GVCs has emerged as a promising method to reduce such errors. The vertical Lagrangianremap method is a suitable framework to build hybrid coordinate models to address the spurious mixing problem while maintaining a dynamical core flexible enough to simulate the wide variety of flow regimes and water mass structures found in the World Ocean. We do note that a practical realization remains a work in progress, and we here touch upon some of the issues.
6.1.1 Vertical Coordinate Choices and Regridding
We generally aim to minimize the amount of flow crossing vertical coordinate surfaces for the purpose of reducing the amount of spurious crosssurface transport. Doing so requires a “smart” choice to the vertical coordinate that reduces the need for regridding and the associated remapping. For example, if the physical flow has relatively small levels of diapcynal mixing, such as for stably stratified flows in the ocean interior, then an isopycnal coordinate is preferred over a geopotential coordinate, thus allowing for the regrid/remap step to be largely absent. The real ocean, however, is not so simple. Rather, a simulation of the global ocean encounters a complex suite of stratification regimes that span surface and bottom boundary layers, interior ocean regimes of stable stratification, and lowlatitude and highlatitude processes. Simulating these multiple regimes is made even more complicated by the nonlinear equation of state for seawater (Chassignet et al., 2003; Hallberg, 2005; IOC et al., 2010; Sun et al., 1999).
Acknowledging the complex nature of the ocean's vertical density stratification, Bleck (2002) designed his “HYCOM” vertical coordinate as a regional hybrid of pressure in the upper ocean, where strong mixing leads to weak if not negative vertical density stratification; potential density referenced to 2,000 dbar (ρ_{2,000}) in the ocean interior, where density stratification is stable; and terrain following near the bottom to represent the bottom boundary layer. Another approach was proposed by Hofmeister et al. (2010), who introduced a prognostic equation for the vertical coordinate that resulted in a vertical coordinate with nontrivial vertical stratification while ensuring lateral coordinate surface smoothness. Other algorithms have been proposed for atmospheric models by Kavćić and Thuburn (2018a, 2018b). At this stage of research, the choice of a hybrid vertical coordinate for realistic ocean climate simulations remains largely an art rather than a science.
6.1.2 Vertical Remapping and Spurious Mixing
As noted in section 4.5, in principle the vertical remap step does not alter the ocean state. However, in practice numerical errors create spurious evolution. Reducing numerical errors requires both a suitable vertical gridding and an accurate remapping algorithm. The remapping scheme from Bleck (2018a) was rather low order and thus led to a nontrivial amount of spurious mixing (Megann et al., 2010). However, White and Adcroft (2008) and White et al. (2009) developed a higherorder conservative scheme that greatly ameliorated the problems with Bleck (2002).
In general, remapping errors degrade buoyancy gradients that in turn affect the mechanical energy, most notably the potential energy. Ilicak et al. (2012) measured changes in potential energy and isolated that portion of the change from spurious mixing. Gibson et al. (2017) went one step further using MOM6 by partitioning the spurious mixing into that arising from lateral processes (e.g., along coordinate advection) and that portion arising from vertical remapping. This ability to separately attribute causes for spurious mixing is natural in models based on vertical Lagrangian remapping given the operator split between lateral processes and vertical remapping.
Although we are not ready to conclude that the spurious mixing problem has been solved, we suggest that the vertical Lagrangianremap method provides a robust and flexible framework for enabling “smart” vertical coordinates and highly accurate interpolation schemes, thus allowing for the reduction of spurious mixing to physically acceptable levels. We thus conjecture that with further advances in vertical coordinate construction, the vertical Lagrangianremap method will facilitate the removal of spurious mixing from the suite of numerical biases that plague realistic ocean climate models. Adcroft et al. (2019) offer evidence to support this optimistic conjecture.
6.2 Conservative WettingDrying
The key algorithmic feature required for conservative wettingdrying, as utilized in HYCOM and MOM6, is to allow the vertical remapping (or vertical advection) to span an arbitrary number of vertical coordinate layers. In so doing the vertical CFL constraint is removed, thus allowing for vanishing/inflating grid layers and the conservation of scalar properties to machine precision. HYCOM and MOM6 arrived at this technology through implementing numerical algorithms historically developed for vanishing and inflating layers in isopycnal models. Alternatively, one can make use of general vertical remapping/advection algorithms within models based on quasiEulerian or vertical ALE methods. Machine precision is required for century to millennial scale climate and earth system applications, in which conservation of mass, potential enthalpy, salt, and biogeochemical tracers is essential. Such precision is ensured by a conservative remapping/advection scheme that allows for vanishing and inflating grid layers.
One application of conservative wetting and drying algorithms concerns the dynamical modification of the landsea boundary through vanishing and reappearing vertical coordinate layers. Models with such capabilities enable mechanistic studies of interactive and coupled ocean/iceshelf processes whereby the iceshelf grounding line evolves (Goldberg et al., 2012a, 2012b). This system is critical to robustly simulate in global climate models aiming to address questions concerning sea level rise from melting land ice. It also facilitates the study of the transition between glacial and interglacial periods whereby massive changes to the landsea boundaries occurred. Finally, conservative wettingdrying algorithms support the study of tidal fluctuations in shallow seas and coastal estuaries, thus enabling a wealth of applications for coastal oceanography.
6.3 Nonhydrostatic Ocean Dynamical Cores
As noted by Adcroft and Hallberg (2006, 2012b), the vertical Lagrangianremap method has only been realized for ocean circulation models making the hydrostatic approximation. However, there is nothing fundamental that precludes using the method for nonhydrostatic modeling. Supporting evidence can be found in the nonhydrostatic applications reviewed by Donea et al. (2004); through its use for the nonhydrostatic atmospheric modeling by Chen et al. (2013) and Kavćić and Thuburn (2018a); by the nonhydrostatic isopycnal ocean model for simulating nonlinear internal solitary gravity waves by Vitousek and Fringer (2014); and by the layered nonhydrostatic approach of Popinet (2020). The key condition is that one must enable regridding at a sufficiently fine time resolution to maintain monotonic vertical grid layering and thus to prevent grid singularities. We thus conjecture that it is only a matter of time before we see a vertical Lagrangianremapbased nonhydrostatic ocean dynamical core for realistic global and regional applications.
Acknowledgments
S. M. G. and R. W. H. acknowledge support from the National Oceanic and Atmospheric Administration Geophysical Fluid Dynamics Laboratory. A. A. acknowledges support by Award NA18OAR4320123 from the National Oceanic and Atmospheric Administration. Over the years we benefitted from discussions, debates, mentoring, and friendship from numerous colleagues, far too many to individually name. However, we would be remiss not to mention our deep gratitude and admiration for Rainer Bleck. He is the pioneer in isopycnal layered modeling as well as vertical Lagrangianremapping algorithms as realized for ocean models. More generally, he is a visionary whose generous and humble nature belie his brilliance at realizing novel and cuttingedge ideas in ocean models. We thank Xi Chen, Raphael Dussein, Robert Helber, Hemant Khatri, Graeme MacGilchrist, Paul Spence, Benjamin Taylor, Marshall Ward, Elizabeth Yankovsky, and Houssam Yassin for very useful comments and suggestions provided to drafts of this paper. We thank the editor, Florian Lemarié, and reviews from Gurvan Madec and two anonymous reviewers, each of whom provided valuable comments and corrections across two rounds of reviews for this very long manuscript. This paper grew from many years of ocean model research and development within NOAA's Geophysical Fluid Dynamics Laboratory (GFDL) and at Princeton University. GFDL/Princeton supports strategic longvision (i.e., multidecadal) research and development efforts to realize stateofthescience tools for both basic and applied research. In particular, the work needed to realize the vertical Lagrangianremap ocean climate model (using the MOM6 ocean code) documented by Adcroft et al. (2019) originates from the PhD thesis work of Hallberg (1995); the quasiEulerian MITgcm code of Adcroft and Campin (2004); the quasiEulerian MOM5 code of Griffies (2012); and the isopycnal General Ocean Layer Dynamics (GOLD) code used by Dunne et al. (2012). This multidecadal effort exemplifies the commitment of GFDL leadership to scientifically sound and robust ocean model research and development. It also offers a sobering reminder that progress in realizing cuttingedge geophysical fluid model tools, suitable for community use in studying the earth climate system, generally comes only via patient and persistent efforts from coordinated teams of dedicated and talented scientists and engineers.
Appendix A: Differential Equations for Fluid Elements
In this appendix we present the fluid mechanical equations of the ocean as expressed by partial differential equations for an infinitesimal fluid element. These equations represent the strong formulation of the ocean equations, with this discussion complementing the finite volume weak formulation from section 2. Here, we conceive of a seawater fluid element as a constant mass region with length scale at or smaller than L_{micro} ≈ 10^{−3} m. This microscale is where molecular viscous forces play an important role in the momentum equation. That is, the Reynolds number based on molecular viscosity is order unity at the microscale. Hence, the equations described in this section are formally appropriate for motions extending down to the microscale. However, we expose certain features of the subgrid scale (SGS) operators required for coarsegrained equations used for most simulations. Although relying on textbooks (e.g., Olbers et al., 2012; Vallis, 2017) for derivation of these equations, we here describe their physical content with an eye toward numerical models.
A1 Partial Differential Equations for the Ocean Fluid
A seawater fluid element maintains a constant mass, yet its matter, momentum, and enthalpy are exchanged with other fluid elements through both reversible and irreversible processes. Partial differential equations describing the evolution of fluid elements are based on Newton's laws of motion, thermodynamics, and mass conservation. Applying Newton's second law to a fluid element leads to the equation for linear momentum, v ρ δV, where δV is the volume of the fluid element, ρ its mass density, and v its velocity. The linear momentum for seawater moving on a rotating and graviting earth is affected by body forces (gravity, centrifugal, and Coriolis) and contact forces (pressure and friction). Conservation of mass for a fluid element, ρ δV, leads to the continuity equation, and mass conservation for trace constituents as exemplified by the mass concentration/fraction of salt, S, leads to the salinity equation. The evolution of potential enthalpy per mass, c_{p} Θ, occurs under the assumption of local thermodynamic equilibrium and is realized by a prognostic equation for Conservative Temperature, Θ, with c_{p} the constant specific heat capacity (McDougall, 2003). Finally, we make use of an empirically determined equation of state for seawater mass density, ρ, with IOC et al. (2010) providing the stateofthescience version (see Roquet et al., 2015, for an efficient numerical implementation).
A2 Linear Momentum Equation
The linear momentum of a fluid element is its velocity times its mass, v (ρ δV), and its time change is given by , which follows since we are interested in the dynamics of constant mass fluid elements so that (which leads to the continuity Equation A2 discussed in section A3). Through Newton's law of motion, we equate time changes in the linear momentum to forces acting on the fluid element. There are two types of forces acting on a continuous media: external/body forces and internal/contact forces. Body forces act throughout the volume of a fluid element whereas contact forces arise from the mechanical interactions between fluid elements and so act on fluid element boundaries.
A2.1 Body Forces From Coriolis and Geopotential (Gravity + Centrifugal)
Observing fluid motion in the rotating planetary reference frame requires body forces from Coriolis and the geopotential. The Coriolis force per volume is written −2 Ω∧ρ v, with Ω the angular velocity of the planet directed along the rotational axis through the north pole. We assume that Ω is constant in time, which is a very accurate assumption for the time scales of order 10^{4} years or less commonly considered in ocean climate studies. For example, adding/removing ice sheets changes Ω by roughly one part in 10^{5}.
The effective gravitational force per volume is given by −ρ ∇Φ, where the geopotential Φ combines accelerations from planetary gravitational (gravitational acceleration between fluid element and the earth) and planetary centrifugal (centrifugal acceleration due to motion on the rotating planet). In many applications we set so that . In these expressions, z is the geopotential vertical position ( defines the geoid as discussed in Gregory et al., 2019), is the local vertical unit vector, and g is a constant effective gravitational acceleration commonly taken to be roughly . The geopotential becomes a space and timedependent field when simulating ocean tides (e.g., Arbic et al., 2018) and when rotational and deformation properties of the planet change due to mass redistributions such as that occurring from ice sheet melt (e.g., Gregory et al., 2019).
A2.2 Contact Forces From Pressure, Viscosity, and Motion
When coarsegrained, the kinetic stress leads to SGS turbulent Reynolds stresses.
Contact forces are in local mechanical equilibrium in accord with Newton's third law (action/reaction law), for example, section 4.3 of Griffies (2004) and section 2.3 of Olbers et al. (2012). Hence, the net contact force acting on a finite fluid region is due only to the contact forces acting on the region boundary. This property of contact forces leads to the notion of a form stress, which arises from the horizontal component of the pressure force acting on a sloped surface such as an interior fluid interface (interfacial form stress), the ocean surface (atmosphere/ocean form stress), or the ocean bottom (topographic form stress). We have more to say about form stress in section B0.1.
A2.3 Eulerian FluxForm Momentum Equation
A3 Mass Conservation for Seawater and Trace Constituents
A4 Barycentric Velocity
This kinematic property is trivially realized for molecular diffusion as given by Equation A17. Correspondingly, the continuity Equation A15 has no SGS operator even after coarse graining; that is, there is no diffusion of seawater mass, and hence, there are no mass sources/sinks in the ocean interior. This property means that a diffusive flux of salt crossing the boundary of a fluid element balances an oppositely directed diffusive flux of freshwater, thus leaving the fluid element with a constant mass but with a nonconstant salt and freshwater content. Nurser and Griffies (2019) explore the associated consequences for surface boundary fluxes of water and salt.
We maintain the kinematic property (A18) for the cell averaged finite volume (weak formulation) of the ocean equations as described in section 3.3. For the continuum equations one can make use of densityweighted averages as proposed by Hesselberg (1926) and McDougall et al. (2002) (see also section 2.8.2 of Olbers et al., 2012).
A5 Conservative Temperature Equation
A6 Evolving the Ocean State
The ocean state is determined by the three components to the velocity, ; the in situ density, ρ; the salinity, S; the Conservative Temperature, Θ; and the pressure, p. Evolution of these seven fields is governed by the six prognostic equations A1–A4 along with the equation of state (A5). The subgrid fluxes and stress tensor are determined by the resolved model fields according to constitutive relations (e.g., the Newtonian relation between stress and strain rate) or parameterizations/closures. Furthermore, boundary fluxes of momentum, matter, and enthalpy are operationally incorporated through Neumann boundary conditions placed on the subgrid fluxes and stresses. Additionally, we incorporate penetrative solar radiation into the Conservative Temperature equation via contributions to its SGS flux vector.
There is no prognostic equation for pressure, with its evolution determined diagnostically. Specifics for its diagnosis depend on the dynamical and/or kinematical approximations. For the unapproximated equation set (A1)–(A5), pressure can be determined by inverting the equation of state after updating ρ, S, and Θ, following the approach used for compressible nonhydrostatic atmospheric models (e.g., Chen et al., 2013). Auclair et al. (2018) also propose a distinct method for ocean modeling. Assuming incompressible Boussinesq kinematics filters acoustic modes to render a threedimensional Poisson equation whose inversion determines the updated pressure (Marshall et al., 1997). Finally, for hydrostatic flow, pressure is diagnosed as a function of density and depth, with this approach familiar from hydrostatic primitive equation ocean models such as the one pioneered by Bryan (1969). Likewise, for hydrostatic isopycnal layer models, the Montgomery potential takes the role of pressure (e.g., Bleck, 1998).
A7 General Comments on SGSs
Numerical simulations are generally made with discrete grid cells of size L_{grid} ≫ L_{micro} ≈ 10^{−3} m. Correspondingly, the equations describing coarse grid simulations involve correlations of nonlinear fluctuations occurring at the finer SGSs. The correlations play a role, sometimes a dominant role, in evolution of the simulated scales of motion. Parameterization of these correlations in terms of simulated motions constitutes an active research problem in theoretical physical oceanography with very important impacts on the integrity of a simulation.
That is, we need no less than 2 π grid points to accurately represent a simulated fluctuation of length scale L_{gradient}. This is a rather general statement that ignores details of numerical algorithms. However, it accords with the condition proposed by Hallberg (2013) for horizontally resolving baroclinic eddies; by Stewart et al. (2017) for vertically resolving baroclinic modes; and by Chen et al. (2018) for the minimum grid points to accurately capture a wave phase speed at different orders of accuracy and grid staggering. See also Soufflet et al. (2016) for an analysis into the effective resolution associated with specific numerical algorithms.
A8 Comments on Mesoscale Eddy Parameterizations
The bolus velocity is horizontally divergent, and it contributes to transport within an isopycnal layer. Importantly, it contributes zero diapycnal transport (McDougall & McIntosh, 2001). Hence, it acts to rearrange matter and tracer within isopycnal layers. When following the notions of Gent and McWilliams (1990) and Gent et al. (1995), the bolus velocity is parameterized according to a downgradient diffusion or smoothing of isopycnal heights, which is equivalent to thickness diffusion away from bathymetry for a vertically uniform diffusivity. Vertical Lagrangianremap models follow the same approach as used by isopycnal models since horizontal advection remains part of the Lagrangian portion of the method. Hence, there is zero diasurface transport arising from the bolus velocity in vertical Lagrangianremap models. In Figure A1 we illustrate the terms appearing in the tracer equation as implemented in vertical Lagrangianremap models, including the bolus velocity.
Appendix B: Forces From Pressure and Subgrid Scales
We here summarize the rudiments of how forces from pressure and subgrid scale stresses are treated in ocean models.
B1 Pressure
The first expression is the volume integral of the “pressure gradient body force” as numerically realized in Bryan (1969). The second expression is the area integral of the contact pressure force acting in a compressive manner on the grid cell boundary. A contact force formulation leads to a finite volume representation of the pressure force as in Lin (1997) for the atmosphere and Adcroft et al. (2008) for the ocean.
B2 Subgrid Scale Stresses
Note that the frictional tensor generally has a zero trace when the trace of the full stress tensor is assumed to be captured by pressure. However, it is notable that the mesoscale eddy parameterization from Anstey and Zanna (2017) and Bachman et al. (2018) introduces a subgrid stress tensor that has a nonzero trace, with the trace interpreted as a dynamical modification to pressure. Holloway (1992), Smagorinsky (1993), Griffies and Hallberg (2000), Large et al. (2001), Smith and McWilliams (2003), Griffies (2004), FoxKemper and Menemenlis (2008), Jochum et al. (2008), Jansen and Held (2014), and Bachman (2017) offer a variety of perspectives on the use of lateral friction operators in numerical ocean models.
Appendix C: Connecting the Integral and Differential Formulations
In this appendix we derive the LeibnizReynolds transport theorem and then make use of the theorem to develop the connection between the partial differential equations for a fluid element (strong formulation from Appendix A) and the integral equations for a finite fluid region (weak formulation from section 2).
C1 LeibnizReynolds Transport Theorem
Leibniz's rule provides the means to differentiate this integral
We emphasize that the boundary term in Equation C3 projects out just the normal component to the boundary velocity; we never make use of the tangential component.
The LeibnizReynolds transport theorem is of great practical utility for fluid mechanics since we are often interested in the evolution of fluid properties within an arbitrary moving domain. As shown in the next subsections, it provides the means to connect the weak and strong formulations of the ocean equations as discussed in section 2. Correspondingly, it forms the starting point in our formulation of ocean model dynamical cores making use of finite volume methods as well as arbitrary EulerianLagrangian and Lagrangian methods.
C2 Mass Conservation
C3 Tracer Conservation
C4 Linear Momentum
Application to the linear momentum follows as for tracers, with the divergence of contact forces directly analogous to the divergence of the flux J. Additionally, linear momentum is affected by sources arising from body forces.
Appendix D: A Tutorial on Generalized Vertical Coordinates
Starr (1945) pioneered the use of generalized vertical coordinates (GVCs) for geophysical fluid theory and models. As reviewed by Griffies, Böning, et al. (2000), there are many reasons to describe lateral motion relative to moving GVC surfaces rather than fixed geopotential surfaces. For example, we may choose isopycnal coordinates in the ocean interior to accurately describe the associated adiabatic “stacked shallow water” dynamics. Coastal applications may motivate terrainfollowing vertical coordinates when focused on boundary constrained flows. Today, there are a number of numerical ocean models that make use of GVCs as the foundation for their dynamical cores (e.g., Adcroft & Campin, 2004; Adcroft et al., 2019; Bleck, 2002; Griffies, 2012; Klingbeil et al., 2018; Madec, 2008; Petersen et al., 2015; Shchepetkin & McWilliams, 2005). Since GVC use is fundamental to these models as well as the vertical Lagrangianremap method, we summarize features most pertinent to the considerations of this paper. A more thorough discussion can be found in Chapter 6 of Griffies (2004).
D1 Geometric Properties
This constraint in turn means that the specific thickness, is singlesigned and bounded away from infinity: z_{s}<∞. The constraint of a nonvanishing sstratification ensures that the GVC surfaces are monotonically stacked in the vertical. This property is fundamental to all GVC models. Regions where physical processes can lead to ssurface overturns require either a regridding procedure to ensure nonsingularity or the appending of another submodel to simulate/parameterize processes in the overturning region (e.g., bulk boundary layers appended to isopycnal models as in Hallberg, 2003).
This orientation contrasts to that arising in a locally orthogonal coordinate representation, whereby gravity also projects into the lateral directions and we locally rotate velocity components to align with sloped GVC surfaces. As noted by Starr (1945), such a locally orthogonal approach is problematic for geophysical fluid modeling. The reason is that the gravitational acceleration is so dominant in geophysical flows so that it is critical to restrict its appearance to a single component of the momentum equation, with that component reducing to the hydrostatic balance for largescale flows.
D2 Layer Thickness
Hence, the specific thickness expands or contracts the layer thickness according to stratification of the vertical coordinate surfaces. At the ocean surface, the top interface of the surface grid cell is defined by the free surface, , whereas the bottom interface of the bottommost grid cell is defined by the bottom topography, (see Figure 2). When extending our considerations to a finite volume grid cell as in section 3, we consider the cell thickness as defined by Equation 19, which is the average of h over the horizontal extent of the grid cell.
D3 Partial Derivatives
We emphasize that the notation, ∇_{s}, is a mere shorthand for the horizontal derivative operator given by Equation D5. This notation is ubiquitous in the literature on GVCs, including isopycnal and terrainfollowing models. Even so, as emphasized by Young (2012), there are certain tensor properties that are not carried by ∇_{s} that have implications for how one treats integrals involving ∇_{s} acting on a horizontal vector.
D4 Diasurface Transport
The diasurface volume flux is positive if there is motion across the surface in the direction defined by . There is zero volume flux when the normal component of the surface velocity matches that of fluid elements. We are only concerned with the flux of fluid normal to the surface, with the flux components in the tangent plane requiring dynamical information that is generally unavailable. In the following, we derive a fundamental kinematic relation that connects the flux, , to the material time derivative of the ssurface.
D4.1 Surface Normal
D4.2 The sSurface Velocity
D4.3 Connecting to Material Time Changes in s
The material time derivative of the GVC surface thus vanishes if no fluid elements cross the surface. That is, if the surface normal velocity exactly matches the fluid element normal velocity, then there is a zero material evolution of s.
As reviewed by Groeskamp et al. (2019), the kinematic identity (D12) is fundamental to water mass analysis. Water mass analysis methods do not assume that slayers are monotonically stacked since the analysis does not require a onetoone invertible mapping between geographical space and water mass space. However, GVC models do require invertibility, as embodied by assuming that the vertical stratification of ssurfaces remains nonzero. Assuming nonzero stratification allows us to go one more step to derive the form of diasurface velocity used in GVC models.
D4.4 Defining the Diasurface Velocity for GVC Models
The expression (D16) is the form for the diasurface velocity commonly found in GVC models. Note that depending on sign of ∂z/∂s, one may choose as a matter of convention to define .
D4.5 Expressions for the Material Time Derivative
D5 Diasurface Subgrid Scale Transport
The SGS flux, , is the “vertical” flux which, when multiplied by the horizontal area element , measures the SGS tracer transport crossing the ssurface in the normal direction.
D6 VectorInvariant Velocity Equation
Rather than the fluxform horizontal momentum Equation 31, many ocean models implement a discrete version of the vectorinvariant velocity equation. They do so for a variety of reasons, one being that the metric terms appearing from expanding the kinetic stress term ∇_{s} · (h u⊗u) with generalized horizontal coordinates (e.g., spherical coordinates) can be awkward (see section 4.4.1 of Griffies, 2004 for details). Another reason is that the appearance of the vorticity and kinetic energy (see below) facilitate development of energyenstrophy conserving finite difference methods such as those pioneered by Sadourny (1975) and Arakawa and Lamb (1981). In contrast to the fluxform momentum equation, the vectorinvariant velocity equation does not admit a finite volume formulation. Instead, it arises from the continuous velocity equation as derived here.
Appendix E: Table of Symbols
Table E1 collects the mathematical symbols used in this paper, and Table E2 lists the physical symbols. These tables assist in crossreferencing symbols used in more than one section of this paper.
Coordinates, operators, and geometry  

Symbol  Meaning  Location defined  Note 
(x, t)  Cartesian spacetime coordinates  Section 1.3  
ϕ  Latitude in radians  Equation A20  π/2 ≤ ϕ ≤ π/2 
z  Geopotential vertical coordinate  Section D0.1  
s  Generalized vertical coordinate  Section D0.1  
D/Dt  Material time derivative  Equations A6 + D17  
∇  3D gradient operator  Section A0.1  
∇_{z}  Horizontal gradient on zsurfaces  Equation D5  
∇_{s}  Horizontal gradient on ssurfaces  Equation D5  
Δ_{x}, Δ_{y}  Horizontal finite difference operators  Equations 16a + 16b  
Δ_{s}  s finite difference operator  Equation 16c  
Outward normal to a surface  Equation A7  
Volume element for integration  Equation 6  
Volume for an element of seawater  Section A0.1  
Area element for horizontal surface  Equation D13  
Horizontal area for a grid cell  Equation 18  
Surface area for an element of seawater  Equation A7  
Surface area for integration  Equation 6  
Volume integral over domain  Equation 6  
Surface integral over closed boundary of  Equation 1  
Lagrangian region moving with fluid velocity  Equation 4 
Physical symbols  

Symbol  Meaning  Location defined  Note 
v  Seawater (barycentric) velocity  Sections A0.1 + A0.3  Velocity used in mass continuity Equation A15 
u  Seawater horizontal (barycentric) velocity  Equation 27  Same for all generalized vertical coordinates 
v^{(b)}  Velocity for a point on a boundary  Equation C4  Distinguishes ALE flavors 
v^{(s)}  Velocity of a point on a ssurface  Section D0.4  
v^{S}  Velocity for a point on a const salinity surface  Equation C11  
Diasurface transport velocity  Section D0.4  in step (A) of vert Lagrangeremap  
w^{grid}  Grid vertical velocity  Section 5.1  Specified by the solution method 
ρ  In situ seawater mass density  Section A0.1  Equation of state (A5) 
S  Salinity  Section A0.3  IOC et al. (2010) + Roquet et al. (2015) 
C  Tracer concentration  Equation 35b  Mass of tracer per mass of seawater 
Θ  Conservative Temperature  Section A0.5  McDougall (2003) + Roquet et al. (2015) 
c_{p}  Constant heat capacity at fixed pressure  Potential enthalpy  McDougall (2003) + IOC et al. (2010) 
J  Subgrid tracer flux  Sections A0.3 + D0.5  For example, for tracer C 
Diasurface subgrid tracer flux  Equation D19  
Cell face averaged flux components  Equations 21a21c  For example,  
κ_{S}  Molecular diffusivity for S  Equation A17  
κ_{Θ}  Molecular diffusivity for Θ  Equation A20  
Φ  Geopotential  Section A0.2  Gravity + planetary centrifugal 
g  Effective gravitational acceleration  Section A0.2  without astronomical tide forces 
Ω  Planetary rotation rate  Section A0.1  Points through north polar axis 
f  Coriolis parameter  Equation 28  
p  Pressure  Sections A0.2 + B0.1  Isotropic compressive force per area 
τ^{form}  Horizontal form stress  Equation B2  
Identity tensor  Equation A1  3 × 3 identity matrix  
Viscous and/or subgrid stress tensor  Sections A0.2 + B0.2  2ndorder tensor; symmetric and trace free  
Rate of strain tensor  Equation A10  2ndorder symmetric tensor  
Viscosity tensor  Equation B7  4thorder tensor  
ν  Kinematic molecular viscosity  Equation A10  ν ≈ 10^{−6} m^{2} s^{−1} for water 
F^{sgs dia}  Subgrid diasurface friction  Equation B6  
F^{sgs lateral}  Subgrid lateral friction  Equation B8  
h  Thickness of a scoordinate layer  Equation D3  
h_{cell}  Thickness of a grid cell  Equation 19 