From d91b65fd6bb61a6a4277a510cf492980368e5205 Mon Sep 17 00:00:00 2001 From: Stephen Griffies Date: Fri, 7 Oct 2022 10:16:50 -0400 Subject: [PATCH 1/4] updates to MOM6 documentation --- docs/ocean.bib | 60 +++++ src/ALE/_ALE.dox | 232 ++++++++++++------ src/ALE/_ALE_timestep.dox | 109 ++++---- src/core/_Governing.dox | 223 ++++++++++++----- src/core/_Notation.dox | 64 +++-- src/parameterizations/vertical/_CVMix_KPP.dox | 2 +- 6 files changed, 494 insertions(+), 196 deletions(-) diff --git a/docs/ocean.bib b/docs/ocean.bib index 2297f25354..a5d7483f9c 100644 --- a/docs/ocean.bib +++ b/docs/ocean.bib @@ -70,3 +70,63 @@ @article{Kasahara1974 title = {Various Vertical Coordinate Systems Used for Numerical Weather Prediction}, journal = {Monthly Weather Rev.} } + +@Article{Griffies_Adcroft_Hallberg2020, +author = "S.M. Griffies and A. Adcroft and R.W. Hallberg", +title = "A primer on the vertical Lagrangian-remap method in + ocean models based on finite volume generalized vertical coordinates", +journal = "Journal of Advances in Modeling Earth Systems", +year = "2020", +volume = "12", +doi = "10.1029/2019MS001954", +} + +@Article{Shao_etal_2020, +author = "A. Shao and A.J. Adcroft and R.W. Hallberg and S.M. Griffies", +title = "A general-coordinate, nonlocal neutral diffusion operator", +journal = "Journal of Advances in Modeling Earth Systems", +year = "2020", +volume = "12", +doi = "10.1029/2019MS001992", +} + +@Article{GM95, +author = "P. R. Gent and J. Willebrand and T. J. McDougall and J. C. McWilliams", +title = "Parameterizing eddy-induced tracer transports in ocean circulation models", +journal = "Journal of Physical Oceanography", +year = "1995", +volume = "25", +pages = "463--474", +doi = "10.1175/1520-0485(1995)025<0463:PEITTI>2.0.CO;2", +} + +@Article{foxkemper_etal2008, +author = "Baylor Fox-Kemper and Raffaele Ferrari and Robert Hallberg", +title = "Parameterization of mixed layer eddies. {I}: {T}heory and diagnosis", +journal = "Journal of Physical Oceanography", +year = "2008", +volume = "38", +pages = "1145--1165", +doi = "10.1175/2007JPO3792.1", +} + +@Article{McDougall_etal_2021, +author = "T. J. McDougall and P.M.\ Barker and R.M.\ Holmes and R.\ Pawlowicz and S.M.\ Grif\/f\/ies and P.J.\ Durack", +title = "The interpretation of temperature and salinity variables in numerical ocean model output, + and the calculation of heat fluxes and heat content", +journal = "Geoscientific Model Development", +year = "2021", +volume = "14", +pages = "6445--6466", +doi = "10.5194/gmd-14-6445-2021", +} + +@article{Young2010, +author = "W. R. Young", +year = "2010", +title = "Dynamic Enthalpy, {Conservative Temperature}, and the Seawater {Boussinesq} Approximation", +journal = "Journal of Physical Oceanography", +volume = "40", +pages = "394--400", +doi = "10.1175/2009JPO4294.1", +} diff --git a/src/ALE/_ALE.dox b/src/ALE/_ALE.dox index 9313ed2aa1..a771692717 100644 --- a/src/ALE/_ALE.dox +++ b/src/ALE/_ALE.dox @@ -1,90 +1,184 @@ -/*! \page ALE ALE +/*! \page ALE Vertical Lagrangian method: conceptual -\section section_ALE Basics of the Vertical Lagrangian-Remap Method in MOM6 +\subsection section_ALE Lagrangian and ALE -As discussed by \cite adcroft2006, there are two general classes +As discussed by Adcroft and Hallberg (2008) \cite adcroft2006 and +Griffies, Adcroft and Hallberg (2020) \cite Griffies_Adcroft_Hallberg2020, +we can conceive of two general classes of algorithms that frame how hydrostatic ocean models are formulated. The two classes differ in how they treat the vertical direction. Quasi-Eulerian methods follow the approach traditionally -used in geopotential coordinate models, whereby vertical motion -is diagnosed via the continuity equation. Quasi-Lagrangian -methods are traditionally used by layered isopycnal models, with -the Lagrangian approach specifying motion that crosses coordinate +used in geopotential coordinate models, whereby vertical motion is +diagnosed via the continuity equation. Quasi-Lagrangian methods are +traditionally used by layered isopycnal models, with the vertical +Lagrangian approach specifying motion that crosses coordinate surfaces. Indeed, such dia-surface flow can be set to zero using -Lagrangian methods for studies of adiabatic dynamics. MOM6 makes -use of the vertical Lagrangian remap method, as pioneered for -ocean modeling by \cite bleck2002, which is a limit case of the -Arbitrary-Lagrangian-Eulerian method (\cite hirt1997). Dia-surface +Lagrangian methods for studies of adiabatic dynamics. MOM6 makes use +of the vertical Lagrangian remap method, as pioneered for ocean +modeling by Bleck (2002) \cite bleck2002 and further documented by +\cite Griffies_Adcroft_Hallberg2020, with this method a limit case of +the Arbitrary-Lagrangian-Eulerian method (\cite hirt1997). Dia-surface transport is implemented via a remapping so that the method can be -summarized as the Lagrangian plus remap approach and is essentially -a one-dimensional version of the incremental remapping of +summarized as the Lagrangian plus remap approach and so it is a +one-dimensional version of the incremental remapping of Dukowicz (2000) \cite dukowicz2000. -The MOM6 implementation of the vertical Lagrangian-remap method makes use -of two general steps. The first evolves the ocean state forward in -time according to a vertical Lagrangian limit with \f$\dot{r}=0\f$. Hence, -the horizontal momentum, thickness, and tracers are time stepped -with the red terms removed in equations \eqref{eq:h-horz-momentum,h-equations,momentum}, -\eqref{eq:h-thickness-equation,h-equations,thickness}, \eqref{eq:h-temperature-equation,h-equations,potential temperature}, -and \eqref{eq:h-salinity-equation,h-equations,salinity}. All advective transport thus -occurs within a layer as defined by constant \f$r\f$-surfaces so that -the volume within each layer is fixed. All other terms are retained in -their full form, including subgrid scale terms that contribute to -the transfer of tracer and momentum into distinct \f$r\f$ layers (e.g., -dia-surface diffusion of tracer and velocity). Maintaining constant -volume within a layer yet allowing for tracers to move between layers -engenders no inconsistency between tracer and thickness evolution. The -reason is that tracer diffusion, even dia-surface diffusion, does -not transfer volume. +\image html ALE_general_schematic.png "Schematic of the 3d Lagrangian regrid/remap method" width=70% +\image latex ALE_general_schematic.png "Schematic of the 3d Lagrangian regrid/remap method" width=0.7\textwidth -The second step in the algorithm comprises the generation of a new +Refer to the above figure taken from Griffies, Adcroft, and Hallberg +(2020) \cite Griffies_Adcroft_Hallberg2020. It shows a schematic of +the Lagrangian-remap method as well as the Arbitrary +Lagrangian-Eulerian (ALE) method. The first panel shows a square fluid +region and square grid used to represent the fluid, along with +rectangular subregions partitioned by grid lines. The second panel +shows the result of evolving the fluid region and evolving the +grid. The grid can evolve according to the fluid flow, as per a +Lagrangian method, or it can evolve according to some specified grid +evolution, as per an ALE method. The right panel depicts the grid +reinitialization onto a target grid (the regrid step). A regrid step +necessitates a corresponding remap step to estimate the ocean state on +the target grid, with conservative remapping required to preserve +integrated scalar contents (e.g., potential enthalpy, salt mass, and +seawater mass). The regrid/remap steps are needed for Lagrangian +methods in order for the grid to retain an accurate representation of +the ocean state. Ideally, the remap step does not affect any changes +to the fluid state; rather, it only modifies where in space the fluid +state is represented. However, any numerical realization incurs +interpolation inaccuracies that lead to unphysical (spurious) state +changes. + +\subsection section_ALE_MOM Vertical Lagrangian regrid/remap method + +We now get a bit more specific to the vertical Lagrangian method. +For this purpose, recall recall the basic dynamical equations (those +equations with a time derivative) of MOM6 discussed in +\ref General_Coordinate +\f{align} +\rho_0 +\left[ \frac{\partial \mathbf{u}}{\partial t} + \frac{( f + \zeta )}{h} \, +\hat{\mathbf{z}} \times h \, \mathbf{u} + \underbrace{ \dot{r} \, +\frac{\partial \mathbf{u}}{\partial r} } +\right] +&= -\nabla_r \, (p + \rho_{0} \, K) - +\rho \nabla_r \, \Phi + \mathbf{\mathcal{F}} +&\mbox{horizontal momentum} +\label{eq:h-horz-momentum-vlm} +\\ +\frac{\partial h}{\partial t} + \nabla_r \cdot \left( h \, \mathbf{u} \right) + +\underbrace{ \delta_r ( z_r \dot{r} ) } + &= 0 +&\mbox{thickness} +\label{eq:h-thickness-equation-vlm} +\\ +\frac{\partial ( \theta \, h )}{\partial t} + \nabla_r \cdot \left( \theta h \, +\mathbf{u} \right) + \underbrace{ \delta_r ( \theta \, z_r \dot{r} ) } +&= +h \mathbf{\mathcal{N}}_\theta^\gamma - \delta_r J_\theta^{(z)} +&\mbox{potential/Conservative temp} +\label{eq:h-temperature-equation-vlm} + \\ +\frac{\partial ( S \, h )}{\partial t} + \nabla_r \cdot \left( S \, h \, +\mathbf{u} \right) + \underbrace{ \delta_r ( S \, z_r \dot{r} ) } + &= +h \mathbf{\mathcal{N}}_S^\gamma - \delta_r J_S^{(z)} +&\mbox{salinity} +\label{eq:h-salinity-equation-vlm} +\f} +The MOM6 implementation of the vertical Lagrangian method makes +use of two general steps. The first evolves the ocean state forward in +time according to a vertical Lagrangian approach with with +\f$\dot{r}=0\f$. Hence, the horizontal momentum, thickness, and +tracers are time stepped with the underbraced terms removed in the +above equations. All advective transport occurs within a layer as +defined by constant \f$r\f$-surfaces so that the volume within each +layer is fixed. All other terms are retained in their full form, +including subgrid scale terms that contribute to the transfer of +tracer and momentum into distinct \f$r\f$ layers (e.g., dia-surface +diffusion of tracer and velocity). Maintaining constant volume within +a layer yet allowing for tracers to move between layers engenders no +inconsistency between tracer and thickness evolution. The reason is +that tracer diffusion, even dia-surface diffusion, does not transfer +volume. + +The second step in the method comprises the generation of a new vertical grid following a prescription, such as whether the grid -should align with isopcynals or constant \f$z^{*}\f$ or a combination. The -ocean state is then vertically remapped to the newly generated vertical -grid. The remapping step incorporates dia-surface transfer of properties, -with such transfer depending on the prescription given for the vertical +should align with isopcynals or constant \f$z^{*}\f$ or a combination. +This second step is known as the regrid step. The ocean state is then +vertically remapped to the newly generated vertical grid. This +remapping step incorporates dia-surface transfer of properties, with +such transfer depending on the prescription given for the vertical grid generation. To minimize discretization errors and the associated -spurious mixing, the remapping step makes use of the high order accurate -methods developed by \cite white2008 and \cite white2009. +spurious mixing, the remapping step makes use of the high order +accurate methods developed by \cite white2008 and \cite white2009. -The underlying algorithm for treatment of the vertical can -be related to operator-splitting of the red terms in equations -\eqref{eq:h-thickness-equation,h-equations,thickness}--\eqref{eq:h-temperature-equation,h-equations,potential temperature}. If we -consider, for simplicity, an Euler-forward update for a time-step \f$\Delta -t\f$, the time-stepping for the continuity and temperature equation can -be summarized as -\f{eqnarray} -\label{html:ale-equations}\notag \\ -h^\dagger &= h^{(n)} - \Delta t \left[ \nabla_r \cdot \left( h \, \mathbf{u} \right) \right] -&\mbox{thickness} \label{eq:ale-thickness-equation} \\ -\theta^\dagger \, h^\dagger &= \theta^{(n)} \, h^{(n)} - \Delta t \left[ \nabla_r \cdot \left( \theta h \, \mathbf{u} \right) - h \boldsymbol{\mathcal{N}}_\theta^\gamma + \delta_r J_\theta^{(z)} \right] -&\;\;\;\;\mbox{potential temp} \label{eq:ale-temperature-equation} \\ -h^{(n+1)} &= h^\dagger - \Delta t \, \delta_r \left( z_r \dot{r} \right) -&\mbox{move grid} \label{eq:ale-new-grid} \\ -\theta^{(n+1)} h^{(n+1)} &= \theta^\dagger h^\dagger - \Delta t \, \delta_r \left( z_r \dot{r} \, \theta^\dagger \right) -&\mbox{remap temperature.} \label{eq:ale-remap-temperature} -\f} +\subsection section_ALE_MOM_numerics Outlining the numerical algorithm -Substituting \eqref{eq:ale-thickness-equation,ale-equations,thickness} into \eqref{eq:ale-new-grid,ale-equations,move grid} -recovers a time-discrete form of \eqref{eq:h-thickness-equation,h-equations,thickness}. The -intermediate quantities indicated by \f$^\dagger\f$-symbols are the result of -the vertical Lagrangian step of the algorithm. What were the red terms in -the continuous-in-time equations are used to evolve the the intermediate -quantities to the final updated quantities each step. In MOM6, equation -\eqref{eq:ale-new-grid,ale-equations,move grid} is essentially used to define the dia-surface -transport \f$z_r \dot{r}\f$ by prescribing \f$h^{(n+1)}\f$. For example, to -recover a z-coordinate model, \f$h^{(n+1)}=\Delta z\f$, and \f$z_r \dot{r}\f$ -becomes the Eulerian vertical velocity, \f$w\f$. +The underlying algorithm for treatment of the vertical can be related +to operator-splitting of the underbraced terms in the above equations. +If we consider, for simplicity, an Euler-forward update for a +time-step \f$\Delta t\f$, the time-stepping for the thickness and +tracer equation (\f$C\f$ is an arbitrary tracer) can be summarized as +(from Table 1 in Griffies, Adcroft and Hallberg (2020) +\cite Griffies_Adcroft_Hallberg2020) +\f{align} +\label{html:ale-equations}\notag +\\ + \delta_{r} w^{\scriptstyle{\mathrm{grid}}} + &= -\nabla_{r} \cdot [h \, \mathbf{u}]^{(n)} + &\mbox{layer motion via convergence of horz advection} +\\ + h^{\dagger} &= h^{(n)} + \Delta t \, \delta_{r} w^{\scriptstyle{\mathrm{grid}}} += h^{(n)} - \Delta t \, \nabla_{r} \cdot [h \, \mathbf{u}]^{(n)} + &\mbox{horz advection thickness update} +\\ + [h \, C]^{\dagger} &= [h \, C]^{(n)} -\Delta t \, \nabla_{r} \cdot [ h \, C \, \mathbf{u} ]^{(n)} + &\mbox{horz advection tracer update} +\\ + h^{(n+1)} &= h^{\scriptstyle{\mathrm{target}}} + &\mbox{regrid to the target grid} +\\ + \delta_{r} w^{(\dot{r})} &= -(h^{\scriptstyle{\mathrm{target}}} - h^{\dagger})/\Delta t + &\mbox{diagnose dia-surface transport} +\\ + [h \, C]^{(n+1)} &= [h \, C]^{\dagger} - \Delta t \, \delta_{r} ( w^{(\dot{r})} \, C^{\dagger}) + &\mbox{remap tracer using dia-surface transport} +\f} +The first three equations constitute the Lagrangian portion of the +algorithm. In particular, the second equation provides an +intermediate or ``predictor'' value for the updated thickness, +\f$h^{\dagger}\f$, resulting from the vertical Lagrangian update. +Similarly, the third equation performs a Lagrangian update of the +thickness-weighted tracer to intermediate values, again operationally +realized by dropping the \f$w^{(\dot{r})}\f$ contribution. +The fourth equation is the regrid step, which is the key step in the +algorithm with the new grid defined by the new thickness +\f$h^{(n+1)}\f$. The new thickness is prescribed by the target values +for the vertical grid, +\f{align} + h^{(n+1)} = h^{\scriptstyle{\mathrm{target}}}. +\f} +The prescribed target grid thicknesses are then used to diagnose the +dia-surface velocity according to +\f{align} + \delta_{r} w^{(\dot{r})} = -(h^{\scriptstyle{\mathrm{target}}} - h^{\dagger})/\Delta t. +\f} +This step, and the remaining step for tracers, constitute the +remapping portion of the algorithm. For example, if the prescribed +coordinate surfaces are geopotentials, then \f$w^{(\dot{r})} = w\f$ +and \f$h^{\scriptstyle{\mathrm{target}}} = h^{(n)}\f$, in which case the +remap step reduces to Cartesian vertical advection. Within the above framework for evolving the ocean state, we make use of a standard split-explicit time stepping method by decomposing the horizontal momentum equation into its fast (depth integrated) and slow -(deviation from depth integrated) components. Furthermore, we follow the -methods of \cite hallberg2009 to ensure that the free surface resulting -from time stepping the depth integrated thickness equation (i.e., the -free surface equation) is consistent with the sum of the thicknesses -that result from time stepping the layer thickness equations for each -of the discretized layers; i.e., \f$\sum_{k} h = H + \eta\f$. +(deviation from depth integrated) components. Furthermore, we follow +the methods of Hallberg and Adcroft (2009) \cite hallberg2009 to +ensure that the free surface resulting from time stepping the depth +integrated thickness equation (i.e., the free surface equation) is +consistent with the sum of the thicknesses that result from time +stepping the layer thickness equations for each of the discretized +layers; i.e., \f$\sum_{k} h = H + \eta\f$. */ diff --git a/src/ALE/_ALE_timestep.dox b/src/ALE/_ALE_timestep.dox index e6da55fda9..8e5af7e8d6 100644 --- a/src/ALE/_ALE_timestep.dox +++ b/src/ALE/_ALE_timestep.dox @@ -1,50 +1,63 @@ -/*! \page ALE_Timestep ALE Timestep - -\section section_ALE_remap Explanation of ALE remapping - -The Arbitrary Lagrangian-Eulerian (ALE) remapping is not a timestep in the traditional -sense, but rather an operation performed to bring the vertical coordinate back to the target -specification. This remapping can be less frequent than the momentum or -thermodynamic timesteps, but must be done before the layer interfaces become entangled -with each other. - -Assuming the target vertical grid is level \f$z\f$-surfaces, the initial state is -shown on the left in the following figure: - -\image html remapping1.png "The initial state with level surface (left) and the perturbed state after a wave has come through (right)." -\image latex remapping1.png "The initial state with level surface (left) and the perturbed state after a wave has come through (right)." - -Some time later, a wave has perturbed the surfaces which move with the -fluid and it has been determined that a remapping operation is needed. The -target vertical grid is still level \f$z\f$-surfaces, so this new target -grid is shown overlaid on the left as regrid: - -\image html remapping2.png "The regrid operation (left) and the remap operation (right)." -\image latex remapping2.png "The regrid operation (left) and the remap operation (right)." - -The complex part of the operation is remapping the wavy field onto the new grid as -shown on the right and again in the final frame after the old deformed coordinate -system has been deleted: - -\image html remapping3.png "The final state after remapping." -\image latex remapping3.png "The final state after remapping." - -Mathematically, the new layer thicknesses, \f$h_k\f$, are computed and then populated -with the new velocities and tracers: - -\f[ - h_k^{\mbox{new}} = \nabla_k z_{\mbox{coord}} -\f] -\f[ - \sum h_k^{\mbox{new}} = \sum h_k^{\mbox{old}} -\f] -\f[ - \vec{u}_k^{\mbox{new}} = \frac{1}{h_k} \int_{z_{k + \frac{1}{2}}}^{z_{k + - \frac{1}{2}} + h_k} \vec{u}^{\mbox{old}}(z')dz' -\f] -\f[ - \theta^{\mbox{new}} = \frac{1}{h_k} \int_{z_{k + \frac{1}{2}}}^{z_{k + - \frac{1}{2}} + h_k} \theta^{\mbox{old}}(z')dz' -\f] +/*! \page ALE_Timestep Vertical Lagrangian method in pictures + +\subsection section_ALE_remap Graphical explanation of vertical Lagrangian method + +Vertical Lagrangian regridding/remapping is not a timestep method in +the traditional sense. Rather, it is a sequence of operations +performed to bring the vertical grid back to a target specification +(the regrid step), and then to remap the ocean state onto this new +grid (the remap step). This regrid/remap process can be chosen to be +less frequent than the momentum or thermodynamic timesteps. We are +motivated to choose less frequent regrid/remap steps to save +computational time and to reduce spurious mixing that occurs due to +truncation errors in the remap step. However, there is a downside to +delaying the regrid/remap. Namely, if delayed too long then the layer +interfaces can become entangled (i.e., no longer monotonic in the +vertical), which is a common problem with purely Lagrangian methods. +On this page we illustrate the regrid/remap steps by making use of +Figure 3 from Griffies, Adcroft, and Hallberg (2020) +\cite Griffies_Adcroft_Hallberg2020. + +For purposes of this example, assume that the target vertical grid is +comprised of geopotential \f$z\f$-surfaces, with the initial ocean +state (e.g., the temperature field) shown on the left in the following +figure. + +\image html remapping1.png "Initial state with level surface (left) and perturbed state after a wave has come through (right)" width=60% +\image latex remapping1.png "Initial state with level surface (left) and perturbed state after a wave has come through (right)" width=0.6\textwidth + +Some time later, assume a wave has perturbed the ocean state. During +the Lagrangian portion of the algorithm, the coordinate surfaces move +vertically with the ocean fluid according to \f$\dot{r}=0\f$. Assume +now that the algorithm has determined that a regrid step is needed, +with the target vertical grid still geopotential \f$z\f$-surfaces, so +this new target grid is shown overlaid on the left as a regrid. + +\image html remapping2.png "The regrid operation (left) and the remap operation (right)" width=60% +\image latex remapping2.png "The regrid operation (left) and the remap operation (right)" width=0.6\textwidth + +The most complex part of the method involves remapping the wavy ocean +field onto the new grid. This step also incurs truncation errors that +are a function of the vertical grid spacing and the numerical method +used to perform the remapping. We illustrate this remap step in the +figure above, as well as in the frame below shown after the old +deformed coordinate grid has been deleted: + +\image html remapping3.png "The final state after regriddinig and remapping" width=30% +\image latex remapping3.png "The final state after regridding and remapping" width=0.3\textwidth + +The new layer thicknesses, \f$h_k\f$, are computed and then the layers +are populated with the new velocities and tracers +\f{align} + % h_k^{\mbox{new}} %= \nabla_k z_{\mbox{coord}} \\ + \sum h_k^{\mbox{new}} &= \sum h_k^{\mbox{old}} +\\ + \mathbf{u}_k^{\mbox{new}} + &= \frac{1}{h_k} + \int_{z_{k + \frac{1}{2}}}^{z_{k + \frac{1}{2}} + h_k} \mathbf{u}^{\mbox{old}}(z') \, \mathrm{d}z' +\\ + \theta^{\mbox{new}} &= \frac{1}{h_k} + \int_{z_{k + \frac{1}{2}}}^{z_{k + \frac{1}{2}} + h_k} \theta^{\mbox{old}}(z') \, \mathrm{d}z' +\f} */ diff --git a/src/core/_Governing.dox b/src/core/_Governing.dox index 646ba52c09..63f870396f 100644 --- a/src/core/_Governing.dox +++ b/src/core/_Governing.dox @@ -1,71 +1,176 @@ /*! \page Governing_Equations Governing Equations -The Boussinesq hydrostatic equations of motion in height coordinates are - -\f{eqnarray} D_t \boldsymbol{u} + f \widehat{\boldsymbol{k}} \times \boldsymbol{u} + \frac{\rho}{\rho_o} \boldsymbol{\nabla}_z \Phi + \frac{1}{\rho_o} \boldsymbol{\nabla}_z p &= \boldsymbol{\mathcal{F}} &\mbox{ momentum} \\ - \rho \, \frac{\partial \Phi}{\partial z} + \frac{\partial p}{\partial z} &= 0 &\mbox{ hydrostatic} \\ - \boldsymbol{\nabla}_z \cdotp \boldsymbol{u} + \frac{\partial w}{\partial z} &= 0 &\mbox{ thickness} \\ - D_t \theta &= \boldsymbol{\mathcal{N}}_\theta^\gamma - \frac{\partial J_\theta^{(z)}}{\partial z} &\mbox{ potential temp} \\ - D_t S &= \boldsymbol{\mathcal{N}}_S^\gamma - \frac{\partial J_S^{(z)}}{\partial z} &\mbox{ salinity} \\ - \rho &= \rho(S, \theta, z) &\mbox{ equation of state.} +MOM6 is a hydrostatic ocean circulation model that time steps either +the non-Boussinesq ocean equations (where the flow velocity is +divergent: \f$\nabla \cdot \mathbf{v} \ne 0\f$), or the Boussinesq +ocean equations (where velocity is non-divergent: \f$\nabla \cdot +\mathbf{v} = 0\f$). We here display the Boussinesq version since +it is most commonly used (as of 2022). We start by casting the +equations in geopotentiial coordinates prior to transforming to the +generalized vertical coordinates used by MOM6. A more thorough +discussion of these equations, and their finite volume realization +appropriate for MOM6, can be found in Griffies, Adcroft and Hallberg (2020) +\cite Griffies_Adcroft_Hallberg2020. + +The hydrostatic Boussinesq ocean equations, written using geopotential +vertical coordinates, are given by +\f{align} + \rho_o \left[ + D_t \mathbf{u} + f \hat{\mathbf{z}} \times \mathbf{u} + \right] + &= -\rho \, \nabla_z \Phi - \nabla_z p + + \rho_o \, \mathbf{\mathcal{F}} + &\mbox{horizontal momentum} +\\ + \rho \, \partial_{z} \Phi + \partial_{z} p &= 0 &\mbox{hydrostatic} +\\ + \nabla_z \cdotp \mathbf{u} + \partial_{z} w + &= 0 + &\mbox{continuity} +\\ + D_t \theta &= \mathbf{\mathcal{N}}_\theta^\gamma + - \partial_{z} J_\theta^{(z)} + &\mbox{potential or Conservative temp} + \\ + D_t S &= \mathbf{\mathcal{N}}_S^\gamma +- \partial_{z} J_S^{(z)} + &\mbox{salinity} +\\ + \rho &= \rho(S, \theta, z) &\mbox{ equation of state} +\\ + \mathbf{v} &= \mathbf{u} + \hat{\mathbf{z}} \, w &\mbox{velocity field.} \f} -where notation is described in \ref Notation, \f$\boldsymbol{\mathcal{F}}\f$ represents the accelerations due to -the divergence of stresses including those provided through boundary interactions. - -The prognostic thermodynamic variables are potential temperature, -\f$\theta\f$, and salinity \f$S\f$, which are related to in situ density -\f$\rho\f$ through the \cite wright1997 equation of state. In the potential -temperature and salinity equations, fluxes due to diabatic, vertically -oriented processes are indicated by \f$J^{(z)}\f$. The tendency due to the -convergence of fluxes oriented along neutral directions is indicated by -\f$\boldsymbol{\mathcal{N}}^\gamma\f$. Our implementation of this neutral -diffusion parameterization is detailed in Shao et al. (personal comm.) - -The total derivative is - -\f{eqnarray} D_t & \equiv \frac{\partial}{\partial t} + \boldsymbol{v} \cdotp \boldsymbol{\nabla} \\ - &= \frac{\partial}{\partial t} + \boldsymbol{u} \cdotp \boldsymbol{\nabla}_z + w \frac{\partial}{\partial z}. +The acceleration term, \f$\mathbf{\mathcal{F}}\f$, in the +horizontal momentum equation includes the acceleration due to the +divergence of internal frictional stresses as well as from bottom and +surface boundary stresses. Other notation is described in \ref +Notation. + +The prognostic temperature, \f$\theta\f$, is either potential +temperature or Conservative Temperature, depending on the chosen +equation of state, and \f$S\f$ is the salinity. We generally follow +the discussion of \cite McDougall_etal_2021 for how to interpret the +prognostic temperature and salinity in ocean models. MOM6 has +historically used the Wright (1997) \cite wright1997 equation of state +to compute the in situ density, \f$\rho\f$. However, there +are other options as documented in \ref Equation_of_State. In the +potential temperature and salinity equations, fluxes due to diabatic +processes are indicated by \f$J^{(z)}\f$. Tendencies due to the +convergence of fluxes oriented along neutral directions are indicated +by \f$\mathbf{\mathcal{N}}^\gamma\f$, with our implementation of +neutral diffusion detailed in Shao et al (2020) \cite +Shao_etal_2020. + +The total or material time derivative operator is given by +\f{align} + D_t &\equiv \partial_{t} + \mathbf{v} \cdotp \nabla + \\ + &= \partial_{t} + \mathbf{u} \cdotp \nabla_z + w \, \partial_{z}, \f} - -The non-divergence of flow allows a total derivative to be re-written in flux form: - -\f{eqnarray} D_t \theta &= \frac{\partial}{\partial t} + \boldsymbol{\nabla} \cdotp ( \boldsymbol{v} \theta ) \\ - &= \frac{\partial}{\partial t} + \boldsymbol{\nabla}_z \cdotp ( \boldsymbol{u} \theta ) + \frac{\partial ( w \theta )}{\partial z}. +where the second equality explosed the horizontal and vertical terms. Using the non-divergence condition +on the three-dimensional velocity allows us to write the material time derivative of an arbitrary scalar field, +\f$\psi\f$, into a flux-form equation +\f{align} D_t \psi &= ( \partial_{t} + \mathbf{u} \cdotp \nabla) \, \psi + \\ + &= \partial_{t} \psi + \nabla \cdotp (\mathbf{v} \, \psi) +\\ + &= \partial_{t} \psi + \nabla_z \cdotp ( \mathbf{u} \, \psi) + \partial_{z} ( w \, \psi). \f} - -The above equations of motion can thus be written as: - -\f{eqnarray} D_t \boldsymbol{u} + f \widehat{\boldsymbol{k}} \times \boldsymbol{u} + \frac{\rho}{\rho_o}\boldsymbol{\nabla}_z \Phi + \frac{1}{\rho_o} \boldsymbol{\nabla}_z p &= \boldsymbol{\mathcal{F}} &\mbox{ momentum}\\ - \rho \, \frac{\partial \Phi}{\partial z} + \frac{\partial p}{\partial z} &= 0 &\mbox{ hydrostatic} \\ - \boldsymbol{\nabla}_z \cdotp \boldsymbol{u} + \frac{\partial w}{\partial z} &= 0 &\mbox{ thickness} \\ - \frac{\partial \theta}{\partial t} + \boldsymbol{\nabla}_z \cdotp ( \boldsymbol{u} \theta ) + \frac{\partial ( w \theta )}{\partial z} &= \boldsymbol{\mathcal{N}}_\theta^\gamma - \frac{\partial J_\theta^{(z)}}{\partial z} &\mbox{ potential temp} \\ - \frac{\partial S}{\partial t} + \boldsymbol{\nabla}_z \cdotp ( \boldsymbol{u} S ) + \frac{\partial ( w S )}{\partial z} &= \boldsymbol{\mathcal{N}}_S^\gamma - \frac{\partial J_S^{(z)}}{\partial z} &\mbox{ salinity} \\ - \rho &= \rho(S, \theta, z) &\mbox{ equation of state.} +Discretizing the flux-form scalar equations means that fluxes +transferring scalars between grid cells act in a conservative manner. +Consequently, the domain integrated scalar (e.g., total seawater volume, total +salt content, total potential enthalpy) is affected only via surface and bottom +boundary transport. Such global conservation properties are +maintained by MOM6 to within computational roundoff, with this level +of precision found to be essential for using MOM6 to study +climate. Making use of the flux-form scalar conservation equations +brings the model equations to the form +\f{align} + \rho_o \left[ + D_t \mathbf{u} + f \hat{\mathbf{z}} \times \mathbf{u} + \right] + &= -\rho \, \nabla_z \Phi - \nabla_z p + + \rho_o \, \mathbf{\mathcal{F}} + &\mbox{horizontal momentum} +\\ + \rho \, \partial_{z} \Phi + \partial_{z} p &= 0 &\mbox{hydrostatic} +\\ + \nabla_z \cdotp \mathbf{u} + \partial_{z} w + &= 0 + &\mbox{continuity} +\\ +\partial_{t} \theta + \nabla_z \cdotp (\mathbf{u} \, \theta) + \partial_{z} (w \, \theta) +&= \mathbf{\mathcal{N}}_\theta^\gamma - \partial_{z} J_\theta^{(z)} +&\mbox{potential or Conservative temp} +\\ +\partial_{t} S + \nabla_z \cdotp (\mathbf{u} \, S) + \partial_{z}(w \, S) +&= \mathbf{\mathcal{N}}_S^\gamma -\partial_{z} J_S^{(z)} + &\mbox{salinity} +\\ +\rho &= \rho(S, \theta, z) &\mbox{equation of state.} \f} -\section vector_invariant_eqns Vector Invariant Equations - -MOM6 solves the momentum equations written in vector-invariant form. - -A vector identity allows the total derivative of velocity to be written in the vector-invariant form: - -\f{eqnarray} D_t \boldsymbol{u} &= \partial_t \boldsymbol{u} + \boldsymbol{v} \cdotp \boldsymbol{\nabla} \boldsymbol{u} \\ - &= \partial_t \boldsymbol{u} + \boldsymbol{u} \cdotp \boldsymbol{\nabla}_z \boldsymbol{u} + w \partial_z \boldsymbol{u} \\ - &= \partial_t \boldsymbol{u} + \left( \boldsymbol{\nabla} - \times \boldsymbol{u} \right) \times \boldsymbol{v} + \boldsymbol{\nabla} \underbrace{\frac{1}{2} \left|\boldsymbol{u}\right|^2}_{\equiv K} . +\subsection vector_invariant_eqns Vector invariant velocity equation + +MOM6 time steps the horizontal velocity equation in its +vector-invariant form. To derive this equation we make use of the +following vector identity +\f{align} + D_t \mathbf{u} + &= + \partial_t \mathbf{u} + \mathbf{v} \cdotp \nabla \mathbf{u} + \\ + &= + \partial_t \mathbf{u} + \mathbf{u} \cdotp \nabla_z \mathbf{u} + w \partial_z \mathbf{u} + \\ + &= + \partial_t \mathbf{u} + \left( \nabla \times \mathbf{u} \right) \times \mathbf{v} + + \nabla \left|\mathbf{u}\right|^2/2 + \\ + &= + \partial_t \mathbf{u} + w \, \partial_{z} \mathbf{u} + + \zeta \, \hat{\mathbf{z}} \times \mathbf{u} + \nabla_{z} K, \f} - -The flux-form equations of motion in height coordinates can thus be written succinctly as: - -\f{eqnarray} \partial_t \boldsymbol{u} + \left( f \widehat{\boldsymbol{k}} + - \boldsymbol{\nabla} \times \boldsymbol{u} \right) \times \boldsymbol{v} + \boldsymbol{\nabla} K - + \frac{\rho}{\rho_o} \boldsymbol{\nabla} \Phi + \frac{1}{\rho_o} \boldsymbol{\nabla} p &= \boldsymbol{\mathcal{F}} &\mbox{ momentum} \\ - \boldsymbol{\nabla}_z \cdotp \boldsymbol{u} + \partial_z w &= 0 &\mbox{ thickness} \\ - \partial_t \theta + \boldsymbol{\nabla}_z \cdotp ( \boldsymbol{u} \theta ) + \partial_z ( w \theta ) &= \boldsymbol{\mathcal{N}}_\theta^\gamma - \frac{\partial J_\theta^{(z)}}{\partial z} &\mbox{ potential temp} \\ - \partial_t S + \boldsymbol{\nabla}_z \cdotp ( \boldsymbol{u} S ) + \partial_z ( w S ) &= \boldsymbol{\mathcal{N}}_S^\gamma - \frac{\partial J_S^{(z)}}{\partial z} &\mbox{ salinity} \\ - \rho &= \rho(S, \theta, z) &\mbox{ equation of state} +where we introduced the vertical component to the relative vorticity +\f{align} + \zeta = \hat{\mathbf{z}} \cdot (\nabla \times \mathbf{u}) + = \partial_{x}v - \partial_{y} u, +\label{eq:relative-vorticity-z} +\f} +as well as the kinetic energy per mass contained in the horizontal flow +\f{align} + K = (u^{2} + v^{2})/2. +\label{eq:kinetic-energy-per-mass} +\f} +It is just the horizontal kinetic energy per mass that appears when +making the hydrostatic approximation, whereas a non-hydrostatic fluid +(such as the MITgcm) includes the contribution from vertical motion. With +these identities we are led to the MOM6 flux-form equations of motion in +geopotential coordinates +\f{align} + \rho_{o} \left[ + \partial_t \mathbf{u} + w \, \partial_{z} \mathbf{u} + + (f + \zeta) \hat{\mathbf{z}} \times \mathbf{u} + \right] + &= -\nabla_{z} (p + K) - \rho \, \nabla_{z} \Phi + \rho_{o} \, \mathbf{\mathcal{F}} + &\mbox{vector-invariant horz velocity} +\\ + \rho \, \partial_{z} \Phi + \partial_{z} p &= 0 &\mbox{hydrostatic} +\\ + \nabla_z \cdotp \mathbf{u} + \partial_{z} w + &= 0 + &\mbox{continuity} + \\ + \partial_t \theta + \nabla_z \cdotp ( \mathbf{u} \, \theta ) + \partial_z ( w \, \theta ) + &= \mathbf{\mathcal{N}}_\theta^\gamma - \partial_{z} J_\theta^{(z)} + &\mbox{potential/Conservative temp} + \\ + \partial_t S + \nabla_z \cdotp ( \mathbf{u} \, S ) + \partial_z (w \, S) + &= \mathbf{\mathcal{N}}_S^\gamma - \partial_{z} J_S^{(z)} + &\mbox{salinity} + \\ + \rho &= \rho(S, \theta, z) &\mbox{equation of state.} \f} -where the horizontal momentum equations and vertical hydrostatic balance equation have been written as a single three-dimensional equation. */ diff --git a/src/core/_Notation.dox b/src/core/_Notation.dox index faecb3b258..ba3ab7f6fe 100644 --- a/src/core/_Notation.dox +++ b/src/core/_Notation.dox @@ -1,35 +1,61 @@ /*! \page Notation Notation for equations -\section Symbols Symbols for variables +\subsubsection Symbols Symbols for variables -\f$z\f$ refers to elevation (or height), increasing upward so that for much of the ocean \f$z\f$ is negative. +\f$z\f$ refers to geopotential elevation (or height), increasing +upward and with \f$z=0\f$ defining the resting ocean surface. Much of +the ocean has \f$z < 0\f$. -\f$x\f$ and \f$y\f$ are the Cartesian horizontal coordinates. +\f$x\f$ and \f$y\f$ are the Cartesian horizontal coordinates. MOM6 + uses generalized orthogonal curvilinear horizontal + coordinates. However, the equations are simpler to write using + Cartesian coordinates, and it is very straightforward to generalize + the horizontal coordinates using the methods in Chapters 20 and 21 of + \cite SMGbook. -\f$\lambda\f$ and \f$\phi\f$ are the geographic coordinates on a sphere (longitude and latitude respectively). +\f$\lambda\f$ and \f$\phi\f$ are the geographic coordinates on a +sphere (longitude and latitude, respectively). -Horizontal components of velocity are indicated by \f$u\f$ and \f$v\f$ and vertical component by \f$w\f$. +Horizontal components of velocity are indicated by \f$u\f$ and \f$v\f$ +and vertical component by \f$w\f$. -\f$p\f$ is pressure and \f$\Phi\f$ is geo-potential: +\f$p\f$ is the hydrostatic pressure. - \f[ \Phi = g z .\f] +\f$\Phi\f$ is the geopotential. In the absence of tides, the +geopotential is given by \f$\Phi = g z,\f$ whereas more general +expressions hold when including astronomical tide forcing. -The thermodynamic state variables are usually salinity, \f$S\f$, and potential temperature, \f$\theta\f$ or the absolute salinity and conservative temperature, depending on the equation of state. \f$\rho\f$ is in-situ density. +The thermodynamic state variables can be salinity, \f$S\f$, and +potential temperature, \f$\theta\f$. Alternatively, one can choose +the Conservative Temperature if using the TEOS10 equation of state +from \cite TEOS2010. -\section vector_notation Vector notation +\f$\rho\f$ is the in-situ density computed as a function +\f$\rho(S,\theta,p)\f$ for non-Boussinesq ocean or +\f$\rho(S,\theta,p=-g \, \rho_o \, z)\f$ for Boussinesq ocean. See +Young (2010) \cite Young2010 or Section 2.4 of Vallis (2017) +\cite GVbook for reasoning behind the simplified pressure +used in the Boussinesq equation of state. -The three-dimensional velocity vector is denoted \f$\boldsymbol{v}\f$ - - \f[\boldsymbol{v} = \boldsymbol{u} + \widehat{\boldsymbol{k}} w ,\f] -where \f$\widehat{\boldsymbol{k}}\f$ is the unit vector pointed in the upward vertical direction and \f$\boldsymbol{u} = (u, v, 0)\f$ is the horizontal -component of velocity normal to the vertical. -The gradient operator without a suffix is three dimensional: +\subsubsection vector_notation Vector notation - \f[\boldsymbol{\nabla} = ( \boldsymbol{\nabla}_z, \partial_z ) .\f] - -but a suffix indicates a lateral gradient along a surface of constant property indicated by the suffix: +The three-dimensional velocity vector is denoted \f$\boldsymbol{v}\f$ +and it is decomposed into its horizontal and vertical components according to + \f[\boldsymbol{v} + = \boldsymbol{u} + \hat{\boldsymbol{z}} w + = \hat{\boldsymbol{x}} u + \hat{\boldsymbol{y}} v + \hat{\boldsymbol{z}} w, + \f] +where \f$\hat{\boldsymbol{z}}\f$ is the unit vector pointed in the +upward vertical direction and \f$\boldsymbol{u} = (u, v, 0)\f$ is the +horizontal component of velocity normal to the vertical. + +The three-dimensional gradient operator is denoted \f$\nabla\f$, and it is decomposed into +its horizontal and vertical components according to + \f[\nabla + = \nabla_z + \hat{\boldsymbol{z}} \partial_z + = \hat{\boldsymbol{x}} \partial_x + \hat{\boldsymbol{y}} \partial_y + \hat{\boldsymbol{z}} \partial_z. + \f] - \f[\boldsymbol{\nabla}_z = \left( \left. \partial_x \right|_z, \left. \partial_y \right|_z, 0 \right) .\f] */ diff --git a/src/parameterizations/vertical/_CVMix_KPP.dox b/src/parameterizations/vertical/_CVMix_KPP.dox index 7a65b6a6a3..72c166c284 100644 --- a/src/parameterizations/vertical/_CVMix_KPP.dox +++ b/src/parameterizations/vertical/_CVMix_KPP.dox @@ -7,7 +7,7 @@ The formulation and implementation of KPP is described in great detail in the [CVMix manual](https://github.com/CVMix/CVMix-description/raw/master/cvmix.pdf) (written by our own Steve Griffies). - \section section_KPP_nutshell KPP in a nutshell + \section section_KPP_nutshell KPP in a nutshell Large et al., \cite large1994, decompose the parameterized boundary layer turbulent flux of a scalar, \f$ s \f$, as \f[ \overline{w^\prime s^\prime} = -K \partial_z s + K \gamma_s(\sigma), \f] From e54ba374c8954dbbd255569e691fd20e50b22157 Mon Sep 17 00:00:00 2001 From: Stephen Griffies Date: Fri, 7 Oct 2022 10:32:02 -0400 Subject: [PATCH 2/4] added this figure for documentation of vertical Lag method --- docs/images/ALE_general_schematic.png | Bin 0 -> 162179 bytes 1 file changed, 0 insertions(+), 0 deletions(-) create mode 100644 docs/images/ALE_general_schematic.png diff --git a/docs/images/ALE_general_schematic.png b/docs/images/ALE_general_schematic.png new file mode 100644 index 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z9I?6i_7h9GdL?P%p^r3f@HLBf<2UX^;>>U`iXTb~)Z_fuQ*2D6@bd9_92FHMCLs|K z5D<`*o*tE)oLsc`uBwXK(9qD{S3p3(+|I6mkN@vK310+!!eHgNbBl!KpWi%#KC5Bk z?k=L7GZdMwrl!_5F=4>^p4YE%dTOey-OkamQYwA;TXjQ&LnPimUqxnIeSLikb90Ot85tApYymkrIl7C1 z|5*aq#9vuq*(hC4`}-sK{e_QEKvPpwv}8?9O~lfuJuL&nlh2<&JJfzt`1@0bH}}ln z{B}Lbf3LSjA`)0tQGN;mHzqHH{JhS zNlB@xky?3`Z!sd;%;*>xR8${MR|X`I{k?>~=KK4p?FEsd1NnTy9 J=$c93{{foHG`au) literal 0 HcmV?d00001 From 7e1117b9304223adccb7d9272c9eeb8d1ecf603d Mon Sep 17 00:00:00 2001 From: Stephen Griffies Date: Fri, 7 Oct 2022 11:00:30 -0400 Subject: [PATCH 3/4] correct some doxygen errors related to subsections, cites --- src/ALE/_ALE.dox | 6 +- src/ALE/_ALE_timestep.dox | 2 +- src/core/_General_coordinate.dox | 200 ++++++++++++++++++++++--------- src/core/_Governing.dox | 6 +- src/core/_Notation.dox | 4 +- 5 files changed, 150 insertions(+), 68 deletions(-) diff --git a/src/ALE/_ALE.dox b/src/ALE/_ALE.dox index a771692717..8239a79a0e 100644 --- a/src/ALE/_ALE.dox +++ b/src/ALE/_ALE.dox @@ -1,6 +1,6 @@ /*! \page ALE Vertical Lagrangian method: conceptual -\subsection section_ALE Lagrangian and ALE +\section section_ALE Lagrangian and ALE As discussed by Adcroft and Hallberg (2008) \cite adcroft2006 and Griffies, Adcroft and Hallberg (2020) \cite Griffies_Adcroft_Hallberg2020, @@ -48,7 +48,7 @@ state is represented. However, any numerical realization incurs interpolation inaccuracies that lead to unphysical (spurious) state changes. -\subsection section_ALE_MOM Vertical Lagrangian regrid/remap method +\section section_ALE_MOM Vertical Lagrangian regrid/remap method We now get a bit more specific to the vertical Lagrangian method. For this purpose, recall recall the basic dynamical equations (those @@ -113,7 +113,7 @@ spurious mixing, the remapping step makes use of the high order accurate methods developed by \cite white2008 and \cite white2009. -\subsection section_ALE_MOM_numerics Outlining the numerical algorithm +\section section_ALE_MOM_numerics Outlining the numerical algorithm The underlying algorithm for treatment of the vertical can be related to operator-splitting of the underbraced terms in the above equations. diff --git a/src/ALE/_ALE_timestep.dox b/src/ALE/_ALE_timestep.dox index 8e5af7e8d6..6aa1ff8f00 100644 --- a/src/ALE/_ALE_timestep.dox +++ b/src/ALE/_ALE_timestep.dox @@ -1,6 +1,6 @@ /*! \page ALE_Timestep Vertical Lagrangian method in pictures -\subsection section_ALE_remap Graphical explanation of vertical Lagrangian method +\section section_ALE_remap Graphical explanation of vertical Lagrangian method Vertical Lagrangian regridding/remapping is not a timestep method in the traditional sense. Rather, it is a sequence of operations diff --git a/src/core/_General_coordinate.dox b/src/core/_General_coordinate.dox index cdaf8a34ea..1f0a7583b9 100644 --- a/src/core/_General_coordinate.dox +++ b/src/core/_General_coordinate.dox @@ -1,76 +1,158 @@ -/*! \page General_Coordinate General coordinate equations +/*! \page General_Coordinate Generalized vertical coordinate equations -Transforming to a vertical coordinate \f$r(z,x,y,t)\f$, with \f$\dot{r} = \frac{\partial r}{\partial t}\f$ ... +The ocean equations discretized by MOM6 are formulated using +generalized vertical coordinates. Motivation for using generalized +vertical coordinates, and a full accounting of the ocean equations +written using these coordinates, can be found in Griffies, Adcroft and +Hallberg (2020) \cite Griffies_Adcroft_Hallberg2020. Here we provide +a brief summary. -The Boussinesq hydrostatic equations of motion in general-coordinate -\f$r\f$ are: +Consider a smooth function of space and time, \f$r(x,y,z,t)\f$, that +has a single-signed and non-zero vertical derivative known as the +specific thickness +\f{align} + \partial z/\partial r = (\partial r/\partial z)^{-1} = \mbox{specific thickness.} +\f} +The specific thickness measures the inverse vertical stratification of +the vertical coordinate surfaces. As so constrained, \f$r\f$ can +uniquely prescribe a positiion in the vertical. Consequently, the +ocean equations can be mapped one-to-one from geopotential vertical +coordinates to generalized vertical coordinate. Upon transforming to +\f$r\f$-coordinates, the material time derivative of \f$r\f$ appears +throughout the equations, playing the role of a pseudo-vertical +velocity, and we make use of the following shorthand for this +derivative +\f{align} +\dot{r} = D_{t} r. +\f} -\f{eqnarray} +The Boussinesq hydrostatic ocean equations take the following form using +generalized vertical coordinates (\f$r\f$-coordinates) +\f{align} \label{html:r-equations}\notag \\ -\rho_0 \left( \frac{\partial \mathbf{u}}{\partial t} + ( f + \zeta ) \, \hat{\mathbf{z}} \times \mathbf{u} + \dot{r} \, \frac{\partial \mathbf{u}}{\partial r} + \nabla_r \, K \right) &= -\nabla_r \, p - \rho \nabla_r \, \Phi + \boldsymbol{\mathcal{F}} -&\mbox{momentum} \label{eq:r-horz-momentum} \\ -\rho \, \frac{\partial \Phi}{\partial r} + \frac{\partial p}{\partial r} &= 0 -&\mbox{hydrostatic} \label{eq:r-hydrostatic-equation} \\ -\frac{\partial z_r }{\partial t} + \nabla_r \cdot \, \left( z_r \, \mathbf{u} \right) + \frac{\partial ( z_r \, \dot{r} ) }{\partial r} &= 0 -&\mbox{thickness} \label{eq:r-non-divergence} \\ -\frac{\partial ( \theta \, z_r ) }{\partial t} + \nabla_r \cdot \left( \theta z_r \, \mathbf{u} \right) + \frac{\partial ( \theta \, z_r \, \dot{r} )}{\partial r} &= z_r \boldsymbol{\mathcal{N}}_\theta^\gamma - \frac{\partial J_\theta^{(z)}}{\partial r} -&\mbox{potential temp} \label{eq:r-temperature-equation} \\ -\frac{\partial ( S \, z_r) }{\partial t} + \nabla_r \cdot \left( S \, z_r \, \mathbf{u} \right) + \frac{\partial ( S \, z_r \, \dot{r} )}{\partial r} &= z_r \boldsymbol{\mathcal{N}}_S^\gamma - \frac{\partial J_S^{(z)}}{\partial r} -&\mbox{salinity} \label{eq:r-salinity-equation} \\ -\rho &= \rho\left( S, \theta, -g \rho_0 z(r) \right) +\rho_o \left[ + \partial_{t} \mathbf{u} + (f + \zeta) \, \hat{\mathbf{z}} \times \mathbf{u} + + \dot{r} \, \partial_{r} \mathbf{u} \right] + &= -\nabla_r \, (p + \rho_{o} \, K) -\rho \nabla_r \Phi + \rho_{o} \, \mathbf{\mathcal{F}} + &\mbox{horizontal momentum} +\label{eq:r-horz-momentum} +\\ +\rho \, \partial_{r} \Phi + \partial_{r}p + &= 0 +&\mbox{hydrostatic} +\label{eq:r-hydrostatic-equation} +\\ + \partial_{t}( z_r) ++ \nabla_r \cdot ( z_r \, \mathbf{u} ) ++ \partial_{r} ( z_r \, \dot{r} ) +&= 0 +&\mbox{specific thickness} +\label{eq:r-non-divergence} +\\ + \partial_{t} ( \theta \, z_r ) ++ \nabla_r \cdot ( \theta z_r \, \mathbf{u} ) ++ \partial_{r} ( \theta \, z_r \, \dot{r} ) +&= z_r \mathbf{\mathcal{N}}_\theta^\gamma +- \partial_{r} J_\theta^{(z)} +&\mbox{potential/Conservative temp} +\label{eq:r-temperature-equation} +\\ +\partial_{t} ( S \, z_r) ++ \nabla_r \cdot ( S \, z_r \, \mathbf{u} ) ++ \partial_{r} ( S \, z_r \, \dot{r} ) +&= z_r \mathbf{\mathcal{N}}_S^\gamma +- \partial_{r} J_S^{(z)} +&\mbox{salinity} +\label{eq:r-salinity-equation} +\\ +\rho &= \rho( S, \theta, -g \rho_0 z ) &\mbox{equation of state.} \f} +The time derivatives appearing in these equations are computed with +the generalized vertical coordinate fixed rather than the +geopotential. It is a common misconception that the horizontal +velocity, \f$\mathbf{u}\f$, is rotated to align with constant \f$r\f$ +surfaces. Such is not the case. Rather, the horizontal velocity, +\f$\mathbf{u}\f$, is precisely the same horizontal velocity used with +geopotential coordinates. However, its evolution has here been +formulated using generalized vertical coordinates. -The time derivatives are now computed with the generalized vertical -coordinate fixed rather than the geopotential. We introduced the -specific thickness, \f$z_r = \partial z/\partial r\f$, which measures the -inverse vertical stratification of the vertical coordinate surfaces. - - Similar to \cite bleck2002, MOM6 is discretized in the vertical by - integrating between surfaces of \f$r\f$ to yield layer equations where the - layer thickness is \f$h = \int z_r dr\f$ and variables are treated as finite - volume averages over each layer: - -\f{eqnarray} -\label{html:h-equations}\notag \\ -\rho_0 \left( \frac{\partial \mathbf{u}}{\partial t} + \frac{( f + \zeta )}{h} \, +As a finite volume model, MOM6 is discretized in the vertical by +integrating between surfaces of constant \f$r\f$. The layer thickness +is a basic term appearing in these equations, which results from +integrating the specific thickness over a layer +\f{align} +h = \int z_r \, \mathrm{d}r. +\f} +Correspondingly, the model variables are treated as finite volume +averages over each layer, with full accounting of this finite volume +approach presented in Griffies, Adcroft and Hallberg (2020) \cite +Griffies_Adcroft_Hallberg2020, and with the semi-discrete model +ocean model equations written as follows. +\f{align} +\rho_0 +\left[ \frac{\partial \mathbf{u}}{\partial t} + \frac{( f + \zeta )}{h} \, \hat{\mathbf{z}} \times h \, \mathbf{u} + \underbrace{ \dot{r} \, -\frac{\partial \mathbf{u}}{\partial r} } + \nabla_r K \right) &= -\nabla_r \, p - -\rho \nabla_r \, \Phi + \boldsymbol{\mathcal{F}} -&\mbox{momentum} \label{eq:h-horz-momentum} \\ -\rho \, \delta_r \Phi + \delta_r p &= 0 -&\mbox{hydrostatic} \label{eq:h-hydrostatic-equation} \\ +\frac{\partial \mathbf{u}}{\partial r} } +\right] +&= -\nabla_r \, (p + \rho_{0} \, K) - +\rho \nabla_r \, \Phi + \mathbf{\mathcal{F}} +&\mbox{horizontal momentum} +\label{eq:h-horz-momentum} +\\ +\rho \, \delta_r \Phi + \delta_r p +&= 0 +&\mbox{hydrostatic} +\label{eq:h-hydrostatic-equation} +\\ \frac{\partial h}{\partial t} + \nabla_r \cdot \left( h \, \mathbf{u} \right) + -\underbrace{ \delta_r ( z_r \dot{r} ) } &= 0 -&\mbox{thickness} \label{eq:h-thickness-equation} \\ +\underbrace{ \delta_r ( z_r \dot{r} ) } + &= 0 +&\mbox{thickness} +\label{eq:h-thickness-equation} +\\ \frac{\partial ( \theta \, h )}{\partial t} + \nabla_r \cdot \left( \theta h \, -\mathbf{u} \right) + \underbrace{ \delta_r ( \theta \, z_r \dot{r} ) } &= -h \boldsymbol{\mathcal{N}}_\theta^\gamma - \delta_r J_\theta^{(z)} -&\mbox{potential temp} \label{eq:h-temperature-equation} \\ +\mathbf{u} \right) + \underbrace{ \delta_r ( \theta \, z_r \dot{r} ) } +&= +h \mathbf{\mathcal{N}}_\theta^\gamma - \delta_r J_\theta^{(z)} +&\mbox{potential/Conservative temp} +\label{eq:h-temperature-equation} +\\ \frac{\partial ( S \, h )}{\partial t} + \nabla_r \cdot \left( S \, h \, -\mathbf{u} \right) + \underbrace{ \delta_r ( S \, z_r \dot{r} ) } &= -h \boldsymbol{\mathcal{N}}_S^\gamma - \delta_r J_S^{(z)} -&\mbox{salinity} \label{eq:h-salinity-equation} \\ +\mathbf{u} \right) + \underbrace{ \delta_r ( S \, z_r \dot{r} ) } +&= +h \mathbf{\mathcal{N}}_S^\gamma - \delta_r J_S^{(z)} +&\mbox{salinity} +\label{eq:h-salinity-equation} +\\ \rho &= \rho\left( S, \theta, -g \rho_0 z(r) \right) &\mbox{equation of state,} \label{eq:h-equation-of-state} \f} - -where \f$\delta_{r} = \mathrm{d}r \, (\partial/\partial r)\f$ is the discrete -vertical difference operator. The pressure gradient accelerations -in the momentum equation \eqref{eq:h-horz-momentum,h-equations,momentum} are written in +where +\f{align} +\delta_{r} = \mathrm{d}r \, (\partial/\partial r) +\f} +is the discrete vertical difference operator. The pressure gradient +accelerations in the momentum equation are written in continuous-in-the-vertical form for brevity; the exact discretization -is detailed in \cite adcroft2008. The MOM6 time-stepping algorithm -integrates the above layer-averaged equations forward allowing the -vertical grid to follow the motion, i.e. \f$\dot{r}=0\f$, so that the underbraced -terms are dropped. This approach is generally known as the Lagrangian -method but here the Lagrangian method is used only in the vertical -direction. After each Lagrangian step, a remap step is applied that -generates a new vertical grid of the user's choosing. The ocean state is -then mapped from the old to the new grid. The physical state is not meant -to change during the remap step, yet truncation errors make remapping -imperfect. We employ high-order accurate reconstructions to minimize -errors introduced during the remap step (\cite white2008, \cite white2009). The -connection between time-stepping and remapping is described in -section \ref ALE_Timestep. +is detailed in \cite adcroft2008 and \cite +Griffies_Adcroft_Hallberg2020. The \f$1/h\f$ and \f$h\f$ appearing in +the horizontal momentum equation are carefully handled in the code to +ensure proper cancellation even when the layer thickness goes to zero; +i.e., l'Hospital's rule is respected. + +The MOM6 time-stepping algorithm integrates the above layer-averaged +equations forward in time allowing the vertical grid to follow the +motion, i.e. \f$\dot{r}=0\f$, so that the underbraced terms are +dropped. This approach is generally known as a Lagrangian method, with +the Lagrangian approach in MOM6 limited to the vertical +direction. After each Lagrangian step, a regrid step is applied that +generates a new vertical grid of the user's choosing. The ocean state +is then remapped from the old to the new grid. The physical state is +not meant to change during the remap step, yet truncation errors make +remapping imperfect. We employ high-order accurate reconstructions to +minimize errors introduced during the remap step (\cite white2008, +\cite white2009). The connection between time-stepping and remapping +is described in section \ref ALE_Timestep. */ diff --git a/src/core/_Governing.dox b/src/core/_Governing.dox index 63f870396f..d1bdeb6f0e 100644 --- a/src/core/_Governing.dox +++ b/src/core/_Governing.dox @@ -59,8 +59,8 @@ potential temperature and salinity equations, fluxes due to diabatic processes are indicated by \f$J^{(z)}\f$. Tendencies due to the convergence of fluxes oriented along neutral directions are indicated by \f$\mathbf{\mathcal{N}}^\gamma\f$, with our implementation of -neutral diffusion detailed in Shao et al (2020) \cite -Shao_etal_2020. +neutral diffusion detailed in Shao et al (2020) +\cite Shao_etal_2020. The total or material time derivative operator is given by \f{align} @@ -111,7 +111,7 @@ brings the model equations to the form \rho &= \rho(S, \theta, z) &\mbox{equation of state.} \f} -\subsection vector_invariant_eqns Vector invariant velocity equation +\section vector_invariant_eqns Vector invariant velocity equation MOM6 time steps the horizontal velocity equation in its vector-invariant form. To derive this equation we make use of the diff --git a/src/core/_Notation.dox b/src/core/_Notation.dox index ba3ab7f6fe..c192929abb 100644 --- a/src/core/_Notation.dox +++ b/src/core/_Notation.dox @@ -1,6 +1,6 @@ /*! \page Notation Notation for equations -\subsubsection Symbols Symbols for variables +\section Symbols Symbols for variables \f$z\f$ refers to geopotential elevation (or height), increasing upward and with \f$z=0\f$ defining the resting ocean surface. Much of @@ -39,7 +39,7 @@ used in the Boussinesq equation of state. -\subsubsection vector_notation Vector notation +\section vector_notation Vector notation The three-dimensional velocity vector is denoted \f$\boldsymbol{v}\f$ and it is decomposed into its horizontal and vertical components according to From a5bb32151543fbc20baecd7e06ec61317dc95bf9 Mon Sep 17 00:00:00 2001 From: Stephen Griffies Date: Fri, 7 Oct 2022 12:51:19 -0400 Subject: [PATCH 4/4] correct some \cite errors --- src/core/_General_coordinate.dox | 8 ++++---- 1 file changed, 4 insertions(+), 4 deletions(-) diff --git a/src/core/_General_coordinate.dox b/src/core/_General_coordinate.dox index 1f0a7583b9..d8d7d393cd 100644 --- a/src/core/_General_coordinate.dox +++ b/src/core/_General_coordinate.dox @@ -86,8 +86,8 @@ h = \int z_r \, \mathrm{d}r. \f} Correspondingly, the model variables are treated as finite volume averages over each layer, with full accounting of this finite volume -approach presented in Griffies, Adcroft and Hallberg (2020) \cite -Griffies_Adcroft_Hallberg2020, and with the semi-discrete model +approach presented in Griffies, Adcroft and Hallberg (2020) +\cite Griffies_Adcroft_Hallberg2020, and with the semi-discrete model ocean model equations written as follows. \f{align} \rho_0 @@ -135,8 +135,8 @@ where is the discrete vertical difference operator. The pressure gradient accelerations in the momentum equation are written in continuous-in-the-vertical form for brevity; the exact discretization -is detailed in \cite adcroft2008 and \cite -Griffies_Adcroft_Hallberg2020. The \f$1/h\f$ and \f$h\f$ appearing in +is detailed in \cite adcroft2008 and +\cite Griffies_Adcroft_Hallberg2020. The \f$1/h\f$ and \f$h\f$ appearing in the horizontal momentum equation are carefully handled in the code to ensure proper cancellation even when the layer thickness goes to zero; i.e., l'Hospital's rule is respected.