[11435] | 1 | |
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[10414] | 2 | \documentclass[../main/NEMO_manual]{subfiles} |
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| 3 | |
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[6997] | 4 | \begin{document} |
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[10501] | 5 | |
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[707] | 6 | % ================================================================ |
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[10501] | 7 | % Chapter 1 Model Basics |
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[707] | 8 | % ================================================================ |
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[9393] | 9 | \chapter{Model Basics} |
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[11543] | 10 | \label{chap:MB} |
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[707] | 11 | |
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[11435] | 12 | \chaptertoc |
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| 13 | |
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[2282] | 14 | \newpage |
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| 15 | |
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[707] | 16 | % ================================================================ |
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| 17 | % Primitive Equations |
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| 18 | % ================================================================ |
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[9393] | 19 | \section{Primitive equations} |
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[11543] | 20 | \label{sec:MB_PE} |
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[707] | 21 | |
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| 22 | % ------------------------------------------------------------------------------------------------------------- |
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[11435] | 23 | % Vector Invariant Formulation |
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[707] | 24 | % ------------------------------------------------------------------------------------------------------------- |
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| 25 | |
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[9393] | 26 | \subsection{Vector invariant formulation} |
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[11543] | 27 | \label{subsec:MB_PE_vector} |
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[707] | 28 | |
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[10354] | 29 | The ocean is a fluid that can be described to a good approximation by the primitive equations, |
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[11435] | 30 | \ie\ the Navier-Stokes equations along with a nonlinear equation of state which |
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[10354] | 31 | couples the two active tracers (temperature and salinity) to the fluid velocity, |
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| 32 | plus the following additional assumptions made from scale considerations: |
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[707] | 33 | |
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[10501] | 34 | \begin{enumerate} |
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| 35 | \item |
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[11335] | 36 | \textit{spherical Earth approximation}: the geopotential surfaces are assumed to be oblate spheriods |
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| 37 | that follow the Earth's bulge; these spheroids are approximated by spheres with |
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[11435] | 38 | gravity locally vertical (parallel to the Earth's radius) and independent of latitude |
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| 39 | \citep[][section 2]{white.hoskins.ea_QJRMS05}. |
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[10501] | 40 | \item |
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| 41 | \textit{thin-shell approximation}: the ocean depth is neglected compared to the earth's radius |
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| 42 | \item |
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| 43 | \textit{turbulent closure hypothesis}: the turbulent fluxes |
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| 44 | (which represent the effect of small scale processes on the large-scale) |
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| 45 | are expressed in terms of large-scale features |
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| 46 | \item |
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| 47 | \textit{Boussinesq hypothesis}: density variations are neglected except in their contribution to |
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| 48 | the buoyancy force |
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[10414] | 49 | \begin{equation} |
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[11543] | 50 | \label{eq:MB_PE_eos} |
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[10501] | 51 | \rho = \rho \ (T,S,p) |
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[707] | 52 | \end{equation} |
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[10501] | 53 | \item |
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| 54 | \textit{Hydrostatic hypothesis}: the vertical momentum equation is reduced to a balance between |
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| 55 | the vertical pressure gradient and the buoyancy force |
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| 56 | (this removes convective processes from the initial Navier-Stokes equations and so |
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| 57 | convective processes must be parameterized instead) |
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[10414] | 58 | \begin{equation} |
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[11543] | 59 | \label{eq:MB_PE_hydrostatic} |
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[10501] | 60 | \pd[p]{z} = - \rho \ g |
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[707] | 61 | \end{equation} |
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[10501] | 62 | \item |
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| 63 | \textit{Incompressibility hypothesis}: the three dimensional divergence of the velocity vector $\vect U$ |
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| 64 | is assumed to be zero. |
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[10414] | 65 | \begin{equation} |
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[11543] | 66 | \label{eq:MB_PE_continuity} |
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[10501] | 67 | \nabla \cdot \vect U = 0 |
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[707] | 68 | \end{equation} |
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[11435] | 69 | \item |
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| 70 | \textit{Neglect of additional Coriolis terms}: the Coriolis terms that vary with the cosine of latitude are neglected. |
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[11335] | 71 | These terms may be non-negligible where the Brunt-Vaisala frequency $N$ is small, either in the deep ocean or |
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[11435] | 72 | in the sub-mesoscale motions of the mixed layer, or near the equator \citep[][section 1]{white.hoskins.ea_QJRMS05}. |
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| 73 | They can be consistently included as part of the ocean dynamics \citep[][section 3(d)]{white.hoskins.ea_QJRMS05} and are |
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| 74 | retained in the MIT ocean model. |
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[10501] | 75 | \end{enumerate} |
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| 76 | |
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| 77 | Because the gravitational force is so dominant in the equations of large-scale motions, |
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[11335] | 78 | it is useful to choose an orthogonal set of unit vectors $(i,j,k)$ linked to the Earth such that |
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[10501] | 79 | $k$ is the local upward vector and $(i,j)$ are two vectors orthogonal to $k$, |
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[11435] | 80 | \ie\ tangent to the geopotential surfaces. |
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[10501] | 81 | Let us define the following variables: $\vect U$ the vector velocity, $\vect U = \vect U_h + w \, \vect k$ |
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[11435] | 82 | (the subscript $h$ denotes the local horizontal vector, \ie\ over the $(i,j)$ plane), |
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[10501] | 83 | $T$ the potential temperature, $S$ the salinity, $\rho$ the \textit{in situ} density. |
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| 84 | The vector invariant form of the primitive equations in the $(i,j,k)$ vector system provides |
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| 85 | the following equations: |
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| 86 | \begin{subequations} |
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[11543] | 87 | \label{eq:MB_PE} |
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[10501] | 88 | \begin{gather} |
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| 89 | \intertext{$-$ the momentum balance} |
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[11543] | 90 | \label{eq:MB_PE_dyn} |
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[10501] | 91 | \pd[\vect U_h]{t} = - \lt[ (\nabla \times \vect U) \times \vect U + \frac{1}{2} \nabla \lt( \vect U^2 \rt) \rt]_h |
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| 92 | - f \; k \times \vect U_h - \frac{1}{\rho_o} \nabla_h p |
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| 93 | + \vect D^{\vect U} + \vect F^{\vect U} \\ |
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| 94 | \intertext{$-$ the heat and salt conservation equations} |
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[11543] | 95 | \label{eq:MB_PE_tra_T} |
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[10501] | 96 | \pd[T]{t} = - \nabla \cdot (T \ \vect U) + D^T + F^T \\ |
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[11543] | 97 | \label{eq:MB_PE_tra_S} |
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[10501] | 98 | \pd[S]{t} = - \nabla \cdot (S \ \vect U) + D^S + F^S |
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| 99 | \end{gather} |
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[707] | 100 | \end{subequations} |
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[10501] | 101 | where $\nabla$ is the generalised derivative vector operator in $(i,j,k)$ directions, $t$ is the time, |
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| 102 | $z$ is the vertical coordinate, $\rho$ is the \textit{in situ} density given by the equation of state |
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[11543] | 103 | (\autoref{eq:MB_PE_eos}), $\rho_o$ is a reference density, $p$ the pressure, |
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[10501] | 104 | $f = 2 \vect \Omega \cdot k$ is the Coriolis acceleration |
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| 105 | (where $\vect \Omega$ is the Earth's angular velocity vector), and $g$ is the gravitational acceleration. |
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| 106 | $\vect D^{\vect U}$, $D^T$ and $D^S$ are the parameterisations of small-scale physics for momentum, |
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| 107 | temperature and salinity, and $\vect F^{\vect U}$, $F^T$ and $F^S$ surface forcing terms. |
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[11543] | 108 | Their nature and formulation are discussed in \autoref{sec:MB_zdf_ldf} and \autoref{subsec:MB_boundary_condition}. |
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[707] | 109 | |
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| 110 | % ------------------------------------------------------------------------------------------------------------- |
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| 111 | % Boundary condition |
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| 112 | % ------------------------------------------------------------------------------------------------------------- |
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[9393] | 113 | \subsection{Boundary conditions} |
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[11543] | 114 | \label{subsec:MB_boundary_condition} |
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[707] | 115 | |
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[10354] | 116 | An ocean is bounded by complex coastlines, bottom topography at its base and |
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| 117 | an air-sea or ice-sea interface at its top. |
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[10501] | 118 | These boundaries can be defined by two surfaces, $z = - H(i,j)$ and $z = \eta(i,j,k,t)$, |
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[11435] | 119 | where $H$ is the depth of the ocean bottom and $\eta$ is the height of the sea surface |
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| 120 | (discretisation can introduce additional artificial ``side-wall'' boundaries). |
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| 121 | Both $H$ and $\eta$ are referenced to a surface of constant geopotential (\ie\ a mean sea surface height) on which $z = 0$. |
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[11543] | 122 | (\autoref{fig:MB_ocean_bc}). |
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[10354] | 123 | Through these two boundaries, the ocean can exchange fluxes of heat, fresh water, salt, and momentum with |
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| 124 | the solid earth, the continental margins, the sea ice and the atmosphere. |
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| 125 | However, some of these fluxes are so weak that even on climatic time scales of thousands of years |
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| 126 | they can be neglected. |
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| 127 | In the following, we briefly review the fluxes exchanged at the interfaces between the ocean and |
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| 128 | the other components of the earth system. |
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[707] | 129 | |
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| 130 | %>>>>>>>>>>>>>>>>>>>>>>>>>>>> |
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[10354] | 131 | \begin{figure}[!ht] |
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[11558] | 132 | \centering |
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[11561] | 133 | \includegraphics[width=0.66\textwidth]{Fig_I_ocean_bc} |
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[11558] | 134 | \caption[Ocean boundary conditions]{ |
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| 135 | The ocean is bounded by two surfaces, $z = - H(i,j)$ and $z = \eta(i,j,t)$, |
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| 136 | where $H$ is the depth of the sea floor and $\eta$ the height of the sea surface. |
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| 137 | Both $H$ and $\eta$ are referenced to $z = 0$.} |
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| 138 | \label{fig:MB_ocean_bc} |
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[10354] | 139 | \end{figure} |
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[707] | 140 | %>>>>>>>>>>>>>>>>>>>>>>>>>>>> |
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| 141 | |
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| 142 | \begin{description} |
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[10354] | 143 | \item[Land - ocean interface:] |
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| 144 | the major flux between continental margins and the ocean is a mass exchange of fresh water through river runoff. |
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| 145 | Such an exchange modifies the sea surface salinity especially in the vicinity of major river mouths. |
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| 146 | It can be neglected for short range integrations but has to be taken into account for long term integrations as |
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| 147 | it influences the characteristics of water masses formed (especially at high latitudes). |
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| 148 | It is required in order to close the water cycle of the climate system. |
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| 149 | It is usually specified as a fresh water flux at the air-sea interface in the vicinity of river mouths. |
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| 150 | \item[Solid earth - ocean interface:] |
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| 151 | heat and salt fluxes through the sea floor are small, except in special areas of little extent. |
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[10414] | 152 | They are usually neglected in the model |
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| 153 | \footnote{ |
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[10354] | 154 | In fact, it has been shown that the heat flux associated with the solid Earth cooling |
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[11435] | 155 | (\ie\ the geothermal heating) is not negligible for the thermohaline circulation of the world ocean |
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[10354] | 156 | (see \autoref{subsec:TRA_bbc}). |
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| 157 | }. |
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| 158 | The boundary condition is thus set to no flux of heat and salt across solid boundaries. |
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| 159 | For momentum, the situation is different. There is no flow across solid boundaries, |
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[11435] | 160 | \ie\ the velocity normal to the ocean bottom and coastlines is zero (in other words, |
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[10354] | 161 | the bottom velocity is parallel to solid boundaries). This kinematic boundary condition |
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| 162 | can be expressed as: |
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[10414] | 163 | \begin{equation} |
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[11543] | 164 | \label{eq:MB_w_bbc} |
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[10501] | 165 | w = - \vect U_h \cdot \nabla_h (H) |
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[10354] | 166 | \end{equation} |
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| 167 | In addition, the ocean exchanges momentum with the earth through frictional processes. |
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| 168 | Such momentum transfer occurs at small scales in a boundary layer. |
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| 169 | It must be parameterized in terms of turbulent fluxes using bottom and/or lateral boundary conditions. |
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| 170 | Its specification depends on the nature of the physical parameterisation used for |
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[11543] | 171 | $\vect D^{\vect U}$ in \autoref{eq:MB_PE_dyn}. |
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| 172 | It is discussed in \autoref{eq:MB_zdf}.% and Chap. III.6 to 9. |
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[10354] | 173 | \item[Atmosphere - ocean interface:] |
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| 174 | the kinematic surface condition plus the mass flux of fresh water PE (the precipitation minus evaporation budget) |
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| 175 | leads to: |
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[10414] | 176 | \[ |
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[11543] | 177 | % \label{eq:MB_w_sbc} |
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[10501] | 178 | w = \pd[\eta]{t} + \lt. \vect U_h \rt|_{z = \eta} \cdot \nabla_h (\eta) + P - E |
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[10414] | 179 | \] |
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[10354] | 180 | The dynamic boundary condition, neglecting the surface tension (which removes capillary waves from the system) |
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[10501] | 181 | leads to the continuity of pressure across the interface $z = \eta$. |
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[10354] | 182 | The atmosphere and ocean also exchange horizontal momentum (wind stress), and heat. |
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| 183 | \item[Sea ice - ocean interface:] |
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| 184 | the ocean and sea ice exchange heat, salt, fresh water and momentum. |
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| 185 | The sea surface temperature is constrained to be at the freezing point at the interface. |
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[10501] | 186 | Sea ice salinity is very low ($\sim4-6 \, psu$) compared to those of the ocean ($\sim34 \, psu$). |
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[10354] | 187 | The cycle of freezing/melting is associated with fresh water and salt fluxes that cannot be neglected. |
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[707] | 188 | \end{description} |
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| 189 | |
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| 190 | % ================================================================ |
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| 191 | % The Horizontal Pressure Gradient |
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| 192 | % ================================================================ |
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[10501] | 193 | \section{Horizontal pressure gradient} |
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[11543] | 194 | \label{sec:MB_hor_pg} |
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[707] | 195 | |
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| 196 | % ------------------------------------------------------------------------------------------------------------- |
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| 197 | % Pressure Formulation |
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| 198 | % ------------------------------------------------------------------------------------------------------------- |
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[9393] | 199 | \subsection{Pressure formulation} |
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[11543] | 200 | \label{subsec:MB_p_formulation} |
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[707] | 201 | |
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[10354] | 202 | The total pressure at a given depth $z$ is composed of a surface pressure $p_s$ at |
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[10501] | 203 | a reference geopotential surface ($z = 0$) and a hydrostatic pressure $p_h$ such that: |
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| 204 | $p(i,j,k,t) = p_s(i,j,t) + p_h(i,j,k,t)$. |
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[11543] | 205 | The latter is computed by integrating (\autoref{eq:MB_PE_hydrostatic}), |
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| 206 | assuming that pressure in decibars can be approximated by depth in meters in (\autoref{eq:MB_PE_eos}). |
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[1224] | 207 | The hydrostatic pressure is then given by: |
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[10414] | 208 | \[ |
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[11543] | 209 | % \label{eq:MB_pressure} |
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[10501] | 210 | p_h (i,j,z,t) = \int_{\varsigma = z}^{\varsigma = 0} g \; \rho (T,S,\varsigma) \; d \varsigma |
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[10414] | 211 | \] |
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[10354] | 212 | Two strategies can be considered for the surface pressure term: |
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| 213 | $(a)$ introduce of a new variable $\eta$, the free-surface elevation, |
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| 214 | for which a prognostic equation can be established and solved; |
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| 215 | $(b)$ assume that the ocean surface is a rigid lid, |
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| 216 | on which the pressure (or its horizontal gradient) can be diagnosed. |
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| 217 | When the former strategy is used, one solution of the free-surface elevation consists of |
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| 218 | the excitation of external gravity waves. |
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| 219 | The flow is barotropic and the surface moves up and down with gravity as the restoring force. |
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| 220 | The phase speed of such waves is high (some hundreds of metres per second) so that |
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[11335] | 221 | the time step has to be very short when they are present in the model. |
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[10501] | 222 | The latter strategy filters out these waves since the rigid lid approximation implies $\eta = 0$, |
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[11435] | 223 | \ie\ the sea surface is the surface $z = 0$. |
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[10354] | 224 | This well known approximation increases the surface wave speed to infinity and |
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[11435] | 225 | modifies certain other longwave dynamics (\eg\ barotropic Rossby or planetary waves). |
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[10354] | 226 | The rigid-lid hypothesis is an obsolescent feature in modern OGCMs. |
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[10501] | 227 | It has been available until the release 3.1 of \NEMO, and it has been removed in release 3.2 and followings. |
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[11335] | 228 | Only the free surface formulation is now described in this document (see the next sub-section). |
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[707] | 229 | |
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| 230 | % ------------------------------------------------------------------------------------------------------------- |
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| 231 | % Free Surface Formulation |
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| 232 | % ------------------------------------------------------------------------------------------------------------- |
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[9393] | 233 | \subsection{Free surface formulation} |
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[11543] | 234 | \label{subsec:MB_free_surface} |
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[707] | 235 | |
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[10354] | 236 | In the free surface formulation, a variable $\eta$, the sea-surface height, |
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| 237 | is introduced which describes the shape of the air-sea interface. |
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| 238 | This variable is solution of a prognostic equation which is established by forming the vertical average of |
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[11543] | 239 | the kinematic surface condition (\autoref{eq:MB_w_bbc}): |
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[10414] | 240 | \begin{equation} |
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[11543] | 241 | \label{eq:MB_ssh} |
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[10501] | 242 | \pd[\eta]{t} = - D + P - E \quad \text{where} \quad D = \nabla \cdot \lt[ (H + \eta) \; \overline{U}_h \, \rt] |
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[707] | 243 | \end{equation} |
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[11543] | 244 | and using (\autoref{eq:MB_PE_hydrostatic}) the surface pressure is given by: $p_s = \rho \, g \, \eta$. |
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[707] | 245 | |
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[10354] | 246 | Allowing the air-sea interface to move introduces the external gravity waves (EGWs) as |
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| 247 | a class of solution of the primitive equations. |
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[11435] | 248 | These waves are barotropic (\ie\ nearly independent of depth) and their phase speed is quite high. |
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[10354] | 249 | Their time scale is short with respect to the other processes described by the primitive equations. |
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[707] | 250 | |
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[10354] | 251 | Two choices can be made regarding the implementation of the free surface in the model, |
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[10501] | 252 | depending on the physical processes of interest. |
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[707] | 253 | |
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[10354] | 254 | $\bullet$ If one is interested in EGWs, in particular the tides and their interaction with |
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| 255 | the baroclinic structure of the ocean (internal waves) possibly in shallow seas, |
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| 256 | then a non linear free surface is the most appropriate. |
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[11543] | 257 | This means that no approximation is made in \autoref{eq:MB_ssh} and that |
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[10354] | 258 | the variation of the ocean volume is fully taken into account. |
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| 259 | Note that in order to study the fast time scales associated with EGWs it is necessary to |
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| 260 | minimize time filtering effects |
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| 261 | (use an explicit time scheme with very small time step, or a split-explicit scheme with reasonably small time step, |
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| 262 | see \autoref{subsec:DYN_spg_exp} or \autoref{subsec:DYN_spg_ts}). |
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[707] | 263 | |
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[10354] | 264 | $\bullet$ If one is not interested in EGW but rather sees them as high frequency noise, |
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| 265 | it is possible to apply an explicit filter to slow down the fastest waves while |
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| 266 | not altering the slow barotropic Rossby waves. |
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| 267 | If further, an approximative conservation of heat and salt contents is sufficient for the problem solved, |
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[11543] | 268 | then it is sufficient to solve a linearized version of \autoref{eq:MB_ssh}, |
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[11123] | 269 | which still allows to take into account freshwater fluxes applied at the ocean surface \citep{roullet.madec_JGR00}. |
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[6140] | 270 | Nevertheless, with the linearization, an exact conservation of heat and salt contents is lost. |
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[707] | 271 | |
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[10354] | 272 | The filtering of EGWs in models with a free surface is usually a matter of discretisation of |
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| 273 | the temporal derivatives, |
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[11123] | 274 | using a split-explicit method \citep{killworth.webb.ea_JPO91, zhang.endoh_JGR92} or |
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| 275 | the implicit scheme \citep{dukowicz.smith_JGR94} or |
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| 276 | the addition of a filtering force in the momentum equation \citep{roullet.madec_JGR00}. |
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[11435] | 277 | With the present release, \NEMO\ offers the choice between |
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[10354] | 278 | an explicit free surface (see \autoref{subsec:DYN_spg_exp}) or |
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[11123] | 279 | a split-explicit scheme strongly inspired the one proposed by \citet{shchepetkin.mcwilliams_OM05} |
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[10354] | 280 | (see \autoref{subsec:DYN_spg_ts}). |
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[707] | 281 | |
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| 282 | % ================================================================ |
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| 283 | % Curvilinear z-coordinate System |
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| 284 | % ================================================================ |
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[9393] | 285 | \section{Curvilinear \textit{z-}coordinate system} |
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[11543] | 286 | \label{sec:MB_zco} |
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[707] | 287 | |
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| 288 | % ------------------------------------------------------------------------------------------------------------- |
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| 289 | % Tensorial Formalism |
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| 290 | % ------------------------------------------------------------------------------------------------------------- |
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[9393] | 291 | \subsection{Tensorial formalism} |
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[11543] | 292 | \label{subsec:MB_tensorial} |
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[707] | 293 | |
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[10354] | 294 | In many ocean circulation problems, the flow field has regions of enhanced dynamics |
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[11435] | 295 | (\ie\ surface layers, western boundary currents, equatorial currents, or ocean fronts). |
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[10354] | 296 | The representation of such dynamical processes can be improved by |
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| 297 | specifically increasing the model resolution in these regions. |
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| 298 | As well, it may be convenient to use a lateral boundary-following coordinate system to |
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| 299 | better represent coastal dynamics. |
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| 300 | Moreover, the common geographical coordinate system has a singular point at the North Pole that |
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| 301 | cannot be easily treated in a global model without filtering. |
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| 302 | A solution consists of introducing an appropriate coordinate transformation that |
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[11123] | 303 | shifts the singular point onto land \citep{madec.imbard_CD96, murray_JCP96}. |
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[10354] | 304 | As a consequence, it is important to solve the primitive equations in various curvilinear coordinate systems. |
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| 305 | An efficient way of introducing an appropriate coordinate transform can be found when using a tensorial formalism. |
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| 306 | This formalism is suited to any multidimensional curvilinear coordinate system. |
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| 307 | Ocean modellers mainly use three-dimensional orthogonal grids on the sphere (spherical earth approximation), |
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| 308 | with preservation of the local vertical. Here we give the simplified equations for this particular case. |
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[11123] | 309 | The general case is detailed by \citet{eiseman.stone_SR80} in their survey of the conservation laws of fluid dynamics. |
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[707] | 310 | |
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[10501] | 311 | Let $(i,j,k)$ be a set of orthogonal curvilinear coordinates on |
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[10354] | 312 | the sphere associated with the positively oriented orthogonal set of unit vectors |
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[10501] | 313 | $(i,j,k)$ linked to the earth such that |
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| 314 | $k$ is the local upward vector and $(i,j)$ are two vectors orthogonal to $k$, |
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[11543] | 315 | \ie\ along geopotential surfaces (\autoref{fig:MB_referential}). |
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[10354] | 316 | Let $(\lambda,\varphi,z)$ be the geographical coordinate system in which a position is defined by |
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| 317 | the latitude $\varphi(i,j)$, the longitude $\lambda(i,j)$ and |
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[10501] | 318 | the distance from the centre of the earth $a + z(k)$ where $a$ is the earth's radius and |
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[11543] | 319 | $z$ the altitude above a reference sea level (\autoref{fig:MB_referential}). |
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[10354] | 320 | The local deformation of the curvilinear coordinate system is given by $e_1$, $e_2$ and $e_3$, |
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| 321 | the three scale factors: |
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[10414] | 322 | \begin{equation} |
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[11543] | 323 | \label{eq:MB_scale_factors} |
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[10354] | 324 | \begin{aligned} |
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[10501] | 325 | e_1 &= (a + z) \lt[ \lt( \pd[\lambda]{i} \cos \varphi \rt)^2 + \lt( \pd[\varphi]{i} \rt)^2 \rt]^{1/2} \\ |
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| 326 | e_2 &= (a + z) \lt[ \lt( \pd[\lambda]{j} \cos \varphi \rt)^2 + \lt( \pd[\varphi]{j} \rt)^2 \rt]^{1/2} \\ |
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| 327 | e_3 &= \lt( \pd[z]{k} \rt) |
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[10354] | 328 | \end{aligned} |
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| 329 | \end{equation} |
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[707] | 330 | |
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[10354] | 331 | % >>>>>>>>>>>>>>>>>>>>>>>>>>>> |
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| 332 | \begin{figure}[!tb] |
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[11558] | 333 | \centering |
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[11561] | 334 | \includegraphics[width=0.66\textwidth]{Fig_I_earth_referential} |
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[11558] | 335 | \caption[Geographical and curvilinear coordinate systems]{ |
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| 336 | the geographical coordinate system $(\lambda,\varphi,z)$ and the curvilinear |
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| 337 | coordinate system $(i,j,k)$.} |
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| 338 | \label{fig:MB_referential} |
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[10354] | 339 | \end{figure} |
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[707] | 340 | %>>>>>>>>>>>>>>>>>>>>>>>>>>>> |
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| 341 | |
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[10501] | 342 | Since the ocean depth is far smaller than the earth's radius, $a + z$, can be replaced by $a$ in |
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[11543] | 343 | (\autoref{eq:MB_scale_factors}) (thin-shell approximation). |
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[10354] | 344 | The resulting horizontal scale factors $e_1$, $e_2$ are independent of $k$ while |
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[10501] | 345 | the vertical scale factor is a single function of $k$ as $k$ is parallel to $z$. |
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[10354] | 346 | The scalar and vector operators that appear in the primitive equations |
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[11543] | 347 | (\autoref{eq:MB_PE_dyn} to \autoref{eq:MB_PE_eos}) can then be written in the tensorial form, |
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[10354] | 348 | invariant in any orthogonal horizontal curvilinear coordinate system transformation: |
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[10414] | 349 | \begin{subequations} |
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[11543] | 350 | % \label{eq:MB_discrete_operators} |
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[10501] | 351 | \begin{gather} |
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[11543] | 352 | \label{eq:MB_grad} |
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[10501] | 353 | \nabla q = \frac{1}{e_1} \pd[q]{i} \; \vect i |
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| 354 | + \frac{1}{e_2} \pd[q]{j} \; \vect j |
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| 355 | + \frac{1}{e_3} \pd[q]{k} \; \vect k \\ |
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[11543] | 356 | \label{eq:MB_div} |
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[10501] | 357 | \nabla \cdot \vect A = \frac{1}{e_1 \; e_2} \lt[ \pd[(e_2 \; a_1)]{\partial i} + \pd[(e_1 \; a_2)]{j} \rt] |
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| 358 | + \frac{1}{e_3} \lt[ \pd[a_3]{k} \rt] |
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| 359 | \end{gather} |
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| 360 | \begin{multline} |
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[11543] | 361 | \label{eq:MB_curl} |
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[10501] | 362 | \nabla \times \vect{A} = \lt[ \frac{1}{e_2} \pd[a_3]{j} - \frac{1}{e_3} \pd[a_2]{k} \rt] \vect i \\ |
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| 363 | + \lt[ \frac{1}{e_3} \pd[a_1]{k} - \frac{1}{e_1} \pd[a_3]{i} \rt] \vect j \\ |
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| 364 | + \frac{1}{e_1 e_2} \lt[ \pd[(e_2 a_2)]{i} - \pd[(e_1 a_1)]{j} \rt] \vect k |
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| 365 | \end{multline} |
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| 366 | \begin{gather} |
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[11543] | 367 | \label{eq:MB_lap} |
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[10501] | 368 | \Delta q = \nabla \cdot (\nabla q) \\ |
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[11543] | 369 | \label{eq:MB_lap_vector} |
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[10501] | 370 | \Delta \vect A = \nabla (\nabla \cdot \vect A) - \nabla \times (\nabla \times \vect A) |
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| 371 | \end{gather} |
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[707] | 372 | \end{subequations} |
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[10501] | 373 | where $q$ is a scalar quantity and $\vect A = (a_1,a_2,a_3)$ a vector in the $(i,j,k)$ coordinates system. |
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[707] | 374 | |
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| 375 | % ------------------------------------------------------------------------------------------------------------- |
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| 376 | % Continuous Model Equations |
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| 377 | % ------------------------------------------------------------------------------------------------------------- |
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[9393] | 378 | \subsection{Continuous model equations} |
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[11543] | 379 | \label{subsec:MB_zco_Eq} |
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[707] | 380 | |
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[10354] | 381 | In order to express the Primitive Equations in tensorial formalism, |
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| 382 | it is necessary to compute the horizontal component of the non-linear and viscous terms of the equation using |
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[11543] | 383 | \autoref{eq:MB_grad}) to \autoref{eq:MB_lap_vector}. |
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[10501] | 384 | Let us set $\vect U = (u,v,w) = \vect U_h + w \; \vect k $, the velocity in the $(i,j,k)$ coordinates system and |
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[10354] | 385 | define the relative vorticity $\zeta$ and the divergence of the horizontal velocity field $\chi$, by: |
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[10501] | 386 | \begin{gather} |
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[11543] | 387 | \label{eq:MB_curl_Uh} |
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[10501] | 388 | \zeta = \frac{1}{e_1 e_2} \lt[ \pd[(e_2 \, v)]{i} - \pd[(e_1 \, u)]{j} \rt] \\ |
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[11543] | 389 | \label{eq:MB_div_Uh} |
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[10501] | 390 | \chi = \frac{1}{e_1 e_2} \lt[ \pd[(e_2 \, u)]{i} + \pd[(e_1 \, v)]{j} \rt] |
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| 391 | \end{gather} |
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[707] | 392 | |
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[11335] | 393 | Using again the fact that the horizontal scale factors $e_1$ and $e_2$ are independent of $k$ and that |
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[10354] | 394 | $e_3$ is a function of the single variable $k$, |
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[11543] | 395 | $NLT$ the nonlinear term of \autoref{eq:MB_PE_dyn} can be transformed as follows: |
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[10501] | 396 | \begin{alignat*}{2} |
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| 397 | &NLT &= &\lt[ (\nabla \times {\vect U}) \times {\vect U} + \frac{1}{2} \nabla \lt( {\vect U}^2 \rt) \rt]_h \\ |
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| 398 | & &= &\lt( |
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| 399 | \begin{array}{*{20}c} |
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| 400 | \lt[ \frac{1}{e_3} \pd[u]{k} - \frac{1}{e_1} \pd[w]{i} \rt] w - \zeta \; v \\ |
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| 401 | \zeta \; u - \lt[ \frac{1}{e_2} \pd[w]{j} - \frac{1}{e_3} \pd[v]{k} \rt] \ w |
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| 402 | \end{array} |
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| 403 | \rt) |
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| 404 | + \frac{1}{2} \lt( |
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| 405 | \begin{array}{*{20}c} |
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| 406 | \frac{1}{e_1} \pd[(u^2 + v^2 + w^2)]{i} \\ |
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| 407 | \frac{1}{e_2} \pd[(u^2 + v^2 + w^2)]{j} |
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| 408 | \end{array} |
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| 409 | \rt) \\ |
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| 410 | & &= &\lt( |
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| 411 | \begin{array}{*{20}c} |
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| 412 | -\zeta \; v \\ |
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| 413 | \zeta \; u |
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| 414 | \end{array} |
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| 415 | \rt) |
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| 416 | + \frac{1}{2} \lt( |
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| 417 | \begin{array}{*{20}c} |
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| 418 | \frac{1}{e_1} \pd[(u^2 + v^2)]{i} \\ |
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| 419 | \frac{1}{e_2} \pd[(u^2 + v^2)]{j} |
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| 420 | \end{array} |
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| 421 | \rt) \\ |
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| 422 | & & &+ \frac{1}{e_3} \lt( |
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| 423 | \begin{array}{*{20}c} |
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| 424 | w \; \pd[u]{k} \\ |
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| 425 | w \; \pd[v]{k} |
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| 426 | \end{array} |
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| 427 | \rt) |
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| 428 | - \lt( |
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| 429 | \begin{array}{*{20}c} |
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| 430 | \frac{w}{e_1} \pd[w]{i} - \frac{1}{2 e_1} \pd[w^2]{i} \\ |
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| 431 | \frac{w}{e_2} \pd[w]{j} - \frac{1}{2 e_2} \pd[w^2]{j} |
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| 432 | \end{array} |
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| 433 | \rt) |
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| 434 | \end{alignat*} |
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[10354] | 435 | The last term of the right hand side is obviously zero, and thus the nonlinear term of |
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[11543] | 436 | \autoref{eq:MB_PE_dyn} is written in the $(i,j,k)$ coordinate system: |
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[10414] | 437 | \begin{equation} |
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[11543] | 438 | \label{eq:MB_vector_form} |
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[10501] | 439 | NLT = \zeta \; \vect k \times \vect U_h + \frac{1}{2} \nabla_h \lt( \vect U_h^2 \rt) |
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| 440 | + \frac{1}{e_3} w \pd[\vect U_h]{k} |
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[707] | 441 | \end{equation} |
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| 442 | |
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[10354] | 443 | This is the so-called \textit{vector invariant form} of the momentum advection term. |
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| 444 | For some purposes, it can be advantageous to write this term in the so-called flux form, |
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[11435] | 445 | \ie\ to write it as the divergence of fluxes. |
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[11543] | 446 | For example, the first component of \autoref{eq:MB_vector_form} (the $i$-component) is transformed as follows: |
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[10501] | 447 | \begin{alignat*}{2} |
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| 448 | &NLT_i &= &- \zeta \; v + \frac{1}{2 \; e_1} \pd[ (u^2 + v^2) ]{i} + \frac{1}{e_3} w \ \pd[u]{k} \\ |
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| 449 | & &= &\frac{1}{e_1 \; e_2} \lt( -v \pd[(e_2 \, v)]{i} + v \pd[(e_1 \, u)]{j} \rt) |
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| 450 | + \frac{1}{e_1 e_2} \lt( e_2 \; u \pd[u]{i} + e_2 \; v \pd[v]{i} \rt) \\ |
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| 451 | & & &+ \frac{1}{e_3} \lt( w \; \pd[u]{k} \rt) \\ |
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| 452 | & &= &\frac{1}{e_1 \; e_2} \lt[ - \lt( v^2 \pd[e_2]{i} + e_2 \, v \pd[v]{i} \rt) |
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| 453 | + \lt( \pd[ \lt( e_1 \, u \, v \rt)]{j} - e_1 \, u \pd[v]{j} \rt) \rt. \\ |
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| 454 | & & &\lt. + \lt( \pd[ \lt( e_2 \, u \, u \rt)]{i} - u \pd[ \lt( e_2 u \rt)]{i} \rt) |
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| 455 | + e_2 v \pd[v]{i} \rt] \\ |
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| 456 | & & &+ \frac{1}{e_3} \lt( \pd[(w \, u)]{k} - u \pd[w]{k} \rt) \\ |
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| 457 | & &= &\frac{1}{e_1 \; e_2} \lt( \pd[(e_2 \, u \, u)]{i} + \pd[(e_1 \, u \, v)]{j} \rt) |
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| 458 | + \frac{1}{e_3} \pd[(w \, u)]{k} \\ |
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| 459 | & & &+ \frac{1}{e_1 e_2} \lt[ - u \lt( \pd[(e_1 v)]{j} - v \, \pd[e_1]{j} \rt) |
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| 460 | - u \pd[(e_2 u)]{i} \rt] |
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| 461 | - \frac{1}{e_3} \pd[w]{k} u \\ |
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| 462 | & & &+ \frac{1}{e_1 e_2} \lt( - v^2 \pd[e_2]{i} \rt) \\ |
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| 463 | & &= &\nabla \cdot (\vect U \, u) - (\nabla \cdot \vect U) \ u |
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| 464 | + \frac{1}{e_1 e_2} \lt( -v^2 \pd[e_2]{i} + u v \, \pd[e_1]{j} \rt) \\ |
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[11335] | 465 | \intertext{as $\nabla \cdot {\vect U} \; = 0$ (incompressibility) it becomes:} |
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[10501] | 466 | & &= &\, \nabla \cdot (\vect U \, u) + \frac{1}{e_1 e_2} \lt( v \; \pd[e_2]{i} - u \; \pd[e_1]{j} \rt) (-v) |
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| 467 | \end{alignat*} |
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[707] | 468 | |
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[817] | 469 | The flux form of the momentum advection term is therefore given by: |
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[10501] | 470 | \begin{equation} |
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[11543] | 471 | \label{eq:MB_flux_form} |
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[10501] | 472 | NLT = \nabla \cdot \lt( |
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| 473 | \begin{array}{*{20}c} |
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| 474 | \vect U \, u \\ |
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| 475 | \vect U \, v |
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| 476 | \end{array} |
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| 477 | \rt) |
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| 478 | + \frac{1}{e_1 e_2} \lt( v \pd[e_2]{i} - u \pd[e_1]{j} \rt) \vect k \times \vect U_h |
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| 479 | \end{equation} |
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[707] | 480 | |
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[10354] | 481 | The flux form has two terms, |
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| 482 | the first one is expressed as the divergence of momentum fluxes (hence the flux form name given to this formulation) |
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| 483 | and the second one is due to the curvilinear nature of the coordinate system used. |
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[11435] | 484 | The latter is called the \textit{metric} term and can be viewed as a modification of the Coriolis parameter: |
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[10414] | 485 | \[ |
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[11543] | 486 | % \label{eq:MB_cor+metric} |
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[10501] | 487 | f \to f + \frac{1}{e_1 e_2} \lt( v \pd[e_2]{i} - u \pd[e_1]{j} \rt) |
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[10414] | 488 | \] |
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[707] | 489 | |
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[10354] | 490 | Note that in the case of geographical coordinate, |
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[11435] | 491 | \ie\ when $(i,j) \to (\lambda,\varphi)$ and $(e_1,e_2) \to (a \, \cos \varphi,a)$, |
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[10501] | 492 | we recover the commonly used modification of the Coriolis parameter $f \to f + (u / a) \tan \varphi$. |
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[707] | 493 | |
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[10354] | 494 | To sum up, the curvilinear $z$-coordinate equations solved by the ocean model can be written in |
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| 495 | the following tensorial formalism: |
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[2282] | 496 | |
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[10501] | 497 | \begin{itemize} |
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| 498 | \item |
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| 499 | \textbf{Vector invariant form of the momentum equations}: |
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| 500 | \begin{equation} |
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[11543] | 501 | \label{eq:MB_dyn_vect} |
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[10501] | 502 | \begin{split} |
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[11543] | 503 | % \label{eq:MB_dyn_vect_u} |
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[10501] | 504 | \pd[u]{t} = &+ (\zeta + f) \, v - \frac{1}{2 e_1} \pd[]{i} (u^2 + v^2) |
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| 505 | - \frac{1}{e_3} w \pd[u]{k} - \frac{1}{e_1} \pd[]{i} \lt( \frac{p_s + p_h}{\rho_o} \rt) \\ |
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| 506 | &+ D_u^{\vect U} + F_u^{\vect U} \\ |
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| 507 | \pd[v]{t} = &- (\zeta + f) \, u - \frac{1}{2 e_2} \pd[]{j} (u^2 + v^2) |
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[11435] | 508 | - \frac{1}{e_3} w \pd[v]{k} - \frac{1}{e_2} \pd[]{j} \lt( \frac{p_s + p_h}{\rho_o} \rt) \\ |
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[10501] | 509 | &+ D_v^{\vect U} + F_v^{\vect U} |
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[10414] | 510 | \end{split} |
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[10501] | 511 | \end{equation} |
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| 512 | \item |
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| 513 | \textbf{flux form of the momentum equations}: |
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[11543] | 514 | % \label{eq:MB_dyn_flux} |
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[10414] | 515 | \begin{multline*} |
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[11543] | 516 | % \label{eq:MB_dyn_flux_u} |
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[10501] | 517 | \pd[u]{t} = + \lt[ f + \frac{1}{e_1 \; e_2} \lt( v \pd[e_2]{i} - u \pd[e_1]{j} \rt) \rt] \, v \\ |
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| 518 | - \frac{1}{e_1 \; e_2} \lt( \pd[(e_2 \, u \, u)]{i} + \pd[(e_1 \, v \, u)]{j} \rt) \\ |
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| 519 | - \frac{1}{e_3} \pd[(w \, u)]{k} - \frac{1}{e_1} \pd[]{i} \lt( \frac{p_s + p_h}{\rho_o} \rt) |
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| 520 | + D_u^{\vect U} + F_u^{\vect U} |
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[10414] | 521 | \end{multline*} |
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| 522 | \begin{multline*} |
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[11543] | 523 | % \label{eq:MB_dyn_flux_v} |
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[10501] | 524 | \pd[v]{t} = - \lt[ f + \frac{1}{e_1 \; e_2} \lt( v \pd[e_2]{i} - u \pd[e_1]{j} \rt) \rt] \, u \\ |
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[11335] | 525 | - \frac{1}{e_1 \; e_2} \lt( \pd[(e_2 \, u \, v)]{i} + \pd[(e_1 \, v \, v)]{j} \rt) \\ |
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[10501] | 526 | - \frac{1}{e_3} \pd[(w \, v)]{k} - \frac{1}{e_2} \pd[]{j} \lt( \frac{p_s + p_h}{\rho_o} \rt) |
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| 527 | + D_v^{\vect U} + F_v^{\vect U} |
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[10414] | 528 | \end{multline*} |
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[11543] | 529 | where $\zeta$, the relative vorticity, is given by \autoref{eq:MB_curl_Uh} and $p_s$, the surface pressure, |
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[10501] | 530 | is given by: |
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| 531 | \[ |
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[11543] | 532 | % \label{eq:MB_spg} |
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[10501] | 533 | p_s = \rho \,g \, \eta |
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| 534 | \] |
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[11543] | 535 | and $\eta$ is the solution of \autoref{eq:MB_ssh}. |
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[707] | 536 | |
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[10501] | 537 | The vertical velocity and the hydrostatic pressure are diagnosed from the following equations: |
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| 538 | \[ |
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[11543] | 539 | % \label{eq:MB_w_diag} |
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[11435] | 540 | \pd[w]{k} = - \chi \; e_3 \qquad |
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[11543] | 541 | % \label{eq:MB_hp_diag} |
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[10501] | 542 | \pd[p_h]{k} = - \rho \; g \; e_3 |
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| 543 | \] |
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[11543] | 544 | where the divergence of the horizontal velocity, $\chi$ is given by \autoref{eq:MB_div_Uh}. |
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[11335] | 545 | |
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[11435] | 546 | \item |
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[11335] | 547 | \textbf{tracer equations}: |
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| 548 | \begin{equation} |
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| 549 | \begin{split} |
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| 550 | \pd[T]{t} = & - \frac{1}{e_1 e_2} \lt[ \pd[(e_2 T \, u)]{i} + \pd[(e_1 T \, v)]{j} \rt] |
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[10501] | 551 | - \frac{1}{e_3} \pd[(T \, w)]{k} + D^T + F^T \\ |
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[11335] | 552 | \pd[S]{t} = & - \frac{1}{e_1 e_2} \lt[ \pd[(e_2 S \, u)]{i} + \pd[(e_1 S \, v)]{j} \rt] |
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| 553 | - \frac{1}{e_3} \pd[(S \, w)]{k} + D^S + F^S \\ |
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| 554 | \rho = & \rho \big( T,S,z(k) \big) |
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| 555 | \end{split} |
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| 556 | \end{equation} |
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[10501] | 557 | \end{itemize} |
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[707] | 558 | |
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[10501] | 559 | The expression of $\vect D^{U}$, $D^{S}$ and $D^{T}$ depends on the subgrid scale parameterisation used. |
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[11543] | 560 | It will be defined in \autoref{eq:MB_zdf}. |
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[10501] | 561 | The nature and formulation of $\vect F^{\vect U}$, $F^T$ and $F^S$, the surface forcing terms, |
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[10354] | 562 | are discussed in \autoref{chap:SBC}. |
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[707] | 563 | |
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[10501] | 564 | \newpage |
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[2282] | 565 | |
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[707] | 566 | % ================================================================ |
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[2282] | 567 | % Curvilinear generalised vertical coordinate System |
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[817] | 568 | % ================================================================ |
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[9393] | 569 | \section{Curvilinear generalised vertical coordinate system} |
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[11543] | 570 | \label{sec:MB_gco} |
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[817] | 571 | |
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[10354] | 572 | The ocean domain presents a huge diversity of situation in the vertical. |
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| 573 | First the ocean surface is a time dependent surface (moving surface). |
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| 574 | Second the ocean floor depends on the geographical position, |
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| 575 | varying from more than 6,000 meters in abyssal trenches to zero at the coast. |
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[10501] | 576 | Last but not least, the ocean stratification exerts a strong barrier to vertical motions and mixing. |
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[10354] | 577 | Therefore, in order to represent the ocean with respect to |
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| 578 | the first point a space and time dependent vertical coordinate that follows the variation of the sea surface height |
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[11435] | 579 | \eg\ an \zstar-coordinate; |
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[10354] | 580 | for the second point, a space variation to fit the change of bottom topography |
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[11435] | 581 | \eg\ a terrain-following or $\sigma$-coordinate; |
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[10354] | 582 | and for the third point, one will be tempted to use a space and time dependent coordinate that |
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[11435] | 583 | follows the isopycnal surfaces, \eg\ an isopycnic coordinate. |
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[2282] | 584 | |
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[11335] | 585 | In order to satisfy two or more constraints one can even be tempted to mixed these coordinate systems, as in |
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[10354] | 586 | HYCOM (mixture of $z$-coordinate at the surface, isopycnic coordinate in the ocean interior and $\sigma$ at |
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[11123] | 587 | the ocean bottom) \citep{chassignet.smith.ea_JPO03} or |
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[10354] | 588 | OPA (mixture of $z$-coordinate in vicinity the surface and steep topography areas and $\sigma$-coordinate elsewhere) |
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[11123] | 589 | \citep{madec.delecluse.ea_JPO96} among others. |
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[2282] | 590 | |
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[10354] | 591 | In fact one is totally free to choose any space and time vertical coordinate by |
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| 592 | introducing an arbitrary vertical coordinate : |
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[10414] | 593 | \begin{equation} |
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[11543] | 594 | \label{eq:MB_s} |
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[10501] | 595 | s = s(i,j,k,t) |
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[2282] | 596 | \end{equation} |
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[10354] | 597 | with the restriction that the above equation gives a single-valued monotonic relationship between $s$ and $k$, |
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| 598 | when $i$, $j$ and $t$ are held fixed. |
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[11543] | 599 | \autoref{eq:MB_s} is a transformation from the $(i,j,k,t)$ coordinate system with independent variables into |
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[10354] | 600 | the $(i,j,s,t)$ generalised coordinate system with $s$ depending on the other three variables through |
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[11543] | 601 | \autoref{eq:MB_s}. |
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[11123] | 602 | This so-called \textit{generalised vertical coordinate} \citep{kasahara_MWR74} is in fact |
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[10354] | 603 | an Arbitrary Lagrangian--Eulerian (ALE) coordinate. |
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[11335] | 604 | Indeed, one has a great deal of freedom in the choice of expression for $s$. The choice determines |
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[10354] | 605 | which part of the vertical velocity (defined from a fixed referential) will cross the levels (Eulerian part) and |
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| 606 | which part will be used to move them (Lagrangian part). |
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[11123] | 607 | The coordinate is also sometime referenced as an adaptive coordinate \citep{hofmeister.burchard.ea_OM10}, |
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[10354] | 608 | since the coordinate system is adapted in the course of the simulation. |
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| 609 | Its most often used implementation is via an ALE algorithm, |
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| 610 | in which a pure lagrangian step is followed by regridding and remapping steps, |
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[11335] | 611 | the latter step implicitly embedding the vertical advection |
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[11123] | 612 | \citep{hirt.amsden.ea_JCP74, chassignet.smith.ea_JPO03, white.adcroft.ea_JCP09}. |
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| 613 | Here we follow the \citep{kasahara_MWR74} strategy: |
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[11335] | 614 | a regridding step (an update of the vertical coordinate) followed by an Eulerian step with |
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[10354] | 615 | an explicit computation of vertical advection relative to the moving s-surfaces. |
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[2282] | 616 | |
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[2376] | 617 | %\gmcomment{ |
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| 618 | %A key point here is that the $s$-coordinate depends on $(i,j)$ ==> horizontal pressure gradient... |
---|
[11335] | 619 | The generalized vertical coordinates used in ocean modelling are not orthogonal, |
---|
[10354] | 620 | which contrasts with many other applications in mathematical physics. |
---|
| 621 | Hence, it is useful to keep in mind the following properties that may seem odd on initial encounter. |
---|
[2282] | 622 | |
---|
[10354] | 623 | The horizontal velocity in ocean models measures motions in the horizontal plane, |
---|
| 624 | perpendicular to the local gravitational field. |
---|
[11335] | 625 | That is, horizontal velocity is mathematically the same regardless of the vertical coordinate, be it geopotential, |
---|
[10354] | 626 | isopycnal, pressure, or terrain following. |
---|
| 627 | The key motivation for maintaining the same horizontal velocity component is that |
---|
| 628 | the hydrostatic and geostrophic balances are dominant in the large-scale ocean. |
---|
[10501] | 629 | Use of an alternative quasi -horizontal velocity, for example one oriented parallel to the generalized surface, |
---|
[10354] | 630 | would lead to unacceptable numerical errors. |
---|
[10501] | 631 | Correspondingly, the vertical direction is anti -parallel to the gravitational force in |
---|
[10354] | 632 | all of the coordinate systems. |
---|
[10501] | 633 | We do not choose the alternative of a quasi -vertical direction oriented normal to |
---|
| 634 | the surface of a constant generalized vertical coordinate. |
---|
[2282] | 635 | |
---|
[10354] | 636 | It is the method used to measure transport across the generalized vertical coordinate surfaces which differs between |
---|
| 637 | the vertical coordinate choices. |
---|
| 638 | That is, computation of the dia-surface velocity component represents the fundamental distinction between |
---|
| 639 | the various coordinates. |
---|
| 640 | In some models, such as geopotential, pressure, and terrain following, this transport is typically diagnosed from |
---|
| 641 | volume or mass conservation. |
---|
| 642 | In other models, such as isopycnal layered models, this transport is prescribed based on assumptions about |
---|
[10501] | 643 | the physical processes producing a flux across the layer interfaces. |
---|
[2282] | 644 | |
---|
[10354] | 645 | In this section we first establish the PE in the generalised vertical $s$-coordinate, |
---|
[10501] | 646 | then we discuss the particular cases available in \NEMO, namely $z$, \zstar, $s$, and \ztilde. |
---|
[2282] | 647 | %} |
---|
| 648 | |
---|
| 649 | % ------------------------------------------------------------------------------------------------------------- |
---|
| 650 | % The s-coordinate Formulation |
---|
| 651 | % ------------------------------------------------------------------------------------------------------------- |
---|
[10501] | 652 | \subsection{\textit{S}-coordinate formulation} |
---|
[2282] | 653 | |
---|
[11543] | 654 | Starting from the set of equations established in \autoref{sec:MB_zco} for the special case $k = z$ and |
---|
[10501] | 655 | thus $e_3 = 1$, we introduce an arbitrary vertical coordinate $s = s(i,j,k,t)$, |
---|
| 656 | which includes $z$-, \zstar- and $\sigma$-coordinates as special cases |
---|
| 657 | ($s = z$, $s = \zstar$, and $s = \sigma = z / H$ or $ = z / \lt( H + \eta \rt)$, resp.). |
---|
[11543] | 658 | A formal derivation of the transformed equations is given in \autoref{apdx:SCOORD}. |
---|
[10501] | 659 | Let us define the vertical scale factor by $e_3 = \partial_s z$ ($e_3$ is now a function of $(i,j,k,t)$ ), |
---|
| 660 | and the slopes in the $(i,j)$ directions between $s$- and $z$-surfaces by: |
---|
[10414] | 661 | \begin{equation} |
---|
[11543] | 662 | \label{eq:MB_sco_slope} |
---|
[10501] | 663 | \sigma_1 = \frac{1}{e_1} \; \lt. \pd[z]{i} \rt|_s \quad \text{and} \quad |
---|
| 664 | \sigma_2 = \frac{1}{e_2} \; \lt. \pd[z]{j} \rt|_s |
---|
[2282] | 665 | \end{equation} |
---|
[11435] | 666 | We also introduce $\omega$, a dia-surface velocity component, defined as the velocity |
---|
[2282] | 667 | relative to the moving $s$-surfaces and normal to them: |
---|
[10414] | 668 | \[ |
---|
[11543] | 669 | % \label{eq:MB_sco_w} |
---|
[11335] | 670 | \omega = w - \, \lt. \pd[z]{t} \rt|_s - \sigma_1 \, u - \sigma_2 \, v |
---|
[10414] | 671 | \] |
---|
[2282] | 672 | |
---|
[11543] | 673 | The equations solved by the ocean model \autoref{eq:MB_PE} in $s$-coordinate can be written as follows |
---|
| 674 | (see \autoref{sec:SCOORD_momentum}): |
---|
[2282] | 675 | |
---|
[10501] | 676 | \begin{itemize} |
---|
| 677 | \item \textbf{Vector invariant form of the momentum equation}: |
---|
| 678 | \begin{multline*} |
---|
[11543] | 679 | % \label{eq:MB_sco_u_vector} |
---|
[10501] | 680 | \pd[u]{t} = + (\zeta + f) \, v - \frac{1}{2 \, e_1} \pd[]{i} (u^2 + v^2) - \frac{1}{e_3} \omega \pd[u]{k} \\ |
---|
[11335] | 681 | - \frac{1}{e_1} \pd[]{i} \lt( \frac{p_s + p_h}{\rho_o} \rt) - g \frac{\rho}{\rho_o} \sigma_1 |
---|
[10501] | 682 | + D_u^{\vect U} + F_u^{\vect U} |
---|
| 683 | \end{multline*} |
---|
| 684 | \begin{multline*} |
---|
[11543] | 685 | % \label{eq:MB_sco_v_vector} |
---|
[10501] | 686 | \pd[v]{t} = - (\zeta + f) \, u - \frac{1}{2 \, e_2} \pd[]{j}(u^2 + v^2) - \frac{1}{e_3} \omega \pd[v]{k} \\ |
---|
[11335] | 687 | - \frac{1}{e_2} \pd[]{j} \lt( \frac{p_s + p_h}{\rho_o} \rt) - g \frac{\rho}{\rho_o} \sigma_2 |
---|
[10501] | 688 | + D_v^{\vect U} + F_v^{\vect U} |
---|
| 689 | \end{multline*} |
---|
| 690 | \item \textbf{Flux form of the momentum equation}: |
---|
| 691 | \begin{multline*} |
---|
[11543] | 692 | % \label{eq:MB_sco_u_flux} |
---|
[10501] | 693 | \frac{1}{e_3} \pd[(e_3 \, u)]{t} = + \lt[ f + \frac{1}{e_1 \; e_2} \lt( v \pd[e_2]{i} - u \pd[e_1]{j} \rt) \rt] \, v \\ |
---|
| 694 | - \frac{1}{e_1 \; e_2 \; e_3} \lt( \pd[(e_2 \, e_3 \, u \, u)]{i} + \pd[(e_1 \, e_3 \, v \, u)]{j} \rt) \\ |
---|
| 695 | - \frac{1}{e_3} \pd[(\omega \, u)]{k} |
---|
| 696 | - \frac{1}{e_1} \pd[]{i} \lt( \frac{p_s + p_h}{\rho_o} \rt) |
---|
[11335] | 697 | - g \frac{\rho}{\rho_o} \sigma_1 + D_u^{\vect U} + F_u^{\vect U} |
---|
[10501] | 698 | \end{multline*} |
---|
| 699 | \begin{multline*} |
---|
[11543] | 700 | % \label{eq:MB_sco_v_flux} |
---|
[10501] | 701 | \frac{1}{e_3} \pd[(e_3 \, v)]{t} = - \lt[ f + \frac{1}{e_1 \; e_2} \lt( v \pd[e_2]{i} - u \pd[e_1]{j} \rt) \rt] \, u \\ |
---|
| 702 | - \frac{1}{e_1 \; e_2 \; e_3} \lt( \pd[( e_2 \; e_3 \, u \, v)]{i} + \pd[(e_1 \; e_3 \, v \, v)]{j} \rt) \\ |
---|
| 703 | - \frac{1}{e_3} \pd[(\omega \, v)]{k} |
---|
| 704 | - \frac{1}{e_2} \pd[]{j} \lt( \frac{p_s + p_h}{\rho_o} \rt) |
---|
[11335] | 705 | - g \frac{\rho}{\rho_o}\sigma_2 + D_v^{\vect U} + F_v^{\vect U} |
---|
[10501] | 706 | \end{multline*} |
---|
| 707 | where the relative vorticity, $\zeta$, the surface pressure gradient, |
---|
| 708 | and the hydrostatic pressure have the same expressions as in $z$-coordinates although |
---|
| 709 | they do not represent exactly the same quantities. |
---|
[11543] | 710 | $\omega$ is provided by the continuity equation (see \autoref{apdx:SCOORD}): |
---|
[10501] | 711 | \[ |
---|
[11543] | 712 | % \label{eq:MB_sco_continuity} |
---|
[10501] | 713 | \pd[e_3]{t} + e_3 \; \chi + \pd[\omega]{s} = 0 \quad \text{with} \quad |
---|
| 714 | \chi = \frac{1}{e_1 e_2 e_3} \lt( \pd[(e_2 e_3 \, u)]{i} + \pd[(e_1 e_3 \, v)]{j} \rt) |
---|
| 715 | \] |
---|
| 716 | \item \textit{tracer equations}: |
---|
| 717 | \begin{multline*} |
---|
[11543] | 718 | % \label{eq:MB_sco_t} |
---|
[10501] | 719 | \frac{1}{e_3} \pd[(e_3 \, T)]{t} = - \frac{1}{e_1 e_2 e_3} \lt( \pd[(e_2 e_3 \, u \, T)]{i} |
---|
| 720 | + \pd[(e_1 e_3 \, v \, T)]{j} \rt) \\ |
---|
| 721 | - \frac{1}{e_3} \pd[(T \, \omega)]{k} + D^T + F^S |
---|
| 722 | \end{multline*} |
---|
| 723 | \begin{multline} |
---|
[11543] | 724 | % \label{eq:MB_sco_s} |
---|
[10501] | 725 | \frac{1}{e_3} \pd[(e_3 \, S)]{t} = - \frac{1}{e_1 e_2 e_3} \lt( \pd[(e_2 e_3 \, u \, S)]{i} |
---|
| 726 | + \pd[(e_1 e_3 \, v \, S)]{j} \rt) \\ |
---|
| 727 | - \frac{1}{e_3} \pd[(S \, \omega)]{k} + D^S + F^S |
---|
| 728 | \end{multline} |
---|
| 729 | \end{itemize} |
---|
[10354] | 730 | The equation of state has the same expression as in $z$-coordinate, |
---|
| 731 | and similar expressions are used for mixing and forcing terms. |
---|
[2282] | 732 | |
---|
| 733 | \gmcomment{ |
---|
[10501] | 734 | \colorbox{yellow}{ to be updated $= = >$} |
---|
| 735 | Add a few works on z and zps and s and underlies the differences between all of them |
---|
| 736 | \colorbox{yellow}{$< = =$ end update} |
---|
| 737 | } |
---|
[2282] | 738 | |
---|
| 739 | % ------------------------------------------------------------------------------------------------------------- |
---|
[10442] | 740 | % Curvilinear \zstar-coordinate System |
---|
[2282] | 741 | % ------------------------------------------------------------------------------------------------------------- |
---|
[10501] | 742 | \subsection{Curvilinear \zstar-coordinate system} |
---|
[11543] | 743 | \label{subsec:MB_zco_star} |
---|
[2282] | 744 | |
---|
[817] | 745 | %>>>>>>>>>>>>>>>>>>>>>>>>>>>> |
---|
[10354] | 746 | \begin{figure}[!b] |
---|
[11558] | 747 | \centering |
---|
[11561] | 748 | \includegraphics[width=0.66\textwidth]{Fig_z_zstar} |
---|
[11558] | 749 | \caption[Curvilinear z-coordinate systems (\{non-\}linear free-surface cases and re-scaled \zstar)]{ |
---|
| 750 | (a) $z$-coordinate in linear free-surface case ; |
---|
| 751 | (b) $z$-coordinate in non-linear free surface case ; |
---|
| 752 | (c) re-scaled height coordinate |
---|
| 753 | (become popular as the \zstar-coordinate \citep{adcroft.campin_OM04}).} |
---|
| 754 | \label{fig:MB_z_zstar} |
---|
[10354] | 755 | \end{figure} |
---|
[817] | 756 | %>>>>>>>>>>>>>>>>>>>>>>>>>>>> |
---|
| 757 | |
---|
[11335] | 758 | In this case, the free surface equation is nonlinear, and the variations of volume are fully taken into account. |
---|
[11435] | 759 | These coordinates systems is presented in a report \citep{levier.treguier.ea_rpt07} available on the \NEMO\ web site. |
---|
[817] | 760 | |
---|
[10442] | 761 | The \zstar coordinate approach is an unapproximated, non-linear free surface implementation which allows one to |
---|
[11123] | 762 | deal with large amplitude free-surface variations relative to the vertical resolution \citep{adcroft.campin_OM04}. |
---|
[10442] | 763 | In the \zstar formulation, |
---|
[10354] | 764 | the variation of the column thickness due to sea-surface undulations is not concentrated in the surface level, |
---|
| 765 | as in the $z$-coordinate formulation, but is equally distributed over the full water column. |
---|
| 766 | Thus vertical levels naturally follow sea-surface variations, with a linear attenuation with depth, |
---|
[11543] | 767 | as illustrated by \autoref{fig:MB_z_zstar}. |
---|
| 768 | Note that with a flat bottom, such as in \autoref{fig:MB_z_zstar}, the bottom-following $z$ coordinate and \zstar are equivalent. |
---|
[10501] | 769 | The definition and modified oceanic equations for the rescaled vertical coordinate \zstar, |
---|
[10354] | 770 | including the treatment of fresh-water flux at the surface, are detailed in Adcroft and Campin (2004). |
---|
| 771 | The major points are summarized here. |
---|
[10501] | 772 | The position (\zstar) and vertical discretization (\zstar) are expressed as: |
---|
[10414] | 773 | \[ |
---|
[11543] | 774 | % \label{eq:MB_z-star} |
---|
[10501] | 775 | H + \zstar = (H + z) / r \quad \text{and} \quad \delta \zstar |
---|
| 776 | = \delta z / r \quad \text{with} \quad r |
---|
[11335] | 777 | = \frac{H + \eta}{H} . |
---|
[10501] | 778 | \] |
---|
[11335] | 779 | Simple re-organisation of the above expressions gives |
---|
| 780 | \[ |
---|
[11543] | 781 | % \label{eq:MB_zstar_2} |
---|
[11435] | 782 | \zstar = H \lt( \frac{z - \eta}{H + \eta} \rt) . |
---|
[11335] | 783 | \] |
---|
[10442] | 784 | Since the vertical displacement of the free surface is incorporated in the vertical coordinate \zstar, |
---|
| 785 | the upper and lower boundaries are at fixed \zstar position, |
---|
[10501] | 786 | $\zstar = 0$ and $\zstar = -H$ respectively. |
---|
[10354] | 787 | Also the divergence of the flow field is no longer zero as shown by the continuity equation: |
---|
[10501] | 788 | \[ |
---|
[11335] | 789 | \pd[r]{t} = \nabla_{\zstar} \cdot \lt( r \; \vect U_h \rt) + \pd[r \; w^*]{\zstar} = 0 . |
---|
[10501] | 790 | \] |
---|
[11335] | 791 | This \zstar coordinate is closely related to the "eta" coordinate used in many atmospheric models |
---|
[10354] | 792 | (see Black (1994) for a review of eta coordinate atmospheric models). |
---|
| 793 | It was originally used in ocean models by Stacey et al. (1995) for studies of tides next to shelves, |
---|
| 794 | and it has been recently promoted by Adcroft and Campin (2004) for global climate modelling. |
---|
[2282] | 795 | |
---|
[10501] | 796 | The surfaces of constant \zstar are quasi -horizontal. |
---|
| 797 | Indeed, the \zstar coordinate reduces to $z$ when $\eta$ is zero. |
---|
[10354] | 798 | In general, when noting the large differences between |
---|
| 799 | undulations of the bottom topography versus undulations in the surface height, |
---|
[10501] | 800 | it is clear that surfaces constant \zstar are very similar to the depth surfaces. |
---|
[10354] | 801 | These properties greatly reduce difficulties of computing the horizontal pressure gradient relative to |
---|
[11543] | 802 | terrain following sigma models discussed in \autoref{subsec:MB_sco}. |
---|
[11335] | 803 | Additionally, since $\zstar = z$ when $\eta = 0$, |
---|
[10354] | 804 | no flow is spontaneously generated in an unforced ocean starting from rest, regardless the bottom topography. |
---|
| 805 | This behaviour is in contrast to the case with "s"-models, where pressure gradient errors in the presence of |
---|
| 806 | nontrivial topographic variations can generate nontrivial spontaneous flow from a resting state, |
---|
| 807 | depending on the sophistication of the pressure gradient solver. |
---|
[10501] | 808 | The quasi -horizontal nature of the coordinate surfaces also facilitates the implementation of |
---|
[11335] | 809 | neutral physics parameterizations in \zstar models using the same techniques as in $z$-models |
---|
[11123] | 810 | (see Chapters 13-16 of \cite{griffies_bk04}) for a discussion of neutral physics in $z$-models, |
---|
[10354] | 811 | as well as \autoref{sec:LDF_slp} in this document for treatment in \NEMO). |
---|
[2282] | 812 | |
---|
[11335] | 813 | The range over which \zstar varies is time independent $-H \leq \zstar \leq 0$. |
---|
| 814 | Hence, all cells remain nonvanishing, so long as the surface height maintains $\eta > -H$. |
---|
[10501] | 815 | This is a minor constraint relative to that encountered on the surface height when using $s = z$ or $s = z - \eta$. |
---|
[2282] | 816 | |
---|
[11335] | 817 | Because \zstar has a time independent range, all grid cells have static increments ds, |
---|
| 818 | and the sum of the vertical increments yields the time independent ocean depth. %k ds = H. |
---|
[10501] | 819 | The \zstar coordinate is therefore invisible to undulations of the free surface, |
---|
[10354] | 820 | since it moves along with the free surface. |
---|
[11335] | 821 | This property means that no spurious vertical transport is induced across surfaces of constant \zstar by |
---|
[10354] | 822 | the motion of external gravity waves. |
---|
[11335] | 823 | Such spurious transport can be a problem in z-models, especially those with tidal forcing. |
---|
| 824 | Quite generally, the time independent range for the \zstar coordinate is a very convenient property that |
---|
| 825 | allows for a nearly arbitrary vertical resolution even in the presence of large amplitude fluctuations of |
---|
[10354] | 826 | the surface height, again so long as $\eta > -H$. |
---|
[2282] | 827 | %end MOM doc %%% |
---|
| 828 | |
---|
[11435] | 829 | \newpage |
---|
[2282] | 830 | |
---|
| 831 | % ------------------------------------------------------------------------------------------------------------- |
---|
| 832 | % Terrain following coordinate System |
---|
| 833 | % ------------------------------------------------------------------------------------------------------------- |
---|
[9393] | 834 | \subsection{Curvilinear terrain-following \textit{s}--coordinate} |
---|
[11543] | 835 | \label{subsec:MB_sco} |
---|
[707] | 836 | |
---|
| 837 | % ------------------------------------------------------------------------------------------------------------- |
---|
| 838 | % Introduction |
---|
| 839 | % ------------------------------------------------------------------------------------------------------------- |
---|
[2282] | 840 | \subsubsection{Introduction} |
---|
[707] | 841 | |
---|
[10354] | 842 | Several important aspects of the ocean circulation are influenced by bottom topography. |
---|
| 843 | Of course, the most important is that bottom topography determines deep ocean sub-basins, barriers, sills and |
---|
| 844 | channels that strongly constrain the path of water masses, but more subtle effects exist. |
---|
| 845 | For example, the topographic $\beta$-effect is usually larger than the planetary one along continental slopes. |
---|
| 846 | Topographic Rossby waves can be excited and can interact with the mean current. |
---|
[11543] | 847 | In the $z$-coordinate system presented in the previous section (\autoref{sec:MB_zco}), |
---|
[10501] | 848 | $z$-surfaces are geopotential surfaces. |
---|
[10354] | 849 | The bottom topography is discretised by steps. |
---|
| 850 | This often leads to a misrepresentation of a gradually sloping bottom and to |
---|
| 851 | large localized depth gradients associated with large localized vertical velocities. |
---|
| 852 | The response to such a velocity field often leads to numerical dispersion effects. |
---|
| 853 | One solution to strongly reduce this error is to use a partial step representation of bottom topography instead of |
---|
[11123] | 854 | a full step one \cite{pacanowski.gnanadesikan_MWR98}. |
---|
[10501] | 855 | Another solution is to introduce a terrain-following coordinate system (hereafter $s$-coordinate). |
---|
[707] | 856 | |
---|
[10354] | 857 | The $s$-coordinate avoids the discretisation error in the depth field since the layers of |
---|
| 858 | computation are gradually adjusted with depth to the ocean bottom. |
---|
| 859 | Relatively small topographic features as well as gentle, large-scale slopes of the sea floor in the deep ocean, |
---|
| 860 | which would be ignored in typical $z$-model applications with the largest grid spacing at greatest depths, |
---|
| 861 | can easily be represented (with relatively low vertical resolution). |
---|
[10501] | 862 | A terrain-following model (hereafter $s$-model) also facilitates the modelling of the boundary layer flows over |
---|
[10354] | 863 | a large depth range, which in the framework of the $z$-model would require high vertical resolution over |
---|
| 864 | the whole depth range. |
---|
| 865 | Moreover, with a $s$-coordinate it is possible, at least in principle, to have the bottom and the sea surface as |
---|
| 866 | the only boundaries of the domain (no more lateral boundary condition to specify). |
---|
| 867 | Nevertheless, a $s$-coordinate also has its drawbacks. Perfectly adapted to a homogeneous ocean, |
---|
| 868 | it has strong limitations as soon as stratification is introduced. |
---|
| 869 | The main two problems come from the truncation error in the horizontal pressure gradient and |
---|
| 870 | a possibly increased diapycnal diffusion. |
---|
[11543] | 871 | The horizontal pressure force in $s$-coordinate consists of two terms (see \autoref{apdx:SCOORD}), |
---|
[707] | 872 | |
---|
[10414] | 873 | \begin{equation} |
---|
[11543] | 874 | \label{eq:MB_p_sco} |
---|
[11335] | 875 | \nabla p |_z = \nabla p |_s - \frac{1}{e_3} \pd[p]{s} \nabla z |_s |
---|
[707] | 876 | \end{equation} |
---|
| 877 | |
---|
[11543] | 878 | The second term in \autoref{eq:MB_p_sco} depends on the tilt of the coordinate surface and |
---|
[11335] | 879 | leads to a truncation error that is not present in a $z$-model. |
---|
[10501] | 880 | In the special case of a $\sigma$-coordinate (i.e. a depth-normalised coordinate system $\sigma = z/H$), |
---|
[11123] | 881 | \citet{haney_JPO91} and \citet{beckmann.haidvogel_JPO93} have given estimates of the magnitude of this truncation error. |
---|
[10354] | 882 | It depends on topographic slope, stratification, horizontal and vertical resolution, the equation of state, |
---|
| 883 | and the finite difference scheme. |
---|
| 884 | This error limits the possible topographic slopes that a model can handle at |
---|
| 885 | a given horizontal and vertical resolution. |
---|
| 886 | This is a severe restriction for large-scale applications using realistic bottom topography. |
---|
| 887 | The large-scale slopes require high horizontal resolution, and the computational cost becomes prohibitive. |
---|
| 888 | This problem can be at least partially overcome by mixing $s$-coordinate and |
---|
[11123] | 889 | step-like representation of bottom topography \citep{gerdes_JGR93*a,gerdes_JGR93*b,madec.delecluse.ea_JPO96}. |
---|
[10354] | 890 | However, the definition of the model domain vertical coordinate becomes then a non-trivial thing for |
---|
| 891 | a realistic bottom topography: |
---|
[11335] | 892 | an envelope topography is defined in $s$-coordinate on which a full or |
---|
[10354] | 893 | partial step bottom topography is then applied in order to adjust the model depth to the observed one |
---|
[11543] | 894 | (see \autoref{subsec:DOM_zgr}. |
---|
[707] | 895 | |
---|
[10501] | 896 | For numerical reasons a minimum of diffusion is required along the coordinate surfaces of |
---|
| 897 | any finite difference model. |
---|
[10354] | 898 | It causes spurious diapycnal mixing when coordinate surfaces do not coincide with isoneutral surfaces. |
---|
| 899 | This is the case for a $z$-model as well as for a $s$-model. |
---|
[10501] | 900 | However, density varies more strongly on $s$-surfaces than on horizontal surfaces in regions of |
---|
[10354] | 901 | large topographic slopes, implying larger diapycnal diffusion in a $s$-model than in a $z$-model. |
---|
| 902 | Whereas such a diapycnal diffusion in a $z$-model tends to weaken horizontal density (pressure) gradients and thus |
---|
| 903 | the horizontal circulation, it usually reinforces these gradients in a $s$-model, creating spurious circulation. |
---|
| 904 | For example, imagine an isolated bump of topography in an ocean at rest with a horizontally uniform stratification. |
---|
| 905 | Spurious diffusion along $s$-surfaces will induce a bump of isoneutral surfaces over the topography, |
---|
| 906 | and thus will generate there a baroclinic eddy. |
---|
| 907 | In contrast, the ocean will stay at rest in a $z$-model. |
---|
| 908 | As for the truncation error, the problem can be reduced by introducing the terrain-following coordinate below |
---|
[11435] | 909 | the strongly stratified portion of the water column (\ie\ the main thermocline) \citep{madec.delecluse.ea_JPO96}. |
---|
[10354] | 910 | An alternate solution consists of rotating the lateral diffusive tensor to geopotential or to isoneutral surfaces |
---|
[11543] | 911 | (see \autoref{subsec:MB_ldf}). |
---|
[10354] | 912 | Unfortunately, the slope of isoneutral surfaces relative to the $s$-surfaces can very large, |
---|
[10501] | 913 | strongly exceeding the stability limit of such a operator when it is discretized (see \autoref{chap:LDF}). |
---|
[707] | 914 | |
---|
[11123] | 915 | The $s$-coordinates introduced here \citep{lott.madec.ea_OM90,madec.delecluse.ea_JPO96} differ mainly in two aspects from |
---|
[10354] | 916 | similar models: |
---|
| 917 | it allows a representation of bottom topography with mixed full or partial step-like/terrain following topography; |
---|
| 918 | It also offers a completely general transformation, $s=s(i,j,z)$ for the vertical coordinate. |
---|
[707] | 919 | |
---|
| 920 | % ------------------------------------------------------------------------------------------------------------- |
---|
[2282] | 921 | % Curvilinear z-tilde coordinate System |
---|
[707] | 922 | % ------------------------------------------------------------------------------------------------------------- |
---|
[10501] | 923 | \subsection{\texorpdfstring{Curvilinear \ztilde-coordinate}{}} |
---|
[11543] | 924 | \label{subsec:MB_zco_tilde} |
---|
[707] | 925 | |
---|
[11123] | 926 | The \ztilde -coordinate has been developed by \citet{leclair.madec_OM11}. |
---|
[11435] | 927 | It is available in \NEMO\ since the version 3.4 and is more robust in version 4.0 than previously. |
---|
[10354] | 928 | Nevertheless, it is currently not robust enough to be used in all possible configurations. |
---|
| 929 | Its use is therefore not recommended. |
---|
[707] | 930 | |
---|
[11435] | 931 | \newpage |
---|
[10414] | 932 | |
---|
[707] | 933 | % ================================================================ |
---|
| 934 | % Subgrid Scale Physics |
---|
| 935 | % ================================================================ |
---|
[9393] | 936 | \section{Subgrid scale physics} |
---|
[11543] | 937 | \label{sec:MB_zdf_ldf} |
---|
[707] | 938 | |
---|
[11335] | 939 | The hydrostatic primitive equations describe the behaviour of a geophysical fluid at space and time scales larger than |
---|
[10354] | 940 | a few kilometres in the horizontal, a few meters in the vertical and a few minutes. |
---|
| 941 | They are usually solved at larger scales: the specified grid spacing and time step of the numerical model. |
---|
[10501] | 942 | The effects of smaller scale motions (coming from the advective terms in the Navier-Stokes equations) |
---|
| 943 | must be represented entirely in terms of large-scale patterns to close the equations. |
---|
[10354] | 944 | These effects appear in the equations as the divergence of turbulent fluxes |
---|
[11435] | 945 | (\ie\ fluxes associated with the mean correlation of small scale perturbations). |
---|
[10354] | 946 | Assuming a turbulent closure hypothesis is equivalent to choose a formulation for these fluxes. |
---|
| 947 | It is usually called the subgrid scale physics. |
---|
| 948 | It must be emphasized that this is the weakest part of the primitive equations, |
---|
| 949 | but also one of the most important for long-term simulations as |
---|
| 950 | small scale processes \textit{in fine} balance the surface input of kinetic energy and heat. |
---|
[707] | 951 | |
---|
[10354] | 952 | The control exerted by gravity on the flow induces a strong anisotropy between the lateral and vertical motions. |
---|
[10501] | 953 | Therefore subgrid-scale physics \textbf{D}$^{\vect U}$, $D^{S}$ and $D^{T}$ in |
---|
[11543] | 954 | \autoref{eq:MB_PE_dyn}, \autoref{eq:MB_PE_tra_T} and \autoref{eq:MB_PE_tra_S} are divided into |
---|
[10501] | 955 | a lateral part \textbf{D}$^{l \vect U}$, $D^{l S}$ and $D^{l T}$ and |
---|
| 956 | a vertical part \textbf{D}$^{v \vect U}$, $D^{v S}$ and $D^{v T}$. |
---|
[10354] | 957 | The formulation of these terms and their underlying physics are briefly discussed in the next two subsections. |
---|
[707] | 958 | |
---|
| 959 | % ------------------------------------------------------------------------------------------------------------- |
---|
| 960 | % Vertical Subgrid Scale Physics |
---|
| 961 | % ------------------------------------------------------------------------------------------------------------- |
---|
[9393] | 962 | \subsection{Vertical subgrid scale physics} |
---|
[11543] | 963 | \label{subsec:MB_zdf} |
---|
[707] | 964 | |
---|
[10354] | 965 | The model resolution is always larger than the scale at which the major sources of vertical turbulence occur |
---|
| 966 | (shear instability, internal wave breaking...). |
---|
| 967 | Turbulent motions are thus never explicitly solved, even partially, but always parameterized. |
---|
| 968 | The vertical turbulent fluxes are assumed to depend linearly on the gradients of large-scale quantities |
---|
[10501] | 969 | (for example, the turbulent heat flux is given by $\overline{T' w'} = -A^{v T} \partial_z \overline T$, |
---|
| 970 | where $A^{v T}$ is an eddy coefficient). |
---|
[10354] | 971 | This formulation is analogous to that of molecular diffusion and dissipation. |
---|
| 972 | This is quite clearly a necessary compromise: considering only the molecular viscosity acting on |
---|
| 973 | large scale severely underestimates the role of turbulent diffusion and dissipation, |
---|
| 974 | while an accurate consideration of the details of turbulent motions is simply impractical. |
---|
| 975 | The resulting vertical momentum and tracer diffusive operators are of second order: |
---|
[10414] | 976 | \begin{equation} |
---|
[11543] | 977 | \label{eq:MB_zdf} |
---|
[10501] | 978 | \begin{gathered} |
---|
| 979 | \vect D^{v \vect U} = \pd[]{z} \lt( A^{vm} \pd[\vect U_h]{z} \rt) \ , \\ |
---|
| 980 | D^{vT} = \pd[]{z} \lt( A^{vT} \pd[T]{z} \rt) \quad \text{and} \quad |
---|
| 981 | D^{vS} = \pd[]{z} \lt( A^{vT} \pd[S]{z} \rt) |
---|
| 982 | \end{gathered} |
---|
[707] | 983 | \end{equation} |
---|
[10354] | 984 | where $A^{vm}$ and $A^{vT}$ are the vertical eddy viscosity and diffusivity coefficients, respectively. |
---|
| 985 | At the sea surface and at the bottom, turbulent fluxes of momentum, heat and salt must be specified |
---|
| 986 | (see \autoref{chap:SBC} and \autoref{chap:ZDF} and \autoref{sec:TRA_bbl}). |
---|
| 987 | All the vertical physics is embedded in the specification of the eddy coefficients. |
---|
| 988 | They can be assumed to be either constant, or function of the local fluid properties |
---|
[11435] | 989 | (\eg\ Richardson number, Brunt-Vais\"{a}l\"{a} frequency, distance from the boundary ...), |
---|
[10354] | 990 | or computed from a turbulent closure model. |
---|
[11435] | 991 | The choices available in \NEMO\ are discussed in \autoref{chap:ZDF}). |
---|
[707] | 992 | |
---|
| 993 | % ------------------------------------------------------------------------------------------------------------- |
---|
| 994 | % Lateral Diffusive and Viscous Operators Formulation |
---|
| 995 | % ------------------------------------------------------------------------------------------------------------- |
---|
[9393] | 996 | \subsection{Formulation of the lateral diffusive and viscous operators} |
---|
[11543] | 997 | \label{subsec:MB_ldf} |
---|
[707] | 998 | |
---|
[10354] | 999 | Lateral turbulence can be roughly divided into a mesoscale turbulence associated with eddies |
---|
| 1000 | (which can be solved explicitly if the resolution is sufficient since |
---|
| 1001 | their underlying physics are included in the primitive equations), |
---|
| 1002 | and a sub mesoscale turbulence which is never explicitly solved even partially, but always parameterized. |
---|
| 1003 | The formulation of lateral eddy fluxes depends on whether the mesoscale is below or above the grid-spacing |
---|
[11435] | 1004 | (\ie\ the model is eddy-resolving or not). |
---|
[707] | 1005 | |
---|
[10354] | 1006 | In non-eddy-resolving configurations, the closure is similar to that used for the vertical physics. |
---|
| 1007 | The lateral turbulent fluxes are assumed to depend linearly on the lateral gradients of large-scale quantities. |
---|
| 1008 | The resulting lateral diffusive and dissipative operators are of second order. |
---|
| 1009 | Observations show that lateral mixing induced by mesoscale turbulence tends to be along isopycnal surfaces |
---|
[11123] | 1010 | (or more precisely neutral surfaces \cite{mcdougall_JPO87}) rather than across them. |
---|
[10354] | 1011 | As the slope of neutral surfaces is small in the ocean, a common approximation is to assume that |
---|
[11435] | 1012 | the `lateral' direction is the horizontal, \ie\ the lateral mixing is performed along geopotential surfaces. |
---|
[10354] | 1013 | This leads to a geopotential second order operator for lateral subgrid scale physics. |
---|
| 1014 | This assumption can be relaxed: the eddy-induced turbulent fluxes can be better approached by assuming that |
---|
| 1015 | they depend linearly on the gradients of large-scale quantities computed along neutral surfaces. |
---|
| 1016 | In such a case, the diffusive operator is an isoneutral second order operator and |
---|
| 1017 | it has components in the three space directions. |
---|
| 1018 | However, |
---|
[11435] | 1019 | both horizontal and isoneutral operators have no effect on mean (\ie\ large scale) potential energy whereas |
---|
[10354] | 1020 | potential energy is a main source of turbulence (through baroclinic instabilities). |
---|
[11335] | 1021 | \citet{gent.mcwilliams_JPO90} proposed a parameterisation of mesoscale eddy-induced turbulence which |
---|
[10354] | 1022 | associates an eddy-induced velocity to the isoneutral diffusion. |
---|
| 1023 | Its mean effect is to reduce the mean potential energy of the ocean. |
---|
| 1024 | This leads to a formulation of lateral subgrid-scale physics made up of an isoneutral second order operator and |
---|
| 1025 | an eddy induced advective part. |
---|
| 1026 | In all these lateral diffusive formulations, |
---|
| 1027 | the specification of the lateral eddy coefficients remains the problematic point as |
---|
| 1028 | there is no really satisfactory formulation of these coefficients as a function of large-scale features. |
---|
[707] | 1029 | |
---|
[10354] | 1030 | In eddy-resolving configurations, a second order operator can be used, |
---|
| 1031 | but usually the more scale selective biharmonic operator is preferred as |
---|
| 1032 | the grid-spacing is usually not small enough compared to the scale of the eddies. |
---|
| 1033 | The role devoted to the subgrid-scale physics is to dissipate the energy that |
---|
| 1034 | cascades toward the grid scale and thus to ensure the stability of the model while |
---|
| 1035 | not interfering with the resolved mesoscale activity. |
---|
| 1036 | Another approach is becoming more and more popular: |
---|
| 1037 | instead of specifying explicitly a sub-grid scale term in the momentum and tracer time evolution equations, |
---|
[11335] | 1038 | one uses an advective scheme which is diffusive enough to maintain the model stability. |
---|
[10354] | 1039 | It must be emphasised that then, all the sub-grid scale physics is included in the formulation of |
---|
[10501] | 1040 | the advection scheme. |
---|
[707] | 1041 | |
---|
[10354] | 1042 | All these parameterisations of subgrid scale physics have advantages and drawbacks. |
---|
[11335] | 1043 | They are not all available in \NEMO. For active tracers (temperature and salinity) the main ones are: |
---|
[10354] | 1044 | Laplacian and bilaplacian operators acting along geopotential or iso-neutral surfaces, |
---|
[11123] | 1045 | \citet{gent.mcwilliams_JPO90} parameterisation, and various slightly diffusive advection schemes. |
---|
[10354] | 1046 | For momentum, the main ones are: Laplacian and bilaplacian operators acting along geopotential surfaces, |
---|
| 1047 | and UBS advection schemes when flux form is chosen for the momentum advection. |
---|
[707] | 1048 | |
---|
[9393] | 1049 | \subsubsection{Lateral laplacian tracer diffusive operator} |
---|
[707] | 1050 | |
---|
[11543] | 1051 | The lateral Laplacian tracer diffusive operator is defined by (see \autoref{apdx:DIFFOPERS}): |
---|
[10414] | 1052 | \begin{equation} |
---|
[11543] | 1053 | \label{eq:MB_iso_tensor} |
---|
[10501] | 1054 | D^{lT} = \nabla \vect . \lt( A^{lT} \; \Re \; \nabla T \rt) \quad \text{with} \quad |
---|
| 1055 | \Re = |
---|
| 1056 | \begin{pmatrix} |
---|
| 1057 | 1 & 0 & -r_1 \\ |
---|
| 1058 | 0 & 1 & -r_2 \\ |
---|
| 1059 | -r_1 & -r_2 & r_1^2 + r_2^2 \\ |
---|
| 1060 | \end{pmatrix} |
---|
[707] | 1061 | \end{equation} |
---|
[10501] | 1062 | where $r_1$ and $r_2$ are the slopes between the surface along which the diffusive operator acts and |
---|
[11435] | 1063 | the model level (\eg\ $z$- or $s$-surfaces). |
---|
[11543] | 1064 | Note that the formulation \autoref{eq:MB_iso_tensor} is exact for |
---|
[10354] | 1065 | the rotation between geopotential and $s$-surfaces, |
---|
| 1066 | while it is only an approximation for the rotation between isoneutral and $z$- or $s$-surfaces. |
---|
[11543] | 1067 | Indeed, in the latter case, two assumptions are made to simplify \autoref{eq:MB_iso_tensor} \citep{cox_OM87}. |
---|
[10354] | 1068 | First, the horizontal contribution of the dianeutral mixing is neglected since the ratio between iso and |
---|
| 1069 | dia-neutral diffusive coefficients is known to be several orders of magnitude smaller than unity. |
---|
| 1070 | Second, the two isoneutral directions of diffusion are assumed to be independent since |
---|
[11543] | 1071 | the slopes are generally less than $10^{-2}$ in the ocean (see \autoref{apdx:DIFFOPERS}). |
---|
[707] | 1072 | |
---|
[10354] | 1073 | For \textit{iso-level} diffusion, $r_1$ and $r_2 $ are zero. |
---|
[10501] | 1074 | $\Re$ reduces to the identity in the horizontal direction, no rotation is applied. |
---|
[6140] | 1075 | |
---|
[10354] | 1076 | For \textit{geopotential} diffusion, |
---|
| 1077 | $r_1$ and $r_2 $ are the slopes between the geopotential and computational surfaces: |
---|
[11543] | 1078 | they are equal to $\sigma_1$ and $\sigma_2$, respectively (see \autoref{eq:MB_sco_slope}). |
---|
[707] | 1079 | |
---|
[10354] | 1080 | For \textit{isoneutral} diffusion $r_1$ and $r_2$ are the slopes between the isoneutral and computational surfaces. |
---|
| 1081 | Therefore, they are different quantities, but have similar expressions in $z$- and $s$-coordinates. |
---|
| 1082 | In $z$-coordinates: |
---|
[10414] | 1083 | \begin{equation} |
---|
[11543] | 1084 | \label{eq:MB_iso_slopes} |
---|
[10501] | 1085 | r_1 = \frac{e_3}{e_1} \lt( \pd[\rho]{i} \rt) \lt( \pd[\rho]{k} \rt)^{-1} \quad |
---|
| 1086 | r_2 = \frac{e_3}{e_2} \lt( \pd[\rho]{j} \rt) \lt( \pd[\rho]{k} \rt)^{-1} |
---|
[707] | 1087 | \end{equation} |
---|
[6289] | 1088 | while in $s$-coordinates $\pd[]{k}$ is replaced by $\pd[]{s}$. |
---|
[707] | 1089 | |
---|
[3294] | 1090 | \subsubsection{Eddy induced velocity} |
---|
[10501] | 1091 | |
---|
[11123] | 1092 | When the \textit{eddy induced velocity} parametrisation (eiv) \citep{gent.mcwilliams_JPO90} is used, |
---|
[1224] | 1093 | an additional tracer advection is introduced in combination with the isoneutral diffusion of tracers: |
---|
[10414] | 1094 | \[ |
---|
[11543] | 1095 | % \label{eq:MB_iso+eiv} |
---|
[10501] | 1096 | D^{lT} = \nabla \cdot \lt( A^{lT} \; \Re \; \nabla T \rt) + \nabla \cdot \lt( \vect U^\ast \, T \rt) |
---|
[10414] | 1097 | \] |
---|
[10501] | 1098 | where $ \vect U^\ast = \lt( u^\ast,v^\ast,w^\ast \rt)$ is a non-divergent, |
---|
[1224] | 1099 | eddy-induced transport velocity. This velocity field is defined by: |
---|
[10544] | 1100 | \begin{gather} |
---|
[11543] | 1101 | % \label{eq:MB_eiv} |
---|
[10544] | 1102 | u^\ast = \frac{1}{e_3} \pd[]{k} \lt( A^{eiv} \; \tilde{r}_1 \rt) \\ |
---|
| 1103 | v^\ast = \frac{1}{e_3} \pd[]{k} \lt( A^{eiv} \; \tilde{r}_2 \rt) \\ |
---|
| 1104 | w^\ast = - \frac{1}{e_1 e_2} \lt[ \pd[]{i} \lt( A^{eiv} \; e_2 \, \tilde{r}_1 \rt) |
---|
[10501] | 1105 | + \pd[]{j} \lt( A^{eiv} \; e_1 \, \tilde{r}_2 \rt) \rt] |
---|
[10544] | 1106 | \end{gather} |
---|
[10354] | 1107 | where $A^{eiv}$ is the eddy induced velocity coefficient |
---|
| 1108 | (or equivalently the isoneutral thickness diffusivity coefficient), |
---|
[10501] | 1109 | and $\tilde r_1$ and $\tilde r_2$ are the slopes between isoneutral and \textit{geopotential} surfaces. |
---|
[11435] | 1110 | Their values are thus independent of the vertical coordinate, but their expression depends on the coordinate: |
---|
[10414] | 1111 | \begin{align} |
---|
[11543] | 1112 | \label{eq:MB_slopes_eiv} |
---|
[10414] | 1113 | \tilde{r}_n = |
---|
[10501] | 1114 | \begin{cases} |
---|
| 1115 | r_n & \text{in $z$-coordinate} \\ |
---|
| 1116 | r_n + \sigma_n & \text{in \zstar- and $s$-coordinates} |
---|
| 1117 | \end{cases} |
---|
| 1118 | \quad \text{where~} n = 1, 2 |
---|
[707] | 1119 | \end{align} |
---|
| 1120 | |
---|
[10354] | 1121 | The normal component of the eddy induced velocity is zero at all the boundaries. |
---|
[10501] | 1122 | This can be achieved in a model by tapering either the eddy coefficient or the slopes to zero in the vicinity of |
---|
| 1123 | the boundaries. |
---|
[11435] | 1124 | The latter strategy is used in \NEMO\ (cf. \autoref{chap:LDF}). |
---|
[707] | 1125 | |
---|
[6289] | 1126 | \subsubsection{Lateral bilaplacian tracer diffusive operator} |
---|
[707] | 1127 | |
---|
[6289] | 1128 | The lateral bilaplacian tracer diffusive operator is defined by: |
---|
[10414] | 1129 | \[ |
---|
[11543] | 1130 | % \label{eq:MB_bilapT} |
---|
[10501] | 1131 | D^{lT}= - \Delta \; (\Delta T) \quad \text{where} \quad |
---|
| 1132 | \Delta \bullet = \nabla \lt( \sqrt{B^{lT}} \; \Re \; \nabla \bullet \rt) |
---|
[10414] | 1133 | \] |
---|
[11543] | 1134 | It is the Laplacian operator given by \autoref{eq:MB_iso_tensor} applied twice with |
---|
[10501] | 1135 | the harmonic eddy diffusion coefficient set to the square root of the biharmonic one. |
---|
[707] | 1136 | |
---|
[6289] | 1137 | \subsubsection{Lateral Laplacian momentum diffusive operator} |
---|
[707] | 1138 | |
---|
[10354] | 1139 | The Laplacian momentum diffusive operator along $z$- or $s$-surfaces is found by |
---|
[11543] | 1140 | applying \autoref{eq:MB_lap_vector} to the horizontal velocity vector (see \autoref{apdx:DIFFOPERS}): |
---|
[10501] | 1141 | \begin{align*} |
---|
[11543] | 1142 | % \label{eq:MB_lapU} |
---|
[10501] | 1143 | \vect D^{l \vect U} &= \nabla_h \big( A^{lm} \chi \big) |
---|
| 1144 | - \nabla_h \times \big( A^{lm} \, \zeta \; \vect k \big) \\ |
---|
| 1145 | &= \lt( \frac{1}{e_1} \pd[ \lt( A^{lm} \chi \rt) ]{i} \rt. |
---|
[11435] | 1146 | - \frac{1}{e_2 e_3} \pd[ \lt( A^{lm} \; e_3 \zeta \rt) ]{j} , |
---|
[10501] | 1147 | \frac{1}{e_2} \pd[ \lt( A^{lm} \chi \rt) ]{j} |
---|
| 1148 | \lt. + \frac{1}{e_1 e_3} \pd[ \lt( A^{lm} \; e_3 \zeta \rt) ]{i} \rt) |
---|
| 1149 | \end{align*} |
---|
[707] | 1150 | |
---|
[10354] | 1151 | Such a formulation ensures a complete separation between the vorticity and horizontal divergence fields |
---|
[11543] | 1152 | (see \autoref{apdx:INVARIANTS}). |
---|
[10354] | 1153 | Unfortunately, it is only available in \textit{iso-level} direction. |
---|
| 1154 | When a rotation is required |
---|
[11435] | 1155 | (\ie\ geopotential diffusion in $s$-coordinates or isoneutral diffusion in both $z$- and $s$-coordinates), |
---|
[10501] | 1156 | the $u$ and $v$-fields are considered as independent scalar fields, so that the diffusive operator is given by: |
---|
| 1157 | \begin{gather*} |
---|
[11543] | 1158 | % \label{eq:MB_lapU_iso} |
---|
[10501] | 1159 | D_u^{l \vect U} = \nabla . \lt( A^{lm} \; \Re \; \nabla u \rt) \\ |
---|
| 1160 | D_v^{l \vect U} = \nabla . \lt( A^{lm} \; \Re \; \nabla v \rt) |
---|
| 1161 | \end{gather*} |
---|
[11543] | 1162 | where $\Re$ is given by \autoref{eq:MB_iso_tensor}. |
---|
[10354] | 1163 | It is the same expression as those used for diffusive operator on tracers. |
---|
| 1164 | It must be emphasised that such a formulation is only exact in a Cartesian coordinate system, |
---|
[11435] | 1165 | \ie\ on a $f$- or $\beta$-plane, not on the sphere. |
---|
[10354] | 1166 | It is also a very good approximation in vicinity of the Equator in |
---|
[11123] | 1167 | a geographical coordinate system \citep{lengaigne.madec.ea_JGR03}. |
---|
[707] | 1168 | |
---|
[11335] | 1169 | \subsubsection{Lateral bilaplacian momentum diffusive operator} |
---|
[707] | 1170 | |
---|
[10354] | 1171 | As for tracers, the bilaplacian order momentum diffusive operator is a re-entering Laplacian operator with |
---|
| 1172 | the harmonic eddy diffusion coefficient set to the square root of the biharmonic one. |
---|
| 1173 | Nevertheless it is currently not available in the iso-neutral case. |
---|
[707] | 1174 | |
---|
[10414] | 1175 | \biblio |
---|
| 1176 | |
---|
[10442] | 1177 | \pindex |
---|
| 1178 | |
---|
[6997] | 1179 | \end{document} |
---|