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r998 r1224 27 27 velocity, plus the following additional assumptions made from scale considerations: 28 28 29 \textit{(1) spherical earth approximation: }the geopotential surfaces are assumed to be spheres so that gravity (local vertical) is parallel to the earth's radius 29 \textit{(1) spherical earth approximation: }the geopotential surfaces are assumed to 30 be spheres so that gravity (local vertical) is parallel to the earth's radius 30 31 31 32 \textit{(2) thin-shell approximation: }the ocean depth is neglected compared to the earth's radius 32 33 33 \textit{(3) turbulent closure hypothesis: }the turbulent fluxes (which represent the effect of small scale processes on the large-scale) are expressed in terms of large-scale features 34 35 \textit{(4) Boussinesq hypothesis:} density variations are neglected except in their contribution to the buoyancy force 36 37 \textit{(5) Hydrostatic hypothesis: }the vertical momentum equation is reduced to a balance between the vertical pressure gradient and the buoyancy force (this removes convective processes from 38 the initial Navier-Stokes equations and so convective processes must be parameterized instead) 39 40 \textit{(6) Incompressibility hypothesis: }the three dimensional divergence of the velocity vector is assumed to be zero. 41 42 Because the gravitational force is so dominant in the equations of large-scale motions, it is useful to choose an orthogonal set of unit vectors (\textbf{i},\textbf{j},\textbf{k}) linked to the earth such that \textbf{k} is the local upward vector and (\textbf{i},\textbf{j}) are two vectors orthogonal to \textbf{k}, $i.e.$ tangent to the geopotential surfaces. Let us define the following variables: \textbf{U} the vector velocity, $\textbf{U}=\textbf{U}_h + w\, \textbf{k}$ (the subscript $h$ denotes the local horizontal vector, $i.e.$ over the (\textbf{i},\textbf{j}) plane), $T$ the potential temperature, $S$ the salinity, \textit{$\rho $} the \textit{in situ} density. The vector invariant form of the primitive equations in the (\textbf{i},\textbf{j},\textbf{k}) vector system provides the following six equations (namely the momentum balance, the hydrostatic equilibrium, the incompressibility equation, the heat and salt conservation equations and an equation of state): 34 \textit{(3) turbulent closure hypothesis: }the turbulent fluxes (which represent the effect 35 of small scale processes on the large-scale) are expressed in terms of large-scale features 36 37 \textit{(4) Boussinesq hypothesis:} density variations are neglected except in their 38 contribution to the buoyancy force 39 40 \textit{(5) Hydrostatic hypothesis: }the vertical momentum equation is reduced to a 41 balance between the vertical pressure gradient and the buoyancy force (this removes 42 convective processes from the initial Navier-Stokes equations and so convective processes 43 must be parameterized instead) 44 45 \textit{(6) Incompressibility hypothesis: }the three dimensional divergence of the velocity 46 vector is assumed to be zero. 47 48 Because the gravitational force is so dominant in the equations of large-scale motions, 49 it is useful to choose an orthogonal set of unit vectors (\textbf{i},\textbf{j},\textbf{k}) linked 50 to the earth such that \textbf{k} is the local upward vector and (\textbf{i},\textbf{j}) are two 51 vectors orthogonal to \textbf{k}, $i.e.$ tangent to the geopotential surfaces. Let us define 52 the following variables: \textbf{U} the vector velocity, $\textbf{U}=\textbf{U}_h + w\, \textbf{k}$ 53 (the subscript $h$ denotes the local horizontal vector, $i.e.$ over the (\textbf{i},\textbf{j}) plane), 54 $T$ the potential temperature, $S$ the salinity, \textit{$\rho $} the \textit{in situ} density. 55 The vector invariant form of the primitive equations in the (\textbf{i},\textbf{j},\textbf{k}) 56 vector system provides the following six equations (namely the momentum balance, the 57 hydrostatic equilibrium, the incompressibility equation, the heat and salt conservation 58 equations and an equation of state): 43 59 \begin{subequations} \label{Eq_PE} 44 60 \begin{equation} \label{Eq_PE_dyn} … … 65 81 \end{equation} 66 82 \end{subequations} 67 where $\nabla$ is the generalised derivative vector operator in $(\bf i,\bf j, \bf k)$ directions, $t$ is the time, $z$ is the vertical coordinate, $\rho $ is the \textit{in situ} density given by the equation of state (\ref{Eq_PE_eos}), $\rho_o$ is a reference density, $p$ the pressure, $f=2 \bf \Omega \cdot \bf k$ is the Coriolis acceleration (where $\bf \Omega$ is the Earth's angular velocity vector), and $g$ is the gravitational acceleration. 68 ${\rm {\bf D}}^{\rm {\bf U}}$, $D^T$ and $D^S$ are the parameterizations of small-scale physics for momentum, temperature and salinity, and ${\rm {\bf F}}^{\rm {\bf U}}$, $F^T$ and $F^S$ surface forcing terms. Their nature and formulation are discussed in \S\ref{PE_zdf_ldf} and page \S\ref{PE_boundary_condition}. 83 where $\nabla$ is the generalised derivative vector operator in $(\bf i,\bf j, \bf k)$ directions, 84 $t$ is the time, $z$ is the vertical coordinate, $\rho $ is the \textit{in situ} density given by 85 the equation of state (\ref{Eq_PE_eos}), $\rho_o$ is a reference density, $p$ the pressure, 86 $f=2 \bf \Omega \cdot \bf k$ is the Coriolis acceleration (where $\bf \Omega$ is the Earth's 87 angular velocity vector), and $g$ is the gravitational acceleration. 88 ${\rm {\bf D}}^{\rm {\bf U}}$, $D^T$ and $D^S$ are the parameterisations of small-scale 89 physics for momentum, temperature and salinity, and ${\rm {\bf F}}^{\rm {\bf U}}$, $F^T$ 90 and $F^S$ surface forcing terms. Their nature and formulation are discussed in 91 \S\ref{PE_zdf_ldf} and page \S\ref{PE_boundary_condition}. 69 92 70 93 . … … 76 99 \label{PE_boundary_condition} 77 100 78 An ocean is bounded by complex coastlines, bottom topography at its base and an air-sea or ice-sea interface at its top. These boundaries can be defined by two surfaces, $z=-H(i,j)$ and $z=\eta(i,j,k,t)$, where $H$ is the depth of the ocean bottom and $\eta$ is the height of the sea surface. Both $H$ and $\eta$ are usually referenced to a given surface, $z=0$, chosen as a mean sea surface (Fig.~\ref{Fig_ocean_bc}). Through these two boundaries, the ocean can exchange fluxes of heat, fresh water, salt, and momentum with the solid earth, the continental margins, the sea ice and the atmosphere. However, some of these fluxes are so weak that even on climatic time scales of thousands of years they can be neglected. In the following, we briefly review the fluxes exchanged at the interfaces between the ocean and the other components of the earth system. 101 An ocean is bounded by complex coastlines, bottom topography at its base and an air-sea 102 or ice-sea interface at its top. These boundaries can be defined by two surfaces, $z=-H(i,j)$ 103 and $z=\eta(i,j,k,t)$, where $H$ is the depth of the ocean bottom and $\eta$ is the height 104 of the sea surface. Both $H$ and $\eta$ are usually referenced to a given surface, $z=0$, 105 chosen as a mean sea surface (Fig.~\ref{Fig_ocean_bc}). Through these two boundaries, 106 the ocean can exchange fluxes of heat, fresh water, salt, and momentum with the solid earth, 107 the continental margins, the sea ice and the atmosphere. However, some of these fluxes are 108 so weak that even on climatic time scales of thousands of years they can be neglected. 109 In the following, we briefly review the fluxes exchanged at the interfaces between the ocean 110 and the other components of the earth system. 79 111 80 112 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 81 113 \begin{figure}[!ht] \label{Fig_ocean_bc} \begin{center} 82 114 \includegraphics[width=0.90\textwidth]{./TexFiles/Figures/Fig_I_ocean_bc.pdf} 83 \caption{The ocean is bounded by two surfaces, $z=-H(i,j)$ and $z=\eta(i,j,k,t)$, where $H$ is the depth of the sea floor and $\eta$ the height of the sea surface. Both $H$ and $\eta $ are referenced to $z=0$.} 115 \caption{The ocean is bounded by two surfaces, $z=-H(i,j)$ and $z=\eta(i,j,k,t)$, where $H$ 116 is the depth of the sea floor and $\eta$ the height of the sea surface. Both $H$ and $\eta $ 117 are referenced to $z=0$.} 84 118 \end{center} \end{figure} 85 119 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> … … 87 121 88 122 \begin{description} 89 \item[Land - ocean interface:] the major flux between continental margins and the ocean is a mass exchange of fresh water through river runoff. Such an exchange modifies the sea surface salinity especially in the vicinity of major river mouths. It can be neglected for short range integrations but has to be taken into account for long term integrations as it influences the characteristics of water masses formed (especially at high latitudes). It is required in order to close the water cycle of the climate system. It is usually specified as a fresh water flux at the air-sea interface in the vicinity of river mouths. 90 \item[Solid earth - ocean interface:] heat and salt fluxes through the sea floor are small, except in special areas of little extent. They are usually neglected in the model \footnote{In fact, it has been shown that the heat flux associated with the solid Earth cooling ($i.e.$the geothermal heating) is not negligible for the thermohaline circulation of the world ocean (see \ref{TRA_bbc}).}. The boundary condition is thus set to no flux of heat and salt across solid boundaries. For momentum, the situation is different. There is no flow across solid boundaries, $i.e.$ the velocity normal to the ocean bottom and coastlines is zero (in other words, the bottom velocity is parallel to solid boundaries). This kinematic boundary condition can be expressed as: 123 \item[Land - ocean interface:] the major flux between continental margins and the ocean is 124 a mass exchange of fresh water through river runoff. Such an exchange modifies the sea 125 surface salinity especially in the vicinity of major river mouths. It can be neglected for short 126 range integrations but has to be taken into account for long term integrations as it influences 127 the characteristics of water masses formed (especially at high latitudes). It is required in order 128 to close the water cycle of the climate system. It is usually specified as a fresh water flux at 129 the air-sea interface in the vicinity of river mouths. 130 \item[Solid earth - ocean interface:] heat and salt fluxes through the sea floor are small, 131 except in special areas of little extent. They are usually neglected in the model 132 \footnote{In fact, it has been shown that the heat flux associated with the solid Earth cooling 133 ($i.e.$the geothermal heating) is not negligible for the thermohaline circulation of the world 134 ocean (see \ref{TRA_bbc}).}. 135 The boundary condition is thus set to no flux of heat and salt across solid boundaries. 136 For momentum, the situation is different. There is no flow across solid boundaries, 137 $i.e.$ the velocity normal to the ocean bottom and coastlines is zero (in other words, 138 the bottom velocity is parallel to solid boundaries). This kinematic boundary condition 139 can be expressed as: 91 140 \begin{equation} \label{Eq_PE_w_bbc} 92 141 w = -{\rm {\bf U}}_h \cdot \nabla _h \left( H \right) 93 142 \end{equation} 94 In addition, the ocean exchanges momentum with the earth through frictional processes. Such momentum transfer occurs at small scales in a boundary layer. It must be parameterized in terms of turbulent fluxes using bottom and/or lateral boundary conditions. Its specification depends on the nature of the physical parameterization used for ${\rm {\bf D}}^{\rm {\bf U}}$ in \eqref{Eq_PE_dyn}. It is discussed in \S\ref{PE_zdf}, page~\pageref{PE_zdf}.% and Chap. III.6 to 9. 95 \item[Atmosphere - ocean interface:] the kinematic surface condition plus the mass flux of fresh water PE (the precipitation minus evaporation budget) leads to: 143 In addition, the ocean exchanges momentum with the earth through frictional processes. 144 Such momentum transfer occurs at small scales in a boundary layer. It must be parameterized 145 in terms of turbulent fluxes using bottom and/or lateral boundary conditions. Its specification 146 depends on the nature of the physical parameterisation used for ${\rm {\bf D}}^{\rm {\bf U}}$ 147 in \eqref{Eq_PE_dyn}. It is discussed in \S\ref{PE_zdf}, page~\pageref{PE_zdf}.% and Chap. III.6 to 9. 148 \item[Atmosphere - ocean interface:] the kinematic surface condition plus the mass flux 149 of fresh water PE (the precipitation minus evaporation budget) leads to: 96 150 \begin{equation} \label{Eq_PE_w_sbc} 97 151 w = \frac{\partial \eta }{\partial t} … … 99 153 + P-E 100 154 \end{equation} 101 The dynamic boundary condition, neglecting the surface tension (which removes capillary waves from the system) leads to the continuity of pressure across the interface $z=\eta$. The atmosphere and ocean also exchange horizontal momentum (wind stress), and heat. 102 \item[Sea ice - ocean interface:] the ocean and sea ice exchange heat, salt, fresh water and momentum. The sea surface temperature is constrained to be at the freezing point at the interface. Sea ice salinity is very low ($\sim4-6 \,psu$) compared to those of the ocean ($\sim34 \,psu$). The cycle of freezing/melting is associated with fresh water and salt fluxes that cannot be neglected. 155 The dynamic boundary condition, neglecting the surface tension (which removes capillary 156 waves from the system) leads to the continuity of pressure across the interface $z=\eta$. 157 The atmosphere and ocean also exchange horizontal momentum (wind stress), and heat. 158 \item[Sea ice - ocean interface:] the ocean and sea ice exchange heat, salt, fresh water 159 and momentum. The sea surface temperature is constrained to be at the freezing point 160 at the interface. Sea ice salinity is very low ($\sim4-6 \,psu$) compared to those of the 161 ocean ($\sim34 \,psu$). The cycle of freezing/melting is associated with fresh water and 162 salt fluxes that cannot be neglected. 103 163 \end{description} 104 164 … … 116 176 \label{PE_p_formulation} 117 177 118 The total pressure at a given depth $z$ is composed of a surface pressure $p_s$ at a reference geopotential surface ($z=0$) and a hydrostatic pressure $p_h$ such that: $p(i,j,k,t)=p_s(i,j,t)+p_h(i,j,k,t)$. The latter is computed by integrating (\ref{Eq_PE_hydrostatic}), assuming that pressure in decibars can be approximated by depth in meters in (\ref{Eq_PE_eos}). The hydrostatic pressure is then given by: 178 The total pressure at a given depth $z$ is composed of a surface pressure $p_s$ at a 179 reference geopotential surface ($z=0$) and a hydrostatic pressure $p_h$ such that: 180 $p(i,j,k,t)=p_s(i,j,t)+p_h(i,j,k,t)$. The latter is computed by integrating (\ref{Eq_PE_hydrostatic}), 181 assuming that pressure in decibars can be approximated by depth in meters in (\ref{Eq_PE_eos}). 182 The hydrostatic pressure is then given by: 119 183 \begin{equation} \label{Eq_PE_pressure} 120 184 p_h \left( {i,j,z,t} \right) 121 = \int_{\varsigma =z}^{\varsigma =0} {g\;\rho \left( {T,S,z} \right)\;d\varsigma } 122 \end{equation} 123 Two strategies can be considered for the surface pressure term: $(a)$ introduce of a new variable $\eta$, the free-surface elevation, for which a prognostic equation can be established and solved; $(b)$ assume that the ocean surface is a rigid lid, on which the pressure (or its horizontal gradient) can be diagnosed. When the former strategy is used, one solution of the free-surface elevation consists of the excitation of external gravity waves. The flow is barotropic and the surface moves up and down with gravity as the restoring force. The phase speed of such waves is high (some hundreds of metres per second) so that the time step would have to be very short if they were present in the model. The latter strategy filters out these waves since the rigid lid approximation implies $\eta=0$, $i.e.$ the sea surface is the surface $z=0$. This well known approximation increases the surface wave speed to infinity and modifies certain other longwave dynamics ($e.g.$ barotropic Rossby or planetary waves). In the present release of \NEMO, both strategies are still available. They are further described in the next two sub-sections. 185 = \int_{\varsigma =z}^{\varsigma =0} {g\;\rho \left( {T,S,\varsigma} \right)\;d\varsigma } 186 \end{equation} 187 Two strategies can be considered for the surface pressure term: $(a)$ introduce of a 188 new variable $\eta$, the free-surface elevation, for which a prognostic equation can be 189 established and solved; $(b)$ assume that the ocean surface is a rigid lid, on which the 190 pressure (or its horizontal gradient) can be diagnosed. When the former strategy is used, 191 one solution of the free-surface elevation consists of the excitation of external gravity waves. 192 The flow is barotropic and the surface moves up and down with gravity as the restoring force. 193 The phase speed of such waves is high (some hundreds of metres per second) so that 194 the time step would have to be very short if they were present in the model. The latter 195 strategy filters out these waves since the rigid lid approximation implies $\eta=0$, $i.e.$ 196 the sea surface is the surface $z=0$. This well known approximation increases the surface 197 wave speed to infinity and modifies certain other longwave dynamics ($e.g.$ barotropic 198 Rossby or planetary waves). In the present release of \NEMO, both strategies are still available. 199 They are further described in the next two sub-sections. 124 200 125 201 % ------------------------------------------------------------------------------------------------------------- … … 129 205 \label{PE_free_surface} 130 206 131 In the free surface formulation, a variable $\eta$, the sea-surface height, is introduced which describes the shape of the air-sea interface. This variable is solution of a prognostic equation which is established by forming the vertical average of the kinematic surface condition (\ref{Eq_PE_w_bbc}): 207 In the free surface formulation, a variable $\eta$, the sea-surface height, is introduced 208 which describes the shape of the air-sea interface. This variable is solution of a 209 prognostic equation which is established by forming the vertical average of the kinematic 210 surface condition (\ref{Eq_PE_w_bbc}): 132 211 \begin{equation} \label{Eq_PE_ssh} 133 212 \frac{\partial \eta }{\partial t}=-D+P-E … … 137 216 and using (\ref{Eq_PE_hydrostatic}) the surface pressure is given by: $p_s = \rho \, g \, \eta$. 138 217 139 Allowing the air-sea interface to move introduces the external gravity waves (EGWs) as a class of solution of the primitive equations. These waves are barotropic because of hydrostatic assumption, and their phase speed is quite high. Their time scale is short with respect to the other processes described by the primitive equations. 140 141 Three choices can be made regarding the implementation of the free surface in the model, depending on the physical processes of interest. 218 Allowing the air-sea interface to move introduces the external gravity waves (EGWs) 219 as a class of solution of the primitive equations. These waves are barotropic because 220 of hydrostatic assumption, and their phase speed is quite high. Their time scale is 221 short with respect to the other processes described by the primitive equations. 222 223 Three choices can be made regarding the implementation of the free surface in the model, 224 depending on the physical processes of interest. 142 225 143 226 $\bullet$ If one is interested in EGWs, in particular the tides and their interaction 144 with the baroclinic structure of the ocean (internal waves) possibly in 145 shallow seas, then a non linear free surface is the most appropriate. This 146 means that no approximation is made in (\ref{Eq_PE_ssh}) and that the variation of the 147 ocean volume is fully taken into account. Note that in order to study the 148 fast time scales associated with EGWs it is necessary to minimize time 149 filtering effects (use an explicit time scheme with very small time step, or 150 a split-explicit scheme with reasonably small time step, see \S\ref{DYN_spg_exp} or 151 \S\ref{DYN_spg_ts}. 227 with the baroclinic structure of the ocean (internal waves) possibly in shallow seas, 228 then a non linear free surface is the most appropriate. This means that no 229 approximation is made in (\ref{Eq_PE_ssh}) and that the variation of the ocean 230 volume is fully taken into account. Note that in order to study the fast time scales 231 associated with EGWs it is necessary to minimize time filtering effects (use an 232 explicit time scheme with very small time step, or a split-explicit scheme with 233 reasonably small time step, see \S\ref{DYN_spg_exp} or \S\ref{DYN_spg_ts}. 152 234 153 235 $\bullet$ If one is not interested in EGW but rather sees them as high frequency … … 163 245 external waves are removed from the system. 164 246 165 The filtering of EGWs in models with a free surface is usually a matter of discretisation of the temporal derivatives, using the time splitting method \citep{Killworth1991, Zhang1992} or the implicit scheme \citep{Dukowicz1994}. In \NEMO, we use a slightly different approach developed by \citet{Roullet2000}: the damping of EGWs is ensured by introducing an additional force in the momentum equation. \eqref{Eq_PE_dyn} becomes: 247 The filtering of EGWs in models with a free surface is usually a matter of discretisation 248 of the temporal derivatives, using the time splitting method \citep{Killworth1991, Zhang1992} 249 or the implicit scheme \citep{Dukowicz1994}. In \NEMO, we use a slightly different approach 250 developed by \citet{Roullet2000}: the damping of EGWs is ensured by introducing an 251 additional force in the momentum equation. \eqref{Eq_PE_dyn} becomes: 166 252 \begin{equation} \label{Eq_PE_flt} 167 253 \frac{\partial {\rm {\bf U}}_h }{\partial t}= {\rm {\bf M}} … … 169 255 - g \ T_c \nabla \left( \widetilde{\rho} \ \partial_t \eta \right) 170 256 \end{equation} 171 where $T_c$, is a parameter with dimensions of time which characterizes the force, $\widetilde{\rho} = \rho / \rho_o$ is the dimensionless density, and $\rm {\bf M}$ represents the collected contributions of the Coriolis, hydrostatic pressure gradient, non-linear and viscous terms in \eqref{Eq_PE_dyn}. 172 173 The new force can be interpreted as a diffusion of vertically integrated volume flux divergence. The time evolution of $D$ is thus governed by a balance of two terms, $-g$ \textbf{A} $\eta$ and $g \, T_c \,$ \textbf{A} $D$, associated with a propagative regime and a diffusive regime in the temporal spectrum, respectively. In the diffusive regime, the EGWs no longer propagate, $i.e.$ they are stationary and damped. The diffusion regime applies to the modes shorter than $T_c$. For longer ones, the diffusion term vanishes. Hence, the temporally unresolved EGWs can be damped by choosing $T_c > \Delta t$. \citet{Roullet2000} demonstrate that (\ref{Eq_PE_flt}) can be integrated with a leap frog scheme except the additional term which has to be computed implicitly. This is not surprising since the use of a large time step has a necessarily numerical cost. Two gains arise in comparison with the previous formulations. Firstly, the damping of EGWs can be quantified through the magnitude of the additional term. Secondly, the numerical scheme does not need any tuning. Numerical stability is ensured as soon as $T_c > \Delta t$. 174 175 When the variations of free surface elevation are small compared to the thickness of the first model layer, the free surface equation (\ref{Eq_PE_ssh}) can be linearized. As emphasized by \citet{Roullet2000} the linearization of (\ref{Eq_PE_ssh}) has consequences on the conservation of salt in the model. With the nonlinear free surface equation, the time evolution of the total salt content is 257 where $T_c$, is a parameter with dimensions of time which characterizes the force, 258 $\widetilde{\rho} = \rho / \rho_o$ is the dimensionless density, and $\rm {\bf M}$ 259 represents the collected contributions of the Coriolis, hydrostatic pressure gradient, 260 non-linear and viscous terms in \eqref{Eq_PE_dyn}. 261 262 The new force can be interpreted as a diffusion of vertically integrated volume flux divergence. 263 The time evolution of $D$ is thus governed by a balance of two terms, $-g$ \textbf{A} $\eta$ 264 and $g \, T_c \,$ \textbf{A} $D$, associated with a propagative regime and a diffusive regime 265 in the temporal spectrum, respectively. In the diffusive regime, the EGWs no longer propagate, 266 $i.e.$ they are stationary and damped. The diffusion regime applies to the modes shorter than 267 $T_c$. For longer ones, the diffusion term vanishes. Hence, the temporally unresolved EGWs 268 can be damped by choosing $T_c > \Delta t$. \citet{Roullet2000} demonstrate that 269 (\ref{Eq_PE_flt}) can be integrated with a leap frog scheme except the additional term which 270 has to be computed implicitly. This is not surprising since the use of a large time step has a 271 necessarily numerical cost. Two gains arise in comparison with the previous formulations. 272 Firstly, the damping of EGWs can be quantified through the magnitude of the additional term. 273 Secondly, the numerical scheme does not need any tuning. Numerical stability is ensured as 274 soon as $T_c > \Delta t$. 275 276 When the variations of free surface elevation are small compared to the thickness of the first 277 model layer, the free surface equation (\ref{Eq_PE_ssh}) can be linearized. As emphasized 278 by \citet{Roullet2000} the linearization of (\ref{Eq_PE_ssh}) has consequences on the 279 conservation of salt in the model. With the nonlinear free surface equation, the time evolution 280 of the total salt content is 176 281 \begin{equation} \label{Eq_PE_salt_content} 177 \frac{\partial }{\partial t}\int\limits_{D\eta } {S\;dv} =\int\limits_S 178 {S\;(-\frac{\partial \eta }{\partial t}-D+P-E)\;ds} 179 \end{equation} 180 where $S$ is the salinity, and the total salt is integrated over the whole ocean volume $D_\eta$ bounded by the time-dependent free surface. The right hand side (which is an integral over the free surface) vanishes when the nonlinear equation (\ref{Eq_PE_ssh}) is satisfied, so that the salt is perfectly conserved. When the free surface equation is linearized, \citet{Roullet2000} show that the total salt content integrated in the fixed volume $D$ (bounded by the surface $z=0$) is no longer conserved: 282 \frac{\partial }{\partial t}\int\limits_{D\eta } {S\;dv} 283 =\int\limits_S {S\;(-\frac{\partial \eta }{\partial t}-D+P-E)\;ds} 284 \end{equation} 285 where $S$ is the salinity, and the total salt is integrated over the whole ocean volume 286 $D_\eta$ bounded by the time-dependent free surface. The right hand side (which is an 287 integral over the free surface) vanishes when the nonlinear equation (\ref{Eq_PE_ssh}) 288 is satisfied, so that the salt is perfectly conserved. When the free surface equation is 289 linearized, \citet{Roullet2000} show that the total salt content integrated in the fixed 290 volume $D$ (bounded by the surface $z=0$) is no longer conserved: 181 291 \begin{equation} \label{Eq_PE_salt_content_linear} 182 \frac{\partial }{\partial t}\int\limits_D {S\;dv} =-\int\limits_S 183 {S\;\frac{\partial \eta }{\partial t}ds} 184 \end{equation} 185 186 The right hand side of (\ref{Eq_PE_salt_content_linear}) is small in equilibrium solutions \citep{Roullet2000}. It can be significant when the freshwater forcing is not balanced and the globally averaged free surface is drifting. An increase in sea surface height \textit{$\eta $} results in a decrease of the salinity in the fixed volume $D$. Even in that case though, the total salt integrated in the variable volume $D_{\eta}$ varies much less, since (\ref{Eq_PE_salt_content_linear}) can be rewritten as 292 \frac{\partial }{\partial t}\int\limits_D {S\;dv} 293 = - \int\limits_S {S\;\frac{\partial \eta }{\partial t}ds} 294 \end{equation} 295 296 The right hand side of (\ref{Eq_PE_salt_content_linear}) is small in equilibrium solutions 297 \citep{Roullet2000}. It can be significant when the freshwater forcing is not balanced and 298 the globally averaged free surface is drifting. An increase in sea surface height \textit{$\eta $} 299 results in a decrease of the salinity in the fixed volume $D$. Even in that case though, 300 the total salt integrated in the variable volume $D_{\eta}$ varies much less, since 301 (\ref{Eq_PE_salt_content_linear}) can be rewritten as 187 302 \begin{equation} \label{Eq_PE_salt_content_corrected} 188 303 \frac{\partial }{\partial t}\int\limits_{D\eta } {S\;dv} … … 191 306 \end{equation} 192 307 193 Although the total salt content is not exactly conserved with the linearized free surface, its variations are driven by correlations of the time variation of surface salinity with the sea surface height, which is a negligible term. This situation contrasts with 194 the case of the rigid lid approximation (following section) in which case freshwater forcing is represented by a virtual salt flux, leading to a spurious source of salt at the ocean surface \citep{Roullet2000}. 308 Although the total salt content is not exactly conserved with the linearized free surface, 309 its variations are driven by correlations of the time variation of surface salinity with the 310 sea surface height, which is a negligible term. This situation contrasts with the case of 311 the rigid lid approximation (following section) in which case freshwater forcing is 312 represented by a virtual salt flux, leading to a spurious source of salt at the ocean 313 surface \citep{Roullet2000}. 195 314 196 315 % ------------------------------------------------------------------------------------------------------------- … … 200 319 \label{PE_rigid_lid} 201 320 202 With the rigid lid approximation, we assume that the ocean surface ($z=0$) is a rigid lid on which a pressure $p_s$ is exerted. This implies that the vertical velocity at the surface is equal to zero. From the continuity equation \eqref{Eq_PE_continuity} and the kinematic condition at the bottom \eqref{Eq_PE_w_bbc} (no flux across the bottom), it can be shown that the vertically integrated flow $H{\rm {\bf \bar {U}}}_h$ is non-divergent (where the overbar indicates a vertical average over the whole water column, i.e. from $z=-H$, the ocean bottom, to $z=0$ , the rigid-lid). Thus, $\rm {\bf \bar {U}}_h$ can be derived from a volume transport streamfunction $\psi$: 321 With the rigid lid approximation, we assume that the ocean surface ($z=0$) is a rigid lid 322 on which a pressure $p_s$ is exerted. This implies that the vertical velocity at the surface 323 is equal to zero. From the continuity equation \eqref{Eq_PE_continuity} and the kinematic 324 condition at the bottom \eqref{Eq_PE_w_bbc} (no flux across the bottom), it can be shown 325 that the vertically integrated flow $H{\rm {\bf \bar {U}}}_h$ is non-divergent (where the 326 overbar indicates a vertical average over the whole water column, i.e. from $z=-H$, 327 the ocean bottom, to $z=0$ , the rigid-lid). Thus, $\rm {\bf \bar {U}}_h$ can be derived 328 from a volume transport streamfunction $\psi$: 203 329 \begin{equation} \label{Eq_PE_u_psi} 204 330 \overline{\vect{U}}_h =\frac{1}{H}\left( \vect{k} \times \nabla \psi \right) 205 331 \end{equation} 206 332 207 As $p_s$ does not depend on depth, its horizontal gradient is obtained by forming the vertical average of \eqref{Eq_PE_dyn} and using \eqref{Eq_PE_u_psi}: 333 As $p_s$ does not depend on depth, its horizontal gradient is obtained by forming the 334 vertical average of \eqref{Eq_PE_dyn} and using \eqref{Eq_PE_u_psi}: 208 335 209 336 \begin{equation} \label{Eq_PE_u_barotrope} … … 214 341 \end{equation} 215 342 216 Here ${\rm {\bf M}} = \left( M_u,M_v \right)$ represents the collected contributions of the Coriolis, hydrostatic pressure gradient, nonlinear and viscous terms in \eqref{Eq_PE_dyn}. The time derivative of $\psi $ is the solution of an elliptic equation which is obtained from the vertical component of the curl of (\ref{Eq_PE_u_barotrope}): 343 Here ${\rm {\bf M}} = \left( M_u,M_v \right)$ represents the collected contributions of the 344 Coriolis, hydrostatic pressure gradient, nonlinear and viscous terms in \eqref{Eq_PE_dyn}. 345 The time derivative of $\psi $ is the solution of an elliptic equation which is obtained from 346 the vertical component of the curl of (\ref{Eq_PE_u_barotrope}): 217 347 \begin{equation} \label{Eq_PE_psi} 218 348 \left[ {\nabla \times \left[ {\frac{1}{H} \vect{\bf k} … … 221 351 \end{equation} 222 352 223 Using the proper boundary conditions, \eqref{Eq_PE_psi} can be solved to find $\partial_t \psi$ and thus using \eqref{Eq_PE_u_barotrope} the horizontal surface pressure gradient. It should be noted that $p_s$ can be computed by taking the divergence of \eqref{Eq_PE_u_barotrope} and solving the resulting elliptic equation. Thus the surface pressure is a diagnostic quantity that can be recovered for analysis purposes. 224 225 A difficulty lies in the determination of the boundary condition on $\partial_t \psi$. The boundary condition on velocity is that there is no flow normal to a solid wall, $i.e.$ the coastlines are streamlines. Therefore \eqref{Eq_PE_psi} is solved with the following Dirichlet boundary condition: $\partial_t \psi$ is constant along each coastline of the same continent or of the same island. When all the coastlines are connected (there are no islands), the constant value of $\partial_t \psi$ along the coast can be arbitrarily chosen to be zero. When islands are present in the domain, the value of the barotropic streamfunction will generally be different for each island and for the continent, and will vary with respect to time. So the boundary condition is: $\psi=0$ along the continent and $\psi=\mu_n$ along island $n$ ($1 \leq n \leq Q$), where $Q$ is the number of islands present in the domain and $\mu_n$ is a time dependent variable. A time evolution equation of the unknown $\mu_n$ can be found by evaluating the circulation of the time derivative of the vertical average (barotropic) velocity field along a closed contour around each island. Since the circulation of a gradient field along a closed contour is zero, from \eqref{Eq_PE_u_barotrope} we have: 353 Using the proper boundary conditions, \eqref{Eq_PE_psi} can be solved to find $\partial_t \psi$ 354 and thus using \eqref{Eq_PE_u_barotrope} the horizontal surface pressure gradient. 355 It should be noted that $p_s$ can be computed by taking the divergence of 356 \eqref{Eq_PE_u_barotrope} and solving the resulting elliptic equation. Thus the surface 357 pressure is a diagnostic quantity that can be recovered for analysis purposes. 358 359 A difficulty lies in the determination of the boundary condition on $\partial_t \psi$. 360 The boundary condition on velocity is that there is no flow normal to a solid wall, 361 $i.e.$ the coastlines are streamlines. Therefore \eqref{Eq_PE_psi} is solved with 362 the following Dirichlet boundary condition: $\partial_t \psi$ is constant along each 363 coastline of the same continent or of the same island. When all the coastlines are 364 connected (there are no islands), the constant value of $\partial_t \psi$ along the 365 coast can be arbitrarily chosen to be zero. When islands are present in the domain, 366 the value of the barotropic streamfunction will generally be different for each island 367 and for the continent, and will vary with respect to time. So the boundary condition is: 368 $\psi=0$ along the continent and $\psi=\mu_n$ along island $n$ ($1 \leq n \leq Q$), 369 where $Q$ is the number of islands present in the domain and $\mu_n$ is a time 370 dependent variable. A time evolution equation of the unknown $\mu_n$ can be found 371 by evaluating the circulation of the time derivative of the vertical average (barotropic) 372 velocity field along a closed contour around each island. Since the circulation of a 373 gradient field along a closed contour is zero, from \eqref{Eq_PE_u_barotrope} we have: 226 374 \begin{equation} \label{Eq_PE_isl_circulation} 227 375 \oint_n {\frac{1}{H}\left[ {{\rm {\bf k}}\times \nabla \left( … … 256 404 \right)_{1\leqslant n\leqslant Q} ={\rm {\bf B}} 257 405 \end{equation} 258 where \textbf{A} is a $Q \times Q$ matrix and \textbf{B} is a time dependent vector. As \textbf{A} is independent of time, it can be calculated and inverted once. The time derivative of the streamfunction when islands are present is thus given by: 406 where \textbf{A} is a $Q \times Q$ matrix and \textbf{B} is a time dependent vector. 407 As \textbf{A} is independent of time, it can be calculated and inverted once. The time 408 derivative of the streamfunction when islands are present is thus given by: 259 409 \begin{equation} \label{Eq_PE_psi_isl_dt} 260 410 \frac{\partial \psi }{\partial t}=\frac{\partial \psi _o }{\partial … … 277 427 \label{PE_tensorial} 278 428 279 In many ocean circulation problems, the flow field has regions of enhanced dynamics ($i.e.$ surface layers, western boundary currents, equatorial currents, or ocean fronts). The representation of such dynamical processes can be improved by specifically increasing the model resolution in these regions. As well, it may be convenient to use a lateral boundary-following coordinate system to better represent coastal dynamics. Moreover, the common geographical coordinate system has a singular point at the North Pole that cannot be easily treated in a global model without filtering. A solution consists of introducing an appropriate coordinate transformation that shifts the singular point onto land \citep{MadecImb1996, Murray1996}. As a consequence, it is important to solve the primitive equations in various curvilinear coordinate systems. An efficient way of introducing an appropriate coordinate transform can be found when using a tensorial formalism. This formalism is suited to any multidimensional curvilinear coordinate system. Ocean modellers mainly use three-dimensional orthogonal grids on the sphere (spherical earth approximation), with preservation of the local vertical. Here we give the simplified equations for this particular case. The general case is detailed by \citet{Eiseman1980} in their survey of the conservation laws of fluid dynamics. 280 281 Let (\textbf{i},\textbf{j},\textbf{k}) be a set of orthogonal curvilinear coordinates on the sphere associated with the positively oriented orthogonal set of unit vectors (\textbf{i},\textbf{j},\textbf{k}) linked to the earth such that \textbf{k} is the local upward vector and (\textbf{i},\textbf{j}) are two vectors orthogonal to \textbf{k}, $i.e.$ along geopotential surfaces (Fig.\ref{Fig_referential}). Let $(\lambda,\varphi,z)$ be the geographical coordinate system in which a position is defined by the latitude $\varphi(i,j)$, the longitude $\lambda(i,j)$ and the distance from the centre of the earth $a+z(k)$ where $a$ is the earth's radius and $z$ the altitude above a reference sea level (Fig.\ref{Fig_referential}). The local deformation of the curvilinear coordinate system is given by $e_1$, $e_2$ and $e_3$, the three scale factors: 429 In many ocean circulation problems, the flow field has regions of enhanced dynamics 430 ($i.e.$ surface layers, western boundary currents, equatorial currents, or ocean fronts). 431 The representation of such dynamical processes can be improved by specifically increasing 432 the model resolution in these regions. As well, it may be convenient to use a lateral 433 boundary-following coordinate system to better represent coastal dynamics. Moreover, 434 the common geographical coordinate system has a singular point at the North Pole that 435 cannot be easily treated in a global model without filtering. A solution consists of introducing 436 an appropriate coordinate transformation that shifts the singular point onto land 437 \citep{MadecImb1996, Murray1996}. As a consequence, it is important to solve the primitive 438 equations in various curvilinear coordinate systems. An efficient way of introducing an 439 appropriate coordinate transform can be found when using a tensorial formalism. 440 This formalism is suited to any multidimensional curvilinear coordinate system. Ocean 441 modellers mainly use three-dimensional orthogonal grids on the sphere (spherical earth 442 approximation), with preservation of the local vertical. Here we give the simplified equations 443 for this particular case. The general case is detailed by \citet{Eiseman1980} in their survey 444 of the conservation laws of fluid dynamics. 445 446 Let (\textbf{i},\textbf{j},\textbf{k}) be a set of orthogonal curvilinear coordinates on the sphere 447 associated with the positively oriented orthogonal set of unit vectors (\textbf{i},\textbf{j},\textbf{k}) 448 linked to the earth such that \textbf{k} is the local upward vector and (\textbf{i},\textbf{j}) are 449 two vectors orthogonal to \textbf{k}, $i.e.$ along geopotential surfaces (Fig.\ref{Fig_referential}). 450 Let $(\lambda,\varphi,z)$ be the geographical coordinate system in which a position is defined 451 by the latitude $\varphi(i,j)$, the longitude $\lambda(i,j)$ and the distance from the centre of 452 the earth $a+z(k)$ where $a$ is the earth's radius and $z$ the altitude above a reference sea 453 level (Fig.\ref{Fig_referential}). The local deformation of the curvilinear coordinate system is 454 given by $e_1$, $e_2$ and $e_3$, the three scale factors: 282 455 \begin{equation} \label{Eq_scale_factors} 283 456 \begin{aligned} … … 295 468 \begin{figure}[!tb] \label{Fig_referential} \begin{center} 296 469 \includegraphics[width=0.60\textwidth]{./TexFiles/Figures/Fig_I_earth_referential.pdf} 297 \caption{the geographical coordinate system $(\lambda,\varphi,z)$ and the curvilinear coordinate system (\textbf{i},\textbf{j},\textbf{k}). } 470 \caption{the geographical coordinate system $(\lambda,\varphi,z)$ and the curvilinear 471 coordinate system (\textbf{i},\textbf{j},\textbf{k}). } 298 472 \end{center} \end{figure} 299 473 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 300 474 301 Since the ocean depth is far smaller than the earth's radius, $a+z$, can be replaced by $a$ in (\ref{Eq_scale_factors}) (thin-shell approximation). The resulting horizontal scale factors $e_1$, $e_2$ are independent of $k$ while the vertical scale factor is a single function of $k$ as \textbf{k} is parallel to \textbf{z}. The scalar and vector operators that appear in the primitive equations (Eqs. \eqref{Eq_PE_dyn} to \eqref{Eq_PE_eos}) can be written in the tensorial form, invariant in any orthogonal horizontal curvilinear coordinate system transformation: 475 Since the ocean depth is far smaller than the earth's radius, $a+z$, can be replaced by 476 $a$ in (\ref{Eq_scale_factors}) (thin-shell approximation). The resulting horizontal scale 477 factors $e_1$, $e_2$ are independent of $k$ while the vertical scale factor is a single 478 function of $k$ as \textbf{k} is parallel to \textbf{z}. The scalar and vector operators that 479 appear in the primitive equations (Eqs. \eqref{Eq_PE_dyn} to \eqref{Eq_PE_eos}) can 480 be written in the tensorial form, invariant in any orthogonal horizontal curvilinear coordinate 481 system transformation: 302 482 \begin{subequations} \label{Eq_PE_discrete_operators} 303 483 \begin{equation} \label{Eq_PE_grad} … … 341 521 \label{PE_zco_Eq} 342 522 343 In order to express the Primitive Equations in tensorial formalism, it is necessary to compute the horizontal component of the non-linear and viscous terms of the equation using \eqref{Eq_PE_grad}) to \eqref{Eq_PE_lap_vector}. Let us set $\vect U=(u,v,w)={\vect{U}}_h +w\;\vect{k}$, the velocity in the $(i,j,k)$ coordinate system and define the relative vorticity $\zeta$ and the divergence of the horizontal velocity field $\chi$, by: 523 In order to express the Primitive Equations in tensorial formalism, it is necessary to compute 524 the horizontal component of the non-linear and viscous terms of the equation using 525 \eqref{Eq_PE_grad}) to \eqref{Eq_PE_lap_vector}. 526 Let us set $\vect U=(u,v,w)={\vect{U}}_h +w\;\vect{k}$, the velocity in the $(i,j,k)$ coordinate 527 system and define the relative vorticity $\zeta$ and the divergence of the horizontal velocity 528 field $\chi$, by: 344 529 \begin{equation} \label{Eq_PE_curl_Uh} 345 530 \zeta =\frac{1}{e_1 e_2 }\left[ {\frac{\partial \left( {e_2 \,v} … … 353 538 \end{equation} 354 539 355 Using the fact that the horizontal scale factors $e_1$ and $e_2$ are independent of $k$ and that $e_3$ is a function of the single variable $k$, the nonlinear term of \eqref{Eq_PE_dyn} can be transformed as follows: 540 Using the fact that the horizontal scale factors $e_1$ and $e_2$ are independent of $k$ 541 and that $e_3$ is a function of the single variable $k$, the nonlinear term of 542 \eqref{Eq_PE_dyn} can be transformed as follows: 356 543 \begin{flalign*} 357 544 &\left[ {\left( { \nabla \times {\rm {\bf U}} } \right) \times {\rm {\bf U}} … … 391 578 \end{flalign*} 392 579 393 The last term of the right hand side is obviously zero, and thus the nonlinear term of \eqref{Eq_PE_dyn} is written in the $(i,j,k)$ coordinate system: 580 The last term of the right hand side is obviously zero, and thus the nonlinear term of 581 \eqref{Eq_PE_dyn} is written in the $(i,j,k)$ coordinate system: 394 582 \begin{equation} \label{Eq_PE_vector_form} 395 583 \left[ {\left( { \nabla \times {\rm {\bf U}} } \right) \times {\rm {\bf U}} … … 401 589 \end{equation} 402 590 403 This is the so-called \textit{vector invariant form} of the momentum advection term. For some purposes, it can be advantageous to write this term in the so-called flux form, $i.e.$ to write it as the divergence of fluxes. For example, the first component of \eqref{Eq_PE_vector_form} (the $i$-component) is transformed as follows: 591 This is the so-called \textit{vector invariant form} of the momentum advection term. 592 For some purposes, it can be advantageous to write this term in the so-called flux form, 593 $i.e.$ to write it as the divergence of fluxes. For example, the first component of 594 \eqref{Eq_PE_vector_form} (the $i$-component) is transformed as follows: 404 595 \begin{flalign*} 405 596 &{ \begin{array}{*{20}l} … … 486 677 \end{multline} 487 678 488 The flux form has two terms, the first one is expressed as the divergence of momentum fluxes (hence the flux form name given to this formulation) and the second one is due to the curvilinear nature of the coordinate system used. The latter is called the \emph{metric} term and can be viewed as a modification of the Coriolis parameter: 679 The flux form has two terms, the first one is expressed as the divergence of momentum 680 fluxes (hence the flux form name given to this formulation) and the second one is due to 681 the curvilinear nature of the coordinate system used. The latter is called the \emph{metric} 682 term and can be viewed as a modification of the Coriolis parameter: 489 683 \begin{equation} \label{Eq_PE_cor+metric} 490 684 f \to f + \frac{1}{e_1 \; e_2} \left( v \frac{\partial e_2}{\partial i} … … 492 686 \end{equation} 493 687 494 Note that in the case of geographical coordinate, $i.e.$ when $(i,j) \to (\lambda ,\varphi )$ and $(e_1 ,e_2) \to (a \,\cos \varphi ,a)$, we recover the commonly used modification of the Coriolis parameter $f \to f+(u/a) \tan \varphi$. 688 Note that in the case of geographical coordinate, $i.e.$ when $(i,j) \to (\lambda ,\varphi )$ 689 and $(e_1 ,e_2) \to (a \,\cos \varphi ,a)$, we recover the commonly used modification of 690 the Coriolis parameter $f \to f+(u/a) \tan \varphi$. 495 691 496 692 To sum up, the equations solved by the ocean model can be written in the following tensorial formalism: … … 545 741 \end{multline} 546 742 \end{subequations} 547 where $\zeta$ is given by \eqref{Eq_PE_curl_Uh} and the surface pressure gradient formulation depends on the one of the free surface: 743 where $\zeta$ is given by \eqref{Eq_PE_curl_Uh} and the surface pressure gradient formulation 744 depends on the one of the free surface: 548 745 549 746 $*$ free surface formulation … … 566 763 \end{equation} 567 764 where ${\vect{M}}= \left( M_u,M_v \right)$ represents the collected contributions of nonlinear, 568 viscosity and hydrostatic pressure gradient terms in \eqref{Eq_PE_dyn_vect} and the overbar indicates a vertical average over the whole water column ($i.e.$ from $z=-H$, the ocean bottom, to $z=0$, the rigid-lid), and where the time derivative of $\psi$ is the solution of an elliptic equation: 765 viscosity and hydrostatic pressure gradient terms in \eqref{Eq_PE_dyn_vect} and the overbar 766 indicates a vertical average over the whole water column ($i.e.$ from $z=-H$, the ocean bottom, 767 to $z=0$, the rigid-lid), and where the time derivative of $\psi$ is the solution of an elliptic equation: 569 768 \begin{multline} \label{Eq_psi_total} 570 769 \frac{\partial }{\partial i}\left[ {\frac{e_2 }{H\,e_1}\frac{\partial}{\partial i} … … 605 804 \end{equation} 606 805 607 The expression of \textbf{D}$^{U}$, $D^{S}$ and $D^{T}$ depends on the subgrid 608 scale parameterization used. It will be defined in \S\ref{PE_zdf}. The nature and formulation of ${\rm {\bf F}}^{\rm {\bf U}}$, $F^T$ and $F^S$, the surface forcing terms, are discussed in Chapter~\ref{SBC}. 806 The expression of \textbf{D}$^{U}$, $D^{S}$ and $D^{T}$ depends on the subgrid scale 807 parameterisation used. It will be defined in \S\ref{PE_zdf}. The nature and formulation of 808 ${\rm {\bf F}}^{\rm {\bf U}}$, $F^T$ and $F^S$, the surface forcing terms, are discussed 809 in Chapter~\ref{SBC}. 609 810 610 811 \newpage … … 617 818 \begin{figure}[!b] \label{Fig_z_zstar} \begin{center} 618 819 \includegraphics[width=1.0\textwidth]{./TexFiles/Figures/Fig_z_zstar.pdf} 619 \caption{(a) $z$-coordinate in linear free-surface case ; (b) $z-$coordinate in non-linear free surface case (c) re-scaled height coordinate (become popular as the \textit{z*-}coordinate \citep{Adcroft_Campin_OM04} ).} 820 \caption{(a) $z$-coordinate in linear free-surface case ; (b) $z-$coordinate in non-linear 821 free surface case (c) re-scaled height coordinate (become popular as the \textit{z*-}coordinate 822 \citep{Adcroft_Campin_OM04} ).} 620 823 \end{center} \end{figure} 621 824 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 622 825 623 826 624 In that case, the free surface equation is nonlinear, and the variations of volume are fully taken into account. These coordinates systems is presented in a report 625 \citep{Levier2007} available on the \NEMO web site. 827 In that case, the free surface equation is nonlinear, and the variations of volume are fully 828 taken into account. These coordinates systems is presented in a report \citep{Levier2007} 829 available on the \NEMO web site. 626 830 627 831 \gmcomment{ 628 The \textit{z*} coordinate approach is an unapproximated, non-linear free surface implementation which allows one to deal with large amplitude free-surface 832 The \textit{z*} coordinate approach is an unapproximated, non-linear free surface implementation 833 which allows one to deal with large amplitude free-surface 629 834 variations relative to the vertical resolution \citep{Adcroft_Campin_OM04}. In 630 835 the \textit{z*} formulation, the variation of the column thickness due to sea-surface … … 642 847 643 848 Since the vertical displacement of the free surface is incorporated in the vertical 644 coordinate \textit{z*}, the upper and lower boundaries are at fixed \textit{z*} position, $\textit{z*} = 0$ and $\textit{z*} = ?H$ respectively. Also the divergence of the flow field is no longer zero as shown by the continuity equation: 849 coordinate \textit{z*}, the upper and lower boundaries are at fixed \textit{z*} position, 850 $\textit{z*} = 0$ and $\textit{z*} = ?H$ respectively. Also the divergence of the flow field 851 is no longer zero as shown by the continuity equation: 645 852 646 853 $\frac{\partial r}{\partial t} = \nabla_{\textit{z*}} \cdot \left( r \; \rm{\bf U}_h \right) … … 662 869 \subsection{Introduction} 663 870 664 Several important aspects of the ocean circulation are influenced by bottom topography. Of course, the most important is that bottom topography determines deep ocean sub-basins, barriers, sills and channels that strongly constrain the path of water masses, but more subtle effects exist. For example, the topographic $\beta$-effect is usually larger than the planetary one along continental slopes. Topographic Rossby waves can be excited and can interact with the mean current. In the $z-$coordinate system presented in the previous section (\S\ref{PE_zco}), $z-$surfaces are geopotential surfaces. The bottom topography is discretised by steps. This often leads to a misrepresentation of a gradually sloping bottom and to large localized depth gradients associated with large localized vertical velocities. The response to such a velocity field often leads to numerical dispersion effects. One solution to strongly reduce this error is to use a partial step representation of bottom topography instead of a full step one \cite{Pacanowski_Gnanadesikan_MWR98}. Another solution is to introduce a terrain-following coordinate system (hereafter $s-$coordinate) 665 666 The $s$-coordinate avoids the discretisation error in the depth field since the layers of computation are gradually adjusted with depth to the ocean bottom. Relatively small topographic features as well as gentle, large-scale slopes of the sea floor in the deep ocean, which would be ignored in typical $z$-model applications with the largest grid spacing at greatest depths, can easily be represented (with relatively low vertical resolution). A terrain-following model (hereafter $s-$model) also facilitates the modelling of the boundary layer flows over a large depth range, which in the framework of the $z$-model would require high vertical resolution over the whole depth range. Moreover, with a $s$-coordinate it is possible, at least in principle, to have the bottom and the sea surface as the only boundaries of the domain (nomore lateral boundary condition to specify). Nevertheless, a $s$-coordinate also has its drawbacks. Perfectly adapted to a homogeneous ocean, it has strong limitations as soon as stratification is introduced. The main two problems come from the truncation error in the horizontal pressure gradient and a possibly increased diapycnal diffusion. The horizontal pressure force in $s$-coordinate consists of two terms (see Appendix~\ref{Apdx_A}), 871 Several important aspects of the ocean circulation are influenced by bottom topography. 872 Of course, the most important is that bottom topography determines deep ocean sub-basins, 873 barriers, sills and channels that strongly constrain the path of water masses, but more subtle 874 effects exist. For example, the topographic $\beta$-effect is usually larger than the planetary 875 one along continental slopes. Topographic Rossby waves can be excited and can interact 876 with the mean current. In the $z-$coordinate system presented in the previous section 877 (\S\ref{PE_zco}), $z-$surfaces are geopotential surfaces. The bottom topography is 878 discretised by steps. This often leads to a misrepresentation of a gradually sloping bottom 879 and to large localized depth gradients associated with large localized vertical velocities. 880 The response to such a velocity field often leads to numerical dispersion effects. 881 One solution to strongly reduce this error is to use a partial step representation of bottom 882 topography instead of a full step one \cite{Pacanowski_Gnanadesikan_MWR98}. 883 Another solution is to introduce a terrain-following coordinate system (hereafter $s-$coordinate) 884 885 The $s$-coordinate avoids the discretisation error in the depth field since the layers of 886 computation are gradually adjusted with depth to the ocean bottom. Relatively small 887 topographic features as well as gentle, large-scale slopes of the sea floor in the deep 888 ocean, which would be ignored in typical $z$-model applications with the largest grid 889 spacing at greatest depths, can easily be represented (with relatively low vertical resolution). 890 A terrain-following model (hereafter $s-$model) also facilitates the modelling of the 891 boundary layer flows over a large depth range, which in the framework of the $z$-model 892 would require high vertical resolution over the whole depth range. Moreover, with a 893 $s$-coordinate it is possible, at least in principle, to have the bottom and the sea surface 894 as the only boundaries of the domain (nomore lateral boundary condition to specify). 895 Nevertheless, a $s$-coordinate also has its drawbacks. Perfectly adapted to a 896 homogeneous ocean, it has strong limitations as soon as stratification is introduced. 897 The main two problems come from the truncation error in the horizontal pressure 898 gradient and a possibly increased diapycnal diffusion. The horizontal pressure force 899 in $s$-coordinate consists of two terms (see Appendix~\ref{Apdx_A}), 667 900 668 901 \begin{equation} \label{Eq_PE_p_sco} … … 671 904 \end{equation} 672 905 673 The second term in \eqref{Eq_PE_p_sco} depends on the tilt of the coordinate surface and introduces a truncation error that is not present in a $z$-model. In the special case of a $\sigma-$coordinate (i.e. a depth-normalised coordinate system $\sigma = z/H$), \citet{Haney1991} and \citet{Beckmann1993} have given estimates of the magnitude of this truncation error. It depends on topographic slope, stratification, horizontal and vertical resolution, the equation of state, and the finite difference scheme. This error limits the possible topographic slopes that a model can handle at a given horizontal and vertical resolution. This is a severe restriction for large-scale applications using realistic bottom topography. The large-scale slopes require high horizontal resolution, and the computational cost becomes prohibitive. This problem can be at least partially overcome by mixing $s$-coordinate and step-like representation of bottom topography \citep{Gerdes1993a,Gerdes1993b,Madec1996}. However, the definition of the model domain vertical coordinate becomes then a non-trivial thing for a realistic bottom topography: a envelope topography is defined in $s$-coordinate on which a full or partial step bottom topography is then applied in order to adjust the model depth to the observed one (see \S\ref{DOM_zgr}. 674 675 For numerical reasons a minimum of diffusion is required along the coordinate surfaces of any finite difference model. It causes spurious diapycnal mixing when coordinate surfaces do not coincide with isoneutral surfaces. This is the case for a $z$-model as well as for a $s$-model. 676 However, density varies more strongly on $s-$surfaces than on horizontal surfaces in regions of large topographic slopes, implying larger diapycnal diffusion in a $s$-model than in a $z$-model. Whereas such a diapycnal diffusion in a $z$-model tends to weaken horizontal density (pressure) gradients and thus the horizontal circulation, it usually reinforces these gradients in a $s$-model, creating spurious circulation. For example, imagine an isolated bump of topography in an ocean at rest with a horizontally uniform stratification. Spurious diffusion along $s$-surfaces will induce a bump of isoneutral surfaces over the topography, and thus will generate there a baroclinic eddy. In contrast, the ocean will stay at rest in a $z$-model. As for the truncation error, the problem can be reduced by introducing the terrain-following coordinate below the strongly stratified portion of the water column (i.e. the main thermocline) \citep{Madec1996}. An alternate solution consists of rotating the lateral diffusive tensor to geopotential or to isoneutral surfaces (see \S\ref{PE_ldf}. Unfortunately, the slope of isoneutral surfaces relative to the $s$-surfaces can very large, strongly exceeding the stability limit of such a operator when it is discretized (see Chapter~\ref{LDF}). 677 678 The $s-$coordinates introduced here \citep{Lott1990,Madec1996} differ mainly in two aspects from similar models: it allows a representation of bottom topography with mixed full or partial step-like/terrain following topography ; It also offers a completely general transformation, $s=s(i,j,z)$ for the vertical coordinate. 906 The second term in \eqref{Eq_PE_p_sco} depends on the tilt of the coordinate surface 907 and introduces a truncation error that is not present in a $z$-model. In the special case 908 of a $\sigma-$coordinate (i.e. a depth-normalised coordinate system $\sigma = z/H$), 909 \citet{Haney1991} and \citet{Beckmann1993} have given estimates of the magnitude 910 of this truncation error. It depends on topographic slope, stratification, horizontal and 911 vertical resolution, the equation of state, and the finite difference scheme. This error 912 limits the possible topographic slopes that a model can handle at a given horizontal 913 and vertical resolution. This is a severe restriction for large-scale applications using 914 realistic bottom topography. The large-scale slopes require high horizontal resolution, 915 and the computational cost becomes prohibitive. This problem can be at least partially 916 overcome by mixing $s$-coordinate and step-like representation of bottom topography \citep{Gerdes1993a,Gerdes1993b,Madec1996}. However, the definition of the model 917 domain vertical coordinate becomes then a non-trivial thing for a realistic bottom 918 topography: a envelope topography is defined in $s$-coordinate on which a full or 919 partial step bottom topography is then applied in order to adjust the model depth to 920 the observed one (see \S\ref{DOM_zgr}. 921 922 For numerical reasons a minimum of diffusion is required along the coordinate surfaces 923 of any finite difference model. It causes spurious diapycnal mixing when coordinate 924 surfaces do not coincide with isoneutral surfaces. This is the case for a $z$-model as 925 well as for a $s$-model. However, density varies more strongly on $s-$surfaces than 926 on horizontal surfaces in regions of large topographic slopes, implying larger diapycnal 927 diffusion in a $s$-model than in a $z$-model. Whereas such a diapycnal diffusion in a 928 $z$-model tends to weaken horizontal density (pressure) gradients and thus the horizontal 929 circulation, it usually reinforces these gradients in a $s$-model, creating spurious circulation. 930 For example, imagine an isolated bump of topography in an ocean at rest with a horizontally 931 uniform stratification. Spurious diffusion along $s$-surfaces will induce a bump of isoneutral 932 surfaces over the topography, and thus will generate there a baroclinic eddy. In contrast, 933 the ocean will stay at rest in a $z$-model. As for the truncation error, the problem can be reduced by introducing the terrain-following coordinate below the strongly stratified portion of the water column 934 ($i.e.$ the main thermocline) \citep{Madec1996}. An alternate solution consists of rotating 935 the lateral diffusive tensor to geopotential or to isoneutral surfaces (see \S\ref{PE_ldf}. 936 Unfortunately, the slope of isoneutral surfaces relative to the $s$-surfaces can very large, 937 strongly exceeding the stability limit of such a operator when it is discretized (see Chapter~\ref{LDF}). 938 939 The $s-$coordinates introduced here \citep{Lott1990,Madec1996} differ mainly in two 940 aspects from similar models: it allows a representation of bottom topography with mixed 941 full or partial step-like/terrain following topography ; It also offers a completely general 942 transformation, $s=s(i,j,z)$ for the vertical coordinate. 679 943 680 944 % ------------------------------------------------------------------------------------------------------------- … … 683 947 \subsection{The \textit{s-}coordinate Formulation} 684 948 685 Starting from the set of equations established in \S\ref{PE_zco} for the special case $k=z$ and thus $e_3=1$, we introduce an arbitrary vertical coordinate $s=s(i,j,k,t)$, which includes $z$-, \textit{z*}- and $\sigma-$coordinates as special cases ($s=z$, $s=\textit{z*}$, and $s=\sigma=z/H$ or $=z/\left(H+\eta \right)$, resp.). A formal derivation of the transformed equations is given in Appendix~\ref{Apdx_A}. Let us define the vertical scale factor by $e_3=\partial_s z$ ($e_3$ is now a function of $(i,j,k,t)$ ), and the slopes in the (\textbf{i},\textbf{j}) directions between $s-$ and $z-$surfaces by : 949 Starting from the set of equations established in \S\ref{PE_zco} for the special case $k=z$ 950 and thus $e_3=1$, we introduce an arbitrary vertical coordinate $s=s(i,j,k,t)$, which includes 951 $z$-, \textit{z*}- and $\sigma-$coordinates as special cases ($s=z$, $s=\textit{z*}$, and 952 $s=\sigma=z/H$ or $=z/\left(H+\eta \right)$, resp.). A formal derivation of the transformed 953 equations is given in Appendix~\ref{Apdx_A}. Let us define the vertical scale factor by 954 $e_3=\partial_s z$ ($e_3$ is now a function of $(i,j,k,t)$ ), and the slopes in the 955 (\textbf{i},\textbf{j}) directions between $s-$ and $z-$surfaces by : 686 956 \begin{equation} \label{Eq_PE_sco_slope} 687 957 \sigma _1 =\frac{1}{e_1 }\;\left. {\frac{\partial z}{\partial i}} \right|_s … … 689 959 \sigma _2 =\frac{1}{e_2 }\;\left. {\frac{\partial z}{\partial j}} \right|_s 690 960 \end{equation} 691 We also introduce $\omega $, a dia-surface velocity component, defined as the velocity relative to the moving $s$-surfaces and normal to them: 961 We also introduce $\omega $, a dia-surface velocity component, defined as the velocity 962 relative to the moving $s$-surfaces and normal to them: 692 963 \begin{equation} \label{Eq_PE_sco_w} 693 964 \omega = w - e_3 \, \frac{\partial s}{\partial t} - \sigma _1 \,u - \sigma _2 \,v \\ … … 716 987 + D_v^{\vect{U}} + F_v^{\vect{U}} \quad 717 988 \end{multline} 718 where the relative vorticity, \textit{$\zeta $}, the surface pressure gradient, and the hydrostatic pressure have the same expressions as in $z$-coordinates although they do not represent exactly the same quantities. $\omega$ is provided by the continuity equation (see Appendix~\ref{Apdx_A}): 989 where the relative vorticity, \textit{$\zeta $}, the surface pressure gradient, and the hydrostatic 990 pressure have the same expressions as in $z$-coordinates although they do not represent 991 exactly the same quantities. $\omega$ is provided by the continuity equation 992 (see Appendix~\ref{Apdx_A}): 719 993 720 994 \begin{equation} \label{Eq_PE_sco_continuity} … … 742 1016 \end{multline} 743 1017 744 The equation of state has the same expression as in $z$-coordinate, and similar expressions are used for mixing and forcing terms. 1018 The equation of state has the same expression as in $z$-coordinate, and similar expressions 1019 are used for mixing and forcing terms. 745 1020 746 1021 \gmcomment{ … … 768 1043 It is usually called the subgrid scale physics. It must be emphasized that 769 1044 this is the weakest part of the primitive equations, but also one of the 770 most important for long-term simulations as small scale processes \textit{in fine} balance771 the surface input of kinetic energy and heat.1045 most important for long-term simulations as small scale processes \textit{in fine} 1046 balance the surface input of kinetic energy and heat. 772 1047 773 1048 The control exerted by gravity on the flow induces a strong anisotropy 774 between the lateral and vertical motions. Therefore subgrid-scale physics \textbf{D}$^{\vect{U}}$, $D^{S}$ and $D^{T}$ in \eqref{Eq_PE_dyn}, \eqref{Eq_PE_tra_T} and \eqref{Eq_PE_tra_S} are divided into a lateral part \textbf{D}$^{l \vect{U}}$, $D^{lS}$ and $D^{lT}$ and a vertical part \textbf{D}$^{vU}$, $D^{vS}$ and $D^{vT}$. The formulation of these terms and their underlying physics are briefly discussed in the next two subsections. 1049 between the lateral and vertical motions. Therefore subgrid-scale physics 1050 \textbf{D}$^{\vect{U}}$, $D^{S}$ and $D^{T}$ in \eqref{Eq_PE_dyn}, 1051 \eqref{Eq_PE_tra_T} and \eqref{Eq_PE_tra_S} are divided into a lateral part 1052 \textbf{D}$^{l \vect{U}}$, $D^{lS}$ and $D^{lT}$ and a vertical part 1053 \textbf{D}$^{vU}$, $D^{vS}$ and $D^{vT}$. The formulation of these terms 1054 and their underlying physics are briefly discussed in the next two subsections. 775 1055 776 1056 % ------------------------------------------------------------------------------------------------------------- … … 785 1065 partially, but always parameterized. The vertical turbulent fluxes are 786 1066 assumed to depend linearly on the gradients of large-scale quantities (for 787 example, the turbulent heat flux is given by $\overline{T'w'}=-A^{vT} \partial_z \overline T$, where $A^{vT}$ is an eddy coefficient). This formulation is 1067 example, the turbulent heat flux is given by $\overline{T'w'}=-A^{vT} \partial_z \overline T$, 1068 where $A^{vT}$ is an eddy coefficient). This formulation is 788 1069 analogous to that of molecular diffusion and dissipation. This is quite 789 1070 clearly a necessary compromise: considering only the molecular viscosity … … 794 1075 \begin{equation} \label{Eq_PE_zdf} 795 1076 \begin{split} 796 {\vect{D}}^{v \vect{U}} 797 &=\frac{\partial }{\partial z}\left( {A^{vm}\frac{\partial {\vect{U}}_h }{\partial z}} \right) \ , \\ D^{vT}&= \frac{\partial }{\partial z}\left( {A^{vT}\frac{\partial T}{\partial z}} \right) \ ,1077 {\vect{D}}^{v \vect{U}} &=\frac{\partial }{\partial z}\left( {A^{vm}\frac{\partial {\vect{U}}_h }{\partial z}} \right) \ , \\ 1078 D^{vT} &= \frac{\partial }{\partial z}\left( {A^{vT}\frac{\partial T}{\partial z}} \right) \ , 798 1079 \quad 799 1080 D^{vS}=\frac{\partial }{\partial z}\left( {A^{vT}\frac{\partial S}{\partial z}} \right) 800 1081 \end{split} 801 1082 \end{equation} 802 where $A^{vm}$ and $A^{vT}$ are the vertical eddy viscosity and diffusivity coefficients, respectively. At the sea surface and at the bottom, turbulent fluxes of momentum, heat and salt must be specified (see Chap.~\ref{SBC} and \ref{ZDF} and \S\ref{TRA_bbl}). All the vertical physics is embedded in the specification of the eddy coefficients. They can be assumed to be either constant, or function of the local fluid properties ($e.g.$ Richardson number, Brunt-Vais\"{a}l\"{a} frequency...), or computed from a turbulent closure model. The choices available in \NEMO are discussed in \S\ref{ZDF}). 1083 where $A^{vm}$ and $A^{vT}$ are the vertical eddy viscosity and diffusivity coefficients, 1084 respectively. At the sea surface and at the bottom, turbulent fluxes of momentum, heat 1085 and salt must be specified (see Chap.~\ref{SBC} and \ref{ZDF} and \S\ref{TRA_bbl}). 1086 All the vertical physics is embedded in the specification of the eddy coefficients. 1087 They can be assumed to be either constant, or function of the local fluid properties 1088 ($e.g.$ Richardson number, Brunt-Vais\"{a}l\"{a} frequency...), or computed from a 1089 turbulent closure model. The choices available in \NEMO are discussed in \S\ref{ZDF}). 803 1090 804 1091 % ------------------------------------------------------------------------------------------------------------- … … 821 1108 lateral diffusive and dissipative operators are of second order. 822 1109 Observations show that lateral mixing induced by mesoscale turbulence tends 823 to be along isopycnal surfaces (or more precisely neutral surfaces \cite{McDougall1987}) rather than across them. 1110 to be along isopycnal surfaces (or more precisely neutral surfaces \cite{McDougall1987}) 1111 rather than across them. 824 1112 As the slope of neutral surfaces is small in the ocean, a common 825 1113 approximation is to assume that the `lateral' direction is the horizontal, … … 834 1122 energy whereas potential energy is a main source of turbulence (through 835 1123 baroclinic instabilities). \citet{Gent1990} have proposed a 836 parameteri zation of mesoscale eddy-induced turbulence which associates an1124 parameterisation of mesoscale eddy-induced turbulence which associates an 837 1125 eddy-induced velocity to the isoneutral diffusion. Its mean effect is to 838 1126 reduce the mean potential energy of the ocean. This leads to a formulation … … 850 1138 the model while not interfering with the solved mesoscale activity. Another approach 851 1139 is becoming more and more popular: instead of specifying explicitly a sub-grid scale 852 term in the momentum and tracer time evolution equations, one uses a advective scheme which is diffusive enough to maintain the model stability. It must be emphasised 1140 term in the momentum and tracer time evolution equations, one uses a advective 1141 scheme which is diffusive enough to maintain the model stability. It must be emphasised 853 1142 that then, all the sub-grid scale physics is in this case include in the formulation of the 854 1143 advection scheme. 855 1144 856 All these parameteri zations of subgrid scale physics present advantages and1145 All these parameterisations of subgrid scale physics present advantages and 857 1146 drawbacks. There are not all available in \NEMO. In the $z$-coordinate 858 1147 formulation, five options are offered for active tracers (temperature and 859 1148 salinity): second order geopotential operator, second order isoneutral 860 operator, \citet{Gent1990} parameterization, fourth order 861 geopotential operator, and various slightly diffusive advection schemes. The same options are available for momentum, except 862 \citet{Gent1990} parameterization which only involves tracers. In the 1149 operator, \citet{Gent1990} parameterisation, fourth order 1150 geopotential operator, and various slightly diffusive advection schemes. 1151 The same options are available for momentum, except 1152 \citet{Gent1990} parameterisation which only involves tracers. In the 863 1153 $s$-coordinate formulation, additional options are offered for tracers: second 864 1154 order operator acting along $s-$surfaces, and for momentum: fourth order … … 881 1171 rotation between geopotential and $s$-surfaces, while it is only an approximation 882 1172 for the rotation between isoneutral and $z$- or $s$-surfaces. Indeed, in the latter 883 case, two assumptions are made to simplify \eqref{Eq_PE_iso_tensor} \citep{Cox1987}. First, the horizontal contribution of the dianeutral mixing is neglected since the ratio between iso and dia-neutral diffusive coefficients is known to be several orders of magnitude smaller than unity. Second, the two isoneutral directions of diffusion are assumed to be independent since the slopes are generally less than $10^{-2}$ in the ocean (see Appendix~\ref{Apdx_B}). 1173 case, two assumptions are made to simplify \eqref{Eq_PE_iso_tensor} \citep{Cox1987}. 1174 First, the horizontal contribution of the dianeutral mixing is neglected since the ratio 1175 between iso and dia-neutral diffusive coefficients is known to be several orders of 1176 magnitude smaller than unity. Second, the two isoneutral directions of diffusion are 1177 assumed to be independent since the slopes are generally less than $10^{-2}$ in the 1178 ocean (see Appendix~\ref{Apdx_B}). 884 1179 885 1180 For \textit{geopotential} diffusion, $r_1$ and $r_2 $ are the slopes between the 886 geopotential and computational surfaces: in $z$-coordinates they are zero ($r_1 = r_2 = 0$) while in $s$-coordinate (including $\textit{z*}$ case) they are equal to $\sigma _1$ and $\sigma _2$, respectively (see \eqref{Eq_PE_sco_slope} ). 887 888 For \textit{isoneutral} diffusion $r_1$ and $r_2$ are the slopes between the isoneutral and computational surfaces. Therefore, they have a same expression in $z$- and $s$-coordinates: 1181 geopotential and computational surfaces: in $z$-coordinates they are zero 1182 ($r_1 = r_2 = 0$) while in $s$-coordinate (including $\textit{z*}$ case) they are 1183 equal to $\sigma _1$ and $\sigma _2$, respectively (see \eqref{Eq_PE_sco_slope} ). 1184 1185 For \textit{isoneutral} diffusion $r_1$ and $r_2$ are the slopes between the isoneutral 1186 and computational surfaces. Therefore, they have a same expression in $z$- and $s$-coordinates: 889 1187 \begin{equation} \label{Eq_PE_iso_slopes} 890 1188 r_1 =\frac{e_3 }{e_1 } \left( {\frac{\partial \rho }{\partial i}} \right) … … 894 1192 \end{equation} 895 1193 896 When the \textit{Eddy Induced Velocity} parametrisation (eiv) \citep{Gent1990} is used, an additional tracer advection is introduced in combination with the isoneutral diffusion of tracers: 1194 When the \textit{Eddy Induced Velocity} parametrisation (eiv) \citep{Gent1990} is used, 1195 an additional tracer advection is introduced in combination with the isoneutral diffusion of tracers: 897 1196 \begin{equation} \label{Eq_PE_iso+eiv} 898 1197 D^{lT}=\nabla \cdot \left( {A^{lT}\;\Re \;\nabla T} \right) 899 1198 +\nabla \cdot \left( {{\vect{U}}^\ast \,T} \right) 900 1199 \end{equation} 901 where ${\vect{U}}^\ast =\left( {u^\ast ,v^\ast ,w^\ast } \right)$ is a non-divergent, eddy-induced transport velocity. This velocity field is defined by: 1200 where ${\vect{U}}^\ast =\left( {u^\ast ,v^\ast ,w^\ast } \right)$ is a non-divergent, 1201 eddy-induced transport velocity. This velocity field is defined by: 902 1202 \begin{equation} \label{Eq_PE_eiv} 903 1203 \begin{split} … … 909 1209 \end{split} 910 1210 \end{equation} 911 where $A^{eiv}$ is the eddy induced velocity coefficient (or equivalently the isoneutral thickness diffusivity coefficient), and $\tilde{r}_1$ and $\tilde{r}_2$ are the slopes between isoneutral and \emph{geopotential} surfaces and thus depends on the coordinate considered: 1211 where $A^{eiv}$ is the eddy induced velocity coefficient (or equivalently the isoneutral 1212 thickness diffusivity coefficient), and $\tilde{r}_1$ and $\tilde{r}_2$ are the slopes 1213 between isoneutral and \emph{geopotential} surfaces and thus depends on the coordinate 1214 considered: 912 1215 \begin{align} \label{Eq_PE_slopes_eiv} 913 1216 \tilde{r}_n = \begin{cases} … … 918 1221 \end{align} 919 1222 920 The normal component of the eddy induced velocity is zero at all the boundaries. this can be achieved in a model by tapering either the eddy coefficient or the slopes to zero in the vicinity of the boundaries. The latter strategy is used in \NEMO (cf. Chap.~\ref{LDF}). 1223 The normal component of the eddy induced velocity is zero at all the boundaries. 1224 This can be achieved in a model by tapering either the eddy coefficient or the slopes 1225 to zero in the vicinity of the boundaries. The latter strategy is used in \NEMO (cf. Chap.~\ref{LDF}). 921 1226 922 1227 \subsubsection{lateral fourth order tracer diffusive operator} … … 928 1233 \end{equation} 929 1234 930 It is the second order operator given by \eqref{Eq_PE_iso_tensor} applied twice with the eddy diffusion coefficient correctly placed. 1235 It is the second order operator given by \eqref{Eq_PE_iso_tensor} applied twice with 1236 the eddy diffusion coefficient correctly placed. 931 1237 932 1238 … … 952 1258 horizontal divergence fields (see Appendix~\ref{Apdx_C}). Unfortunately, it is not 953 1259 available for geopotential diffusion in $s-$coordinates and for isoneutral 954 diffusion in both $z$- and $s$-coordinates ($i.e.$ when a rotation is required). In these two cases, the $u$ and $v-$fields are considered as independent scalar fields, so that the diffusive operator is given by: 1260 diffusion in both $z$- and $s$-coordinates ($i.e.$ when a rotation is required). 1261 In these two cases, the $u$ and $v-$fields are considered as independent scalar 1262 fields, so that the diffusive operator is given by: 955 1263 \begin{equation} \label{Eq_PE_lapU_iso} 956 1264 \begin{split} … … 959 1267 \end{split} 960 1268 \end{equation} 961 where $\Re$ is given by \eqref{Eq_PE_iso_tensor}. It is the same expression as those used for diffusive operator on tracers. It must be emphasised that such a formulation is only exact in a Cartesian coordinate system, $i.e.$ on a $f-$ or $\beta-$plane, not on the sphere. It is also a very good approximation in vicinity of the Equator in a geographical coordinate system \citep{Lengaigne_al_JGR03}. 1269 where $\Re$ is given by \eqref{Eq_PE_iso_tensor}. It is the same expression as 1270 those used for diffusive operator on tracers. It must be emphasised that such a 1271 formulation is only exact in a Cartesian coordinate system, $i.e.$ on a $f-$ or 1272 $\beta-$plane, not on the sphere. It is also a very good approximation in vicinity 1273 of the Equator in a geographical coordinate system \citep{Lengaigne_al_JGR03}. 962 1274 963 1275 \subsubsection{lateral fourth order momentum diffusive operator} 964 1276 965 As for tracers, the fourth order momentum diffusive operator along $z$ or $s$-surfaces is a re-entering second order operator \eqref{Eq_PE_lapU} or \eqref{Eq_PE_lapU} with the eddy viscosity coefficient correctly placed: 1277 As for tracers, the fourth order momentum diffusive operator along $z$ or $s$-surfaces 1278 is a re-entering second order operator \eqref{Eq_PE_lapU} or \eqref{Eq_PE_lapU} 1279 with the eddy viscosity coefficient correctly placed: 966 1280 967 1281 geopotential diffusion in $z$-coordinate:
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