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1\documentclass[../main/NEMO_manual]{subfiles}
2
3\begin{document}
4% ================================================================
5% Chapter 1 Model Basics
6% ================================================================
7
8\chapter{Model Basics}
9\label{chap:PE}
10
11\minitoc
12
13\newpage
14
15% ================================================================
16% Primitive Equations
17% ================================================================
18\section{Primitive equations}
19\label{sec:PE_PE}
20
21% -------------------------------------------------------------------------------------------------------------
22%        Vector Invariant Formulation
23% -------------------------------------------------------------------------------------------------------------
24
25\subsection{Vector invariant formulation}
26\label{subsec:PE_Vector}
27
28
29The 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
31couples the two active tracers (temperature and salinity) to the fluid velocity,
32plus the following additional assumptions made from scale considerations:
33
34\textit{(1) spherical earth approximation:} the geopotential surfaces are assumed to be spheres so that
35gravity (local vertical) is parallel to the earth's radius
36
37\textit{(2) thin-shell approximation:} the ocean depth is neglected compared to the earth's radius
38
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
41large-scale features
42
43\textit{(4) Boussinesq hypothesis:} density variations are neglected except in their contribution to
44the buoyancy force
45
46\textit{(5) Hydrostatic hypothesis:} the vertical momentum equation is reduced to a balance between
47the vertical pressure gradient and the buoyancy force
48(this removes convective processes from the initial Navier-Stokes equations and so
49convective processes must be parameterized instead)
50
51\textit{(6) Incompressibility hypothesis:} the three dimensional divergence of the velocity vector is assumed to
52be zero.
53
54Because the gravitational force is so dominant in the equations of large-scale motions,
55it is useful to choose an orthogonal set of unit vectors (\textbf{i},\textbf{j},\textbf{k}) linked to
56the 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.
58Let us define the following variables: \textbf{U} the vector velocity, $\textbf{U}=\textbf{U}_h + w\, \textbf{k}$ 
59(the subscript $h$ denotes the local horizontal vector, $i.e.$ over the (\textbf{i},\textbf{j}) plane),
60$T$ the potential temperature, $S$ the salinity, \textit{$\rho $} the \textit{in situ} density.
61The vector invariant form of the primitive equations in the (\textbf{i},\textbf{j},\textbf{k}) vector system
62provides the following six equations
63(namely the momentum balance, the hydrostatic equilibrium, the incompressibility equation,
64the heat and salt conservation equations and an equation of state):
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}}
74  \end{equation}
75  \begin{equation}
76    \label{eq:PE_hydrostatic}
77    \frac{\partial p }{\partial z} = - \rho \ g
78  \end{equation}
79  \begin{equation}
80    \label{eq:PE_continuity}
81    \nabla \cdot {\bf U}=  0
82  \end{equation}
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
86  \end{equation}
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
90  \end{equation}
91  \begin{equation}
92    \label{eq:PE_eos}
93    \rho = \rho \left( T,S,p \right)
94  \end{equation}
95\end{subequations}
96where $\nabla$ 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,
102temperature and salinity, and ${\rm {\bf F}}^{\rm {\bf U}}$, $F^T$ and $F^S$ surface forcing terms.
103Their nature and formulation are discussed in \autoref{sec:PE_zdf_ldf} and \autoref{subsec:PE_boundary_condition}.
104
105
106
107% -------------------------------------------------------------------------------------------------------------
108% Boundary condition
109% -------------------------------------------------------------------------------------------------------------
110\subsection{Boundary conditions}
111\label{subsec:PE_boundary_condition}
112
113An ocean is bounded by complex coastlines, bottom topography at its base and
114an air-sea or ice-sea interface at its top.
115These boundaries can be defined by two surfaces, $z=-H(i,j)$ and $z=\eta(i,j,k,t)$,
116where $H$ is the depth of the ocean bottom and $\eta$ is the height of the sea surface.
117Both $H$ and $\eta$ are usually referenced to a given surface, $z=0$, chosen as a mean sea surface
118(\autoref{fig:ocean_bc}).
119Through these two boundaries, the ocean can exchange fluxes of heat, fresh water, salt, and momentum with
120the solid earth, the continental margins, the sea ice and the atmosphere.
121However, some of these fluxes are so weak that even on climatic time scales of thousands of years
122they can be neglected.
123In the following, we briefly review the fluxes exchanged at the interfaces between the ocean and
124the other components of the earth system.
125
126%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
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}
137%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
138
139
140\begin{description}
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.
150  They are usually neglected in the model
151  \footnote{
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:
161  \begin{equation}
162    \label{eq:PE_w_bbc}
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:
174  \[
175    % \label{eq:PE_w_sbc}
176    w = \frac{\partial \eta }{\partial t}
177    + \left. {{\rm {\bf U}}_h } \right|_{z=\eta } \cdot  \nabla _h \left( \eta \right)
178    + P-E
179  \]
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.
188\end{description}
189
190
191%\newpage
192
193% ================================================================
194% The Horizontal Pressure Gradient
195% ================================================================
196\section{Horizontal pressure gradient }
197\label{sec:PE_hor_pg}
198
199% -------------------------------------------------------------------------------------------------------------
200% Pressure Formulation
201% -------------------------------------------------------------------------------------------------------------
202\subsection{Pressure formulation}
203\label{subsec:PE_p_formulation}
204
205The total pressure at a given depth $z$ is composed of a surface pressure $p_s$ at
206a reference geopotential surface ($z=0$) and a hydrostatic pressure $p_h$ such that:
207$p(i,j,k,t)=p_s(i,j,t)+p_h(i,j,k,t)$.
208The latter is computed by integrating (\autoref{eq:PE_hydrostatic}),
209assuming that pressure in decibars can be approximated by depth in meters in (\autoref{eq:PE_eos}).
210The hydrostatic pressure is then given by:
211\[
212  % \label{eq:PE_pressure}
213  p_h \left( {i,j,z,t} \right)
214  = \int_{\varsigma =z}^{\varsigma =0} {g\;\rho \left( {T,S,\varsigma} \right)\;d\varsigma }
215\]
216Two strategies can be considered for the surface pressure term:
217$(a)$ introduce of a  new variable $\eta$, the free-surface elevation,
218for which a prognostic equation can be established and solved;
219$(b)$ assume that the ocean surface is a rigid lid,
220on which the pressure (or its horizontal gradient) can be diagnosed.
221When the former strategy is used, one solution of the free-surface elevation consists of
222the excitation of external gravity waves.
223The flow is barotropic and the surface moves up and down with gravity as the restoring force.
224The phase speed of such waves is high (some hundreds of metres per second) so that
225the time step would have to be very short if they were present in the model.
226The latter strategy filters out these waves since the rigid lid approximation implies $\eta=0$,
227$i.e.$ the sea surface is the surface $z=0$.
228This well known approximation increases the surface wave speed to infinity and
229modifies certain other longwave dynamics ($e.g.$ barotropic Rossby or planetary waves).
230The rigid-lid hypothesis is an obsolescent feature in modern OGCMs.
231It has been available until the release 3.1 of  \NEMO, and it has been removed in release 3.2 and followings.
232Only the free surface formulation is now described in the this document (see the next sub-section).
233
234% -------------------------------------------------------------------------------------------------------------
235% Free Surface Formulation
236% -------------------------------------------------------------------------------------------------------------
237\subsection{Free surface formulation}
238\label{subsec:PE_free_surface}
239
240In the free surface formulation, a variable $\eta$, the sea-surface height,
241is introduced which describes the shape of the air-sea interface.
242This variable is solution of a prognostic equation which is established by forming the vertical average of
243the kinematic surface condition (\autoref{eq:PE_w_bbc}):
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]
249\end{equation}
250and using (\autoref{eq:PE_hydrostatic}) the surface pressure is given by: $p_s = \rho \, g \, \eta$.
251
252Allowing the air-sea interface to move introduces the external gravity waves (EGWs) as
253a class of solution of the primitive equations.
254These waves are barotropic because of hydrostatic assumption, and their phase speed is quite high.
255Their time scale is short with respect to the other processes described by the primitive equations.
256
257Two choices can be made regarding the implementation of the free surface in the model,
258depending on the physical processes of interest.
259
260$\bullet$ If one is interested in EGWs, in particular the tides and their interaction with
261the baroclinic structure of the ocean (internal waves) possibly in shallow seas,
262then a non linear free surface is the most appropriate.
263This means that no approximation is made in (\autoref{eq:PE_ssh}) and that
264the variation of the ocean volume is fully taken into account.
265Note that in order to study the fast time scales associated with EGWs it is necessary to
266minimize time filtering effects
267(use an explicit time scheme with very small time step, or a split-explicit scheme with reasonably small time step,
268see \autoref{subsec:DYN_spg_exp} or \autoref{subsec:DYN_spg_ts}).
269
270$\bullet$ If one is not interested in EGW but rather sees them as high frequency noise,
271it is possible to apply an explicit filter to slow down the fastest waves while
272not altering the slow barotropic Rossby waves.
273If further, an approximative conservation of heat and salt contents is sufficient for the problem solved,
274then it is sufficient to solve a linearized version of (\autoref{eq:PE_ssh}),
275which still allows to take into account freshwater fluxes applied at the ocean surface \citep{Roullet_Madec_JGR00}.
276Nevertheless, with the linearization, an exact conservation of heat and salt contents is lost.
277
278The filtering of EGWs in models with a free surface is usually a matter of discretisation of
279the temporal derivatives,
280using a split-explicit method \citep{Killworth_al_JPO91, Zhang_Endoh_JGR92} or
281the implicit scheme \citep{Dukowicz1994} or
282the addition of a filtering force in the momentum equation \citep{Roullet_Madec_JGR00}.
283With the present release, \NEMO offers the choice between
284an explicit free surface (see \autoref{subsec:DYN_spg_exp}) or
285a split-explicit scheme strongly inspired the one proposed by \citet{Shchepetkin_McWilliams_OM05}
286(see \autoref{subsec:DYN_spg_ts}).
287
288%\newpage
289
290% ================================================================
291% Curvilinear z-coordinate System
292% ================================================================
293\section{Curvilinear \textit{z-}coordinate system}
294\label{sec:PE_zco}
295
296
297% -------------------------------------------------------------------------------------------------------------
298% Tensorial Formalism
299% -------------------------------------------------------------------------------------------------------------
300\subsection{Tensorial formalism}
301\label{subsec:PE_tensorial}
302
303In 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).
305The representation of such dynamical processes can be improved by
306specifically increasing the model resolution in these regions.
307As well, it may be convenient to use a lateral boundary-following coordinate system to
308better represent coastal dynamics.
309Moreover, the common geographical coordinate system has a singular point at the North Pole that
310cannot be easily treated in a global model without filtering.
311A solution consists of introducing an appropriate coordinate transformation that
312shifts the singular point onto land \citep{Madec_Imbard_CD96, Murray_JCP96}.
313As a consequence, it is important to solve the primitive equations in various curvilinear coordinate systems.
314An efficient way of introducing an appropriate coordinate transform can be found when using a tensorial formalism.
315This formalism is suited to any multidimensional curvilinear coordinate system.
316Ocean modellers mainly use three-dimensional orthogonal grids on the sphere (spherical earth approximation),
317with preservation of the local vertical. Here we give the simplified equations for this particular case.
318The general case is detailed by \citet{Eiseman1980} in their survey of the conservation laws of fluid dynamics.
319
320Let (\textit{i},\textit{j},\textit{k}) be a set of orthogonal curvilinear coordinates on
321the 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}).
325Let $(\lambda,\varphi,z)$ be the geographical coordinate system in which a position is defined by
326the latitude $\varphi(i,j)$, the longitude $\lambda(i,j)$ and
327the 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}).
329The local deformation of the curvilinear coordinate system is given by $e_1$, $e_2$ and $e_3$,
330the three scale factors:
331\begin{equation}
332  \label{eq:scale_factors}
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}
341
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}
351%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
352
353Since 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).
355The resulting horizontal scale factors $e_1$, $e_2$  are independent of $k$ while
356the vertical scale factor is a single function of $k$ as \textbf{k} is parallel to \textbf{z}.
357The 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,
359invariant in any orthogonal horizontal curvilinear coordinate system transformation:
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}
398\end{subequations}
399where $q$ is a scalar quantity and ${\rm {\bf A}}=(a_1,a_2,a_3)$ a vector in the $(i,j,k)$ coordinate system.
400
401% -------------------------------------------------------------------------------------------------------------
402% Continuous Model Equations
403% -------------------------------------------------------------------------------------------------------------
404\subsection{Continuous model equations}
405\label{subsec:PE_zco_Eq}
406
407In order to express the Primitive Equations in tensorial formalism,
408it 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}.
410Let us set $\vect U=(u,v,w)={\vect{U}}_h +w\;\vect{k}$, the velocity in the $(i,j,k)$ coordinate system and
411define the relative vorticity $\zeta$ and the divergence of the horizontal velocity field $\chi$, by:
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]
417\end{equation}
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]
423\end{equation}
424
425Using 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 $k$,
427the nonlinear term of \autoref{eq:PE_dyn} can be transformed as follows:
428\begin{flalign*}
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        &
431\end{flalign*}
432\begin{flalign*}
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)        &
447\end{flalign*}
448\begin{flalign*}
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)        &
473\end{flalign*}
474
475The last term of the right hand side is obviously zero, and thus the nonlinear term of
476\autoref{eq:PE_dyn} is written in the $(i,j,k)$ coordinate system:
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}
485\end{equation}
486
487This is the so-called \textit{vector invariant form} of the momentum advection term.
488For 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.
490For example, the first component of \autoref{eq:PE_vector_form} (the $i$-component) is transformed as follows:
491\begin{flalign*}
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  }         &
507\end{flalign*}
508\begin{flalign*}
509  &{
510    \begin{array}{*{20}l}
511      \qquad =\frac{1}{e_1 \; e_2}  \left\{
512      -\left(         v^\frac{\partial e_2                                }{\partial i}
513      +e_2 \,v    \frac{\partial v                                   }{\partial i}     \right)
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}
518      -u       \frac{\partial \left( {e_2 u}         \right)}{\partial i}  \right)
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  }      &
526\end{flalign*}
527\begin{flalign*}
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}
533      +        \frac{\partial \left( {e_1 \,u\,v} \right)}{\partial j}  \right)
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(
537      -u \left(   \frac{\partial \left( {e_1 v   } \right)}{\partial j}
538      -v\,\frac{\partial e_1 }{\partial j}             \right)
539      -u       \frac{\partial \left( {e_2 u   } \right)}{\partial i}
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  }      &
545\end{flalign*}
546\begin{flalign*}
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(
552      -v^2     \frac{\partial e_2 }{\partial i}
553      +uv   \,    \frac{\partial e_1 }{\partial j}    \right) \\
554    \end{array}
555  }      &
556\end{flalign*}
557as $\nabla \cdot {\rm {\bf U}}\;=0$ (incompressibility) it comes:
558\begin{flalign*}
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  }      &
566\end{flalign*}
567
568The flux form of the momentum advection term is therefore given by:
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
586\end{multline}
587
588The flux form has two terms,
589the first one is expressed as the divergence of momentum fluxes (hence the flux form name given to this formulation)
590and the second one is due to the curvilinear nature of the coordinate system used.
591The latter is called the \emph{metric} term and can be viewed as a modification of the Coriolis parameter:
592\[
593  % \label{eq:PE_cor+metric}
594  f \to f + \frac{1}{e_1\;e_2\left(  v \frac{\partial e_2}{\partial i}
595    -u \frac{\partial e_1}{\partial j}  \right)
596\]
597
598Note that in the case of geographical coordinate,
599$i.e.$ when $(i,j) \to (\lambda ,\varphi )$ and $(e_1 ,e_2) \to (a \,\cos \varphi ,a)$,
600we recover the commonly used modification of the Coriolis parameter $f \to f+(u/a) \tan \varphi$.
601
602To sum up, the curvilinear $z$-coordinate equations solved by the ocean model can be written in
603the following tensorial formalism:
604
605\vspace{+10pt}
606$\bullet$ \textbf{Vector invariant form of the momentum equations} :
607
608\begin{subequations}
609  \label{eq:PE_dyn_vect}
610  \[
611    % \label{eq:PE_dyn_vect_u}
612    \begin{split}
613      \frac{\partial u}{\partial t}
614      = +   \left( {\zeta +f} \right)\,v
615      -   \frac{1}{2\,e_1}           \frac{\partial}{\partial i} \left(  u^2+v^2   \right)
616      -   \frac{1}{e_3    }  w     \frac{\partial u}{\partial k}      &      \\
617      -   \frac{1}{e_1    }            \frac{\partial}{\partial i} \left( \frac{p_s+p_h }{\rho_o}    \right)
618      &+   D_u^{\vect{U}}  +   F_u^{\vect{U}}      \\ \\
619      \frac{\partial v}{\partial t} =
620      -   \left( {\zeta +f} \right)\,u
621      -   \frac{1}{2\,e_2 }        \frac{\partial }{\partial j}\left(  u^2+v^2  \right)
622      -   \frac{1}{e_3     }   w  \frac{\partial v}{\partial k}     &      \\
623      -   \frac{1}{e_2     }        \frac{\partial }{\partial j}\left( \frac{p_s+p_h }{\rho_o}  \right)
624      &+  D_v^{\vect{U}}  +   F_v^{\vect{U}}
625    \end{split}
626  \]
627\end{subequations}
628
629
630\vspace{+10pt}
631$\bullet$ \textbf{flux form of the momentum equations} :
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}
642      +        \frac{\partial \left( {e_1 \,v\,u} \right)}{\partial j}  \right)
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}
655      +        \frac{\partial \left( {e_1 \,v\,v} \right)}{\partial j}  \right)
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*}
660\end{subequations}
661where $\zeta$, the relative vorticity, is given by \autoref{eq:PE_curl_Uh} and
662$p_s $, the surface pressure, is given by:
663\[
664  % \label{eq:PE_spg}
665  p_s =  \rho \,g \,\eta
666\]
667with $\eta$ is solution of \autoref{eq:PE_ssh}.
668
669The vertical velocity and the hydrostatic pressure are diagnosed from the following equations:
670\[
671  % \label{eq:w_diag}
672  \frac{\partial w}{\partial k}=-\chi \;e_3
673\]
674\[
675  % \label{eq:hp_diag}
676  \frac{\partial p_h }{\partial k}=-\rho \;g\;e_3
677\]
678where the divergence of the horizontal velocity, $\chi$ is given by \autoref{eq:PE_div_Uh}.
679
680\vspace{+10pt}
681$\bullet$ \textit{tracer equations} :
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\[
697  % \label{eq:rho}
698  \rho =\rho \left( {T,S,z(k)} \right)
699\]
700
701The expression of \textbf{D}$^{U}$, $D^{S}$ and $D^{T}$ depends on the subgrid scale parameterisation used.
702It will be defined in \autoref{eq:PE_zdf}.
703The nature and formulation of ${\rm {\bf F}}^{\rm {\bf U}}$, $F^T$ and $F^S$, the surface forcing terms,
704are discussed in \autoref{chap:SBC}.
705
706
707\newpage 
708
709% ================================================================
710% Curvilinear generalised vertical coordinate System
711% ================================================================
712\section{Curvilinear generalised vertical coordinate system}
713\label{sec:PE_gco}
714
715The ocean domain presents a huge diversity of situation in the vertical.
716First the ocean surface is a time dependent surface (moving surface).
717Second the ocean floor depends on the geographical position,
718varying from more than 6,000 meters in abyssal trenches to zero at the coast.
719Last but not least, the ocean stratification exerts a strong barrier to vertical motions and mixing.
720Therefore, in order to represent the ocean with respect to
721the first point a space and time dependent vertical coordinate that follows the variation of the sea surface height
722$e.g.$ an $z$*-coordinate;
723for the second point, a space variation to fit the change of bottom topography
724$e.g.$ a terrain-following or $\sigma$-coordinate;
725and for the third point, one will be tempted to use a space and time dependent coordinate that
726follows the isopycnal surfaces, $e.g.$ an isopycnic coordinate.
727
728In order to satisfy two or more constrains one can even be tempted to mixed these coordinate systems, as in
729HYCOM (mixture of $z$-coordinate at the surface, isopycnic coordinate in the ocean interior and $\sigma$ at
730the ocean bottom) \citep{Chassignet_al_JPO03} or
731OPA (mixture of $z$-coordinate in vicinity the surface and steep topography areas and $\sigma$-coordinate elsewhere)
732\citep{Madec_al_JPO96} among others.
733
734In fact one is totally free to choose any space and time vertical coordinate by
735introducing an arbitrary vertical coordinate :
736\begin{equation}
737  \label{eq:PE_s}
738  s=s(i,j,k,t)
739\end{equation}
740with the restriction that the above equation gives a single-valued monotonic relationship between $s$ and $k$,
741when $i$, $j$ 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
743the $(i,j,s,t)$ generalised coordinate system with $s$ depending on the other three variables through
744\autoref{eq:PE_s}.
745This so-called \textit{generalised vertical coordinate} \citep{Kasahara_MWR74} is in fact
746an Arbitrary Lagrangian--Eulerian (ALE) coordinate.
747Indeed, choosing an expression for $s$ is an arbitrary choice that determines
748which part of the vertical velocity (defined from a fixed referential) will cross the levels (Eulerian part) and
749which part will be used to move them (Lagrangian part).
750The coordinate is also sometime referenced as an adaptive coordinate \citep{Hofmeister_al_OM09},
751since the coordinate system is adapted in the course of the simulation.
752Its most often used implementation is via an ALE algorithm,
753in which a pure lagrangian step is followed by regridding and remapping steps,
754the later step implicitly embedding the vertical advection
755\citep{Hirt_al_JCP74, Chassignet_al_JPO03, White_al_JCP09}.
756Here we follow the \citep{Kasahara_MWR74} strategy:
757a regridding step (an update of the vertical coordinate) followed by an eulerian step with
758an explicit computation of vertical advection relative to the moving s-surfaces.
759
760%\gmcomment{
761
762%A key point here is that the $s$-coordinate depends on $(i,j)$ ==> horizontal pressure gradient...
763
764the generalized vertical coordinates used in ocean modelling are not orthogonal,
765which contrasts with many other applications in mathematical physics.
766Hence, it is useful to keep in mind the following properties that may seem odd on initial encounter.
767
768The horizontal velocity in ocean models measures motions in the horizontal plane,
769perpendicular to the local gravitational field.
770That is, horizontal velocity is mathematically the same regardless the vertical coordinate, be it geopotential,
771isopycnal, pressure, or terrain following.
772The key motivation for maintaining the same horizontal velocity component is that
773the hydrostatic and geostrophic balances are dominant in the large-scale ocean.
774Use of an alternative quasi-horizontal velocity, for example one oriented parallel to the generalized surface,
775would lead to unacceptable numerical errors.
776Correspondingly, the vertical direction is anti-parallel to the gravitational force in
777all of the coordinate systems.
778We do not choose the alternative of a quasi-vertical direction oriented normal to
779the surface of a constant generalized vertical coordinate.
780
781It is the method used to measure transport across the generalized vertical coordinate surfaces which differs between
782the vertical coordinate choices.
783That is, computation of the dia-surface velocity component represents the fundamental distinction between
784the various coordinates.
785In some models, such as geopotential, pressure, and terrain following, this transport is typically diagnosed from
786volume or mass conservation.
787In other models, such as isopycnal layered models, this transport is prescribed based on assumptions about
788the physical processes producing a flux across the layer interfaces.
789
790
791In this section we first establish the PE in the generalised vertical $s$-coordinate,
792then we discuss the particular cases available in \NEMO, namely $z$, $z$*, $s$, and $\tilde z$
793%}
794
795% -------------------------------------------------------------------------------------------------------------
796% The s-coordinate Formulation
797% -------------------------------------------------------------------------------------------------------------
798\subsection{\textit{S-}coordinate formulation}
799
800Starting from the set of equations established in \autoref{sec:PE_zco} for the special case $k=z$ and thus $e_3=1$,
801we introduce an arbitrary vertical coordinate $s=s(i,j,k,t)$,
802which 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.).
804A formal derivation of the transformed equations is given in \autoref{apdx:A}.
805Let us define the vertical scale factor by $e_3=\partial_s z$  ($e_3$ is now a function of $(i,j,k,t)$ ),
806and the slopes in the (\textbf{i},\textbf{j}) directions between $s-$ and $z-$surfaces by:
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
812\end{equation}
813We also introduce  $\omega $, a dia-surface velocity component, defined as the velocity
814relative to the moving $s$-surfaces and normal to them:
815\[
816  % \label{eq:PE_sco_w}
817  \omega  = w - e_3 \, \frac{\partial s}{\partial t} - \sigma_1 \,u - \sigma_2 \,v    \\
818\]
819
820The equations solved by the ocean model \autoref{eq:PE} in $s-$coordinate can be written as follows
821(see \autoref{sec:A_momentum}):
822
823 \vspace{0.5cm}
824$\bullet$ Vector invariant form of the momentum equation :
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^\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*}
845
846 \vspace{0.5cm}
847$\bullet$ Flux form of the momentum equation :
848\begin{multline*}
849  % \label{eq:PE_sco_u_flux}
850  \frac{1}{e_3} \frac{\partial \left(  e_3\,\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\,\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_\,u\,v} \right)}{\partial i}
870    +       \frac{\partial \left( {e_1 \; e_\,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*}
876
877where the relative vorticity, \textit{$\zeta $}, the surface pressure gradient,
878and the hydrostatic pressure have the same expressions as in $z$-coordinates although
879they do not represent exactly the same quantities.
880$\omega$ is provided by the continuity equation (see \autoref{apdx:A}):
881\[
882  % \label{eq:PE_sco_continuity}
883  \frac{\partial e_3}{\partial t} + e_3 \; \chi + \frac{\partial \omega }{\partial s} = 0
884  \qquad \text{with }\;\;
885  \chi =\frac{1}{e_1 e_2 e_3 }\left[ {\frac{\partial \left( {e_2 e_3 \,u}
886        \right)}{\partial i}+\frac{\partial \left( {e_1 e_3 \,v} \right)}{\partial
887        j}} \right]
888\]
889
890 \vspace{0.5cm}
891$\bullet$ tracer equations:
892\begin{multline*}
893  % \label{eq:PE_sco_t}
894  \frac{1}{e_3} \frac{\partial \left(  e_3\,\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*}
899
900\begin{multline*}
901  % \label{eq:PE_sco_s}
902  \frac{1}{e_3} \frac{\partial \left(  e_3\,\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*}
907
908The equation of state has the same expression as in $z$-coordinate,
909and similar expressions are used for mixing and forcing terms.
910
911\gmcomment{
912\colorbox{yellow}{ to be updated $= = >$}
913Add 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% -------------------------------------------------------------------------------------------------------------
921\subsection{Curvilinear \textit{z*}--coordinate system}
922\label{subsec:PE_zco_star}
923
924%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
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}
936%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
937
938
939In that case, the free surface equation is nonlinear, and the variations of volume are fully taken into account.
940These coordinates systems is presented in a report \citep{Levier2007} available on the \NEMO web site.
941
942%\gmcomment{
943The \textit{z*} coordinate approach is an unapproximated, non-linear free surface implementation which allows one to
944deal with large amplitude free-surface variations relative to the vertical resolution \citep{Adcroft_Campin_OM04}.
945In the \textit{z*} formulation,
946the variation of the column thickness due to sea-surface undulations is not concentrated in the surface level,
947as in the $z$-coordinate formulation, but is equally distributed over the full water column.
948Thus vertical levels naturally follow sea-surface variations, with a linear attenuation with depth,
949as illustrated by figure fig.1c.
950Note that with a flat bottom, such as in fig.1c, the bottom-following $z$ coordinate and \textit{z*} are equivalent.
951The definition and modified oceanic equations for the rescaled vertical coordinate  \textit{z*},
952including the treatment of fresh-water flux at the surface, are detailed in Adcroft and Campin (2004).
953The major points are summarized here.
954The position ( \textit{z*}) and vertical discretization (\textit{z*}) are expressed as:
955\[
956  % \label{eq:z-star}
957  H +  \textit{z*} = (H + z) / r \quad \text{and} \ \delta \textit{z*} = \delta z / r \quad \text{with} \ r = \frac{H+\eta} {H}
958\] 
959Since the vertical displacement of the free surface is incorporated in the vertical coordinate \textit{z*},
960the upper and lower boundaries are at fixed  \textit{z*} position,
961$\textit{z*} = 0$ and  $\textit{z*} = -H$ respectively.
962Also the divergence of the flow field is no longer zero as shown by the continuity equation:
963\[ 
964  \frac{\partial r}{\partial t} = \nabla_{\textit{z*}} \cdot \left( r \; \rm{\bf U}_h \right)
965  \left( r \; w\textit{*} \right) = 0 
966\] 
967%}
968
969
970% from MOM4p1 documentation
971
972To overcome problems with vanishing surface and/or bottom cells, we consider the zstar coordinate
973\[
974  % \label{eq:PE_}
975  z^\star = H \left( \frac{z-\eta}{H+\eta} \right)
976\]
977
978This 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).
980It was originally used in ocean models by Stacey et al. (1995) for studies of tides next to shelves,
981and it has been recently promoted by Adcroft and Campin (2004) for global climate modelling.
982
983The surfaces of constant $z^\star$ are quasi-horizontal.
984Indeed, the $z^\star$ coordinate reduces to $z$ when $\eta$ is zero.
985In general, when noting the large differences between
986undulations of the bottom topography versus undulations in the surface height,
987it is clear that surfaces constant $z^\star$ are very similar to the depth surfaces.
988These properties greatly reduce difficulties of computing the horizontal pressure gradient relative to
989terrain following sigma models discussed in \autoref{subsec:PE_sco}.
990Additionally, since $z^\star$ when $\eta = 0$,
991no flow is spontaneously generated in an unforced ocean starting from rest, regardless the bottom topography.
992This behaviour is in contrast to the case with "s"-models, where pressure gradient errors in the presence of
993nontrivial topographic variations can generate nontrivial spontaneous flow from a resting state,
994depending on the sophistication of the pressure gradient solver.
995The quasi-horizontal nature of the coordinate surfaces also facilitates the implementation of
996neutral 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,
998as well as \autoref{sec:LDF_slp} in this document for treatment in \NEMO).
999
1000The range over which $z^\star$ varies is time independent $-H \leq z^\star \leq 0$.
1001Hence, all cells remain nonvanishing, so long as the surface height maintains $\eta > ?H$.
1002This is a minor constraint relative to that encountered on the surface height when using $s = z$ or $s = z - \eta$.
1003
1004Because $z^\star$ has a time independent range, all grid cells have static increments ds,
1005and the sum of the ver tical increments yields the time independent ocean depth. %k ds = H.
1006The $z^\star$ coordinate is therefore invisible to undulations of the free surface,
1007since it moves along with the free surface.
1008This proper ty means that no spurious vertical transport is induced across surfaces of constant $z^\star$ by
1009the motion of external gravity waves.
1010Such spurious transpor t can be a problem in z-models, especially those with tidal forcing.
1011Quite generally, the time independent range for the $z^\star$ coordinate is a very convenient property that
1012allows for a nearly arbitrary ver tical resolution even in the presence of large amplitude fluctuations of
1013the surface height, again so long as $\eta > -H$.
1014
1015%end MOM doc %%%
1016
1017
1018
1019\newpage
1020
1021% -------------------------------------------------------------------------------------------------------------
1022% Terrain following  coordinate System
1023% -------------------------------------------------------------------------------------------------------------
1024\subsection{Curvilinear terrain-following \textit{s}--coordinate}
1025\label{subsec:PE_sco}
1026
1027% -------------------------------------------------------------------------------------------------------------
1028% Introduction
1029% -------------------------------------------------------------------------------------------------------------
1030\subsubsection{Introduction}
1031
1032Several important aspects of the ocean circulation are influenced by bottom topography.
1033Of course, the most important is that bottom topography determines deep ocean sub-basins, barriers, sills and
1034channels that strongly constrain the path of water masses, but more subtle effects exist.
1035For example, the topographic $\beta$-effect is usually larger than the planetary one along continental slopes.
1036Topographic Rossby waves can be excited and can interact with the mean current.
1037In the $z-$coordinate system presented in the previous section (\autoref{sec:PE_zco}),
1038$z-$surfaces are geopotential surfaces.
1039The bottom topography is discretised by steps.
1040This often leads to a misrepresentation of a gradually sloping bottom and to
1041large localized depth gradients associated with large localized vertical velocities.
1042The response to such a velocity field often leads to numerical dispersion effects.
1043One solution to strongly reduce this error is to use a partial step representation of bottom topography instead of
1044a full step one \cite{Pacanowski_Gnanadesikan_MWR98}.
1045Another solution is to introduce a terrain-following coordinate system (hereafter $s-$coordinate).
1046
1047The $s$-coordinate avoids the discretisation error in the depth field since the layers of
1048computation are gradually adjusted with depth to the ocean bottom.
1049Relatively small topographic features as well as  gentle, large-scale slopes of the sea floor in the deep ocean,
1050which would be ignored in typical $z$-model applications with the largest grid spacing at greatest depths,
1051can easily be represented (with relatively low vertical resolution).
1052A terrain-following model (hereafter $s-$model) also facilitates the modelling of the boundary layer flows over
1053a large depth range, which in the framework of the $z$-model would require high vertical resolution over
1054the whole depth range.
1055Moreover, with a $s$-coordinate it is possible, at least in principle, to have the bottom and the sea surface as
1056the only boundaries of the domain (no more lateral boundary condition to specify).
1057Nevertheless, a $s$-coordinate also has its drawbacks. Perfectly adapted to a homogeneous ocean,
1058it has strong limitations as soon as stratification is introduced.
1059The main two problems come from the truncation error in the horizontal pressure gradient and
1060a possibly increased diapycnal diffusion.
1061The horizontal pressure force in $s$-coordinate consists of two terms (see \autoref{apdx:A}),
1062
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
1067\end{equation}
1068
1069The second term in \autoref{eq:PE_p_sco} depends on the tilt of the coordinate surface and
1070introduces a truncation error that is not present in a $z$-model.
1071In 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.
1073It depends on topographic slope, stratification, horizontal and vertical resolution, the equation of state,
1074and the finite difference scheme.
1075This error limits the possible topographic slopes that a model can handle at
1076a given horizontal and vertical resolution.
1077This is a severe restriction for large-scale applications using realistic bottom topography.
1078The large-scale slopes require high horizontal resolution, and the computational cost becomes prohibitive.
1079This problem can be at least partially overcome by mixing $s$-coordinate and
1080step-like representation of bottom topography \citep{Gerdes1993a,Gerdes1993b,Madec_al_JPO96}.
1081However, the definition of the model domain vertical coordinate becomes then a non-trivial thing for
1082a realistic bottom topography:
1083a envelope topography is defined in $s$-coordinate on which a full or
1084partial step bottom topography is then applied in order to adjust the model depth to the observed one
1085(see \autoref{sec:DOM_zgr}.
1086
1087For numerical reasons a minimum of diffusion is required along the coordinate surfaces of any finite difference model.
1088It causes spurious diapycnal mixing when coordinate surfaces do not coincide with isoneutral surfaces.
1089This is the case for a $z$-model as well as for a $s$-model.
1090However, density varies more strongly on $s-$surfaces than on horizontal surfaces in regions of
1091large topographic slopes, implying larger diapycnal diffusion in a $s$-model than in a $z$-model.
1092Whereas such a diapycnal diffusion in a $z$-model tends to weaken horizontal density (pressure) gradients and thus
1093the horizontal circulation, it usually reinforces these gradients in a $s$-model, creating spurious circulation.
1094For example, imagine an isolated bump of topography in an ocean at rest with a horizontally uniform stratification.
1095Spurious diffusion along $s$-surfaces will induce a bump of isoneutral surfaces over the topography,
1096and thus will generate there a baroclinic eddy.
1097In contrast, the ocean will stay at rest in a $z$-model.
1098As for the truncation error, the problem can be reduced by introducing the terrain-following coordinate below
1099the strongly stratified portion of the water column ($i.e.$ the main thermocline) \citep{Madec_al_JPO96}.
1100An alternate solution consists of rotating the lateral diffusive tensor to geopotential or to isoneutral surfaces
1101(see \autoref{subsec:PE_ldf}).
1102Unfortunately, the slope of isoneutral surfaces relative to the $s$-surfaces can very large,
1103strongly exceeding the stability limit of such a operator when it is discretized (see \autoref{chap:LDF}).
1104
1105The $s-$coordinates introduced here \citep{Lott_al_OM90,Madec_al_JPO96} differ mainly in two aspects from
1106similar models:
1107it allows a representation of bottom topography with mixed full or partial step-like/terrain following topography;
1108It also offers a completely general transformation, $s=s(i,j,z)$ for the vertical coordinate.
1109
1110
1111\newpage
1112
1113% -------------------------------------------------------------------------------------------------------------
1114% Curvilinear z-tilde coordinate System
1115% -------------------------------------------------------------------------------------------------------------
1116\subsection{\texorpdfstring{Curvilinear $\tilde{z}$--coordinate}{}}
1117\label{subsec:PE_zco_tilde}
1118
1119The $\tilde{z}$-coordinate has been developed by \citet{Leclair_Madec_OM11}.
1120It is available in \NEMO since the version 3.4.
1121Nevertheless, it is currently not robust enough to be used in all possible configurations.
1122Its use is therefore not recommended.
1123
1124
1125\newpage 
1126
1127% ================================================================
1128% Subgrid Scale Physics
1129% ================================================================
1130\section{Subgrid scale physics}
1131\label{sec:PE_zdf_ldf}
1132
1133The primitive equations describe the behaviour of a geophysical fluid at space and time scales larger than
1134a few kilometres in the horizontal, a few meters in the vertical and a few minutes.
1135They are usually solved at larger scales: the specified grid spacing and time step of the numerical model.
1136The 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.
1137These effects appear in the equations as the divergence of turbulent fluxes
1138($i.e.$ fluxes associated with the mean correlation of small scale perturbations).
1139Assuming a turbulent closure hypothesis is equivalent to choose a formulation for these fluxes.
1140It is usually called the subgrid scale physics.
1141It must be emphasized that this is the weakest part of the primitive equations,
1142but also one of the most important for long-term simulations as
1143small scale processes \textit{in fine} balance the surface input of kinetic energy and heat.
1144
1145The control exerted by gravity on the flow induces a strong anisotropy between the lateral and vertical motions.
1146Therefore 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
1148a lateral part  \textbf{D}$^{l \vect{U}}$, $D^{lS}$ and $D^{lT}$ and
1149a vertical part  \textbf{D}$^{vU}$, $D^{vS}$ and $D^{vT}$.
1150The formulation of these terms and their underlying physics are briefly discussed in the next two subsections.
1151
1152% -------------------------------------------------------------------------------------------------------------
1153% Vertical Subgrid Scale Physics
1154% -------------------------------------------------------------------------------------------------------------
1155\subsection{Vertical subgrid scale physics}
1156\label{subsec:PE_zdf}
1157
1158The model resolution is always larger than the scale at which the major sources of vertical turbulence occur
1159(shear instability, internal wave breaking...).
1160Turbulent motions are thus never explicitly solved, even partially, but always parameterized.
1161The 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 $\overline{T'w'}=-A^{vT} \partial_z \overline T$,
1163where $A^{vT}$ is an eddy coefficient).
1164This formulation is analogous to that of molecular diffusion and dissipation.
1165This is quite clearly a necessary compromise: considering only the molecular viscosity acting on
1166large scale severely underestimates the role of turbulent diffusion and dissipation,
1167while an accurate consideration of the details of turbulent motions is simply impractical.
1168The resulting vertical momentum and tracer diffusive operators are of second order:
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}
1177\end{equation}
1178where $A^{vm}$ and $A^{vT}$ are the vertical eddy viscosity and diffusivity coefficients, respectively.
1179At 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}).
1181All the vertical physics is embedded in the specification of the eddy coefficients.
1182They 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...),
1184or computed from a turbulent closure model.
1185The choices available in \NEMO are discussed in \autoref{chap:ZDF}).
1186
1187% -------------------------------------------------------------------------------------------------------------
1188% Lateral Diffusive and Viscous Operators Formulation
1189% -------------------------------------------------------------------------------------------------------------
1190\subsection{Formulation of the lateral diffusive and viscous operators}
1191\label{subsec:PE_ldf}
1192
1193Lateral turbulence can be roughly divided into a mesoscale turbulence associated with eddies
1194(which can be solved explicitly if the resolution is sufficient since
1195their underlying physics are included in the primitive equations),
1196and a sub mesoscale turbulence which is never explicitly solved even partially, but always parameterized.
1197The formulation of lateral eddy fluxes depends on whether the mesoscale is below or above the grid-spacing
1198($i.e.$ the model is eddy-resolving or not).
1199
1200In non-eddy-resolving configurations, the closure is similar to that used for the vertical physics.
1201The lateral turbulent fluxes are assumed to depend linearly on the lateral gradients of large-scale quantities.
1202The resulting lateral diffusive and dissipative operators are of second order.
1203Observations 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.
1205As the slope of neutral surfaces is small in the ocean, a common approximation is to assume that
1206the `lateral' direction is the horizontal, $i.e.$ the lateral mixing is performed along geopotential surfaces.
1207This leads to a geopotential second order operator for lateral subgrid scale physics.
1208This assumption can be relaxed: the eddy-induced turbulent fluxes can be better approached by assuming that
1209they depend linearly on the gradients of large-scale quantities computed along neutral surfaces.
1210In such a case, the diffusive operator is an isoneutral second order operator and
1211it has components in the three space directions.
1212However,
1213both horizontal and isoneutral operators have no effect on mean ($i.e.$ large scale) potential energy whereas
1214potential energy is a main source of turbulence (through baroclinic instabilities).
1215\citet{Gent1990} have proposed a parameterisation of mesoscale eddy-induced turbulence which
1216associates an eddy-induced velocity to the isoneutral diffusion.
1217Its mean effect is to reduce the mean potential energy of the ocean.
1218This leads to a formulation of lateral subgrid-scale physics made up of an isoneutral second order operator and
1219an eddy induced advective part.
1220In all these lateral diffusive formulations,
1221the specification of the lateral eddy coefficients remains the problematic point as
1222there is no really satisfactory formulation of these coefficients as a function of large-scale features.
1223
1224In eddy-resolving configurations, a second order operator can be used,
1225but usually the more scale selective biharmonic operator is preferred as
1226the grid-spacing is usually not small enough compared to the scale of the eddies.
1227The role devoted to the subgrid-scale physics is to dissipate the energy that
1228cascades toward the grid scale and thus to ensure the stability of the model while
1229not interfering with the resolved mesoscale activity.
1230Another approach is becoming more and more popular:
1231instead of specifying explicitly a sub-grid scale term in the momentum and tracer time evolution equations,
1232one uses a advective scheme which is diffusive enough to maintain the model stability.
1233It must be emphasised that then, all the sub-grid scale physics is included in the formulation of
1234the advection scheme.
1235
1236All these parameterisations of subgrid scale physics have advantages and drawbacks.
1237There are not all available in \NEMO. For active tracers (temperature and salinity) the main ones are:
1238Laplacian and bilaplacian operators acting along geopotential or iso-neutral surfaces,
1239\citet{Gent1990} parameterisation, and various slightly diffusive advection schemes.
1240For momentum, the main ones are: Laplacian and bilaplacian operators acting along geopotential surfaces,
1241and UBS advection schemes when flux form is chosen for the momentum advection.
1242
1243\subsubsection{Lateral laplacian tracer diffusive operator}
1244
1245The lateral Laplacian tracer diffusive operator is defined by (see \autoref{apdx:B}):
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)
1256\end{equation}
1257where $r_1 \;\mbox{and}\;r_2 $ are the slopes between the surface along which the diffusive operator acts and
1258the model level ($e. g.$ $z$- or $s$-surfaces).
1259Note that the formulation \autoref{eq:PE_iso_tensor} is exact for
1260the rotation between geopotential and $s$-surfaces,
1261while it is only an approximation for the rotation between isoneutral and $z$- or $s$-surfaces.
1262Indeed, in the latter case, two assumptions are made to simplify \autoref{eq:PE_iso_tensor} \citep{Cox1987}.
1263First, the horizontal contribution of the dianeutral mixing is neglected since the ratio between iso and
1264dia-neutral diffusive coefficients is known to be several orders of magnitude smaller than unity.
1265Second, the two isoneutral directions of diffusion are assumed to be independent since
1266the slopes are generally less than $10^{-2}$ in the ocean (see \autoref{apdx:B}).
1267
1268For \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.
1270
1271For \textit{geopotential} diffusion,
1272$r_1$ and $r_2 $ are the slopes between the geopotential and computational surfaces:
1273they are equal to $\sigma_1$ and $\sigma_2$, respectively (see \autoref{eq:PE_sco_slope}).
1274
1275For \textit{isoneutral} diffusion $r_1$ and $r_2$ are the slopes between the isoneutral and computational surfaces.
1276Therefore, they are different quantities, but have similar expressions in $z$- and $s$-coordinates.
1277In $z$-coordinates:
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} \,
1282\end{equation}
1283while in $s$-coordinates $\pd[]{k}$ is replaced by $\pd[]{s}$.
1284
1285\subsubsection{Eddy induced velocity}
1286When the \textit{eddy induced velocity} parametrisation (eiv) \citep{Gent1990} is used,
1287an additional tracer advection is introduced in combination with the isoneutral diffusion of tracers:
1288\[
1289  % \label{eq:PE_iso+eiv}
1290  D^{lT}=\nabla \cdot \left( {A^{lT}\;\Re \;\nabla T} \right)
1291  +\nabla \cdot \left( {{\vect{U}}^\ast \,T} \right)
1292\]
1293where ${\vect{U}}^\ast =\left( {u^\ast ,v^\ast ,w^\ast } \right)$ is a non-divergent,
1294eddy-induced transport velocity. This velocity field is defined by:
1295\[
1296  % \label{eq:PE_eiv}
1297  \begin{split}
1298    u^\ast  &= +\frac{1}{e_3       }\frac{\partial }{\partial k}\left[ {A^{eiv}\;\tilde{r}_1 } \right] \\
1299    v^\ast  &= +\frac{1}{e_3       }\frac{\partial }{\partial k}\left[ {A^{eiv}\;\tilde{r}_2 } \right] \\
1300    w^\ast &=  -\frac{1}{e_1 e_2 }\left[
1301      \frac{\partial }{\partial i}\left( {A^{eiv}\;e_2\,\tilde{r}_1 } \right)
1302      +\frac{\partial }{\partial j}\left( {A^{eiv}\;e_1\,\tilde{r}_2 } \right)      \right]
1303  \end{split}
1304\]
1305where $A^{eiv}$ is the eddy induced velocity coefficient
1306(or equivalently the isoneutral thickness diffusivity coefficient),
1307and $\tilde{r}_1$ and $\tilde{r}_2$ are the slopes between isoneutral and \emph{geopotential} surfaces.
1308Their values are thus independent of the vertical coordinate, but their expression depends on the coordinate:
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
1317\end{align}
1318
1319The normal component of the eddy induced velocity is zero at all the boundaries.
1320This can be achieved in a model by tapering either the eddy coefficient or
1321the slopes to zero in the vicinity of the boundaries.
1322The latter strategy is used in \NEMO (cf. \autoref{chap:LDF}).
1323
1324\subsubsection{Lateral bilaplacian tracer diffusive operator}
1325
1326The lateral bilaplacian tracer diffusive operator is defined by:
1327\[
1328  % \label{eq:PE_bilapT}
1329  D^{lT}= - \Delta \left( \;\Delta T \right)
1330  \qquad \text{where} \;\; \Delta \bullet = \nabla \left( {\sqrt{B^{lT}\,}\;\Re \;\nabla \bullet} \right)
1331\]
1332It is the Laplacian operator given by \autoref{eq:PE_iso_tensor} applied twice with
1333the harmonic eddy diffusion coefficient set to the square root of the biharmonic one.
1334
1335
1336\subsubsection{Lateral Laplacian momentum diffusive operator}
1337
1338The Laplacian momentum diffusive operator along $z$- or $s$-surfaces is found by
1339applying \autoref{eq:PE_lap_vector} to the horizontal velocity vector (see \autoref{apdx:B}):
1340\[
1341  % \label{eq:PE_lapU}
1342  \begin{split}
1343    {\rm {\bf D}}^{l{\rm {\bf U}}}
1344    &= \quad \  \nabla _h \left( {A^{lm}\chi } \right)
1345    \ - \ \nabla _h \times \left( {A^{lm}\,\zeta \;{\rm {\bf k}}} \right)     \\
1346    &=   \left(
1347      \begin{aligned}
1348        \frac{1}{e_1      } \frac{\partial \left( A^{lm} \chi          \right)}{\partial i}
1349        &-\frac{1}{e_2 e_3}\frac{\partial \left( {A^{lm} \;e_3 \zeta} \right)}{\partial j}  \\
1350        \frac{1}{e_2      }\frac{\partial \left( {A^{lm} \chi         } \right)}{\partial j}
1351        &+\frac{1}{e_1 e_3}\frac{\partial \left( {A^{lm} \;e_3 \zeta} \right)}{\partial i}
1352      \end{aligned}
1353    \right)
1354  \end{split}
1355\]
1356
1357Such a formulation ensures a complete separation between the vorticity and horizontal divergence fields
1358(see \autoref{apdx:C}).
1359Unfortunately, it is only available in \textit{iso-level} direction.
1360When a rotation is required
1361($i.e.$ geopotential diffusion in $s-$coordinates or isoneutral diffusion in both $z$- and $s$-coordinates),
1362the $u$ and $v-$fields are considered as independent scalar fields, so that the diffusive operator is given by:
1363\[
1364  % \label{eq:PE_lapU_iso}
1365  \begin{split}
1366    D_u^{l{\rm {\bf U}}} &= \nabla .\left( {A^{lm} \;\Re \;\nabla u} \right) \\
1367    D_v^{l{\rm {\bf U}}} &= \nabla .\left( {A^{lm} \;\Re \;\nabla v} \right)
1368  \end{split}
1369\]
1370where $\Re$ is given by \autoref{eq:PE_iso_tensor}.
1371It is the same expression as those used for diffusive operator on tracers.
1372It 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.
1374It is also a very good approximation in vicinity of the Equator in
1375a geographical coordinate system \citep{Lengaigne_al_JGR03}.
1376
1377\subsubsection{lateral bilaplacian momentum diffusive operator}
1378
1379As for tracers, the bilaplacian order momentum diffusive operator is a re-entering Laplacian operator with
1380the harmonic eddy diffusion coefficient set to the square root of the biharmonic one.
1381Nevertheless it is currently not available in the iso-neutral case.
1382
1383\biblio
1384
1385\end{document}
1386
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