[10414] | 1 | \documentclass[../main/NEMO_manual]{subfiles} |
---|
| 2 | |
---|
[6997] | 3 | \begin{document} |
---|
[707] | 4 | % ================================================================ |
---|
[10414] | 5 | % Chapter 1 Model Basics |
---|
[707] | 6 | % ================================================================ |
---|
| 7 | |
---|
[9393] | 8 | \chapter{Model Basics} |
---|
[9407] | 9 | \label{chap:PE} |
---|
[10414] | 10 | |
---|
[707] | 11 | \minitoc |
---|
| 12 | |
---|
[2282] | 13 | \newpage |
---|
| 14 | |
---|
[707] | 15 | % ================================================================ |
---|
| 16 | % Primitive Equations |
---|
| 17 | % ================================================================ |
---|
[9393] | 18 | \section{Primitive equations} |
---|
[9407] | 19 | \label{sec:PE_PE} |
---|
[707] | 20 | |
---|
| 21 | % ------------------------------------------------------------------------------------------------------------- |
---|
| 22 | % Vector Invariant Formulation |
---|
| 23 | % ------------------------------------------------------------------------------------------------------------- |
---|
| 24 | |
---|
[9393] | 25 | \subsection{Vector invariant formulation} |
---|
[9407] | 26 | \label{subsec:PE_Vector} |
---|
[707] | 27 | |
---|
| 28 | |
---|
[10354] | 29 | The ocean is a fluid that can be described to a good approximation by the primitive equations, |
---|
| 30 | $i.e.$ the Navier-Stokes equations along with a nonlinear equation of state which |
---|
| 31 | couples the two active tracers (temperature and salinity) to the fluid velocity, |
---|
| 32 | plus the following additional assumptions made from scale considerations: |
---|
[707] | 33 | |
---|
[10354] | 34 | \textit{(1) spherical earth approximation:} the geopotential surfaces are assumed to be spheres so that |
---|
| 35 | gravity (local vertical) is parallel to the earth's radius |
---|
[707] | 36 | |
---|
[10354] | 37 | \textit{(2) thin-shell approximation:} the ocean depth is neglected compared to the earth's radius |
---|
[707] | 38 | |
---|
[10354] | 39 | \textit{(3) turbulent closure hypothesis:} the turbulent fluxes |
---|
| 40 | (which represent the effect of small scale processes on the large-scale) are expressed in terms of |
---|
| 41 | large-scale features |
---|
[707] | 42 | |
---|
[10354] | 43 | \textit{(4) Boussinesq hypothesis:} density variations are neglected except in their contribution to |
---|
| 44 | the buoyancy force |
---|
[707] | 45 | |
---|
[10354] | 46 | \textit{(5) Hydrostatic hypothesis:} the vertical momentum equation is reduced to a balance between |
---|
| 47 | the vertical pressure gradient and the buoyancy force |
---|
| 48 | (this removes convective processes from the initial Navier-Stokes equations and so |
---|
| 49 | convective processes must be parameterized instead) |
---|
[707] | 50 | |
---|
[10354] | 51 | \textit{(6) Incompressibility hypothesis:} the three dimensional divergence of the velocity vector is assumed to |
---|
| 52 | be zero. |
---|
[707] | 53 | |
---|
[10354] | 54 | Because the gravitational force is so dominant in the equations of large-scale motions, |
---|
| 55 | it is useful to choose an orthogonal set of unit vectors (\textbf{i},\textbf{j},\textbf{k}) linked to |
---|
| 56 | the earth such that \textbf{k} is the local upward vector and (\textbf{i},\textbf{j}) are two vectors orthogonal to |
---|
| 57 | \textbf{k}, $i.e.$ tangent to the geopotential surfaces. |
---|
| 58 | Let us define the following variables: \textbf{U} the vector velocity, $\textbf{U}=\textbf{U}_h + w\, \textbf{k}$ |
---|
[1224] | 59 | (the subscript $h$ denotes the local horizontal vector, $i.e.$ over the (\textbf{i},\textbf{j}) plane), |
---|
[10354] | 60 | $T$ the potential temperature, $S$ the salinity, \textit{$\rho $} the \textit{in situ} density. |
---|
| 61 | The vector invariant form of the primitive equations in the (\textbf{i},\textbf{j},\textbf{k}) vector system |
---|
| 62 | provides the following six equations |
---|
| 63 | (namely the momentum balance, the hydrostatic equilibrium, the incompressibility equation, |
---|
| 64 | the heat and salt conservation equations and an equation of state): |
---|
[10414] | 65 | \begin{subequations} |
---|
| 66 | \label{eq:PE} |
---|
| 67 | \begin{equation} |
---|
| 68 | \label{eq:PE_dyn} |
---|
| 69 | \frac{\partial {\rm {\bf U}}_h }{\partial t}= |
---|
| 70 | -\left[ {\left( {\nabla \times {\rm {\bf U}}} \right)\times {\rm {\bf U}} |
---|
| 71 | +\frac{1}{2}\nabla \left( {{\rm {\bf U}}^2} \right)} \right]_h |
---|
| 72 | -f\;{\rm {\bf k}}\times {\rm {\bf U}}_h |
---|
| 73 | -\frac{1}{\rho_o }\nabla _h p + {\rm {\bf D}}^{\rm {\bf U}} + {\rm {\bf F}}^{\rm {\bf U}} |
---|
[707] | 74 | \end{equation} |
---|
[10414] | 75 | \begin{equation} |
---|
| 76 | \label{eq:PE_hydrostatic} |
---|
| 77 | \frac{\partial p }{\partial z} = - \rho \ g |
---|
[707] | 78 | \end{equation} |
---|
[10414] | 79 | \begin{equation} |
---|
| 80 | \label{eq:PE_continuity} |
---|
| 81 | \nabla \cdot {\bf U}= 0 |
---|
[707] | 82 | \end{equation} |
---|
[10414] | 83 | \begin{equation} |
---|
| 84 | \label{eq:PE_tra_T} |
---|
| 85 | \frac{\partial T}{\partial t} = - \nabla \cdot \left( T \ \rm{\bf U} \right) + D^T + F^T |
---|
[707] | 86 | \end{equation} |
---|
[10414] | 87 | \begin{equation} |
---|
| 88 | \label{eq:PE_tra_S} |
---|
| 89 | \frac{\partial S}{\partial t} = - \nabla \cdot \left( S \ \rm{\bf U} \right) + D^S + F^S |
---|
[707] | 90 | \end{equation} |
---|
[10414] | 91 | \begin{equation} |
---|
| 92 | \label{eq:PE_eos} |
---|
| 93 | \rho = \rho \left( T,S,p \right) |
---|
[707] | 94 | \end{equation} |
---|
| 95 | \end{subequations} |
---|
[10354] | 96 | where $\nabla$ is the generalised derivative vector operator in $(\bf i,\bf j, \bf k)$ directions, $t$ is the time, |
---|
| 97 | $z$ is the vertical coordinate, $\rho $ is the \textit{in situ} density given by the equation of state |
---|
| 98 | (\autoref{eq:PE_eos}), $\rho_o$ is a reference density, $p$ the pressure, |
---|
| 99 | $f=2 \bf \Omega \cdot \bf k$ is the Coriolis acceleration |
---|
| 100 | (where $\bf \Omega$ is the Earth's angular velocity vector), and $g$ is the gravitational acceleration. |
---|
| 101 | ${\rm {\bf D}}^{\rm {\bf U}}$, $D^T$ and $D^S$ are the parameterisations of small-scale physics for momentum, |
---|
| 102 | temperature and salinity, and ${\rm {\bf F}}^{\rm {\bf U}}$, $F^T$ and $F^S$ surface forcing terms. |
---|
| 103 | Their nature and formulation are discussed in \autoref{sec:PE_zdf_ldf} and \autoref{subsec:PE_boundary_condition}. |
---|
[707] | 104 | |
---|
| 105 | |
---|
[10354] | 106 | |
---|
[707] | 107 | % ------------------------------------------------------------------------------------------------------------- |
---|
| 108 | % Boundary condition |
---|
| 109 | % ------------------------------------------------------------------------------------------------------------- |
---|
[9393] | 110 | \subsection{Boundary conditions} |
---|
[9407] | 111 | \label{subsec:PE_boundary_condition} |
---|
[707] | 112 | |
---|
[10354] | 113 | An ocean is bounded by complex coastlines, bottom topography at its base and |
---|
| 114 | an air-sea or ice-sea interface at its top. |
---|
| 115 | These boundaries can be defined by two surfaces, $z=-H(i,j)$ and $z=\eta(i,j,k,t)$, |
---|
| 116 | where $H$ is the depth of the ocean bottom and $\eta$ is the height of the sea surface. |
---|
| 117 | Both $H$ and $\eta$ are usually referenced to a given surface, $z=0$, chosen as a mean sea surface |
---|
| 118 | (\autoref{fig:ocean_bc}). |
---|
| 119 | Through these two boundaries, the ocean can exchange fluxes of heat, fresh water, salt, and momentum with |
---|
| 120 | the solid earth, the continental margins, the sea ice and the atmosphere. |
---|
| 121 | However, some of these fluxes are so weak that even on climatic time scales of thousands of years |
---|
| 122 | they can be neglected. |
---|
| 123 | In the following, we briefly review the fluxes exchanged at the interfaces between the ocean and |
---|
| 124 | the other components of the earth system. |
---|
[707] | 125 | |
---|
| 126 | %>>>>>>>>>>>>>>>>>>>>>>>>>>>> |
---|
[10354] | 127 | \begin{figure}[!ht] |
---|
| 128 | \begin{center} |
---|
| 129 | \includegraphics[width=0.90\textwidth]{Fig_I_ocean_bc} |
---|
| 130 | \caption{ \protect\label{fig:ocean_bc} |
---|
| 131 | The ocean is bounded by two surfaces, $z=-H(i,j)$ and $z=\eta(i,j,t)$, |
---|
| 132 | where $H$ is the depth of the sea floor and $\eta$ the height of the sea surface. |
---|
| 133 | Both $H$ and $\eta$ are referenced to $z=0$. |
---|
| 134 | } |
---|
| 135 | \end{center} |
---|
| 136 | \end{figure} |
---|
[707] | 137 | %>>>>>>>>>>>>>>>>>>>>>>>>>>>> |
---|
| 138 | |
---|
[817] | 139 | |
---|
[707] | 140 | \begin{description} |
---|
[10354] | 141 | \item[Land - ocean interface:] |
---|
| 142 | the major flux between continental margins and the ocean is a mass exchange of fresh water through river runoff. |
---|
| 143 | Such an exchange modifies the sea surface salinity especially in the vicinity of major river mouths. |
---|
| 144 | It can be neglected for short range integrations but has to be taken into account for long term integrations as |
---|
| 145 | it influences the characteristics of water masses formed (especially at high latitudes). |
---|
| 146 | It is required in order to close the water cycle of the climate system. |
---|
| 147 | It is usually specified as a fresh water flux at the air-sea interface in the vicinity of river mouths. |
---|
| 148 | \item[Solid earth - ocean interface:] |
---|
| 149 | heat and salt fluxes through the sea floor are small, except in special areas of little extent. |
---|
[10414] | 150 | They are usually neglected in the model |
---|
| 151 | \footnote{ |
---|
[10354] | 152 | In fact, it has been shown that the heat flux associated with the solid Earth cooling |
---|
| 153 | ($i.e.$the geothermal heating) is not negligible for the thermohaline circulation of the world ocean |
---|
| 154 | (see \autoref{subsec:TRA_bbc}). |
---|
| 155 | }. |
---|
| 156 | The boundary condition is thus set to no flux of heat and salt across solid boundaries. |
---|
| 157 | For momentum, the situation is different. There is no flow across solid boundaries, |
---|
| 158 | $i.e.$ the velocity normal to the ocean bottom and coastlines is zero (in other words, |
---|
| 159 | the bottom velocity is parallel to solid boundaries). This kinematic boundary condition |
---|
| 160 | can be expressed as: |
---|
[10414] | 161 | \begin{equation} |
---|
| 162 | \label{eq:PE_w_bbc} |
---|
[10354] | 163 | w = -{\rm {\bf U}}_h \cdot \nabla _h \left( H \right) |
---|
| 164 | \end{equation} |
---|
| 165 | In addition, the ocean exchanges momentum with the earth through frictional processes. |
---|
| 166 | Such momentum transfer occurs at small scales in a boundary layer. |
---|
| 167 | It must be parameterized in terms of turbulent fluxes using bottom and/or lateral boundary conditions. |
---|
| 168 | Its specification depends on the nature of the physical parameterisation used for |
---|
| 169 | ${\rm {\bf D}}^{\rm {\bf U}}$ in \autoref{eq:PE_dyn}. |
---|
| 170 | It is discussed in \autoref{eq:PE_zdf}.% and Chap. III.6 to 9. |
---|
| 171 | \item[Atmosphere - ocean interface:] |
---|
| 172 | the kinematic surface condition plus the mass flux of fresh water PE (the precipitation minus evaporation budget) |
---|
| 173 | leads to: |
---|
[10414] | 174 | \[ |
---|
| 175 | % \label{eq:PE_w_sbc} |
---|
[10354] | 176 | w = \frac{\partial \eta }{\partial t} |
---|
| 177 | + \left. {{\rm {\bf U}}_h } \right|_{z=\eta } \cdot \nabla _h \left( \eta \right) |
---|
[707] | 178 | + P-E |
---|
[10414] | 179 | \] |
---|
[10354] | 180 | The dynamic boundary condition, neglecting the surface tension (which removes capillary waves from the system) |
---|
| 181 | leads to the continuity of pressure across the interface $z=\eta$. |
---|
| 182 | The atmosphere and ocean also exchange horizontal momentum (wind stress), and heat. |
---|
| 183 | \item[Sea ice - ocean interface:] |
---|
| 184 | the ocean and sea ice exchange heat, salt, fresh water and momentum. |
---|
| 185 | The sea surface temperature is constrained to be at the freezing point at the interface. |
---|
| 186 | Sea ice salinity is very low ($\sim4-6 \,psu$) compared to those of the ocean ($\sim34 \,psu$). |
---|
| 187 | The cycle of freezing/melting is associated with fresh water and salt fluxes that cannot be neglected. |
---|
[707] | 188 | \end{description} |
---|
| 189 | |
---|
| 190 | |
---|
[2376] | 191 | %\newpage |
---|
[2282] | 192 | |
---|
[707] | 193 | % ================================================================ |
---|
| 194 | % The Horizontal Pressure Gradient |
---|
| 195 | % ================================================================ |
---|
[9393] | 196 | \section{Horizontal pressure gradient } |
---|
[9407] | 197 | \label{sec:PE_hor_pg} |
---|
[707] | 198 | |
---|
| 199 | % ------------------------------------------------------------------------------------------------------------- |
---|
| 200 | % Pressure Formulation |
---|
| 201 | % ------------------------------------------------------------------------------------------------------------- |
---|
[9393] | 202 | \subsection{Pressure formulation} |
---|
[9407] | 203 | \label{subsec:PE_p_formulation} |
---|
[707] | 204 | |
---|
[10354] | 205 | The total pressure at a given depth $z$ is composed of a surface pressure $p_s$ at |
---|
| 206 | a reference geopotential surface ($z=0$) and a hydrostatic pressure $p_h$ such that: |
---|
| 207 | $p(i,j,k,t)=p_s(i,j,t)+p_h(i,j,k,t)$. |
---|
| 208 | The latter is computed by integrating (\autoref{eq:PE_hydrostatic}), |
---|
| 209 | assuming that pressure in decibars can be approximated by depth in meters in (\autoref{eq:PE_eos}). |
---|
[1224] | 210 | The hydrostatic pressure is then given by: |
---|
[10414] | 211 | \[ |
---|
| 212 | % \label{eq:PE_pressure} |
---|
| 213 | p_h \left( {i,j,z,t} \right) |
---|
| 214 | = \int_{\varsigma =z}^{\varsigma =0} {g\;\rho \left( {T,S,\varsigma} \right)\;d\varsigma } |
---|
| 215 | \] |
---|
[10354] | 216 | Two strategies can be considered for the surface pressure term: |
---|
| 217 | $(a)$ introduce of a new variable $\eta$, the free-surface elevation, |
---|
| 218 | for which a prognostic equation can be established and solved; |
---|
| 219 | $(b)$ assume that the ocean surface is a rigid lid, |
---|
| 220 | on which the pressure (or its horizontal gradient) can be diagnosed. |
---|
| 221 | When the former strategy is used, one solution of the free-surface elevation consists of |
---|
| 222 | the excitation of external gravity waves. |
---|
| 223 | The flow is barotropic and the surface moves up and down with gravity as the restoring force. |
---|
| 224 | The phase speed of such waves is high (some hundreds of metres per second) so that |
---|
| 225 | the time step would have to be very short if they were present in the model. |
---|
| 226 | The latter strategy filters out these waves since the rigid lid approximation implies $\eta=0$, |
---|
| 227 | $i.e.$ the sea surface is the surface $z=0$. |
---|
| 228 | This well known approximation increases the surface wave speed to infinity and |
---|
| 229 | modifies certain other longwave dynamics ($e.g.$ barotropic Rossby or planetary waves). |
---|
| 230 | The rigid-lid hypothesis is an obsolescent feature in modern OGCMs. |
---|
| 231 | It has been available until the release 3.1 of \NEMO, and it has been removed in release 3.2 and followings. |
---|
| 232 | Only the free surface formulation is now described in the this document (see the next sub-section). |
---|
[707] | 233 | |
---|
| 234 | % ------------------------------------------------------------------------------------------------------------- |
---|
| 235 | % Free Surface Formulation |
---|
| 236 | % ------------------------------------------------------------------------------------------------------------- |
---|
[9393] | 237 | \subsection{Free surface formulation} |
---|
[9407] | 238 | \label{subsec:PE_free_surface} |
---|
[707] | 239 | |
---|
[10354] | 240 | In the free surface formulation, a variable $\eta$, the sea-surface height, |
---|
| 241 | is introduced which describes the shape of the air-sea interface. |
---|
| 242 | This variable is solution of a prognostic equation which is established by forming the vertical average of |
---|
| 243 | the kinematic surface condition (\autoref{eq:PE_w_bbc}): |
---|
[10414] | 244 | \begin{equation} |
---|
| 245 | \label{eq:PE_ssh} |
---|
| 246 | \frac{\partial \eta }{\partial t}=-D+P-E |
---|
| 247 | \quad \text{where} \ |
---|
| 248 | D=\nabla \cdot \left[ {\left( {H+\eta } \right) \; {\rm{\bf \overline{U}}}_h \,} \right] |
---|
[707] | 249 | \end{equation} |
---|
[9407] | 250 | and using (\autoref{eq:PE_hydrostatic}) the surface pressure is given by: $p_s = \rho \, g \, \eta$. |
---|
[707] | 251 | |
---|
[10354] | 252 | Allowing the air-sea interface to move introduces the external gravity waves (EGWs) as |
---|
| 253 | a class of solution of the primitive equations. |
---|
| 254 | These waves are barotropic because of hydrostatic assumption, and their phase speed is quite high. |
---|
| 255 | Their time scale is short with respect to the other processes described by the primitive equations. |
---|
[707] | 256 | |
---|
[10354] | 257 | Two choices can be made regarding the implementation of the free surface in the model, |
---|
[1224] | 258 | depending on the physical processes of interest. |
---|
[707] | 259 | |
---|
[10354] | 260 | $\bullet$ If one is interested in EGWs, in particular the tides and their interaction with |
---|
| 261 | the baroclinic structure of the ocean (internal waves) possibly in shallow seas, |
---|
| 262 | then a non linear free surface is the most appropriate. |
---|
| 263 | This means that no approximation is made in (\autoref{eq:PE_ssh}) and that |
---|
| 264 | the variation of the ocean volume is fully taken into account. |
---|
| 265 | Note that in order to study the fast time scales associated with EGWs it is necessary to |
---|
| 266 | minimize time filtering effects |
---|
| 267 | (use an explicit time scheme with very small time step, or a split-explicit scheme with reasonably small time step, |
---|
| 268 | see \autoref{subsec:DYN_spg_exp} or \autoref{subsec:DYN_spg_ts}). |
---|
[707] | 269 | |
---|
[10354] | 270 | $\bullet$ If one is not interested in EGW but rather sees them as high frequency noise, |
---|
| 271 | it is possible to apply an explicit filter to slow down the fastest waves while |
---|
| 272 | not altering the slow barotropic Rossby waves. |
---|
| 273 | If further, an approximative conservation of heat and salt contents is sufficient for the problem solved, |
---|
| 274 | then it is sufficient to solve a linearized version of (\autoref{eq:PE_ssh}), |
---|
| 275 | which still allows to take into account freshwater fluxes applied at the ocean surface \citep{Roullet_Madec_JGR00}. |
---|
[6140] | 276 | Nevertheless, with the linearization, an exact conservation of heat and salt contents is lost. |
---|
[707] | 277 | |
---|
[10354] | 278 | The filtering of EGWs in models with a free surface is usually a matter of discretisation of |
---|
| 279 | the temporal derivatives, |
---|
| 280 | using a split-explicit method \citep{Killworth_al_JPO91, Zhang_Endoh_JGR92} or |
---|
| 281 | the implicit scheme \citep{Dukowicz1994} or |
---|
| 282 | the addition of a filtering force in the momentum equation \citep{Roullet_Madec_JGR00}. |
---|
| 283 | With the present release, \NEMO offers the choice between |
---|
| 284 | an explicit free surface (see \autoref{subsec:DYN_spg_exp}) or |
---|
| 285 | a split-explicit scheme strongly inspired the one proposed by \citet{Shchepetkin_McWilliams_OM05} |
---|
| 286 | (see \autoref{subsec:DYN_spg_ts}). |
---|
[707] | 287 | |
---|
[6140] | 288 | %\newpage |
---|
[707] | 289 | |
---|
| 290 | % ================================================================ |
---|
| 291 | % Curvilinear z-coordinate System |
---|
| 292 | % ================================================================ |
---|
[9393] | 293 | \section{Curvilinear \textit{z-}coordinate system} |
---|
[9407] | 294 | \label{sec:PE_zco} |
---|
[707] | 295 | |
---|
| 296 | |
---|
| 297 | % ------------------------------------------------------------------------------------------------------------- |
---|
| 298 | % Tensorial Formalism |
---|
| 299 | % ------------------------------------------------------------------------------------------------------------- |
---|
[9393] | 300 | \subsection{Tensorial formalism} |
---|
[9407] | 301 | \label{subsec:PE_tensorial} |
---|
[707] | 302 | |
---|
[10354] | 303 | In many ocean circulation problems, the flow field has regions of enhanced dynamics |
---|
| 304 | ($i.e.$ surface layers, western boundary currents, equatorial currents, or ocean fronts). |
---|
| 305 | The representation of such dynamical processes can be improved by |
---|
| 306 | specifically increasing the model resolution in these regions. |
---|
| 307 | As well, it may be convenient to use a lateral boundary-following coordinate system to |
---|
| 308 | better represent coastal dynamics. |
---|
| 309 | Moreover, the common geographical coordinate system has a singular point at the North Pole that |
---|
| 310 | cannot be easily treated in a global model without filtering. |
---|
| 311 | A solution consists of introducing an appropriate coordinate transformation that |
---|
| 312 | shifts the singular point onto land \citep{Madec_Imbard_CD96, Murray_JCP96}. |
---|
| 313 | As a consequence, it is important to solve the primitive equations in various curvilinear coordinate systems. |
---|
| 314 | An efficient way of introducing an appropriate coordinate transform can be found when using a tensorial formalism. |
---|
| 315 | This formalism is suited to any multidimensional curvilinear coordinate system. |
---|
| 316 | Ocean modellers mainly use three-dimensional orthogonal grids on the sphere (spherical earth approximation), |
---|
| 317 | with preservation of the local vertical. Here we give the simplified equations for this particular case. |
---|
| 318 | The general case is detailed by \citet{Eiseman1980} in their survey of the conservation laws of fluid dynamics. |
---|
[707] | 319 | |
---|
[10354] | 320 | Let (\textit{i},\textit{j},\textit{k}) be a set of orthogonal curvilinear coordinates on |
---|
| 321 | the sphere associated with the positively oriented orthogonal set of unit vectors |
---|
| 322 | (\textbf{i},\textbf{j},\textbf{k}) linked to the earth such that |
---|
| 323 | \textbf{k} is the local upward vector and (\textbf{i},\textbf{j}) are two vectors orthogonal to \textbf{k}, |
---|
| 324 | $i.e.$ along geopotential surfaces (\autoref{fig:referential}). |
---|
| 325 | Let $(\lambda,\varphi,z)$ be the geographical coordinate system in which a position is defined by |
---|
| 326 | the latitude $\varphi(i,j)$, the longitude $\lambda(i,j)$ and |
---|
| 327 | the distance from the centre of the earth $a+z(k)$ where $a$ is the earth's radius and |
---|
| 328 | $z$ the altitude above a reference sea level (\autoref{fig:referential}). |
---|
| 329 | The local deformation of the curvilinear coordinate system is given by $e_1$, $e_2$ and $e_3$, |
---|
| 330 | the three scale factors: |
---|
[10414] | 331 | \begin{equation} |
---|
| 332 | \label{eq:scale_factors} |
---|
[10354] | 333 | \begin{aligned} |
---|
| 334 | e_1 &=\left( {a+z} \right)\;\left[ {\left( {\frac{\partial \lambda}{\partial i}\cos \varphi } \right)^2 |
---|
| 335 | +\left( {\frac{\partial \varphi }{\partial i}} \right)^2} \right]^{1/2} \\ |
---|
| 336 | e_2 &=\left( {a+z} \right)\;\left[ {\left( {\frac{\partial \lambda }{\partial j}\cos \varphi } \right)^2+ |
---|
| 337 | \left( {\frac{\partial \varphi }{\partial j}} \right)^2} \right]^{1/2} \\ |
---|
| 338 | e_3 &=\left( {\frac{\partial z}{\partial k}} \right) \\ |
---|
| 339 | \end{aligned} |
---|
| 340 | \end{equation} |
---|
[707] | 341 | |
---|
[10354] | 342 | % >>>>>>>>>>>>>>>>>>>>>>>>>>>> |
---|
| 343 | \begin{figure}[!tb] |
---|
| 344 | \begin{center} |
---|
| 345 | \includegraphics[width=0.60\textwidth]{Fig_I_earth_referential} |
---|
| 346 | \caption{ \protect\label{fig:referential} |
---|
| 347 | the geographical coordinate system $(\lambda,\varphi,z)$ and the curvilinear |
---|
| 348 | coordinate system (\textbf{i},\textbf{j},\textbf{k}). } |
---|
| 349 | \end{center} |
---|
| 350 | \end{figure} |
---|
[707] | 351 | %>>>>>>>>>>>>>>>>>>>>>>>>>>>> |
---|
| 352 | |
---|
[10354] | 353 | Since the ocean depth is far smaller than the earth's radius, $a+z$, can be replaced by $a$ in |
---|
| 354 | (\autoref{eq:scale_factors}) (thin-shell approximation). |
---|
| 355 | The resulting horizontal scale factors $e_1$, $e_2$ are independent of $k$ while |
---|
| 356 | the vertical scale factor is a single function of $k$ as \textbf{k} is parallel to \textbf{z}. |
---|
| 357 | The scalar and vector operators that appear in the primitive equations |
---|
| 358 | (\autoref{eq:PE_dyn} to \autoref{eq:PE_eos}) can be written in the tensorial form, |
---|
| 359 | invariant in any orthogonal horizontal curvilinear coordinate system transformation: |
---|
[10414] | 360 | \begin{subequations} |
---|
| 361 | % \label{eq:PE_discrete_operators} |
---|
| 362 | \begin{equation} |
---|
| 363 | \label{eq:PE_grad} |
---|
| 364 | \nabla q=\frac{1}{e_1 }\frac{\partial q}{\partial i}\;{\rm {\bf |
---|
| 365 | i}}+\frac{1}{e_2 }\frac{\partial q}{\partial j}\;{\rm {\bf j}}+\frac{1}{e_3 |
---|
| 366 | }\frac{\partial q}{\partial k}\;{\rm {\bf k}} \\ |
---|
| 367 | \end{equation} |
---|
| 368 | \begin{equation} |
---|
| 369 | \label{eq:PE_div} |
---|
| 370 | \nabla \cdot {\rm {\bf A}} |
---|
| 371 | = \frac{1}{e_1 \; e_2} \left[ |
---|
| 372 | \frac{\partial \left(e_2 \; a_1\right)}{\partial i } |
---|
| 373 | +\frac{\partial \left(e_1 \; a_2\right)}{\partial j } \right] |
---|
| 374 | + \frac{1}{e_3} \left[ \frac{\partial a_3}{\partial k } \right] |
---|
| 375 | \end{equation} |
---|
| 376 | \begin{equation} |
---|
| 377 | \label{eq:PE_curl} |
---|
| 378 | \begin{split} |
---|
| 379 | \nabla \times \vect{A} = |
---|
| 380 | \left[ {\frac{1}{e_2 }\frac{\partial a_3}{\partial j} |
---|
| 381 | -\frac{1}{e_3 }\frac{\partial a_2 }{\partial k}} \right] \; \vect{i} |
---|
| 382 | &+\left[ {\frac{1}{e_3 }\frac{\partial a_1 }{\partial k} |
---|
| 383 | -\frac{1}{e_1 }\frac{\partial a_3 }{\partial i}} \right] \; \vect{j} \\ |
---|
| 384 | &+\frac{1}{e_1 e_2 } \left[ {\frac{\partial \left( {e_2 a_2 } \right)}{\partial i} |
---|
| 385 | -\frac{\partial \left( {e_1 a_1 } \right)}{\partial j}} \right] \; \vect{k} |
---|
| 386 | \end{split} |
---|
| 387 | \end{equation} |
---|
| 388 | \begin{equation} |
---|
| 389 | \label{eq:PE_lap} |
---|
| 390 | \Delta q = \nabla \cdot \left( \nabla q \right) |
---|
| 391 | \end{equation} |
---|
| 392 | \begin{equation} |
---|
| 393 | \label{eq:PE_lap_vector} |
---|
| 394 | \Delta {\rm {\bf A}} = |
---|
| 395 | \nabla \left( \nabla \cdot {\rm {\bf A}} \right) |
---|
| 396 | - \nabla \times \left( \nabla \times {\rm {\bf A}} \right) |
---|
| 397 | \end{equation} |
---|
[707] | 398 | \end{subequations} |
---|
[817] | 399 | where $q$ is a scalar quantity and ${\rm {\bf A}}=(a_1,a_2,a_3)$ a vector in the $(i,j,k)$ coordinate system. |
---|
[707] | 400 | |
---|
| 401 | % ------------------------------------------------------------------------------------------------------------- |
---|
| 402 | % Continuous Model Equations |
---|
| 403 | % ------------------------------------------------------------------------------------------------------------- |
---|
[9393] | 404 | \subsection{Continuous model equations} |
---|
[9407] | 405 | \label{subsec:PE_zco_Eq} |
---|
[707] | 406 | |
---|
[10354] | 407 | In order to express the Primitive Equations in tensorial formalism, |
---|
| 408 | it is necessary to compute the horizontal component of the non-linear and viscous terms of the equation using |
---|
| 409 | \autoref{eq:PE_grad}) to \autoref{eq:PE_lap_vector}. |
---|
| 410 | Let us set $\vect U=(u,v,w)={\vect{U}}_h +w\;\vect{k}$, the velocity in the $(i,j,k)$ coordinate system and |
---|
| 411 | define the relative vorticity $\zeta$ and the divergence of the horizontal velocity field $\chi$, by: |
---|
[10414] | 412 | \begin{equation} |
---|
| 413 | \label{eq:PE_curl_Uh} |
---|
| 414 | \zeta =\frac{1}{e_1 e_2 }\left[ {\frac{\partial \left( {e_2 \,v} |
---|
| 415 | \right)}{\partial i}-\frac{\partial \left( {e_1 \,u} \right)}{\partial j}} |
---|
| 416 | \right] |
---|
[707] | 417 | \end{equation} |
---|
[10414] | 418 | \begin{equation} |
---|
| 419 | \label{eq:PE_div_Uh} |
---|
| 420 | \chi =\frac{1}{e_1 e_2 }\left[ {\frac{\partial \left( {e_2 \,u} |
---|
| 421 | \right)}{\partial i}+\frac{\partial \left( {e_1 \,v} \right)}{\partial j}} |
---|
| 422 | \right] |
---|
[707] | 423 | \end{equation} |
---|
| 424 | |
---|
[10354] | 425 | Using the fact that the horizontal scale factors $e_1$ and $e_2$ are independent of $k$ and that |
---|
| 426 | $e_3$ is a function of the single variable $k$, |
---|
| 427 | the nonlinear term of \autoref{eq:PE_dyn} can be transformed as follows: |
---|
[707] | 428 | \begin{flalign*} |
---|
[10414] | 429 | &\left[ {\left( { \nabla \times {\rm {\bf U}} } \right) \times {\rm {\bf U}} |
---|
| 430 | +\frac{1}{2} \nabla \left( {{\rm {\bf U}}^2} \right)} \right]_h & |
---|
[707] | 431 | \end{flalign*} |
---|
| 432 | \begin{flalign*} |
---|
[10414] | 433 | &\qquad=\left( {{ |
---|
| 434 | \begin{array}{*{20}c} |
---|
| 435 | {\left[ { \frac{1}{e_3} \frac{\partial u }{\partial k} |
---|
| 436 | -\frac{1}{e_1} \frac{\partial w }{\partial i} } \right] w - \zeta \; v } \\ |
---|
| 437 | {\zeta \; u - \left[ { \frac{1}{e_2} \frac{\partial w}{\partial j} |
---|
| 438 | -\frac{1}{e_3} \frac{\partial v}{\partial k} } \right] \ w} \\ |
---|
| 439 | \end{array} |
---|
| 440 | }} \right) |
---|
| 441 | +\frac{1}{2} \left( {{ |
---|
| 442 | \begin{array}{*{20}c} |
---|
| 443 | { \frac{1}{e_1} \frac{\partial \left( u^2+v^2+w^2 \right)}{\partial i}} \hfill \\ |
---|
| 444 | { \frac{1}{e_2} \frac{\partial \left( u^2+v^2+w^2 \right)}{\partial j}} \hfill \\ |
---|
| 445 | \end{array} |
---|
| 446 | }} \right) & |
---|
[707] | 447 | \end{flalign*} |
---|
| 448 | \begin{flalign*} |
---|
[10414] | 449 | & \qquad =\left( {{ |
---|
| 450 | \begin{array}{*{20}c} |
---|
| 451 | {-\zeta \; v} \hfill \\ |
---|
| 452 | { \zeta \; u} \hfill \\ |
---|
| 453 | \end{array} |
---|
| 454 | }} \right) |
---|
| 455 | +\frac{1}{2}\left( {{ |
---|
| 456 | \begin{array}{*{20}c} |
---|
| 457 | {\frac{1}{e_1 }\frac{\partial \left( {u^2+v^2} \right)}{\partial i}} \hfill \\ |
---|
| 458 | {\frac{1}{e_2 }\frac{\partial \left( {u^2+v^2} \right)}{\partial j}} \hfill \\ |
---|
| 459 | \end{array} |
---|
| 460 | }} \right) |
---|
| 461 | +\frac{1}{e_3 }\left( {{ |
---|
| 462 | \begin{array}{*{20}c} |
---|
| 463 | { w \; \frac{\partial u}{\partial k}} \\ |
---|
| 464 | { w \; \frac{\partial v}{\partial k}} \\ |
---|
| 465 | \end{array} |
---|
| 466 | }} \right) |
---|
| 467 | -\left( {{ |
---|
| 468 | \begin{array}{*{20}c} |
---|
| 469 | {\frac{w}{e_1}\frac{\partial w}{\partial i} -\frac{1}{2e_1}\frac{\partial w^2}{\partial i}} \hfill \\ |
---|
| 470 | {\frac{w}{e_2}\frac{\partial w}{\partial j} -\frac{1}{2e_2}\frac{\partial w^2}{\partial j}} \hfill \\ |
---|
| 471 | \end{array} |
---|
| 472 | }} \right) & |
---|
[707] | 473 | \end{flalign*} |
---|
| 474 | |
---|
[10354] | 475 | The last term of the right hand side is obviously zero, and thus the nonlinear term of |
---|
[9407] | 476 | \autoref{eq:PE_dyn} is written in the $(i,j,k)$ coordinate system: |
---|
[10414] | 477 | \begin{equation} |
---|
| 478 | \label{eq:PE_vector_form} |
---|
| 479 | \left[ {\left( { \nabla \times {\rm {\bf U}} } \right) \times {\rm {\bf U}} |
---|
| 480 | +\frac{1}{2} \nabla \left( {{\rm {\bf U}}^2} \right)} \right]_h |
---|
| 481 | =\zeta |
---|
| 482 | \;{\rm {\bf k}}\times {\rm {\bf U}}_h +\frac{1}{2}\nabla _h \left( {{\rm |
---|
| 483 | {\bf U}}_h^2 } \right)+\frac{1}{e_3 }w\frac{\partial {\rm {\bf U}}_h |
---|
| 484 | }{\partial k} |
---|
[707] | 485 | \end{equation} |
---|
| 486 | |
---|
[10354] | 487 | This is the so-called \textit{vector invariant form} of the momentum advection term. |
---|
| 488 | For some purposes, it can be advantageous to write this term in the so-called flux form, |
---|
| 489 | $i.e.$ to write it as the divergence of fluxes. |
---|
| 490 | For example, the first component of \autoref{eq:PE_vector_form} (the $i$-component) is transformed as follows: |
---|
[707] | 491 | \begin{flalign*} |
---|
[10414] | 492 | &{ |
---|
| 493 | \begin{array}{*{20}l} |
---|
| 494 | \left[ {\left( {\nabla \times \vect{U}} \right)\times \vect{U} |
---|
| 495 | +\frac{1}{2}\nabla \left( {\vect{U}}^2 \right)} \right]_i % \\ |
---|
| 496 | % \\ |
---|
| 497 | = - \zeta \;v |
---|
| 498 | + \frac{1}{2\;e_1 } \frac{\partial \left( {u^2+v^2} \right)}{\partial i} |
---|
| 499 | + \frac{1}{e_3}w \ \frac{\partial u}{\partial k} \\ \\ |
---|
| 500 | \qquad =\frac{1}{e_1 \; e_2} \left( -v\frac{\partial \left( {e_2 \,v} \right)}{\partial i} |
---|
| 501 | +v\frac{\partial \left( {e_1 \,u} \right)}{\partial j} \right) |
---|
| 502 | +\frac{1}{e_1 e_2 }\left( +e_2 \; u\frac{\partial u}{\partial i} |
---|
| 503 | +e_2 \; v\frac{\partial v}{\partial i} \right) |
---|
| 504 | +\frac{1}{e_3} \left( w\;\frac{\partial u}{\partial k} \right) \\ |
---|
| 505 | \end{array} |
---|
| 506 | } & |
---|
[707] | 507 | \end{flalign*} |
---|
| 508 | \begin{flalign*} |
---|
[10414] | 509 | &{ |
---|
| 510 | \begin{array}{*{20}l} |
---|
| 511 | \qquad =\frac{1}{e_1 \; e_2} \left\{ |
---|
| 512 | -\left( v^2 \frac{\partial e_2 }{\partial i} |
---|
[707] | 513 | +e_2 \,v \frac{\partial v }{\partial i} \right) |
---|
[10414] | 514 | +\left( \frac{\partial \left( {e_1 \,u\,v} \right)}{\partial j} |
---|
| 515 | -e_1 \,u \frac{\partial v }{\partial j} \right) \right. \\ |
---|
| 516 | \left. \qquad \qquad \quad |
---|
| 517 | +\left( \frac{\partial \left( {e_2 u\,u} \right)}{\partial i} |
---|
[707] | 518 | -u \frac{\partial \left( {e_2 u} \right)}{\partial i} \right) |
---|
[10414] | 519 | +e_2 v \frac{\partial v }{\partial i} |
---|
| 520 | \right\} |
---|
| 521 | +\frac{1}{e_3} \left( |
---|
| 522 | \frac{\partial \left( {w\,u} \right) }{\partial k} |
---|
| 523 | -u \frac{\partial w }{\partial k} \right) \\ |
---|
| 524 | \end{array} |
---|
| 525 | } & |
---|
[707] | 526 | \end{flalign*} |
---|
| 527 | \begin{flalign*} |
---|
[10414] | 528 | & |
---|
| 529 | { |
---|
| 530 | \begin{array}{*{20}l} |
---|
| 531 | \qquad =\frac{1}{e_1 \; e_2} \left( |
---|
| 532 | \frac{\partial \left( {e_2 \,u\,u} \right)}{\partial i} |
---|
[707] | 533 | + \frac{\partial \left( {e_1 \,u\,v} \right)}{\partial j} \right) |
---|
[10414] | 534 | +\frac{1}{e_3 } \frac{\partial \left( {w\,u } \right)}{\partial k} \\ |
---|
| 535 | \qquad \qquad \quad |
---|
| 536 | +\frac{1}{e_1 e_2 } \left( |
---|
[707] | 537 | -u \left( \frac{\partial \left( {e_1 v } \right)}{\partial j} |
---|
[10414] | 538 | -v\,\frac{\partial e_1 }{\partial j} \right) |
---|
[707] | 539 | -u \frac{\partial \left( {e_2 u } \right)}{\partial i} |
---|
[10414] | 540 | \right) |
---|
| 541 | -\frac{1}{e_3 } \frac{\partial w}{\partial k} u |
---|
| 542 | +\frac{1}{e_1 e_2 }\left( -v^2\frac{\partial e_2 }{\partial i} \right) |
---|
| 543 | \end{array} |
---|
| 544 | } & |
---|
[707] | 545 | \end{flalign*} |
---|
| 546 | \begin{flalign*} |
---|
[10414] | 547 | &{ |
---|
| 548 | \begin{array}{*{20}l} |
---|
| 549 | \qquad = \nabla \cdot \left( {{\rm {\bf U}}\,u} \right) |
---|
| 550 | - \left( \nabla \cdot {\rm {\bf U}} \right) \ u |
---|
| 551 | +\frac{1}{e_1 e_2 }\left( |
---|
[707] | 552 | -v^2 \frac{\partial e_2 }{\partial i} |
---|
| 553 | +uv \, \frac{\partial e_1 }{\partial j} \right) \\ |
---|
[10414] | 554 | \end{array} |
---|
| 555 | } & |
---|
[707] | 556 | \end{flalign*} |
---|
| 557 | as $\nabla \cdot {\rm {\bf U}}\;=0$ (incompressibility) it comes: |
---|
| 558 | \begin{flalign*} |
---|
[10414] | 559 | &{ |
---|
| 560 | \begin{array}{*{20}l} |
---|
| 561 | \qquad = \nabla \cdot \left( {{\rm {\bf U}}\,u} \right) |
---|
| 562 | + \frac{1}{e_1 e_2 } \left( v \; \frac{\partial e_2}{\partial i} |
---|
| 563 | -u \; \frac{\partial e_1}{\partial j} \right) \left( -v \right) |
---|
| 564 | \end{array} |
---|
| 565 | } & |
---|
[707] | 566 | \end{flalign*} |
---|
| 567 | |
---|
[817] | 568 | The flux form of the momentum advection term is therefore given by: |
---|
[10414] | 569 | \begin{multline} |
---|
| 570 | \label{eq:PE_flux_form} |
---|
| 571 | \left[ |
---|
| 572 | \left( {\nabla \times {\rm {\bf U}}} \right) \times {\rm {\bf U}} |
---|
| 573 | +\frac{1}{2} \nabla \left( {{\rm {\bf U}}^2} \right) |
---|
| 574 | \right]_h \\ |
---|
| 575 | = \nabla \cdot \left( {{ |
---|
| 576 | \begin{array}{*{20}c} |
---|
| 577 | {\rm {\bf U}} \, u \hfill \\ |
---|
| 578 | {\rm {\bf U}} \, v \hfill \\ |
---|
| 579 | \end{array} |
---|
| 580 | }} |
---|
| 581 | \right) |
---|
| 582 | +\frac{1}{e_1 e_2 } \left( |
---|
| 583 | v\frac{\partial e_2}{\partial i} |
---|
| 584 | -u\frac{\partial e_1}{\partial j} |
---|
| 585 | \right) {\rm {\bf k}} \times {\rm {\bf U}}_h |
---|
[707] | 586 | \end{multline} |
---|
| 587 | |
---|
[10354] | 588 | The flux form has two terms, |
---|
| 589 | the first one is expressed as the divergence of momentum fluxes (hence the flux form name given to this formulation) |
---|
| 590 | and the second one is due to the curvilinear nature of the coordinate system used. |
---|
| 591 | The latter is called the \emph{metric} term and can be viewed as a modification of the Coriolis parameter: |
---|
[10414] | 592 | \[ |
---|
| 593 | % \label{eq:PE_cor+metric} |
---|
| 594 | f \to f + \frac{1}{e_1\;e_2} \left( v \frac{\partial e_2}{\partial i} |
---|
| 595 | -u \frac{\partial e_1}{\partial j} \right) |
---|
| 596 | \] |
---|
[707] | 597 | |
---|
[10354] | 598 | Note that in the case of geographical coordinate, |
---|
| 599 | $i.e.$ when $(i,j) \to (\lambda ,\varphi )$ and $(e_1 ,e_2) \to (a \,\cos \varphi ,a)$, |
---|
| 600 | we recover the commonly used modification of the Coriolis parameter $f \to f+(u/a) \tan \varphi$. |
---|
[707] | 601 | |
---|
[10354] | 602 | To sum up, the curvilinear $z$-coordinate equations solved by the ocean model can be written in |
---|
| 603 | the following tensorial formalism: |
---|
[2282] | 604 | |
---|
[817] | 605 | \vspace{+10pt} |
---|
[2282] | 606 | $\bullet$ \textbf{Vector invariant form of the momentum equations} : |
---|
[817] | 607 | |
---|
[10414] | 608 | \begin{subequations} |
---|
| 609 | \label{eq:PE_dyn_vect} |
---|
| 610 | \[ |
---|
| 611 | % \label{eq:PE_dyn_vect_u} |
---|
| 612 | \begin{split} |
---|
| 613 | \frac{\partial u}{\partial t} |
---|
| 614 | = + \left( {\zeta +f} \right)\,v |
---|
| 615 | - \frac{1}{2\,e_1} \frac{\partial}{\partial i} \left( u^2+v^2 \right) |
---|
| 616 | - \frac{1}{e_3 } w \frac{\partial u}{\partial k} & \\ |
---|
| 617 | - \frac{1}{e_1 } \frac{\partial}{\partial i} \left( \frac{p_s+p_h }{\rho_o} \right) |
---|
| 618 | &+ D_u^{\vect{U}} + F_u^{\vect{U}} \\ \\ |
---|
| 619 | \frac{\partial v}{\partial t} = |
---|
| 620 | - \left( {\zeta +f} \right)\,u |
---|
| 621 | - \frac{1}{2\,e_2 } \frac{\partial }{\partial j}\left( u^2+v^2 \right) |
---|
| 622 | - \frac{1}{e_3 } w \frac{\partial v}{\partial k} & \\ |
---|
| 623 | - \frac{1}{e_2 } \frac{\partial }{\partial j}\left( \frac{p_s+p_h }{\rho_o} \right) |
---|
| 624 | &+ D_v^{\vect{U}} + F_v^{\vect{U}} |
---|
| 625 | \end{split} |
---|
| 626 | \] |
---|
[707] | 627 | \end{subequations} |
---|
| 628 | |
---|
[2282] | 629 | |
---|
| 630 | \vspace{+10pt} |
---|
| 631 | $\bullet$ \textbf{flux form of the momentum equations} : |
---|
[10414] | 632 | \begin{subequations} |
---|
| 633 | % \label{eq:PE_dyn_flux} |
---|
| 634 | \begin{multline*} |
---|
| 635 | % \label{eq:PE_dyn_flux_u} |
---|
| 636 | \frac{\partial u}{\partial t}= |
---|
| 637 | + \left( { f + \frac{1}{e_1 \; e_2} |
---|
| 638 | \left( v \frac{\partial e_2}{\partial i} |
---|
| 639 | -u \frac{\partial e_1}{\partial j} \right)} \right) \, v \\ |
---|
| 640 | - \frac{1}{e_1 \; e_2} \left( |
---|
| 641 | \frac{\partial \left( {e_2 \,u\,u} \right)}{\partial i} |
---|
[817] | 642 | + \frac{\partial \left( {e_1 \,v\,u} \right)}{\partial j} \right) |
---|
[10414] | 643 | - \frac{1}{e_3 }\frac{\partial \left( { w\,u} \right)}{\partial k} \\ |
---|
| 644 | - \frac{1}{e_1 }\frac{\partial}{\partial i}\left( \frac{p_s+p_h }{\rho_o} \right) |
---|
| 645 | + D_u^{\vect{U}} + F_u^{\vect{U}} |
---|
| 646 | \end{multline*} |
---|
| 647 | \begin{multline*} |
---|
| 648 | % \label{eq:PE_dyn_flux_v} |
---|
| 649 | \frac{\partial v}{\partial t}= |
---|
| 650 | - \left( { f + \frac{1}{e_1 \; e_2} |
---|
| 651 | \left( v \frac{\partial e_2}{\partial i} |
---|
| 652 | -u \frac{\partial e_1}{\partial j} \right)} \right) \, u \\ |
---|
| 653 | \frac{1}{e_1 \; e_2} \left( |
---|
| 654 | \frac{\partial \left( {e_2 \,u\,v} \right)}{\partial i} |
---|
[994] | 655 | + \frac{\partial \left( {e_1 \,v\,v} \right)}{\partial j} \right) |
---|
[10414] | 656 | - \frac{1}{e_3 } \frac{\partial \left( { w\,v} \right)}{\partial k} \\ |
---|
| 657 | - \frac{1}{e_2 }\frac{\partial }{\partial j}\left( \frac{p_s+p_h }{\rho_o} \right) |
---|
| 658 | + D_v^{\vect{U}} + F_v^{\vect{U}} |
---|
| 659 | \end{multline*} |
---|
[817] | 660 | \end{subequations} |
---|
[10354] | 661 | where $\zeta$, the relative vorticity, is given by \autoref{eq:PE_curl_Uh} and |
---|
| 662 | $p_s $, the surface pressure, is given by: |
---|
[10414] | 663 | \[ |
---|
| 664 | % \label{eq:PE_spg} |
---|
| 665 | p_s = \rho \,g \,\eta |
---|
| 666 | \] |
---|
[10354] | 667 | with $\eta$ is solution of \autoref{eq:PE_ssh}. |
---|
[707] | 668 | |
---|
| 669 | The vertical velocity and the hydrostatic pressure are diagnosed from the following equations: |
---|
[10414] | 670 | \[ |
---|
| 671 | % \label{eq:w_diag} |
---|
| 672 | \frac{\partial w}{\partial k}=-\chi \;e_3 |
---|
| 673 | \] |
---|
| 674 | \[ |
---|
| 675 | % \label{eq:hp_diag} |
---|
| 676 | \frac{\partial p_h }{\partial k}=-\rho \;g\;e_3 |
---|
| 677 | \] |
---|
[9407] | 678 | where the divergence of the horizontal velocity, $\chi$ is given by \autoref{eq:PE_div_Uh}. |
---|
[707] | 679 | |
---|
[817] | 680 | \vspace{+10pt} |
---|
| 681 | $\bullet$ \textit{tracer equations} : |
---|
[10414] | 682 | \[ |
---|
| 683 | % \label{eq:S} |
---|
| 684 | \frac{\partial T}{\partial t} = |
---|
| 685 | -\frac{1}{e_1 e_2 }\left[ { \frac{\partial \left( {e_2 T\,u} \right)}{\partial i} |
---|
| 686 | +\frac{\partial \left( {e_1 T\,v} \right)}{\partial j}} \right] |
---|
| 687 | -\frac{1}{e_3 }\frac{\partial \left( {T\,w} \right)}{\partial k} + D^T + F^T |
---|
| 688 | \] |
---|
| 689 | \[ |
---|
| 690 | % \label{eq:T} |
---|
| 691 | \frac{\partial S}{\partial t} = |
---|
| 692 | -\frac{1}{e_1 e_2 }\left[ {\frac{\partial \left( {e_2 S\,u} \right)}{\partial i} |
---|
| 693 | +\frac{\partial \left( {e_1 S\,v} \right)}{\partial j}} \right] |
---|
| 694 | -\frac{1}{e_3 }\frac{\partial \left( {S\,w} \right)}{\partial k} + D^S + F^S |
---|
| 695 | \] |
---|
| 696 | \[ |
---|
| 697 | % \label{eq:rho} |
---|
| 698 | \rho =\rho \left( {T,S,z(k)} \right) |
---|
| 699 | \] |
---|
[707] | 700 | |
---|
[10354] | 701 | The expression of \textbf{D}$^{U}$, $D^{S}$ and $D^{T}$ depends on the subgrid scale parameterisation used. |
---|
| 702 | It will be defined in \autoref{eq:PE_zdf}. |
---|
| 703 | The nature and formulation of ${\rm {\bf F}}^{\rm {\bf U}}$, $F^T$ and $F^S$, the surface forcing terms, |
---|
| 704 | are discussed in \autoref{chap:SBC}. |
---|
[707] | 705 | |
---|
[2282] | 706 | |
---|
[707] | 707 | \newpage |
---|
[10414] | 708 | |
---|
[707] | 709 | % ================================================================ |
---|
[2282] | 710 | % Curvilinear generalised vertical coordinate System |
---|
[817] | 711 | % ================================================================ |
---|
[9393] | 712 | \section{Curvilinear generalised vertical coordinate system} |
---|
[9407] | 713 | \label{sec:PE_gco} |
---|
[817] | 714 | |
---|
[10354] | 715 | The ocean domain presents a huge diversity of situation in the vertical. |
---|
| 716 | First the ocean surface is a time dependent surface (moving surface). |
---|
| 717 | Second the ocean floor depends on the geographical position, |
---|
| 718 | varying from more than 6,000 meters in abyssal trenches to zero at the coast. |
---|
| 719 | Last but not least, the ocean stratification exerts a strong barrier to vertical motions and mixing. |
---|
| 720 | Therefore, in order to represent the ocean with respect to |
---|
| 721 | the first point a space and time dependent vertical coordinate that follows the variation of the sea surface height |
---|
| 722 | $e.g.$ an $z$*-coordinate; |
---|
| 723 | for the second point, a space variation to fit the change of bottom topography |
---|
| 724 | $e.g.$ a terrain-following or $\sigma$-coordinate; |
---|
| 725 | and for the third point, one will be tempted to use a space and time dependent coordinate that |
---|
| 726 | follows the isopycnal surfaces, $e.g.$ an isopycnic coordinate. |
---|
[2282] | 727 | |
---|
[10354] | 728 | In order to satisfy two or more constrains one can even be tempted to mixed these coordinate systems, as in |
---|
| 729 | HYCOM (mixture of $z$-coordinate at the surface, isopycnic coordinate in the ocean interior and $\sigma$ at |
---|
| 730 | the ocean bottom) \citep{Chassignet_al_JPO03} or |
---|
| 731 | OPA (mixture of $z$-coordinate in vicinity the surface and steep topography areas and $\sigma$-coordinate elsewhere) |
---|
| 732 | \citep{Madec_al_JPO96} among others. |
---|
[2282] | 733 | |
---|
[10354] | 734 | In fact one is totally free to choose any space and time vertical coordinate by |
---|
| 735 | introducing an arbitrary vertical coordinate : |
---|
[10414] | 736 | \begin{equation} |
---|
| 737 | \label{eq:PE_s} |
---|
| 738 | s=s(i,j,k,t) |
---|
[2282] | 739 | \end{equation} |
---|
[10354] | 740 | with the restriction that the above equation gives a single-valued monotonic relationship between $s$ and $k$, |
---|
| 741 | when $i$, $j$ and $t$ are held fixed. |
---|
| 742 | \autoref{eq:PE_s} is a transformation from the $(i,j,k,t)$ coordinate system with independent variables into |
---|
| 743 | the $(i,j,s,t)$ generalised coordinate system with $s$ depending on the other three variables through |
---|
| 744 | \autoref{eq:PE_s}. |
---|
| 745 | This so-called \textit{generalised vertical coordinate} \citep{Kasahara_MWR74} is in fact |
---|
| 746 | an Arbitrary Lagrangian--Eulerian (ALE) coordinate. |
---|
| 747 | Indeed, choosing an expression for $s$ is an arbitrary choice that determines |
---|
| 748 | which part of the vertical velocity (defined from a fixed referential) will cross the levels (Eulerian part) and |
---|
| 749 | which part will be used to move them (Lagrangian part). |
---|
| 750 | The coordinate is also sometime referenced as an adaptive coordinate \citep{Hofmeister_al_OM09}, |
---|
| 751 | since the coordinate system is adapted in the course of the simulation. |
---|
| 752 | Its most often used implementation is via an ALE algorithm, |
---|
| 753 | in which a pure lagrangian step is followed by regridding and remapping steps, |
---|
| 754 | the later step implicitly embedding the vertical advection |
---|
| 755 | \citep{Hirt_al_JCP74, Chassignet_al_JPO03, White_al_JCP09}. |
---|
| 756 | Here we follow the \citep{Kasahara_MWR74} strategy: |
---|
| 757 | a regridding step (an update of the vertical coordinate) followed by an eulerian step with |
---|
| 758 | an explicit computation of vertical advection relative to the moving s-surfaces. |
---|
[2282] | 759 | |
---|
[2376] | 760 | %\gmcomment{ |
---|
[2282] | 761 | |
---|
[2376] | 762 | %A key point here is that the $s$-coordinate depends on $(i,j)$ ==> horizontal pressure gradient... |
---|
[2282] | 763 | |
---|
[10354] | 764 | the generalized vertical coordinates used in ocean modelling are not orthogonal, |
---|
| 765 | which contrasts with many other applications in mathematical physics. |
---|
| 766 | Hence, it is useful to keep in mind the following properties that may seem odd on initial encounter. |
---|
[2282] | 767 | |
---|
[10354] | 768 | The horizontal velocity in ocean models measures motions in the horizontal plane, |
---|
| 769 | perpendicular to the local gravitational field. |
---|
| 770 | That is, horizontal velocity is mathematically the same regardless the vertical coordinate, be it geopotential, |
---|
| 771 | isopycnal, pressure, or terrain following. |
---|
| 772 | The key motivation for maintaining the same horizontal velocity component is that |
---|
| 773 | the hydrostatic and geostrophic balances are dominant in the large-scale ocean. |
---|
| 774 | Use of an alternative quasi-horizontal velocity, for example one oriented parallel to the generalized surface, |
---|
| 775 | would lead to unacceptable numerical errors. |
---|
| 776 | Correspondingly, the vertical direction is anti-parallel to the gravitational force in |
---|
| 777 | all of the coordinate systems. |
---|
| 778 | We do not choose the alternative of a quasi-vertical direction oriented normal to |
---|
| 779 | the surface of a constant generalized vertical coordinate. |
---|
[2282] | 780 | |
---|
[10354] | 781 | It is the method used to measure transport across the generalized vertical coordinate surfaces which differs between |
---|
| 782 | the vertical coordinate choices. |
---|
| 783 | That is, computation of the dia-surface velocity component represents the fundamental distinction between |
---|
| 784 | the various coordinates. |
---|
| 785 | In some models, such as geopotential, pressure, and terrain following, this transport is typically diagnosed from |
---|
| 786 | volume or mass conservation. |
---|
| 787 | In other models, such as isopycnal layered models, this transport is prescribed based on assumptions about |
---|
| 788 | the physical processes producing a flux across the layer interfaces. |
---|
[2282] | 789 | |
---|
[2376] | 790 | |
---|
[10354] | 791 | In this section we first establish the PE in the generalised vertical $s$-coordinate, |
---|
[2376] | 792 | then we discuss the particular cases available in \NEMO, namely $z$, $z$*, $s$, and $\tilde z$. |
---|
[2282] | 793 | %} |
---|
| 794 | |
---|
| 795 | % ------------------------------------------------------------------------------------------------------------- |
---|
| 796 | % The s-coordinate Formulation |
---|
| 797 | % ------------------------------------------------------------------------------------------------------------- |
---|
[9393] | 798 | \subsection{\textit{S-}coordinate formulation} |
---|
[2282] | 799 | |
---|
[10354] | 800 | Starting from the set of equations established in \autoref{sec:PE_zco} for the special case $k=z$ and thus $e_3=1$, |
---|
| 801 | we introduce an arbitrary vertical coordinate $s=s(i,j,k,t)$, |
---|
| 802 | which includes $z$-, \textit{z*}- and $\sigma-$coordinates as special cases |
---|
| 803 | ($s=z$, $s=\textit{z*}$, and $s=\sigma=z/H$ or $=z/\left(H+\eta \right)$, resp.). |
---|
| 804 | A formal derivation of the transformed equations is given in \autoref{apdx:A}. |
---|
| 805 | Let us define the vertical scale factor by $e_3=\partial_s z$ ($e_3$ is now a function of $(i,j,k,t)$ ), |
---|
| 806 | and the slopes in the (\textbf{i},\textbf{j}) directions between $s-$ and $z-$surfaces by: |
---|
[10414] | 807 | \begin{equation} |
---|
| 808 | \label{eq:PE_sco_slope} |
---|
| 809 | \sigma_1 =\frac{1}{e_1 }\;\left. {\frac{\partial z}{\partial i}} \right|_s |
---|
| 810 | \quad \text{, and } \quad |
---|
| 811 | \sigma_2 =\frac{1}{e_2 }\;\left. {\frac{\partial z}{\partial j}} \right|_s |
---|
[2282] | 812 | \end{equation} |
---|
| 813 | We also introduce $\omega $, a dia-surface velocity component, defined as the velocity |
---|
| 814 | relative to the moving $s$-surfaces and normal to them: |
---|
[10414] | 815 | \[ |
---|
| 816 | % \label{eq:PE_sco_w} |
---|
| 817 | \omega = w - e_3 \, \frac{\partial s}{\partial t} - \sigma_1 \,u - \sigma_2 \,v \\ |
---|
| 818 | \] |
---|
[2282] | 819 | |
---|
[10354] | 820 | The equations solved by the ocean model \autoref{eq:PE} in $s-$coordinate can be written as follows |
---|
| 821 | (see \autoref{sec:A_momentum}): |
---|
[2282] | 822 | |
---|
| 823 | \vspace{0.5cm} |
---|
[6140] | 824 | $\bullet$ Vector invariant form of the momentum equation : |
---|
[10414] | 825 | \begin{multline*} |
---|
| 826 | % \label{eq:PE_sco_u_vector} |
---|
| 827 | \frac{\partial u }{\partial t}= |
---|
| 828 | + \left( {\zeta +f} \right)\,v |
---|
| 829 | - \frac{1}{2\,e_1} \frac{\partial}{\partial i} \left( u^2+v^2 \right) |
---|
| 830 | - \frac{1}{e_3} \omega \frac{\partial u}{\partial k} \\ |
---|
| 831 | - \frac{1}{e_1} \frac{\partial}{\partial i} \left( \frac{p_s + p_h}{\rho_o} \right) |
---|
| 832 | + g\frac{\rho }{\rho_o}\sigma_1 |
---|
| 833 | + D_u^{\vect{U}} + F_u^{\vect{U}} \quad |
---|
| 834 | \end{multline*} |
---|
| 835 | \begin{multline*} |
---|
| 836 | % \label{eq:PE_sco_v_vector} |
---|
| 837 | \frac{\partial v }{\partial t}= |
---|
| 838 | - \left( {\zeta +f} \right)\,u |
---|
| 839 | - \frac{1}{2\,e_2 }\frac{\partial }{\partial j}\left( u^2+v^2 \right) |
---|
| 840 | - \frac{1}{e_3 } \omega \frac{\partial v}{\partial k} \\ |
---|
| 841 | - \frac{1}{e_2 }\frac{\partial }{\partial j}\left( \frac{p_s+p_h }{\rho_o} \right) |
---|
| 842 | + g\frac{\rho }{\rho_o }\sigma_2 |
---|
| 843 | + D_v^{\vect{U}} + F_v^{\vect{U}} \quad |
---|
| 844 | \end{multline*} |
---|
[6140] | 845 | |
---|
| 846 | \vspace{0.5cm} |
---|
[9414] | 847 | $\bullet$ Flux form of the momentum equation : |
---|
[10414] | 848 | \begin{multline*} |
---|
| 849 | % \label{eq:PE_sco_u_flux} |
---|
| 850 | \frac{1}{e_3} \frac{\partial \left( e_3\,u \right) }{\partial t}= |
---|
| 851 | + \left( { f + \frac{1}{e_1 \; e_2 } |
---|
| 852 | \left( v \frac{\partial e_2}{\partial i} |
---|
| 853 | -u \frac{\partial e_1}{\partial j} \right)} \right) \, v \\ |
---|
| 854 | - \frac{1}{e_1 \; e_2 \; e_3 } \left( |
---|
| 855 | \frac{\partial \left( {e_2 \, e_3 \, u\,u} \right)}{\partial i} |
---|
| 856 | + \frac{\partial \left( {e_1 \, e_3 \, v\,u} \right)}{\partial j} \right) |
---|
| 857 | - \frac{1}{e_3 }\frac{\partial \left( { \omega\,u} \right)}{\partial k} \\ |
---|
| 858 | - \frac{1}{e_1} \frac{\partial}{\partial i} \left( \frac{p_s + p_h}{\rho_o} \right) |
---|
| 859 | + g\frac{\rho }{\rho_o}\sigma_1 |
---|
| 860 | + D_u^{\vect{U}} + F_u^{\vect{U}} \quad |
---|
| 861 | \end{multline*} |
---|
| 862 | \begin{multline*} |
---|
| 863 | % \label{eq:PE_sco_v_flux} |
---|
| 864 | \frac{1}{e_3} \frac{\partial \left( e_3\,v \right) }{\partial t}= |
---|
| 865 | - \left( { f + \frac{1}{e_1 \; e_2} |
---|
| 866 | \left( v \frac{\partial e_2}{\partial i} |
---|
| 867 | -u \frac{\partial e_1}{\partial j} \right)} \right) \, u \\ |
---|
| 868 | - \frac{1}{e_1 \; e_2 \; e_3 } \left( |
---|
| 869 | \frac{\partial \left( {e_2 \; e_3 \,u\,v} \right)}{\partial i} |
---|
| 870 | + \frac{\partial \left( {e_1 \; e_3 \,v\,v} \right)}{\partial j} \right) |
---|
| 871 | - \frac{1}{e_3 } \frac{\partial \left( { \omega\,v} \right)}{\partial k} \\ |
---|
| 872 | - \frac{1}{e_2 }\frac{\partial }{\partial j}\left( \frac{p_s+p_h }{\rho_o} \right) |
---|
| 873 | + g\frac{\rho }{\rho_o }\sigma_2 |
---|
| 874 | + D_v^{\vect{U}} + F_v^{\vect{U}} \quad |
---|
| 875 | \end{multline*} |
---|
[6140] | 876 | |
---|
[10354] | 877 | where the relative vorticity, \textit{$\zeta $}, the surface pressure gradient, |
---|
| 878 | and the hydrostatic pressure have the same expressions as in $z$-coordinates although |
---|
| 879 | they do not represent exactly the same quantities. |
---|
| 880 | $\omega$ is provided by the continuity equation (see \autoref{apdx:A}): |
---|
[10414] | 881 | \[ |
---|
| 882 | % \label{eq:PE_sco_continuity} |
---|
| 883 | \frac{\partial e_3}{\partial t} + e_3 \; \chi + \frac{\partial \omega }{\partial s} = 0 |
---|
| 884 | \qquad \text{with }\;\; |
---|
| 885 | \chi =\frac{1}{e_1 e_2 e_3 }\left[ {\frac{\partial \left( {e_2 e_3 \,u} |
---|
| 886 | \right)}{\partial i}+\frac{\partial \left( {e_1 e_3 \,v} \right)}{\partial |
---|
| 887 | j}} \right] |
---|
| 888 | \] |
---|
[2282] | 889 | |
---|
| 890 | \vspace{0.5cm} |
---|
[6140] | 891 | $\bullet$ tracer equations: |
---|
[10414] | 892 | \begin{multline*} |
---|
| 893 | % \label{eq:PE_sco_t} |
---|
| 894 | \frac{1}{e_3} \frac{\partial \left( e_3\,T \right) }{\partial t}= |
---|
| 895 | -\frac{1}{e_1 e_2 e_3 }\left[ {\frac{\partial \left( {e_2 e_3\,u\,T} \right)}{\partial i} |
---|
| 896 | +\frac{\partial \left( {e_1 e_3\,v\,T} \right)}{\partial j}} \right] \\ |
---|
| 897 | -\frac{1}{e_3 }\frac{\partial \left( {T\,\omega } \right)}{\partial k} + D^T + F^S \qquad |
---|
| 898 | \end{multline*} |
---|
[2282] | 899 | |
---|
[10414] | 900 | \begin{multline*} |
---|
| 901 | % \label{eq:PE_sco_s} |
---|
| 902 | \frac{1}{e_3} \frac{\partial \left( e_3\,S \right) }{\partial t}= |
---|
| 903 | -\frac{1}{e_1 e_2 e_3 }\left[ {\frac{\partial \left( {e_2 e_3\,u\,S} \right)}{\partial i} |
---|
| 904 | +\frac{\partial \left( {e_1 e_3\,v\,S} \right)}{\partial j}} \right] \\ |
---|
| 905 | -\frac{1}{e_3 }\frac{\partial \left( {S\,\omega } \right)}{\partial k} + D^S + F^S \qquad |
---|
| 906 | \end{multline*} |
---|
[2282] | 907 | |
---|
[10354] | 908 | The equation of state has the same expression as in $z$-coordinate, |
---|
| 909 | and similar expressions are used for mixing and forcing terms. |
---|
[2282] | 910 | |
---|
| 911 | \gmcomment{ |
---|
| 912 | \colorbox{yellow}{ to be updated $= = >$} |
---|
| 913 | Add a few works on z and zps and s and underlies the differences between all of them |
---|
| 914 | \colorbox{yellow}{ $< = =$ end update} } |
---|
| 915 | |
---|
| 916 | |
---|
| 917 | |
---|
| 918 | % ------------------------------------------------------------------------------------------------------------- |
---|
| 919 | % Curvilinear z*-coordinate System |
---|
| 920 | % ------------------------------------------------------------------------------------------------------------- |
---|
[9393] | 921 | \subsection{Curvilinear \textit{z*}--coordinate system} |
---|
[9407] | 922 | \label{subsec:PE_zco_star} |
---|
[2282] | 923 | |
---|
[817] | 924 | %>>>>>>>>>>>>>>>>>>>>>>>>>>>> |
---|
[10354] | 925 | \begin{figure}[!b] |
---|
| 926 | \begin{center} |
---|
| 927 | \includegraphics[width=1.0\textwidth]{Fig_z_zstar} |
---|
| 928 | \caption{ \protect\label{fig:z_zstar} |
---|
| 929 | (a) $z$-coordinate in linear free-surface case ; |
---|
| 930 | (b) $z-$coordinate in non-linear free surface case ; |
---|
| 931 | (c) re-scaled height coordinate (become popular as the \textit{z*-}coordinate |
---|
| 932 | \citep{Adcroft_Campin_OM04} ). |
---|
| 933 | } |
---|
| 934 | \end{center} |
---|
| 935 | \end{figure} |
---|
[817] | 936 | %>>>>>>>>>>>>>>>>>>>>>>>>>>>> |
---|
| 937 | |
---|
| 938 | |
---|
[10354] | 939 | In that case, the free surface equation is nonlinear, and the variations of volume are fully taken into account. |
---|
| 940 | These coordinates systems is presented in a report \citep{Levier2007} available on the \NEMO web site. |
---|
[817] | 941 | |
---|
[2282] | 942 | %\gmcomment{ |
---|
[10354] | 943 | The \textit{z*} coordinate approach is an unapproximated, non-linear free surface implementation which allows one to |
---|
| 944 | deal with large amplitude free-surface variations relative to the vertical resolution \citep{Adcroft_Campin_OM04}. |
---|
| 945 | In the \textit{z*} formulation, |
---|
| 946 | the variation of the column thickness due to sea-surface undulations is not concentrated in the surface level, |
---|
| 947 | as in the $z$-coordinate formulation, but is equally distributed over the full water column. |
---|
| 948 | Thus vertical levels naturally follow sea-surface variations, with a linear attenuation with depth, |
---|
| 949 | as illustrated by figure fig.1c. |
---|
| 950 | Note that with a flat bottom, such as in fig.1c, the bottom-following $z$ coordinate and \textit{z*} are equivalent. |
---|
| 951 | The definition and modified oceanic equations for the rescaled vertical coordinate \textit{z*}, |
---|
| 952 | including the treatment of fresh-water flux at the surface, are detailed in Adcroft and Campin (2004). |
---|
| 953 | The major points are summarized here. |
---|
| 954 | The position ( \textit{z*}) and vertical discretization (\textit{z*}) are expressed as: |
---|
[10414] | 955 | \[ |
---|
| 956 | % \label{eq:z-star} |
---|
| 957 | H + \textit{z*} = (H + z) / r \quad \text{and} \ \delta \textit{z*} = \delta z / r \quad \text{with} \ r = \frac{H+\eta} {H} |
---|
| 958 | \] |
---|
[10354] | 959 | Since the vertical displacement of the free surface is incorporated in the vertical coordinate \textit{z*}, |
---|
| 960 | the upper and lower boundaries are at fixed \textit{z*} position, |
---|
| 961 | $\textit{z*} = 0$ and $\textit{z*} = -H$ respectively. |
---|
| 962 | Also the divergence of the flow field is no longer zero as shown by the continuity equation: |
---|
[10406] | 963 | \[ |
---|
[10414] | 964 | \frac{\partial r}{\partial t} = \nabla_{\textit{z*}} \cdot \left( r \; \rm{\bf U}_h \right) |
---|
| 965 | \left( r \; w\textit{*} \right) = 0 |
---|
[10406] | 966 | \] |
---|
[2282] | 967 | %} |
---|
[817] | 968 | |
---|
| 969 | |
---|
[2282] | 970 | % from MOM4p1 documentation |
---|
[817] | 971 | |
---|
[10354] | 972 | To overcome problems with vanishing surface and/or bottom cells, we consider the zstar coordinate |
---|
[10414] | 973 | \[ |
---|
| 974 | % \label{eq:PE_} |
---|
| 975 | z^\star = H \left( \frac{z-\eta}{H+\eta} \right) |
---|
| 976 | \] |
---|
[817] | 977 | |
---|
[10354] | 978 | This coordinate is closely related to the "eta" coordinate used in many atmospheric models |
---|
| 979 | (see Black (1994) for a review of eta coordinate atmospheric models). |
---|
| 980 | It was originally used in ocean models by Stacey et al. (1995) for studies of tides next to shelves, |
---|
| 981 | and it has been recently promoted by Adcroft and Campin (2004) for global climate modelling. |
---|
[2282] | 982 | |
---|
[10354] | 983 | The surfaces of constant $z^\star$ are quasi-horizontal. |
---|
| 984 | Indeed, the $z^\star$ coordinate reduces to $z$ when $\eta$ is zero. |
---|
| 985 | In general, when noting the large differences between |
---|
| 986 | undulations of the bottom topography versus undulations in the surface height, |
---|
| 987 | it is clear that surfaces constant $z^\star$ are very similar to the depth surfaces. |
---|
| 988 | These properties greatly reduce difficulties of computing the horizontal pressure gradient relative to |
---|
| 989 | terrain following sigma models discussed in \autoref{subsec:PE_sco}. |
---|
| 990 | Additionally, since $z^\star$ when $\eta = 0$, |
---|
| 991 | no flow is spontaneously generated in an unforced ocean starting from rest, regardless the bottom topography. |
---|
| 992 | This behaviour is in contrast to the case with "s"-models, where pressure gradient errors in the presence of |
---|
| 993 | nontrivial topographic variations can generate nontrivial spontaneous flow from a resting state, |
---|
| 994 | depending on the sophistication of the pressure gradient solver. |
---|
| 995 | The quasi-horizontal nature of the coordinate surfaces also facilitates the implementation of |
---|
| 996 | neutral physics parameterizations in $z^\star$ models using the same techniques as in $z$-models |
---|
| 997 | (see Chapters 13-16 of \cite{Griffies_Bk04}) for a discussion of neutral physics in $z$-models, |
---|
| 998 | as well as \autoref{sec:LDF_slp} in this document for treatment in \NEMO). |
---|
[2282] | 999 | |
---|
[10354] | 1000 | The range over which $z^\star$ varies is time independent $-H \leq z^\star \leq 0$. |
---|
| 1001 | Hence, all cells remain nonvanishing, so long as the surface height maintains $\eta > ?H$. |
---|
| 1002 | This is a minor constraint relative to that encountered on the surface height when using $s = z$ or $s = z - \eta$. |
---|
[2282] | 1003 | |
---|
[10354] | 1004 | Because $z^\star$ has a time independent range, all grid cells have static increments ds, |
---|
[10414] | 1005 | and the sum of the ver tical increments yields the time independent ocean depth. %k ds = H. |
---|
[10354] | 1006 | The $z^\star$ coordinate is therefore invisible to undulations of the free surface, |
---|
| 1007 | since it moves along with the free surface. |
---|
| 1008 | This proper ty means that no spurious vertical transport is induced across surfaces of constant $z^\star$ by |
---|
| 1009 | the motion of external gravity waves. |
---|
| 1010 | Such spurious transpor t can be a problem in z-models, especially those with tidal forcing. |
---|
| 1011 | Quite generally, the time independent range for the $z^\star$ coordinate is a very convenient property that |
---|
| 1012 | allows for a nearly arbitrary ver tical resolution even in the presence of large amplitude fluctuations of |
---|
| 1013 | the surface height, again so long as $\eta > -H$. |
---|
[2282] | 1014 | |
---|
| 1015 | %end MOM doc %%% |
---|
| 1016 | |
---|
| 1017 | |
---|
| 1018 | |
---|
[10414] | 1019 | \newpage |
---|
| 1020 | |
---|
[2282] | 1021 | % ------------------------------------------------------------------------------------------------------------- |
---|
| 1022 | % Terrain following coordinate System |
---|
| 1023 | % ------------------------------------------------------------------------------------------------------------- |
---|
[9393] | 1024 | \subsection{Curvilinear terrain-following \textit{s}--coordinate} |
---|
[9407] | 1025 | \label{subsec:PE_sco} |
---|
[707] | 1026 | |
---|
| 1027 | % ------------------------------------------------------------------------------------------------------------- |
---|
| 1028 | % Introduction |
---|
| 1029 | % ------------------------------------------------------------------------------------------------------------- |
---|
[2282] | 1030 | \subsubsection{Introduction} |
---|
[707] | 1031 | |
---|
[10354] | 1032 | Several important aspects of the ocean circulation are influenced by bottom topography. |
---|
| 1033 | Of course, the most important is that bottom topography determines deep ocean sub-basins, barriers, sills and |
---|
| 1034 | channels that strongly constrain the path of water masses, but more subtle effects exist. |
---|
| 1035 | For example, the topographic $\beta$-effect is usually larger than the planetary one along continental slopes. |
---|
| 1036 | Topographic Rossby waves can be excited and can interact with the mean current. |
---|
| 1037 | In the $z-$coordinate system presented in the previous section (\autoref{sec:PE_zco}), |
---|
| 1038 | $z-$surfaces are geopotential surfaces. |
---|
| 1039 | The bottom topography is discretised by steps. |
---|
| 1040 | This often leads to a misrepresentation of a gradually sloping bottom and to |
---|
| 1041 | large localized depth gradients associated with large localized vertical velocities. |
---|
| 1042 | The response to such a velocity field often leads to numerical dispersion effects. |
---|
| 1043 | One solution to strongly reduce this error is to use a partial step representation of bottom topography instead of |
---|
| 1044 | a full step one \cite{Pacanowski_Gnanadesikan_MWR98}. |
---|
| 1045 | Another solution is to introduce a terrain-following coordinate system (hereafter $s-$coordinate). |
---|
[707] | 1046 | |
---|
[10354] | 1047 | The $s$-coordinate avoids the discretisation error in the depth field since the layers of |
---|
| 1048 | computation are gradually adjusted with depth to the ocean bottom. |
---|
| 1049 | Relatively small topographic features as well as gentle, large-scale slopes of the sea floor in the deep ocean, |
---|
| 1050 | which would be ignored in typical $z$-model applications with the largest grid spacing at greatest depths, |
---|
| 1051 | can easily be represented (with relatively low vertical resolution). |
---|
| 1052 | A terrain-following model (hereafter $s-$model) also facilitates the modelling of the boundary layer flows over |
---|
| 1053 | a large depth range, which in the framework of the $z$-model would require high vertical resolution over |
---|
| 1054 | the whole depth range. |
---|
| 1055 | Moreover, with a $s$-coordinate it is possible, at least in principle, to have the bottom and the sea surface as |
---|
| 1056 | the only boundaries of the domain (no more lateral boundary condition to specify). |
---|
| 1057 | Nevertheless, a $s$-coordinate also has its drawbacks. Perfectly adapted to a homogeneous ocean, |
---|
| 1058 | it has strong limitations as soon as stratification is introduced. |
---|
| 1059 | The main two problems come from the truncation error in the horizontal pressure gradient and |
---|
| 1060 | a possibly increased diapycnal diffusion. |
---|
| 1061 | The horizontal pressure force in $s$-coordinate consists of two terms (see \autoref{apdx:A}), |
---|
[707] | 1062 | |
---|
[10414] | 1063 | \begin{equation} |
---|
| 1064 | \label{eq:PE_p_sco} |
---|
| 1065 | \left. {\nabla p} \right|_z =\left. {\nabla p} \right|_s -\frac{\partial |
---|
| 1066 | p}{\partial s}\left. {\nabla z} \right|_s |
---|
[707] | 1067 | \end{equation} |
---|
| 1068 | |
---|
[10354] | 1069 | The second term in \autoref{eq:PE_p_sco} depends on the tilt of the coordinate surface and |
---|
| 1070 | introduces a truncation error that is not present in a $z$-model. |
---|
| 1071 | In the special case of a $\sigma-$coordinate (i.e. a depth-normalised coordinate system $\sigma = z/H$), |
---|
| 1072 | \citet{Haney1991} and \citet{Beckmann1993} have given estimates of the magnitude of this truncation error. |
---|
| 1073 | It depends on topographic slope, stratification, horizontal and vertical resolution, the equation of state, |
---|
| 1074 | and the finite difference scheme. |
---|
| 1075 | This error limits the possible topographic slopes that a model can handle at |
---|
| 1076 | a given horizontal and vertical resolution. |
---|
| 1077 | This is a severe restriction for large-scale applications using realistic bottom topography. |
---|
| 1078 | The large-scale slopes require high horizontal resolution, and the computational cost becomes prohibitive. |
---|
| 1079 | This problem can be at least partially overcome by mixing $s$-coordinate and |
---|
| 1080 | step-like representation of bottom topography \citep{Gerdes1993a,Gerdes1993b,Madec_al_JPO96}. |
---|
| 1081 | However, the definition of the model domain vertical coordinate becomes then a non-trivial thing for |
---|
| 1082 | a realistic bottom topography: |
---|
| 1083 | a envelope topography is defined in $s$-coordinate on which a full or |
---|
| 1084 | partial step bottom topography is then applied in order to adjust the model depth to the observed one |
---|
| 1085 | (see \autoref{sec:DOM_zgr}. |
---|
[707] | 1086 | |
---|
[10354] | 1087 | For numerical reasons a minimum of diffusion is required along the coordinate surfaces of any finite difference model. |
---|
| 1088 | It causes spurious diapycnal mixing when coordinate surfaces do not coincide with isoneutral surfaces. |
---|
| 1089 | This is the case for a $z$-model as well as for a $s$-model. |
---|
| 1090 | However, density varies more strongly on $s-$surfaces than on horizontal surfaces in regions of |
---|
| 1091 | large topographic slopes, implying larger diapycnal diffusion in a $s$-model than in a $z$-model. |
---|
| 1092 | Whereas such a diapycnal diffusion in a $z$-model tends to weaken horizontal density (pressure) gradients and thus |
---|
| 1093 | the horizontal circulation, it usually reinforces these gradients in a $s$-model, creating spurious circulation. |
---|
| 1094 | For example, imagine an isolated bump of topography in an ocean at rest with a horizontally uniform stratification. |
---|
| 1095 | Spurious diffusion along $s$-surfaces will induce a bump of isoneutral surfaces over the topography, |
---|
| 1096 | and thus will generate there a baroclinic eddy. |
---|
| 1097 | In contrast, the ocean will stay at rest in a $z$-model. |
---|
| 1098 | As for the truncation error, the problem can be reduced by introducing the terrain-following coordinate below |
---|
| 1099 | the strongly stratified portion of the water column ($i.e.$ the main thermocline) \citep{Madec_al_JPO96}. |
---|
| 1100 | An alternate solution consists of rotating the lateral diffusive tensor to geopotential or to isoneutral surfaces |
---|
| 1101 | (see \autoref{subsec:PE_ldf}). |
---|
| 1102 | Unfortunately, the slope of isoneutral surfaces relative to the $s$-surfaces can very large, |
---|
[9407] | 1103 | strongly exceeding the stability limit of such a operator when it is discretized (see \autoref{chap:LDF}). |
---|
[707] | 1104 | |
---|
[10354] | 1105 | The $s-$coordinates introduced here \citep{Lott_al_OM90,Madec_al_JPO96} differ mainly in two aspects from |
---|
| 1106 | similar models: |
---|
| 1107 | it allows a representation of bottom topography with mixed full or partial step-like/terrain following topography; |
---|
| 1108 | It also offers a completely general transformation, $s=s(i,j,z)$ for the vertical coordinate. |
---|
[707] | 1109 | |
---|
[2282] | 1110 | |
---|
[10414] | 1111 | \newpage |
---|
| 1112 | |
---|
[707] | 1113 | % ------------------------------------------------------------------------------------------------------------- |
---|
[2282] | 1114 | % Curvilinear z-tilde coordinate System |
---|
[707] | 1115 | % ------------------------------------------------------------------------------------------------------------- |
---|
[9347] | 1116 | \subsection{\texorpdfstring{Curvilinear $\tilde{z}$--coordinate}{}} |
---|
[9407] | 1117 | \label{subsec:PE_zco_tilde} |
---|
[707] | 1118 | |
---|
[6289] | 1119 | The $\tilde{z}$-coordinate has been developed by \citet{Leclair_Madec_OM11}. |
---|
[10354] | 1120 | It is available in \NEMO since the version 3.4. |
---|
| 1121 | Nevertheless, it is currently not robust enough to be used in all possible configurations. |
---|
| 1122 | Its use is therefore not recommended. |
---|
[707] | 1123 | |
---|
[6289] | 1124 | |
---|
[707] | 1125 | \newpage |
---|
[10414] | 1126 | |
---|
[707] | 1127 | % ================================================================ |
---|
| 1128 | % Subgrid Scale Physics |
---|
| 1129 | % ================================================================ |
---|
[9393] | 1130 | \section{Subgrid scale physics} |
---|
[9407] | 1131 | \label{sec:PE_zdf_ldf} |
---|
[707] | 1132 | |
---|
[10354] | 1133 | The primitive equations describe the behaviour of a geophysical fluid at space and time scales larger than |
---|
| 1134 | a few kilometres in the horizontal, a few meters in the vertical and a few minutes. |
---|
| 1135 | They are usually solved at larger scales: the specified grid spacing and time step of the numerical model. |
---|
| 1136 | The effects of smaller scale motions (coming from the advective terms in the Navier-Stokes equations) must be represented entirely in terms of large-scale patterns to close the equations. |
---|
| 1137 | These effects appear in the equations as the divergence of turbulent fluxes |
---|
| 1138 | ($i.e.$ fluxes associated with the mean correlation of small scale perturbations). |
---|
| 1139 | Assuming a turbulent closure hypothesis is equivalent to choose a formulation for these fluxes. |
---|
| 1140 | It is usually called the subgrid scale physics. |
---|
| 1141 | It must be emphasized that this is the weakest part of the primitive equations, |
---|
| 1142 | but also one of the most important for long-term simulations as |
---|
| 1143 | small scale processes \textit{in fine} balance the surface input of kinetic energy and heat. |
---|
[707] | 1144 | |
---|
[10354] | 1145 | The control exerted by gravity on the flow induces a strong anisotropy between the lateral and vertical motions. |
---|
| 1146 | Therefore subgrid-scale physics \textbf{D}$^{\vect{U}}$, $D^{S}$ and $D^{T}$ in |
---|
| 1147 | \autoref{eq:PE_dyn}, \autoref{eq:PE_tra_T} and \autoref{eq:PE_tra_S} are divided into |
---|
| 1148 | a lateral part \textbf{D}$^{l \vect{U}}$, $D^{lS}$ and $D^{lT}$ and |
---|
| 1149 | a vertical part \textbf{D}$^{vU}$, $D^{vS}$ and $D^{vT}$. |
---|
| 1150 | The formulation of these terms and their underlying physics are briefly discussed in the next two subsections. |
---|
[707] | 1151 | |
---|
| 1152 | % ------------------------------------------------------------------------------------------------------------- |
---|
| 1153 | % Vertical Subgrid Scale Physics |
---|
| 1154 | % ------------------------------------------------------------------------------------------------------------- |
---|
[9393] | 1155 | \subsection{Vertical subgrid scale physics} |
---|
[9407] | 1156 | \label{subsec:PE_zdf} |
---|
[707] | 1157 | |
---|
[10354] | 1158 | The model resolution is always larger than the scale at which the major sources of vertical turbulence occur |
---|
| 1159 | (shear instability, internal wave breaking...). |
---|
| 1160 | Turbulent motions are thus never explicitly solved, even partially, but always parameterized. |
---|
| 1161 | The vertical turbulent fluxes are assumed to depend linearly on the gradients of large-scale quantities |
---|
| 1162 | (for example, the turbulent heat flux is given by $\overline{T'w'}=-A^{vT} \partial_z \overline T$, |
---|
| 1163 | where $A^{vT}$ is an eddy coefficient). |
---|
| 1164 | This formulation is analogous to that of molecular diffusion and dissipation. |
---|
| 1165 | This is quite clearly a necessary compromise: considering only the molecular viscosity acting on |
---|
| 1166 | large scale severely underestimates the role of turbulent diffusion and dissipation, |
---|
| 1167 | while an accurate consideration of the details of turbulent motions is simply impractical. |
---|
| 1168 | The resulting vertical momentum and tracer diffusive operators are of second order: |
---|
[10414] | 1169 | \begin{equation} |
---|
| 1170 | \label{eq:PE_zdf} |
---|
| 1171 | \begin{split} |
---|
| 1172 | {\vect{D}}^{v \vect{U}} &=\frac{\partial }{\partial z}\left( {A^{vm}\frac{\partial {\vect{U}}_h }{\partial z}} \right) \ , \\ |
---|
| 1173 | D^{vT} &= \frac{\partial }{\partial z}\left( {A^{vT}\frac{\partial T}{\partial z}} \right) \ , |
---|
| 1174 | \quad |
---|
| 1175 | D^{vS}=\frac{\partial }{\partial z}\left( {A^{vT}\frac{\partial S}{\partial z}} \right) |
---|
| 1176 | \end{split} |
---|
[707] | 1177 | \end{equation} |
---|
[10354] | 1178 | where $A^{vm}$ and $A^{vT}$ are the vertical eddy viscosity and diffusivity coefficients, respectively. |
---|
| 1179 | At the sea surface and at the bottom, turbulent fluxes of momentum, heat and salt must be specified |
---|
| 1180 | (see \autoref{chap:SBC} and \autoref{chap:ZDF} and \autoref{sec:TRA_bbl}). |
---|
| 1181 | All the vertical physics is embedded in the specification of the eddy coefficients. |
---|
| 1182 | They can be assumed to be either constant, or function of the local fluid properties |
---|
| 1183 | ($e.g.$ Richardson number, Brunt-Vais\"{a}l\"{a} frequency...), |
---|
| 1184 | or computed from a turbulent closure model. |
---|
| 1185 | The choices available in \NEMO are discussed in \autoref{chap:ZDF}). |
---|
[707] | 1186 | |
---|
| 1187 | % ------------------------------------------------------------------------------------------------------------- |
---|
| 1188 | % Lateral Diffusive and Viscous Operators Formulation |
---|
| 1189 | % ------------------------------------------------------------------------------------------------------------- |
---|
[9393] | 1190 | \subsection{Formulation of the lateral diffusive and viscous operators} |
---|
[9407] | 1191 | \label{subsec:PE_ldf} |
---|
[707] | 1192 | |
---|
[10354] | 1193 | Lateral turbulence can be roughly divided into a mesoscale turbulence associated with eddies |
---|
| 1194 | (which can be solved explicitly if the resolution is sufficient since |
---|
| 1195 | their underlying physics are included in the primitive equations), |
---|
| 1196 | and a sub mesoscale turbulence which is never explicitly solved even partially, but always parameterized. |
---|
| 1197 | The formulation of lateral eddy fluxes depends on whether the mesoscale is below or above the grid-spacing |
---|
[817] | 1198 | ($i.e.$ the model is eddy-resolving or not). |
---|
[707] | 1199 | |
---|
[10354] | 1200 | In non-eddy-resolving configurations, the closure is similar to that used for the vertical physics. |
---|
| 1201 | The lateral turbulent fluxes are assumed to depend linearly on the lateral gradients of large-scale quantities. |
---|
| 1202 | The resulting lateral diffusive and dissipative operators are of second order. |
---|
| 1203 | Observations show that lateral mixing induced by mesoscale turbulence tends to be along isopycnal surfaces |
---|
| 1204 | (or more precisely neutral surfaces \cite{McDougall1987}) rather than across them. |
---|
| 1205 | As the slope of neutral surfaces is small in the ocean, a common approximation is to assume that |
---|
| 1206 | the `lateral' direction is the horizontal, $i.e.$ the lateral mixing is performed along geopotential surfaces. |
---|
| 1207 | This leads to a geopotential second order operator for lateral subgrid scale physics. |
---|
| 1208 | This assumption can be relaxed: the eddy-induced turbulent fluxes can be better approached by assuming that |
---|
| 1209 | they depend linearly on the gradients of large-scale quantities computed along neutral surfaces. |
---|
| 1210 | In such a case, the diffusive operator is an isoneutral second order operator and |
---|
| 1211 | it has components in the three space directions. |
---|
| 1212 | However, |
---|
| 1213 | both horizontal and isoneutral operators have no effect on mean ($i.e.$ large scale) potential energy whereas |
---|
| 1214 | potential energy is a main source of turbulence (through baroclinic instabilities). |
---|
| 1215 | \citet{Gent1990} have proposed a parameterisation of mesoscale eddy-induced turbulence which |
---|
| 1216 | associates an eddy-induced velocity to the isoneutral diffusion. |
---|
| 1217 | Its mean effect is to reduce the mean potential energy of the ocean. |
---|
| 1218 | This leads to a formulation of lateral subgrid-scale physics made up of an isoneutral second order operator and |
---|
| 1219 | an eddy induced advective part. |
---|
| 1220 | In all these lateral diffusive formulations, |
---|
| 1221 | the specification of the lateral eddy coefficients remains the problematic point as |
---|
| 1222 | there is no really satisfactory formulation of these coefficients as a function of large-scale features. |
---|
[707] | 1223 | |
---|
[10354] | 1224 | In eddy-resolving configurations, a second order operator can be used, |
---|
| 1225 | but usually the more scale selective biharmonic operator is preferred as |
---|
| 1226 | the grid-spacing is usually not small enough compared to the scale of the eddies. |
---|
| 1227 | The role devoted to the subgrid-scale physics is to dissipate the energy that |
---|
| 1228 | cascades toward the grid scale and thus to ensure the stability of the model while |
---|
| 1229 | not interfering with the resolved mesoscale activity. |
---|
| 1230 | Another approach is becoming more and more popular: |
---|
| 1231 | instead of specifying explicitly a sub-grid scale term in the momentum and tracer time evolution equations, |
---|
| 1232 | one uses a advective scheme which is diffusive enough to maintain the model stability. |
---|
| 1233 | It must be emphasised that then, all the sub-grid scale physics is included in the formulation of |
---|
| 1234 | the advection scheme. |
---|
[707] | 1235 | |
---|
[10354] | 1236 | All these parameterisations of subgrid scale physics have advantages and drawbacks. |
---|
| 1237 | There are not all available in \NEMO. For active tracers (temperature and salinity) the main ones are: |
---|
| 1238 | Laplacian and bilaplacian operators acting along geopotential or iso-neutral surfaces, |
---|
| 1239 | \citet{Gent1990} parameterisation, and various slightly diffusive advection schemes. |
---|
| 1240 | For momentum, the main ones are: Laplacian and bilaplacian operators acting along geopotential surfaces, |
---|
| 1241 | and UBS advection schemes when flux form is chosen for the momentum advection. |
---|
[707] | 1242 | |
---|
[9393] | 1243 | \subsubsection{Lateral laplacian tracer diffusive operator} |
---|
[707] | 1244 | |
---|
[9407] | 1245 | The lateral Laplacian tracer diffusive operator is defined by (see \autoref{apdx:B}): |
---|
[10414] | 1246 | \begin{equation} |
---|
| 1247 | \label{eq:PE_iso_tensor} |
---|
| 1248 | D^{lT}=\nabla {\rm {\bf .}}\left( {A^{lT}\;\Re \;\nabla T} \right) \qquad |
---|
| 1249 | \mbox{with}\quad \;\;\Re =\left( {{ |
---|
| 1250 | \begin{array}{*{20}c} |
---|
| 1251 | 1 \hfill & 0 \hfill & {-r_1 } \hfill \\ |
---|
| 1252 | 0 \hfill & 1 \hfill & {-r_2 } \hfill \\ |
---|
| 1253 | {-r_1 } \hfill & {-r_2 } \hfill & {r_1 ^2+r_2 ^2} \hfill \\ |
---|
| 1254 | \end{array} |
---|
| 1255 | }} \right) |
---|
[707] | 1256 | \end{equation} |
---|
[10354] | 1257 | where $r_1 \;\mbox{and}\;r_2 $ are the slopes between the surface along which the diffusive operator acts and |
---|
| 1258 | the model level ($e. g.$ $z$- or $s$-surfaces). |
---|
| 1259 | Note that the formulation \autoref{eq:PE_iso_tensor} is exact for |
---|
| 1260 | the rotation between geopotential and $s$-surfaces, |
---|
| 1261 | while it is only an approximation for the rotation between isoneutral and $z$- or $s$-surfaces. |
---|
| 1262 | Indeed, in the latter case, two assumptions are made to simplify \autoref{eq:PE_iso_tensor} \citep{Cox1987}. |
---|
| 1263 | First, the horizontal contribution of the dianeutral mixing is neglected since the ratio between iso and |
---|
| 1264 | dia-neutral diffusive coefficients is known to be several orders of magnitude smaller than unity. |
---|
| 1265 | Second, the two isoneutral directions of diffusion are assumed to be independent since |
---|
| 1266 | the slopes are generally less than $10^{-2}$ in the ocean (see \autoref{apdx:B}). |
---|
[707] | 1267 | |
---|
[10354] | 1268 | For \textit{iso-level} diffusion, $r_1$ and $r_2 $ are zero. |
---|
| 1269 | $\Re $ reduces to the identity in the horizontal direction, no rotation is applied. |
---|
[6140] | 1270 | |
---|
[10354] | 1271 | For \textit{geopotential} diffusion, |
---|
| 1272 | $r_1$ and $r_2 $ are the slopes between the geopotential and computational surfaces: |
---|
[10414] | 1273 | they are equal to $\sigma_1$ and $\sigma_2$, respectively (see \autoref{eq:PE_sco_slope}). |
---|
[707] | 1274 | |
---|
[10354] | 1275 | For \textit{isoneutral} diffusion $r_1$ and $r_2$ are the slopes between the isoneutral and computational surfaces. |
---|
| 1276 | Therefore, they are different quantities, but have similar expressions in $z$- and $s$-coordinates. |
---|
| 1277 | In $z$-coordinates: |
---|
[10414] | 1278 | \begin{equation} |
---|
| 1279 | \label{eq:PE_iso_slopes} |
---|
| 1280 | r_1 =\frac{e_3 }{e_1 } \left( \pd[\rho]{i} \right) \left( \pd[\rho]{k} \right)^{-1} \, \quad |
---|
| 1281 | r_2 =\frac{e_3 }{e_2 } \left( \pd[\rho]{j} \right) \left( \pd[\rho]{k} \right)^{-1} \, |
---|
[707] | 1282 | \end{equation} |
---|
[6289] | 1283 | while in $s$-coordinates $\pd[]{k}$ is replaced by $\pd[]{s}$. |
---|
[707] | 1284 | |
---|
[3294] | 1285 | \subsubsection{Eddy induced velocity} |
---|
[10354] | 1286 | When the \textit{eddy induced velocity} parametrisation (eiv) \citep{Gent1990} is used, |
---|
[1224] | 1287 | an additional tracer advection is introduced in combination with the isoneutral diffusion of tracers: |
---|
[10414] | 1288 | \[ |
---|
| 1289 | % \label{eq:PE_iso+eiv} |
---|
| 1290 | D^{lT}=\nabla \cdot \left( {A^{lT}\;\Re \;\nabla T} \right) |
---|
| 1291 | +\nabla \cdot \left( {{\vect{U}}^\ast \,T} \right) |
---|
| 1292 | \] |
---|
[10354] | 1293 | where ${\vect{U}}^\ast =\left( {u^\ast ,v^\ast ,w^\ast } \right)$ is a non-divergent, |
---|
[1224] | 1294 | eddy-induced transport velocity. This velocity field is defined by: |
---|
[10414] | 1295 | \[ |
---|
| 1296 | % \label{eq:PE_eiv} |
---|
| 1297 | \begin{split} |
---|
| 1298 | u^\ast &= +\frac{1}{e_3 }\frac{\partial }{\partial k}\left[ {A^{eiv}\;\tilde{r}_1 } \right] \\ |
---|
| 1299 | v^\ast &= +\frac{1}{e_3 }\frac{\partial }{\partial k}\left[ {A^{eiv}\;\tilde{r}_2 } \right] \\ |
---|
| 1300 | w^\ast &= -\frac{1}{e_1 e_2 }\left[ |
---|
| 1301 | \frac{\partial }{\partial i}\left( {A^{eiv}\;e_2\,\tilde{r}_1 } \right) |
---|
| 1302 | +\frac{\partial }{\partial j}\left( {A^{eiv}\;e_1\,\tilde{r}_2 } \right) \right] |
---|
| 1303 | \end{split} |
---|
| 1304 | \] |
---|
[10354] | 1305 | where $A^{eiv}$ is the eddy induced velocity coefficient |
---|
| 1306 | (or equivalently the isoneutral thickness diffusivity coefficient), |
---|
| 1307 | and $\tilde{r}_1$ and $\tilde{r}_2$ are the slopes between isoneutral and \emph{geopotential} surfaces. |
---|
| 1308 | Their values are thus independent of the vertical coordinate, but their expression depends on the coordinate: |
---|
[10414] | 1309 | \begin{align} |
---|
| 1310 | \label{eq:PE_slopes_eiv} |
---|
| 1311 | \tilde{r}_n = |
---|
| 1312 | \begin{cases} |
---|
| 1313 | r_n & \text{in $z$-coordinate} \\ |
---|
| 1314 | r_n + \sigma_n & \text{in \textit{z*} and $s$-coordinates} |
---|
| 1315 | \end{cases} |
---|
| 1316 | \quad \text{where } n=1,2 |
---|
[707] | 1317 | \end{align} |
---|
| 1318 | |
---|
[10354] | 1319 | The normal component of the eddy induced velocity is zero at all the boundaries. |
---|
| 1320 | This can be achieved in a model by tapering either the eddy coefficient or |
---|
| 1321 | the slopes to zero in the vicinity of the boundaries. |
---|
| 1322 | The latter strategy is used in \NEMO (cf. \autoref{chap:LDF}). |
---|
[707] | 1323 | |
---|
[6289] | 1324 | \subsubsection{Lateral bilaplacian tracer diffusive operator} |
---|
[707] | 1325 | |
---|
[6289] | 1326 | The lateral bilaplacian tracer diffusive operator is defined by: |
---|
[10414] | 1327 | \[ |
---|
| 1328 | % \label{eq:PE_bilapT} |
---|
| 1329 | D^{lT}= - \Delta \left( \;\Delta T \right) |
---|
| 1330 | \qquad \text{where} \;\; \Delta \bullet = \nabla \left( {\sqrt{B^{lT}\,}\;\Re \;\nabla \bullet} \right) |
---|
| 1331 | \] |
---|
[10354] | 1332 | It is the Laplacian operator given by \autoref{eq:PE_iso_tensor} applied twice with |
---|
[6140] | 1333 | the harmonic eddy diffusion coefficient set to the square root of the biharmonic one. |
---|
[707] | 1334 | |
---|
[817] | 1335 | |
---|
[6289] | 1336 | \subsubsection{Lateral Laplacian momentum diffusive operator} |
---|
[707] | 1337 | |
---|
[10354] | 1338 | The Laplacian momentum diffusive operator along $z$- or $s$-surfaces is found by |
---|
[9407] | 1339 | applying \autoref{eq:PE_lap_vector} to the horizontal velocity vector (see \autoref{apdx:B}): |
---|
[10414] | 1340 | \[ |
---|
| 1341 | % \label{eq:PE_lapU} |
---|
| 1342 | \begin{split} |
---|
| 1343 | {\rm {\bf D}}^{l{\rm {\bf U}}} |
---|
| 1344 | &= \quad \ \nabla _h \left( {A^{lm}\chi } \right) |
---|
| 1345 | \ - \ \nabla _h \times \left( {A^{lm}\,\zeta \;{\rm {\bf k}}} \right) \\ |
---|
| 1346 | &= \left( |
---|
| 1347 | \begin{aligned} |
---|
| 1348 | \frac{1}{e_1 } \frac{\partial \left( A^{lm} \chi \right)}{\partial i} |
---|
| 1349 | &-\frac{1}{e_2 e_3}\frac{\partial \left( {A^{lm} \;e_3 \zeta} \right)}{\partial j} \\ |
---|
| 1350 | \frac{1}{e_2 }\frac{\partial \left( {A^{lm} \chi } \right)}{\partial j} |
---|
| 1351 | &+\frac{1}{e_1 e_3}\frac{\partial \left( {A^{lm} \;e_3 \zeta} \right)}{\partial i} |
---|
| 1352 | \end{aligned} |
---|
| 1353 | \right) |
---|
| 1354 | \end{split} |
---|
| 1355 | \] |
---|
[707] | 1356 | |
---|
[10354] | 1357 | Such a formulation ensures a complete separation between the vorticity and horizontal divergence fields |
---|
| 1358 | (see \autoref{apdx:C}). |
---|
| 1359 | Unfortunately, it is only available in \textit{iso-level} direction. |
---|
| 1360 | When a rotation is required |
---|
| 1361 | ($i.e.$ geopotential diffusion in $s-$coordinates or isoneutral diffusion in both $z$- and $s$-coordinates), |
---|
| 1362 | the $u$ and $v-$fields are considered as independent scalar fields, so that the diffusive operator is given by: |
---|
[10414] | 1363 | \[ |
---|
| 1364 | % \label{eq:PE_lapU_iso} |
---|
| 1365 | \begin{split} |
---|
| 1366 | D_u^{l{\rm {\bf U}}} &= \nabla .\left( {A^{lm} \;\Re \;\nabla u} \right) \\ |
---|
| 1367 | D_v^{l{\rm {\bf U}}} &= \nabla .\left( {A^{lm} \;\Re \;\nabla v} \right) |
---|
| 1368 | \end{split} |
---|
| 1369 | \] |
---|
[10354] | 1370 | where $\Re$ is given by \autoref{eq:PE_iso_tensor}. |
---|
| 1371 | It is the same expression as those used for diffusive operator on tracers. |
---|
| 1372 | It must be emphasised that such a formulation is only exact in a Cartesian coordinate system, |
---|
| 1373 | $i.e.$ on a $f-$ or $\beta-$plane, not on the sphere. |
---|
| 1374 | It is also a very good approximation in vicinity of the Equator in |
---|
| 1375 | a geographical coordinate system \citep{Lengaigne_al_JGR03}. |
---|
[707] | 1376 | |
---|
[6289] | 1377 | \subsubsection{lateral bilaplacian momentum diffusive operator} |
---|
[707] | 1378 | |
---|
[10354] | 1379 | As for tracers, the bilaplacian order momentum diffusive operator is a re-entering Laplacian operator with |
---|
| 1380 | the harmonic eddy diffusion coefficient set to the square root of the biharmonic one. |
---|
| 1381 | Nevertheless it is currently not available in the iso-neutral case. |
---|
[707] | 1382 | |
---|
[10414] | 1383 | \biblio |
---|
| 1384 | |
---|
[6997] | 1385 | \end{document} |
---|
| 1386 | |
---|